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[ [ "When facts fail: Bias, polarisation and truth in social networks" ], [ "Abstract Online social networks provide users with unprecedented opportunities to engage with diverse opinions.", "At the same time, they enable confirmation bias on large scales by empowering individuals to self-select narratives they want to be exposed to.", "A precise understanding of such tradeoffs is still largely missing.", "We introduce a social learning model where most participants in a network update their beliefs unbiasedly based on new information, while a minority of participants reject information that is incongruent with their preexisting beliefs.", "This simple mechanism generates permanent opinion polarization and cascade dynamics, and accounts for the aforementioned tradeoff between confirmation bias and social connectivity through analytic results.", "We investigate the model's predictions empirically using US county-level data on the impact of Internet access on the formation of beliefs about global warming.", "We conclude by discussing policy implications of our model, highlighting the downsides of debunking and suggesting alternative strategies to contrast misinformation." ], [ "Introduction", "We currently live in a paradoxical stage of the information age.", "The more we gain access to unprecedented amounts of knowledge thanks to digital technologies, the less our societies seem capable of discerning what is true from what is false, even in the presence of overwhelming evidence in support of a particular position.", "For example, large segments of our societies do not believe in the reality of climate change [1] or believe in the relationship between vaccinations and autism [2].", "As recent studies indicate, over two-thirds of US adults get information from online and social media, with the proportion growing annually [3], [4].", "Hence, the impact such media have in shaping societal narratives cannot be understated.", "Online media empower their users to choose the news sources they want to be exposed to.", "This, in turn, makes it easier to restrict exposure only to narratives that are congruent to pre-established viewpoints [5], [6], [7], [8], and this positive feedback mechanism is further exacerbated by the widespread use of personalized news algorithms [9].", "In other words, confirmation bias [10], [11] is enabled at unprecedented scales [12].", "Another major impact of digital technologies has been the increase in connectivity fostered by the growth of online social networks, which plays a double-edged role.", "On the one hand, it can compound the effects of confirmation bias, as users are likely to re-transmit the same information they are selectively exposed to, leading to fragmented societies that break down into online “echo chambers” where the same opinions keep being bounced around [12], [13].", "On the other hand, it also translates into a potentially increased heterogeneity of the information and viewpoints users are exposed to [14], [15].", "Online social networks can therefore both improve and restrict the diversity of information individuals engage with, and their net effect is still very much debated.", "Empirical research is still in its infancy, with evidence for both positive and negative effects being found [15], [16], [17].", "The theoretical literature is lagging somewhat further behind.", "While there exist a plethora of models related to information diffusion and opinion formation in social networks, a sound theoretical framework accounting for the emergence of the phenomena that are relevant to modern information consumption (rather than explicitly introducing them ad hoc), is still largely lacking.", "In bounded confidence models [18], [19], [20] agents only interact with others sharing similar opinions, and thus are characterized by a form of confirmation bias.", "In such models polarization is a natural outcome assuming agents are narrow enough in their choice of interaction partners [21].", "However, these models tend to lack behavioural micro-foundations and a clear mechanism to link information diffusion to opinion formation, making it hard to draw conclusions about learning and accuracy amongst agents.", "Social learning models [22], [23], [24] provide a broader, empirically grounded, and analytically tractable framework to understand information aggregation [25].", "Their main drawback, however, is that by design they tend to produce long run population consensus, hence fail to account for any form of opinion heterogeneity or polarization [26].", "Polarization can be generated by introducing “stubborn” agents that remain fully attached to their initial opinions rather than interacting and learning from their neighbors [27], [28], a mechanism reminiscent of confirmation bias.", "However, the conditions under which polarization occurs are very strict, as populations converge towards consensus as soon as stubborn agents accept even a negligible fraction of influence from their neighbors [26].", "A similar phenomenon is explored in social physics literature where it is referred to as networks with “zealots”, which similarly impede consensus, such as in [29].", "A key distinction in the model we introduce in the following is that all agents are free to vary their opinions over time, resulting in cascade dynamics that separate consensus and polarization regimes.", "Overall, while it is clear from the literature that some notion of “bias” in networks is a key requirement for us to reproduce realistic dynamics of opinion formation, it is still difficult to provide a unified framework that can account for information aggregation, polarization and learning.", "The purpose of the present paper is to develop a framework that naturally captures the effect of large-scale confirmation bias on social learning, and to examine how it can drastically change the way a networked, decentralized, society processes information.", "We are able to provide analytic results at all scales of the model.", "At the macroscopic scale, we determine under what conditions the model ends up in a polarised state or cascades towards a consensus.", "At the mesoscopic scale, we are able to provide an intuitive chracterization of the trade-off between bias and connectivity in the context of such dynamics, and explain the role echo chambers play in such outcomes.", "At the microscopic scale, we are able to study the full distribution of each agent's available information and subsequent accuracy, and demonstrate that small amounts of bias can have positive effects on learning by preserving information heterogeneity.", "Our model unveils a stylized yet rich phenomenology which, as we shall discuss in our final remarks, has substantial correspondence with the available empirical evidence.", "We consider a model of a social network described by a graph $G = (V,E)$ consisting of a set of agents $V$ (where $\\vert V \\vert $ = $n$ ), and the edges between them $E$ .", "Each agent seeks to learn the unobservable ground truth about a binary statement $X$ such as, e.g., “global warming is / is not happening” or “gun control does / does not reduces crime”.", "The value $X = +1$ represents the statement's true value, whose negation is $X=-1$ .", "Following standard social learning frameworks [25], at the beginning of time ($t = 0$ ), each agent $i$ ($i = 1, \\ldots , n$ ) independently receives an initial signal $s_i = \\pm 1$ , which is informative of the underlying state, i.e.", "$p=\\mathrm {Prob}(s_i=+1 | X=+1)=1-\\mathrm {Prob}(s_i=-1 | X=+1) > 1/2$ .", "Signals can be thought of as news, stories, quotations, etc., that support or detract from the ground truth.", "The model evolves in discrete time steps, and at each time step $t > 0$ all agents synchronously share with their neighbors the full set of signals they have accrued up to that point.", "For example, the time $t=1$ information set of an agent $i$ with two neighbors $j$ and $\\ell $ will be $s_i(t=1) = \\lbrace s_i,s_j,s_\\ell \\rbrace $ , their time $t=2$ set will be $s_i(t=2) = \\lbrace s_i,s_i,s_i,s_j,s_j,s_\\ell ,s_\\ell ,s_{d = 2}\\rbrace $ , where $s_{d = 2}$ denotes the set of all signals incoming from nodes at distance $d = 2$ (i.e., $j$ 's and $\\ell $ 's neighbors), and so on.", "Furthermore, we define the following: $x_i(t) = \\frac{N_i^+(t)}{N_i^+(t)+N_i^-(t)} \\ ,$ where $N_i^+(t)$ and $N_i^-(t)$ denote, respectively, the number of positive and negative signals accrued by $i$ up to time $t$ .", "We refer to this quantity as an agent's signal mix, and we straightforwardly generalize it to any set of agents $C \\subseteq V$ , i.e., we indicate the fraction of positive signals in their pooled information sets at time $t$ as $x_C(t)$ .", "The list of all agents' signal mixes at time $t$ is vectorised as $x(t)$ .", "Each agent forms a posterior belief of the likelihood of the ground truth given their information sets using Bayes' rule.", "This is done under a bounded rationality assumption, as the agents fail to accurately model the statistical dependence between the signals they receive, substituting it with a naive updating rule that assumes all signals in their information sets to be independent ($\\mathrm {Prob}(X|s_i(t)) = \\mathrm {Prob}(s_i(t)|X)\\mathrm {Prob}(X) / \\mathrm {Prob}(\\textbf {s}_i(t))$ , where $\\mathrm {Prob}(s_i(t)|X)$ is computed as a factorization over probabilities associated to individual signals, i.e.", "$\\mathrm {Prob}(s_i(t)|X) = \\prod _c \\mathrm {Prob}\\left(s_i^{(c)}(t)|X \\right)$ , where $s_i^{(c)}(t)$ denotes the $c$ th component of the vector), which is a standard assumption of social learning models [25].", "Under such a framework (and uniform priors), the best guess an agent can make at any time over the statement $X$ given their information set is precisely equal to their orientation $y_i(t)$ , where $y_i(t) = +1$ for $x_i(t) > 1/2$ and $y_i(t) = -1$ for $x_i(t)<1/ 2$ (without loss of generality, in the following we shall choose network structures that rule out the possibility of $x_i(t) = 1/2$ taking place).", "The orientations of all $n$ agents at time $t$ are vectorized as $y(t)$ ; the fraction of positively oriented agents in a group of nodes $C \\subseteq V$ is denoted as $y_C(t)$ .", "The polarization $z_C(t) = \\min (y_C(t),1-y_C(t))$ of the group $C$ is then defined as the fraction of agents in that group that have the minority orientation.", "Note that polarization equals zero when there is full consensus and all agents are either positively or negatively oriented.", "It is maximized when there are exactly half the group in each orientation.", "It is useful to think of $x(t)$ , $y(t)$ and $z_V(t)$ as respectively representing the pool of available signals, the conclusions agents draw on the basis of the available signals, and a summary measure of the heterogeneity of agents' conclusions.", "In the context of news diffusion, for example, they would represent the availability of news of each type across agents, the resulting agents' opinions on some topic, and the extent to which those opinions have converged to a consensus.", "We distinguish between two kinds of agents in the model: unbiased agents and biased agents.", "Both agents share signals and update their posterior beliefs through Bayes' rule, as described in the previous section.", "However, they differ in how they acquire incoming signals.", "Unbiased agents accept the set of signals provided by their neighbours without any distortion.", "On the other hand, biased agents exercise a model of confirmation bias [30], [11], and are able to distort the information sets they accrue.", "We denote the two sets of agents as $\\mathcal {U}$ and $\\mathcal {B}$ , respectively.", "To describe the behaviour of these biased agents we use a slight variation of the confirmation bias model introduced by Rabin and Shrag [31].", "We refer to an incoming signal $s$ as congruent to $i$ if it is aligned with $i$ 's current orientation, i.e.", "if $s = y_i(t)$ , and incongruent if $s = - y_i(t)$ .", "When biased agents are presented with incongruent signals, they reject them with a fixed probability $q$ and replace them with a congruent signal, which they add to their information set and propagate to their neighbors.", "We refer to $q$ as the confirmation bias parameter.", "Denote the set of positively (negatively) oriented biased agents at time $t$ as ${\\mathcal {B}^+}(t)$ (${\\mathcal {B}^-}(t)$ ), and the corresponding fraction as $y_{\\mathcal {B}}(t) = \\vert {\\mathcal {B}^+}(t) \\vert / \\vert \\mathcal {B} \\vert $ .", "Note that this is an important departure from “stubborn agent” models, as such biased agents do have a non-zero influence from their neighbours, and they can change their beliefs over time as they aggregate information.", "An intuitive interpretation of what this mechanism is intended to model is as follows: biased agents are empowered to reject incoming signals they disagree with, and instead refer to preferred sources of information to find signals that are congruent with their existing viewpoint (see Fig.", "REF ).", "This mechanism models both active behaviour, where agents deliberately choose to ignore or contort information that contradicts their beliefs (mirroring the “backfire effect” evidenced both in psychological experiments [32], [33] and in online social network behaviour [34]), and passive behaviour, where personalized news algorithms filter out incongruent information and select other information which coheres with the agents' beliefs [15].", "In the following, we shall denote the fraction of biased agents in a network as $f$ .", "We shall refer to networks where $f=0$ as unbiased networks, and to networks where $f > 0$ as biased networks.", "For the bulk of the analytic results in the paper, we assume that the social network $G$ is an undirected $k$ -regular network.", "The motivation for this is two-fold.", "Firstly, empirical research [35] suggests that for online social networks such as Facebook (where social connections are symmetric), heterogenous network features such as hubs do not play a disproportionately significant role in the diffusion of information.", "Intuitively, while social networks themselves might be highly heterogenous, the network of information transmission is a lot more restricted, as individuals tend to discuss topics with a small group such as immediate friends and family.", "Secondly, utilizing a simple $k$ -regular network allows for considerable analytical tractability.", "However, one can show that our main results can be easily extended to hold under a variety of network topologies characterized by degree heterogeneity.", "The assumption of regular network structure (coupled with the aforementioned synchronous belief update dynamics) allows the information sets of all agents to grow at the same rate, and as a result the evolution of the signal mix $x(t)$ can be mapped to a DeGroot averaging process for unbiased networks [36]: $x(t) = A \\ x(t-1)$ , where $A$ is an $n \\times n$ matrix with entries $a_{ij} = 1/(k+1)$ for each pair $(i,j)$ of connected nodes.", "For biased networks, one can demonstrate (see Section S1 of the Supplementary Materials) that the above confirmation bias mechanics can be reproduced by introducing a positive and negative “ghost” node which maintain respective signal mixes of 1 and 0.", "Biased agents sample each signal from their orientation-aligned ghost node nodes with probability $q$ , and from their neighbourhood with probability $(1-q)$ .", "Furthermore, while the process is stochastic, we also show that it converges to a deterministic process with a simple update matrix described as follows.", "In Section S3 of the Supplementary Materials we discuss in detail the correspondence and convergence between the stochastic and deterministic processes.", "Biased agents down-weight their connections to neighbours by a factor $(1-q)$ and place the remaining fraction $kq/(k+1)$ of their outgoing weight on the corresponding ghost node.", "With these positions the updating process now simply reads $\\hat{x}(t) = \\hat{A}(t) \\hat{x}(t-1)$ , where $\\hat{A}(t)$ is an $(n+2) \\times (n+2)$ asymmetric matrix whose entries in its $n \\times n$ upper-left block are as those in $A$ , except $\\hat{a}_{ij} = (1-q)/(k+1)$ when $i \\in \\mathcal {B} = \\mathcal {B}^+ \\cup \\mathcal {B}^-$ .", "The positive (negative) ghost node corresponds to node $n+1$ ($n+2$ ) of the augmented matrix $\\hat{A}(t)$ , and we shall label it as $+$ ($-$ ) for convenience, i.e., we shall have $\\hat{a}_{i+} = kq/(k+1)$ ($\\hat{a}_{i-} = kq/(k+1)$ ) for $i \\in \\mathcal {B}^+$ ($i \\in \\mathcal {B}^-$ ) and $\\hat{a}_{++} = \\hat{a}_{--} = 1$ .", "Similarly, $\\hat{x}(t)$ denotes an augmented signal mix vector where $\\hat{x}_+(t) = \\hat{x}_{n+1}(t) = 1$ , and $\\hat{x}_-(t) = \\hat{x}_{n+2}(t) = 0$ .", "The time dependence of the matrix $\\hat{A}(t)$ is due to the fact that whenever a biased agent switches orientation its links to the ghost nodes change.", "This happens whenever the agent's signal mix $x_i(t)$ (see Eq.", "REF ) goes from below to above $1/2$ or vice versa, due to an overwhelming amount of incongruent incoming signals from its neighbors.", "For example, when switching from being positively to negatively oriented, a biased agent $i$ will change its links as follows: $\\hat{a}_{i+} = kq/(k+1) \\rightarrow \\hat{a}_{i+} = 0$ , and $\\hat{a}_{i-} = 0 \\rightarrow \\hat{a}_{i-} = kq/(k+1)$ .", "In the following, all quantities pertaining to biased networks will be denoted with a $\\hat{}$ symbol.", "We provide a sketch of the above mapping in Fig.", "REF .", "There is an appealing intuition to this interpretation: biased agents have a “preferred” information source they sample from in lieu of incongruent information provided from their peers.", "If their beliefs change, their preferred information source can change.", "Figure: Sketch of an unbiased (left) and biased (right) network with three nodes.", "Below each sketch is the update matrix of the corresponding (deterministic) DeGroot model.", "The network on the right shows that the stochastic “signal distortion” behaviour can be approximated with a biased agent (ℬ\\mathcal {B}) reducing the weight it places on each of its neighbors to (1-q)/(k+1)(1-q)/(k+1), and placing the remaining weight kq/(k+1)kq/(k+1) on an external positively oriented “ghost” node.", "When a biased agents changes orientation, it switches such an edge to an external negatively oriented node instead.Our main focus will be on the long-run properties of the dynamics introduced above.", "In this respect, it is important to establish whether the agents actually reach an equilibrium over their signal mixes and orientations, or whether they continue to oscillate.", "It is straightforward to demonstrate that unbiased networks always converge to a limiting steady state for their signal mixes and orientations, which follows directly from the correspondence between such networks and DeGroot models [23].", "The convergence of biased networks is much less trivial to prove due to the non-linear dynamics introduced by the confirmation bias mechanics.", "It can be shown however that convergence holds under fairly general conditions, and in Section S2 of the Supplementary Materials we demonstrate such convergence under a number of network topologies.", "Building on this, we are able to establish the following results on $k$ -regular networks of any size (in the following, and throughout the rest of the paper, we shall denote the steady state value of a variable $v$ as $v^*$ ).", "For biased $k$ -regular networks, denote $t^*$ as the time after which biased agents cease switching their orientation.", "Define $\\hat{y}_{\\mathcal {B}}^*$ as the steady state fraction of positively oriented biased agents.", "Then the following holds (see Section S3 of the Supplementary Materials).", "(1) The signal mix vector $\\hat{x}(t)$ converges to some $\\hat{x}^* = \\hat{A}^*\\hat{x}(0)$ for both biased and unbiased networks, where $\\hat{A}^*$ is a steady-state matrix of influence weights which can be computed explicitly (see Section S3 of the Supplementary Materials).", "(2) Unbiased networks achieve consensus, and converge to influence weights of $a^*_{ij} = 1/n$ for all pairs $(i,j)$ .", "This ensures that, for all $i \\in V$ , $x^*_i=x^*_V=\\bar{x}(0)$ , where $\\bar{x}(0) = \\sum _{i=1}^n s_i$ is the intial average signal mix.", "(3) Biased networks where $\\hat{y}_{\\mathcal {B}}^* = 0,1$ achieve consensus, and converge to influence weights $\\hat{a}^*_{ij} = 0$ for all pairs $(i, j) \\in V$ , $\\hat{a}^*_{i+} = \\hat{y}_{\\mathcal {B}}^*$ and $\\hat{a}^*_{i-} = 1-\\hat{y}_{\\mathcal {B}}^*$ for all $i \\in V$ .", "(4) Biased networks where $ 0 < \\hat{y}_{\\mathcal {B}}^* < 1$ do not achieve consensus, and converge to influence weights $\\hat{a}^*_{ij} = 0$ for all $(i, j) \\in V$ , and $\\hat{a}^*_{i+} + \\hat{a}^*_{i-}=1$ for all $i \\in V$ .", "From the above, we can conclude that while unbiased networks efficiently aggregate the information available to them at $t=0$ , the outcome of the information aggregation process in biased networks ends up being entirely determined by the long-run orientations of biased agents.", "We shall devote the following sections to examine the consequences of the model in greater detail through mean field approximations coupled with numerical verifications on finite networks.", "We begin by studying the signal mix of unbiased agents in biased networks ($\\hat{x}_\\mathcal {U}^*$ ) to provide a like for like comparison with the fully unbiased networks.", "In the context of the diffusion of news, the global signal mix can be thought of as a model of the long term balance of news of different types that survive following the diffusion dynamics.", "For unbiased networks, it is demonstrated in [37] that $\\hat{x}_\\mathcal {U}^* = \\bar{x}(0)$ .", "That is, the steady state signal mix in unbiased networks precisely reflects the original, unbiased informative signals injected into the network.", "Determining the steady state signal mix of biased networks entails considering the interactions between three subpopulations - the unbiased agents $\\mathcal {U}$ , positively biased agents ${\\mathcal {B}^+}$ , and negatively biased agents $\\mathcal {B}^-$ .", "One can show (see Section S4 of the Supplementary Materials) that this can be approximated as: $ \\begin{pmatrix}\\hat{x}_\\mathcal {U}^* \\\\\\hat{x}_{{\\mathcal {B}^+}}^* \\\\\\hat{x}_{\\mathcal {B}^-}^*\\end{pmatrix}=\\begin{pmatrix}\\hat{y}_{\\mathcal {B}}(t^*) \\\\(1-q)\\hat{y}_{\\mathcal {B}}(t^*) +q \\\\(1-q)\\hat{y}_{\\mathcal {B}}(t^*)\\end{pmatrix} \\ .$ Let us now consider the situation under which $t^* = 0$ , i.e.", "where the initial orientation of each biased agent does not change, and is therefore equal to the initial signal it receives.", "Intuitively, this will occur for large $q$ which allows for biased agents to reject the majority of incongruent signals they receive (shortly we demonstrate in fact this generally occurs for $q>1/2$ ).", "Given Eq.", "REF , we can therefore calculate the steady state signal mix of any subset of agents based on our knowledge of the distribution of the initial signals.", "In this scenario, the average signal mix $\\hat{x}_\\mathcal {U}^*$ of unbiased agents is determined by the initial proportion of positively oriented biased agents $y_{\\mathcal {B}^+}(0)$ , which is the mean of $fn$ i.i.d.", "Bernoulli variables with probability $p$ (which, we recall, denotes the probability of an initially assigned signal being informative).", "One can compare this to unbiased networks ($f=0$ ), where the long run average signal mix is $x_V^* = \\bar{x}(0)$ , and is hence the mean of $n$ i.i.d.", "Bernoulli variables with probability $p$ .", "Applying the central limit theorem we see that injecting a fraction $f$ of biased agents therefore amplifies the variance of the long run global signal mix by a factor of $f^{-1}$ with respect to the unbiased case: $ x_V^* \\sim \\mathcal {N} \\left(p,\\frac{p(1-p)}{n} \\right) \\ \\rightarrow \\ \\hat{x}_\\mathcal {U}^* \\sim \\mathcal {N} \\left(p,\\frac{p(1-p)}{fn} \\right) \\ .$ This means that the “wisdom of unbiased crowds” is effectively undone by small biased populations, and the unbiased network's variability is recovered for $f \\rightarrow 1$ , and not for $f \\rightarrow 0^+$ , as one might intuitively expect.", "Consider now the general case where biased agents can, in principle, switch orientation a few times before settling on their steady state orientation.", "Using mean-field methods one can determine the general conditions under which a cascade in these orientation changes can be expected (see Section S4 of the Supplementary Materials) but here we only provide some intuition.", "As $q$ is lower, it is easier for an initial majority camp of biased agents to convert the minority camp of biased agents.", "As the conversion of the minority camp begins, this triggers a domino effect as newly converted biased agents add to the critical mass of the majority camp and are able to overwhelm the minority orientation.", "This mechanism allows us to derive analytic curves in the parameter space to approximate the steady state outcome of the unbiased agent population's average signal mix based on the orientations of the biased agents at time 0: $ \\hat{x}^*_\\mathcal {U} = \\left\\lbrace \\begin{array}{ll}\\hat{y}_{\\mathcal {B}}(0) & \\text{for} \\ \\frac{1-2q}{2(1-q)} \\le \\hat{y}_{\\mathcal {B}}(0) \\le \\frac{1}{2(1-q)} \\\\1 & \\text{for} \\ \\hat{y}_{\\mathcal {B}}(0) > \\frac{1-2q}{2(1-q)} \\\\0 & \\text{for} \\ \\hat{y}_{\\mathcal {B}}(0) < \\frac{1}{2(1-q)} .", "\\\\\\end{array}\\right.$ The above result is sketched in Fig.", "REF , and we have verified that it matches numerical simulations even for heterogenous networks.", "For $1 / 2 < q \\le 1$ biased agents can convert at least half of the incongruent signals they receive to their preferred type, meaning that biased agents of either orientation cannot be eradicated from the network, which preserves signals of both types in the steady state.", "For sufficiently small values of $q$ , on the other hand, small variations in the initial biased population translate to completely opposite consensus, and only by increasing the confirmation bias $q$ , paradoxically, the model tends back to a balance of signals that resembles the initially available information.", "Figure: Average steady state signal mix of unbiased agents (x ^ 𝒰 * \\hat{x}_{\\mathcal {U}}^*) as a function of the time t=0t=0 fraction of positively oriented biased agents (y ^ ℬ (0)\\hat{y}_\\mathcal {B}(0)) and confirmation bias qq.", "The color gradient denotes the average long run signal mix for unbiased agents from 0 to 1.", "The top-left and bottom-left regions are characterized, respectively, by a global signal mix of 1 and 0 respectively, and are separated by a discontinuous transition from a region characterized by a steady state that maintains a mixed set of signals.", "That is, in the top-left region almost all negative signals have been removed from the network, leaving almost entirely positive signals in circulation (and vice versa for the bottom-left region).", "In the remaining region, signal mixes of both types survive in the long run, and the balance between positive and negative signals reflects the fraction of positively oriented biased agents.", "The lower qq falls, the easier it is to tip the network into a total assimilation of a single signal type.", "Results are shown for simulations on kk-regular (top left), Erdős-Rényi (top right), Barabasi-Albert (bottom left) and Small-world (bottom right) networks.", "Analytic predictions (given by Eq.", "()) are denoted by solid red lines.", "The parameters used in the simulations were n=10 3 n = 10^3, p=0.51p = 0.51, k=6k = 6 (which corresponds to an average degree in all cases), f=0.4f = 0.4.Putting the above results together, we note that biased networks with small $f$ and $q$ are, surprisingly, the most unstable.", "Indeed, such networks sit on a knife-edge between two extremes where one signal type flourishes and the other is totally censored.", "In this context, the model indicates that confirmation bias helps preserve a degree of information heterogeneity, which, in turn, ensures that alternative viewpoints and information are not eradicated.", "In subsequent sections we consider a normative interpretation of this effect in the context of accuracy and learning.", "So far we have derived the statistical properties of the average steady state signal mix across all unbiased agents.", "We now aim to establish how these signals are distributed across individual agents.", "Throughout the following, assume the global steady state signal mix $\\hat{x}_\\mathcal {U}^*$ has been determined.", "In the limit of large $n$ and $k$ , $x_i^*$ for $i \\in \\mathcal {U}$ is normally distributed with mean $\\hat{x}_\\mathcal {U}^*$ and variance $ \\sigma ^2(\\hat{x}_\\mathcal {U}^*)$ that can be approximated as follows (see Section S4 of the Supplementary Materials): $ \\sigma ^2(\\hat{x}_\\mathcal {U}^*) \\approx \\frac{f q^2}{k} (\\hat{x}^*_\\mathcal {U}(1-\\hat{x}^*_\\mathcal {U})) \\ ,$ and this result is quite accurate even when compared with simulations for small $n$ and $k$ , as demonstrated in Figure REF .", "Figure: Distribution of individual unbiased agents' steady state signal mixes (points) vs analytic predictions (dashed lines).", "As discussed in the main text, the model predicts such distribution to be a Gaussian (for both large nn and kk) with mean equal to x ^ 𝒰 * \\hat{x}^*_{\\mathcal {U}} and variance given by Eq.", "(5).", "x ^ 𝒰 * \\hat{x}^*_{\\mathcal {U}} is kept fixed to demonstrate the effect of varying ff and kk.", "As shown in the case for k/f=120k/f = 120, ff and kk trade off, and scaling both by the same constant results in the same distribution.", "The parameters used in the simulations were n=10 4 n = 10^4, x ^ 𝒰 * =0.55\\hat{x}^*_{\\mathcal {U}} = 0.55, q=0.6q = 0.6.This result further shows that the presence of biased agents is effectively responsible for the polarization of unbiased agents in the steady state.", "Indeed, both a larger biased population and higher confirmation bias - i.e.", "higher $f$ or $q$ , respectively - result in an increased variance and steady state polarization $\\hat{z}_\\mathcal {U}^*$ , since a larger variance $\\sigma ^2(x_\\mathcal {U}^*)$ implies larger numbers of agents displaying the minority orientation.", "This is illustrated in the top left panel of Fig.", "REF .", "On the other hand, a larger degree $k$ contrasts this effect by creating more paths to transport unbiased information.", "It is worth pointing out that the variance in Eq.", "REF does not decay with $n$ , showing that steady state polarization persists even in the large $n$ limit.", "We refer to this as the bias-connectivity trade-off, and the intuition behind this result is illustrated in Figure REF .", "Figure: For illustration of the bias-connectivity trade-off, consider a Cayley 3-tree structure and let q=1q=1.", "When f=0f=0, each unbiased node has 3 independent sources of novel information, one from each branch associated with a neighbour.", "As ff increases, unbiased neighbours are steadily replaced with biased agents that act as gatekeepers and restrict the flow of novel information from their branches, resulting in an equivalence with fewer branches overall.", "Increasing social connectivity kk bypasses biased neighbours and enables the discovery of novel information.", "This can also be interpreted as agents being encased in “echo chambers” as bias grows, and circumventing these chambers as connectivity increases.Further intuition for this result can be found at the mesoscopic level of agent clusters, where we see the emergence of natural “echo chambers” in the model.", "We define an echo chamber $C$ as a subset of unbiased agents such that: $C = \\lbrace i \\in \\mathcal {U}: \\partial _i \\in \\mathcal {B}\\cup C\\ \\wedge \\partial _i \\cap C \\ne \\emptyset \\rbrace $ , where $\\partial _i$ denotes the neighbourhood of agent $i$ .", "In other words, an echo chamber is a set of connected unbiased agents such that all nodes are either connected to other nodes in the echo chamber or to biased agents.", "Therefore, biased agents form the echo chamber's boundary, which we refer to as $\\partial _C$ .", "Echo chambers in our model represent groups of unbiased agents that are completely surrounded by biased agents who effectively modulate the information that can flow in and out of these groups.", "Echo chambers allow us to examine the qualitative effect of confirmation bias ($f, q$ ) and connectivity $k$ .", "Let us label the fraction of unbiased agents enclosed in an echo chamber as $\\eta _C$ .", "Leveraging some simple results from percolation theory[38] we can show that $\\eta _C$ increases with $f$ and decreases with $k$ , as the creation of more pathways that bypass biased agents effectively breaks up echo chambers.", "Furthermore, the equilibrium signal mix of unbiased agents inside echo chambers is well approximated by a weighted average between the signal mix of the biased agents surrounding them ($x^*_{\\partial _C}$ ) and the signal mix $x_{\\mathcal {U}}^*$ of the whole population: $x_C^* = q \\ x^*_{\\partial C} + (1-q) x_{\\mathcal {U}}^*$ .", "The confirmation bias parameter $q$ therefore determines the “permeability” of echo chambers to the information flow from the broader network.", "Hence, unbiased agents enclosed in echo chambers are likely to be exceedingly affected by the views of the small set of biased agents surrounding them, and, as such, to hold information sets that are unrepresentative of the information available to the broader network.", "In doing so, we can envision these echo chambers as effective “building blocks” of the overall polarization observed in the network.", "Up until now, we have not attempted to make any normative interpretations of the ground truth $X=+1$ .", "In the following, we shall refer to unbiased agents whose steady state orientation is positive (negative) as accurate (inaccurate) agents, and we shall define the overall accuracy $\\mathcal {A}(G)$ of a network $G$ as the expected fraction of accurate agents in the steady state.", "This allows us to investigate how biased and unbiased networks respond to changes in the reliability of the available information, which ultimately depends on the prevalence of positive or negative signals (modulated by the parameter $p = \\mathrm {Prob} ( s = +1 | X = +1 )$ ), which, loosely speaking, can be interpreted as “real” and “fake” news.", "The accuracy of unbiased networks obtains a neat closed form that can be approximated as $\\mathcal {A}(G | f=0) \\approx \\mathrm {erfc} ( (1-2p) \\sqrt{n / 2} ) / 2$ (see [37]).", "For $f>0$ , we compute the expected accuracy as the expected fraction of accurate agents with respect to a certain global signal mix.", "This reads: $ \\mathcal {A}(G | f>0) = \\frac{1}{2} \\int _0^1 \\mathrm {d} x_\\mathcal {U}^* \\ P(x_\\mathcal {U}^*) \\ \\mathrm {erfc} \\left(\\frac{1/2 - x_\\mathcal {U}^*}{\\sqrt{2}\\sigma _{x_\\mathcal {U}^*}} \\right) \\ ,$ where $P(x_\\mathcal {U}^*)$ is the distribution of the average signal mix across unbiased agents (see Eq.", "REF ) (we take the simplifying case of $q > 1/2$ , but this can easily be extended to the case for $q \\le 1/2$ using Eq.", "REF ), and where we have used the previously mentioned Gaussian approximation for the distribution of individual signal mixes (whose variance $\\sigma _{x_\\mathcal {U}^*}^2$ is given by Eq.", "REF ).", "Figure: polarization z ^ 𝒰 * =1-y ^ 𝒰 +* \\hat{z}_\\mathcal {U}^* = 1-\\hat{y}^{+*}_\\mathcal {U} of the unbiased agent population as a function of the average signal mix x ^ 𝒰 * \\hat{x}^*_\\mathcal {U} in the steady state calculated as erfc (x ^ 𝒰 * -1/2)/(2σ(x ^ 𝒰 * )))/2\\mathrm {erfc} (\\hat{x}^*_\\mathcal {U} - 1/2) / (\\sqrt{2}\\sigma (\\hat{x}^*_\\mathcal {U})))/2, with σ(x ^ 𝒰 * )\\sigma (\\hat{x}^*_\\mathcal {U}) given by Eq.", "(top left panel).", "Expected accuracy (Eq. )", "as a function of the initial signals' informativeness pp (top right panel) and of the fraction ff of biased agents (bottom left panel).", "behaviour of the accuracy-maximizing value f * f^* and of the corresponding accuracy 𝒜(G|f=f * )\\mathcal {A}(G | f = f^*) as functions of kk (bottom right panel).", "In the first three panels the model's parameter n=10 3 n=10^3, q=1q = 1, k=8k=8, while the parameters in the last panel are n=10 3 n=10^3, q=1q = 1, p=0.53p = 0.53.", "In all cases we assume X=+1X=+1 without loss of generality.The top right panel in Fig.", "REF contrasts biased and unbiased networks, and shows how the former remain very inefficient in aggregating information compared to the latter, even as the reliability of the signals ($p$ ) improve.", "However, accuracy in biased networks is non-monotonic with respect to $f$ .", "As shown in the bottom left panel in Fig.", "REF , accuracy reaches a maximum in correspondence of an optimal value $f^*$ (see Section S5 of the Supplementary Materials for a comparison with numerical simulations).", "Intuitively, this is because for small values of $f$ , as already discussed, the model can converge to the very inaccurate views of a small set of biased agents.", "As $f$ grows, the views of the two biased camps tend to cancel each other out, and the signal set will match more closely the balance of the original distribution of signals (Eq.", "REF ).", "However, in doing so large values of $f$ lead to increased polarization (Eq.", "REF ), where accurate and inaccurate agents coexist.", "The trade-off between balance and polarization is optimised at $f^*$ .", "It is also interesting to note that, as shown in the bottom right panel of Fig.", "REF , the optimal fraction of biased agents $f^*$ and the corresponding maximum accuracy $\\mathcal {A}(G | f = f^*)$ both increase monotonically with the degree.", "This indicates that as networks are better connected, they can absorb a greater degree of confirmation bias without affecting accuracy.", "We now seek to test some of the model's predictions against real world data.", "Clearly, a full validation of the model will require an experimental setup, but a simple test case on existing data can demonstrate the utility of the framework in disambiguating the competing effects of bias and connectivity.", "We employ the model to investigate the effect of online media in the process of opinion formation using survey data.", "Empirical literature on this phenomenon has been mixed, with different analyses reaching completely opposite conclusions, e.g., showing that Internet access increases [39], decreases [40] and has no effect [41] on opinion polarization.", "In Section S6 of the Supplementary Materials we briefly review how our model can help better understand some of the inconsistencies between these results.", "Our position is that the effect of Internet access can be split into the effect it has on social connectivity and social discussion ($k$ ) and the residual effect it has on enabling active and passive confirmation bias behaviours ($f$ ).", "As per Eq.", "REF , assuming the majority of the population accurately learns the ground truth ($x_\\mathcal {U}^*>1/2$ ), increases in social discussion should improve consensus around the truth and reduce the fraction of inaccurate agents.", "However, when controlling for the improvement in social connectivity, we should expect an increase in Internet access to have the opposite effect.", "We utilise data from the Yale Programme on Climate Change Communication [42], which provides state and county level survey data on opinions on global warming, as well as information about the propensity to discuss climate change with friends and family, which proxies connectivity $k$ .", "We combine this with FCC reports on county level broadband internet penetration, which proxies for $f$ after controlling for the considerable effect this has on social connectivity.", "We also account for a range of covariates (income, age, education, etc) and make use of an instrumental variable approach to account for simultaneous causality.", "We then attempt to predict the fraction of each county's population that correctly learns that “global warming is happening” (see Section S6 of the Supplementary Materials for details on assumptions and results).", "As predicted by our model, we find the accurate fraction of the population to have statistically significant positive relationships with $k$ , and a negative relationship with $f$ .", "We find such relationships to account for $65\\%$ of the variance in the data.", "This indicates that, after controlling for the improvements on social connectivity, Internet access does indeed increase polarization and reduces a population's ability to accurately learn.", "While simple, this analysis illustrates the value of our model: by explicitly accounting for the separate effects of large-scale online communication (confirmation bias and connectivity), it can shed light on the mixed empirical results currently available in the literature.", "In Section S6 of the Supplementary Materials we explore this further by reviewing some of these empirical results and showing how our model provides useful further interpretations of available findings.", "It should be emphasized that this result is merely an initial exploration of how our model can provide some testable predictions to empirical data, as opposed to a detailed effort to understand the effect of Internet access on global warming beliefs.", "Having said that, the initial results are encouraging, and we hope the clarity of the analytic results of our model pave the way for testing variations of the idea of biased information aggregation in a range of outcomes and settings.", "We introduced a model of social learning in networked societies where only a fraction of the agents update beliefs unbiasedly based on the arrival of new information.", "The model only provides a stylized representation of the real-world complexity underpinning the propagation of information and the ensuing opinion formation process.", "Its value stands in the transparency of the assumptions made, and in the fact that it allows us to “unpack” blanket terms such as, e.g., social media and Internet penetration, by assigning specific parameters to their different facets, such as connectivity ($k$ ) and the level of confirmation bias it enables in a society ($f, q$ ).", "This, in turn, yields quantitative testable predictions that contribute to shed light on the mixed results that the empirical literature has so far collected on the effects online media have in shaping societal debates.", "Our model indicates the possibility that the “narratives” (information sets) biased societies generate can be entirely determined by the composition of their sub-populations of biased reasoners.", "This is reminiscent of the over-representation in public discourse of issues that are often supported by small but dedicated minorities, such as GMO opposition [43], and of the domination of political news sharing on Facebook by heavily partisan users [15]; it also resonates with recent experimental results showing that committed minorities can overturn established social conventions [44].", "The model indicates that societies that contain only small minorities of biased individuals ($f \\rightarrow 0^+$ ) may be much more prone to producing long run narratives that deviate significantly from their initially available information set (see Eq.", "REF ) than societies where the vast majority of the agents actively propagate biases.", "This resonates, for example, with Gallup survey data about vaccine beliefs in the US population, where only $6\\%$ of respondents report their belief in the relationship between vaccines and autism, but more than $50\\%$ report to be unsure about it and almost $75\\%$ report to have heard about the disadvantages of vaccinations [45].", "Similarly, the model suggests that mild levels of confirmation bias ($q \\ll 1$ ) may prove to be the most damaging in this regard, as they cause societies to live on a knife-edge where small fluctuations in the information set initially available to the biased agent population can completely censor information signals from opposing viewpoints (see Fig.", "REF ).", "All in all, the model suggests that a lack of confirmation bias can ensure that small biased minorities much more easily hijack and dictate public discourse.", "The model suggests that as the prevalence of biased agents grows, the available balance of information improves and society is more likely to maintain a long term narrative that is representative of all the information available.", "On the other hand, it suggests that such societies may grow more polarised.", "When we examine the net effect of this trade off between bias and polarization through an ensemble approach, our model suggests that the expected accuracy of a society may initially improve with the growth of confirmation bias, then reach a maximum at a value $f^*$ before marginal returns to confirmation bias are negative, i.e.", "confirmation bias experiences an “optimal” intermediate value.", "The model suggests that such value and its corresponding accuracy should increase monotonically with a society's connectivity, meaning that more densely connected societies can support a greater amount of biased reasoners (and healthy debate between biased camps) before partitioning into echo chambers and suffering from polarization." ], [ "Update dynamics as random matrix $A(t)$ .", "Consider the set of signals ${s}_i(t)$ possessed by a positively oriented agent $i$ at time $t$ (i.e., $i \\in \\mathcal {B}^+$ ).", "This will consist of a set of signals retained from the previous time step, ${s}_i(t-1)$ , and a set of biased signals ${s}^\\prime _i(t)$ constructed from the signals available from the nodes $j \\in \\partial _i$ at the end of time $(t-1)$ .", "Let ${s}^*_{i}(t) = \\bigcup _{j} {s}_{j}(t-1)$ be the set of the unbiased signals available to node $i$ at time $t$ , i.e.", "the set of nodes $i$ will receive from her neighbors before applying the confirmation bias function.", "Let $s_i^{*(a)}$ ($a = 1, \\ldots , k(k+1)^{t-1}$ ) be a generic signal in the set ${s}^*_{i}(t)$ .", "After the application of the confirmation bias function, this will be turned into a signal $s^{\\prime (a)}_{i}(t) \\in {s}^\\prime _i(t)$ such that $s^{\\prime (a)}_{i}(t) = \\pm s^{*(a)}_{i}(t)$ according to the following probabilities: $\\mathrm {Prob}(s^{\\prime (a)}_{i} = +1|s^{*(a)}_{i} = -1) &=& q \\\\ \\nonumber \\mathrm {Prob}(s^{\\prime (a)}_{i} = -1|s^{*(a)}_{i} = -1) &=& (1-q) \\\\ \\nonumber \\mathrm {Prob}(s^{\\prime (a)}_{i} = +1|s^{*(a)}_{i} = +1) &=& 1 \\\\ \\nonumber \\mathrm {Prob}(s^{\\prime (a)}_{i} = -1|s^{*(a)}_{i} = +1) &=& 0 \\ .$ According to the above rules, agent $i$ checks the value of the new incoming signal, and flips it with probability $q$ if it is incongruent with respect to her current orientation.", "This is entirely equivalent to node $i$ sampling with probability $q$ from the set ${s}^*_{i}(t)$ , and with probability $1-q$ from an equally sized set of positive signals belonging to a positively oriented “ghost” node.", "Let us consider the number $N_i^+(t)$ of positive signals possessed by agent $i$ at time $t$ .", "Due to the above rules, its time evolution will be such that $N_i^+(t) = N_i^+(t-1) + \\sum _{j \\in \\partial _i} \\left( N_j^+(t-1) + w_i(t) N_j^-(t-1) \\right) \\ ,$ where $w_i(t) \\in [0,1]$ is a random variable denoting the fraction of negative signals successfully distorted by $i$ of those received by its neighbours at time $t$ , with distribution such that $w_i(t) N^-_{\\partial _i}(t) \\sim \\mathrm {Bin}(N^-_{\\partial _i}(t),q)$ , where $N^-_{\\partial _i}(t)$ is the number of negative signals received by $i$ from her neighbourhood at time $t$ .", "When considering agent $i$ 's signal mixWe recall that the signal mix, as per Eq.", "(1) of the main paper, is defined as the fraction of positive signals possessed by an agent at a certain time, i.e., $x_i(t) = N_i^+(t)/(N_i^+(t) + N_i^-(t)) = N_i^+(t) / (k+1)^t.$, the above translates to $ x_i(t) = \\frac{1}{k+1} \\left( x_i(t-1) + \\left(1-w_i(t) \\right) \\sum _{j \\in \\partial _i} x_j(t-1) + w_i(t) k \\right) \\ .$ Similarly, for a negatively oriented biased agent (i.e., $i \\in \\mathcal {B}^-$ ) we have $ x_i(t) = \\frac{1}{k+1} \\left( x_i(t-1) + \\left(1-w_i(t) \\right) \\sum _{j \\in \\partial _i} x_j(t-1) \\right) \\ ,$ with $w_i(t) N^+_{\\partial _i}(t) \\sim \\mathrm {Bin}(N^+_{\\partial _i}(t),q)$ .", "Combining Eqs.", "(REF ) and (REF ) with the time evolution for the signal mix of unbiased agents, which reads $ x_i(t) = \\frac{1}{k+1} \\left( x_i(t-1) + \\sum _{j \\in \\partial _i} x_j(t-1) \\right) \\ ,$ we can see that the time evolution for the vector of signal mixes $x(t)$ can be written as $ \\hat{x}(t) = \\hat{A}(t) \\hat{x}(t-1) \\ ,$ where $\\hat{x} = [x^T, 1, 0]^T$ where the latter terms represent the (fixed) signal mixes of the ghost nodes and the $\\hat{x}(t)$ the signal mixes of the original set.", "$\\hat{A}(t)$ is an $(n+2) \\times (n+2)$ random matrix with entries with a block structure as follows: $\\hat{A}(t)=\\left[\\begin{array}{c|c}Q(t) & R(t) \\\\\\hline 0 & I\\end{array}\\right] \\ ,$ where $Q(t)$ is an $(n \\times n)$ matrix representing the original graph structure with $Q_{ii}(t) = \\frac{1}{k+1}$ , $Q_{ij}(t) = \\frac{1}{k+1}$ where $i$ is a unbiased agent connected to $j$ , $Q_{ij}(t) = \\frac{1-w_i(t)}{k+1}$ where $i$ is a biased agent connected to $j$ , and 0 otherwise.", "$R(t)$ is an $(n \\times 2)$ matrix representing connections from biased agents to their preferred ghost node (which we index by $+$ and $-$ ).", "$R_{i+}(t) = \\frac{w_i(t)k}{k+1}$ if $i \\in \\mathcal {B}^+$ and 0 otherwise.", "Analogous weights exist for negatively biased agents to the negative ghost node.", "$I$ is a $(2 \\times 2)$ identity matrix representing the weights of ghost nodes to themselves.", "0 is the $(2 \\times n)$ block of zeros representing the (lack of) edges outbound from the ghost nodes.", "Finally, it is worth noting that the above formulation consisting of two ghost nodes is fully equivalent to a formulation where each biased agent has a “personalized” ghost node that reflects their positive or negative orientation appropriately.", "In this case $\\hat{A}(t)$ is an $(n+fn) \\times (n+fn)$ matrix with an extra $fn$ ghost nodes added, one for each biased agent.", "However, while this formulation has a more favourable interpretation in terms of “content personalization”, it is less convenient analytically, so for the reminder of the Supplementary Information the simplified ghost node formulation will be utilised." ], [ "Almost sure convergence of $\\hat{A}(t)$ .", "We now proceed to show that stochastic weights $w_i(t)$ appearing in the matrix $\\hat{A}(t)$ of (REF ) converge almost surely to $q$ when $t \\rightarrow \\infty $ as long as at least one signal of each type is held by at least one node in the network.", "As such the random matrix $\\hat{A}(t)$ converges almost surely to a fixed matrix $\\hat{A} = \\mathbb {E}(\\hat{A}(t))$ .", "Let us consider $i \\in \\mathcal {B}^+$ .", "As established in the previous section, $w_i(t)$ is simply the fraction of negative signals held by node $i$ 's neighbours that $i$ successfully flips to positive at time $t$ .", "Let us also recall that $N^-_{\\partial _i}(t)$ represents this set of negative signals available from all $j \\in \\partial _i$ , and that each one is independently flipped to positive with probability $q$ .", "If we can establish that $N^{-}_{\\partial _i}(t)$ grows indefinitely as $t \\rightarrow \\infty $ , the Strong Law of Large Numbers (SLLN) can then be invoked to establish the desired result.", "Since $N^-_{\\partial _i}(t) = \\sum _{j \\in \\partial _i} N_j^{-}(t)$ , then if $i$ 's neighbours possess an increasing and unbounded number of negative signals over time, then $N^-_{\\partial _i}(t)$ will also be increasing and unbounded.", "As such, each $w_i(t)$ will converge almost surely to $q$ .", "Consider an arbitrary $j \\in \\partial _i$ .", "Note that since information sets are retained by agents at every time step, we can immediately rule out the possibility of that the number of negative signals held by agent $j$ shrinks over time, and we merely need to show that her set of negative signals does not remain constant over time.", "Let us assume that at least one negative signal has been injected into the network at $t=0$ , and that one agent $\\ell $ possesses such negative signal.", "In a strongly connected network (such as the $k$ -regular network we consider in the main paper), there exists at least one directed path from $k$ to $j$ of length $d$ .", "Let us indicate the probability of a negative signal successfully being transmitted from an agent $a$ to an agent $b$ along such path as $p_{ab}$ .", "We note that $p_{ab} = 1-q$ if $b \\in \\mathcal {B}^+$ and $p_{ab} = 1$ otherwise.", "Therefore, the probability of the signal successfully reaching $j$ in $d$ time steps is: $p_{\\ell j} = \\prod _{(a,b)} p_{ab} \\ge (1-q)^d > 0 \\ ,$ Which allows us to conclude that at each time step $t > d$ there exists a strictly positive probability that a negative signal is added to $j$ 's information set.", "This, in turn, implies that the set of negative signals obtained by $j$ will grow without bound for $t \\rightarrow \\infty $ , which establishes our result.", "Since this occurs for each $w_i$ , we can conclude also that $\\hat{A}(t) \\xrightarrow{} \\hat{A}$ , as well as the block submatrices $Q(t) \\xrightarrow{} Q$ and $R(t) \\xrightarrow{} R$ .", "The edges of these fixed matrices are identical to the structure outlined in the previous section except $w_i$ is replaced with $q$ .", "Finally it is worth noting that this convergence result depends only on the strong connectedness of $\\mathcal {G}$ and not on the edges from the biased agents to the ghost nodes.", "This is important as this means that even as the orientations of the biased agents change (which is reflected in the rewiring of these ghost node edges), the almost sure convergence is not interrupted.", "In this section, we show that biased agents cannot continue to switch orientation indefinitely, and instead settle into a fixed set of orientations given sufficient time.", "Recall that a biased agent $i \\in \\mathcal {B}$ switches her orientation $y_i(t)$ when her information sets switches from a majority of positive signals ($x_i(t) > 1/2$ ) to a majority of negative signals ($x_i(t) < 1/2$ ), or vice versa.", "We begin by arguing that in some network topologies there exists some $t$ after which biased agents cease switching their orientation.", "For convenience, we define a network as settled at $t^*$ if for all $t > t^*$ , $y_i(t^*) = y_i(t)$ for all $i \\in \\mathcal {B}$ .", "To do this, we first consider an “adversarial” toy example designed to maximise the likelihood of indefinite switching, and show that assuming perpetual switching leads to a contradiction even in this case.", "We then go on to show how other, more complex, network topologies are also guaranteed to settle.", "We limit to two topologies for brevity but these results can be extended.", "Alongside the extensive evidence from numerical simulations, we argue that the model is likely to settle for any arbitrary graph." ], [ "Two node network.", "Consider a network with two nodes, labeled 1 and 2 respectively, both of which are biased agents.", "Each node has a self-weight of $y$ and a weight of $(1-y)$ on its sole neighbourThis setting generalizes the one introduced in Eqs.", "(REF ) and (REF ), which is recovered for $k=1$ and $y = 1/2$ ..", "This schematic is illustrated in Figure REF .", "As has been established in , the signal distortion dyanmics can be mimicked by introducing two ghost nodes that represent a source of positive and negative signals respectively.", "The weights associated with these ghost nodes are random variables that converge almost surely to $q$ as $t \\rightarrow \\infty $ .", "Figure: Left: A schematic of the two node symmetric network.", "Right: A schematic of the two node symmetric network where ghost nodes are introduced to mimic the effect of the biased signals.In what follows, we show that this simplified model settles (i.e., both biased agents settle at a finite time on a pair of orientations that they do not thereafter change).", "For the purposes of illustration, for the moment let us consider the asymptotic case where the random weights have converged to a deterministic set of weights ($q$ ).", "The outline of this proof (and subsequent ones on alternative network structures) is to establish that in order for a biased agent $i$ to switch orientation, their neighbours must have signal mixes sufficiently far from $i$ 's that they can cause $i$ to switch orientation despite the fact that $i$ 's ghost node biases her learning to maintain “inertia” in the current orientation.", "However, at the same time, the network structure ensures that nodes tend to converge closely to their neighbourhood, which eventually prevents switching from occurring.", "The proof follows by contradiction.", "Suppose that the model never stabilizes, i.e., that at least one of the biased agents keeps switching perpetually.", "Suppose node 1 switches at arbitrary times $\\lbrace T\\rbrace = \\ldots < t_{n-2} < t_{n-1} < t_{n} < \\ldots $ .", "We do not assume for now that times in $T$ are over consecutive time steps, the gap between them can be as large as intended (see Fig.", "REF ).", "Figure: By assumption, node 1 continues to switch orientation at arbitrary time steps t n-2 ,t n-1 ,t n t_{n-2}, t_{n-1}, t_{n} by crossing the threshold signal mix x i =1 2x_i = \\frac{1}{2}.", "The threshold is denoted by a dashed line.", "If node 1 switches, its neighbour (node 2, white) must cross sufficiently distant thresholds, denoted by the red dashed lines.", "Furthermore, the switches must be simultaneous, or else the switching terminates perpetually.", "The regions 𝒪 u ,ℐ u ,ℐ l ,𝒪 l \\mathcal {O}_u, \\mathcal {I}_u, \\mathcal {I}_l, \\mathcal {O}_l are outlined.Consider some arbitrary $t_n$ , where $x_1$ switches from $x_1(t_n - 1)<1/2$ to $x_1(t_n) > 1/2$ .", "Using the model's update rule (see (REF )) we can note: $ x_1(t_n) = yx_1(t_{n}-1) + (1-q)(1-y)x_2(t_{n}-1) > \\frac{1}{2} \\ ,$ and using the fact that $x_1(t_{n}-1)< 1/2$ we get , $\\frac{y}{2} + (1-q)(1-y)x_2(t_{n}-1) \\ge \\frac{1}{2}$ which in turn implies $x_2(t_{n}-1) \\ge \\frac{1}{2(1-q)} > \\frac{1}{2} \\ .$ By following the same reasoning one can show that an $x_1$ switch in the opposite direction would imply $ x_2(t_{n}-1) \\le \\frac{1-2q}{2(1-q)} = \\frac{1}{2}-\\frac{q}{2(1-q)} < \\frac{1}{2} \\ .$ Therefore, for node 1 to switch endlessly, then node 2 must also do so, and cannot start from an arbitrary point, but rather has to either be above $1 / (2(1-q))$ or below $(1-2q)/(2(1-q)$ at time $t_n-1$ for $x_1$ to cross the $1/2$ line at time $t_n$ from below or above, respectively.", "For the sake of convenience we introduce the following regions $\\mathcal {O}_l &=& \\left[0,\\frac{1-2q}{2(1-q)} \\right) \\\\ \\nonumber \\mathcal {I}_l &=& \\left[\\frac{1-2q}{2(1-q)}, \\frac{1}{2} \\right) \\\\ \\nonumber \\mathcal {I}_u &=& \\left[\\frac{1}{2},\\frac{1}{2(1-q)} \\right) \\\\ \\nonumber \\mathcal {O}_u &=& \\left(\\frac{1}{2(1-q)},1 \\right] \\ ,$ Where the subscripts indicate whether the interval lies in the upper or lower hemisphere (above and below $1/2$ , denoted by the subscripts $u$ and $l$ ).", "We also denote the “inner” region $\\mathcal {I} = \\mathcal {I}_l \\cup \\mathcal {I}_u$ and the “outer” region $\\mathcal {O} = [0,1] / \\mathcal {I}$ defined by the above boundaries.", "According to the above considerations, for node 1 to switch orientation to negative at $t_n$ , then $x_2(t_n-1) \\in \\mathcal {O}_l$ , and for node 1 to switch to positive at $t_n$ , then $x_2(t_n-1) \\in \\mathcal {O}_u$ .", "These regions are highlighted in REF .", "Clearly, the size of $\\mathcal {I}$ grows with $q$ (and $\\mathcal {O}$ shrinks with $q$ ).", "It is worth noting that for $q>1/2$ , the inner region's boundaries exceed $[0,1]$ , i.e., orientation switches are impossible.", "The intuition behind this is that if a node is able to flip more than half of the incongruent signals coming its way, it will never include enough incogruent signals in her information set to switch orientation.", "We further note that if a node and its neighbour are ever in the same orientation, then any future switches are impossible.", "Indeed, if two nodes share the same orientation, they are both linked to the same ghost node.", "As such, the set of available signals for each node is only its neighbour and its ghost node.", "Regardless of the value of $q$ , there is no way for either node to accumulate sufficient incongruent signals to switch orientation.", "All in all, it follows that both node 1 and node 2 must switch at the same time step whenever a switch occurs.", "This result is illustrated in REF .", "We now show that if $x_2$ lies in the outer region $\\mathcal {O}$ , it will converge to the inner region $\\mathcal {I}$ .", "Furthermore, once it enters the inner region, it cannot leave it.", "Also, this ceases the switching of the node 1, since its switching requires $x_2$ to alternate between the upper and lower hemispheres of the outer region.", "As proved above, at any given time step node 1 and its neighbour 2 can either both switch orientation, or both maintain their current orientation.", "We will consider both possibilities.", "Assume the former first, in which case we can show the two nodes must grow closer together.", "Suppose that at time $t_n$ , $x_1(t_n)>1/2$ , and $x_2(t_n)<1/2$ .", "At $t_n+1$ , this orientation switches so $x_2(t_n+1)>x_1(t_n+1)$ .", "Making use of Eqs.", "(REF ) and (REF ), we can write $ x_2(t_n+1) - x_1(t_n+1) &=& \\left[(1-y)(1-q)-y \\right] (x_1(t_n) - x_2(t_n)) - (1-y)q \\\\ \\nonumber &<& \\delta (x_1(t) - x_2(t)) \\ ,$ where $\\delta = \\left[(1-y)(1-q)-y \\right] < 1$ .", "Therefore, when the node switches orientation with their neighbour, they must converge strictly closer.We require $\\delta < 1$ and not $|\\delta |<1$ .", "While $\\delta <-1$ would violate the convergence criterion, it would also imply $x_2(t+1)<x_1(t+1)$ , leading to a contradiction.", "We now consider the logical disjunct.", "Suppose instead that a switch does not occur, and at times $t_n$ , $t_n+1$ we have $x_1(t_n),x_1(t_n+1)>1/2$ , and $x_2(t_n),x_2(t_n+1)<1/2$ .", "Therefore, we can write $ x_1(t_n+1) - x_2(t_n+1) = \\left[y-(1-y)(1-q)) \\right] (x_1(t_n) - x_2(t_n)) + (1-y)q.$ If the two nodes are to move closer in this time step, then we must have $x_1(t_n+1) - x_2(t_n+1) < (1-\\mu )(x_1(t_n) - x_2(t_n))$ for some $\\mu \\in (0,1)$ .", "Using this in (REF ) we obtain the following sufficient condition for convergence: $ x_1(t_n) - x_2(t_n) > \\frac{q}{2-q-\\frac{\\mu }{1-y}} \\ .$ Finally, note that if $x_1(t_n) > 1/2$ and $x_2(t_n) \\in \\mathcal {O}_l$ , then: $ x_1(t_n) - x_2(t_n) > \\frac{1}{2} - \\frac{1-2q}{2(1-q)} > \\frac{q}{2-q-\\frac{\\mu }{1-y}}$ for an arbitrarily small $\\mu $ .", "Thus, if $x_2(t_n) \\in \\mathcal {O}_l$ , then the two nodes are sufficiently far apart that the condition in (REF ) holds, and the two nodes must converge closer together.", "The parallel argument can be made for the opposite starting orientations.", "Even if a switch does not occur, then the nodes will converge strictly closer.", "Indeed, We have established that if $x_2(t) \\in \\mathcal {O}$ , then at each time step the distance $|x_1(t)-x_2(t)|$ must strictly shrink.", "As such, the nodes will eventually become close enough that $x_2(t) \\in \\mathcal {I}$ , and switching of node 1 ceases.", "We complete the proof by showing that once node 2's signal mix has entered the inner region $\\mathcal {I}$ , it cannot leave it.", "We have established already that nodes must have opposing orientations at all times.", "Let us consider the case where $x_2(t_n) \\in \\mathcal {I}_u$ and $x_1(t_n)< 1/2$ .", "Suppose by contradiction that in time step $t_n+1$ node 2 is able to “escape” $\\mathcal {I}$ from below, going from below $1/(2(1-q))$ to above such value (i.e., to $\\mathcal {O}_u$ ).", "This implies $\\frac{1}{2(1-q)} &<& x_2(t_n+1) = y x_2(t_n) + (1-y)(1-q)x_1(t_n) + (1-y)q \\\\ \\nonumber &<& \\frac{y}{2(1-q)} + \\frac{1}{2}(1-y)(1-q) + (1-y)q \\ ,$ which leads to $1+q < (1-q)^{-1}$ , i.e.", "to the impossible result $q^2 < 0$ .", "Therefore, node 2 cannot go from $\\mathcal {I}$ to $\\mathcal {O}_u$ .", "Finally we also know that it cannot go from $\\mathcal {I}$ to $\\mathcal {O}_l$ as this would require both nodes to switch orientation, which is ruled out because $x_2(t) \\in \\mathcal {I}$ .", "A parallel argument can be made if the orientations are reversed.", "Thus, the two node symmetric network will always converge to a region of the signal mix space where the nodes' signal mixes are too close to support any switch of orientation, arriving at the desired result.", "The above proof can be easily replicated after relaxing the simplifying asymptotic assumption that $q$ is fixed.", "This can be done by reintroducing the time-dependent random weights $w_i(t)$ ($i = 1,2$ ), and recalling that, due to their almost sure convergence to $q$ , for any $\\epsilon > 0$ there exists a time $t^*$ such that for all $t > t^*$ and for all $i$ $q-\\epsilon < w_i(t) < q+\\epsilon \\ .$ Adjusting the bounds used in the convergence proof to include the above time evolution allows to obtain the same result." ], [ "Star network.", "Let us now consider a $k$ -star network of biased agents, with the central node labeled as 0 and branch nodes labeled as $1, \\ldots , k$Strictly speaking, the signal diffusion mechanism would need to be modified for non-regular graphs to allow for signal diffusion to be equivalent to node averaging.", "More complex regular structures can also be shown to converge, but a star graph permits us to show how convergence holds even with a strikingly different topology.", "We proceed with the star graph for the purpose of illustration.. As before, allow $y$ to be the self-weight of each node and $q$ the confirmation bias parameter.", "Assume for simplicity that the central node has a weight of $(1-y) / k$ on each branch node.", "Firstly, note that if any branch node switches indefinitely, then the central node 0 must also switch indefinitely (or else there would be no “driving force” causing the branch nodes to switch).", "So, let us focus on showing that it is impossible for the central node to do so.", "The logic of the two-node network proof can be followed almost exactly by replacing $x_1(t)$ and $x_2(t)$ with $x_0(t)$ and $\\sum _{j=1}^k x_j(t) / k$ , respectively.", "The first set of results up to (REF ) follow precisely given the substitution of terms above.", "We use this to establish once again that for $x_0(t)$ to switch indefinitely $\\sum _{j=1}^k x_j(t)/k$ must oscillate between $1/(2(1-q))$ and $(1-2q)/(2(1-q))$ , i.e., between the upper and lower hemispheres of the outer region $\\mathcal {O}$ .", "Furthermore, whenever the central node switches orientation, the branch nodes' average signal mix must also change from above to below $1/2$ (or vice versa), even if none of the branch nodes in particular switch orientation.", "The next steps follow closely those of the two-nodes network.", "Suppose firstly that the central node switches orientation (and the branch nodes' average must also shift accordingly).", "Suppose that at time $t_n$ , $x_0(t_n)> 1/2$ , and $\\sum _{j=1}^k x_j(t_n)/k<1/2$ .", "At $t_n+1$ , this orientation switches so that $\\sum _{j=1}^k x_j(t_n+1)/k >x_0(t_n+1)$ .", "Adapting Eqs.", "(REF ) and (REF ) to the present case, we have $\\frac{1}{k} \\sum _{j=1}^k x_j(t_n+1) - x_0(t_n+1) &=& \\left[\\frac{1}{k}\\sum _{j=1}^k \\left(x_j(t_n) + (1-y)(1-q) x_0(t_n) + (1-y)q g_j(t_n) \\right) \\right] \\\\ \\nonumber &-& \\left[y x_0(t_n) + (1-y)(1-q) \\frac{1}{k} \\sum _{j=1}^k x_j(t_n) + (1-y)q \\right] \\ ,$ where we have introduced a new indicator variable such that $g_j(t) = 1$ is node $j$ is positively oriented at time $t$ , and $g_j(t) = 0$ otherwise.", "Let $g(t) = \\sum _{j=1}^k g_j(t) / k$ be the fraction of positively oriented branch nodes.", "We can then simplify the above: $\\frac{1}{k} \\sum _{j=1}^k x_j(t_n+1) - x_0(t_n+1) &=& \\left[ (1-y)(1-q)-y \\right] \\left(x_0(t_n) - \\frac{1}{k} \\sum _{j=1}^k x_j(t_n) \\right) - (1-y)q(1-g(t_n)) \\\\ \\nonumber &<& \\left[ (1-y)(1-q)-y \\right] \\left(x_0(t_n) - \\frac{1}{k} \\sum _{j=1}^k x_j(t_n) \\right) \\ ,$ where we used the fact that $(1-y)q(1-g(t_n)) > 0$ .", "This can be rewritten as $\\frac{1}{k} \\sum _{j=1}^k x_j(t_n+1) - x_0(t+1) < \\delta \\left(x_0(t_n) - \\frac{1}{k} \\sum _{j=1}^k x_j(t_n) \\right) \\ ,$ where $\\delta = \\left[ (1-y)(1-q)-y \\right] < 1$ , which re-establishes the result of (REF ): if the central node flips, it must converge strictly closer to the branch nodes.", "Figure: A network structure consisting of a central node 0 and k=6k = 6 branch nodes.", "All nodes are biased agents for the purposes of the toy example.Next, we establish that in the time steps where the central node does not switch orientation, the centre and branches still converge as long as the branch average is within $\\mathcal {O}$ .", "The reasoning follows the one of the previous section exactly given the appropriate substitutions, and we can replace the condition in (REF ) with: $ x_0(t_n) - \\frac{1}{k} \\sum _{j=1}^k x_j(t) > \\frac{q(1-g(t_n))}{2-q-\\frac{\\mu }{1-y}} \\ .$ Recall that $x_0(t_n) > 1 / 2$ and $\\sum _{j=1}^k x_j(t) / k < (1-2q)(2(1-q)) \\in \\mathcal {O}_l$ , therefore: $x_0(t_n) - \\frac{1}{k} \\sum _{j=1}^k x_j(t) > \\frac{1}{2} - \\frac{1-2q}{2(1-q)} > \\frac{q}{2-q-\\frac{\\mu }{1-y}} \\ge \\frac{q(1-g(t))}{2-q-\\frac{\\mu }{1-y}}$ for an arbitrarily small $\\mu > 0$ .", "Hence, even if a switch does not occur, then the nodes will converge strictly closer.", "The final steps of the proof mirror those that follow (REF ) of the previous section, except a factor of $g(t_n)$ dampens the ability of the branch nodes to escape the inner region even further.", "As such, we establish that even on a star network structure, the biased agents cannot switch their orientation endlessly, and must eventually converge." ], [ "Simulated dynamics and convergence criteria.", "As has been established in the previous sections, settling is guaranteed under some simple network topologies chosen specifically to hinder convergence.", "We round out the argument by noting that settling also occurs in simulations for the $k$ -regular network employed throughout the paper and in the following proofs.", "In what follows, we establish criteria for the case of a fixed $q$ .", "Analogous criteria can be easily established for the case of stochastic convergent weights $w_i(t)$ instead, although without much adding much insight.", "Furthermore, in practice the stochasticity rapidly settles in numerical simulations, meaning that convergence can be safely studied using the asymptotic fixed $q$ assumption.", "In order to guarantee that a network has in fact settled over the course of a simulation, we identify a “settling” rule for the signal mix ${\\hat{x}}(t)$ .", "As we demonstrate in the following section, if one assumes that the biased agents at time $t$ no longer switch orientations, one can calculate the steady state that would arise from this configuration of biased agents.", "Call this $\\hat{x}^*({\\hat{x}}(t))$ .", "We can show that if the signal mix ${\\hat{x}}(t)$ is sufficiently close to its corresponding steady state $\\hat{x}^*({\\hat{x}}(t))$ it will converge uniformly to that steady state without any further changes to any agent's orientation.", "Define the difference between a signal mix and its steady state: $\\epsilon (t) = \\hat{x}(t) - \\hat{x}^* \\ .$ Recalling that the model's dynamics is such that ${\\hat{x}}(t) = \\hat{A}{\\hat{x}}(t-1) \\ $ we then have ${\\hat{x}}^* + \\epsilon (t) = \\hat{A}{\\hat{x}}^* + \\hat{A}\\epsilon (t-1) = {\\hat{x}}^* + \\hat{A}\\epsilon (t-1) \\ ,$ and therefore: $\\epsilon (t) = \\hat{A}\\epsilon (t-1) \\ .$ Finally, define $\\epsilon ^*(t) = \\max _i(|\\epsilon _i(t)|) = ||\\epsilon (t)||_{\\infty }$ .", "Then for any arbitrary biased node: $\\epsilon _i(t+1) = \\hat{a}_{ii}\\epsilon _i(t) + (1-q)\\sum _j \\hat{a}_{ij}\\epsilon _j(t) \\ ,$ where $\\hat{a}_{ij}$ is the weight between node $i$ and $j$ in matrix $\\hat{A}$ , and we use the fact that the ghost nodes are always at their exact steady state, so their $\\epsilon _G = 0$ .", "Then taking the absolute distance and using the triangular inequality: $&& |\\epsilon _i(t+1)| \\le \\hat{a}_{ii}|\\epsilon _i(t)| + (1-q)\\sum _j \\hat{a}_{ij}|\\epsilon _j(t)| \\\\ \\nonumber && \\le \\hat{a}_{ii}|\\epsilon ^*(t)| + (1-q)\\sum _j \\hat{a}_{ij}|\\epsilon ^*(t)| = (1 - q(1 - \\hat{a}_{ii}))|\\epsilon ^*(t)| < |\\epsilon ^*(t)| \\ ,$ and similarly for an unbiased agent, we can show: $|\\epsilon _i(t+1)| \\le |\\epsilon ^*(t)| \\ .$ In short, for each steady state once the current signal mixes are within some $\\epsilon $ -cube of the steady state, they must remain within that $\\epsilon $ -cube.", "Furthermore, biased agents at each time step must converge strictly closer to the steady state.", "A larger $q$ or smaller self-weight ($\\hat{a}_{ii}$ ) will cause faster convergence.", "Finally, we can also note because the network is strongly connected, there are some $r \\ge 1$ steps between each unbiased and a biased node, and so it can be shown that in a finite number of steps all nodes must converge strictly closer to the steady state than the maximum threshold of the $\\epsilon $ -cube.", "Given all the above, we can now explicitly outline a “stable” region.", "Denote: $\\epsilon _s = \\min _i \\left(|x^*_i - \\frac{1}{2}| \\right) \\ .$ That is, the closest any of the steady states are to the threshold.", "If the current signal mixes can get within the $\\epsilon _s$ -ball of the steady state, then there can be no crossing the $0.5$ threshold and as such the steady state cannot move - this is a sufficient condition for settling to be guaranteed.", "Using this condition, we are able to demonstrate settling occurs for a wide range of parameters for $k$ -regular networks.", "We tested the condition on 1000 iterations each of the following parameter sets: $n=10^3$ , $p=0.5$ , $k =\\lbrace 3, 4, 5, 10, 100, 999\\rbrace $ , $f = \\lbrace .05, .1, .2, .5, 1\\rbrace $ , $q=\\lbrace .05, .1, .25, .5, .75.", "1\\rbrace $ .", "The settling criteria was successfully reached for every single run of the model, establishing extremely high confidence that the $k$ -regular biased information aggregation model always settles.", "So far, we have established that the random update matrix $A(t)$ converges almost surely to a fixed update matrix $A$ .", "Furthermore, we have demonstrated with extremely high confidence that biased agents settle in their orientation after some finite time.", "As such, for biased $k$ -regular networks, assume that there exists some time $t^*$ after which biased agents cease switching their orientation.", "Define $\\hat{y}_{\\mathcal {B}}^*$ as the steady state fraction of positively oriented biased agents.", "Then the following holds.", "(1) The signal mix vector $\\hat{x}(t)$ converges to some $\\hat{x}^* = \\hat{A}^*\\hat{x}(0)$ for both biased and unbiased networks, where $\\hat{A}^*$ is a steady-state matrix of influence weights which can be computed explicitly.", "(2) Unbiased networks achieve consensus, and converge to influence weights of $a^*_{ij} = 1/n$ for all pairs $(i,j)$ .", "This ensures that, for all $i \\in V$ , $x^*_i=x^*_V=\\bar{x}(0)$ , where $\\bar{x}(0) = \\sum _{i=1}^n s_i$ is the intial average signal mix.", "(3) Biased networks where $\\hat{y}_{\\mathcal {B}}^* = 0,1$ achieve consensus, and converge to influence weights $\\hat{a}^*_{ij} = 0$ for all pairs $(i, j) \\in V$ , $\\hat{a}^*_{i+} = \\hat{y}_{\\mathcal {B}}^*$ and $\\hat{a}^*_{i-} = 1-\\hat{y}_{\\mathcal {B}}^*$ for all $i \\in V$ .", "(4) Biased networks where $ 0 < \\hat{y}_{\\mathcal {B}}^* < 1$ do not achieve consensus, and converge to influence weights $\\hat{a}^*_{ij} = 0$ for all $(i, j) \\in V$ , and $\\hat{a}^*_{i+} + \\hat{a}^*_{i-}=1$ for all $i \\in V$ .", "We note that results regarding unbiased networks (part of (1) and all of (2)) are already well established results (see, for example, [36]) and are listed purely for comparison with biased networks.", "We focus on proving the remainder of the results.", "The results follow from the structure of $\\hat{x}^* = \\hat{A}^*\\hat{x}(0) = \\lim \\limits _{t \\rightarrow \\infty }\\prod _{\\tau =0}^{t}\\hat{A}(\\tau ){\\hat{x}}(0)$ .", "We proceed by demonstrating that despite the stochasticity in the random update mechanism $\\hat{A}(\\tau )$ , the steady state converges to a fixed vector $\\hat{x}^*$ .", "First note that for $\\tau > t^*$ the biased agents will have ceased switching their orientation, and the random update matrix $\\hat{A}(\\tau )$ will have a fixed underlying structure $\\hat{A} = \\mathbb {E}(\\hat{A}(\\tau ))$ .", "The proof will follow by demonstrating that $\\lim \\limits _{t \\rightarrow \\infty }\\prod _{\\tau =0}^{t}\\hat{A}(\\tau ) = \\lim \\limits _{t \\rightarrow \\infty }\\hat{A}^t$ .", "That is, the products of random matrices converges to the products of their expectation.", "Firstly recall the block structure of $\\hat{A}(\\tau )$ : $\\hat{A}(\\tau )=\\left[\\begin{array}{c|c}Q(\\tau ) & R(\\tau ) \\\\\\hline 0 & I\\end{array}\\right] \\ ,$ with dimensions (clockwise from top-left): $(n \\times n)$ , $(n \\times 2)$ , $(2 \\times 2)$ , $(2 \\times 2)$ .", "Important properties of the blocks include: $&& Q(\\tau ) = Q + \\epsilon _Q(\\tau ) \\xrightarrow{} Q \\\\ \\nonumber && R(\\tau ) = R + \\epsilon _R(\\tau ) \\xrightarrow{} R \\ .$ The properties above indicate that the blocks converge to their deterministic counterparts almost surely.", "This allows us to state that for any $\\epsilon $ and matrix norm $||.||$ , there is guaranteed some $t^{\\prime } \\ge t^*$ such that for all $\\tau > t^{\\prime }$ , $||Q-Q(\\tau )|| = ||\\epsilon _Q(\\tau )|| <\\epsilon $ .", "Also, the matrix $Q$ is such that $&& \\sum _j Q_{ij} < 1 \\ \\ \\forall i \\in \\mathcal {B} \\\\ \\nonumber && \\sum _i Q_{ij} < 1 \\ \\ \\forall j \\in \\partial \\mathcal {B} \\ .$ The properties respectively indicate that the limit matrix $Q$ is both row and column sub-stochastic.", "Row sub-stochasticity follows from the outgoing edges from the set of biased agents ($\\mathcal {B}$ ).", "For the $k$ -regular graphs that are the focus of our analysis this can be specifically shown to be $\\frac{(1-q)k+1}{k+1}$ .", "Column sub-stochasticity follows from the neighbours of the biased agents ($\\partial \\mathcal {B}$ ) having incoming connections necessarily less than 1.", "We now define the product of the random matrices $\\hat{A}(\\tau )$ as: $\\prod _{\\tau =0}^{t}\\hat{A}(\\tau ) = \\tilde{A}(t,0) = \\left[\\begin{array}{c|c}\\tilde{Q}(t,0) & \\tilde{R}(t,0) \\\\\\hline 0 & I\\end{array}\\right] \\ ,$ where $\\tilde{Q}(t,0$ and $\\tilde{R}(t,0)$ are placeholder terms for the the random block matrix products which arise through products of the random matrices $\\hat{A}(\\tau )$ .", "Consider also the deterministic analog to this expression: $\\prod _{\\tau =0}^{t}\\hat{A} = \\hat{A}^t = \\dot{A}(t,0) = \\left[\\begin{array}{c|c}\\dot{Q}(t,0) & \\dot{R}(t,0) \\\\\\hline 0 & I\\end{array}\\right] \\ .$ This formulation defines a random and analogous deterministic sequence for each of the blocks, denoted by $\\tilde{A}(t,0)$ and $\\dot{A}(t,0)$ respectively.", "Firstly, we demonstrate that $\\lim \\limits _{t \\rightarrow \\infty }\\dot{Q}(t,0) = \\lim \\limits _{t \\rightarrow \\infty }Q^t = \\textbf {0}$ .", "Consider the 2-norm $||.||_2$ .", "We consider first the deterministic matrix $Q$ .", "Recall that since $Q$ is doubly sub-stochastic, $Q^T Q$ is necessarily sub-stochastic and therefore: $||Q|| = (1 - \\delta ) < 1 \\ ,$ for some $\\delta \\in (0,1)$ .", "It follows that: $\\lim \\limits _{t \\rightarrow \\infty }||Q^t|| \\le \\lim \\limits _{t \\rightarrow \\infty }||Q||^t = \\lim \\limits _{t \\rightarrow \\infty }(1 - \\delta )^t = 0 \\ ,$ therefore $\\lim \\limits _{t \\rightarrow \\infty }Q^t = \\textbf {0}$ (making use of the fact that $||X|| = 0 \\iff X = \\textbf {0}$ ).", "Now consider the term of interest $\\tilde{Q}(t,0)$ : $||\\tilde{Q}(t,0)|| = ||\\prod _0^t Q(\\tau )|| \\le \\prod _0^t||Q(\\tau )|| \\ .$ Note that $||Q(\\tau )|| \\le 1$ for all $\\tau $ .", "However we can show that almost all $||Q(\\tau )|| < 1$ : $||Q(\\tau )|| = ||Q + \\epsilon _Q(\\tau )|| \\le ||Q|| + ||\\epsilon _Q(\\tau )||= (1 - \\delta ) + ||\\epsilon _Q(\\tau )||$ We select some $t^{\\prime }$ such that for all $\\tau > t^{\\prime }, ||\\epsilon _Q(\\tau )|| = \\mu < \\delta $ for some $\\mu > 0$ .", "Therefore: $ ||Q(\\tau )|| \\le (1-\\delta +\\mu ) = (1-\\delta ^*) < 1 \\ ,$ where $\\delta ^* = \\delta +\\mu $ .", "We can now conclude: $\\lim \\limits _{t \\rightarrow \\infty }||\\tilde{Q}(t,0)|| \\le \\lim \\limits _{t \\rightarrow \\infty }\\prod _{t^*}^t (1-\\delta ^*) = \\lim \\limits _{t \\rightarrow \\infty }(1 - \\delta ^*)^t = 0 \\ .$ We now show that $\\lim \\limits _{t \\rightarrow \\infty }\\tilde{R}(t,0) = \\lim \\limits _{t \\rightarrow \\infty }\\dot{R}(t,0) = (I-Q)^{-1}R$ .", "Consider firstly the deterministic sequence, which can be defined through the following iterative relationship: $ \\dot{R}(t,0) = Q \\dot{R}(t-1,0) + R \\ .$ Note again that $\\dot{R}(t,0)$ refers to the $t$ -th term in a deterministic sequence whereas $Q$ and $R$ are specific block matrices.", "The expression (REF ) can be straightforwardly solved in the limit: $\\lim \\limits _{t \\rightarrow \\infty }\\dot{R}(t,0) = (I-Q)^{-1}R \\ .$ Consider now the random sequence, which can be defined analogously: $ \\tilde{R}(t,0) = Q(t) \\tilde{R}(t-1,0) + R(t) \\ .$ Note here $\\tilde{R}(t,0)$ refers to the $t$ -th term in a random sequence and $Q(t)$ and $R(t)$ are random block matrices that occur at time $\\tau = t$ .", "In order to proceed we define: $ \\tilde{R}(t,0) = \\dot{R}(t,0) + E(t) \\ ,$ where here $E(t)$ is an error term capturing the difference between the terms of the deterministic and random sequences at time $\\tau = t$ .", "We substitute (REF ) into (REF ): $ \\tilde{R}(t,0) = Q(t) (\\dot{R}(t-1,0) + E(t-1)) + R(t) \\ .$ We now substitute the definition of $Q(\\tau )$ and $R(\\tau )$ into (REF ): $\\tilde{R}(t,0) = Q \\dot{R}(t-1,0) + R + Q E(t-1) + \\epsilon _Q(t)(\\dot{R}(t-1,0)+E(t-1)) + \\epsilon _R(t) \\ .$ Note that we can substitute (REF ) for the two leading terms on the RHS: $\\tilde{R}(t,0) - R(t,0) = Q E(t-1) + \\epsilon _Q(t)(\\dot{R}(t-1,0)+E(t-1)) + \\epsilon _R(t) \\ .$ We can now take the 2-norm $||.||_2$ : $||E(t)|| = ||\\tilde{R}(t,0) - \\dot{R}(t,0)|| = ||Q E(t-1) + \\epsilon _Q(t)(\\dot{R}(t-1,0)+E(t-1)) + \\epsilon _R(t)|| \\\\\\le ||Q|| \\ ||E(t-1)|| + ||\\epsilon _Q(t)|| \\ ||\\tilde{R}(t-1)|| + ||\\epsilon _R(t)|| \\ .$ We can now substitute in (REF ) and once again make use of the fact that for any $0<\\epsilon $ we can define $t^{\\prime }$ such that $||\\epsilon _Q(t)||$ , $||\\epsilon _R(t)|| < \\epsilon $ .", "We also note that $||\\tilde{R}(t-1)|| < n$ (where $n$ is the size of the network), therefore $||E(t)|| \\le (1-\\delta )||E(t-1)|| + \\epsilon (n+1) \\ ,$ and therefore for a sufficiently small $\\epsilon $ , there is a corresponding $t^{\\prime }$ such that for $t > t^{\\prime }$ : $||E(t)|| \\le (1-\\delta ^*)||E(t-1)|| < ||E(t-1)|| \\ .$ Finally we get: $\\lim \\limits _{t \\rightarrow \\infty } ||\\tilde{R}(t,0)-\\dot{R}(t,0)|| = \\lim \\limits _{t \\rightarrow \\infty }||E(t)|| = 0 \\ ,$ which allows us to conclude that $\\lim \\limits _{t \\rightarrow \\infty } \\tilde{R}(t,0) = \\lim \\limits _{t \\rightarrow \\infty } \\dot{R}(t,0) = (I-Q)^{-1}R$ .", "We can combine these results to conclude: $\\hat{A}^* = \\lim \\limits _{t \\rightarrow \\infty } \\prod _{\\tau =0}^{t}\\hat{A}(\\tau )= \\lim \\limits _{t \\rightarrow \\infty } \\tilde{A}(t,0)= \\left[\\begin{array}{c|c}\\lim \\limits _{t \\rightarrow \\infty } \\tilde{Q}(t,0) & \\lim \\limits _{t \\rightarrow \\infty } \\tilde{R}(t,0) \\\\\\hline 0 & I\\end{array}\\right]= \\left[\\begin{array}{c|c}0 & (I-Q)^{-1}R \\\\\\hline 0 & I\\end{array}\\right] \\ .$ Given that $\\hat{x}^* = \\hat{A}^*\\hat{x}(0)$ we can conclude Result (1) - that the signal mixes do converge.", "In particular: $\\hat{x}^* = \\hat{A}^* \\hat{x}(0)= \\left[\\begin{array}{c|c|c}0 & (I-Q)^{-1}R^+ & (I-Q)^{-1}R^- \\\\\\hline 0 & 1 & 0 \\\\\\hline 0 & 0 & 1\\end{array}\\right]\\left[\\begin{array}{c}\\hat{x}(0) \\\\\\hline 1 \\\\\\hline 0\\end{array}\\right]= \\left[\\begin{array}{c}(I-Q)^{-1}R^+ \\\\\\hline 1 \\\\\\hline 0\\end{array}\\right] \\ ,$ that is, the steady state signal mixes of the agents not a function of the initial signals $\\hat{x}(0)$ .", "The signal mixes are instead entirely a function of the steady state orientations of the biased agents, encoded by the vector $R^+$ , the edges from the positive biased agents to the positive ghost nodes.", "Our remaining conclusions follow summarily from this.", "If all biased agents are negative ($\\hat{y}_\\mathcal {B}^* = 0$ ) $R^+$ is 0 and $x^*$ is 0 for all agents.", "Inversely if all biased agents are positive ($\\hat{y}_\\mathcal {B}^* = 1$ ), $x^*$ is 1 for all agents.", "For any other configuration of biased agents, the steady state is determined by the closed form $(I-Q)^{-1}R^+$ .", "In this scenario $z_V^* > 0$ trivially as some biased nodes will be of the minority orientation.", "However, more crucially $z_R^* \\ge 0$ .", "That is, unbiased agents are no longer guaranteed to converge despite having no bias mechanism themselves.", "We investigate this and other properties of the unbiased agents in more detail in the next section." ], [ "Section S4: Steady state signal mix distribution", "We now seek to approximate the distribution of signal mixes of the agents once the steady state is reached.", "We will first approximate the average steady state signal mix of each sub-population in the network, followed by the steady state signal mix variance, and finally the full distribution itself.", "We will do this for the $k$ -regular network case used in the body of the paper, and show via numerical simulations that it also captures the model's dynamics on more heterogeneous networks.", "Let us note that the results given in the following are the empirical distribution of the signal mixes for a given run of the model, as opposed to an ensemble over all possible runs of the model." ], [ "Steady state expected signal mix.", "As detailed in , the model converges to a steady state ${\\hat{\\textbf {x}}}^*$ which is entirely contingent on the settled orientation of the biased agents in the network.", "In what follows, we will calculate an approximation for the model's steady state expected signal mix conditional on a given fraction of positively oriented biased agents $f^+(t) = \\hat{y}_{\\mathcal {B}}(t) f$ .", "We will then show how under some reasonable assumptions the “settled” value of $f^+(t)$ can be approximated from the initial orientation $f^+(0)$ .", "Consider an agent $i$ picked uniformly at random at time $t$ from the unbiased, positively oriented biased, negatively oriented biased sub-populations.", "Let us denote the signal mixes of agents belonging to such sub-populations as $\\hat{x}_{i_{\\mathcal {U}}}(t)$ , $\\hat{x}_{i_{\\mathcal {B}^+}}(t)$ and $\\hat{x}_{i_{\\mathcal {B}^-}}(t)$ , respectively.", "We are interested in establishing the expected steady state values for each of these quantities, denoted as $\\hat{x}_{\\mathcal {U}}^* = \\lim _{t \\rightarrow \\infty } \\mathbb {E} [\\hat{x}_{i_{\\mathcal {U}}}(t)] = \\lim _{t \\rightarrow \\infty } \\hat{x}_{\\mathcal {U}}(t) \\ ,$ with analogous definitions for $\\hat{x}_{\\mathcal {B}^+}^*$ and $\\hat{x}_{\\mathcal {B}^-}^*$ .", "We begin by considering the sub-population of unbiased agents at some finite $t$ .", "We note the following: $ \\hat{x}_{\\mathcal {U}} (t+1) = \\mathbb {E} [\\hat{x}_{i_{\\mathcal {U}}}(t+1)] = \\mathbb {E} \\left[\\frac{\\hat{x}_{i_{\\mathcal {U}}}(t)+\\sum _{j \\in \\partial _i}\\hat{x}_j(t)}{k+1} \\right] =\\frac{\\hat{x}_{\\mathcal {U}}(t)+k\\mathbb {E} [\\hat{x}_j(t)]}{k+1} \\ ,$ where $\\mathbb {E}[\\hat{x}_j(t)]$ refers to the expected signal mix of a randomly picked agent $j$ from the entire population, which of course consists of the three aforementioned sub-populations.", "Therefore, we have $ \\mathbb {E} [\\hat{x}_j(t)] &=& (1-f) \\mathbb {E} [\\hat{x}_j(t) | j \\in U] + f^+(t) \\mathbb {E} [\\hat{x}_j(t) | j \\in \\mathcal {B}^+(t)] + f^-(t) \\mathbb {E} [\\hat{x}_j(t) | j \\in \\mathcal {B}^-(t)] \\\\ \\nonumber &=& (1-f) \\hat{x}_{\\mathcal {U}}(t) + f^+(t) \\hat{x}_{\\mathcal {B}^+}(t) + f^-(t) \\hat{x}_{\\mathcal {B}^-}(t) \\ .$ Plugging the above in (REF ) we get $\\hat{x}_{\\mathcal {U}}(t+1) = \\frac{1}{k+1} \\left[ (1+k(1-f))\\hat{x}_{\\mathcal {U}}(t) + kf^+\\hat{x}_{\\mathcal {B}^+}(t) + kf^-\\hat{x}_{\\mathcal {B}^-}(t) \\right] \\ .$ Repeating the above steps for positively oriented biased agents we get $\\hat{x}_{\\mathcal {B}^+}(t+1) &=& \\mathbb {E} [\\hat{x}_{i_{\\mathcal {B}^+}}(t+1)] = \\mathbb {E} \\left[\\frac{\\hat{x}_{i_{\\mathcal {B}^+}}(t)+\\sum _{j \\in \\partial _i} \\left((1-w_i(t))\\hat{x}_j(t) + w_i(t) \\right)}{k+1} \\right] \\\\ \\nonumber &=& \\frac{\\hat{x}_{\\mathcal {B}^+}(t)+(1-q)k\\mathbb {E}[\\hat{x}_j(t)] + kq}{k+1} \\ .$ where we have explicitly referenced the random variable $w_i(t)$ representing the fraction of successfully distorted negative signals (see ), and made use of the fact that $\\mathbb {E}[w_i(t)] = q$ .", "We can use (REF ) again and write $ \\hat{x}_{\\mathcal {B}^+}(t+1) = \\frac{1}{k+1} \\left[ k(1-f)(1-q)\\hat{x}_{\\mathcal {U}}(t) + ((1-q)kf^+(t) + 1)\\hat{x}_{\\mathcal {B}^+}(t) + (1-q)kf^-(t)\\hat{x}_{\\mathcal {B}^-}(t) + kq \\right] \\ ,$ and similarly for negatively oriented biased agents: $\\hat{x}_{\\mathcal {B}^-}(t+1) = \\frac{1}{k+1} \\left[ k(1-f)(1-q)\\hat{x}_{\\mathcal {U}}(t) + (1-q)kf^+(t)\\hat{x}_{\\mathcal {B}^+}(t) + ((1-q)kf^-+1)(t)\\hat{x}_{\\mathcal {B}^-}(t) \\right] \\ .$ We have therefore established the update rule for the expected signal mix of the three sub-populations at any time $t$ .", "We collate this update rule into a matrix form for convenience: $ \\xi (t+1) = \\frac{1}{k+1}(F(t)+I_3) \\xi (t) + b \\ ,$ where $\\xi (t) = [\\hat{x}_{\\mathcal {U}}(t), \\hat{x}_{\\mathcal {B}^+}(t), \\hat{x}_{\\mathcal {B}^-}(t)]^T \\ , \\qquad b = \\left[0, \\frac{kq}{k+1}, 0 \\right]^T$ and $F(t)=k\\begin{bmatrix}(1-f) & f^+(t) & f^-(t) \\\\(1-q)(1-f) & (1-q)f^+(t) & (1-q)f^-(t) \\\\(1-q)(1-f) & (1-q)f^+(t) & (1-q)f^-(t)\\end{bmatrix} .$ If we further simplify notation by defining $\\hat{F}(t) = (F(t)+I_3)/(k+1)$ , we get to the following compact expression for (REF ): $\\xi (t+1) = \\hat{F}(t) \\xi (t+1) + b \\ .$ The long-run evolution of the signal mixes can be determined from the above equation if the evolution of $f^+(t)$ (and, consequently, of $f^-(t)$ ) in the matrix $\\hat{F}(t)$ is known.", "Assume for the moment we are at some time $t^*$ at which the system has settled, i.e., biased agents will keep their orientations intact and therefore will not cause the value of $f^+(t)$ to change for $t > t^*$ .", "In this case we can write: $\\lim _{t \\rightarrow \\infty }\\xi (t) = \\lim _{t \\rightarrow \\infty } \\hat{F}(t^*)^t \\xi (t^*) + (\\hat{F}(t^*)-I_3)^{-1}b \\ .$ It can be shown easily that, due to its double substochasticity, we have $\\lim _{t \\rightarrow \\infty }\\hat{F}(t^*)^t = 0$ , and therefore $\\lim _{t \\rightarrow \\infty }\\xi (t) = (\\hat{F}(t^*)-I_3)^{-1}b \\ .$ The above limit allows to calculate the steady state signal mixes for all sub-populations explicitly: $ \\lim _{t \\rightarrow \\infty }\\xi (t) =\\begin{bmatrix}\\hat{x}_{\\mathcal {U}}^* \\\\\\hat{x}_{\\mathcal {B}^+}^* \\\\\\hat{x}_{\\mathcal {B}^-}^*\\end{bmatrix} = \\begin{bmatrix}f^+(t^*) / f\\\\(1-q) f^+(t^*) / f + q\\\\(1-q) f^+(t^*) / f\\end{bmatrix} =\\begin{bmatrix}\\hat{y}_{\\mathcal {B}}(t^*)\\\\(1-q)\\hat{y}_{\\mathcal {B}}(t^*) + q\\\\(1-q)\\hat{y}_{\\mathcal {B}}(t^*)\\end{bmatrix} .$ In the next Section we will approximate this result to the case where biased agents have not settled their orientation yet." ], [ "Predicting the trajectory of biased agents' orientations", "Biased agents change their orientation when they receive a stream of incongruent signals that overcome their ability to distort them using confirmation bias.", "There are two points in the evolution of the model where this is possible.", "Firstly, this may happen in the early stages of the evolution, where the information sets held by the agents are relatively small and the stochasticity of the model can induce changes in orientation.", "Secondly, this may happen in the long run, where sustained changes in orientation can be brought along when one of the two camps of biased agents becomes able to systematically bias the available information.", "This leads to the composition of signals experienced by each node to change consistently in one direction, which can cause large scale switches in orientation, which in turn triggers a domino effect, as newly switched nodes will accelerate the rate at which signals are distorted.", "Let us capture this notion more formally.", "Consider the expected long term signal mix of each sub-population assuming the biased agents have settled ((REF )).", "Suppose the positively oriented biased agents have an expected steady state signal mix $\\hat{x}^*_{\\mathcal {B}^+} < 1/2$ .", "If such steady state value is to be reached, then some positively oriented biased agents' signal mixes must fall below $1 / 2$ , thereby switching orientation to negative.", "If this happens, then $\\hat{y}_{\\mathcal {B}}(t)$ falls and the steady state signal mix for all agents strictly decreasesThis can be proven rigorously with the results from the previous Section: the steady state mix is $(I-Q)^{-1}R^+$ .", "$R^+$ is a vector with 0 for each negative biased agent, and $(I-Q)^{-1}$ is element-wise $> 0$ .", "A biased agent switching to positive turns a previous zero element of $R^+$ to positive, and adds another strictly positive vector to the steady state signal mix.", "The same argument is made in reverse for a positive to negative switch.", "This, in turn, means more positively oriented biased agents switch orientation to reach their steady state, and so forth until all such agents switch to a negative orientation, yielding $\\hat{y}_{\\mathcal {B}} = 0$ .", "A corresponding outcome can be determined for negatively oriented biased agents all being converted.", "We can therefore determine, for any given $t^*$ , the approximate conditions under which we expect all positively oriented biased agents to switch their orientations to negative in the eventual steady state.", "Setting $\\hat{x}^*_{\\mathcal {B}^+} < 1/2$ in (REF ) we have $ (1-q)\\hat{y}_{\\mathcal {B}}(t^*) + q < \\frac{1}{2} \\qquad \\Longrightarrow \\qquad \\hat{y}_{\\mathcal {B}}(t^*) < \\frac{1}{2-2q} \\ ,$ and, correspondingly, for all negatively oriented biased agents to be tipped to positive we have the following condition: $ \\hat{y}_{\\mathcal {B}}(t^*) > \\frac{1-2q}{2-2q} \\ .$ Let us now consider the case $t_0 = 0$ , which means we are approximating the expected trajectory of the entire system given a starting fraction of positively oriented biased agents $f^{+*}(0) / f = \\hat{y}_{\\mathcal {B}}(0)$ .", "We then have the following approximate result for the steady state signal mix of the unbiased sub-population: $ \\hat{x}^*_{\\mathcal {U}} = \\hat{y}^*_{\\mathcal {B}} \\approx \\left\\lbrace \\begin{array}{ll}\\hat{y}_{\\mathcal {B}}(0) & \\text{for} \\ \\frac{1}{2(1-q)} \\le \\hat{y}_{\\mathcal {B}}(0) \\le \\frac{1-2q}{2(1-q)} \\\\1 & \\text{for} \\ \\hat{y}_{\\mathcal {B}}(0) > \\frac{1-2q}{2(1-q)} \\\\0 & \\text{for} \\ \\hat{y}_{\\mathcal {B}}(0) < \\frac{1}{2(1-q)} \\ , \\\\\\end{array}\\right.$ where the latter two conditions derive from Eqs.", "(REF ) and (REF ), while the first condition is the same reported in (REF ) adapted for the case $t_0 = 0$ ." ], [ "Steady state signal mix variance.", "In the previous Section we have provided approximations for the first moment of the steady state signal mixes of the unbiased agents, as well as those of the two biased agent sub-populations.", "We have also approximated the long term “settled” fractions of positively and negatively oriented biased agents.", "We noted that for $\\hat{y}_{\\mathcal {B}}(0) > \\frac{1-2q}{2(1-q)}$ ($\\hat{y}_{\\mathcal {B}}(0) < \\frac{1}{2(1-q)}$ ), the steady state signal mix is likely to asymptotically reach 1 (0).", "Under these conditions, all agents eventually trivially possess the same signal $+1$ ($-1$ ).", "Therefore the distribution of signal mixes tends asymptotically to a Dirac distribution on 1 (0).", "We therefore proceed with the assumption that $\\frac{1}{2(1-q)} \\le \\hat{y}_{\\mathcal {B}}(0) \\le \\frac{1-2q}{2(1-q)}$ , and as such use (REF ) to approximate: $\\lim _{t \\rightarrow \\infty }\\xi (t) =\\begin{bmatrix}\\hat{y}_{\\mathcal {B}}(t^*)\\\\(1-q)\\hat{y}_{\\mathcal {B}}(t^*) + q\\\\(1-q)\\hat{y}_{\\mathcal {B}}(t^*)\\end{bmatrix} \\approx \\begin{bmatrix}\\hat{y}_{\\mathcal {B}}(0)\\\\(1-q)\\hat{y}_{\\mathcal {B}}(0) + q\\\\(1-q)\\hat{y}_{\\mathcal {B}}(0)\\end{bmatrix}$ Correspondingly, we note $f^{+*} = \\hat{y}_{\\mathcal {B}}(t^*) = \\hat{y}_{\\mathcal {B}}(0)$ .", "We would now like to characterise the distribution of signal mixes for each sub-population at the steady state beyond its first moment.", "We begin with an approximation of the variance, under the asymptotic limit of large populations $n \\rightarrow \\infty $ .", "For convenience we define the steady state signal mix variance of any sub-population $G$ as $\\sigma ^2_{G}$ .", "In the following, we will provide approximate expressions for the steady state signal mix variances $\\sigma ^2_{\\mathcal {U}}$ , $\\sigma ^2_{\\mathcal {B}^+}$ , and $\\sigma ^2_{\\mathcal {B}^-}$ , and for the overall variance $\\sigma ^2_{V}$ .", "Consider an agent $i$ picked uniformly at random from the entire population.", "The variance of such an agent's steady state signal mix $\\mathrm {Var}[\\hat{x}_i^*]$ represents the variance across the entire population $\\sigma ^2_V$ .", "From the law of total variance, this can be broken down as follows $ \\sigma ^2_{V} = \\mathrm {Var}[\\hat{x}_i^*] = \\mathbb {E} \\big [ \\mathrm {Var}[\\hat{x}_i^*] \\ | \\ i \\in \\lbrace \\mathcal {U}, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace \\big ] + \\mathrm {Var} \\big [ \\mathbb {E} \\left[ \\hat{x}^*_i \\ | \\ i \\in \\lbrace U, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace \\right] \\big ] \\ ,$ where $ \\mathbb {E} \\big [ \\mathrm {Var}[\\hat{x}_i^*] \\ | \\ i \\in \\lbrace \\mathcal {U}, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace \\big ] = (1-f) \\sigma ^2_{\\mathcal {U}} + f^{+*} \\sigma ^2_{\\mathcal {B}^+} + f^{-*} \\sigma ^2_{\\mathcal {B}^-} \\ ,$ and $ \\mathrm {Var} \\big [ \\mathbb {E} \\left[ \\hat{x}^*_i \\ | \\ i \\in \\lbrace \\mathcal {U}, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace \\right] \\big ] &=&(1-f) \\bigg \\lbrace \\frac{f^{+*}}{f} - \\mathbb {E} \\Big [ \\mathbb {E} [ \\hat{x}^*_i \\ | \\ i \\in \\lbrace \\mathcal {U}, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace ] \\Big ] \\bigg \\rbrace ^2 \\\\ \\nonumber &+& f^{+*} \\bigg \\lbrace (1-q)\\frac{f^{+*}}{f}+q - \\mathbb {E} \\Big [ \\mathbb {E} [ \\hat{x}^*_i \\ | \\ i \\in \\lbrace \\mathcal {U}, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace ] \\Big ] \\bigg \\rbrace ^2 \\\\ \\nonumber &+& f^{-*} \\bigg \\lbrace (1-q)\\frac{f^{+*}}{f} - \\mathbb {E} \\Big [ \\mathbb {E} [ \\hat{x}^*_i \\ | \\ i \\in \\lbrace \\mathcal {U}, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace ] \\Big ] \\bigg \\rbrace ^2 \\ .$ Noting that $\\mathbb {E} \\Big [ \\mathbb {E} [ \\hat{x}^*_i \\ | \\ i \\in \\lbrace \\mathcal {U}, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace ] \\Big ] = (1-f) \\frac{f^{+*}}{f}+ f^{+*} \\left((1-q)\\frac{f^{+*}}{f}+q \\right) + f^{-*} \\left((1-q)\\frac{f^{+*}}{f} \\right) = \\frac{f^{+*}}{f}$ we can considerably simplify (REF ): $\\mathrm {Var} \\big [ \\mathbb {E} \\left[ \\hat{x}^*_i \\ | \\ i \\in \\lbrace \\mathcal {U}, \\mathcal {B}^+, \\mathcal {B}^-\\rbrace \\right] \\big ] = \\frac{q^2 f^{+*} f^{-*}}{f} = f q^2 \\hat{x}^*_{\\mathcal {U}} (1-\\hat{x}^*_{\\mathcal {U}}) \\ ,$ where in the last step we have used the fact that $\\hat{x}^*_{\\mathcal {U}} = f^{+*} / f$ , as per (REF ).", "(REF ) provides a compact expression for the second contribution for the overall variance $\\sigma ^2_V$ in (REF ).", "We now turn to the first term ((REF )).", "In order to be able to calculate it, we must compute the variance of each sub-population.", "Let us begin with the unbiased agent sub-population: $ \\sigma ^2_{\\mathcal {U}} = \\mathrm {Var}[\\hat{x}^*_{\\mathcal {U}}] = \\mathrm {Var} \\left[ \\frac{1}{k}\\sum _{j \\in \\partial _i} \\hat{x}^*_j \\right] =\\frac{\\sigma ^2_{V}}{k} + \\frac{2}{k^2}\\sum _{(j,\\ell ) \\in \\partial _i} \\mathrm {Cov} [\\hat{x}^*_j,\\hat{x}^*_\\ell ] = \\frac{\\sigma ^2_{V}}{k} + \\mathcal {O} \\left(\\frac{1}{k^2} \\right) \\ ,$ where in the last term we have assumed the covariance term to decay as $k^{-2}$ , which will be proved in the next Section.", "In analogy with the above results, for the biased agent sub-population (either positively or negatively oriented) we have: $ \\sigma ^2_{\\mathcal {B}} = \\mathrm {Var} [\\hat{x}^*_{i_B}] = \\frac{(1-q)^2 \\sigma ^2_{V}}{k} + \\mathcal {O} \\left(\\frac{1}{k^2} \\right) \\ .$ Substituting the two expressions above in (REF ), and combining the result with the one obtained in (REF ), we finally obtain the following result for the overall variance in (REF ): $\\sigma ^2_{V} = \\frac{1-fq(2-q)}{k} \\sigma ^2_{V} + f q^2 \\hat{x}^*_{\\mathcal {U}} (1-\\hat{x}^*_{\\mathcal {U}}) + \\mathcal {O}\\left(\\frac{1}{k^2}\\right) \\ .$ Solving for $\\sigma ^2_{V}$ we get $\\sigma ^2_{V} = \\frac{k f q^2 \\hat{x}^*_{\\mathcal {U}} (1-\\hat{x}^*_{\\mathcal {U}})}{k+fq(2-q)-1} + \\frac{k}{k+fq(2-q)-1} \\mathcal {O}\\left(\\frac{1}{k^2}\\right) \\approx f q^2 \\hat{x}^*_{\\mathcal {U}} (1-\\hat{x}^*_{\\mathcal {U}}) \\ ,$ where we have used the fact that $fq(2-q)-1 \\in [-1,0]$ , and therefore $fq(2-q)-1 \\ll k$ even for moderate connectivity.", "Finally, we can specialize the above result to the three sub-populations via Eqs.", "(REF ) and (REF ): $ \\sigma ^2_{\\mathcal {U}} &\\approx & \\frac{f q^2 \\hat{x}^*_{\\mathcal {U}} (1-\\hat{x}^*_{\\mathcal {U}})}{k}\\\\\\sigma ^2_{\\mathcal {B}_\\pm } &\\approx & \\frac{f \\left(q(1-q) \\right)^2 \\hat{x}^*_{\\mathcal {U}} (1-\\hat{x}^*_{\\mathcal {U}})}{k} \\ .$" ], [ "Explicit neighbourhood covariance expressions.", "In this Section we establish that the covariance term appearing in (REF ) can indeed be assumed to be of order $k^{-2}$ .", "The first thing to note is that the term $\\mathrm {Cov}[\\hat{x}^*_j,\\hat{x}^*_\\ell ]$ for two generic agents can be bounded above by the covariance $\\mathrm {Cov}[\\hat{x}^*_j,\\hat{x}^*_\\ell | j,\\ell \\in U]$ between unbiased agents' steady state signal mixes.", "To see this, suppose $j,\\ell \\in \\mathcal {B}^+$ : $\\mathrm {Cov}[\\hat{x}^*_j,\\hat{x}^*_\\ell | j,\\ell \\in \\mathcal {B}^+] &=& \\mathrm {Cov} \\left[ \\frac{(1-q)\\sum _{h \\in \\partial _j} \\hat{x}^*_h + qk}{k},\\frac{(1-q)\\sum _{m \\in \\partial _\\ell } \\hat{x}^*_m + qk}{k} \\right] \\\\ \\nonumber &=& \\frac{(1-q)^2}{k^2} \\mathrm {Cov} \\left[\\sum _{h \\in \\partial _j} \\hat{x}^*_h,\\sum _{m \\in \\partial _\\ell } \\hat{x}^*_m \\right]< \\frac{1}{k^2} \\mathrm {Cov} \\left[ \\sum _{h \\in \\partial _j} \\hat{x}^*_h,\\sum _{m \\in \\partial _\\ell } \\hat{x}^*_m \\right] \\\\ \\nonumber &=& \\mathrm {Cov}[\\hat{x}^*_j,\\hat{x}^*_\\ell | j,\\ell \\in U] \\ .$ Figure: The covariance of ii's neighbours, jj and kk (grey), can be decomposed into the covariance between its neighbours (black and white).", "This consists of the covariance between neighbours that are two steps apart (black to white) as well as those that have distance four steps apart (white to white).Let $\\mathrm {Cov}(\\hat{x}^*_j,\\hat{x}^*_l|(j,l) \\in U) = \\sigma _2$ , denote the least upper bound for the covariance between two nodes of distance 2 apart.", "Similarly, let $\\sigma _d$ be the same for nodes of distance $d$ apart.", "In the remainder of this section we are seeking to establish a relationship between these upper bounds and in doing so recursively determine the upper bound at $d=2$ .", "It is worth reiterating that we are approximating the variance of the steady state signal mixes at the asymptotic limit $n \\rightarrow \\infty $ .", "Given this assumption, a $k$ -regular tree will approximate a Cayley tree.", "A useful consequence of this assumption is that the network is globally tree-like, and loops vanish in the limit.", "As such, only a single path exists between any two nodes.", "Therefore, in (REF ) we have: $\\sigma ^2_{\\mathcal {U}} = \\frac{\\sigma ^2_V}{k} + \\frac{2}{k^2}\\sum _{(j,\\ell ) \\in \\partial _i}\\mathrm {Cov}[\\hat{x}^*_j,\\hat{x}^*_\\ell ] \\le \\frac{\\sigma ^2_V}{k} +\\frac{2}{k^2}\\sum _{(j,\\ell ) \\in \\partial _i} \\sigma _2 = \\frac{\\sigma ^2_V}{k} +\\frac{\\sigma _2}{k}(k-1) \\ .$ Therefore we can establish that if $\\sigma _2 = \\mathcal {O}(k^{-2})$ , the whole expression will be of order $\\mathcal {O}(k^{-2})$ .", "To do this note: $\\mathrm {Cov}[\\hat{x}_j^*, \\hat{x}_\\ell ^*] &=& \\frac{1}{k^2} \\mathrm {Cov}\\left[\\hat{x}_i^* + \\sum _{m \\in \\partial _j/i}\\hat{x}_m^* , \\hat{x}_i^* + \\sum _{n \\in \\partial _\\ell /i}\\hat{x}_n^*\\right] = \\frac{1}{k^2} \\left(\\mathrm {Cov}[\\hat{x}_i^*,\\hat{x}_i^*]+ \\sum _{(m,n) \\in [\\partial _j\\times \\partial _\\ell ] / (i,i)} \\mathrm {Cov}[\\hat{x}_m^*,\\hat{x}_n^*] \\right) \\\\ \\nonumber &=& \\frac{1}{k^2} \\left( \\mathrm {Cov}[\\hat{x}_i^*,\\hat{x}_i^*] + \\sum _{m \\in \\partial _j/i} \\mathrm {Cov}[\\hat{x}_i^*,\\hat{x}_m^*] + \\sum _{n \\in \\partial _\\ell /i} \\mathrm {Cov}[\\hat{x}_i^*,\\hat{x}_n^*] + \\sum _{(m,n) \\in [\\partial _j/i \\times \\partial _\\ell /i]} \\mathrm {Cov}[\\hat{x}_m^*,\\hat{x}_n^*] \\right) \\ .$ In this last step, we explicitly break down the covariance sum into the covariance between the neighbours of $j$ and $l$ , which exist at various distances to one another.", "This is illustrated in REF .", "We can group these covariance pairs by their distance and bound them using our defined bounds $\\sigma _d$ : $\\mathrm {Cov}[\\hat{x}_j^*, \\hat{x}_\\ell ^*] \\le \\frac{1}{k^2} \\left( \\sigma ^2_{\\mathcal {U}}+ 2(k-1)\\sigma _2 + (k-1)^2 \\sigma _4 \\right) \\ .$ This allows us to recursively define $\\sigma _2 = \\frac{1}{k^2} \\left(\\sigma ^2_{\\mathcal {U}}+ 2(k-1)\\sigma _2 + (k-1)^2 \\sigma _4 \\right) \\ .$ Re-arranging this expression we get: $\\sigma _2 = \\frac{1}{(k-1)^2+1} \\sigma ^2_{\\mathcal {U}}+ \\frac{(k-1)^2}{(k-1)^2-1} \\sigma _4 = \\frac{1}{(k-1)^2+1} \\sigma _0+ \\frac{(k-1)^2}{(k-1)^2-1} \\sigma _4 \\ .$ In the final step, we have replaced $\\sigma ^2_{\\mathcal {U}}$ with $\\sigma _0$ - which emphasizes that this term is merely the covariance of a node with a node at distance 0 (i.e.", "its own variance), and $\\sigma ^2_{\\mathcal {U}}$ is the largest possible variance expression amongst the biased and unbiased nodes.", "We can easily (though quite tediously) repeat this process for the covariance of nodes at any distance $d$ to establish: $\\sigma _d = \\frac{1}{(k-1)^2+1} \\sigma _{d-2}+ \\frac{(k-1)^2}{(k-1)^2-1} \\sigma _{d+2} \\ .$ This linear recurrence relation can be solved with the boundary conditions that $\\sigma _0 = \\sigma ^2_{\\mathcal {U}}$ and $\\lim _{d \\rightarrow \\infty } \\sigma _d = 0$In other words, nodes at infinitely long distances have a covariance that decays to zero.", "This establishes: $\\sigma _d = \\frac{\\sigma _{\\mathcal {U}}^2}{(k-1)^d} \\ ,$ and therefore: $\\sigma _2 = \\frac{\\sigma _{\\mathcal {U}}^2}{(k-1)^2} = \\mathcal {O}(k^{-2}) \\ .$ Therefore, to finalise (REF ): $\\sigma ^2_{\\mathcal {U}} = \\frac{1}{k}\\sigma ^2_V + \\mathcal {O} \\left(\\frac{1}{k^2}(k^2-k)\\sigma _2\\right) = \\frac{1}{k}\\sigma ^2_V+\\mathcal {O}(k^{-2}) \\ .$" ], [ "Steady state signal mix normality.", "We finally proceed to demonstrate that the distribution of signal mixes is approximately normal when $k$ and $n$ are large.", "Assume firstly that $n \\rightarrow \\infty $ , which ensures that the model's $k$ -regular network becomes a Cayley tree with no loops.", "Let us also assume that $k = \\epsilon n$ for some $\\epsilon > 0$ , ensuring $k$ also grows arbitrarily large, but still can be arbitrarily smaller than $n$ .", "As we have already established in the previous section, the covariance between the steady state signal mixes of unbiased agents at distance 2 decays as $k^{-2}$ , which implies that such signals mixes become asymptotically independent in the aforementioned limits.", "Therefore, the steady state signal mix of an unbiased agent $\\hat{x}_{i_{\\mathcal {U}}}^* = \\sum _{j \\in \\partial _i} \\hat{x}^*_j / k$ becomes the sum of an infinitely large set of independent random variables.", "Furthermore, the variables will follow one of three distributions, depending on which sub-population the agent's neighbors belong to: $\\hat{x}_{i_{\\mathcal {U}}}^*=\\frac{1}{k}\\sum _{j\\in \\partial _i} \\hat{x}_j^* =\\frac{1}{k} \\left(\\sum _{j\\in \\partial _i \\cap U} \\hat{x}_j^* + \\sum _{j\\in \\partial _i \\cap \\mathcal {B}^+} \\hat{x}_j^* + \\sum _{j\\in \\partial _i \\cap \\mathcal {B}^-} \\hat{x}_j^* \\right) \\ .$ Each of the above contributions is a sum of an infinitely large set of independent and identically distributed variables, which implies that each of them is normally distributed.", "This, in turn, implies that the steady state signal mixes of the unbiased agents (and, by generalisation, of the biased agents) is asymptotically normal.", "From the results obtained for the first two moments of the signal mix distributions in the previous sections (see Eqs.", "(REF ) and (REF )), we can conclude that when $n \\rightarrow \\infty $ and $k = \\epsilon n$ we have $ \\hat{x}_{\\mathcal {U}} &\\overset{d}{\\rightarrow }& \\mathcal {N} \\left(\\hat{x}^*_{\\mathcal {U}},\\frac{f q^2 \\hat{x}^*_{\\mathcal {U}}(1-\\hat{x}^*_{\\mathcal {U}})}{k} \\right) \\\\ \\nonumber \\hat{x}_{\\mathcal {B}^+} &\\overset{d}{\\rightarrow }& \\mathcal {N} \\left((1-q)\\hat{x}^*_{\\mathcal {U}} + q, \\frac{f \\left(q(1-q) \\right)^2 \\hat{x}^*_{\\mathcal {U}} (1-\\hat{x}^*_{\\mathcal {U}})}{k} \\right) \\\\ \\nonumber \\hat{x}_{\\mathcal {B}^-} &\\overset{d}{\\rightarrow }& \\mathcal {N} \\left((1-q)\\hat{x}^*_{\\mathcal {U}} , \\frac{f \\left(q(1-q) \\right)^2 \\hat{x}^*_{\\mathcal {U}} (1-\\hat{x}^*_{\\mathcal {U}})}{k} \\right) \\ .$ The normality of the distribution for the unbiased agents is demonstrated in Figure REF .", "We can now aggregate our results in order to approximate the accuracy of a social network.", "As described in the main body of the paper, the accuracy of a network $\\mathcal {A}(G)$ is the expected fraction of accurate unbiased agents in the steady state, i.e.", "accuracy quantifies the probability $\\mathrm {Prob}(y_{i_{\\mathcal {U}}}=+1)$ that a randomly picked unbiased agent in a random realisation of the model will correctly learn the ground truthThe definition of accuracy could very easily be extended to all agents instead of just unbiased agents, but we retain discussion to unbiased agents for simplicity.", "This is of course a complex outcome determined by a dynamic series of processes worth recapping.", "First, the model will generate initial signals for all agents, both biased and unbiased.", "All agents will share their signals, but biased agents will selectively sample incoming signals based on their current orientation.", "Over time, biased agents are able to influence the set of signals in the system, and the system converges towards a steady state where each agent possesses an equilibrium mix of signals.", "Accurate agents are those whose equilibrium signal mix is contains more positive than negative signals, i.e.", "$x_i^* > 1 / 2$ .", "In the previous Sections we have calculated the distribution of the initial signals, as well as the approximate steady state signal mix for a given set of initial signals (see Equation REF ).", "We have also approximated the steady state individual signal mix distributions (see Eq.", "REF ), and as such we can approximate the fraction of accurate unbiased agents for a given steady state, which reads $ A(\\mathcal {G}) = \\frac{1}{2} \\int _{0}^{1} dx_{\\mathcal {U}}^* \\ P(x_{\\mathcal {U}}^*) \\ \\mathrm {erfc} \\left(\\frac{\\frac{1}{2}-x_{\\mathcal {U}}^*}{\\sqrt{2} \\ \\sigma _{\\mathcal {U}}} \\right) \\ ,$ where $P(x^*_{\\mathcal {U}})$ is the distribution of the average signal mix across unbiased agents as determined by (REF ) (see Eq.", "(3) of the main paper), while the complementary error function quantifies the fraction of unbiased agents whose steady state signal mix is above $1/2$ , and are therefore accurate, under the normal approximation outlined in the previous section.", "Both ingredients employed in (REF ) have been obtained based on a number of approximations and asymptotic assumptions.", "We checked how such approximations hold against numerical simulations of the model's dynamics.", "The results are shown in Fig.", "REF both for $k$ -regular and Erdős-Rényi networks.", "As can be seen, the average accuracy obtained across independent numerical simulations of the model closely matches the expected value obtained with (REF ), even for relatively low network size and average degree (the results reported were obtained for $n = 10^4$ and $k = 8$ ).", "The wider error bars for lower $f$ reflect the expected outcome that most runs of the model will result in total consensus on either $X = \\pm 1$ (and therefore all accurate or all inaccurate agents), whereas as $f$ grows, the agents are highly polarised and the fraction of accurate and inaccurate agents will be relatively constant.", "Figure: Non-monotonic changes in expected accuracy as ff increases.", "The model's prediction are compared to numerical results obtained with simulations on both kk-regular (light blue) and Erdős-Rényi (purple) networks.", "The parameters used in the simulations were n=10 4 n = 10^4, p=0.53p = 0.53, k=8k = 8, q=1q = 1." ], [ "Theory and model interpretation.", "Our model is stylized, and therefore largely agnostic as to a particular interpretation of its parameters.", "Nevertheless, it is quite well suited to provide an initial exploration on a number of issue.", "In this Section, we shall test the model's ability to shed light on the impact that Internet access has on shaping popular opinion on specific issues (global warming in this case).", "In order to do this, we first specify how we are going to relate our model's parameters to real-world measurable quantities.", "There are two convenient (and pragmatically equivalent) interpretations of the model in the context of Internet use.", "Consider the agent-specific ghost node interpretation, where each ghost node attached to a biased agent represents an aggregation of the “filter bubble\" (passive algorithmic affects) and “selective exposure\" (actively selecting information in a biased way) effects.", "An increase in Internet access therefore translates to an increase in access to these self-confirmatory effects, and corresponds to changing unbiased agents into biased agents (an increase in $f$ ).", "Alternatively, one could consider a scenario where the fraction of biased agents is fixed, in which case an increase in Internet would improve their ability to obtain self-confirmatory information (an increase in $q$ ).", "For the purposes of this exploration, however, the two effects are equivalent, and for convenience we only retain the interpretation where $f$ increases.", "As far as the interpretation of the degree variable $k$ is concerned, the important distinction to make here is that we are not interested in “social networks” as a catch-all term for the number of family and friends one has.", "Rather, given the model, we are interested in the degree to which individuals actively exchange information with their underlying social network with regards to the topic of interest.", "Therefore, for $k$ we wish to measure the volume of active social information diffusion in a given population.", "As per (REF ), one of our model's main results is that $f$ and $k$ work in opposite directions when it comes to polarisationStrictly speaking the result refers to the variance in information sets, but we exploit the monotonic relationship between information variance $\\sigma _{x^*_{\\mathcal {U}}}^2$ and polarization $z_{\\mathcal {U}}^*$ for the remainder of this section - an increase in confirmatory behaviours increases polarisation and is equivalent to a reduction in social information.", "Furthermore, if the majority of the population accurately learns the ground truth ($x_{\\mathcal {U}}^*>1/2$ ), reductions in polarisation can be translated to an increase in consensus on the truth, as a smaller fraction of the population will arrive at inaccurate beliefs.", "Translated to current research on the role of the Internet, we attempt to use our model to shed light on what has been thought of as the dichotomous effects of Internet access on social learning and polarisation.", "On one hand it has been argued that Internet access improves exposure to diverse information via social networks [46], [47], [48], whereas on another it has been argued that Internet access enables confirmation bias on a previously unprecedented scale [39], [12].", "These contradictory effects may be in part responsible for the range of conflicting results obtained in recent research, and in our closing remarks we revisit some of these existing results in the context of our model." ], [ "Data sources and measuring variables", "In order to test the model's predictions in the aforementioned context, we gathered data from the Yale Programme on Climate Change Communication 2016 Opinion Maps [49], which provides state and county level survey data on opinions on global warming, as well as behaviours such as the propensity to discuss climate change with friends and family.", "We combined this with FCC 2016 county level data on residential high speed Internet access [50].", "Finally, we also used a supplemental source in the data aggregated by the Joint Economic Council's Social Capital Project [51], a government initiative aiming to measure social capital at a county level by aggregating a combination of state and county level data from sources such as the American Community Survey, the Current Population Survey, and the IRS.", "In this context, we measured accuracy as the estimated fraction of the population believing that “global warming is happening”.", "We refer to this as “GW Accuracy”.", "In other words, we are attempting to examine the degree to which social information and access to confirmatory bias mechanisms affect the ability of individuals to accurately learn an objective, measurable and uncontroversial ground truth (that global temperatures are rising).", "Internet access is measured by the FCC's data on county-level high speed broadband penetration amongst residentials (in [39], the authors utilise another instrumental variable approach to argue that increased broadband penetration does in fact increase Internet use).", "In Table REF we demonstrate preliminary ordinary least squares regression results by regressing GW Accuracy on Internet access, accounting for a range of covariates such as median age, median income, county population size and the fraction of adults with college degrees.", "The results indicate that even after controlling for relevant covariates, the net effect of Internet access on accuracy is positiveOne may note that the impact of median income on this regression, and all subsequent results.", "is negative.", "We have verified this result through a number of additional checks.", "It appears that the inclusion of college education heavily affects this coefficient, implying that the effect of income on global warming beliefs is heavily mediated by access to education.", "We also performed some further checks by including dummy variables for political partisanship using county level voting results for the 2016 presidential elections.", "While political partisanship provides additional explanatory power over and above the current set of variables, the coefficient for income when including it is still negative.", "Unpacking the exact nature of this relationship would require a broader range of economic and political factors, which is clearly outside the scope this initial analysis, so we exclude partisanship and continue with the original model, allowing the coefficients to be taken at face value.", "(and by interpretation, the effect of polarisation on this particular ground truth is negative).", "However, this alone is insufficient as research indicates Internet access is likely to improve the degree to which individuals can communicate information to friends and family, which in our model is precisely the variable $k$ .", "The Yale Climate Change data includes a measure estimating the fraction of the county population that discusses global warming regularly with family and friends (“Social Discussion”).", "To sense check this, Table REF (column 2) demonstrates that increased Internet access does indeed improve the ability to discuss matters with friends and family, even after controlling for relevant covariates, which is consistent with a broad set of empirical research on the topic (see [52] for a review).", "Therefore, this allows us to construct our final model in Table 1(3) where we regress GW Accuracy on both Social Discussion and Internet access (and the covariates).", "We can now interpret the coefficient on Internet access as the residual effect of Internet access after controlling for the effect it has on Social Discussion.", "One way of thinking about this is to consider all causal pathways from Internet access to belief formation - some fraction of them will be via improved access to social and discussion networks (communication platforms, online social networks, and forums), and the remaining fraction will be non-social (algorithmic effects, filter bubbles, online news media, selective exposure, etc).", "By accounting for the former effects by observing the discussion network size in Social Discussion, the residual effect of internet access will aggregate all these other effects.", "This lines up with the interpretation of $f$ in our model - Internet users will have access to these effects (“biased agents”) and non-Internet users will not.", "The results confirm our hypothesis - Social Discussion ($k$ ) and residual Internet Access ($f$ ) act in opposite directions when it comes to learning the ground truth, even after conditioning on a range of covariates.", "It is worth unpacking these results in detail.", "The direct effect of a 1 percentage point increase in Internet access on global warming accuracy is negativeTable 1, Column 3, Row 2.", "($-2.400$ ).", "The direct effect on social discussion is extremely positiveTable 1, Column 2, Row 2.", "($3.736$ ), which leads to a corresponding improvement in accuracyTable 1, Column 3, Row 1. of $1.057 \\times 3.736 \\approx 3.95$ .", "The net effect, of course, is positive ($3.95 - 2.40 = 1.55$ ), as indicated in the original, simple regressionTable 1, Column 1, Row 2..", "However, breaking down the causal mechanism into its constituent elements - direct internet use effects vs socially mediated internet effects - allows us the capture the nuance of what is actually happening.", "0.9 Table: Initial Regression Results1" ], [ "Accounting for simultaneous causality.", "A clear shortcoming of the above analysis is the fact that the variable “Social Discussion” is likely to have a reverse causal relationship with the outcome variable of “GW Accuracy”.", "That is, the more likely individuals are to believe global warming is happening, the more likely they are to discuss this topic with friends and family.", "In order to account for this, we will take an instrumental variable approach.", "That is, we need some instrument that can account for independent variation in discussion with family and friends, which is otherwise unlikely to affect the belief in global warming.", "We note as before that $k$ can be interpreted as the fraction of the “underlying social network” that is activated to transmit social information related to the topic of global warming.", "We are therefore interested in a variable that can measure the pre-existing strength of these underlying social networks.", "To do so, we make use of the Social Capital Project, a government research programme by the Joint Economic Committee that attempts to measure Social Capital at a state and county level throughout the US.", "Social Capital as defined in this study (and numerous othersi.e.", "Putnam [53] (1995, p.19), “...social capital refers to connections among individuals' social networks and the norms of reciprocity and trustworthiness that arise from them”.)", "refers broadly to something “related to social relationships, social networks, and civil society”.", "More specifically, it is measured with an intention to reflect communities with “an abundance of close, supportive relationships” [51].", "The index itself measures a spectrum of factors, and in particular a “Community Health” subindex.", "The subindex is calculated as the leading principal component across a variety of state and county-level measures of community engagement (where people ostensibly meet and socialise with friends and family), including religious congregations, non-religious non-profit activities, public meeting attendance, working with neighbours to fix things, attending a meeting where politics was discussed, etc.", "This index is then validated by examining bivariate correlations with a battery of county level benchmarks and measures of social dysfunction.", "The strength of this instrument is established in Table REF (column 1), where a first stage least squares regression is run to show that improvements in Community Health do translate to improved discussion with friends and family (controlling for covariates).", "The validity is established through a series of additional checks.", "Factors such as religious attendance, public meetings, etc.", "are unlikely to have a causal effect on people's beliefs about global warming independent of them being a medium to allow for social discussion of these topics.", "The only other reasonable and plausibly significant causal channel is if these factors are caused by or cause an increase membership in social groups (for instance, political parties) that are strongly associated with reduced belief in global warming.", "In particular, it is well-established that members of the Republican Party have a reduced belief in the existence of Global Warming [54].", "To check this, we examined the bivariate correlation between Community Health and the percentage of GOP votes cast in the 2016 presidential election.", "The results were weak, with a correlation of only $0.14$ , meaning only $1.8\\%$ of the variation in the measures were explained by the relationship.", "Having established the strength and validity of the instrument, we demonstrate the results from the two stage least squares regression results in Table REF (column 2).", "We can see the qualitative results of the simpler model have been preserved, with the effects predictably attenuated.", "However, the results are still significant, and corroborate our theory.", "After separating out the social and confirmatory effects of Internet access, we can see the impact on Accuracy (and Polarisation) both occur in the direction that we predict.", "Once again, let us unpack the results.", "The direct effect of a 1 percentage point increase in internet access on global warming accuracy is negativeTable 2, Column 2, Row 3.", "($-1.712$ ).", "The direct effect on social discussion is extremely positiveTable 2, Column 1, Row 2.", "($3.143$ ), which leads to a corresponding improvement in accuracyTable 2, Column 2, Row 2. of $0.872 \\times 3.143 \\approx 2.74$ .", "The net effect, of course, is positive ($2.74 - 1.71 = 1.03$ ).", "Once again, breaking down the causal mechanism into its constituent elements - direct internet use effects vs socially mediated internet effects - allows us the capture the nuance of what is actually happening.", "It appears, for the topic of global warming, the net impact of Internet access on social learning is positive.", "Increase in Internet access has a direct negative impact on learning (via $f$ , or $q$ ).", "However, it leads to a significant positive impact on social discussion ($k$ ), and the net result of this is positive.", "This result remains robust even after controlling for a battery of relevant covariates.", "It should be emphasized that this result is merely an initial exploration of how our model can provide some testable predictions to empirical data, as opposed to a detailed effort to understand the effect of Internet access on global warming beliefs.", "Having said that, the initial results are encouraging, and we hope the clarity of the analytic results of our model pave the way for testing variations of the idea of biased information aggregation in a range of outcomes and settings.", "0.9 Table: IV Regression Results1" ], [ "Making sense of broader empirical results.", "We have seen so far that our model can help us decompose the effect of internet access on learning in the specific case of global warming facts.", "We now see if the model can help us better understand the seemingly conflicting findings we have found in existing research as indicated above.", "It should be said that the following interpretations are meant only to be indicative of how our model can help shape our theoretical understanding of empirical phenomena, rather than a detailed exploration of the specific empirical questions these papers explore.", "In [41], the authors argue that internet access has not had an effect on political polarisation because the demographic with the lowest increase in internet use - the elderly - has had the highest increase in political polarisation.", "However, it is also well established that older people have smaller network sizes than younger people [55] and growing evidence of demographic shifts suggest that older people are increasingly living alone [56].", "This translates to a direct fall in $k$ for such populations, and without a corresponding increase in $k$ provided by internet access, we would in fact expect to see higher polarisation in such a group.", "In [39], the authors argue that an increase in internet access leads to an increase in political polarisation.", "Firstly, it is worth noting that the overall effect size is very small - increasing the number of broadband providers by $10\\%$ increases political polarisation by $0.003$ points (on a scale between 0 and 1).", "This is consistent with notion that social connectivity will dampen the direct effect of biased media, and it is possible one could uncouple the effect of the internet on social connectivity as opposed to enabling confirmation bias with some proxy measure for social connectivity.", "What is also noteworthy is that the researchers included the level of “political interest” per county as a mediating variable in parts of the analysis.", "So for example, if we allow $f$ to represent the fraction of respondents in each county with such strong partisan interest, then $q$ could represent the level of bias these agents can display due to access to partisan media on the internet.", "Under this interpretation we can make sense of the interaction terms in the regression results - the effect of internet access on polarisation was considerably higher for counties where political interest is higher, which is exactly what we predict from the product ($fq^2$ ) in (REF ).", "In [40], the author argues that internet access leads to a decrease in political polarisation.", "This study looks solely at Twitter networks over time (but shows how they relate to political polarisation data offline).", "The author finds that more diverse Twitter networks lead to reduced polarisation over time.", "Again, our model predicts the following - since everyone is already on Twitter in this scenario, the fractions $f$ and $q$ are untouched.", "However, the author notes that more diverse networks are directly correlated with larger networks - a larger $k$ .", "It follows therefore that these users with reduced polarisation experienced an increased $k$ without a corresponding change in $f$ or $q$ , and the results follow.", "All in all, our biased learning model has proven to provide useful insight into a long-standing debate about an important empirical topic.", "We show that it allows us compress a large and complex set of causal mechanisms in the literature down to the effect of three terms of interest - the prevalence of biased agents ($f$ ), degree of bias ($q$ ), and social connectivity ($k$ ).", "In doing so, we were able to shed insights on the mechanisms at play when it came to internet access, and provide the beginnings of a more uniform understanding of what previously conflicting data has suggested to date." ] ]

[ [ "Gravitational waves and the Sagnac effect" ], [ "Abstract Light propagating in opposite directions around the same loop in general shows a relative phase shift when recombined.", "This phenomenon is known as the Sagnac effect after Georges Sagnac who, in 1913, demonstrated with an interferometer on a rotating table that the phase shift depended on the angular velocity of the table.", "In previous work we have given a very general formula for the Sagnac effect, valid in full general relativity.", "The relativistic effect not only contains the `classical' contribution from the rotation of the laboratory but also contributions due its acceleration and due to incoming gravitational waves.", "Here, we point out a major consequence of this gravitational effect which may have implications for third generation gravitational wave detectors.", "We describe an `antenna' design which picks out specific components of the Weyl tensor describing the incident gravitational waves." ], [ "Introduction", "In general relativity, a laboratory is modeled as a time-like world-line to measure the proper time $t$ passing in the lab together with a set of space-like mutually orthogonal vectors $(\\mathbf {e}_1,\\mathbf {e}_2,\\mathbf {e}_3)$ attached at each point of the world-line [1].", "This reference frame indicates the orientation of the lab in space-time at each instant of time.", "The lab may rotate, it may be accelerated and it may travel through an arbitrarily curved region of space-time.", "Introducing (generalized) Fermi coordinates $(t,x^1,x^2,x^3)$ adapted to the lab [2], [3] one can describe the geometry of space-time by means of its metric $g_{ab}$ with respect to these coordinates by $\\begin{aligned}g_{00} &= 1 - 2 a_l x^l + 3 (a_mx^m)^2 + \\omega _{im} \\omega ^i{}_n x^m x^n + R_{m0n0} x^m x^n + O(x^3),\\\\g_{0k} &= \\omega _{kl} x^l + \\frac{2}{3} R_{m0nk} x^m x^n+ O(x^3),\\\\g_{kl} &= -\\delta _{kl} + \\frac{1}{3} R_{mlnk} x^m x^n+ O(x^3).\\end{aligned}$ Here, $a^i$ , $\\omega ^i{}_k$ and $R_{m0n0}$ etc.", "are components of the acceleration and the angular velocity of the lab and of the Riemann tensor of the space-time.", "These quantities depend on $t$ and the expressions are valid up to the given order in the spatial coordinates.", "Suppose two photons travel around a closed loop $C$ which has no self-intersections.", "It can be given in parametrised form as $x^i(s)$ .", "The photons start at the same time at the point $Q=x^i(0)$ , returning back to $Q$ at different times depending on the travel direction.", "In [4] we derived the general formula for the difference in the arrival times of the photons.", "This formula is not immediately useful since it involves the solution of a differential equation along the path.", "However, with the very reasonable assumption that the travel time of the photons is negligibly small compared to the time scales of changes in the lab motion and the surrounding curvature one can derive the succinct formula $\\Delta T = -2 \\int _C \\frac{g_{0i}}{g_{00}}\\,\\mathrm {d}x^i.$ Using the Stokes theorem we can recast this line integral as a surface integral over a surface $S$ which is bounded by the curve $C$ .", "Inserting the expression for the metric in terms of the Fermi coordinates one obtains three terms contributing to the time difference.", "To discuss them we use the usual 3-vector notation $\\mathbf {a}$ and ${\\omega }$ for the acceleration and the angular velocity and we use the position vector $\\mathbf {x}$ and the vector $\\mathbf {n}$ normal to the surface $S$ .", "Then, the first term becomes $\\Delta _\\omega T = 4 \\int _S {\\omega }\\cdot \\mathbf {n}\\, \\mathrm {d}^2 S.$ This is the classical Sagnac effect as first described by Sagnac [5], [6] expressed as the `rotation flux' through the surface $S$ .", "It is proportional to the magnitude of the angular velocity but it also depends on its direction in relation to the surface $S$ and therefore to the curve $C$ .", "In fact, by considering different shapes of $C$ one can construct different `antennas', i.e., configurations with different directional dependence.", "For instance, the curve which is described by the seam of a tennis ball is insensitive to rotations around the two axes piercing the opposite lobes but can detect rotations around the third axis.", "This contribution is translation invariant.", "The second term depends on the rotation as well as the acceleration: $\\Delta _a T = 4 \\int _S (\\mathbf {a}\\cdot {\\omega }) (\\mathbf {n}\\cdot \\mathbf {x}) - 3 (\\mathbf {a}\\cdot \\mathbf {x}) ({\\omega }\\cdot \\mathbf {n})\\,\\mathrm {d}^2S.$ The appearance of $\\mathbf {x}$ shows that this contribution is not translation invariant, the time difference also depends on the position of the loop.", "This term can, at least in principle, be used to detect the acceleration of the lab in relation to the rotation axis and the orientation of the curve.", "The third term is due to the curvature of the space-time and has two separate parts.", "One of the pieces is caused by the Weyl tensor and, hence, is related to gravitational waves while the other part comes from the Ricci tensor and, therefore due to the Einstein equation, is caused by the matter content of the space-time: $\\Delta _RT = 4 \\int _S \\mathbf {n}\\cdot \\mathbb {B} \\cdot \\mathbf {x}\\, \\mathrm {d}^2S - \\frac{16\\pi G}{c^4} \\int _S \\mathbf {n}\\cdot (\\mathbf {x}\\times \\mathbf {j})\\, \\mathrm {d}^2S.$ In the first term, $\\mathbb {B} =(B_{ij})$ is a symmetric, trace-free $3\\times 3$ -matrix which describes the magnetic part of the Weyl tensor.", "It contains the information of gravitational waves propagating in the three spatial directions together with their two independent polarisation states.", "In the second part we find the momentum density $\\mathbf {j}$ of the matter so that this term is caused by the flux of the material angular momentum density through $S$ .", "Both parts of the gravitational contribution depend on the position of the curve.", "For the remainder we focus on the gravitational wave part.", "To give a very simple example we consider a closed path without self-intersections in a plane with normal vector $\\mathbf {n}$ through a point $\\mathbf {x}_0$ .", "We choose the axes so that $\\mathbf {n}= \\mathbf {e}_3$ , i.e., it points in the positive $z$ -direction.", "Then the points on the plane can be written in the form $\\mathbf {x}= \\mathbf {x}_0 + u \\mathbf {e}_1 + v \\mathbf {e}_2.$ and the curve itself is specified by a parametrization $(u(t),v(t))$ .", "The time difference for photons traveling along that path is then obtained from the integral $\\Delta _RT = 4 \\int _S \\mathbf {n}\\cdot \\mathbb {B} \\cdot \\mathbf {x}\\, \\mathrm {d}^2S = 4 A (\\mathbf {n}\\cdot \\mathbb {B}\\cdot \\mathbf {x}_0 + \\mathbf {n}\\cdot \\mathbb {B}\\cdot \\mathbf {e}_1 u_0 + \\mathbf {n}\\cdot \\mathbb {B}\\cdot \\mathbf {e}_2 v_0 )$ where $A$ is the (signed) area enclosed by the curve.", "Here, we have defined $u_0=A^{-1}\\int _S u \\mathrm {d}u \\mathrm {d}v$ and similarly for $v_0$ .", "These define the `center of mass' of the area surrounded by the curve.", "Choosing this point as $\\mathbf {x}_0$ we can obtain the simple formula $\\Delta _RT = 4 A (\\mathbf {n}\\cdot \\mathbb {B}\\cdot \\mathbf {x}_0 )$ for the time difference along a simple path in a plane with normal vector $\\mathbf {n}$ passing through the point $\\mathbf {x}_0$ which is the center of mass of the area $A$ enclosed by the curve.", "For this simple system the time difference depends on the location $\\mathbf {x}_0$ of the loop.", "However, we can combine such loops and obtain more complicated configurations with better behaviour.", "One example is shown in Fig.", "REF .", "Other possibilities exist.", "Figure: A simple configuration of loops for a Sagnac detector for gravitational wavesThe configuration consists of two identical loops which are almost closed and connected in such a way that a photon which travels counterclockwise in the upper loop around the $z$ -axis will loop around in the clockwise direction in the lower loop.", "In the ideal situation, the vertical strip can be made arbitrarily small compared to the area enclosed by each loop.", "The loops are positioned so that the upper loop is centred around $\\mathbf {x}_0 = (a,b,c)$ and the lower one is centred at $\\mathbf {x}_1 = (a,b,-c)$ .", "The orientation of the loops has the consequence that the normal vector is oriented along the positive $z$ -axis in the upper part and points in the opposite direction in the lower part of the path.", "The net effect of the time difference between two counter propagating photons can then be obtained by simply adding the contributions from each loop taking into account their different orientations.", "The result is $\\Delta T = 4 A (\\mathbf {n}\\cdot \\mathbb {B}\\cdot \\mathbf {x}_0 + \\mathbf {n}\\cdot \\mathbb {B}\\cdot \\mathbf {x}_1) = 4 A B_{33} h.$ Here, we introduced the height $h=2c$ of the configuration and we denote by $A$ the area of one loop.", "This “antenna” is sensitive to exactly one component of the magnetic part of the Weyl tensor.", "As a bonus, we find that it is not sensitive at all to the classical Sagnac effect when the vertical area is made zero.", "In a similar way we can construct a configuration which is sensitive to an off-diagonal element of the Weyl tensor, see Fig.", "REF .", "Figure: A loop configuration to detect an off-diagonal element of the magnetic part of the Weyl tensorHere, we have assumed the loops to lie in the same plane.", "They are traversed with different orientations so that $\\mathbf {n}=\\mathbf {e}_3$ for one loop and $\\mathbf {n}=-\\mathbf {e}_3$ for the other.", "The loops are centered around the points $\\mathbf {x}_0=(-a,0,0)$ and $\\mathbf {x}_1=(a,0,0)$ , respectively, and connected along two crossing lines.", "For this system we obtain for the time difference between two counter propagating photons the value $\\Delta T = 4 A (\\mathbf {n}\\cdot \\mathbb {B}\\cdot \\mathbf {x}_0 + \\mathbf {n}\\cdot \\mathbb {B}\\cdot \\mathbf {x}_1) = 4 A B_{13} L$ where we have introduced $L=2a$ , the distance between the centres of the two loops.", "This shows that we can — at least in principle — devise configurations which are able to pick up all the components of the Weyl tensor.", "In particular, a combination of such loops can be arranged in such a way that the antenna is also sensitive to the polarisation of the incident gravitational wave.", "As a final remark, we point out that this way of detecting gravitational waves is, in some sense, dual to the detectors in current use which are based on a Michelson type interferometer.", "These detect the wave form due to the geodesic deviation equation which is driven by the electric part of the Weyl tensor.", "This, and the particular design of Sagnac antennas may be of interest to the current discussion of third generation gravitational wave detectors, see [7].", "The author is grateful to Prof. Ezra Newman and Prof. Rainer Weiss for comments." ] ]

[ [ "Computing delay Lyapunov matrices and H2 norms for large-scale problems" ], [ "Abstract A delay Lyapunov matrix corresponding to an exponentially stable system of linear time-invariant delay differential equations can be characterized as the solution of a boundary value problem involving a matrix valued delay differential equation.", "This boundary value problem can be seen as a natural generalization of the classical Lyapunov matrix equation.", "We present a general approach for computing delay Lyapunov matrices and H2 norms for systems with multiple discrete delays, whose applicability extends towards problems where the matrices are large and sparse, and the associated positive semidefinite matrix (the ``right-hand side' for the standard Lyapunov equation), has a low rank.", "In contract to existing methods that are based on solving the boundary value problem directly, our method is grounded in solving standard Lyapunov equations of increased dimensions.", "It combines several ingredients: i) a spectral discretization of the system of delay equations, ii) a targeted similarity transformation which induces a desired structure and sparsity pattern and, at the same time, favors accurate low rank solutions of the corresponding Lyapunov equation, and iii) a Krylov method for large-scale matrix Lyapunov equations.", "The structure of the problem is exploited in such a way that the final algorithm does not involve a preliminary discretization step, and provides a fully dynamic construction of approximations of increasing rank.", "Interpretations in terms of a projection method directly applied to a standard linear infinite-dimensional system equivalent to the original time-delay system are also given.", "Throughout the paper two didactic examples are used to illustrate the properties of the problem, the challenges and methodological choices, while numerical experiments are presented at the end to illustrate the effectiveness of the algorithm." ], [ "Introduction", "We consider a linear system with multiple discrete delays, $\\dot{x}(t)=A_0 x(t)+\\sum _{i=1}^m A_i x(t-\\tau _i),$ where $x(t)\\in \\mathbb {R}^n$ is the state variable at time $t$ , $A_i\\in \\mathbb {R}^{n\\times n} are the system matrices$ and $\\tau _i,\\ i=1,\\ldots ,m$ , represent time-delays, ordered such that $0<\\tau _1<\\cdots <\\tau _m.$ Throughout the paper we assume that the zero solution of (REF ) is exponentially stable, or equivalently, that all its characteristic roots, i.e., the solutions of equation $\\det \\left(\\lambda I-A_0-\\sum _{i=1}^m A_i e^{-\\lambda \\tau _i}\\right)=0,$ are confined to the open left half plane [18], [19].", "The fundamental solution of (REF ), which we denote by $K:\\mathbb {R}\\rightarrow \\mathbb {R}^{n\\times n}$ , is defined as the function satisfying $\\left\\lbrace \\begin{array}{ll}\\dot{K}(t)=A_0 K(t)+\\sum _{i=1}^m A_i K(t-\\tau _i),\\ \\ & \\mathrm {for\\ almost\\ all\\ } t\\ge 0,\\\\K(0)=I, &\\\\K(t)=0,\\ & \\mathrm {for}\\ t<0.\\end{array}\\right.$ The delay Lyapunov matrix for (REF ), associated with a positive semidefinite matrix, whose rank revealing decomposition reads as $B B^T$ , where $B\\in \\mathbb {R}^{n\\times r}$ is of full rank $r$ , is defined as a function $P:\\ \\mathbb {R}\\rightarrow \\mathbb {R}^{n\\times n}$ such that $P(t)=\\int _0^{\\infty } K(s) B B^T K^T(s+t) ds.$ Following from the exponential stability condition of (REF ), the delay Lyapunov matrix can be characterized as the unique solution of the matrix valued “boundary” value problem $ \\left\\lbrace \\begin{array}{rcl}\\dot{P}(t)&=&P(t)A_0^T+\\sum _{k=1}^m P(t-\\tau _k)A_k^T,\\ \\ t\\ge 0, \\\\P(-t)&=&P^T(t), \\\\{-B B^T}&{=}&{P(0)A_0^T+A_0 P(0)} + \\sum _{k=1}^m\\left(P(-\\tau _k)A_k^T+A_kP(\\tau _k)\\right),\\end{array}\\right.$ see [13].", "There is also a dual formulation: with a positive semi-definite $(n\\times n)$ -matrix $C^T C $ we can associate Lyapunov matrix $Q(t)=\\int _0^{\\infty } K^T(s) C^T C K(s+t)ds,$ which corresponds to the unique solution of $ \\left\\lbrace \\begin{array}{rcl}\\dot{Q}(t)&=& Q(t) A_0+\\sum _{k=1}^m Q(t-\\tau _k) A_k,\\ \\ t\\ge 0, \\\\Q(-t)&=&Q^T(t), \\\\{-C^TC}&{=}&{Q(0)A_0+A_0^T Q(0)} + \\sum _{k=1}^m\\left(Q(-\\tau _k)A_k+A_k^T Q(\\tau _k)\\right).\\end{array}\\right.$ Note that in the delay-free case the third equation in (REF ) and the third one in (REF ) reduce to a standard pair of primal and dual Lyapunov matrix equations.", "The delay Lyapunov matrix is a building block for the construction of Lyapunov functionals of complete type, which are associated with necessary and sufficient stability conditions, see [13] for an excellent review.", "It should be remarked that in the literature on complete type Lyapunov functionals, the Lyapunov matrix is usually denoted by $U(t)$ , which corresponds to $Q(t)$ in our adopted notation.", "Another comment is that in several works the Lyapunov matrix is alternatively defined directly as the solution of boundary value problem (REF ) or (REF ).", "In this way it can also be defined for an exponentially unstable system (provided the delay systems has no pair of eigenvalues $(\\lambda _1,\\lambda _2)$ such that $\\lambda _1+\\lambda _2=0$ , see [14], [13]), at the price that the aforementioned connection with the fundamental solution is lost.", "The Lyapunov matrix also plays a major role in the characterization of the $\\mathcal {H}_2$ norm for system $\\left\\lbrace \\begin{array}{lll}\\dot{x}(t)&=&A_0 x(t)+\\sum _{i=1}^m A_i x(t-\\tau _i)+B u(t), \\\\y(t)&=&C x(t),\\end{array}\\right.$ where $u\\in \\mathbb {C}^r$ is the input, $y\\in \\mathbb {C}^{s}$ is the output and $B\\in \\mathbb {R}^{n\\times r}$ , respectively $C\\in \\mathbb {R}^{s\\times n}$ are the input, respectively and out matrix of the model.", "The transfer function of the system (REF ) is given by $\\Upsilon (s)=C\\left(sI-A_0-\\sum _{i=1}^m A_i e^{-s\\tau _i}\\right)^{-1} B.$ The $\\mathcal {H}_2$ norm of $\\Upsilon $ is defined in the frequency domain as $\\Vert \\Upsilon \\Vert _2=\\frac{1}{2\\pi }\\left(\\int _{-\\infty }^{\\infty } \\mathrm {Tr}\\left( \\Upsilon ^*(\\imath \\omega ) \\Upsilon (\\imath \\omega )\\right)d\\omega \\right)^{\\frac{1}{2}},$ while an equivalent definition in the time-domain is given by $\\Vert \\Upsilon \\Vert _2= \\left(\\int _{0}^{\\infty } \\mathrm {Tr}\\left( h^T(t) h(t)\\right)dt\\right)^{\\frac{1}{2}},$ with $h$ the impulse response.", "The following proposition expresses the $\\mathcal {H}_2$ norm in terms of the delay Lyapunov matrix, whose proof trivially follows from the identity $h(t)=CK(t) B.$ [12] The $\\mathcal {H}_2$ norm of (REF ) satisfies $\\Vert \\Upsilon \\Vert _2^2=\\mathrm {Tr}\\left(C P(0) C^T\\right),$ where $P(t)$ is the delay Lyapunov matrix associated with matrix $B B^T$ .", "The aim of this paper is to present a novel method for computing delay Lyapunov matrices and $\\mathcal {H}_2$ norms with the following properties: it is generally applicable, in the sense that there are no restrictions on the number and values of the delays, and the delay Lyapunov matrix can be easily computed or extended a posteriori beyond the interval $[-\\tau _m,\\ \\tau _m]$ ; the number of operations scales favorably with respect to the dimension $n$ of the system matrices, particularly if the matrices are sparse, targeting (discretizations of) partial differential equations (PDEs) with delay, provided that the rank $r$ of $B$ is small compared to $n$ In the description of the results we restrict ourselves to the computation of Lyapunov matrix $P(t)$ , since $Q(t)$ can be obtained from Lyapunov matrix $P(t)$ associated with a “transposed\" system, inferred from the substitutions $A_k\\leftarrow A_k^T,\\ k=0,\\ldots ,m $ and $B\\leftarrow C^T$ .", "The latter directly follows from a comparison between (REF ) and (REF ).", "The characterization (REF ) provides a natural way to compute the Lyapunov matrix, and the $\\mathcal {H}_2$ norm via formula (REF ).", "However, there are major challenges.", "First, when making the leap from ordinary to delay differential equations, the algebraic Lyapunov matrix equation is replaced by a matrix-valued boundary value problem with delay.", "Second, bringing the equation in triangular form using a Schur decomposition, which forms the basis of the celebrated Bartels-Stewart algorithm for the matrix Lyapunov equation, is no longer possible.", "Third, it has been shown in [12] that function $\\mathbb {R} \\ni t\\mapsto P(t)$ may be non-smooth.", "The function is continuous, but it may be not be differentiable at $t=0$ .", "On the interval $[0,\\ \\infty )$ , to which we restrict in this paper because of the second condition in (REF ), it is continuously differentiable, yet the second derivative might be discontinuous at $t=\\tau _i,\\ i=1,\\ldots ,m$ , as we shall illustrate in the next section.", "In the present literature two approaches for solving (REF ) can be identified.", "The first one, the so-called direct approach, is based on approximating the solution on an interval by a matrix polynomial or a piecewise matrix polynomial and, besides imposing the boundary conditions and continuity requirements, determining the coefficients by collocation conditions for the differential equation, see, e.g., [8], [12].", "With $N$ the number of collocation points, this results in a linear system of equations in $\\mathcal {O}(n^2 N)$ variables.", "The convergence of the obtained approximations to the solution as a function of $N$ might be slowed down by the lack of smoothness of the solution discussed above, see [12] for a detailed analysis.", "The second approach can be interpreted as a shooting method.", "It is applicable only if the time-delays are commensurate, i.e., $\\tau _i= n_i h$ for some $h>0$ and $n_i\\in \\mathbb {N},\\ i=1,\\ldots ,m$ , and it exploits that, in this case, the solution of (REF ) is piecewise smooth (more precisely, smooth on intervals of form $(i h,\\ (i+1)h),\\ i\\in \\mathbb {Z}$ ).", "Then (REF ) can be reformulated as a standard boundary value problem for an ordinary differential equation of dimensions $2n^2 n_m$ on the interval $[0,\\ h]$ .", "For the latter boundary value problem, the transition between starting and end time can be determined explicitly in the form of the action of a matrix exponential (the so-called semi-analytic approach [13], [6]) or by a numerical time-integration scheme [11].", "The common factor that leads to a poor scalability of the above vectorization based approaches with respect to the dimension $n$ is that they rely on solving a system of equations in $n^2$ variables, possibly multiplied with a large factor, hence when using a direct solver the number of elementary operations amounts to $\\mathcal {O}(n^6)$ operations.", "To the best of the authors' knowledge the only available method that allows to address large-scale problems is the one presented in [11] for the single delay case, which falls under the umbrella of shooting methods, with the transition map determined by time-integration.", "The key idea behind this approach, which has been shown to be effective for problems with $n$ up to $\\approx 1000$ , is to solve the linear system of equations arising from the shooting method using a preconditioned Krylov method, where the preconditioner is determined from the corresponding problem without delay.", "The latter allows an application of the preconditioner using $\\mathcal {O}(n^3)$ operations.", "This approach is complementary to the presented approach, which has the distinctive feature that it grounded in solving standard Lypunov matrix equations.", "In Section  we present a spectral discretization of equation (REF ) into an ordinary equation of dimensions $(N+1)n$ , where $N$ determines the resolution of the discretization.", "This allows us to obtain approximations of delay Lyapunov matrices and $\\mathcal {H}_2$ norms from solving standard Lyapunov matrix equations.", "We also show how using a transformation a favorable structure can be imposed.", "The main results are obtained in Section , where among others projections on Krylov spaces are used to approximate the solutions of these Lyapunov equations, resulting in a dynamic construction of Lyapunov matrix approximations.", "Note that Krylov methods constitute an established approach for solving large scale matrix Lyapunov equations, see, e.g., [22], [21], [5] and the references therein.", "We will show that several methodological choices can be made in such a way that the overall algorithm does not depend any more on parameter $N$ (the only condition is that it is sufficiently large with respect to the number of iterations for building the Krylov space).", "This property is at the basis of an interpretation of in terms of a projection method applied directly to a linear infinite-dimensional system equivalent to the original delay system.", "In this sense the algorithm complements the set of “discretizaton free\" algorithms for solving nonlinear eigenvalue problems and associated problems in [9], [10].", "Numerical experiments are reported in Section  and some concluding remark are formulated in Section .", "Preliminary results regarding the computation of $\\mathcal {H}_2$ norms have been presented in [20]." ], [ "Finite-dimensional approximation", "In Section 2.1 we outline how to discretize (REF ) (and as a consequence (REF )) using a spectral method [24], resulting in a system described by ordinary differential equations.", "For sake of conciseness, the derivation is slightly different from [2], in the sense that the connection of (REF ) with an abstract infinite-dimensional linear system is not explicitly made.", "Subsequently, we outline how an approximation of the delay Lyapunov matrix can be obtained from this discretization.", "In Section 2.2 we discuss and illustrate properties of the obtained approximations.", "In Section REF we refermulate the expressions for the delay Lyapunov approximations in a form that is more suitable for the application of a Krylov method." ], [ "A spectral discretization", "Given a positive integer $N$ , we consider a mesh $\\Omega _N$ of $N+1$ distinct points in the interval $[-\\tau _{m},\\ 0]$ : $\\Omega _N=\\left\\lbrace \\theta _{N,i},\\ i=1,\\ldots ,N+1\\right\\rbrace ,$ where $-\\tau _{m}\\le \\theta _{N,1}<\\ldots <\\theta _{N,N}<\\theta _{N,N+1}=0.$ Throughout the paper we choose the nonzero mesh points as scaled and shifted zeros of the Chebyshev polynomial of the second kind and order $N$ , i.e.", "the mesh points are specified as $\\theta _{N,i}=\\frac{\\tau _m}{2}(\\alpha _{N,i}-1),\\ \\ \\alpha _{N,i}=-\\cos \\frac{\\pi i}{N+1},\\ i=1,\\ldots ,N+1.$ Denoting with $l_{N,k}$ the Lagrange polynomials corresponding to $\\Omega _N$ , i.e., real valued polynomials of degree $N$ satisfying $l_{N,k}(\\theta _{N,i})=\\left\\lbrace \\begin{array}{ll}1 & i=k,\\\\0 & i\\ne k,\\end{array}\\right.$ and letting $x_k,\\ k=1,\\ldots ,N+1$ functions from $\\mathbb {R}$ to $\\mathbb {R}^n$ , we approximate the “piece of trajectory\" $x(t+\\theta ),\\ \\theta \\in [-\\tau _m,\\ 0]$ as follows, $x(t+\\theta )\\approx \\sum _{k=1}^{N+1} l_{N,k}(\\theta ) x_k(t),\\ \\ \\theta \\in [-\\tau _m,\\ 0],$ which induces on its term the approximation $\\left\\lbrace \\begin{array}{rcl}x_1(t) &\\approx & x(t+\\theta _{N,1}),\\\\&\\vdots & \\\\x_{N}(t)&\\approx &x(t+\\theta _{N,N}),\\\\x_{N+1}(t)& \\approx &x(t).\\end{array}\\right.$ Along a solution of (REF ), $x$ is differentiable almost everywhere, hence for almost all $t\\ge 0,\\ \\theta \\in [-\\tau _m,\\ 0]$ we can express $\\frac{\\partial x(t+\\theta )}{\\partial t}=\\frac{\\partial x(t+\\theta )}{\\partial \\theta }.$ Requiring that the right-hand side of (REF ) satisfies this identity for (collocation points) $\\theta _{N,1},\\ldots ,\\theta _{N,N}$ brings us to the equations $\\dot{x}_i(t)= \\sum _{k=1}^{N+1} \\dot{l}_{N,k}(\\theta _i) x_k(t),\\ \\ \\ i=1,\\ldots ,N.$ Next, substituting the right-hand side of (REF ) into (REF ) yields $\\left\\lbrace \\begin{array}{lll}\\dot{x}_{N+1}(t) &=& A_0 x_{N+1}(t)+ \\left(\\sum _{i=1}^m \\sum _{k=1}^{N+1} A_i l_{N,k}(-\\tau _i) \\right) x_k(t)+ B u(t),\\\\y(t) &=& C x_{N+1}(t).\\end{array}\\right.$ Letting $z(t)=[x_1^T(t)\\ \\cdots \\ x_{N+1}^T(t)]^T\\in \\mathbb {R}^{(N+1)n\\times 1}$ , Equations (REF ) and (REF ) can be written as $\\left\\lbrace \\begin{array}{l}\\dot{z}(t)= \\mathcal {A}_N z(t)+B_N u(t), \\\\y(t)= C_N z(t),\\end{array}\\right.$ where $\\begin{array}{l}\\mathcal {A}_N=\\left[\\begin{array}{lll}d_{1,1} &\\hdots & d_{1,N+1} \\\\\\vdots & & \\vdots \\\\d_{N,1} &\\hdots & d_{N,N+1} \\\\a_{1} & \\hdots & a_{N+1}\\\\\\end{array}\\right],\\ \\ B_N=\\left[\\begin{array}{c}0\\\\ \\vdots \\\\ 0 \\\\1 \\end{array}\\right]\\otimes B,\\\\C_N=[0\\ \\cdots \\ 0\\ 1]\\otimes C\\end{array}$ and $\\left\\lbrace \\begin{array}{llll}d_{i,k}&=&\\dot{l}_{N,k}(\\theta _{N,i}) I_n,\\ \\ \\ \\ & i\\in \\lbrace 1,\\ldots ,N\\rbrace ,\\ k\\in \\lbrace 1,\\ldots ,N+1\\rbrace , \\\\a_{k}&=&A_0 l_{N,k}(0)+\\sum _{i=1}^m A_il_{N,k}(-\\tau _i),\\ \\ \\ & k\\in \\lbrace 1,\\ldots ,N+1\\rbrace .\\end{array}\\right.$ The advantage of approximation (REF ) is that it is in the form of a standard state space representation, for which many analysis and control design techniques exist.", "We refer to [26] where (REF ) is at the basis of a design method for fixed-order $\\mathcal {H}_2$ optimal controller.", "According to (REF ), it is natural to relate the initial condition in the definition of the fundamental solution $K$ , see (REF ), with initial condition $z(0)=E_N$ of (REF ), where $E_N=[0\\ \\cdots \\ 0\\ 1]^T \\otimes I_n.$ This allows us to approximate the fundamental matrix $K(t)$ by $K_N(t)$ , defined as $K_N(t)=E_N^T e^{\\mathcal {A}_N t} E_N,$ which by (REF ) leads us on its turn to an approximation $\\mathcal {P}_N$ of $P$ , $\\begin{array}{lll}\\mathcal {P}_N(t)&= &\\int _{0}^{\\infty } K_N(s) B B^T K(s+t) ds\\\\&=&\\int _{0}^{\\infty } E_N^T e^{\\mathcal {A}_N s} B_N B_N^T e^{\\mathcal {A}_N^T(s+t)}E_N ds.\\end{array}$ Similarly, we can approximate $\\Upsilon $ in (REF ) by the transfer function of (REF ), given by $\\Upsilon _N(s)=C_N\\left(sI-\\mathcal {A}_N\\right)^{-1}B_N.$ The following proposition provides a computational expression for $\\mathcal {P}_N$ in terms of a Lyapunov matrix equation.", "The arguments in the proof are well known but we include them to make the paper self contained.", "If matrix $\\mathcal {A}_N$ is Hurwitz, we can express $\\mathcal {P}_N(t)=E_N^T P_N e^{\\mathcal {A}_N^T t} E_N, $ where $P_N$ satisfies the Lyapunov equation $\\mathcal {A}_N P_N+P_N \\mathcal {A}_N^T+B_N B_N^T=0.$ We can write (REF ) as $\\mathcal {P}_N(t)= E_N^T \\tilde{P}_N e^{\\mathcal {A}_N^T t} E_N$ , where $\\tilde{P}_N=\\int _{0}^{\\infty } e^{\\mathcal {A}_N s} B_N B_N^T e^{\\mathcal {A}_N^T s} ds.$ We have $\\begin{array}{lll}\\mathcal {A}_N \\tilde{P}_N+ \\tilde{P}_N \\mathcal {A}_N^T &=& \\int _{0}^{\\infty } \\frac{d}{ds}\\left(e^{\\mathcal {A}_N s} B_N B_N^T e^{\\mathcal {A}_N^T s} \\right)ds\\\\&=& -B_N B_N^T,\\end{array}$ the latter following from the Hurwitz property of $\\mathcal {A}_N$ .", "Since for the same reason the solution to Lyapunov equation (REF ) uniquely exists, we conclude $\\tilde{P}_N=P_N$ .", "The next proposition expresses that the approximation (REF ) of the delay Lyapunov matrix, and the approximation of the transfer function, are consistent with respect to property (REF ).", "Function (REF ) and transfer function (REF ) satisfy $\\Vert \\Upsilon _N\\Vert _2^2= \\mathrm {tr} \\left(C \\mathcal {P}_N(0) C^T\\right).$" ], [ "Properties", "We discuss properties of approximations (REF ) and (REF ) which are instrumental to the developments in the next sections, and which further shed a light on the difficulty of the problem of computing the delay Lyapunov matrix.", "They are illustrated by means of the didactic example $\\dot{x}(t)=\\frac{1}{2} x(t)-x(t-1)+u(t),\\ y(t)=x(t).$" ], [ "Time domain", "The function $t\\mapsto K(t)$ is in general not analytic on $(0,\\ \\infty )$ , due to the propagation of the discontinuity at $t=0$ .", "If function $K$ has a discontinuity in its $k$ -th derivative ($k=0$ for a discontinuity in the function) at some time $\\hat{t}\\ge 0$ , then the function has, in the generic case, a discontinuity in its $(k+1)$ -th derivative at time instants $\\hat{t}+\\tau _i,\\ i=1,\\ldots ,m$ .", "The increase of regularity is called the smoothing property of solutions [7].", "Via definition (REF ) the non-smoothness of $K$ propagates to the function $t\\ge 0\\mapsto P(t)$ (we restrict to non-negative $t$ because of the so-called symmetry property $P(-t)=P(t)^T$ ).", "In Section 4 of [12] it has been shown that the function $P$ is in general not infinitely many times differentiable for $t\\in S,$ where $S=\\left\\lbrace \\vec{\\tau }\\cdot \\vec{z}:\\ \\vec{z}\\in \\mathbb {Z}^m,\\ \\vec{\\tau }\\cdot \\vec{z}>0 \\right\\rbrace ,$ where $\\vec{\\tau }=(\\tau _1,\\ldots ,\\tau _m)$ and $\\vec{z}=(z_1,\\ldots ,z_m)$ .", "In the commensurate delay case, where $\\vec{\\tau }=h \\vec{n}$ with $n\\in \\mathbb {N}^m$ and $\\gcd (\\vec{n})=1$ , we have $S=\\left\\lbrace k h:\\ k=0,1,2,\\ldots \\right\\rbrace $ .", "In case of non-commensurate delays, set $S$ is dense in $[0,\\infty )$ .", "In both cases, function $P$ is continuous, $\\dot{P} $ are continuous on $(0, \\infty )$ , while $\\ddot{P}$ is continuous for all $t\\in (0,\\infty )$ except for $t=\\tau _i,\\ i\\in \\lbrace 1,\\ldots ,m\\rbrace $ , but still of bounded variation.", "For more details we refer to [12].", "As an illustration we plot the functions $K$ and $P$ , corresponding to (REF ), in Figure REF .", "Figure: Plot of functions KK and PP for system ().", "The circles correspond to time-instants where the function is not infinitely many times differentiable.", "Function KK (function PP) exhibits a discontinuity in its kk-th derivative ((k+1)(k+1)-th derivative) at t=kt=k, for all k∈ℕk\\in \\mathbb {N}.In Figure REF we plot for system (REF ) the normalized errors $\\frac{\\max _{t\\in [0,\\ t_{\\max }]} |P(t)-\\mathcal {P}_N(t)|}{\\max _{t\\in [0,\\ t_{\\max }]} |P(t)|}$ for $t_{\\max }=2$ and $\\frac{|P(0)-\\mathcal {P}_N(0)|}{|P(0)|},$ as a function of $N$ .", "Note that, as $B=C=1$ for system (REF ), expression (REF ) corresponds to the normalized error on the squared $\\mathcal {H}_2$ norm if the latter is approximated by $\\Vert \\Upsilon _N\\Vert _2^2$ , see (REF ).", "We observe the following rates of convergence: $\\mathcal {O}\\left(N^{-2}\\right)$ for (REF ), and $\\mathcal {O}\\left(N^{-3}\\right)$ for (REF ).", "In all other experiments we observed the same rates of convergence.", "The seemingly slow convergence, $\\mathcal {O}\\left(N^{-2}\\right)$ for the maximum error of $P$ on a compact interval, is expected in view of the smoothness properties of function $P$ .", "As we have seen, $P$ has discontinuities in its second derivative at $t=\\tau _i,\\ i=1,\\ldots ,m$ (with $\\ddot{P}$ of bounded variation), while function $\\mathcal {P}_N$ , defined by (REF ), is analytic on $\\mathbb {R}$ .", "Thus, we are approximating a non-smooth function by a series of smooth functions.", "Note that we would obtain the same rate of convergence when approximating $P$ on an interval by a series of polynomials interpolating in a Chebyshev mesh [23].", "As $P$ is analytic in the interval $(0,\\tau _1)$ and we only consider nonnegative $t$ , the convergence rate is better at $t=0$ .", "We refer to [26], where an extensive argumentation for the rate $\\mathcal {O}(N^{-3})$ for the $\\mathcal {H}_2$ norm approximation induced by $\\Vert \\Upsilon _N\\Vert _2$ is given.", "We recall that the lack of smoothness of $P$ also affects solution schemes based on solving the boundary value problem (REF ) directly [12].", "Figure: Normalized error () (blue curve) and () (green curve) as a function of NN for system ().", "The dashed lines indicate the rates 𝒪N -2 \\mathcal {O}\\left(N^{-2}\\right) and 𝒪N -3 \\mathcal {O}\\left(N^{-3}\\right)." ], [ "Frequency domain", "With the choice of the Chebyshev mesh (REF ) the asymptotic convergence of the individual eigenvalues of $\\mathcal {A}_N$ to corresponding characteristic roots is fast.", "More specifically, in [2] it is proven that spectral accuracy (approximation error $O(N^{-N})$ ) is obtained.", "An additional property of using mesh (REF ) for discretizing (REF ), observed in extensive numerical experiments, is that the eigenvalues of $\\mathcal {A}_N$ , which have not yet converged to corresponding characteristic roots of (REF ), are very often located to the left of the eigenvalues that have already converged, which is important with respect to preservation of stability.", "These properties are illustrated for system (REF ) in Figure REF .", "Finally, since the effect of the spectral discretization can be interpreted in terms of a rational approximation of functions $\\lambda \\rightarrow \\exp (-\\lambda \\tau _i),\\ i=1,\\ldots ,m$ around zero, see [28], convergence is almost invariably reached first for the smallest characteristic roots in modulus if $N$ is gradually increased.", "Due to the characteristic shape on the spectrum of delay equation (exhibiting infinite root chains extending in the left half plane, along which the imaginary part grows exponentially as a function of the real part, see [27] for a detailed description), the rightmost, stability determining roots, are typically among the smallest characteristic roots.", "Figure: (left) all eigenvalues of 𝒜 N \\mathcal {A}_N, corresponding to system (), for N=30N=30 (black circles).", "(right) Zoom of the right-part of the spectrum of 𝒜 N \\mathcal {A}_N, supplemented with the characteristic roots of the delay equation (blue stars).", "Its null solution is exponentially stable, with rightmost characteristic roots -0.1629±0.9725j-0.1629 \\pm 0.9725j.With respect to the approximation of the transfer function, the following moment matching property is proven in [17], which is in fact independent of the choice of the mesh points in (REF ).", "The transfer functions (REF ) and (REF ) satisfy, $\\left.\\frac{d^i \\Upsilon _N(s)}{ds^i}\\right|_{s=0}=\\left.\\frac{d^i \\Upsilon (s)}{ds^i}\\right|_{s=0},\\ \\ i=0,\\ldots , N,$ and $\\left.\\frac{d^i \\Upsilon _N(s^{-1})}{ds^i}\\right|_{s=0}=\\left.\\frac{d^i \\Upsilon (s^{-1})}{ds^i}\\right|_{s=0},\\ \\ i=0,1,$ that is, the moments of $\\Upsilon (s)$ and $\\Upsilon _N(s)$ at zero match up to the $N$ th moment, and the moments at infinity match up to the first moment.", "By Property (REF ), which corresponds to Hermite interpolation at $s=0$ , the region in the complex plane where the approximation is accurate extends from the origin as $N$ is increased, consistently with the convergence behavior of characteristic root approximations sketched in the right pane of Figure REF .", "At the same time, the asymptotic delay rate of the transfer function for $\\omega \\rightarrow \\infty $ , which is described by $CB/\\omega $ , is captured by property (REF ).", "Note that higher-order moments of (REF ) at infinity are not well defined, which is related to the property that $s=\\infty $ is an essential singularity of $\\Upsilon $ .", "As a consequence, the overall approximation error is mainly due to a mismatch in the mid-frequency range.", "This is illustrated in Figure REF , where we compare the transfer function of (REF ) and its approximation of form (REF ).", "Figure: (left) Modulus of the transfer function of () (blue curve) and the corresponding approximation () for N=5N=5 (red curve), evaluated on the imaginary axis, i.e.", "for s=ıω,ω≥0s=\\imath \\omega ,\\ \\omega \\ge 0.", "(right) Approximation error on the imaginary axis for N=5N=5 (red curve), N=30N=30 (blue curve) and N=100N=100 (black curve).The right pane in Figure REF gives a complementary explanation, besides the smoothness properties of the function $t\\mapsto P(t)$ , why the convergence of $\\mathcal {P}_N(0)$ to $P(0)$ has exhibits a low rate of convergence $\\mathcal {O}(N^{-3})$ , compared to the spectral convergence of the eigenvalues of $\\mathcal {A}_N$ : unlike an individual pole and the $\\mathcal {H}_{\\infty }$ norm, the $\\mathcal {H}_2$ norm is a global characteristic of the transfer function, in the sense that an accurate computation involves approximating the transfer function well over whole the imaginary axis." ], [ "A reformulation of the discretized problem", "The following main theorem reformulates expressions (REF )-(REF ) in terms of a matrix $G_N$ similar to $\\mathcal {A}_N^{-1}$ , giving the Lyapunov equation a favorable structure that will be exploited by the algorithms presented in Section .", "Assume that $\\mathcal {A}_N$ is Hurwitz and let $G_N=\\Sigma _N^{-1}\\Pi _N,$ where $\\Pi _N=\\frac{\\tau _{m}}{4}\\left[\\begin{array}{rrrrrrr}\\frac{4}{\\tau _m} &\\frac{4}{\\tau _m} & \\frac{4}{\\tau _m}&\\cdots &&\\cdots &\\frac{4}{\\tau _m} \\\\2& 0&-1 &&&& \\\\&\\frac{1}{2} &0 &-\\frac{1}{2}&&&\\\\&&\\frac{1}{3}&0&-\\frac{1}{3}&&\\\\&&&\\ddots &\\ddots &\\ddots &\\\\&&& & \\frac{1}{N-1} &0& -\\frac{1}{N-1}\\\\&&& & &\\frac{1}{N} &0\\end{array}\\right]\\otimes I$ and $\\Sigma _N=\\left[\\begin{array}{cccc}R_0 & R_1& \\cdots & R_N \\\\& I_n & & \\\\& & \\ddots & \\\\& & & I_n \\\\\\end{array}\\right],$ with $R_i=A_0 T_i(1)+\\sum _{k=1}^m A_kT_i\\left(-2\\frac{\\tau _k}{\\tau _{m}}+1\\right),\\ i=0,\\ldots ,N$ and $T_i$ the Chebyshev polynomial of the first kind and order $i,\\ i=0,1,2,\\ldots $ .", "Moreover, let $H_N=\\left[\\begin{array}{c}R_0^{-1}\\left(I-\\frac{\\tau _m}{2}R_1\\right)R_0^{-1} B\\\\\\frac{\\tau _m}{2}R_0^{-1} B \\\\0\\\\\\vdots \\\\0\\end{array}\\right]$ and $F_N=[ R_0\\ R_1\\ \\ \\cdots \\ R_N].$ Then we can express $\\mathcal {P}_N$ in (REF ) as $\\mathcal {P}_N(t)= F_N Q_N e^{G_N^{-T}t} F_N^T$ where $Q_N$ satisfies the Lyapunov equation $G_N Q_N+Q_N G_N^T+H_N H_N^T=0.$ Moreover, system (REF ) is equivalent to $\\left\\lbrace \\begin{array}{l}G_N \\dot{\\eta }(t)= \\eta (t)+H_N u(t), \\\\y(t)= C F_N \\eta (t),\\end{array}\\right.$ and we can express $\\Upsilon _N(s)=C F_N(s G_N-I)^{-1} H_N.$ In [17] it has been shown that $\\mathcal {A}_N= (S_N\\otimes I) G_N^{-1} (S_N^{-1}\\otimes I),$ where matrix $S_N\\in \\mathbb {R}^{(N+1)\\times (N+1)}$ maps coefficients of a polynomial of degree $N$ in the Chebyshev basis $\\left\\lbrace T_i\\left(2 \\frac{t}{\\tau _m}+1\\right):\\ i=0,\\ldots ,N\\right\\rbrace $ onto the corresponding coefficients in the Lagrange basis, $\\lbrace l_{N,i}(t):\\ i=1,\\ldots ,N+1\\rbrace ,$ defined on the mesh (REF ).", "Substituting (REF ) into (REF ) yields $\\mathcal {P}_N(t)=\\int _{0}^{\\infty } E_N^T (S_N\\otimes I) e^{G_N^{-1} s} (S_N^{-1}\\otimes I) B_N\\\\B_N^T (S_N^{-T}\\otimes I)e^{G_N^{-T}(s+t)}(S_N^T\\otimes I) E_Nds.$ In the proof of Theorem 3.2 of [17] it has been shown that $(S_N^{-1}\\otimes I)B_N=c_N\\otimes B,\\ \\ E_N^T (S_N\\otimes I)=\\mathbf {1}_N^T\\otimes I,$ with $c_N=\\left\\lbrace \\begin{array}{ll}\\frac{2}{N+1}\\ [0\\ 1\\ 0\\ 1\\ \\cdots \\ 0\\ 1]^T\\otimes B, & N\\ \\mathrm {odd}, \\\\\\frac{2}{N+1}\\ [\\frac{1}{2}\\ 0\\ 1\\ 0\\ 1\\ \\cdots \\ 0\\ 1]^T\\otimes B, & N\\ \\mathrm {even}, \\\\\\end{array}\\right.$ and $\\mathbf {1}_N=[1\\ 1\\ \\cdots \\ 1]^T.", "$ Using these expressions, as well as the identity $e^{G_N^{-1}t}= G_N^{-1} e^{G_N^{-1}t} G_N$ , we can write (REF ) as $\\mathcal {P}_N(t)=\\int _{0}^{\\infty } (\\mathbf {1}_N^T\\otimes I)G_N^{-1} e^{G_N^{-1} s} G_N (c_N\\otimes B)\\\\(c_N^T\\otimes B^T)G_N^T e^{G_N^{-T}(s+t)} G_N^{-T}(\\mathbf {1}_N\\otimes I) ds.", "$ A straightforward computation shows that $(\\mathbf {1}_N^T\\otimes I) G_N^{-1}=F_N,\\ \\ \\ G_N (c_N\\otimes B)=\\hat{H}_N,$ with $\\hat{H}_N=\\left[\\begin{array}{c} R_0^{-1} B\\\\0\\\\ \\vdots \\\\ 0 \\end{array}\\right].$ As a consequence, we can write $\\mathcal {P}_N(t)= F_N \\left(\\int _{0}^{\\infty } e^{G_N^{-1} s} \\hat{H}_N \\hat{H}_N^T e^{G_N^{-T}s}~ds\\right) e^{G_N^{-T}t} F_N^T.$ Denoting the integral in (REF ) by $\\hat{Q}_N$ , we can express the latter, relying on the assumption that $\\mathcal {A}_N$ and $G_N^{-1}$ are Hurwitz, as the solution of the Lyapunov equation $G_{N}^{-1} \\hat{Q}_N+ \\hat{Q}_N G_N^{-T}+ \\hat{H}_N \\hat{H}_N^T=0.$ Pre-multiplying this equations with $G_N$ and post-multiplying with $G_N^T$ yields $\\hat{Q}_N G_N^T+ G_N \\hat{Q}_N + G_N \\hat{H}_N \\hat{H}_N^T G_N^T=0.$ Since we have $G_N\\hat{H}_N=H_N$ , it follows that $\\hat{Q}_N=Q_N$ , where $Q_N$ uniquely solves (REF ).", "Hence, (REF ) corresponds to (REF ) and (REF ).", "Finally,expression (REF ) constitutes the assertion of Theorem 3.2 of [17].", "Matrices $\\Sigma _N$ and $\\Pi _N$ have a sparse structure that can be exploited.", "In what follows a key role will be played by the following property.", "Assume that $N_1,N_2\\in \\mathbb {N}$ with $N_1<N_2$ .", "Then the matrices $\\Sigma _{N_1},\\Pi _{N_1},F_{N_1},H_{N_1}$ in Theorem REF are submatrices of $\\Sigma _{N_2},\\Pi _{N_2},F_{N_2},H_{N_2}$ .", "The price to pay for the discretization of the delay equation and the standard state space representation (REF ), which on their turn led us to delay Lyapunov matrix approximations in explicit form, namely (REF )-(REF ) and (REF )-(REF ), is an increase of dimension from $n$ to $(N+1)n$ .", "At the same time relatively of high value of $N$ are expected for an accurate approximation, as motivated in Section REF .", "If $Nn$ is large and matrix $B_N B_N^T$ , respectively $H_N H_N^T$ , has low rank (in the sense of $r<< Nn$ ), computing a low-rank approximation of $P_N$ , respectively $Q_N$ , may be beneficial.", "In this section we construct an approximation inferred from the projection of the Lyapunov equation on a Krylov space of dimension $kr$ .", "Before we present the construction in Sections REF -REF , we use another didactic example to motivate important methodological choices regarding i) the relation between parameters $N$ and $k$ , ii) the choice of the Krylov space, and iii) the system matrix / Lyapunov equation to be projected on this space.", "We discuss some implementation aspects in Section REF and conclude with an interpretation in terms of projecting an infinite-dimension system linear in Section REF .", "The main contributions are contained in Sections REF -REF .", "The Arnoldi process of Section REF and the construction of the reduced model in Sections REF extend results presented in [9], [17] to the multiple-input setting.", "Since the technical derivations involve many steps, we included Figure REF at the end of the section in order to keep an overview of the main steps and corresponding notations." ], [ "Motivation of methodological choices", "We consider system ${\\small \\begin{array}{lll}\\dot{x}(t) &=&\\left[\\begin{array}{rrr}-0.08 & -0.03 & 0.2\\\\0.2 & -0.04 & -0.005\\\\-0.06 & 0.2 & -0.07\\end{array}\\right]x(t)+\\left[\\begin{array}{rrr}-0.0471 & -0.0504 & -0.0602\\\\-0.0942 & -0.1008 & -0.1204\\\\0.0471 & 0.0504 & 0.0602\\end{array}\\right] x(t-5)\\\\&&+\\left[\\begin{array}{r}1\\\\1 \\\\1 \\end{array}\\right]u(t),\\ \\ y(t)=\\left[\\begin{array}{rrr}1 & 0 & 0 \\end{array}\\right]x(t).\\end{array}}$ For $N=50, 100, 150$ and 200 we computed matrices $P_N$ and $Q_N$ , solving Lyapunov equations (REF ) and (REF ).", "We display in Figure REF (above) their ordered singular values, normalized such that the leading singular value equals to one.", "We also show, in the lower figure, the leading singular value of both matrices as a function of $N$ .", "Figure: (above) Normalized eigenvalues (λ i (·)\\lambda _i(\\cdot ) denoting the ii-th, eigenvalue in decreasing order) of matrix P N P_N and Q N Q_N, computed for system ().", "(below) Spectral norm of P N P_N (red) and Q N Q_N (blue) as a function of NN.This experiment indicates that the solution of Lyapunov equation (REF ), inferred from the representation (REF ), is more amendable for a low-rank approximation.", "Concerning the input-output behavior, Proposition REF expresses that functions $\\Upsilon $ and $\\Upsilon _N$ match $N+1$ moments at zero and two at infinity.", "To have these matching moments carried over by a projection of (REF ) on a right Krylov space, one needs in general a subspace of dimension $N+2$ .", "At the same time, if more than $N+1$ moments at zero are preserved by the projection, or more than two at infinity, the highest order moments won't match anymore with those of the original transfer function (REF ).", "This can be interpreted as an instance of “over-fitting\" in the sense that particularities of the discretization (REF ) are captured by the projection, which are not present in the original delay equation and related to the discretization error.", "Similar conclusions can be made from the experiment related to the upper right pane of Figure REF .", "On a compact interval for index $i$ , the eigenvalue functions of $Q_N$ uniformly converge for $N\\rightarrow \\infty $ to the limit function indicated in black color, which is related to the original (non-discretized) delay equation (we come back to this in Section REF ).", "Important to observe is that, for a given value of $N$ , less than $N$ singular values are related to the limit behavior.", "This indicates that, at least for a best rank-$k$ approximation of $Q_N$ , the choice $k>N$ could lead to a similar instance of over-fitting.", "All the above elments motivate us to assure $N$ being sufficiently large, such that the dimension of the subspace $k$ satisfies $ k\\le N$ and, preferably $k<<N$ .", "The typical spectrum distribution of delay equations, with rightmost characteristic roots close to the origin, the properties of the spectral discretization, illustrated in Figure REF , and the above reasoning with respect to matching moments, suggest to build a Krylov space using matrix $\\mathcal {A}_N^{-1}$ , $\\mathcal {K}_{k}(A_{N}^{-1},B_N)=\\operatorname{span}\\left\\lbrace B_N,\\mathcal {A}_N^{-1} B_N,\\ldots ,A_{N}^{-(k-1)}B_N\\right\\rbrace .$ Letting the columns of $V_{N,k}$ be an orthogonal basis for this Krylov space, we depict in Figure REF the approximation error on the smallest characteristic roots for (REF ), obtained as the reciprocal of the eigenvalues of $V_{N,k}^T \\mathcal {A}_N^{-1} V_{N,k},$ and as the eigenvalues of $V_{N,k}^T \\mathcal {A}_N V_{N,k},$ for $N=30$ , $N=60$ and in both cases $k=N$ (such that (REF ) is taken into account).", "The plots illustrates a property observed in many experiments, that it is beneficial to project matrix $\\mathcal {A}_N^{-1}$ on the Krylov space, compared to projecting $\\mathcal {A}_N$ .", "This observation can be explained by a better separation of the targeted characteristic roots after an inversion of the spectrum.", "Figure: Absolute error on the smallest characteristic roots of (), obtained from () (blue) and () (red).", "In the left pane we consider N=k=30N=k=30, in the right pane N=k=60N=k=60.The preference for building a Krylov space for $\\mathcal {A}_N^{-1}$ and for projecting this matrix, the property that $G_N$ is similar to $\\mathcal {A}_N^{-1}$ , and, last but not least, the typically faster decay of singular values of $Q_N$ than those of $P_N$ naturally lead us to the representation (REF )-(REF ) of the discretized system and associated approximation of the delay Lyapunov matrix.", "In addition, matrices $\\Sigma _N$ and $\\Pi _N$ have a sparse structure that can be exploited.", "In particular, the property expressed in Proposition REF along with condition (REF ) will allow us to ultimately arrive at a method that does not rely on an a-priori choice of critical parameter $N$ , similar to the infinite-Arnoldi method for eigenvalue computations [9]." ], [ "Dynamic construction of a Krylov space", "We fix integer $k$ and assume $N$ large enough such that (REF ) holds.", "We consider the block Krylov space $\\mathcal {K}_{k}(G_{N},b)=\\operatorname{span}\\lbrace b,G_{N} b,\\ldots ,G_{N}^{k-1}b\\rbrace ,$ where $b$ is a block vector of size $(N+1) n\\times r$ , having the structure $b=[x_0^T\\ 0\\ \\cdots 0]^T,\\ \\ $ with $ x_0\\in \\mathbb {R}^{n\\times r}$ to be specified in Section REF .", "The block Arnoldi algorithm builds the Krylov sequence, block vector by block vector, where these vectors are orthogonalized.", "Due to the special structure of $b$ and the fact that $G_N$ is a block Hessenberg matrix, whose blocks have size $n\\times n$ , the block vectors $G_{N} b,\\ldots , G_{N}^{k-1}b$ only have their first $2n$ , $3n,\\ \\ldots ,\\ k n$ block rows different from zero.", "Moreover, in computing the matrix vector products with (REF ), only sub-matrices of $G_N$ are needed.", "Hence, in the computation of the Krylov space, we can restrict to storing only the nonzero part of the block vectors and using the relevant part of $G_N$ .", "This leads us to the following procedure.", "Apply Algorithm REF for computing a basis of $\\mathcal {K}_k(G_{k-1},[x_0^T\\ 0 \\cdots \\ 0]^T)$ .", "There we use notation common for Arnoldi iterations: we let $\\underline{\\mathcal {H}}_i\\in \\mathbb {R}^{(i+1)r\\times r i}$ denote the constructed rectangular block Hessenberg matrix and $\\mathcal {H}_i\\in \\mathbb {R}^{r i\\times r i}$ the corresponding $i\\times i$ upper blocks.", "A basis for $\\mathcal {K}_k(G_N,[x_0^T\\ 0 \\ \\cdots \\ 0]^T)$ is spanned by the columns of $V_{N,k}=\\left[\\begin{array}{cccc} \\mathbf {V}_k^T & 0 & \\cdots &0 \\end{array}\\right]^T\\in \\mathbb {R}^{ (N+1)n\\times k r},$ while, due to the structure of $G_N$ , expression $\\mathcal {H}_k= V_{N,k}^T\\ G_N\\ V_{N,k},$ holds, i.e., $\\mathcal {H}_k$ can be considered as an orthogonal projection of $G_N$ on a $k$ -dimensional Krylov subspace, for any $N$ satisfying (REF ).", "A structure exploiting block Arnoldi algorithm [1] $x_0\\in \\mathbb {R}^{n\\times r}$ of full column rank, number of iterations $k$ Let $x_0= Q_0 \\tilde{R}_0$ be the reduced QR factorization of $x_0$ .", "Set $\\mathbf {V}_1=Q_0$ and let $\\underline{\\mathcal {H}}_0$ be the empty matrix $i=1,2,\\ldots , k$ Let $W_i=G_{i} \\left[\\begin{array}{c}Q_{i-1}\\\\0 \\end{array}\\right]$ Compute $H_i=[\\mathbf {V}_i^T\\ 0]\\ W_i$ and then $\\hat{W}_i = W_i - \\left[\\begin{array}{c}\\mathbf {V}_i\\\\0\\end{array}\\right] H_i$ (orthogonalization) Compute $\\hat{W}_i= Q_{i} \\tilde{R}_i$ as the reduced QR factorization of $\\hat{W}_i$ (normalization) Let $\\underline{\\mathcal {H}}_i = \\left[\\begin{array}{cc}\\underline{\\mathcal {H}}_{i-1} & H_i \\\\ 0 & \\tilde{R}_i \\end{array}\\right] \\in \\mathbb {R}^{(i+1)r\\times i r}$ Expand $\\mathbf {V}_i$ into $\\mathbf {V}_{i+1} =\\left[\\begin{array}{c|c} \\begin{array}{c}\\mathbf {V}_i \\\\ 0 \\end{array} & Q_i\\end{array}\\right]$ Output: matrix $\\mathbf {V}_{k}$ , whose columns are an orthogonal basis for       $\\mathcal {K}_k(G_{k-1},[x_0^T\\ 0 \\cdots \\ 0]^T)$ , $\\mathcal {H}_{k},\\ \\underline{\\mathcal {H}}_k$ , satisfying $\\mathcal {H}_{k}=\\mathbf {V}_k^T G_{k-1} \\mathbf {V}_k$ ." ], [ "Dynamic approximation of the transfer function", "We now arrive at the derivation of an approximation of $\\Upsilon _N(s)$ , defined by (REF ) or, equivalently, (REF ), having a prescribed order $k r$ , once again under the condition that (REF ) is satisfied.", "For this we construct the Krylov space (REF ) and project matrices $F_N, G_N, H_N$ , defined in Theorem REF , on this Krylov space.", "An orthogonal projection yields the following approximation of $\\Upsilon _N(s)$ : $\\mathbf {\\Upsilon }_{k}(s)=\\mathbf {F}_k\\ (s\\mathbf {G}_k-I)^{-1}\\ \\mathbf {H}_k,$ where $\\begin{array}{lllll}\\mathbf {F}_k&=& C F_{N}\\ V_{N,k}&=&C F_{k-1} \\mathbf {V}_k, \\\\\\mathbf {G}_k&=&V_{N,k}^T\\ G_N\\ V_{N,k}&=&\\mathcal {H}_{k}, \\\\\\mathbf {H}_k&=&V_{N,k}^T\\ H_{N}&=&\\mathbf {V}_k^T H_{k-1},\\ \\ \\end{array}$ matrix $\\mathbf {V}_k$ and $\\mathcal {H}_k$ refer to the output of Algorithm REF and $V_{N,k}$ is given by (REF ).", "The matrices of the reduced model (REF ) do not depend on $N$ .", "Furthermore, matrices $\\mathbf {F}_{k}$ and $\\mathbf {H}_{k}$ are submatrices of $\\mathbf {F}_{k+1}$ and $\\mathbf {H}_{k+1}$ .", "Therefore, they can be constructed in a dynamic way when doing iterations of Algorithm REF , as is the case with the Hessenberg matrix $\\mathbf {G}_k=\\mathcal {H}_k$ .", "With a particular choice of the vector $x_0$ in (REF ), the transfer function (REF ) satisfies the following moment matching property with the (original) transfer function (REF ) of the time-delay system (REF ).", "[17] Let $N,k\\in \\mathbb {N}$ with $N\\ge k\\ge 2$ and let the Krylov space (REF ) be constructed from $x_0=R_0^{-1} B.$ Then transfer function (REF ) satisfies $\\left.\\frac{d^i \\mathbf {\\Upsilon }_{k}(s)}{ds^i}\\right|_{s=0}=\\left.\\frac{d^i \\Upsilon (s)}{ds^i}\\right|_{s=0},\\ \\ i=0,\\ldots , k-2$ and $\\left.\\frac{d^i \\mathbf {\\Upsilon }_{k}(s^{-1})}{ds^i}\\right|_{s=0}=\\left.\\frac{d^i \\Upsilon (s^{-1})}{ds^i}\\right|_{s=0},\\ \\ i=0,1.$ Note that Proposition REF concerns the matching of moments with the transfer function of the original delay system (REF ).", "This is due to to the property that the moments, preserved by projection of the discretized system, are precisely matching moments between the discretized system and the delay system, by Proposition REF ." ], [ "Dynamic approximation of the delay Lyapunov matrix", "The evaluation of $\\mathcal {P}_N(t)$ , defined by (REF ), relies on solving Lyapunov equation (REF ).", "An established way to solve large-scale Lyapunov equations consists of computing a low-rank approximation obtained from the projection of the Lyapunov equation on a Krylov space, see, e.g., [21] and the references therein.", "To determine an appropriate Krylov space, it is useful to express $Q_N$ in terms of matrix exponentials, $Q_N=\\int _0^{\\infty } e^{G_N^{-1} s} \\left(G_N^{-1} H_N\\right)\\ \\left(H_N^T G_N^{-T}\\right) e^{G_N^{-T}s}ds.$ Hence, a low rank approximation of $Q_N$ can be induced by approximating the action of $e^{G_N^{-1} t}$ on vector(s) $(G_N^{-1} H_N)$ in a low-dimensional space.", "This motivates us to include $G_N^{-1} H_N= \\left[\\begin{array}{c}R_0^{-1}B \\\\ 0 \\\\ \\vdots \\\\ 0\\end{array}\\right]$ in the Krylov space.", "Furthermore, since the rightmost characteristic roots of a delay equation are typically very well approximated by the dominant eigenvalues of $G_N$ (equivalently, the smallest eigenvalues of $\\mathcal {A}_N$ in modulus), while the largest eigenvalues of $A_N$ have no correspondence with characteristic roots (see the arguments in Section REF and the illustration in Figure REF ), approximating the dominant eigenspace of $G_N$ should be favored, which brings us once again to Krylov space (REF ) with starting vector $x_0=R_0^{-1} B$ .", "Replacing $Q_N$ in (REF ) by $V_{N,k} \\mathbf {Q_k} V_{N,k}^T$ and requiring the residual to be orthogonal with respect to the Krylov space, we arrive at the projected Lyapunov equation $\\mathbf {G}_{k} \\mathbf {Q}_{k}+ \\mathbf {Q}_{k} \\mathbf {G}_{k}^{T}+\\mathbf {H}_{k} \\mathbf {H}_{k}^T=0.$ Hence, under assumption that $\\mathbf {G}_k$ is invertible we can approximate $\\begin{array}{lll}Q_N &\\approx & V_{N,k} \\mathbf {Q}_{k} V_{N,k}^T\\\\&=& \\int _0^{\\infty }V_{N,k} e^{ s \\mathbf {G}_{k}^{-1} } (\\mathbf {G}_{k}^{-1} \\mathbf {H}_{k})\\ (\\mathbf {H}_{k}^T \\mathbf {G}_{k}^{-T})e^{ s \\mathbf {G}_{k}^{-T} } V_{N,k}^T\\ ds.\\end{array}$ Let us now compare approximation (REF ) with expression (REF ).", "By construction of the Krylov space we have $G_N^{-1} H_N=V_{N,k} \\beta $ for some matrix $\\beta $ of appropriate dimensions.", "As a consequence, $H_N=G_N V_{N,k}\\beta \\ \\Rightarrow \\ \\mathbf {H}_k= \\mathbf {G}_k\\beta .$ Thus, the approximation of $Q_N$ as in (REF ) can be interpreted in terms of the approximation $e^{t G_N^{-1}} (G_N^{-1} H_N)= e^{t G_N^{-1}} \\left(V_{N,k}\\right)\\beta \\approx V_{N,k} e^{t\\mathbf {G}_{k}^{-1}} \\beta .$ Substituting the right-hand side of (REF ) into (REF ) we get $\\begin{array}{ll}\\mathcal {P}_N(t) &\\approx F_N V_{N,k} \\mathbf {Q}_k V_{N,k}^T e^{G_N^{-T}t} F_N^T\\\\&= F_N V_{N,k} \\mathbf {Q}_{k} \\left(e^{t G_N^{-1}} V_{N,k}\\right)^T F_N^T.\\end{array}$ To approximate $e^{t G_N^{-1}} V_{N,k}$ we use the same principle underlying (REF ).", "More precisely, we build a Krylov space, $\\mathrm {span}\\left\\lbrace V_{N,k},\\ G_N V_{N,k},\\ \\ldots , G_N^{k} V_{N,k} \\right\\rbrace $ .", "Since the columns of $V_{N,k}$ already span a Krylov space, this can be done by doing $k$ more iterations of Algorithm REF , provided condition (REF ) on $N$ is strengthened to $2k\\le N.$ It results in a basis $V_{N,2k}$ such that $V_{N,k}=V_{N,2k} \\left[\\begin{array}{c}I\\\\ 0\\end{array}\\right]$ , hence, we can approximate $\\left(e^{t G_N^{-1}} V_{N,k}\\right) \\approx V_{N,2k} e^{t \\mathbf {G}_{2k}^{-1}} \\left[\\begin{array}{c}I\\\\ 0\\end{array}\\right].$ Finally, combining (REF ) and (REF ) we arrive at the following approximation of $\\mathcal {P}_N(t)$ and thus of the Lyapunov matrix $P(t)$ , $\\mathbf {P}_k(t)=[R_0\\ R_1\\ \\cdots R_{k-1}] \\mathbf {V}_{k} \\mathbf {Q}_{k} \\left[I\\ 0\\right]e^{t \\mathbf {G}_{2k}^{-T}} \\mathbf {V}_{2k}^T\\left[\\begin{array}{c} R_0^T\\\\ R_1^T \\\\ \\vdots \\\\ R_{2k-1}^T\\end{array}\\right],$ where $\\mathbf {Q}_k$ satisfies (REF ).", "This brings us to Algorithm REF .", "[h] Construction of a (uniformly) low-rank approximation of the the delay Lyapunov matrix [1] $B\\in \\mathbb {R}^{n\\times r}$ of full column rank, parameter $k$ determining number of Arnoldi iterations Set $x_0=R_0^{-1} B$ and perform $2k$ iterations of Algorithm REF , resulting in $\\mathbf {V}_{2k}$ and $\\mathbf {G}_{2k}=\\mathcal {H}_{2k}$ ; set $\\mathbf {G}_k= \\left[\\begin{array}{cc}I_{kr} & 0\\end{array}\\right] \\mathbf {G}_{2k} \\left[\\begin{array}{c}I_{kr}\\\\ 0\\end{array}\\right].$ Construct matrices $\\mathbf {H}_k=\\mathbf {V}_k^T H_{k-1}$ and $\\mathbf {L}_k=[R_0\\ R_1 \\cdots R_{2k-1}] \\mathbf {V}_{2k}$ .", "Solve Lyapunov equation (REF ) for $\\mathbf {Q}_k$ .", "Output: matrices $\\mathbf {L}_k,\\ \\mathbf {Q}_k,\\ \\mathbf {G}_{2k}$ from which $\\mathbf {P}_k$ can be constructed        according to (REF ).", "Finally we note that the low-order approximation (REF ) of transfer function $\\Upsilon $ and the approximation (REF ) of Lyapunov matrix $P(t)$ of rank smaller or equal to $kr$ are still consistent, in view of Proposition REF and Proposition REF .", "We can express $\\left\\Vert \\mathbf {\\Upsilon }_{k}\\right\\Vert _2^2=\\mathrm {Tr}\\left(C \\mathbf {P}_k(0) C^T\\right)$ From (REF ) we directly have $\\begin{array}{lll}\\mathrm {Tr}\\left( C \\mathbf {P}_k(0) C^T\\right) &=& \\mathrm {Tr}\\left( C F_N \\mathbf {V}_{N,k} \\mathbf {Q}_k \\mathbf {V}_{N,k}^T F_N^T C^T\\right)\\\\&=& \\mathrm {Tr}\\left(\\mathbf {F}_k \\mathbf {Q}_k \\mathbf {F}_k^T\\right)\\end{array}$ The latter expression, combined with (REF ), characterize the $\\mathcal {H}_2$ norm of $\\mathbf {\\Upsilon }_k$ ." ], [ "Implementation aspects and computational complexity", "Algorithm REF is fully dynamic, in the sense that by increasing iteration count $k$ , matrices $\\mathbf {V}_k$ , $\\mathbf {G}_k$ , $\\mathbf {L}_k$ , etc., only need to be extended or updated, hence, the iteration can be resumed if the accuracy is deemed insufficient.", "If $k$ is not chosen a-priori, this brings us to discuss stopping criteria.", "The most reliable approach consists of testing the residual for boundary value problem (REF ) at a set of time-instants in the interval under consideration.", "Substituting (REF ) in (REF ) and letting the columns of $\\mathcal {W}_k$ be on orthogonal basis for the column space of $\\left[\\mathbf {L}_{k}\\ A_0 \\mathbf {L}_{k}\\ \\cdots \\ A_m\\mathbf {L}_{k} \\right],$ every term in the equations has its column, respectively row range contained in those of $\\mathcal {W}_k$ , respectively $\\mathcal {W}_k^T$ .", "As a consequence the Euclidean norm of the residual at a given time-instant can be expressed in terms of the residual for a boundary value problem where the size of the matrices is determined by the rank of $\\mathcal {W}_k$ .", "The construction of matrix $\\mathcal {W}_k$ however introduces a significant additional computational cost.", "To our experience a good indicator of convergence consists of determining the residual for Lyapunov equation (REF ).", "Recall that $G_N Q_N $ is approximated by $G_N V_{N,k} \\mathbf {Q}_k V_{N,k}^T=V_{N,k+1}\\underline{\\mathcal {H}}_k \\left[\\mathbf {Q}_k\\ 0\\right] V_{N,k+1}^T.$ At the same time we have $H_N=V_{N,k}V_{N,k}^T H_N=V_{N,k} \\mathbf {H}_k= V_{N,k+1} \\left[ \\begin{array}{c} \\mathbf {H}_k \\\\ 0 \\end{array}\\right].$ Since the columns of $V_{N,k+1}$ are orthogonal, the residual of (REF ), $R_{N,k}$ satisfies $\\Vert R_{N,k}\\Vert _2=\\left\\Vert \\underline{\\mathcal {H}}_k \\left[\\mathbf {Q}_k\\ 0\\right]+\\left[\\begin{array}{c}\\mathbf {Q}_k^T\\\\ 0\\end{array}\\right]\\underline{\\mathcal {H}}_k^T +\\left[\\begin{array}{c} \\mathbf {H}_k \\\\0 \\end{array}\\right]\\left[\\begin{array}{ccc} \\mathbf {H}_k^T &0 \\end{array}\\right]\\right\\Vert _2.$ Note that the residual norm can be expressed in terms of projected matrices and is independent of $N$ .", "What concerns the computational complexity, the core of Algorithm REF consists of doing $2k$ iterations of Algorithm REF .", "Expressed in terms of operations on vectors of length $n$, the computational complexity is as follows: Table: NO_CAPTIONIt is important to point out that all backwards solves are with the same matrix ($R_0$ ), inherent to an Arnoldi type algorithm.", "Hence, the first step in our implementation consists of computing a (sparse) LU factorization of matrix $R_0=\\sum _{i=0}^m A_i$ .", "For the remaining steps of Algoirthm REF the dominant cost in most cases consists of solving Lyapunov equation (REF ) for $\\mathbf {Q}_k$ , whose complexity is described by $\\mathcal {O}(r^3 k^3)$ operations for the adopted Bartels-Stewart algorithm.", "In addition, our implementation fully exploits the property that, due to the special structure of $G_k$ and the starting vector of the Arnoldi iteration, $\\mathbf {V}_k$ can be represented in the form $\\mathbf {V}_k= \\left( I_k\\otimes W_k\\right)\\left[\\begin{array} {llll}v_{1,1} & v_{1,2} & \\cdots & v_{1,k} \\\\0 & v_{2,2} & \\dots & v_{2,k} \\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & 0 & v_{k,k}\\end{array}\\right],$ where both factors are orthogonal matrices, matrix $W_k$ has dimensions $n\\times s$ with $s\\le kr$ and $v_{i,j}\\in \\mathbb {R}^{s\\times r},\\ i,j=1,\\ldots ,k$ .", "Furthermore, both factors can be dynamically constructed.", "These properties are fundamental in the so-called tensor infinite Arnoldi method and CORK framework (COmpact Rational Krylov algorithms) for nonlinear eigenvalue problems [25], [10], on their turn generalizing [1] for quadratic eigenvalue problems.", "We refer to these references for more details on representation (REF ).", "Obviously, for large $n$ its use leads to a significant reduction in the memory requirements, but it is also beneficial in terms of computational complexity, as argued in [10]." ], [ "Interpretation in terms of projections of an infinite-dimensional system", "The spectral discretization in Section , resulting in a finite-dimensional approximation of dimension $(N+1)n$ , played a major role in the technical derivation of Algorithm REF .", "However, eventually the role of parameter $N$ is marginal: the execution of Algorithm REF (and Algorithm REF on which it relies), as well as the discussed stopping criteria, do not rely on a choice of $N$ ; the algorithms are dynamic in the sense that the iterative processes can always be resumed; Proposition REF connects moments of transfer functions $\\mathbf {\\Upsilon }_k$ and $\\Upsilon $ directly.", "As a matter of fact it is only implicitly assumed that $N$ is sufficiently large (such that (REF ) holds).", "A limit argument, for $N\\rightarrow \\infty $ , provides some intuition for the existence of an interpretation of Algorithm REF as an algorithm acting on an infinite-dimensional linear system equivalent to (REF ).", "This is also suggested by Figure REF , where the singular value functions of $Q_N$ uniformly converge on compact intervals to the limit function displayed in black color.", "In what follows we make a connection with an infinite-dimensional linear system concrete.", "We reconsider system (REF ) and define $v(\\theta ,t)=x(t+\\theta ),\\ \\ \\theta \\in [-\\tau _m,\\ 0],\\ \\ t\\ge 0.$ Solutions of (REF ), starting at $t=0$ , are continuous for $t\\ge 0$ , and they satisfy the advection PDE $\\left\\lbrace \\begin{array}{ll}\\frac{\\partial v}{\\partial t} (\\theta ,t)-\\frac{\\partial v}{\\partial \\theta } (\\theta ,t)=0, & \\theta \\in [-\\tau _m,\\ 0),\\ \\ t\\ge \\tau _m,\\\\\\frac{\\partial v}{\\partial t}(0,t)=A_0 v(0,t)+\\sum _{i=1}^m A_i v(-\\tau _i,t)+ B u(t),\\ & t\\ge \\tau _m,\\\\\\end{array}\\right.$ see [15].", "Let us represent $v(\\theta ,t)$ in a Chebyshev series in variable $\\theta $ on the interval $[-\\tau _m,\\ 0]$ , $\\begin{array}{l}v(\\theta ,t)=\\sum _{j=0}^\\infty c_j(t) T_j\\left( \\frac{2\\theta }{\\tau _m}+1\\right),\\ \\theta \\in [-\\tau _m,0].\\end{array}$ The second equation in (REF ) then becomes $\\begin{array}{lll}\\sum _{j=0}^{\\infty } \\dot{c}_j(t)&=&A_0 \\left(\\sum _{j=0}^\\infty c_j(t)\\right) +\\sum _{i=1}^m A_i \\left(\\sum _{j=0}^\\infty c_j(t) T_j\\left( -\\frac{2\\tau _i}{\\tau _m}+1\\right)\\right)\\\\&=&\\sum _{j=0}^{\\infty } c_j(t) \\left(A_0+\\sum _{i=1}^m A_i T_j\\left( -\\frac{2\\tau _i}{\\tau _m}+1\\right) \\right).\\end{array}$ In the same way the first equation in (REF ) becomes $\\sum _{j=0}^{\\infty } \\dot{c}_j(t) T_j\\left( \\frac{2\\theta }{\\tau _m}+1\\right)=\\sum _{j=1}^{\\infty } c_j(t) \\frac{2j}{\\tau _m} U_{j-1} \\left( \\frac{2\\theta }{\\tau _m}+1\\right),$ where we employed the property $\\dot{T}_{j+1}(\\theta )=(j+1) U_j(\\theta ),$ with $U_j$ the Chebyshev polynomial of the second kind and order $j$ , for $j\\ge 0$ .", "For $j\\ge 2$ we can substitute expression $T_j\\left( \\frac{2\\theta }{\\tau _m}+1\\right)=\\frac{1}{2} \\left( U_j\\left( \\frac{2\\theta }{\\tau _m}+1\\right)- U_{j-2}\\left( \\frac{2\\theta }{\\tau _m}+1\\right)\\right)$ in (REF ), as well as $T_1\\left( \\frac{2\\theta }{\\tau _m}+1\\right)=\\frac{1}{2} U_1\\left( \\frac{2\\theta }{\\tau _m}+1\\right),\\ \\ T_0\\left( \\frac{2\\theta }{\\tau _m}+1\\right)=U_0\\left( \\frac{2\\theta }{\\tau _m}+1\\right).$ Multiplying subsequently left and right hand side of (REF ) with $U_{i-1} \\left( \\frac{2\\theta }{\\tau _m}+1\\right)\\sqrt{1-\\left( \\frac{2\\theta }{\\tau _m}+1\\right)^2},$ taking the integral in $\\theta $ from $-\\tau _m$ to zero, and considering the orthogonality properties of Chebyshev polynomials of the second kind, we arrive at $\\begin{array}{l}\\dot{c}_0(t)-\\frac{1}{2}\\dot{c}_2(t) =\\frac{2}{\\tau _m} c_1,\\\\\\frac{1}{2} \\left( \\dot{c}_{i-1}(t)-\\dot{c}_{i+1}(t) \\right) =\\frac{2i}{\\tau _m} c_i,\\ \\ i\\ge 2.\\end{array}$ Letting $\\mathbf {c}=\\left[c_0^T\\ c_1^T\\ \\cdots \\right]^T$ , $\\mathbf {e_1}=[1 \\ 0\\ \\cdots ]^T$ and $\\mathbf {1}=[1\\ 1\\ \\cdots ]^T$ , differential equations (REF ) and (REF ) can be written as $ \\left\\lbrace \\begin{array}{lll}\\Pi _\\infty \\dot{\\mathbf {c}}(t) &=&\\Sigma _\\infty \\mathbf {c}(t) +\\left(\\mathbf {e_1}\\otimes B\\right) u(t), \\\\y(t)&=&\\left(\\mathbf {1}^T \\otimes C\\right) \\mathbf {c}(t),\\end{array}\\right.$ with $\\Pi _\\infty =\\frac{\\tau _{m}}{4}\\left[\\begin{array}{rrrrrr}\\frac{4}{\\tau _m} &\\frac{4}{\\tau _m} & \\frac{4}{\\tau _m}&\\cdots &\\cdots &\\cdots \\\\2& 0&-1 &&& \\\\&\\frac{1}{2} &0 &-\\frac{1}{2}&&\\\\&&\\frac{1}{3}&0&-\\frac{1}{3}&\\\\&&&\\ddots &\\ddots &\\ddots \\\\\\end{array}\\right]\\otimes I$ and $\\Sigma _\\infty =\\left[\\begin{array}{cccc}R_0 & R_1& \\cdots & \\\\& I_n & & \\\\& & \\ddots &\\end{array}\\right].$ System (REF )-(REF ) can be interpreted as alternative representation of (REF ), and of the original delay equation (REF ).", "At the same time, system (REF ), obtained after a spectral discretization and at the basis of the approach spelled out in the previous sections, is equivalent to $\\left\\lbrace \\begin{array}{rcl}\\Pi _N\\dot{c}_N(t)&=&\\Sigma _N c_N(t)+(\\mathbf {1}_N\\otimes B) u(t),\\\\y(t)&=&(\\mathbf {1}_N^T\\otimes C) x(t)\\end{array}\\right.$ since $G_N\\Sigma _N^{-1} (\\mathbf {1}_N\\otimes B)=H_N,\\ \\ (\\mathbf {1}_N^T\\otimes C) G_N^{-1}=F_N.$ System (REF ) can be obtained from (REF ) by truncating the state to the first $N+1$ block components (or, equivalently applying a Galerkin projection on the range of $\\left( [I_{n(N+1)} \\ 0 \\cdots \\ 0]^T\\right)$ .", "Algorithms REF -REF only rely on the use of submatrices of $\\Sigma _N,\\Pi _N$ , at top-left position (recall definition (REF ) of $G_N$ ), which are on their turn “submatrices\" of $\\Pi _{\\infty }$ and $\\Sigma _\\infty $ .", "Therefore, these algorithms can be interpreted as applied to infinite-dimensional system (REF ) directly.", "We note that, for case of approximating characteristic roots by the reciprocal of eigenvalues of $\\mathbf {G}_k$ , a related interpretation of Algorithm REF is given in [9], in terms of an operator eigenvalue problem.", "Finally, an overview of the developments throughout the Sections - is given by Figure  REF Figure: Overview of different steps in the derivations and corresponding notations.", "If only the ℋ 2 \\mathcal {H}_2 norm needs to be approximated, a Kyrlov space of dimension kk such that k≤Nk\\le N is sufficient.With the relation between kk and NN satisfied, the delay Lyapunov matrix and ℋ 2 \\mathcal {H}_2 norm approximations, obtained after projection of the discretized system, do not depend on the value of NN, only on kk.", "This leads to the interpretation spelled out in Section  and illustrated with the curves arrows." ], [ "Experiments", "We first consider the model for a heat exchanger described in [27], for which the controller (based on a combination of static state feedback and proportional integral (PI) control) has been determined by optimizing the spectral abscissa using the method of [16].", "The closed-loop system is described by a delay equation of form (REF ) with $n=5$ state variables and $m=7$ delays.", "The non-zero elements of matrices $A_i,\\ i=0,\\ldots ,7$ , are specified as in the following table, ${\\small \\begin{array}{l|l}A_0&\\ \\ \\ (2,1):\\ \\frac{1}{3},\\ (2,2):\\ -\\frac{2}{3},\\ (3,3):\\ -\\frac{1}{3}\\ (5,4):\\ -1\\\\A_1&\\ \\ \\ (4,3):\\ 0.0324\\\\A_2&\\ \\ \\ (1,1):\\ -0.07142857143\\\\A_3&\\ \\ \\ (4,4):\\ -0.04\\\\A_4&\\ \\ \\ (2,4):\\ \\frac{1}{3}\\\\A_5&\\ \\ \\ (1,1):\\ -0.01219364644,\\ (1,2):\\ -0.05460277319,\\ (1,3):\\ -0.1005215423 \\\\& \\ \\ \\ (1,4):\\ -0.1290047174,\\ (1,5):\\ 0.005063395489\\\\A_6&\\ \\ \\ (3,2):\\ 0.3133333333\\\\A_7&\\ \\ \\ (1,2):\\ 0.01714285714,\\end{array}}$ while input matrix $B$ , output matrix $C$ and the delay values are given by ${\\small B=\\left[\\begin{array}{c}0.0278571429\\\\0\\\\0\\\\0\\\\0\\end{array}\\right],\\ \\ C=I,\\ \\left[\\begin{array}{c}\\tau _1\\\\\\tau _2\\\\\\tau _3\\\\\\tau _4\\\\\\tau _5\\\\\\tau _6\\\\\\tau _7\\end{array}\\right]=\\left[\\begin{array}{c}2.8\\\\6.5\\\\9.2\\\\13\\\\13.2\\\\18\\\\40\\end{array}\\right].", "}$ In Figure REF we plot the normalized error on the Lyapunov matrix, $\\frac{\\max _{t\\in [0,\\ t_{\\max }]} |P(t)-\\mathbf {P}_k(t)|}{\\max _{t\\in [0,\\ t_{\\max }]} |P(t)|},$ for $t_{\\max }=50$ as a function of $k$ , computed using Algorithm REF .", "We also show the normalized error on the $\\mathcal {H}_2$ norm, $\\frac{\\left|\\Vert \\Upsilon \\Vert _2-\\Vert \\mathbf {\\Upsilon }_k\\Vert _2\\right|}{\\Vert \\Upsilon \\Vert _2}.$ Finally, the evolution of selected elements of the Lyapunov matrix $P(t)$ is shown in Figure REF .", "Even though the dimension $n$ is small, the advantage of using a projection method is significant.", "To illustrate this, when choosing $k=100$ the application of Algorithm REF involves the solution of a matrix Lyapunov equation of dimension $100\\times 100$ , leading to an error on the $\\mathcal {H}_2$ norm approximation smaller than $2\\ 10^{-8}$ , see Figure REF .", "At the same time, when discritizing the delay equation into an ordinary equation as in Section 2, with $N=19$ , and computing an $\\mathcal {H}_2$ norm approximation via (REF ), one also has to solve a Lyapunov equation of size $100\\times 100$ , but the error is then around $10^{-6}$ .", "The underlying reason is that the former approach can be interpreted in terms of a much more accurate discretization with $N>99$ points, followed by 100 steps of an Arnoldi iteration (see Figure REF ).", "Figure: Normalized errors () with t max =50t_{\\max }=50 (blue curve) and () (green curve) as a function of kk for system () with matrices and delays ()-().", "The dashed lines indicate the rates 𝒪k -2 \\mathcal {O}\\left(k^{-2}\\right) and 𝒪k -3 \\mathcal {O}\\left(k^{-3}\\right).Figure: Some elements of P(t)P(t) as a function of tt for system () with matrices and delays()-().For the second and third example we consider models described by partial differential equations (PDE) $ \\frac{\\partial v(x,t)}{\\partial t}=\\frac{\\partial ^2 v(x,t)}{\\partial x^2}-\\frac{1}{4}x\\ v(x,t-1)$ and $ \\frac{\\partial v(x,t)}{\\partial t}=\\frac{\\partial ^2 v(x,t)}{\\partial x^2}-2\\sin (x)v(x,t)+2\\sin (x)v(\\pi -x,t-1),$ with in both cases $v(0,t)=v(\\pi ,t)=0$ .", "The equations, which are variants of examples in [3], can be interpreted as heat equations describing in the temperature in a rod, controlled with distributed delayed feedback.", "In (REF ) the feedback is proportional and localized, in (REF ) it of Pyragas type and non-localized.", "We discretize differential equations (REF )-(REF ) in space using central differences.", "For (REF ), for instance, this resulting a systems of the form (REF ) with matrices $A_0=\\left(\\frac{n-1}{\\pi } \\right)^2\\left[\\begin{array}{ccccc}-2 & 1 & & & \\\\1 & -2 & 1 & & \\\\&\\ddots &\\ddots &\\ddots & \\\\& &1 & -2 & 1\\\\&&&1 &-2\\end{array}\\right] -2\\Delta _0$ and $A_1=2\\Delta _{-1}$ .", "Here $A_0,\\ A_1\\in \\mathbb {R}^{n\\times n}$ , and $\\Delta _0$ is a diagonal matrix containing the elements of the vector $\\left(0,\\sin \\left(\\frac{1}{n-1}\\pi \\right),\\cdots , \\sin \\left(\\frac{n-2}{n-1}\\pi \\right),0\\right)$ on its diagonal, while $\\Delta _{-1}$ is the anti-diagonal vector based on the same vector.", "For both (REF ) and (REF ) we we take $n=10000$ and output matrix $C=(1,1,\\ldots ,1)/\\Vert (1,1\\ldots ,1)\\Vert _2$ , i.e., the output is the average temperature of the rod.", "We further assume $B=C^T$ .", "In Figure REF we display the normalized error (REF ) on the Lyapunov matrix for the interval $[0,\\ t_{\\max }]=[0,\\ 3],$ as well as the normalized error on the associated $\\mathcal {H}_2$ norm approximation, as a function of $k$ .", "To shed a light on the computation time, for system (REF ) and $k=100$ the computation time for the delay Lyapunov matrix, respectively $\\mathcal {H}_2$ normAs can be seen from (REF ) only $\\mathbf {V}_k$ needs to be available to evaluate $\\mathbf {P}_k(t)$ at $t=0$ ., was 42 seconds, respectively $4.8$ seconds, using MATLAB R2017b on a laptop with Intel Core i7 2.80 GHz processor and 16GB RAM.", "Figure: Normalized errors (), with t max =3t_{\\max }=3, and () as a function of kk, with matrices obtained form the spatial discretization of () (blue curves) and () (green curves), such that n=10000n=10000.Let us now comment on the convergence behavior shown in Figures REF and REF .", "The experiments carried out for $k r>>n,$ which is natural if $n$ is small as for the first presented example, indicate an asymptotic rate of $\\mathcal {O}(k^3)$ , respectively $\\mathcal {O}(k^2)$ for the $\\mathcal {H}_2$ norm, respectively the delay Lyapunov matrix approximation.", "These rates are similar to those obtained by the spectral discretization in Section 2 (as a function on $N$ ), hence, the projection step does not result in a slowing down of the asymptotic convergence rate (recall the arguments in Section REF where the rates are, among others, related to the lack of smoothness of $P(\\cdot )$ ), even though it is highly advantageous from the point of view of computational complexity.", "Some intuition behind this observation is given by Theorems REF and REF : by construction precisely the matching moment between $\\Upsilon $ and $\\Upsilon _N$ carry over to the projected transfer function $\\mathbf {\\Upsilon }_k$ .", "In experiments with very large $n$ , we have $kr << n$ for a realistic range for $k$ values as in the second and third example, and the observed decay rate is slower, which is illustrated by a comparison between Figure REF and Figure REF .", "A possible explanation is that unlike the previous case a low-rank approximation of Lyapunov matrix $P(t)\\in \\mathbb {R}^{n\\times n}$ is enforced by construction.", "Inherent to the projection approach, the efficiency of the computational approach depends on whether or not accurate low rank approximations exist, whose determining factors are not well understood, and the projected system matrix $\\mathbf {G}_{2k}$ must be stability preserving (this is the case for most problems and it was an important consideration in the methodological choices, but not always - a counter example is the 2nd example in [11] for $n=1023$ , where spurious roots are observed in the right half plane).", "The latter is not necessarily a strong limitation for the $\\mathcal {H}_2$ norm computation, since the $\\mathcal {L}_2$ norm of the low-order, projected transfer function $\\mathbf {\\Gamma }_k$ can still be computed using other techniques different from solving the Lyapunov equation directly.", "All these issues, and related fixes are subject for further investigation." ], [ "Concluding remarks", "A novel algorithm for computing delay Lyaopunov matrices and $\\mathcal {H}_2$ norms has been presented, which is the first algorithm generally applicable to linear time-delay systems with multiple delays and at the same time having favorable scaling properties with respect to dimension $n$ (the examples with $n=10000$ in Section  indicates the potential of the approach).", "Furthermore, the algorithm is dynamic in nature, in the sense that the computations can be resumed if the accuracy is judged insufficient.", "The algorithm results in approximations of the delay Lyapunov matrix in an explicit form given by (REF ).", "Computing delay Lyapunov matrices induces a lot of challenges and complication compared to solving classical Lyapunov matrix equations (making the leap from an algebraic equation to matrix valued boundary problem (REF ) with a non-smooth solution).", "At the same time the research is in still an initial phase, with to the best of our knowledge, for the moment only two methods available applicable to large problems, the presented one and the one of [11], which are fundamentally different.", "Therefore we hope that the methodology, results and observations trigger further research on the topic.", "Finally we come back to the assumption of exponential stability of (REF ).", "It implies that computing the Lyapunov matrix (when alternatively defined as the solution of (REF ) and not via the fundamental solution), with the presented method is not useful in the context of verifying recent stability conditions, precisely expressed precisely in terms of the delay Lyapunov matrix (see, e.g., [4] and the references therein).", "Yet, the overall algorithm starts with iterations of Algorithm REF , which corresponds to the Infinite-Arnoldi algorithm [9] for eigenvalue computations and which does require an exponentially stable system.", "Consequently, from the output of the first step, more precisely from the spectrum of $\\mathbf {G}_{2k}$ , we directly obtain a certificate whether or not the system is exponentially stable." ], [ "Acknowledgements", "The first author thanks V.L.", "Kharitonov for an invitation to give a talk in a session on Lyapunov matrices at the 14th IFAC Workshop on Time-Delay System, which was the starting point of this work.", "The research was supported by the project C14/17/072 of the KU Leuven Research Council, by the project G0A5317N of the Research Foundation-Flanders (FWO - Vlaanderen), and by the project UCoCoS, funded by the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No 675080." ] ]

[ [ "A Bayesian Approach to Restricted Latent Class Models for\n Scientifically-Structured Clustering of Multivariate Binary Outcomes" ], [ "Abstract In this paper, we propose a general framework for combining evidence of varying quality to estimate underlying binary latent variables in the presence of restrictions imposed to respect the scientific context.", "The resulting algorithms cluster the multivariate binary data in a manner partly guided by prior knowledge.", "The primary model assumptions are that 1) subjects belong to classes defined by unobserved binary states, such as the true presence or absence of pathogens in epidemiology, or of antibodies in medicine, or the \"ability\" to correctly answer test questions in psychology, 2) a binary design matrix $\\Gamma$ specifies relevant features in each class, and 3) measurements are independent given the latent class but can have different error rates.", "Conditions ensuring parameter identifiability from the likelihood function are discussed and inform the design of a novel posterior inference algorithm that simultaneously estimates the number of clusters, design matrix $\\Gamma$, and model parameters.", "In finite samples and dimensions, we propose prior assumptions so that the posterior distribution of the number of clusters and the patterns of latent states tend to concentrate on smaller values and sparser patterns, respectively.", "The model readily extends to studies where some subjects' latent classes are known or important prior knowledge about differential measurement accuracy is available from external sources.", "The methods are illustrated with an analysis of protein data to detect clusters representing auto-antibody classes among scleroderma patients." ], [ "Introduction", "Let $\\mathbf {Y}$ be a $N\\times L$ binary data matrix of $N$ observations with $L$ dimensions or features.", "Such multivariate binary data frequently arise as noisy measurements of presence or absence of a list of unobservable or latent binary variables $$ called states.", "Suppose we seek to cluster such data subject to the hypothesis that a cluster is likely to be defined by individuals who share a relatively small number of states.", "That is, there exist subgroups of $$ vectors that take values on a relatively small number of elements in $\\lbrace 0,1\\rbrace ^M$ with $M\\le L$ ; let the subgroups be denoted by $$ .", "We propose a method for estimating scientifically-structured clusters (SSC).", "Our method is most useful for a large dimension $L$ with an unknown number of clusters.", "Structured clustering for multivariate binary data has a number of potential advantages.", "If the underlying clusters differ from one another only at subsets of features, SSC can more accurately estimate these clusters than standard clustering methods such as latent class analysis and hierarchical clustering.", "SSC also results in more interpretable clusters.", "Consider three examples from medicine, psychology and epidemiology that motivate scientifically-structured clustering.", "Example 1 is to estimate subgroups of autoimmune disease patients using autoantibody data that have the potential to predict homogenous disease trajectories [30].", "The observed binary responses are imperfect indicators of the presence or absence of specific autoantibody combinations detected in patient sera.", "Inherent limitation of the lab technique used to identify these autoantibodies (immunoprecipitation, IP) and biological and biochemical variability cause discrepancies between the expected presence/absence of each antibody and the observed values from IP assays.", "In addition, autoantigens (the specific proteins targeted by autoantibodies) frequently exist as multi-protein complexes, which we will refer to as “machines\" in this paper [46].", "The medical goals are to define the “machines\" by their component proteins, and infer whether or not each patient has each machine, using the imprecise IP data.", "The second example relates to cognitive diagnosis in psychological and educational assessment.", "The binary outcomes indicate a subject's responses to many diagnostic questions (“items\").", "The measurements reflect the person's long-term “true\" responses to these items, indicating a student's knowledge for correctly answering a test question absent guessing or other errors.", "These “true\" or “ideal\" responses are further assumed to define a smaller number of binary latent skills that indicate the presence or absence of the particular knowledge (called “states\" in the psychology literature).", "For example, teachers assess whether the student possesses basic arithmetic skills (e.g., addition, multiplication); and psychiatrists diagnose whether patients have certain mental disorders based on a subject's survey responses [31].", "Each question or item is designed to measure a particular subset of latent states, where such item-latent-state correspondence may be known, partially known or unknown.", "Example 3 is to estimate the causes of childhood pneumonia from a list of more than 30 different species of pathogens including viruses, bacteria and fungi [42].", "The imperfect binary outcomes indicate whether or not each pathogen was detected by the polymerase chain reaction (PCR) or cell culture from two compartments: the nasopharyngeal (NP) cavity and blood.", "The binary latent states of scientific interest are the true presence or absence of the pathogens in a child's lung, the site of infection that can seldom be directly observed in practice.", "This example differs from Example 1 in that the correspondence between each of the compartment-technology-pathogen diagnostic measurements (“features\") and the latent lung infection (“state\") is known because each measurement is designed to detect one specific pathogen and hence is expected to have higher positive rates in classes infected by that pathogen.", "In addition, the two measurements (NP with PCR and blood with cell culture) are known to have different error rates [23], [52].", "In each of these examples, the clustering of observations and subject-specific prediction of $\\lbrace _i\\rbrace $ comprise the scientific targets for inference.", "Our examples can be distinguished by: a) whether the latent state variables ($_i$ ) are constrained or unconstrained to take values from a pre-specified subset $$ where classes are defined by distinct values of $_i$ , b) whether it is known, partially known, or unknown about the binary design matrix $\\Gamma $ that specifies for each latent class the set of relevant features having the highest positive response probability than other classes; and c) the form of the conditional distribution of measurements given latent states and the design matrix ($\\Gamma $ ) and response probabilities ($\\Lambda $ ): $[_i \\mid _i, \\Gamma , \\Lambda ]$ .", "This paper discusses a family of latent class models, referred to as restricted latent class models or RLCMs [55] specified by the three components listed above.", "The model formulation includes as special cases: probabilistic boolean matrix decomposition [47], subset clustering models [26], and partially latent class models [52] among others discussed in detail in Section REF .", "The focus is on estimating clusters based on multivariate binary data that exhibit differential errors depending on the true latent class.", "The design matrix $\\Gamma $ is assumed to be generated from a low-dimensional latent state vector $_i$ .", "However, in many applications, the number of clusters and/or the set of latent states $()$ are not known in a priori and must be inferred from data.", "We discuss large-sample identifiability conditions for RLCM likelihood-based inference to motivate our posterior algorithm design.", "However, in finite samples, the likelihood function can be relatively flat before asymptotics concentrate the likelihood around the major mode.", "To improve finite-sample estimation efficiency at the expense of some bias, we specify sparsity-inducing priors that propagate into the posterior distribution to encourage few clusters with sparse latent state patterns.", "We begin this paper with a unified survey of restricted latent class models drawing on the previous work of [52], [53], [26], [55].", "The second objective is to present novel Markov chain Monte Carlo (MCMC) algorithms for Bayesian RLCMs with discrete component parameters building on the sampling techniques of [29], [39] and [5].", "Section presents the model formulation including the likelihood, prior distribution and theoretical identifiability results.", "In Section , we present our MCMC algorithm to efficiently estimate posterior distributions for clusters.", "Section REF compares via simulation the proposed clustering method to three common alternatives.", "Section REF illustrates the methods with analysis of the autoantibody data for Example 1.", "The paper concludes with a discussion of model extensions and limitations." ], [ "Model", "Let $\\mathbf {Y}_i = (Y_{i1}, \\ldots , Y_{iL})^\\top \\in \\lbrace 0,1\\rbrace ^L$ represent a $L$ -dimensional multivariate binary response for subject $i=1, \\ldots , N$ ; Let $\\mathbf {Y}$ collect data from all subjects.", "We assume each observation is associated with an unobserved or latent state vector $_i\\in $ , where $\\subset \\lbrace 0,1\\rbrace ^M$ is a set of $M$ -dimension binary vectors.", "Given a pre-specified dimension of latent states $M$ , we first specify the likelihood $[_i \\mid _i, \\Gamma , \\Lambda ]$ via restricted latent class models (RLCM) and then, among others, a prior distribution for $H=\\lbrace _i, i=1, \\ldots , N\\rbrace $ that groups subjects by their binary patterns $\\lbrace _i\\rbrace $ (Supplementary Material A.1 extends the prior on $H$ to $M=\\infty $ ).", "Let $\\tilde{K}=||$ represent the number of groups with non-zero population prevalence.", "Although it is no greater than $2^M$ , $\\tilde{K}$ can be unknown; And when $\\tilde{K}$ is known and $\\tilde{K} < 2^M$ , $$ can be unknown.", "The two steps jointly specify a so-called mixture of finite mixture model for $\\lbrace _i\\rbrace $ [39].", "In our setting, the salient feature of scientific import is the discrete component parameters $\\lbrace _i\\rbrace \\subset $ that requires additional handling in the posterior algorithm (Section ).", "Section REF discusses special cases of the RLCM relevant to the motivating examples.", "By taking $N$ to infinity in the likelihood, Section REF further studies theoretical limits of identifying unknown model parameters." ], [ "Latent Class Model", "For the traditional latent class model (LCM) [19], we assume that the latent state vectors $\\lbrace _i\\rbrace $ take values from a set of binary patterns $=\\lbrace \\tilde{}_k, k=1, \\ldots , \\tilde{K}\\rbrace $ , where $\\tilde{K} = ||$ is the number of distinct patterns.", "Latent classes differ in their latent state patterns.", "Given an observation's latent states $_i$ , we assume the probability of observing a positive response of feature $\\ell $ for subject $i$ is $(Y_{i\\ell }=1\\mid \\Lambda ) = \\lambda _{i\\ell }$ , $\\ell =1, \\ldots , L$ , where $\\Lambda = \\lbrace \\lambda _{i\\ell }\\rbrace $ is a $N\\times L$ matrix of response probabilities; For $\\Lambda $ and other matrices in this paper, we will use $\\Lambda _{\\star \\ell }$ and $\\Lambda _{i\\star }$ to denote the $\\ell $ -th column and $i$ -th row, respectively.", "A more useful, non-saturated model lets the response probability $\\lambda _{i\\ell }$ depend on the subject's latent state vector $_i$ via $\\lambda _{i\\ell } = \\lambda _\\ell (_i)$ where $\\lambda _\\ell : \\rightarrow [0,1]$ .", "Because $_i$ can be one of $\\tilde{K}$ elements in $$ , the classes have at most $\\tilde{K}$ distinct response probabilities, referred to as between-class differential measurement errors.", "The LCM has a conditional independence assumption whereby the measurements from distinct dimensions are independent of one another given the latent class and response probabilities in that class, i.e.", "$Y_{i\\ell } \\perp Y_{i\\ell ^{\\prime }} \\mid _i, \\lambda _{\\ell }(\\cdot )$ .", "Fitting LCM is to attribute, for example, a positive marginal association observed between two dimensions $\\ell $ and $\\ell ^{\\prime }$ to their similar response probabilities that define the latent classes.", "Taken together, LCMs specify the conditional probability of observing a multivariate binary outcome $\\mathbf {y}\\in \\lbrace 0,1\\rbrace ^L$ by $(\\mathbf {Y}_i = \\mathbf {y} \\mid \\mathbf {\\eta }_i, {\\lambda _\\ell (\\cdot )}) = \\prod _{\\ell =1}^L(\\lambda _{i\\ell })^{y_{i\\ell }}(1-\\lambda _{i\\ell })^{1-y_{i\\ell }}, \\text{~where~} \\lambda _{i\\ell }=\\lambda _{\\ell }(_i).$ Because $_i$ is not observed, it is integrated out of (REF ) with respect to its distribution $(_i = \\mid _{\\tilde{K}}) = \\pi _{} >0$ , for $\\in $ .", "Based on $N$ independent observations, the LCM likelihood takes the form of “mixture of Bernoulli products\": $\\prod _{i=1}^N\\sum _{\\in }\\pi _{} \\left\\lbrace _i \\mid _i = , {\\lambda _\\ell ()}, \\ell = 1, \\ldots , L\\right\\rbrace $ .", "Given $\\ell $ , traditional LCMs impose no structure upon the response probability vectors except that they differ among classes almost surely: $\\lambda _\\ell ()\\ne \\lambda _\\ell (^{\\prime })$ for latent classes $\\ne ^{\\prime }$ .", "Let $\\tilde{Z}_i \\in \\lbrace 1, \\ldots , \\tilde{K}\\rbrace $ indicate the unobserved class assignment for observation $i$ .", "An equivalent and more familiar formulation $\\lambda _{i\\ell } = \\lambda _{\\ell }(\\tilde{Z}_i)$ results.", "The LCM approximates any multivariate discrete distribution for sufficiently large $\\tilde{K}$ [11] and, up to class relabeling, is generically identified whenever $L\\ge 2{\\tilde{K}}+1$ [2].", "Fitted LCM results will show the estimated response probability profiles that differ by class and can be interpreted as population heterogeneity in particular scientific contexts.", "Estimation of clusters in finite mixture models often makes use of $\\lbrace Z_i\\rbrace $ , for example, by maximizing the plugged-in conditional posterior $\\hat{Z}_i = \\arg \\max _{k=1, \\ldots , \\tilde{K}} (Z_i = k \\mid \\mathbf {Y},\\hat{}_{\\tilde{K}})$ or a least-square estimate of clusters based on distance from pairwise co-coclustering posterior probabilities $\\hat{\\pi }_{i,i^{\\prime }} = (Z_i=Z_{i^{\\prime }}\\mid \\mathbf {Y})$ [9]." ], [ "Motivation for Scientifically-Structured Classes", "The traditional LCM does not incorporate important prior scientific knowledge about how clusters (classes) structurally differ.", "In Example 1, autoimmune disease patients may differ in their antibody protein presence or absence patterns at $L=50$ protein landmarks over a grid of molecular weights.", "The focus is on estimating groups of patients who differ in their immune responses to unknown machines.", "We formulate this biological prior knowledge by introducing the following model parameters: i) An $M$ by $L$ machine matrix $Q$ where $Q_{m\\ell } = 1$ indicates presence of landmark autoantigen protein $\\ell $ in machine $m$ ; We refer to the rows in $Q$ as “machine profiles\".", "In addition, feature $\\ell $ with $\\sum _m Q_{m\\ell }=0$ indicates landmark autoantigen protein $\\ell $ is not targeted as part of any machine.", "ii) A patient-specific vector of length $M$ that represents the presence or absence of $M$ machines ($_i = (\\eta _{i1}, \\ldots , \\eta _{iM})^\\top $ ).", "For example, In Figure REF , for $M=3$ , a subject with $_i = (1,0,1)^\\top $ has Machines 1 and 3 (middle panel).", "The two machines produced her antibody proteins (left panel) subject to further errors.", "Given $Q$ and $_i$ , we can represent the presence or absence of antibody proteins deterministically, for example, by $\\Gamma _{i \\star } = ^\\top _i Q$ under a row-orthogonal $Q$ as illustrated in Figure REF .", "For feature $\\ell $ with $\\sum _{m}Q_{m\\ell }=0$ , we have $\\Gamma _{i\\ell }=0$ for all subjects.", "iii) Positive rate parameters, the true- $(= \\lbrace \\theta _{\\ell } = (Y_{i\\ell }=1 \\mid \\Gamma _{i\\ell }=1)\\rbrace )$ and false- positive rates $(= \\lbrace \\psi _{\\ell }=(Y_{i\\ell }=1 \\mid \\Gamma _{i\\ell }=0)\\rbrace )$ .", "Two sources of stochastic variations contribute to the discrepancy between the expected presence of autoantibody ($\\Gamma _{i\\ell }=1)$ and the observed presence ($Y_{i\\ell }=1$ ) or absence ($Y_{i\\ell }=0$ ): selective immunological non-response to certain autoantigen proteins in a machine and experimental errors.", "In a priori, we assume high true- and low false- positive rates ($\\theta _\\ell > \\psi _\\ell $ ) because GEA method is robust for detecting immunoprecipitated antibodies.", "In summary, i) and ii) incorporate the prior knowledge that antibody proteins are produced in groups against autoantigen proteins coded by the rows of $Q$ and iii) is the measurement likelihood function that assigns probabilities to observed data accounting for stochastic variations.", "The other two examples in Section can be parameterized in the same way with known or unknown $Q$ (Section REF ).", "Given subjects with $\\Gamma _{i\\ell }=1$ or 0, the response probability $\\lambda _{i\\ell } = \\theta _\\ell $ or $\\psi _\\ell $ regardless of $i$ 's class membership (e.g., true presence of antibody protein in serum no matter which machine it comes).", "Consequently, unlike traditional LCM, a new model where not all features exhibit difference in response probabilities $\\lbrace \\lambda _{i\\ell }, i = 1, \\ldots , N\\rbrace $ is needed.", "Using separate class-specific estimates of $\\lbrace \\lambda _{\\ell }(1), \\ldots ,\\lambda _{\\ell }(\\tilde{K})\\rbrace $ for features without actual between-class differential errors can be imprecise and will result in inferior clustering performance (Figure REF , d).", "RLCMs provide a general framework for specifying class response probability profiles to respect scientific structures through which we show achieve better clustering performance.", "Figure: Binary matrix factorization generates composite autoantibody signatures that are further subject to misclassification.", "The signature Γ i☆ =η i ⊤ 𝐐\\Gamma _{i\\star }= \\mathbf {\\eta }_i^\\top \\mathbf {Q} assembles three orthogonal machines with 3, 4 and 3 landmark proteins, respectively.", "The highlighted individual is expected to mount immune responses against antigens in Machines 1 and 3.", "See texts after model ()." ], [ "Restricted LCMs to Incorporate Scientifically-Structured Classes", "RLCMs assume equality among a subset of response probabilities across classes.", "That is for some $\\ell $ , RLCMs assume $\\lambda _{i\\ell } = \\lambda _{i^{\\prime }\\ell }$ for some subjects in distinct latent classes ($_i \\ne _{i^{\\prime }}$ ).", "The set of RLCM parameters therefore comprises a Lebesgue measure zero set in the parameter space for the traditional unconstrained LCM.", "The restrictions on response probabilities in RLCMs are specified by introducing a binary design matrix $\\Gamma = \\lbrace \\Gamma _{,\\ell }\\rbrace \\in \\lbrace 0,1\\rbrace ^{\\tilde{K}\\times L}$ with latent classes and dimensions in the rows and columns, respectively.", "$\\Gamma _{, \\ell } =1$ represents a positive ideal response for which subjects in latent class $$ will have the highest response probability at dimension $\\ell $ ; 0 for a negative ideal response for which subjects in latent class $$ will have a lower response probability.", "If $\\Gamma _{,\\ell }=\\Gamma _{^{\\prime },\\ell }=1$ for two latent classes $$ and $^{\\prime }$ , it is assumed that the $\\ell $ -th dimension is observed with identical positive response probabilities: $\\lambda _{i\\ell } = \\lambda _{i^{\\prime }\\ell }$ .", "On the other hand, there can be more than one response probability if $\\Gamma _{,\\ell }=0$ .", "That is, no equality constraint upon the response probabilities $(\\lambda _{,\\ell }=\\lambda _{^{\\prime },\\ell })$ is required for two latent classes with $\\Gamma _{,\\ell }=\\Gamma _{^{\\prime },\\ell }=0$ .", "In this paper, we focus on $Q$ -restricted LCM where the design matrix $\\Gamma $ is determined by the latent state vectors $$ and an $M$ by $L$ binary matrix $Q$ , i.e., $\\Gamma _{,\\ell } = \\Gamma (, Q_{\\star \\ell }),\\forall \\in , \\ell = 1, \\ldots , L, $ where the mapping or design matrix $\\Gamma (\\cdot , \\cdot ): \\times \\lbrace 0,1\\rbrace ^M \\rightarrow \\lbrace 0,1\\rbrace ^{\\tilde{K}\\times L}$ needs to be specified in the context of the particular scientific study (e.g., $\\Gamma _{i \\ell } = _i^\\top Q_{\\star \\ell }$ as in Figure REF ).", "We now introduce scientific structures through the restriction of response probabilities.", "Let $_{\\ell } = \\lbrace \\in : \\Gamma _{, \\ell } = 1\\rbrace $ where $_{\\ell }$ collects latent classes with the highest response probability for dimension $\\ell $ according to $\\Gamma $ [21].", "If $_\\ell \\ne \\emptyset $ , we restrict the response probabilities at feature $\\ell $ by $\\max _{\\in _\\ell } \\lambda _{, \\ell } = \\min _{\\in _\\ell } \\lambda _{, \\ell } > \\lambda _{^{\\prime }, \\ell }, \\ell = 1, \\ldots , L, ^{\\prime } \\notin _\\ell .$ Further, there can exist a class $\\in $ that gives rise to all-zero ideal responses $\\Gamma _{\\star }=\\mathbf {0}_{1\\times L}$ .", "To make the notation clear, in Example 1, $Y_{i\\ell }$ represents the observed presence/absence of protein $\\ell $ on the immunoprecipitation gel for patient $i$ , $_i$ indicates this patient's latent class and which protein complexes (“machines\") among the rows of $Q$ are present in patient $i$ 's class, $Q_{m\\star }, m=1,\\ldots , M,$ indicates which proteins comprise Machine $m$ , and $\\Gamma _{,\\ell }$ indicates whether or not protein $\\ell $ is present from any machines in latent class $$ .", "The class with no machine has all zeros in its row of $\\Gamma $ .", "The probability of observing a protein given it is present is its true positive rate or sensitivity.", "The probability of observing the protein given it is absent is its false positive rate or one minus its specificity.", "The sensitivity for a given protein is assumed to be the same regardless from which machine(s) it comes; the specificities are allowed to vary across proteins.", "Finally, the true positive rates are assumed to be larger than the false positive rates.", "Finally, we specify the measurement likelihood through parameterization of the response probabilities $\\lambda _{i\\ell } = \\lambda ^R_\\ell \\left(_{i};_\\ell , Q_{\\star \\ell } \\right) \\in [0,1], $ where $\\lambda ^R_\\ell $ specifies the response probability at feature $\\ell $ with “restriction\" (REF ).", "The restriction is prescribed by $\\Gamma _{_i, \\ell }$ which is further determined by $_i$ and $Q_{\\star \\ell }$ in (REF ).", "$\\lambda ^R_\\ell $ also depends on $_i$ and unknown real-value parameters $_\\ell $ according to particular parametric models; See model (REF ) below for an example.", "In what follows, we use $\\Gamma _{i\\ell }$ to denote $\\Gamma _{_i, \\ell }$ unless otherwise noted.", "Motivated by our applications, we present an equivalent formulation for $\\lambda ^R_{\\ell }$ that separate true and false positive rates.", "Let $K^+_\\ell = \\#\\lbrace \\lambda _{i\\ell }: _i \\in _\\ell \\rbrace $ ($K^-_\\ell = \\#\\lbrace \\lambda _{i\\ell }: _i \\notin _\\ell \\rbrace $ ) be the number of distinct response probability levels at feature $\\ell =1, \\ldots , L$ .", "In RLCMs, we have $K^+_\\ell = 1$ and $K^-_\\ell \\ge 1$ , $\\ell = 1, \\ldots , L$ ( see Table S1 in Supplementary Materials that tabulate the number of distinct response probabilities at dimension $\\ell $ , $(K^+_\\ell , K^-_\\ell )$ , for other variants of LCMs).", "Let $\\theta _l$ be the maximum response probability at feature $\\ell $ and $\\mathbf {\\psi }_\\ell = \\lbrace \\psi _{l1}, \\ldots , \\psi _{l,K^-_\\ell }\\rbrace $ be the rest of response probabilities, respectively.", "Given $_i = \\notin _\\ell $ , let $v_i = v(_i, \\ell )$ , where $v(\\cdot , \\cdot )$ : $(_i,\\ell ) \\mapsto v$ is the integer-valued function that selects among $\\mathbf {\\psi }_\\ell $ her associated response probability $\\psi _{\\ell , v_i}$ at feature $\\ell $ .", "The parameters $\\theta _\\ell $ and $_\\ell $ may be further parameterized by $(_\\ell ,Q_{\\star \\ell })$ as in (REF ).", "For models with $K^-_\\ell =1$ , $v(\\cdot )=1$ ; Otherwise, $\\nu (\\cdot ,\\cdot )$ depends on $$ (the set of possible pattern of $_i$ ), the specific functional form of $\\lambda ^R_\\ell (\\cdot )$ and parameter values of $(_\\ell , Q_{\\star \\ell })$ in a RLCM (see the example (REF ) in Section REF ; The traditional LCM results by setting $Q=1_{M\\times L}$ and under $K^++K^-=\\tilde{K}$ for each $\\ell $ ).", "In this paper, because we focus on models with the structure in (REF ), we can equivalently represent the response probability parameters $\\lambda ^R_\\ell $ in (REF ) by $\\lambda ^R_\\ell (_i; _\\ell , Q_{\\star \\ell }) =\\left\\lbrace \\theta _\\ell \\right\\rbrace ^{\\Gamma _{_i,\\ell }}\\cdot \\left\\lbrace _{\\ell ,v(_i,\\ell )}\\right\\rbrace ^{1-\\Gamma _{_i,\\ell }} \\in [0,1],$ where $_\\ell = \\lbrace =\\left\\lbrace \\theta _\\ell \\rbrace , \\Psi = \\lbrace _\\ell \\rbrace \\right\\rbrace $ with constraints $\\theta _\\ell > \\psi _{\\ell , v}, \\forall v=1, \\ldots , K^-_\\ell $ .", "RLCMs therefore let observations with latent state patterns in $_\\ell $ take identical and the highest probability than other classes in $^c_\\ell $ .", "Other classes in $_\\ell ^c$ respond with lower probabilities at dimension $\\ell $ .", "We now discuss some examples." ], [ "Examples of RLCMs in the Literature", "Special cases of restricted LCMs result when $K^+_\\ell =K^-_\\ell =1$ .", "For example, a class of models assumes the response probabilities $\\lambda _{i\\ell } & = \\theta _{\\ell }^{\\Gamma _{i\\ell }}(\\psi _{\\ell })^{1-\\Gamma _{i\\ell }}, ~~~~\\Gamma _{i\\ell }= 1-\\prod _{m=1}^M(1-\\eta _{im})^{Q_{m\\ell }}.", "$ Consider $N$ subjects each responding to $L$ items where $Q_{m\\ell }=1$ means item $\\ell $ requires positive latent state $m$ , otherwise $Q_{m\\ell }=0$ .", "This model, referred to as partially latent class models in disease epidemiology [52] or Deterministic In and Noisy Or (DINO) in cognitive diagnostic models [50] that needs just one required state ($\\lbrace m: Q_{m\\ell }=1\\rbrace $ ) for a positive ideal response $\\Gamma _{i\\ell }=1$ .", "Imposing constant and symmetric error rates $\\theta _{\\ell }=\\psi _{\\ell }$ , $\\ell = 1, \\ldots , L$ , the one-layer model of [47] results.", "The model can also be viewed as Boolean matrix factorization [38] by noting that $\\Gamma _{i\\ell }= \\vee _{m=1}^M \\eta _{im}Q_{m\\ell }$ where the logical “OR\" operator $``\\vee \"$ outputs one if any argument equals one.", "The rows in ${Q}$ are basis patterns for compactly encoding the $L$ dimensional $\\Gamma _{i\\star }$ vector by $M(\\ll L)$ bits.", "BMF further reduces to nonnegative matrix factorization [34] $\\Gamma =HQ$ where $H=\\lbrace \\eta _{im}\\rbrace $ if ${Q}$ has orthogonal rows.", "See Supplementary Materials A.2 for a connection to subset clustering in [26].", "A second two-parameter example results by assuming $\\Gamma _{i\\ell }= \\prod _{m=1}^M(\\eta _{im})^{Q_{m\\ell }}$ [31].", "This model, referred to as Deterministic In and Noise And (DINA) gate model in the cognitive diagnostic literature, assumes a conjunctive (noncompensatory) relationship among latent states $m=1, \\ldots , M$ .", "That is, it is necessary to possess all the attributes (states) indicated by non-zero elements in $Q_{\\star \\ell }$ to be capable of providing a positive ideal response $\\Gamma _{i\\ell }=1$ .", "The model also imposes the assumption that possessing additional unnecessary attributes does not compensate for the lack of the necessary ones.", "These two-parameter models are equivalent upon defining $\\eta _{im}^*=1-\\eta _{im}$ , $\\Gamma _{i\\ell }^*=1-\\Gamma _{i\\ell }$ , $\\psi ^*_{\\ell }=1-\\psi _{\\ell }$ and $\\psi ^*_{\\ell }=1-\\theta _{\\ell }$ [6].", "There are several other examples in this category as discussed by [54].", "Two-parameter models assume that “$\\Gamma _{,\\ell }=\\Gamma _{^{\\prime },\\ell }=0$ implies identical response probabilities $\\lambda _{,\\ell } = \\lambda _{^{\\prime },\\ell } = \\psi _{\\ell }$ \", regardless of the distinct patterns $\\ne ^{\\prime }$ .", "In practice, deviation from such assumptions occurs if $$ has more nonzero elements than $^{\\prime }$ and alters the response probabilities, i.e., $K^-_\\ell >1$ .", "Multi-parameter models where $K^-_\\ell >K^+_\\ell =1$ , popular in multidimensional item response theory, is readily specified for example by assuming an all-effect model: $\\lambda _{i\\ell } = \\lambda ^R_{\\ell }(_i; \\mathbf {\\beta }_\\ell , Q_{\\star \\ell })= \\textrm {expit}\\left\\lbrace \\mathbf {\\beta }_\\ell ^\\top (_i, Q_{\\star \\ell })\\right\\rbrace = $ $& \\textrm {expit} \\left\\lbrace \\beta _{\\ell 0}+\\sum _{m=1}^M\\beta _{\\ell m}(Q_{m\\ell }\\eta _{im})+\\sum _{m < m^{\\prime } }\\beta _{\\ell m m^{\\prime }}(Q_{m\\ell }\\eta _{im})(Q_{m^{\\prime }\\ell }\\eta _{im^{\\prime }})+\\ldots +\\beta _{\\ell 12\\ldots M}\\prod _{m}(Q_{m\\ell }\\eta _{im})\\right\\rbrace $ that includes higher order interactions among latent states required by an item [25]; Here $\\text{ expit}(x)=\\frac{\\exp (x)}{1+\\exp (x)}$ .", "When $\\prod _{m=m_1, \\ldots , m_s}Q_{m\\ell }=0$ , this saturated model needs no $\\beta _{\\ell ,m_1\\ldots m_s}$ term.", "Setting second or higher order terms to zero, an additive main-effect model results.", "The effects of latent states need not be additive.", "For example, ${\\sf log}(\\lambda _{i\\ell }) = \\beta _{\\ell 0}+\\sum _{m=1}^M \\beta _{\\ell m}Q_{m\\ell }\\eta _{im}$ specifies a multiplicative model that penalizes the absence of an required latent state $m$ if $Q_{m\\ell }=1$ .", "Table S1 in Supplementary Materials summarizes these and other variants of LCMs by specifications of the latent state space, design matrix, and measurement processs." ], [ "Identifiability", "There are two sources of indeterminancy in restricted LCMs: invariance of the likelihood function to permutation of the ordering of the latent states and over-parameterized models.", "The permutation invariance manifests itself as a multimodal posterior distribution.", "Where $Q$ is unknown, we address the permutation invariance by labeling the latent states, one dimension at a time, by the non-zero patterns of the corresponding rows in an estimated $Q$ .", "We address the over-parameterization by introducing prior distributions that encourage in a priori few clusters hence a small number of parameters via mixture of finite mixture models [39].", "It helps to show identifiability results or lack thereof to motivate such sparsity-inducing priors.", "Given $\\tilde{K}$ and $M$ , identifiability conditions characterize the theoretical limits of recovering the unknown model parameters ($Q$ , $\\Lambda $ , $_{\\tilde{K}}$ ) from the likelihood for all or a subset of the parameter space.", "We first discuss the identifiability of $Q$ because it is needed for interpreting latent states (see Section REF ) and for estimating both $H$ and $_{\\tilde{K}}$ .", "Based on the likelihood $[_i \\mid _{\\tilde{K}}, \\Lambda , \\Gamma =\\Gamma (Q)]$ with a given $Q$ and a saturated $$ (or “full diversity\": $\\pi _{}>0, \\forall \\in =\\lbrace 0,1\\rbrace ^M$ ), [54] studied sufficient conditions for strict identifiability of $\\Lambda $ and $_{\\tilde{K}}$ over the entire parameter space in RLCMs.", "Under weaker conditions upon the design matrix $\\Gamma $ (instead of $Q$ ) and possibly non-saturated $$ , [21] established conditions that guarantee partial identifiability for general RLCMs which means the likelihood function is flat over a subset of the parameter space.", "When $Q$ -matrix is completely unknown, it is possible to identify $\\lbrace _{\\tilde{K}}, \\Lambda , Q\\rbrace $ just using likelihood $[_i \\mid _{\\tilde{K}}, \\Lambda , \\Gamma =\\Gamma (Q)]$ .", "In particular, [6] provided sufficient conditions for the special cases of DINA and DINO models (see Section REF ); [55] further generalized them to general RLCM: ($Q$ , $\\Lambda $ , $_{\\tilde{K}}$ ) are strictly identifiable (up to row reordering of $Q$ ) in RLCMs with saturated $$ if the following two conditions hold: C1) The true ${Q}$ can be written as a block matrix ${Q}=[{I}_M; {I}_M; \\tilde{{Q}}]$ after necessary column and row reordering, where $\\tilde{{Q}}$ is a $M\\times (L-2M)$ binary matrix and C2) $(\\Lambda _{,\\ell }, \\ell > 2M)^\\top \\ne (\\Lambda _{^{\\prime },\\ell }, \\ell > 2M)^\\top $ for any $\\ne ^{\\prime }$ and $\\succeq ^{\\prime }$ , where $\\mathbf {a}\\succeq \\mathbf {b}$ for $\\mathbf {a} = \\lbrace a_j\\rbrace $ and $\\mathbf {b}=\\lbrace b_j\\rbrace $ if and only if $a_j\\ge b_j$ holds element-wise.", "Because condition (C2) depends on $Q$ , $\\Lambda $ and row and column permutations, the number of operations to check (C2) increases exponentially with $M$ , $((L-2M)2^MM)$ , for a saturated $$ with $2^M$ patterns of latent state vectors.", "We instead use condition (C3) that just depends on $Q$ and that is invariant to row or column permutations: C3) Each latent state is associated to at least three items, $\\sum _{\\ell =1}^L Q_{m\\ell } \\ge 3$ for all $m$ .", "Condition (C3) enables convenient restrictions in MCMC sampling and takes just $(LM)$ operations to check.", "For special cases of RLCM, the DINA and DINO models (Section REF ) with a saturated $$ , Conditions (C1) and (C3) suffice to identify ($Q$ , $\\Lambda $ , $_{\\tilde{K}}$ ) [6].", "Posterior algorithms typically restrict MCMC sampling of non-identified parameters by identifiability conditions to prevent aggregation of posterior probability mass from multiple modes.", "For example, in factor analysis of multivariate continuous data, one can restrict the loading matrices in lower triangular forms [16].", "Alternatively, one may first perform MCMC sampling with weak and simple-to-check constraints without fully ensuring identifiability and just check afterwards whether the parameters are conditionally identifiable.", "One then performs necessary deterministic transformations on parameters that may only be identified up to equivalent classes to pick coherent and economical representatives, for example, by relabeling sampled mixture components at each iteration or varimax rotations of factor loading matrices in classical Gaussian factor analysis [45].", "We initialize the sampling chain from the set defined by simple identifiability conditions (C1) and (C3) and only check afterwards at each iteration whether the parameters are conditionally identifiable according to conditions (C1) and (C2) that are stronger and computationally more expensive.", "The relabeling of the latent states is done by inspecting the non-zero patterns in the rows of $Q$ (Step 7, Supplementary Material C.1).", "In applications where $Q$ is unknown with $M<L/2$ , we focus on the set of $Q$ -matrices that satisfy both (C1) and (C3): $\\mathcal {Q} = \\lbrace {Q} \\in \\lbrace 0,1\\rbrace ^{M\\times L}: {Q}=P_1{Q}^\\dagger P_2, ~{Q}^\\dagger =[{I}_M; {I}_M; \\tilde{{Q}}], ~\\tilde{{Q}}\\mathbf {1}_{L-2M}\\succeq \\mathbf {1}_{L-2M}\\rbrace ,$ where ${P}_1$ and $P_2$ are $M$ - and $L$ -dimensional permutation matrices for rows and columns, respectively.", "The constraint $\\mathcal {Q}$ also greatly facilitates posterior sampling by focusing on a small subset of binary matrices.", "In fact, among all $M$ by $L$ binary matrices, the fraction of $Q\\in \\mathcal {Q}$ is at most $\\frac{{L \\atopwithdelims ()2M}\\left[2^{(L-2M)M}\\right]}{2^{L\\cdot M}}$ and quickly decay as the number of machines $M$ increases.", "In some applications it may also simplify posterior inference by exploiting further assumptions upon $Q$ for example partially known $Q$ or non-overlapping (i.e., orthogonal) rows of $Q$ .", "See Supplementary Materials A.3 and A.4 for other identifiability considerations that motivate our posterior algorithms." ], [ "Priors", "Given $M$ , we specify the prior for $H=\\lbrace _i\\rbrace $ with cluster structure among $N$ subjects in five steps: 1) Generate the vector of probabilities of a subject $i$ belonging to each of $K$ clusters $_K=(\\pi _1, \\ldots , \\pi _K)^\\top $ where $K$ is possibly unknown and sampled from its prior $p_K(\\cdot )$ ; 2) Partition observations by indicators $Z_i\\overset{i.i.d}{\\sim } {\\sf Categorical}(_K)$ ; Suppose we obtain $T$ distinct $\\lbrace Z_i\\rbrace $ values; 3) Draw the vector of marginal probabilities of each latent state being active $= \\lbrace p_m\\rbrace $ ; 4) Draw from $[^*_j \\mid ,M]$ , for clusters labeled $j=1, \\ldots , T$ , where “$^*$ \" indicates cluster-specific quantities; 5) Combine $\\lbrace ^*_j\\rbrace $ and $\\lbrace Z_i\\rbrace $ to obtain subject-specific latent states $_i = ^*_{Z_i}$ , $i=1, \\ldots , N$ ." ], [ "Prior for Partitioning Observations", "Though used interchangeably by many authors, we first make a distinction between a “component\" that represents one of the true mixture components in the specification of a mixture model and a “cluster\" that represents one element in any partition of observations.", "Let $K$ be the number of mixture components in the population and $T$ the number of clusters in the sample [39].", "To establish notation, let $Z_i \\in \\lbrace 1, 2, \\ldots , K\\rbrace $ be the subject-specific component indicators, $E_z= \\lbrace i: Z_i =z\\rbrace $ the set of subjects in component $j$ , $\\mathcal {C} = \\lbrace C_j: |C_j|>0\\rbrace $ the partition of $N$ subjects induced by $=\\lbrace Z_i, i=1, \\ldots , N\\rbrace $ ; Note the partition $$ is invariant to component relabeling.", "Let $T =|\\mathcal {C}|$ be the number of clusters formed by the $N$ subjects; it may differ from $K$ , the number of components for the population.", "Further let $C \\in \\mathcal {C}$ denote one of the clusters in partition $\\mathcal {C}$ ; let $j$ be the index associated with cluster $C_j$ , for $j \\in \\lbrace 1, \\ldots , T\\rbrace $ .", "Let $\\mathcal {C}_{-i} = \\lbrace C_j \\setminus \\lbrace i\\rbrace : |C_j \\setminus \\lbrace i\\rbrace |>0\\rbrace $ be the partition of subjects excluding subject $i$ .", "For simplicity, let $\\mathbf {Y}_{C} = \\lbrace \\mathbf {Y}_{i}, i\\in C\\rbrace $ be the collection of data in a cluster $C\\in \\mathcal {C}$ .", "Finally, let $_i$ be the latent state vector for subject $i=1, \\ldots , N$ , and $^*_j$ be the latent state vectors for cluster $j=1, \\ldots , T$ .", "We assume the indicators $$ are drawn as follows: ${\\sf Number~of~components:~~} K & \\sim p_K, \\\\{\\sf Mixing~weights:~~} _K & \\sim {\\sf Dirichlet}(\\gamma , \\ldots , \\gamma ),\\\\{\\sf Cluster~indicators:~~} Z_i & \\sim {\\sf Categorical}\\lbrace _K=(\\pi _1, \\ldots , \\pi _{K})\\rbrace , i = 1, \\ldots , N,$ where $p_K$ is a probability mass function over non-zero integers $\\lbrace 1,2, \\ldots \\rbrace $ and $\\gamma >0$ is the hyperparameter for symmetric $K$ -dimensional Dirichlet distribution.", "Note that though $\\tilde{K}\\le 2^M$ , $K$ is not upper bounded (unless constrained through the support of $p_K$ ).", "The prior of partition $\\mathcal {C}$ induced by (REF -) is $p(\\mathcal {C}\\mid \\gamma , p_K(\\cdot )) = V_N(T)\\prod _{C\\in \\mathcal {C}} \\gamma ^{(|C|)}, $ where $V_N(T) = \\sum _{k=1}^\\infty \\frac{k_{(T)}}{(\\gamma k)^{(N)}}p_K(k)$ , $T=|\\mathcal {C}|$ is the number of blocks/partitions for $N$ subjects and by convention $k^{(n)} = k\\cdot (k+1)\\cdots (k+n-1)$ , $k_{(n)} = k\\cdot (k-1)\\cdots (k-n+1)$ , and $k^{(0)}=k_{(0)}=1$ , $k_{(n)}=0$ if $k<n$ [39]." ], [ "Prior for $H^*$", "Given $\\lbrace Z_i\\rbrace $ , we draw the latent state vector $^*_j\\in \\lbrace 0,1\\rbrace ^M$ for which $Z_i=j$ indicates, referred to as “component-specific parameters\" in mixture models.", "We discuss priors for these discrete component parameters according as $$ is known or not.", "Pre-specified $$ .", "In applications such as Example 3, pre-specifying $$ is appealing when the scientific interest lies in itemized characterization of the population fractions for each element of $$ .", "Given $$ , the cluster membership indicators $\\lbrace Z_i\\rbrace $ take value from $\\lbrace 1, \\ldots , T\\rbrace $ where $T= \\tilde{K}$ .", "Existing approaches then assign to each cluster one of $\\lbrace ^*_1, \\ldots , ^*_T\\rbrace $ by enumerating the distinct known elements in $$ .", "For example, see [6] for $=\\lbrace 0,1\\rbrace ^M$ , $\\tilde{K}=2^M$ .", "[52] analyzed data from Example 3 and specified $= \\lbrace _1, \\ldots , _M,\\mathbf {0}_M\\rbrace $ among pneumonia cases that represents latent states as the lung infection caused by pathogen $1, 2, \\ldots , M$ or none-of-the-above and $_i=\\mathbf {0}_M$ among observed controls.", "Absent the uncertainty in $$ , simpler posterior sampling algorithms result.", "In practice, to avoid misleading estimates based on a pre-specified $$ subject to potential misspecification, analysts may conservatively specify $=\\lbrace 0,1\\rbrace ^M$ .", "However, $_i=^*_{Z_i}$ then take its value from a space that grows exponentially with $M$ (e.g., $M=30$ in Example 3).", "Consequently, upon fitting the model for inferring $\\pi _{k}, k=1,\\ldots , \\tilde{K}(=2^M)$ , although many elements in $$ may receive low posterior probabilities, none is exactly zero.", "Important elements in $$ are commonly selected by ad hoc thresholding.", "In addition, pre-specifying $\\subsetneqq \\lbrace 0,1\\rbrace ^M$ does not address the question of what are the distinct latent state patterns $\\tilde{}^*_j$ in the data.", "Unknown $$.", "Absent knowledge of $$ , we draw in a priori the component-specific parameters $H^*=\\lbrace \\eta _{jm}^*\\rbrace $ in two steps for regularizing $^*_j$ towards sparsity: ${\\sf probability~of~an~active~state:~~} p_{m} \\mid \\alpha _1, \\alpha _2 & \\sim {\\sf Beta}(\\alpha _1\\alpha _2/M, \\alpha _2), \\\\{\\sf latent~states:~~} \\eta ^*_{jm} \\mid p_m & \\sim {\\sf Bernoulli}(p_m), j=1, \\ldots , T,$ for $m=1, \\ldots , M$ .", "Note that it is possible that $^*_j = ^*_{j^{\\prime }}$ for some $j,j^{\\prime }=1, \\ldots , T$ where equality holds element-wise.", "For example, $^*_{Z_i}$ may equal $^*_{Z_i^{\\prime }}$ even if $Z_i\\ne Z_{i^{\\prime }}$ .", "Because we are interested in estimating distinct $^*_j$ 's that represent distinct values of scientific latent constructs, we will merge such clusters $j$ and $j^{\\prime }$ into one, referred to as a “scientific cluster\"; We denote it by $\\tilde{}$ .", "We also denote the unique values in $H^* = \\lbrace ^*_j, j=1, \\ldots , T\\rbrace $ by $\\tilde{H}^{*}=\\lbrace \\tilde{}^*_j, j=1, \\ldots , \\tilde{T}\\rbrace $ .", "Supplementary Material A.5 and A.6 further remarks on the induced priors on the partitions $$ and $\\tilde{}$ .", "The $K$ introduced in the prior specification is to make it not upper bounded and therefore differs from $\\tilde{K}$ .", "The latter represents the number of distinct latent state vectors in the population and must be no greater than $2^M$ .", "$\\tilde{}_k, k= 1, \\ldots , \\tilde{K}$ represent the set of true distinct latent state vectors in the population; while $^*_j, j=1, \\ldots , T$ ($T\\le K$ ) represent the realized latent state vectors that are possibly duplicated in the data generating process () or the posterior sampling.", "With unconstrained $K$ , we are able to build on the algorithm of [39] that does not bound the number of mixture components.", "The resulting algorithm works for general mixture of finite mixture models with discrete component distributions (Section ).", "By Beta-Bernoulli conjugacy, we integrate $[H^* \\mid ][\\mid \\alpha _1, \\alpha _2]$ over $$ to obtain the marginal prior: $pr(H^*) = \\prod _{m=1}^M \\frac{(\\alpha _1\\alpha _2/M)\\Gamma (s_m +\\alpha _1\\alpha _2/M)\\Gamma (T-s_m+\\alpha _2)}{\\Gamma (T+\\alpha _2+\\alpha _1/M)},$ where $\\Gamma (\\bullet )$ is the Gamma function and $s_m = \\sum _{m=1}^T \\eta ^*_{jm}$ , $j=1, \\ldots , M$ .", "Holding $\\alpha _2$ constant, the average number of positives among $^*_j$ decreases with $\\alpha _1$ ; Holding $\\alpha _1$ constant, the latent state vectors, $^*_j$ and $^*_{j^{\\prime }}$ , $j\\ne j^{\\prime }$ , become increasingly similar as $\\alpha _2$ decreases.", "In fact, the probability of two subjects with distinct cluster indicators $Z_i$ and $Z_{i^{\\prime }}$ have identical $m$ -th latent state, $[\\eta ^*_{im} = \\eta ^*_{i^{\\prime }m} \\mid Z_i = j, Z_{i^{\\prime }}=j^{\\prime }, j\\ne j^{\\prime }, \\alpha _1, \\alpha _2] = \\lbrace p_{m}^2+(1-p_m)^2 \\mid \\alpha _1,\\alpha _2\\rbrace = 1-2\\frac{\\alpha _1}{\\alpha _1+M}\\left(1-\\frac{\\alpha _1\\alpha _2+M}{\\alpha _1\\alpha _2+\\alpha _2M+M}\\right)$ approaches one when $\\alpha _2$ goes to zero.", "In what follows, $\\alpha _2$ is set to 1 which offers good clustering results in simulations and data analyses.", "Finally in applications where no pooling across $j$ is needed, one can set $p_m=0.5$ to specify uniform distribution over all possible patterns over $=\\lbrace 0,1\\rbrace ^M$ ." ], [ "Priors for Other Model Parameters", "We focus on the situation where $Q$ is completely unknown.", "Let $Q$ be uniformly distributed over the constrained space in $\\lbrace 0,1\\rbrace ^{M\\times L}$ defined by (REF ).", "In applications where $Q$ is not fully identifiable and/or encouraged to be different among its rows in finite samples, we specify sparsity priors for each column of $Q$ to encourage proteins to be specific to a small number of machines (see Supplementary Material A.6).", "We specify the priors for response probabilities $\\Lambda =\\lbrace \\lambda _{i\\ell }\\rbrace $ in (REF ) to satisfy the monotonic constraints in (REF ) as follows $\\psi _{\\ell ,v} & {\\sim } {\\sf Beta}(N_{\\psi } a_\\psi , N_{\\psi }(1-a_\\psi )), v = 1, \\ldots , K^-_\\ell , \\text{~constrained to~}\\Delta = \\left\\lbrace \\lbrace _{\\ell }\\rbrace : \\psi _{\\ell ,1} < \\ldots < \\psi _{\\ell ,K^-_\\ell }\\right\\rbrace , \\nonumber \\\\\\theta _{1}, \\ldots , \\theta _{L} & \\sim {\\sf Beta}(N_{\\theta } a_\\theta ,N_{\\theta }(1-a_\\theta ))\\lbrace (\\max _{1\\le v \\le K^-_\\ell }{\\psi _{\\ell , v}}, 1)\\rbrace , a_\\psi \\sim {\\sf Beta}(a_0, b_0), \\text{~and~} a_\\theta \\sim {\\sf Beta}(a_0^{\\prime }, b_0^{\\prime }),\\nonumber $ for $\\ell = 1, \\ldots , L$ , where $K^-_\\ell \\ge 1$ is the number of response probability parameters for latent classes $$ with $\\Gamma _{, \\ell }=0$ defined in (REF ) and the truncation of $\\theta _{\\ell }$ follows from the definition of RLCM (REF ).", "With ($a_\\theta $ , $a_\\psi $ ) unknown, the hierarchical priors on $$ and $\\lbrace _v\\rbrace $ propagate into the posterior and have the effect of shrinking the parameters towards a population value by sharing information across dimensions; ($N_\\theta $ , $N_{\\psi }$ ) can further be sampled in the posterior algorithm or fixed.", "When multi-parameter RLCMs specify particular parametric forms of the response probability for feature $\\ell $ (e.g., in (REF )), other sets of priors on the parameters may be readily incorporated into posterior sampling by modifying Step 4 in Supplementary Material C.1.", "Finally, we specify prior for hyperparameter $\\alpha _1$ in (REF ).", "One may specify a prior conjugate to $[H^* \\mid \\alpha _1]$ by $\\alpha _1 \\overset{d}{\\sim } {\\sf Gamma}(e_0,f_0)$ (shape and inverse scale parameterization with mean $e_0/f_0$ and variance $e_0/f_0^2$ ).", "Posterior sampling for non-conjugate prior for $\\alpha _1$ can also be carried out by sampling over a dense grid upon bounded reparameterization (see Step 5 in Supplementary Material C.1).", "Taken together, the likelihood and priors give the joint distribution of data $\\mathbf {Y}=\\lbrace \\mathbf {Y}_i\\rbrace $ , the true and false positive rates $$ and $$ , ${Q}$ matrix, and latent state vectors ${H} = \\lbrace _i\\rbrace $ (see Supplementary Material A.8)." ], [ "Posterior Inference", "We design posterior sampling algorithms to address three questions, 1) how many scientific clusters $(\\tilde{T})$ in the sample (data); 2) what are the latent state vectors $\\lbrace \\tilde{}^*_j, j=1, \\ldots , \\tilde{T}\\rbrace $ in the sample; and 3) what are the subjects' latent states $_i$ and the scientific clusters $\\tilde{}$ .", "Given $Q$ , $$ and $\\Psi $ , RLCM as a mixture model has discrete component-specific parameters $_i \\in $ .", "This is to be contrasted with mixture models with a continuous base measure from which component parameters are drawn to differ from one another with probability one.", "Therefore, when sampled conditional on other parameters, the discrete component parameters $\\lbrace ^*_j,j=1, \\ldots , T\\rbrace $ may be duplicated.", "Because we are interested in estimating scientific clusters with distinct latent states, we post-process the posterior samples by merging clusters in $$ associated with identical $^*_{j}$ at each MCMC iteration.", "Given $M$ , no more than $2^M$ distinct latent state vectors $\\tilde{}_j^*$ results after merging.", "More generally, for inference based on mixture of finite mixture (MFM) models with discrete component parameters, (REF ) uses a prior over all non-negative integers to remove the otherwise hard constraint $K = \\tilde{K} \\le 2^M$ (would be so if we force distinct latent states in the prior) and greatly simplify the design of posterior algorithms (see Remark REF ).", "We use Markov chain Monte Carlo (MCMC) algorithm for posterior inference which by design simulate samples that approximate the joint posterior distribution of unknown parameters and latent variables: $(, H^*, Q, , \\Psi , \\alpha _1)$ .", "See Supplementary Material C.1 for more details of the sampling algorithms and convergence checks.", "We discuss information from data that updates the clusters $$ .", "Gibbs updates of the partitions.", "Given our focus on estimating clusters, we choose to directly sample $\\mathcal {C}$ from its posterior without the need for considering component labels or empty components.", "A key step is to sample $$ based on an urn process that begins with one cluster comprised of all subjects (or a warm start informed by crude initial clusters) and re-assigns each subject to an old or new cluster [39].", "In sampling $\\lbrace Z_i\\rbrace $ one subject at a time, the full conditional distribution $[Z_i \\mid \\mathbf {Z}_{-i}, \\mathbf {Y}, , , Q, ]$ given cluster assignments for the rest $\\mathbf {Z}_{-i}=\\lbrace Z_{i^{\\prime }},i^{\\prime }\\ne i\\rbrace $ , other model parameters and data is proportional to the product of the conditional prior $pr(Z_i \\mid \\mathbf {Z}_{-i}, \\gamma )$ and the complete data likelihood integrated over latent states $[\\mathbf {Y} \\mid , , \\Psi , Q, ]$ (equivalent to conditional upon partition $\\mathcal {C}$ ignoring the labels).", "Because of exchangeability among subjects, we view subject $i$ as the last observation to be updated during a Gibbs step which assigns subject $i$ to an existing cluster $C \\in \\mathcal {C}_{-i}$ or a new cluster on its own with probabilities: $(Z_i = j \\mid -) & \\propto {\\left\\lbrace \\begin{array}{ll}(|C|+\\gamma )\\cdot \\frac{g(C\\cup \\lbrace i\\rbrace )}{g(C)}, &{\\sf ~if~} C\\in \\mathcal {C}_{-i}, j =1, \\ldots , |\\mathcal {C}_{-i}|, {\\sf ~or~} \\\\\\gamma \\frac{V_{N}(t+1)}{V_N(t)}\\cdot g(C), & {\\sf ~if~}C=\\lbrace i\\rbrace , j= |\\mathcal {C}_{-i}|+1,\\end{array}\\right.", "}$ where $g(C) = g(C; , \\Psi , Q, ) = \\prod _{\\ell =1}^L pr(\\lbrace Y_{i\\ell }: i \\in C\\rbrace \\mid , \\Psi , Q, )$ is the marginal likelihood for data in cluster $C$ (see (S4) in Supplementary Material B for an illustration using model (REF )).", "If adding subject $i$ to any existing cluster fits poorly with data $\\mathbf {Y}_{C}$ , i.e., knowing $\\mathbf {Y}_{C}$ tells little about $_i$ , low marginal likelihood ratio $\\frac{g(C\\cup \\lbrace i\\rbrace )}{g(C)g(\\lbrace i\\rbrace )}$ will result for any $C\\in _{-i}$ .", "The Gibbs update will favor forming a cluster of its own $\\lbrace i\\rbrace $ .", "Posterior summaries.", "We summarize the posterior distribution of partitions $[\\mid \\mathbf {Y}]$ by computing the empirical frequencies $\\hat{ \\pi }_{ii^{\\prime }}$ for every pair of subjects being clustered together, referred to as the posterior co-clustering probabilities $\\pi _{ii^{\\prime }}=(Z_i=Z_{i^{\\prime }} \\mid \\mathbf {Y})$ , for subjects $i$ , $i^{\\prime }=1, \\ldots , N$ .", "We compute a simple least square (LS) clustering $\\hat{}^{(LS)}$ on the basis of the squared distance from the posterior co-clustering probabilities, $\\arg \\min _{b} \\sum _{i,i^{\\prime }} \\left\\lbrace \\delta (Z_i^{(b)}, Z_{i^{\\prime }}^{(b)}) - \\hat{\\pi }_{ii^{\\prime }}\\right\\rbrace ^2$ , where $\\delta (a,a^{\\prime })=1$ if $a=a^{\\prime }$ and zero otherwise [9].", "RLCM has the salient feature of subject-specific discrete latent states $_i$ .", "However, the interpretation of $_i$ depends on $Q$ which is of scientific interest on its own in many applications.", "Based on the posterior samples obtained from a model with an unknown $Q$ , we select the iteration(s) $b^*$ with the mininum loss, $ \\min _{b^*} \\Vert Q^{(b^*)\\top } Q^{(b^*)}-\\frac{1}{B}\\sum _{b=1}^B Q^{(b)\\top } Q^{(b)}\\Vert _F$ where $\\Vert A\\Vert _F=\\sqrt{\\sum a^2_{ij}}$ is the matrix Frobenius norm.", "$Q^\\top Q$ is a $L$ by $L$ matrix invariant to relabeling of latent states.", "The $(\\ell , \\ell ^{\\prime })$ -th element of $Q^\\top Q$ represents the number of activated states at feature $\\ell $ when $\\ell =\\ell ^{\\prime }$ and the number of co-activated states at feature pair ($\\ell $ , $\\ell ^{\\prime }$ ) when $\\ell \\ne \\ell ^{\\prime }$ .", "Minimization of the least squares criterion therefore selects an iteration closest to the posterior means of all the co-activation counts.", "Turning to the inference of $_i$ , although in the original MCMC chain the subset of the $H^{*(b)}$ and $^{(b)}$ samples drawn along with $Q^{(b^*)}$ usefully approximate $[_i = ^*_{Z_i}, i=1, \\ldots , N\\mid Q=Q^{(b*)}, \\mathbf {Y}]$ , inferences of their functions enjoy reduced Monte Carlo errors through refitting a model with $Q=Q^{(b*)}$ that generate more posterior samples.", "Section REF further illustrates these use of the posterior summaries through detailed analyses of data from Example 1." ], [ "Results", "We illustrate the utility of RLCM on both simulated and real data.", "We focus on scenarios where $Q$ is unknown.", "First, we assess the performance of RLCM on estimating clusters under simulation scenarios corresponding to distinct levels of measurement errors, feature dimensions, sparsity levels of each machine, sample sizes, and population fractions of latent state patterns.", "Here the goal is to show that the proposed Bayesian RLCM performs clustering as well as or better than common alternative binary-data clustering methods.", "We first analyze a single randomly generated data set to highlight the differences among the methods.", "We then investigate the frequentist property of Bayesian RLCM in cluster estimation and compare it to other methods through repeated application of each method to replication data sets.", "Finally, data from Example 1 is analyzed, focusing on the posterior inferences of clusters, cluster-specific latent states and the estimated $Q$ -matrix." ], [ "Simulated Examples to Study Model Performance", "Simulation 1: More accurate clustering through feature selection in scientifically structured classes.", "$N=50$ independent observations are generated from an $L=100$ dimension multivariate binary distribution with $M=3$ machines.", "Here we randomly generated an $M$ by $L$ matrix $Q$ where each row has on average $s=20\\%$ non-zero elements.", "That is, $Q_{m\\ell }\\overset{i.i.d}{\\sim }{\\sf Bernouli}(0.2), \\ell =1, \\ldots , L$ ; In the rare event where a randomly generated $Q\\notin $ (identifiability constraint (REF )), we randomly permute pairs of elements in $Q_{m\\star }$ until $Q\\in $ .", "We draw latent states for each observation independently according to $_i \\overset{d}{\\sim }{\\sf Categorical}\\left(_0 = (1/6,1/6,1/6, 1/6,1/12,1/12,1/12, 1/12)\\right)$ where $_0 = \\lbrace (_i = {\\sf (0,0,0),(1,0,0),(0,1,0),(1,1,0),(0,0,1),(1,0,1),(0,1,1),(1,1,1)})\\rbrace .$ Here we focus on the two-parameter model ((REF ), DINO) which will be applied to Example 1 in Section REF .", "We assume the response probabilities shift between two levels $\\theta _\\ell = 0.8$ and $\\psi _\\ell =0.15$ .", "The distinct subsets of features where shifts occur define eight classes $\\tilde{K}=8=(2^M)$ , which upon enumeration by observation gives an $N$ by $L$ design matrix $\\Gamma $ .", "The resulting data $\\mathbf {Y}$ , the design matrix $\\Gamma $ , as well as the clusters obtained using complete-linkage, Hamming distance hierarchical clustering (HC), standard eight-class Bayesian latent class analysis (LCA, e.g., [14]), subset clustering analysis [26] and our Bayesian RLCM with unknown number of clusters fitted with truncation level $M^\\dagger =5$ can be seen in Figure REF .", "Specifically, for Bayesian LCA, RLCM and subset clustering [26], we plot the posterior co-clustering probability matrix $\\lbrace \\hat{\\pi }_{i,i^{\\prime }}\\rbrace $ for $N$ observations; For HC, we indicate co-clustering by filled cells.", "The true clusters are separated (dashed grids) and ordered according to the truth.", "Filled blocks on the main diagonal indicate perfect recovery of the true clusters.", "In this setting, HC is sensitive to noise and tends to split a true cluster (blank cells within the main diagonal blocks) or group observations from different true clusters (blue cells in the off-diagonal blocks).", "Unlike the Bayesian LCA and the subset clustering, the Bayesian RLCM automatically selects and filter subsets of features that distinguish eight classes (through scientific structures in (REF )) hence has superior clustering performance producing clusters that agrees quite well with the truth.", "This advantage of Bayesian RLCM relative to alternatives is maintained under data replications (see Simulation 2).", "Figure: In the 100-dimension multivariate binary data example, the eight classes differ with respect to subsets of measured features.", "Bayesian restricted latent class analysis accounts for measurement errors, selects the relevant feature subsets and filters the subsets by a low-dimensional model () and therefore yields superior clustering results.Compared to traditional all-feature methods under large dimensions, through the inference of all-zero columns of $Q$ ($\\lbrace \\ell : \\Gamma _{, \\ell }=0, \\forall \\in \\rbrace $ ), Bayesian RLCM removes irrelevant features hence reduces the impact of noise at less important features and in the current setting has better clustering performance (see Supplementary Material E for additional simulated examples on this point).", "Simulation 2: Assess clustering performance under various parameter settings.", "We simulated $R=60$ replication data sets for each of $1,920$ combinations of (#features, sample size, true positive rate,  false positive rate,  population fractions,  sparsity level of the rows of $Q$): $(L,N,\\theta _0,\\psi _0,_0, s )\\in \\lbrace 50,100,200,400\\rbrace \\otimes \\lbrace 50,100,200\\rbrace \\otimes \\lbrace 0.8,0.9\\rbrace \\otimes \\lbrace 0.05,0.15\\rbrace \\otimes \\lbrace _a = (\\frac{1}{8}, \\ldots , \\frac{1}{8}), _b = (\\frac{1}{6},\\ldots , \\frac{1}{6},\\frac{1}{12},\\ldots , \\frac{1}{12})\\rbrace \\otimes \\lbrace 10\\%,20\\%\\rbrace $ .", "The parameter values are designed to mimic what would be expected in Examples 1-3.", "We use adjusted Rand index [27] to assess the agreement between two clusterings, e.g,.", "the estimated and the true clusters.", "aRI is defined by ${\\sf aRI}(\\mathcal {C}, \\mathcal {C}^{\\prime }) = \\frac{\\sum _{r,c}{n_{rc}\\atopwithdelims ()2}-\\left[\\sum _r {n_{r\\cdot }\\atopwithdelims ()2}\\sum _c {n_{\\cdot c}\\atopwithdelims ()2}\\right]/{N\\atopwithdelims ()2}}{0.5\\left[\\sum _r {n_{r\\cdot }\\atopwithdelims ()2}+\\sum _c {n_{\\cdot c}\\atopwithdelims ()2}\\right]-\\left[\\sum _r {n_{r\\cdot }\\atopwithdelims ()2}\\sum _c {n_{\\cdot c}\\atopwithdelims ()2}\\right]/{N\\atopwithdelims ()2}},$ where $n_{rc}$ represents the number of observations placed in the $r$ th cluster of the first partition $\\mathcal {C}$ and in the $c$ th cluster of the second partition $\\mathcal {C}^{\\prime }$ , $\\sum _{r,c}{n_{rc}\\atopwithdelims ()2} (\\le 0.5\\left[\\sum _r {n_{r\\cdot }\\atopwithdelims ()2}+\\sum _c {n_{\\cdot c}\\atopwithdelims ()2}\\right])$ is the number of observation pairs placed in the same cluster in both partitions and $\\sum _r {n_{r\\cdot }\\atopwithdelims ()2}$ and $\\sum _c {n_{\\cdot c}\\atopwithdelims ()2}$ calculates the number of pairs placed in the same cluster for the first and the same cluster for second partition, respectively.", "aRI is bounded between $-1$ and 1 and corrects for chance agreement.", "It equals one for identical clusterings and is on average zero for two random partitions; larger values indicate better agreements between the two clustering methods.", "First we apply Bayesian RLCM to each replication data set and focus on studying its performance in recovering the true clusters (boxes with solid lines in Figure REF ).", "The clustering performance varies by the sparsity level $(s)$ in each machine, level of measurement errors $(\\theta _\\ell , \\psi _\\ell )$ , population fractions of latent classes $\\lbrace \\pi _{}, \\in \\rbrace $ and sample sizes $(N)$ .", "Given $s$ , a larger $L$ means a larger number of relevant features per machine and leads to better cluster recovery.", "In Figure S2 of Supplementary Materials (Figure REF here shows its 8 subplots), increasing $L$ from 50 to 400 (from the top to the bottom row), the mean aRI (averaged over replications) increases, e.g., in the first column, from $0.7$ to $0.98$ at the sparsity level $s=10\\%$ , $0.88$ to $0.99$ under $s=20\\%$ .", "More generally, clustering performance improves by increasing the sparsity level in each machine from $s=10\\%$ to $20\\%$ (compare the 1st and 3rd, 2nd and 4th RLCM boxplots with solid lines in each panel of Figure REF ).", "In the context of Example 1, given a fixed number of protein landmarks $L$ , patients will be more accurately clustered if each machine comprises more component proteins.", "This observation is also consistent with simulation studies conducted in the special case of $Q=I_L$ [26].", "We obtain more accurate cluster estimates under larger discrepancies between $\\theta _\\ell $ and $\\psi _\\ell $ .", "For $\\theta _0$ fixed at $0.8$ or $0.9$ , the mean aRI averaged over replications is higher under $\\psi _0=0.05$ than $\\psi _0=0.15$ over all combinations of the rest of parameters.", "Under the non-uniform population fraction $_0=_b$ , the clustering performance by Bayesian RLCM is similar or slightly worse than under a uniformly distributed population ($_a$ ).", "Finally, we observe mixed relative performances at distinct sample sizes as a result of two competing factors: more precise estimation of measurement error parameters under large sample sizes that improve clustering and a larger space of clusterings under a larger $N$ .", "Figure REF also shows better clustering performance of Bayesian RLCM (boxes with solid lines) relative to the three common alternatives (boxes with dotted lines).", "The Bayesian RLCM on average most accurately recovers the clusters compared to other methods.", "Bayesian RLCM produces the highest aRIs compared to others which are in many settings perfect (close to one).", "For example, the ratio of the mean aRIs (averaged over replications) for Bayesian RLCM relative to subset clustering is $2.06$ , $2.04$ , $1.88$ , $1.71$ for the sample-size-to-dimension ratios $N/P=1, 0.5,0.25,0.125$ , respectively (the leftmost group of four boxplots in Column 1, Figure S2 of Supplementary Materials $\\psi _0=0.05$ , $s=10\\%$ , $_0=_a$ ); The relative advantage of Bayesian RLCM and HC narrows under a higher false positive rate ($\\psi _0=0.15$ ) as shown by the smaller aRI ratios $1.23$ , $1.62$ , $1.49$ , $1.16$ (the leftmost group of four boxplots in Column Two, Figure S2).", "We remark on the performance of other three methods.", "Over all parameter settings investigated here, the traditional LCA performed the worst in the recovery of true clusters (aRI $< 0.68$ ).", "The likelihood function of subset clustering is a special case of RLCM that assumes a non-parsimonious $Q=I_L$ and therefore loses power for detecting clusters compared to RLCM that estimates a structured $Q$ with multiple non-zero elements in its rows.", "HC is fast and recovers the true clusters reasonably well (ranked second or first among the four methods more than two thirds of the parameter settings here; See Figure S3 in Supplementary Materials).", "The performance of HC is particularly good under a low level of measurement errors ($\\psi _0=0.05$ ) and a large number of relevant features per machine and sometimes performs much better than traditional LCA and subset clustering (e.g., $L=200$ , $N=50$ , $\\theta _\\ell =0.8$ , $\\psi _\\ell =0.05$ in Figure S2, Supplementary Materials).", "The HC studied here requires a pre-specified number of clusters to cut the dendrogram at an appropriate level and produces clusters that require separate methods for uncertainty assessment [48].", "The proposed Bayesian RLCM, in contrast, enjoys superior clustering performance and provides direct internal assessment of the uncertainty of clusters and measurement error parameters through the posterior distribution.", "Figure: Based on R=60R=60 replications for each parameter setting, Bayesian RLCM (boxplots with solid lines) most accurately recovers the true clusters compared to subset clustering (Hoff, 2005) hierarchical clustering (HC) and traditional Bayesian latent class analysis (LCA) (from the left to the right in each group of four boxplots).", "See Figure S2 in Suppmentary Materials for an expanded version over more parameter settings." ], [ "GEA Data, Preprocessing and Informative Priors", "Example 1 is about estimating autoimmune disease patient clusters via reconstructing components of protein complexes.", "Autoantibodies are the immune system's response to specific cellular protein complexes or “machines\".", "We seek to identify components of the machines and to quantify the variations in their occurrence among individuals.", "The binary responses $_i$ indicate the observed presence of autoantibodies at equi-spaced molecular weight landmarks as produced via a preprocessing method [51] implemented using publicly available software R package “spotgear\" (https://github.com/zhenkewu/spotgear).", "We ran 4 GEA gels, each loaded with IPs performed using sera from 19 different patients, and one reference lane.", "All sera were from scleroderma patients with cancer, and were all negative for the three most common autoantibodies found in scleroderma (anti-RNA polymerase III, anti-topoisomerase I, and anti-centromere).", "The IPs were loaded in random order on each gel; the reference sample is comprised of known molecules of defined sizes (molecular weights) and was always loaded in the first lane.", "The left panel in Figure REF shows for each sample lane (labeled in the left margin; excluding the reference lanes) the binary responses indicating the observed presence or absence of autoantibodies at $L=50$ landmarks.", "Patients differ in their antibody protein presence or absence patterns at the protein landmarks.", "Eleven out of $L=50$ aligned landmarks are absent among the patients tested.", "The rest of the landmarks are observed with prevalences between $1.3\\%$ and $94.7\\%$ .", "We apply two-parameter RLCM (REF ) with unknown $M(<L/2=50)$ and $Q$ , $$ , $$ .", "The GEA technologies are known to be highly specific and sensitive for nearly all proteins studied in this assay so we specify the priors for the true and false positive rates by ${\\sf Beta}(a_{\\theta \\ell },b_{\\theta \\ell })$ and ${\\sf Beta}(a_{\\psi \\ell },b_{\\psi \\ell })$ , $\\ell = 1, \\ldots , L$ respectively.", "We set $a_{\\theta \\ell }=9$ , $b_{\\theta \\ell }=1$ , $a_{\\psi \\ell }=1$ , $b_{\\psi \\ell }=99$ and conducted sensitivity analyses varying these hyperparameter values.", "Because proteins of distinct weights may have systematically different measurement errors, we choose not to share measurement error rates across dimension in this analysis.", "In our analysis, we sampled many $Q$ across iterations of MCMC.", "Because the interpretation of $_i$ depends on the row patterns in $Q$ , we condition on the least square clustering ($\\hat{}^{(LS)}$ ) and refit the model to obtain the least square $Q$ (Section ).", "The prior of $H^*$ (Section REF ) prevents overfitting by encouraging a small number of active latent states ($\\lbrace m: \\sum _{i}\\eta _{im}\\ne 0\\rbrace $ ) for small $\\alpha _1$ which in this analysis we draw its posterior samples for inference.", "In this application, the scientists had previously identified and independently verified through additional protein chemistry the importance of a small subset of protein bands in determining clusters.", "They proposed that these proteins should be grouped together.", "We therefore fitted the Bayesian RLCM without further splitting these partial clusters $^{(0)}$ so that the number of scientific clusters visited by the MCMC chain has an upper bound $\\tilde{T}^{(b)}\\le |^{(0)}|+N-\\sum _{j=1}^{|^{(0)}|}C^{(0)}_j$ , where $C^{(0)}_j$ counts the number of observations in the initial cluster $j$ .", "We fitted models and compared the results under multiple “working\" truncation levels $M^\\dagger =8,9,\\ldots , 15$ and obtained identical clustering results." ], [ "GEA Results", "Figure REF shows: the observations grouped by the RLCM-estimated clusters (not merged) $\\hat{}^{(LS)}$ (left), the estimated $Q$ -matrix $\\hat{Q}(\\hat{}^{(LS)})$ (right), and the marginal posterior probabilities of the machines $(\\eta _{im}=1 \\mid \\hat{}^{(LS)}, \\hat{Q}(\\hat{}^{(LS)}), \\mathbf {Y})$ (middle).", "The matrix $Q$ is estimated from the observed marginal associations (positive or negative) among the protein landmarks.", "Landmark protein pairs observed with positive association tend to be placed in the same estimated machine.", "For example, Landmarks 4, 7 and 8 appear together in Machine 5.", "Subjects either have all three landmarks or none at all, which induces strong positive pairwise associations among these landmarks.", "Indeed, the estimated log odds ratio (LOR) is $3.13$ (standard error $1.16$ ) for Landmark 4 versus 7, $2.21$ (s.e., $0.98$ ) for Landmark 4 versus 8, and $2.92$ (s.e.", "$1.2$ ) for Landmark 7 versus 8.", "The observed negative marginal associations between two landmarks suggest existence of machines with discordant landmarks.", "For example, Landmarks 10 and 27 are rarely estimated to be present or absent together in a subject as a result of 1) estimated machines with discordant landmarks and 2) subject-specific machine assignments.", "First, the model estimated that Landmark 10 (in Machine Set A: 1, 3 and 4) belongs to machines not having Landmark 27 (it is in Machine Set B: 2).", "Second, with high posterior probabilities, most observations have machines from one of, not both Set A and B hence creating discordance (high posterior probability $(\\Gamma _{i,10}\\ne \\Gamma _{i,27}\\mid \\mathbf {Y})$ ).", "In the presence of observation errors, strong negative marginal association results (observed LOR for Landmark 10 versus 27: $-1.98$ , s.e.", "$0.8$ ).", "Figure: Results for GEA data in Example 1.Left) Aligned data matrix for band presence or absence; row for 76 serum lanes, reordered into optimal estimated clusters (not merged) ^ (LS) \\hat{}^{(LS)} separated by gray horizontal lines “—–”; columns for L=50L=50 protein landmarks.", "A blue vertical line “ |\\mathbf {\\vert }\" indicates a band;Middle) lane-machine matrix for the probability of a lane (serum sample) having a particular machine.", "The blue cells correspond to high probability of having a machine in that column.", "Smaller probabilities are shown in lighter blue;.Right) The estimated machine profiles.", "Here seven estimated machines are shown, each with component proteins shown by a blue bar “ |\\vert \".Our algorithm also directly infers the number of scientific clusters in the data given an initial partial clustering $^{(0)}$ .", "The marginal posterior of the number of scientific clusters $\\tilde{T}$ can be approximated by empirical samples of $\\lbrace \\tilde{T}^{(b)}\\rbrace $ which result in a posterior median of 12 ($95\\%$ credible interval: $(8,16)$ ; Figure S4 in Supplementary Materials).", "The advantage of Bayesian RLCM is the posterior inference about both the clusters and the distinct latent state variables $_i$ interpreted based on the inferred $Q$ matrix.", "The middle panel of Figure REF shows that clusters differ in their marginal posterior probabilities of having each of the estimated machines.", "Among 76 subjects analyzed, 23 of them have greater than $95\\%$ marginal posterior probabilities of having both Machine 4 and 6.", "A group of seven observations are enriched with Machine 4 and 7 which as expected from the raw band patterns have distinctive combination of Landmarks 35, 40 and 49 (33, 27 and 18 kDa bands, respectively).", "Such inference about $_i$ is not available to us based on hierarchical clustering or traditional latent class models.", "We also fitted a Bayesian RLCM without the partial clusters $^{(0)}$ identified in prior work by the scientists.", "We estimated lower true positive rates so that it is more likely to observe negative protein landmarks within clusters partially identified by having a machine with a protein at that landmark.", "This makes the findings more difficult to interpret.", "As discussed in the simulation studies, clustering performance of Bayesian RLCM is poorer under lower sparsity levels $s=10\\%$ .", "As our scientific team recruits and tests more serum samples from their scleroderma patient cohort, samples with novel antibodies will improve inference about the measurement error parameters.", "This highlights the importance of using available prior knowledge about the measurement technologies in inferring latent states in finite samples [52].", "Figure S5 in Supplementary Materials compares for each landmark the prior and posterior distributions of the true and false positive rates.", "The discrepancies observed at many landmarks suggest the learning of measurement error parameters from the data.", "Other landmarks have similar prior and posterior distributions as a result of nearly flat likelihood function or absence of protein at that landmark so learning based only on likelihood is impossible.", "We performed posterior predictive checking to assess model fit [15].", "At each MCMC iteration, given the posterior sample of model parameters (without conditioning on the best clustering $\\hat{C}^{(LS)}$ or the best $\\hat{Q}$ ), we simulated a data set of the same size as the original set.", "For each replicated data set, we compute the marginal means and marginal pairwise log odds ratios ($0.5$ adjustment for zero counts).", "Across all replications, we compute the $95\\%$ posterior predictive confidence intervals (PPCI) defined by the $2.5\\%$ and $97.5\\%$ quantiles of the PPD.", "All the observed marginal means are covered by their respective PPCIs; The $95\\%$ PPCIs cover all but 24 of ${L \\atopwithdelims ()2}=1,225$ landmark pairs of observed pairwise log odds ratios (see Figure S6 and S7 in Supplementary Materials).", "The proposed model adequately fits the GEA data.", "There are potential improvements in our analysis.", "The posterior predictive probabilities (PPP) of observing a more extreme log odds ratio in future data $({\\sf LOR_{1,2}}(\\mathbf {Y}^{\\sf rep}) < {\\sf LOR_{1,2}}(\\mathbf {Y}) \\mid \\mathbf {Y})$ are between $0.004$ and $0.024$ .", "Most of these misfits of marginal log odds ratio occurred for landmark pairs with an observed marginal two-way table with small cell counts.", "Because the Bayesian RLCM treats the zeros as random, if these zero cells correspond to impossible combinations of proteins, or structural zeros, it may overestimate the probability for these cells; See [36] for a truncated extension of traditional latent class models that can be adapted to address the structural zero issue.", "On the other hand, the neighboring Landmarks 1 and 2 have an observed log odds ratio of $-1.17$ (s.e.", "$0.48$ ) with PPP $0.011$ .", "The two landmarks compete for being aligned with an observed band during pre-processing [51] hence creating negative dependence even within a latent class.", "Deviation from local independence can be further accounted for by explicitly modeling local dependence structure, discussed elsewhere, e.g., by nesting subclasses within each class [53]." ], [ "Discussion", "Modern scientific technologies give rise to measurements of varying precision and accuracy that are better targeted at the underlying state variables than ever before.", "In this paper we have discussed Bayesian restricted latent class model for analyzing multivariate binary data in the presence of between-class differential errors.", "The focus has been on the clustering of observations with unknown number of clusters, uncertainty assessment of the clustering and the prediction of individual latent states.", "The proposed method is motivated by clustering autoimmune disease patients based on their antibody presence or absence in sera where it is scientifically meaningful to restrict the values of response probabilities among latent classes.", "We have compared the proposed method with variants of latent class models through their specifications in Table S1 in Supplementary Materials and illustrated its advantage through simulations relative to three commonly used binary-data clustering.", "The Bayesian RLCM performs what we have called scientifically-structured clustering.", "It automatically selects subset of features for each latent class and filters them through a low dimensional model to improve our ability to accurately estimate clusters.", "Though the present paper focused on demonstrating the method through an example in medicine, the developed method and algorithms apply to many problems including Example 2 and 3 (Section ).", "RLCMs decompose the variation among multivariate binary responses into structure that reflects prior scientific knowledge and stochastic variation without a known explanation.", "In Example 1, it is certainly likely that there is some variability related to the vagaries of the measurement assay.", "However, it is also highly likely that there are systematic biological and biochemical processes not included in the structural part because they are unknown to us today.", "RLCM analyses can be a useful tool in the effort to uncover the unknown structure.", "One approach would be to show that the latent classes are diagnostic of specific diseases.", "Another is that we might uncover a novel mechanism by defining distinct patterns of the same autoantigen machine in patients with the same disease or potentially in patients with different diseases that target the same machines.", "This paper has focused on developing and applying RLCMs and algorithms to identify clusters and estimate subject-specific latent states.", "However, applied to public health research (e.g., pneumonia etiology research in Example 3), RLCM analyses more often focus on population quantities such as $\\Pi =\\lbrace \\pi _{}, \\sum _{j}\\pi _{} = 1, _{}\\ge 0, \\in \\lbrace 0,1\\rbrace ^M\\rbrace $ an $M$ -way contingency table characterizing the population frequencies of the latent state vector $_i$ .", "Further research into flexible and parsimonious parameterization of $\\Pi $ and its regression formulation in RLCMs are warranted.", "For example, quadratic exponential family [57] with negative second-order natural parameters assigns higher probabilities for $$ comprised of few ones or use another level of latent Gaussian variables to induce flexible dependence among $_i$ [56].", "We are currently studying a few potentially useful model extensions.", "First, nested partially LCMs [53] incorporate local dependence and multiple sensitivity parameters $(K^+>1)$ that would improve the utility of Bayesian RLCMs as well.", "Second, because the algorithm involves iterating over subjects to find clusters in (REF ), the computational time increases with the number of subjects $N$ .", "Divide-Cluster-Combine schemes that estimate clusters in subsamples which are then combined may improve the computational speed at the expense of the approximation introduced by the multi-stage clustering [40].", "Finally, in applications where the clustering of multivariate binary data comprises an important component of a hierarchical Bayesian model with multiple components, the posterior uncertainty in clustering propagates into other parts of the model and can be integrated into posterior inference of other model parameters [28]." ], [ "Software Availability", " All model estimations are performed by an R package “rewind\", which is freely available at https://github.com/zhenkewu/rewind.", "The supplementary materials contain referenced figures, a table, remarks, and further technical details, e.g., on identifiability and sampling algorithms, as well as additional simulations and extended data analysis results.", "The research is supported in part by a gift from the Jerome L. Greene Foundation and by the Patient-Centered Outcomes Research Institute (PCORI) Award (ME-1408-20318), National Institutes of Health (NIH) grants R01 AR073208, P30 AR070254 and P30 CA-046592 (ZW, Cancer Center Support Grant (CCSG) Development Funds from University of Michigan Comprehensive Cancer Center (UMCCC)).", "We also thank Gongjun Xu, Peter Hoff and Jian Kang for their insightful comments.", "Supplementary Materials for “A Bayesian Approach to Restricted Latent Class Models for Scientifically-Structured Clustering of Multivariate Binary Outcomes\" -1 The supplementary materials contain referenced remarks, figures and a table in Main Paper, and further technical details, e.g., on identifiability and sampling algorithms, as well as additional simulations and extended data analysis results.", "In particular, Section A contains remarks, Section B illustrates the calculation of marginal likelihood central to the posterior sampling of clusters ((15) in Main Paper), Section C details the posterior algorithms for pre-specified $M$ (Section C.1) and infinite $M$ (Section C.2), respectively.", "Section D briefly summarizes useful theoretical identifiability conditions for RLCMs based on [21].", "Section E illustrates through simulations the benefit of removing irrelevant features.", "Finally, Section F collects a table for variants of LCMs as well as figures for model results on the data analysis in Main Paper." ], [ "On Extending Prior of $H$ to {{formula:53861d99-fb39-44a4-af6e-c34775268260}}", "In Main Paper, we have focused on models with a finite number of latent states with $M=M^\\dagger $ typically set to a number that is large enough for the particular applications.", "In the MCMC sampling (Supplementary Material REF ), not all of the “working\" $M^\\dagger $ states will be used by the observations.", "The active number of states is usually strictly smaller than $M^\\dagger $ based on simulations.", "We extend to infinite $M$ to obtain a prior for $H^*$ under infinite dimension of latent state vectors ($_i$ ).", "We take $M$ in (14) in Main Paper to infinity and obtain infinite-column prior for $H$ (through a prior on $H^*$ in Section 2.6.2 in Main Paper); This construction defines the infinite Indian Buffet process [17].", "Supplementary Material REF provides posterior sampling algorithms for dealing with an infinite number of latent states by a novel slice sampler without the need of truncation [49]." ], [ "RLCM Connection to {{cite:5b76e9a1c603e12c0a08159b0dd88931833d3b87}}", "Setting $Q=I_{L\\times L}$ and $_i \\in = \\lbrace 0,1\\rbrace ^L$ (i.e., $M=L$ ) gives “mixture of Bernouli products\" with each latent class (defined by $_i$ ) having relevant features at possibly overlapping subsets of features $_{} = \\lbrace \\ell : \\Gamma _{,\\ell }=1\\rbrace $ , $\\in $ [26].", "[26] assumes the positive response probability $\\lambda _{i\\ell } = \\left\\lbrace \\theta _{\\ell ,v}\\right\\rbrace ^{\\Gamma _{i\\ell }}(\\psi _{\\ell })^{1-\\Gamma _{i\\ell }}$ , where $\\Gamma _{i\\ell }= \\eta _{i\\ell }$ given $Q=I_{L\\times L}$ and the multiple true positive rates $\\lbrace \\theta _{\\ell ,v}\\rbrace $ are greater than a single false positive rate $\\psi _\\ell $ , for $\\ell =1, \\ldots , L$ .", "This model can be written into a RLCM form with $K^+=1$ and $K^-\\ge 1$ by reparametrization: $\\Gamma ^*_{i\\ell }=1-\\Gamma _{i\\ell }$ , $\\psi ^*_{\\ell ,v} = 1-\\theta _{\\ell ,v}$ and $\\theta ^*_{\\ell }=1-\\psi _{\\ell }$ and relabeling of the outcomes $Y^*_{i\\ell }=1-Y_{i\\ell }$ .", "Indeed, the positive response probability under relabeling and reparameterization is $\\lambda ^*_{i\\ell } = (Y^*_{i\\ell }=1 \\mid - ) = 1-(Y_{i\\ell }=1 \\mid - ) = 1-\\lambda _{i\\ell } = \\left\\lbrace \\psi ^*_{\\ell ,v}\\right\\rbrace ^{1-\\Gamma ^*_{i\\ell }}(\\theta ^*_{\\ell })^{\\Gamma ^*_{i\\ell }}$ ." ], [ "Additional Identifiability Considerations for Designing Posterior Algorithms", "We now turn to inferring subject-specific latent state vectors $ H = \\lbrace _i\\rbrace $ based on complete-data likelihood $[\\lbrace _i\\rbrace \\mid H, \\Lambda ,Q]$ .", "Even given $Q$ , conditions for identifying $H$ exist but may fall short of ensuring consistent estimation of $H$ because the number of unknowns in $H$ diverges as the sample size increases.", "For example, it requires extra conditions that the number of measurements $L$ increases with the sample size [7].", "In finite samples and dimensions, we address this issue in a Bayesian framework by in a priori encouraging $H$ to be of low complexity, i.e., few clusters of distinct and sparse latent state vectors $\\lbrace _i\\rbrace $ , which combined with data likelihood will by design tend to concentrate the posterior at such low-complexity $H$ .", "In addition, when the latent space $\\subsetneqq \\lbrace 0,1\\rbrace ^M$ , general identifiability theory for $Q$ depends on the identifiability of $\\Gamma $ , the structure of which then determines the set of $Q$ s that are identifiable from the observed data distribution.", "Some RLCMs motivate our posterior algorithm design.", "For example, in two-parameter RLCMs, if two latent states are either always present or absent at the same time (“partners\"), it is impossible for the likelihood alone to distinguish it from a model that combines the two latent states.", "In our posterior algorithm, we therefore merge such “partner\" latent states if present at some iterations and the corresponding rows in $Q$ (Step 3, Supplementary Material REF ).", "As another example, two latent states can form a hierarchical structure, that is, one latent state cannot be present unless the other is.", "Suppose the second latent state require the first latent state, then $Q_{2\\ast }$ values at $\\lbrace \\ell : Q_{1\\ell }=1\\rbrace $ can be zero or one without altering the model likelihood.", "The sparsity priors on $H$ and the rows of $Q$ constraining $\\sum _{\\ell } Q_{m\\ell }$ therefore concentrate the posterior distributions of $H$ and $Q$ towards low-dimensional latent states and a smaller number of rows in $Q$ (Section 2.6.2 in Main Paper)." ], [ "Prior information about $\\Lambda $ .", "In applications where prior information about a subset of response probabilities $\\Lambda $ is available, it is essential to integrate the informative priors into model estimation if strict or generic identifiabilities do not hold [22], [52].", "The sufficient conditions (C1) and (C2) in Main Paper ensure identifiability of $Q$ with completely unknown $(\\Lambda , _{\\tilde{K}})$ .", "Otherwise, absent likelihood-based identifiability of $Q$ and other parameters, prior information about $\\Lambda $ alleviates the non-identifiability issue by concentrating the posterior at parameter values that better explain the observed data in light of the informative priors.", "In general non-identified models, the uncertainty in the prior will propagate into the posterior and will not vanish even as the sample size approaches infinity [32]." ], [ "Prior for Partition $$", "The prior distribution $p(\\mathcal {C}\\mid \\gamma , p_K(\\cdot ))$ is an exchangeable partition probability function [44], because it only symmetrically depends on the sizes of each block of the partition $\\lbrace |_j|: _j \\in \\rbrace $ .", "[39] also derives an urn process for generating partitions $\\mathcal {C}_1, \\mathcal {C}_2, \\ldots , $ such that the probability mass function for $_N$ is given by $p(\\mathcal {C}\\mid \\gamma , p_K(\\cdot )) = V_N(T)\\prod _{C\\in \\mathcal {C}} \\gamma ^{(|C|)},$ ; we will use this urn process for Gibbs updates of $\\lbrace Z_i\\rbrace $ one subject at a time in (17) in Main Paper.", "Note that the mapping from $\\mathbf {Z}$ to $$ is many-to-one with each $$ corresponding to ${ K \\atopwithdelims ()T}T!$ distinct $\\mathbf {Z}$ that differ by relabeling.", "Starting from a prior for partition $$ then followed by drawing component-specific parameters from their prior distributions is particularly fruitful in product partition models [24]." ], [ "On Merging Clusters with Identical Discrete Latent States", "At each MCMC iteration, two observations falling in distinct clusters ($Z_{i}\\ne Z_{i^{\\prime }}$ ) might have identical latent states, i.e., $^*_{Z_i} = ^*_{Z_i^{\\prime }}$ where the equality holds elementwise.", "At each iteration, we use unique multivariate binary vectors among all subjects $H = \\lbrace _i = ^*_{Z_i}, i=1,\\ldots , N\\rbrace $ to define “scientific clusters\" $\\tilde{\\mathcal {C}}$ through merging clusters associated with identical latent states.", "That is, $\\tilde{\\mathcal {C}}= \\left\\lbrace \\lbrace i: _i = \\tilde{}^*_j\\rbrace , j = 1, \\ldots , \\tilde{T} \\right\\rbrace $ where $\\lbrace \\tilde{}^*_j, j=1,\\ldots ,\\tilde{T}\\rbrace $ collects $\\tilde{T}(\\le T)$ unique patterns among $\\lbrace ^*_j, j=1, \\ldots , T\\rbrace $ .", "Let $\\mathcal {M}: \\lbrace ^*_{Z_i}, i=1, \\ldots , N\\rbrace \\mapsto \\tilde{\\mathcal {C}}$ represent this merge operation, i.e., $\\tilde{} = (\\lbrace ^*_j\\rbrace ,\\lbrace Z_i\\rbrace )$ .", "As detailed in Section 3 in Main Paper, we first build on Gibbs updates (15) and split-merge updates [29] to efficiently sample $\\lbrace Z_i\\rbrace $ from its posterior distribution.", "Given $\\lbrace Z_i\\rbrace $ , we then update $H^*=\\lbrace ^*_j\\rbrace $ and merge clusters $$ to obtain $\\tilde{}$ via the mapping $$ .", "Define partial ordering $``\\preceq \"$ over partitions $_1 \\preceq _2$ if for any $C_1 \\in _1$ , one can find a $C_2 \\in _2$ satisfying $C_1\\subseteq C_2$ .", "We have $\\preceq \\tilde{}$ , i.e., $\\tilde{\\mathcal {C}}$ is coarser than $\\mathcal {C}$ .", "Our procedure for obtaining clusters $\\tilde{}$ differs from mixture models where distinct $Z_i$ values with probability one correspond to distinct component parameters sampled from a continuous base measure [39].", "$\\tilde{\\mathcal {C}} = \\mathcal {C}$ is implicitly assumed in [26] under a Dirichlet process mixture model.", "We specify priors on $K$ that represents the distinct values that $\\lbrace Z_i\\rbrace $ can take and a prior on $H^* = \\lbrace ^*_j,j=1, \\ldots , T\\rbrace $ , which together induce a prior for $\\tilde{\\mathcal {C}}$ via $p(\\tilde{\\mathcal {C}}\\mid \\alpha _1, \\gamma ) & = \\sum _{: \\preceq \\tilde{} }p(\\tilde{}\\mid , \\alpha )\\cdot p(\\mid \\gamma ) \\\\& =\\sum _{: \\preceq \\tilde{}} {2^M \\atopwithdelims ()\\tilde{T}} (\\tilde{T})!", "\\left\\lbrace \\int p(H^*\\mid , )p(\\mid \\alpha _1) \\mathrm {d}\\right\\rbrace \\cdot p(\\mid \\gamma )\\cdot T!,$ where $= \\lbrace S_1, \\ldots , S_T\\rbrace $ is a ordered partition of $N$ subjects, obtained by randomly ordering parts or blocks of $$ uniformly over $T!$ possible choices and $p(\\mid \\gamma ) \\cdot T!", "= p(\\mid \\gamma )$ .", "The prior for the number of components $K$ serves to regularize the number of clusters $T = |\\mathcal {C}|$ among observed subjects (see [39]).", "Because $\\tilde{}$ is coarser than $\\mathcal {C}$ , a exponentially decaying prior on $K$ then encourages a small number of scientific clusters $\\tilde{}$ among $N$ subjects which results in using fewer component specific parameters to fit finite samples and improves estimation of unknown $H^*$ and $Q$ ." ], [ "On Prior for $Q$", "In applications where $Q$ is not fully identifiable or encouraged to be different among its rows, we specify sparsity priors for each column of $Q$ to encourage proteins to be specific to a small number of machines.", "That is, $(Q_{m\\ell } \\mid \\lbrace Q_{m^{\\prime },\\ell },m^{\\prime }\\ne m\\rbrace , \\zeta ) = 1/\\left\\lbrace 1+\\exp \\left\\lbrace - \\zeta \\sum _{1\\le m^{\\prime } < m^{\\prime \\prime } \\le M^*}Q_{m^{\\prime }\\ell }Q_{m^{\\prime \\prime }\\ell }\\right\\rbrace \\right\\rbrace $ , where $ \\zeta $ is the canonical parameter characterizing the strength and direction of interactions among $m$ .", "We either fix $\\zeta $ to be a negative number, or specify a hyperprior for $\\zeta $ ; In this paper, we fix $\\zeta =0$ ." ], [ "Joint Distribution", "The joint distribution of data $\\mathbf {Y}=\\lbrace \\mathbf {Y}_i\\rbrace $ , true and false positive rates $$ and $$ , ${Q}$ matrix, and latent state vectors ${H} = \\lbrace _i\\rbrace $ , denoted by ${pr(,{H = H(H^*,)},{Q}, , )}$ , is $&\\left\\lbrace \\prod _{i=1}^N \\prod _{\\ell =1}^L \\left[\\Gamma _{_i, \\ell } \\theta _{\\ell }^{Y_{i\\ell }}(1-\\theta _{\\ell })^{1-Y_{i\\ell }}+(1-\\Gamma _{_i, \\ell })\\psi _{\\ell , v_i}^{Y_{i\\ell }}(1-\\psi _{\\ell , v_i})^{1-Y_{i\\ell }}\\right]\\right\\rbrace \\nonumber \\\\&\\times \\prod _{\\ell =1}^L\\left[{\\sf TruncatedBeta}(\\theta _\\ell ; a_\\theta ,b_\\theta , (\\max _{1\\le v \\le K^-_\\ell }{\\psi _{\\ell v}}, 1)) \\prod _{v}{\\sf Beta}(\\psi _{\\ell v}; a_\\psi ,b_\\psi )\\mathbf {1}\\lbrace _\\ell \\in \\Delta \\rbrace \\right] \\cdot \\nonumber \\\\&\\times f(\\alpha _1)\\cdot {\\sf IBP}_M({H}^*;\\alpha _1, K)\\cdot (; \\gamma , p_K(\\cdot )), $ where $f(\\alpha _1)$ is the density function of the hyperprior of truncated IBP (to at most $M$ columns) parameter $\\alpha _1$ and $(; \\gamma , p_K(\\cdot ))$ is the prior in the space of partitions of observations." ], [ "On Posterior Summary Given a Pre-specified Q", "In applications where $Q$ is known (Example 3), we infer for each subject the probability of having a latent state pattern $$ , $(_i = \\mid \\mathbf {Y})$ , as estimated by the relative frequency of the event $_i = $ across MCMC iterations: $\\frac{1}{B}\\sum _{b=1}^B \\mathbf {1}\\lbrace _i^{(b)}=\\rbrace , \\forall \\in $ where $b$ indexes the stored MCMC samples obtained in Supplementary Material REF .", "Similarly, the posterior distribution for the total number of positive latent states $(\\sum _{m=1}^M\\eta _{im}=z \\mid \\mathbf {Y})$ is estimated by the empirical frequencies $\\frac{1}{B}\\sum _{b=1}^B \\mathbf {1}\\lbrace \\sum _{m=1}^M_{im}^{(b)}=z\\rbrace $ , $z=0, \\ldots , M$ , which in Example 3 represents the number of pathogens infecting the lung of a pneumonia child.", "To characterize the differential importance of each latent state among clusters, we also compute the posterior probability for $m$ -th state being positive $(^*_{(j)m}=1 \\mid \\lbrace Y_i\\rbrace )$ , $j=1, \\ldots , J^{\\prime }$ , for $J^{\\prime }$ largest clusters across MCMC iteration.", "Note that given $Q$ , no merging or relabeling is required as in Step 3 and 7 in Supplementary Material REF .", "The number of scientific clusters $\\tilde{K}$ can also be summarized by its empirical frequencies based on posterior samples." ], [ "Marginal Likelihood $g(C)$", "To illustrate the calculation of marginal likelihood $g(C)$ , we focus on two-parameter DINO model; see Remark for extensions to general restricted LCMs.", "Given assignment of subjects to clusters $$ , the model likelihood in a cluster $C_j\\in $ is $pr\\left(\\lbrace \\mathbf {Y}_i, i\\in C_j\\rbrace \\mid ^*_{j},\\Theta ,\\Psi , Q \\right) & = \\prod _{\\ell : \\xi _{j\\ell }=0} \\psi _{\\ell }^{n_{j\\ell 1}} \\left(1-\\psi _{\\ell }\\right)^{n_{j\\ell 0}}\\cdot \\prod _{\\ell : \\xi _{j\\ell }=1} \\theta _{\\ell }^{n_{j\\ell 1}} \\left(1-\\theta _{\\ell }\\right)^{n_{j\\ell 0}},$ where $n_{j\\ell 1} = \\sum _{i: Z_i=j} Y_{i\\ell }$ and $n_{j\\ell 0} = \\sum _{i: Z_i = j} (1-Y_{i\\ell })$ are the number of positive and negative responses at dimension $\\ell $ for subjects in cluster $C_j$ , and $\\xi _{j\\ell } = \\Gamma _{^*_j, \\ell } = 1-\\prod _{m=1}^M(1-\\eta ^*_{jm})^{Q_{m\\ell }}$ indicates the true status for $\\ell =1, \\ldots , L$ and the product over $\\ell $ is due to conditional independence given a cluster.", "We obtain the marginal likelihood $g(C)$ for cluster $C_j$ by integrating out latent states $^*_{j}$ in (REF ): $g(C) & = \\sum _{\\in \\lbrace 0,1\\rbrace ^M}pr\\left(\\lbrace \\mathbf {Y}_i, i\\in C_j\\rbrace \\mid ,\\Theta ,\\Psi , Q \\right)(^*_j = \\mid ),$ where $(^*_j = \\mid ) = \\prod _{m=1}^M p_m^{\\eta _m}(1-p_m)^{1-\\eta _m}$ .", "Note that $g(C)$ factorizes with respect to $\\ell $ when $M=L$ and $Q=I_{L \\times L}$ that leads to $\\xi _{j\\ell }=\\eta ^*_{j\\ell }$ .", "Computational considerations.", "One of the computational costs results from the summation under a large $M$ in (REF ), or “add\" operation over $\\in \\lbrace 0,1\\rbrace ^M$ .", "The factorization with respect to $\\ell $ allows the summations to be done for each $\\ell $ separately and therefore reduces the number of “add\" operations from $(2^M)$ to $(M)$ [26].", "More generally, $g(C)$ also factorizes with respect to blocks that partition $\\lbrace 1, \\ldots , M\\rbrace $ , $\\lbrace _u, u=1, \\ldots , U\\rbrace $ with $\\cup {_u}=\\lbrace 1, \\ldots , M\\rbrace $ when the corresponding row blocks of $Q$ are orthogonal ($\\check{Q}_u = \\vee _{m\\in _u}Q_{m\\star }$ , $u=1, \\ldots , U$ are orthogonal), resulting in reduced “add\" operations $(2^{\\max _u |_u|}L)$ .", "Given $Q$ , we use Reverse Cuthill-McKee (RCM) algorithm [8] for the $M$ by $M$ matrix $QQ^\\top $ to simultaneously rearrange its rows and columns to obtain this block structure.", "To generalize (REF ) from two-parameter models to general restricted LCMs, simply replace the first product with $\\prod _{\\ell : \\Gamma _{^*_j, \\ell }=0}\\left(\\psi _{\\ell , v(^*_i, \\ell )}\\right)^{n_{j\\ell 1}}\\left(1-\\psi _{\\ell , v(^*_i, \\ell )}\\right)^{n_{j\\ell 0}}$ ." ], [ "Pre-specified Latent State Dimension $M < \\infty $", "When the number of components $K$ is unknown, one class of techniques updates component-specific parameters along with $K$ .", "For example, the reversible-jump MCMC [20] works by an update to $K$ along with proposed updates to the model parameters which together are then accepted or rejected.", "However, designing good proposals for high-dimensional component parameters can be non-trivial.", "Alternative approaches include direct sampling of $K$[41], [37].", "Here we build on the algorithm of [39] for sampling clusters with discrete component parameters $^*_j$ .", "We focus on model (6) in Main Paper to illustrate the posterior algorithm.", "Initialization.", "Initialize all model parameters from prior distributions.", "When a $Q_{m\\star }$ is initialized to have redundant ones under high true positive rates, the likelihood of a sparse observation $_i$ is much lower under $\\eta _{im}=0$ than under $\\eta _{im}=1$ .", "Consequently, the sampling chain will visit $\\eta _{im}=0$ , i.e., inactive latent state $m$ , with high probability.", "To better initialize active latent states, we therefore use a more stringent data-driven initialization for $Q_{\\star \\ell }$ by $Q_{m\\ell } \\overset{d}{\\sim } {\\sf Bernoulli}(p), m = 1, \\ldots , M,$ only if many observations are positive at dimension $\\ell $ : $N^{-1}\\sum _{i}Y_{i\\ell } > \\tau _1$ , where $p$ and $\\tau _1$ can be prespecified.", "In our simulations and data analysis, we set $p=0.1$ and $\\tau _1=0.3$ .", "Split-merge update clusters $$ .", "The one-subject-at-a-time, Gibbs-type update is typically slow in exploring a large space of clusterings.", "In fact, the number of ways to partition $N$ subjects is $B_N$ , referred to as the Bell number and can be computed through the iterative formula $B_{N+1}= \\sum _{n=0}^N {N \\atopwithdelims ()n} B_{n}$ with $B_0=B_1=1$ resulting in $B_{50}>2^{157}$ .", "We remedy this by adding split-merge updates designed for conjugate models [29] that alter the cluster memberships for many subjects at once.", "Because the Gibbs update (15) in Main Paper assigns clusters one subject at a time and updates clusters in a local fashion resulting in potential slow mixing of the sampling chain for $$ , we use global updates to create or remove clusters for multiple subjects at a time that are likely to be accepted according to a Metropolis-Hastings ratio.", "We adapt an existing recipe designed for models with priors conjugate to the component-specific parameters [29], which uses split-merge updates to make global changes to cluster configuration followed by further refinement of clusters via Gibbs update one subject at a time.", "Given $$ , $$ , $Q$ and $\\mathbf {Y}$ , a single split-merge update comprises the following steps: Randomly choose two observations $i$ and $j$ from $N$ subjects; Let $S$ be the indices of subjects either belonging to $C_{Z_i}$ or $C_{Z_{j}}$ .", "Perform $r=5$ steps of intermediate Gibbs scan (17) restricted to observations in the same clusters as $i$ or $j$ .", "That is, use (17) to update observation $k\\in S\\setminus \\lbrace i,j\\rbrace $ with the constraint that $Z_k \\in \\lbrace Z_i,Z_j\\rbrace $ ; At the end of intermediate Gibbs scan, we obtain $\\mathbf {Z}^{\\sf \\scriptsize launch}$ .", "In this step, one assigns a subject $k$ in $S\\setminus \\lbrace i,j\\rbrace $ to either the cluster of $i$ or $j$ with probability $& (Z_k = z \\mid \\mathbf {Z}_{-k}, \\mathbf {Y}, {\\sf ~other~parameters~}) \\nonumber \\\\& = \\frac{(|C_{z}|+\\gamma )g(C_{z} \\cup \\lbrace k\\rbrace )/g(C_{z})}{(|C_{Z_i}|+\\gamma )g(C_{Z_i} \\cup \\lbrace k\\rbrace )/g(C_{Z_i})+(|C_{Z_j}|+\\gamma )g(C_{Z_j} \\cup \\lbrace k\\rbrace )/g(C_{Z_j})}, z\\in \\lbrace Z_i,Z_j\\rbrace ,$ Perform a final Gibbs scan restricted to observations $S\\setminus \\lbrace i,j\\rbrace $ using (REF ) and obtain updated clusters as the proposal states to be used in a Metroplis-Hasting step which we denote by $\\mathbf {Z}^{\\sf \\scriptsize cand}$ .", "We compute the proposal densities $q(\\mathbf {Z}^{\\sf \\scriptsize cand} \\mid \\mathbf {Z})$ and $q( \\mathbf {Z} \\mid \\mathbf {Z}^{\\sf \\scriptsize cand})$ ; For the non-trivial cases, the proposal densities depend on the random launch state $\\mathbf {Z}^{\\sf \\scriptsize launch}$ and are products of Gibbs update densities in (REF ).", "Accept or reject the proposed clustering $\\mathbf {Z}^{\\sf \\scriptsize cand} $ with acceptance probability computed from prior ratio (based on two sets of clusters induced by $\\mathbf {Z}^{\\sf \\scriptsize cand}$ vs $\\mathbf {Z}^{\\sf \\scriptsize launch}$ ), likelihood ratio (given clusters $\\mathbf {Z}^{\\sf \\scriptsize cand}$ vs $\\mathbf {Z}^{\\sf \\scriptsize launch}$ and other population parameters), ratio of proposal densities (from 1c).", "See [29] for the general recipe of computing the acceptance probability.", "Perform one complete Gibbs scan (17) of $\\mathbf {Z}$ for all individuals to refine the current state of cluster indicators.", "The above is referred to as $(5,1,1)$ split-merge update where 5 intermediate Gibbs scans are used to reach launch states $\\mathbf {Z}^{\\sf \\scriptsize launch}$ , one Metroplis-Hasting step to accept or reject a candidate clustering $\\mathbf {Z}^{\\sf \\scriptsize cand}$ , and one final complete Gibbs scan for all observations to refine the newly obtained cluster [29].", "Update individual machine usage profiles $H=\\lbrace {\\eta }_{im}\\rbrace $ .", "Because subjects within a cluster share latent states $_i = ^*_j$ , $i\\in \\lbrace i: Z_i = j\\rbrace $ for cluster $j=1, \\ldots , T$ , we sample from $[{}^*_{j} \\mid {\\sf others}] \\propto &~ \\prod _{m=1}^M\\lbrace p_m\\rbrace ^{\\eta ^*_{jm}}\\lbrace 1-p_m\\rbrace ^{1-\\eta ^*_{jm}}\\cdot \\prod _{\\ell : \\xi _{j\\ell }=0} \\psi _{\\ell }^{n_{j\\ell 1}} \\left(1-\\psi _{\\ell }\\right)^{n_{j\\ell 0}}\\cdot \\prod _{\\ell : \\xi _{j\\ell }=1} \\theta _{\\ell }^{n_{j\\ell 1}} \\left(1-\\theta _{\\ell }\\right)^{n_{j\\ell 0}},\\nonumber $ where $\\xi _{j\\ell }=\\Gamma _{^*_j, \\ell }$ indicates the active or inactive status at dimension $\\ell $ in cluster $C_j$ , $=\\lbrace p_m\\rbrace $ are within-cluster prevalence of $M$ latent states and $n_{j\\ell 1}=\\sum _{i: Z_i = j}Y_{i\\ell }$ and $n_{j\\ell 0}=\\sum _{i: Z_i = j}(1-Y_{i\\ell })$ .", "Because $_j^{*}\\in \\lbrace 0,1\\rbrace ^M$ , it is important to move around in this space fast.", "We currently use multinomial sampling in simplex $\\Delta ^{2^M-1}$ , which can be improved by either Hamming ball sampler or parallel tempering.", "We remark on “partner latent states\" that motivate merging a subset of rows in $Q^{(b)}$ .", "Let $H^{(b)} = \\lbrace \\eta ^{(b)}_{im}\\rbrace $ be an $N$ by $M$ binary matrix that collects latent states for all subjects at iteration $t$ .", "Let $M^{(b)}_{\\sf eff} = \\sum _{m=1}^{M} \\lbrace \\mathbf {1}^\\top _N H_{\\star m}^{(b)}\\ne 0\\rbrace $ be the number of nonzero columns in ${H}$ at $t$ -th MCMC iteration.", "The identifiability conditions apply only to the first $M^{(b)}_{\\sf eff}$ rows of $Q$ .", "Condition (C1) and (C3) hold at each iteration regardless of the value of $M^{(b)}_{\\sf eff}$ because $Q\\in \\mathcal {Q}$ truncated to first $M^{(b)}_{\\sf eff}$ rows remains in $\\mathcal {Q}$ .", "At each iteration, conditions (C1) and (C3) also hold if we collapse two identical columns $(m,m^{\\prime })$ of ${H}^{(b)}$ to combine two partner machines that are present or absent together among subjects ($\\eta _{im}^{(b)} = \\eta _{im^{\\prime }}^{(b)}$ , $i=1, \\ldots , N$ ); We set $H^{(b)}_{\\star m^{\\prime }}= \\mathbf {0}_{N}$ and the other row ${Q}^{(b)}_{m\\ell }= \\max \\lbrace Q^{(b)}_{m\\ell },Q^{(b)}_{m^{\\prime }\\ell }\\rbrace $ , $\\ell = 1, \\ldots , L$ .", "It is easy to verify that this scheme preserves conditions (C1) and (C3) and readily generalizes to cases where more than two columns of $H^{(b)}$ are identical.", "In the population, the diversity assumption $=\\lbrace 0,1\\rbrace ^M$ does not hold if two latent states always positive together.", "When external knowledge is available for two “partner\" states with separate known rows in $Q$ , it can be readily integrated into posterior sampling.", "Sample false positive rates from $[\\psi _\\ell \\mid {\\sf others} ] ~{\\sim } ~ {\\sf Beta}\\left(\\sum _{i}(1-\\xi _{i\\ell })Y_{i\\ell }+a_\\psi ,\\sum _{i}(1-\\xi _{i\\ell })(1-Y_{i\\ell })+b_\\psi \\right)\\lbrace (0,\\theta _\\ell )\\rbrace , \\ell = 1, \\ldots , L.$ Sample true positive rates from $[\\theta _\\ell \\mid {\\sf others} ]~{\\sim } ~{\\sf Beta}\\left(\\sum _{i}{\\xi _{i\\ell }Y_{i\\ell }}+a_\\theta ,\\sum _{i}\\xi _{i\\ell }(1-Y_{i\\ell })+b_\\theta \\right)\\lbrace (\\psi _\\ell ,1)\\rbrace , \\ell =1,\\ldots , L.$ We also implemented in “rewind\" specified upper bounds for $\\lbrace \\psi _\\ell \\rbrace $ and lower bounds for $\\lbrace \\theta _\\ell \\rbrace $ when needed.", "Update hyperparameter $\\alpha $ .", "Suppose the hyperprior for $\\alpha $ is $p(\\alpha )$ .", "Then by the marginal distribution of $H^*$ from finite-$M$ IBP [17], we reparametrize in terms of $\\beta = \\frac{\\alpha }{\\alpha +1} \\in (0,1)$ and obtain $[\\beta \\mid H^*] \\propto p(\\beta ) \\cdot \\left(\\frac{\\beta }{1-\\beta }\\right)^M \\prod _{m=1}^M\\frac{\\Gamma (s_m+\\beta /\\lbrace M(1-\\beta )\\rbrace )}{\\Gamma (T+1+\\beta /\\lbrace M(1-\\beta )\\rbrace ))},$ which can be sampled from a dense grid over $(0,1)$ and $s_m = \\sum _{j=1}^T \\eta ^*_{jm}$ is the number of clusters that $m$ -th latent state is positive.", "We use Beta distribution $\\beta \\sim {\\sf Beta}(a_\\beta ,b_\\beta )$ where $a_\\beta =b_\\beta =1$ in our simulations and data analyses.", "Update prevalence parameters $=\\lbrace p_1, \\ldots , p_m\\rbrace $ from $[\\mathbf {p}\\mid {\\sf others}] & \\propto ~ \\prod _{m=1}^M(p_m)^{n^*_{m1}}(1-p_m)^{n^*_{m0}} {\\sf Beta}(p_m; \\alpha /M,1),$ which we sample independently $p_m\\sim {\\sf Beta}(n^*_{m1}+\\alpha /M, n^*_{m0}+1)$ , $m=1, \\ldots , M$ .", "Update machine matrix $Q$ via constrained Gibbs sampler.", "Update to $Q_{m\\ell }^{(b)}$ , $\\ell = 1, 2, \\ldots , L$ , $m=1, 2, \\ldots , M$ under two mutually exclusive scenarios: Keep $Q_{m\\ell }^{(t-1)}$ if one of the three criteria holds: 1) $Q^{(t-1)}_{\\star \\ell } = \\mathbf {e}_m$ , 2) ${1}_{L}^\\top Q^{(t-1)}_{m,\\star }=3$ and $Q_{m\\ell }=1$ or 3) $Q_{m\\ell }^{(t-1)}=0$ , $Q_{\\star \\ell }^{(t-1)}=\\mathbf {e}_m$ and there are only two $\\mathbf {e}_m$ in the columns of $Q$ .", "Otherwise, flip $Q_{m\\ell }^{(t-1)}$ to a different value $z$ with probability $p(z\\mid {\\sf others})/(1-p(z\\mid {\\sf others}))$ , where $p(z\\mid {\\sf others})$ is the full conditional distribution $pr(Q_{m\\ell }=z \\mid {\\sf others}) &\\propto & \\prod _{i=1}^N pr\\left(Y_{i\\ell }\\mid \\lbrace \\mathbf {\\eta }_i\\rbrace , Q^{(b)}_{\\sf new},Q_{m\\ell }=z, Q_{\\sf old}^{(t-1)}, \\theta _\\ell , \\psi _\\ell \\right)\\nonumber \\\\& = & \\prod _{i: \\xi _{i\\ell }=1} \\theta _\\ell ^{n^{\\prime }_{1\\ell 1}}(1-\\theta _\\ell )^{n^{\\prime }_{1\\ell 0}}\\cdot \\prod _{i: \\xi _{i\\ell }=0} \\psi _\\ell ^{n^{\\prime }_{0\\ell 1}}(1-\\psi _\\ell )^{n^{\\prime }_{0\\ell 0}}, z=0,1, \\nonumber $ where $n^{\\prime }_{1\\ell 1} = \\sum _{i=1}^N \\xi _{i\\ell }Y_{i\\ell }$ , $n^{\\prime }_{1\\ell 0} = \\sum _{i=1}^N \\xi _{i\\ell }(1-Y_{i\\ell })$ , $n^{\\prime }_{0\\ell 1} = \\sum _{i=1}^N (1-\\xi _{i\\ell })Y_{i\\ell }$ , $n^{\\prime }_{0\\ell 0} = \\sum _{i=1}^N (1-\\xi _{i\\ell })(1-Y_{i\\ell })$ , and $Q^{(b)}_{\\sf new}$ and $Q_{\\sf old}^{(t-1)}$ represent entries of $Q$ that have and have not been updated, respectively.", "Permute the rows of $Q^{(b)}$ by natural ordering of binary codes $\\lbrace Q_{m\\star }, m=1, \\ldots , M\\rbrace $ represented in binary system.", "We order the rows of $Q^{(b)}$ by decreasing order of $M$ -dimensional vector $Q^{(b)}\\mathbf {v}$ where $\\mathbf {v} = (2^{L-1},2^{L-2}, \\ldots , 1)^\\top $ .", "We only do so after all the MCMC iterations.", "Condition (C1) guarantees that once $Q^\\top $ is written in left-ordered form [17], the bottom row of ${Q}$ corresponds to a row with a positive ideal response at the smallest dimension $\\ell _{(1)}=\\arg \\min _\\ell \\lbrace Q_{m\\ell }=1, \\forall m,\\ell \\rbrace $ , which if shared by more than one row, then the row having a postive ideal response at the second lowest dimension $\\ell _{(2)}=\\arg \\min _{\\ell :\\ell > \\ell _{(1)}} \\lbrace Q_{m\\ell }=1, \\forall m,\\ell \\rbrace $ is placed at the bottom row; this scheme of ordering the rows of $Q$ will always succeed according to (C1).", "Finally, suppose at iteration $s$ , the MCMC algorithm produces latent states unused by any observation: $^{\\sf non,(b)}=\\lbrace m^{\\prime }: \\sum _i \\eta ^{(b)}_{im^{\\prime }} =0\\rbrace $ .", "We reset to zeros the subset of rows of $Q$ corresponding to the unused latent states at an iteration.", "Given the sampled $^{(b)}_{i}$ , the corresponding set of rows $Q^{(b)}_{^{\\sf non}}=\\lbrace Q^{(b)}_{m\\star }, m\\in ^{\\sf non,(b)}\\rbrace $ does not enter likelihood.", "We re-initiate $Q^{(b)}_{^{\\sf non}}$ which upon sequential Gibbs scans create new machines that may enter and improve the likelihood at the next iteration.", "In our experiments, resetting $Q^{(b)}_{m\\star }$ side-steps the difficulty of splitting a sampled machine that is populated with too many ones.", "Resetting is also practically easier to implement compared to a fine-tuned split-merge algorithm applied to the rows of $Q$ in tandem with simulated annealing which are designed for a more complex time series segmentation tasks [13].", "Convergence checks.", "In simulations and data analysis, we ran three MCMC chains each with a burn-in period of $10,000$ iterations followed by $10,000$ iterations stored for posterior inference.", "We look for potential non-convergence in terms of Gelman-Rubin statistic [4] that compares between-chain and within-chain variances for each model parameter where a large difference ($R_c>1.1$ ) indicates non-convergence; We also used Geweke's diagnostic [16] that compare the observed mean for each unknown variable using the first $10\\%$ and the last $50\\%$ of the stored samples where a large $Z$ -score indicates non-convergence ($|Z|>2$ ).", "In our simulations and data analyses, we observed fast convergence (many satisfied convergence criteria within $2,000$ iterations) that led to well recovered clusters and $Q$ matrices (results not shown here)." ], [ "Algorithm under $M=\\infty $", "This section presents the algorithm without the need to pre-specify the exact or an upper bound of the number of factors $M$ .", "The algorithm adapts the slice sampler for infinite factor model [49] which performs adaptive truncation of the infinite model to finite dimensions and avoids approximation of the Indian Buffet Process (IBP) prior for $H^*$ .", "The algorithm builds on the semi-ordered representation of the IBP, where the probabilities of active states are non-ordered and the probabilities of inactive states truncated to a random number $M^0$ are ordered.", "We use this algorithm to infer the number of active states.", "0.", "Initialize the number of active states $M^+$ , the random truncation level for inactive states $M^0=0$ .", "Initialize $Q$ with an appropriate $M^*=M^++M^0$ by $L$ binary matrix; Initialize the IBP hyperparameter $\\alpha $ ; Initialize $p$ of length $M^*$ to be the vector of the probabilities for each state being used (if the initial $M^0=0$ as recommended, then $p$ needs not be ordered).", "Initiate $H^*$ as $(T_{\\max }+3)$ by $M_{\\max }$ matrix with all zeros, where $T_{\\max }$ and $M_{\\max }$ are the guessed maximum number of clusters and truncated number of states the algorithm will visit across iterations.", "Neither $T_{\\max }$ nor $M_{\\max }$ is introduced to approximate any probabilistic distribution: one can increase both numbers as appropriate at the expense of extra memory.", "Repeat steps 1 to 10 below for iterations $b=1, \\ldots , B$ : 1.", "Gibbs update cluster indicators $=\\lbrace Z_i, i=1, \\ldots , N\\rbrace $ and the cluster-specific sizes $|_j|, j=1, \\ldots , t$ , where $t$ is the number of unique values in $$ 2.", "For Iteration 1, update $H^*$ elementwise for $t\\cdot M^*$ elements corresponding to the currently non-empty clusters and the current truncation level $M^*$ for the number of factors; Otherwise, update $H^*$ by the full conditional distribution given other parameters including the slice variable $s$ : $& pr(\\eta ^*_{jm} = z \\mid {\\sf others}) \\propto \\frac{p_m}{p^+_{\\min }}\\times \\nonumber \\\\& \\prod _{m=1}^M\\lbrace p_m\\rbrace ^{\\eta ^*_{jm}}\\lbrace 1-p_m\\rbrace ^{1-\\eta ^*_{jm}}\\cdot \\prod _{\\ell : \\xi _{j\\ell }=0} \\psi _{\\ell }^{n_{j\\ell 1}} \\left(1-\\psi _{\\ell }\\right)^{n_{j\\ell 0}}\\cdot \\prod _{\\ell : \\xi _{c\\ell }=1} \\theta _{\\ell }^{n_{j\\ell 1}} \\left(1-\\theta _{\\ell }\\right)^{n_{j\\ell 0}},\\nonumber $ for $z=0,1$ , $m=1, \\ldots , M^+$ , where $p^+_{\\min }=p^+_{\\min }(H^*, \\lbrace p_{m},m=1, 2, \\ldots , \\rbrace )=\\min _{1\\le m \\le M^+}\\lbrace p_m\\rbrace $ depends on $\\eta ^*_{jm}$ and is the normalizing constant for the uniform distribution of the slice variable: $pr(s \\mid H^*,\\lbrace p_{m},m=1,2,\\ldots ,\\rbrace )=\\mathbf {1}_{\\lbrace 0\\le s \\le p^+_{\\min }\\rbrace }/p^+_{\\min }$ .", "For example, given $s$ one must set to zero any column $m\\in \\lbrace 1,\\ldots , M^+\\rbrace $ in $H^*$ , $\\lbrace \\eta ^*_{jm},j=1, \\ldots , t\\rbrace $ whenever $p_m < s$ .", "3.", "Update $Q$ matrix ($M^*$ by $L$ ) as in Step 6 in Section REF ; 4.", "Update the number of active factors ($M^+$ ) by finding the number of columns in $H^*$ with non-zero column sums.", "5.", "Update unordered $\\lbrace p_m,m=1, \\ldots , M^+\\rbrace $ by $p_m\\sim {\\sf Beta}(\\sum _{j=1}^t \\eta ^*_{jm}, 1+t-\\sum _{j=1}^t \\eta ^*_{jm})$ , $m=1, \\ldots , M^+$ ; 6.", "Update slice variable $s\\sim {\\sf Uniform}(0,\\min _m p_m)$ ; 7.", "Starting from $m=1$ , sample $p^0_{(m)} \\mid p^0_{(m-1)} \\sim \\exp \\lbrace \\alpha \\sum _{j=1}^t(1-p_{(m)}^0)^j\\rbrace (p^0_{(m)})^\\alpha (1-p^0_{(m)})^N\\cdot \\mathbf {1}_{\\lbrace 0\\le p^0_{(m)}\\le p^0_{(m-1)}\\rbrace },$ until $p^0_{(M^0+1)}<s$ , where $p^0_{(0)}=1$ .", "Use adaptive rejection sampling [18] to sample from this distribution iteratively for $m=1, \\ldots , M^0$ , where $M^0>0$ only when $p^0_{(1)} > s$ ; 8.", "If $M^0>0$ , update $p$ by concatenating the old $p$ and $p^0$ ; update $M^*=M^++M^0$ ; 9.", "Pad $H^*$ with $M^0$ columns of zeros to its right; Subset the rows of $Q$ to those $M^+$ factors and pad it with $M^0$ extra rows sampled from an appropriate initialization sampler; 10.", "Update other parameters $$ , $$ , $\\alpha $ as in Section REF ." ], [ "Likelihood-based identifiability conditions given $\\tilde{K}$ , {{formula:cdc858f6-287a-49ed-9cb7-3f9c64d8d299}} and {{formula:aa7c8ba2-f221-47a0-8841-65e67dabd0a6}} (or {{formula:9ab565ac-a478-4aca-afee-2818000c4178}} )", "Given $\\Gamma $ , [21] established that the separability of $\\Gamma $ is sufficient and necessary for identifying $_{\\tilde{K}}$ under two-parameter models for known conditional response probabilities $\\Lambda $ ; If $\\Gamma $ is inseparable, $_{\\tilde{K}}$ is identified up to equivalent classes defined by identical rows in $\\Gamma $ (in this paper, we transposed $\\Gamma $ used in [21]).", "When $\\Lambda $ is unknown, [21] established sufficient conditions for $_{\\tilde{K}}$ -partial identifiability (strictly identify $\\Lambda $ but identify $_{\\tilde{K}}$ up to equivalent classes defined by identical rows in $\\Gamma $ ).", "For $Q$ -restricted two-parameter models, if $$ is saturated and $\\Gamma $ is separable, then these conditions become minimal, i.e.", "sufficient and necessary conditions: 1) $\\ge 3$ items per latent state and 2) $Q=[I_M,Q_1^\\top ]$ where $Q_1$ has distinct columns.", "For multi-parameter models, separability of $\\Gamma $ is sufficient for identifying $_{\\tilde{K}}$ given known $\\Lambda $ .", "$_{\\tilde{K}}$ will be strictly identifiable given two technical conditions [21] - Condition (C3) implies separability of $\\Gamma $ which could be true for $Q$ -RLCM induced $\\Gamma $ with unsaturated $$ and without single-attribute items in $Q$ .", "They also established “generic identifiability\" results for $\\Lambda $ and $_{\\tilde{K}}$ when $\\Gamma $ is inseparable: as long as one can flip entries to satisfy two technical conditions.", "The notion of “generic identifiability\" is introduced, because the identifiability results for multi-parameter models hold except on a Lebesgue measure-zero set where the models are reduced to two-parameter models.", "For the special cases of $Q$ -restricted model (saturated), the two technical conditions do not require the $Q$ -matrix to contain an identity submatrix and provides a flexible new condition for generic identifiability under various $Q$ -matrix structures; the results are generically identifiable up to label swapping among those latent classes that have the same row vectors in the $\\Gamma $ -matrix." ], [ "Additional simulated example: removing irrelevant features reduces the noise and improves cluster estimation", "When $Q$ is unknown, the proposed method for scientifically structured clustering includes an additional step for sampling $Q$ .", "A zero column in $Q$ , say column $\\ell $ , indicates irrelevance of $\\ell $ -th dimension because all positive observations at that dimension will be false positives.", "By estimating which columns are zeros, our algorithm removes irrelevant features when clustering observations.", "Clustering multivariate binary data on a subset of features reduces the impact of noise introduced by less important features and therefore can be superior to all-feature clustering methods such as the standard latent class analysis.", "For example, in model (2) with $Q = I_{L\\times L}$ , irrelevant features $ ^c=\\lbrace \\ell : \\Gamma _{\\star \\ell }=\\mathbf {0}\\rbrace $ ideally would not enter likelihood ratio calculations when assigning observations to clusters.", "Indeed, let $R_{kk^{\\prime }}(=)$ be the log relative probabilities of assigning an observation $_i$ to cluster $k$ ($^{(k)}_{-i}$ ) versus $k^{\\prime }$ ($^{(k^{\\prime })}_{-i}$ ) given other parameters and clustering $_{-i}$ can be Taylor approximated by $R_{kk^{\\prime }}(_i) & \\approx \\log \\frac{|^{(k)}_{-i}|+\\gamma }{|^{(k^{\\prime })}_{-i}|+\\gamma }+\\sum _{\\ell =1}^L p_\\ell \\log \\left(\\frac{\\hat{\\theta }_{(k)\\ell }}{\\hat{\\theta }_{(k^{\\prime })\\ell }}\\right)^{Y_{i\\ell }} \\left(\\frac{1-\\hat{\\theta }_{(k)\\ell }}{1-\\hat{\\theta }_{(k^{\\prime })\\ell }}\\right)^{1-Y_{i\\ell }},$ where the terms corresponding to irrelevant features become negligible if $\\hat{\\theta }_{(k)\\ell } \\approx \\psi _{\\ell }$ .", "The response probabilities at irrelevant dimensions ($\\lbrace \\psi _\\ell : \\ell \\in ^c\\rbrace $ ) are nevertheless estimated with error and contribute to noise in assigning each observation to an existing cluster.", "$R_{kk^{\\prime }}() >0, =0, <0$ indicate assignment of observation $$ to cluster $k$ more, equally and less likely than to cluster $k^{\\prime }$ , respectively.", "Consider a triple of observations ($_1, _2, _3$ ) where the first (cluster $k^{\\prime }$ ) and the rest (cluster $k$ ) belong to two distinct clusters, respectively.", "The probability of clustering $_1$ and $_2$ into their respective true clusters is $p_{12} = (1-{\\sf expit}\\lbrace R_{kk^{\\prime }}(_1)\\rbrace ){\\sf expit}\\lbrace R_{kk^{\\prime }}(_2)\\rbrace $ ; the probability of assigning $_2$ and $_3$ into the same true cluster is $p_{23}={\\sf expit}\\lbrace R_{kk^{\\prime }}(_2)\\rbrace {\\sf expit}\\lbrace R_{kk^{\\prime }}(_3)\\rbrace $ .", "Here we have used lower case $_i$ to represent the sub-vector of $_i$ that entered the calculation in (REF ).", "We simulated $L_1=5$ relevant dimensions and $L_2=40$ irrelevant dimensions $^c = \\lbrace 6, \\ldots , 45\\rbrace $ .", "To mimic the noisy estimates of the response probabilities in cluster $k$ and $k^{\\prime }$ , we simulated $\\hat{\\theta }_{(k)\\ell } = (\\log , \\log )$ and $\\hat{\\theta }_{(k^{\\prime })\\ell } = (\\log ^{\\prime }, \\log ^{\\prime })$ where $r_{\\ell 1}, \\ldots , r_{\\ell , L_1}\\overset{d}{\\sim }{\\sf Beta}(0.1N_k,0.9N_k)$ , $r^{\\prime }_{\\ell 1}, \\ldots , r^{\\prime }_{\\ell , L_1}\\overset{d}{\\sim }{\\sf Beta}(0.9N_{k^{\\prime }},0.1N_{k^{\\prime }})$ and $\\epsilon _{\\ell 1}, \\ldots , r_{\\ell , L_2}\\overset{\\sf iid}{\\sim }{\\sf Beta}(0.1N_k,0.9N_k)$ and $\\epsilon ^{\\prime }_{\\ell 1}, \\ldots , \\epsilon ^{\\prime }_{\\ell , L_2}\\overset{\\sf iid}{\\sim }{\\sf Beta}(0.1N_{k^{\\prime }},0.9N_{k^{\\prime }})$ .", "We set $N_k = N_{k^{\\prime }}=20$ .", "Given $\\lbrace \\hat{\\theta }_{(k)\\ell }\\rbrace $ and $\\lbrace \\hat{\\theta }_{(k^{\\prime })\\ell }\\rbrace $ , we draw observations from two classes that have response probability profiles ($_2$ and $_3$ from $\\lbrace \\theta _{(k)\\ell }, \\ell =1, \\ldots , L\\rbrace =(\\underbrace{0.9,\\ldots , 0.9}_{L_1}, \\underbrace{0.1, \\ldots , 0.1}_{L_2})$ and $_1$ from $\\lbrace \\theta _{(k^{\\prime })\\ell }, \\ell =1, \\ldots , L\\rbrace =(\\underbrace{0.1,\\ldots , 0.1}_{L_1}, \\underbrace{0.1, \\ldots , 0.1}_{L_2})$ ).", "Based on $R=100$ replications, Figure REF shows $R=100$ values of $p_{12}$ (left) and $R=100$ values of $p_{23}$ (right) computed by setting $\\lbrace _i, i=1, 2,3\\rbrace $ to be the irrelevant, all and relevant features in the data vector $\\lbrace _i,i=1,2,3\\rbrace $ , respectively.", "By selecting relevant features, the model improves our ability to separate observations from distinct clusters and group observations that belong to the same cluster.", "On the left panel, the all-feature $p_{12}$ values are pulled towards zero (towards left) that favors assigning $_2$ to cluster $k$ and $_1$ to cluster $k^{\\prime }$ .", "On the right panel, the all-feature $p_{23}$ values are pulled towards one (towards right) that favors clustering $_2$ and $_3$ together in the true cluster ($k$ ).", "In practice, the relevant features are of course to be inferred from data, by their observed marginal independence from the rest of the measured features.", "The improvements of clustering using subset clustering with inferred subsets can be seen from in Figure 2 in Main Paper by the superior clustering performance in (f) under feature selection compared to (e) obtained without selecting features.", "Figure: Removing irrelevant features improves estimation of clusters.", "Left) 100 random pairs of observations drawn from distinct clusters; the probability of them not being clustered correctly is lowered (pulled towards zero) once the irrelevant features are removed.", "Right) 100 random pairs of observations drawn from the same cluster; the probability of co-clustering to the correct cluster is increased towards one once the irrelevant features are removed." ] ]

[ [ "Functions of perturbed pairs of noncommuting contractions" ], [ "Abstract We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem for which functions $f$ we have Lipschitz type estimates in Schatten--von Neumann norms.", "We prove that if $f$ belongs to the Besov class $(B_{\\infty,1}^1)_+({\\Bbb D}^2)$ of analytic functions in the bidisk, then we have a Lipschitz type estimate for functions $f(T,R)$ of pairs of not necessarily commuting contractions $(T,R)$ in the Schatten--von Neumann norms $\\boldsymbol{S}_p$ for $p\\in[1,2]$.", "On the other hand, we show that for functions in the Besov space $(B_{\\infty,1}^1)_+({\\Bbb D}^2)$, there are no Lipschitz such type estimates for $p>2$ as well as in the operator norm." ], [ "The purpose of this paper is to study the behavior of functions $f(T,R)$ of (not necessarily commuting) contractions $T$ and $R$ under perturbation.", "We are going to obtain Lipschitz type estimates in the Sachatten–von Neumann norms ${S}_p$ , $1\\le p\\le 2$ , for functions $f$ in the Besov class $\\big (B_{\\infty ,1}^1\\big )_+(2)$ of analytic functions.", "Note that functions $f(T,R)$ of noncommuting contractions can be defined in terms of double operator integrals with respect to semi-spectral measures, see § below.", "This paper can be considered as a continuation of the results of [23]–[29], [2]–[5], [7], [8], [21], [1], [32] and [20] for functions of perturbed self-adjoint operators, contractions, normal operators, dissipative operators, functions of collections of commuting operators and functions of collections of noncommuting operators.", "Recall that a Lipschitz function $f$ on ${R}$ does not have to be operator Lipschitz, i.e., the condition $|f(x)-f(y)|\\le \\operatorname{const}|x-y|$ , $x,\\,y\\in {R}$ , does not imply that $\\Vert f(A)-f(B)\\Vert \\le \\operatorname{const}\\Vert A-B\\Vert $ for arbitrary self-adjoint operators (bounded or unbounded, does not matter) $A$ and $B$ .", "This was first established in [15].", "It turned out that functions in the (homogeneous) Besov space $B_{\\infty ,1}^1({R})$ are operator Lipschitz; this was established in [23] and [25] (see [22] for detailed information about Besov classes).", "We refer the reader to the recent survey [5] for detailed information on operator Lipschitz functions.", "In particular, [5] presents various sufficient conditions and necessary conditions for a function on ${R}$ to be operator Lipschitz.", "It is well known that if $f$ is an operator Lipschitz function on ${R}$ , and $A$ and $B$ are self-adjoint operators such that the difference $A-B$ belongs to the Schatten–von Neumann class ${S}_p$ , $1\\le p<\\infty $ , then $f(A)-f(B)\\in {S}_p$ and $\\Vert f(A)-f(B)\\Vert _{{S}_p}\\le \\operatorname{const}\\Vert A-B\\Vert _{{S}_p}$ .", "Moreover, the constant on the right does not depend on $p$ .", "In particular, this is true for functions $f$ in the Besov class $B_{\\infty ,1}^1({R})$ , i.e., $\\Vert f(A)-f(B)\\Vert _{{S}_p}\\le \\operatorname{const}\\Vert f\\Vert _{B_{\\infty ,1}^1}\\Vert A-B\\Vert _{{S}_p},\\quad 1\\le p\\le \\infty .$ However, it was discovered in [2] (see also [16]) that the situation becomes quite different if we replace the class of Lipschitz functions with the class $\\Lambda _\\alpha ({R})$ of Hölder functions of order $\\alpha $ , $0<\\alpha <1$ .", "Namely, the inequality $|f(x)-f(y)|\\le \\operatorname{const}|x-y|^\\alpha $ , $x,\\,y\\in {R}$ , implies that $\\Vert f(A)-f(B)\\Vert \\le \\operatorname{const}\\Vert A-B\\Vert ^\\alpha $ for arbitrary self-adjoint operators $A$ and $B$ .", "Moreover, it was shown in [3] that if $A-B\\in {S}_p$ , $p>1$ , and $f\\in \\Lambda _a({R})$ , then $f(A)-f(B)\\in {S}_{p/\\alpha }$ and $\\Vert f(A)-f(B)\\Vert _{{S}_{p/a}}\\le \\operatorname{const}\\Vert A-B\\Vert _{{S}_p}^\\alpha $ for arbitrary self-adjoint operators $A$ and $B$ .", "Analogs of the above results for functions of normal operators, functions of contractions, functions of dissipative operators and functions of commuting collections of self-adjoint operators were obtained in [24], [4], [8], [21].", "Note that it was shown in [32] that for $p\\in (1,\\infty )$ , inequality (REF ) holds for arbitrary Lipschitz (not necessarily operator Lipschitz) functions $f$ with constant on the right that depends on $p$ .", "An analog of this result for functions of commuting self-adjoint operators was obtained in [20].", "In [1] similar problems were considered for functions of two noncommuting self-adjoint operators (such functions can be defined in terms of double operator integrals, see [1]).", "It was shown in [1] that for functions $f$ on ${R}^2$ in the (homogeneous) Besov class $B_{\\infty ,1}^1({R}^2)$ and for $p\\in [1,2]$ , the following Lipschitz type estimate holds: $\\Vert f(A_1,B_1)-f(A_2,B_2)\\Vert _{{S}_p}\\le \\operatorname{const}\\max \\big \\lbrace \\Vert A_1-A_2\\Vert _{{S}_p},\\Vert B_1-B_2\\Vert _{{S}_p}\\big \\rbrace $ for arbitrary pairs $(A_1,B_1)$ and $(A_2,B_2)$ of (not necessarily commuting) self-adjoint operators.", "However, it was shown in [1] that for $p>2$ there is no such Lipschitz type estimate in the ${S}_p$ norm as well as in the operator norm.", "Moreover, it follows from the construction given in [1] that for $p\\in (2,\\infty ]$ and for positive numbers $\\varepsilon ,\\,\\sigma ,\\,M$ , there exists a function $f$ in $L^\\infty ({R}^2)$ with Fourier transform supported in $[-\\sigma ,\\sigma ]\\times [-\\sigma ,\\sigma ]$ such that $\\max \\big \\lbrace \\Vert A_1-A_2\\Vert _{{S}_p},\\Vert B_1-B_2\\Vert _{{S}_p}\\big \\rbrace <\\varepsilon $ while $\\Vert f(A_1,B_1)-f(A_2,B_2)\\Vert _{{S}_p}>M.$ Here we use the notation $\\Vert \\cdot \\Vert _{{S}_\\infty }$ for operator norm.", "This implies that unlike in the case of commuting operators, there cannot be any Hölder type estimates in the norm of ${S}_p$ , $p>2$ , for Hölder functions $f$ of order $\\alpha $ .", "Moreover, for $p>2$ , there cannot be any estimate for $\\Vert f(A_1,B_1)-f(A_2,B_2)\\Vert _{{S}_p}$ for functions in the Besov class $B_{\\infty ,q}^s({R})$ for any $q>0$ and $s>0$ .", "On the other hand, it was observed by the anonymous referee of [1] that unlike in the case of commuting self-adjoint operators, there is no Lipschitz type estimates for $\\Vert f(A_1,B_1)-f(A_2,B_2)\\Vert _{{S}_2}$ for Lipschitz functions $f$ on ${R}^2$ , see [1].", "Finally, let us mention that in the case of functions of triples of noncommuting operators there are no such Lipschitz type estimates for functions in the Besov class $B_{\\infty ,1}^1({R}^3)$ in the norm of ${S}_p$ for any $p\\in [1,\\infty ]$ .", "This was established in [29].", "In § we give an introduction to double and triple operator integrals and we define functions $f(T,R)$ of noncommuting contractions.", "We define the Haagerup and Haagerup-like tensor products of three copies of the disk-algebra ${\\rm C}_{\\rm A}$ and we define triple operator integrals whose integrands belong to such tensor products.", "Lipschitz type estimates in Schatten–von Neumann norm will be obtained in §.", "We show that for $p\\in [1,2]$ and for a function $f$ on 2 in the analytic Besov space $\\big (B_{\\infty ,1}^1\\big )_+(2)$ , the following Lipschitz type inequality holds: $\\big \\Vert f(T_1,R_1)-f(T_0,R_0)\\big \\Vert _{{S}_p}\\le \\operatorname{const}\\max \\big \\lbrace \\Vert T_1-T_0\\Vert _{{S}_p},\\Vert R_1-R_0\\Vert _{{S}_p}\\big \\rbrace $ for arbitrary pairs $(T_0,T_1)$ and $(R_0,R_1)$ of contractions.", "Recall that similar inequality was established in [1] for functions of self-adjoint operators.", "However, to obtain this inequality for functions of contractions, we need new algebraic formulae.", "Moreover, to obtain this inequality for functions of contractions, we offer an approach that does not use triple operator integrals.", "To be more precise, we reduce the inequality to the case of analytic polynomials $f$ and we integrate over finite sets, in which case triple operator integrals become finite sums.", "We establish explicit representations of the operator differences $f(T_1,R_1)-f(T_0,R_0)$ for analytic polynomials $f$ in terms of finite sums of elementary tensors which allows us to estimate the ${S}_p$ norms.", "However, we still use triple operator integrals to obtain in § explicit formulae for the operator differences for arbitrary functions $f$ in $\\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "In § we study differentiability properties in Schatten–von Neumann norms of the function $t\\mapsto f\\big (T(t),R(t)\\big )$ for $f$ in $\\big (B_{\\infty ,1}^1\\big )_+(2)$ and contractive valued functions $t\\mapsto T(t)$ and $t\\mapsto R(t)$ .", "We obtain explicit formulae for the derivative in terms of triple operator integrals.", "Again, to prove the existence of the derivative, we do not need triple operator integrals.", "As in the case of functions of pairs self-adjoint operators (see [1]), there are no Lipschitz type estimates in the norm of ${S}_p$ , $p>2$ , for functions of pairs of not necessarily commuting contractions $f(T,R)$ , $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "This will be established in §.", "Note that the construction differs from the construction in the case of self-adjoint operators given in [1].", "In § we state some open problems and in § we give an introduction to Besov classes on polydisks.", "We use the notation ${m}$ for normalized Lebesgue measure on the unit circle $ and the notation $ m2$ for normalized Lebesgue measure on $ 2$.$ For simplicity we assume that we deal with separable Hilbert spaces." ], [ "In this section we give a brief introduction to Besov spaces on the torus.", "To define Besov spaces on the torus $d$ , we consider an infinitely differentiable function $w$ on ${R}$ such that $w\\ge 0,\\quad \\operatorname{supp}w\\subset \\left[\\frac{1}{2},2\\right],\\quad \\mbox{and} \\quad w(s)=1-w\\left(\\frac{s}{2}\\right)\\quad \\mbox{for}\\quad s\\in [1,2].$ Let $W_n$ , $n\\ge 0$ , be the trigonometric polynomials defined by $W_n(\\zeta )\\stackrel{\\mathrm {def}}{=}\\sum _{j\\in {Z}^d}w\\left(\\frac{|j|}{2^n}\\right)\\zeta ^j,\\quad n\\ge 1,\\quad W_0(\\zeta )\\stackrel{\\mathrm {def}}{=}\\sum _{\\lbrace j:|j|\\le 1\\rbrace }\\zeta ^j,$ where $\\zeta =(\\zeta _1,\\cdots ,\\zeta _d)\\in d,\\quad j=(j_1,\\cdots ,j_d),\\quad \\mbox{and}\\quad |j|=\\big (|j_1|^2+\\cdots +|j_d|^2\\big )^{1/2}.$ For a distribution $f$ on $d$ we put $f_n=f*W_n,\\quad n\\ge 0.$ It is easy to see that $f=\\sum _{n\\ge 0}f_n;$ the series converges in the sense of distributions.", "We say that $f$ belongs the Besov class $B_{p,q}^s(d)$ , $s>0$ , $1\\le p,\\,q\\le \\infty $ , if $\\big \\lbrace 2^{ns}\\Vert f_n\\Vert _{L^p}\\big \\rbrace _{n\\ge 0}\\in \\ell ^q.$ The analytic subspace $\\big (B_{p,q}^s\\big )_+(d)$ of $B_{p,q}^s(d)$ consists of functions $f$ in $B_{p,q}^s(d)$ for which the Fourier coefficients $\\widehat{f}(j_1,\\cdots ,j_d)$ satisfy the equalities: $\\widehat{f}(j_1,\\cdots ,j_d)=0\\quad \\mbox{whenever}\\quad \\min _{1\\le k\\le d}j_k<0.$ We refer the reader to [22] for more detailed information about Besov spaces." ], [ "3.1.", "Double operator integrals.", "In this section we give a brief introduction to double and triple operator integrals with respect to semi-spectral measures.", "Double operator integrals with respect to spectral measures are expressions of the form $\\iint \\Phi (x,y)\\,dE_1(x)Q\\,dE_2(y),$ where $E_1$ and $E_2$ are spectral measures, $Q$ is a linear operator and $\\Phi $ is a bounded measurable function.", "They appeared first in [14].", "Later Birman and Solomyak developed in [9]–[11] a beautiful theory of double operator integrals.", "Double operator integrals with respect to semi-spectral measures were defined in [24], see also [5] (recall that the definition of a semi-spectral measure differs from the definition of a spectral measure by replacing the condition that it takes values in the set of orthogonal projections with the condition that it takes values in the set of nonnegative contractions, see [5] for more detail).", "For the double operator integral to make sense for an arbitrary bounded linear operator $T$ , we have to impose an additional assumption on $\\Phi $ .", "The natural class of such functions $\\Phi $ is called the class of Schur multipliers, see [23].", "There are various characterizations of the class of Schur multipliers.", "In particular, $\\Phi $ is a Schur multiplier if and only if it belongs to the Haagerup tensor product $L^\\infty (E_1)\\otimes _{\\rm h}\\!L^\\infty (E_2)$ of $L^\\infty (E_1)$ and $L^\\infty (E_2)$ , i.e., it admits a representation of the form $\\Phi (x,y)=\\sum _jj(x)\\psi _j(y),$ where the $j$ and $\\psi _j$ satisfy the condition $\\sum _j|j|^2\\in L^\\infty (E_1)\\quad \\mbox{and}\\quad \\sum _j|\\psi _j|^2\\in L^\\infty (E_2).$ In this case $\\iint \\Phi (x,y)\\,dE_1(x)Q\\,dE_2(y)=\\sum _j\\Big (\\int j\\,dE_1\\Big )Q\\Big (\\int \\psi _j\\,dE_2\\Big );$ the series converges in the weak operator topology.", "The right-hand side of this equality does not depend on the choice of a representation of $\\Phi $ in (REF ).", "One can also consider double operator integrals of the form (REF ) in the case when $E_1$ and $E_2$ are semi-spectral measures.", "In this case, as in the case of spectral measures, formula (REF ) still holds under the same assumption (REF ).", "It is easy to see that if $\\Phi $ belongs to the projective tensor product $L^\\infty (E_1)\\widehat{\\otimes }L^\\infty (E_2)$ of $L^\\infty (E_1)$ and $L^\\infty (E_2)$ , i.e., $\\Phi $ admits a representation of the form (REF ) with $j$ and $\\psi _j$ satisfying $\\sum _j\\Vert j\\Vert _{L^\\infty (E_1)}\\Vert \\psi _j\\Vert _{L^\\infty (E_2)}<\\infty ,$ then $\\Phi $ is a Schur multiplier and (REF ) holds.", "3.2.", "The semi-spectral measures of contractions.", "Recall that if $T$ is a contraction (i.e., $\\Vert T\\Vert \\le 1$ ) on a Hilbert space $, then by the Sz.-Nagy dilation theorem(see \\cite {SNF}), $ T$ has a unitary dilation, i.e., there exist a Hilbert space $ K$ that contains $ and a unitary operator $U$ on ${K}$ such that $T^n=P_Ủ^n\\big |\\quad n\\ge 0,$ where $P_ is the orthogonal projection onto $ .", "Among all unitary dilations of $T$ one can always select a minimal unitary dilation (in a natural sense) and all minimal unitary dilations are isomorphic, see [33].", "The existence of a unitary dilation allows us to construct the natural functional calculus $f\\mapsto f(T)$ for functions $f$ in the disk-algebra ${\\rm C}_{\\rm A}$ defined by $f(T)=P_f̉(U)\\big |P_(\\int _f̰(\\zeta )\\,dE_U(\\zeta )\\Big |\\quad f\\in {\\rm C}_{\\rm A}.$ where $E_U$ is the spectral measure of $U$ .", "Consider the operator set function ${E}_T$ defined on the Borel subsets of the unit circle $ by$${E}_T(=P_Ẻ_U(\\big |\\quad $$Then $ ET$ is a {\\it semi-spectral measure}.", "It can be shown that it does not depend on the choice of a unitary dilation.The semi-spectral measure $ ET$ is called the {\\it semi-spectral measure} of$ T$.$ 3.3.", "Functions of noncommuting contractions.", "Let $f$ be a function on the torus 2 that belongs to the Haagerup tensor product ${\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}$ , i.e., $f$ admits a representation of the form $f(\\zeta ,)=\\sum _jj(\\zeta )\\psi _j(),\\quad \\zeta ,~\\in $ where $j$ , $\\psi _j$ are functions in ${\\rm C}_{\\rm A}$ such that $\\sup _{\\zeta \\in \\sum _j|j(\\zeta )|^2<\\infty \\quad \\mbox{and}\\quad \\sup _{\\in \\sum _j|\\psi _j()|^2<\\infty .For a pair (T,R) of (not necessarily commuting contractions),the operator f(T,R) is defined as the double operator integral\\iint _{f(\\zeta ,)\\,d{E}_T(\\zeta )\\,d{E}_R()=\\iint _{f(\\zeta ,)\\,d{E}_T(\\zeta )I\\,d{E}_R().", "}Note that if f\\in \\big (B_{\\infty ,1}^1\\big )_+(2), then f\\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}, and so we can take functions f(T,R) of contractions for an arbitrary function f in \\big (B_{\\infty ,1}^1\\big )_+(2).Indeed, if f is an analytic polynomial in two variables of degree at most N in each variable, then we can represent f in the formf(\\zeta ,)=\\sum _{j=0}^N\\zeta ^j\\left(\\sum _{k=0}^N\\widehat{f}(j,k)^k\\right).Thus f belongs to the projective tensor product {\\rm C}_{\\rm A}\\hat{\\otimes }{\\rm C}_{\\rm A} and\\begin{eqnarray}\\Vert f\\Vert _{{\\rm C}_{\\rm A}\\hat{\\otimes }{\\rm C}_{\\rm A}}\\le \\sum _{j=0}^N\\sup _\\left|\\sum _{k=0}^N\\widehat{f}(j,k)^k\\right|\\le (1+N)\\Vert f\\Vert _{L^\\infty }\\end{eqnarray}It follows easily from (\\ref {Bperf}) that every function f of Besov class \\big (B_{\\infty ,1}^1\\big )_+(2) belongs to{\\rm C}_{\\rm A}\\hat{\\otimes }{\\rm C}_{\\rm A}, and so the operator f(T,R) is well defined.", "Clearly,\\begin{eqnarray}f(T,R)=\\sum _{n\\ge 0}\\sum _{j=0}^{2^{n+1}}T^j\\left(\\sum _{k=0}^{2^{n+1}}\\widehat{f}_n(j,k)R^k\\right),\\end{eqnarray}where f_n is the polynomial defined by (\\ref {fnWn}).", "It follows immediately from(\\ref {tenzprepol}) and (\\ref {Bperf}) that the series converges absolutely in the operator norm.", "Note thatformula (\\ref {opfunnekom}) can be used as a definition of the functions f(T,R) of noncommuting contractions in the case when f\\in \\big (B_{\\infty ,1}^1\\big )_+(2).", "}\\medskip }{\\bf 3.4.~Triple operator integrals.~Haagerup tensor products.}", "There are several approaches to multiple operator integrals.", "Triple operator integrals are expressions of the formW_\\Phi \\stackrel{\\mathrm {def}}{=}\\iiint \\Phi (x,y,z)\\,dE_1(x)X\\,dE_2(y)Y\\,dE_3(z),where \\Phi is a bounded measurable function, E_1, E_2 and E_3 are spectral measures, and X and Y are bounded linear operators on Hilbert space.", "}In \\cite {Pe4} triple (and more general, multiple) operator integrals were definedfor functions $ $ in the integral projective product\\mbox{$L^\\infty (E_1)\\otimes _{\\rm i}L^\\infty (E_2)\\otimes _{\\rm i}L^\\infty (E_3)$}.", "For suchfunctions $$, the following Schatten--von Neumann properties hold:$$\\left\\Vert \\iiint \\Phi \\,dE_1X\\,dE_2Y\\,dE_3\\right\\Vert _{{S}_r}\\le \\Vert \\Phi \\Vert _{L^\\infty \\otimes _{\\rm i}L^\\infty \\otimes _{\\rm i}L^\\infty }\\Vert X\\Vert _{{S}_p}\\Vert Y\\Vert _{{S}_q},\\quad \\frac{1}{r}=\\frac{1}{p}+\\frac{1}{q},$$whenever $ 1/p+1/q1$.", "Later in \\cite {JTT} triple (and multiple) operator integralswere defined for functions $$ in the Haagerup tensor product\\mbox{$L^\\infty (E_1)\\otimes _{\\rm h}L^\\infty (E_2)\\otimes _{\\rm h}L^\\infty (E_3)$}.However, it turns out that under the assumption$ LhLhL$,the conditions $ XSp$ and $ YSq$ imply that$  dE1X dE2Y dE3Sr$, $ 1/r=1/p+1/q$,only under the conditions that $ p2$ and $ q2$, see \\cite {AP5} (see also\\cite {ANP}).", "Moreover, the following inequality holds:$$\\left\\Vert \\iiint \\Phi \\,dE_1X\\,dE_2Y\\,dE_3\\right\\Vert _{{S}_r}\\le \\Vert \\Phi \\Vert _{L^\\infty \\otimes _{\\rm h}L^\\infty \\otimes _{\\rm h}L^\\infty }\\Vert X\\Vert _{{S}_p}\\Vert Y\\Vert _{{S}_q},\\quad \\frac{1}{r}=\\frac{1}{p}+\\frac{1}{q},$$whenever $ p2$ and $ q2$, see \\cite {AP5}.$ Note also that to obtain Lipschitz type estimates for functions of noncommuting self-adjoint operators in [1], we had to use triple operator integrals with integrands $\\Phi $ that do not belong to the Haagerup tensor product $L^\\infty \\!\\otimes _{\\rm h}\\!L^\\infty \\!\\otimes _{\\rm h}\\!L^\\infty $ .", "That is why we had to introduce in [1] Haagerup-like tensor products of the first kind and of the second kind.", "In this paper we are going to use triple operator integrals with integrands being continuous functions on 3 that belong to Haagerup and Haagerup-like tensor products of three copies of the disk-algebra ${\\rm C}_{\\rm A}$ .", "We briefly define such tensor products and discuss inequalities we are going to use in the next section.", "Definition 1.", "We say that a continuous function $\\Phi $ on 3 belongs to the Haagerup tensor product ${\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}$ if $\\Phi $ admits a representation $\\Phi (\\zeta ,,\\varkappa )=\\sum _{j,k\\ge 0}\\alpha _j(\\zeta )\\beta _{jk}()\\gamma _k(\\varkappa ),\\qquad \\zeta ,\\,,\\,\\varkappa \\in $ where $\\alpha _j$ , $\\beta _{jk}$ and $\\gamma _k$ are functions in ${\\rm C}_{\\rm A}$ such that $\\sup _{\\zeta \\in \\left(\\sum _{j\\ge 0}|\\alpha _j(\\zeta )|^2\\right)^{1/2}\\sup _{\\in \\big \\Vert \\lbrace \\beta _{jk}()\\rbrace _{j,k\\ge 0}\\big \\Vert _{\\mathcal {B}}\\,\\sup _{\\varkappa \\in \\left(\\sum _{k\\ge 0}|\\gamma _k(\\varkappa )|^2\\right)^{1/2}<\\infty .", "}Here \\Vert \\cdot \\Vert _\\mathcal {B} stands for the operator norm of a matrix (finite or infinite) on the space \\ell ^2 or on a finite-dimensional Euclidean space.", "By definition, thenorm of \\Phi in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A} is the infimum of the left-hand side of (\\ref {noHaatepr}) over all representations of \\Phi in the form of (\\ref {Haatepre}).", "}\\medskip }Suppose that \\Phi \\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}and both (\\ref {Haatepre}) and (\\ref {noHaatepr}) hold.", "Let T_1, T_2 and T_3be contractions with semi-spectral measures {E}_{T_1}, {E}_{T_2}and {E}_{T_3}.", "Then for bounded linear operators X and Y, we candefine the triple operator integral\\begin{eqnarray}W_\\Phi =\\iiint \\Phi \\,d{E}_{T_1}X\\,d{E}_{T_2}Y\\,d{E}_{T_3}\\end{eqnarray}as{\\begin{@align*}{1}{-1}W_\\Phi &\\stackrel{\\mathrm {def}}{=}\\sum _{j,k}\\Big (\\int \\alpha _j(\\zeta )\\,d{E}_{T_1}(\\zeta )\\Big )X\\Big (\\int \\beta _{jk}()\\,d{E}_{T_2}()\\Big )Y\\Big (\\int \\gamma _k(\\varkappa )\\,d{E}_{T_3}(\\varkappa )\\Big )\\\\[.2cm]&=\\sum _{j,k}\\alpha _j(T_1)X\\beta _{jk}(T_2)Y\\gamma _k(T_3).\\end{@align*}}It is easy to verify that the series converges in the weak operator topologyif we consider partial sums over rectangles.", "It can be shown in the same way as in the case of triple operator integrals with respect to spectral measures that the sum on the right does not depend on the choice of a representation of \\Phi in the form of (\\ref {Haatepre}), see Theorem 3.1 of \\cite {ANP}.$ We are going to use Lemma 3.2 of [6].", "Suppose that $\\lbrace Z_j\\rbrace _{j\\ge 0}$ is a sequence of bounded linear operators on Hilbert space such that $\\left\\Vert \\sum _{j\\ge 0}Z_j^*Z_j\\right\\Vert ^{1/2}\\le M\\quad \\mbox{and}\\quad \\left\\Vert \\sum _{j\\ge 0}Z_jZ_j^*\\right\\Vert ^{1/2}\\le M.$ Let $Q$ be a bounded linear operator.", "Consider the row ${\\rm R}_{\\lbrace Z_j\\rbrace }(Q)$ and the column ${\\rm C}_{\\lbrace Z_j\\rbrace }(Q)$ defined by ${\\rm R}_{\\lbrace Z_j\\rbrace }(Q)\\stackrel{\\mathrm {def}}{=}\\big (Z_0Q\\:Z_1Q\\:Z_2Q\\:\\cdots \\big )$ and ${\\rm C}_{\\lbrace Z_j\\rbrace }(Q)\\stackrel{\\mathrm {def}}{=}\\left(\\begin{matrix}QZ_0\\\\QZ_1\\\\QZ_2\\\\\\vdots \\end{matrix}\\right).$ Then by Lemma 3.2 of [6], for $p\\in [2,\\infty ]$ , the following inequalities hold: $\\big \\Vert {\\rm R}_{\\lbrace Z_j\\rbrace }(Q)\\big \\Vert _{{S}_p}\\le M\\Vert Q\\Vert _{{S}_p}\\quad \\mbox{and}\\quad \\big \\Vert {\\rm C}_{\\lbrace Z_j\\rbrace }(Q)\\big \\Vert _{{S}_p}\\le M\\Vert Q\\Vert _{{S}_p}$ whenever $Q\\in {S}_p$ .", "It is easy to verify that under the above assumptions $W_\\Phi ={\\rm R}_{\\lbrace \\alpha _j(T_1)\\rbrace }(X)\\,B\\,{\\rm C}_{\\lbrace \\gamma _j(T_3)\\rbrace }(Y),$ where $B$ is the operator matrix $\\lbrace \\beta _{jk}(T_2)\\rbrace _{j,k\\ge 0}$ .", "Lemma 3.1 Under the above hypotheses, $\\Vert B\\Vert \\le \\sup _{\\in \\big \\Vert \\lbrace \\beta _{jk}()\\rbrace _{j,k\\ge 0}\\big \\Vert _{\\mathcal {B}}.", "}$ Proof.", "Let $U$ be a unitary dilation of the contraction $T_2$ on a Hilbert space ${K}$ , ${K}\\supset .", "Clearly, we can consider the space $ 2($ as a subspace of $ 2(K)$.", "It is easy to see that$$\\lbrace \\beta _{jk}(T_2)\\rbrace _{j,k\\ge 0}=P_{\\ell ^2(}\\lbrace \\beta _{jk}(U)\\rbrace _{j,k\\ge 0}\\big |\\ell ^2(,$$where $ P2($ is the orthogonal projection onto $ 2($.", "The result follows from the inequality$ {jk(U)}j,k0{jk()}j,k0B$,which is a consequence of the spectral theorem.", "$$$ It follows from Lemma 3.2 of [30] that under the above assumptions, inequalities (REF ) hold for $Z_j=\\alpha _j(T_1)$ , $j\\ge 0$ , with $M=\\sup _{\\zeta \\in \\left(\\sum _{j\\ge 0}|\\alpha _j(\\zeta )|^2\\right)^{1/2} and forZ_j=\\gamma _j(T_3), j\\ge 0, withM=\\sup _{\\zeta \\in \\left(\\sum _{j\\ge 0}|\\gamma _j(\\zeta )|^2\\right)^{1/2}.This together with Lemma (\\ref {norBjk}) and inequalities (\\ref {RCAjSp})implies that under the above assumptions,\\begin{multline}\\big \\Vert {\\rm R}_{\\lbrace \\alpha _j(T_1)\\rbrace }(X)\\,B\\,{\\rm C}_{\\lbrace \\gamma _j(T_3)\\rbrace }(Y)\\big \\Vert _{{S}_r}\\\\[.2cm]\\le \\sup _{\\zeta \\in \\left(\\sum _{j\\ge 0}|\\alpha _j(\\zeta )|^2\\right)^{1/2}\\sup _{\\in \\big \\Vert \\lbrace \\beta _{jk}()\\rbrace _{j,k\\ge 0}\\big \\Vert _{\\mathcal {B}}\\,\\sup _{\\varkappa \\in \\left(\\sum _{k\\ge 0}|\\gamma _k(\\varkappa )|^2\\right)^{1/2}}whenever p\\ge 2, q\\ge 2 and 1/r=1/p+1/q.", "}The following theorem is an analog of the corresponding result for triple operator integrals with respect to spectral measures, see \\cite {AP5}.", "It follows immediately from (\\ref {nervodlyapr}).", "}\\end{multline}\\begin{thm}Let T_1, T_2 and T_3 be contractions, and letX\\in {S}_p and Y\\in {S}_q, 2\\le p\\le \\infty , 2\\le q\\le \\infty .Suppose that\\Phi \\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}.", "ThenW_\\Phi \\in {S}_r, 1/r=1/p+1/q, and\\end{thm}\\left\\Vert \\iiint \\Phi \\,d{E}_{T_1}X\\,d{E}_{T_2}Y\\,d{E}_{T_3}\\right\\Vert _{{S}_r}\\le \\Vert \\Phi \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}}\\Vert X\\Vert _{{S}_p}\\Vert Y\\Vert _{{S}_q}.Recall that by {S}_\\infty we mean the class of bounded linear operators.", "}\\medskip }{\\bf 3.5.", "Haagerup-like tensor products.", "}We define here Haagerup-like tensor products of disk-algebras by analogy withHaagerup-like tensor products of $ L$ spaces, see \\cite {ANP}.$ Definition 2.", "A continuous function $\\Phi $ on 3 is said to belong to the Haagerup-like tensor product ${\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}$ of the first kind if it admits a representation $\\Phi (\\zeta ,,\\varkappa )=\\sum _{j,k\\ge 0}\\alpha _j(\\zeta )\\beta _{k}()\\gamma _{jk}(\\varkappa ),\\qquad \\zeta ,\\,,\\,\\varkappa \\in $ where $\\alpha _j$ , $\\beta _k$ and $\\gamma _{jk}$ are functions in ${\\rm C}_{\\rm A}$ such that $\\sup _{\\zeta \\in \\left(\\sum _{j\\ge 0}|\\alpha _j(\\zeta )|^2\\right)^{1/2}\\sup _{\\in \\left(\\sum _{k\\ge 0}|\\beta _k()|^2\\right)^{1/2}\\,\\sup _{\\varkappa \\in \\big \\Vert \\lbrace \\gamma _{jk}(\\varkappa )\\rbrace _{j,k\\ge 0}\\big \\Vert _{\\mathcal {B}}<\\infty .", "}}Clearly, \\Phi \\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}if and only if the function(z_1,z_2,z_3)\\mapsto \\Phi (z_3,z_1,z_2)belongs to the Haagerup tensor product{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}.", "}\\medskip $ Similarly, we can define the Haagerup-like tensor product ${\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}$ of the second kind.", "Definition 3.", "A continuous function $\\Phi $ on 3 is said to belong to the Haagerup-like tensor product ${\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}$ of the second kind if it admits a representation $\\Phi (\\zeta ,,\\varkappa )=\\sum _{j,k\\ge 0}\\alpha _{jk}(\\zeta )\\beta _{j}()\\gamma _k(\\varkappa ),\\qquad \\zeta ,\\,,\\,\\varkappa \\in $ where $\\alpha _{jk}$ , $\\beta _j$ and $\\gamma _k$ are functions in ${\\rm C}_{\\rm A}$ such that $\\sup _{\\zeta \\in \\big \\Vert \\lbrace \\alpha _{jk}(\\zeta )\\rbrace _{j,k\\ge 0}\\big \\Vert _{\\mathcal {B}}\\,\\sup _{\\in \\left(\\sum _{j\\ge 0}|\\beta _j()|^2\\right)^{1/2}\\,\\sup _{\\varkappa \\in \\left(\\sum _{k\\ge 0}|\\gamma _k(\\varkappa )|^2\\right)^{1/2}<\\infty .", "}Let us first consider the situation when \\Phi is defined by (\\ref {tensprpe})or by (\\ref {tensprvt}) with summation over a finite set.In this case triple operator integrals of the form (\\ref {tropin})can be defined for arbitrary bounded linear operators X and Y and for arbitrary contractions T_1, T_2 and T_3.", "}Suppose that\\begin{eqnarray}\\Phi (\\zeta ,,\\varkappa )=\\sum _{j\\in F_1}\\sum _{k\\in F_2}\\alpha _j(\\zeta )\\beta _{k}()\\gamma _{jk}(\\varkappa ),\\qquad \\zeta ,\\,,\\,\\varkappa \\in \\qquad \\alpha _j,\\:\\beta _k,\\:\\gamma _{jk}\\in {\\rm C}_{\\rm A},\\end{eqnarray}where F_1 and F_2 are finite sets.", "We put\\begin{eqnarray}\\iiint \\Phi \\,d{E}_{T_1}X\\,d{E}_{T_2}Y\\,d{E}_{T_3}\\stackrel{\\mathrm {def}}{=}\\sum _{j\\in F_1}\\sum _{k\\in F_2}\\alpha _j(T_1)X\\beta _{k}(T_2)Y\\gamma _{jk}(T_3).\\end{eqnarray}Suppose now that\\begin{eqnarray}\\Phi (\\zeta ,,\\varkappa )=\\sum _{j\\in F_1}\\sum _{k\\in F_2}\\alpha _{jk}(\\zeta )\\beta _{j}()\\gamma _k(\\varkappa ),\\qquad \\zeta ,\\,,\\,\\varkappa \\in \\qquad \\alpha _{jk},\\:\\beta _j,\\:\\gamma _k\\in {\\rm C}_{\\rm A},\\end{eqnarray}where F_1 and F_2 are finite sets.Then we put\\begin{eqnarray}\\iiint \\Phi \\,d{E}_{T_1}X\\,d{E}_{T_2}Y\\,d{E}_{T_3}\\stackrel{\\mathrm {def}}{=}\\sum _{j\\in F_1}\\sum _{k\\in F_2}\\alpha _{jk}(T_1)X\\beta _{j}(T_2)Y\\gamma _k(T_3).\\end{eqnarray}}The following estimate is a very special case of Theorem \\ref {pervto} below.", "However, we have stated it separately because its proof is elementary and does not require the definition of triple operator integrals with integrands in Haagerup-like tensor products.$ Theorem 3.2 Let $X$ and $Y$ be bounded linear operators and let $T_1$ , $T_2$ and $T_3$ are contractions.", "Suppose that $F_1$ and $F_2$ are finite sets.", "The following statements hold: (i) Let $\\Phi $ be given by (REF ).", "Suppose that $q\\ge 2$ and $1/r\\stackrel{\\mathrm {def}}{=}1/p+1/q\\in [1/2,1]$ .", "If $X\\in {S}_p$ and $Y\\in {S}_q$ , then the sum on the right of () belongs to ${S}_r$ and $\\left\\Vert \\sum _{j\\in F_1}\\sum _{k\\in F_2}\\alpha _j(T_1)X\\beta _{k}(T_2)Y\\gamma _{jk}(T_3)\\in {S}_r\\right\\Vert _{{S}_r}\\le \\\\[.2cm]\\;\\;\\;\\sup _{\\zeta \\in \\left(\\sum _{j\\in F_1}|\\alpha _j(\\zeta )|^2\\right)^{1/2}\\!\\!\\sup _{\\in \\left(\\sum _{k\\in F_2}|\\beta _k()|^2\\right)^{{\\bf 1}/2}\\!\\!\\sup _{\\varkappa \\in \\big \\Vert \\big \\lbrace \\gamma _{jk}(\\varkappa )\\big \\rbrace _{j\\in F_1,k\\in F_2}\\big \\Vert _\\mathcal {B}\\Vert X\\Vert _{{S}_p}\\Vert Y\\Vert _{{S}_q}.", "}}{\\em (ii)} Let \\Phi be given by {\\em (\\ref {Fitipa2})}.Suppose thatq\\ge 2 and 1/r\\stackrel{\\mathrm {def}}{=}1/p+1/q\\in [1/2,1].", "If X\\in {S}_p and Y\\in {S}_q, thenthe sum on the right of {\\em (\\ref {konsumvtti})} belongs to{S}_r and\\begin{multline*}\\left\\Vert \\sum _{j\\in F_1}\\sum _{k\\in F_2}\\alpha _{jk}(T_1)X\\beta _{j}(T_2)Y\\gamma _k(T_3)\\in {S}_r\\right\\Vert _{{S}_r}\\le \\\\[.2cm]\\sup _{\\zeta \\in \\big \\Vert \\big \\lbrace \\alpha _{jk}(\\zeta )\\big \\rbrace _{j\\in F_1,k\\in F_2}\\big \\Vert _\\mathcal {B}\\sup _{\\in \\left(\\sum _{j\\in F_1}|\\beta _j()|^2\\right)^{1/2}\\!\\!\\sup _{\\varkappa \\in \\left(\\sum _{k\\in F_2}|\\gamma _k(\\varkappa )|^2\\right)^{{\\bf 1}/2}\\!\\!\\Vert X\\Vert _{{S}_p}\\Vert Y\\Vert _{{S}_q}.", "}}}{\\bf Proof.", "}Let us prove (i).", "The proof of (ii) is the same.", "We are going to use a duality argument.", "Suppose that Q\\in {S}_{r^{\\prime }} and \\Vert Q\\Vert _{{S}_{r^{\\prime }}}\\le 1,1/r+1/r^{\\prime }=1.", "We have\\begin{multline*}\\sup _Q\\left|\\operatorname{trace}\\left(Q\\sum _{j\\in F_1}\\sum _{k\\in F_2}\\alpha _{jk}(T_1)X\\beta _{j}(T_2)Y\\gamma _k(T_3)\\right)\\right|\\\\[.2cm]=\\sup _Q\\left|\\operatorname{trace}\\left(\\sum _{j\\in F_1}\\sum _{k\\in F_2}\\gamma _k(T_3)Q\\alpha _{jk}(T_1)X\\beta _{j}(T_2)\\right)Y\\right|\\\\[.2cm]\\le \\Vert Y\\Vert _{{S}_q}\\sup _Q\\left\\Vert \\sum _{j\\in F_1}\\sum _{k\\in F_2}\\gamma _k(T_3)Q\\alpha _{jk}(T_1)X\\beta _{j}(T_2)\\right\\Vert _{{S}_{q^{\\prime }}}.\\end{multline*}Th result follows now from (\\ref {WPhi}) and (\\ref {nervodlyapr}).", "\\blacksquare \\end{multline*}\\medskip }{\\bf 3.6.", "Triple operator integrals with integrands in Haagerup-liketensor products.", "}We define triple operator integrals with integrands in{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A} by analogy with triple operator integrals with respect to spectral measures, see \\cite {ANP} and \\cite {AP5}.Let \\Phi \\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A} and let p\\in [1,2].Suppose that T_1, T_2 and T_3 are contractions.", "Foran operator X of class {S}_p and fora bounded linear operator Y, we define the triple operator integral\\begin{eqnarray}\\stackrel{1}{W}_\\Phi \\stackrel{\\mathrm {def}}{=}\\iint \\!\\!\\upintop \\nolimits \\Phi (\\zeta ,,\\varkappa )\\,d{E}_{T_1}(\\zeta )X\\,d{E}_{T_2}()Y\\,d{E}_{T_3}(\\varkappa )\\end{eqnarray}as the following continuous linear functional on {S}_{p^{\\prime }},1/p+1/p^{\\prime }=1 (on the class of compact operators in the case p=1):Q\\mapsto \\operatorname{trace}\\left(\\left(\\iiint \\Phi (\\zeta ,,\\varkappa )\\,dE_{T_2}()Y\\,dE_{T_3}(\\varkappa )Q\\,dE_{T_1}(\\zeta )\\right)X\\right).Note that the triple operator integral\\iiint \\Phi (\\zeta ,,\\varkappa )\\,dE_{T_2}()Y\\,dE_{T_3}(\\varkappa )Q\\,dE_{T_1}(\\zeta )is well defined as the integrand belongs to the Haagerup tensor product{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}.$ Again, we can define triple operator integrals with integrands in ${\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}$ by analogy with the case of spectral measures, see [1] and [6].", "Let $\\Phi \\in {\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}$ and let $T_1$ , $T_2$ and $T_3$ be contractions.", "Suppose that $X$ is a bounded linear operator and $Y\\in {S}_p$ , $1\\le p\\le 2$ .", "The triple operator integral $\\stackrel{2}{W}_\\Phi \\stackrel{\\mathrm {def}}{=}\\upintop \\nolimits \\!\\!\\!\\iint \\Phi (\\zeta ,,\\varkappa )\\,d{E}_{T_1}(\\zeta )X\\,d{E}_{T_2}()Y\\,d{E}_{T_3}(\\varkappa )$ is defined as the continuous linear functional $Q\\mapsto \\operatorname{trace}\\left(\\left(\\iiint \\Phi (\\zeta ,,\\varkappa )\\,dE_3(\\varkappa )Q\\,dE_1(\\zeta )X\\,dE_2()\\right)Y\\right)$ on ${S}_{p^{\\prime }}$ (on the class of compact operators if $p=1$ ).", "As in the case of spectral measures (see [6]), the following theorem can be proved: Theorem 3.3 Suppose that $T_1$ , $T_2$ and $T_3$ are contractions, and let $X\\in {S}_p$ and $Y\\in {S}_q$ .", "The following statements hold: (1) Let $\\Phi \\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}$ .", "Suppose $q\\ge 2$ and $1/r\\stackrel{\\mathrm {def}}{=}1/p+1/q\\in [1/2,1]$ .", "If $X\\in {S}_p$ and $Y\\in {S}_q$ , then the operator $\\stackrel{1}{W}_\\Phi $ in (REF ) belongs to ${S}_r$ and $\\left\\Vert \\stackrel{1}{W}_\\Phi \\right\\Vert _{{S}_r}\\le \\Vert \\Phi \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\,\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}{\\rm C}_{\\rm A}}\\Vert X\\Vert _{{S}_p}\\Vert Y\\Vert _{{S}_q};$ (2) Let $\\Phi \\in {\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}$ .", "Suppose that $p\\ge 2$ and $1/r\\stackrel{\\mathrm {def}}{=}1/p+1/q\\in [1/2,1]$ .", "If $X\\in {S}_p$ and $Y\\in {S}_q$ , then the operator $\\stackrel{2}{W}_\\Phi $ in (REF ) belongs to ${S}_r$ and $\\Big \\Vert \\stackrel{2}{W}_\\Phi \\Big \\Vert _{{S}_r}\\le \\Vert \\Phi \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}{\\rm C}_{\\rm A}}\\Vert X\\Vert _{{S}_p}\\Vert Y\\Vert _{{S}_q}.$" ], [ "In this section we obtain Lipschitz type estimates in the Schatten–von Neumann classes ${S}_p$ for $p\\in [1,2]$ for functions of contractions.", "To obtain such estimates, we are going to use an elementary approach and obtain elementary formulae that involve only finite sums.", "Later we will need explicit expressions for operator differences, which will be obtained in the next section in terms of triple operator integrals.", "Such formulae will be used in §  to obtain formulae for operator derivatives.", "Suppose that $f$ is a function that belongs to the Besov space $\\big (B_{\\infty ,1}^1\\big )_+(2)$ of analytic functions (see §).", "As we have observed in Subsection 3.3, we can define functions $f(T,R)$ for (not necessarily commuting) contractions $T$ and $R$ on Hilbert space by formula ().", "For a differentiable function $f$ on $, we use the notation$ Df$ for the divided difference:$$(Df)(\\zeta ,)\\stackrel{\\mathrm {def}}{=}\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\frac{f(\\zeta )-f()}{\\zeta -}},&\\zeta \\ne \\\\[.4cm]f^{\\prime }(\\zeta ),&\\zeta =,\\end{array}\\right.\\qquad \\zeta ,\\;\\in $$For a differentiable function $ f$ on $ 2$, we define the divided differences$ D[1]f$ and $ D[2]f$ by$$\\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\stackrel{\\mathrm {def}}{=}\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\frac{f(\\zeta _1,)-f(\\zeta _2,)}{\\zeta _1-\\zeta _2}},&\\zeta _1\\ne \\zeta _2,\\\\[.4cm]\\displaystyle {\\frac{\\partial f}{\\partial \\zeta }\\Big |_{\\zeta =\\zeta _1}},&\\zeta _1=\\zeta _2,\\end{array}\\right.\\qquad \\zeta _1,\\;\\zeta _2,\\:\\in $$and$$\\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\stackrel{\\mathrm {def}}{=}\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\frac{f(\\zeta ,_1)-f(\\zeta ,_2)}{_1-_2}},&_1\\ne _2,\\\\[.4cm]\\displaystyle {\\frac{\\partial f}{\\partial }\\Big |_{=_1}},&_1=_2,\\end{array}\\right.\\qquad \\zeta ,\\;_1,\\:_2\\in $$$ We need several elementary identities.", "Let $\\Pi _m$ be the set of $m$ th roots of 1: $\\Pi _m\\stackrel{\\mathrm {def}}{=}\\lbrace \\xi \\in \\xi ^m=1\\rbrace $ and let $\\Upsilon _m(\\zeta )\\stackrel{\\mathrm {def}}{=}\\frac{\\zeta ^{m}-1}{m(\\zeta -1)}=\\frac{1}{m}\\sum _{k=0}^{m-1}\\zeta ^k,\\quad \\zeta \\in $ The following elementary formulae are well known.", "We give proofs for completeness.", "Lemma 4.1 Let $f$ and $g$ be analytic polynomials in one variable of degree less than $m$ .", "Then $\\int _f̰\\overline{g}\\,d{m}=\\frac{1}{m}\\sum _{\\xi \\in \\Pi _m}f(\\xi )\\overline{g(\\xi )}.$ In particular, $\\int _f|^2\\,d{m}=\\frac{1}{m}\\sum _{\\xi \\in \\Pi _m}|f(\\xi )|^2.$ Proof.", "It suffices to consider the case where $f(z)=z^j$ and $g(z)=z^k$ with $0\\le j, k<m$ .", "Then $-m<j-k<m$ and $\\sum _{\\xi \\in \\Pi _m}\\xi ^j\\,\\overline{\\xi }^k=\\left\\lbrace \\begin{array}{ll}0,&j\\ne k\\\\[.2cm]m,&j=k.\\end{array}\\right.\\quad \\blacksquare $ Corollary 4.2 $\\sum _{\\xi \\in \\Pi _m}|\\Upsilon _m(\\zeta \\bar{\\xi })|^2=1,\\quad \\zeta \\in $ In the same way we can obtain similar formulae for polynomials in several variables.", "We need only the case of two variables.", "Lemma 4.3 Let $f$ and $g$ be polynomials in two variables of degree less than $m$ in each variable.", "Then $\\int _{2} f\\overline{g}\\,d{m}_2=\\frac{1}{m^2}\\sum _{\\xi ,\\eta \\in \\Pi _m}f(\\xi ,\\eta )\\overline{g(\\xi ,\\eta )}.$ In particular, $\\int _{2} |f|^2\\,d{m}_2=\\frac{1}{m^2}\\sum _{\\xi ,\\eta \\in \\Pi _m}|f(\\xi ,\\eta )|^2.$ Proof.", "It suffices to consider the case when $f(\\zeta ,)=\\zeta ^{j_1}^{j_2}$ and $g(\\zeta ,)=\\zeta ^{k_1}^{k_2}$ with $0\\le j_1, j_2, k_1, k_2<m$ .", "Then $-m<j_1-k_1, j_2-k_2<m$ and $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\xi ^{j_1}\\eta ^{j_2}\\,\\overline{\\xi }^{k_1}\\,\\overline{\\eta }^{k_2}=\\left\\lbrace \\begin{array}{ll}0,&(j_1,j_2)\\ne (k_1,k_2)\\\\[.2cm]m^2,&(j_1,j_2)=(k_1,k_2).\\end{array}\\right.\\quad \\blacksquare $ Suppose now that $(T_0,R_0)$ and $(T_1,R_1)$ are pairs of not necessarily commuting contractions.", "Theorem 4.4 Let $f$ be an analytic polynomial in two variable of degree at most $m$ in each variable.", "Then $f(T_1,R_1)-f(T_0,R_1)=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)\\,(D^{[1]}f)(\\xi ,\\eta ,R_1)$ and $f(T_0,R_1)-f(T_0,R_0)=\\sum _{\\xi ,\\eta \\in \\Pi _m}(D^{[2]}f)(T_0,\\xi ,\\eta )\\,\\Upsilon _m(\\overline{\\xi }R_1)(R_1-R_0)\\,\\Upsilon _m(\\overline{\\eta }R_0).$ We are going to establish (REF ).", "The proof of (REF ) is similar.", "We need the following lemma.", "Lemma 4.5 Let $ be an analytic polynomial in one variable of degree at most $ m$.", "Then$$T_1)-T_0)=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)(D(\\xi ,\\eta ).$$$ Proof of the lemma.", "Let $0\\le j, j_0, k, k_0<m$ .", "Then $\\sum _{\\xi ,\\eta \\in \\Pi _m}(\\overline{\\xi }T_1)^{j_0}\\,(\\overline{\\eta }T_0)^{k_0}\\xi ^j\\eta ^k=\\left\\lbrace \\begin{array}{ll}\\displaystyle {m^2T_1^jT_0^k},&(j_0,k_0)=(j,k),\\\\[.4cm]0,&(j_0,k_0)\\ne (j,k).\\end{array}\\right.$ Thus, $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)\\,\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k=T_1^jT_0^k$ if $0\\le j, k<n$ .", "Hence, $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)T_1\\,\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k=T_1^{j+1}T_0^k$ and $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)\\,T_0\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k=T_1^{j}T_0^{k+1}.$ It follows that $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k=T_1^j(T_1-T_0)T_0^k$ whenever $0\\le j, k<m$ .", "Let $\\sum \\limits _{s=0}^ma_sz^s$ .", "It is easy to see that $(D(z,w)=\\sum _{j,k\\ge 0, j+k<m}a_{j+k+1}z^jw^k.$ Hence, $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)(D(\\xi ,\\eta )\\\\=\\sum _{j,k\\ge 0, j+k<m}a_{j+k+1}\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k\\\\=\\sum _{j,k\\ge 0, j+k<m}a_{j+k+1}T_1^j(T_1-T_0)T_0^k=T_1)-T_0).\\quad \\blacksquare $ Proof of Theorem REF .", "Clearly, it suffices to prove (REF ) in the case when $f(z_1,z_2)=z_1)z_2^j$ , where $ is a polynomial of onevariable of degree at most $ n$ and $ 0jm$.Clearly, in this case$$f(T_1,R_1)-f(T_0,R_1)=\\big (T_1)-T_0)\\big )R_1^j.$$On the other hand,$$(Df^{[1]})(\\xi ,\\eta ,R_1)=(D(\\xi ,\\eta )R_1^j.$$Identity (\\ref {perra}) follows now from Lemma \\ref {odnape}.", "$$$ For $K\\in L^2(2)$ , we denote by $I_K$ the integral operator on $L^2($ with kernel function $K$ , i.e., $(I_K(\\zeta )=\\int _K̰(\\zeta ,))\\,d{m}(),\\quad L^2(.$ The following lemma allows us to evaluate the operator norm $\\Vert I_K\\Vert _{\\mathcal {B}(L^2)}$ of this operator for polynomials $K$ of degree less than $m$ in each variable in terms of the operator norms of the matrix $\\lbrace K(\\zeta ,\\eta )\\rbrace _{\\zeta ,\\eta \\in \\Pi _m}$ .", "Lemma 4.6 Let $K$ be an analytic polynomial in two variables of degree less than $m$ in each variable.", "Then $\\Vert \\lbrace K(\\xi ,\\eta )\\rbrace _{\\xi ,\\eta \\in \\Pi _m}\\Vert _{\\mathcal {B}}=m\\Vert I_K\\Vert _{\\mathcal {B}(L^2)}.$ Proof.", "It is easy to see that $\\Vert I_K\\Vert _{\\mathcal {B}(L^2)}=\\sup _{\\Vert \\varphi \\Vert _{L^2}\\le 1,\\Vert \\psi \\Vert _{L^2}\\le 1}\\left|\\iint _{K(\\zeta ,)\\,\\overline{\\varphi (\\zeta )\\psi ()}\\,d{m}(\\zeta )\\,d{m}()\\\\=\\sup _{\\Vert \\varphi \\Vert _{L^2}\\le 1,\\Vert \\psi \\Vert _{L^2}\\le 1}\\left|\\iint _{K(\\zeta ,)\\,\\overline{\\varphi _m(z)\\psi _m()}\\,d{m}(\\zeta )\\,d{m}(),}where \\right.\\varphi _m(z)=\\sum \\limits _{k=0}^{m-1}\\widehat{\\varphi }(k)z^k and \\psi _m(z)=\\sum \\limits _{k=0}^{m-1}\\widehat{\\psi }(k)z^k.Hence,\\Vert I_K\\Vert _{\\mathcal {B}(L^2)}=\\sup \\left|\\iint _{K(\\zeta ,w)\\,\\overline{\\varphi (z)\\psi (w)}\\,d{m}(\\zeta )\\,d{m}(),where the supremum is taken over all polynomials \\varphi and \\psi in one variable of degree less than m and such that\\Vert \\varphi \\Vert _{L^2}\\le 1, \\Vert \\psi \\Vert _{L^2}\\le 1.", "Next, by Lemma \\ref {137},for arbitrary polynomials \\varphi and \\psi with \\deg \\varphi <m and \\deg \\psi <m,we have\\iint _{K(\\zeta ,)\\,\\overline{\\varphi (z)\\psi (w)}\\,d{m}(\\zeta )\\,d{m}()=\\frac{1}{m^2}\\sum _{\\xi ,\\eta \\in \\Pi _m}K(\\xi ,\\eta )\\,\\overline{\\varphi (\\xi )\\psi (\\eta )}.It remains to observe that by Lemma \\ref {136},\\Vert \\varphi \\Vert _{L^2}\\le 1 if and only if \\sum \\limits _{\\xi \\in \\Pi _m}|\\varphi (\\xi )|^2\\le mand the same is true for \\psi .\\blacksquare }\\begin{thm}Let g be a polynomial in one variable of degree at most m. Then\\Vert \\lbrace (Dg)(\\xi ,\\eta )\\rbrace _{\\xi ,\\eta \\in \\Pi _m}\\Vert _{\\mathcal {B}}\\le m\\Vert g\\Vert _{L^\\infty }.\\end{thm}}{\\bf Proof.", "}The result follows from Lemma \\ref {intopera} and the inequality\\right.\\Vert I_{Dg}\\Vert _{\\mathcal {B}(L^2)}\\le \\Vert g\\Vert _{L^\\infty },which is a consequence of the fact that \\Vert I_{Dg}\\Vert _{\\mathcal {B}(L^2)} is equal to the norm of the Hankel operator H_{\\bar{g}} on the Hardy class H^2, see\\cite {Pe5}, Ch.", "1, Th.", "1.10.\\blacksquare }\\right.$ Corollary 4.7 Let $f$ be a trigonometric polynomial of degree at most $m$ in each variable and let $p\\in [1,2]$ .", "Suppose that $T_1,\\,R_1,T_0,\\,R_0$ are contractions such that $T_1-T_0\\in {S}_p$ and $R_1-R_0\\in {S}_p$ .", "Then $\\Vert f(T_1,R_1)-f(T_0,R_0)\\Vert _{{S}_p}\\le 2m\\Vert f\\Vert _{L^\\infty }\\max \\big \\lbrace \\Vert T_1-T_0\\Vert _{{S}_p},\\Vert R_1-R_0\\Vert _{{S}_p}\\big \\rbrace .$ Proof.", "Let us estimate $\\Vert f(T_1,R_1)-f(T_0,R_1)\\Vert _{{S}_p}$ .", "The norm $\\Vert f(T_0,R_1)-f(T_0,R_0)\\Vert _{{S}_p}$ can be estimated in the same way.", "The result is a consequence of formula (REF ), Theorem REF , Theorem REF and Corollare REF .", "$\\blacksquare $ Corollary REF allows us to establish a Lipschitz type inequality for functions in $\\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Theorem 4.8 Let $1\\le p\\le 2$ and let $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Suppose that $T_1,\\,R_1,T_0,\\,R_0$ are contractions such that $T_1-T_0\\in {S}_p$ and $R_1-R_0\\in {S}_p$ .", "Then $\\Vert f(T_1,R_1)-f(T_0,R_0)\\Vert _{{S}_p}\\le \\operatorname{const}\\Vert f\\Vert _{B_{\\infty ,1}^1}\\max \\big \\lbrace \\Vert T_1-T_0\\Vert _{{S}_p},\\Vert R_1-R_0\\Vert _{{S}_p}\\big \\rbrace .$ Proof.", "Indeed, the result follows immediately from Corollary REF and inequality (REF ).", "$\\blacksquare $" ], [ "In this section we obtain an explicit formula for the operator differences $f(T_1,R_1)-f(T_0,R_0)$ , $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , in terms of triple operator integrals.", "Theorem 5.1 Let $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Then $D^{[1]}f\\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\quad \\mbox{and}\\quad D^{[2]}f\\in {\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}.$ Lemma 5.2 Let $f$ be an analytic polynomial in two variables of degree at most $m$ in each variable.", "Then $\\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\zeta _1\\overline{\\xi })\\,\\Upsilon _m(\\zeta _2\\overline{\\eta })\\big (D^{[1]}f\\big )(\\xi ,\\eta ,)$ and $\\big (D^{[2]}f\\big )(\\zeta ,_1,_2)=\\sum _{\\xi ,\\eta \\in \\Pi _m}(D^{[2]}f)(\\zeta ,\\xi ,\\eta )\\Upsilon _m(_1\\overline{\\xi })\\Upsilon _m(_2\\overline{\\eta }).$ Proof.", "Both formulae (REF ) and (REF ) can be verified straightforwardly.", "However, we deduce them from Theorem REF .", "Formula (REF ) follows immediately from formula (REF ) if we consider the special case when $T_0$ , $T_1$ and $R_1$ are the operators on the one-dimensional space of multiplication by $\\zeta _2$ , $\\zeta _1$ and $$ .", "Similarly, formula (REF ) follows immediately from formula (REF ).", "$\\blacksquare $ Corollary 5.3 Under the hypotheses of Lemma REF, $\\big \\Vert D^{[1]}f\\big \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}{\\rm C}_{\\rm A}}\\le m\\Vert f\\Vert _{L^\\infty }\\quad \\mbox{and}\\quad \\big \\Vert D^{[2]}f\\big \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}}\\le m\\Vert f\\Vert _{L^\\infty }.$ Proof.", "The result is a consequence of Lemma REF , Theorem REF , Corollary REF and Definitions 2 and 3 in §.", "$\\blacksquare $ Proof of Theorem REF .", "The result follows immediately from Corollary REF and inequality (REF ).", "$\\blacksquare $ Theorem 5.4 Let $p\\in [1,2]$ .", "Suppose that $T_0,\\,R_0,\\,T_1,\\,R_1$ are contractions such that $T_1-T_0\\in {S}_p$ and $R_1-R_0\\in {S}_p$ .", "Then for $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , the following formula holds: $f(T_1,R_1)&-f(T_0,R_0)\\nonumber \\\\[.2cm]&=\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T_1}(\\zeta _1)(T_1-T_0)\\,dE_{T_2}(\\zeta _2)\\,dE_{R_1}(),\\nonumber \\\\[.2cm]&+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T_2}(\\zeta )\\,dE_{R_1}(_1)(R_1-R_0)\\,dE_{R_2}(_2).$ Proof.", "Suppose first that $f$ is an analytic polynomial in two variables of degree at most $m$ in each variable.", "In this case equality (REF ) is a consequence of Theorem REF , Lemma REF and the definition of triple operator integrals given in Subsection 3.5.", "In the general case we represent $f$ by the series (REF ) and apply (REF ) to each $f_n$ .", "The result follows from (REF ).", "$\\blacksquare $" ], [ "In this section we study differentiability properties of the map $t\\mapsto f\\big (T(t),R(t)\\big )$ in the norm of ${S}_p$ , $1\\le p\\le 2$ , for functions $t\\mapsto T(t)$ and $t\\mapsto R(t)$ that take contractive values and are differentiable in ${S}_p$ .", "We say that an operator-valued function $\\Psi $ defined on an interval $J$ is differentiable in ${S}_p$ if $\\Phi (s)-\\Phi (t)\\in {S}_p$ for any $s,\\,t\\in J$ , and the limit $\\lim _{h\\rightarrow {0}}\\frac{1}{h}\\big (\\Psi (t+h)-\\Psi (t)\\big )\\stackrel{\\mathrm {def}}{=}\\Phi ^{\\prime }(t)$ exists in the norm of ${S}_p$ for each $t$ in $J$ .", "Theorem 6.1 Let $p\\in [1,2]$ and let $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Suppose that $t\\mapsto T(t)$ and $t\\mapsto R(t)$ are operator-valued functions on an interval $J$ that take contractive values and are differentiable in ${S}_p$ .", "Then the function (REF ) is differentiable on $J$ in ${S}_p$ and $\\frac{d}{dt}&f\\big (T(t),R(t)\\big )\\Big |_{t=s}\\\\[.2cm]&=\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T(s)}(\\zeta _1)T^{\\prime }(s)\\,dE_{T(s)}(\\zeta _2)\\,dE_{R(s)}()\\nonumber \\\\[.2cm]&+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T(s)}(\\zeta )\\,dE_{R(s)}(_1)R^{\\prime }(s)\\,dE_{R(s)}(_2),$ $s\\in J$ .", "Proof.", "As before, it suffices to prove the result in the case when $f$ is an analytic polynomial of degree at most $m$ in each variable.", "Suppose that $f$ is such a polynomial.", "Put $F(t)\\stackrel{\\mathrm {def}}{=}f\\big (T(t),R(t)\\big )$ .", "We have $F(s+h)&-F(s)\\\\[.2cm]&=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m\\big (\\overline{\\xi }T(s+h)\\big )\\big (T(s+h)-T(s)\\big )\\Upsilon _m\\big (\\overline{\\eta }T(s)\\big )\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s+h)\\big )\\\\[.2cm]&+\\sum _{\\xi ,\\eta \\in \\Pi _m}\\big (D^{[2]}f\\big )\\big (T(s),\\xi ,\\eta \\big )\\Upsilon _m\\big (\\overline{\\xi }R(s+h)\\big )\\big (R(s+h)-R(s)\\big )\\Upsilon _m\\big (\\overline{\\eta }R(s)\\big ).$ Clearly, $\\lim _{h\\rightarrow 0}\\frac{1}{h}\\big (T(s+h)-T(s)\\big )=T^{\\prime }(s)\\quad \\mbox{and}\\quad \\lim _{h\\rightarrow 0}\\frac{1}{h}\\big (R(s+h)-R(s)\\big )=R^{\\prime }(s)$ in the norm of ${S}_p$ .", "On the other hand, it is easy to see that $\\lim _{h\\rightarrow 0}\\Upsilon _m\\big (\\overline{\\xi }T(s+h)\\big )=\\Upsilon _m\\big (\\overline{\\xi }T(s)\\big ),\\quad \\lim _{h\\rightarrow 0}\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s+h)\\big )=\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s)\\big )$ and $\\lim _{h\\rightarrow 0}\\Upsilon _m\\big (\\overline{\\xi }R(s+h)\\big )=\\Upsilon _m\\big (\\overline{\\xi }R(s)\\big )$ in the operator norm.", "Hence, $F^{\\prime }(s)&=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m\\big (\\overline{\\xi }T(s)\\big )T^{\\prime }(s)\\Upsilon _m\\big (\\overline{\\eta }T(s)\\big )\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s)\\big )\\\\[.2cm]&+\\sum _{\\xi ,\\eta \\in \\Pi _m}\\big (D^{[2]}f\\big )\\big (T(s),\\xi ,\\eta \\big )\\Upsilon _m\\big (\\overline{\\xi }R(s)\\big )R^{\\prime }(s)\\Upsilon _m\\big (\\overline{\\eta }R(s)\\big ).$ It follows now from Lemma REF and from the definition of triple operator integrals given in § that the right-hand side is equal to $\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T(s)}(\\zeta _1)T^{\\prime }(s)\\,dE_{T(s)}(\\zeta _2)\\,dE_{R(s)}()\\\\[.2cm]+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T(s)}(\\zeta )\\,dE_{R(s)}(_1)R^{\\prime }(s)\\,dE_{R(s)}(_2)$ which completes the proof.", "$\\blacksquare $" ], [ "In this section we show that unlike in the case $p\\in [1,2]$ , there are no Lipschitz type estimates in the norm of ${S}_p$ in the case when $p>2$ for functions $f(T,R)$ , $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , of not noncommuting contractions.", "In particular, there are no such Lipschitz type estimates for functions $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ in the operator norm.", "Moreover, we show that for $p>2$ , such Lipschitz type estimates do not hold even for functions $f$ in $\\big (B_{\\infty ,1}^1\\big )_+(2)$ and for pairs of noncommuting unitary operators.", "Recall that similar results were obtained in [1] for functions of noncommuting self-adjoint operators.", "However, in this paper we use a different construction to obtain results for functions of unitary operators.", "Lemma 7.1 For each matrix $\\lbrace a_{\\xi \\, \\eta }\\rbrace _{\\xi , \\eta \\in \\Pi _m}$ , there exists an analytic polynomial $f$ in two variables of degree at most $2m-2$ in each variable such that $f(\\xi ,\\eta )=a_{\\xi \\, \\eta }$ for all $\\xi , \\eta \\in \\Pi _m$ and $\\Vert f\\Vert _{L^\\infty (2)}\\le \\sup \\limits _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|$ .", "Proof.", "Put $f(z,w)\\stackrel{\\mathrm {def}}{=}\\sum _{\\xi , \\eta \\in \\Pi _m}a_{\\xi \\, \\eta }\\Upsilon _m^2(z\\overline{\\xi })\\Upsilon _m^2(w\\overline{\\eta }).$ Clearly, $f(\\xi ,\\eta )=a_{\\xi \\, \\eta }$ for all $\\xi , \\eta \\in \\Pi _m$ and $|f(z,w)|&\\le \\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|\\sum _{\\xi , \\eta \\in \\Pi _m}|\\Upsilon _m(z\\overline{\\xi })|^2|\\Upsilon _m(w\\overline{\\eta })|^2\\\\[.2cm]&=\\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|\\sum _{\\xi \\in \\Pi _m}|\\Upsilon _m(z\\overline{\\xi })|^2\\sum _{\\eta \\in \\Pi _m}|\\Upsilon _m(w\\overline{\\eta })|^2=\\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|$ by Corollary REF .", "$\\blacksquare $ Lemma 7.2 For each $m\\in N$ , there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable, and unitary operators $U_1$ , $U_2$ and $V$ such that $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}>\\pi ^{-1}m^{\\frac{3}{2}-\\frac{1}{p}} \\Vert f\\Vert _{L^\\infty (2)}\\Vert U_1-U_2\\Vert _{{S}_p}$ for every $p>0$ .", "Proof.", "One can select orthonormal bases $\\lbrace g_\\xi \\rbrace _{\\xi \\in \\Pi _m}$ and $\\lbrace h_\\eta \\rbrace _{\\eta \\in \\Pi _m}$ in an $m$ -dimensional Hilbert space $ such that$ |(g,h)|=m-12$ for all $ ,m$.Indeed, let $ be the subspace of $L^2($ of analytic polynomials of degree less than $m$ .", "We can put $g_\\xi \\stackrel{\\mathrm {def}}{=}\\sqrt{m}\\Upsilon _m(z\\overline{\\xi })$ and $h_\\eta =z^k$ , where $\\eta =e^{2\\pi {\\rm i}k/m}$ , $0\\le k\\le m-1$ .", "Consider the rank one projections $\\lbrace P_\\xi \\rbrace _{\\xi \\in \\Pi _m}$ and $\\lbrace Q_\\eta \\rbrace _{\\eta \\in \\Pi _m}$ defined by $P_\\xi v = (v,g_\\xi )g_\\xi $ , $\\xi \\in \\Pi _m$ , and $Q_\\eta v = (v,h_\\eta )h_\\eta $ , $\\eta \\in \\Pi _m$ .", "We define the unitary operators $U_1$ , $U_2$ , and $V$ by $U_1=\\sum _{\\xi \\in \\Pi _m}\\xi P_\\xi ,\\quad U_2=e^{\\frac{\\pi {\\rm i}}{m}}U_1\\quad \\mbox{and}\\quad V=\\sum _{\\eta \\in \\Pi _m}\\eta Q_\\eta .$ By Lemma REF , there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable such that $f(\\xi ,\\eta )=\\sqrt{m}(g_\\xi ,h_\\eta )$ for all $\\xi , \\eta \\in \\Pi _m$ , $f(\\xi ,\\eta )=0$ for all $\\xi \\in \\Pi _{2m}\\setminus \\Pi _m$ , $\\eta \\in \\Pi _m$ and $\\Vert f\\Vert _{L^\\infty (2)}=1$ .", "Clearly, $f(U_2,V)={0}$ and $f(U_1,V)=\\sum \\limits _{\\xi ,\\eta \\in \\Pi _m}f(\\xi ,\\eta )P_\\xi Q_\\eta $ .", "We have $( f(U_1,V)h_\\eta , g_\\xi )=f(\\xi ,\\eta )(h_\\eta , g_\\xi )=\\frac{1}{\\sqrt{m}}.$ Hence, $\\operatorname{rank}f(U_1,V)=1$ and $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}=\\Vert f(U_1,V)\\Vert _{{S}_p}=\\Vert f(U_1,V)\\Vert _{{S}_2}=\\sqrt{m}.$ It remains to observe that $\\Vert U_1-U_2\\Vert _{{S}_p}=\\big |1-e^{\\frac{\\pi {\\rm i}}{m}}\\big |m^{\\frac{1}{p}}<\\pi m^{\\frac{1}{p}-1}$ .", "$\\blacksquare $ Remark.", "If we replace the polynomial $f$ constructed in the proof of Lemma REF with the polynomial $g$ defined by $g(z_1,z_2)=z_1^{4m-2}z_2^{4m-2}f(z_1,z_2),$ it will obviously satisfy the same inequality: $\\Vert g(U_1,V)-g(U_2,V)\\Vert _{{S}_p}>\\pi ^{-1}m^{\\frac{3}{2}-\\frac{1}{p}} \\Vert g\\Vert _{L^\\infty (2)}\\Vert U_1-U_2\\Vert _{{S}_p}.$ It is easy to deduce from (REF ) that for such polynomials $g$ $c_1m\\Vert g\\Vert _{L^\\infty (2)}\\le \\Vert g\\Vert _{B^\\infty _{\\infty ,1}}\\le c_2m\\Vert g\\Vert _{L^\\infty (2)}$ for some constants $c_1$ and $c_2$ .", "This together with (REF ) implies the following result: Theorem 7.3 Let $M>0$ and $2<p\\le \\infty $ .", "Then there exist unitary operators $U_1$ , $U_2$ , $V$ and an analytic polynomial $f$ in two variables such that $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}>M\\Vert f\\Vert _{B^1_{\\infty ,1}(2)}\\Vert U_1-U_2\\Vert _{{S}_p}.$" ], [ "In this section we state open problems for functions of noncommuting contractions.", "Functions of triples of contractions.", "Recall that it was shown in [29] that for $f\\in B_{\\infty ,1}^1({R})$ , there are no Lipschitz type estimates in the norm of ${S}_p$ for any $p>0$ for functions $f(A,B,C)$ of triples of noncommuting self-adjoint operators.", "We conjecture that the same must be true in the case of functions of triples of not necessarily commuting contractions.", "Note that the construction given in [29] does not generalize to the case of functions of contractions.", "Lipschitz functions of noncommuting contractions.", "Recall that an unknown referee of [1] observed that for Lipschitz functions $f$ on the real line there are no Lipschitz type estimates for functions $f(A,B)$ of noncommuting self-adjoint operators in the Hilbert–Schmidt norm.", "The construction is given in [1].", "We conjecture that the same result must hold in the case of functions of noncommuting contractions.", "Lipschitz type estimates for $p>2$ and Hölder type estimates.", "It follows from results of [1] that in the case of functions of noncommuting self-adjoint operators for any $s>0$ , $q>0$ and $p>2$ , there exist pairs of self-adjoint operators $(A_0,A_1)$ and $(B_0,B_1)$ and a function $f$ in the homogeneous Besov space $B_{\\infty ,q}^s({R})$ such that $\\Vert f(A_1,B_1)-f(A_0,B_0)\\Vert _{{S}_p}$ can be arbitrarily large while $\\max \\lbrace \\Vert A_1-A_0\\Vert _{{S}_p},\\Vert B_1-B_0\\Vert _{{S}_p}\\rbrace $ can be arbitrarily small.", "In particular, the condition $f\\in B_{\\infty ,q}^s({R})$ does not imply any Lipschitz or Hölder type estimates in the norm of ${S}_p$ , $p>2$ , for any positive $s$ and $q$ .", "It is easy to see that in the case of contractions the situation is different: for any $q>0$ and $p\\ge 1$ , there exists $s>0$ such that the condition $f\\in B_{\\infty ,q}^s$ guarantees a Lipschitz type estimate for functions of not necessarily commuting contractions in ${S}_p$ .", "It would be interesting to find optimal conditions on $f$ that would guarantee Lipschitz or Hölder type estimates in ${S}_p$ for a given $p$ .", "Table: NO_CAPTION" ], [ "In this section we obtain Lipschitz type estimates in the Schatten–von Neumann classes ${S}_p$ for $p\\in [1,2]$ for functions of contractions.", "To obtain such estimates, we are going to use an elementary approach and obtain elementary formulae that involve only finite sums.", "Later we will need explicit expressions for operator differences, which will be obtained in the next section in terms of triple operator integrals.", "Such formulae will be used in §  to obtain formulae for operator derivatives.", "Suppose that $f$ is a function that belongs to the Besov space $\\big (B_{\\infty ,1}^1\\big )_+(2)$ of analytic functions (see §).", "As we have observed in Subsection 3.3, we can define functions $f(T,R)$ for (not necessarily commuting) contractions $T$ and $R$ on Hilbert space by formula ().", "For a differentiable function $f$ on $, we use the notation$ Df$ for the divided difference:$$(Df)(\\zeta ,)\\stackrel{\\mathrm {def}}{=}\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\frac{f(\\zeta )-f()}{\\zeta -}},&\\zeta \\ne \\\\[.4cm]f^{\\prime }(\\zeta ),&\\zeta =,\\end{array}\\right.\\qquad \\zeta ,\\;\\in $$For a differentiable function $ f$ on $ 2$, we define the divided differences$ D[1]f$ and $ D[2]f$ by$$\\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\stackrel{\\mathrm {def}}{=}\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\frac{f(\\zeta _1,)-f(\\zeta _2,)}{\\zeta _1-\\zeta _2}},&\\zeta _1\\ne \\zeta _2,\\\\[.4cm]\\displaystyle {\\frac{\\partial f}{\\partial \\zeta }\\Big |_{\\zeta =\\zeta _1}},&\\zeta _1=\\zeta _2,\\end{array}\\right.\\qquad \\zeta _1,\\;\\zeta _2,\\:\\in $$and$$\\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\stackrel{\\mathrm {def}}{=}\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\frac{f(\\zeta ,_1)-f(\\zeta ,_2)}{_1-_2}},&_1\\ne _2,\\\\[.4cm]\\displaystyle {\\frac{\\partial f}{\\partial }\\Big |_{=_1}},&_1=_2,\\end{array}\\right.\\qquad \\zeta ,\\;_1,\\:_2\\in $$$ We need several elementary identities.", "Let $\\Pi _m$ be the set of $m$ th roots of 1: $\\Pi _m\\stackrel{\\mathrm {def}}{=}\\lbrace \\xi \\in \\xi ^m=1\\rbrace $ and let $\\Upsilon _m(\\zeta )\\stackrel{\\mathrm {def}}{=}\\frac{\\zeta ^{m}-1}{m(\\zeta -1)}=\\frac{1}{m}\\sum _{k=0}^{m-1}\\zeta ^k,\\quad \\zeta \\in $ The following elementary formulae are well known.", "We give proofs for completeness.", "Lemma 4.1 Let $f$ and $g$ be analytic polynomials in one variable of degree less than $m$ .", "Then $\\int _f̰\\overline{g}\\,d{m}=\\frac{1}{m}\\sum _{\\xi \\in \\Pi _m}f(\\xi )\\overline{g(\\xi )}.$ In particular, $\\int _f|^2\\,d{m}=\\frac{1}{m}\\sum _{\\xi \\in \\Pi _m}|f(\\xi )|^2.$ Proof.", "It suffices to consider the case where $f(z)=z^j$ and $g(z)=z^k$ with $0\\le j, k<m$ .", "Then $-m<j-k<m$ and $\\sum _{\\xi \\in \\Pi _m}\\xi ^j\\,\\overline{\\xi }^k=\\left\\lbrace \\begin{array}{ll}0,&j\\ne k\\\\[.2cm]m,&j=k.\\end{array}\\right.\\quad \\blacksquare $ Corollary 4.2 $\\sum _{\\xi \\in \\Pi _m}|\\Upsilon _m(\\zeta \\bar{\\xi })|^2=1,\\quad \\zeta \\in $ In the same way we can obtain similar formulae for polynomials in several variables.", "We need only the case of two variables.", "Lemma 4.3 Let $f$ and $g$ be polynomials in two variables of degree less than $m$ in each variable.", "Then $\\int _{2} f\\overline{g}\\,d{m}_2=\\frac{1}{m^2}\\sum _{\\xi ,\\eta \\in \\Pi _m}f(\\xi ,\\eta )\\overline{g(\\xi ,\\eta )}.$ In particular, $\\int _{2} |f|^2\\,d{m}_2=\\frac{1}{m^2}\\sum _{\\xi ,\\eta \\in \\Pi _m}|f(\\xi ,\\eta )|^2.$ Proof.", "It suffices to consider the case when $f(\\zeta ,)=\\zeta ^{j_1}^{j_2}$ and $g(\\zeta ,)=\\zeta ^{k_1}^{k_2}$ with $0\\le j_1, j_2, k_1, k_2<m$ .", "Then $-m<j_1-k_1, j_2-k_2<m$ and $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\xi ^{j_1}\\eta ^{j_2}\\,\\overline{\\xi }^{k_1}\\,\\overline{\\eta }^{k_2}=\\left\\lbrace \\begin{array}{ll}0,&(j_1,j_2)\\ne (k_1,k_2)\\\\[.2cm]m^2,&(j_1,j_2)=(k_1,k_2).\\end{array}\\right.\\quad \\blacksquare $ Suppose now that $(T_0,R_0)$ and $(T_1,R_1)$ are pairs of not necessarily commuting contractions.", "Theorem 4.4 Let $f$ be an analytic polynomial in two variable of degree at most $m$ in each variable.", "Then $f(T_1,R_1)-f(T_0,R_1)=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)\\,(D^{[1]}f)(\\xi ,\\eta ,R_1)$ and $f(T_0,R_1)-f(T_0,R_0)=\\sum _{\\xi ,\\eta \\in \\Pi _m}(D^{[2]}f)(T_0,\\xi ,\\eta )\\,\\Upsilon _m(\\overline{\\xi }R_1)(R_1-R_0)\\,\\Upsilon _m(\\overline{\\eta }R_0).$ We are going to establish (REF ).", "The proof of (REF ) is similar.", "We need the following lemma.", "Lemma 4.5 Let $ be an analytic polynomial in one variable of degree at most $ m$.", "Then$$T_1)-T_0)=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)(D(\\xi ,\\eta ).$$$ Proof of the lemma.", "Let $0\\le j, j_0, k, k_0<m$ .", "Then $\\sum _{\\xi ,\\eta \\in \\Pi _m}(\\overline{\\xi }T_1)^{j_0}\\,(\\overline{\\eta }T_0)^{k_0}\\xi ^j\\eta ^k=\\left\\lbrace \\begin{array}{ll}\\displaystyle {m^2T_1^jT_0^k},&(j_0,k_0)=(j,k),\\\\[.4cm]0,&(j_0,k_0)\\ne (j,k).\\end{array}\\right.$ Thus, $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)\\,\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k=T_1^jT_0^k$ if $0\\le j, k<n$ .", "Hence, $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)T_1\\,\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k=T_1^{j+1}T_0^k$ and $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)\\,T_0\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k=T_1^{j}T_0^{k+1}.$ It follows that $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k=T_1^j(T_1-T_0)T_0^k$ whenever $0\\le j, k<m$ .", "Let $\\sum \\limits _{s=0}^ma_sz^s$ .", "It is easy to see that $(D(z,w)=\\sum _{j,k\\ge 0, j+k<m}a_{j+k+1}z^jw^k.$ Hence, $\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)(D(\\xi ,\\eta )\\\\=\\sum _{j,k\\ge 0, j+k<m}a_{j+k+1}\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\overline{\\xi }T_1)(T_1-T_0)\\,\\Upsilon _m(\\overline{\\eta }T_0)\\xi ^j\\eta ^k\\\\=\\sum _{j,k\\ge 0, j+k<m}a_{j+k+1}T_1^j(T_1-T_0)T_0^k=T_1)-T_0).\\quad \\blacksquare $ Proof of Theorem REF .", "Clearly, it suffices to prove (REF ) in the case when $f(z_1,z_2)=z_1)z_2^j$ , where $ is a polynomial of onevariable of degree at most $ n$ and $ 0jm$.Clearly, in this case$$f(T_1,R_1)-f(T_0,R_1)=\\big (T_1)-T_0)\\big )R_1^j.$$On the other hand,$$(Df^{[1]})(\\xi ,\\eta ,R_1)=(D(\\xi ,\\eta )R_1^j.$$Identity (\\ref {perra}) follows now from Lemma \\ref {odnape}.", "$$$ For $K\\in L^2(2)$ , we denote by $I_K$ the integral operator on $L^2($ with kernel function $K$ , i.e., $(I_K(\\zeta )=\\int _K̰(\\zeta ,))\\,d{m}(),\\quad L^2(.$ The following lemma allows us to evaluate the operator norm $\\Vert I_K\\Vert _{\\mathcal {B}(L^2)}$ of this operator for polynomials $K$ of degree less than $m$ in each variable in terms of the operator norms of the matrix $\\lbrace K(\\zeta ,\\eta )\\rbrace _{\\zeta ,\\eta \\in \\Pi _m}$ .", "Lemma 4.6 Let $K$ be an analytic polynomial in two variables of degree less than $m$ in each variable.", "Then $\\Vert \\lbrace K(\\xi ,\\eta )\\rbrace _{\\xi ,\\eta \\in \\Pi _m}\\Vert _{\\mathcal {B}}=m\\Vert I_K\\Vert _{\\mathcal {B}(L^2)}.$ Proof.", "It is easy to see that $\\Vert I_K\\Vert _{\\mathcal {B}(L^2)}=\\sup _{\\Vert \\varphi \\Vert _{L^2}\\le 1,\\Vert \\psi \\Vert _{L^2}\\le 1}\\left|\\iint _{K(\\zeta ,)\\,\\overline{\\varphi (\\zeta )\\psi ()}\\,d{m}(\\zeta )\\,d{m}()\\\\=\\sup _{\\Vert \\varphi \\Vert _{L^2}\\le 1,\\Vert \\psi \\Vert _{L^2}\\le 1}\\left|\\iint _{K(\\zeta ,)\\,\\overline{\\varphi _m(z)\\psi _m()}\\,d{m}(\\zeta )\\,d{m}(),}where \\right.\\varphi _m(z)=\\sum \\limits _{k=0}^{m-1}\\widehat{\\varphi }(k)z^k and \\psi _m(z)=\\sum \\limits _{k=0}^{m-1}\\widehat{\\psi }(k)z^k.Hence,\\Vert I_K\\Vert _{\\mathcal {B}(L^2)}=\\sup \\left|\\iint _{K(\\zeta ,w)\\,\\overline{\\varphi (z)\\psi (w)}\\,d{m}(\\zeta )\\,d{m}(),where the supremum is taken over all polynomials \\varphi and \\psi in one variable of degree less than m and such that\\Vert \\varphi \\Vert _{L^2}\\le 1, \\Vert \\psi \\Vert _{L^2}\\le 1.", "Next, by Lemma \\ref {137},for arbitrary polynomials \\varphi and \\psi with \\deg \\varphi <m and \\deg \\psi <m,we have\\iint _{K(\\zeta ,)\\,\\overline{\\varphi (z)\\psi (w)}\\,d{m}(\\zeta )\\,d{m}()=\\frac{1}{m^2}\\sum _{\\xi ,\\eta \\in \\Pi _m}K(\\xi ,\\eta )\\,\\overline{\\varphi (\\xi )\\psi (\\eta )}.It remains to observe that by Lemma \\ref {136},\\Vert \\varphi \\Vert _{L^2}\\le 1 if and only if \\sum \\limits _{\\xi \\in \\Pi _m}|\\varphi (\\xi )|^2\\le mand the same is true for \\psi .\\blacksquare }\\begin{thm}Let g be a polynomial in one variable of degree at most m. Then\\Vert \\lbrace (Dg)(\\xi ,\\eta )\\rbrace _{\\xi ,\\eta \\in \\Pi _m}\\Vert _{\\mathcal {B}}\\le m\\Vert g\\Vert _{L^\\infty }.\\end{thm}}{\\bf Proof.", "}The result follows from Lemma \\ref {intopera} and the inequality\\right.\\Vert I_{Dg}\\Vert _{\\mathcal {B}(L^2)}\\le \\Vert g\\Vert _{L^\\infty },which is a consequence of the fact that \\Vert I_{Dg}\\Vert _{\\mathcal {B}(L^2)} is equal to the norm of the Hankel operator H_{\\bar{g}} on the Hardy class H^2, see\\cite {Pe5}, Ch.", "1, Th.", "1.10.\\blacksquare }\\right.$ Corollary 4.7 Let $f$ be a trigonometric polynomial of degree at most $m$ in each variable and let $p\\in [1,2]$ .", "Suppose that $T_1,\\,R_1,T_0,\\,R_0$ are contractions such that $T_1-T_0\\in {S}_p$ and $R_1-R_0\\in {S}_p$ .", "Then $\\Vert f(T_1,R_1)-f(T_0,R_0)\\Vert _{{S}_p}\\le 2m\\Vert f\\Vert _{L^\\infty }\\max \\big \\lbrace \\Vert T_1-T_0\\Vert _{{S}_p},\\Vert R_1-R_0\\Vert _{{S}_p}\\big \\rbrace .$ Proof.", "Let us estimate $\\Vert f(T_1,R_1)-f(T_0,R_1)\\Vert _{{S}_p}$ .", "The norm $\\Vert f(T_0,R_1)-f(T_0,R_0)\\Vert _{{S}_p}$ can be estimated in the same way.", "The result is a consequence of formula (REF ), Theorem REF , Theorem REF and Corollare REF .", "$\\blacksquare $ Corollary REF allows us to establish a Lipschitz type inequality for functions in $\\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Theorem 4.8 Let $1\\le p\\le 2$ and let $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Suppose that $T_1,\\,R_1,T_0,\\,R_0$ are contractions such that $T_1-T_0\\in {S}_p$ and $R_1-R_0\\in {S}_p$ .", "Then $\\Vert f(T_1,R_1)-f(T_0,R_0)\\Vert _{{S}_p}\\le \\operatorname{const}\\Vert f\\Vert _{B_{\\infty ,1}^1}\\max \\big \\lbrace \\Vert T_1-T_0\\Vert _{{S}_p},\\Vert R_1-R_0\\Vert _{{S}_p}\\big \\rbrace .$ Proof.", "Indeed, the result follows immediately from Corollary REF and inequality (REF ).", "$\\blacksquare $" ], [ "In this section we obtain an explicit formula for the operator differences $f(T_1,R_1)-f(T_0,R_0)$ , $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , in terms of triple operator integrals.", "Theorem 5.1 Let $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Then $D^{[1]}f\\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\quad \\mbox{and}\\quad D^{[2]}f\\in {\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}.$ Lemma 5.2 Let $f$ be an analytic polynomial in two variables of degree at most $m$ in each variable.", "Then $\\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\zeta _1\\overline{\\xi })\\,\\Upsilon _m(\\zeta _2\\overline{\\eta })\\big (D^{[1]}f\\big )(\\xi ,\\eta ,)$ and $\\big (D^{[2]}f\\big )(\\zeta ,_1,_2)=\\sum _{\\xi ,\\eta \\in \\Pi _m}(D^{[2]}f)(\\zeta ,\\xi ,\\eta )\\Upsilon _m(_1\\overline{\\xi })\\Upsilon _m(_2\\overline{\\eta }).$ Proof.", "Both formulae (REF ) and (REF ) can be verified straightforwardly.", "However, we deduce them from Theorem REF .", "Formula (REF ) follows immediately from formula (REF ) if we consider the special case when $T_0$ , $T_1$ and $R_1$ are the operators on the one-dimensional space of multiplication by $\\zeta _2$ , $\\zeta _1$ and $$ .", "Similarly, formula (REF ) follows immediately from formula (REF ).", "$\\blacksquare $ Corollary 5.3 Under the hypotheses of Lemma REF, $\\big \\Vert D^{[1]}f\\big \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}{\\rm C}_{\\rm A}}\\le m\\Vert f\\Vert _{L^\\infty }\\quad \\mbox{and}\\quad \\big \\Vert D^{[2]}f\\big \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}}\\le m\\Vert f\\Vert _{L^\\infty }.$ Proof.", "The result is a consequence of Lemma REF , Theorem REF , Corollary REF and Definitions 2 and 3 in §.", "$\\blacksquare $ Proof of Theorem REF .", "The result follows immediately from Corollary REF and inequality (REF ).", "$\\blacksquare $ Theorem 5.4 Let $p\\in [1,2]$ .", "Suppose that $T_0,\\,R_0,\\,T_1,\\,R_1$ are contractions such that $T_1-T_0\\in {S}_p$ and $R_1-R_0\\in {S}_p$ .", "Then for $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , the following formula holds: $f(T_1,R_1)&-f(T_0,R_0)\\nonumber \\\\[.2cm]&=\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T_1}(\\zeta _1)(T_1-T_0)\\,dE_{T_2}(\\zeta _2)\\,dE_{R_1}(),\\nonumber \\\\[.2cm]&+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T_2}(\\zeta )\\,dE_{R_1}(_1)(R_1-R_0)\\,dE_{R_2}(_2).$ Proof.", "Suppose first that $f$ is an analytic polynomial in two variables of degree at most $m$ in each variable.", "In this case equality (REF ) is a consequence of Theorem REF , Lemma REF and the definition of triple operator integrals given in Subsection 3.5.", "In the general case we represent $f$ by the series (REF ) and apply (REF ) to each $f_n$ .", "The result follows from (REF ).", "$\\blacksquare $" ], [ "In this section we study differentiability properties of the map $t\\mapsto f\\big (T(t),R(t)\\big )$ in the norm of ${S}_p$ , $1\\le p\\le 2$ , for functions $t\\mapsto T(t)$ and $t\\mapsto R(t)$ that take contractive values and are differentiable in ${S}_p$ .", "We say that an operator-valued function $\\Psi $ defined on an interval $J$ is differentiable in ${S}_p$ if $\\Phi (s)-\\Phi (t)\\in {S}_p$ for any $s,\\,t\\in J$ , and the limit $\\lim _{h\\rightarrow {0}}\\frac{1}{h}\\big (\\Psi (t+h)-\\Psi (t)\\big )\\stackrel{\\mathrm {def}}{=}\\Phi ^{\\prime }(t)$ exists in the norm of ${S}_p$ for each $t$ in $J$ .", "Theorem 6.1 Let $p\\in [1,2]$ and let $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Suppose that $t\\mapsto T(t)$ and $t\\mapsto R(t)$ are operator-valued functions on an interval $J$ that take contractive values and are differentiable in ${S}_p$ .", "Then the function (REF ) is differentiable on $J$ in ${S}_p$ and $\\frac{d}{dt}&f\\big (T(t),R(t)\\big )\\Big |_{t=s}\\\\[.2cm]&=\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T(s)}(\\zeta _1)T^{\\prime }(s)\\,dE_{T(s)}(\\zeta _2)\\,dE_{R(s)}()\\nonumber \\\\[.2cm]&+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T(s)}(\\zeta )\\,dE_{R(s)}(_1)R^{\\prime }(s)\\,dE_{R(s)}(_2),$ $s\\in J$ .", "Proof.", "As before, it suffices to prove the result in the case when $f$ is an analytic polynomial of degree at most $m$ in each variable.", "Suppose that $f$ is such a polynomial.", "Put $F(t)\\stackrel{\\mathrm {def}}{=}f\\big (T(t),R(t)\\big )$ .", "We have $F(s+h)&-F(s)\\\\[.2cm]&=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m\\big (\\overline{\\xi }T(s+h)\\big )\\big (T(s+h)-T(s)\\big )\\Upsilon _m\\big (\\overline{\\eta }T(s)\\big )\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s+h)\\big )\\\\[.2cm]&+\\sum _{\\xi ,\\eta \\in \\Pi _m}\\big (D^{[2]}f\\big )\\big (T(s),\\xi ,\\eta \\big )\\Upsilon _m\\big (\\overline{\\xi }R(s+h)\\big )\\big (R(s+h)-R(s)\\big )\\Upsilon _m\\big (\\overline{\\eta }R(s)\\big ).$ Clearly, $\\lim _{h\\rightarrow 0}\\frac{1}{h}\\big (T(s+h)-T(s)\\big )=T^{\\prime }(s)\\quad \\mbox{and}\\quad \\lim _{h\\rightarrow 0}\\frac{1}{h}\\big (R(s+h)-R(s)\\big )=R^{\\prime }(s)$ in the norm of ${S}_p$ .", "On the other hand, it is easy to see that $\\lim _{h\\rightarrow 0}\\Upsilon _m\\big (\\overline{\\xi }T(s+h)\\big )=\\Upsilon _m\\big (\\overline{\\xi }T(s)\\big ),\\quad \\lim _{h\\rightarrow 0}\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s+h)\\big )=\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s)\\big )$ and $\\lim _{h\\rightarrow 0}\\Upsilon _m\\big (\\overline{\\xi }R(s+h)\\big )=\\Upsilon _m\\big (\\overline{\\xi }R(s)\\big )$ in the operator norm.", "Hence, $F^{\\prime }(s)&=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m\\big (\\overline{\\xi }T(s)\\big )T^{\\prime }(s)\\Upsilon _m\\big (\\overline{\\eta }T(s)\\big )\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s)\\big )\\\\[.2cm]&+\\sum _{\\xi ,\\eta \\in \\Pi _m}\\big (D^{[2]}f\\big )\\big (T(s),\\xi ,\\eta \\big )\\Upsilon _m\\big (\\overline{\\xi }R(s)\\big )R^{\\prime }(s)\\Upsilon _m\\big (\\overline{\\eta }R(s)\\big ).$ It follows now from Lemma REF and from the definition of triple operator integrals given in § that the right-hand side is equal to $\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T(s)}(\\zeta _1)T^{\\prime }(s)\\,dE_{T(s)}(\\zeta _2)\\,dE_{R(s)}()\\\\[.2cm]+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T(s)}(\\zeta )\\,dE_{R(s)}(_1)R^{\\prime }(s)\\,dE_{R(s)}(_2)$ which completes the proof.", "$\\blacksquare $" ], [ "In this section we show that unlike in the case $p\\in [1,2]$ , there are no Lipschitz type estimates in the norm of ${S}_p$ in the case when $p>2$ for functions $f(T,R)$ , $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , of not noncommuting contractions.", "In particular, there are no such Lipschitz type estimates for functions $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ in the operator norm.", "Moreover, we show that for $p>2$ , such Lipschitz type estimates do not hold even for functions $f$ in $\\big (B_{\\infty ,1}^1\\big )_+(2)$ and for pairs of noncommuting unitary operators.", "Recall that similar results were obtained in [1] for functions of noncommuting self-adjoint operators.", "However, in this paper we use a different construction to obtain results for functions of unitary operators.", "Lemma 7.1 For each matrix $\\lbrace a_{\\xi \\, \\eta }\\rbrace _{\\xi , \\eta \\in \\Pi _m}$ , there exists an analytic polynomial $f$ in two variables of degree at most $2m-2$ in each variable such that $f(\\xi ,\\eta )=a_{\\xi \\, \\eta }$ for all $\\xi , \\eta \\in \\Pi _m$ and $\\Vert f\\Vert _{L^\\infty (2)}\\le \\sup \\limits _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|$ .", "Proof.", "Put $f(z,w)\\stackrel{\\mathrm {def}}{=}\\sum _{\\xi , \\eta \\in \\Pi _m}a_{\\xi \\, \\eta }\\Upsilon _m^2(z\\overline{\\xi })\\Upsilon _m^2(w\\overline{\\eta }).$ Clearly, $f(\\xi ,\\eta )=a_{\\xi \\, \\eta }$ for all $\\xi , \\eta \\in \\Pi _m$ and $|f(z,w)|&\\le \\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|\\sum _{\\xi , \\eta \\in \\Pi _m}|\\Upsilon _m(z\\overline{\\xi })|^2|\\Upsilon _m(w\\overline{\\eta })|^2\\\\[.2cm]&=\\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|\\sum _{\\xi \\in \\Pi _m}|\\Upsilon _m(z\\overline{\\xi })|^2\\sum _{\\eta \\in \\Pi _m}|\\Upsilon _m(w\\overline{\\eta })|^2=\\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|$ by Corollary REF .", "$\\blacksquare $ Lemma 7.2 For each $m\\in N$ , there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable, and unitary operators $U_1$ , $U_2$ and $V$ such that $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}>\\pi ^{-1}m^{\\frac{3}{2}-\\frac{1}{p}} \\Vert f\\Vert _{L^\\infty (2)}\\Vert U_1-U_2\\Vert _{{S}_p}$ for every $p>0$ .", "Proof.", "One can select orthonormal bases $\\lbrace g_\\xi \\rbrace _{\\xi \\in \\Pi _m}$ and $\\lbrace h_\\eta \\rbrace _{\\eta \\in \\Pi _m}$ in an $m$ -dimensional Hilbert space $ such that$ |(g,h)|=m-12$ for all $ ,m$.Indeed, let $ be the subspace of $L^2($ of analytic polynomials of degree less than $m$ .", "We can put $g_\\xi \\stackrel{\\mathrm {def}}{=}\\sqrt{m}\\Upsilon _m(z\\overline{\\xi })$ and $h_\\eta =z^k$ , where $\\eta =e^{2\\pi {\\rm i}k/m}$ , $0\\le k\\le m-1$ .", "Consider the rank one projections $\\lbrace P_\\xi \\rbrace _{\\xi \\in \\Pi _m}$ and $\\lbrace Q_\\eta \\rbrace _{\\eta \\in \\Pi _m}$ defined by $P_\\xi v = (v,g_\\xi )g_\\xi $ , $\\xi \\in \\Pi _m$ , and $Q_\\eta v = (v,h_\\eta )h_\\eta $ , $\\eta \\in \\Pi _m$ .", "We define the unitary operators $U_1$ , $U_2$ , and $V$ by $U_1=\\sum _{\\xi \\in \\Pi _m}\\xi P_\\xi ,\\quad U_2=e^{\\frac{\\pi {\\rm i}}{m}}U_1\\quad \\mbox{and}\\quad V=\\sum _{\\eta \\in \\Pi _m}\\eta Q_\\eta .$ By Lemma REF , there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable such that $f(\\xi ,\\eta )=\\sqrt{m}(g_\\xi ,h_\\eta )$ for all $\\xi , \\eta \\in \\Pi _m$ , $f(\\xi ,\\eta )=0$ for all $\\xi \\in \\Pi _{2m}\\setminus \\Pi _m$ , $\\eta \\in \\Pi _m$ and $\\Vert f\\Vert _{L^\\infty (2)}=1$ .", "Clearly, $f(U_2,V)={0}$ and $f(U_1,V)=\\sum \\limits _{\\xi ,\\eta \\in \\Pi _m}f(\\xi ,\\eta )P_\\xi Q_\\eta $ .", "We have $( f(U_1,V)h_\\eta , g_\\xi )=f(\\xi ,\\eta )(h_\\eta , g_\\xi )=\\frac{1}{\\sqrt{m}}.$ Hence, $\\operatorname{rank}f(U_1,V)=1$ and $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}=\\Vert f(U_1,V)\\Vert _{{S}_p}=\\Vert f(U_1,V)\\Vert _{{S}_2}=\\sqrt{m}.$ It remains to observe that $\\Vert U_1-U_2\\Vert _{{S}_p}=\\big |1-e^{\\frac{\\pi {\\rm i}}{m}}\\big |m^{\\frac{1}{p}}<\\pi m^{\\frac{1}{p}-1}$ .", "$\\blacksquare $ Remark.", "If we replace the polynomial $f$ constructed in the proof of Lemma REF with the polynomial $g$ defined by $g(z_1,z_2)=z_1^{4m-2}z_2^{4m-2}f(z_1,z_2),$ it will obviously satisfy the same inequality: $\\Vert g(U_1,V)-g(U_2,V)\\Vert _{{S}_p}>\\pi ^{-1}m^{\\frac{3}{2}-\\frac{1}{p}} \\Vert g\\Vert _{L^\\infty (2)}\\Vert U_1-U_2\\Vert _{{S}_p}.$ It is easy to deduce from (REF ) that for such polynomials $g$ $c_1m\\Vert g\\Vert _{L^\\infty (2)}\\le \\Vert g\\Vert _{B^\\infty _{\\infty ,1}}\\le c_2m\\Vert g\\Vert _{L^\\infty (2)}$ for some constants $c_1$ and $c_2$ .", "This together with (REF ) implies the following result: Theorem 7.3 Let $M>0$ and $2<p\\le \\infty $ .", "Then there exist unitary operators $U_1$ , $U_2$ , $V$ and an analytic polynomial $f$ in two variables such that $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}>M\\Vert f\\Vert _{B^1_{\\infty ,1}(2)}\\Vert U_1-U_2\\Vert _{{S}_p}.$" ], [ "In this section we state open problems for functions of noncommuting contractions.", "Functions of triples of contractions.", "Recall that it was shown in [29] that for $f\\in B_{\\infty ,1}^1({R})$ , there are no Lipschitz type estimates in the norm of ${S}_p$ for any $p>0$ for functions $f(A,B,C)$ of triples of noncommuting self-adjoint operators.", "We conjecture that the same must be true in the case of functions of triples of not necessarily commuting contractions.", "Note that the construction given in [29] does not generalize to the case of functions of contractions.", "Lipschitz functions of noncommuting contractions.", "Recall that an unknown referee of [1] observed that for Lipschitz functions $f$ on the real line there are no Lipschitz type estimates for functions $f(A,B)$ of noncommuting self-adjoint operators in the Hilbert–Schmidt norm.", "The construction is given in [1].", "We conjecture that the same result must hold in the case of functions of noncommuting contractions.", "Lipschitz type estimates for $p>2$ and Hölder type estimates.", "It follows from results of [1] that in the case of functions of noncommuting self-adjoint operators for any $s>0$ , $q>0$ and $p>2$ , there exist pairs of self-adjoint operators $(A_0,A_1)$ and $(B_0,B_1)$ and a function $f$ in the homogeneous Besov space $B_{\\infty ,q}^s({R})$ such that $\\Vert f(A_1,B_1)-f(A_0,B_0)\\Vert _{{S}_p}$ can be arbitrarily large while $\\max \\lbrace \\Vert A_1-A_0\\Vert _{{S}_p},\\Vert B_1-B_0\\Vert _{{S}_p}\\rbrace $ can be arbitrarily small.", "In particular, the condition $f\\in B_{\\infty ,q}^s({R})$ does not imply any Lipschitz or Hölder type estimates in the norm of ${S}_p$ , $p>2$ , for any positive $s$ and $q$ .", "It is easy to see that in the case of contractions the situation is different: for any $q>0$ and $p\\ge 1$ , there exists $s>0$ such that the condition $f\\in B_{\\infty ,q}^s$ guarantees a Lipschitz type estimate for functions of not necessarily commuting contractions in ${S}_p$ .", "It would be interesting to find optimal conditions on $f$ that would guarantee Lipschitz or Hölder type estimates in ${S}_p$ for a given $p$ .", "Table: NO_CAPTION" ], [ "In this section we obtain an explicit formula for the operator differences $f(T_1,R_1)-f(T_0,R_0)$ , $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , in terms of triple operator integrals.", "Theorem 5.1 Let $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Then $D^{[1]}f\\in {\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\quad \\mbox{and}\\quad D^{[2]}f\\in {\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}.$ Lemma 5.2 Let $f$ be an analytic polynomial in two variables of degree at most $m$ in each variable.", "Then $\\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m(\\zeta _1\\overline{\\xi })\\,\\Upsilon _m(\\zeta _2\\overline{\\eta })\\big (D^{[1]}f\\big )(\\xi ,\\eta ,)$ and $\\big (D^{[2]}f\\big )(\\zeta ,_1,_2)=\\sum _{\\xi ,\\eta \\in \\Pi _m}(D^{[2]}f)(\\zeta ,\\xi ,\\eta )\\Upsilon _m(_1\\overline{\\xi })\\Upsilon _m(_2\\overline{\\eta }).$ Proof.", "Both formulae (REF ) and (REF ) can be verified straightforwardly.", "However, we deduce them from Theorem REF .", "Formula (REF ) follows immediately from formula (REF ) if we consider the special case when $T_0$ , $T_1$ and $R_1$ are the operators on the one-dimensional space of multiplication by $\\zeta _2$ , $\\zeta _1$ and $$ .", "Similarly, formula (REF ) follows immediately from formula (REF ).", "$\\blacksquare $ Corollary 5.3 Under the hypotheses of Lemma REF, $\\big \\Vert D^{[1]}f\\big \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}{\\rm C}_{\\rm A}}\\le m\\Vert f\\Vert _{L^\\infty }\\quad \\mbox{and}\\quad \\big \\Vert D^{[2]}f\\big \\Vert _{{\\rm C}_{\\rm A}\\!\\otimes ^{\\rm h}{\\rm C}_{\\rm A}\\!\\otimes _{\\rm h}\\!", "{\\rm C}_{\\rm A}}\\le m\\Vert f\\Vert _{L^\\infty }.$ Proof.", "The result is a consequence of Lemma REF , Theorem REF , Corollary REF and Definitions 2 and 3 in §.", "$\\blacksquare $ Proof of Theorem REF .", "The result follows immediately from Corollary REF and inequality (REF ).", "$\\blacksquare $ Theorem 5.4 Let $p\\in [1,2]$ .", "Suppose that $T_0,\\,R_0,\\,T_1,\\,R_1$ are contractions such that $T_1-T_0\\in {S}_p$ and $R_1-R_0\\in {S}_p$ .", "Then for $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , the following formula holds: $f(T_1,R_1)&-f(T_0,R_0)\\nonumber \\\\[.2cm]&=\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T_1}(\\zeta _1)(T_1-T_0)\\,dE_{T_2}(\\zeta _2)\\,dE_{R_1}(),\\nonumber \\\\[.2cm]&+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T_2}(\\zeta )\\,dE_{R_1}(_1)(R_1-R_0)\\,dE_{R_2}(_2).$ Proof.", "Suppose first that $f$ is an analytic polynomial in two variables of degree at most $m$ in each variable.", "In this case equality (REF ) is a consequence of Theorem REF , Lemma REF and the definition of triple operator integrals given in Subsection 3.5.", "In the general case we represent $f$ by the series (REF ) and apply (REF ) to each $f_n$ .", "The result follows from (REF ).", "$\\blacksquare $" ], [ "In this section we study differentiability properties of the map $t\\mapsto f\\big (T(t),R(t)\\big )$ in the norm of ${S}_p$ , $1\\le p\\le 2$ , for functions $t\\mapsto T(t)$ and $t\\mapsto R(t)$ that take contractive values and are differentiable in ${S}_p$ .", "We say that an operator-valued function $\\Psi $ defined on an interval $J$ is differentiable in ${S}_p$ if $\\Phi (s)-\\Phi (t)\\in {S}_p$ for any $s,\\,t\\in J$ , and the limit $\\lim _{h\\rightarrow {0}}\\frac{1}{h}\\big (\\Psi (t+h)-\\Psi (t)\\big )\\stackrel{\\mathrm {def}}{=}\\Phi ^{\\prime }(t)$ exists in the norm of ${S}_p$ for each $t$ in $J$ .", "Theorem 6.1 Let $p\\in [1,2]$ and let $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ .", "Suppose that $t\\mapsto T(t)$ and $t\\mapsto R(t)$ are operator-valued functions on an interval $J$ that take contractive values and are differentiable in ${S}_p$ .", "Then the function (REF ) is differentiable on $J$ in ${S}_p$ and $\\frac{d}{dt}&f\\big (T(t),R(t)\\big )\\Big |_{t=s}\\\\[.2cm]&=\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T(s)}(\\zeta _1)T^{\\prime }(s)\\,dE_{T(s)}(\\zeta _2)\\,dE_{R(s)}()\\nonumber \\\\[.2cm]&+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T(s)}(\\zeta )\\,dE_{R(s)}(_1)R^{\\prime }(s)\\,dE_{R(s)}(_2),$ $s\\in J$ .", "Proof.", "As before, it suffices to prove the result in the case when $f$ is an analytic polynomial of degree at most $m$ in each variable.", "Suppose that $f$ is such a polynomial.", "Put $F(t)\\stackrel{\\mathrm {def}}{=}f\\big (T(t),R(t)\\big )$ .", "We have $F(s+h)&-F(s)\\\\[.2cm]&=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m\\big (\\overline{\\xi }T(s+h)\\big )\\big (T(s+h)-T(s)\\big )\\Upsilon _m\\big (\\overline{\\eta }T(s)\\big )\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s+h)\\big )\\\\[.2cm]&+\\sum _{\\xi ,\\eta \\in \\Pi _m}\\big (D^{[2]}f\\big )\\big (T(s),\\xi ,\\eta \\big )\\Upsilon _m\\big (\\overline{\\xi }R(s+h)\\big )\\big (R(s+h)-R(s)\\big )\\Upsilon _m\\big (\\overline{\\eta }R(s)\\big ).$ Clearly, $\\lim _{h\\rightarrow 0}\\frac{1}{h}\\big (T(s+h)-T(s)\\big )=T^{\\prime }(s)\\quad \\mbox{and}\\quad \\lim _{h\\rightarrow 0}\\frac{1}{h}\\big (R(s+h)-R(s)\\big )=R^{\\prime }(s)$ in the norm of ${S}_p$ .", "On the other hand, it is easy to see that $\\lim _{h\\rightarrow 0}\\Upsilon _m\\big (\\overline{\\xi }T(s+h)\\big )=\\Upsilon _m\\big (\\overline{\\xi }T(s)\\big ),\\quad \\lim _{h\\rightarrow 0}\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s+h)\\big )=\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s)\\big )$ and $\\lim _{h\\rightarrow 0}\\Upsilon _m\\big (\\overline{\\xi }R(s+h)\\big )=\\Upsilon _m\\big (\\overline{\\xi }R(s)\\big )$ in the operator norm.", "Hence, $F^{\\prime }(s)&=\\sum _{\\xi ,\\eta \\in \\Pi _m}\\Upsilon _m\\big (\\overline{\\xi }T(s)\\big )T^{\\prime }(s)\\Upsilon _m\\big (\\overline{\\eta }T(s)\\big )\\big (D^{[1]}f\\big )\\big (\\xi ,\\eta ,R(s)\\big )\\\\[.2cm]&+\\sum _{\\xi ,\\eta \\in \\Pi _m}\\big (D^{[2]}f\\big )\\big (T(s),\\xi ,\\eta \\big )\\Upsilon _m\\big (\\overline{\\xi }R(s)\\big )R^{\\prime }(s)\\Upsilon _m\\big (\\overline{\\eta }R(s)\\big ).$ It follows now from Lemma REF and from the definition of triple operator integrals given in § that the right-hand side is equal to $\\iint \\!\\!\\upintop \\nolimits \\big (D^{[1]}f\\big )(\\zeta _1,\\zeta _2,)\\,dE_{T(s)}(\\zeta _1)T^{\\prime }(s)\\,dE_{T(s)}(\\zeta _2)\\,dE_{R(s)}()\\\\[.2cm]+\\upintop \\nolimits \\!\\!\\!\\iint \\big (D^{[2]}f\\big )(\\zeta ,_1,_2)\\,dE_{T(s)}(\\zeta )\\,dE_{R(s)}(_1)R^{\\prime }(s)\\,dE_{R(s)}(_2)$ which completes the proof.", "$\\blacksquare $" ], [ "In this section we show that unlike in the case $p\\in [1,2]$ , there are no Lipschitz type estimates in the norm of ${S}_p$ in the case when $p>2$ for functions $f(T,R)$ , $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ , of not noncommuting contractions.", "In particular, there are no such Lipschitz type estimates for functions $f\\in \\big (B_{\\infty ,1}^1\\big )_+(2)$ in the operator norm.", "Moreover, we show that for $p>2$ , such Lipschitz type estimates do not hold even for functions $f$ in $\\big (B_{\\infty ,1}^1\\big )_+(2)$ and for pairs of noncommuting unitary operators.", "Recall that similar results were obtained in [1] for functions of noncommuting self-adjoint operators.", "However, in this paper we use a different construction to obtain results for functions of unitary operators.", "Lemma 7.1 For each matrix $\\lbrace a_{\\xi \\, \\eta }\\rbrace _{\\xi , \\eta \\in \\Pi _m}$ , there exists an analytic polynomial $f$ in two variables of degree at most $2m-2$ in each variable such that $f(\\xi ,\\eta )=a_{\\xi \\, \\eta }$ for all $\\xi , \\eta \\in \\Pi _m$ and $\\Vert f\\Vert _{L^\\infty (2)}\\le \\sup \\limits _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|$ .", "Proof.", "Put $f(z,w)\\stackrel{\\mathrm {def}}{=}\\sum _{\\xi , \\eta \\in \\Pi _m}a_{\\xi \\, \\eta }\\Upsilon _m^2(z\\overline{\\xi })\\Upsilon _m^2(w\\overline{\\eta }).$ Clearly, $f(\\xi ,\\eta )=a_{\\xi \\, \\eta }$ for all $\\xi , \\eta \\in \\Pi _m$ and $|f(z,w)|&\\le \\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|\\sum _{\\xi , \\eta \\in \\Pi _m}|\\Upsilon _m(z\\overline{\\xi })|^2|\\Upsilon _m(w\\overline{\\eta })|^2\\\\[.2cm]&=\\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|\\sum _{\\xi \\in \\Pi _m}|\\Upsilon _m(z\\overline{\\xi })|^2\\sum _{\\eta \\in \\Pi _m}|\\Upsilon _m(w\\overline{\\eta })|^2=\\sup _{\\xi , \\eta \\in \\Pi _m}|a_{\\xi \\, \\eta }|$ by Corollary REF .", "$\\blacksquare $ Lemma 7.2 For each $m\\in N$ , there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable, and unitary operators $U_1$ , $U_2$ and $V$ such that $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}>\\pi ^{-1}m^{\\frac{3}{2}-\\frac{1}{p}} \\Vert f\\Vert _{L^\\infty (2)}\\Vert U_1-U_2\\Vert _{{S}_p}$ for every $p>0$ .", "Proof.", "One can select orthonormal bases $\\lbrace g_\\xi \\rbrace _{\\xi \\in \\Pi _m}$ and $\\lbrace h_\\eta \\rbrace _{\\eta \\in \\Pi _m}$ in an $m$ -dimensional Hilbert space $ such that$ |(g,h)|=m-12$ for all $ ,m$.Indeed, let $ be the subspace of $L^2($ of analytic polynomials of degree less than $m$ .", "We can put $g_\\xi \\stackrel{\\mathrm {def}}{=}\\sqrt{m}\\Upsilon _m(z\\overline{\\xi })$ and $h_\\eta =z^k$ , where $\\eta =e^{2\\pi {\\rm i}k/m}$ , $0\\le k\\le m-1$ .", "Consider the rank one projections $\\lbrace P_\\xi \\rbrace _{\\xi \\in \\Pi _m}$ and $\\lbrace Q_\\eta \\rbrace _{\\eta \\in \\Pi _m}$ defined by $P_\\xi v = (v,g_\\xi )g_\\xi $ , $\\xi \\in \\Pi _m$ , and $Q_\\eta v = (v,h_\\eta )h_\\eta $ , $\\eta \\in \\Pi _m$ .", "We define the unitary operators $U_1$ , $U_2$ , and $V$ by $U_1=\\sum _{\\xi \\in \\Pi _m}\\xi P_\\xi ,\\quad U_2=e^{\\frac{\\pi {\\rm i}}{m}}U_1\\quad \\mbox{and}\\quad V=\\sum _{\\eta \\in \\Pi _m}\\eta Q_\\eta .$ By Lemma REF , there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable such that $f(\\xi ,\\eta )=\\sqrt{m}(g_\\xi ,h_\\eta )$ for all $\\xi , \\eta \\in \\Pi _m$ , $f(\\xi ,\\eta )=0$ for all $\\xi \\in \\Pi _{2m}\\setminus \\Pi _m$ , $\\eta \\in \\Pi _m$ and $\\Vert f\\Vert _{L^\\infty (2)}=1$ .", "Clearly, $f(U_2,V)={0}$ and $f(U_1,V)=\\sum \\limits _{\\xi ,\\eta \\in \\Pi _m}f(\\xi ,\\eta )P_\\xi Q_\\eta $ .", "We have $( f(U_1,V)h_\\eta , g_\\xi )=f(\\xi ,\\eta )(h_\\eta , g_\\xi )=\\frac{1}{\\sqrt{m}}.$ Hence, $\\operatorname{rank}f(U_1,V)=1$ and $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}=\\Vert f(U_1,V)\\Vert _{{S}_p}=\\Vert f(U_1,V)\\Vert _{{S}_2}=\\sqrt{m}.$ It remains to observe that $\\Vert U_1-U_2\\Vert _{{S}_p}=\\big |1-e^{\\frac{\\pi {\\rm i}}{m}}\\big |m^{\\frac{1}{p}}<\\pi m^{\\frac{1}{p}-1}$ .", "$\\blacksquare $ Remark.", "If we replace the polynomial $f$ constructed in the proof of Lemma REF with the polynomial $g$ defined by $g(z_1,z_2)=z_1^{4m-2}z_2^{4m-2}f(z_1,z_2),$ it will obviously satisfy the same inequality: $\\Vert g(U_1,V)-g(U_2,V)\\Vert _{{S}_p}>\\pi ^{-1}m^{\\frac{3}{2}-\\frac{1}{p}} \\Vert g\\Vert _{L^\\infty (2)}\\Vert U_1-U_2\\Vert _{{S}_p}.$ It is easy to deduce from (REF ) that for such polynomials $g$ $c_1m\\Vert g\\Vert _{L^\\infty (2)}\\le \\Vert g\\Vert _{B^\\infty _{\\infty ,1}}\\le c_2m\\Vert g\\Vert _{L^\\infty (2)}$ for some constants $c_1$ and $c_2$ .", "This together with (REF ) implies the following result: Theorem 7.3 Let $M>0$ and $2<p\\le \\infty $ .", "Then there exist unitary operators $U_1$ , $U_2$ , $V$ and an analytic polynomial $f$ in two variables such that $\\Vert f(U_1,V)-f(U_2,V)\\Vert _{{S}_p}>M\\Vert f\\Vert _{B^1_{\\infty ,1}(2)}\\Vert U_1-U_2\\Vert _{{S}_p}.$" ], [ "In this section we state open problems for functions of noncommuting contractions.", "Functions of triples of contractions.", "Recall that it was shown in [29] that for $f\\in B_{\\infty ,1}^1({R})$ , there are no Lipschitz type estimates in the norm of ${S}_p$ for any $p>0$ for functions $f(A,B,C)$ of triples of noncommuting self-adjoint operators.", "We conjecture that the same must be true in the case of functions of triples of not necessarily commuting contractions.", "Note that the construction given in [29] does not generalize to the case of functions of contractions.", "Lipschitz functions of noncommuting contractions.", "Recall that an unknown referee of [1] observed that for Lipschitz functions $f$ on the real line there are no Lipschitz type estimates for functions $f(A,B)$ of noncommuting self-adjoint operators in the Hilbert–Schmidt norm.", "The construction is given in [1].", "We conjecture that the same result must hold in the case of functions of noncommuting contractions.", "Lipschitz type estimates for $p>2$ and Hölder type estimates.", "It follows from results of [1] that in the case of functions of noncommuting self-adjoint operators for any $s>0$ , $q>0$ and $p>2$ , there exist pairs of self-adjoint operators $(A_0,A_1)$ and $(B_0,B_1)$ and a function $f$ in the homogeneous Besov space $B_{\\infty ,q}^s({R})$ such that $\\Vert f(A_1,B_1)-f(A_0,B_0)\\Vert _{{S}_p}$ can be arbitrarily large while $\\max \\lbrace \\Vert A_1-A_0\\Vert _{{S}_p},\\Vert B_1-B_0\\Vert _{{S}_p}\\rbrace $ can be arbitrarily small.", "In particular, the condition $f\\in B_{\\infty ,q}^s({R})$ does not imply any Lipschitz or Hölder type estimates in the norm of ${S}_p$ , $p>2$ , for any positive $s$ and $q$ .", "It is easy to see that in the case of contractions the situation is different: for any $q>0$ and $p\\ge 1$ , there exists $s>0$ such that the condition $f\\in B_{\\infty ,q}^s$ guarantees a Lipschitz type estimate for functions of not necessarily commuting contractions in ${S}_p$ .", "It would be interesting to find optimal conditions on $f$ that would guarantee Lipschitz or Hölder type estimates in ${S}_p$ for a given $p$ .", "Table: NO_CAPTION" ] ]

[ [ "Guiding Deep Learning System Testing using Surprise Adequacy" ], [ "Abstract Deep Learning (DL) systems are rapidly being adopted in safety and security critical domains, urgently calling for ways to test their correctness and robustness.", "Testing of DL systems has traditionally relied on manual collection and labelling of data.", "Recently, a number of coverage criteria based on neuron activation values have been proposed.", "These criteria essentially count the number of neurons whose activation during the execution of a DL system satisfied certain properties, such as being above predefined thresholds.", "However, existing coverage criteria are not sufficiently fine grained to capture subtle behaviours exhibited by DL systems.", "Moreover, evaluations have focused on showing correlation between adversarial examples and proposed criteria rather than evaluating and guiding their use for actual testing of DL systems.", "We propose a novel test adequacy criterion for testing of DL systems, called Surprise Adequacy for Deep Learning Systems (SADL), which is based on the behaviour of DL systems with respect to their training data.", "We measure the surprise of an input as the difference in DL system's behaviour between the input and the training data (i.e., what was learnt during training), and subsequently develop this as an adequacy criterion: a good test input should be sufficiently but not overtly surprising compared to training data.", "Empirical evaluation using a range of DL systems from simple image classifiers to autonomous driving car platforms shows that systematic sampling of inputs based on their surprise can improve classification accuracy of DL systems against adversarial examples by up to 77.5% via retraining." ], [ "Introduction", "Deep Learning (DL) [24] systems have achieved significant progress in many domains including image recognition [22], [13], [38], speech recognition [17], and machine translation [20], [37].", "Based on their capability to match or even surpass human performance, DL systems are increasingly being adopted as part of larger systems in both safety and security critical domains such as autonomous driving [6], [10], and malware detection [12].", "Such adoption of DL systems calls for new challenges, as it is critically important that these larger systems are both correct and predictable.", "Despite their impressive experimental performances, DL systems are known to exhibit unexpected behaviours under certain circumstances.", "For example, in a reported incident, an autonomous driving vehicle expected another vehicle to yield in one of the rarer circumstances, and crashed into the other vehicle when the expectation proved incorrect [3].", "There is an urgent need to verify and validate behaviours of DL systems.", "However, a significant part of existing software testing technique is not directly applicable to DL systems.", "Most notably, traditional white-box testing techniques that aim to increase structural coverage [4] is not very useful for DL systems, as their behaviour is not explicitly encoded in their control flow structures.", "A number of novel approaches towards testing and verification of DL systems have been recently proposed to fill in the gap [19], [34], [40], [27].", "Most of these techniques share two assumptions.", "The first assumption is essentially a generalisation of the essence of metamorphic testing [11]: if two inputs to a DL system are similar with respect to some human sense, the outputs should also be similar.", "For example, DeepTest [40] checks whether an autonomous driving system behaves in the same way when the input image is transformed as if the same scene is under a different weather condition.", "The second assumption, also based in more traditional software testing results [15], is that the more diverse a set of input is, the more effective testing of a DL system one can perform.", "For example, DeepXplore [34] presented the Neuron Coverage (the ratio of neurons whose activation values were above a predefined threshold) as the measure of diversity of neuron behaviour, and subsequently showed that inputs violating the first assumption will also increase the neuron coverage.", "While the recently introduced techniques have made significant advances over manual ad hoc testing of DL systems, there is a major limitation.", "The coverage criteria proposed so far are not sufficiently fine grained, in a sense that all of them simply count neurons whose activation values satisfy certain conditions.", "While this aggregation by counting does allow the tester to quantify the test effectiveness of a given input set, it conveys little information about individual inputs.", "For example, it is not immediately clear when an input with higher NC should be considered better than another with lower NC, and why: certain inputs may naturally activate more neurons above the threshold than others, and vice versa.", "Another example is the $k$ -Multisection Neuron Coverage [27], which partitions the ranges of activation values of neurons, observed during training, into $k$ buckets, and count the number of total buckets covered by a set of inputs.", "When measured for a single input, the coverage will be either $\\frac{1}{k}$ if the input activates each neuron with a value from one of the $k$ buckets, or smaller than that if some neurons activate outside the range observed during training.", "Again, the information about how far such activations go beyond observed range is lost during aggregation, making it hard to evaluate the relative value of each input.", "For a test adequacy criterion to be practically useful, it should be able to guide the selection of individual inputs, eventually resulting in improvements of the accuracy of the DL system under investigation.", "To overcome these limitations, we propose a new test adequacy for DL systems, called Surprise Adequacy for DL systems (SADL).", "Intuitively, a good test input set for a DL system should be systematically diversified to include inputs ranging from those similar to training data to those significantly different and adversarial.Experiments show benefits of diversity for general testing [15] and benefits of a `scale of distances' of test inputs for robustness testing introduced in [35].", "At individual input granularity, SADL measures how surprising the input is to a DL system with respect to the data the system was trained with: the actual measure of surprise can be either based on the likelihood of the system having seen a similar input during training (here with respect to probability density distributions extrapolated from the training process using kernel density estimation [41]), or the distance between vectors representing the neuron activation traces of the given input and the training data (here simply using Euclidean distance).", "Subsequently, the Surprise Adequacy (SA) of a set of test inputs is measured by the range of individual surprise values the set covers.", "We show that SADL is sufficiently fine rained by training adversarial example classifiers based on SADL values that can produce higher accuracy compared to the state of the art.", "We also show that sampling inputs according to SADL for retraining DL systems can result in higher accuracy, thus showing that SADL is an independent variable that can positively affect the effectiveness of DL system testing.", "The technical contributions of this paper are as follows: We propose SADL, a fine grained test adequacy metric that measures the surprise of an input, i.e., the difference in the behaviour of a DL system between a given input and the training data.", "Two concrete instances of SADL are proposed based on different ways to quantify surprise.", "Both are shown to be correlated with existing coverage criteria for DL systems.", "We show that SADL is sufficiently fine grained in capturing the behaviour of DL systems by training a highly accurate adversarial example classifier.", "Our adversarial example classifier shows as much as 100% and 94.53% ROC-AUC score when applied to MNIST [25] and CIFAR-10 [21] dataset, respectively.", "We show that SADL metrics can be used to sample effective test input sets.", "When retraining DL systems using additional adversarial examples, sampling additional inputs with broader SA values can improve the accuracy after retraining by up to 77.5%.", "We undertake all our experiments using publicly available DL systems ranging from small benchmarks (MNIST and CIFAR-10) to a large system for autonomous driving vehicles (Dave-2 [6] and Chauffeur [1]).", "The remaining of this paper is organised as follows.", "Section  introduces Surprise Adequacy for DL systems, SADL: two variants of SADL are presented along with algorithms that measure them.", "Section  sets out the research questions and Section  describes the experimental set-up of the empirical evaluations.", "Section  presents the results from empirical evaluations.", "Section  addresses threats to validity.", "Section  presents related work, and Section  concludes." ], [ "Surprise Adequacy for Deep Learning Systems", "All existing test adequacy criteria for DL systems aim to measure the diversity of an input set.", "Neuron Coverage (NC) [34] posits that the higher the number of neurons that are activated above a predefined threshold, the more diverse input the DL system has been executed with.", "DeepGauge [27] proposed a range of finer grained adequacy criteria including $k$ -Multisection Neuron Coverage, which measures the ratio of activation value buckets that have been covered across all neurons, and Neuron Boundary Coverage, which measures the ratio of neurons that are activated beyond the ranges observed during training.", "We argue that diversity in testing of DL systems is more meaningful when it is measured with respect to the training data, as DL systems are likely to be more error prone for inputs that are unfamiliar, i.e., diverse.", "Furthermore, while neuron activation above thresholds, or beyond observed ranges, may be closely related to diversity of the given input, they do not measure to what degree the activations of the network for one input differs from the activations for another input.", "They are fundamentally discretisations and do not utilize the fact that neuron activations are continuous quantities.", "In contrast, our aim is to define an adequacy criterion that quantitatively measures behavioural differences observed in a given set of inputs, relative to the training data." ], [ "Activation Trace and Surprise Adequacy", "Let $\\mathbf {N} = \\lbrace n_1, n_2, \\ldots \\rbrace $ be a set of neurons that constitutes a DL system $\\mathbf {D}$ , and let $X = \\lbrace x_1, x_2, \\ldots \\rbrace $ be a set of inputs.", "We denote the activation value of a single neuron $n$ with respect to an input $x$ as $\\alpha _n(x)$ .", "For an ordered (sub)set of neurons, let $N \\subseteq \\mathbf {N}$ , $\\alpha _N(x)$ denote a vector of activation values, each element corresponding to an individual neuron in $N$ : the cardinality of $\\alpha _N(x)$ is equal to $|N|$ .", "We call $\\alpha _N(x)$ the Activation Trace (AT) of $x$ over neurons in $N$ .", "Similarly, let $A_N(X)$ be a set of activation traces, observed over neurons in $N$ , for a set of inputs $X$ : $A_N(X) = \\lbrace \\alpha _N(x) \\mid x \\in X\\rbrace $ .", "We note that the activation trace is trivially available after each execution of the network for a given input.", "Since behaviours of DL systems are driven along the data-flow and not control-flow, we assume that activation traces observed over all $\\mathbf {N}$ with respect to $X$ , $A_\\mathbf {N}(X)$ , fully captures the behaviours of the DL system under investigation when executed using $X$ .For the sake of simplicity, we assume that it is possible to get the complete activation traces from all the neurons in a DL system.", "For network architectures with loops, such as Recurrent Neural Nets (RNNs) [18], it is possible to unroll the loops up to a predefined bound [40].", "Surprise Adequacy (SA) aims to measure the relative novelty (i.e., surprise) of a given new input with respect to the inputs used for training.", "Given a training set $\\mathbf {T}$ , we first compute $A_\\mathbf {N}(\\mathbf {T})$ by recording activation values of all neurons using every input in the training data set.", "Subsequently, given a new input $x$ , we measure how surprising $x$ is when compared to $\\mathbf {T}$ by comparing the activation trace of $x$ to $A_\\mathbf {N}(\\mathbf {T})$ .", "This quantitative similarity measure is called Surprise Adequacy (SA).", "We introduce two variants of SA, each with different way of measuring the similarity between $x$ and $A_\\mathbf {N}(\\mathbf {T})$ .However, the main idea is general and other, specific variants would result if using other similarity functions.", "Note that certain types of DL tasks allow us to focus on parts of the training set $\\mathbf {T}$ to get more precise and meaningful measurement of SA.", "For example, suppose we are testing a classifier with a new input $x$ , which is classified by the DL system under investigation as the class $c$ .", "In this case, the surprise of $x$ is more meaningfully measured against $A_\\mathbf {N}(T_c)$ , in which $T_c$ is the subset of $\\mathbf {T}$ where members are classified as $c$ .", "Basically, the input might be surprising as an example of class $c$ even if not surprising in relation to the full set of training examples." ], [ "Likelihood-based Surprise Adequacy", "Kernel Density Estimation (KDE) [41] is a way of estimating the probability density function of a given random variable.", "The resulting density function allows the estimation of relative likelihood of a specific value of the random variable.", "Likelihood-based SA (LSA) uses KDE to estimate the probability density of each activation value in $A_\\mathbf {N}(\\mathbf {T})$ , and obtains the surprise of a new input with respect to the estimated density.", "This is an extension of existing work that uses KDE to detect adversarial examples [14].", "To reduce dimensionality and computational cost, we only consider the neurons in a selected layer $N_{L} \\subseteq \\mathbf {N}$ , which yields a set of activation traces, $A_{N_{L}}(\\mathbf {X})$ .", "To further reduce the computational cost, we filter out neurons whose activation values show variance lower than a pre-defined threshold, $t$ , as these neurons will not contribute much information to KDE.", "The cardinality of each trace will be $|N_{L}|$ .", "Given a bandwidth matrix $H$ and Gaussian kernel function $K$ , the activation trace of the new input $x$ , and $x_i \\in \\mathbf {T}$ , KDE produces density function $\\hat{f}$ as follows: $\\hat{f}(x) = \\frac{1}{|A_{N_{L}}(\\mathbf {T})|}\\sum _{x_i \\in \\mathbf {T}}{K_{H}(\\alpha _{N_{L}}(x)-\\alpha _{N_{L}}(x_i))}$ Since we want to measure the surprise of the input $x$ , we need a metric that increases when probability density decreases (i.e., the input is rarer compared to the training data), and vice versa (i.e., the input is similar to the training data).", "Adopting common approach of converting probability density to a measure of rareness [39], [26], we define LSA to be the negative of the log of density: $LSA(x) = -log(\\hat{f}(x))$ Note that extra information about input types can be used to make LSA more precise.", "For example, given a DL classifier $\\mathbf {D}$ , we expect inputs that share the same class label will have similar ATs.", "We can exploit this by computing LSA per class, replacing $\\mathbf {T}$ with $\\lbrace x \\in \\mathbf {T} \\mid \\mathbf {D}(x) = c\\rbrace $ for class $c$ .", "We use per-class LSA for DL classifiers in our empirical evaluation.", "Figure: An example of Distance-based SA.", "Black dots represent ATs of trainingdata inputs, whereas grey dots represent ATs of new inputs, x 1 x_1 andx 2 x_2.", "Compared to distances from x 1a x_{1a} and x 2a x_{2a} to classc 2 c_2, AT of x 1 x_1 is farther out from class c 1 c_1 than that of x 2 x_2, i.e.,a 1 b 1 >a 2 b 2 \\frac{a_1}{b_1} > \\frac{a_2}{b_2} (see Equations , , and ).", "Consequently, we decide that x 1 x_1 ismore surprising than x 2 x_2 w.r.t.", "class c 1 c_1." ], [ "Distance-based Surprise Adequacy", "An alternative to LSA is simply to use the distance between ATs as the measure of surprise.", "Here, we define Distance-based Surprise Adequacy (DSA) using the Euclidean distance between the AT of a new input $x$ and ATs observed during training.", "Being a distance metric, DSA is ideally suited to exploit the boundaries between inputs, as can be seen in the classification example in Figure REF .", "By comparing the distances $a_1$ and $a_2$ (i.e., distance between the AT of a new input and the reference point, which is the nearest AT of training data in $c_1$ ) to distances $b_1$ and $b_2$ (i.e., distance to $c_2$ measured from the reference point), we get a sense of how close to the class boundary the new inputs are.", "We posit that, for classification problems, inputs that are closer to class boundaries are more surprising and valuable in terms of test input diversity.", "On the other hand, for tasks without any boundaries between inputs, such as prediction of appropriate steering angle for autonomous driving car, DSA may not be easily applicable.", "With no class boundaries, an AT of a new input being far from that of another training input does not guarantee that the new input is surprising, as the AT may still be located in crowded parts of the AT space.", "Consequently, we only apply DSA for classification tasks, for which it can be more effective than LSA (see Section REF and REF for more details).", "Let us assume that a DL system $\\mathbf {D}$ , which consists of a set of neurons $\\mathbf {N}$ , is trained for a classification task with a set of classes $C$ , using a training dataset $\\mathbf {T}$ .", "Given the set of activation traces $A_\\mathbf {N}(\\mathbf {T})$ , a new input $x$ , and a predicted class of the new input $c_x \\in C$ , we define the reference point $x_a$ to be the closest neighbour of $x$ that shares the same class.", "The distance between $x$ and $x_a$ follows from the definition: $\\begin{split}&x_a = \\mathop {\\mathrm {argmin}}_{\\mathbf {D}(x_i) = c_x}\\Vert \\alpha _\\mathbf {N}(x) - \\alpha _\\mathbf {N}(x_i)\\Vert ,\\\\&dist_a = \\Vert \\alpha _\\mathbf {N}(x) - \\alpha _\\mathbf {N}(x_a)\\Vert \\end{split}$ Subsequently, from $x_a$ , we find the closest neighbour of $x_a$ in a class other than $c_x$ , $x_b$ , and the distance $dist_b$ , as follows: $\\begin{split}&x_b = \\mathop {\\mathrm {argmin}}_{\\mathbf {D}(x_i) \\in C \\setminus \\lbrace c_x\\rbrace }\\Vert \\alpha _\\mathbf {N}(x_a) - \\alpha _\\mathbf {N}(x_i)\\Vert ,\\\\&dist_b = \\Vert \\alpha _\\mathbf {N}(x_a) - \\alpha _\\mathbf {N}(x_b)\\Vert \\end{split}$ Intuitively, DSA aims to compare the distance from the AT of a new input $x$ to known ATs belonging to its own class, $c_x$ , to the known distance between ATs in class $c_x$ and ATs in other classes in $C \\setminus \\lbrace c_x\\rbrace $ .", "If the former is relatively larger than the latter, $x$ would be a surprising input for class $c_x$ to the classifying DL system $\\mathbf {D}$ .", "While there are multiple ways to formalise this we select a simple one and calculate DSA as the ratio between $dist_a$ and $dist_b$ .", "Investigation of more complicated formulations is left as future work.", "$DSA(x) = \\frac{dist_a}{dist_b}$" ], [ "Surprise Coverage", "Given a set of inputs, we can also measure the range of SA values the set covers, called Surprise Coverage (SC).", "Since both LSA and DSA are defined in continuous spaces, we use bucketing to discretise the space of surprise and define both Likelihood-based Surprise Coverage (LSC) and Distance-based Surprise Coverage (DSC).", "Given an upper bound of $U$ , and buckets $B = \\lbrace b_1, b_2, ... , b_n\\rbrace $ that divide $(0, U]$ into $n$ SA segments, SC for a set of inputs $X$ is defined as follows: $SC(X) = \\frac{|\\lbrace b_i \\mid \\exists x \\in X : SA(x) \\in (U\\cdot \\frac{i-1}{n}, U\\cdot \\frac{i}{n}]\\rbrace |}{n}$ A set of inputs with high SC is a diverse set of inputs ranging from similar to those seen during training (i.e., low SA) to very different from what was seen during training (i.e., high SA).", "We argue that an input set for a DL system should not only be diversified, but systematically diversified considering SA.", "Recent results also validate this notion by showing that more distant test inputs were more likely to lead to exceptions but might not be as relevant for testing [35].", "While we use the term cover and coverage, the implications of SA based coverage is different from the traditional structural coverage.", "First, unlike most of the structural coverage criteria, there is no finite set of targets to cover, as in statement or branch coverage: an input can, at least in theory, be arbitrarily surprising.", "However, an input with arbitrarily high SA value may simply be irrelevant, or at least less interesting, to the problem domain (e.g., an image of a traffic sign will be irrelevant to the testing of animal photo classifiers).", "As such, SC can only be measured with respect to pre-defined upper bound, in the same way the theoretically infinite path coverage is bounded by a parameter [44].", "Second, SC does not render itself to a combinatorial set cover problem, which the test suite minimisation is often formulated into [43].", "This is because a single input yields only a single SA value and cannot belong to multiple SA buckets.", "The sense of redundancy with respect to SC as a coverage criteria is weaker than that of structural coverage, for which a single input can cover multiple targets.", "While we aim to show that SA can guide the better selection of inputs, rigorous study of optimisation of test suites for DL systems remains a future work.", "However, as we show with our empirical studies, SC can still guide test input selection.", "Table: List of datasets and models used in the study." ], [ "Research Questions", "Our empirical evaluation is designed to answer the following research questions.", "RQ1.", "Surprise: Is SADL capable of capturing the relative surprise of an input of a DL system?", "We provide answers to RQ1 from different angles.", "First, we compute the SA of each test input included in the original dataset, and see if a DL classifier finds inputs with higher surprise more difficult to correctly classify.", "We expect more surprising input to be harder to correctly classify.", "Second, we evaluate whether it is possible to detect adversarial examples based on SA values, as we expect adversarial examples to be more surprising as well as to cause different behaviours of DL systems.", "Using different techniques, multiple sets of adversarial examples are generated and compared by their SA values.", "Finally, we train adversarial example classifiers using logistic regression on SA values.", "For each adversarial attack strategy, we generate 10,000 adversarial examples using 10,000 original test images provided by MNIST and CIFAR-10.", "Using 1,000 original test images and 1,000 adversarial examples, all chosen randomly, we train the logistic regression classifiers.", "Finally, we evaluate the trained classifiers using the remaining 9,000 original test images and 9,000 adversarial examples.", "If SA values correctly capture the behaviour of DL systems, we expect the SA based classifiers to successfully detect adversarial examples.", "We use Area Under Curve of Receiver Operator Characteristics (ROC-AUC) for evaluation as it captures both true and false positive rates [8].", "RQ2.", "Layer Sensitivity: Does the selection of layers of neurons used for SA computation have any impact on how accurately SA reflects the behaviour of DL systems?", "Bengio et al.", "suggest that deeper layers represent higher level features of the input [5]: subsequent work that introduced KDE based adversarial example detection technique [14] assumes the deepest (i.e., the last hidden) layer to contain the most information helpful for detection.", "We evaluate this assumption in the context of SA by calculating LSA and DSA of all individual layers, and subsequently by comparing adversarial example classifiers trained on SA from each layer.", "RQ3.", "Correlation: Is SC correlated to existing coverage criteria for DL systems?", "In addition to capturing input surprise, we want SC to be consistent with existing coverage criteria based on counting aggregation.", "If not, there is a risk that SC is in fact measuring something other than input diversity.", "For this, we check whether SC is correlated with other criteria.", "We control the input diversity by cumulatively adding inputs generated by different method (i.e., different adversarial example generation techniques or input synthesis techniques), execute the studied DL systems with these input, and compare the observed changes of various coverage criteria including SC and four existing ones: DeepXplore's Neuron Coverage (NC) [40] and three Neuron-level Coverages (NLCs) introduced by DeepGauge [27]: $k$ -Multisection Neuron Coverage (KMNC), Neuron Boundary Coverage (NBC), and Strong Neuron Activation Coverage (SNAC).", "For MNIST and CIFAR-10, we start from the original test data provided by the dataset (10,000 images), and add 1,000 adversarial examples, generated by FGSM, BIM-A, BIM-B, JSMA, and C&W, at each step.", "For Dave-2, we start from the original test data (5,614 images) and add 700 synthetic images generated by DeepXplore at each step.", "For Chauffeur, each step adds 1,000 synthetic images (Set1 to Set3), each produced by applying random number of DeepTest transformations.", "RQ4.", "Guidance: Can SA guide retraining of DL systems to improve their accuracy against adversarial examples and synthetic test inputs generated by DeepXplore?", "To evaluate whether SADL can guide additional training of existing DL systems with the aim of improved accuracy against adversarial examples, we ask whether SA can guide the selection of input for additional training.", "From the adversarial examples and synthesised inputs for these modelsWe could not resume training of Chauffeur model for additional five epochs, which is why it is absent from RQ4., we choose four sets of 100 images from four different SA ranges.", "Given $U$ as the upper bound used in RQ3 to compute the SC, we divide the range of SA $[0, U]$ into four overlapping subsets: the first subset including the low 25% SA values ($[0, \\frac{U}{4}]$ ), the second including the lower half ($[0, \\frac{2U}{4}]$ ), the third including the lower 75% ($[0,\\frac{3U}{4}]$ ), and finally the entire range, $[0, U]$ .", "These four subsets are expected to represent increasingly more diverse sets of inputs.", "We set the range $R$ to one of these four, randomly sample 100 images from each $R$ , and train existing models for five additional epochs.", "Finally, we measure each model's performance (accuracy for MNIST and CIFAR-10, MSE for Dave-2) against the entire adversarial and synthetic inputs, respectively.", "We expect retraining with more diverse subset will result in higher performance." ], [ "Experimental Setup", "We evaluate SADL on four different DL systems using (a) the original test sets, (b) adversarial examples generated by five attack strategies, and (c) synthetic inputs generated by DeepXplore [34] and DeepTest [40].", "This section describes the studied DL systems and the input generation methods." ], [ "Datasets and DL Systems", "Table REF lists the subject datasets and models of DL systems.", "MNIST [25] and CIFAR-10 [21] are widely used datasets for machine learning research, each of which is a collection of images in ten different classes.", "For MNIST, we adopt the widely studied five layer Convolutional Neural Network (ConvNet) with max-pooling and dropout layers and train it to achieve 99.31% accuracy on the provided test set.", "Similarly, the adopted model for CIFAR is a 12-layer ConvNet with max-pooling and dropout layers, trained to achieve 82.27% accuracy on the provided test set.", "For evaluation of SADL for DL systems in safety critical domains, we use the Udacity self-driving car challenge dataset [2], which contains a collection of camera images from the driving car.", "As its aim is to predict steering wheel angle, the model accuracy is measured using Mean Squared Error (MSE) between actual and predicted steering angles.", "We use a pre-trained Dave-2 model [6], which is a public artefact provided by DeepXploreDeepXplore is available from: https://github.com/peikexin9/deepxplore., and a pre-trained Chauffeur model [1], made publicly available by the Udacity self-driving car challenge.", "Dave-2 consists of nine layers including five convolutional layers, and achieves 0.09 in MSE.", "Chauffeur consists of both a ConvNet and an LSTM sub-model, and achieves 0.10 in MSE." ], [ "Adversarial Examples and Synthetic Inputs", "SADL is evaluated using both adversarial examples and synthetic test inputs.", "Adversarial examples are crafted by applying, to the original input, small perturbations imperceptible to humans, until the DL system under investigation behaves incorrectly [16].", "We rely on adversarial attacks to generate input images for MNIST and CIFAR-10: these generated images are more likely to reveal robustness issues in the DL systems than the test inputs provided by the original datasets.", "We use five widely studied attack strategies to evaluate SADL: Fast Gradient Sign Method (FGSM) [16], Basic Iterative Method (BIM-A, BIM-B) [23], Jacobian-based Saliency Map Attack (JSMA) [33], and Carlini&Wagner (C&W) [9].", "Our implementation of these strategies is based on cleverhans [32] and a framework of Ma et al. [30].", "For Dave-2 and Chauffeur, we use the state-of-the-art synthetic input generation algorithms, DeepXplore [34] and DeepTest[40].", "Both algorithms are designed to synthesise new test input from existing ones with the aim of detecting erroneous behaviours in autonomous driving vehicle.", "For Dave-2, we use DeepXplore's input generation via joint optimization algorithm, whose aim is to generate inputs that lead multiple DL systems trained independently, but using the same training data, to disagree with each other.", "Using Dave-2 and its two variants, Dave-dropout and Dave-norminit, we collect synthetic inputs generated by lighting effect (Light), occlusion by a single black rectangle (SingleOcc), and occlusion by multiple black rectangles (MultiOcc).", "For Chauffeur, we synthesise new inputs by iteratively applying random transformations provided by DeepTest to original input images: translation, scale, shear, rotation, contrast, brightness, and blur.At the time of our experiments, the publicly available version of DeepTest did not internally support realistic image transformations such as fog and rain effects.", "Table: Configurations for RQ3." ], [ "Configurations", "For all research questions, the default activation variance threshold for LSA is set to $10^{-5}$ , and the bandwidth for KDE is set using Scott's Rule [36].", "The remaining of this Section details RQ specific configurations.", "For RQ1, we use the activation_2 layer for MNIST, and activation_6 for CIFAR-10, when computing LSA values.", "Computation of LSA based on all neurons is computationally infeasible due to precision loss.", "For RQ2, we set the activation variance threshold for layers activation_7 and activation_8 of CIFAR-10 to $10^{-4}$ , which reduces the number of neurons used for the computation of LSA and, consequently, the computational cost.", "For computation of other coverage criteria in RQ3, we use the configurations in Table REF .", "The threshold of NC is set to 0.5.", "For NLCs, we all set the number of sections ($k$ ) to 1,000.", "For LSC and DSC, we manually choose the layer, the number of buckets ($n$ ), and the upper bound ($ub$ ).", "For RQ4, the layers chosen for MNIST and CIFAR-10 are activation_3 and activation_5 respectively.", "We perform 20 runs of retraining for each subject and report the statistics.", "All experiments were performed on machines equipped with Intel i7-8700 CPU, 32GB RAM, running Ubuntu 16.04.4 LTS.", "MNIST and CIFAR-10 are implemented using Keras v.2.2.0.", "Figure: Accuracy of test inputs in MNIST and CIFAR-10 dataset, selected fromthe input with the lowest SA, increasingly including inputs with higher SA,and vice versa (i.e., from the input with the highest SA to inputs with lower SA)." ], [ "Input Surprise (RQ1)", "Figure REF shows how the classification accuracy changes when we classify sets of images of growing sizes from the test inputs included in the MNIST and CIFAR-10 dataset.", "The sets of images corresponding to the red dots (Ascending SA) start with images with the lowest SA, and increasingly include images with higher SA in the ascending order of SA; the sets of images corresponding to the blue dots grow in the opposite direction (i.e., from images with the highest SA to lower SA).", "As a reference, the green dots show the mean accuracy of randomly growing sets across 20 repetitions.", "It is clear that including images with higher LSA values, i.e., more surprising images, leads to lower accuracy.", "For visual confirmation on another dataset, we also chose sets of inputs synthesised for Chauffeur by DeepTest, from three distinct levels of LSA values: Figure REF shows that the higher the LSA values are, the harder it is to recognise images visually.", "Both quantitatively and visually, the observed trend supports our claim that SADL captures input surprise: even for unseen inputs, SA can measure how surprising the given input is, which is directly related to the performance of the DL system.", "Figure: Synthetic images for Chauffeur model generated by DeepTest.Images with higher LSA values tend to be harder to recognise and interpret visually.Figure REF shows plots of sorted DSA values of 10,000 adversarial examples, generated by each of the five techniques, as well as the original test inputs.", "Figure REF contains similar plots based on LSA values of 2,000 randomly selected adversarial examples and the original test set, from different layers of MNIST and CIFAR-10.", "For both MNIST and CIFAR-10, the test inputs provided with the datasets (represented in blue colour) tend to be the least surprising, whereas the majority of adversarial examples are clearly separated from the test inputs by their higher SA values.", "This supports our claim that SADL can capture the differences in DL system's behaviours for adversarial examples.", "Figure: Sorted DSA values of adversarial examples for MNIST and CIFAR-10.Figure: Sorted LSA of randomly selected 2,000 adversarial examples for MNIST and CIFAR-10 from different layersFinally, Table REF shows the ROC-AUC results of DSA-based classification using all neurons in MNIST and CIFAR-10.LSA-based classification is only possible for subsets of neurons due to the computational cost of KDE; hence we introduce the results of LSA-based classification when answering the impact of layer selection for RQ2.", "The results show that the gap in DSA values observed in Figure REF can be used to classify adversarial examples with high accuracy.", "For the relatively simpler MNIST model, the DSA-based classifier can detect adversarial examples with ROC-AUC ranging from 96.97% to 99.38%.", "The DSA-based classification for the more complicated CIFAR-10 model shows lower ROC-AUC values, but answers to RQ2 suggest that DSA from specific layers can produce significantly higher accuracy (see Section REF ).", "Table: ROC-AUC of DSA-based classification of adversarial examples for MNIST and CIFAR-10Based on three different analyses, the answer to RQ1 is that SADL can capture the relative surprise of inputs.", "Inputs with higher SA are harder to correctly classify; adversarial examples show higher SA values and can be classified based on SA accordingly." ], [ "Impact of Layer Selection (RQ2)", "Table REF shows the ROC-AUC of classification of adversarial examples, resulting in each row corresponding to a classifier trained on LSA and DSA from a specific layer in MNIST, respectively.", "Rows are ordered by their depth, i.e., activation_3 is the deepest and the last hidden layer in MNIST.", "The highest ROC-AUC values for each attack strategy are typeset in bold.", "For MNIST, there is no clear evidence that the deepest layer is the most effective.", "Table: ROC-AUC results of SA per layers on MNIST.The cases for which ROC-AUC is 100% can be explained by Figure REF : LSA values from activation_1 of MNIST, for example, show a clear separation between the original test inputs and FGSM, BIM-A, or BIM-B: by choosing an appropriate threshold, it is possible to completely separate test inputs from adversarial examples.", "Similarly, the plot of LSA from activation_3 of MNIST shows that C&W LSA line crossing with that of the original test data (i.e., C&W adversarial examples are less surprising than the original test data): this results in the low ROC-AUC value of 37.96%.", "Table REF contains the ROC-AUC values of LSA- and DSA-based classifiers, trained on each layer of the CIFAR-10 model: for each attack strategy, the highest ROC-AUC values are typeset in bold.", "Interestingly, LSA and DSA show different trends with CIFAR-10.", "With LSA, there is no strong evidence that the deepest layer produces the most accurate classifiers.", "However, with DSA, the deepest layer produces the most accurate classifiers for three out of five attack strategies (BIM-B, JSMA, and C&W), while the second deepest layer produces the most accurate classifier for BIM-A.", "More importantly, per-layer DSA values produce much more accurate classification results than all neuron DSA values, as can be seen in the comparison between Table REF and Table REF & REF .", "Identical models have been used to produce results in Tables above.", "Table: ROC-AUC results of SA per layers on CIFAR-10.Based on these results, we answer RQ2 that DSA is sensitive to the selection of layers it is computed from, and benefits from choosing the deeper layer.", "However, for LSA, there is no clear evidence supporting the deeper layer assumption.", "The layer sensitivity varies across different adversarial example generation strategies." ], [ "Correlation between SC and Other Criteria (RQ3)", "Table REF shows how different coverage criteria respond to increasing diversity levels.", "Columns represent steps, at each of which more inputs are added to the original test set.", "If the increase in coverage at a step is less than 0.1 percentage point when compared to the previous step, the value is underlined.", "The threshold of 0.1 percentage point is based on the finest step change possible for LSC, DSC, as well as KMNC, as all three use bucketing with $k = 1,000$ .", "We acknowledge that the threshold is arbitrary, and provide it only as a supporting aid.", "Figure REF presents visualisation of results from CIFAR-10 and Chauffeur.", "Note that DSC cannot be computed for these two DL systems, as they are not classifiers (see Section REF ).", "Overall, most of the studied criteria increase as additional inputs are added at each step.", "The notable exception is NC, which plateaus against many steps.", "This is in line with results in existing work [27].", "There exists an interplay between the type of added inputs and how different criteria respond: SNAC, KMNC, and NBC show significant increases with the addition of BIM-B examples to CIFAR-10, but change little when C&W inputs are added.", "However, only SNAC and NBC exhibit a similar increase with the addition of input Set 1 for Chauffeur, while KMNC increases more steadily.", "Overall, with the exception of NC, we answer RQ3 that SC is correlated with other coverage criteria introduced so far.", "Table: Changes in various coverage criteria against increasing input diversity.We put additional inputs into the original test inputs and observe changes in coverage values.Figure: Visualisation of CIFAR-10 and Chauffeur in Table .As additional sets of inputs (xx-axis) are added to the original test set,various coverage criteria (yy-axis) increase." ], [ "Retraining Guidance (RQ4)", "Table REF shows the impact of SA-based guidance for retraining of MNIST, CIFAR-10, and Dave-2 models.", "The column $R$ from $\\frac{1}{4}$ to $\\frac{4}{4}$ represents the increasingly wider ranges of SA from which the inputs for additional training are sampled; rows with $R = \\emptyset $ show performance of the DL system before retraining.", "Overall, there are 23 retraining configurations (2 SA types $\\times $ 2 DL systems $\\times $ 5 adversarial attack strategies, and 1 SA type $\\times $ 1 DL system $\\times $ three input synthesis methods), each of which is evaluated against four SA ranges with 20 repetitions.", "Columns $\\mu $ and $\\sigma $ contain the mean and standard deviation of observed performance metric (i.e., the highest accuracy for MNIST and CIFAR-10, the lowest MSE for Dave-2).", "The best performance is typeset in bold.", "The full range, $\\frac{4}{4}$ , produces the best retraining performance for 13 configurations, followed by $\\frac{2}{4}$ (5 configurations), $\\frac{3}{4}$ (3 configurations), and $\\frac{1}{4}$ (3 configurations).", "Note that for the configuration of CIFAR-10 and BIM-B, both ranges $\\frac{2}{4}$ and $\\frac{2}{4}$ produces the same and the best retraining performance.", "The largest improvement is observed when retraining MNIST against FGSM using DSA: the accuracy of the $\\frac{4}{4}$ range shows 77.5% increase from that of $\\frac{1}{4}$ (i.e., from 15.60% to 28.69%).", "While retraining MNIST against BIM-B using DSA shows even greater improvement (from 9.40% to 40.94%), we suspect this is an outlier as the accuracy for ranges $\\frac{1}{4}$ and $\\frac{2}{4}$ are significantly smaller when compared to other configurations.", "While our observations are limited to the DL systems and input generation techniques studied here, we answer RQ4 that SA can provide guidance for more effective retraining against adversarial examples based on our interpretation of the observed trend.", "Table: Retraining guided by SA: we sample 100 inputs from four increasingly wider ranges of SA:[0,U 4][0, \\frac{U}{4}], [0,2U 4][0, \\frac{2U}{4}], [0,3U 4][0, \\frac{3U}{4}], and [0,U][0, U],and retrain for five additional epochs using the samples as the training data,and measure the accuracy and MSE against the entire adversarial and synthetic inputs.Sampling from wider ranges improves the retraining accuracy." ], [ "Threats to Validity", "The primary threat to internal validity of this study is the correctness of implementation of the studied DL systems, as well as the computation of SA values.", "We have used publicly available architectures and pre-trained models as our subjects to avoid incorrect implementation.", "SA computation depends on a widely used computation library, SciPy, which has stood the public scrutiny.", "Threats to external validity mostly concerns the number of the models and input generation techniques we study here.", "It is possible that SADL is less effective against other DL systems.", "While we believe the core principle of measuring input surprise is universally applicable, only further experimentations can reduce this particular risk.", "Finally, threats to construct validity asks whether we are measuring the correct factors to draw our conclusion.", "For all studied DL systems, activation traces are immediate artefacts of their executions and the meaning of output accuracy is well established, minimising the risk of this threat." ], [ "Related Work", "Adversarial examples pose significant threats to the performance of DL systems [7].", "There are existing work in the machine learning community on detection of such inputs.", "Feinman et al.", "[14] first introduced the KDE as a means of similarity measurement, with the aim of detecting adversarial examples.", "SADL improves upon the existing work by a number of different ways.", "First, we generalise the concept of Surprise Adequacy (SA) and introduce Distance-based SA.", "Second, our evaluation is in the context of DL system testing.", "Third, our evaluation of SADL includes more complicated and practical DL systems, as well as testing techniques such as DeepXplore and DeepTest.", "Finally, we show that the choice of neurons has limited impact on LSA.", "A range of techniques has been recently proposed to test and verify DL systems.", "The existing techniques are largely based on two assumptions.", "The first assumption is a variation of metamorphic testing [11], [31], [42].", "Suppose a DL system $N$ produces an output $o$ when given $i$ as the input, i.e., $N(i) = o$ .", "Then we expect $N(i^{\\prime }) \\simeq o$ when $i^{\\prime } \\simeq i$ .", "Huang et al.", "[19] proposed a verification technique that can automatically generate counter-examples that violate this assumption.", "Pei et al.", "introduced DeepXplore [34], a white-box technique that generates test inputs that cause disagreement among a set of DL systems, i.e., $N_m(i) \\ne N_n(i)$ for independently trained DL systems $N_m$ and $N_n$ .", "Tian et al.", "presented DeepTest, whose metamorphic relations include both simple geometric perturbations as well as realistic weather effects [40].", "The second assumption is that the more diverse a set of input is, the more effective it will be for testing and validating DL systems.", "Pei et al.", "proposed Neuron Coverage (NC), which measures the ratio of neurons whose activation values are above a predefined threshold [34].", "It has been shown that adding test inputs that violate the first assumption increases the diversity measured through NC.", "Similarly, DeepGauge introduced a set of multi-granularity coverage criteria that are thought to reflect behaviours of DL systems in finer granularity [27].", "While these criteria capture input diversity, all of them are essentially count of neurons unlike SA, and therefore cannot be directly linked to behaviours of DL systems.", "We show that SA is closely related to the behaviours by training accurate adversarial example classifiers based on SA.", "Apart from coverage criteria, other concepts in traditional software testing have been reformulated and applied to testing of DL systems.", "Ma et al.", "proposed DeepCT, which views ranges of neuron activation values as parameter choices and applies Combinatorial Interaction Testing (CIT) to measure interaction coverage [29].", "SC is different from DeepCT as SADL aims to quantify the amount of surprise, rather than simply to detect surprise via increase in coverage.", "DeepMutation applies the principle of mutation testing to DL systems by mutating training data, test data, as well as the DL system itself, based on source and model level mutation operators [28]." ], [ "Conclusion", "We propose SADL, a surprise adequacy framework for DL systems that can quantitatively measure relative surprise of each input with respect to the training data, which we call Surprise Adequacy (SA).", "Using SA, we also develop Surprise Coverage (SC), which measures the coverage of discretised input surprise ranges, rather than the count of neurons with specific activation traits.", "Our empirical evaluation shows that SA and SC can capture the surprise of inputs accurately and are good indicators of how DL systems will react to unknown inputs.", "SA is correlated with how difficult a DL system finds an input, and can be used to accurately classify adversarial examples.", "SC can be used to guide selection of inputs for more effective retraining of DL systems for adversarial examples as well as inputs synthesised by DeepXplore." ] ]

[ [ "Optically driven collective spin excitations and magnetization dynamics\n in the N\\'eel-type skyrmion host GaV$_4$S$_8$" ], [ "Abstract GaV$_4$S$_8$ is a multiferroic semiconductor hosting magnetic cycloid (Cyc) and N\\'eel-type skyrmion lattice (SkL) phases with a broad region of thermal and magnetic stability.", "Here, we use time-resolved magneto-optical Kerr spectroscopy and micro-magnetic simulations to demonstrate the coherent generation of collective spin excitations in the Cyc and SkL phases driven by an optically-induced modulation of uniaxial anisotropy.", "Our results shed light on spin-dynamics in anisotropic materials hosting skyrmions and pave a new pathway for the optical control of their magnetic order." ], [ "3 Optically driven collective spin excitations and magnetization dynamics in the Néel-type skyrmion host GaV4S8 P. Padmanabhan ppadmana@ph2.uni-koeln.de Physics Institute II, University of Cologne, 50937 Cologne, Germany F. Sekiguchi Physics Institute II, University of Cologne, 50937 Cologne, Germany R. B. Versteeg Physics Institute II, University of Cologne, 50937 Cologne, Germany E. Slivina Physics Institute II, University of Cologne, 50937 Cologne, Germany V. Tsurkan Institute of Applied Physics, MD 2028, Chisinau, Republic of Moldova Department of Physics, Budapest University of Technology and Economics and MTA-BME LendÃŒlet Magneto-optical Spectroscopy Research Group, 1111 Budapest, Hungary S. Bordács Department of Physics, Budapest University of Technology and Economics and MTA-BME LendÃŒlet Magneto-optical Spectroscopy Research Group, 1111 Budapest, Hungary Hungarian Academy of Sciences, Premium Postdoctoral Program, 1051 Budapest, Hungary I. Kézsmárki Department of Physics, Budapest University of Technology and Economics and MTA-BME LendÃŒlet Magneto-optical Spectroscopy Research Group, 1111 Budapest, Hungary Experimental Physics V, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany P. H. M. van Loosdrecht pvl@ph2.uni-koeln.de Physics Institute II, University of Cologne, 50937 Cologne, Germany GaV4S8 is a multiferroic semiconductor hosting magnetic cycloid (Cyc) and Néel-type skyrmion lattice (SkL) phases with a broad region of thermal and magnetic stability.", "Here, we use time-resolved magneto-optical Kerr spectroscopy and micro-magnetic simulations to demonstrate the coherent generation of collective spin excitations in the Cyc and SkL phases driven by an optically-induced modulation of uniaxial anisotropy.", "Our results shed light on spin-dynamics in anisotropic materials hosting skyrmions and pave a new pathway for the optical control of their magnetic order.", "The optical manipulation of topologically-nontrivial phases in quantum materials [1], [2], [3], [4], [5], [6] is an emerging area within condensed-matter physics [7], with efforts aimed at uncovering novel phases and exploring their non-equilibrium properties.", "Seminal examples, in this regard, include the realization of Floquet-Bloch states resulting from photon–surface-state hybridization in topological insulators [8], [9] and helicity-dependent control of topological-surface currents [10].", "Interest has also extended to magnetic topological defects known as skyrmions (Sks) [11], fueled by their importance in memory technology [12], [13], [14], spintronics [15], and emergent electromagnetism [16], [17], [18].", "Recently, optical stimulus has been successfully used to write and erase individual Sks [19], confirm their topological robustness [20], and identify new metastable Sk states [21].", "Skyrmions can be broadly classified into two varieties by their internal structure.", "Whirl-like Bloch-type Sks are typically found in chiral magnets [19], [22], [3] and are generally stable over a relatively narrow temperature range in bulk crystals.", "Néel-type Sks, where spins rotate in radial planes from their cores to their peripheries, have been identified in bulk lacunar spinels [23], [24], tetragonal oxides [25], and thin-film heterostructures [26].", "Notably, these systems all posses a polar, rather than chiral, structure and exhibit axial symmetry.", "Moreover, Néel-type skyrmion lattice (SkL) states in bulk crystals of these polar magnets show an enhanced thermal stability.", "This stems from their orientational confinement primarily due to the pattern of the Dzyaloshinskii-Moriya interaction (DMI).", "The ultrafast-optical manipulation of magnetic states can generally be accomplished through mechanisms that leverage direct spin-photon [27], [28], [29] coupling as well as those that exploit the thermal response of the host material [30], [31].", "At present, however, SkL states have only been coherently excited opto-magnetically using the inverse-Faraday effect in the insulating Bloch-type–SkL-host Cu2OSeO3 [32], owing to its strong linear–magneto-optical response [33].", "Another possible avenue is the optical modulation of magnetic interactions (e.g.", "the uniaxial anisotropy), which has proven successful in driving spin precessions in a variety of magnetic materials [34], [35], [36], [37], [38].", "Within this context, lacunar spinels, possessing large uniaxial anisotropies of the easy-axis or easy-plane varieties [23], [24], [39], [40], represent attractive targets for the optical control of SkLs mediated by the energetic exchange between the electronic, lattice, and spin subsystems.", "In this letter, we report on ultrafast time-resolved magneto-optical Kerr effect (TR-MOKE) measurements that demonstrate the generation of coherent collective excitations of the magnetic cycloid (Cyc) and SkL states in the lacunar spinel GaV4S8.", "Our results reveal GHz oscillations of the magnetization, driven by a laser-induced thermal modulation of the uniaxial anisotropy.", "Additionally, we observe a photo-induced enhancement of the magnetization that originates from the light-driven switching between the Cyc and ferromagnetic phases.", "These experiments establish an alternative route towards the optical control of the dynamic magnetic character of novel spin textures, leveraging the intimate coupling between the lattice and spin degrees of freedom.", "Figure: (a) Pump-induced change in the normalized Kerr rotation angle of theprobe pulse below (blue) and above (red) T C T_{C}, (b) the absolutevalue of the maximum demagnetization as a function of temperatureand normalized to the value at T=11.4KT=11.4\\text{ K}, and (c) the differentialreflectivity trace taken at T=10KT=10\\text{ K}.GaV4S8 is a multiferroic narrow-gap semiconductor belonging to a family of lacunar spinels consisting of an FCC lattice of tetrahedral (GaS4)5- and cubane (V4S4)5+ clusters, the latter carrying $S=1/2$ spins.", "Below $T_{JT}=44\\text{ K}$ , a rhombohedral ($C_{3v}$ ) distortion appears, with the rhombohedral axis oriented along one of the cubic body diagonals [41], [42].", "For $T<T_{C}\\approx 13\\text{ K}$ and moderate external fields, the material is an easy-axis ferromagnet with spins oriented along the rhombohedral axis.", "At lower fields, a complex magnetic-phase diagram emerges consisting of Cyc and SkL ground states due to the competition between the Heisenberg-exchange interaction, the DMI particular to the $C_{3v}$ point-group, and the magnetocrystalline anisotropy [43], [23].", "Unlike SkLs in chiral magnets, the Néel-type SkL in GaV4S8 is pinned to the plane perpendicular to the rhombohedral axis [23], [44].", "This is primarily due to the Lifshitz-invariants that comprise the DMI term, which energetically favor magnetic modulations with $\\mathbf {q}$ -vectors perpendicular to the rhombohedral axis.", "Due to this and the uniaxial magnetocrystalline anisotropy, the field-stability range of the Cyc and SkL phases depend on the orientation of the magnetic field with respect to the rhombohedral axis, since different domains often coexist in these crystals.", "As a result, several Cyc and SkL phases can be supported simultaneously, owing to the different projections of the external field along the easy axis for the four different domains.", "In this work, the external magnetic field was oriented perpendicular to an as-grown $\\left(001\\right)$ surface of a GaV4S8 crystal.", "This ensured that all the rhombohedral domains were magnetically equivalent, thereby hosting Cyc and SkL phases over the same field ranges [23].", "We employed TR-MOKE spectroscopy to probe the magnetization dynamics of the ferromagnetic, Cyc, and SkL states.", "The pump and probe pulses ($30\\text{ fs}$ , $1.57\\text{ eV}$ ) were modulated at $100\\text{ kHz}$ and $1.87\\text{ kHz}$ and focused to $50\\text{ $\\mu $m}$ and $35\\text{ $\\mu $m}$ diameters, respectively, with an on-sample pump fluence of $0.67\\text{ $\\mu $J/cm}^{2}$ .", "The magnetic field and sample temperature were controlled with a superconducting-magnetic cryostat equipped with a variable-temperature insert.", "To detect the Kerr-rotation (KR) in the reflected probe beam, we used a polarization-sensitive bridge, the differential signal from which we measured directly at the intermodulation frequencies via a phase-sensitive-detection scheme.", "This allowed for rapid data acquisition, avoiding the response-time issues associated with cascaded lock-in-amplifier configurations.", "Figure: Time-derivative of the TR-MOKE signal at T=12KT=12\\text{ K} at variousexternal magnetic fields.", "The inset shows the temperature dependenceof the magnetization for H ext =30mTH_{ext}=30\\text{ mT} in cyan and 50mT50\\text{ mT}in orange.", "The peak in the former, marked by the arrow, occurs atthe boundary of the Cyc phase.Figure REF (a) shows the pump-induced change to the normalized KR angle ($\\Delta \\theta _{k}$ ) of the probe pulse for the ferromagnetic phase in blue and the paramagnetic phase in red.", "As seen in Figure REF (b), the magnitude of the demagnetization step dramatically increases below $T_{C}$ , consistent with the onset of ferromagnetic order.", "We found that the demagnetization occurs over $\\sim 200\\text{ ps}$ for all external fields and sample temperatures that coincide with the ferromagnetic phase.", "This timescale is consistent with other semiconducting ferromagnets and can be attributed to the slow thermalization of the spin system due to its coupling to the lattice and isolation from the electronic bath [45], [46].", "This is supported by the differential-reflectivity trace shown in Figure REF (c), which contains contributions from electron-electron, electron-phonon, and phonon-phonon scattering [47], [48], all of which reach quasi-equilibrium within a few picoseconds.", "The change in $\\Delta \\theta _{k}$ occurs on a much longer timescale, demonstrated by its relatively small variation during the first few picoseconds.", "Accordingly, the thermalization of the phonon bath is a nearly instantaneous event for the spins and the time scale of the demagnetization is primarily governed by the strength of the magnon-phonon coupling [49], [50].", "The slow response of the spin-system to changes in the lattice is further exemplified by the magnetization dynamics just below $T_{C}$ .", "Figure REF shows the time-derivative of $\\Delta \\theta _{k}$ at $T=12\\text{ K}$ for different external magnetic fields ($H_{ext}$ ).", "For $H_{ext}\\ge 35\\text{ mT}$ , we observe demagnetization dynamics consistent with the pump-induced response of the ferromagnetic phase.", "However, for $H_{ext}\\le 30\\text{ mT}$ , the signal increases following the pump pulse.", "This photo-induced enhancement of the magnetization originates from the temperature dependence of the magnetization ($M$ ) across the magnetic phase boundaries [51].", "The inset of Figure REF shows $M(T)$ normal to the $\\left(001\\right)$ surface for $H_{ext}=30\\text{ mT}$ and $50\\text{ mT}$ .", "As indicated by the arrow in the inset, the $30\\text{-mT}$ $M(T)$ curve is peaked at the Cyc/ferromagnetic boundary above $12\\text{ K}$ .", "Accordingly, the enhancement of the magnetization results from the comparatively slow response of the spins to the impulsive heating of the lattice, pushing the system along the $M(T)$ peak following the pump excitation.", "At higher fields, the peak in $M(T)$ vanishes, as shown in the $50\\text{-mT}$ curve in the inset of Figure REF , corresponding to the restoration of conventional demagnetization behavior.", "Figure: (a) Time derivative of the TR-MOKE signal at T=10KT=10\\text{ K} fordifferent external magnetic fields spanning the Cyc, SkL, and ferromagneticphases and (b) the Fourier transforms of the 20mT20\\text{ mT} (blue)and 50mT50\\text{ mT} (green) traces in (a).", "The dashed line in (a)marks the end of the first oscillatory period.", "The green lines in(a) are calculated using Eqs.", "() and () with β=0.03\\beta =0.03 and β=0.1\\beta =0.1for the 20mT20\\text{ mT} and 50mT50\\text{ mT} curves, respectively.To probe the dynamics of the Cyc and SkL phases, we fixed the sample temperature at $10\\text{ K}$ and collected TR-MOKE traces for several external magnetic fields.", "The results are shown as black lines in Figure REF (a), where we plot the derivative of $\\Delta \\theta _{k}$ to suppress the incoherent magnetization dynamics.", "For small fields, we observe GHz oscillations in the signal, as seen in the $20-75\\text{ mT}$ traces.", "With increasing field, the oscillatory frequency changes, as noted by the dashed line that marks the end of the first oscillatory period.", "For $H_{ext}>75\\text{ mT}$ , the oscillatory structure vanishes.", "Figure REF (b) shows Fourier transforms of two traces in Figure REF (a), one representative of the Cyc phase ($20\\text{ mT}$ ) and the other of the SkL phase ($50\\text{ mT}$ ).", "In the Cyc phase, we see a single peak centered at approximately $6\\text{ GHz}$ .", "Comparing this to ESR measurements [39], we identify this as the low frequency Cyc eigenmode.", "In the SkL phase, we observe a strong peak centered at $3.75\\text{ GHz}$ and a weaker peak at $7.50\\text{ GHz}$ .", "These frequencies are consistent with the SkL breathing mode and the counter-clockwise (CCW) rotation mode, respectively [39].", "Notably, the clockwise (CW) rotation mode is absent in our measurements.", "We now address two fundamental questions: (1) what is the underlying mechanism driving the coherent collective spin excitations and (2) why are only certain modes excited?", "To answer the first, we note that the presence of coherent Cyc and SkL excitations were found to be invariant to the incident pump polarization.", "This is typically a fingerprint of a thermal process that does not involve a direct coupling between the pump-photon field and the spin system [30].", "Thermal mechanisms of this type have been explored in the study of laser-induced spin-precessions in materials such as TmFeO3 [34], Co films [35], and GaMnAs [36], [37].", "Being a polar semiconductor, the electron and optical phonon subsystems in GaV4S8 are strongly coupled, leading to a substantial increase in the lattice temperature following the pump pulse [47].", "This can, in turn, lead to a modulation of the magnetocrystalline anisotropy [30].", "Though such an effect typically requires a large pump fluence, this constraint is eased in GaV4S8 due to the strong temperature variation of its first uniaxial anisotropy constant ($K_{1}$ ) below $T_{C}$ [40].", "Therefore, the laser-induced heating of the sample significantly modulates the effective field acting on the magnetic system through the anisotropy contribution, driving the magnetic excitations of the SkL and Cyc states.", "Owing to the relatively long time required for heat to dissipate from the photo-excited volume through diffusion [47], this can be interpreted as a step-like modulation of $K_{1}$ within the experimental window.", "To justify the above description, we used the finite-difference time-domain method to solve the Landau-Lifshitz-Gilbert equation for the SkL state using the Mumax3 code [52].", "The effective field acting on the magnetic system is given by $\\mathbf {H}_{eff} & = & \\mathbf {H}_{ext}+\\mathbf {H}_{ani}+\\mathbf {H}_{d}+\\mathbf {H}_{DMI}+\\mathbf {H}_{exch}\\nonumber \\\\& = & H_{ext}\\hat{\\mathbf {e}}_{ext}+\\frac{2K_{1}}{\\mu _{0}M_{S}}\\left(\\hat{\\mathbf {e}}_{u}\\cdot \\text{\\textbf {m}}\\right)\\hat{\\mathbf {e}}_{u}+\\mathbf {H}_{d}\\nonumber \\\\& & \\;+\\frac{2D}{\\mu _{0}M_{S}}\\left[\\mathcal {L}_{xz}^{\\left(x\\right)}+\\mathcal {L}_{yz}^{\\left(y\\right)}\\right]+\\frac{2A_{ex}}{\\mu _{0}M_{S}}\\nabla ^{2}\\mathbf {m}$ where $A_{ex}=0.0588\\text{ pJ/m}$ is the exchange stiffness, $M_{S}=28.8\\text{ kA/m}$ is the saturation magnetization, $K_{1}=10\\text{ kJ/m}^{3}$ is the (steady state) anisotropy constant, $D=0.043\\text{ mJ/m}^{2}$ is the DMI constant, $\\mathcal {L}_{jk}^{\\left(i\\right)}=m_{j}\\partial _{i}m_{k}-m_{k}\\partial _{i}m_{j}$ are Lifshitz-invariants corresponding to $C_{3v}$ symmetry, $\\hat{\\mathbf {e}}_{u}$ is a unit vector in the direction of the easy axis, $\\hat{\\mathbf {e}}_{ext}$ is a unit vector in the direction of the applied field, $\\mathbf {m}=\\mathbf {M}/M_{S}$ , and $\\mathbf {H}_{d}$ is the demagnetizing field.", "The material parameters were estimated from literature and match the experimental periodicity of the SkL state [23], [40].", "Here, $\\hat{\\mathbf {x}}^{\\prime }\\parallel \\left[100\\right]$ , $\\hat{\\mathbf {y}}^{\\prime }\\parallel \\left[010\\right]$ , $\\hat{\\mathbf {z}}^{\\prime }\\parallel \\left[001\\right]$ , $\\hat{\\mathbf {x}}\\parallel \\left[11\\overline{2}\\right]$ , $\\hat{\\mathbf {y}}\\parallel \\left[1\\overline{1}0\\right]$ , and $\\hat{\\mathbf {z}}=\\hat{\\mathbf {e}}_{u}\\parallel \\left[111\\right]$ .", "We introduced a time dependence in the effective magnetic field through a step-like decrease of $K_{1}$ by 1% of its steady state value, consistent with our estimate of the pump-induced heating of the lattice.", "The simulated system consisted of a $128\\times 64$ grid of $0.8\\text{ nm}^{3}$ cuboids with periodic boundary conditions along $\\hat{\\mathbf {x}}$ and $\\hat{\\mathbf {y}}$ , initialized with one unit cell of a triangular Néel-type SkL with the SkL-plane normal to $\\hat{\\mathbf {z}}$ .", "The stability of this state was established by slowly field cooling the system in the presence of an external magnetic field and fluctuating thermal field [52], [53].", "The lattice parameters were then determined by minimizing the total energy.", "For the results discussed below, the external field parallel to $\\hat{\\mathbf {z}}$ was $125\\text{ mT}$ and the field along $\\hat{\\mathbf {x}}$ was varied, resulting in a tilting of the external field with respect to $\\hat{\\mathbf {e}}_{u}$ by an angle $\\alpha $ .", "Figure: (a) The Fourier transforms of the simulated m ˙ z \\dot{m}_{z}, m ˙ x \\dot{m}_{x},and m ˙ z ' \\dot{m}_{z^{\\prime }} for various values of α\\alpha spanned by theexternal magnetic field and the rhombohedral axis and (b) the simulatedm z (t)m_{z}(t), m x (t)m_{x}(t), and m z ' (t)m_{z^{\\prime }}(t) for α=54.7 ∘ \\alpha =54.7{}^{\\circ }.Figure REF (a) shows the Fourier transforms of $\\dot{m}_{z}$ , $\\dot{m}_{x}$ , and $\\dot{m}_{z^{\\prime }}$ for various values of $\\alpha $ .", "For $\\alpha =0^{\\circ }$ , we see a single resonance peak that manifests only in the $z$ -component, corresponding to the breathing mode.", "This is because the modulation of $\\mathbf {H}_{eff}$ is entirely along the $z$ -direction (i.e.", "normal to the SkL plane) and can therefore only couple to the breathing mode [54].", "As $\\alpha $ is increased, however, we see the gradual appearance of two additional peaks appearing in the $z$ -, $x$ -, and $z^{\\prime }$ -components of the magnetization.", "The appearance of the new resonances is due to the the core shift characteristic of Néel-type SkLs subject to oblique external fields [55].", "This results in a deformation of the Sk texture, reducing the six-fold rotational symmetry of the SkL to a two-fold rotation, thereby introducing a time-dependent component to the effective field in the plane of the SkL, which activates the rotational modes [54].", "The tilting of the net magnetization and deformation of the SkL are relatively small, which accounts for the weakness of the CCW mode in our experimental results where $\\alpha =54.7^{\\circ }$ .", "Further, we see that the third simulated resonance peak is relatively weak, a fact that is supported by the absence of the CW mode in our measurements.", "Finally, we note that the simulated resonances were blue-shifted with respect to the experimental results.", "This deviation is most likely due to the strong sensitivity of the mode frequencies to the values of $D$ and $A_{ex}$ [53].", "Nevertheless, the order of the simulated resonances is consistent with previous ESR measurements [39] as well as our experimental observations.", "We now construct a phenomenological model of the experimental TR-MOKE traces.", "From Figure REF (b), we see that the magnetization dynamics resulting from a modulation of $K_{1}$ are comprised of decaying sinusoidal oscillations superimposed on a step-like offset.", "This reflects the transient reorientation of $\\mathbf {m}$ due to the reduced anisotropy following the optical pump.", "This type of response can be modeled by a damped harmonic oscillator driven by a step-like force, in this case, representing the optically-induced modulation of the uniaxial anisotropy.", "The incoherent de/remagnetization dynamics can be described by the sum of two exponentials convolved with a step-like function representing the response time of the spin-system to the lattice.", "For both the oscillatory and incoherent parts, we use the same step-like function.", "Finally, we model the magnetization dynamics as the sum of the incoherent and oscillatory contributions, taking $\\Delta \\theta \\left(t\\right)=A\\cdot I_{i}\\left(t\\right)+B\\cdot I_{o}\\left(t\\right)$ where the incoherent ($I_{i}$ ) and oscillatory ($I_{o}$ ) parts of the signal are given by the solutions to $\\frac{d^{2}I_{o}}{dt^{2}}+2\\gamma \\omega _{0}\\frac{dI_{o}}{dt}+\\omega _{o}^{2}I_{o}=g\\left(t\\right)\\cdot e^{-t/\\tau }\\:\\text{,}$ $I_{i}\\left(t\\right)=\\left(h\\ast g\\right)\\left(t\\right)$ where $h\\left(t\\right)=e^{-t/\\tau _{1}}-\\beta e^{-t/\\tau _{2}}$ and $g\\left(t\\right)=\\left[\\text{erf}\\left(\\alpha t\\right)+1\\right]$ .", "Here, $\\tau _{1}$ and $\\tau _{2}$ are the demagnetization and remagnetization time-constants, $\\beta $ is a scaling parameter, $\\alpha $ controls the spin-response time, and $\\tau $ is the rate at which $K_{1}$ returns to the pre-time-zero value.", "Estimating $\\tau _{1}=130\\text{ ps}$ , $\\tau _{2}=\\tau =2600\\text{ ps}$ , and $\\alpha =0.05$ from the experimental results, we obtain the curves plotted in green in Figure REF (a).", "The agreement between this model and the experimental results illustrates that the measured magnetization dynamics reflect the competition between incoherent and oscillatory signals.", "In summary, we have demonstrated the ultrafast optical generation of coherent collective spin excitations of the Cyc and SkL phases in GaV4S8, driven by an optically-induced modulation of the uniaxial magnetocrystalline anisotropy.", "This indirect coupling between the optical pulse and the spin system is mediated by the lattice and represents a new mechanism by which magnetic excitations can be generated in skyrmion-host compounds with strong anisotropy.", "Furthermore, the peculiar nature of the magnetic ordering at the phase boundaries of GaV4S8 allows for a transient enhancement of the magnetization driven by the optically-induced heating of the lattice.", "This study underscores the intimate coupling between the spin and lattice subsystems in GaV4S8, and may provide a framework for the optical control of topological magnetic objects in semiconductors.", "P.P., F.S., R.B.V., E.S., and P.H.M.vL.", "acknowledge financial support from the the Deutsche Forschungsgemeinschaft (DFG) through SFB-1238 (Project B05).", "V.T.", "and I.K.", "acknowledge financial support from the DFG via the Transregional Research Collaboration TRR 80: From Electronic Correlations to Functionality (Augsburg-Munich-Stuttgart) and via the Skyrmionics Priority Program SPP2137.", "S.B.", "acknowledges financial support from the National Research, Development and Innovation Office – NKFIH, ANN 122879, and the BME-Nanotechnology and Materials Science FIKP grant of EMMI (BME FIKP-NAT)." ] ]

[ [ "Dynamic structure factor of superfluid He-4 from Quantum Monte Carlo:\n Maximum Entropy revisited" ], [ "Abstract We use the Maximum Entropy Method (MaxEnt) to estimate the dynamic structure factor of superfluid He-4 at T=1 K, by inverting imaginary-time density correlation functions computed by a Quantum Monte Carlo (QMC) simulation.", "Our procedure consists of a Metropolis random walk in the space of all possible spectral images, sampled from a probability density which includes the entropic prior, in the context of the so-called \"classic\" MaxEnt.", "Comparison with recent work by other authors shows that, contrary to what is often stated, sharp features in the reconstructed image are not \"washed out\" by the entropic prior if the underlying QMC data have sufficient accuracy.", "Only spurious features that tend to appear in a straightforward chi-square minimization are suppressed." ], [ "Introduction", "Quantum Monte Carlo simulations are among the most reliable tools to investigate the physics of quantum many-body systems in thermal equilibrium.", "In particular, thermodynamic properties of interacting Bose assemblies, such as superfluid $^4$ He, can be calculated quite accurately [1].", "At least in principle, QMC also allows one to obtain dynamical properties, at least within the linear response approximation; for, one can compute correlation functions in imaginary time, from which spectral functions can be inferred through an inverse Laplace transformation.", "Unfortunately, the inversion is mathematically ill-posed, and because QMC data are inevitably affected by statistical uncertainties, an unambiguous determination of the spectral function is usually not possible.", "In some cases, prior knowledge about the physics of the system may constrain the set of possible solutions, allowing for a reliable reconstruction; for example, one may know that the spectral function is dominated by one or two well-defined peaks, and simply fit the QMC data accordingly (see, for instance, Ref.", "saccani).", "In the general case, however, when no such knowledge is available, a large number of very different images will be consistent with the QMC data.", "Thus, one will typically resort to some kind of “regularization” scheme (RS), aimed at retaining only those images whose non-trivial structure is truly warranted by the data.", "Consequently, any RS will inevitably tend to soften some of the sharpest features; for example, distinct, isolated peaks will be broadened, to reflect the inherent uncertainty arising from the finite precision of the data and the ill-posedness of the problem [3].", "A popular RS, in the context of inversion of QMC data, is the Maximum Entropy method (MaxEnt) [4], [5], which has been applied to the determination of spectral functions of various lattice many-body Hamiltonians [6], [7], [8], [9], [10] as well as of the dynamic structure factor in normal and superfluid $^4$ He [11].", "In general, MaxEnt has yielded quantitatively reliable results for some of the main aspects of the reconstructed images, i.e., the positions of the peaks, and therefore the determination of the excitation spectrum; on the other hand, the quantitative accuracy of predictions concerning, e.g., the widths of the peaks, and the ensuing ability to resolve adjacent peaks, was less satisfactory, although in most cases the limiting factor was the quality of the QMC data, rather than the RS adopted to extract the images.", "Alternative RS have been proposed in the course of the years, the context of QMC simulations [12], [13], [14], [15], [16], displaying some advantages over others for specific applications, but no comprehensive, systematic comparison has yet been carried out (at least to our knowledge).", "In recent years, the problem of extraction of the dynamic structure factor of superfluid $^4$ He from imaginary-time correlations computed by QMC has been independently revisited by two groups [17], [18], who proposed RSs not making use of MaxEnt's entropic prior.", "In both cases, their procedure essentially amounts to $\\chi ^2$ -fitting [19], supplemented by averaging over a set of comparable images, in order to suppress some of the spurious structure that inevitably arises on carrying out $\\chi ^2$ minimization in the presence of an ill-posed problem.", "Both works make the claim that their proposed approaches are superior to MaxEnt, in that the resulting images are sharper and in better agreement with experimental data.", "In this paper, we revisit the use of MaxEnt for the same problem, in order to assess quantitatively the claims made in Refs.", "gift,ferre.", "Specifically, we estimate the dynamic structure factor $S({\\bf q},\\omega )$ for superfluid $^4$ He, by computing imaginary-time density correlations by QMC, and by using MaxEnt to carry out the inversion.", "Our methodology is similar to that of Ref.", "massimo1996, i.e., it consists of a Metropolis random walk in the space of spectral images, sampled from a probability density proportional to the standard maximum likelihood estimator, multiplied by the entropic prior (see below).", "This procedure allows us to assign an uncertainty in the value of $S({\\bf q},\\omega )$ , as the standard deviation of the values recorded for the different frequencies in the course of the random walk.", "Compared to Ref.", "massimo1996, our present study obviously benefits from two decades of advances, both in computing hardware as well as in the QMC methodology utilized to generate the imaginary-time data.", "As a result, our statistical uncertainties are much smaller than those of the 1996 work, comparable to those of the data used in Refs.", "gift,ferre, which is a necessary condition in order to carry out a meaningful and fair comparison.", "Based on the results presented here, we contend that MaxEnt does not prevent sharp features from appearing in the reconstructed spectral functions, as long as the accuracy of the QMC data justifies their inclusion.", "Indeed, the spectral images shown here are of comparable (or better) quality than those offered in Refs.", "gift,ferre.", "Ultimately, the sharpness of the spectral image almost exclusively hinges on the accuracy of the QMC data; by promoting smoothness, the entropic prior serves in our view a useful, noise-reducing purpose.", "It is worth noting that a general scheme capable of tackling this kind of problem can be applied in other, rather different contexts, e.g., the determination of ground state expectation values in QMC transient estimate calculations [20].", "These are typically carried out for Fermi systems, which are affected by the infamous “sign” problem, resulting in an exponential increase with imaginary time of the statistical error (see, for instance, Ref.", "bm).", "The remainder of this paper is organized as follows: in section we describe the model of the system and the QMC calculations carried out in this work; in Sec.", "we describe in detail our inversion method; we present and discuss our results in Sec.", "and finally outline our conclusions in Sec.", "." ], [ "Model and QMC calculation", "In this section we describe the QMC calculation of the imaginary-time correlation function which is then inverted to obtain the dynamic structure factor.", "The system is described as an ensemble of $N$ point-like, identical particles with mass $m$ equal to that of a He atom and with spin $S=0$ , thus obeying Bose statistics.", "It is enclosed in a cubic cell, with periodic boundary conditions in the three directions.", "The quantum-mechanical many-body Hamiltonian reads as follows: $\\hat{H} = - \\lambda \\sum _{i}\\nabla ^2_{i}+\\sum _{i<j}v(r_{ij})$ where the first (second) sum runs over all particles (pairs of particles), $\\lambda \\equiv \\hbar ^2/2m=6.0596415$ KÅ$^{2}$ , $r_{ij}\\equiv |{\\bf r}_i-{\\bf r}_j|$ and $v(r)$ is a pair potential which describes the interaction between two atoms.", "We make use in this study of the accepted Aziz pair potential [22], which has been utilized in most simulation studies of superfluid helium.", "A more accurate model would also include interactions among triplets of atoms; however, published numerical work has given strong indications that three-body corrections, while significantly affecting the estimation of the pressure, have a relatively small effect on the structure and dynamics of the system, of interest here [23].", "We carried out QMC simulations of the system described by Eq.", "(REF ) at temperature $T = 1$ K, using the continuous-space Worm Algorithm [1].", "Since this technique is by now fairly well-established, and extensively described in the literature, we shall not review it here.", "A canonical variant of the algorithm was utilized, in which the total number of particles $N$ is held fixed [24], [25].", "The quantity of interest here is the dynamic structure factor $S({\\bf q},\\omega )$ , which describes density fluctuations of wave vector q.", "For superfluid $^4$ He it has been extensively studied experimentally by neutron scattering (for a review, see, for instance, Ref.", "glyde).", "It is a direct probe of the elementary excitations (phonons and rotons) that underlie the physical behavior of the system at low temperature [27], [28], [29].", "$S({\\bf q},\\omega )$ is non-negative function satisfying the relation [30] $\\langle \\omega \\rangle =\\int _0^\\infty \\ d\\omega \\ \\omega \\ S({\\bf q},\\omega )\\ (1-e^{-\\beta \\omega })=\\frac{{q}^2}{2m}$ known as f-sum rule (we henceforth set $\\hbar =1$ , the Boltzmann constant $k_B=1$ and define $\\beta =1/T$ ).", "There is no known QMC scheme allowing for the direct calculation of $S({\\bf q},\\omega )$ .", "However, it can be shown (see, for instance, Ref.", "massimo1996) that $F(\\mathbf {q},\\tau ) = \\int _0^{\\infty }\\ d\\omega \\ (e^{-\\omega \\tau }+e^{-\\omega (\\beta -\\tau )})\\ S(\\mathbf {q},\\omega )$ where $0\\le \\tau \\le \\beta $ and $F({\\bf q},\\tau )$ is the imaginary-time auto-correlation function $F(\\mathbf {q},\\tau )=\\frac{1}{N}\\ \\langle \\hat{\\rho }_{\\mathbf {q}}(\\tau )\\ \\hat{\\rho }_{\\mathbf {q}}^\\dagger (0)\\rangle $ where $\\langle ...\\rangle $ stands for thermal average, and with $\\rho _{\\mathbf {q}}({\\tau }) = \\sum _{j=1}^N e^{i \\mathbf {q} \\cdot \\mathbf {r}_{j}},$ where the $\\lbrace {\\bf r}_j\\rbrace $ , $j=1,2,...N$ are the positions of the $N$ $^4$ He atoms at imaginary time $\\tau $ along the many-particle path.", "The quantity $F({\\bf q},\\tau )$ is what is actually computed by QMC, for a discrete set of values of $\\tau $ ; $S({\\bf q},\\omega )$ is inferred from $F({\\bf q},\\tau )$ through a numerical inversion of eq.", "REF .", "The details of this procedure are outlined in Sec.", ".", "The QMC simulation is standard; we adopted the usual the short-time approximation to the imaginary-time propagator accurate to fourth order in the time step $\\epsilon $ (see, for instance, Ref.", "jltp).", "All of the results presented here are extrapolated to the $\\epsilon \\rightarrow 0$ limit; just like for other observables, the numerical estimates of the quantities of interest here, namely the imaginary-time correlation functions described below, computed with a value of the time step $\\epsilon =$ (1/640) K$^{-1}$ are indistinguishable from the extrapolated ones, within the statistical uncertainties of the calculation.", "Calculations were carried out at two different densities, namely 0.021834 Å$^{-3}$ , which is that at saturated vapor pressure (SVP) [32], and 0.0260 Å$^{-3}$ , which is very close to the freezing density (at a pressure of approximately 25 bars).", "All calculations were carried out at $T=1$ K. The experimental and theoretical data we compare our results against are at temperatures that range from 0 K to 1.3 K. All such temperatures are well below the lambda transition, and at that level the excitations are essentially independent of temperature (see, for instance, Refs.", "gibbs,dietrich1972).", "We took advantage of space and time symmetry to improve statistics; a rough estimate of the statistical error on the generic value of $F({\\bf q},\\tau )$ is given by $5\\times 10^{-4}\\ F({\\bf q},0)$ .", "The bulk of the results shown here were obtained on a system comprising $N=64$ particles, a number which is not particularly large but that allows us to collect good statistics in a given simulation time; experience with previous work [11] suggests that this system size is sufficient to extract information at the wave vectors of interest here (see below).", "However, we have also repeated the simulation with $N=256$ particles, and found no statistically significant difference in the values of $F({\\bf q},\\tau )$ , within the statistical errors of our calculation.", "$F({\\bf q},0)\\equiv S_{\\bf q}$ is known as the static structure factor, which is experimentally accessible and it is related via a Fourier transformation to the atomic pair correlation function.", "The values of $S_{\\bf q}$ obtaind here are in quantitative agreement with previous calculations, i.e., in excellent agreement with experiment (see Ref.", "rmp).", "Figure: Color online.", "Typical F(𝐪,τ)F(\\mathbf {q},\\tau ) results computed in a simulation of superfluid 4 ^4He at T=1T=1 K at density 0.021834 Å -3 ^{-3}.", "Results shown here are for the wave vectors q=1.075q=1.075 Å -1 ^{-1} (bottom curve), q=1.756q=1.756 Å -1 ^{-1} (middle curve) and q=1.964q=1.964 Å -1 ^{-1} (top curve).", "When not shown, statistical errors are smaller than the size of the symbols.Typical results for $F({\\bf q},\\tau )$ are shown in Fig.", "REF ; because $F({\\bf q},\\tau )=F({\\bf q},\\beta -\\tau )$ (see, for instance, Ref.", "lovesey), one need only compute this quantity in the $0\\le \\tau \\le \\beta /2$ interval." ], [ "MaxEnt Inversion", "The problem with the numerical inversion of eq.", "REF , aimed at obtaining $S({\\bf q},\\omega )$ from the values of $F({\\bf q},\\tau )$ computed by QMC, lies in the fact that the integral kernel exponentially suppresses the contribution at high frequency of the spectral function to $F({\\bf q},\\tau )$ ; consequently, $F(\\mathbf {q},\\tau )$ is minimally affected by the high frequency behavior of $S({\\bf q},\\omega )$ .", "Because $F(\\mathbf {q},\\tau )$ is the result of QMC simulations, and therefore possesses finite statistical uncertainties, there will be typically a large set of physically different spectral functions consistent with the numerical data for $F(\\mathbf {q},\\tau )$ .", "Most of these solutions are unphysical and/or bear little resemblance to the actual $S({\\bf q},\\omega )$ .", "The goal is that of finding a systematic and robust way to weed out spurious solutions, and retaining only a relatively small subset of physical ones, from which at least the most important physical features of $S({\\bf q},\\omega )$ may be reliably extracted.", "As mentioned above, $F({\\bf q},\\tau )$ is computed for the discrete set of imaginary times $l\\delta \\tau $ , $l=0,1,...,L$ , with $2L\\delta \\tau =\\beta $ .", "In order to simplify the notation, for a given value of q we define $\\mathbf {F}\\equiv \\lbrace F_0,...,F_L\\rbrace $ , with $F_l\\equiv F({\\bf q},l\\delta \\tau )$ .", "Each entry $F_l$ is affected by a statistical uncertainty $\\sigma _l$ , estimated by careful binning analyses of data (see, for instance, Ref.", "flyvbjerg1989) collected over sufficiently long simulations.", "We begin by approximating the integral on the right hand side of eq.", "REF with a sum, i.e., turn eq.", "REF into a system of algebraic equations that can be expressed in compact matrix form $\\mathbf {F} = \\mathbf {K} \\mathbf {S},$ having defined $K_{lj} = [e^{-jl \\delta \\omega \\delta \\tau } + e^{-j(2L-l)\\delta \\omega \\delta \\tau }]\\ \\delta \\omega ,$ $\\mathbf {S} \\equiv \\lbrace S_1,...,S_M \\rbrace $ , $S_j\\equiv S({\\bf q}, j\\delta \\omega )$ , and $M\\delta \\omega =\\omega _M$ , $\\omega _M$ chosen large enough that $S({\\bf q},\\omega )$ can be set to zero for $\\omega > \\omega _M$ , and $\\delta \\omega $ small enough to achieve the desired frequency resolution.", "In this study, $\\omega _M$ is between 100 and 300 K, whereas $M$ is between 150 and 400.", "An important observation is that typically $M>L$ , i.e., the system (REF ) is underdetermined, and therefore, in general, no unique solution can be found, quite irrespective of the ill-posedness of the problem and of statistical errors of the computed imaginary-time correlation functions [37].", "We take the same approach as in Ref.", "massimo1996, based on “classic” MaxEnt (see, for instance, Ref.", "bayesian) and define our “optimal” solution as ${\\mathbf {S}}_{\\circ } \\equiv \\int d\\alpha \\ \\int {D}{\\mathbf {S}}\\ {\\mathbf {S}}\\ {\\cal F}(\\alpha ,{\\mathbf {S}})$ where ${D}{\\mathbf {S}}\\equiv dS_1dS_2...dS_M$ , and ${\\cal F}({\\alpha ,\\mathbf {S}})=\\frac{e^{-\\chi ^2({\\mathbf {S}})/2}}{Z_Q}\\ \\frac{e^{\\alpha {\\cal S}({\\mathbf {S}})}}{Z_{\\cal S}(\\alpha )}\\ \\rho ({\\mathbf {S}})$ is a prior probability assigned to the generic image $\\mathbf {S}$ .", "Here, $\\alpha $ is a non-negative regularization parameter, to which we come back below; $Z_Q$ and $Z_S(\\alpha )\\propto \\alpha ^{-M/2}$ are normalization constants; $\\chi ^2({\\mathbf {\\bar{S}}}) = (\\mathbf {F}-\\mathbf {\\bar{F}})^T\\mathbf {C}^{-1} (\\mathbf {F}-\\mathbf {\\bar{F}})$ is the standard measure of goodness of fit, with ${\\mathbf {\\bar{F}}}={\\mathbf {K}}{\\mathbf {\\bar{S}}}$ and we make the diagonal approximation [38] for the covariance matrix $\\mathbf {C}$ , i.e., $C_{ij}=\\sigma ^2_{i}\\delta _{ij},$ and ${\\cal S}(\\mathbf {S}) =-\\sum _{i=1}^M \\ f_i\\ ln\\biggl ( {Mf_i}\\biggr ),$ with $f_i=S_i/(\\sum _j S_j)$ , is Jaynes' entropy of the image ${\\mathbf {S}}$ [39], [40]; and finally, $\\rho (\\mathbf {S}) \\propto {\\rm exp} \\left(-\\frac{[\\langle \\omega \\rangle -\\omega _\\mathbf {q}]^2}{2\\eta ^2\\omega ^2_\\mathbf {q}}\\right)$ where $\\langle \\omega \\rangle $ is defined in eq.", "REF , $\\omega _{\\bf q}=q^2/(2m)$ and $\\eta $ is adjusted to enforce that relation (REF ) be satisfied to the desired degree of accuracy (typically $\\eta \\le 0.01$ ).", "The prior probability (REF ) ascribes greater weight to those spectral functions that are consistent with the data, and therefore have a low value of $\\chi ^2$ and fulfill the $f$ -sum rule, while at the same time are smoother in character.", "In other words, sharp features such as isolated peaks should not be included unless consistency with the data requires it.", "The parameter $\\alpha $ can be used to “tune” the relative importance of the entropic prior in ${\\cal F}(\\mathbf {S})$ ; in the limit $\\alpha \\rightarrow 0$ , one is performing conventional $\\chi ^2$ -fitting; on the other hand, as $\\alpha $ grows the entropic prior becomes increasingly important.", "The question arises of how to choose the value of $\\alpha $ .", "In “historic” MaxEnt, one adjusts $\\alpha $ so that on average, the value of $\\chi ^2\\sim L$ .", "As mentioned above, we adopt the “classic” MaxEnt approach, in which $\\alpha $ is treated as a random variable, and assigned a prior probability distribution $p(\\alpha )$ , which is incorporated in the normalization constant $Z_S(\\alpha )$ .", "We evaluate the multidimensional integral in eq.", "REF by Monte Carlo, just as in Ref.", "massimo1996.", "Specifically, we perform a random walk in $\\lbrace \\mathbf {S},\\alpha \\rbrace $ -space, using the Metropolis algorithm to sample the probability density given by eq.", "REF .", "We achieve that through few elementary moves, designed to satisfy the usual detailed balance condition.", "Specifically, we randomly attempt either one of the following: the displacement of an elementary amount of area, equal to $\\gamma \\ \\delta S$ , where $0\\le \\gamma \\le 1$ is a uniform random number, from a randomly selected channel $j$ to another one, randomly selected among $j-p,...j-1,j+1,... j+p$ .", "the addition or subtraction of $\\gamma \\ \\delta S^\\prime $ from a randomly selected channel $j$ .", "the change of $\\alpha $ by an amount $(1/2-\\gamma ) \\ \\delta \\alpha $ .", "Proposed moves are accepted or rejected based on the usual Metropolis test, making use of eq.", "REF in the acceptance ratio [41].", "The parameters $\\delta S$ , $\\delta S^\\prime , \\delta \\alpha $ and $p$ are adjusted to ensure a 50% acceptance rate.", "The move attempting to change the value of $\\alpha $ is typically attempted every $\\sim M$ attempts to perform either one of the first two moves.", "Figure: Posterior probability for the regularization parameter α\\alpha (top) and for thethe goodness-of-fit parameter χ 2 \\chi ^2 (bottom), obtained from the Metropolis random walk in {𝐒,α}\\lbrace \\mathbf {S},\\alpha \\rbrace -space as described in the text.", "This particular result refers to the q=1.756q=1.756 Å -1 ^{-1} case.The posterior probability of $\\alpha $ , $Pr[\\alpha ]$ as well as the $\\chi ^2$ distribution $Pr[\\chi ^2]$ , are obtained from the random walk, just as in Ref.", "massimo1996.", "Fig.", "REF shows a typical result.", "The optimal image $\\mathbf {S}_\\circ $ (eq.", "REF ), determined as an average over the images generated in the random walk, is affected by a statistical error, that can be estimated in the standard way, and can be rendered sufficiently small upon using a relatively modest amount of CPU time.", "More significant, however, given the inherent uncertainty of the inversion, is the standard deviation associated with the fluctuation of the values $S_i$ around their averages; we report it below, when illustrating our results, as it furnishes in our view a fair assessment of the range of variation of the solution." ], [ "Results", "Fig.", "REF shows results for $S({\\bf q},\\omega )$ for the roton wave vector ($q=1.963$ Å$^{-1}$ ) at $T=1$ K and at saturated vapor pressure (SVP).", "Squares represent the values of $\\mathbf {S}_\\circ $ defined through eq.", "REF , computed by means of the Monte Carlo Metropolis procedure described in Sec.", ".", "The statistical errors on the values of $\\mathbf {S}_\\circ $ are smaller than the sizes of the symbols.", "Also shown in the figure are experimental data [43] from Ref.", "andersen1994 at $T=1.3$ K and for the wave vector $q=1.90$ Å$^{-1}$ .", "Agreement between theory and experiment seems fairly good; not only the position, but also the width of the peak is rather well reproduced, unlike in previous applications of MaxEnt [11].", "This result shows that MaxEnt does not prevent the reconstructed spectral image from developing sharp features, if the quality of the underlying QMC data justifies their inclusion.", "In the presence of greater statistical uncertainties, on the other hand, MaxEnt implies a more conservative choice, namely one in which smoother images are privileged.", "As mentioned above, the statistical errors on $\\mathbf {S}_\\circ $ are comparable to, or smaller than the sizes of the symbols, and can always be rendered negligible with modest computing resources.", "Obviously, however, the issue arises of assessing systematic errors, which are inherent to this image reconstruction problem.", "In other words, how far off can the optimal image $\\mathbf {S}_\\circ $ be expected to be from the actual spectral function?", "The Metropolis procedure adopted here allows us to offer an estimate of that through the standard deviation of the values of $\\mathbf {S}_\\circ $ for each and every value of the energy.", "In Fig.", "REF we show one such standard deviation, corresponding to the energy interval $\\omega _m$ in which $\\mathbf {S}_\\circ $ takes on its highest value.", "Although not shown in the figure for clarity, $\\mathbf {S}_\\circ $ for the two energy intervals adjacent to $\\omega _m$ have comparable standard deviations, whereas the standard deviation for all other values is much smaller (of the order of symbol sizes in Fig.", "REF ).", "This is generally found to be the case, i.e., the (typically relatively few) values of $\\mathbf {S}_\\circ $ for which it is most important, are affected by the largest uncertainty.", "Thus, at least for the roton wave vector MaxEnt yields a reasonably accurate estimate of the position and the width of the peak, with some remaining uncertainty regarding its height.", "Figure: Color online.", "S(𝐪,ω)S(\\mathbf {q},\\omega ), defined as 𝐒 ∘ \\mathbf {S}_\\circ (eq. )", "and computed as illustrated in the text, for superfluid 4 ^4He at T=1T=1 K for the roton wave vector at SVP (diamonds, q=1.963q=1.963 Å -1 ^{-1}) and at 25 bars (circles, q=2.081q=2.081 Å -1 ^{-1}).", "Statistical errors on S(𝐪,ω)S(\\mathbf {q},\\omega ) are comparable to the sizes of the symbols for both curves.It is interesting to note that, despite the uncertainty, nevertheless relative comparisons of data obtained with the procedure illustrated here are still meaningful.", "For example, Fig.", "REF shows $S({\\bf q},\\omega )$ for the roton wave vector at two different pressures, namely SVP and 25 bars.", "The roton minimum shifts from $\\sim 1.9$ Å$^{-1}$ at SVP to $\\sim 2.1$ Å$^{-1}$ at 25 bars [44].", "Our results show that the position of the peak moves to lower energy and the peak itself gains strength, in remarkable quantitative agreement with experimental observation [33].", "Figure: Color online.", "S(𝐪,ω)S(\\mathbf {q},\\omega ) in superfluid 4 ^4He for the roton wave vector (q=1.963q=1.963 Å -1 ^{-1}) calculated through the inversion of QMC data based on four different methods.", "Hexagons show the result of the inversion using MaxEnt (eq.", "), whereas squares show that with α=0\\alpha =0 (which amounts to standard χ 2 \\chi ^2 fitting).", "Stars show the result of the inversion using GIFT for the wave vector q=1.977q=1.977 Å -1 ^{-1} at T=0T=0 K. Dark circles show the result of χ 2 \\chi ^2-minimization using simulated annealing (SA) for the wave vector q=1.91q=1.91 Å -1 ^{-1} at T=0.8T=0.8 K.In Fig.", "REF , we compare our results with those of other authors who made use of different approaches (not based on MaxEnt) to tackle the inversion of QMC data [45].", "The wave vectors are not identical but are reasonably close to the roton minimum in all cases; all calculations are carried out in the low temperature limit (see caption of Fig.", "REF for details).", "There is nearly perfect agreement between our image and that of Ref.", "gift, especially if the standard deviation of our result is taken into account.", "On the other hand, the spectral image obtained in Ref.", "ferre is much broader, with a significantly lower peak.", "It is interesting to compare these curves with that arising from $\\chi ^2$ -fitting carried out in the context of our procedure, namely by simply setting $\\alpha =0$ .", "In this case, the average value of $\\chi ^2$ is $\\sim 0.2\\ L$ , i.e., slightly lower than that obtained with finite $\\alpha $ .", "However, as can be seen in Fig.", "REF , the peak is significantly higher (in fact its height exceeds that of the experimental result by almost a factor two) and also narrower than what is observed experimentally.", "This is consistent with the general notion that “brute force” $\\chi ^2$ minimization, while yielding sharp features, is all too likely to result in unphysical behavior.", "The use of the entropic prior emphasizes the contribution from smoother images (still consistent with the QMC data), which in this case results in better agreement with experiment.", "Figure: Color online.", "S(𝐪,ω)S(\\mathbf {q},\\omega ) in superfluid 4 ^4He for the wave vector q=1.756q=1.756 Å -1 ^{-1} calculated through the inversion of QMC data based on three different methods.", "Hexagons show the result of the inversion using MaxEnt (eq.", ").", "Stars show the result of the inversion using GIFT for the wave vector q=1.755q=1.755 Å -1 ^{-1} at T=0T=0 K. Dark circles show the result of χ 2 \\chi ^2-minimization using simulated annealing (SA) for the wave vector q=1.76q=1.76 Å -1 ^{-1} at T=1.2T=1.2 K. Diamonds show experimental data from Ref.", "andersen1994 (only the coherent part is shown) at T=1.3T=1.3 K for the wave vector q=1.70q=1.70 Å -1 ^{-1}.Let us now consider a second wave vector, namely $q=1.756$ Å$^{-1}$ .", "In Fig.", "REF , we compare again the result of our MaxEnt inversion with those of Refs.", "gift,ferre, as well as experimental data from Ref.", "andersen1994.", "Our procedure yields a spectral image in much closer agreement with experiment than the other two.", "In particular, both the shape of the curve and the location of the main peak are in excellent agreement with experiment, taking into account the slight difference in wave vectors [46] and the resolution of our spectral image.", "On the other hand, the spectral image reported in Ref.", "ferre is once again much too broad compared to the experimentally observed one, while that of Ref.", "gift is considerably sharper.", "Figure: Color online.", "S(𝐪,ω)S(\\mathbf {q},\\omega ) in superfluid 4 ^4He for the maxon wave vector (q=1.075q=1.075 Å -1 ^{-1}).", "Squares show the result calculated through the inversion of QMC data based on MaxEnt (eq.", ").", "Stars show the result of the inversion using GIFT , , calculated for the wave vector q=1.107q=1.107 Å -1 ^{-1} at T=0T=0 K. Circles show experimental data from Ref.", "andersen1994 (only the coherent part is shown) at T=1.3T=1.3 K for the wave vector q=1.10q=1.10 Å -1 ^{-1}.", "Statistical errors are of the order of the symbol sizes.", "The error bar on the square data point close to the peak represents a typical standard deviation.Finally, let us examine results at a third wave vector, namely $q=1.075$ Å$^{-1}$ , which corresponds to the maxon.", "In this case, our spectral image features a single peak, which is however nowhere near as sharp as in the experimentally observed dynamic structure factor [42], as shown in Fig.", "REF .", "The considerably greater difficulty in extracting sharp features for this wave vector is a direct consequence of the behavior in imaginary time for $F({\\bf q},\\tau )$ , namely the much faster decay in the maxon case (Fig.", "REF ).", "Indeed, we find that the difficulty of reconstructing $S({\\bf q},\\omega )$ from QMC data is particularly severe for wave vectors near the maxon.", "For our procedure to recover sharp features at this wave vector, it appears that the underlying QMC data should possess errors that are significantly smaller than those which we could achieve within this project.", "This illustrates the difficulty of an a priori, even semi-quantitative assessment of the required precision of the QMC data.", "Interestingly, the procedure illustrated in Ref.", "gift does yield a sharp peak in this case as well, of width comparable to that of the experimental image, and $\\sim 30$ % greater height (data from Ref.", "ferre for this wave vector were not available).", "However, the position of the peak itself is off, compared to experiment, by roughly as much as that estimated by MaxEnt (in the case of GIFT the peak is detected at higher energy).", "Thus, although the shape of the GIFT image is certainly closer to the experimental result, in quantitative terms (e.g., position of the peak and area in the experimentally observed peak region), a comparison between the two results may not be so straightforward; in particular, one ought to think of situations in which this procedure is to be used in a predictive way, i.e., no experimental data are available for comparison.", "Thus, we conclude that for this particular wave vector the precision required in the QMC data, in order to achieve a spectral image reconstruction of quality comparable to that of the other two wave vectors, is significantly greater than that afforded by the computational resources available to this project.", "It is incorrect to attribute the lack of sharpness of the reconstructed spectral image in this case to the inversion scheme utilized, which proves equally or more effective than the alternatives at other wave vectors." ], [ "Conclusions", "We have revisited the use of MaxEnt to extract the dynamic structure factor of superfluid $^4$ He from imaginary-time density correlation functions computed by QMC.", "This method was first applied to this problem over two decades ago, yielding results that were deemed “only qualitatively interesting”, as the sharper features of the experimentally measured spectra were not fully recovered.", "In recent years, alternative schemes [17], [18] have been proposed to tackle the same problem; although they are based on different numerical optimization strategies, these schemes ultimately amount to $\\chi ^2$ -fitting.", "We adopted in this work a procedure similar to that first proposed in Ref.", "massimo1996, i.e, we performed a random walk in the space of spectral images, using the entropic prior in the context of “classic” MaxEnt.", "Our study benefits from the availability of new QMC data obtained using state-of-the-art techniques and obviously far more powerful computing resources than those available two decades ago.", "The accuracy of our QMC data is, to the best of our determination, comparable to that of the data used in Refs.", "gift,ferre.", "Our spectral images are of quality at least comparable (and often superior) to that of those yielded by the methods proposed in Refs.", "gift,ferre.", "In particular, spectral images provided in Ref.", "ferre are too broad, and compare poorly to experiment, whereas those of Ref.", "gift are at times much sharper than the experimental ones.", "We show that the use of the entropic prior does not cause the reconstructed spectral images to be unphysically smooth and featureless.", "Rather, it is the precision of the underlying QMC data that determines by itself whether the reconstructed spectra should display sharp peaks or not.", "In general, the elimination of the entropy from the inversion process indeed promotes sharper features, but we argue that that often comes at the expense of accuracy, as such sharpness is ultimately not warranted by the data.", "This means that some sharp features might appear at incorrect locations, or even be downright spurious.", "One is therefore left with no real justification to choose a “sharper” over a more conservative, smoother image, if both are consistent with the data (a posteriori validation based on agreement with available experiments for one particular physical system being a dubious criterion to compare different methodologies)." ], [ "acknowledgments", "This work was supported by the Natural Sciences and Engineering Research Council of Canada.", "One of us (MB) wishes to acknowledge the hospitality of the International Centre for Theoretical Physics in Trieste, Italy, where parts of the research work were carried out.", "The authors thank D. E. Galli for providing GIFT data.", "Useful conversations with S. Moroni are also gratefully acknowledged." ] ]

[ [ "Avoiding Disparity Amplification under Different Worldviews" ], [ "Abstract We mathematically compare four competing definitions of group-level nondiscrimination: demographic parity, equalized odds, predictive parity, and calibration.", "Using the theoretical framework of Friedler et al., we study the properties of each definition under various worldviews, which are assumptions about how, if at all, the observed data is biased.", "We argue that different worldviews call for different definitions of fairness, and we specify the worldviews that, when combined with the desire to avoid a criterion for discrimination that we call disparity amplification, motivate demographic parity and equalized odds.", "We also argue that predictive parity and calibration are insufficient for avoiding disparity amplification because predictive parity allows an arbitrarily large inter-group disparity and calibration is not robust to post-processing.", "Finally, we define a worldview that is more realistic than the previously considered ones, and we introduce a new notion of fairness that corresponds to this worldview." ], [ "Introduction", "Researchers in the field of fair machine learning have proposed numerous tests for fairness, which focus on some quantitative aspect of a model that can be operationalized and checked using empirical, statistical, or program analytic methods.", "These tests abstract away more subtle issues that are difficult to operationalize or too contentious to decide algorithmically, such as which groups or attributes should be protected and which cases should be treated as exceptions to general rules.", "Our work sheds light on some of the possible assumptions behind and motivations for three common empirical tests that check for discrimination against groups.", "The simplest of these tests, demographic parity, checks whether the model gives the favorable outcome to two given groups of people at equal rates.", "This test is an abstraction of the legal notion of disparate impact, or indirect discrimination, which in certain circumstances requires that some approximation of demographic parity hold.", "Like disparate impact, demographic parity does not depend upon the intentions of the modeler, and it can flag a model that does not directly use the protected attribute if it instead uses another attribute that is correlated with the protected one.", "However, demographic parity abstracts away disparate impact's exceptions for cases where there is sufficient justification for a disparity in outcomes, such as a business necessity [13], [1].", "By completely abstracting away such exceptions, demographic parity may lead to models so inaccurate as to become useless, such as when predicting physical strength while requiring demographic parity on gender.", "This impossibility of accuracy motivates moving away from demographic parity to tests that take the ground truth into account, allowing a degree of accuracy.", "One such test, called equalized odds by Hardt et al.", "[14], requires equal false positive and false negative rates for each protected group.", "Another commonly used test is predictive parity [4], which roughly corresponds to the requirement of equal predictive values.", "Like demographic parity, both of these tests can be seen as abstractions of disparate impact in that they too examine disparities in outcomes, not how or why they were reached.", "In contexts where accuracy can be considered a business necessity, these tests arguably provide a more refined abstraction of disparate impact than demographic parity does.", "However, in some cases, the “ground truth” may be tainted by past discrimination, and consulting it will help perpetuate the discrimination.", "In this work, we handle the issue of biased ground truth by adopting the framework of Friedler et al.", "[11], who make a distinction between the observed ground truth and the construct, which is the attribute that is truly relevant for prediction.", "Because the construct is usually unmeasurable, Friedler et al.", "introduce and analyze two assumptions, or worldviews, about the construct: Under the We're All Equal (WAE) worldview, there is no association between the construct and the protected attribute, and under the WYSIWYG worldview, the observations accurately reflect the construct.", "By using the construct, we specify a natural criterion for discrimination.", "This criterion, disparity amplification, deals with the disparity in positive classification rates, which is a widely accepted measure of discriminatory effect in both law [9] and computer science [2], [3], [16], [31], [10], [30].", "It stipulates that a disparity in the output of the model is justified by a commensurate disparity in the construct, thereby allowing accurate models even when the base rates are different for different protected groups, as equalized odds and predictive parity do.", "In addition, because it uses the construct, it does not depend upon the possibly biased ground truth.", "The use of the construct often makes it impossible to test for disparity amplification; we argue that its value instead comes from its ability to organize the space of empirical tests.", "In particular, one of our main contributions is our argument that the WAE and WYSIWYG worldviews, when combined with the desire to avoid disparity amplification, motivate demographic parity and equalized odds, respectively.", "We thus shed light on why people may disagree about which empirical test of discrimination to apply in a particular setting: Even if they agree on the need to avoid disparity amplification, they may disagree about the correct worldview to apply in that setting.", "We also show that, regardless of the worldview and the base rates of the observed ground truth, predictive parity does not impose any restrictions on the extent to which a model amplifies disparity.", "Since equalized odds and predictive parity are incompatible  [6], [4], [18], this is an argument for the use of equalized odds instead of predictive parity.", "Furthermore, we compare our approach to that of Zafar et al.", "[29] in their work on disparate mistreatment, or disparate misclassification rates, showing that the definition of disparity amplification can be modified to apply in their setting.", "Although the WAE and WYSIWYG worldviews are useful for theoretical analysis, they are unlikely to be true in practice.", "To remedy this issue, we introduce a family of hybrid worldviews that is parametrized by a measure of how biased the observed data is against a protected group of people.", "This allows us to model many real-world situations by simply adjusting the parameter.", "We then create a parametrized test for discrimination that corresponds to the new family of worldviews, showing how one can apply the analysis in our paper to real-world scenarios.", "Our most fundamental contribution is the introduction of a framework in which to motivate empirical tests in terms of construct-based criteria of discrimination and worldviews.", "Disparity amplification is not the only relevant notion of discrimination, nor is it suitable in every context.", "Indeed, there are many other aspects of discrimination that we do not address in this paper, such as intentional discrimination [1], individual fairness [8], proxy discrimination [7], delayed outcomes [21], and affirmative action [17].", "Future work may use our approach to tease out the assumptions implicit in these tests.", "We view the discussed tests and disparity amplification as diagnostics that can lead to further investigations of potentially discriminatory behavior in a model.", "As a result, we do not provide an algorithm for ensuring that a model does not have disparity amplification since, in our view, doing so would be treating the symptom rather than the cause.", "Such algorithms can eliminate one aspect of discrimination, but may in the process create a model that is obviously discriminatory from another angle.", "When a model does not satisfy a notion of nondiscrimination, it should be a starting point for investigation as to why.", "While it could be that the learning algorithm is corrupt, it could also be due to a mismatch between the construct and the observed data, or a need for better features.", "Thus, no single algorithm will be appropriate in all cases." ], [ "Related Work", "Our work is most similar in structure to that of Heidari et al.", "[15], who propose a unifying framework that reformulates some existing fairness definitions through the lens of equality of opportunity from political philosophy [23], [24].", "They then propose a new fairness definition that is inspired by this lens.", "Although we also present a unifying framework, our unification is through the lens of constructs and worldviews.", "Friedler et al.", "[11] introduced the concept of the construct in fair machine learning.", "Although they also use the construct in their definition of nondiscrimination, their definition uses the Gromov–Wasserstein distance and as a result is more difficult to compute and reason about.", "One benefit of their approach is that it enables their treatment of fairness at both the individual level and the group level.", "By contrast, we consider group nondiscrimination only, and this allows us to draw a parallel between the worldviews and the existing empirical tests of discrimination.", "Barocas and Selbst [1] discuss in detail the potential legal issues with discrimination in machine learning.", "One widely consulted legal standard for detecting disparate impact is the four-fifths rule [9].", "The four-fifths rule is a guideline that checks whether the ratio of the rates of favorable outcomes for different demographic groups is at least four-fifths.", "This guideline can be considered a relaxation of demographic parity, which would instead require that the ratio of the positive classification rates be exactly one.", "The four-fifths rule has inspired the work of Feldman et al.", "[10] and Zafar et al.", "[30], who deal with a generalization of the four-fifths rule, called the $p$ % rule, in their efforts to remove disparate impact.", "On the other hand, many others [2], [3], [16], [31] consider the difference, rather than the ratio, of the positive classification rates.", "Our discrimination criterion is a generalization of this difference-based measure, but it differs from the others in that it uses the construct rather than the observed data.", "Other works in the field of fair machine learning deal with aspects of discrimination that are not well described by positive classification rates.", "Dwork et al.", "[8] formally define individual fairness and give examples of cases where models are blatantly unfair at the individual level even though they satisfy demographic parity.", "Datta et al.", "[7] tackle the issue that some parts of a model could be discriminatory even if the model, when taken as a whole, does not appear to have discriminatory effect.", "Liu et al.", "[21] consider the delayed impact of a decision, noting that a “positive” loan decision can harm borrowers if they eventually default on the loan.", "Zafar et al.", "[29] point out a that a model can have a higher misclassification rate for one protected group than another, and they propose a method for mitigating this form of discrimination.", "Hardt et al.", "[14] characterize nondiscrimination through equalized odds, which requires that two measures of misclassification, false positive and false negative rates, be equal for all protected groups.", "Finally, predictive parity, Chouldechova [4] points out, is widely accepted in the “educational and psychological testing and assessment literature”.", "In this paper, we prove that equalized odds, but not predictive parity, is sometimes a useful way to avoid disparity amplification.", "We refer the reader to a survey by Romei and Ruggieri [25] for a discussion of other measures of discrimination.", "As mentioned previously, discriminatory effects can be justified if there is a sufficient reason.", "For prediction tasks, it is natural to think of accuracy as a sufficient justification.", "Zafar et al.", "[30] handle this by solving an optimization problem to maximize fairness subject to some accuracy constraints.", "This reflects the idea that a classifier is justified in sacrificing fairness for accuracy.", "To a lesser extent, equalized odds and predictive parity can also be thought of as motivated by the dual desires for accuracy and fairness.", "Our approach to justification is also motivated by these desires, but we use the construct and say that a classifier is justified in predicting the construct correctly." ], [ "Notation", "In the framework introduced by Friedler et al.", "[11], there are three spaces that describe the target attribute of a prediction model.", "The construct space represents the value of the attribute that is truly relevant for the prediction task.", "This value is usually unmeasurable, so prediction models in a supervised learning problem are instead trained with a related measurable label, whose values reside in the observed space.", "Finally, the prediction space (called decision space by Friedler et al.)", "describes the output of the model.", "We will use $Y^{\\prime }$ , $Y$ , and $\\hat{Y}$ as the random variables representing values from the construct, observed, and prediction spaces, respectively.", "(See Figure REF .)", "In addition, we will use $Z$ to denote the protected attribute at hand, and we will assume that $Z \\in \\lbrace 0, 1\\rbrace $ .", "For example, if $Z$ is gender, the values 0 and 1 could represent male and female, respectively.", "Although the input features $X = (X_1, \\ldots , X_n)$ are also critical for both the training and the prediction of the model, they are rarely used in this paper.", "Example 1.", "Some jurisdictions have started to use machine learning models to predict how much risk a criminal defendant poses [20].", "Judges are then allowed to consider the risk score as one of many factors when making bail or sentencing decisions [27].", "Using the three-space framework of Friedler et al.", "[11], we can represent the risk score output by the model as $\\hat{Y}$ .", "The model would be trained with the observation $Y$ , which in this case may be recorded data about past criminal defendants and their failures to appear in court (bail) or recidivism (sentencing).", "These models would also be trained with features $X$ from the input space, such as age and criminal history.", "For sentencing decisions, presumably we want to know whether the defendant will commit another crime in the future, regardless of whether the defendant will be caught committing the crime.", "Therefore, we argue that the recorded recidivism rate $Y$ is merely a proxy for the actual reoffense rate $Y^{\\prime }$ , which is the relevant attribute for the prediction task.", "There is evidence that black Americans are arrested at a higher rate than white Americans for the same crime [22], so it is reasonable to suspect that $Y$ is a racially biased proxy for $Y^{\\prime }$ .", "Example 2.", "Universities want the students that they admit to the university to be successful in the university ($Y^{\\prime }$ ).", "Because success is a vague term that encompasses many factors, a model that predicts success in university would instead be trained with a more concrete measure, such as graduating within six years ($Y$ ).", "This model may take inputs such as a student's high-school grades and standardized test scores ($X$ ), and will output a prediction of how likely the student is to graduate within six years ($\\hat{Y}$ ).", "Admissions officers can then use this prediction to guide their decision about whether to admit the student.", "It is important to note that the models in the above examples do not make the final decision and that human judgments are a major part of the decision process.", "However, we are concerned about the fairness of the model rather than that of the entire decision process.", "Thus, we focus on $\\hat{Y}$ , the output of the model, rather than the final decision made using it.", "Figure: Three relevant spaces for prediction models.The space of input features X=(X 1 ,...,X n )X = (X_1, \\ldots , X_n) is not depicted here.The observed space and the prediction space are measurable, and the existing empirical tests (Definitions , , ) impose constraints on the relationship between the two spaces.On the other hand, the construct space is usually unmeasurable, so we must assume a particular worldview (e.g., Worldview  or ) about how the construct space relates to the observed space, if at all.Then, we can define disparity amplification and construct accuracy, which relate the construct space to the prediction space." ], [ "Preliminary Definitions", "In this work, we use two notions of distance between two random variables that measure how different the random variables are.", "When the random variables are categorical, we use the total variation distance.", "Definition 1 (Total Variation Distance) Let $Y_0$ and $Y_1$ be categorical random variables with finite supports $\\mathcal {Y}_0$ and $\\mathcal {Y}_1$ .", "Then, the total variation distance between $Y_0$ and $Y_1$ is $ d_{\\mathrm {tv}}(Y_0, Y_1) = \\frac{1}{2} \\sum _{y \\in \\mathcal {Y}_0 \\cup \\mathcal {Y}_1} \\big |\\Pr [Y_0{=}y] - \\Pr [Y_1{=}y]\\big |.", "$ In the special case where $Y_0, Y_1 \\in \\lbrace 0, 1\\rbrace $ , the total variation distance can also be expressed as $|\\Pr [Y_0{=}1] - \\Pr [Y_1{=}1]|$ .", "When the random variables are numerical, our notion of distance takes into account the magnitude of the difference in the numerical values.", "The following definition assumes that the random variables are continuous, but a similar definition is applicable when they are discrete.", "Definition 2 (Earthmover Distance) Let $Y_0$ and $Y_1$ be continuous numerical random variables with probability density functions $p_0$ and $p_1$ defined over support $\\mathcal {Y}$ .", "Furthermore, let $\\Gamma $ be the set of joint probability density functions $\\gamma (u, v)$ such that $\\int _{\\mathcal {Y}} \\gamma (u, v) \\, dv = p_0(u)$ for all $u \\in \\mathcal {Y}$ and $\\int _{\\mathcal {Y}} \\gamma (u, v) \\, du = p_1(v)$ for all $v \\in \\mathcal {Y}$ .", "Then, the earthmover distance between $Y_0$ and $Y_1$ is $ d_{\\mathrm {em}}(Y_0, Y_1) = \\inf _{\\gamma \\in \\Gamma } \\int _{\\mathcal {Y}} \\int _{\\mathcal {Y}} \\gamma (u, v) \\, d(u, v) \\, du \\, dv, $ where $d$ is a distance metric defined over $\\mathcal {Y}$ .", "The joint probability density function $\\gamma $ has marginal distributions that correspond to $Y_0$ and $Y_1$ .", "Intuitively, if we use the graphs of the probability density functions $p_0$ and $p_1$ to represent mounds of sand, $\\gamma $ corresponds to a transportation plan that dictates how much sand to transport in order to reshape the $p_0$ mound into the $p_1$ mound.", "In particular, the value of $\\gamma (u, v)$ is the amount of sand to be transported from $u$ to $v$ .", "The distance $d(u, v)$ can then be interpreted as the cost of transporting one unit of sand from $u$ to $v$ , and the earthmover distance is simply the cost of the transportation plan $\\gamma $ that incurs the least cost.", "Now we define Lipschitz continuity.", "Definition 3 Let $f: \\mathcal {Y}\\rightarrow \\mathbb {R}$ be a function, and let $d$ be a distance metric defined over $\\mathcal {Y}$ .", "$f$ is $\\rho $ -Lipschitz continuous if, for all $u, v \\in \\mathcal {Y}$ , $ |f(u) - f(v)| \\le \\rho \\cdot d(u, v).$" ], [ "Existing Empirical Tests of Discrimination", "Many fairness definitions for prediction models have been proposed previously, and here we restate three of them.", "Because much of the prior work does not make the distinction between the construct space and the observed space, there is some ambiguity about whether $Y^{\\prime }$ or $Y$ is the appropriate variable to use these definitions.", "Given that these works suggest that these definitions can be computed, we interpret them to be empirical tests that can help verify whether a model is fair.", "As a result, none of these definitions include the construct $Y^{\\prime }$ .", "In all three definitions, the probabilities are taken over random draws of data points from the data distribution, as well as any randomness used by the model.", "Definition 4 (Demographic Parity Test) A model passes the demographic parity test if, for all $\\hat{y}$ , $ \\Pr [\\hat{Y}{=}\\hat{y}\\mid Z{=}0] = \\Pr [\\hat{Y}{=}\\hat{y}\\mid Z{=}1].", "$ Definition 5 (Equalized Odds Test [14]) A model passes the equalized odds test if, for all $y$ and $\\hat{y}$ , $ \\Pr [\\hat{Y}{=}\\hat{y}\\mid Y{=}y, Z{=}0] = \\Pr [\\hat{Y}{=}\\hat{y}\\mid Y{=}y, Z{=}1].", "$ Definition 6 (Predictive Parity Test [4]) A model passes the predictive parity test if, for all $y$ and $\\hat{y}$ , $ \\Pr [Y{=}y\\mid \\hat{Y}{=}\\hat{y}, Z{=}0] = \\Pr [Y{=}y\\mid \\hat{Y}{=}\\hat{y}, Z{=}1].", "$" ], [ "Worldviews", "Our intuitive notion of discrimination involves the relationship between the construct space and the prediction space.", "For example, consider the context of recidivism prediction described in Example 1.", "Suppose that one group of people is much more likely to be arrested for the same crime than another group.", "Then, the disparity in arrest rates can cause the recorded recidivism rate $Y$ to be biased, and a model trained using such $Y$ would likely learn to discriminate as a result.", "If in fact the two groups have equal reoffense rates $Y^{\\prime }$ , it would hardly be considered justified that one group tends to be given longer sentences as a result of the bias in $Y$ .", "However, because $Y^{\\prime }$ is typically unmeasurable, in practice we do not know whether $Y^{\\prime }$ is the same for both groups.", "Therefore, to reason about discrimination using the construct space, we must make assumptions about the construct space.", "Two such assumptions, or worldviews, have previously been introduced by Friedler et al.", "[11] and are described below.", "Our versions of these worldviews are simpler than the original because they are exact, whereas the original versions allow deviations by a parameter $\\epsilon $ .", "Worldview 1 (We're All Equal) Under the We're All Equal (WAE) worldview, every group is identical with respect to the construct space.", "More formally, $Y^{\\prime }$ is independent of $Z$ , i.e., $Y^{\\prime }\\perp Z$ .", "Worldview 2 (WYSIWYG) Under the What You See Is What You Get (WYSIWYG) worldview, the observed space accurately reflects the construct space.", "More formally, $Y^{\\prime }= Y$ ." ], [ "Using Worldviews to Motivate Empirical Tests", "In this section, we introduce our construct-based criterion of discrimination and use it to analyze which worldviews motivate the existing empirical tests of discrimination.", "We first begin with the case where $Y^{\\prime }$ and $\\hat{Y}$ are categorical (but not necessarily binary), and in Section  we generalize the definition to numerical $Y^{\\prime }$ ." ], [ "Disparity Amplification", "When $\\hat{Y}$ is binary, the size of a model's discriminatory effect is commonly measured by $|\\Pr [\\hat{Y}{=}1 \\mid Z{=}0] - \\Pr [\\hat{Y}{=}1 \\mid Z{=}1]|$ , or the difference in positive classification rates.", "Output disparity (Definition REF ) is a generalization of this measure for the case of non-binary categorical $\\hat{Y}$ .", "Definition 7 (Output Disparity) Let the output $\\hat{Y}$ of a model be categorical.", "The output disparity of the model is the quantity $d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1)$ .", "However, not all output disparities are bad in every context.", "In particular, because we want the model to accurately reflect the construct, we allow an output disparity insofar as it can be explained by the inter-group disparity in $Y^{\\prime }$ .", "This happens when $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) \\le d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1).$ Since a model can have issues with discrimination that are not well characterized by output disparity (discussed below), Equation REF is not the conclusive definition of nondiscrimination.", "Therefore, we use the logical negation of Equation REF as a criterion for one particular discrimination concern, which occurs when an output disparity is not explained by $Y^{\\prime }$ .", "Definition 8 (Disparity Amplification) Let $Y^{\\prime }$ and $\\hat{Y}$ be categorical.", "Then, a model exhibits disparity amplification if $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) > d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1).$ Of course, there are forms of discrimination that are not well described by output disparity alone.", "For example, a model could have a higher misclassification rate for one group of people [29], and Definition REF is not well suited for detecting such errors.", "In addition, even if Definition REF does not show a violation (i.e., Equation REF does not hold) for the entire model, it is possible that some part of the model is a proxy for the protected attribute and that it causes a discriminatory effect.", "In their work on proxy use, Datta et al.", "[7] show that the input/output behavior of the model does not give enough information to decide whether a model uses a proxy of the protected attribute.", "As a result, we would have to look at the internal details of the model to determine whether any part of the model is discriminatory.", "On the other hand, Definition REF is intended to be a general, model-agnostic way to incriminate, but not necessary absolve, a model." ], [ "Construct Accuracy", "As mentioned in Section REF , we want the output of the model to accurately reflect the value of $Y^{\\prime }$ .", "However, the simple accuracy measure $\\Pr [Y^{\\prime }= Y]$ incentivizes the model to become more accurate on the larger protected group at the expense of becoming less accurate on the smaller protected group.", "Therefore, we instead measure accuracy as the average of the accuracy on the two groups.", "Definition 9 (Construct Accuracy) The construct accuracy of a model is $ \\textstyle \\frac{1}{2} \\big (\\Pr [Y^{\\prime }{=}\\hat{Y}\\mid Z{=}0] + \\Pr [Y^{\\prime }{=}\\hat{Y}\\mid Z{=}1]\\big ).$ Definition 10 (Construct Optimality) A model is construct optimal if its construct accuracy is 1, i.e., its output $\\hat{Y}$ and the construct $Y^{\\prime }$ are always equal.", "Because the construct $Y^{\\prime }$ usually cannot be measured, construct accuracy usually cannot be measured or directly optimized for.", "Even when it can measured, construct optimality would be rare since the quality of the features, data, or machine learning algorithm may preclude perfection.", "As with disparity amplification, we introduce construct accuracy not to empirically measure it, but as a theoretical tool for analyzing discrimination.", "In particular, note that the equality in Equation REF holds for every construct optimal model.", "In other words, a construct optimal model displays the maximum amount of output disparity allowed by Definition REF .", "On the other hand, if the output disparity is greater than the disparity in $Y^{\\prime }$ , the model must be amplifying a disparity in a way that cannot be justified by the desire to achieve construct optimality." ], [ "Criteria for Motivation", "If an empirical test does not guarantee the lack of disparity amplification, it is insufficient as an anti-discrimination measure.", "On the other hand, if the test disallows a construct optimal model, the test may be too strict in a way that lowers the utility of the model.", "Therefore, to argue that a worldview motivates an empirical test, we will prove the following two statements: (a) Every model that passes the empirical test does not have disparity amplification, and (b) every optimal model passes the empirical test.", "We apply this reasoning to demographic parity (Definition REF ) and equalized odds (Definition REF ), showing that the WAE and WYSIWYG worldviews, respectively, motivate these empirical tests.", "More formally, we will prove statements (a) and (b) for every joint distribution of $Y^{\\prime }$ , $Y$ , $\\hat{Y}$ , and $Z$ that is consistent with the worldview.", "Table REF summarizes these results.", "Table: Summary of the results in Section .We say that a worldview motivates an empirical test if it precludes disparity amplification (Definition ) but does not preclude a perfectly predictive model.The We're All Equal (WAE) worldview motivates the demographic parity test, and if the worldview does not hold, the demographic parity test tends to lower the utility of the model.The WYSIWYG worldview motivates the equalized odds test, and if the worldview does not hold, the equalized odds test allowed models that have disparity amplification.Finally, regardless of the worldview, the predictive parity test does not effectively prevent disparity amplification.Here, we assume that WAE and WYSIWYG do not hold simultaneously." ], [ "Demographic Parity and WAE", "Theorem 1 A model that passes the demographic parity test does not have disparity amplification under Definition REF .", "Moreover, if the WAE worldview holds, every construct optimal model satisfies demographic parity.", "By the definition of demographic parity, the left-hand side of Equation REF is $d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) = 0$ .", "Since the total variation distance is always nonnegative, demographic parity ensures the lack of disparity amplification.", "If the WAE worldview holds, we have $Y^{\\prime }\\perp Z$ , so every optimal model satisfies $\\hat{Y}\\perp Z$ .", "This implies demographic parity by Definition REF .", "The first part of Theorem REF shows that we can guarantee that a model will not have disparity amplification by training it to pass the demographic parity test.", "However, this does not mean that demographic parity is appropriate for every situation.", "First, we remind the reader that the lack of disparity amplification does not mean that the model will be free of all issues related to discrimination.", "In particular, disparity amplification is only designed to catch the type of discrimination akin to disparate impact.", "If the WAE worldview holds, demographic parity is the only way to avoid disparity amplification, so it makes sense to enforce demographic parity.", "On the other hand, if the WAE worldview does not hold, enforcing demographic parity may introduce other forms of discrimination.", "For example, the U.S. Supreme Court held in Ricci v. DeStefano [26] that the prohibition against intentional discrimination can sometimes override the consideration of disparate impact, ruling that an employer unlawfully discriminated by discarding the results of a bona fide job-related test because of a racial performance gap.", "Second, demographic parity can unnecessarily lower the utility of a model.", "If the WAE worldview does not hold, $d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ is positive, and Theorem REF shows that any model that satisfies demographic parity must be suboptimal.", "In fact, the more we deviate from the WAE worldview, the lower the maximum possible construct accuracy becomes.", "Theorem 2 If a model satisfies demographic parity, the construct accuracy of the model is at most $1 - \\frac{1}{2} d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ .", "Moreover, there exists a distribution of $\\hat{Y}$ that satisfies demographic parity and attains this construct accuracy.", "To prove this theorem, we will use Lemma REF .", "Lemma 3 Let $Y_0$ and $Y_1$ be categorical random variables with finite supports $\\mathcal {Y}_0$ and $\\mathcal {Y}_1$ .", "Then, $ \\sum _{y \\in \\mathcal {Y}_0 \\cup \\mathcal {Y}_1} \\min \\Big (\\Pr [Y_0{=}y], \\Pr [Y_1{=}y]\\Big ) = 1 - d_{\\mathrm {tv}}(Y_0, Y_1).", "$ [Proof of Lemma REF ] For brevity, let $p_y = \\Pr [Y_0{=}y]$ and $q_y = \\Pr [Y_1{=}y]$ .", "We can then rewrite the total variation distance in terms of $\\max $ and $\\min $ .", "$2 d_{\\mathrm {tv}}(Y_0, Y_1) &= \\textstyle \\sum _{y \\in \\mathcal {Y}_0 \\cup \\mathcal {Y}_1} |p_y - q_y| \\\\&= \\textstyle \\sum _{y \\in \\mathcal {Y}_0 \\cup \\mathcal {Y}_1} \\big (\\max (p_y, q_y) - \\min (p_y, q_y)\\big ).$ In addition, we have $ \\sum _{y \\in \\mathcal {Y}_0 \\cup \\mathcal {Y}_1} \\Big (\\max (p_y, q_y) + \\min (p_y, q_y)\\Big ) = \\sum _{y \\in \\mathcal {Y}_0 \\cup \\mathcal {Y}_1} (p_y + q_y) = 2.", "$ Subtracting the first equation from the second gives us $\\sum \\min (p_y, q_y) = 1 - d_{\\mathrm {tv}}(Y_0, Y_1)$ , which is what we want.", "[Proof of Theorem REF ] We first prove the upper bound on the construct accuracy.", "Let $\\mathcal {Y}^{\\prime }$ and $\\hat{\\mathcal {Y}}$ be the supports of $Y^{\\prime }$ and $\\hat{Y}$ , respectively.", "Then, by the law of total probability we have $ \\Pr [Y^{\\prime }{=}y^{\\prime }, \\hat{Y}{=}y^{\\prime }\\mid Z{=}z] \\le \\min (\\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}z], \\Pr [\\hat{Y}{=}y^{\\prime }\\mid Z{=}z]) $ for all $y^{\\prime }\\in \\mathcal {Y}^{\\prime }\\cup \\hat{\\mathcal {Y}}$ and $z \\in \\lbrace 0, 1\\rbrace $ .", "We then sum this over $y^{\\prime }$ and apply Lemma REF to get $&\\Pr [Y^{\\prime }{=}\\hat{Y}\\mid Z{=}z] \\\\&= \\textstyle \\sum _{y^{\\prime }\\in \\mathcal {Y}^{\\prime }\\cup \\hat{\\mathcal {Y}}} \\Pr [Y^{\\prime }{=}y^{\\prime }, \\hat{Y}{=}y^{\\prime }\\mid Z{=}z] \\\\&\\le \\textstyle \\sum _{y^{\\prime }\\in \\mathcal {Y}^{\\prime }\\cup \\hat{\\mathcal {Y}}} \\min \\big (\\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}z], \\Pr [\\hat{Y}{=}y^{\\prime }\\mid Z{=}z]\\big ) \\\\&= 1 - d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}z, \\hat{Y}|Z{=}z) \\\\&= 1 - d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}z, \\hat{Y}),$ where the last equality follows from our assumption that the model satisfies demographic parity.", "Therefore, the construct accuracy can be bounded as $&\\textstyle \\frac{1}{2} \\big (\\Pr [Y^{\\prime }{=}\\hat{Y}\\mid Z{=}0] + \\Pr [Y^{\\prime }{=}\\hat{Y}\\mid Z{=}1]\\big ) \\\\&\\textstyle \\le \\frac{1}{2} \\big (1 - d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, \\hat{Y}) + 1 - d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}1, \\hat{Y})\\big ) \\\\&\\textstyle \\le 1 - \\frac{1}{2} d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1),$ where the last inequality is an application of the triangle inequality.", "Now we construct a random variable $\\hat{Y}$ that satisfies demographic parity and attains this bound.", "When $Z{=}0$ , we simply let $\\hat{Y}= Y^{\\prime }$ , making the first term in Equation REF equal to 1.", "When $Z{=}1$ , we constrain the marginal distribution of $(\\hat{Y}|Z{=}1)$ to be the same as that of $(\\hat{Y}|Z{=}0) = (Y^{\\prime }|Z{=}0)$ , and we make the joint distribution of $(Y^{\\prime }|Z{=}1)$ and $(\\hat{Y}|Z{=}1)$ a maximal coupling [19].", "Then, by the theorem in [19], such $\\hat{Y}$ attains the value of $1 - d_{\\mathrm {tv}}(\\hat{Y}|Z{=}1, Y^{\\prime }{=}1) = 1 - d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ for the second term of Equation REF .", "This means that the construct accuracy, which is the average of the two terms, is $1 - \\frac{1}{2} d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ , which is what we want.", "Moreover, $(\\hat{Y}|Z{=}1)$ and $(\\hat{Y}|Z{=}0)$ have the same distribution, so $\\hat{Y}$ satisfies demographic parity.", "Theorems REF and REF demonstrate that the demographic parity test is best suited for a setting where the WAE worldview holds." ], [ "Equalized Odds and WYSIWYG", "We now argue that a similar relationship exists between the equalized odds test and the WYSIWYG worldview.", "Theorem 4 If the WYSIWYG worldview holds, a model that passes the equalized odds test does not have disparity amplification under Definition REF .", "Moreover, if the WYSIWYG worldview holds, every construct optimal model satisfies equalized odds.", "Let $\\mathcal {Y}^{\\prime }$ and $\\hat{\\mathcal {Y}}$ be the supports of $Y^{\\prime }$ and $\\hat{Y}$ , respectively.", "Applying the WYSIWYG worldview to the definition of equalized odds, we get $\\Pr [\\hat{Y}{=}\\hat{y}\\mid Y^{\\prime }{=}y^{\\prime }, Z{=}0] = \\Pr [\\hat{Y}{=}\\hat{y}\\mid Y^{\\prime }{=}y^{\\prime }, Z{=}1] = \\Pr [\\hat{Y}{=}\\hat{y}\\mid Y^{\\prime }{=}y^{\\prime }]$ for all $y^{\\prime }\\in \\mathcal {Y}^{\\prime }$ and $\\hat{y}\\in \\hat{\\mathcal {Y}}$ .", "Therefore, we have $\\textstyle d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) \\\\\\multicolumn{1}{l}{\\textstyle = \\frac{1}{2} \\sum _{\\hat{y}\\in \\hat{\\mathcal {Y}}} \\big |\\Pr [\\hat{Y}{=}\\hat{y}\\mid Z{=}0] - \\Pr [\\hat{Y}{=}\\hat{y}\\mid Z{=}1]\\big |} \\\\\\multicolumn{1}{l}{\\textstyle = \\frac{1}{2} \\sum _{\\hat{y}\\in \\hat{\\mathcal {Y}}} \\Big |\\sum _{y^{\\prime }\\in \\mathcal {Y}^{\\prime }} \\Pr [\\hat{Y}{=}\\hat{y}\\mid Y^{\\prime }{=}y^{\\prime }]} \\\\\\multicolumn{1}{r}{\\textstyle \\cdot \\big (\\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}0] - \\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}1]\\big )\\Big |} \\\\\\multicolumn{1}{l}{\\textstyle \\le \\frac{1}{2} \\sum _{\\hat{y}\\in \\hat{\\mathcal {Y}}} \\sum _{y^{\\prime }\\in \\mathcal {Y}^{\\prime }} \\Pr [\\hat{Y}{=}\\hat{y}\\mid Y^{\\prime }{=}y^{\\prime }]} \\\\\\multicolumn{1}{r}{\\textstyle \\cdot \\big |\\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}0] - \\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}1]\\big |} \\\\\\multicolumn{1}{l}{\\textstyle = \\frac{1}{2} \\sum _{y^{\\prime }\\in \\mathcal {Y}^{\\prime }} \\Big (\\big |\\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}0] - \\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}1]\\big |} \\\\\\multicolumn{1}{r}{\\textstyle \\cdot \\sum _{\\hat{y}\\in \\hat{\\mathcal {Y}}} \\Pr [\\hat{Y}{=}\\hat{y}\\mid Y^{\\prime }{=}y^{\\prime }]\\Big )} \\\\\\multicolumn{1}{l}{\\textstyle = \\frac{1}{2} \\sum _{y^{\\prime }\\in \\mathcal {Y}^{\\prime }} \\big |\\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}0] - \\Pr [Y^{\\prime }{=}y^{\\prime }\\mid Z{=}1]\\big |} \\\\\\multicolumn{1}{l}{\\textstyle = d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1).", "\\hfill }$ This concludes the proof of the first statement.", "For an optimal model, we have $\\hat{Y}= Y^{\\prime }= Y$ by the WYSIWYG worldview.", "Because $Y$ fully determines the value of $\\hat{Y}$ , Definition REF implies that every optimal model satisfies equalized odds.", "On the other hand, our intuition is that when the observation process is biased, and WYSIWYG does not hold, treating the observation $Y$ as accurate, as implicit with equalized odds, may lead to a failure to pass our construct-based criterion.", "We prove as much: Theorem 5 If the WYSIWYG worldview does not hold, a model passing the equalized odd test can still have disparity amplification.", "We show that there exists a joint distribution of $Y^{\\prime }$ , $Y$ , $\\hat{Y}$ , and $Z$ such that a model with equalized odds still has disparity amplification.", "Many models with equalized odds have nonzero output disparity, i.e., $d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) > 0$ .", "Consider any such model.", "Since the WYSIWYG worldview does not hold, we have no guarantee that $Y^{\\prime }$ will resemble $Y$ in any way.", "Therefore, the equalized odds requirement does not restrict the distribution of $Y^{\\prime }$ , and the model can have disparity amplification if $d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ is small enough." ], [ "Predictive Parity", "Under the WYSIWYG worldview, the predictive parity test conveniently holds for optimal models, but also implies that $d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) \\ge d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ , as can be seen from switching $Y^{\\prime }$ and $\\hat{Y}$ in the proof of the first part of Theorem REF .", "The inequality here is in the opposite direction of that in Equation REF , so the predictive parity test does not place any upper bound on the output disparity of $\\hat{Y}$ and guarantees that it is equal to that of $Y^{\\prime }$ or amplified beyond this limit.", "In fact, the following theorem shows that, regardless of the worldview and the base rates of $Y$ , even a model with almost the maximum output disparity can still pass the predictive parity test.", "Theorem 6 Let $Y$ be a categorical random variable with finite support such that $\\Pr [Y{=}y\\mid Z{=}z]$ is positive for all $y$ and $z$ .", "Then, for any sufficiently small $\\epsilon > 0$ , there exists a model that passes the predictive parity test such that $d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) = 1 - \\epsilon $ .", "The main idea behind the proof is that the model simply outputs the value of $Z$ .", "However, because predictive parity is not well-defined if $\\Pr [\\hat{Y}{=}\\hat{y}, Z{=}z] = 0$ for any $\\hat{y}$ and $z$ , we must allow the model to output the other value with some very small probability.", "More specifically, we construct a model such that $ \\Pr [\\hat{Y}{=}\\hat{y}\\mid Z{=}z] = {\\left\\lbrace \\begin{array}{ll}1 - \\frac{\\epsilon }{2}, & \\text{if } \\hat{y}= z \\\\\\frac{\\epsilon }{2}, & \\text{if } \\hat{y}\\ne z.\\end{array}\\right.}", "$ We can choose which values our constructed model outputs, so assume without loss of generality that $\\hat{Y}\\in \\lbrace 0, 1\\rbrace $ .", "Let $\\mathcal {Y}$ be the support of $Y$ .", "By the predictive parity test, we have $\\Pr [Y{=}y\\mid \\hat{Y}{=}\\hat{y}, Z{=}0] = \\Pr [Y{=}y\\mid \\hat{Y}{=}\\hat{y}, Z{=}1] = \\Pr [Y{=}y\\mid \\hat{Y}{=}\\hat{y}]$ for all $y\\in \\mathcal {Y}$ and $\\hat{y}\\in \\lbrace 0, 1\\rbrace $ .", "Let $p_{y\\hat{y}} = \\Pr [Y{=}y\\mid \\hat{Y}{=}\\hat{y}]$ .", "Our goal is to find the values of $p_{y0}$ and $p_{y1}$ that are consistent with the fixed observed probabilities $\\Pr [Y{=}y\\mid Z{=}0]$ and $\\Pr [Y{=}1 \\mid Z{=}1]$ .", "By the law of total probability, our model must satisfy $\\begin{pmatrix} \\Pr [Y{=}y\\mid Z{=}0] \\\\ \\Pr [Y{=}y\\mid Z{=}1] \\end{pmatrix}= \\begin{pmatrix} 1 - \\frac{\\epsilon }{2} & \\frac{\\epsilon }{2} \\\\ \\frac{\\epsilon }{2} & 1 - \\frac{\\epsilon }{2} \\end{pmatrix}\\begin{pmatrix} p_{y0} \\\\ p_{y1} \\end{pmatrix}.", "$ Solving for $p_{y0}$ and $p_{y1}$ , we see that they converge to $\\Pr [Y{=}y\\mid Z{=}0]$ and $\\Pr [Y{=}y\\mid Z{=}1]$ , respectively, as $\\epsilon $ approaches zero.", "By assumption, these probabilities are positive.", "Since $\\mathcal {Y}$ is finite, this means that there exists a small enough $\\epsilon > 0$ such that $p_{y0}, p_{y1} > 0$ for all $y\\in \\mathcal {Y}$ .", "Moreover, it is easy to verify that $\\sum _{y\\in \\mathcal {Y}} p_{y0} = \\sum _{y\\in \\mathcal {Y}} p_{y1} = 1$ , making them valid probability distributions.", "Now, when given $Y{=}y$ and $Z{=}z$ , our model can output $\\hat{Y}{=}\\hat{y}$ with probability $ \\Pr [\\hat{Y}{=}\\hat{y}\\mid Y{=}y, Z{=}z] = \\frac{p_{y\\hat{y}} \\cdot \\Pr [\\hat{Y}{=}\\hat{y}\\mid Z{=}z]}{\\Pr [Y{=}y\\mid Z{=}z]}, $ where $\\Pr [\\hat{Y}{=}\\hat{y}\\mid Z{=}z]$ is either $\\frac{\\epsilon }{2}$ or $1 - \\frac{\\epsilon }{2}$ depending on whether $\\hat{y}= z$ .", "Because the predictive parity test allows models, such as the one we constructed in the above proof, that clearly amplify disparity, it is unsuitable for ensuring nondiscrimination as characterized by output disparity.", "As a result, in the rest of the paper we focus on the equalized odds test rather than the predictive parity test.", "We leave as future work the identification of a discrimination criterion and a worldview that together motivate the predictive parity test." ], [ "Connection to Misclassification", "Here, we show that the definition of disparity amplification is closely related to that given by Zafar et al.", "[29] in their treatment of disparate misclassification rates.", "First, we motivate the issue of disparate misclassification rates with an example.", "Let $Y^{\\prime }$ and $Z$ be independent and uniformly random binary variables.", "If $\\hat{Y}= Y^{\\prime }\\oplus Z$ , where $\\oplus $ is the bitwise XOR, both protected groups are given the positive label exactly half of the time, so there is no output disparity.", "However, one group always receives the correct classification and the other always receives the incorrect classification, so the disparity in the misclassification rates is as large as it can be.", "This shows that a lack of disparity amplification does not imply a lack of disparity in misclassification rates.", "Conversely, a lack of disparity in misclassification rates does not imply a lack of disparity amplification.", "To see this, modify the above example so that $\\hat{Y}= Z$ instead.", "In this case, both groups have half of its members misclassified since $Z$ is independent of $Y^{\\prime }$ , so they have the same overall misclassification rate.", "On the other hand, we have $d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) = d_{\\mathrm {tv}}(Y^{\\prime }, Y^{\\prime }) = 0$ and $d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) = d_{\\mathrm {tv}}(Z|Z{=}0, Z|Z{=}1) = 1$ .", "Thus, $\\hat{Y}$ has disparity amplification.", "However, we can still find a connection between misclassification parity and disparity amplification.", "Let $C = {1}(Y^{\\prime }= \\hat{Y})$ , and replace $\\hat{Y}$ with $C$ in the definition of output disparity (Definition REF ).", "Since $C$ is binary, the resulting expression $d_{\\mathrm {tv}}(C|Z{=}0, C|Z{=}1)$ is simply the difference in the misclassification rates.", "We would like to compare this value to some measure of disparity in the construct space.", "Since our standard measure of $d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ does not necessarily justify inter-group differences in $C$ , it may not be a correct measure to use.", "Exploring what measures provide justification for disparate misclassification rates is interesting future work." ], [ "Hybrid Worldviews", "So far, we have assumed either the WAE or the WYSIWYG worldview.", "While these worldviews are interesting from a theoretical perspective, in practice it is unlikely that these worldviews hold.", "In this section, we propose a family of more realistic worldviews for the case where $Y^{\\prime }$ and $Y$ are categorical.", "As we have depicted in Figure REF , worldviews describe the relationship between the construct and observed spaces.", "Because our definition of disparity amplification has to do with inter-group disparities, here we focus specifically on the inter-group disparities in $Y^{\\prime }$ and $Y$ .", "Note that the WAE worldview has the effect of assuming that none of the disparity in $Y$ is explained by $Y^{\\prime }$ .", "By contrast, under the WYSIWYG worldview, all of the disparity in $Y$ is explained by $Y^{\\prime }$ .", "Described below is the $\\alpha $ -Hybrid worldview, which is a family of worldviews that occupy the space between the two extremes of WAE and WYSIWYG.", "Worldview 3 ($\\alpha $ -Hybrid) Let $\\alpha \\in [0, 1]$ .", "Under the $\\alpha $ -Hybrid worldview, exactly an $\\alpha $ fraction of the disparity in $Y$ is explained by $Y^{\\prime }$ .", "More formally, $ d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) = \\alpha \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1)$ It is easy to see that the WAE worldview is equivalent to the 0-Hybrid worldview.", "On the other hand, the relationship between the WYSIWYG and 1-Hybrid worldviews is only unidirectional.", "Although the WYSIWYG worldview implies the 1-Hybrid worldview, there are plenty of ways to satisfy $d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) = d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1)$ even when the equality $Y^{\\prime }= Y$ does not hold.", "If we wanted to make the relationship bidirectional, we could instead have assumed that $Y^{\\prime }$ can be broken down into two components, one of which satisfies WAE and the other WYSIWYG.", "However, this would mean that every component of $Y^{\\prime }$ is either equal with respect to $Z$ (WAE) or measurable (WYSIWYG), whereas in practice many inter-group disparities in the construct space are not easily measurable.", "Therefore, to make the $\\alpha $ -Hybrid worldview more realistic, we sacrifice one direction of the relationship between the WYSIWYG and 1-Hybrid worldviews.", "Now we introduce the $\\alpha $ -disparity test and prove that it corresponds to the $\\alpha $ -Hybrid worldview.", "Unlike the demographic parity and equalized odds tests, the $\\alpha $ -disparity test is parametrized and therefore can be applied to various real-world situations.", "Definition 11 ($\\alpha $ -Disparity Test) A model passes the $\\alpha $ -disparity test if $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) \\le \\alpha \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1).$ Theorem 7 If the $\\alpha $ -Hybrid worldview holds, a model that passes the $\\alpha $ -disparity test does not have disparity amplification under Definition REF .", "Moreover, if the $\\alpha $ -Hybrid worldview holds, every construct optimal model satisfies the $\\alpha $ -disparity test.", "To prove the first part of the theorem, we simply combine the inequality guaranteed by the $\\alpha $ -disparity test (Equation REF ) with the equation that defines the $\\alpha $ -Hybrid worldview (Equation REF ).", "Then, we get $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) \\le \\alpha \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1) = d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1), $ which is what we want.", "For the second part of the theorem, an optimal model has $Y^{\\prime }= \\hat{Y}$ , so we can substitute the $Y^{\\prime }$ in Equation REF with $\\hat{Y}$ to get $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) = \\alpha \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1).", "$ This is simply the equality in Equation REF , so we are done.", "The $\\alpha $ -disparity test is closely related to demographic parity and equalized odds.", "0-disparity is satisfied if and only if the output disparity is zero, so it is equivalent to demographic parity.", "In addition, we can easily adapt the proof of Theorem REF to show that equalized odds implies 1-disparity.", "However, because equalized odds imposes a condition for each possible value of $Y$ , 1-disparity does not imply equalized odds.", "Although it may thus seem that equalized odds is stronger and better than 1-disparity, recent results by Corbett-Davies and Goel [5] show that the threshold rule, which they argue is optimal, does not lead to equalized odds in general.", "Therefore, there is a trade-off between the stronger fairness guarantee provided by equalized odds and the higher utility that is attainable under 1-disparity.", "Of course, the 1-disparity test has the additional benefit that it can be generalized to other values of $\\alpha $ .", "We end this section with Theorems REF and REF , which describe the consequences of enforcing the $\\alpha $ -disparity test with a wrong value of $\\alpha $ .", "These theorems are close analogues of Theorems REF and REF , respectively.", "Theorem 8 If the $\\alpha $ -Hybrid worldview holds, a model that passes the $\\alpha ^{\\prime }$ -disparity test, with $\\alpha > \\alpha ^{\\prime }$ , has a construct accuracy at most $1 - \\frac{1}{2} (\\alpha - \\alpha ^{\\prime }) \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1)$ .", "By the reasoning in the proof of Theorem REF , we have for all $z \\in \\lbrace 0, 1\\rbrace $ , $ \\Pr [Y^{\\prime }{=}\\hat{Y}\\mid Z{=}z] \\le 1 - d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}z, \\hat{Y}|Z{=}z), $ which can be rewritten as $ \\Pr [Y^{\\prime }{\\ne }\\hat{Y}\\mid Z{=}z] \\ge d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}z, \\hat{Y}|Z{=}z).", "$ Thus, the construct inaccuracy of the model is $&\\textstyle \\frac{1}{2} \\big (\\Pr [Y^{\\prime }{\\ne }\\hat{Y}\\mid Z{=}0] + \\Pr [Y^{\\prime }{\\ne }\\hat{Y}\\mid Z{=}1]\\big ) \\\\&\\textstyle \\ge \\frac{1}{2} \\big (d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, \\hat{Y}|Z{=}0) + d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}1, \\hat{Y}|Z{=}1)\\big ) \\\\&\\textstyle \\ge \\frac{1}{2} \\big (d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) - d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1)\\big ) \\\\&\\textstyle \\ge \\frac{1}{2} (\\alpha - \\alpha ^{\\prime }) \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1),$ where the second inequality is an application of the triangle inequality and the third follows from the definitions of the $\\alpha $ -Hybrid worldview and the $\\alpha ^{\\prime }$ -disparity test.", "Therefore, the construct accuracy, which is one minus the construct inaccuracy, is at most $1 - \\frac{1}{2} (\\alpha - \\alpha ^{\\prime }) \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1)$ .", "Theorem 9 If the $\\alpha $ -Hybrid worldview holds, a model that passes the $\\alpha ^{\\prime }$ -disparity test, with $\\alpha < \\alpha ^{\\prime }$ , can still have disparity amplification.", "The $\\alpha ^{\\prime }$ -disparity test ensures that $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) \\le \\alpha ^{\\prime } \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1), $ and if equality holds here, we have $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) = \\alpha ^{\\prime } \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1) > \\alpha \\cdot d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1) $ whenever $d_{\\mathrm {tv}}(Y|Z{=}0, Y|Z{=}1) \\ne 0$ .", "By the $\\alpha $ -Hybrid worldview, the rightmost quantity equals $d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ , making the above inequality exactly that of disparity amplification (Equation REF )." ], [ "A More General Notion of Disparity Amplification", "In this section, we present a more general definition of disparity amplification that is a broader discrimination criterion and is applicable to numerical $Y^{\\prime }$ .", "Definition REF allows an output disparity if there exists an equally large disparity in $Y^{\\prime }$ , but it does not explicitly reflect the fact that we care about how the model came to exhibit the disparity.", "The only reason why we allow the disparity is that $Y^{\\prime }$ is the right attribute to use.", "Thus, if the model does not use $Y^{\\prime }$ at all, then there should be no output disparity.", "More formally, we want that if $Y^{\\prime }\\perp \\hat{Y}$ , then $\\hat{Y}\\perp Z$ .", "Definition REF generalizes this requirement and, unlike Definition REF , is applicable for both categorical and numerical $Y^{\\prime }$ at the expense of limiting $\\hat{Y}$ to be binary.", "The generalization deals with cases where $\\hat{Y}$ is not completely independent of $Y^{\\prime }$ by measuring how much $\\hat{Y}$ depends upon $Y^{\\prime }$ .", "For binary $\\hat{Y}$ , this dependence is captured by the likelihood function $\\ell (y^{\\prime }) = \\Pr [\\hat{Y}{=}1 \\mid Y^{\\prime }{=}y^{\\prime }]$ , and we use the Lipschitz continuity of this function to measure the dependence.", "Definition 12 (Disparity Amplification, Stronger) For $\\hat{Y}\\in \\lbrace 0, 1\\rbrace $ and $\\ell (y^{\\prime }) = \\Pr [\\hat{Y}{=}1 \\mid Y^{\\prime }{=}y^{\\prime }]$ , let $\\rho ^*_{\\!\\ell }$ be the smallest nonnegative $\\rho $ such that $\\ell $ is $\\rho $ -Lipschitz continuous.Technically, $\\rho ^*_{\\!\\ell }$ should be the infimum of all $\\rho $ such that $\\ell $ is $\\rho $ -Lipschitz continuous, but it is not difficult to show then that $\\ell $ is in fact $\\rho ^*_{\\!\\ell }$ -Lipschitz continuous.", "Then, a model exhibits disparity amplification if $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) > \\rho ^*_{\\!\\ell }\\cdot d_{\\mathrm {em}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1).$ $\\rho ^*_{\\!\\ell }$ characterizes how much impact $Y^{\\prime }$ can have on the output of the model.", "If the impact is small, we can conclude that the model is not using $Y^{\\prime }$ much, so not much output disparity can be explained by $Y^{\\prime }$ .", "On the other hand, if a small change in $Y^{\\prime }$ can cause a large change in the probability distribution of $\\hat{Y}$ , then even a large output disparity can possibly be due to a small inter-group difference in $Y^{\\prime }$ .", "In fact, the use of $\\rho ^*_{\\!\\ell }$ makes Definition REF invariant to scaling in $Y^{\\prime }$ .", "If a numerical $Y^{\\prime }$ is increased by some factor, $\\rho ^*_{\\!\\ell }$ will decrease by the same factor, so the quantity on the right-hand side of Equation REF will not change.", "We now give two arguments that Definition REF is the correct refinement of the previous definition (Definition REF ).", "First, we show that the new definition is a broader discrimination criterion than the previous one.", "The previous definition assumes that $Y^{\\prime }$ is categorical, and in this case a natural distance metric for its support $\\mathcal {Y}^{\\prime }$ is the indicator $d(u, v) = {1}(u \\ne v)$ .", "With this distance metric, we can relate the total variation distance used in the right-hand side of Equation REF with the earthmover distance used in Equation REF .", "Theorem 10 Let the construct $Y^{\\prime }$ be categorical with support $\\mathcal {Y}^{\\prime }$ , which has distance metric $d(u, v) = {1}(u \\ne v)$ .", "If a model has disparity amplification under Definition REF , the model has disparity amplification under Definition REF as well.", "We proceed by showing that $\\rho ^*_{\\!\\ell }\\cdot d_{\\mathrm {em}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) \\le d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ .", "Since the likelihood function $\\ell $ in Definition REF is always between 0 and 1, we have $|\\ell (u) - \\ell (v)| \\le 1 = d(u, v)$ when $u \\ne v$ , so $\\ell $ is 1-Lipschitz continuous.", "Therefore $\\rho ^*_{\\!\\ell }\\le 1$ , and it suffices to show that $d_{\\mathrm {em}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) \\le d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)$ .", "By [12], we get $d_{\\mathrm {em}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)&\\le \\Big (\\max _{u,v \\in \\mathcal {Y}^{\\prime }} d(u, v)\\Big ) \\cdot d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) \\\\&= d_{\\mathrm {tv}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1),$ so we are done.", "Second, we show that Theorems REF and REF still hold under the refined definition of disparity amplification.", "Since the definitions of optimality and the empirical tests have not changed, we focus strictly on the nondiscrimination portions of the theorems.", "Theorem 11 A model that passes the demographic parity test does not have disparity amplification under Definition REF .", "The proof of Theorem REF is very similar to that of Theorem REF and will thus be omitted.", "Theorem 12 If the WYSIWYG worldview holds, then a model that passes the equalized odds test does not have disparity amplification under Definition REF .", "We present the proof for the case where $Y^{\\prime }$ is continuous, but the proof for the discrete case is very similar.", "Let $p_0$ and $p_1$ be the probability density functions of $Y^{\\prime }|Z{=}0$ and $Y^{\\prime }|Z{=}1$ , respectively.", "By Kantorovich duality [28], we have $ d_{\\mathrm {em}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) \\\\\\ge \\int _{\\mathcal {Y}^{\\prime }} \\phi (v) \\, p_1(v) \\, dv - \\int _{\\mathcal {Y}^{\\prime }} \\psi (u) \\, p_0(u) \\, du$ for all $\\phi $ and $\\psi $ such that $\\phi (v) - \\psi (u) \\le d(u,v)$ for all $u, v \\in \\mathcal {Y}^{\\prime }$ .", "We set $\\phi (v) = \\psi (v) = \\ell (v) / \\rho ^*_{\\!\\ell }$ , where $\\ell $ and $\\rho ^*_{\\!\\ell }$ are defined as in Definition REF .", "Then, $\\phi (v) - \\psi (u) = (\\ell (v) - \\ell (u))/\\rho ^*_{\\!\\ell }\\le d(u, v)$ by Lipschitz continuity.", "Thus, Equation REF applies and implies that $ \\rho ^*_{\\!\\ell }\\cdot d_{\\mathrm {em}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) \\\\\\ge \\int _{\\mathcal {Y}^{\\prime }} \\ell (v) \\, p_1(v) \\, dv - \\int _{\\mathcal {Y}^{\\prime }} \\ell (u) \\, p_0(u) \\, du.$ By the WYSIWYG worldview and equalized odds, we have $\\ell (y) = \\Pr [\\hat{Y}{=}1 \\mid Y^{\\prime }{=}y] = \\Pr [\\hat{Y}{=}1 \\mid Y^{\\prime }{=}y, Z{=}0] = \\Pr [\\hat{Y}{=}1 \\mid Y^{\\prime }{=}y, Z{=}1]$ .", "Therefore, we can use the law of total probability to rewrite the first term on the right-hand side of Equation REF as $\\Pr [\\hat{Y}{=}1 \\mid Z{=}1]$ , and similarly the second term becomes $\\Pr [\\hat{Y}{=}1 \\mid Z{=}0]$ .", "If we let $\\phi (v) = \\psi (v) = -\\ell (v) / \\rho ^*_{\\!\\ell }$ in Equation REF instead, we get $\\rho ^*_{\\!\\ell }\\cdot d_{\\mathrm {em}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1) \\ge \\Pr [\\hat{Y}{=}1 \\mid Z{=}0] - \\Pr [\\hat{Y}{=}1 \\mid Z{=}1]$ .", "Finally, combining this inequality with the previous one gives us $\\rho ^*_{\\!\\ell }\\cdot d_{\\mathrm {em}}(Y^{\\prime }|Z{=}0, Y^{\\prime }|Z{=}1)&\\ge \\Big |\\Pr [\\hat{Y}{=}1 \\mid Z{=}0] - \\Pr [\\hat{Y}{=}1 \\mid Z{=}1]\\Big | \\\\&= d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1),$ which is what we want.", "We now briefly discuss the tightness of the above result.", "In the extreme example where $\\ell $ is a step function over real-valued $y^{\\prime }$ , $\\rho ^*_{\\!\\ell }$ is infinite, so we trivially have a lack of disparity amplification under Definition REF .", "Therefore, in order to receive meaningful fairness guarantees from Theorem REF , we must make sure that $\\rho ^*_{\\!\\ell }$ is not too large.", "One way to achieve this is to apply the function $\\ell $ to the construct space and reason about the transformed construct space.", "If any transformation of the construct space results in a finding of disparity amplification under Definition REF , then it is evidence that there could be a problem with the model with respect to discrimination.", "Let $\\tilde{y}^{\\prime }= \\ell (y^{\\prime })$ be a value in the transformed construct space, and $\\tilde{\\ell }$ denote the likelihood function on this space.", "Then, $ \\tilde{\\ell }(\\tilde{y}^{\\prime }) = \\Pr [\\hat{Y}{=}1 \\mid \\tilde{Y}^{\\prime }{=}\\tilde{y}^{\\prime }] = \\Pr [\\hat{Y}{=}1 \\mid Y^{\\prime }{=}y^{\\prime }] = \\ell (y^{\\prime }) = \\tilde{y}^{\\prime }, $ so the transformation ensures that $\\rho ^*_{\\!\\tilde{\\ell }} = 1$ ." ], [ "Connection to the $\\alpha $ -Disparity Test", "When $Y^{\\prime }$ and $Y$ are numerical, a natural extension of the $\\alpha $ -disparity test (Definition REF ) is $ d_{\\mathrm {tv}}(\\hat{Y}|Z{=}0, \\hat{Y}|Z{=}1) \\le \\rho ^*_{\\!\\ell }\\cdot \\alpha \\cdot d_{\\mathrm {em}}(Y|Z{=}0, Y|Z{=}1).$ For this to work, Worldview REF would have to change to use the earthmover distance rather than the total variation distance.", "Since the earthmover distance is defined over a distance metric, the parameter $\\alpha $ is not very meaningful unless $Y^{\\prime }$ and $Y$ have the same scale.", "As a result, here we consider the case where $Y^{\\prime }$ and $Y$ are defined over the same metric space $(\\mathcal {Y}, d)$ .", "Unfortunately, Equation REF is still not an empirical test because $\\rho ^*_{\\!\\ell }$ is defined in terms of $Y^{\\prime }$ .", "Although it is tempting to redefine $\\rho ^*_{\\!\\ell }$ in terms of $Y$ , it is possible for $Y^{\\prime }$ and $Y$ to have vastly different likelihood functions while having the same disparity, so this new empirical test will not guarantee the lack of disparity amplification under Definition REF .", "We leave as future work the discovery of an empirical test for numerical $Y^{\\prime }$ and $Y$ that corresponds to the $\\alpha $ -Hybrid worldview." ], [ "Conclusion", "We showed that demographic parity and equalized odds are related through our construct-based discrimination criterion of disparity amplification, arguing that the difference between the two empirical tests boils down to one's worldview.", "In addition, we proved that predictive parity allows a model with an arbitrarily large output disparity regardless of the worldview and the observed base rates.", "Our work differs from much of the prior work in that we consider the construct as separate from the observed data.", "In particular, we interpreted the existing fairness definitions as acting on the observed data, whereas the discrimination criterion was viewed as a property of the construct.", "This bifurcation allowed us to handle the following issues simultaneously: (a) prohibitions against disparate impact have exceptions such as a business necessity, but (b) due to past discrimination, the observed data can be biased in an unjustified way.", "It is the second of these points that motivates our use of worldviews to characterize how biased the observed data is.", "To illustrate how this might work in practice, let us revisit the examples in Section .", "In Example 1, there are reasons to believe that the observed recidivism rate is a racially biased measurement of the actual reoffense rate.", "In Example 2, for various socioeconomic reasons, some protected groups may have disproportionately many people who take longer than six years to graduate but are eventually considered successful in the university.", "The $\\alpha $ -Hybrid worldview can characterize these real-world scenarios, and the value of $\\alpha $ reflects one's beliefs about how much more biased the observed data is than the construct.", "Then, a practitioner can apply the $\\alpha $ -disparity test as a substitute for demographic parity or equalized odds, with the value of $\\alpha $ determined through social research and public dialogue." ], [ "Acknowledgments", "This material is based upon work supported by the National Science Foundation under Grant No. 1704985.", "Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation." ] ]

[ [ "On $C^{0}$ Interior Penalty Method for Fourth Order Dirichlet Boundary\n Control Problem and a New Error Analysis for Fourth Order Elliptic Equation\n with Cahn-Hilliard Boundary Condition" ], [ "Abstract In this paper, we revisit the $L_2$-norm error estimate for $C^0$-interior penalty analysis of Dirichlet boundary control problem governed by biharmonic operator.", "In this work, we have relaxed the interior angle condition of the domain from $120$ degrees to $180$ degrees, therefore this analysis can be carried out for any convex domain.", "The theoretical findings are illustrated by numerical experiments.", "Moreover, we propose a new analysis to derive the error estimates for the biharmonic equation with Cahn-Hilliard type boundary condition under minimal regularity assumption." ], [ "Introduction", "Let $\\Omega \\subset \\mathbb {R}^2$ be a bounded polygonal domain.", "If $D\\subset \\bar{\\Omega }$ then the $L_{2}(D)$ norm and inner product are denoted by $\\Vert \\cdot \\Vert _{D}$ and $(\\cdot ,\\cdot )_{D}$ respectively and when $\\Omega =D$ then they are denoted by $\\Vert \\cdot \\Vert $ and $(\\cdot ,\\cdot )$ respectively for the rest of the article.", "The other symbols, unless mentioned otherwise coincides with the standard Sobolev space notations.", "Let $Q$ denotes the following function space: $Q=\\lbrace p\\in H^2(\\Omega ):~\\partial p/\\partial n=0 \\;\\text{on}\\;\\ \\partial \\Omega \\rbrace .$ We consider the following optimal control problem: $\\min _{p\\in Q} J(u,p),$ subject to $&u=u_f+p,\\\\ &\\Delta ^2u=f\\;\\text{in}\\;\\Omega ,\\\\&u=p,\\;\\;\\partial u/\\partial n=0\\;\\text{on}\\;\\partial \\Omega ,$ where $f\\in L_{2}(\\Omega )$ denotes the external force and $J$ denotes the cost functional, given by $J(u,p)=\\frac{1}{2}\\Vert u-u_{d}\\Vert ^2+\\frac{\\alpha }{2}|p|^2_{H^2(\\Omega )}.$ In this connection we mention that $u_d\\in L_{2}(\\Omega )$ and $\\alpha >0$ stand for the desired state and regularization parameter respectively.", "This paper revisits the $L_2$ norm error estimate for a fourth order Dirichlet boundary control problem discussed in [35] and derives it under less stringent angle condition, additionally an alternative analysis to derive the energy norm estimate for elliptic Cahn-Hilliard equation is proposed under minimal regularity assumption, compared to [11].", "Classical non conforming methods and $C^0$ interior penalty (IP) methods have been two popular schemes to approximate the solutions of higher order equations within the finite element framework.", "In this connection we refer to the works of [10], [8], [12], [13], [14], [15], [16], [18], [19], [25], [28], [36], [37], [39], and references there in.", "These methods are computationally more efficient compared to the one of conforming finite element method.", "For the interested readers we refer to [21] for a discontinuous mixed formulation based analysis of fourth order problem.", "In this regard we would like to remark that, mixed schemes are complicated in general, and have its restrictions (solutions to discrete scheme may converge to a wrong solution for a fourth order problem if the solution is not $H^3$ regular).", "We notice that finite element error analysis for higher order optimal control problems could be found relatively less in the literature.", "In [3] a mixed finite element (Hermann- Miyoshi mixed formulation) analysis is proposed for a fourth order interior control problem.", "In this work, an optimal order $a\\; priori$ error estimates for the optimal control, optimal state and adjoint state are derived, followed by a super convergence result for the optimal control.", "For a $C^0$ interior penalty method based analysis of a fourth order interior control problem we refer to [5].", "In this work an optimal order $a\\; priori$ error estimate and a super convergence result is derived for the optimal on a general polygonal domain and subsequently a residual based $a\\; posteriori$ error estimates are derived for the construction of an efficient adaptive algorithm.", "In [4] abstract frameworks for both $a\\; priori$ and $a\\; posteriori$ error analysis of fourth order interior and Neumann boundary control problems are proposed.", "The analysis of this paper is applicable for second and sixth order problems as well.", "We continue our discussions on higher order Dirichlet boundary control problems.", "In this connection we note that the analysis of Dirichlet boundary control problem is more subtle compared to interior and Neumann boundary control problems.", "This is due to the fact that the control does not appear naturally in the formulation for Dirichlet boundary control problems.", "For the $C^{0}$ -IP analysis of an energy space based fourth order Dirichlet boundary control problem, we refer to [35], where the control variable is sought from the energy space $H^{3/2}(\\partial \\Omega )$ (the definition of the space $H^{3/2}(\\partial \\Omega )$ is given in Section ).", "In this work, an optimal order a priori energy norm error estimate is derived and subsequently an optimal order $L_2$ norm error estimate is derived with the help of a dual problem.", "But the derivation of $L_{2}$ norm error estimate for the optimal control involves a quite restrictive assumption on the domain, which says that, the interior angles of the domain should be less than 120o (in order to assure $H^{5/2+\\epsilon }(\\Omega )$ regularity for the optimal control).", "In this work we revisit this problem and extend the angle condition to $180^{o}$ .", "Moreover the technique used to prove an additional regularity result for the optimal control (Lemma REF ) motivates us to propose an alternative analysis to derive an optimal order error estimate for the solution of Cahn-Hilliard equation of elliptic type.", "The contributions of this paper can be summarized as follows.", "With the help of a crucial lemma (Lemma REF ) which establishes an equivalent form of the Hessian bilinear form over the space $Q$ , an optimal order $L_{2}$ norm estimate for the optimal control is derived when the domain is convex.", "An alternative error analysis for biharmonic equation with Cahn-Hilliard type boundary condition under minimal regularity assumption is derived.", "The rest of the article is organized as follows.", "In Section , we introduce the $C^0$ interior penalty method and define some general notations and concepts ($e.", "g.$ enriching operators) which are used in the later discussions of the article.", "We start Section , by showing the equality of two bilinear forms over the space of control variables, which plays a very crucial role in establishing the $L_{2}$ norm estimate for the optimal control.", "Subsequently we discuss corresponding optimality system for our model problem and also discuss the equivalence of this problem with the corresponding energy space based Dirichlet boundary control problem, where the control is sought from $H^{3/2}(\\partial \\Omega )$ space (the definition of $H^{3/2}(\\partial \\Omega )$ is given therein).", "We conclude this section with the discrete optimlity system.", "In Section we propose an alternative approach for the $a\\;priori$ error analysis for the numerical approximation of Cahn-Hilliard equation of elliptic type under minimal regularity assumptions.", "We discuss the optimal order energy norm estimates under minimal regularity assumption for the optimal control and subsequently for optimal state, adjoint state variable in Section .", "We derive the optimal order $L_2$ norm estimate for the optimal control variable in Section .", "We conclude the article with Section .", "We will follow the standard notion of spaces and operators that can be found for example in [17], [22] and [23].", "If $S\\subset \\bar{\\Omega }$ then the space of all square integrable functions defined over $S$ are denoted by $L_{2}(S)$ .", "When $m>0$ is an integer then by $H^{m}(S)$ we denote the space of $L_{2}(S)$ functions whose distributional derivative upto $m$ -th order is in $L_{2}(S)$ .", "If $s>0$ is a real number then there exists an integer $m>0$ such that $m-1<s<m$ .", "There $H^{s}(S)$ denotes the space of all $H^{m-1}(S)$ functions which belong to the fractional order Sobolev space $H^{s-m+1}(S)$ .", "When $\\Omega =S$ then the $L_{2}(\\Omega )$ inner product is denoted either by $(\\cdot ,\\cdot )$ , or by its usual integral representation .", "If $S=\\Omega $ then $L_{2}(\\Omega )$ norm is denoted by $\\Vert \\cdot \\Vert $ , else it is denoted by $\\Vert \\cdot \\Vert _{S}$ .", "In this context we mention that $H^{-s}(S)$ denotes the dual of $H^{s}_{0}(S)$ , and this duality is denoted by $\\langle \\cdot ,\\cdot \\rangle _{-s,s,S}$ for positive fractional $s$ ." ], [ "Quadratic $C^{0}$ Interior Penalty Method", "In this section we introduce the $C^0$ interior penalty method for this problem.", "Let $\\mathcal {T}_{h}$ be a simplicial, regular triangulation of $\\Omega .$ (See [22]).", "A generic triangle is denoted by $T$ and its diameter by $h_{T}$ .", "We define the mesh discretization parameter $h$ by $h=\\max _{T\\in \\mathcal {T}_{h}}h_{T}.$ The finite element spaces are given by, $&V_{h}=\\lbrace v_{h}\\in H^{1}_{0}(\\Omega ): v_{h}|_{T}\\in P_{2}(T)\\;\\forall T\\in \\mathcal {T}_{h}\\rbrace ,\\\\&Q_{h}=\\lbrace p_{h}\\in H^{1}(\\Omega ): p_{h}|_{T}\\in P_{2}(T)\\;\\forall T\\in \\mathcal {T}_{h}\\rbrace ,$ where $P_{2}(T)$ denotes the space of polynomials of degree less than or equal to two restricted to $T$ .", "For this triangulation a generic edge, length of it and the set of all edges are denoted by $e$ , $|e|$ and $\\mathcal {E}_{h}$ respectively.", "Note that $\\mathcal {E}_{h}$ is union of the set of all interior edges or $\\mathcal {E}^{i}_{h}$ and set of all boundary edges or $\\mathcal {E}^{b}_{h}$ .", "Any $e\\in \\mathcal {E}_h^i$ , could be written as $e=\\partial T_+\\cap \\partial T_-$ , for two adjacent triangles $T_{+}$ and $T_{-}$ .", "$n_-$ represents the unit normal of $e$ pointing from $T_-$ to $T_+$ and set $n_+=-n_-$ .", "For any $s>\\frac{3}{2}$ the set of piecewise $H^s$ , globally $H^{1}$ functions are denoted by $H^{s}(\\Omega ,\\mathcal {T}_h)$ .", "Let $v\\in H^2(\\Omega ,\\mathcal {T}_h)$ , the jump of normal derivative of $v$ on $e$ is defined by $[\\hspace{-1.5pt}[\\partial v/\\partial n]\\hspace{-1.5pt}] =\\left.", "\\nabla v_+\\right|_{e}\\cdot n_+ +\\left.", "\\nabla v_-\\right|_{e}\\cdot n_-,$ where $v_\\pm =v\\big |_{T_\\pm }$ .", "For all $v$ with $\\Delta v\\in H^1(\\Omega ,\\mathcal {T}_h)$ , we define mean and jump of the following second order quantity across $e$ by $\\lbrace \\hspace{-3.0pt}\\lbrace \\Delta v\\rbrace \\hspace{-3.0pt}\\rbrace =\\frac{1}{2}\\left(\\Delta v_{+}+\\Delta v_{-} \\right),$ and $[\\hspace{-1.5pt}[\\Delta v]\\hspace{-1.5pt}] =\\left(\\Delta v_{+}-\\Delta v_{-}\\right),$ respectively.", "For the convenience of notation, jump and average are defined on boundary edges as well.", "For any $e\\in \\mathcal {E}_h^b$ , there is an element $T\\in \\mathcal {T}_h$ such that $e=\\partial T\\cap \\partial \\Omega $ .", "Let $n_e$ be the unit normal of $e$ that points outside $T$ .", "For any $v\\in H^2(T)$ , we set on $e$ $[\\hspace{-1.5pt}[\\partial v/\\partial n]\\hspace{-1.5pt}]= \\nabla v \\cdot n_e,$ and for any $v$ with $\\Delta v\\in H^{1}(T)$ , we set $\\lbrace \\hspace{-3.0pt}\\lbrace \\Delta v\\rbrace \\hspace{-3.0pt}\\rbrace =\\Delta v.$ We define below several mesh dependent quantities (bilinear form, norms, semi-norms) as they are needed in our analysis.", "Begin with a mesh dependent bilinear form $a_{h}(\\cdot ,\\cdot )$ defined on $Q_{h}\\times Q_{h}$ by, $a_{h}(p_{h},r_{h})=&\\sum _{T\\in \\mathcal {T}_{h}}\\int _{T}\\Delta p_{h} \\Delta r_{h} dx+\\sum _{e\\in \\mathcal {E}_{h}}\\int _{e}\\lbrace \\hspace{-3.0pt}\\lbrace \\Delta p_{h}\\rbrace \\hspace{-3.0pt}\\rbrace [\\hspace{-1.5pt}[\\partial r_{h}/\\partial n_{e}]\\hspace{-1.5pt}]\\;ds+\\\\&\\sum _{e\\in \\mathcal {E}_{h}}\\int _{e}\\lbrace \\hspace{-3.0pt}\\lbrace \\Delta r_{h}\\rbrace \\hspace{-3.0pt}\\rbrace [\\hspace{-1.5pt}[\\partial p_{h}/\\partial n_{e}]\\hspace{-1.5pt}]\\;ds+\\sum _{e\\in \\mathcal {E}_{h}}\\frac{\\sigma }{|e|}\\int _{e}[\\hspace{-1.5pt}[\\partial p_{h}/\\partial n_{e}]\\hspace{-1.5pt}][\\hspace{-1.5pt}[\\partial r_{h}/\\partial n_{e}]\\hspace{-1.5pt}]\\;ds,$ without loss of generality we can assume the penalty parameter $\\sigma \\ge 1$ .", "Define the following mesh dependent norms and semi-norms on $Q_{h}$ by: $\\Vert p_{h}\\Vert ^{2}_{h}=\\sum _{T\\in \\mathcal {T}_{h}}\\Vert \\Delta p_{h}\\Vert _{T}^{2}+\\sum _{e\\in \\mathcal {E}_{h}}\\frac{\\sigma }{|e|}\\Vert [\\hspace{-1.5pt}[\\partial p_{h}/\\partial n_{e}]\\hspace{-1.5pt}]\\Vert _{e}^{2}\\;\\forall p_{h}\\in Q_{h}.$ Note that (REF ) defines a norm on $V_{h}$ whereas it is a semi-norm on $Q_{h}.$ The energy norm on $Q_{h}$ is defined by $|\\Vert p_{h}\\Vert |^{2}_{h}=\\Vert p_{h}\\Vert ^{2}_{h}+\\Vert p_{h}\\Vert ^{2}\\;\\forall p_{h}\\in Q_{h}.$ An alternative mesh dependent norm on $V_{h}$ (resp semi-norm on $Q_{h}$ ) by, $\\Vert p_{h}\\Vert ^{2}_{Q_{h}}=\\sum _{T\\in \\mathcal {T}_{h}}\\Vert \\Delta p_{h}\\Vert ^{2}_{T}+\\sum _{e\\in \\mathcal {E}_{h}}|e| \\Vert \\lbrace \\hspace{-3.0pt}\\lbrace \\Delta p_{h}\\rbrace \\hspace{-3.0pt}\\rbrace \\Vert _{e}^{2}+\\sum _{e\\in \\mathcal {E}_{h}}\\frac{\\sigma }{|e|}\\Vert [\\hspace{-1.5pt}[\\partial p_{h}/\\partial n_{e}]\\hspace{-1.5pt}]\\Vert _{e}^{2}\\;\\forall p_{h}\\in Q_{h}.$ We note that (REF ) defines a semi-norm on $Q_{h}$ , but it is a norm on $V_{h}$ .", "It is introduced in [11].", "It is clear that with the help of trace inequality for the finite element spaces we can show that there exists constants $C,c>0$ such that $c\\Vert p_h\\Vert _{h}\\le \\Vert p_h\\Vert _{Q_h}\\le C\\Vert p_h\\Vert _h\\;\\forall p_h\\in Q_h.$ Additionally $a_{h}$ is coercive and bounded on $Q_{h}$ with respect to $\\Vert \\cdot \\Vert _{h}$ $i.\\;e.$ $a_h(p_h,p_h)\\ge c\\Vert p_h\\Vert ^2_h\\;\\forall p_h\\in Q_h.$ $|a_h(x_h,y_h)|\\le C\\Vert x_h\\Vert _h\\Vert y_h\\Vert _h\\;\\forall x_h,y_h\\in Q_h.$ For details we refer the reader to [8].", "Given, $f\\in L_{2}(\\Omega )$ , $p_{h}\\in Q_{h}$ , let $v_{h}(f,p_{h})\\in V_{h} $ be the unique solution of the following equation: $a_{h}(v_{h}(f,p_{h}),w_{h})=(f,w_{h})-a_{h}(p_{h},w_{h})\\;\\forall w_{h}\\in V_{h}.$" ], [ "Enriching Operators", "We introduce a smoothing operator $E_h: Q_h\\rightarrow \\tilde{Q}_h,$ where $\\tilde{Q}_h$ is a conforming finite element discretization of control space $Q$ .", "The construction of $\\tilde{Q}_{h}$ and $E_{h}$ are described below.", "Various estimates satisfied by this operator play important roles in our forthcoming analysis.", "The construction of $\\tilde{Q}_{h}$ is briefly given as follows.", "Let $W_{h}$ be the Hsieh-Clough-Tocher macro finite element space associated with the triangulation $\\mathcal {T}_{h}$ , [22].", "Functions in $W_{h}$ belong to $C^{1}(\\bar{\\Omega })$ , and on each triangle they are piecewise cubic polynomials with respect to the partition obtained by connecting the centroid of the triangle to its vertices.", "Such functions are determined by their derivatives up to first order at the vertices and their normal derivatives at the midpoint of the edges.", "Let $\\tilde{Q}_{h}$ be defined by, $\\tilde{Q}_{h}=Q\\cap W_{h}.$ The smoothing operator $E_{h}$ which is also known as enriching operator is defined as follows.", "Given any $p_{h}\\in Q_{h}$ , we can define the macro finite element function $E_{h}(p_{h})$ by specifying its degrees of freedom (dofs), which are either its values at the vertices of $\\mathcal {T}_{h}$ , the values of its first order partial derivatives at the vertices, or the values of its normal derivatives at the midpoints of the edges of $\\mathcal {T}_{h}$ .", "Let $x_i$ be a degree of freedom in $\\mathcal {T}_{h}$ , if $x_i$ is a corner point of $\\Omega $ then we define $\\nabla E_{h}(p_{h})(x_i)=0,$ and if $x_i\\in \\partial \\Omega $ but $x_i$ is not a corner point of $\\Omega $ then we define $\\frac{\\partial }{\\partial n}E_{h}(p_{h})(x_i)=0,$ otherwise we assign these dofs of $E_{h}(p_{h})$ by averaging.", "The above defined enriching operators satisfy some approximation properties which are given by the following lemma: Lemma 2.1 Let $v \\in V_h$ .", "It holds that $\\sum _{T \\in \\mathcal {T}_h}\\left(h_T^{-4}\\Vert E_hv-v\\Vert ^2_{T}+h_T^{-2}\\Vert \\nabla (E_hv-v)\\Vert ^2_{T}\\right)&\\le C\\Big (\\sum _{e \\in \\mathcal {E}_h}\\frac{1}{|e|}\\Big \\Vert \\left[\\hspace{-3.5pt}\\left[\\frac{\\partial v}{\\partial n}\\right]\\hspace{-3.5pt}\\right]\\Big \\Vert ^2_{e}\\Big ) \\;\\forall \\; v \\in Q_h,$ and $\\sum _{T \\in \\mathcal {T}_h}|E_hv-v|^2_{H^2(T)}\\le C\\Big (\\sum _{e \\in \\mathcal {E}_h}\\frac{1}{|e|}\\Big \\Vert \\left[\\hspace{-3.5pt}\\left[\\frac{\\partial v}{\\partial n}\\right]\\hspace{-3.5pt}\\right]\\Big \\Vert ^2_{e}\\Big )\\; \\forall \\; v \\in Q_h.$ For its proof we refer the reader to [11]." ], [ "Auxiliary Results", "In this section we prove the agreement of two bilinear forms over the space $Q$ , which plays a key role in obtaining the $L_{2}$ norm estimate under an improved regularity assumption.", "Subsequently we state the existence and uniqueness results for the solution to the optimal control problem and derive the corresponding optimality system for it.", "At the end of this section we remark that this problem is equivalent to its corresponding Dirichlet control problem [35].", "We begin with defining a bilinear form $a:Q\\times Q\\rightarrow \\mathbb {R}$ by, $&a(u,v)=\\int _{\\Omega }\\Delta u\\Delta v\\;dx,$ The following lemma proves the equality of two bilinear forms over $Q$ .", "Lemma 3.1 Given $p,\\;q\\in Q$ we have $a(q,p)=\\int _{\\Omega } D^2q:D^2p\\;dx$ where $D^2q:D^2p$ is defined by $\\sum _{i,j=1,2}\\frac{\\partial ^{2} q}{\\partial x_i\\partial x_j}\\frac{\\partial ^2p}{\\partial x_{i}\\partial x_{j}}.$ Consider $E_hI_h(p)$ .", "Where $E_{h}I_{h}(p)$ denotes the enrichment of $I_{h}(p)$ defined in subsection REF and $I_{h}(p)$ denotes the Lagrange interpolation of $p$ onto the finite element space $Q_h$ [17].", "Introduce a new function space $X$ defined by $X=\\lbrace \\phi \\in H^{1}(\\Omega ): \\;\\Delta \\phi \\in L_{2}(\\Omega )\\rbrace $ endowed with the inner product $(\\cdot ,\\cdot )_{X}$ given by $(\\phi _{1},\\phi _{2})_{X}=(\\phi _{1},\\phi _{2})_{H^{1}(\\Omega )}+(\\Delta \\phi _{1},\\Delta \\phi _{2}),$ which is known to be a Hilbert space [20].", "In this context we mention that $(\\cdot ,\\cdot )_{H^{1}(\\Omega )}$ denotes the standard $H^{1}(\\Omega )$ inner product.", "Approximation properties of $I_h$ [17], Lemma REF and triangle inequality yield $\\Vert E_hI_h(p)\\Vert _{X}\\le C\\Vert p\\Vert _{H^2(\\Omega )}$ .", "Banach Alaoglu theorem implies the existence of a subsequence of $\\lbrace E_hI_h(p)\\rbrace $ (still denoted by $\\lbrace E_hI_h(p)\\rbrace $ for notational convenience) converging weakly to some $z\\in X$ .", "Continuity of first normal trace operator from $H(div,\\Omega )$ to $H^{-\\frac{1}{2}}(\\partial \\Omega )$ [26] implies the closedness of $kernel(\\frac{\\partial }{\\partial n})$ .", "For convenience of notation we denote it by $Z$ .", "Therefore $Z$ is complete and hence weakly closed, which implies $z\\in Z$ .", "Given $\\phi \\in Z$ , consider the problem given by $&-\\Delta \\psi =-\\Delta \\phi ~\\text{in} ~\\Omega ,\\\\&\\partial \\psi /\\partial n=0~\\text{on}~\\partial \\Omega .$ Elliptic regularity estimates imply $|\\psi |_{H^{1+s}(\\Omega )}\\le C\\Vert |\\phi \\Vert _{X}$ and hence $|\\phi |_{H^{1+s}(\\Omega )} \\le C\\Vert \\phi \\Vert _{X}\\; \\text{or}\\\\ \\;\\Vert \\phi \\Vert _{H^{1+s}(\\Omega )}\\le C\\Vert \\phi \\Vert _{X}\\;\\forall \\phi \\in Z$ , for some $s>0$ depending upon the interior angle of the domain [43].", "Positiveness of $s$ implies the compact embedding of $H^{1+s}(\\Omega )$ in $H^1(\\Omega )$ [20], which in turn implies the compact embedding of $Z$ in $H^1(\\Omega )$ .", "Therefore $E_hI_h(p)$ converges strongly to $z$ in $H^1(\\Omega ).$ A combination of the approximation properties of enriching operators (Lemma REF , [11]), trace inequality for $H^1(\\Omega )$ functions [17] and the $H^{2}$ regularity of $p$ implies the strong convergence of $E_hI_h(p)$ to $p$ in $H^1(\\Omega )$ .", "The uniqueness of limit implies $z=p$ .", "Hence $E_hI_h(p)$ converges weakly to $p$ in $Z.$ For any $\\eta \\in Z$ $a( \\eta ,E_hI_h(p))=(\\eta ,E_hI_h(p))_{X}-(\\eta ,E_hI_h(p))_{H^1(\\Omega )}.$ Note that since $E_hI_h(p)$ converges to $p$ weakly in $Z$ and $H^1(\\Omega )$ then $(\\eta ,E_hI_h(p))_{X}$ converges to $(\\eta ,p)_{X}$ and $(\\eta ,E_hI_h(p))_{H^{1}(\\Omega )}$ converges to $(\\eta ,p)_{H^1(\\Omega )}$ .", "Thus, $a( \\eta ,E_hI_h(p))\\rightarrow a( \\eta , p).$ Since $\\Vert E_hI_h(p)\\Vert _{H^2(\\Omega )}\\le C\\Vert p\\Vert _{H^{2}(\\Omega )}$ , the subsequence considered in the previous case ($i.\\;e.$ for the space $X$ which was still denoted by $\\lbrace E_hI_h(p)\\rbrace $ ) must have a weakly convergent subsequence denoted by $\\lbrace E_hI_h(p)\\rbrace $ (again for notational convenience!)", "converges weakly to some $w^{^{\\prime }}\\in H^{2}(\\Omega )$ .", "Then $E_hI_h(p)$ converges strongly to $w^{^{\\prime }}$ in $H^1(\\Omega )$ .", "But a combination of Lemma REF , trace inequality for $H^1(\\Omega )$ functions, and regularity of $p$ implies that $E_hI_h(p)$ converges strongly to $p$ with respect to $H^{1}(\\Omega )$ norm.", "Now from the uniqueness of the limit we have $w^{^{\\prime }}=p$ .", "Hence $\\int _{\\Omega }D^{2}\\eta :D^{2}E_hI_h(p))\\;dx\\rightarrow \\int _{\\Omega }D^2\\eta :D^2p\\;dx ~\\text{as}~ h\\rightarrow 0$ We now aim to show that for $p\\in Q,~ a(q,p)=\\int _{\\Omega } D^2q: D^2p\\;dx$ , with $q$ being the optimal control.", "There exists a sequence $\\lbrace \\phi _m\\rbrace \\subseteq C^{\\infty }(\\bar{\\Omega })$ with $\\phi _{m}$ converges to $q$ in $H^{2}(\\Omega )$ .", "Applying Green's formula we get $\\int _{\\Omega }[D^{2}\\phi _{m}:D^2E_hI_h(p)-(\\Delta \\phi _m,\\Delta E_hI_h(p))]\\;dx &=-\\int _{\\partial \\Omega }\\frac{\\partial ^2\\phi _m}{\\partial n\\partial t}\\frac{\\partial E_hI_h(p)}{\\partial t}\\;ds\\\\&=-\\sum _{k=1}^{l}\\int _{\\Gamma _k}\\frac{\\partial ^2\\phi _m}{\\partial n\\partial t}\\frac{\\partial E_hI_h(p)}{\\partial t}\\;ds.$ Combination of $\\frac{\\partial E_hI_h(p)}{\\partial t}\\in C(\\partial {\\Omega })$ , $\\frac{\\partial E_{h}I_{h}(p)}{\\partial t}$ being piecewise polynomial and $\\frac{\\partial E_{h}I_{h}(p)(x_{i})}{\\partial t}=0$ for corner points $x_{i}$ yields $&\\int _{\\Omega }D^{2}\\phi _{m}:D^{2}E_hI_h(p)\\;dx-a(\\phi _m, E_hI_h(p))=\\sum _{k=1}^{l}\\int _{\\Gamma _k}\\frac{\\partial \\phi _m}{\\partial n}\\frac{\\partial ^2 E_hI_h(p)}{\\partial t^2}\\;ds$ if an integration by parts is applied to the right hand side of (REF ).", "Taking limit on both sides w.r.t.", "$m$ we find: $\\int _{\\Omega }D^{2}q:D^{2}E_hI_h(p)\\;dx-a(q, E_hI_h(p))&=\\sum _{k=1}^{l}\\int _{\\Gamma _k}\\frac{\\partial q}{\\partial n}\\frac{\\partial ^2 E_hI_h(p)}{\\partial t^2} \\;ds\\\\&=0.$ Hence $a(q,E_hI_h(p))-\\int _{\\Omega }D^2 q:D^2 E_hI_h(p)\\;dx=0.$ Since $q\\in Q$ we conclude that $a(q,p)=\\int _{\\Omega }D^2 q:D^2 p\\;dx.$ The bilinear form $a$ defined in (REF ) is coercive on $V (=H^{2}_{0}(\\Omega ))$ and continuous on $Q\\times Q$ , see [17].", "For a given $f\\in L_{2}(\\Omega )$ , $p\\in Q$ , an application of Lax-Milgram lemma [17], [22] gives the existence of an unique $u_{f}\\in V$ such that, $a(u_{f},v)=(f,v)-a(p,v)\\;\\forall v\\in V.$ Therefore $u=u_{f}+p$ is the weak solution to the following Dirichlet problem: $&\\Delta ^{2}u=f\\;\\;\\;\\;\\text{in} \\;\\;\\;\\;\\;\\;\\;\\;\\Omega \\\\&u=p,\\;\\;\\frac{\\partial u}{\\partial n}=0\\;\\;\\;\\; \\text{on} \\;\\;\\;\\;\\partial \\Omega .$ In connection to the above discussion the optimal control problem described in (REF ) can be recasted as the following: $\\min _{p\\in Q} j(p),$ where $j(p)=\\frac{1}{2}\\Vert u_f+p-u_{d}\\Vert ^{2}+\\frac{\\alpha }{2}|p|_{H^{2}(\\Omega )}^{2}.$ The following proposition provides existence and uniqueness of the solution to the optimal control problem and the corresponding optimality system.", "Proposition 3.2 There exists a unique solution $(u,q)\\in Q\\times Q$ for the above described Dirichlet optimal control problem (REF ).", "Furthermore there exists an adjoint state $\\phi \\in V$ , and the triplet $(u,q,\\phi )\\in Q\\times Q\\times V$ satisfying the following system, which is known as the optimality or Karush Kuhn Tucker (KKT) system: $&u=u_{f}+q,\\;\\;\\;\\;u_{f}\\in V, \\\\&a(u_{f},v)=(f,v)-a(q,v)\\;\\forall v\\in V,\\\\&a(v,\\phi )=(u-u_{d},v)\\;\\forall v\\in V,\\\\&\\alpha a(q,p)=a(p,\\phi )-(u-u_{d},p)\\;\\forall p\\in Q.$ The proof follows from the similar arguments as in [2] and Lemma REF .", "The following remark shows the equivalence of this problem to its corresponding energy space based Dirichlet boundary control problem, where the control is sought from $H^{3/2}(\\partial \\Omega )$ space.", "Remark 3.3 The trace theory for polygonal domains says that the first trace of $Q$ into $\\prod _{i=1}^{m}H^{3/2}(\\Gamma _{i})$ is not onto, but surjective onto a subspace of $\\prod _{i=1}^{m}H^{3/2}(\\Gamma _{i})$ [23], which is denoted here by $H^{3/2}(\\partial \\Omega )$ .", "For any $p \\in H^{3/2}(\\partial \\Omega )$ , its $H^{3/2}(\\partial \\Omega )$ semi-norm can be equivalently defined by the Dirichlet norm: $|p|_{H^{3/2}(\\partial \\Omega )} := |u_{p}|_{H^{2}(\\Omega )} =\\min _{w\\in Q,w=p\\;\\;on\\;\\partial \\Omega }|w|_{H^{2}(\\Omega )},$ where the minimizer $u_{p}\\in Q$ satisfies the following equation: $&u_{p}=z+p,$ such that, $&\\int _{\\Omega }D^2z:D^2v\\;dx=-\\int _{\\Omega }D^2p:D^2v\\;dx\\;\\forall \\;v\\in V.$ Hence $\\int _{\\Omega }D^2u_{q}:D^2v\\;dx=0\\;\\forall \\;v\\in V.$ Therefore Lemma REF gives $a(u_{q},v)=0.$ From () we have, $a(q,v)=0\\; \\forall \\;v\\in V.$ Hence $q=u_{q}.$ Therefore the minimum energy in the minimization problem (REF ) is realized with an equivalent $H^{3/2}(\\partial \\Omega )$ norm of the optimal control $q$ ." ], [ "Discrete Optimality System", ".", "For $f\\in L_{2}(\\Omega )$ , $p_{h}\\in Q_{h}$ , let $v_{h}(f,p_{h})\\in V_{h} $ be the unique solution of the following equation: $a_{h}(v_{h}(f,p_{h}),w_{h})=(f,w_{h})-a_{h}(p_{h},w_{h})\\;\\forall w_{h}\\in V_{h}.$ $\\mathbf {Discrete\\;system}.$ A $C^{0}IP$ discretization of the continuous optimality system consists of finding $u_h\\in Q_h$ , $\\phi _h\\in V_h$ and $q_h\\in Q_h$ such that $&u_{h}=u^{h}_{f}+q_{h},\\quad u_h^f\\in V_h,\\\\&a_{h}(u_{f}^{h},v_{h})=(f,v_{h})-a_{h}(q_{h},v_{h})\\;\\forall v_{h}\\in V_{h},\\\\&a_{h}(\\phi _{h},v_{h})=(u_{h}-u_{d},v_{h})\\;\\forall v_{h}\\in V_{h},\\\\&\\alpha a_{h}(q_{h},p_{h})=a_{h}(\\phi _{h},p_{h})-(u_{h}-u_{d},p_{h})\\;\\forall p\\in Q_{h}.$ It is easy to check that if $f=u_{d}=0$ , then $u_{f}^{h}=q_{h}=\\phi _{h}=0$ .", "This implies that this discrete system is uniquely solvable.", "For $p_{h}\\in Q_{h}$ $u^{h}_{p_{h}}\\in Q_{h}$ is defined as follows: $u^{h}_{p_{h}}=w_{h}+p_{h},$ where $w_{h}\\in V_{h}$ solves the following equation, $a_{h}(w_{h},v_{h})=-a_{h}(p_{h},v_{h})\\;\\forall \\;v_{h}\\in V_{h}.$" ], [ "Energy Norm Estimate", "In this section we state the error estimate results for optimal control, optimal state and adjoint state $q$ , $u$ and $\\phi $ respectively in energy norm.", "We skip the proofs here, as they follow from similar arguments as in [35].", "These estimates are derived under the minimal regularity assumptions.", "The following theorems state these results.", "Theorem 4.1 For the optimal control $q$ , the following optimal order estimates holds: $\\Vert |q-q_{h}|\\Vert _{h}\\le Ch^{min(\\gamma ,1)}(\\Vert q\\Vert _{H^{2+\\gamma _1}(\\Omega )}+\\Vert \\phi \\Vert _{H^{2+\\gamma _2}(\\Omega )}+\\Vert f\\Vert )+(\\sum _{T\\in \\mathcal {T}_{h}}h_{T}^{4}\\Vert u-u_{d}\\Vert ^{2}_{T})^{1/2},$ and $\\Vert q-q_{h}\\Vert +\\Vert q-q_{h}\\Vert _{Q_{h}}\\le &Ch^{min(\\gamma ,1)}(\\Vert q\\Vert _{H^{2+\\gamma _1}(\\Omega )}+\\Vert \\nabla (\\Delta q)\\Vert +\\Vert \\phi \\Vert _{H^{2+\\gamma _2}(\\Omega )}+\\\\&\\Vert f\\Vert )+(\\sum _{T\\in \\mathcal {T}_{h}}h_{T}^{4}\\Vert u-u_{d}\\Vert ^{2}_{T})^{1/2},$ where $\\gamma =\\text{min}\\lbrace \\gamma _1,\\gamma _2\\rbrace $ the minimum of the regularity index between optimal control $q$ and adjoint state $\\phi $ .", "The generic constant C depends only on the shape regularity of the triangulation.", "We mention a few points on the second estimate in Theorem REF .", "Firstly we refer to Lemma REF of Section , from which we have $\\Delta q\\in H^{1}(\\Omega )$ , this is needed to derive $\\Vert q-I_{h}(q)\\Vert _{Q_{h}}$ along with triangle wise trace inequality for $H^{1}$ functions and standard interpolation error estimates [17].", "Secondly, in the proof of Theorem 4.1 of [35] essentially $\\Vert |I_{h}(q)-q_{h}\\Vert |_{h}$ is derived, which is equivalent to derive $\\Vert I_{h}(q)-q_{h}\\Vert +\\Vert I_{h}(q)-q_{h}\\Vert _{Q_{h}}$ .", "Combining first and second points we obtain the second estimate of Theorem REF .", "This estimate is used in the derivation of $L_{2}$ norm error estimate for the optimal control.", "Theorem 4.2 For the optimal state $u$ and the adjoint state $\\phi $ the following optimal order estimates hold: $&\\Vert |u-u_{h}\\Vert |_{h}\\le Ch^{ min(\\gamma ,1)}(\\Vert f|\\Vert +\\Vert q\\Vert _{H^{2+\\gamma _1}(\\Omega )}+\\Vert \\phi \\Vert _{H^{2+\\gamma _2}(\\Omega )}),\\\\&|\\Vert \\phi -\\phi _{h}|\\Vert _{h}\\le Ch^{min(\\gamma ,1)}(\\Vert f\\Vert +\\Vert q\\Vert _{H^{2+\\gamma _1}(\\Omega )}+\\Vert \\phi \\Vert _{H^{2+\\gamma _2}(\\Omega )}),$ where $\\gamma _1$ , $\\gamma _2$ and $\\gamma $ are as in Theorem REF ." ], [ "The $L_{2}$ -Norm Estimate", "In this section, we derive the $L_{2}$ norm error estimate for the optimal control on a convex domain.", "This restriction does not restrict optimal control to attain its minimal regularity [11], but helps adjoint state to gain $H^{3}$ regularity[8], which is required to derive the estimate.", "Density of $C^{\\infty }(\\bar{\\Omega })\\times C^{\\infty }(\\bar{\\Omega })$ in $H(div,\\Omega )$ space (with respect to the natural norm defined on $H(div,\\Omega )$ [26]) enables us to write $\\int _{\\Omega }\\nabla (\\Delta )\\phi .\\nabla \\psi \\;dx+\\int _{\\Omega }\\Delta ^{2}\\phi \\psi \\;dx=\\langle \\frac{\\partial \\Delta \\phi }{\\partial n},\\psi \\rangle _{-\\frac{1}{2},\\frac{1}{2},\\Omega }\\;\\forall \\,\\psi \\in H^{1}(\\Omega ).$ Choosing test functions in () from $\\mathcal {D}(\\Omega )$ we obtain $\\Delta ^2\\phi =u-u_d\\;\\text{ in } \\;\\Omega ,$ in the sense of distributions.", "Further density of $\\mathcal {D}(\\Omega )$ in $L_{2}(\\Omega )$ yields $\\Delta ^2\\phi =u-u_d,\\;\\text{a.\\;e.\\; in } \\;\\Omega .$ Combining (REF ) and (REF ), we obtain $\\int _{\\Omega }\\nabla (\\Delta )\\phi .\\nabla \\psi \\;dx+\\int _{\\Omega }(u-u_d)\\psi \\;dx=\\langle \\frac{\\partial \\Delta \\phi }{\\partial n},\\psi \\rangle _{-\\frac{1}{2},\\frac{1}{2},\\Omega }\\;\\forall \\psi \\in H^{1}(\\Omega ).$ Additionally if $\\psi \\in Q$ in (REF ), then $a(\\phi ,\\psi )-\\int _{\\Omega }(u-u_{d})\\psi \\;dx=-\\langle \\frac{\\partial \\Delta \\phi }{\\partial n},\\psi \\rangle _{-\\frac{1}{2},\\frac{1}{2},\\Omega }.$ We need the following auxiliary result in our error analysis: Lemma 5.1 Consider the variational problem, $(\\nabla w,\\nabla p)=-\\frac{1}{\\alpha }\\langle \\partial (\\Delta \\phi )/\\partial n ,p\\rangle \\;\\forall p\\in H^{1}(\\Omega ).$ There exists a unique solution $w\\in H^1(\\Omega )$ to the above variational problem upto an additive constant.", "The proof is a consequence of the fact that $H^1(\\Omega )$ -semi norm defines a norm on the quotient space $H^{1}(\\Omega )/\\mathbb {R}$ .", "Though we have assumed the domain to be convex for this section but the derivation of the following regularity property does not need it.", "It helps to establish a relation between the optimal control and adjoint state.", "Lemma 5.2 For the optimal control $q$ , we have $\\Delta q \\in H^{1}(\\Omega )$ and $\\nabla (\\Delta q)\\in H(div,\\Omega )$ .", "From Lemma REF , () and (REF ) we get, $a(q,p)=-\\frac{1}{\\alpha }\\langle \\partial (\\Delta \\phi )/\\partial n,p\\rangle _{-1/2,1/2,\\partial \\Omega }\\;\\forall p\\in Q.$ Lemma REF proves the existence of a unique weak solution $w\\in H^{1}(\\Omega )$ (up to an additive constant) of the following variational problem: $(\\nabla w,\\nabla p)=\\frac{1}{\\alpha }\\langle \\partial (\\Delta \\phi )/\\partial n,p\\rangle _{-1/2,1/2,\\partial \\Omega }\\;\\forall p\\in H^{1}(\\Omega ).$ If $p\\in Q$ in the above equation we have, $(w,\\Delta p)=-\\frac{1}{\\alpha }\\langle \\partial (\\Delta \\phi )/\\partial n,p\\rangle _{-1/2,1/2,\\partial \\Omega }\\;\\forall p\\in H^{1}(\\Omega ).$ Subtracting (REF ) from (REF ) we obtain, $(w-\\Delta q,\\Delta p)=0\\;\\forall p\\in Q.$ An application of elliptic regularity theory for Poisson equation with Neumann boundary conditionon convex polygonal domains along with the fact that $q\\in H^{2}(\\Omega )$ imply that $w-\\Delta q$ belongs to the orthogonal complement of $\\tilde{L}_{2}(\\Omega )$ , where $\\tilde{L}_{2}(\\Omega )=\\lbrace \\xi \\in L_{2}(\\Omega ): \\int _{\\Omega }\\xi =0\\rbrace .$ Therefore, $\\Delta q=w+constant.$ Hence $\\Delta q\\in H^{1}(\\Omega ).$ Choosing test functions from $\\mathcal {D}(\\Omega )$ in () and using (REF ) together with integration by parts in the sense of distributions, we obtain $\\Delta ^{2}q=0\\;\\;\\;\\text{in}\\;\\;\\Omega .$ Further density of $\\mathcal {D}(\\Omega )$ in $L_{2}(\\Omega )$ yields $\\Delta ^{2}q=0\\;\\;\\;\\text{a.\\;e.\\;in}\\;\\;\\Omega .$ These prove our claim.", "Density of $C^{\\infty }(\\bar{\\Omega })\\times C^{\\infty }(\\bar{\\Omega })$ in $H(div,\\Omega )$ implies $(\\Delta ^{2}q,p)+(\\nabla (\\Delta q),\\nabla p)=\\langle \\partial (\\Delta q)/\\partial n,p\\rangle _{-1/2,1/2,\\partial \\Omega },$ but $\\Delta ^2 q=0$ in $\\Omega $ , gives $(\\nabla (\\Delta q),\\nabla p)=\\langle \\partial (\\Delta q)/\\partial n,p\\rangle _{-1/2,1/2,\\partial \\Omega }.$ Applying integration by parts formula, we find $a(q,p)=-\\langle \\frac{\\partial \\Delta q}{\\partial n},p\\rangle _{-\\frac{1}{2},\\frac{1}{2},\\partial \\Omega }\\;\\forall p\\in Q.$ Thus from (REF ), () and (REF ) we get $\\langle \\frac{\\partial \\Delta \\phi }{\\partial n},p\\rangle _{-\\frac{1}{2},\\frac{1}{2},\\partial \\Omega }=\\alpha \\langle \\frac{\\partial \\Delta q}{\\partial n},p\\rangle _{-\\frac{1}{2},\\frac{1}{2},\\partial \\Omega }\\;\\forall p\\in Q.$ The following result is proved in [35] but still we are providing a proof for the convenience of the reader.", "It establishes a relation between optimal control and adjoint state.", "Lemma 5.3 If $p\\in H^{\\frac{1}{2}}(\\partial \\Omega )$ then, $\\alpha \\langle \\frac{\\partial \\Delta q}{\\partial n},p \\rangle _{-1/2,1/2,\\partial \\Omega }=\\langle \\frac{\\partial \\Delta \\phi }{\\partial n},p \\rangle _{-1/2,1/2,\\partial \\Omega }.$ Let $p\\in H^{1/2}(\\partial \\Omega )$ .", "For a Lipschitz domain $\\Omega $ , we know the space $\\lbrace u|_{\\partial \\Omega }: u\\in C^{\\infty }(\\mathbb {R}^{2})\\rbrace $ is dense in $H^{1/2}(\\partial \\Omega )$ [33].", "Therefore there exists a sequence $\\lbrace \\psi _{n}\\rbrace \\subset C^{\\infty }(\\partial \\Omega )$ such that $\\psi _{n}\\rightarrow p$ in $H^{1/2}(\\partial \\Omega )$ .", "Let $u_{n}$ be the weak solution of the following PDE: $&\\Delta ^{2}u_{n}=0\\text{ in }\\Omega ,\\\\&u_{n}=\\psi _{n}\\text{ on }\\partial \\Omega ,\\\\&\\partial u_{n}/\\partial n=0\\text{ on }\\partial \\Omega .$ Clearly, $u_{n}\\in Q$ and $u_{n}|_{\\partial \\Omega }=\\psi _{n}$ .", "Consider, $|\\langle \\partial (\\Delta \\phi )/\\partial n-\\alpha \\partial (\\Delta q)/\\partial n,p\\rangle _{-1/2,1/2,\\partial \\Omega }|&=|\\langle \\partial (\\Delta \\phi )/\\partial n-\\alpha \\partial (\\Delta q)/\\partial n,p-\\psi _{n}\\rangle _{-1/2,1/2,\\partial \\Omega }|\\\\&\\le \\Vert \\partial (\\Delta \\phi )/\\partial n-\\alpha \\partial (\\Delta q)/\\partial n\\Vert _{H^{-1/2}(\\partial \\Omega )}\\Vert p-\\psi _{n}\\Vert _{H^{1/2}(\\partial \\Omega )}\\\\&\\le \\epsilon ,\\;\\forall \\epsilon >0.$ Hence, we get (REF ).", "Theorem 5.4 For the optimal control $q$ , we obtain the following estimate: $\\Vert q-q_{h}\\Vert \\le Ch^{2\\beta }(\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert \\nabla (\\Delta q)\\Vert +\\Vert f\\Vert +\\Vert \\phi \\Vert _{H^{3}(\\Omega )}).$ The key ingredient to derive the $L_2$ norm estimate is a duality argument.", "Introduce the following auxiliary optimal control problem: Find $r\\in Q$ such that $j(r)=\\min _{p\\in Q}\\;j(p),$ where $j(p)$ is defined by, $j(p)=\\frac{1}{2}\\Vert u_{p}-(q-q_{h})\\Vert ^{2}+\\frac{\\alpha }{2}|p|^{2}_{H^{2}(\\Omega )}\\;\\forall \\;p\\in Q,$ $u_{p}=w+p$ and $w\\in V$ satisfies the following equation: $\\int _{\\Omega }D^{2}w:D^{2}v\\;dx=-\\int _{\\Omega }D^2p:D^{2}v\\;dx\\;\\forall \\;v\\in V.$ The standard theory of PDE constrained optimal control problems provide us with the existence of an unique solution of the above optimal control problem.", "We denote it by $r$ .", "For a detailed discussion on this topic we refer to [7], [24].", "Clearly $r$ satisfies the following optimality condition: $\\alpha \\int _{\\Omega }D^{2}r:D^{2}p\\;dx+\\int _{\\Omega }u_{r}u_{p}\\;dx=(q-q_{h},u_{p})\\;\\forall p\\in Q.$ From Lemma REF we obtain $\\alpha a(r,p)+(u_{r},u_{p})=(q-q_{h},u_{p})\\;\\forall p\\in Q.", "$ This implies, $\\alpha a(r,p)+(u_{r},p)-a(\\xi ,p)=(q-q_{h},p)\\;\\forall p\\in Q,$ with $\\xi \\in H^{2}_{0}(\\Omega )$ satisfies the following equation: $a(\\xi ,v)=(u_{r}-(q-q_{h}),v)\\;\\forall v\\in H^{2}_{0}(\\Omega ).$ Elliptic regularity theory for clamped plate problems imply $\\xi \\in H^{3}(\\Omega )\\cap H^{2}_{0}(\\Omega )$ .", "From (REF ), we obtain $\\Delta ^{2}\\xi =u_r-(q-q_h)\\quad \\text{in}\\quad \\Omega ,$ in the sense of distributions.", "Density of $\\mathcal {D}(\\Omega )$ in $L_2(\\Omega )$ gives $&\\Delta ^{2}\\xi =u_{r}-(q-q_{h})\\quad \\text{a.\\;e.\\;in} \\quad \\Omega ,\\\\&\\xi =0;~~~~~~~~~\\frac{\\partial \\xi }{\\partial n}=0 \\quad \\text{on} \\quad \\partial \\Omega .$ Therefore $\\nabla (\\Delta \\xi )\\in H(div,\\Omega )$ , which implies $\\frac{\\partial (\\Delta \\xi )}{\\partial n}\\in H^{-1/2}(\\partial \\Omega )$ .", "Using density of $C^{\\infty }(\\bar{\\Omega })\\times C^{\\infty }(\\bar{\\Omega })$ in $H(div,\\Omega )$ , we get $\\int _{\\Omega }\\nabla (\\Delta \\xi )\\nabla p\\;dx+\\int _{\\Omega }\\Delta ^{2}\\xi p\\;dx=\\langle \\partial (\\Delta \\xi )/\\partial n,p\\rangle _{-1/2,1/2,\\partial \\Omega }\\;\\forall p\\in H^{1}(\\Omega ).$ Hence integration by parts and (REF ) imply $\\alpha a(r,p)=-\\langle \\frac{\\partial (\\Delta \\xi )}{\\partial n},p\\rangle _{-1/2,1/2,\\partial \\Omega }\\;\\forall p\\in Q.$ Choosing test functions from $\\mathcal {D}(\\Omega )$ in (REF ), we get $\\Delta ^{2}r=0,$ in the sense of distributions.", "Consequently using density of $\\mathcal {D}(\\Omega )$ in $L_2(\\Omega )$ , we conclude that $\\Delta ^2r=0\\;\\text{a.\\;e.\\;in}\\;\\Omega .$ Arguments similar to the ones used for proving Lemma REF , and (REF ) yields $\\nabla (\\Delta r)\\in H(div,\\Omega )$ .", "Therefore $\\frac{\\partial (\\Delta r)}{\\partial n}\\in H^{-1/2}(\\partial \\Omega )$ .", "This implies, $\\alpha \\langle \\frac{\\partial (\\Delta r)}{\\partial n},p\\rangle _{-1/2,1/2,\\partial \\Omega }=\\langle \\frac{\\partial (\\Delta \\xi )}{\\partial n},p\\rangle _{-1/2,1/2,\\partial \\Omega }\\;\\forall p\\in Q.$ By similar arguments used in Lemma REF , we deduce $\\alpha \\langle \\frac{\\partial (\\Delta r)}{\\partial n},p\\rangle _{-1/2,1/2,\\partial \\Omega }=\\langle \\frac{\\partial (\\Delta \\xi )}{\\partial n},p\\rangle _{-1/2,1/2,\\partial \\Omega }\\;\\forall p\\in H^{1/2}(\\partial \\Omega ).$ Density of $C^{\\infty }(\\bar{\\Omega })\\times C^{\\infty }(\\bar{\\Omega })$ in $H(div,\\Omega )$ [26] implies the following integration by parts formula: $\\int _{\\Omega }\\nabla (\\Delta \\xi )\\nabla p\\;dx+\\int _{\\Omega }\\Delta ^{2}\\xi p\\;dx=\\langle \\partial (\\Delta \\xi )/\\partial n,p\\rangle _{-1/2,1/2,\\partial \\Omega }\\;\\forall p\\in H^{1}(\\Omega ).$ To derive the $L_2$ norm error estimate $\\langle q\\rangle + Q_{h}$ is used as test function space.", "Choosing $p_h\\in \\langle q\\rangle +Q_{h}$ in the last equation, an application of triangle-wise integration by parts produces $&a_{h}(\\xi ,p_{h})+\\langle \\frac{\\partial (\\Delta \\xi )}{\\partial n},p_{h}\\rangle =\\int _{\\Omega }(u_{r}-(q-q_{h}))p_{h}\\;dx.$ From (REF ), we get $& a_{h}(\\xi ,p_{h})+\\langle \\alpha \\frac{\\partial (\\Delta r)}{\\partial n},p_{h}\\rangle -\\int _{\\Omega }u_{r}p_{h}\\;dx=-\\int _{\\Omega }(q-q_{h})p_{h}\\;dx.$ Since $\\langle \\alpha \\frac{\\partial (\\Delta r)}{\\partial n},p_{h}\\rangle _{-\\frac{1}{2},\\frac{1}{2},\\partial \\Omega }=\\langle \\frac{\\partial (\\Delta r)}{\\partial n},u^{h}_{p_{h}}\\rangle _{-\\frac{1}{2},\\frac{1}{2},\\partial \\Omega }$ , we find $a_{h}(\\xi ,p_{h})&-\\int _{\\Omega }u_{r}(p_{h}-u^{h}_{p_{h}})\\;dx-\\int _{\\Omega }u_{r}u^{h}_{p_{h}}\\;dx+\\alpha \\langle \\frac{\\partial (\\Delta r)}{\\partial n},u^{h}_{p_{h}}\\rangle _{-\\frac{1}{2},\\frac{1}{2}\\partial \\Omega }\\\\&=-\\int _{\\Omega }(q-q_{h})(p_{h}-u^{h}_{p_{h}})\\;dx-\\int _{\\Omega }(q-q_{h})u^{h}_{p_{h}}\\;dx.$ We observe $\\langle \\frac{\\partial (\\Delta r)}{\\partial n},u^h_{p_h}\\rangle _{-\\frac{1}{2},\\frac{1}{2},\\partial \\Omega }=-a_h(r,u^h_{p_h})$ .", "Hence, $& a_{h}(\\xi ,p_{h})-\\int _{\\Omega }u_{r}(p_{h}-u^{h}_{p_{h}})\\;dx-\\int _{\\Omega }u_{r}u^{h}_{p_{h}}\\;dx-\\alpha a_{h}(r,u^{h}_{p_{h}})\\\\&=-\\int _{\\Omega }(q-q_{h})(p_{h}-u^{h}_{p_{h}})\\;dx-\\int _{\\Omega }(q-q_{h})u^{h}_{p_{h}}\\;dx,$ therefore $a_{h}(\\xi ,p_{h}-u^{h}_{p_{h}})& -\\int _{\\Omega }(u_{r}-(q-q_{h}))(p_{h}-u^{h}_{p_{h}})\\;dx\\\\&+a_{h}(\\xi ,u^{h}_{p_{h}})-\\int _{\\Omega }u_{r}u^{h}_{p_{h}}\\;dx-\\alpha a_{h}(r,p_{h})=-\\int _{\\Omega }(q-q_{h})u^{h}_{p_{h}}\\;dx.$ Since $\\xi $ satisfies the discrete formulation (similar to the proof in [8]), we have $\\alpha a_{h}(r,p_{h})+\\int _{\\Omega }u_{r}u^{h}_{p_{h}}\\;dx=a_{h}(\\xi ,u^{h}_{p_{h}})+\\int _{\\Omega }(q-q_{h})u^{h}_{p_{h}}\\;dx.$ Choosing $p_{h}=q-q_{h}$ in (REF ), we find $&\\alpha a_{h}(r,q-q_{h})+\\int _{\\Omega }u_{r}u^{h}_{q-q_{h}}\\;dx=a_{h}(\\xi ,u^{h}_{q-q_{h}})+\\int _{\\Omega }(q-q_{h})u^{h}_{q-q_{h}}\\;dx,$ and $& a_{h}(\\xi ,u^{h}_{q-q_{h}})+\\int _{\\Omega }(q-q_{h})(u^{h}_{q}-u_{q}+u_{q}-u^{h}_{q_{h}})\\;dx\\\\&=\\alpha a_{h}(r,q-q_{h})+\\int _{\\Omega }u_{r}(u^{h}_{q}-u_{q}+u_{q}-u^{h}_{q_{h}})\\;dx.$ Thus, $\\Vert q-q_{h}\\Vert ^{2}=&-\\int _{\\Omega }(q-q_{h})(u^{h}_{q}-u_{q})\\;dx-a_{h}(\\xi ,u^{h}_{q-q_{h}})\\\\&+\\alpha a_{h}(r,q-q_{h})+\\int _{\\Omega }u_{r}(u^{h}_{q}-u_{q})\\;dx+\\int _{\\Omega }u_{r}(u_{q}-u^{h}_{q_{h}})\\;dx\\\\ =&-\\int _{\\Omega }(q-q_{h})(u^{h}_{q}-u_{q})\\;dx-a_{h}(\\xi ,u^{h}_{q-q_{h}})\\\\&+\\alpha a_{h}(r-r_{h},q-q_{h})+\\alpha a_{h}(r_{h},q-q_{h})+\\int _{\\Omega }u_{r}(u^{h}_{q}-u_{q})\\;dx\\\\&+\\int _{\\Omega }u_{r}(u_{q}-u^{h}_{q_{h}})\\;dx\\\\ =&-\\int _{\\Omega }(q-q_{h})(u^{h}_{q}-u_{q})\\;dx-a_{h}(\\xi ,u^{h}_{q-q_{h}})\\\\&+\\alpha a_{h}(r-r_{h},q-q_{h})+a_{h}(\\phi ,r_{h})-\\int _{\\Omega }(u-u_{d})r_{h}\\;dx-a_{h}(\\phi _{h},r_{h})\\\\&+\\int _{\\Omega }(u_{h}-u_{d})r_{h}\\;dx+\\int _{\\Omega }u_{r}(u^{h}_{q}-u_{q})\\;dx+\\int _{\\Omega }u_{r}(u_{q}-u^{h}_{q_{h}}))\\;dx\\\\ =&-\\int _{\\Omega }(q-q_{h})(u^{h}_{q}-u_{q})\\;dx-a_{h}(\\xi ,u^{h}_{q-q_{h}})\\\\&+\\alpha a_{h}(r-r_{h},q-q_{h})+a_{h}(\\phi -\\phi _{h},r_{h})-\\int _{\\Omega }(u-u_{d})r_{h}\\;dx\\\\&+\\int _{\\Omega }(u_{h}-u_{d})r_{h}\\;dx+\\int _{\\Omega }u_{r}(u^{h}_{q}-u_{q})\\;dx+\\int _{\\Omega }u_{r}(u_{q}-u^{h}_{q_{h}})\\;dx\\\\ =&-\\int _{\\Omega }(q-q_{h})(u^{h}_{q}-u_{q})\\;dx-a_{h}(\\xi ,u^{h}_{q-q_{h}})\\\\&+\\alpha a_{h}(r-r_{h},q-q_{h})+a_{h}(\\phi -\\phi _{h},r_{h}-r)+a_{h}(\\phi -\\phi _{h},r)\\\\&-\\int _{\\Omega }(u-u_{h})r_{h}\\;dx+\\int _{\\Omega }u_{r}(u^{h}_{q}-u_{q})\\;dx+\\int _{\\Omega }u_{r}(u_{q}-u^{h}_{q_{h}})\\;dx\\\\ =&-\\int _{\\Omega }(q-q_{h})(u^{h}_{q}-u_{q})\\;dx-a_{h}(\\xi ,u^{h}_{q-q_{h}})\\\\&+\\alpha a_{h}(r-r_{h},q-q_{h})+a_{h}(\\phi -\\phi _{h},r_{h}-r)+a_{h}(\\phi -\\phi _{h},r)\\\\&-\\int _{\\Omega }(u_{f}-u_{f}^{h})r_{h}\\;dx+\\int _{\\Omega }(u_{q}-u^{h}_{q_{h}})(r-r_{h})\\;dx+\\int _{\\Omega }u_{r}(u^{h}_{q}-u_{q})\\;dx.$ Now estimate each term one by one on the right hand side of (REF ).", "Begin with, $-\\int _{\\Omega }(q-q_{h})(u^{h}_{q}-u_{q})\\;dx$ i.e.", "we aim to estimate $\\Vert u^{h}_{q}-u_{q}\\Vert $ .", "The following duality argument is used to get the estimate $\\Vert u^{h}_{q}-u_{q}\\Vert =\\sup _{w\\in L^{2}(\\Omega ),w\\ne 0}\\frac{(u^{h}_{q}-u_{q},w)}{\\Vert w\\Vert }.$ Consider $&\\Delta ^{2}\\phi _{w}=w\\text{ in }\\Omega ,\\\\&\\phi _{w}=0,\\quad \\frac{\\partial \\phi _{w}}{\\partial n}=0 \\text{ on }\\partial \\Omega .$ Let $P_{h}(w)$ be the $C^{0}$ interior penalty approximation of the solution of (REF ).", "Hence $&(u^{h}_{q}-u_{q},w)=a_{h}(\\phi _{w},u^{h}_{q}-u_{q})=a_{h}(\\phi _{w}-P_{h}(\\phi _w),u^{h}_{q}-u_{q})\\\\&\\le C\\Vert \\phi _{w}-P_{h}(\\phi _{w})\\Vert _{Q_{h}}\\Vert u^{h}_{q}-u_{q}\\Vert _{Q_{h}}\\le Ch\\Vert w\\Vert \\; \\Vert u^{h}_{q}-u_{q}\\Vert _{Q_{h}}.$ We note $\\Vert u^h_q-u_q\\Vert _{Q_{h}}\\le \\Vert u^h_q-u^h_{q_h}\\Vert _{Q_{h}}+\\Vert u^h_{q_h}-q\\Vert _{Q_h}.$ Now estimate $\\Vert u^{h}_{q}-u^h_{q_h}\\Vert _{Q_h}$ .", "$u^h_{q-q_h}$ is defined by $u^h_{q-q_h}=v_{0h}+q-q_h$ , where $v_{0h}$ is defined by: find $v_{0h}\\in V_{h}$ by, $a_{h}(v_{0h},v_{h})=-a_{h}(q-q_{h},v_{h})\\; \\forall v_{h}\\in V_{h}$ Therefore $&c_{1}\\Vert v_{0h}\\Vert ^2_{h}=-a_{h}(q-q_{h},v_{0h})=-[\\sum _{T\\in \\mathcal {T}_{h}}\\int _{T}\\Delta (q-q_{h})\\Delta v_{0h}\\;dx+\\sum _{e\\in \\mathcal {E}_{h}}\\int _{e}\\lbrace \\hspace{-3.0pt}\\lbrace \\Delta (q-q_{h})\\rbrace \\hspace{-3.0pt}\\rbrace ,\\\\&[\\hspace{-1.5pt}[\\partial v_{0h}/\\partial n]\\hspace{-1.5pt}]\\;ds+\\sum _{e\\in \\mathcal {E}_{h}}\\int _{e}\\lbrace \\hspace{-3.0pt}\\lbrace \\Delta v_{0h}\\rbrace \\hspace{-3.0pt}\\rbrace [\\hspace{-1.5pt}[\\partial (q-q_{h})/\\partial n]\\hspace{-1.5pt}]\\;ds+\\\\&\\sum _{e\\in \\mathcal {E}_{h}}\\sigma /|e|\\int _{e}[\\hspace{-1.5pt}[\\partial (q-q_{h})/\\partial n]\\hspace{-1.5pt}][\\hspace{-1.5pt}[\\partial v_{0h}/\\partial n]\\hspace{-1.5pt}]]\\;ds\\\\&\\le [\\sum _{T\\in \\mathcal {T}_{h}}\\Vert \\Delta (q-q_{h})\\Vert ^2_{T}+\\sum _{e\\in \\mathcal {E}_{h}}|e|\\Vert \\left\\lbrace \\hspace{-5.0pt}\\left\\lbrace \\Delta (q-q_{h})\\right\\rbrace \\hspace{-5.0pt}\\right\\rbrace \\Vert _{e}^2+\\sigma \\sum _{e\\in \\mathcal {E}_{h}}\\Vert [\\hspace{-1.5pt}[\\partial (q-q_{h})/\\partial n]\\hspace{-1.5pt}]\\Vert _{e}^2]^{1/2}\\\\&[\\sum _{T\\in \\mathcal {T}_{h}}\\Vert \\Delta v_{0h}\\Vert _{T}^{2}+(\\sigma +2)\\sum _{e\\in \\mathcal {E}_{h}}\\frac{1}{|e|}\\Vert [\\hspace{-1.5pt}[\\partial v_{0h}/\\partial n]\\hspace{-1.5pt}]\\Vert _{e}^2+\\sum _{e\\in \\mathcal {E}_{h}}|e|\\Vert \\lbrace \\hspace{-3.0pt}\\lbrace \\Delta v_{0h}\\rbrace \\hspace{-3.0pt}\\rbrace \\Vert _{e}^2]^{1/2}\\\\ &\\le C_{2}|e|^{\\beta }[\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert f\\Vert _{\\Omega }+\\Vert \\nabla (\\Delta q)\\Vert _{\\Omega }+|e|^{-\\beta }(\\sum _{T\\in \\mathcal {T}_{h}}h^{4}\\Vert u-u_{d}\\Vert ^{2}_{T})^{1/2}]\\Vert v_{0h}\\Vert _{Q_{h}}$ Now using the equivalence of $\\Vert .\\Vert _{Q_{h}}$ and $\\Vert .\\Vert _{h}$ on the finite dimensional space $V_{h}$ we get the following estimate for $\\Vert v_{0h}\\Vert _{Q_{h}}$ : $\\Vert v_{0h}\\Vert _{Q_{h}}\\le C_{3}|e|^{\\beta }[\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert \\nabla (\\Delta q)\\Vert +\\Vert f\\Vert +(h^{2-\\beta }\\Vert u-u_{d}\\Vert ))]$ Therefore from Theorem REF and (REF ) we obtain the following estimate for $u^{h}_{q}-u_{q}$ $ \\Vert u^{h}_{q}-q\\Vert _{Q_{h}}\\le C_{4}|e|^{\\beta } [\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert f\\Vert _{\\Omega }+\\Vert \\nabla (\\Delta q)\\Vert _{\\Omega }+|e|^{-\\beta }(\\sum _{T\\in \\mathcal {T}_{h}}h^{4}\\Vert u-u_{d}\\Vert ^{2}_{T})^{1/2}]$ Hence we have $\\Vert u^{h}_{q}-u_{q}\\Vert \\le C_{5}h^{1+\\beta }(\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert f\\Vert +\\Vert \\nabla (\\Delta q)\\Vert +h^{2-\\beta }\\Vert u-u_{d}\\Vert ))$ We now estimate $a_{h}(\\xi ,u^{h}_{q-q_{h}})$ .", "Let $\\xi _{h}$ be the Lagrange interpolation of $\\xi .$ $&a_{h}(\\xi ,u^{h}_{q-q_{h}})=a_{h}(\\xi -\\xi _{h},u^{h}_{q-q_{h}})=\\sum _{T\\in \\mathcal {T}_{h}} \\int _{T}\\Delta (\\xi -\\xi _{h})\\Delta u^{h}_{q-q_{h}}\\;dx+\\sum _{e\\in \\mathcal {E}_{h}} \\int _{e}\\lbrace \\hspace{-3.0pt}\\lbrace \\Delta (\\xi -\\xi _{h})\\rbrace \\hspace{-3.0pt}\\rbrace [\\hspace{-1.5pt}[\\partial u^{h}_{q-q_{h}}/\\partial n]\\hspace{-1.5pt}]\\;ds+\\\\&\\sum _{e\\in \\mathcal {E}_{h}}\\int _{e}\\lbrace \\hspace{-3.0pt}\\lbrace \\Delta u^{h}_{q-q_{h}}\\rbrace \\hspace{-3.0pt}\\rbrace [\\hspace{-1.5pt}[\\partial (\\xi -\\xi _{h})/\\partial n]\\hspace{-1.5pt}]\\;ds+\\sum _{e\\in \\mathcal {E}_{h}}\\int _{e}\\frac{\\sigma }{|e|}[\\hspace{-1.5pt}[\\partial (\\xi -\\xi _{h})/\\partial n]\\hspace{-1.5pt}][\\hspace{-1.5pt}[\\partial {u^{h}_{q-q_{h}}/\\partial n}]\\hspace{-1.5pt}]\\;ds\\\\&=[\\sum _{T\\in \\mathcal {T}_{h}}\\Vert \\Delta (\\xi -\\xi _{h})\\Vert _{T}^{2}+\\sum _{e\\in \\mathcal {E}_{h}}|e|\\Vert \\left\\lbrace \\hspace{-5.0pt}\\left\\lbrace \\Delta (\\xi -\\xi _{h})\\right\\rbrace \\hspace{-5.0pt}\\right\\rbrace \\Vert _{e}^{2}+\\sum _{e\\in \\mathcal {E}_{h}}|e|^{-1}\\Vert [\\hspace{-1.5pt}[\\partial (\\xi -\\xi _{h})/\\partial n]\\hspace{-1.5pt}]\\Vert ^{2}_{e}+\\\\ &\\sum _{e\\in \\mathcal {E}_{h}}\\frac{1}{|e|}\\Vert [\\hspace{-1.5pt}[\\partial (\\xi -\\xi _{h})/\\partial n]\\hspace{-1.5pt}]\\Vert ^{2}_{e}]^{1/2}[\\sum _{T\\in \\mathcal {T}_{h}}\\Vert \\Delta u^{h}_{q-q_{h}}\\Vert _{T}^{2}+\\sum _{e\\in \\mathcal {E}_{h}}|e|\\Vert \\lbrace \\hspace{-3.0pt}\\lbrace \\Delta u^{h}_{q-q_{h}}\\rbrace \\hspace{-3.0pt}\\rbrace \\Vert ^{2}_{e} \\\\&+(\\sigma +1)\\sum _{e\\in \\mathcal {E}_{h}}\\frac{1}{|e|}\\Vert [\\hspace{-1.5pt}[\\partial u^{h}_{q-q_{h}}/\\partial n]\\hspace{-1.5pt}]\\Vert _{e}^{2}]^{1/2}$ Using scaling argument and trace inequality we obtain estimate for $\\Vert \\Delta u^{h}_{q-q_{h}}\\Vert _{e}$ and therefore finally we get the following estimate $|a_{h}(\\xi ,u^{h}_{q-q_{h}})|\\le Ch^{1+\\beta }[\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert \\nabla (\\Delta q)\\Vert +h^{2-\\beta }\\Vert u-u_{d}\\Vert ]\\Vert \\xi \\Vert _{H^{3}(\\Omega )}$ Putting $p=r$ in (REF ) and using the fact that $u_{r}=r$ , we get that $\\Vert r\\Vert _{\\Omega }\\le \\Vert q-q_{h}\\Vert _{\\Omega }$ .", "Elliptic regularity estimate for (REF ) gives that $\\Vert \\xi \\Vert _{H^{3}(\\Omega )}\\le C\\Vert q-q_{h}\\Vert $ .", "Hence we find $a_{h}(\\xi ,u^{h}_{q-q_{h}})\\le Ch^{1+\\beta }\\Vert q-q_{h}\\Vert (\\Vert \\nabla (\\Delta q)\\Vert _{\\Omega }+\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+ h^{2-\\beta }\\Vert u-u_{d}\\Vert ).$ We now show that $\\Vert r\\Vert _{H^{2+\\beta }(\\Omega )}\\le C\\Vert q-q_{h}\\Vert .$ First we show that $|r|_{H^{2+\\beta }(\\Omega )}\\le C\\Vert q-q_{h}\\Vert .$ To this end, we use similar arguments used in [11].", "Let $z=\\alpha r-\\xi $ .", "Then $z$ satisfies the following variational equality: $a(z,v)=a(\\alpha r-\\xi ,v)\\;\\forall v\\in Q.$ Using integration by parts on right hand side of the above equation, we obtain $a(z,v)=-(\\Delta ^{2}\\xi ,v)\\;\\forall v\\in Q.$ From elliptic regularity theory for the biharmonic equation with Cahn-Hilliard boundary condition [11], we obtain $z\\in H^{2+\\beta }(\\Omega ).$ Further $|z|_{H^{2+\\beta }(\\Omega )}&\\le \\Vert \\Delta ^{2}\\xi \\Vert \\\\&\\le \\Vert u_{r}-(q-q_{h})\\Vert \\\\&\\le \\Vert r-(q-q_{h})\\Vert \\\\&\\le C\\Vert q-q_{h}\\Vert .$ Taking $p=r$ in (REF ), we obtain $|r|_{H^{2}(\\Omega )}+\\Vert r\\Vert \\le C\\Vert q-q_{h}\\Vert .$ Then using integration by parts, we get $|r|_{H^{1}(\\Omega )}\\le C(|r|_{H^{2}(\\Omega )}+\\Vert r\\Vert ) \\le C\\Vert q-q_{h}\\Vert .$ Hence from above estimates, we obtain $\\Vert r\\Vert _{H^{2+\\beta }(\\Omega )}\\le C\\Vert q-q_{h}\\Vert .$ To estimate the third term of the right hand side of (REF ), we proceed the same way as above and obtain $a(r-r_{h},q-q_{h})\\le C\\Vert r-r_{h}\\Vert _{Q_{h}}\\Vert q-q_{h}\\Vert _{Q_{h}}\\le & C|e|^{2\\beta }(\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert f\\Vert +\\Vert \\phi \\Vert _{H^{3}(\\Omega )}+\\\\& h^{2-\\beta }\\Vert u-u_{d}\\Vert )(\\Vert r\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert \\nabla (\\Delta r)\\Vert )$ Since $\\nabla (\\Delta r)\\in H(div,\\Omega )$ and $C^{\\infty }(\\bar{\\Omega })\\times C^{\\infty }(\\bar{\\Omega })$ is dense in $H(div,\\Omega )$ with respect to the natural norm induced on $H(div,\\Omega )$ the following integration by parts formula holds true $\\int _{\\Omega }\\nabla (\\Delta r)\\cdot \\nabla p\\;dx+\\int _{\\Omega }\\Delta ^{2}rp\\;dx=\\langle \\frac{\\partial \\Delta r}{\\partial n},p\\rangle _{-\\frac{1}{2},\\frac{1}{2},\\partial \\Omega }.", "$ With the help of (REF ), (REF ) we obtain $\\int _{\\Omega }\\nabla (\\Delta r)\\cdot \\nabla p\\;dx=\\frac{1}{\\alpha }\\langle \\frac{\\partial \\Delta \\xi }{\\partial n},p\\rangle _{-\\frac{1}{2},\\frac{1}{2},\\partial \\Omega }\\;\\forall p\\in H^{1}(\\Omega ).$ We note if we put $p=1$ in (REF ) we obtain $\\int _{\\Omega }u_{r}-(q-q_{h})=0$ , from (REF ) we conclude that (REF ) satisfies the compatibility condition, if we treat $\\Delta r$ as the unknown variable.", "We take $p=\\Delta r-\\frac{1}{|\\Omega |}\\int _{\\Omega }\\Delta r\\;dx$ in (REF ), an use of trace and Poincare friedrich's inequality gives $\\Vert \\nabla (\\Delta r)\\Vert \\le C(\\Vert \\frac{\\partial \\Delta \\xi }{\\partial n}\\Vert _{H^{-1/2}(\\partial \\Omega )}.$ Using (REF ) we obtain $\\Vert \\frac{\\partial \\Delta \\xi }{\\partial n}\\Vert _{H^{-1/2}(\\partial \\Omega )}\\le C\\Vert \\Delta ^{2}\\xi \\Vert +\\Vert \\nabla (\\Delta \\xi )\\Vert )$ The solution to (REF ) satisfies $\\Vert \\xi \\Vert _{H^{3}(\\Omega )}\\le C\\Vert u_{r}-(q-q_{h})\\Vert $ and $\\Delta ^{2}\\xi =u_{r}-(q-q_{h})$ .", "Since $u_r=r$ we have from (REF ), (REF ) and(REF ) that $\\Vert \\nabla (\\Delta r)\\Vert $ $\\le \\Vert q-q_{h}\\Vert $ .", "Therefore we have $a(r-r_{h},q-q_{h})\\le C h^{2\\beta }[\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+\\Vert f\\Vert +\\Vert \\phi \\Vert _{H^{3}(\\Omega )}+h^{2-\\beta }\\Vert u-u_{d}\\Vert ]\\Vert q-q_{h}\\Vert $ Using the same arguments we obtain following estimate for $a_{h}(\\phi -\\phi _{h},r-r_{h})$ .", "$a_{h}(\\phi -\\phi _{h},r-r_{h})\\le Ch^{1+\\beta }\\Vert q-q_{h}\\Vert [\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}\\Vert f\\Vert +\\Vert \\phi \\Vert _{H^{3}(\\Omega )}+h^{2-\\beta }\\Vert u-u_{d}\\Vert ]$ Note that as $\\phi \\in H^{2}_{0}(\\Omega ),\\;\\phi _{h}\\in H^{1}_{0}(\\Omega )$ from (REF ) and (REF ) we obtain $a_{h}(\\phi -\\phi _{h},r)=0.$ Rest of the terms in (REF ) can be estimated easily and they are given as follows: $-\\int _{\\Omega }(u_{f}-u^{h}_{f})r_{h}\\;dx\\le C\\Vert u_{f}-u^{h}_{f}\\Vert \\;\\Vert r\\Vert _{H^{2+\\beta }(\\Omega )}&\\le C h^{2}\\Vert f\\Vert \\;\\Vert q-q_{h}\\Vert .\\\\&(by\\; Aubin- Nitche\\; duality \\;argument)$ $-\\int _{\\Omega }(u_{q}-u^{h}_{q_{h}})(r-r_{h})\\;dx\\le Ch^{2+2\\beta }(\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+ \\Vert f\\Vert +\\Vert \\phi \\Vert _{H^{3}(\\Omega )})\\Vert q-q_{h}\\Vert .$ Further from (REF ), we get $\\int _{\\Omega }u_{r}(u^{h}_{q}-u_{q})\\;dx&\\le \\Vert u^{h}_{q}-u_{q}\\Vert \\Vert r\\Vert _{H^{2+\\beta }(\\Omega )}\\\\&\\le Ch^{1+\\beta }(\\Vert q\\Vert _{H^{2+\\beta }(\\Omega )}+ \\Vert f\\Vert +\\Vert \\phi \\Vert _{H^{3}(\\Omega )})\\Vert q-q_{h}\\Vert .$ Finally from (REF ), (REF ) and (REF )-(REF ), we obtain the desired estimate." ], [ "Alternative Approach of Error Analysis", "This section is devoted to the discussion of an alternative approach energy norm estimate for the solution of elliptic Cahn-Hilliard equation under minimal regularity assumption.", "For simplicity consider the equation $&\\Delta ^2\\psi =g_1\\;\\text{in}\\; \\Omega ,\\\\&\\partial \\psi /\\partial n=0,\\;\\partial (\\Delta \\psi )/\\partial n=g_2\\; \\text{on}\\; \\partial \\Omega ,$ where $g_1\\in L_{2}(\\Omega )$ and $g_{2}\\in L_{2}(\\partial \\Omega )$ with the compatibility condition $\\int _{\\Omega }g_1\\;dx=\\int _{\\partial \\Omega }g_2\\;ds$ .", "The solution to this equation is unique upto adding a constant.", "In variational formulation (REF ) can be written as $a(\\psi ,p)=(g_1,p)-(g_{2},p)_{\\partial \\Omega }\\;\\forall p\\in \\;Q.$ For $\\psi $ satisfying (REF ) satisfies the following elliptic regularity estimate $\\Vert \\psi \\Vert _{H^{2+\\beta }(\\Omega )}\\le C(\\Vert g_{1}\\Vert _{\\Omega }+\\Vert g_{2}\\Vert _{\\partial \\Omega }),$ where $0<\\beta <1$ denotes the regularity index for the solution to (REF ) which depends upon the interior angles of the domain [11].", "We prove the following lemma which is crucial to obtain the optimal order energy norm error estimate for the solution of (REF ).", "We begin by recalling the Hilbert space $Z$ defined in Lemma REF .", "Lemma 6.1 The control space $Q$ is dense in $Z.$ From the discussions in Lemma REF we have $Z$ is a closed subspace of $X.$ Let $Z^{\\perp }$ denotes the orthogonal complement of $Z$ in $X$ .", "Then we have $X=Z\\oplus Z^{\\perp }$ , where $\\oplus $ denotes the direct sum of two orthogonal subspaces.", "Clearly $H^{2}(\\Omega )\\cap Z=Q.$ We aim to show that $Q$ is dense in $Z$ .", "Choose $p\\in Z$ arbitrarily.", "Density of $H^2(\\Omega )$ in $X$ [20] produces the existence of a sequence $\\lbrace p_n\\rbrace \\subset H^2(\\Omega )$ with $\\lbrace p_n\\rbrace $ converging to $p$ in $X$ .", "Let $p_{n}=p_{n}^{\\prime }+p_{n}^{\\prime \\prime }$ , with $p_{n}^{\\prime }\\in Z$ and $p_{n}^{\\prime \\prime }\\in Z^{\\perp }$ .", "Consider $\\Vert p_{n}^{\\prime \\prime }\\Vert ^{2}_{X}=(p_{n}^{\\prime \\prime },p^{\\prime \\prime }_{n})_{X}=(p_{n}^{\\prime \\prime },p_{n}^{\\prime }+p_{n}^{\\prime \\prime }-p)_{X}\\le \\Vert p_{n}^{\\prime \\prime }\\Vert _{X}\\Vert p-(p_{n}^{\\prime }+p_{n}^{\\prime \\prime })\\Vert _{X}.$ This gives $\\Vert p_{n}^{\\prime \\prime }\\Vert _{X}\\rightarrow 0$ .", "Hence $\\Vert p_{n}^{\\prime }-p\\Vert _{X}\\rightarrow 0$ , which concludes our proof.", "The following lemma helps us to prove an additional regularity for the solution of (REF ).", "Lemma 6.2 Let $\\psi \\in Q$ be the solution to (REF ).", "Then $\\Delta \\psi \\in H^{1}(\\Omega )$ , hence $\\nabla (\\Delta \\psi )\\in H(div,\\Omega )$ .", "Theory of Fredholm alternative provides the existence of an unique weak solution $w\\in H^{1}(\\Omega )$ , up to an additive constant to the following variational formulation: $(\\nabla w,\\nabla p)=(g_{1},p)-(g_{2},p)_{\\partial \\Omega }\\;\\forall p\\in H^{1}(\\Omega ).$ If $p\\in Q$ , an integration by parts gives, $(w,\\Delta p)=.", "(g_{1},p)-(g_{2},p)_{\\partial \\Omega }\\; \\forall p\\in Q$ Subtracting (REF ) from (REF ) we obtain, $(w-\\Delta \\Psi ,\\Delta p)=0\\;\\forall p\\in Q.$ Application of Lemma REF implies $w-\\Delta \\psi $ belongs to the orthogonal complement of $\\tilde{L}_{2}(\\Omega )$ , where $(\\tilde{L}_{2}(\\Omega )=\\lbrace \\xi \\in L_{2}(\\Omega ): \\int _{\\Omega }\\xi =0\\rbrace .$ Therefore, $\\Delta \\psi =w+constant.$ Which implies $\\Delta \\psi \\in H^{1}(\\Omega )$ and subsequently from (REF ) we obtain the regularity result.", "We know the solution to (REF ) is unique upto an additive constant.", "Let $\\psi _{1}\\in Q$ be a solution to (REF ) and $c$ be a corner point to $\\Omega $ .", "Define $\\psi _{2}\\in Q$ to be $\\psi _{2}(x)=\\psi _{1}(x)-\\psi _{1,c}(x)\\;\\forall x\\in \\bar{\\Omega }.$ where the the constant function $\\psi _{1,c}(x)=\\psi _{1}(c)\\quad \\forall x\\in \\bar{\\Omega }.$ Then $\\psi _2(c)=0$ .", "Hence $\\psi _2$ satisfies (REF ).", "Define $Q^*=\\lbrace p\\in Q: p(c)=0\\rbrace .$ In this connection consider the following variational problem, find $\\psi _{2}^{\\prime }\\in Q^*$ such that, $a(\\psi _{2}^{\\prime },p)=(g_{1},p)-(g_2,p)_{\\partial \\Omega }\\;\\forall p\\in Q^*.$ We know that the solution to (REF ) is unique [11] and $\\psi _{2}\\in Q^*$ satisfies (REF ), so $\\psi _{2}=\\psi _{2}^{\\prime }$ .", "Therefore $\\psi _{2}^{\\prime }$ posses the regularity property described in Lemma REF .", "An application of Lemma REF gives us that $\\Delta \\psi _{2}\\in H^{1}(\\Omega )$ and $\\nabla (\\Delta \\psi _{2})\\in H(div,\\Omega )$ .", "This implies $\\int _{\\Omega }\\nabla (\\Delta )\\psi _{2}.\\nabla \\phi \\;dx+\\int _{\\Omega }\\Delta ^{2}\\psi _{2}\\phi \\;dx=\\langle \\frac{\\partial \\Delta \\psi _{2}}{\\partial n},\\phi \\rangle _{-\\frac{1}{2},\\frac{1}{2},\\Omega }\\;\\forall \\,\\phi \\in H^{1}(\\Omega ).$ We now enter the discretization.", "We consider the finite element space same as taken in [11].", "Putting $p_{h}\\in Q_{h}^{*}$ in the last equation and performing trianglewise integration by parts we obtain $a_h(\\psi _{2},p_h)=(g_1,p_h)-(g_2,p_h)_{\\partial \\Omega }\\; \\forall p_h\\in Q_{h}^{*},$ where $Q_{h}^{*}$ is defined by: $Q_{h}^{*}=\\lbrace p_{h}\\in Q_{h}|p_{h}(c)=0\\rbrace .$ Note that since $\\Delta \\psi _{2}\\in H^1(\\Omega )$ , the above equation makes sense.", "In light of the above discussions Consider the following discrete problem: Find $\\psi _{h}\\in Q_{h}^{*}$ such that $a_h(\\psi _{h},p_h)=(g_1,p_h)-(g_2,p_h)_{\\partial \\Omega }\\; \\forall p_h\\in Q_{h}^{*}.$ Now we state and prove the main result of this section, which gives the error estimate for for the solution to the solutions of (REF ) and (REF ) in energy norm.", "Theorem 6.3 Let $\\eta $ be the solution to (REF ) and $\\eta _{h}$ to (REF ).", "Then the following optimal order error estimate holds: $\\Vert \\eta -\\eta _{h}\\Vert _{h}\\le Ch^{\\beta }(\\Vert g_{1}\\Vert _{\\Omega }+\\Vert g_{2}\\Vert _{\\partial \\Omega }).$ Coercivity of the bilinear form $a_{h}$ with respect to $\\Vert \\cdot \\Vert _{h}$ norm on $Q_{h}^{*}$ gives the existence of an unique solution to (REF ).", "Now subtracting (REF ) from (REF ) we obtain the following Galerkin orthogonality $a_h(\\eta -\\eta _{h},p_h)\\; \\forall p_{h}\\in Q_{h}^{*}.$ Let $I_{h}(\\eta )$ denote the Lagrange interpolation of $\\eta $ .", "Then using (REF ) we have $&\\Vert \\eta -\\eta _{h}\\Vert \\le \\Vert \\eta -I_{h}(\\eta )\\Vert _{h}+\\Vert I_{h}(\\eta )-\\eta _{h}\\Vert _{h}\\\\&\\text{Consider} \\;\\Vert I_{h}(\\eta )-\\eta _{h}\\Vert _{h}.\\\\&\\Vert I_{h}(\\eta )-\\eta _{h}\\Vert _{h}=\\sup _{\\phi _{h}\\in Q_{h}^{*},\\phi _{h}\\ne 0}\\frac{a_{h}(I_{h}(\\eta )-\\eta _{h},\\phi _{h})}{\\Vert \\phi _{h}\\Vert _{h}}=\\sup _{\\phi _{h}\\in Q_{h}^{*},\\phi _{h}\\ne 0}\\frac{a_{h}(I_{h}(\\eta )-\\eta ,\\phi _{h})}{\\Vert \\phi _{h}\\Vert _{h}}\\le C\\Vert \\eta -I_{h}(\\eta )\\Vert _{Q_{h}}.$ With the help of the above equation, (REF ) and (REF ) the desired result is obtained." ], [ "Conclusion", "In this article we have derived the $L_{2}$ norm estimate for the solution of a Dirichlet control problem on more general domain than the one was studied in [35].", "Additionally getting motivated from the technique of deriving an additional regularity result for the optimal control (lemma REF ) we have proposed an alternative approach for the error analysis of biharmonic equation of Cahn Hilliard type boundary condition under minimal regularity assumption.", "In order to prove these results we have derived an equality of two well known bilinear forms arising in the context of weak formulation of bi harmonic equations and a density result which may be of theoretical interest." ], [ "Acknowledgements", "The author would like to thank Prof. Neela Nataraj and Dr. Mayukh Mukherjee for their invaluable suggestions and cooperation.", "The author would also like to thank Indian Institute of Technology Bombay for various financial helps." ] ]

[ [ "A note on the existence results for Schrodinger-Maxwell system with\n super-criticalnonlinearitie" ], [ "Abstract The paper considers the Schrodinger-Maxwell system with supercritical nonlinearitie.", "We prove the existence of at least one non-trivial weak solution.", "This result is already known for the subcritical case.", "In this paper, we extend it to the supercritical values of p as well.", "We use a new variational principle to prove our result." ], [ " The paper considers the following Schrödinger-Maxwell system with supercritical nonlinearitie, ${\\left\\lbrace \\begin{array}{ll}-\\Delta u+K(x) \\phi u =|u|^{p-1}u+h(x), \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{in} \\ \\ \\Omega ,\\\\-\\Delta \\phi = K(x)u^{2}, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{in} \\ \\ \\Omega ,\\\\\\phi = u=0, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{in} \\ \\ \\partial \\Omega ,\\end{array}\\right.", "}$ where $\\Omega \\subset \\mathbb {R}^{3}$ is a bounded domain with smooth boundary, $1<p\\ \\ \\mbox{and} \\ \\ K,h\\in {L^{\\infty }}\\left(\\Omega \\right)$ .", "We prove the existence of at least one non-trivial weak solution.", "This result is already known for the subcritical case.", "In this paper, we extend it to the supercritical values of $p$ as well.", "We use a new variational principle to prove our result.", "section1-3.5ex plus -1ex minus -.2ex2.3ex plus .2exIntroduction and main results In the present paper we study the existence of solution for the following electrostatic nonlinear Schrödinger-Maxwell equations also known as nonlinear Schrödinger-Poisson system ${\\left\\lbrace \\begin{array}{ll}-\\Delta u+K(x) \\phi u =|u|^{p-1}u+h(x), \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{in} \\ \\ \\Omega ,\\\\-\\Delta \\phi = K(x)u^{2}, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{in} \\ \\ \\Omega ,\\\\\\phi = u=0, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{in} \\ \\ \\partial \\Omega ,\\end{array}\\right.", "}$ where $\\Omega \\subset \\mathbb {R}^{N}$ , ($N=3$ ), is a bounded domain with smooth boundary, $1<p$ and $K,h\\in L^{\\infty }(\\Omega )$ .", "Similar system arises in many mathematical physics contexts while looking for existence of standing waves for the nonlinear Schrödinger equations interacting with an unknown electrostatic field.", "For more details on the physics aspect we refer the reader to [6], [9].", "In recent years, a number of papers have contributed to investigate the existence of solutions of (REF ).", "We can cite [1], [2], [3], [4], [7], [8], [9], [11], [12], [16] and the references therein.", "For the case where $\\Omega $ is a bounded domain, we would like to cite the papers of Ruiz and Siciliano [17] and Siciliano[18].", "In all those papers, the solutions found are in the case where $1<p<5$ .", "In the unbounded case, Ambrosetti and Ruiz [2] and Ruiz [16] considered problem ${\\left\\lbrace \\begin{array}{ll}-\\Delta u+V(x)u+\\mu \\phi u =|u|^{p-1}u, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{in} \\ \\ \\mathbb {R}^{3},\\\\-\\Delta \\phi = 4 \\pi ^{2}u^{2}, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{in} \\ \\ \\mathbb {R}^{3}.\\end{array}\\right.", "}$ By working in the radial functions subspace of $H^{1}(\\mathbb {R}^{3})$ and taking $1<p<5$ and $V(x)=1$ , they were able to obtain the existence and multiplicity results.", "In [12], Jiang and Zhou have treated the problem (REF ) where $\\Omega =\\mathbb {R}^{3}$ , $K=\\lambda >0, \\ \\ 1<p<6$ and $V$ change sign.", "With further assumptions on $V$ , the authors have proved that problem (REF ) has at least a positive solution.", "If $0<p<1$ , Bahrouni and Ounaies [4] has treated system (REF ) where $\\Omega =\\mathbb {R}^{3}$ .", "By using the variational method, they have proved that problem (REF ) has infinitely many solutions.", "We also refer to [5], [13], [17], [18], [19].", "Motivated by papers above, we are interested in finding solution for system (REF ), by assuming only that $p>1$ .", "Our methodology is based on a new variational principle established in [14], [15].", "In order to state our main results, we give the following assumptions: $\\left(K\\right) \\ \\ K\\in L^{\\infty }\\left(\\Omega \\right)$ and $K(x)\\ge 0, \\ \\ \\forall x \\in \\Omega $ .", "$\\left(H\\right) \\ \\ h\\in L^{\\infty }\\left(\\Omega \\right)$ and $h(x)> 0, \\ \\ \\forall x \\in \\Omega .$ Now we can state our result.", "Theorem 0.1 Assume that $(H)$ and $(K)$ hold.", "Suppose that $p>1$ .", "Then, there exists $m>0$ such that if $\\Vert h\\Vert _{L^{N}(\\Omega )}\\le m$ , problem (REF ) admits at least one nontrivial solution.", "The remainder of our paper is organized as follows.", "In section 2, some preliminary results are presented.", "While section 3 is dedicated to the proof of Theorems REF .", "section1-3.5ex plus -1ex minus -.2ex2.3ex plus .2exVariational settings and preliminary results First, we give some notations.", "For $1\\le m< +\\infty , \\ \\ L^{m}\\left(\\Omega \\right)$ is the usual Lebesgue space with the norm $\\left\\Vert u\\right\\Vert _{L^{m}(\\Omega )}=\\displaystyle \\left(\\int _{\\Omega } \\left| u \\right|^{m}dx\\right)^{\\frac{1}{m}}.$ Hereafter, the space $E=H^{1}_{0}\\left(\\Omega \\right)\\cap L^{p+1}\\left(\\Omega \\right)$ is endowed with the following norm $\\left\\Vert u\\right\\Vert =\\left\\Vert \\nabla u\\right\\Vert _{2}+\\left\\Vert u\\right\\Vert _{p+1}.$ We shall now recall some results for the Sobolev space required in the sequel (see [10], [14]).", "Lemma 0.2 Let $\\Omega $ be a bounded $C^{0,1}$ domain in $\\mathbb {R}^{N}$ .", "Then: $i)$ If $0\\le m< k-\\frac{N}{p}<m+1$ , the space $W^{k,p}(\\Omega )$ is continuously imbedded in $C^{m,\\alpha }(\\overline{\\Omega })$ , $\\alpha =k-\\frac{N}{p}-m$ , and compactly imbedded in $C^{m,\\beta }(\\overline{\\Omega })$ for any $\\beta <\\alpha .$ $ii)$ $u\\rightarrow \\Vert \\Delta u\\Vert _{L^{N}(\\Omega )}$ is an equivalent norm on $H^{1}_{0}(\\Omega )\\frown W^{2,N}(\\Omega )=E\\frown W^{2,N}(\\Omega )$ .", "An important fact involving system (REF ) is that this class of system can be transformed into a Schrödinger equation (see, for instance [9], [16]), with a nonlocal term.", "By the Lax-Milgram Theorem, given $u\\in E$ , there exists a unique $\\phi _{u}\\in H^{1}_{0}(\\Omega ) $ such tha $-\\Delta \\phi _{u}=K(x)u^{2}.$ By using standard arguments, we have that $\\phi _{u}$ verifies the following properties ( see [9], [16]); Lemma 0.3 For any $u\\in E$ , we have $1)$ there exists $C>0$ such that $\\Vert \\phi _{u}\\Vert \\le C\\Vert u\\Vert ^{2}$ .", "$2)$ $\\phi _{u}\\ge 0, \\ \\ \\phi _{tu}=t^{2}\\phi _{u}, \\forall \\ \\ t\\ge 0 \\ \\ \\mbox{and} \\ \\ u\\in E$ .", "$3)$ If $u_{n}\\rightharpoonup u$ in $E$ , then $\\phi _{u_{n}}\\rightharpoonup \\phi _{u}$ in $E$ and $\\displaystyle \\lim _{n\\rightarrow +\\infty }\\int _{\\Omega }\\phi _{u_{n}}{u_{n}}^{2}dx=\\int _{\\Omega }\\phi _{u}u^{2}dx$ .", "$4)$ If $u\\in W^{2,N}(\\Omega ) \\frown H^{1}_{0}(\\Omega )$ , then $\\phi _{u}\\in W^{2,N}(\\Omega ) \\frown H^{1}_{0}(\\Omega )$ .", "So, the functional $I: E\\rightarrow \\mathbb {R}$ , $I(u)=\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u\\right|^{2}dx+\\frac{1}{4}\\displaystyle \\int _{\\Omega }K(x)\\phi _{u}u^{2}dx-\\frac{1}{p+1}\\displaystyle \\int _{\\Omega }\\left|u\\right|^{p+1}dx-\\displaystyle \\int _{\\Omega } h(x)u dx, \\ \\ \\forall u\\in E$ is $C^{1}$ on $E$ and $\\left\\langle I^{^{\\prime }}(u),\\varphi \\right\\rangle =\\displaystyle \\int _{\\Omega }\\nabla u\\nabla \\varphi dx+ \\displaystyle \\int _{\\Omega }K(x)\\phi _{u}u \\varphi dx-\\displaystyle \\int _{\\Omega }\\left|u\\right|^{p-1}u \\varphi dx-\\displaystyle \\int _{\\Omega } h(x)\\varphi dx,$ for all $u\\in E$ and $\\varphi \\in H^{1}_{0}\\left(\\Omega \\right)$ .", "It is also known that $\\left(u,\\phi \\right)\\in E \\times H^{1}_{0}\\left(\\Omega \\right)$ is a solution of (REF ) if and only if $u\\in E$ is a critical point of the functional $I$ , and $\\phi =\\phi _{u}$ , see for instance [6].", "let us recall that a Palais-Smale sequence for the functional I, for short we write (PS)- sequence, is a sequence $\\left(u_{n}\\right)$ such that $\\left(I(u_{n})\\right) \\ \\ \\mbox{is bounded in E and} \\ \\ \\left\\Vert I^{^{\\prime }}(u_{n})\\right\\Vert _{E^{^{\\prime }}}\\rightarrow 0.$ I is said to satisfy the Palais-Smale condition if any (PS)-sequence possesses a convergent subsequence in $E$ .", "Now, we recall some important definitions and results from [10].", "Let $E$ be a real Banach space.", "Let $\\psi : E\\rightarrow \\mathbb {R}\\smile \\lbrace \\infty \\rbrace $ be a proper (i.e.", "$Dom(\\psi ))=\\lbrace u\\in E;\\psi (u)<\\infty \\rbrace \\ne \\emptyset $ ) convex function.", "The subdifferential $\\partial \\psi $ of $\\psi $ is defined to be the following set-valued operator: if $u\\in Dom(\\psi )$ , set $\\partial \\psi (u)=\\lbrace u^{^{\\prime }}\\in E^{^{\\prime }}; \\langle u^{^{\\prime }},v-u\\rangle +\\psi (u)\\le \\psi (v), \\ \\ \\forall v\\in E\\rbrace ,$ and if $u \\notin Dom(\\psi )$ , set $\\partial \\psi (u)=\\emptyset $ .", "If $\\psi $ is Gâteaux differentiable at $u$ , denote by $D\\phi (u)$ the Gâteaux derivative of $\\psi $ at $u$ .", "In this case $\\partial \\psi (u)=\\lbrace D\\psi (u)\\rbrace $ .", "The restriction of $\\psi $ to $K\\subset E$ is denoted by $\\psi _{K}$ and defined by $\\psi _{K}(u)=\\psi (u) \\ \\ \\mbox{if} \\ \\ u\\in K \\ \\ \\mbox{and} \\ \\ \\psi _{K}(u)=+\\infty \\ \\ \\mbox{if} \\ \\ u\\notin K.$ Let $J$ be a function on $E$ satisfying the following hypothesis: $(R)$ : $J=\\psi -\\phi $ , where $\\phi \\in C^{1}(E,\\mathbb {R})$ and $\\psi :E \\rightarrow (-\\infty , +\\infty ]$ is proper, convex and lower semi continuous.", "Definition 0.4 A point $u\\in E$ is said to be a critical point of $I=\\psi -\\phi $ if $u\\in Dom(\\psi )$ and if it satisfies the inequality $\\langle D \\phi (u),u-v\\rangle +\\psi (v)-\\psi (u)\\ge 0, \\ \\ \\forall v\\in E,$ where $D\\phi (u)$ stands for the derivative of $\\phi $ at $u.$ Lemma 0.5 If $I$ satisfies $(R)$ , then each local minimum of $I$ is necessarily a critical point of $I$ .", "See [14].", "See [14].", "Now, we define the functionals $\\phi ,\\psi :E \\rightarrow \\mathbb {R}$ by $\\phi (u)=-\\frac{1}{4}\\displaystyle \\int _{\\Omega }K(x)\\phi _{u}u^{2}dx+\\frac{1}{p+1}\\displaystyle \\int _{\\Omega }\\left|u\\right|^{p+1}dx\\\\+\\displaystyle \\int _{\\Omega }h(x)udx,$ and $\\psi (u)=\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u\\right|^{2}dx \\ \\ \\mbox{and} \\ \\ I_{K}(u)=\\psi _{K}(u)-\\phi (u).$ section1-3.5ex plus -1ex minus -.2ex2.3ex plus .2exProof of Theorem REF We now give the following variational principle version applicable to problem (REF ).", "Theorem 0.6 Let $K\\subset E$ be a convex and weakly closed subset of $E$ .", "If the following two assertions hold: $(i)$ The functional $I_{K}$ has a critical point $u_{1}\\in E$ as in Definition REF , and; $(ii)$ there exists $u_{2}\\in K$ such that $\\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla \\varphi dx=-\\displaystyle \\int _{\\Omega }K(x)\\phi _{u_{1}}u_{1} \\varphi dx+\\displaystyle \\int _{\\Omega }\\left|u_{1}\\right|^{p-1}u_{1} \\varphi dx+\\displaystyle \\int _{\\Omega }h(x)\\varphi dx, \\ \\ \\forall \\varphi \\in E.$ Then $u_{1}\\in K$ is a weak solution of system (REF ).", "Since $u_{1}$ is a critical point of $I_{K}$ , then $\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla v\\right|^{2}dx-\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{1}\\right|^{2}dx\\ge \\displaystyle \\int _{\\Omega }(-K(x)\\phi _{u_{1}}u_{1}+\\left|u_{1}\\right|^{p-1}u_{1}+h(x))(v-u_{1})dx, \\ \\ \\forall v\\in K.$ Invoking assumption $(ii)$ in the theorem, we deduce that $\\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla (u_{1}-u_{2})dx=\\displaystyle \\int _{\\Omega }(-K(x)\\phi _{u_{1}}u_{1}+\\left|u_{1}\\right|^{p-1}u_{1}+h(x))(u_{1}-u_{2})dx.$ Now by substituting $v=u_{2}$ in (REF ) and taking into account (REF ), we obtain $\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{2}\\right|^{2}dx-\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{1}\\right|^{2}dx &\\ge \\frac{1}{4}\\displaystyle \\int _{\\Omega }(-K(x)\\phi _{u_{1}}u_{1}+\\left|u_{1}\\right|^{p-1}u_{1}+h(x))(u_{2}-u_{1})dx \\nonumber \\\\&=\\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla (u_{2}-u_{1})dx.$ On the other hand, in view of the convexity of $\\psi $ , we infer that $\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{1}\\right|^{2}dx-\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{2}\\right|^{2}dx\\ge \\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla (u_{1}-u_{2})dx.$ Using the above pieces of informations, we obtain that $\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{2}\\right|^{2}dx-\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{1}\\right|^{2}dx= \\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla (u_{2}-u_{1})dx.$ This shows that $ \\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{2}-\\nabla u_{1}\\right|^{2}dx=0.$ Thus, $u_{2}=u_{1},$ for a.e.", "$x\\in \\Omega .$ This ends the proof.", "Since $u_{1}$ is a critical point of $I_{K}$ , then $\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla v\\right|^{2}dx-\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{1}\\right|^{2}dx\\ge \\displaystyle \\int _{\\Omega }(-K(x)\\phi _{u_{1}}u_{1}+\\left|u_{1}\\right|^{p-1}u_{1}+h(x))(v-u_{1})dx, \\ \\ \\forall v\\in K.$ Invoking assumption $(ii)$ in the theorem, we deduce that $\\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla (u_{1}-u_{2})dx=\\displaystyle \\int _{\\Omega }(-K(x)\\phi _{u_{1}}u_{1}+\\left|u_{1}\\right|^{p-1}u_{1}+h(x))(u_{1}-u_{2})dx.$ Now by substituting $v=u_{2}$ in (REF ) and taking into account (REF ), we obtain $\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{2}\\right|^{2}dx-\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{1}\\right|^{2}dx &\\ge \\frac{1}{4}\\displaystyle \\int _{\\Omega }(-K(x)\\phi _{u_{1}}u_{1}+\\left|u_{1}\\right|^{p-1}u_{1}+h(x))(u_{2}-u_{1})dx \\nonumber \\\\&=\\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla (u_{2}-u_{1})dx.$ On the other hand, in view of the convexity of $\\psi $ , we infer that $\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{1}\\right|^{2}dx-\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{2}\\right|^{2}dx\\ge \\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla (u_{1}-u_{2})dx.$ Using the above pieces of informations, we obtain that $\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{2}\\right|^{2}dx-\\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{1}\\right|^{2}dx= \\displaystyle \\int _{\\Omega }\\nabla u_{2}\\nabla (u_{2}-u_{1})dx.$ This shows that $ \\frac{1}{2}\\displaystyle \\int _{\\Omega }\\left|\\nabla u_{2}-\\nabla u_{1}\\right|^{2}dx=0.$ Thus, $u_{2}=u_{1},$ for a.e.", "$x\\in \\Omega .$ This ends the proof.", "We shall use the above theorem to prove our main result in Theorem REF .", "The convex subset $K\\subset E$ required in TheoremREF is defined as follows $K(r)=\\lbrace u\\in E: \\Vert u\\Vert _{W^{2,N}(\\Omega )}\\le r\\rbrace ,$ for some $r>0$ to be determined later.", "Lemma 0.7 Let $r>0$ be fixed.", "The set $\\lbrace u\\in E: \\Vert u\\Vert _{W^{2,N}(\\Omega )}\\le r\\rbrace ,$ is a weakly closed in $E$ .", "See [14].", "See [14].", "In the sequel, we need the following technical lemmas.", "Lemma 0.8 Let $r>0$ be fixed.", "Then, there exists $C_{1},C_{2}>0$ such that $\\Vert -K(x)\\phi _{u}u+|u|^{p-1}u+h(x)\\Vert _{L^{N}(\\Omega )}\\le C_{1}r^{3}+C_{2}r^{p}+\\Vert h\\Vert _{L^{N}(\\Omega )}, \\ \\ \\forall u\\in K(r).$ Let $u\\in K(r)$ .", "Then, using Lemmas REF and REF and Hölder's inequality, we get $\\Vert -K(x)\\phi _{u}u+|u|^{p-1}u\\Vert _{L^{N}(\\Omega )}&\\le \\Vert K\\Vert _{\\infty } \\Vert \\phi _{u}u\\Vert _{L^{N}(\\Omega )}+\\Vert |u|^{p-1}u\\Vert _{L^{N}(\\Omega )}\\\\&\\le \\Vert K\\Vert _{\\infty }\\Vert u\\Vert _{L^{2N}(\\Omega )}\\Vert \\phi _{u}\\Vert _{L^{2N}(\\Omega )}+\\Vert u\\Vert _{L^{Np}(\\Omega )}^{p}\\\\&\\le c_{1}\\Vert u\\Vert _{W^{2,N}(\\Omega )}\\Vert \\phi _{u}\\Vert _{W^{2,N}(\\Omega )}+c_{2}\\Vert u\\Vert _{W^{2,N}(\\Omega )}^{p }\\\\& \\le C_{1}\\Vert u\\Vert _{W^{2,N}(\\Omega )}^{3}+C_{2}\\Vert |u|^{p-1}u\\Vert _{L^{N}(\\Omega )}\\\\ &\\le C_{1} r^{3}+C_{2}r^{p},$ where $c_{1},c_{2},C_{1},C_{2}>0$ .", "This ends the proof.", "Let $u\\in K(r)$ .", "Then, using Lemmas REF and REF and Hölder's inequality, we get $\\Vert -K(x)\\phi _{u}u+|u|^{p-1}u\\Vert _{L^{N}(\\Omega )}&\\le \\Vert K\\Vert _{\\infty } \\Vert \\phi _{u}u\\Vert _{L^{N}(\\Omega )}+\\Vert |u|^{p-1}u\\Vert _{L^{N}(\\Omega )}\\\\&\\le \\Vert K\\Vert _{\\infty }\\Vert u\\Vert _{L^{2N}(\\Omega )}\\Vert \\phi _{u}\\Vert _{L^{2N}(\\Omega )}+\\Vert u\\Vert _{L^{Np}(\\Omega )}^{p}\\\\&\\le c_{1}\\Vert u\\Vert _{W^{2,N}(\\Omega )}\\Vert \\phi _{u}\\Vert _{W^{2,N}(\\Omega )}+c_{2}\\Vert u\\Vert _{W^{2,N}(\\Omega )}^{p }\\\\& \\le C_{1}\\Vert u\\Vert _{W^{2,N}(\\Omega )}^{3}+C_{2}\\Vert |u|^{p-1}u\\Vert _{L^{N}(\\Omega )}\\\\ &\\le C_{1} r^{3}+C_{2}r^{p},$ where $c_{1},c_{2},C_{1},C_{2}>0$ .", "This ends the proof.", "Lemma 0.9 Assume that $C_{1}$ and $C_{2}$ are given in Lemma REF .", "Then, there is $r_{1}>0$ such that $C_{1}r^{3}+C_{2}r^{p}\\le \\frac{r}{2},\\ \\ \\forall r\\in (0,r_{1}].$ Moreover, if $\\Vert h\\Vert _{L^{N}(\\Omega )}\\le \\frac{r_{1}}{2}$ , we have $C_{1}r_{1}^{3}+C_{2}r_{1}^{p}+\\Vert h\\Vert _{L^{N}(\\Omega )} \\le r_{1}.$ The proof follows by a straightforward computation.", "The proof follows by a straightforward computation.", "Lemma 0.10 Suppose that conditions of Theorem REF are fulfilled.", "Let $r_{1}$ be given in Lemma REF .", "Moreover, assume that $\\Vert h\\Vert _{L^{N}(\\Omega )}\\le \\frac{r_{1}}{2}$ .", "Then for each $u \\in K(r_{1})$ there exists $v\\in K(r_{1})$ such that $\\displaystyle \\int _{\\Omega }\\nabla v\\nabla \\varphi dx+ \\displaystyle \\int _{\\Omega }K(x)\\phi _{u}u \\varphi dx=\\displaystyle \\int _{\\Omega }\\left|u\\right|^{p-1}u \\varphi dx+\\displaystyle \\int _{\\Omega }h(x)\\varphi dx,$ for all $u\\in E$ and $\\varphi \\in H^{1}_{0}\\left(\\Omega \\right).$ In particular, $v\\in W^{2,N}(\\Omega )\\frown H^{1}_{0}(\\Omega )$ , and $-\\Delta v=-K(x)\\phi _{u}u+|u|^{p-1}u+h(x), \\ \\ \\mbox{for a.e} \\ \\ x\\in \\Omega .$ Using a standard argument, there exists $v\\in H^{1}_{0}(\\Omega )$ which satisfies (REF ).", "Since the right hand side is an element in $L^{N}(\\Omega )$ , it follows from the standard regularity results that $v\\in W^{2,N}(\\Omega )\\frown H^{1}_{0}(\\Omega )$ and (REF ) holds.", "Therefore, using Lemmas REF and REF , we deduce that $ \\Vert v\\Vert _{W^{2,N}(\\Omega )}=\\Vert \\Delta v\\Vert _{L^{N}(\\Omega )}&=\\Vert -K(x)\\phi _{u}u+|u|^{p-1}u+h(x)\\Vert _{L^{N}(\\Omega )}\\\\&\\le r_{1},$ the lemma is proven.", "Using a standard argument, there exists $v\\in H^{1}_{0}(\\Omega )$ which satisfies (REF ).", "Since the right hand side is an element in $L^{N}(\\Omega )$ , it follows from the standard regularity results that $v\\in W^{2,N}(\\Omega )\\frown H^{1}_{0}(\\Omega )$ and (REF ) holds.", "Therefore, using Lemmas REF and REF , we deduce that $ \\Vert v\\Vert _{W^{2,N}(\\Omega )}=\\Vert \\Delta v\\Vert _{L^{N}(\\Omega )}&=\\Vert -K(x)\\phi _{u}u+|u|^{p-1}u+h(x)\\Vert _{L^{N}(\\Omega )}\\\\&\\le r_{1},$ the lemma is proven.", "Proof of Theorem REF completed: Let $r_{1}>0$ be as in Lemma REF and define $K= K(r_{1}).$ We suppose that $\\Vert h\\Vert _{L^{N}(\\Omega )} \\le \\frac{r_{1}}{2}$ .", "Consider the following minimizing problem $\\beta =\\displaystyle \\inf _{u\\in E}I_{K}(u).$ Hence, by definition of $\\psi _{K}$ , we deduce that $\\beta =\\displaystyle \\inf _{u\\in K}I_{K}(u).$ On the other hand, using Lemma REF , we infer that $\\beta >-\\infty .$ Take $0< e\\in K$ .", "For $t\\in [0,1]$ , we have that $te\\in K$ and therefore $I_{K}(te)\\le t(t \\displaystyle \\int _{\\Omega }|\\nabla e|^{2}dx+t^{3}\\displaystyle \\int _{\\Omega } \\phi _{e}e^{2}dx-t^{p}\\displaystyle \\int _{\\Omega } |e|^{p+1}dx-\\displaystyle \\int _{\\Omega } h(x)e dx).$ Since $h,e>0$ , we can conclude that $\\beta <0$ .", "Now suppose that $(u_{n})$ is a sequence in $E$ such that $I_{K}(u_{n})\\rightarrow \\beta $ .", "So the sequence is bounded and we can conclude by the definition of $I_{K}$ that the sequence is bounded in $W^{2,N}(\\Omega )$ .", "Using standard results in Sobolev spaces, after passing to a subsequence if necessary, there exists $u_{1}\\in K$ such that $u_{n}\\rightharpoonup u_{1}$ in $W^{2,N}(\\Omega )$ and strongly in $E$ .", "Therefore, $\\beta =I_{K}(u_{1})<0.$ Then, by Lemma REF , we conclude that $u_{1}$ is a nontrivial critical point of $I_{K}$ .", "Now, by Lemma REF together with the fact that $u_{1}\\in K(r_{2})$ we obtain that there exists $u_{2}\\in K$ such that $-\\Delta u_{2}=-\\phi _{u_{1}}u_{1}+|u_{1}|^{p-1}u_{1}+h.$ Combining the above pieces of informations and applying Theorem REF , we conclude that $u_{1}$ is a nontrivial solution of problem (REF )." ] ]

[ [ "The Impact of a Major Security Event on an Open Source Project: The Case\n of OpenSSL" ], [ "Abstract Context: The Heartbleed vulnerability brought OpenSSL to international attention in 2014.", "The almost moribund project was a key security component in public web servers and over a billion mobile devices.", "This vulnerability led to new investments in OpenSSL.", "Objective: The goal of this study is to determine how the Heartbleed vulnerability changed the software evolution of OpenSSL.", "We study changes in vulnerabilities, code quality, project activity, and software engineering practices.", "Method: We use a mixed methods approach, collecting multiple types of quantitative data and qualitative data from web sites and an interview with a developer who worked on post-Heartbleed changes.", "We use regression discontinuity analysis to determine changes in levels and slopes of code and project activity metrics resulting from Heartbleed.", "Results: The OpenSSL project made tremendous improvements to code quality and security after Heartbleed.", "By the end of 2016, the number of commits per month had tripled, 91 vulnerabilities were found and fixed, code complexity decreased significantly, and OpenSSL obtained a CII best practices badge, certifying its use of good open source development practices.", "Conclusions: The OpenSSL project provides a model of how an open source project can adapt and improve after a security event.", "The evolution of OpenSSL shows that the number of known vulnerabilities is not a useful indicator of project security.", "A small number of vulnerabilities may simply indicate that a project does not expend much effort to finding vulnerabilities.", "This study suggests that project activity and CII badge best practices may be better indicators of code quality and security than vulnerability counts." ], [ "Introduction", "The OpenSSL project came to the attention of the world with the Heartbleed vulnerability on April 7, 2014.", "The open source cryptographic library was widely used to secure communications using the Transport Layer Security (TLS) protocol.", "An estimated 24-55% of popular web sites using TLS were exposed to Heartbleed attacks [6], which allowed attackers to remotely access private keys and passwords.", "The number of client devices impacted was much larger, as OpenSSL was the default cryptographic library for Android devices, and more than one billion Android devices shipped in 2014 [34].", "At the time of Heartbleed, the OpenSSL project was largely inactive, except for an ever-growing number of unaddressed issues.", "The project had no full time developers, and the OpenSSL Software Foundation received about $2,000 per year in donations [21].", "There were no policies for handling issues or vulnerabilities.", "OpenSSL had no release plan, and the project was still supporting version 0.9.8, which had been released in 2005.", "Project source code was complex and difficult to understand and maintain, while the project team was small with static membership [13].", "The goal of this case study is to understand how the OpenSSL project responded to the Heartbleed vulnerability.", "We build this understanding with a mixed methods approach.", "We collect qualitative data from web sites and a developer interview, and we use quantitative data in the form of software metrics, including both project activity and code metrics.", "Our choice of metrics was guided in part by Lehman's laws of software evolution [19].", "In particular, we analyze the first (continuing change) law using project activity metrics, the second law (increasing complexity), using code complexity metrics, and the sixth law (continuing growth) using code size metrics.", "We use regression discontinuity analysis to determine changes in levels and slopes of software metrics resulting from Heartbleed.", "We believe this is the first study of the impact of a major security incident on software evolution.", "Our primary contribution is an understanding of how a project can recover from a major security incident.", "We learn that the number of reported vulnerabilities is a poor indicator of security.", "Project activity and software engineering practices required by the CII best practices badge https://bestpractices.coreinfrastructure.org/ may be better indicators of project security.", "Finally, we provide a replication package that includes both the data used in this paper and the code used to collect and analyze the data [48].", "Code to build the tables and figures in this paper is also included.", "Table: OpenSSL Versions" ], [ "Context", "The first version of OpenSSL was released on December 23, 1998.", "It was based on a fork of the SSLeay project.", "The project name comes from the Secure Sockets Layer (SSL) protocol, which has been deprecated in favor of Transport Layer Security.", "OpenSSL consists of two libraries, which provide support for TLS and general cryptographic algorithms respectively, and a command line tool, openssl, which can be used for encryption, decryption, and certificate generation.", "Table REF summarizes the major versions of OpenSSL, including lifespan, minor releases, code size (number of statements), and number of vulnerabilities reported per release.", "The version number format is three digits followed by an optional letter, indicating minor releases.", "Prior to Heartbleed, OpenSSL had no release policy.", "Therefore, earlier versions have no end of life date.", "Lifespan is the period between the first and last release of a numerical version.", "Vulnerability reporting started in 2002, so earlier versions have no vulnerabilities associated with them.", "Vulnerabilities often affect multiple versions of OpenSSL, so the total number of vulnerabilities is smaller than the sum of the vulnerabilities in the table.", "Heartbleed focused intense scrutiny on OpenSSL.", "Two approaches to OpenSSL were advocated: replacement or repair.", "Two forks were created shortly after Heartbleed as potential replacements.", "Google's fork, BoringSSL [18], focused on supporting Android and Chrome.", "The OpenBSD project's fork was called LibreSSL [2] and focused on improving security and code quality.", "The repair approach was supported by the Core Infrastructure Initiative (CII), which was started by the Linux Foundation as a response to Heartbleed.", "The purpose of the CII was to fund and support open-source projects that are critical to the functioning of the Internet [7].", "OpenSSL was among the first projects funded.", "CII funded two full-time developers and a code audit.", "An additional two developers were funded by donations [13].", "CII supported the project's first face-to-face meeting in late 2014, during which project members drafted major policies, including a release strategy, coding style guide, and security policy." ], [ "Related Work", "Lehman's laws [19] provide a framework for understanding software evolution that has been widely studied.", "Previous studies have operationalized these laws terms of software metrics in multiple ways [29], [10], [20].", "We examine the first, second, and sixth laws in this work.", "Multiple studies using a variety of metrics have found that these three laws hold for most but not all open source projects studied [9], [31], [52], [14], [29].", "To study Lehman's second law, we use code complexity metrics.", "McCabe developed his cyclomatic complexity metric as a quantitative measure of which software modules are difficult to maintain or test [23].", "Halstead developed complexity metrics for similar purposes [12].", "Midha et al.", "[26] found that the number of bugs in software increased with increasing cyclomatic complexity, while Gill and Kemerer [8] found that complexity density (the ratio of cyclomatic complexity to lines of code) was a useful predictor of maintenance productivity.", "Halstead's and McCabe's complexity metrics have been used as predictors in defect prediction [24], [25] and vulnerability prediction [37] models.", "We use code size metrics to study Lehman's sixth law.", "Code size metrics have been used in multiple studies of Lehman's laws [10], [20].", "Jimenez et al.", "describe characteristics of vulnerable files in OpenSSL using code size and code complexity metrics [16], but do not use them to study software evolution.", "We compute both code size and code complexity metrics using the cqmetrics tool that Spinellis used to study the evolution of the Unix operating system [41].", "In addition to code size and complexity, we examine code style and language feature use.", "Following a consistent coding style may be an important aspect of readability and maintainability of software [30], [38].", "Programming languages like Java [30] and Python [47] have style guides, while organizations like Google  publish style guides for a variety of languages.", "Smit et al.", "[38] found that in the absence of automated style checkers, the number of style violations grows in a linear relationship with code size.", "While the idea that use of goto is harmful [5] has been widely heard by programmers, an empirical study of GitHub projects suggested that use of goto for specific purposes like error handling was considered good practice by open source developers [28].", "On the other hand, use of C preprocessor conditionals for portability across different architectures has continued to receive blame for making code difficult to read and maintain [40], [27].", "While there are many studies of open source evolution, few studies examine the impact of an external change on software evolution.", "We use a regression discontinuity design [33], [15] (RDD) to assess the impact of Heartbleed on the evolution of OpenSSL.", "This methodology has recently begun to be used in empirical software engineering analyses of time series data.", "Zhao et al.", "used RDD to evaluate the impact of adopting continuous integration on other software development practices [53], while Trockman et al.", "used RDD to test whether repository badges were reliable signals of software quality [46].", "Zimmermann and Artís evaluated the impact of switching bug trackers on a single project with a RDD model [54].", "Durumeric et al.", "measured the reaction to Heartbleed, finding that Alexa Top 100 sites patched within 48 hours, while less popular sites took longer to deploy patches [6].", "Kupsch and Miller discuss the difficulty software security tools have in finding vulnerabilities like Heartbleed [17].", "Wheeler describes software engineering practices and technologies that could find vulnerabilities like Heartbleed, including simplifying the code, fuzzing with address checking, and thorough security-focused testing [50].", "We will examine how OpenSSL adopted some of those approaches below." ], [ "Research Questions", "We begin our study by examining the security of OpenSSL.", "We examine the number and severity of vulnerability reports over time to understand how the Heartbleed vulnerability affected vulnerability reporting.", "Our first research question is: [boxsep=0mm] Research Question 1: How did the number and severity of reported vulnerabilities change after Heartbleed?", "We expect to see an increased number of vulnerability reports after Heartbleed, as more effort was devoted to finding security issues.", "Such an increase is an indicator of improving project security as vulnerabilities are found and remediated.", "OpenSSL has been under development for over two decades, a substantial span of time in which open source development has greatly changed.", "Lehman's sixth law of software evolution states that software continually grows in functionality to maintain user satisfaction [19].", "Adding new functionality typically requires adding more code to a project.", "Therefore, we assess the size of OpenSSL to see if this law holds after a major security incident.", "It is worth noting that both of the post-Heartbleed forks of OpenSSL began development by removing older cryptographic algorithms and support for a variety of computing environments [18], [2].", "Our second research question is: [boxsep=0mm] Research Question 2: How did OpenSSL change in size after Heartbleed?", "The quality of OpenSSL code before Heartbleed was perceived as poor [2].", "It was difficult to read and maintain, discouraging new contributors from working on the project [13].", "Therefore, we want to measure code quality using metrics that impact the readability and maintainability of code, such as code complexity metrics.", "Lehman's second law of software evolution states that program complexity increases over time unless work is done to prevent that [19].", "With two decades of history, there has been ample time for OpenSSL's code to increase in complexity.", "While some complexity is necessary for cryptographic code, unnecessary complexity can accumulate over time.", "The complexity of OpenSSL code was implicated as one of the causes of the Debian project accidentally breaking the OpenSSL pseudo-random number generator in 2006 [4].", "[boxsep=0mm] Research Question 3: How did the complexity of OpenSSL source code change after Heartbleed?", "As code style and the use of certain programming language features can affect the readability and maintainability of software, we also study these characteristics of the OpenSSL code base.", "We examine the consistency of stylistic choices like bracket placement and indentation, and we study the use of certain language features in our study of coding style, such as the C preprocessor and the goto statement.", "[boxsep=0mm] Research Question 4: How did the coding style of OpenSSL source code change after Heartbleed?", "Lehman's first law focuses on continuing change of a software project.", "One of the major problems with OpenSSL at the time of Heartbleed was insufficient developer activity to address technical debt.", "The project team was also small and included no full time developers, while team membership was static [13].", "The number of contributions from outside developers was small.", "There were no guidelines for contributing to the project, while the source code was difficult to understand and maintain, discouraging new contributors.", "The time for the developers to respond to issues was high, and most issues were not addressed.", "The number of open, unaddressed issues had grown steadily, reaching almost 1500 at the time of Heartbleed [36].", "To measure the change in project activity, we count the number of commits before and after Heartbleed and measure the number of commits per month.", "We also want to determine whether the OpenSSL project was able to grow its development team and accept a substantial number of outside contributions in the wake of Heartbleed, so we measure the number of authors.", "[boxsep=0mm] Research Question 5: How did the number of commits and the number of authors change after Heartbleed?", "We also study the software engineering practices that led to changes in project activity, code quality, and security.", "As the only requirements and design documents available for OpenSSL are for the forthcoming 3.0.0 release, we focus on implementation and testing activities that have been identified as best practices by the CII Best Practices Badge project .", "[boxsep=0mm] Research Question 6: How did software engineering practices change after Heartbleed?", "Finally, we want to determine if the changes in OpenSSL were sustained well after the discovery of the Heartbleed vulnerability.", "To address this question, we will not only examine trends in our code and activity metrics through the end of 2019 but we will also examine external assessments of the OpenSSL project.", "[boxsep=0mm] Research Question 7: Are the changes in OpenSSL's security, source code, and software engineering practices sustained five years after Heartbleed?" ], [ "Data", "We collected data on the OpenSSL project from a variety of sources, including the project web site, vulnerabilities list, and GitHub repository.", "We also interviewed one OpenSSL developer.", "Vulnerabilities: Vulnerability data was collected from the OpenSSL vulnerabilities list https://www.openssl.org/news/vulnerabilities.html.", "We count vulnerabilities using unique Common Vulnerabilities and Exposures (CVE) identifiers.", "Code Metrics: The source code of OpenSSL was obtained from the project's GitHub repository https://github.com/openssl/openssl.", "We used the cqmetrics package https://github.com/dspinellis/cqmetrics to compute code metrics, including size, complexity, language feature use, and style metrics.", "This open source tool was chosen in part because of its prior use in computing metrics on many versions of Unix released over several decades [41].", "For our monthly time series data, we compute code metrics on the first commit made during a month.", "Project Activity: Metrics on project activity, such as the number of authors and commits were obtained from the project's GitHub repository using PyDriller [39].", "We compute authors per month as the number of unique author names in commits made during a month.", "Software Engineering Practices: We reviewed the OpenSSL web site, mailing lists, and GitHub repository for information on changes in software engineering practices.", "We collected data on unit testing from the project GitHub repository and from test coverage data reported via coveralls.io.", "Interviews: Our interview requests received a single response from an OpenSSL team member who worked on the project immediately after Heartbleed.", "We interviewed Rich Salz of Akamai, who joined the project a couple of months after Heartbleed.", "He helped change software engineering processes, including moving to a GitHub pull request work flow, transitioning issue tracking to GitHub, and adopting continuous integration." ], [ "Methods", "We use data visualization and statistical modeling to discover changes in code metrics and project activity.", "In particular, we use a regression discontinuity design [33], [15] approach to analyze time series of code and project activity metrics.", "RDD allows us to determine whether changes in metrics occurred and to measure the effect size of those changes.", "Regression discontinuity design is a rigorous quasi-experimental approach for analyzing the casual effect of an intervention.", "It is based on the idea that in the absence of the intervention, the observed trend before the intervention would continue afterwards.", "In such a design, observations are assigned to an intervention condition based on a cutoff score.", "In time series analysis, the cutoff is a date.", "The cutoff date can be determined by visually inspecting the time series for discontinuities or by the date of a specific event.", "The effect of the intervention is estimated as a discontinuity between the intervention groups before and after the cutoff date.", "When using a linear regression model, the intervention effect can include both a change in level and a change in slope at the cutoff.", "Regression discontinuity can be performed with a global or local regression approach.", "In global regression approaches, the entire data set is used to fit the model.", "In local regression models, the model is fitted using a subset of the time series, with some data on each side on the cutoff.", "This amount is called the bandwidth.", "RDD works better with an equal amount of data before and after the cutoff.", "While the global approach offers greater precision, it includes data far from the cutoff, which may be influenced by trends other than the intervention being studied.", "Given the extensive history of OpenSSL before Heartbleed, we choose a local regression approach.", "We use the following regression discontinuity model equation to estimate changes in level and trend in code metrics after the beginning of major post-Heartbleed work: $y_i = \\alpha + \\tau D + \\beta _1 (t_i - c) + \\beta _2 D (t_i - c) + \\epsilon _i$ where D is a function that represents the discontinuity $D = {\\left\\lbrace \\begin{array}{ll}0 & t_i < c \\\\1 & t_i \\ge c \\\\\\end{array}\\right.", "}$ In the equation above, $c$ is the cutoff date, when code metrics showed substantial changes.", "The coefficient $\\alpha $ represents the level of the regression line before the cutoff, while $\\tau $ represents the change in level after the cutoff.", "The sum $\\alpha + \\tau $ is the level of the regression line after the discontinuity.", "The response variable $y_i$ is the value of a particular code metric at time $t_i$ .", "The variable $t_i$ represents time in months.", "The coefficient $\\beta _1$ is the slope of the regression line before the cutoff, while $\\beta _2$ is the change in slope after the cutoff.", "The sum $\\beta _1 + \\beta _2$ is the slope of the regression line after the cutoff.", "The variable $\\epsilon _i$ represents the error at time $t_i$ .", "In our analysis of code metrics, we build RDD models with a cutoff of February 2015.", "This cutoff was initially identified visually.", "It is clear to see in the plots of nesting depth and style inconsistency in Figure 2, which approach a step function that transitions from one interval to the other in this month.", "While Heartbleed was the initial impetus for changing OpenSSL, large changes to the code base began in February 2015.", "Investigation of the project web site and mailing lists reveal that the time between Heartbleed and our cutoff date was spent building the team, developing policies, and planning how to change the code base.", "Confirming our choice of cutoff, the OpenSSL project blog published an article about reformatting the entire code base to meet the project's new coding style guidelines in February 2015 [3].", "Additional code cleanup was performed in the following months, described in another blog entry published in July 2015 [35].", "We choose a bandwidth of 25 months on each side of the cutoff date, so that we have a sufficient number of data points for our model without including data points that are so far from the cutoff that they are influenced by factors other than Heartbleed." ], [ "Vulnerabilities", "Our first research question focuses on security vulnerabilities.", "There have been 177 vulnerabilities reported in OpenSSL through the end of 2019.", "Figure REF shows the trend of the annual number of vulnerabilities.", "The date of Heartbleed is indicated by a dashed line.", "While 66 (37.3%) vulnerabilities were reported in the approximately sixteen years before Heartbleed, 110 (62.1%) vulnerabilities were reported in the five years after Heartbleed.", "We can see three eras of vulnerability reporting for the OpenSSL project in Figure REF : the pre-Heartbleed era, the high vulnerability reporting era from 2014 to 2016, and the modern era from 2017 to the present.", "Vulnerability statistics for the three eras are detailed in Table REF .", "Figure: OpenSSL Vulnerabilities Reported by YearThe pre-Heartbleed era has the lowest number of vulnerabilities per year but with slightly higher severity scores (CVSSv2) than the other two eras.", "There is a dramatic increase in the number of vulnerabilities reported per year, from 4 to 30.33, in the high vulnerability era, with severities remaining on par with the pre-Heartbleed era.", "A majority (51.4%) of all vulnerabilities were reported in the three year period between 2014 and 2016.", "Only five of the 91 vulnerabilities found in the 2014-2016 time period were found in code written after Heartbleed, so this era produced a substantial improvement in the security of OpenSSL.", "In the modern era starting in 2017, the number of vulnerabilities reported per-year is almost twice as high as the annual count before Heartbleed, but the mean severity of vulnerabilities has declined, as has variability around that mean.", "However, the pre-Heartbleed mean number of vulnerabilities per year is computed across many more years than the mean for the modern era.", "If we restrict our view to the four years before Heartbleed, the mean number of vulnerabilities reported per-year is 7.0, which is quite close the modern era mean of 7.33.", "The mean CVSSv2 score for the four pre-Heartbleed years is 5.45 and the standard deviation is 0.84, so we still observe a decline in vulnerability severity between the late pre-Heartbleed era and the modern era.", "Table: OpenSSL Vulnerabilities" ], [ "Code Size", "In this and the following three sections, we analyze monthly time series of the software metrics shown in Figure 2.", "We compute monthly code metrics by checking out the version of the code available from the first commit made during the month and running cqmetrics on that version.", "Project activity metrics are computed using all commits during the month.", "To study RQ2, we analyze multiple code size metrics.", "Code size, both in terms of the number of files and number of statements, starts dropping shortly after Heartbleed, as can be seen in Figure REF .", "However, much larger changes begin during the code cleanup in February 2015, which we chose as the cutoff date for our RDD models.", "Table REF shows a significant decrease of 12,260 statements (10.8% of pre-cutoff size) after code cleanup began in February 2015.", "We see a similar decrease in the number of files.", "Both models were good fits as measured using the adjusted $R^2$ metric, which is given in parentheses after the model name in Table REF .", "To better understand these size changes, we examined 126 commits made in the 25 months after Heartbleed, in which the only file changes were deletions.", "A total of 315 C source and header files were deleted in those commits.", "We found multiple mentions of obsolete or old files, including support for obsolete platforms, old insecure protocols like SSLv2, and unmaintained demo code in developer commit messages of these commits.", "These messages point to the removal of files being an improvement of code quality, especially since support for old protocols exposed OpenSSL to protocol downgrade attacks like DROWN [1].", "In Figure REF , we can see code size as measured in terms of the number of files and statements decreasing from Heartbleed until about the release of version 1.1.0 in August 2016.", "Version 1.1.0 was the only major release of OpenSSL to have a smaller size at the time of release than its predecessors.", "Code sizes measured by number of statements for each version are provided in Table REF .", "Table: Code Size Metrics Regression Discontinuity ModelsAfter February 2015, the number of functions had decreased significantly by 737 (9.3%), but the number of functions being added per month was almost eight times the pre-cutoff number.", "To understand how the number of functions was increasing rapidly while the numbers of statements and files were declining, we examine mean function length.", "Function length transitions from an essentially flat pre-Heartbleed trend to a decreasing trend (-0.072 lines/month) after Heartbleed.", "This evidence suggests that the OpenSSL project began to factor their code into smaller functions after Heartbleed.", "Many programmers find that smaller functions are easier to understand [22]." ], [ "Code Complexity", "Complex code is difficult to understand and maintain, so unnecessary complexity should be avoided.", "RQ3 focuses on changes to OpenSSL's code complexity.", "We examine three types of code complexity metrics: nesting complexity, cyclomatic complexity, and Halstead complexity.", "Nesting complexity is the number of levels of nesting in source code.", "Table: Code Complexity Metrics Regression Discontinuity ModelsWe study mean and maximum values of nesting complexity and mean values of cyclomatic and Halstead complexity.", "Mean complexity metrics are computed as a mean of the per-file mean values of these metrics for each month in the dataset.", "We construct four RDD models, which are summarized in Table REF .", "Figure REF shows both data and model fit for these four metrics.", "The maximum depth of nesting in OpenSSL source files dropped suddenly from 13 to 7 in February 2015.", "This change affected 94 files, when the OpenSSL project reformatted its code base [3].", "The RDD model of mean nesting depth shows a drop of approximately half a level from 1.29 to 0.79 in February 2015.", "Mean cyclomatic complexity dropped by 7.3% after February 2015.", "The trend of cyclomatic complexity evolution changes from a small increasing slope before intervention to a larger decreasing slope after the cutoff.", "The evolution of Halstead complexity follows the same pattern, with a drop of 6.6% and a similar change in slope.", "These changes suggest that the post-Heartbleed code cleanup was effective at reducing the complexity of OpenSSL code and thus improving readability and maintainability." ], [ "Coding Style", "RQ4 focuses on coding style.", "Poor coding style can make code difficult to understand and maintain, increasing the likelihood of programmer errors and decreasing the likelihood of attracting contributors to the project.", "Prior to the Heartbleed vulnerability, the OpenSSL project had no coding style guide, and different sections of code used a range of different styles [3].", "In January 2015, the project published a coding style guide [42].", "In early February 2015, project members reformatted the source code of the versions supported at the time (0.9.8 through 1.0.2) to ensure style consistency throughout the project.", "Table: Code Style and Language Feature Regression Discontinuity ModelsThe cqmetrics tool computes a coding style inconsistency metric based on $n = 19$ style rules selected from sources such as the Google, FreeBSD, and GNU coding style documents [41].", "For each way to format a particular C construct, such as placing a space after the while keyword, cqmetrics computes the sum of how many times the formatting rule is followed, $a_i$ , and the sum of how many times the rule is not followed, $b_i$ .", "The style inconsistency is the ratio of the smaller of two sums with the total times the rule could be applied in either way.", "$SI = \\frac{ \\sum ^{n}_{i=1} min(a_i, b_i) }{ \\sum ^{n}_{i=1} a_i + b_i }$ This style inconsistency metric shows a dramatic drop immediately after the reformatting, from 57.045 to 4.088, as can be seen in Figure REF .", "The slope prior to reformatting was slightly downwards, starting in mid-2014 after Heartbleed, while the slope after reformatting is slowly upwards, showing a gradual increase in deviations from the coding style as new contributions are made.", "As part of coding style, we examine the use of both goto and the C preprocessor.", "The density of goto statements does not change substantially after the code reformatting and cleanup in 2015.", "However, the density of preprocessor statements decreases by about 10% from 0.168 to 0.152 at the same point, with potentially problematic conditional preprocessor statements decreasing 23% from 0.014 to 0.011.", "The substantial improvements in coding style consistency and reduction in C preprocessor use described above suggest that the OpenSSL project focused on improving the readability and maintainability of their code after Heartbleed." ], [ "Project Activity", "RQ5 addresses project activity.", "The OpenSSL project has become much more active since the discovery of Heartbleed.", "Contributors made 13,238 git commits (52.7%) in the 68 months after Heartbleed.", "Only 11,905 commits (47.3%) were made in the 197 months before Heartbleed.", "The difference in the number of commit authors is even larger, with only 51 commit authors (8.8%) before Heartbleed and 554 authors (96.0%) after Heartbleed.", "Note that the percentages do not sum to 100%, as a small subset of authors were active both before and after Heartbleed.", "The tenfold growth in the number of unique authors does not equate to a tenfold growth in the number of commits after Heartbleed.", "This can be explained in part by the substantial number of authors who contributed only a single commit.", "Only ten (19.6%) of the 51 pre-Heartbleed authors contributed just one commit.", "After Heartbleed, 331 (59.7%) of the 554 authors contributed only a single commit.", "The large growth in both total and single commit authors suggests that the OpenSSL project has become much more attractive to outside contributors after Heartbleed.", "In addition to looking at total numbers of commits and authors, we perform a time series analysis of the number of commits and the number of authors per month.", "We compute the number of authors per month by counting the number of unique authors who appear in git commit author fields during a month.", "Authors are identified by name, not by e-mail address.", "We fit RDD models to both time series.", "Project activity begins changing immediately after Heartbleed, so we use the month of Heartbleed (April 2014) as the cutoff date.", "Due to the high variance in both of these metrics after Heartbleed, we use a bandwidth of 50 months on each side of the Heartbleed month rather than 25 months used above.", "This high variance is reflected in the lower adjusted $R^2$ values for these models compared to most of our previous models, as shown in Table REF .", "Figure REF shows the model fit and data points.", "While project activity occurs in spurts, there are clear and substantial changes in activity after Heartbleed.", "The change in number of commits per month (122.44) is much larger than the number of monthly commits before Heartbleed (46.22).", "Similarly, the change in number of unique authors per month (13.65) is much larger than the number of monthly authors before Heartbleed (7.06).", "Table: Project Activity Regression Discontinuity Models" ], [ "Software Engineering Practices", "RQ6 focuses on OpenSSL's software engineering practices.", "In June 2014, shortly after Heartbleed, the OpenSSL project published a project roadmap [43], identifying current issues, objectives, and forthcoming features.", "Project issues included a backlog of bug reports, some of which had been open for years, incomplete and incorrect documentation, code complexity, inconsistent coding style, a lack of code review, and the absence of a release strategy and a security policy.", "The developer we interviewed indicated that the OpenSSL team did not use code metrics to direct their software engineering efforts.", "The CII began funding OpenSSL in 2014, enabling the developer team to grow rapidly.", "By December 2014, team size had increased from two main developers to fifteen project members and four full time funded developers [13].", "The project began formalizing decision making and published a vulnerability handling policy.", "Project development moved to GitHub.", "According to the developer we interviewed, the motivation for using GitHub was to increase transparency and attract more developers.", "OpenSSL published its first release strategy in December 2014, establishing end of life dates and planning for future versions.", "Version 1.0.2 was created as a long term support release with a backwards compatible API, while version 1.1.0 was planned to improve design and code while breaking compatibility with earlier versions.", "The project published bylaws in February 2017, describing project roles, including the OpenSSL Management Committee (OMC) and Committers [44].", "Starting in July 2014, the OpenSSL project required that code submissions be reviewed and approved by a core team member.", "The updated 2017 committer policy required that all code submissions to be reviewed and approved by at least two committers, one of whom must be an OMC member [45].", "The CII's Best Practices Badge project provides a guide to good open source development practices.", "The badge project was established soon after Heartbleed to provide a method for open source developers to certify that their projects follow best practices.", "Attainment of a badge requires meeting 66 criteria in six categories: basics, change control, reporting, quality, security, and analysis .", "Prior to the changes made in response to Heartbleed, OpenSSL had completed 62% of badge requirements.", "The OpenSSL project attained its CII badge in February 2016, by enabling TLS for its web site, protecting downloads of OpenSSL with TLS, publishing processes for reporting vulnerabilities and contributing code, using static and dynamic analysis before public releases, and using continuous integration.", "The OpenSSL project began using Travis for continuous integration in August 2015.", "In additional to building the software and performing unit tests, OpenSSL's Travis configuration reports test coverage to coveralls.io and runs the flake8 static analysis tool on the library's python scripts.", "Continuous integration has been shown to change other software development practices, such as code contribution processes, issue handling, and testing [53].", "After adopting continuous integration, OpenSSL changed in all of those areas, adopting a new code contribution policy, migrating issue tracking from Request Tracker to GitHub [36], and increasing unit testing.", "Figure: OpenSSL Unit Tests by VersionOpenSSL testing practices changed considerably after Heartbleed.", "The project adopted a new framework to make it easier to write unit tests.", "The number of unit tests increased from 42 in version 1.0.1 (the most recent version as of Heartbleed) to 152 in 1.1.1 (the current version).", "Figure REF displays the growth in unit tests by major version, including the trend for the number of tests at the time of initial release and the number of tests for the most recent release.", "It is important to note that the time between initial and final release of a version is much larger for older versions.", "Version 0.9.8 adds three tests from first to final release, which happened over the course of ten years and 35 minor releases.", "Version 1.1.1 also adds three tests, but over a much shorter time period: three minor releases and a single year.", "Modern versions (1.1.0 and 1.1.1) added approximately one test per minor release, while older versions rarely added tests for minor releases.", "The OpenSSL project began measuring test coverage in 2016.", "Test coverage grew from 54.6% in 2016 to 64.2% in 2019 as measured by coveralls.io.", "There is no coverage data prior to 2016, but the increase in the number of unit tests from the last pre-Heartbleed version (1.0.1), which had 47 unit tests, to the version released in 2016 (1.1.0), which had 85 unit tests on initial release, suggests that code coverage was lower before test coverage statistics were collected.", "OpenSSL has incorporated fuzz testing in its unit tests, and Google's OSS Fuzz project https://google.github.io/oss-fuzz/ also tests OpenSSL.", "While only sixteen OpenSSL vulnerabilities identify the technique or tool used to find the vulnerability, all sixteen vulnerabilities identify fuzz testing as the technique.", "Nine of the sixteen identify OSS Fuzz as the tool used, while five identify libFuzzer and two identify TLS-Attacker.", "The OpenSSL project's code is regularly scanned by a security-oriented static analysis tool as part of the Coverity Scan project https://scan.coverity.com/.", "When OpenSSL was initially scanned in 2006, commit messages reported fixing several bugs reported by Coverity.", "Coverity related commit messages continue through January 2, 2009.", "The next mention of Coverity in a commit message occurs on May 5, 2014, a month after Heartbleed.", "Similar commit messages continue through the end of 2019, so static analysis has continued to be used since 2014.", "Our interviewee indicated that Coverity scan results are primarily used when preparing a new release of OpenSSL." ], [ "Sustainability of Changes", "Our final research question, RQ7, focuses on the durability of the changes made in response to Heartbleed.", "Improvements in the OpenSSL project during the immediate post-Heartbleed era can be seen in the project's attainment of a CII Best Practices badge.", "The badge score improved from 62% to 105%.", "While the project has not updated its badge application since 2016, we were able to use the application to manually verify that OpenSSL has continued to follow the best practices identified in its application, including continuous integration, static analysis, and fuzz testing.", "When we examine the evolution of code metrics, we find that most improvements have been sustained.", "Code complexity, as measured by nesting depth, cyclomatic complexity, and Halstead complexity, has continued to decline since 2016.", "The maximum nesting depth remains at the level it dropped to after the code cleanup.", "Style inconsistency has begun to grow at a slow rate, but the December 2019 value of 7.09 is still far below the April 2014 value of 58.9.", "Growth of the code base resumed in 2016, as features like TLS 1.3 were added, showing that code shrinkage inspired by Heartbleed was a temporary change in software evolution.", "Figure REF shows the post-Heartbleed evolution of a sample of metrics, including code size in statements, mean nesting complexity, mean cyclomatic complexity, and style inconsistency.", "Figure: Four Code Metrics after HeartbleedThe security audit commissioned by the CII was performed on version 1.1.0 of OpenSSL in 2016 by the Open Crypto Audit Project (OCAP) [51].", "The audit included both code review and fuzz testing.", "The audit reported several potential bugs in OpenSSL, including two possible code execution vulnerabilities and two possible denial of service (DoS) vulnerabilities.", "Other issues were reported by the audit team as low severity or difficult to exploit.", "Only one OpenSSL vulnerability report identifies the OCAP audit team as the reporter.", "The Open Source Technology Improvement Fund (OSTIF) performed an audit of OpenSSL 1.1.1, with a focus on the new TLS 1.3 protocol and changes made to the Pseudo Random Number Generator (PRNG) [32].", "Their report was published in 2019.", "The OSTIF audit combined manual code review with fuzz testing.", "OSTIF found two DoS vulnerabilities prior to the release date, enabling the OpenSSL team to fix them before release.", "The report identified some areas where code quality could be improved by checking return values and implementing global checks for NULL values." ], [ "Threats to Validity", "Construct Validity: We use multiple widely used metrics for measuring code size and complexity to avoid bias in measuring those code characteristics.", "The style inconsistency metric we use has been used in other studies too [41].", "Internal Validity: One of the most important validity threats to RDD time series analysis is the presence of an event near the cutoff date that influences the observed changes.", "We mitigated this threat by thoroughly examining OpenSSL blog entries, commit messages, and email archives in the months before and after our cutoff dates for such events.", "Another threat to internal validity is that only one OpenSSL developer consented to be interviewed, which may bias our qualitative data.", "External Validity: As this work is a case study of a single project, we cannot generalize our conclusions to other open source projects.", "We therefore leave the question of how open source projects can successfully react to security events to future work." ], [ "Conclusions", "The Heartbleed vulnerability brought dramatic changes to OpenSSL, transforming an almost moribund project to an active project with substantial improvements in code quality and security.", "OpenSSL remains the most commonly used TLS library on public web servers five years after Heartbleed, according to IPv4 scan data collected by censys.io.", "These improvements provide a model for how open source projects can adapt and improve after a major security event.", "We found substantial and sustained improvements in code quality.", "Code complexity declined sharply during the major code cleanup activity in 2015, and both cyclomatic and Halstead complexity have continued to decline.", "The code cleanup made coding style much more consistent.", "While style inconsistency has slowly increased since the cleanup, it remains much lower than before Heartbleed.", "The number of vulnerability reports dramatically increased for three years after Heartbleed before returning to previous levels.", "Only five of the 91 vulnerabilities found in those three years were in post-Heartbleed code, so this represents a substantial improvement in security.", "Positive results from two external code audits also suggest that the security of OpenSSL has greatly improved.", "This means that vulnerability count is not a useful indicator of project security.", "Low vulnerability counts may just indicate that a project is devoting little effort to finding vulnerabilities.", "Our results suggest that project activity and practices may be better predictors of project security.", "The CII badge may be a good indicator of project security, since it requires good open source development practices.", "To understand how generalizable these recommendations are, we plan to compare the software evolution of OpenSSL with that of related projects, such as BoringSSL and GnuTLS.", "We also plan to examine the impact of the CII badge and project activity on the development practices and security across multiple open source projects to validate these ideas.", "We have provided a replication package for this paper [48], which includes the software metrics and project activity data used in this paper.", "The package includes data collection scripts and the code used to create the models and generate the figures and tables for this paper.", "Documentation on how to use the scripts is also provided." ] ]

[ [ "Quantum gravity on polygons and $\\Bbb R\\times \\Bbb Z_n$ FLRW model" ], [ "Abstract We fully solve the quantum geometry of $\\Bbb Z_n$ as a polygon graph with arbitrary metric lengths on the edges, finding a $*$-preserving quantum Levi-Civita connection which is unique for $n\\ne 4$.", "As a first application, we numerically compute correlation functions for Euclideanised quantum gravity on $\\Bbb Z_n$ for small $n$.", "We then study an FLRW model on $\\Bbb R\\times\\Bbb Z_n$, finding the same expansion rate as for the classical flat FLRW model in 1+2 dimensions.", "We also look at particle creation on $\\Bbb R\\times \\Bbb Z_n$ and find an additional $m=0$ adiabatic no particle creation expansion as well as the particle creation spectrum for a smoothed step expansion." ], [ "Introduction", "This paper is the third in a sequence[1], [2] in which we apply the recently developed formalism of `quantum Riemannian geometry'[3], [4], [5], [6], [7] to quantum gravity on discrete spacetimes, as a new approach compared to previous lattice and other discrete schemes [8], [9], [10], [11].", "It is also very different from loop quantum cosmology such as [9] but has in common the idea of hugely simplifying the gravitational degrees of freedom to the point of something calculable.", "In the first of the two papers, we solved quantum gravity on a square[2] but with differential structure given by the group ${\\mathbb {Z}}_2\\times {\\mathbb {Z}}_2$ , which is different from the case of a polygon based on the group ${\\mathbb {Z}}_n$ that we treat now, even when $n=4$ .", "The metric data in our approach consists of generic `square length' data assigned to each edge of the polygon, while the group structure affects the choice of 2-forms and higher forms, but also provides a natural basis for the 1-forms in the same way that one implicitly fixes a differential structure classically by use of translation-invariant differentials ${\\rm d}x^\\mu $ .", "In [1], we similarly set up but could not solve quantum gravity on the lattice line ${\\mathbb {Z}}$ , but did manage computations for cosmological particle creation on a universe with only a lattice time direction.", "The connection and Einstein-Hilbert action now on ${\\mathbb {Z}}_n$ will turn out, which is the main result of Section REF , to be a modulo $n$ version of the ones for ${\\mathbb {Z}}$ .", "This is not surprising but required significant proof and in fact turns out to be not the only possibility for $n=4$ .", "The main difference, however, is that now, for finite $n$ depending only on computing power, Euclidean quantum gravity is fully computable, as we demonstrate in Section REF for $n\\le 6$ .", "We switched here to a Euclidean interpretation since a periodic time direction makes no sense physically, but we note that Euclidean quantum gravity is still of interest for classical compact Riemannian manifolds without boundary, see [12].", "Also note that the overall normalisation of the metric does not enter into the connection $\\nabla $ nor into the quantum gravity action, but from a positivity point of view, since our basis of invariant one-forms obey $e^+{}^*=-e^-$ , the inverse metric in Section REF is actually negative definite in the sense $(e^+,e^+{}^*)=-{1\\over a}<0$ for the associated hermitian metric evaluated on the diagonal and with `square-length' function $a>0$ .", "Section REF looks carefully at the $n\\rightarrow \\infty $ limit of the polygon and identifies it as a central extension in the sense of [13], [14] of the classical calculus on a circle, with an extra `normal' direction $\\Theta _0$ .", "Our second main result is then a detailed study of the FLRW model on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ with ${\\mathbb {R}}$ classical, including cosmological particle creation following the approach of Parker [15], [16], [17], [18].", "This is an important part of quantum theory on curved spacetime and relates also to Bekenstein-Hawking radiation, see [19], [20].", "We first find in Section REF that quantum metrics on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ are forced to have the block form $g=\\mu {\\rm d}t\\otimes {\\rm d}t+ h_{ab}e^a\\otimes e^b$ and, moreover, $h_{ab}$ has to have a specific form where the time dependence enters uniformly in the spatial metric.", "Section REF focusses on the FLRW cosmology case of a uniform metric on ${\\mathbb {Z}}_n$ expanded by a time-dependent factor $R(t)$ , so $ g=-{\\rm d}t\\otimes {\\rm d}t- R^2(t)(e^+\\otimes e^-+e^-\\otimes e^+).$ The negative sign in the second term is required by our positivity remark above and we then find that the Friedmann equations for $R(t)$ in our discrete case actually come out the same as for the usual flat FLRW model in two spatial dimensions, which is in line with our cotangent bundle on ${\\mathbb {Z}}_n$ being necessarily 2-dimensional, not 1-dimensional.", "Section REF provides some elementary checks for QFT in the constant $R$ case, then Section REF covers the cosmological particle creation for varying $R(t)$ .", "We first consider the classical geometry case of ${\\mathbb {R}}\\times S^1$ , which sets up the formalism and for which we did not find a suitable treatment elsewhere, then the modifications for ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ .", "Of interest are the adiabatic no particle creation possibilities for $R(t)$ aside from the obvious constant $R$ case; for ${\\mathbb {R}}\\times S^1$ there is a further possibility with $m\\rightarrow \\infty $ , but for ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ we find a second further possibility with $m\\rightarrow 0$ .", "The particle creation calculation itself is done only for `in' and `out' regimes of constant $R$ , with results a little different in the ${\\mathbb {Z}}_n$ case due to the periodic nature of the spatial momentum compared to the $S^1$ case, see Figure REF .", "Since this is the third in a sequence of papers, we keep the remaining general remarks, as well as the recap of the formalism in the preliminaries Section , to a minimum.", "Suffice it to say that our particular approach to discrete quantum gravity has its roots in quantum spacetime or the idea that space and time coordinates are noncommutative or `quantum'.", "This was speculated on since the early days of quantum theory but has also emerged by now as a better-than-classical effective theory that includes some quantum gravity effects.", "It was first discussed in modern times in [21] in the context of non-commutativity of phase space and quantum Born reciprocity or observable-state duality, where it led to the bicrossproduct class of quantum groups (rather different from the other main class, the $q$ -deformation ones, arising from integrable systems).", "An important model here was the bicrossproduct model Minkowski spacetime $[x_i,t]=\\imath \\lambda _p x_i$ in [22], with quantum group Poincaré symmetry having a bicrossproduct form (as well as a construction[23] by contraction from $U_q(so_{3,2})$ ).", "Other early works were [24], which did not itself propose a closed spacetime algebra, its adaptation [25] with classical (not quantum) symmetry and the proposal [26] of the angular momentum algebra as a quantum spacetime.", "We refer to [2] for more details and literature.", "What is important is that, as argued back in [21], a true theory of quantum gravity effects on spacetime also needed models where the spacetime (and indeed the entire position-momentum space) was both curved and quantum, and for this one needed an actual formalism for that.", "What emerged, somewhat different in character from `noncommutative geometry à la Connes'[27] coming out of cyclic cohomology, K-theory and `spectral triples' as abstract Dirac operators, was a more constructive `quantum groups approach' motivated by quantum groups and their homogeneous spaces as examples but ultimately working for any algebra $A$ equipped with differential structure, over any field.", "The starting point here is to specify the differential structure as a bimodule $\\Omega ^1$ of `1-forms' (this means we can multiply them from either side by elements of $A$ ) equipped with an exterior derivative ${\\rm d}: A\\rightarrow \\Omega ^1$ obeying the Leibniz rule.", "We then define a metric as an invertible element $g\\in \\Omega ^1\\otimes _A\\Omega ^1$ with some kind of symmetry condition and a quantum Levi-Civita connection (QLC) in these terms is a bimodule connection[28], [29] $\\nabla :\\Omega ^1\\rightarrow \\Omega ^1\\otimes _A\\Omega ^1$ which is metric compatible and torsion free.", "For each quantum Riemannian geometry, one can compute a Laplacian $\\Delta =(\\ ,\\ )\\nabla {\\rm d}: A\\rightarrow A$ and, with a little more `lifting' data, a Ricci tensor in $\\Omega ^1\\otimes _A\\Omega ^1$ and a Ricci scalar $S\\in A$ , see Section  and [3] for more details and references.", "Along with this new formalism has come a new generation of examples.", "Notably, the above bicrossproduct model spacetime in 1+1 dimensions turned out [5], [6] from this point of view to admit two main classes of translation invariant 2D differential structures and each of these to admit a 1-parameter moduli of curved quantum Riemannian geometries.", "Finally, which is the critical thing for discrete quantum gravity, this emergent quantum Riemannian geometry, since it works for any algebra, can just as well be applied to functions on a discrete set.", "Here the algebra is commutative but it turns out that differentials $\\Omega ^1$ on this algebra are the same thing as graphs with the given set as vertices, and cannot commute with functions for consistency of the Leibniz rule.", "So this is a very different regime from deformation-type quantum spacetimes, but the thinking is the same; a better model of spacetime that includes some quantum gravity effects, reflected now in discrete positions and noncommutative differentials.", "More details are in Section  and [1], [2], [3], [4].", "The algebra $A$ can also perfectly well be finite-dimensional, so in our case this immediately includes calculable `baby quantum gravity' models where spacetime is a small graph.", "What is critical for this to have any validity is that the theory is not ad-hoc to the discrete case but simply one extreme end of a single functorial framework that include continuum geometries and their deformations at the other end.", "Section  concludes with some directions for further work.", "We work in units with $\\hbar =c=1$ ." ], [ "Preliminaries", "As mentioned, it is important that we have a single formalism that includes both classical and discrete cases as well as the mixed case of ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ needed in the paper.", "For each layer of Riemannian geometry, we briefly recall the general set up over a unital algebra $A$ as in [3], for orientation purposes, then give details for the discrete graph case where $A$ is functions on a discrete set, which was also the setting of [1], [2]." ], [ "Differentials and metrics", "As explained in the introduction, the first step is a graded exterior algebra $(\\Omega ,{\\rm d})$ where $\\Omega ^0=A$ is the algebra of `functions' and ${\\rm d}$ increases the differential form degree by 1, obeys a graded-Leibniz rule and ${\\rm d}^2=0$ .", "We also require $\\Omega $ to be generated by $A,{\\rm d}A$ .", "If one fixes $\\Omega ^1$ first then there is a unique `maximal prolongation' $\\Omega $ of which one can chose a quotient if one wants.", "In our case, we will be interested in the commutative algebra $A=C(X)$ of complex functions on a discrete set $X$ with pointwise product.", "Then choosing $\\Omega ^1$ is equivalent to assigning arrows to make a graph with vertex set $X$ .", "Denoting a vector space basis of $\\Omega ^1$ by $\\lbrace \\omega _{x\\rightarrow y}\\rbrace $ labelled by arrows $x\\rightarrow y$ , the bimodule products and exterior derivative are $ f.\\omega _{x\\rightarrow y}=f(x)\\omega _{x\\rightarrow y},\\quad \\omega _{x\\rightarrow y}.f=f(y)\\omega _{x\\rightarrow y},\\quad {\\rm d}f=\\sum _{x\\rightarrow y}(f(y)-f(x))\\omega _{x\\rightarrow y}.$ We will be interested in the case where the graph is bidirected i.e., for every arrow $x\\rightarrow y$ there is an arrow $y\\rightarrow x$ .", "In other words, the data is just a usual undirected graph which we understand as arrows both ways in the above formulae.", "A metric as a tensor $g\\in \\Omega ^1\\otimes _A\\Omega ^1$ then has the form $ g=\\sum _{x\\rightarrow y}g_{x\\rightarrow y}\\omega _{x\\rightarrow y}\\otimes \\omega _{y\\rightarrow x}\\in \\Omega ^1\\otimes _{C(X)}\\Omega ^1$ for nonzero weights $g_{x\\rightarrow y}$ for every edge, as is dictated by being central [3] [4].", "Here a term $\\omega _{x\\rightarrow y}\\otimes \\omega _{y^{\\prime }\\rightarrow x^{\\prime }}$ needs $y^{\\prime }=y$ to be nonzero since we could left-multiply the second factor by $\\delta _{y^{\\prime }}$ which does not change it, or move $\\delta _{y^{\\prime }}$ over to the first factor where it acts from the right by $\\delta _{y^{\\prime },y}$ .", "And it needs $x^{\\prime }=x$ to be central so that right multiplication by $\\delta _{x^{\\prime }}$ on the second factor, which does not change it, can coincide with left-multiplication on the first factor, which gives $\\delta _{x^{\\prime },x}$ .", "That centrality is needed for bimodule invertibility was a key result in [5].", "Canonically, a metric is `quantum symmetric' if $\\wedge (g)=0$ for the wedge product of $\\Omega $ .", "Specific to graphs, we also have a slightly different notion that $g$ is edge-symmetric if $g_{x\\rightarrow y}=g_{y\\rightarrow x}$ for all $x\\rightarrow y$ , i.e., does not depend on the direction of travel.", "As in [1] for the line graph, we will see that this variant also works better when we apply it to the polygon.", "Next, it is useful to endow $X$ with a group structure and look for $\\Omega ^1$ which is left and right translation invariant.", "These will be the Cayley graph for an Ad-stable set of generators ${\\mathcal {C}}\\subseteq {\\mathbb {G}}\\setminus \\lbrace e\\rbrace $ (where $e$ is the identity of a group ${\\mathbb {G}}$ ), with arrows of the form $x\\rightarrow xa$ for $a\\in {\\mathcal {C}}$ .", "In this case, one has a basis of invariant 1-forms $ e^a=\\sum _{x\\rightarrow x a} \\omega _{x\\rightarrow xa}$ with $\\Omega ^1=A.\\lbrace e^a\\rbrace $ , bimodule relations and derivative $ e^a f= R_a(f)e^a,\\quad {\\rm d}f=\\sum _a ({\\partial }_a f)e^a,\\quad {\\partial }_a=R_a-{\\rm id},\\quad R_a(f)(x)=f(xa)$ defined by the right translation operators $R_a$ as stated.", "These formulae now makes sense even when $X$ is infinite as long as ${\\mathcal {C}}$ is finite.", "Moreover $\\Omega $ is canonically generated by functions and basic 1-forms with the above as well as certain `braided-anticommutation relations' between the $\\lbrace e^a\\rbrace $ .", "In the case of an Abelian group (which is all we will need), this is just the usual Grassmann algebra on the $e^a$ , i.e., they anticommute and we also have ${\\rm d}e_a=0$ in this case." ], [ "Connections", "A connection in quantum Riemannian geometry is a map $\\nabla :\\Omega ^1\\rightarrow \\Omega ^1\\otimes _A\\Omega ^1$ .", "Given a quantum vector field in the form of a right module map $X:\\Omega ^1\\rightarrow A$ , we can evaluate this against the first output to obtain a covariant derivative $\\nabla _X:\\Omega ^1\\rightarrow \\Omega ^1$ , but the connection itself is defined independently of any vector field.", "It is required to obey two Leibniz rules as follows.", "From the left, we ask for $ \\nabla (a\\omega )={\\rm d}a\\otimes \\omega + a\\nabla \\omega $ for all $a\\in A,\\omega \\in \\Omega ^1$ .", "From the right, we similarly ask[28], [29] for $ \\nabla (\\omega a)=(\\nabla \\omega )a+\\sigma (\\omega \\otimes {\\rm d}a);\\quad \\sigma :\\Omega ^1\\otimes _A\\Omega ^1\\rightarrow \\Omega ^1\\otimes _A\\Omega ^1$ for some `bimodule map' $\\sigma $ (i.e.", "commuting with the action of $A$ from either side, so `tensorial' in a strong sense.)", "In the case we need of a Cayley graph calculus on a group, we see that $\\nabla $ just needs to be specified on the $e^a$ provided this is consistent with its extension to $\\Omega ^1$ by the two Leibniz rules.", "We write $ \\nabla e^a=-\\Gamma ^a{}_{bc}e^b\\otimes e^c,\\quad \\sigma (e^a\\otimes e^b)=\\sigma ^{ab}{}_{mn}e^m\\otimes e^n$ for coefficients in $A$ with a certain compatibility between these tensors for a bimodule connection.", "In general, torsion-free amounts to $\\wedge \\nabla -{\\rm d}=0$ as maps from $\\Omega ^1\\rightarrow \\Omega ^2$ and needs in the case of an abelian group the additional relations $ \\Gamma ^a{}_{bc}=\\Gamma ^a{}_{cb},\\quad \\sigma ^{ab}{}_{mn}e^m\\wedge e^n+ e^a\\wedge e^b=0.$ Next, any bimodule connection extends canonically to a connection on tensor products.", "This implies a meaning to $\\nabla g=0$ , namely if $g=g^1\\otimes g^2$ , say, then this is $ \\nabla g^1\\otimes g^2+(\\sigma (g^1\\otimes (\\ ))\\otimes {\\rm id})\\nabla g^2=0.$ In the discrete Cayley graph setting, we write $g=h_{ab}e^a\\otimes e^b$ , where centrality needs $ h_{ab}=\\delta _{a^{-1},b}h_a$ for some functions $h_a$ .", "In these terms, (i) edge symmetry and, in the case of the Grassmann algebra, quantum symmetry (ii) appear as $ (i)\\quad h_a=R_a(h_{a^{-1}}),\\quad (ii)\\quad h_a=h_{a^{-1}}.$" ], [ "*-structures, inner calculi and structure constants", "For physics, there should also be a $*$ -involution on $A$ , which in our examples is just pointwise complex conjugation, and everything should be unitary or `real' in the sense of $*$ -preserving.", "We require this to extend to $\\Omega $ with an extra minus signs for swapping two odd elements and to commute with ${\\rm d}$ .", "For the metric and connection, `reality' means $ g^\\dagger =g,\\quad \\nabla \\circ *= \\sigma \\circ \\dagger \\circ \\nabla , $ which also implies $\\dagger \\circ \\sigma =\\sigma ^{-1}\\circ \\dagger $ with $(\\omega \\otimes \\eta )^\\dagger =\\eta ^*\\otimes \\omega ^*$ for $\\omega ,\\eta \\in \\Omega ^1$ .", "In the Cayley graph case, $e^a{}^*=-e^{a^{-1}}$ is the natural choice.", "Then reality of the metric is $\\overline{h_{ab}}=h_{b^{-1},a^{-1}}$ , which means the metric functions $h_a$ are real valued.", "For $\\Gamma $ , the formula depends on $\\sigma $ and is more complicated.", "Finally, when the calculus is inner in the sense of a 1-form $\\Theta $ which generates ${\\rm d}$ by graded-commutator ${\\rm d}=[\\Theta ,\\ \\rbrace $ , it is shown in [4] that $ \\nabla \\omega = \\Theta \\otimes \\omega + (\\alpha -\\sigma _\\Theta )(\\omega )$ for some bimodule map $\\alpha :\\Omega ^1\\rightarrow \\Omega ^1\\otimes _A\\Omega ^1$ and some bimodule map $\\sigma $ , with $\\sigma _\\Theta =\\sigma ((\\ )\\otimes \\Theta )$ .", "To be torsion free, we require the condition on $\\sigma $ as above and $\\wedge \\alpha =0$ .", "To be metric compatible, we need $ \\Theta \\otimes g+(\\alpha \\otimes {\\rm id})g+(\\sigma \\otimes {\\rm id})({\\rm id}\\otimes (\\alpha -\\sigma _\\Theta ))g=0.$ To be `real', we need the condition on $\\sigma $ above and $\\alpha \\circ *=\\sigma \\circ \\dagger \\circ \\alpha $ .", "A Cayley graph calculus is inner with $\\Theta =\\sum _a e^a$ .", "In this case, to be bimodule maps, we need $\\sigma ^{ab}{}_{mn}=0$ unless $ab=mn$ in the group and $\\alpha (e^a)=\\alpha ^a{}_{mn}e^m\\otimes e^n$ needs $\\alpha ^a{}_{mn}=0$ unless $a=mn$ in the group, see [3][4].", "The indices here range over elements of the generating set ${\\mathcal {C}}$ of the calculus and are not being multiplied in the 4-index and 3-index tensors $\\sigma ^{ab}{}_{mn},\\alpha ^a{}_{mn}$ .", "We will need this a little more explicitly than currently in the literature.", "Lemma 2.1 Let $\\Omega ({\\mathbb {G}})$ be a Cayley graph calculus and cf.", "[3], [4], write a bimodule connection on $\\Omega ^1$ in the form $ \\sigma ^{ab}{}_{mn}=\\delta ^a{}_n\\delta ^b{}_m+\\delta ^b{}_{a^{-1}mn}\\tau ^{a}{}_{mn},\\quad \\Gamma ^a{}_{bc}=\\tau ^a{}_{bc}-\\delta ^a{}_{bc}\\alpha _{bc}$ for coefficient functions $\\tau ^a{}_{bc}=0$ unless $a^{-1}bc\\in {\\mathcal {C}}$ and $\\alpha _{bc}=0$ unless $bc\\in {\\mathcal {C}}$ .", "(1) For ${\\mathbb {G}}$ abelian, the condition for torsion freeness is $\\tau ^a{}_{bc},\\alpha _{bc}$ symmetric in $b,c$ .", "(2) The conditions for `reality' of the connection (to be $*$ -preserving) are $ \\alpha _{bc}+R_{bc}(\\overline{\\alpha _{c^{-1}b^{-1}}}) +\\sum _n R_{nbcn^{-1}}(\\overline{\\alpha _{c^{-1}b^{-1}n^{-1},n}})\\tau ^{n^{-1}}{}_{bc}=0,$ $\\tau ^{a^{-1}}{}_{cd}+R_{cd}(\\overline{\\tau ^a{}_{c^{-1}d^{-1}}})+\\sum _n R_{cd}(\\overline{\\tau ^a{}_{c^{-1}d^{-1}n,n^{-1}}})\\tau ^n{}_{cd}=0 $ for all $a,b,c,d$ .", "(3) The conditions for metric compatibility with an edge-symmetric metric are $ h_{mn}\\alpha _{mn}+R_n(h_{n^{-1}}\\alpha _{m,n^{-1}m^{-1}})- \\sum _a R_{a^{-1}}(h_a\\alpha _{amn,n^{-1}m^{-1}}) -R_n(h_{n^{-1}}\\tau ^{n^{-1}}{}_{m,n^{-1}m^{-1}})=0, $ $ \\delta ^p{}_{n^{-1}}{\\partial }_m h_n=h_{p^{-1}}\\tau ^{p^{-1}}{}_{mn}-\\sum _a R_{a^{-1}}(h_a\\tau ^a{}_{amn,p})\\tau ^{a^{-1}}{}_{mn}$ for all $m,n,p$ .", "(1) The first formula displayed is basically in [4] (or see [3]) in the inner case with $\\Theta =\\sum _a e^a$ , merely put in terms of the components of $\\Gamma $ and after subtracting off the flip map from $\\sigma $ and imposing the bimodule properties of the maps $\\alpha ,\\sigma $ (hence written in terms of $\\tau $ ).", "It is easy to see that $\\wedge \\alpha =0$ and $\\wedge ({\\rm id}+\\sigma )=0$ for the Grassmann algebra case reduce to symmetry in the lower indices (this technique is used in [3] but is in any case straightforward).", "Note that $e\\notin {\\mathcal {C}}$ so $\\Gamma ^a{}_{bc}$ has value $-\\alpha _{bc}:=-\\alpha _{b,c}$ when $a=bc$ and $\\tau ^a{}_{bc}:=\\tau ^a{}_{b,c}$ otherwise.", "We omit the commas when there are only two elements not being multiplied.", "(2) The condition for $\\alpha $ is immediate from $\\sigma \\circ \\dagger \\circ \\alpha =\\alpha \\circ *$ evaluated on $e^a$ with $e^a{}^*=-e^{a^{-1}}$ .", "The condition $\\sigma \\circ \\dagger \\circ \\sigma =\\dagger $ is easily seen (as in the proof of [3] for $\\alpha =0$ ) to be $ \\sum _{m,n}R_{n^{-1}m^{-1}}(\\overline{\\sigma ^{ab}{}_{mn}})\\sigma ^{n^{-1}m^{-1}}{}_{cd}=\\delta ^{b^{-1}}{}_c\\delta ^{a^{-1}}{}_d,$ which we now evaluate for the stated form of $\\sigma $ .", "(3) Metric compatibility is $ \\nabla (h_{ab}e^a)\\otimes e^b-\\sigma (h_{ab}e^a\\otimes \\Gamma ^b{}_{cd}e^c)\\otimes e^d=0,$ which expands out using the Leibniz rules and the form of the metric to $ \\delta _{p,n^{-1}}{\\partial }_m h_n-h_{p^{-1}}\\Gamma ^{p^{-1}}{}_{mn}-h_a R_a(\\Gamma ^{a^{-1}}{}_{bp})\\sigma ^{ab}{}_{mn}=0$ In the edge-symmetric case, this becomes $ \\delta _{p,n^{-1}}{\\partial }_m h_n-h_{p^{-1}}\\Gamma ^{p^{-1}}{}_{mn}- R_a(h_{a^{-1}}\\Gamma ^{a^{-1}}{}_{bp})\\sigma ^{ab}{}_{mn}=0.$ We now insert the form of $\\Gamma ,\\sigma $ to obtain the condition stated in the mutually exclusive cases $p=n^{-1}m^{-1}$ and $p\\ne n^{-1}m^{-1}$ (where the terms shown do not contribute when $p=n^{-1}m^{-1}$ due to the conditions on $\\tau $ and $e\\notin {\\mathcal {C}}$ , so we do not need to write that this $p$ is excluded).", "We will apply this to ${\\mathbb {G}}={\\mathbb {Z}}_n$ and ${\\mathcal {C}}=\\lbrace \\pm 1\\rbrace $ , denoting the corresponding basis indices for brevity as $\\pm $ .", "The Cayley graph is then the polygon with arrows in both directions." ], [ "Curvature", "Given a left connection $\\nabla $ on an algebra with differential calculus (it does not even need to be a bimodule one), we have Riemann curvature $ R_\\nabla :\\Omega ^1\\rightarrow \\Omega ^2\\otimes _A\\Omega ^1,\\quad R_\\nabla =({\\rm d}\\otimes {\\rm id}-{\\rm id}\\wedge \\nabla )\\nabla .$ For example, in the inner case of a connection defined by maps $\\sigma ,\\alpha $ as above, this is $ R_{\\nabla }\\omega = \\Theta \\wedge \\Theta \\otimes \\omega - (\\wedge \\otimes {\\rm id})({\\rm id}\\otimes (\\alpha -\\sigma _\\Theta ))(\\alpha - \\sigma _\\Theta )\\omega $ for all $\\omega \\in \\Omega ^1$ .", "Next, given a bimodule `lift' map $i:\\Omega ^2\\rightarrow \\Omega ^1\\otimes _A\\Omega ^1$ such that $\\wedge \\circ i={\\rm id}$ , we define Ricci and the Ricci scalar $S$ relative to it as $ {\\rm Ricci} = ((\\ ,\\ ) \\otimes {\\rm id})({\\rm id}\\otimes i \\otimes {\\rm id})({\\rm id}\\otimes R_\\nabla )g,\\quad S = (\\ ,\\ ){\\rm Ricci}.", "$ This is a `working definition' rather than part of a fully developed theory (for which an understanding of conservation laws and the stress-energy tensor would be needed).", "In the Cayley graph case of Lemma REF , there is a canonical $\\Omega $ and with it a canonical $i$ in [3], which for an abelian group is just $ i(e^a\\wedge e^b)={1\\over 2}(e^a\\otimes e^b- e^b\\otimes e^a)$ on the Grassmann algebra generators (extended as a bimodule map).", "Thus, once we have found a QLC for our quantum metric, the route to the scalar curvature needed for the Einstein-Hilbert action is canonical at least for Abelian groups such as ${\\mathbb {Z}}_n$ ." ], [ "Quantization of ${\\mathbb {Z}}_n$", "Here we consider the general theory above for the case of an $n$ -gon for $n\\ge 3$ .", "A metric is a free assignment of a `square-length' to each edge and Section REF solves the quantum Riemannian geometry to find the quantum Levi-Civita connection.", "Section REF then constructs Euclidean quantum gravity on the polygon." ], [ "Quantum Riemannian geometry on ${\\mathbb {Z}}_n$", "Just as it is useful in classical geometry to use local coordinates where the differential structure is the standard one for ${\\mathbb {R}}^n$ , it is similarly useful to regard the $n$ -gon as the group ${\\mathbb {G}}={\\mathbb {Z}}_n$ for its differential structure as explained in Section .", "Here the calculus $\\Omega ^1 ({\\mathbb {Z}}_n)$ with generators ${\\mathcal {C}}= \\lbrace 1,-1\\rbrace $ has corresponding left-invariant basis $e^+,e^-$ given by $e^+ = \\sum _{i=0}^{n-1} \\omega _{i \\rightarrow i+1}; \\quad e^- = \\sum _{i=0}^{n-1} \\omega _{i \\rightarrow i-1},$ where $i\\in {\\mathbb {Z}}_n$ runs over the vertices.", "The $n=2$ case is different and was already solved for its quantum Riemannian geometry in [2].", "Since the $e^\\pm $ are a basis over the algebra, a bimodule invertible quantum metric must take the central form $ g = a e^{+} \\otimes e^{-} + b e^{-} \\otimes e^{+}$ for non-vanishing functions $a,b \\in {\\mathbb {R}}({\\mathbb {Z}}_n)$ , with inverse metric $ (e^+,e^+) = (e^-,e^-) = 0, \\quad (e^+,e^-) = 1/R_+(b), \\quad (e^-,e^+) = 1/R_{-}(a).$ We write $R_\\pm =R_{\\pm 1}$ for the shift operators.", "We also have an inner element $\\Theta = e^+ + e^-$ and the canonical $*$ -structure $(e^+)^{*} = -e^-; (e^-)^{*} =-e^+$ .", "On the other hand, from the graph perspective, the relevant Cayley graph of ${\\mathbb {Z}}_n$ with the above generators is a polygon of $n$ sides where the values of the functions $a,b$ are directed edge weights according to Figure REF .", "From this, it is clear that the edge-symmetric case, where each side of the polygon has weight independent of the direction, requires $b = R_{-}a$ .", "Proceeding in this case, the quantum metric is therefore $ g=ae^+\\otimes e^-+R_-(a)e^-\\otimes e^+,\\quad (e^+,e^-) = {1\\over a}, \\quad (e^-,e^+) = {1\\over R_{-}a}$ as governed by one nonzero function $a$ .", "For convenience, we define functions on ${\\mathbb {Z}}_n$ , $ \\rho = {R_{+}(a)\\over a}.$ Figure: A quantum metric on ℤ n {\\mathbb {Z}}_n is given by metric coefficient functions a,ba,b or equivalently by directed edge weights g i→i±1 g_{i\\rightarrow i\\pm 1}.Proposition 3.1 For $n\\ge 3$ , there is a $*$ -preserving QLC for any given edge-symmetric metric (REF ) on $\\Omega ^1({\\mathbb {Z}}_n)$ .", "This is the unique for $n\\ne 4$ and coincides with the restriction to periodic metrics mod $n$ of the unique such connection on ${\\mathbb {Z}}$ in [1], namely $ \\sigma (e^{+} \\otimes e^{+}) = \\rho e^{+} \\otimes e^{+}, \\quad \\sigma (e^{+} \\otimes e^{-}) = e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{+}) = e^{+} \\otimes e^{-}, \\quad \\sigma (e^{-} \\otimes e^{-}) = R_{-}^2\\rho ^{-1} e^{-} \\otimes e^{-}$ with the geometric structures $\\nabla e^+ &= (1-\\rho ) e^{+} \\otimes e^{+}, \\quad \\nabla e^- = (1-R_{-}^2\\rho ^{-1}) e^{-} \\otimes e^{-}, \\\\R_{\\nabla }e^+ &= \\partial _{-}\\rho e^+\\wedge e^{-} \\otimes e^{+}, \\quad R_{\\nabla }e^- = -\\partial _{+}(R_{-}^2\\rho ^{-1}) e^+\\wedge e^{-} \\otimes e^{-}, \\\\{\\rm Ricci}&= \\frac{1}{2}\\left(\\partial _{-}(R_{-}\\rho ) e^{-} \\otimes e^{+} - \\partial _{-}\\rho ^{-1} e^{+} \\otimes e^{-} \\right), \\\\S &= \\frac{1}{2}\\left( -\\frac{\\partial _{-}\\rho ^{-1}}{a} + \\frac{\\partial _{-}(R_{-}\\rho )}{R_{-}a} \\right), \\quad \\Delta f = -\\frac{R_{-}{\\rho } +1}{a}({\\partial }_++{\\partial }_-)f.$ (For $n=4$ , there is a second $*$ -preserving QLC given below.)", "Due to the grading restrictions for a bimodule map, the most general $\\sigma $ for $n\\ne 4$ has the form $ \\sigma (e^{+} \\otimes e^{+}) = \\sigma _0 e^{+} \\otimes e^{+} , \\quad \\sigma (e^{+} \\otimes e^{-}) = \\sigma _1 e^{+} \\otimes e^{-} + \\sigma _2 e^{-} \\otimes e^{+}, \\nonumber \\\\ \\sigma (e^{-} \\otimes e^{+}) = \\sigma _3 e^{+} \\otimes e^{-} + \\sigma _4 e^{-} \\otimes e^{+}, \\quad \\sigma (e^{-} \\otimes e^{-}) = \\sigma _5 e^{-} \\otimes e^{-}$ (where the $\\sigma _i$ are functional parameters) while for $n=4$ we can have additional terms leading to another solution (given below).", "Similarly, for $n\\ne 3$ we can only have the map $\\alpha = 0$ while for $n=3$ we may have additional terms leading to non $*$ -preserving solutions in the Appendix.", "Taking the displayed main form of $\\sigma $ and $\\alpha =0$ , torsion freeness $\\wedge ({\\rm id}+\\sigma )=0$ amounts to $ \\sigma _2 = \\sigma _1 + 1,\\quad \\sigma _3 = \\sigma _4 + 1, $ while metric compatibility is $R_{+}(a) = aR_{+}(\\sigma _3)\\sigma _0, \\quad a = aR_{+}(\\sigma _4)\\sigma _1 + R_{-}(a)R_{-}(\\sigma _0)\\sigma _3, \\nonumber \\\\R_{-}(a) = aR_{+}(\\sigma _5)\\sigma _2 + R_{-}(a) R_{-}(\\sigma _1)\\sigma _4, \\quad R_{-}^2(a) = R_{-}(a)R_{-}(\\sigma _2)\\sigma _5, \\nonumber \\\\0 = aR_{ 1}(\\sigma _5)\\sigma _1 + R_{-}(a)R_{-}(\\sigma _1)\\sigma _3, \\quad 0 = aR_{+}(\\sigma _4)\\sigma _2 + R_{-}(a)R_{-}(\\sigma _0)\\sigma _4.$ It is then a matter of solving these, which was done using SAGE[30].", "Among the solutions, we find a unique one that is $*$ -preserving.", "The others are described for completeness in the Appendix.", "Note that the form of $\\Delta $ in comparison to the usual lattice Laplacian makes it clear that $a$ has units of length${}^2$ [1], [2].", "That the restriction of the unique $*$ -preserving QLC on ${\\mathbb {Z}}$ in [1] to periodic metrics gives a $*$ -preserving QLC is not surprising, but that this gives all $*$ -preserving solutions for $n\\ne 4$ is a nontrivial result of solving the equations as described.", "For $n=4$ , similar methods lead to a further 2-parameter moduli of $*$ -preserving connections of the form $ \\sigma (e^{+} \\otimes e^{+}) &= \\gamma e^{-} \\otimes e^{-} ,\\quad \\sigma (e^{+} \\otimes e^{-}) = - e^{+} \\otimes e^{-}, \\\\ \\sigma (e^{-} \\otimes e^{+}) & = -e^{-} \\otimes e^{+},\\quad \\sigma (e^{-} \\otimes e^{-}) = {R_+a\\over R_-(a\\gamma )} e^{+} \\otimes e^{+},$ where $\\gamma = (\\gamma _0, \\gamma _1, \\bar{\\gamma }^{-1}_0, \\bar{\\gamma }^{-1}_1)$ specifies a function on the four points of ${\\mathbb {Z}}_4$ (in order) in terms of two complex parameters $\\gamma _0,\\gamma _1$ , such that $R_+^2\\gamma =\\bar{\\gamma }^{-1}$ .", "The associated quantum geometric structures are $\\nabla e^+ &= e^{+} \\otimes e^{+} + e^{-} \\otimes e^{+} + e^{+} \\otimes e^{-} - \\gamma e^{-} \\otimes e^{-},\\\\\\nabla e^- &= e^{-} \\otimes e^{-} + e^{+} \\otimes e^{-} + e^{-} \\otimes e^{+} - re^{+} \\otimes e^{+},\\\\R_\\nabla e^+ &= \\left( R_-r -1\\right) e^+\\wedge e^- \\otimes e^+,\\quad R_\\nabla e^- = \\left(1- r \\right) e^+\\wedge e^- \\otimes e^-, \\\\{\\rm Ricci} &= {1\\over 2}\\left( R_+r -1 \\right) e^{+} \\otimes e^{-} + {1\\over 2}\\left( R_+^2r-1 \\right) e^{-} \\otimes e^{+},\\\\S &= {1\\over 2a}\\left( ({R_-\\rho } )(R_+^2r - 1 ) + R_+r -1 \\right),\\\\\\Delta f&=0,$ where we use the shorthand $ r:= {R_+(a)\\over R_-(a)}|\\gamma |^2.$ This is the $*$ -preserving case of the general $n=4$ solution (i) in the Appendix." ], [ "The circle limit of the ${\\mathbb {Z}}_n$ quantum geometry", "We now turn to the matter of the classical limit $n\\rightarrow \\infty $ of the quantum geometry on ${\\mathbb {Z}}_n$ .", "Given that $\\Omega ^1({\\mathbb {Z}}_n)$ is 2-dimensional, we can not expect exactly a classical circle in the limit.", "To put the quantum geometry in a more convenient form, we first (Fourier) transform to a new variable $s$ , where $s\\in {\\mathbb {Z}}_n)$ is defined by $ s(i)=q^i,\\quad q=e^{2\\pi \\imath \\over n},\\quad {\\mathbb {Z}}_n)\\cong _n:=s]/(s^n-1)$ In this new description, our same algebra $A$ is generated by $s$ with the relation $s^{n}=1$ .", "Also note that ${\\rm d}s^{-1}=- s^{-1}({\\rm d}s)s^{-1}$ is independent of ${\\rm d}s$ until we specify the commutation relations of ${\\rm d}s$ with $s$ .", "We thus define two left-invariant 1-forms $ f^+:=s^{-1}{\\rm d}s,\\quad f^-:=s{\\rm d}s^{-1}.$ For the $n\\rightarrow \\infty $ limit, we can now just drop the $s^n=1$ relation so that $A=s,s^{-1}]$ , the algebraic circle with $s^*=s^{-1}$ .", "On can think of this as $s=e^{\\imath \\theta }$ in terms of an angle coordinate $\\theta $ .", "Its classical differential calculus has ${\\rm d}s$ central and hence one left-invariant 1-form $\\bar{f}^+=\\imath {\\rm d}\\theta =-\\bar{f}^-$ , and the standard constant metric is $ {\\rm d}\\theta \\otimes {\\rm d}\\theta = -\\bar{f}^+\\otimes \\bar{f}^+.$ We are not in this classical case.", "We set $[m]_q:=(1-q^m)/(1-q)$ as the usual $q$ -deformed integer.", "Proposition 3.2 In these new coordinates, the $f^\\pm $ form a Grassmann algebra and $ f^-s=-s f^+,\\quad f^+s=s(f^-+(q+q^{-1}) f^+),\\quad {\\rm d}s^m=-{q[m]_q s^m\\over (q+1)}\\left(q[-1-m]_qf^++ [1-m]_qf^-\\right),$ while the $*$ -operation and the element that makes the calculus inner are $ f^\\pm {}^*=-f^\\pm ,\\quad \\Theta ={q\\over (q-1)^2}\\Theta _0;\\quad \\Theta _0=f^++f^-$ and the constant $a=1$ metric $g=e^+\\otimes e^-+e^-\\otimes e^+$ is $ g={g_0 \\over (q-q^{-1})^2};\\quad g_0= -2 f^+\\otimes f^++\\Theta _0\\otimes f^++f^+\\otimes \\Theta _0+ {2q\\over (q-1)^2}\\Theta _0\\otimes \\Theta _0.$ Moreover, the above does not require $s^n=1$ , i.e.", "applies equally well to the algebraic circle $s,s^{-1}]$ with $q$ a real or modulus 1 free parameter.", "Working in our original calculus $\\Omega ({\\mathbb {Z}}_n)$ and $s,q$ the function and the root of unity specified in (REF ), we compute that $ f^-=s{\\rm d}s^{-1}=(q^{-1}-1)e^++ (q-1)e^-,\\quad f^+=s^{-1}{\\rm d}s=(q-1)e^++(q^{-1}-1)e^-$ which inverts for $n>2$ as $ e^+={f^-+q f^+\\over (q-q^{-1})(q-1)},\\quad e^-={f^++q f^-\\over (q-q^{-1})(q-1)}.$ As they are linear combinations, the $f^\\pm $ are closed and form a Grassmann algebra since the $e^\\pm $ do.", "We have $e^\\pm s= R_\\pm (s)e^\\pm =q^{\\pm 1}s e^\\pm $ which implies the relations shown for $f^\\pm $ .", "Finally, ${\\rm d}s^m=({\\partial }_+ s^m)e^++({\\partial }_- s^m)e^-=(q^m-1)e^++ (q^{-m}-1)e^-$ which translates to the formula shown in terms of $f^\\pm $ .", "The $*$ structure also matches but is in any case required by $f^+{}^*=(s^{-1}{\\rm d}s)^*=({\\rm d}s^{-1})s=s^{-1}f^-s=-f^+$ and similary for $f^-$ .", "We also have $\\Theta =e^++e^-$ and $g$ as stated when written as above in terms of $f^\\pm $ .", "The quantum Levi-Civita connection now appears equivalently as $\\nabla f^\\pm =0$ .", "Moreover, these formulae do not directly reference $n$ and one can check directly that they give a $*$ -differential calculus even without the relation $s^n=1$ , i.e.", "on the algebraic circle.", "Now $q$ is a free parameter but a check shows that we still need it real or modulus one for a $*$ -calculus.", "The end result of Proposition REF is a novel, 2-dimensional, $q$ -deformed calculus on the algebraic circle.", "In the $q$ real case, we can quotient it by a relation such as $f^+=-qf^-$ , which is equivalent to the relation $e^-=0$ and gives the standard 1-dimensional $q$ -deformed calculus on the circle [3] with ${\\rm d}s.s=q s{\\rm d}s$ or ${\\rm d}s^m=[m]_q s^{m-1}{\\rm d}s$ .", "In this quotient, we would have $g=0$ (this quotient calculus in fact admits no quantum metric due to the centrality requirement, making it unsuitable for our purposes).", "Corollary 3.3 In the limit $q\\rightarrow 1$ , the above $q$ -deformed calculus on the circle algebra $s,s^{-1}]$ tends to a noncommutative 2D calculus with $ f^-s=-s f^+,\\quad f^+s=s(f^-+2 f^+),\\quad {\\rm d}s^m={m s^m\\over 2}\\left((m+1)f^++ (m-1)f^-\\right), \\quad f^\\pm {}^*=-f^\\pm $ In this limit, the 1-form $\\Theta _0$ is closed and graded-central and the classical calculus on $S^1$ is then given by the quotient where we set $\\Theta _0=0$ .", "Conversely, this 2D calculus is a central extension in the sense of [13], [14] of the classical calculus on $S^1$ by $\\Theta _0$ .", "Most of this is immediate.", "For the last sentence, note that if $f$ is a function of $s$ then we can write the differential in the corollary equivalently as $ {\\rm d}f= s{{\\rm d}f\\over {\\rm d}s}f^++ {s^2\\over 2}{{\\rm d}^2 f\\over {\\rm d}s^2}\\Theta _0,$ where the first term is the expected left-invariant derivative associated to $f^+$ and the second is a higher order derivative associated to an `extra direction' $\\Theta _0$ .", "This has the structure of a central extension of the classical calculus on $S^1$ in the sense of [3][13], [14] according to the canonical Riemannian structure of $S^1$ and a second order operator with respect to it.", "The central extension here is defined by a deformed $\\bullet $ product where $s\\bullet \\bar{f}^+=s\\bar{f}^+$ is undeformed for left multiplication on the classical left-invariant 1-form $\\bar{f}^+=s^{-1}{\\rm d}s=\\imath {\\rm d}\\theta $ .", "From the other side, we set $\\bar{f}^+\\bullet s=\\bar{f}^+ s+ (\\bar{f}^+,{\\rm d}s)\\Theta _0= s\\bar{f}^++s(\\bar{f}^+,\\bar{f}^+)\\Theta _0=s\\bullet \\bar{f}^+ + s\\Theta _0$ , which is the stated commutation relation for $f^+$ if we take the classical constant metric on $S^1$ with normalisation $(\\bar{f}^+,\\bar{f}^+)=1$ .", "As $\\Theta _0$ commutes with functions, this determines the correct commutation relation for $f^-$ also.", "The second order operator defines ${\\rm d}$ and here is $s^2{{\\rm d}^2\\over {\\rm d}s^2}$ , which is the Laplacian plus a vector field as an example of the general set up [3][13].", "Next, we note that the rescaled metric $g_0$ in Proposition REF has a part with a $q\\rightarrow 1$ limit plus a singular term proportional to $\\Theta _0\\otimes \\Theta _0$ .", "Hence, if $\\pi _{class}$ denotes taking $q\\rightarrow 1$ and simultaneously projecting to the classical calculus, we have $ \\pi _{class}(g_0)= -2\\bar{f}^+\\otimes \\bar{f}^+=2{\\rm d}\\theta \\otimes {\\rm d}\\theta ,$ provided that in this process, the killing of $\\Theta _0$ takes precedence over setting $q\\rightarrow 1$ in the singular term.", "It is not clear how to make this precise (one cannot simply set $\\Theta _0=0$ first without destroying the structure of the $q$ -deformed calculus).", "Aside from this technical detail, we still have the trivial flat QLC $\\nabla f^\\pm =0$ and the projection is covariantly constant with respect to this and the usual classical connection.", "We have focussed here on the limit of the constant metric on ${\\mathbb {Z}}_n$ , but one can analyse general metrics in the similar way.", "Also, in the $q\\rightarrow 1$ limit as in Corollary REF , one can directly analyse the possible generalised (not necessarily quantum-symmetric) metrics, e.g.", "the ones with constant coefficients have the form $ g={\\rm Re}(z) (f^+\\otimes f^++f^-\\otimes f^-)+ z f^+\\otimes f^-+ \\bar{z} f^-\\otimes f^+$ for a complex parameter $z$ in order to be central and obey the reality condition.", "If we then impose quantum symmetry, we are forced to a real multiple of $\\Theta _0\\otimes \\Theta _0$ , which is indeed the only component of the flat metric $g$ if we fully scale out the singularity visible in Proposition REF and then set $q\\rightarrow 1$ .", "This is a `purely quantum' metric in the 2D calculus in Corollary REF , in that it projects to zero in the classical calculus on $S^1$ .", "We have already seen that the extra direction $\\Theta _0$ of the calculus in Corollary REF arises as the residue of the element $\\Theta $ that makes the $q$ -deformed calculus on the circle inner, which is a purely quantum phenomenon.", "It can also be viewed as defining a central extension of the classical calculus on $S^1$ with associated `partial derivative' the second order operator in (REF ).", "A third point of view is given by moving to `cartesian coordinates' $ x={1\\over 2}(s+s^{-1}),\\quad y={1\\over 2\\imath }(s-s^{-1});\\quad x^2+y^2=1$ from which we compute $ 2(x{\\rm d}x+ y{\\rm d}y)=s^{-1}{\\rm d}s+s{\\rm d}s^{-1} =\\Theta _0.$ Thus, $\\Theta _0$ can be thought of as something like the normal to the circle viewed in the plane, similarly to the picture for the extra direction for the 3D calculus on the fuzzy sphere in [31].", "Finally, in cohomological terms, one can check that the noncommutative de Rham cohomology ring $H_{\\rm dR}({\\mathbb {Z}}_n)$ is the Grassmann algebra generated by $e^\\pm $ i.e.", "dimensions $1:2:1$ and spanned by $e^\\pm $ in degree 1.", "The same is true in terms of the $f^\\pm $ for finite $n$ , which latter description holds also for $n\\rightarrow \\infty $ ; $H_{\\rm dR}({\\mathbb {Z}})$ is generated by $f^\\pm $ in the case of the corollary.", "This is the same as the cohomology of a torus, so it is tempting to think of the quantum geometry as a circle thickened into a torus, at least in a cohomological sense.", "The geometric picture, as we have seen, is a little like this with an extra direction related to the normal to the circle (rather than an actual torus)." ], [ "Euclideanised quantum gravity on ${\\mathbb {Z}}_n$", "As for the integer line graph[1], the two-dimensional cotangent bundle on ${\\mathbb {Z}}_n$ required by the quantum geometry now admits the possibility of curvature.", "We envision that there could be various applications of such curved discrete geometries, but here we focus on just one, namely Euclideanised quantum gravity on ${\\mathbb {Z}}_n$ .", "For integration on ${\\mathbb {Z}}_n$ needed in the action, we take a sum over ${\\mathbb {Z}}_n$ with a weight $a$ (in the commutative case, this would be $\\sqrt{|\\det g|}$ ), which has the merit that then the action is $ S_g={1\\over 2}\\sum _{{\\mathbb {Z}}_n}(R_-\\rho {\\partial }_- R_-\\rho )={1\\over 2}\\sum _{{\\mathbb {Z}}_n}\\rho {\\partial }_-\\rho ={1\\over 2}\\sum _{{\\mathbb {Z}}_n}\\rho {\\partial }_+\\rho ={1\\over 4}\\sum _{{\\mathbb {Z}}_n}\\rho ({\\partial }_++{\\partial }_-)\\rho , $ where ${\\partial }_++{\\partial }_-$ is the usual lattice double-differential on ${\\mathbb {Z}}_n$ .", "This has the same form as for a scalar field except that $\\rho $ is a positive function, as already observed for ${\\mathbb {Z}}$ in [1].", "We consider two approaches, depending on what we regard as our underlying field, and in both cases maintaining ${\\mathbb {Z}}_n$ symmetry in the result.", "(i) As suggested by the form of the action, we can take $ \\rho _0={a(1)\\over a(0)},\\quad \\cdots ,\\quad \\rho _{n-2}={a(n-1)\\over a(n-2)},\\quad \\rho _{n-1}={a(0)\\over a(n-1)}$ as $n$ dynamical variables subject to the constraint $\\rho _0\\cdots \\rho _{n-1}=1$ .", "We think of the constraint as a hypersurface in ${\\mathbb {R}}_{>0}^n$ , which induces a metric ${\\mathfrak {g}}_\\rho $ on the hypersurface, and we use the Riemannian measure in this.", "Thus, we can take $\\rho _0,\\cdots ,\\rho _{n-2}$ as local coordinates and measure ${\\mathcal {D}}\\rho =(\\prod _{i=0}^{n-2}{\\rm d}\\rho _i)\\sqrt{\\det ({\\mathfrak {g}}_\\rho )}$ .", "The measure here maintains the ${\\mathbb {Z}}_n$ symmetry as ultimately independent of the choice of coordinates.", "Explicitly, for $n=3$ , we take $\\rho _0,\\rho _1$ as coordinates and the constrained surface in ${\\mathbb {R}}_{>0}^3$ is $\\rho _2=1/(\\rho _0 \\rho _1)$ .", "The coordinate tangent vectors and induced metric are $ v_0=(1,0, -{1\\over \\rho _0^2\\rho _1}),\\quad v_1=(0,1,-{1\\over \\rho _0\\rho _1^2});$ $ {\\mathfrak {g}}_\\rho =(v_i\\cdot v_j)=\\begin{pmatrix} 1+{1\\over \\rho _0^4\\rho _2}& {1\\over \\rho _0^3\\rho _1^3}\\\\ {1\\over \\rho _0^3\\rho _1^3}& 1+ {1\\over \\rho _0^2\\rho _1^4}\\end{pmatrix},\\quad \\det ({\\mathfrak {g}}_\\rho )=1+{1\\over \\rho _0^4\\rho _1^2}+{1\\over \\rho _0^2\\rho _1^4}.$ Hence the partition function is $ Z= \\int _0^\\infty {\\rm d}\\rho _0\\int _0^\\infty {\\rm d}\\rho _1\\sqrt{\\det ({\\mathfrak {g}}_\\rho )}\\, e^{- {1\\over 2 {G}}(\\rho _0^2+\\rho _1^2+\\rho _2^2-\\rho _0\\rho _1-\\rho _1\\rho _2-\\rho _2\\rho _0)};\\quad \\rho _2:={1\\over \\rho _0\\rho _1}$ These integrals can be done numerically and appear to converge for all values ${G}>0$ of the coupling constant (the numerical results need ${G}$ not too small for working precision but this case can be analysed separately).", "We are interested in expectation values ${\\langle }\\rho _{i_1}\\cdots \\rho _{i_m}{\\rangle }$ , where we insert $\\rho _{i_1}\\cdots \\rho _{i_m}$ in the integrand and take the ratio with $Z$ .", "Some results obtained from this theory for $n=3$ are plotted in Figure REF .", "Numerical evidence is limited due to convergence accuracy issues, but it seems clear that expectation values of products of $\\rho _i$ tend to 1 and hence $\\Delta \\rho _i\\rightarrow 0$ as ${G}\\rightarrow 0$ , as might be expected.", "As in [2], this should be thought of as the weak gravity limit given that fluctuations expressed in $\\rho $ enter the action relative to ${G}$ .", "Meanwhile, it appears as ${G}\\rightarrow \\infty $ that $ {\\Delta \\rho _i\\over {\\langle }\\rho _i{\\rangle }}\\sim 1.11,\\quad {{\\langle }\\rho _i^2{\\rangle }\\over {\\langle }\\rho _i{\\rangle }^2}\\sim 2.23,\\quad {{\\langle }\\rho _i\\rho _j{\\rangle }\\over {\\langle }\\rho _i{\\rangle }{\\langle }\\rho _j{\\rangle }}\\sim 0.845$ for $i\\ne j$ .", "The asymptotic values here are from plotting out to ${G}=500$ , but would need to be confirmed analytically due to potential numerical convergence issues.", "The first of these limits, if confirmed, would be a similar phenomenon of a uniform the relative metric uncertainty in [2] in the `strong gravity' limit.", "The correlations are real and relative correlation between two distinct vertices of the triangle is lower than the relative self-correlation, which is in line with the $n=3$ case of the relative quantisation in Figure REF .", "Figure: Euclidean quantum gravity vevs on ℤ 3 {\\mathbb {Z}}_3 for gauge invariant variables ρ i \\rho _i(ii) We can take (as more usual) the metric coefficients as the underlying field, so in our case the edge `square-lengths' $a=(a_0,\\cdots ,a_{n-1})\\in {\\mathbb {R}}_{>0}^n$ .", "Assuming Lebesgue measure, the partition function is $ Z=\\int _0^\\infty (\\prod _i{\\rm d}a_i) e^{{S_g\\over {G}}}=\\int _0^L (\\prod _i{\\rm d}a_i) e^{ {1\\over 2 {G}}\\sum _{{\\mathbb {Z}}_n}\\rho {\\partial }_+\\rho }$ and we introduce a field strength upper bound $L$ to control divergences as in [2].", "One can then look at ratios independent of $L$ or indeed consider a formal renormalisation process.", "On the other hand, the divergences come from the global scaling symmetry $a_i\\mapsto \\lambda a_i$ for $\\lambda \\in {\\mathbb {R}}_{>0}$ of the action (since this depends only on the ratios $\\rho $ ) and therefore another approach would be to `factor out' the geometric mean as a new variable which we do not integrate over, keeping only the ratios relative to this as the dynamic degrees of freedom.", "This is again in the spirit of [2], except that we proceed multiplicatively.", "Thus, we let $A=(\\prod _i a_i)^{1\\over n}$ be the geometric mean and $b_i:=a_i/A$ , which by construction obey $b_0\\cdots b_{n-1}=1$ .", "These are similar to the $\\rho _i$ variables in forming the corresponding hypersurface in ${\\mathbb {R}}_{>0}^n$ , but the action is different and the measure is also different since it is inherited from the Lebesgue measure on the $a_i$ .", "Again, we will look at this explicitly for $n=3$ .", "Then the action is $ S_g={1\\over 2}\\left({b_0\\over b_1}+{b_1\\over b_2}+ {b_2\\over b_0}- ({b_1\\over b_0})^2-({b_2\\over b_1})^2-({b_0\\over b_2})^2\\right);\\quad b_2={1\\over b_0 b_1},$ while the Jacobean for the change of variables from $a_0,a_1,a_2$ to $b_0,b_1,A$ gives us $ {\\rm d}a_0\\, {\\rm d}a_1\\, {\\rm d}a_2={3 A^2\\over b_0 b_1}{\\rm d}b_0\\, {\\rm d}b_1\\, {\\rm d}A.$ Omitting the now decoupled integration over $A$ as an (infinite) constant, we have effectively $ Z= \\int _0^\\infty {\\rm d}b_0\\int _0^\\infty {\\rm d}b_1{1\\over b_0 b_1} e^{ {1\\over 2 {G}b_0^2 b_1^4}(-1+(1+b_0^3)b_1^3+( -1+b_0^3-b_0^6) b_1^6)}.$ The graphical expectation values against ${G}$ look qualitatively similar to those of $\\rho _i$ in Figure REF , but one also has ${\\langle }b_i{\\rangle }={\\langle }b_i b_j{\\rangle }$ for $i\\ne j$ , albeit this is specific to $n=3$ .", "Larger $n>3$ can proceed entirely similarly and one has $1<{\\langle }b_i{\\rangle }< {\\langle }b_i b_{i+1}{\\rangle }$ .", "One can also then see that the $i$ -step correlations ${\\langle }b_0b_i{\\rangle }$ (or between any two points differing by $i$ ) decrease as $i$ increases from $i=0$ to reach a minimum (as expected) half way around the polygon.", "This is based on numerical data for small $n$ as shown in Figure REF .", "The data for $n=6$ are already noisy due to numerical convergence issues, but suggest that for large $n$ the ${\\langle }b_0b_i{\\rangle }$ may be approximated by $\\alpha -\\beta \\sin ({\\pi i\\over n})$ for positive $\\alpha >\\beta $ depending on $G$ and $n$ .", "This is broadly similar to the form of correlation functions for a scalar field ${\\langle }\\phi _0\\phi _i{\\rangle }$ in a lattice box in [1], but without the overall $\\imath $ there.", "Figure: Euclidean quantum gravity correlations 〈b 0 b i 〉{\\langle }b_0b_i{\\rangle } plotted against ii for 3≤n≤63\\le n\\le 6 and suitable G{G}." ], [ "Quantum geometric cosmological models on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$", "In this section, we first start with an analysis of quantum metrics and QLCs on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ , where ${\\mathbb {R}}$ is a classical time and ${\\mathbb {Z}}_n$ is a discrete space.", "We find that the full `strongly tensorial' bimodule properties for an invertible quantum metric force us to the block diagonal case, without taking this as an assumption.", "Existence of a QLC further dictates its form, again without taking this as an assumption, and we then find a unique $*$ -preserving one.", "We then focus on the case where the ${\\mathbb {Z}}_n$ geometry is flat (modelling an actual geometric circle) but possibly time-dependent as in FLRW cosmology." ], [ "Quantum metric and QLC on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$", "We consider a general metric on the product $\\mathbb {R}\\times {\\mathbb {G}}$ where ${\\mathbb {R}}$ has a variable $t$ and we are interested in the finite group ${\\mathbb {G}}={\\mathbb {Z}}_n$ with $e^a=e^\\pm $ , but we do not need to specialise at this stage.", "We consider metrics of the form $g=\\mu {\\rm d}t\\otimes {\\rm d}t+ h_{ab}e^a\\otimes e^b+ n_a(e^a\\otimes {\\rm d}t+{\\rm d}t\\otimes e^a)$ for $\\mu , h_{ab}, n_a$ in $A=\\infty ({\\mathbb {R}})\\otimes \\mathbb {C}({\\mathbb {G}})$ but note right away that if we take the tensor product calculus where the continuous variable and its differential $t,{\\rm d}t$ graded commute with functions and forms on ${\\mathbb {G}}$ then centrality of the metric needed for a bimodule inverse dictates that $n_a=0$ .", "We therefore proceed in this case.", "Similarly, we look for general QLCs of the form $ \\nabla {\\rm d}t=-\\Gamma {\\rm d}t\\otimes {\\rm d}t+ c_a(e^a\\otimes {\\rm d}t+{\\rm d}t\\otimes e^a)+ d_{ab}e^a\\otimes e^b,$ $ \\nabla e^a=-\\Gamma ^a{}_{bc}e^b\\otimes e^c+ \\gamma ^a{}_b(e^b\\otimes {\\rm d}t+{\\rm d}t\\otimes e^b)+ f^a{\\rm d}t\\otimes {\\rm d}t$ and note that for the tensor form of calculus along with the natural choice where $\\sigma ({\\rm d}t\\otimes \\ ),\\sigma (\\ \\otimes {\\rm d}t)$ are the flip on the basic 1-forms ${\\rm d}t, e^a$ , requiring the above to be a bimodule connection compatible with the relations of each algebra forces us to $ c_a=0,\\quad f^a=0,\\quad \\gamma ^a{}_b=\\gamma _a\\delta _{a,b},\\quad d_{a,b}=d_a\\delta _{a,b^{-1}}$ for some functions $\\gamma _a$ .", "We therefore proceed in this case.", "Next, for zero torsion, we need that $ d_{ab}=d_{ba},\\quad \\Gamma ^a{}_{bc}=\\Gamma ^a{}_{cb},\\quad \\wedge ({\\rm id}+\\sigma )(e^a\\otimes e^b)=0$ (which means $\\sigma $ restricted to the $\\lbrace e^a\\rbrace $ has the form studied before for a torsion free bimodule connection on an inner calculus, but note the calculus as a whole is not inner).", "And for $\\nabla g=0$ , we obtain 8 equations which we compute under our assumptions above for a central metric and bimodule connection, with $\\dot{\\mu }={\\partial \\over \\partial t}\\mu $ , ${\\rm d}t^{\\otimes 3}:&\\quad {\\dot{\\mu }\\over 2}-\\mu \\Gamma = 0, \\\\{\\rm d}t\\otimes {\\rm d}t\\otimes e^a:&\\quad 0=0,\\\\{\\rm d}t\\otimes e^a\\otimes {\\rm d}t:&\\quad 0=0,\\\\e^a\\otimes {\\rm d}t\\otimes {\\rm d}t:&\\quad {\\partial }_a\\mu =0,\\\\{\\rm d}t\\otimes e^a\\otimes e^b:&\\quad h_{cb}\\gamma ^c{}_a+h_{ac}R_a(\\gamma ^c{}_b)+ \\dot{h}_{ab} =0,\\\\e^a\\otimes {\\rm d}t\\otimes e^b:&\\quad h_{cb}\\gamma ^c{}_a+\\mu d_{ab} =0, \\\\e^a\\otimes e^b\\otimes {\\rm d}t:&\\quad \\mu d_{ab}+h_{mp}R_m(\\gamma ^p{}_n)\\sigma ^{mn}{}_{ab}=0, \\\\e^m\\otimes e^n\\otimes e^p:&\\quad {\\partial }_m h_{np}-h_{ap}\\Gamma ^a{}_{mn}-h_{ac}R_a(\\Gamma ^c{}_{bp})\\sigma ^{ab}{}_{mn}=0.$ The first and last of the 8 equations are just that $\\Gamma $ is a QLC on the line and $\\sigma ,\\Gamma ^a{}_{bc}$ a QLC on ${\\mathbb {G}}$ .", "The 4th equation tells us that $\\mu $ is constant on ${\\mathbb {G}}$ .", "If we write the metric as $h_{ab}=h_a\\delta _{a,b^{-1}}$ for functions $h_a$ etc., then the 6th equation tells us $ d_a=-{h_{a}\\gamma _{a}\\over \\mu }$ and the 5th and 7th equations reduce to $ \\dot{h}_a+h_a\\gamma _a+R_a(h_{a^{-1}}\\gamma _{a^{-1}})=0,\\quad \\sum _p R_{p^{-1}}(h_p\\gamma _p)\\sigma ^{p^{-1},p}{}_{a,b}=h_a\\gamma _a\\delta _{a,b^{-1}}.$ Finally, we impose $*$ -structure ${\\rm d}t^*={\\rm d}t$ and suppose that the connection on ${\\mathbb {G}}$ is also $*$ -preserving for $e^a{}^*=-e^{a^{-1}}$ as usual.", "The extended metric then obeys the quantum reality condition if $\\mu $ is real, which we suppose henceforth, and the metric on ${\\mathbb {G}}$ is `real' in the required sense (which amounts to $h_a$ real-valued).", "Then the additional condition for our extended $\\nabla $ to be $*$ -preserving comes down to $\\Gamma $ real and $ \\bar{\\gamma }_a=R_a\\gamma _{a^{-1}}, \\quad \\sum _a \\bar{d}_a\\sigma (e^a\\otimes e^{a^{-1}})=\\sum _a d_{a^{-1}}e^{a^{-1}}\\otimes e^a, $ where the 1st part comes from $\\nabla e^a{}^*$ and the 2nd from $\\nabla {\\rm d}t^*$ .", "Next, we use (REF ) and that $h_a$ are real and edge-symmetric to deduce from the 1st part that $\\bar{d}_a=R_a d_{a^{-1}}$ .", "Then since $d_a$ are constant on ${\\mathbb {G}}$ , we have $\\bar{d}_a=d_{a^{-1}}$ and our condition to be $*$ -preserving is $ \\bar{\\gamma }_a=R_a\\gamma _{a^{-1}}, \\quad \\sum _a d_{a^{-1}}(\\sigma (e^a\\otimes e^{a^{-1}})-e^{a^{-1}}\\otimes e^a)=0.$ Since $\\mu $ has to be a constant on ${\\mathbb {G}}$ , it is some function of $t$ alone.", "Generically, we can absorb this in a change of the variable $t$ , so we proceed for simplicity with $\\mu =-1$ for a cosmological type solution.", "Theorem 4.1 For $\\sigma ,\\nabla ^{{\\mathbb {Z}}_n}$ the $*$ -preserving QLC on ${\\mathbb {Z}}_n$ in Propostion REF , a quantum metric on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ admitting a $*$ -preserving QLC has the form $g=-{\\rm d}t\\otimes {\\rm d}t- a e^+\\otimes e^--R_-a e^-\\otimes e^+$ up to choice of the $t$ parametrization, such that ${\\partial }_-\\dot{a}=0$ , i.e., $a$ has the form $ a(t,i) = \\alpha (t) + \\beta (i)$ for some functions $\\alpha ,\\beta $ with $\\sum _i\\beta (i)=0$ .", "In these terms, there is a unique $*$ -preserving QLC with scalar curvature and Laplacian $2S =& - \\ddot{\\alpha }\\left(\\frac{ 1}{\\alpha + \\beta } + \\frac{1}{\\alpha + R_-\\beta } \\right) + \\frac{\\dot{\\alpha }^2}{4} \\left( \\frac{1}{(\\alpha + \\beta )^2} + \\frac{1}{(\\alpha + R_-\\beta )^2} \\right) \\\\&+ \\frac{ s }{ (\\alpha +\\beta )^2(\\alpha +R_+\\beta ) } + R_-\\left(\\frac{ s }{(\\alpha +\\beta )^2(\\alpha +R_-\\beta )}\\right),\\\\\\Delta f=&-{\\partial }_t^2+\\left({1\\over \\alpha +\\beta }+{1\\over \\alpha +R_-\\beta }\\right)(-{\\dot{\\alpha }\\over 2}{\\partial }_t f+ \\Delta _{{\\mathbb {Z}}_n}f),$ where $s:= (\\alpha +R_+\\beta )(\\alpha +R_-\\beta )-(\\alpha +\\beta )^2=\\alpha (\\Delta _{{\\mathbb {Z}}_n}\\beta )+({\\partial }_+\\beta ){\\partial }_-\\beta - \\beta ^2 $ in terms of the usual Laplacian $\\Delta _{{\\mathbb {Z}}_n}\\beta =({\\partial }_++{\\partial }_-)\\beta =R_+\\beta + R_-\\beta -2\\beta $ on ${\\mathbb {Z}}_n$ .", "We use the general analysis above applied in the specific case of ${\\mathbb {Z}}_n$ .", "Also, for the purpose of the proof, it is convenient to have a shorthand notation $a_+=a$ and $a_-=R_-a$ , so that $h_\\pm =a_\\pm $ for our particular metric.", "Then the 2nd of (REF ) holds automatically as $\\sigma (e^\\pm \\otimes e^\\mp )=e^\\mp \\otimes e^\\pm $ and $a_\\pm \\gamma _\\pm =d_\\pm (t)$ are constants on ${\\mathbb {Z}}_n$ for a solution, while the 1st of (REF ) is that $\\dot{a}_\\pm =- d_+- d_-$ , which requires ${\\partial }_-\\dot{a}=0$ as stated.", "We assume the QLC on ${\\mathbb {Z}}_n$ at each $t$ for the metric functions $a=a(t,i)$ .", "The flip form of $\\sigma (e^\\pm \\otimes e^\\mp )$ for this also means that the 2nd part of (REF ) is automatic and we just need $\\bar{\\gamma }_\\pm =R_\\pm \\gamma _{\\mp }$ , or equivalently $\\bar{d}_\\pm =d_\\mp $ , for a $*$ -preserving connection.", "This means that $ d_+= -{\\dot{a}\\over 2} + \\imath b,\\quad d_-=\\bar{d}_+=-{\\dot{a}\\over 2}- \\imath b;\\quad \\gamma _\\pm = -{\\dot{a}\\over 2 a_\\pm } \\pm {\\imath b\\over a_\\pm }$ for any real-valued function $b(t)$ .", "The unique solution with real coefficients for $\\nabla $ in our basis is $b=0$ and gives the $*$ -preserving QLC $ \\nabla {\\rm d}t = {\\dot{a}\\over 2}(e^+\\otimes e^-+e^-\\otimes e^+),\\quad \\nabla e^\\pm =\\nabla ^{{\\mathbb {Z}}_n}e^\\pm - {\\dot{a}\\over 2 a_\\pm }(e^\\pm \\otimes {\\rm d}t+{\\rm d}t\\otimes e^\\pm ).", "$ The $\\sigma $ for this when one argument is ${\\rm d}t$ is the flip.", "We then proceed to compute the curvature of this QLC, $R_\\nabla e^\\pm &= R^{{\\mathbb {Z}}_n}_\\nabla e^\\pm - \\left(\\dot{\\Gamma }^\\pm {}_{ab} - \\Gamma ^\\pm {}_{ab}R_a({\\dot{a}\\over 2a_b}) + {\\dot{a}\\over 2a_\\pm {}}\\Gamma ^\\pm {}_{ab} \\right){\\rm d}t\\wedge e^a\\otimes e^b - \\Gamma ^\\pm {}_{ab}R_a({\\dot{a}\\over 2a_b}) e^a\\wedge e^b\\otimes {\\rm d}t\\\\&\\quad \\pm ({\\dot{a}\\over 2 a_\\pm })^2 a_\\pm e^+\\wedge e^-\\otimes e^\\pm -{\\dot{a}\\over 2}{\\partial }_b({1\\over a_\\pm })e^b\\wedge e^\\pm \\otimes {\\rm d}t + {\\dot{a}\\over 2}{\\partial }_b({1\\over a_\\pm }){\\rm d}t\\wedge e^b\\otimes e^\\pm \\\\&\\quad - \\left( { {\\partial }\\over {\\partial }t}({\\dot{a}\\over 2a_\\pm {}}) + ({\\dot{a}\\over 2a_\\pm {}})^2 \\right){\\rm d}t \\wedge e^\\pm {} \\otimes {\\rm d}t, \\\\R_\\nabla {\\rm d}t&={\\ddot{a}\\over 2}{\\rm d}t\\wedge (e^+\\otimes e^-+e^-\\otimes e^+)+{\\dot{a}\\over 2}e^+\\wedge \\Gamma ^-{}_{-b}e^-\\otimes e^b+{\\dot{a}\\over 2}e^-\\wedge \\Gamma ^+{}_{+b}e^+\\otimes e^b\\\\&\\quad +\\sum _\\pm ({\\dot{a}\\over 2a_\\pm })^2a_{\\pm }e^\\pm \\wedge (e^\\mp \\otimes {\\rm d}t+{\\rm d}t\\otimes e^\\mp ),$ in terms of the Christoffel symbols on ${\\mathbb {Z}}_n$ .", "The Ricci tensor and the Ricci scalar $S$ are then ${\\rm Ricci} =&\\ {\\rm Ricci}^{{\\mathbb {Z}}_n} + {\\ddot{a}\\over 4} (e^{+} \\otimes e^{-} + e^{-} \\otimes e^{+}) + {1\\over 2}\\left( R_+(\\dot{\\Gamma }^-_{--}) -{\\dot{a}\\over 2} (R_+(\\Gamma ^-_{--}) + 1) {\\partial }_-\\left( {1\\over a} \\right) \\right) {\\rm d}t \\otimes e^- \\\\&+ {1\\over 2}\\left( R_-(\\dot{\\Gamma }^+_{++}) - {\\dot{a}\\over 2}(R_-(\\Gamma ^+_{++}) +1){\\partial }_+\\left({1\\over a_-}\\right) \\right) {\\rm d}t \\otimes e^+ + {\\dot{a}\\over 4}\\left( (R_-(\\Gamma ^+_{+-}) + 1){\\partial }_-\\left({1\\over a_-}\\right) \\right) e^-\\otimes {\\rm d}t \\\\&- {\\dot{a}\\over 4}\\left( (R_+(\\Gamma ^-_{+-}) + 1){\\partial }_-\\left( {1\\over R_+(a)} \\right) \\right) e^+\\otimes {\\rm d}t + {1\\over 2}\\left( {\\partial }_t\\left( {\\dot{a}\\over 2a} + {\\dot{a}\\over 2a_-} \\right) + \\left({\\dot{a}\\over 2a}\\right)^2 + \\left({\\dot{a}\\over 2a_-}\\right)^2 \\right){\\rm d}t \\otimes {\\rm d}t, \\\\S =& -S^{{\\mathbb {Z}}_n} - {\\ddot{a}\\over 2}\\left( {1\\over a} + {1\\over a_-} \\right) + {1\\over 2} \\left({\\dot{a}\\over 2a}\\right)^2 + {1\\over 2}\\left({\\dot{a} \\over 2a_-}\\right)^2$ (where we have used that $\\Gamma ^\\pm _{+-} = \\Gamma ^{\\pm }_{-+}$ ).", "We now insert values for the QLC in Proposition REF to obtain $R_\\nabla e^\\pm &= \\pm \\left( -{\\partial }_\\pm \\left({a_\\pm \\over a_\\mp }\\right) + \\left( {\\dot{a}\\over 2a_{\\pm }}\\right)^{2}a_\\pm \\right) e^+ \\wedge e^{-} \\otimes e^{\\pm }+ {\\dot{a}\\over 2 a^2_\\pm }{\\partial }_\\pm \\left(a_\\pm \\right) {\\rm d}t\\wedge e^{\\pm } \\otimes e^{\\pm } \\nonumber \\\\&\\quad +{ \\dot{a}\\over 2} {\\partial }_\\mp \\left({1\\over a_\\pm } \\right)( e^\\pm \\wedge e^\\mp \\otimes {\\rm d}t + {\\rm d}t\\wedge e^{\\mp } \\otimes e^{\\pm }) \\nonumber \\\\&\\quad + \\left( -{ \\ddot{a} \\over 2a_\\pm } + \\left( { \\dot{a} \\over 2a_\\pm } \\right)^2 \\right) {\\rm d}t \\wedge e^\\pm \\otimes {\\rm d}t, \\\\R_\\nabla {\\rm d}t&= \\sum _{\\pm } \\left( {\\ddot{a}\\over 2a_\\pm } - \\left( \\dot{a}\\over 2a_\\pm \\right)^2 \\right)a_\\pm {\\rm d}t \\wedge e^{\\pm } \\otimes e^{\\mp } + \\sum _\\pm {\\dot{a}\\over 2a_\\pm }{\\partial }_-( a) e^+ \\wedge e^{-} \\otimes e^{\\mp }\\nonumber \\\\ &\\quad + {\\dot{a}^2\\over 4} {\\partial }_-\\left( 1\\over a^2 \\right) e^+ \\wedge e^- \\otimes {\\rm d}t$ and as a result, ${\\rm Ricci}&= {1\\over 2}\\sum _\\pm \\left(\\big ( { \\ddot{a } \\over 2} +{\\partial }_\\pm \\big ({a_\\mp \\over a_\\pm }\\big ) \\big )e^{\\pm } \\otimes e^{\\mp } - {\\dot{a}\\over 2a^2_\\mp }{\\partial }_\\pm (a_\\mp ) {\\rm d}t \\otimes e^\\pm + {\\dot{a}\\over 2}{\\partial }_\\pm \\big ( {1\\over a_\\pm } \\big ) e^\\pm \\otimes {\\rm d}t \\right) \\nonumber \\\\&\\quad - {1\\over 2} \\left(-{\\ddot{a}\\over 2} \\left( {1 \\over a } + {1\\over a_-} \\right) + \\left( {\\dot{a}\\over 2a } \\right)^2 +\\left( {\\dot{a}\\over 2a_-} \\right)^2 \\right) {\\rm d}t \\otimes {\\rm d}t,\\\\S &={1\\over 2}\\left(- \\ddot{a} \\left({1\\over a } + {1 \\over a_-} \\right) + \\left({\\dot{a}\\over 2a }\\right)^2 + \\left({\\dot{a} \\over 2a_-}\\right)^2 - {1\\over a }{\\partial }_+ \\left({a_-\\over a }\\right) - {1\\over a_-}{\\partial }_- \\left({a\\over a_-}\\right) \\right).$ We now note that the requirement ${\\partial }_-\\dot{a}=0$ is equivalent to $a$ being of the form stated.", "Clearly, such a form obeys this condition as $\\dot{a}=\\alpha $ is constant on ${\\mathbb {Z}}_n$ .", "Conversely, given $a(t,i)$ obeying the condition, we let $\\alpha (t)={1\\over n}\\sum _ia(t,i)$ be the average value and $\\beta =a-\\alpha $ .", "The latter averages to zero and has zero time derivative by the assumption on $a$ , hence depends only on $i$ .", "We now insert this specific form into the curvature calculations to obtain ${\\rm Ricci} =& \\left( \\frac{\\ddot{\\alpha }}{4} - \\frac{s }{(\\alpha +\\beta )(\\alpha +R_+\\beta )}\\right) e^+\\otimes e^-+\\left( \\frac{\\ddot{\\alpha }}{4} - R_-\\left( \\frac{s }{(\\alpha +\\beta )(\\alpha +R_-\\beta )}\\right) \\right) e^-\\otimes e^+ \\nonumber \\\\&-{\\dot{\\alpha }\\over 4}R_-\\left( \\frac{\\partial _+\\beta }{(\\alpha + \\beta )^2} \\right) {\\rm d}t \\otimes e^+- \\frac{\\partial _+\\beta }{(\\alpha + \\beta )(\\alpha + R_+\\beta )} e^+ \\otimes {\\rm d}t \\nonumber \\\\&-{\\dot{\\alpha }\\over 4} \\frac{\\partial _-\\beta }{(\\alpha + \\beta )^2} {\\rm d}t \\otimes e^--R_-\\left( \\frac{\\partial _-\\beta }{(\\alpha + \\beta )(\\alpha + R_-\\beta )} \\right) e^- \\otimes {\\rm d}t \\nonumber \\\\&+\\left( \\frac{\\ddot{\\alpha }}{4} \\left(\\frac{2\\alpha + \\beta + R_-\\beta }{(\\alpha + \\beta )(\\alpha + R_-\\beta )}\\right) + \\frac{\\dot{\\alpha }^2}{4}\\left( \\frac{(\\alpha +\\beta +R_-\\beta )^2-(\\alpha ^2+2\\beta R_-\\beta )}{(\\alpha +\\beta )^2(\\alpha + R_-\\beta )^2} \\right) \\right){\\rm d}t \\otimes {\\rm d}t$ and the scalar curvature as stated.", "Without loss of generality, we have fixed $\\sum _i\\beta (i)=0$ since this could be shifted into the value of $\\alpha $ .", "We also have the geometric Laplacian $ \\Delta f = -\\Delta ^{{\\mathbb {Z}}_n} f - \\left({1\\over a} + {1\\over a_-}\\right){\\dot{a} \\over 2}{\\partial }_t f - {\\partial }_t^{2}f=- \\left({1\\over a} + {1\\over a_-}\\right)({\\dot{a} \\over 2}{\\partial }_t f -\\Delta _{{\\mathbb {Z}}_n}f)-{\\partial }_t^2f,$ which simplifies as stated.", "We are using $\\Delta ^{{\\mathbb {Z}}_n}$ for the Laplacian in Propostion REF and $\\Delta _{{\\mathbb {Z}}_n}$ with lower label for the standard finite difference Laplacian.", "In this theorem, $\\alpha (t)>0$ is the average `radius' of the ${\\mathbb {Z}}_n$ geometry, evolving with time, while $\\beta (i)$ as a fluctuation as we go around ${\\mathbb {Z}}_n$ and we see that this has to be `frozen' (does not depend on time) in order for the metric to admit a quantum geometry.", "It is striking that this includes the FLRW-type models studied in the remaining section in the class forced by the quantum geometry.", "Note that we also need to restrict to $ {\\rm min}_i\\beta (i)>-{\\rm inf}_t\\alpha (t)$ so that $a(t,i)$ is everywhere positive.", "Although we will not study it here, we are now in position to start thinking about quantum gravity on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ in a functional integral approach.", "Given the identified restrictions, this would presumably have the form of a partition function $ Z=\\int {\\mathcal {D}}\\alpha \\prod _{i=0}^{n-2}\\int \\!\\!\\!", "{\\rm d}\\beta (i)\\, J_\\beta e^{{\\imath \\over {G}}\\int _{-\\infty }^\\infty {\\rm d}t\\sum _{ {\\mathbb {Z}}_n}\\mu S[\\alpha ,\\beta ] }$ for some measure $\\mu (t,i)$ .", "Classically, the latter would come from the metric coefficients and, for example, we might take something of the form $\\mu =\\sqrt{(\\alpha +\\beta )(\\alpha +R_-\\beta )}$ in line with the case of ${\\mathbb {Z}}_n$ alone in Section REF .", "It is not clear what would be the right choice, however.", "For the integral over functions $\\lbrace \\alpha (t)\\rbrace $ , there would be the usual issues to make this rigorous (as some kind of continuous product of integrals).", "The new feature is that these should be restricted to values $\\alpha (t)>0$ and for a given configuration $\\lbrace \\alpha (t)\\rbrace $ , we should limit the lower bound on the $\\int {\\rm d}\\beta (i)$ integrations according to (REF ).", "Finally, we presumably would want, to maintain the ${\\mathbb {Z}}_n$ symmetry, a Jacobian which we have denoted $J_\\beta $ to reflect the geometry of the constraint $\\sum \\beta (i)=0$ .", "The choice of $\\mu $ and the constrained integration are both issues that we already encountered for ${\\mathbb {Z}}_n$ in Section REF but are now significantly more complicated.", "We also should now aim for a physical theory given the Lorentzian signature, hence the $\\imath $ in the action." ], [ "Equations of state in FLRW model on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$", "For the remainder of the paper, we focus on the cosmological FLRW model case where $a=R^2(t)$ with no fluctuation $\\beta (i)$ over ${\\mathbb {Z}}_n$ and hence $ g=-{\\rm d}t\\otimes {\\rm d}t- R^2(t) e^+\\otimes _s e^-,$ where $e^+ \\otimes _s e^- = e^+ \\otimes e^- +e^- \\otimes e^+$ .", "In this case, the results above simplify to $\\nabla {\\rm d}t &= R\\dot{R}e^+\\otimes _s e^-,\\quad \\nabla e^\\pm =- {\\dot{R}\\over R}e^\\pm \\otimes _s{\\rm d}t,\\\\R_\\nabla e^\\pm &=-{\\ddot{R}\\over R}{\\rm d}t\\wedge e^\\pm \\otimes {\\rm d}t \\pm \\left({\\dot{R}\\over R}\\right)^2 R^2 e^+\\wedge e^-\\otimes e^\\pm ,\\quad R_\\nabla {\\rm d}t= \\ddot{R}R{\\rm d}t\\wedge e^+\\otimes _s e^-,\\\\{\\rm Ricci}&= {\\ddot{R}\\over R} {\\rm d}t\\otimes {\\rm d}t+{1\\over 2}\\left({\\dot{R}^2\\over R^2}+{ \\ddot{R}\\over R} \\right) R^2 e^+\\otimes _s e^- ,\\quad S=-2{\\ddot{R}\\over R} -\\left({\\dot{R}\\over R}\\right)^2.$ Although a general scheme for a noncommutative Einstein tensor is not known, in the present model it seems sufficient to define it in the usual way, in which case ${\\rm Eins} = {\\rm Ricci} - {1\\over 2}Sg = - {1\\over 2 }\\left( {\\dot{R}\\over R} \\right)^2 {\\rm d}t\\otimes {\\rm d}t-{ R\\ddot{R}\\over 2} e^+ \\otimes _s e^-.$ Lemma 4.2 The divergence $\\nabla \\cdot =((\\ ,\\ )\\otimes {\\rm id})\\nabla $ of a 1-1 tensor of the form $ T=f {\\rm d}t\\otimes {\\rm d}t - p R^2 e^+ \\otimes _s e^-$ defined by functions $f,p$ on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ , and for metric defined as above by $R(t)$ , is $\\nabla \\cdot T = - \\left(\\dot{f} + 2{\\dot{R}\\over R} (f+p)\\right){\\rm d}t + {\\partial }_b p e^b.$ In particular, the Einstein tensor (REF ) is conserved in the sense $\\nabla \\cdot {\\rm Eins}=0$ .", "The Leibniz rule for the action of the connection produces $\\nabla (f {\\rm d}t&\\otimes {\\rm d}t - p R^2 e^+ \\otimes _s e^-)\\nonumber \\\\&= {\\rm d}f\\otimes {\\rm d}t \\otimes {\\rm d}t - {\\rm d}p\\otimes R^2 e^+ \\otimes _s e^- +f\\nabla ({\\rm d}t \\otimes {\\rm d}t) - p\\nabla (R^2 e^+ \\otimes _s e^-)\\nonumber \\\\&= {\\rm d}f\\otimes {\\rm d}t \\otimes {\\rm d}t - {\\rm d}p\\otimes R^2 e^+ \\otimes _s e^- + (f+p)\\nabla ({\\rm d}t \\otimes {\\rm d}t)\\nonumber \\\\&= \\dot{f} {\\rm d}t\\otimes {\\rm d}t \\otimes {\\rm d}t - \\dot{p} {\\rm d}t \\otimes R^2 e^+ \\otimes _s e^- + {\\partial }_b f e^b \\otimes {\\rm d}t \\otimes {\\rm d}t + {\\partial }_b p e^b\\otimes R^2 e^+ \\otimes _s e^-\\nonumber \\\\&\\quad +R\\dot{R}(f+p)\\left( e^+ \\otimes _s e^-\\otimes {\\rm d}t + e^-\\otimes {\\rm d}t \\otimes e^+ + e^+\\otimes {\\rm d}t \\otimes e^- \\right)$ on using metric compatibility whereby $\\nabla ({\\rm d}t \\otimes {\\rm d}t) = -\\nabla ( R^2 e^+ \\otimes _s e^-)$ and then evaluating the former with $\\sigma =$ flip on ${\\rm d}t$ .", "Now applying $(,)\\otimes {\\rm id}$ with the inverse metric, we arrive at the stated result for the divergence.", "For ${\\rm Eins}$ in (REF ), the coefficients are constant on ${\\mathbb {Z}}_n$ , so there is no $e^\\pm $ term in $\\nabla \\cdot {\\rm Eins}$ .", "For the ${\\rm d}t$ term it is easy to verify that $\\dot{f} + 2{\\dot{R}\\over R} (f+p)=0$ automatically for the effective values of the specific coefficients $f,p$ in (REF ) defined by $R(t)$ .", "Next, recall from Section REF that our formulation of Ricci is -1/2 of the usual value, hence Einstein's equation for us should be written as $ {\\rm Eins}+4\\pi {G}T=0$ and from (REF ) we see that this holds if $T$ has the form for dust of pressure $p$ and densisty $f$ , namely $ T= p g + (f+p) {\\rm d}t\\otimes {\\rm d}t= f {\\rm d}t\\otimes {\\rm d}t- p R^2 e_+\\otimes _s e_-$ for pressure and density $p=-{1\\over 8 \\pi {G}}\\left({\\ddot{R}\\over R}\\right),\\quad f={1\\over 8 \\pi {G}}\\left({\\dot{R}\\over R} \\right)^2.$ Note that $T$ is automatically conserved by the same calculation as for the Einstein tensor and this does not give any constraint on $R(t)$ .", "Setting $ \\mathit {H}:= {\\dot{R}\\over R}, $ conservation is equivalent to the continuity equation $ \\dot{f} =- 2\\mathit {H}(f+p), $ which also holds automatically.", "The standard consideration in cosmology at this point is to assume an equation of state $p = \\omega f$ for a real parameter $\\omega $ , in which case the continuity equation becomes ${{\\rm d}f\\over {\\rm d}R}=-2f(1+\\omega )$ so that $f \\propto R^{-2(1+\\omega )}$ .", "Given this form of the density $f$ , our assumption $p=\\omega f$ can be solved for $\\omega \\ne -1$ to give $R(t) = R_0\\left( 1 + \\sqrt{8\\pi G f_0}(1+w)t \\right)^{{1\\over 1+w}}$ for initial radius and pressure $R_0,f_0$ .", "Here $\\omega >-1$ leads to an expanding universe.", "Recall that one usually takes $\\omega =0,1/3$ for cold dust and radiation respectively.", "If we add a cosmological constant so that ${\\rm Eins} -{1\\over 2}g\\Lambda +4\\pi {G}T=0$ , this is equivalent to a modified stress energy tensor given as before but with modified $ f^\\Lambda =f+{ \\Lambda \\over 8 \\pi {G}},\\quad p^\\Lambda =p-{ \\Lambda \\over 8 \\pi {G}}=\\omega f^\\Lambda - {1+\\omega \\over 8\\pi {G}}\\Lambda .", "$ The effective equation of state now leads to $ R(t)=R_0\\left( \\frac{\\cosh ({\\rm arccosh} (\\sqrt{-\\frac{\\Lambda }{8 \\pi {G}f_0}})+\\sqrt{\\Lambda } (1+\\omega ) t)}{\\sqrt{-\\frac{\\Lambda }{8 \\pi {G}f_0}}}\\right)^{1\\over 1+\\omega }$ with reasonable behaviour for $f_0>0$ (with $f$ remaining positive) and real $\\Lambda $ but a limited range of $t$ when $\\Lambda <0$ .", "For comparison, note that the classical Einstein tensor on ${\\mathbb {R}}\\times S^1$ with $g=-{\\rm d}t\\otimes {\\rm d}t+ R^2(t){\\rm d}x\\otimes {\\rm d}x$ vanishes as for any 2-manifold and $T=f{\\rm d}t\\otimes {\\rm d}t+ p R^2(t) {\\rm d}x\\otimes {\\rm d}x=p g + (f+p){\\rm d}t\\otimes {\\rm d}t$ admits only zero pressure and density if we want Einstein's equation.", "One can also add a cosmological constant, in which case we need $p=-{\\Lambda \\over 8\\pi {G}}$ and $f={\\Lambda \\over 8\\pi {G}}$ and $\\omega =-1$ .", "This is therefore not the right comparable.", "Proposition 4.3 The results (REF )-(REF ) for $R(t)$ (as well as for $f(t)$ ) for the FLRW model on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ are the same as for the classical flat FLRW-model on ${\\mathbb {R}}\\times {\\mathbb {R}}^2$ .", "The flat FLRW model in 1+2 dimensions is an easy exercise starting with the metric $g = -{\\rm d}t\\otimes {\\rm d}t + R^2(t) ( {\\rm d}x \\otimes {\\rm d}x + {\\rm d}y \\otimes {\\rm d}y)$ to compute the Ricci tensor (in our conventions, which is $-{1\\over 2}$ of the usual values) as ${\\rm Ricci} = { \\ddot{R}\\over R} {\\rm d}t \\otimes {\\rm d}t - \\frac{1}{2}\\left( {\\ddot{R}\\over R} + {\\dot{R}^2 \\over R^2} \\right)R^2({\\rm d}x \\otimes {\\rm d}x + {\\rm d}y \\otimes {\\rm d}y)$ and the same scalar curavture $S$ as in ().", "The Einstein tensor is therefore $ {\\rm Eins}= - {1\\over 2}\\left( {\\dot{R}\\over R} \\right)^2 {\\rm d}t\\otimes {\\rm d}t+{R\\ddot{R}\\over 2}({\\rm d}x \\otimes {\\rm d}x + {\\rm d}y \\otimes {\\rm d}y)$ by a similar calculation as for (REF ).", "The stress tensor for dust being similarly $f{\\rm d}t\\otimes {\\rm d}t+ p R^2({\\rm d}x \\otimes {\\rm d}x + {\\rm d}y \\otimes {\\rm d}y)$ means that the Einstein equations give $p,f$ by the same expressions (REF ) as before.", "The Friedmann equations are therefore the same as we solved.", "This is perhaps not too surprising given that $\\Omega ^1$ on ${\\mathbb {Z}}_n$ is 2-dimensional, indeed $-e^+\\otimes _s e^-$ plays the same role as the classical spatial metric ${\\rm d}x \\otimes {\\rm d}x + {\\rm d}y \\otimes {\\rm d}y$ .", "We also recall by way of comparison that the standard $k=0$ Friedmann equations for the FLRW model ${\\mathbb {R}}\\times {\\mathbb {R}}^3$ has the well-known solution, $ R(t) = R_0(1 + \\sqrt{6\\pi {G}f_0}(w+1)t)^{\\frac{2}{3(w+1)}}$ without cosmological constant and can also be solved with it, as $ R(t) = R_0 \\left(\\frac{\\cosh {\\left( {\\rm arccosh\\left( \\sqrt{ -\\frac{\\Lambda }{8\\pi {G}f_0}} \\right)} + \\sqrt{\\frac{3\\Lambda }{4}}(w+1)t \\right)}}{\\sqrt{-\\frac{\\Lambda }{8\\pi {G}f_0}}}\\right)^{\\frac{2}{3(w+1)}}.$ As usual, the case of $R(t)$ independent of time is a solution for the Einstein vacuum equation with ${\\rm Ricci} = 0$ .", "It is easy to see that there are no other solutions of interest with ${\\rm Ricci} \\propto g$ or ${\\rm Eins} \\propto g$ .", "On the other hand, we do have the following.", "Proposition 4.4 The equation ${\\rm Ricci}-\\lambda S g=0$ with time-varying $R(t)$ and constant $\\lambda $ has a unique solution of the form $ \\lambda ={1\\over 3},\\quad R(t)=R_0 e^{\\mu t}$ for some growth constant $\\mu \\ne 0$ and initial $R_0>0$ .", "Considering the equation ${\\rm Ricci} = \\lambda gS$ , where $\\lambda $ is an arbitrary real constant, we have two equations; one related to $e^{\\pm } \\otimes e^{\\mp }$ is $\\frac{\\ddot{R}}{R} + \\left(\\frac{2\\lambda }{1-4\\lambda } + 1\\right)\\left(\\frac{\\dot{R}}{R}\\right)^2 = 0$ and other related to ${\\rm d}t \\otimes {\\rm d}t$ is $\\frac{\\ddot{R}}{R} + \\left( \\frac{\\lambda -1}{1-2\\lambda } + 1\\right)\\left(\\frac{\\dot{R}}{R}\\right)^2 = 0.$ This requires $\\lambda = {1\\over 3}$ and $\\frac{\\ddot{R}}{R} = \\left(\\frac{\\dot{R}}{R}\\right)^2$ , which has the solution claimed." ], [ "Quantum field theory on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$", "Here we consider quantum field theory in the flat case where $R$ is a constant.", "The corresponding Laplacian operator and the Klein-Gordon equation are $ \\Delta = {2\\over R^2} \\left( \\partial _+ + \\partial _- \\right) - \\partial _t^2 ; \\quad (-\\Delta + m^2)\\phi =0.$ We write $q=e^{2\\pi \\imath \\over n}$ , where $\\imath $ denotes the imaginary unit, and Fourier transform on ${\\mathbb {Z}}_n$ by considering solutions of the form $\\phi (t,i)=q^{ik}e^{-\\imath w_k t}$ , where $i$ denotes the position in ${\\mathbb {Z}}_n$ .", "This is labelled by a discrete momentum $k=0,\\cdots ,n-1$ with associated `mass on-shell' expression $w_k^2 = {8\\over R^2} \\sin ^2{\\left({\\pi \\over n} k\\right)} +m^2.$ We then consider the corresponding operator-valued fields starting with $ \\phi _i= \\sum _{k = 0}^{n-1} {1\\over \\sqrt{2 w_k}} {(}q^{ik}a_k + q^{-ik}a_k^{\\dag }{)}, $ where now $a_k, a^\\dag _k$ are self-adjoint operators and $a_k\\mathinner {|{0}\\rangle } = 0$ , with $\\mathinner {|{k}\\rangle }$ eigenvectors of the corresponding Hamiltonian $ H = \\sum _{k = 0}^{n-1} w_k( a_ka_k^{\\dag } +{n\\over 2}).", "$ From the commutators $[H,a_k]=-w_k a_k$ and $[H,a_k^\\dag ]=w_k a_k^\\dag $ , and using the Heisenberg representation for the time evolution of the field, we obtain $ \\phi _i(t) = e^{\\imath Ht}\\phi _i e^{- \\imath Ht} = \\sum _{k = 0}^{n-1} {1\\over \\sqrt{2 w_k}} {(} q^{ik-\\imath w_kt}a_k + q^{-ik+\\imath w_kt}a_k^{\\dag } {)} $ with the time-ordered correlation function $\\mathinner {\\langle {0}|} T[\\phi _i(t_a) \\phi _j(t_b)]\\mathinner {|{0}\\rangle } = \\sum _{k = 0}^{n-1}{1\\over w_k} \\cos {\\left({2\\pi \\over n}k(i-j)\\right)}e^{-\\imath w_k |t_a - t_b|}.$ Next we check that we obtain the same correlation function via a formal path integral approach with the $\\imath \\epsilon $ -prescription.", "The partition functional integral $Z[J]$ with source $J$ is defined as $ Z[J] = {\\int {\\mathcal {D}}\\phi \\, e^{ {1\\over \\beta }S[\\phi ] + {1\\over \\beta }\\int \\sum _{i=0}^{n-1} J_i(t)\\phi _i(t) } \\over \\int {\\mathcal {D}}\\phi \\, e^{ {1\\over \\beta }S[\\phi ] } } = {\\int {\\mathcal {D}}\\phi \\, e^{ {1\\over 2\\beta } \\int dt \\sum _{i = 0}^{n-1} \\left( \\phi _i(t)(\\Delta - m^2 + \\imath \\epsilon )\\phi _i(t) + 2J_i(t)\\phi _i(t) \\right)}\\over \\int {\\mathcal {D}}\\phi \\, e^{ {1\\over 2\\beta } \\int dt \\sum _{i = 0}^{n-1} \\left( \\phi _i(t)(\\Delta - m^2 + \\imath \\epsilon )\\phi _i(t) \\right)} }, $ where $\\beta $ is a dimensionless coupling constant.", "We diagonalize the action $S[\\phi ]$ using Fourier transform to write $ \\phi _i(t) = \\sum _{k = 0}^{n-1}\\int _{-\\infty }^\\infty {dw\\over 2\\pi } \\tilde{\\phi }_k(w) q^{ik} e^{\\imath wt}; \\quad J_i(t) = \\sum _{k = 0}^{n-1}\\int _{-\\infty }^\\infty {dw\\over 2\\pi } \\tilde{J}_k(w) q^{ik} e^{\\imath wt}, $ which produces the action $ S[\\tilde{\\phi }] = \\int _{-\\infty }^{\\infty } {dw\\over 2\\pi }{1\\over 2\\beta } \\sum _{k=0}^{n-1} \\left( \\tilde{\\phi ^{\\prime }}_{-k}(-w) (-w^2 + w_k^2)\\tilde{\\phi ^{\\prime }}_k(w) + \\tilde{J}_{-k}(-w) {1\\over -w^2 + w_k^2} \\tilde{J}_k(w)\\right), $ where $\\tilde{\\phi ^{\\prime }}_k(w) = \\tilde{\\phi }_k(w) - (-w^2 + w^2_k)^{-1}\\tilde{J}_k(w)$ .", "The first term in terms of the new variables gives a Gaussian integral, which we ignore as an overall factor independent of the source.", "Using $ \\tilde{J}_k(w) ={1\\over n} \\int dt \\sum _{i=0}^{n-1} J_i(t) q^{-ik}e^{\\imath wt}, $ the functional integral becomes $Z[J] = e^{{1\\over \\beta }\\int dt^{\\prime } dt^{\\prime \\prime } J_i(t^{\\prime }) \\imath \\Delta _f(i,t^{\\prime };j,t^{\\prime \\prime })J_j(t^{\\prime \\prime })}, $ where the Feynman propagator is $\\Delta _f(i,t^{\\prime };j,t^{\\prime \\prime }) &= \\sum _{k = 0}^{n-1}q^{k(i-j)} \\int {dw\\over 2\\pi } {e^{-\\imath w(t^{\\prime }-t^{\\prime \\prime })} \\over (-w+w_k-\\imath \\epsilon {)}(w + w_k+ \\imath \\epsilon )}\\nonumber \\\\&= \\sum _{k = 0}^{n-1}{1\\over w_k} \\cos {\\left({2\\pi \\over n}k(i-j)\\right)}e^{-\\imath w_k |t_a - t_b|}.$ Finally, by construction, we have $ \\mathinner {\\langle {0}|} T[\\phi _i(t_a) \\phi _j(t_b)]\\mathinner {|{0}\\rangle } = {\\beta ^2\\over \\imath ^2} {\\partial \\over \\partial J_i(t_a)} {\\partial \\over \\partial J_j(t_b)}Z[J]=\\Delta _f(i,t^{\\prime };j,t^{\\prime \\prime }), $ which therefore gives the same result as obtained by Hamiltonian quantisation.", "This is as expected, but provides a useful check that our methodology makes sense at least in the flat case of constant $R$ ." ], [ "Particle creation in FLRW model on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$", "Here we follow the procedure developed by Parker [15], [16], [17], [18] to study cosmological particle creation, adapted now to an FLRW model on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ with an expanding quantum metric (REF )." ], [ "Model case of ${\\mathbb {R}}\\times S^1$ .", "We start with the classical background geometry case of ${\\mathbb {R}}\\times S^1$ , which is presumably known but sets up the procedure and our notations.", "Here the metric has the usual 2D FLRW form $ g = -{\\rm d}t\\otimes {\\rm d}t + R^2(t) {\\rm d}x\\otimes {\\rm d}x, $ where $R(t)$ is an arbitrary positive function.", "Thus the Klein-Gordon equation for the field $\\phi $ is $ \\left(g^{\\mu \\nu }\\nabla _\\mu \\nabla _\\nu -m^2\\right)\\phi = 0 $ or in explicit form $\\ddot{\\phi } + \\frac{\\dot{R}}{R}\\dot{\\phi } - \\frac{1}{R^2}\\partial ^2_x\\phi + m^2\\phi = 0.$ We impose the periodic boundary condition $\\phi (t,x + L) = \\phi (t,x)$ , where $L$ is a dimensionless parameter for the normalisation of the box geometry.", "We then expand the field in terms of a Fourier series $\\phi (t,x) = \\sum _k (A_k f_k(t,x) + A_k^* f^*_k(t,x) ),$ where $f_k(t,x) = \\frac{1}{\\sqrt{LR}}e^{\\imath xk}h_k(t)$ and $k = 2l\\pi /L$ for $l$ an integer.", "Here $k/R$ is the physical momentum and $l$ the corresponding `integer momentum' on a circle.", "Then $\\phi $ obeys (REF ) provided $\\ddot{h}_k(t) + \\left( \\frac{k^2}{R^2} + m^2 \\right)h_k(t) + \\left(\\frac{1}{4}\\left( \\frac{\\dot{R}}{R} \\right)^2 - \\frac{1}{2}\\frac{\\ddot{R}}{R} \\right)h_k(t) = 0$ for each momentum mode.", "We will be particularly interested in the adiabatic limit, where $R$ varies slowly with respect to the time in such way that $\\dot{R}/R\\rightarrow 0, \\ddot{R}/ R\\rightarrow 0$ .", "The solutions to (REF ) in this approximation are $h_k(t) \\sim (w_k) ^{-\\frac{1}{2}} \\left(\\alpha _k e^{\\imath \\int ^t w_k(t^{\\prime }) {\\rm d}t^{\\prime }} + \\beta _ke^{-\\imath \\int ^t w_k(t^{\\prime }) {\\rm d}t^{\\prime }} \\right),$ where $\\alpha _k$ and $\\beta _k$ are complex constants that satisfy $|\\alpha _k|^2 - |\\beta _k|^2 = 1$ and $w_k(t) = \\sqrt{ m^2+\\frac{k^2}{R^2(t)} }.$ In order to have an exact solution, we now let $\\alpha _k$ and $\\beta _k$ be functions of time such that $h_k(t) = (w_k(t)) ^{-\\frac{1}{2}} \\left(\\alpha _k(t) e^{\\imath \\int ^t w_k(t^{\\prime }) {\\rm d}t^{\\prime }} + \\beta _k(t)e^{-\\imath \\int ^t w_k(t^{\\prime }) {\\rm d}t^{\\prime }} \\right)$ and $|\\alpha _k(t)|^2 - |\\beta _k(t)|^2= 1$ for all $t$ .", "Equivalently, we can rewrite the expansion of the field as $\\phi (t,x) = \\sum _k (a_k(t) g_k(t,x) + a_k^*(t) g^*_k(t,x) ),$ where now $ g_k(t,x) = \\frac{R^{-\\frac{1}{2}}}{\\sqrt{Lw_k}} e^{\\imath (xk - \\int ^t w_k(t^{\\prime }){\\rm d}t^{\\prime } )} $ and $a_k(t) = \\alpha _k(t) ^*A_k + \\beta _k (t) A^*_k.$ In order to follow the usual procedure of canonical quantisation, we next define the conjugate momentum as $ \\pi (t,x) = R\\dot{\\phi }(t,x), $ promote the field $\\phi (t,x)$ and the momentum $\\pi (t,x)$ to operators $\\hat{\\phi }(t,x), \\hat{\\pi }(t,x)$ respectively, and impose the commutators relations $[\\hat{\\phi }(t,x),\\hat{\\phi }(t,x^{\\prime })] = [\\hat{\\pi }(t,x),\\hat{\\pi }(t,x^{\\prime })] = 0, \\quad [\\hat{\\phi }(t,x), \\hat{\\pi }(t,x^{\\prime })] = \\imath \\delta (x-x^{\\prime }).$ This requires that $A_k$ and $A_k^*$ in (REF ) are promoted to operators $A_k$ and $A_k^\\dagger $ with the usual commutation relations $[ A_{k^{\\prime }}, A_k ] = [A_{k´}^\\dagger ,A_{k^{\\prime }}^\\dagger ] = 0, \\quad [A_{k^{\\prime }},A_k^\\dagger ] = \\delta _{k,k^{\\prime }}.$ It then follows from these and a conserved quantity (see [15]), that the operator versions of (REF ) obey $[a_k(t), a_{k^{\\prime }}(t)] = [a^\\dagger _k(t), a^\\dagger _{k^{\\prime }}(t)] = 0, \\quad [a_k(t), a^\\dagger _{k^{\\prime }}(t)] = \\delta _{k,k^{\\prime }}.$ Now note that for any function $W_k(t)$ with at least derivatives to second order, the function $H(t):= W_k(t){}^{-{1\\over 2}}(\\alpha _ke^{\\imath \\int ^t dt^{\\prime } W_k(t^{\\prime })} + \\beta _k e^{-\\imath \\int ^t dt^{\\prime } W_k(t^{\\prime })})$ for any constants $\\alpha _k,\\beta _k$ is an exact solution of the equation $ \\ddot{H}(t) + \\left[ W_k^2 - W_k^{\\frac{1}{2}}\\frac{d^2}{dt^2}W_k^{-\\frac{1}{2}} \\right]H(t) = 0.", "$ Hence, if we can solve for $W_k(t)$ such that $W_k^2 = W_k^{\\frac{1}{2}}\\frac{d^2}{dt^2}W_k^{-\\frac{1}{2}} + w_k^2 + \\sigma $ holds, where $ \\sigma = \\frac{1}{4}\\left( \\frac{\\dot{R}}{R} \\right)^2 - \\frac{1}{2}\\frac{\\ddot{R}}{R}, $ then $H(t)$ provides exact solutions $h_k(t)$ of (REF ) for each $k$ .", "We can then expand $W_k$ as a sum of terms $W_k = w^{(0)} + w^{(1)}+ w^{(2)} +\\dots ,$ where the superfix denotes the adiabatic order.", "Putting this into (REF ) and just keeping the elements of order zero, we have $w^{(0)}= w_k$ .", "Just keeping the elements of first order tell us that $w^{(1)}=0$ , while for elements of second adiabatic order we require $ w^{(2)} = \\frac{(w^{(0)})^{-\\frac{1}{2}}}{2} \\frac{d^2}{dt^2}\\left((w^{(0)})^{- \\frac{1}{2}}\\right) + \\frac{\\sigma }{2w^{(0)}}.$ We can continue this procedure to any desired order to find odd $w^{(i)}=0$ and even $w^{(i)}$ determined from lower even ones.", "The form of the functions $\\alpha _k(t)$ and $\\beta _k(t)$ can be obtained when we impose (REF ).", "From its temporal derivative, one is led to the ansatz $\\alpha _k(t) = - \\dot{\\beta _{k}} (t)e^{-2\\imath \\int ^t dt^{\\prime } W_k(t^{\\prime })},\\quad \\beta _k(t) = -\\dot{\\alpha _{k}} (t)e^{2\\imath \\int ^t dt^{\\prime } W_k(t^{\\prime })}$ as justified by consistency with (REF ), given (REF ).", "For a more explicit form of these coefficients, see [32].", "A special case of interest here is when the $w_k^{(i)} $ vanish for all the orders bigger that zero (and all $k$ ).", "In this case, the operator $a_k(t)$ defined in (REF ) is independent of time, the number of particles is constant and there is no particle creation.", "From the above remarks, it is sufficient that $w_k^{(2)}=0$ , which amounts to $\\frac{1}{4}\\frac{m^2\\left(4\\frac{k^2}{R^2} - m^2 \\right)}{\\left( \\frac{k^2}{R^2} + m^2 \\right)^2}\\left(\\frac{\\dot{R}}{R}\\right)^2 + \\frac{1}{2} \\frac{m^2}{(\\frac{k^2}{R^2} + m^2) } \\frac{\\ddot{R}}{R} = 0.$ The only way that this can hold for all time and $k$ is in the infinite mass limit $m\\rightarrow \\infty $ (cf.", "[15]), where it reduces to an FLRW-like equation $ \\frac{1}{2}\\frac{\\ddot{R}}{R}= \\frac{1}{4}\\left(\\frac{\\dot{R}}{R}\\right)^2$ with solution $R\\propto t^{2}$ .", "As well as the obvious flat Minkowski case of constant $R$ , this represents a further possibility for no particle creation.", "For an actual particle creation computation, it is convenient to move to a new time variable $\\eta $ such that ${\\rm d}\\eta =\\frac{ {\\rm d}t}{R(t)},$ in which case our metric becomes conformally flat as $g = C(\\eta )(- {\\rm d}\\eta \\otimes {\\rm d}\\eta + {\\rm d}x \\otimes {\\rm d}x),$ where $C(\\eta ) = R^2(t)$ is now regarded as a function of $\\eta $ .", "Following the same steps as before but using this metric puts the wave equation (REF ) on spatial momentum modes in the simpler form $\\frac{{\\rm d}^2h_k(\\eta ) }{{\\rm d}\\eta ^2} + w_k(\\eta ) h_k(\\eta ) = 0,$ where $w_k(\\eta ) = \\sqrt{C(\\eta ) m^2 + k^2}$ as a modification of (REF ).", "We now consider particle creation under the assumption that $R$ and hence $C$ has a constant constant value $C(\\eta )=R^2_{in}$ for early times $\\eta <\\eta _{in}$ , say, and a constant value $C(\\eta )=R^2_{out}$ for late times $\\eta >\\eta _{out}$ , with $\\eta _{in} < \\eta _{out}$ .", "For these early and late times, we let $w_k^{\\rm in}= \\sqrt{R^2_{in} m^2+k^2 } ; \\quad w_k^{\\rm out}= \\sqrt{R^2_{out} m^2+k^2 }$ as functions of $k$ .", "The fields at early and late times behave exactly as flat Minkowski space-time with the corresponding frequency or effective mass, with solutions of (REF ) at early and late times provided by $h_k^{\\rm in}(\\eta )= (w_k^{\\rm in})^{-\\frac{1}{2}} e^{\\imath w_k^{\\rm in} \\eta },\\quad h_k^{\\rm out} (\\eta )=(w_k^{\\rm out})^{-\\frac{1}{2}} e^{\\imath w_k^{\\rm out} \\eta }.$ Now suppose that we start with $h_k^{\\rm in}(\\eta )$ at early times, i.e.", "$h_k(\\eta )$ for $\\alpha _k(\\eta _{in})=1$ and $\\beta _k(\\eta _{in})=0$ in the analogue of (REF ), and extend this by solving (REF ) to late times.", "There we expand it as the Bogolyubov transformation $ h_k^{\\rm in}= \\alpha _{k} h_k^{\\rm out} + \\beta _{k} h_k^{\\rm out}{}^*$ valid at late times and for some complex constants $\\alpha _k$ , $\\beta _k$ .", "Comparing with the analogue of (REF ) at late times, these constants up to phases are just the evolved values $\\alpha _k(\\eta _{out}),\\beta _k(\\eta _{out})$ in the general scheme.", "(The phases come from $e^{\\imath \\int _{\\eta _{in}}^{\\eta _{out}}w_k(\\eta ){\\rm d}\\eta }$ and are not relevant in what follows.)", "Finally, we fix a vacuum $|0{\\rangle }$ as characterised by $A_k\\mathinner {|{0}\\rangle } = 0$ and consider the number operator $N_k(\\eta )=a_k^\\dagger ( \\eta ) a_k( \\eta )$ is it evolves in time, where we use the analogue of (REF ) as our solution evolves.", "Starting now with $\\alpha _k( \\eta _{in})=1$ , $\\beta _k( \\eta _{in})=0$ in defining $a_k,a_k^\\dagger $ , we have of course $ \\mathinner {\\langle {0}|}N_k(\\eta _{in})\\mathinner {|{0}\\rangle }=0$ at early times, but in this same state at late times we have the possibility of particle creation according to ${\\langle }N_k{\\rangle }:=\\mathinner {\\langle {0}|}N_k(\\eta _{out})\\mathinner {|{0}\\rangle } = |\\beta _{k}(\\eta _{out})|^2=|\\beta _k|^2.$ This completes the general scheme, which is also well-known from several other points of view.", "To proceed further we need to fix a particular $C(\\eta )$ , and the standard choice for purposes of calculation is to interpolate the initial and final values as $C(\\eta ) = \\frac{R^2_{in}+R^2_{out}}{2} + \\frac{R^2_{out}-R^2_{in}}{2}\\tanh (\\mu \\eta ),$ where $\\mu $ is a positive constant parameter.", "Equation (REF ) can then be solved with hypergeometric functions that have the correct asymptotic limit for late and early times.", "Comparison with (REF ) gives (see [20]), $\\alpha _{k} = \\left(\\frac{w_{k}^{\\rm out}}{w_{k}^{\\rm in}}\\right)^{1/2}\\frac{\\Gamma (1-\\imath \\frac{ w_{k}^{\\rm in}}{\\mu }) \\Gamma (-\\imath \\frac{ w_{k}^{\\rm out}}{\\mu } )}{\\Gamma (-\\imath \\frac{ w_{k}^+}{\\mu }) \\Gamma (1-\\imath \\frac{ w_{k}^+}{\\mu } )},\\\\\\beta _{k} = \\left(\\frac{w_{k}^{\\rm out}}{w_{k}^{\\rm in}}\\right)^{1/2}\\frac{\\Gamma (1-\\imath \\frac{ w_{k}^{\\rm in}}{\\mu }) \\Gamma (\\imath \\frac{ w_{k}^{\\rm out}}{\\mu } )}{\\Gamma (\\imath \\frac{ w_{k}^-}{\\mu }) \\Gamma (1+\\imath \\frac{ w_{k}^-}{\\mu } )}, $ where $w_k^{\\pm } = \\frac{1}{2}(w_k^{out}\\pm w_k^{in}).$ These values result in $|\\alpha _k|^2 = \\frac{\\sinh ^2{\\left(\\pi \\frac{w_k^+}{\\mu }\\right)} }{\\sinh {(\\pi \\frac{w_k^{in}}{\\mu } )} \\sinh {(\\pi \\frac{w_k^{out}}{\\mu })}},\\quad |\\beta _k|^2 = \\frac{\\sinh ^2{\\left(\\pi \\frac{w_k^-}{\\mu }\\right)} }{\\sinh {(\\pi \\frac{w_k^{in}}{\\mu } )} \\sinh {(\\pi \\frac{w_k^{out}}{\\mu })}}, $ which, as one can check, obeys the unitarity condition (REF ).", "Figure REF includes a plot of $\\mathinner {\\langle {N_k}\\rangle } = |\\beta _k|^2$ as a function of $k$ , or rather of the associated integer momentum $l$ ." ], [ "Adaptation to ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ .", "We now repeat the previous analysis for the polygon case with $n$ sides and time-varying metric (REF ).", "We have the Laplacian $ \\Delta = - {\\partial }^2_t -2\\frac{\\dot{R}}{R}{\\partial }_t+\\frac{2}{R^2}(\\partial _{+} + \\partial _{-})$ from Theorem REF with $\\beta =0$ .", "The Klein-Gordon equation $(-\\Delta + m^2)\\phi = 0$ is $\\left(-\\frac{2}{R^2} (\\partial _+ + \\partial _-) + \\frac{1}{R^2}\\partial _t(R^2\\partial _t) +m^2\\right)\\phi = 0.$ Next, we expand the field in terms of a Fourier series $\\phi (t,i) = \\sum _k (A_k f_k(t,i) + A_k^* f^*_k(t,i) )$ in place of (REF ), where now $f_k(t,i) = \\frac{1}{R(t)}q^{ik}h_k(t)$ and $k$ is an integer mod $n$ .", "For the modes $f_k$ to obey (REF ), the $h_k$ have to solve $\\ddot{h}_k(t) + \\left( m^2 + \\frac{8}{R^2}\\sin ^2{\\left(\\frac{\\pi }{n}k\\right) } \\right)h_k(t) - \\frac{\\ddot{R}}{R} h_k(t) = 0.$ The corresponding on-shell frequency is therefore $w_k(t) = \\sqrt{ m^2 + \\frac{8}{R^2(t)}\\sin ^2{\\left(\\frac{\\pi }{n}k\\right) } }$ instead of (REF ).", "We again consider an exact solution of the form $h_k(t) = (w_k(t)) ^{-\\frac{1}{2}} \\left(\\alpha _k(t) e^{\\imath \\int ^t w_k(t^{\\prime }) {\\rm d}t^{\\prime }} + \\beta _k(t)e^{-\\imath \\int ^t w_k(t^{\\prime }) {\\rm d}t^{\\prime }} \\right).$ Analogously to the previous case, we can re-write the expansion of the field as $\\phi (t,i) = \\sum _k (a_k(t) g_k(t,i) + a_k^*(t) g^*_k(t,i) ),$ where $ g_k(t,i) = {R^{-1}\\over \\sqrt{w}_k} q^{ik} e^{-\\imath \\int ^t w_k(t^{\\prime }){\\rm d}t^{\\prime } } $ and the operator $a_k(t)$ has the same form as (REF ).", "The quantisation procedure and analysis then proceeds as before.", "Our previous expressions for $W_k(t),\\alpha _k(t), \\alpha _k$ are still valid, but we have to take into account that the zero adiabatic order term $w_k$ is different and that now $ \\sigma = -\\frac{\\ddot{R}}{R} $ as the factor in (REF ).", "For our first result, we look at when the $w_k^{(2)}$ correction vanishes so that there is no particle creation.", "In place of (REF ), we now require $\\frac{ \\frac{4}{R^2}\\sin ^2{\\left( \\frac{\\pi }{n}k\\right)}(\\frac{4}{R^2}\\sin ^2{\\left( \\frac{\\pi }{n}k\\right)} + 3m^2) }{\\left( \\frac{8}{R^2}\\sin ^2{\\left(\\frac{\\pi }{n}k\\right)} + m^2 \\right)^2 } \\left(\\frac{\\dot{R}}{R}\\right)^2 + \\frac{ \\frac{4}{R^2} \\sin ^2{\\left(\\frac{\\pi }{n}k\\right)} +m^2}{\\left(\\frac{8}{R^2}\\sin ^2{\\left(\\frac{\\pi }{n}k\\right)} + m^2\\right) }\\frac{\\ddot{R}}{R} = 0.$ This can happen for all time and all $k$ in the infinite mass limit $m\\rightarrow \\infty $ if $ \\ddot{R}= 0 $ with solution $R\\propto t$ .", "However, we also have a new possibility when $m\\rightarrow 0$ , with $ \\frac{\\ddot{R}}{R} = -\\frac{1}{2}\\left(\\frac{\\dot{R}}{R}\\right)^2$ and solution $R\\propto t^{\\frac{2}{3}}$ .", "Thus we have not one but two additional possibilities for no particle creation beyond the constant Minkowski metric case.", "For our second result, we want to analyse particle creation for the ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ model in an analogous way to the case when space is a circle.", "Thus, we make the same change of variable (REF ) in the metric (REF ) to write $g = C(\\eta )(-{\\rm d}\\eta \\otimes {\\rm d}\\eta - e^+\\otimes _s e^-),$ where $C(\\eta ) = R^2(t)$ , and the corresponding connection is $\\nabla {\\rm d}\\eta = \\frac{\\dot{R}}{R} (-{\\rm d}\\eta \\otimes {\\rm d}\\eta + e^+\\otimes _s e^-), \\quad \\nabla e^{\\pm } = -\\frac{\\dot{R}}{R}e^\\pm \\otimes _s {\\rm d}\\eta .$ Figure: Number operator for ℤ 100 {\\mathbb {Z}}_{100} against kk compared to S 1 S^1 with length scale factor L=100/2L=100/\\sqrt{2}, plotted against integer momentum ll where k=2πl/Lk=2\\pi l/L.", "In both cases, R in m=1R_{\\rm in}\\, m= 1, R out m=5R_{\\rm out}\\, m=\\sqrt{5} and μ=100\\mu =100 for the interpolation parameter.Using the quantum geometric Laplacian for this connection, we require $\\frac{{\\rm d}^2h_k(\\eta ) }{{\\rm d}\\eta ^2} + \\left(C(\\eta ) m^2 + 8\\sin ^2{\\left(\\frac{\\pi }{n}k\\right) } \\right)h_k(\\eta ) = 0$ analogously to (REF ), but now in place (REF ) we have $w_k(\\eta ) = \\sqrt{ C(\\eta ) m^2 + 8\\sin ^2{\\left(\\frac{\\pi }{n}k\\right) } }.$ The rest of the procedure follows in the same way with the same considerations, and in particular (REF ) is still valid but with (REF ) instead of (REF ).", "Figure REF shows the expected value of the number operator $\\mathinner {\\langle { N_k}\\rangle }$ as a function of $k$ as well as comparing to the circle case.", "The big difference of course is that the ${\\mathbb {Z}}_n$ has to be periodic in $k$ since this is only defined mod $n$ ." ], [ "Concluding Remarks", "In Section REF , we completely solved the quantum Riemannian geometry on a polygon ${\\mathbb {Z}}_n$ in the sense of arbitrary square-lengths $a(i)$ on the edges.", "As is typical for discrete calculi, the increasing and decreasing derivatives are closely related but nevertheless linearly independent so that $\\Omega ^1$ is 2-dimensional – in effect, the polygon acquires an extra `normal' direction (a remnant of a quantum geometry effect) and now admits curvature.", "Clearly, one could look beyond to discrete tori ${\\mathbb {Z}}_{n_1}\\times \\cdots \\times {\\mathbb {Z}}_{n_m}$ and as well as to electromagnetism both in flat and curved metrics on the ${\\mathbb {Z}}_{n_i}$ factors.", "Also interesting could be quantum geodesics even on one copy ${\\mathbb {Z}}_n$ , using the new formalism of [33].", "We also exploited the functorial nature of the formalism to take the continuum limit of the discrete geometry on ${\\mathbb {Z}}_n$ in Section REF , first converting to a $q$ -deformed geometry on the reduced circle $s,s^{-1}]$ with $s^n=1$ and $q^n=1$ , and then dropping the restriction on $s$ while sending $q\\rightarrow 1$ .", "We arrived in Corollary REF at a central extension by a 1-form $\\Theta _0$ of the classical differential forms on an algebraic circle, which can then be embedded in a $C^\\infty (S^1)$ version with $s=e^{\\imath \\theta }$ using a formalism in [3][13], [14].", "We demonstrated how the continuum metric could also emerge, focussing on the constant $a=1$ case to illustrate the remaining issues.", "Specifically, the discrete metric had to be rescaled and expanded at the $q$ -deformed level as $ g_0=(q-q^{-1})^2(e^+\\otimes e^-+e^-\\otimes e^+)= - 2 f^+\\otimes f^++ O(\\Theta _0)$ where $f^+=s^{-1}{\\rm d}s$ projects by setting $\\Theta _0\\rightarrow 0$ onto the 1-dimensional classical circle, so the first term projects to $2{\\rm d}\\theta \\otimes {\\rm d}\\theta $ .", "The scale factor $(q-q^{-1})^2= -4 \\sin ^2({2\\pi \\over n})$ in the ${\\mathbb {Z}}_n$ case is negative, which explains why, counterintuitively from the graph point of view, the physical metric needed an overall minus sign in later sections.", "However, some of the coefficients in the $O(\\Theta _0)$ terms are singular as $q\\rightarrow 1$ and we had to assume that they remain killed by $\\Theta _0\\rightarrow 0$ .", "To resolve this would need some significant functional analysis in order to formulate the limiting process more carefully, which was beyond our scope here.", "It would also be interesting to extend these ideas to more complicated models where a family of discrete approximations of a Riemannian manifold $M$ may limit to a one-higher dimension central extension of the classical geometry of $C^\\infty (M)$ of the type in [13] (where the central extension formalism was used as a wave-operator approach to a noncommutative black hole).", "The discrete quantum geometry in our approach works in principle for any graph[4], not only Cayley graphs on a discrete group, but while there is always a `maximal prolongation'[3] candidate for $\\Omega ^2$ and higher forms, for a reasonable continuum limit we will need to cut this down according to the manifold that we are approximating and so as to be able to solve for a quantum Levi-Civita connection.", "A first step would be to construct quantum geometries for general metrics on some other interesting graphs beyond the group case, which remains substantially open.", "We then, in Section REF , computed Euclideanised quantum gravity expectation values on ${\\mathbb {Z}}_n$ for small $n$ .", "In the spirit of ${\\mathbb {Z}}_2\\times {\\mathbb {Z}}_2$ in [2], we did this in two versions: the full quantisation and one for only fluctuations relative to an average field value.", "The polygon case is very different in that the full quantisation in terms of the ratios $\\rho _i=a(i+1)/a(i)$ that enter into the action appears to be finite, but numerical work for $n=3$ gave us a strikingly similar phenomenon for the uncertainty $\\Delta \\rho _i\\sim 1.1{\\langle }\\rho _i{\\rangle }$ (compared to $\\Delta a={\\langle }a{\\rangle }/\\sqrt{8}$ in [2]).", "It was speculated in [2] that this could be indicative of some kind of vacuum energy.", "The metric correlation functions on ${\\mathbb {Z}}_n$ were also substantial enough now to be interesting.", "These were computed more fully in the relative theory, where we found it useful to work with $b_i=a(i)/A$ , with $A$ the geometric mean of the $a$ field values rather than the additive one as in [2].", "These results, in Figure REF , are somewhat similar to correlations for a scalar field lattice box in [1], but now in a real positive version, which both reassures us that the model is giving reasonable answers and gives a flavour of what to expect for quantum gravity in our approach.", "Clearly, more baby models should be computed to develop our intuition further.", "As discussed in [2], our approach is not immediately comparable with other computable approaches such as [8], [9], [10], [11].", "We also looked in Section REF at the quantum geometry on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ , including a first look at quantum gravity now with a time direction ${\\mathbb {R}}$ .", "The most striking result is that centrality of the quantum metric forces the shift vectors to vanish so that the quantum metric is block diagonal with the metric on ${\\mathbb {Z}}_n$ free as before but scaled to an average value which can depend on time, see Theorem REF .", "This is another example of the phenomenon in [5] that not every classical metric can be the limit of a quantum geometry due to the centrality constraint.", "The general phenomenon here appears at the Poisson level [3][34] as quantisability equations for classical metrics.", "These are not Einstein's equations but they do involve curvature constraints and may provide the beginning of a mechanism for how the former might yet emerge as a quantum geometry consistency condition.", "Noncommutativity can also force the quantum calculus to be higher dimensional (as in our case) which is a further constraint studied in [14] but not yet analysed at the Poisson level.", "Returning to our model, the quantum differential structure on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ (this is independent of any metric) can also be expected to limit the possible diffeomorphisms in the classical limit since at the discrete level it will not be so easy to transform a discrete coordinate to a continuous one, and this may also relate indirectly to the absence of shift vectors in the allowed metric.", "In fact, diffeomorphism-invariance in GR enters in two ways, as freedom in the choice of coordinates, and as an active symmetry generated by Lie derivatives.", "The first aspect is taken care of in quantum geometry as the equations and concepts are all coordinate-independent – we are free to describe our algebras and differential forms with whatever generators and bases we prefer.", "This means there is a large but classical automorphism group behind any model.", "In quantum geometry, one has quantum vector fields as right $A$ -module maps $\\Omega ^1\\rightarrow A$ but these now generate something of a different character.", "Classically, functions and vector fields together generate the algebra of differential operators on a smooth manifold, which forms a Hopf algebroid as an infinitesimally-generated version of the path groupoid[35].", "The quantum version is studied in [3] and specifically in a quantum Hopf algebroid version in [36], which includes results for the finite group case.", "Another starting point is the notion of a universal measuring bialgebra of any algebra $A$[37], which is much bigger than the classical automorphism group and which has in principle a differential algebra version.", "Key to the latter is the concept of a differentiable (co)action, with some related first results in [38].", "The physical application of these concepts to quantum geometry and their role in quantum gravity, however, remains very much to be explored.", "The rest of the paper focussed on the special case of the ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ FLRW-type cosmological model as a background quantum geometry, where the metric on ${\\mathbb {Z}}_n$ is constant but with an overall variable $R(t)$ factor.", "The Friedmann equations for $R(t)$ turned out to be the same as those for the standard flat 1+2 dimensional FLRW model, which is perhaps not too surprising given that the calculus on ${\\mathbb {Z}}_n$ is 2-dimensional.", "For a natural model with spatial curvature, one could next take a non-Abelian group such as $S_3$ or a fuzzy sphere $\\lambda [S^2]$ as in [31] for the spatial sector, to be considered elsewhere.", "In the spirit of Connes' approach to internal symmetries of particle physics by tensoring a classical spacetime by a finite-dimensional algebra such as matrices or quaternions[39], one could also consider one of these in place of ${\\mathbb {Z}}_n$ , but now from an FLRW perspective.", "Note also that for the equations of state for the FLRW model on ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ in Section REF , we considered only the standard form of stress energy tensor for an incompressible fluid.", "Stress tensors in quantum geometry remain poorly understood, with no general theory.", "In particular, one can check that the obvious choice $ T={\\rm d}\\phi \\otimes {\\rm d}\\phi - {1\\over 2}(({\\rm d}\\phi ,{\\rm d}\\phi )+ m^2 \\phi ^2) g$ is not conserved for a free scalar field obeying the Klein-Gordon equation for the geometric Laplacian (REF ).", "Therefore, it remains to consider further what would be natural as stress tensor for a scalar field, even in our FLRW-type background.", "Similarly, the Einstein tensor remains poorly understood and while the usual formula ${\\rm Ricci}-{1\\over 2}g S$ in terms of the Ricci tensor and scalar was sufficient for our model in Section REF , and has also been used in other approaches such as [40], this is not derived as part of a noncommutative geometric calculus of variations and hence not directly connected to an Einstein-Hilbert action built from the Ricci scalar.", "Such a calculus of variations appears to be a hard problem and our approach here is to continue to explore both sides so as to first gain experience from specific models, i.e.", "quantum gravity using a functional integral approach and ideas for the Einstein tensor on a quantum geometry background.", "The continuum limit even of the ${\\mathbb {Z}}_n$ model, once better understood as discussed above and with general metrics, will provide further input here.", "A long term goal is to have a parallel Hamiltonian quantisation formalism for quantum gravity, and again this would be interesting even for our baby ${\\mathbb {Z}}_n$ model.", "We then analysed quantum field theory and particle creation in the ${\\mathbb {R}}\\times {\\mathbb {Z}}_n$ FLRW case, taking as model the set up of Parker[15], [16], [17], [18], [32] applied to ${\\mathbb {R}}\\times S^1$ .", "The main difference compared to the circle case is that we found adiabatic no particle creation solutions for $R(t)$ at $m=0$ , not only at $m=\\infty $ .", "Another difference of course is that the particle creation ${\\langle }N_k{\\rangle }$ from constant `in' to constant `out' metrics is periodic in the discrete momentum $k$ rather than decaying as $k\\rightarrow \\infty $ as it would on $S^1$ , see Figure REF .", "This is not surprising since the discrete momentum on ${\\mathbb {Z}}_n$ differs fundamentally in being periodic mod $n$ .", "In principle, one could consider particle creation between the new $m=0$ solutions, but this would need new ideas beyond the ones used (we would not be able to just adapt the circle case).", "The fact that the comparable here is particle creation on ${\\mathbb {R}}\\times S^1$ and not on something 1+2-dimensional reflects that scalar quantum fields themselves are not directly sensitive to the 2D nature of the calculus on ${\\mathbb {Z}}_n$ , a situation that we might expect to change for particle creation of higher spin fields.", "Finally, while we have focussed on the quantum field theory, one could consider the quantum mechanics limit.", "In the flat warm-up case of Section REF and following the usual steps of factoring into a wave in the time direction and a slowly varying factor, and adding a potential $V(t,i)$ , gives the Schrödinger-like equation $\\imath {\\partial }_t \\psi (t,i) = -\\frac{1}{R^2 m}({\\partial }_+ + {\\partial }_-)\\psi (t,i) + V(t,i)\\psi (t,i).$ The free particle plane-waves are clearly $\\psi _k(t,i)=e^{-\\imath E_k t}e^{{2\\pi \\imath \\over n}ik}$ with energy spectrum $E_k = \\frac{4}{mR^2}\\sin ^2{(\\frac{\\pi }{n}k)}$ , for $k=0, \\dots , n-1$ so that the trace of the free Hamiltonian is $\\sum _{k=0}^{n-1}E_k= \\frac{2n}{mR^2}$ , compared to the circle case where the trace diverges.", "This discrete-space quantum mechanics could be studied further with specific potentials $V(t,i)$ .", "In summary, we have indicated several directions for further work building on the results in the present paper.", "Stepping back, the machinery of quantum Riemannian geometry[3] can be applied in principle to almost any unital algebra in a step by step fashion and hence explored in a similar way for other algebras of interest.", "We have already mentioned [31] as another model, and we refer to the conclusions of [2] for further discussion of different algebras that could be interesting." ], [ "Non $*$ -preserving solutions", "We have rightly focussed in the Section REF on the unitary or $*$ -preserving quantum geometries over $ on $ Zn$.", "However, the underlying classificationwas done by computer algebra and works over any field of characteristic zero.", "For completeness, we list the remaining solutions which over $ would not obey the unitarity or `reality' condition (REF ).", "These could be useful in other contexts over ${\\mathbb {R}}$ or applied to other fields, for example to obtain `digital' quantum geometries over ${\\mathbb {F}}_2$ in the setting of [7] (in this case there could be other solutions also, as the field then has non-zero characteristic).", "For $n\\ge 3$ odd, there are two further independent solutions: ${\\rm (i)}\\qquad \\qquad \\qquad \\sigma (e^{+} \\otimes e^{+}) &= -\\rho e^{+} \\otimes e^{+}, \\quad \\sigma (e^{-} \\otimes e^{+}) = -e^{+} \\otimes e^{-} -2e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{+} \\otimes e^{-}) &= e^{-} \\otimes e^{+}, \\quad \\sigma (e^{-} \\otimes e^{-}) = R_{-}^2(\\rho ^{-1}) e^{-} \\otimes e^{-}, $ giving the geometric structures $\\nabla e^+ &= (1+\\rho ) e^{+} \\otimes e^{+},\\quad \\nabla e^- = (1-R_{-}^{2}(\\rho ^{-1})) e^{-} \\otimes e^{-} + 2(e^{+} \\otimes e^{-} + e^{-} \\otimes e^{+}) \\\\R_{\\nabla }e^+ &= - \\partial _{-}(\\rho ) e^+\\wedge e^{-} \\otimes e^{+}, \\\\R_{\\nabla }e^- &= - \\partial _{-}(R_{-}(\\rho ^{-1})) e^+\\wedge e^{-} \\otimes e^{-} - 2(1-R_{-}(\\rho ))e^+\\wedge e^{-} \\otimes e^{+},\\\\{\\rm Ricci} &= \\frac{1}{2}\\left( -\\partial _{-}(R_{-}(\\rho )) e^{-} \\otimes e^{+} + 2(1-\\rho )e^{+} \\otimes e^{+} + \\partial _{-}(\\rho ^{-1}) e^{+} \\otimes e^{-} \\right), \\\\S &= \\frac{1}{2}\\left( \\frac{\\partial _{-}(\\rho ^{-1})}{a} - \\frac{\\partial _{-}(R_{-}(\\rho ))}{R_{-}a} \\right), \\\\\\Delta f &= \\frac{1}{a}(R_{-}{f} - R_{+}({f}))(R_{-}({\\rho }) +1).$ For $n=3$, we may freely add a map $\\alpha $ given by $\\alpha (e^-)=\\lambda R_+(a)e^+\\otimes e^+$ to $\\nabla e^-$ for a free parameter $\\lambda $ , and $\\alpha (e^+)=0$ , so no change to $\\nabla e^+$ .", "This agrees with the triangle analysis in [3] aside from a different definition of $\\rho $ .", "${\\rm (ii)}\\qquad \\qquad \\qquad \\sigma (e^{+} \\otimes e^{+}) &= \\rho e^{+} \\otimes e^{+}, \\quad \\sigma (e^{+} \\otimes e^{-}) = -2 e^{+} \\otimes e^{-} - e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{+}) &= e^{+} \\otimes e^{-}, \\quad \\sigma (e^{-} \\otimes e^{-}) = -R_{-}^2(\\rho ^{-1}) e^{-} \\otimes e^{-},$ giving the geometric structures $\\nabla e^+ &= (1-\\rho )e^{+} \\otimes e^{+} + 2(e^+\\otimes e^- + e^{-} \\otimes e^{+}),\\quad \\nabla e^- = (1+R_{-}^{2}(\\rho ^{-}))e^{-} \\otimes e^{-}, \\\\R_{\\nabla }e^+ &= -\\partial _{-}\\rho e^+\\wedge e^{-} \\otimes e^{+} + 2(1-R_{-}(\\rho ^{-1}))e^+\\wedge e^{-} \\otimes e^{-}, \\\\R_{\\nabla }e^- &= -\\partial _{-}(R_{-}(\\rho ^{-1}))e^+\\wedge e^{-} \\otimes e^{-}, \\\\{\\rm Ricci} &= \\frac{1}{2}\\left( -\\partial _{-}(R_{-}(\\rho )) e^{-} \\otimes e^{+} + 2(1-R_{-}^2(\\rho ^{-1}) )e^{-} \\otimes e^{-} + \\partial _{-}(\\rho ^{-1}) e^{+} \\otimes e^{-} \\right), \\\\S &= \\frac{1}{2}\\left( \\frac{\\partial _{-}(\\rho ^{-1})}{a} - \\frac{\\partial _{-}(R_{-}(\\rho ))}{R_{-}a} \\right), \\\\\\Delta f &= \\frac{1}{a}(R_{+}(f) - R_{-}(f))(R_{-}(\\rho ) +1).$ For $n=3$, we may freely add a map $\\alpha $ given by $\\alpha (e^+)=\\lambda R_+(a)e^-\\otimes e^-$ to $\\nabla e^+$ for a free parameter $\\lambda $ , and $\\alpha (e^-)=0$ , so no change to $\\nabla e^-$ .", "This again agrees with the triangle analysis in [3] aside from a different definition of $\\rho $ .", "For $n\\ge 4$ even, there are two further independent solutions each with a free nonzero parameter $q$ , from which we define a function $Q = q^{(-1)^{i}} = \\begin{pmatrix} q \\\\ q^{-1} \\\\ \\vdots \\end{pmatrix}.$ Then ${\\rm (i)}\\qquad \\qquad \\qquad \\sigma (e^{+} \\otimes e^{+}) &= \\rho e^{+} \\otimes e^{+}, \\quad \\sigma (e^{+} \\otimes e^{-}) = (Q-1) e^{+} \\otimes e^{-} + Qe^{-} \\otimes e^{+},\\\\\\sigma (e^{-} \\otimes e^{+}) &= e^{+} \\otimes e^{-}, \\quad \\sigma (e^{-} \\otimes e^{-}) = R_{-}^2(\\rho ^{-1})Q e^{-} \\otimes e^{-}, $ giving the geometric structures $\\nabla e^+ &= (1-\\rho )e^{+} \\otimes e^{+} + (1-Q)(e^{-} \\otimes e^{+} + e^{+} \\otimes e^{-}),\\quad \\nabla e^- = (1-R_{-}^2(\\rho ^{-1})Q) e^{-} \\otimes e^{-}, \\\\R_{\\nabla }e^+ &= \\partial _{-}(\\rho R_{+}(Q)) e^+\\wedge e^{-} \\otimes e^{+} + (R_{+}(Q-1)R_{-}(\\rho ^{-1}) - (Q-1))e^+\\wedge e^{-} \\otimes e^{-}, \\\\R_{\\nabla }e^- &= \\partial _{-}( R_{-}(\\rho ^{-1})R_{+}(Q)) e^+\\wedge e^{-} \\otimes e^{+}, \\\\{\\rm Ricci} &= \\frac{1}{2}\\left(\\partial _{-}(R_{-}(\\rho )Q) e^{-} \\otimes e^{+} + \\partial _{+}(R_{+}(Q)R_{-}(\\rho ^{-1})) e^{+} \\otimes e^{-} +((Q-1)R_{-}^2(\\rho ^{-1}) -R_{-}(Q-1))e^{-} \\otimes e^{-} \\right), \\\\S &= \\frac{1}{2a}\\left(\\partial _{+}(R_{+}(Q)R_{-}(\\rho ^{-1})) - R_-(\\rho )\\partial _{-}(R_{-}(\\rho )Q) \\right),\\\\\\Delta f&= -\\left( \\frac{1}{R_{-}(a)} + \\frac{1}{a} \\right) ( \\partial _{-}f + Q\\partial _{+}f).$ ${\\rm (ii)}\\qquad \\qquad \\qquad \\sigma (e^{+} \\otimes e^{+}) &= \\rho Q e^{+} \\otimes e^{+}, \\quad \\sigma (e^{-} \\otimes e^{-}) = R_{-}^2(\\rho ^{-1}) e^{-} \\otimes e^{-}, \\\\\\sigma (e^{+} \\otimes e^{-}) &= e^{-} \\otimes e^{+}, \\quad \\sigma (e^{-} \\otimes e^{+}) = Qe^{+} \\otimes e^{-} +(Q-1)e^{-} \\otimes e^{+},$ giving the geometric structures $\\nabla e^+ &= (1-\\rho Q) e^{+} \\otimes e^{+},\\quad \\nabla e^- = (1-R_{-}^2(\\rho ^{-1})) e^{-} \\otimes e^{-} + (1-Q)( e^{+} \\otimes e^{-} + e^{-} \\otimes e^{+}), \\\\R_{\\nabla }e^+ &= \\partial _{-}(\\rho Q) e^+\\wedge e^{-} \\otimes e^{-}, \\\\R_{\\nabla }e^- &= (-R_{+}(Q-1)R_{-}(\\rho ) + Q-1)e^+\\wedge e^{-} \\otimes e^{+} + \\partial _{-}(Q R_{-}(\\rho ^{-1})) e^+\\wedge e^{-} \\otimes e^{-}, \\\\{\\rm Ricci} &= \\frac{1}{2}\\left(\\partial _{-}(R_{-}(\\rho Q)) e^{-} \\otimes e^{-} - (\\partial _{-}(R_{+}(Q)\\rho ^{-1}) e^{+} \\otimes e^{-} + (\\rho (Q-1) - R_{+}(Q-1))e^{+} \\otimes e^{+} \\right), \\\\S &= -\\frac{1}{2 a} \\partial _{-}(R_{+}(Q) \\rho ^{-1}), \\\\\\Delta f&= -\\left( \\frac{1}{R_{-}(a)} + \\frac{1}{a} \\right) ( Q\\partial _{-}f + \\partial _{+}f).$ For $n=4$, we have a further more general form for the generalised braiding $ \\sigma (e^{+} \\otimes e^{+}) = \\sigma _0 e^{+} \\otimes e^{+} + \\sigma _6 e^{-} \\otimes e^{-} , \\quad \\sigma (e^{+} \\otimes e^{-}) = \\sigma _1 e^{+} \\otimes e^{-} + \\sigma _2 e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{+}) = \\sigma _3 e^{+} \\otimes e^{-} + \\sigma _4 e^{-} \\otimes e^{+}, \\quad \\sigma (e^{-} \\otimes e^{-}) = \\sigma _5 e^{-} \\otimes e^{-}+ \\sigma _7 e^{+} \\otimes e^{+}$ for which the conditions for zero torsion are the same as before but metric compatibility now has a more complicated form due to the two extra parameters $\\sigma _6,\\sigma _7$ .", "The QLCs turn out to fall into 10 families of which 3 are the ones with $\\sigma _6 = \\sigma _7 = 0$ already covered above.", "In addition we have (i) a 4-parameter solution with a free nonzero function $\\gamma =(\\gamma _0, \\gamma _1, \\gamma _2, \\gamma _3)$ and $ \\sigma (e^{+} \\otimes e^{+}) &= \\gamma e^{-} \\otimes e^{-} ,\\quad \\sigma (e^{+} \\otimes e^{-}) = -e^{+} \\otimes e^{-}, \\\\ \\sigma (e^{-} \\otimes e^{+}) & = -e^{-} \\otimes e^{+},\\quad \\sigma (e^{-} \\otimes e^{-}) = R_-(\\gamma ^{-1})R_+(\\rho ^{\\prime }) e^{+} \\otimes e^{+}, \\\\\\nabla e^+ &= e^{+} \\otimes e^{+} + e^{-} \\otimes e^{+} + e^{+} \\otimes e^{-} - \\gamma e^{-} \\otimes e^{-},\\\\\\nabla e^- &= e^{-} \\otimes e^{-} + e^{+} \\otimes e^{-} + e^{-} \\otimes e^{+} - R_-(\\gamma ^{-1})R_{+}(\\rho ^{\\prime })e^{+} \\otimes e^{+},$ where $ \\rho ^{\\prime }={1\\over \\rho R_{+}\\rho }.$ This is $*$ -preserving if and only if $\\gamma $ has the 2-parameter form such that $R_+^2(\\gamma )=\\bar{\\gamma }^{-1}$ as in the main text.", "(ii) a 3-parameter solution with parameter $\\beta $ and functions $\\gamma =(p, q, p, q), \\quad \\delta ={pq-1\\over R_+(\\gamma )-1}=(pq-1)({1\\over q-1}, {1\\over p-1},{1\\over q-1},{1\\over p-1}),$ $ \\sigma (e^{+} \\otimes e^{+}) = \\rho (1-\\delta ) e^{+} \\otimes e^{+} + \\beta (\\gamma -1) \\rho ^{\\prime } e^{-} \\otimes e^{-}, \\quad \\sigma (e^{+} \\otimes e^{-}) = (\\gamma -1)e^{+} \\otimes e^{-} + \\gamma e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{+}) = (1-\\delta ) e^{+} \\otimes e^{-} - \\delta e^{-} \\otimes e^{+}, \\quad \\sigma (e^{-} \\otimes e^{-}) = -{ \\delta \\over \\beta R_+^2\\rho ^{\\prime } }e^{+} \\otimes e^{+} + {\\gamma \\over R^{2}_{+}\\rho }e^{-} \\otimes e^{-},$ where $\\rho ^{\\prime } = (\\frac{\\rho _0}{\\rho _2},\\rho _0\\rho _{1},1,\\rho _0\\rho _3),$ giving the QLC $\\nabla e^+ &= (1 - \\rho (1-\\delta )) e^{+} \\otimes e^{+} + (1-\\gamma )(e^{-} \\otimes e^{+} + e^{+} \\otimes e^{-}) + \\beta \\rho ^{\\prime }(1-\\gamma ) e^{-} \\otimes e^{-}, \\\\\\nabla e^- &= (1-{\\gamma \\over R^{2}_{+}\\rho })e^{-} \\otimes e^{-} + \\delta (e^{+} \\otimes e^{-} +e^{-} \\otimes e^{+}) + {\\delta \\over \\beta R_+^2\\rho ^{\\prime }}e^{+} \\otimes e^{+}.\\\\$ (iii) a 3-parameter solution with parameters $ \\beta $ and functions $\\gamma = ( p, 0, q, 0), \\quad \\delta = (1,{q\\over p},1,{p\\over q}),$ , $ \\sigma (e^{+} \\otimes e^{+}) &= R_-\\left({\\gamma \\over \\gamma -1}\\right) \\rho e^{+} \\otimes e^{+}+ { \\beta \\delta \\rho ^{\\prime } \\over 1-R_-(\\gamma ) } e^{-} \\otimes e^{-} , \\\\ \\sigma (e^{+} \\otimes e^{-}) &= ( \\gamma -1)e^{+} \\otimes e^{-} + \\gamma e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{+}) &= R_{+}\\left({\\gamma \\over \\gamma -1}\\right)e^{+} \\otimes e^{-} + \\frac{1}{R_{+}(\\gamma -1)}e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{-}) &= { R_-(\\delta ) \\over \\beta R_+^2(\\rho ^{\\prime })}(1-\\gamma )e^{+} \\otimes e^{+} + R^{2}_{+}({\\gamma \\over \\rho })e^{-} \\otimes e^{-},$ where $ \\rho ^{\\prime }= ({\\rho _0\\over \\rho _2}, \\rho _0\\rho _1 , 1 , \\rho _0\\rho _3), $ giving the QLC $\\nabla e^+ &= (1+R_-({ \\gamma \\over 1-\\gamma })\\rho ) e^{+} \\otimes e^{+} + (1-\\gamma )(e^{-} \\otimes e^{+} + e^{+} \\otimes e^{-} ) - {\\beta \\delta \\rho ^{\\prime } \\over 1 - R_-(\\gamma )} e^{-} \\otimes e^{-}, \\\\\\nabla e^- &= \\left(1-R^{2}_{+}\\left({\\gamma \\over \\rho }\\right)\\right)e^{-} \\otimes e^{-} + {1 \\over 1-R_{+}(\\gamma )}(e^{+} \\otimes e^{-}+e^{-} \\otimes e^{+}) - { R_-(\\delta ) \\over \\beta R_+^2\\rho ^{\\prime }}(1-\\gamma )e^{+} \\otimes e^{+}.$ (iv) a 3-parameter solution with parameters $\\beta $ and the functions $\\gamma = ( 0, p, 0, q ), \\quad \\delta = ({p\\over q},1,{q\\over p},1),$ , $ \\sigma (e^{+} \\otimes e^{+}) &= \\rho R_-({\\gamma \\over \\gamma -1}) e^{+} \\otimes e^{+}+ { \\beta \\delta \\rho ^{\\prime } \\over 1-R_-(\\gamma ) } e^{-} \\otimes e^{-} , \\\\ \\sigma (e^{+} \\otimes e^{-}) &= ( \\gamma -1)e^{+} \\otimes e^{-} + \\gamma e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{+}) &= R_{+}(\\frac{\\gamma }{\\gamma -1})e^{+} \\otimes e^{-} + \\frac{1}{R_{+}(\\gamma -1)}e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{-}) &= { R_-(\\delta ) \\over \\beta R_+^2(\\rho ^{\\prime })}(1-\\gamma )e^{+} \\otimes e^{+} + R^{2}_{+}({\\gamma \\over \\rho })e^{-} \\otimes e^{-},$ where $ \\rho ^{\\prime }= ({\\rho _0\\over \\rho _2} , \\rho _0\\rho _1 , 1, \\rho _0\\rho _3 ), $ giving the QLC $\\nabla e^+ &= (1+R_-({\\gamma \\over 1-\\gamma })\\rho ) e^{+} \\otimes e^{+} + (1-\\gamma )(e^{-} \\otimes e^{+} + e^{+} \\otimes e^{-} ) - {\\beta \\delta \\rho ^{\\prime } \\over 1 - R_-(\\gamma )} e^{-} \\otimes e^{-}, \\\\\\nabla e^- &= (1-R^{2}_{+}({\\gamma \\over \\rho }))e^{-} \\otimes e^{-} + {1\\over 1-R_{+}(\\gamma )}(e^{-} \\otimes e^{+}+e^{-} \\otimes e^{+}) - { R_-(\\delta ) \\over \\beta R_+^2\\rho ^{\\prime }}(1-\\gamma )e^{+} \\otimes e^{+}.$ (v) a 2-parameter solution with parameter $\\beta $ and $Q= ( q, q^{-1}, q, q^{-1} )$ as usual, $ \\sigma (e^{+} \\otimes e^{+}) = \\rho e^{+} \\otimes e^{+}, \\quad \\sigma (e^{+} \\otimes e^{-}) = (Q-1)e^{+} \\otimes e^{-} + Qe^{-} \\otimes e^{+},\\\\ \\sigma (e^{-} \\otimes e^{+}) = e^{+} \\otimes e^{-}, \\quad \\sigma (e^{-} \\otimes e^{-}) = \\beta \\rho ^{\\prime } e^{+} \\otimes e^{+} + R^{2}_{+}(\\rho ^{-1})Qe^{-} \\otimes e^{-},$ where $ \\rho ^{\\prime }= ( 1, -\\frac{\\rho _1\\rho _2}{q} , {\\rho _2\\over \\rho _0} , -\\frac{\\rho _2\\rho _3}{q} ), $ giving the QLC $\\nabla e^+ &= (1-\\rho )e^{+} \\otimes e^{+} + (1-Q)(e^{+} \\otimes e^{-} + e^{-} \\otimes e^{+}), \\\\\\nabla e^- &= (1-R^{2}_{+}(\\rho ^{-1})Q)e^{-} \\otimes e^{-} -\\beta \\rho ^{\\prime } e^{+} \\otimes e^{+}.$ (vi) a 2-parameter solution with parameter $\\beta $ and $Q= ( q, q^{-1}, q, q^{-1} )$ as usual, $ \\sigma (e^{+} \\otimes e^{-}) & = e^{-} \\otimes e^{+},\\quad \\sigma (e^{-} \\otimes e^{+}) = Qe^{+} \\otimes e^{-} + (Q-1)e^{-} \\otimes e^{+},\\\\ \\sigma (e^{+} \\otimes e^{+}) &= \\rho Qe^{+} \\otimes e^{+},\\quad \\sigma (e^{-} \\otimes e^{-}) = \\beta \\rho ^{\\prime }e^{+} \\otimes e^{+} + R^{2}_{+}(\\rho ^{-1})e^{-} \\otimes e^{-},$ where $ \\rho ^{\\prime }= ( 1, -\\frac{\\rho _1\\rho _2}{q} , {\\rho _2\\over \\rho _0} , -\\frac{\\rho _2\\rho _3}{q} ), $ giving the QLC $\\nabla e^+ &= (1-\\rho Q)e^{+} \\otimes e^{+}, \\\\\\nabla e^- &= (1-R^{2}_{+}(\\rho ^{-1}))e^{-} \\otimes e^{-} + (1-Q)(e^{+} \\otimes e^{-} +e^{-} \\otimes e^{+}) - \\beta \\rho ^{\\prime } e^{+} \\otimes e^{+}.$ (vii) a 2-parameter solution with parameter $\\beta $ and $Q= ( q, q^{-1}, q, q^{-1} )$ as usual, $ \\sigma (e^{+} \\otimes e^{+}) = - \\rho ^{\\prime }\\rho Q e^{+} \\otimes e^{+}+ \\beta \\rho ^{\\prime \\prime }e^{-} \\otimes e^{-} , \\quad \\sigma (e^{+} \\otimes e^{-}) = e^{-} \\otimes e^{+}, \\\\ \\sigma (e^{-} \\otimes e^{+}) = - \\rho ^{\\prime }Qe^{+} \\otimes e^{-} - ( \\rho ^{\\prime }Q+1)e^{-} \\otimes e^{+}, \\quad \\sigma (e^{-} \\otimes e^{-}) = R^{2}_{+}(\\rho ^{-1})e^{-} \\otimes e^{-},$ where $ \\rho ^{\\prime } = (\\rho _1\\rho _0, \\rho ^{-1}_0\\rho ^{-1}_1, \\rho _1\\rho _0, \\rho ^{-1}_0\\rho ^{-1}_1),\\quad \\rho ^{\\prime \\prime } = ({\\rho _0\\over \\rho _2}q,1,q, {\\rho _3\\over \\rho _1}), $ giving the QLC $\\nabla e^+ &= (1+ \\rho ^{\\prime }\\rho Q)e^{+} \\otimes e^{+} - \\beta \\rho ^{\\prime \\prime }e^{-} \\otimes e^{-}, \\\\\\nabla e^- &= (1 - R^{2}_{+}(\\rho ^{-1}))e^{-} \\otimes e^{-} + (1+ \\rho ^{\\prime }Q)(e^{+} \\otimes e^{-} + e^{-} \\otimes e^{+}).$ Note that ${\\mathbb {Z}}_4$ here is a different group from ${\\mathbb {Z}}_2\\times {\\mathbb {Z}}_2$ treated in [2][3], even though in both cases the graph is a square.", "This means that, although $\\Omega ^1$ and the metric can be made to match up and hence the metric compatibility part of the QLC condition is the same, $\\Omega ^2$ and hence the condition for torsion freeness are different.", "This work [2] also treats the ${\\mathbb {Z}}_2$ case." ] ]

[ [ "A Normalized Fully Convolutional Approach to Head and Neck Cancer\n Outcome Prediction" ], [ "Abstract In medical imaging, radiological scans of different modalities serve to enhance different sets of features for clinical diagnosis and treatment planning.", "This variety enriches the source information that could be used for outcome prediction.", "Deep learning methods are particularly well-suited for feature extraction from high-dimensional inputs such as images.", "In this work, we apply a CNN classification network augmented with a FCN preprocessor sub-network to a public TCIA head and neck cancer dataset.", "The training goal is survival prediction of radiotherapy cases based on pre-treatment FDG PET-CT scans, acquired across 4 different hospitals.", "We show that the preprocessor sub-network in conjunction with aggregated residual connection leads to improvements over state-of-the-art results when combining both CT and PET input images." ], [ "Introduction", "Cancer treatment planning remains a long process for the patient: from pre-treatment staging to post-therapy follow-up, many factors could have changed that can impact the effectiveness of the treatment.", "One of the key decisions of the physician is the choice of line of therapy, for which automatic outcome prediction can be beneficial.", "In head and neck cancer cases, positron emission tomography with fluorodeoxyglucose integrated with computed tomography (FDG PET-CT) for diagnosis and treatment planning [1] can be used as inputs to deep learning-based medical image analysis models.", "In previous works using this Cancer Imaging Archive (TCIA) Head-Neck-PET-CT dataset , random forests were used to classify overall survival (OS) based on a combination of both PET and CT extracted radiomics features and clinical information .", "More recently, an end-to-end convolutional neural network (CNN) was used to successfully predict radiotherapy outcomes using only the planning CT scans as input using the same dataset [3].", "In this work, we show that combining PET and CT image inputs improves binary classification performance.", "For this purpose, a CNN architecture was implemented using residual connections , which have been shown to reduce vanishing gradient problems through identity shortcuts and thus allowing deeper models, and aggregated convolutions, which have been shown to reduce the number of parameters in a layer without loss of performance.", "The combination of these two methods lead to a reduction in model size compared to previous state-of-the-art work by [3].", "Furthermore, a fully-convolutional network (FCN) sub-network is used as a preprocessor.", "showed previously that this technique acts as an image normalization on medical images for liver tumor segmentation.", "We hypothesize that reusing this method for multi-modality inputs for classification, in combination with the aforementioned modifications, will lead to overall performance improvements when predicting patient survival from pre-treatment FDG PET-CT." ], [ "Methodology and Experiments", "The Head-Neck-PET-CT dataset used in this study consisted of 298 head and neck cancer patients acquired from 4 different institutions in Quebec.", "Each patient had a pre-radiotherapy FDG PET-CT scan.", "Both PET and CT volumes were converted to 2D images using largest primary GTV lesion area slice selection.", "Images were normalized to 0 mean and unit standard deviation.", "The PET image was up-scaled to 512$\\times $ 512 before being concatenated to the CT image to form the 512$\\times $ 512$\\times $ 2 input image.", "The dataset presented only 56 (19%) cases with the OS target (death).", "Thus a resampling strategy with data augmentation (random flip, random shifts of 40%, random rotations of 20 deg) was used to rebalance the dataset.", "Our end-to-end binary classification model consists of two parts: a FCN sub-network and a fully convolutional classifier shown in Figure .", "The FCN consists of 4 downsampling blocks followed by 4 upsampling blocks.", "Each block is composed of a 3x3 convolution layer with SeLU activation and uses strided convolutions to modify the output dimension.", "Output features of each downsampling block are concatenated with input features of each corresponding upsampling blocks.", "Figure: Proposed model architecture (top).", "The input consists of a 2 channel PET-CT image that is initially passed through a FCN (lower left).", "Downsampling uses convolutions with stride 2 while upsampling uses transposed convolutions.", "The output is then fed to an 18-layer deep CNN.", "Aggregated residual convolutional blocks (lower right) are repeated twice before being downsampled by setting the stride to 2.", "Classification is performed by taking the output vector with 256 features through a fully connected layer with softmax activation.The CNN classifier is inspired by the ResNeXt architecture, with 2 3x3 bottleneck layers with a filter growth factor of 2 and cardinality of 32 and residual connections around each.", "Downsampling is done using strided convolutions in the first layer of each block.", "Global average pooling and a fully connected layer are used to output the binary survival class.", "Training is performed using categorical cross-entropy loss for classification (0: survival, 1: death) using the Adam optimizer with a learning rate of 0.0006 for 100 epochs and a batch size of 8.", "No early-stopping techniques were used.", "The code is implemented in Keras[2] and trained on a GeForce RTX 2080 Ti for 1 hour." ], [ "Results and Conclusions", "Binary classification results are monitored using the area under the receiver-operating characteristic curve (AUC).", "Table shows the progression of modifications from the CNN as presented by [3] up to our proposed model, using the different input modality.", "The best performing model using the CNN architecture proposed by [3] used the masked CT input which is the same model input as was used in their report.", "The performance observed is lower (0.67 $<$ 0.70) due to different initial seed conditions.", "With the addition of the FCN preprocessor, the performance drops to 0.59 for PET, 0.65 for CT, 0.63 for masked CT but improves to 0.70 for PET+CT.", "This shows that the FCN works by mapping features from multiple domains (modalities) to a latent embedding that benefits classification performance.", "Our proposed CNN with aggregated convolutions and residual connections (AggResCNN) performs better than previous state-of-the-art deep learning methods (0.74 $>$ 0.70) and previous radiomics-based random forest methods (0.74 $>$ 0.65) when considering the area under the receiver-operating characteristic curve (AUC) and using both PET and CT as inputs With the addition of the FCN preprocessor, this performance improves to 0.76 AUC.", "The final proposed FCN+AggResCNN thus improves over previous methods (0.76 $>$ 0.70 $>$ 0.65) on the same dataset while having less total parameters (683650 $<$ 930146).", "Furthermore, the model trained on masked CT performs better than the whole CT for the CNN case (0.67 $>$ 0.57) while the reverse is true for the FCN+CNN case (0.63 $<$ 0.65).", "Similarly, the masked CT performance is greater than the full CT image for the AggResCNN (0.69 $>$ 0.65), while the reverse is true for the FCN+AggResCNN (0.67 $<$ 0.70).", "This suggests that the FCN learns features on information outside the GTV boundaries to be effective.", "In every case, classification performance on just the PET input is worse than the rest, which suggests that this imaging modality only contributes auxiliary information that complements the CT image.", "The cause of this apparent worse performance can be attributed both to the lower image resolution and the absence of image information in PET away from the tumor lesion, as healthy tissues don't absorb the radioactive tracing agent.", "Table: Classification performance of the proposed models compared to state of the art results on the Head and Neck FDG PET-CT TCIA Dataset.", "The first model consists of a random forest for radiomics features selection followed by another random forest for classification.", "The second model consists of a CNN with 3 convolutions/PReLU/max-pooling layers and 2 fully connected layers.", "The last three models show ablation results of our proposed FCN 18-ResNeXt trained on PET and CT images.Finally, the proposed architecture using aggregated convolutions and residual connections improves over state-of-the-art deep learning methods and radiomics based machine learning method by incorporating additional imaging data in the form of PET images.", "With the adapted architecture, the model can extract more features that help binary survival outcome prediction all while lowering the memory cost of the model using state-of-the-art convolutional techniques.", "Thus, after training on both CT and PET images, our proposed model has overall less total parameters (683 650 $<$ 930146) and improves in AUC by 6 percentage points over the state of the art." ] ]

[ [ "copent: Estimating Copula Entropy and Transfer Entropy in R" ], [ "Abstract Statistical independence and conditional independence are two fundamental concepts in statistics and machine learning.", "Copula Entropy is a mathematical concept defined by Ma and Sun for multivariate statistical independence measuring and testing, and also proved to be closely related to conditional independence (or transfer entropy).", "As the unified framework for measuring both independence and causality, CE has been applied to solve several related statistical or machine learning problems, including association discovery, structure learning, variable selection, and causal discovery.", "The nonparametric methods for estimating copula entropy and transfer entropy were also proposed previously.", "This paper introduces copent, the R package which implements these proposed methods for estimating copula entropy and transfer entropy.", "The implementation detail of the package is introduced.", "Three examples with simulated data and real-world data on variable selection and causal discovery are also presented to demonstrate the usage of this package.", "The examples on variable selection and causal discovery show the strong ability of copent on testing (conditional) independence compared with the related packages.", "The copent package is available on the Comprehensive R Archive Network (CRAN) and also on GitHub at https://github.com/majianthu/copent." ], [ "Introduction", "Statistical independence and conditional independence are two fundamental concepts in statistics and machine learning.", "The research on mathematical tool for their measurement date back to the early days of the statistics discipline.", "The most widely used tool is correlation coefficients proposed by Pearson [22].", "However, it is only applicable to linear cases with Gaussian assumptions.", "The other popular tool for statistical independence is Mutual Information (MI) in information theory [3], which is defined for bivariate cases.", "Copula is the theory on representation of dependence relationships [20], [10].", "According to Sklar theorem [26], any probabilistic distribution can be represented as a copula function with marginal functions as its inputs.", "Based on this representation, Ma and Sun [18] proposed a mathematical concept for statistical independence measurement, named Copula Entropy (CE).", "They also proved the equivalence between MI and CE.", "CE enjoys several properties which an ideal statistical independence measure should have, such as multivariate, symmetric, non-negative (0 iff independence), invariant to monotonic transformation, and equivalent to correlation coefficient in Gaussian cases.", "The nonparametric method for estimating CE was also proposed in [18], which is composed of two simple steps: estimating empirical copula function with rank statistic and estimating CE with the k-Nearest Neighbour (kNN) method proposed in [11].", "Transfer Entropy (TE) (or conditional independence) [25] is the fundamental concept for testing causality or conditional independence, which generalizes Granger Causality to more broader nonlinear cases.", "Since it is model-free, TE has great potential application in different areas.", "However, estimating TE is a hard problem if without assumptions.", "Recently, we proved that TE can be represented with CE only [14].", "According to this representation, the nonparametric method for estimating TE via CE is proposed in [14].", "In summary, CE provides a unified theoretical framework for measuring both statistical independence and conditional independence.", "In this framework, statistical independence and conditional independence (causality) can be measured with only CE [18], [14].", "As a fundamental tool, CE has been applied to solve several basic problems, including association discovery [13], structure learning [17], variable selection [15], and causal discovery [14].", "There are two similar theoretical frameworks for testing both independence and conditional independence based on kernel tricks in machine learning [5], [32] and distance covariance/correlation [29], [27], [31].", "Both frameworks can be considered as nonlinear generalization of traditional (partial) correlation, and both have non-parametric estimation methods.", "The kernel-base framework is based on the idea, called kernel mean embedding, that test correlation [5] or partial correlation [32] by transforming distributions into RKHS with kernel functions.", "The other framework is based on a concept called distance correlation defined with characteristic function [29], [27].", "With this concept, Wang et al.", "[31] defined a concept for conditional independence testing, called conditional distance correlation, with characteristic function for conditional function.", "Compared with these two frameworks, the framework based on CE is much sound theoretically due to the rigorous definition of CE and TE and much efficient computationally due to the simple estimation methods.", "This paper introduces copent [16], the R [24] package which implements the nonparametric method for estimating CE and TE proposed in [18], [14], and now is available on CRAN at https://CRAN.R-project.org/package=copent.", "The latest release of the package is available on GitHub at https://github.com/majianthu/copent.", "The copent package in Python [30] is also provided on the Python Package Index (PyPI) at https://pypi.org/project/copent.", "As the implementation of the nonparametric estimation of CE and TE, the copent package has great potentials in real applications of CE and TE, as demonstrated with the examples in this paper.", "There are several R packages which implement the estimation of other popular statistical independence measures, such as energy for distance correlation [29], [28], dHSIC for multivariate Hilbert-Schmidt Independence Criterion (HSIC) [6], [23], HHG for Heller-Heller-Gorfine Tests of Independence [7], [8], independence for Hoeffding's D [9] and Bergsma-Dassios T* sign covariance [2], and Ball for ball correlation [21].", "In this paper, we will compare them with the copent package on the variable selection problem with real-world data in an example in Section REF .", "Several R packages for testing conditional independence are also available on CRAN, including CondIndTests on kernel-based test [32], and cdcsis on conditional distance correlation [31].", "These two packages are the implementations of the methods in the other two frameworks mentioned above.", "In the example on causal discovery in Section REF , we will compare them with the method for estimating TE implemented in the copent package.", "This paper is organized as follows: the theory, estimation, and applications of CE and TE are introduced in Section , Section presents the implementation details of the copent package with an open dataset, and then three examples on simple simulation experiment, variable selection and causal discovery are presented to further demonstrate the usage of the copent package and to compare our package with the related packages in Section , and Section summarizes the paper.", "Copula theory unifies representation of multivariate dependence with copula function [20], [10].", "According to Sklar theorem [26], multivariate density function can be represented as a product of its marginals and copula density function which represents dependence structure among random variables.", "This section is to define an association measure with copula.", "For clarity, please refer to [18] for notations.", "With copula density, Copula Entropy is define as follows [18]: Definition 1 (Copula Entropy) Let $\\mathbf {X}$ be random variables with marginals $\\mathbf {u}$ and copula density $c(\\mathbf {u})$ .", "CE of $\\mathbf {X}$ is defined as $H_c(\\mathbf {X})=-\\int _{\\mathbf {u}}{c(\\mathbf {u})\\log {c(\\mathbf {u})}}d\\mathbf {u}.$ In information theory, MI and entropy are two different concepts [3].", "In [18], Ma and Sun proved that MI is actually a kind of entropy, negative CE, stated as follows: Theorem 1 MI of random variables is equivalent to negative CE: $I(\\mathbf {X})=-H_c(\\mathbf {X}).$ Theorem REF has simple proof [18] and an instant corollary (Corollary REF ) on the relationship between information containing in joint probability density function, marginals and copula density.", "Corollary 1 $H(\\mathbf {X})=\\sum _{i}{H(X_i)}+H_c(\\mathbf {X})$ The above results cast insight into the relationship between entropy, MI, and copula through CE, and therefore build a bridge between information theory and copula theory.", "CE itself provides a theoretical concept of statistical independence measure.", "Conditional independence is another fundamental concept in statistics with wide applications.", "TE is a statistical measure for causality, which is essentially conditional independence testing.", "It is defined by Schreiber [25] as follows: Definition 2 (Transfer Entropy) Let $x_t,y_t$ be two time series observations at time $t=1,\\ldots ,N$ of the processes $X_t,Y_t$ .", "The transfer entropy $T_{Y \\rightarrow X}$ from $Y$ to $X$ is defined as $T_{Y \\rightarrow X} = \\sum {p(x_{t+1},x_t,y_t)\\log {\\frac{p(x_{t+1}|x_t,y_t)}{p(x_{t+1}|x_t))}}}.$ It can be written as conditional MI between $x_{t+1}$ and $y_t$ conditioned on $x_t$ : $T_{Y \\rightarrow X} = I(x_{t+1};y_t|x_t).", "$ In [14], Ma proved that CE is closely related with TE, stated as the following proposition: Proposition 1 TE can be represented with only CE as follows: $T_{Y \\rightarrow X}=-H_c(x_{t+1},x_t,y_t)+H_c(x_{t+1},x_t)+H_c(y_t,x_t).$ This proposition can be easily proved by expanding the definition of TE [14].", "This result gives the way of measuring causality with CE.", "Therefore, we developed a theoretical framework for measuring both statistical independence and conditional independence with only CE." ], [ "Estimating CE and TE", "It is widely considered that estimating MI is notoriously difficult.", "Under the blessing of Theorem REF , Ma and Sun [18] proposed a non-parametric method for estimating CE (MI) from data which is composed of only two simple steps: Estimating Empirical Copula Density (ECD); Estimating CE.", "For Step 1, if given data samples $\\lbrace \\mathbf {x}_1,\\ldots ,\\mathbf {x}_T\\rbrace $ i.i.d.", "generated from random variables $\\mathbf {X}=\\lbrace x_1,\\ldots ,x_N\\rbrace ^T$ , one can easily estimate ECD as follows: $F_i(x_i)=\\frac{1}{T}\\sum _{t=1}^{T}{\\chi (\\mathbf {x}_{t}^{i}\\le x_i)},$ where $i=1,\\ldots ,N$ and $\\chi $ represents for indicator function.", "Let $\\mathbf {u}=[F_1,\\ldots ,F_N]$ , and then one can derives a new samples set $\\lbrace \\mathbf {u}_1,\\ldots ,\\mathbf {u}_T\\rbrace $ as data from ECD $c(\\mathbf {u})$ .", "Once ECD is estimated, Step 2 is essentially a problem of entropy estimation which can be tackled by many existing methods.", "Among those methods, the kNN method [11] was suggested in [18], which leads to a non-parametric way of estimating CE.", "As a model-free measure of causality, TE has great potential applications in many areas.", "However, estimating TE is also widely considered as a hard problem.", "Proposition REF presents a representation of TE with only CE.", "This representation suggests a method for estimating TE via CE [14], which composes of two steps: Estimating three CE terms in (REF ); Calculating TE with the estimated CE terms.", "Here, Step 1 is proposed to be done with the above nonparametric method for estimating CE, and hence we proposed a simple and elegant nonparametric method for estimating TE.", "This nonparametric method makes it possible for applying TE to real problem without any assumptions on the underlying dynamical systems." ], [ "Applications", "CE has been applied to solve several typical statistical problems, including: Association Measuring [13].", "CE is used as an association measure, which enjoys many advantages over the traditional association measure, such as Pearson's correlation coefficient.", "Structure Learning [17].", "Based on dependence relationship between random variables measured by CE, a graph can be derived with the maximal spanning tree algorithm.", "Variable Selection [15].", "For regression or classification tasks, variables can be selected based on statistical independence strength between variables and target variable measured by CE.", "Due to the merits of CE, such selection is both model-free and tuning-free.", "An example in Section REF demonstrates the variable selection method with CE.", "Causal Discovery [14].", "To discover causal relationships from observational data, transfer entropy can be estimated via CE non-parametrically to measure causality as proposed in Section REF .", "Such estimation makes no assumption on the underlying mechanism and can be applied to any cases provided time series data are available.", "An example in Section REF demonstrates the method.", "The copent package contains five functions as listed in Table REF .", "The function copent() is the main function which implements the method for estimating CE and the other two functions construct_empirical_copula() and entknn() are called by copent() as two steps of the estimation method.", "The function ci() implements the method for conditional independence testing, which calls the function copent().", "The function transent() implements the method for estimating TE [14], which calls the function ci() since estimating TE is essentially conditional independence testing.", "Table: The functions in the package.", "k,dtype represent the arguments for k th k^{th} nearest neighbour, and distance type respectively.To illustrate the implementation and usage of the copent() function, we use the “airquality” dataset in R as a working dataset, which contains daily air quality measurements in New York, May to September 1973.", "R> library(copent) R> data(\"airquality\") R> x1 = airquality[,1:4] The function construct_empirical_copula() estimates empirical copula from data with rank statistic.", "After the four numerical measurements are loaded, the corresponding empirical copula function can be derived by the function construct_empirical_copula() as follows: R> xc1 = constructempiricalcopula(x1) The estimated empirical copula of the four measurements is illustrated in Figure REF .", "Figure: The joint distribution of the four measurements (upper panels) and the estimated empirical copula (lower panels).The function entknn() implements the kNN method for estimating entropy proposed in [11].", "It is based on the following estimation equation: $\\hat{H}(X) = -\\psi (k) + \\psi (N) + \\log {c_d} + \\frac{d}{N}\\sum _{i=1}^{N}{\\log {\\epsilon (i)}}.$ Here, $\\psi ()$ is the digamma function; the $c_d$ is the volume of the d-dimensional unit ball, for which two cases are implemented: $c_d=1$ for the maximum norm and $c_d={\\pi ^{d/2}}/{\\Gamma (1+d/2)}/{2^d}$ for Euclidean norm; and $\\epsilon ()$ is twice the distance from the sample to its k-th nearest neighbor.", "In the package, the function entknn() has three arguments, two of which are k and dtype, k-th neighbor and distance type (maximum norm or Euclidean norm) which are used for computing the last two terms in the above estimation equation.", "Now we can use the function entknn() to estimate the entropy of empirical copula of these four measurements: R> entknn(xc1) [1] -0.03305222 Here we use the default value of k and dtype because of the good estimation performance of the kNN method for estimating entropy.", "The main function copent() implements the method in Section REF .", "As shown above, it simply call the function construct_empirical_copula() to derive empirical copula function from data and then use the estimated empirical copula as input of the function entknn() to estimate CE.", "For user's convenience, the function copent() returns negative value of CE.", "Here, the negative CE of the four measurement can be easily estimated with copent(): R> copent(x1) [1] 0.03305222 With the main function copent(), we can easily implement the other two functions based on their theoretical relationships with CE.", "The function ci() for testing conditional independence (x,y) conditioned on z is implemented based on (REF ) with two steps: first estimating three CEs terms of (x,y,z),(y,z) and (x,z) by calling copent() and then calculating the result from the estimated terms.", "Since TE is essentially conditional independence, the function transent() for estimating TE from y to x with time lag lag is then implemented as conditional independence testing with two simple steps: first preparing the data of $x_{t+lag}$ , $x_t$ ,and $y_t$ from x,y according to lag and then call the function ci() on the prepared data according to the relationship between TE and conditional independence (REF ).", "We will demonstrate the usage of these two functions in Section REF ." ], [ "Examples", "To further demonstrate the usage and advantages of the copent package, three examples are presented in this section: the first one based on simulated data and the second and third one based on real-world data for variable selection and causal discovery respectively.", "We will compare our package with the related packages in the last two examples." ], [ "Simulation Example", "This demonstration example is based on the simulated data.", "We generate the simulated data with the mnormt [1] package.", "R> library(copent) R> library(mnormt) First, 500 data samples are generated from bivariate Gaussian distribution.", "Without loss of generality, the correlation coefficient $\\rho $ is set as 0.75.", "R> rho = 0.75 R> sigma = matrix(c(1,rho,rho,1),2,2) R> x = rmnorm(500,c(0,0),sigma) The negative CE of bivariate Gaussian can be calculated analytically as $-\\log (1-\\rho ^2)/2$ : R> truevalue = -0.5 * log(1- rho2) R> truevalue [1] 0.4133393 With the function copent(), the estimated value is: R> copent(x) [1] 0.4039309" ], [ "Example on Variable Selection", "The second example The code for this example is available at https://github.com/majianthu/aps2020.", "is about the application of the package on variable selection [15].", "The data used here is the heart disease dataset in the UCI machine learning repository [4], which contains 4 databases about heart disease diagnosis collected from four locations.", "The dataset includes 920 samples totally, of which only 899 samples without missing values are used in the example.", "Each sample is with 76 raw attributes, of which the attribute `num' is the diagnosis of patients' disease and 13 other attributes are recommended by professional clinicians as relevant [19].", "The aim of the example is to select a subset of attributes with statistical independence criteria for building the model for predicting disease status.", "The performance of the methods will be measured by the number of the selected attributes out of 13 recommended ones.", "Besides CE, several other independence measures are also considered as contrasts in the example, including Hilbert-Schmidt Independence Criterion (HSIC) [6], [23], Distance Correlation [29], [28], Heller-Heller-Gorfine Tests of Independence [7], [8], Hoeffing's D Test [9], Bergsma-Dassios T* sign covariance [2], and Ball Correlation [21].", "The R packages used as the implementation of the above measures are dHSIC, energy, HHG, independence, and Ball respectively.", "The example first loads the related packages: library(copent) Copula Entropy library(energy) Distance Correlation library(dHSIC) Hilbert-Schmidt Independence Criterion library(HHG) Heller-Heller-Gorfine Tests of Independence library(independence) Hoeffding's D test or Bergsma-Dassios T* sign covariance library(Ball) Ball correlation And then load the data samples for the UCI repository with the following codes: scanheartdata <-function(filename1, nl = 0) data1 = scan(filename1, nlines = nl, what = c(as.list(rep(0,75)),list(\"\"))) l = length(data1[[1]]) data1m = matrix(unlist(data1), l, 76) matrix(as.numeric(data1m[,1:75]), l, 75) load heart disease data (899 samples) dir = \"http://archive.ics.uci.edu/ml/machine-learning-databases/heart-disease/\" h1 = scanheartdata(paste(dir,\"cleveland.data\",sep=\"\"), 282*10) h2 = scanheartdata(paste(dir,\"hungarian.data\",sep=\"\")) h3 = scanheartdata(paste(dir,\"switzerland.data\",sep=\"\")) h4 = scanheartdata(paste(dir,\"long-beach-va.data\",sep=\"\")) heart1 = as.matrix( rbind(h1,h2,h3,h4) ) m = dim(heart1)[1] n = dim(heart1)[2] The above codes load the 899 samples from four datasets, of which the Cleveland dataset has only 282 samples without missing values.", "With the following codes, we estimates the strength of statistical independence between 75 attributes and disease diagnosis 'num' (#58).", "l = 50 ce58 = rep(0,n) for (i in 1:n) for (j in 1:l) data2 = heart1[,c(i,58)] data2[,1] = data2[,1] + max(abs(data2[,1])) * 0.000001 * rnorm(m) data2[,2] = data2[,2] + max(abs(data2[,2])) * 0.000001 * rnorm(m) ce58[i] = ce58[i] + copent(data2) ce58 = ce58 / l To avoid numerical instability of the estimation algorithm, we add a slight Gaussian noise on the data.", "As consequents, we run the estimation for 50 times to average the uncertainty caused by the added noise.", "In this way, we can estimate CE from any data with both continuous and discrete values.", "The code for the other measures is as following: dcor58 = rep(0,n) Distance Correlation dhsic58 = rep(0,n) Hilbert-Schmidt Independence Criterion hhg58 = rep(0,n) Heller-Heller-Gorfine Tests ind58 = rep(0,n) Hoeffding's D test or Bergsma-Dassios T* sign covariance ball58 = rep(0,n) Ball correlation for (i in 1:n) dcor58[i] = dcor(heart1[,i],heart1[,58]) dhsic58[i] = dhsic(heart1[,i],heart1[,58])$dHSICDx = as.matrix(dist((heart1[,i]), diag = TRUE, upper = TRUE))Dy = as.matrix(dist((heart1[,58]), diag = TRUE, upper = TRUE))hhg58[i] = hhg.test(Dx,Dy, nr.perm = 1000)ind58[i] = hoeffding.D.test(heart1[,i],heart1[,58])$ Dn ind58[i] = hoeffding.refined.test(heart1[,i],heart1[,58])$Rn#ind58[i] = tau.star.test(heart1[,i],heart1[,58])$ Tn ball58[i] = bcor(heart1[,i],heart1[,58]) To compare the performance of all the measures, we check the interpretability of the selected attributes with them.", "The recommended variables are taken as references for the selection since they are recommended by professional researchers as clinical relevant.", "We choose the #16 attribute carefully as threshold of selection for every measure's results.", "Such selected results of different measures are shown in Figure REF .", "Figure: dHSICFigure: Ball CorrelationThe variables selected for different measures are summarized in Table REF .", "It can be learned that CE selected 11 out of 13 recommended attributes, better than all the other methods did, which means CE make much interpretable models with biomedical meaningful attributes.", "This result shows clearly the advantage of the copent package as a tool for variable selection problem.", "Table: Selected variables by different measures." ], [ "Example on Causal Discovery", "The third example The code for this example is available at https://github.com/majianthu/transferentropy.", "is based on the Beijing PM2.5 dataset on the UCI machine learning repository [4], which is about air pollution at Beijing.", "This hourly data set contains the PM2.5 data of US Embassy in Beijing.", "Meanwhile, meteorological data from Beijing Capital International Airport are also included.", "The data has been analyzed at month scale [12].", "With this data, we try to discover the causal relationships between meteorological factors and PM2.5 by estimating transfer entropy via CE with the method proposed in [14].", "We also compare our method with the kernel-based methods on conditional independence testing [32] in the package CondIndTests, and conditional distance correlation [31] in the package cdcsis.", "The example first loads the related packages: library(copent) Copula Entropy library(CondIndTests) kernel based test library(cdcsis) conditional distance correlation Then the data is loaded from the UCI repository.", "We select only a part of data as the working set.", "For illustration purpose, the factors on PM2.5 and pressure are chosen in this example.", "Meanwhile, to avoid tackling missing values, only a continuous part of 501 hours data without missing values are used.", "dir = \"https://archive.ics.uci.edu/ml/machine-learning-databases/00381/\" prsa2010 = read.csv(paste(dir,\"PRSAdata2010.1.1-2014.12.31.csv\",sep=\"\")) data = prsa2010[2200:2700,c(6,9)] We consider causal relationship from pressure to PM2.5 with time lag from 1 hour to 24 hour.", "By setting time lag lag as 1 hour, we prepare the working set as follows: lag = 1 pm25a = data[1:(501-lag),1] pm25b = data[(lag+1):501,1] pressure = data[1:(501-lag),2] where pm25a and pm25b is the PM2.5 time series for `now' and `1 hour later', and pressure is the pressure time series for `now'.", "So the TE from pressure to PM2.5 with 1 hour lag can be easily estimated with the function ci() for conditional independence testing: te1[lag] = ci(pm25b,pressure,pm25a) Or, the estimation can also be done without preparing the working data set by simply calling the function transent() on the original data with the time lag argument lag: te1[lag] = transent(data[,0],data[,1],lag) The same conditional independence can also be estimated with the kernel-based and distance-based methods from the prepared data as follows: kci1[lag] = KCI(pm25b,pressure,pm25a)$testStatisticcdc1[lag] = cdcor(pm25b,pressure,pm25a)$ By setting the time lag lag from 1 to 24, we get the estimated values of three methods as illustrated in Figure REF .", "It can be learned from the Figure that the change of TE strength is well estimated which can be interpreted with meteorological meanings [14].", "Particularly, the trend of the estimated TE has mainly two phases: the sharp increasing phase and the slow increasing phase.", "The estimation with conditional distance correlation presents similar results as ours while the results with kernel-based method does not show such trend, which is different from ours and that of the kernel-based method.", "Figure: Kernel-base Conditional Independence" ], [ "Summary", "CE is a fundamental concept for multivariate statistical independence measuring and testing and TE is a model-free concept for measuring causality.", "It has been proved that TE can be represented with only CE.", "Therefore, CE provides a unified theoretical framework for measuring both independence and conditional independence (or TE).", "It has been applied to solve several related statistical or machine learning problems.", "We have proposed the nonparametric methods for estimating CE and TE previously.", "In this paper, copent, the R package implementing the proposed methods, is introduced with implementation details.", "Three examples with simulated data and two UCI datasets on variable selection and causal discovery illustrate the usage of this package.", "The examples on variable selection and causal discovery show the strong ability of the copent package on testing (conditional) independence compared with the related packages.", "The copent package in R is available on the CRAN and also on GitHub at https://github.com/majianthu/copent." ], [ "Computational details", "The results in this paper were obtained using R 3.6.3 with the datasets 3.6.3, copent 0.2, energy 1.7-7, dHSIC 2.1, HHG 2.3.2, independence 1.0.1, Ball 1.3.10, cdcsis 2.0.3, CondIndTests 0.1.5, and mnormt 1.5-7 packages.", "R itself and all packages used are available from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/.", "The UCI Heart Disease dataset and Beijing PM2.5 dataset were accessed at March 6th, 2021.", "The code of the copent package was first developed during the author's PhD study at Tsinghua University." ] ]

[ [ "Competing Marangoni and Rayleigh convection in evaporating binary\n droplets" ], [ "Abstract For a small sessile or pendant droplet it is generally assumed that gravity does not play any role once the Bond number is small.", "This is even assumed for evaporating binary sessile or pendant droplets, in which convective flows can be driven due to selective evaporation of one component and the resulting concentration and thus surface tension differences at the air-liquid interface.", "However, recent studies have shown that in such droplets gravity indeed can play a role and that natural convection can be the dominant driving mechanism for the flow inside evaporating binary droplets (Edwards et al., Phys.", "Rev.", "Lett.", "121, 184501 (2018); Li et al., Phys.", "Rev.", "Lett.", "122, 114501 (2019)).", "In this study, we derive and validate a quasi-stationary model for the flow inside evaporating binary sessile and pendant droplets, which successfully allows to predict the prevalence and the intriguing interaction of Rayleigh and/or Marangoni convection on the basis of a phase diagram for the flow field expressed in terms of the Rayleigh and Marangoni numbers." ], [ "Introduction", "Evaporating droplets frequently occur in nature and applications, be it a rain droplet evaporating on a leaf, a droplet on a hot surface in spray cooling, a droplet of insecticides sprayed on a leaf or an inkjet-printed ink droplet on paper.", "Many of such droplets are multicomponent, i.e.", "consisting of a mixture of liquids.", "From the physical point of view, an evaporating multicomponent droplet in a host gas is paradigmatic for combined multi-phase and multi-component flow including a phase transition.", "Scientifically, this process encompasses the various fields of fluid mechanics, thermodynamics and also aspects from the field of chemistry.", "The evaporation dynamics is also relevant for the deposit left behind the evaporation of a particle-laden droplet.", "Here, pioneering work was done by [4] around 20 years ago, when they identified the coffee-stain effect – i.e.", "the phenomenon of finding a typical ring structure of deposited particles after the evaporation of a coffee droplet – and successfully explained it by the combination of a non-uniform evaporation rate along the droplet interface and a pinned contact line.", "In applications, one usually wants to prevent such coffee-stain effect, e.g.", "for obtaining a homogeneous deposition pattern in inkjet printing [31], [21], [16], [36].", "For reviews on evaporating pure droplets we refer to [2] and [11].", "Preventing the coffee-stain effect can be achieved by altering the flow inside the droplet during the drying process by inducing gradients in the acting forces.", "Focussing on the interfacial forces first, a tangential gradient of the surface tension along the liquid-gas interface leads to the well-known Marangoni effect, i.e.", "a tangential traction that drives the liquid towards positions of higher surface tension [35], [32].", "By that, the entire flow in the droplet can be altered from the typical outwards flow towards the contact line to a recirculating flow driven by a persistent Marangoni effect [18].", "For the case of a pure droplet, the necessary gradient in surface tension can be generated by thermal effects, e.g.", "either self-induced by latent heat of evaporation or externally imposed by heating or cooling the substrate [13], [39], [9], [43].", "The other mechanism to induce Marangoni flow is known as solutal Marangoni effect, which is usually much stronger.", "For solutal Marangoni flow, the droplet must consist of more than one component, e.g.", "a solvent and one or more surfactants [42], [30], [22] or a solvent and possibly multiple co-solvents [37], [3], [44], [25] or dissolved salts [40], [29].", "For a recent perspective review on droplets consisting of more than one component, we refer to [27].", "The difference in the volatilities of the individual constituents leads to preferential evaporation of one or the other component and thereby compositional gradients are induced.", "Since the surface tension is a function of the composition and due to the non-uniform evaporation profile, a surface tension gradient along the liquid-gas interface can build up and result in a similar Marangoni circulation as in the thermally-driven case.", "The nature of the resulting flow can be quite different, mostly depending on whether the evaporation process leads to an overall decreasing or increasing surface tension, i.e.", "whether the more volatile component has a higher or lower surface tension than the less volatile component.", "In a binary droplet consisting e.g.", "of water and glycerol, with water being more volatile and having the higher surface tension, the overall surface tension decreases during the preferential evaporation of water and the resulting Marangoni flow is usually regular, axisymmetric and directed towards the position of the lowest evaporation rate of water, i.e.", "towards the contact line for contact angles above ${90}{}$ and towards the apex for contact angles below ${90}{}$ [7], [6].", "On the contrary, e.g.", "in case of a binary droplet consisting of water and ethanol, where the overall surface tension increases due to the predominant evaporation of ethanol, the typical Marangoni effect is way more violent and chaotic [3], [1].", "Here, in particular, the axial symmetry of the droplet is usually broken, leading to a complicated scenario of initially chaotic flow driven by the solutal Marangoni effect and followed by either thermal Marangoni flow or the typical coffee-stain flow, when almost only water is left [8].", "Remarkably, the presence of a strong Marangoni effect can also have a significant influence on the shape and wetting behaviour of droplets [46], [19].", "Finally, the evaporation of mixture droplets can show a variety of additional intriguing phenomena, e.g.", "multiple phase changes and microdroplet nucleation in ternary droplets like ouzo [44], [45], and phase segregation in binary droplets [25] or rather homogeneous deposition patterns by an interplay of Marangoni flow, surfactants and polymers [20].", "As highlighted above, besides the gradient in the surface tension, i.e.", "in the interfacial forces, also gradients in the mass density, i.e.", "in the bulk force due to gravity, can influence the flow by natural convection.", "Similar to the surface tension, the mass density is a function of the temperature and, in the case of mixtures, of the composition, so that thermally and solutally driven natural convection can be realized in evaporating droplets.", "Flow driven by natural convection is one of the most important fields of fluid mechanics, as e.g.", "in Rayleigh-Bénard systems, however, these are usually investigated at large spatial dimensions.", "For small droplets, on the other hand, one would naively expect that the flow in case of thermal or solutal gradients is predominantly driven by the Marangoni effect, since a small droplet is associated with a small Bond number and hence surface tension effects would dominate over gravity.", "As a consequence, most studies on droplet evaporation focus on the Marangoni effect, but disregard the presence of natural convection by this argument.", "Recent studies, however, showed that even for small droplets with small Bond numbers, the internal flow can be decisively determined by natural convection and not by Marangoni flow [10], [23].", "This has been even found at the later stages of water-ethanol droplets, which initially show a very intense chaotic Marangoni flow [10].", "Obviously, these findings give rise to the following question: Under what circumstances which kind of flow pattern can be found in an evaporating binary droplet, i.e.", "when is the flow dominated by the Marangoni effect and when by natural convection?", "In this manuscript, we answer this questions by carefully investigating both kinds of driving forces and their mutual interaction.", "The corresponding effects can be quantified by non-dimensional numbers, namely the Marangoni number for flow due to surface tension gradients and the Rayleigh (or Archimedes/Grashof) number for the natural convection.", "By considering quasi-stationary instants during the drying process, these numbers successfully allow to predict the flow inside the droplets on the basis of phase diagrams in the $\\mbox{\\it Ra}$ -$\\mbox{\\it Ma}$ parameter space.", "We also validated these phase diagrams with full simulations and corresponding experiments.", "The paper is organised as follows: We will first present the complete set of dynamical equations describing the evaporation of a binary mixture droplet.", "In section , these equations are solved to discuss an illustrative example case.", "We will then introduce the quasi-stationary approximation in section and discuss the phase diagrams obtained by this model in section .", "The paper ends with a conclusion and a comparison with experimental data in the appendix." ], [ "Governing Equations", "The evaporation of a mixture droplet is a multi-phase and free interface problem with multi-component dynamics in both the liquid and gas phase.", "For a binary droplet, the liquid is constituted by two components, $\\alpha =\\text{A},\\,\\text{B}$ , whereas the gas phase is in general a ternary gas mixture of the ambient medium, e.g.", "air, and the vapours of the components $\\text{A}$ and $\\text{B}$ .", "When the droplet evaporates at a temperature $T$ far below the boiling point and in absence of forced or strong natural convection, the diffusive dynamics in the gas phase can be approximated by the quasi-stationary Laplace equation [5], [17], [34], [7] $\\nabla ^2c_{\\alpha }=0$ for the vapour concentrations $c_{\\alpha }$ , i.e.", "the partial mass densities.", "The corresponding boundary conditions are given by the vapour-liquid equilibrium according to Raoult's law at the liquid-gas interface and the ambient vapour concentration at infinity, i.e.", "$c_{\\alpha }&=c_{\\alpha }^{\\text{eq}}=c_{\\alpha }^{\\text{pure}}\\gamma _{\\alpha }x_{\\alpha } &\\text{at the liquid-gas interface and} \\\\c_{\\alpha }&=c_{\\alpha }^\\infty &\\text{far away from the droplet}\\,,$ where $c_{\\alpha }^{\\text{pure}}$ is the saturation vapour concentration in case of the pure liquid $\\alpha $ and $x_{\\alpha }$ is the liquid mole fraction.", "The activity coefficients $\\gamma _{\\alpha }$ account for thermodynamic non-idealities and are functions of the composition, i.e.", "of $x_{\\alpha }$ .", "Neglecting the small contribution of the Stefan flow at temperatures sufficiently below the boiling point, the evaporation rates $j_{\\alpha }$ are given by the diffusive fluxes at the liquid-gas interface, i.e.", "$j_{\\alpha }=-D_{\\alpha }^{\\text{vap}}\\partial _nc_{\\alpha } \\,.$ While the dynamics in the gas phase can be considered in the diffusive and quasi-stationary limit, convection can be dominant in the liquid phase, which can be attributed to the typical diffusion coefficients, namely $D_{\\alpha }^{\\text{vap}}\\sim {e-5}{^2}$ in the gas phase and $D\\sim {e-9}{^2}$ in the liquid phase.", "Therefore, the liquid phase has to be described by the full convection-diffusion equation for the liquid mass fraction $y_{\\alpha }$ , which is expressed for the component A only due to the identity $y_{\\text{A}}+y_{\\text{B}}=1$ , i.e.", "$\\rho \\left(\\partial _t y_{\\text{A}} + uy_{\\text{A}} \\right) = \\left(\\rho D y_{\\text{A}}\\right) \\,.$ The liquid density $\\rho $ and the mutual diffusivity $D$ are in general functions of the composition, i.e.", "of $y_{\\text{A}}$ .", "The mass transfer rates $j_{\\alpha }$ due to evaporation induce a change in composition at the liquid-gas interface.", "Using the mass transfer expression $j_{\\alpha }=\\rho y_{\\alpha }(u_\\alpha -u_{\\text{I}})n$ with $u_\\alpha $ denoting the velocity of component $\\alpha $ and $u_{\\text{I}}$ and $n$ denoting the interface velocity and normal, respectively, this compositional change can be expressed by a Robin boundary condition for Eq.", "(REF ) in the frame co-moving with the interface in normal direction, namely $-\\rho D{}y_{\\text{A}}n= y_{\\text{B}}j_{\\text{A}}- y_{\\text{A}}j_{\\text{B}}= (1-y_{\\text{A}})j_{\\text{A}}-y_{\\text{A}}j_{\\text{B}}\\,.$ Finally, the flow in the droplet is given by the Navier-Stokes equations $\\rho \\left(\\partial _tu+uu\\right)&=-p + \\left( \\mu \\left(u + (u)^\\text{t} \\right) \\right) + \\rho g e_z \\\\\\partial _t \\rho + (\\rho u)&=0\\,.$ Here, we have chosen the $z$ -axis to point towards the apex of the droplet, i.e.", "a sessile droplet and a pendant droplet can be realized by negative and positive values for $g$ , respectively.", "Note that the viscosity $\\mu $ and the mass density $\\rho $ are in general functions of the composition $y_{\\text{A}}$ .", "A dependency on the temperature is disregarded in the following due to the fact that thermal effects at lower temperatures are usually considerably inferior to the impact of solutal gradients.", "For the contact line dynamics, we are focussing here on a pinned contact line, i.e.", "evaporation in the constant radius mode (CR-mode, [33], [41]).", "To resolve the incompatibility of a no-slip boundary condition at the substrate and the evaporative mass loss at the contact line, we impose a Navier-slip boundary condition with a small slip-length in the nanometre scale instead.", "This effectively resembles a no-slip boundary condition in the main part of the droplet-substrate interface, but still allows for a consistent mass transfer at the contact line.", "The free liquid-gas interface is subject to the kinematic boundary condition considering the mass transfer, i.e.", "$\\left(u-u_{\\text{I}}\\right)n=\\frac{1}{\\rho }(j_{\\text{A}}+j_{\\text{B}})$ and furthermore to the Laplace pressure in normal direction $-p + \\mu n\\left(u + (u)^\\text{t} \\right)\\cdot n = \\sigma \\kappa \\,,$ where the traction in the gas phase has been neglected due to the viscosity ratio.", "Here, $\\sigma $ is the local surface tension, $\\kappa $ the curvature of the interface and $p$ denotes the pressure difference with respect to the ambient gas pressure.", "Finally, also the Marangoni shear stress in tangential direction has to be considered: $\\mu n\\left(u + (u)^\\text{t} \\right) \\cdot t = _{t} \\sigma t \\,.$ Here, $_{t}=(\\mathbf {1}-nn)$ is the surface gradient operator.", "A sketch of the model is depicted in  REF ." ], [ "Numerical solution of the dynamical equations for an instructive example", "In order to solve the given set of equations numerically, we have generalised the sharp-interface arbitrary Lagrangian-Eulerian finite element method described in [6] by considering the gravitational force, and also validated it by a more general reimplementation of the same model with the finite element package oomph-liboomph-lib.maths.man.ac.uk [15], which allows for interface deformations and considers the general continuity equation ().", "The latter method has been successfully validated against various experiments [25], [23], [12], [24].", "In  REF , a simulation of a sessile glycerol-water droplet (initially 5 wt.% glycerol) with an initial volume of 1 and an initial contact angle of 120 evaporating at a constant temperature of 22 and a relative humidity of 20 is shown.", "The contact line remains pinned during drying and glycerol (liquid B) is assumed to be non-volatile due to its low volatility compared to water (liquid A), i.e.", "$c_{\\text{B}}=c_{\\text{B}}^\\infty =0$ and $j_{\\text{B}}=0$ .", "For more details about these kind of simulations we refer to [8], [6], where however we did not consider of the influence of gravity.", "Figure: Simulation of a 1 glycerol-water droplet revealing rich flow patterns during the evaporation process.", "The water vapour mass fraction is shown in the gas phase, whereas the glycerol mass fraction (left) and the velocity magnitude (right) are shown inside the droplet.", "(a) Initially, both Rayleigh and Marangoni convection support the flow from the apex to the contact line.", "(b) Although the contact angle is still above θ>90\\theta >{90}{}, a Marangoni-induced counter-rotating vortex (black) emerges close to the interface, whereas the bulk flow is driven by natural convection (white).", "(c) Due to the increased evaporation rate at the contact line for θ<90\\theta <{90}{}, the Marangoni-driven vortex grows in size until (d) the vortex driven by natural convection disappears.", "See supplementary movie 1 for the entire simulation.Initially, in  REF (a), one can see a single vortex in the entire droplet, directed from the apex towards the contact line.", "This vortex is generated for two reasons, namely Marangoni convection and natural convection (Rayleigh convection).", "Due to the enhanced evaporation rate of water at the apex at the high contact angle, the water content is predominantly reduced at the top of the droplet, resulting in a lower surface tension compared to the region near the contact line.", "This drives a Marangoni flow towards the contact line.", "Since glycerol is more dense than water, the glycerol-rich outer shell of the droplet also sinks down due to natural convection, which also results in a flow from the apex to the contact line due to the spherical geometry.", "Hence, both mechanisms support recirculating flow in the same direction.", "Remarkably, in  REF (b), the situation changes.", "The contact angle is still above 90, i.e.", "still having the highest water evaporation rate at the top of the droplet.", "According to the afore-mentioned discussion, one would still naively expect the same kind of single vortex flow.", "However, the simulation clearly shows two vortices, one in the bulk driven by natural convection (white) and another one close to the interface, which is driven by Marangoni flow in the opposite direction (black).", "The reason why the Marangoni flow is reversed, i.e.", "why there is more water at the top of the droplet although the evaporation rate of water is still dominant at the apex, is the fact that there is enhanced water replenishment by diffusion at the apex, which compensates for the rather small difference in the evaporation rates at the top and near the contact line.", "This can be seen by the rather steep concentration gradient in normal direction at the apex compared to the region near the contact line.", "The reason of the steep concentration gradient in normal direction close at the apex is the upward directed convective water replenishment from the bulk, which is governed by the internal vortex driven by natural convection.", "This means that sufficiently strong natural convection in the bulk can reverse the Marangoni flow at the interface, although one would not anticipate this by just considering the profile of the evaporation rate at this contact angle.", "Upon further evaporation, in  REF (c), the contact angle falls below 90, resulting in a higher water evaporation rate near the contact line.", "Hence, less water is present at the contact line as compared to the apex due two facts, namely the effect of preferential evaporation at the contact line and the lower replenishing diffusive flux of water from the bulk at the contact line.", "Thereby, the Marangoni flow gets enhanced compared to the situation in  REF (b) and the relative size of the Marangoni-induced vortex at the interface grows at the expense of the counter-rotating bulk vortex by natural convection.", "Finally, in  REF (d), the contact angle becomes rather small so that the Marangoni flow at the interface is even stronger due to the enhanced non-uniformity of the evaporation rate.", "Furthermore, the influence of natural convection also diminishes rather quickly, i.e.", "with cubic power in terms of the length scale according to the Rayleigh number (see later for its definition), due to the reduced volume of the droplet.", "This results in the depicted situation, i.e.", "that the flow direction within the entire droplet is completely determined by the Marangoni effect.", "In a nutshell, one can infer from the direct numerical simulation results in  REF that there can be multiple flow scenarios during the drying of a single binary droplet, driven by an interplay of natural (i.e.", "Rayleigh) convection and Marangoni convection.", "One also clearly sees that, for a particle laden droplet, the coffee-stain effect would not occur as there is no noticeable flow towards the contact line (which for pure evaporating droplets transports the suspended particles to the rim of the pinned droplet) as compared to the strongly recirculating flow due to Marangoni flow and gravity." ], [ "Quasi-stationary approximation of the dynamical equations", "After discussing some possible flow scenarios by considering a representative numerical example in the previous section, we will now focus on a simplification of the full model described in section .", "We generalise again from the particular case of a water-glycerol droplet to the general case of a binary droplet, where both liquids A and B are allowed to evaporate.", "The goal is to find the simplest model possible that allows to predict the expected flow scenario in the droplet by a minimum number of non-dimensional quantities." ], [ "Evaporation Numbers", "As shown in the example simulation, the liquid recirculates multiple times during the evaporation process due to the fast flow in the droplet.", "Hence, the typical liquid velocity $u$ is much larger than the normal interface movement velocity $u_{\\text{I}}$ .", "Moreover, this leads to a rather well-mixed droplet, i.e.", "with typical compositional deviations of about a few percent in terms of mass fractions.", "These observations allow for several simplifications of the model.", "First of all, the liquid composition is expanded into two terms, i.e.", "$y_{\\text{A}}(x,t)=y_{\\text{A},0}+y$ , namely the spatially averaged composition $y_{\\text{A},0}$ , which slowly evolves over the entire drying time, and the small local composition deviations $y(x,t)$ .", "Since the composition-dependent liquid properties are usually rather smooth functions of the composition, this separation can be transferred to a first order Taylor expansion of the liquid properties, i.e.", "$\\rho &=\\rho _0+y\\:\\partial _{y_\\text{A}}\\rho \\,, &\\sigma &=\\sigma _0+y\\:\\partial _{y_\\text{A}}\\sigma \\,, \\nonumber \\\\\\mu &=\\mu _0+y\\:\\partial _{y_\\text{A}}\\mu \\,, &D&=D_{0}+y\\:\\partial _{y_\\text{A}}D\\,, \\\\c_{\\text{A}}^{\\text{eq}}&=c_{\\text{A},0}^{\\text{eq}}+y\\:\\partial _{y_\\text{A}}c_{\\text{A}}^{\\text{eq}} \\,, &c_{\\text{B}}^{\\text{eq}}&=c_{\\text{B},0}^{\\text{eq}}+y\\:\\partial _{y_\\text{A}}c_{\\text{B}}^{\\text{eq}} \\,.", "\\nonumber $ Since the averaged composition $y_{\\text{A},0}$ evolves slowly, this expansion can be done at any specific time of interest during the evaporation process.", "In particular, this means that the coefficients of the Taylor expansions (REF ) can be treated as constants during some time close to the considered instant.", "This allows us to introduce the following non-dimensionalized scales $x=V^{1/3}\\tilde{x}\\,,\\qquad t=\\frac{V^{2/3}}{D_{0}}\\tilde{t}\\,,\\qquad u=\\frac{D_{0}}{V^{1/3}}\\tilde{u} \\,,$ where the spatial scale is chosen in that way, that the nondimensionalized droplet volume $\\tilde{V}$ becomes unity.", "In a next step, the vapour fields are decomposed in a similar manner as (REF ), namely in a normalized contribution $\\tilde{c}_0$ which is one at the interface and zero at infinity and a contribution $\\tilde{c}_\\Delta $ accounting for the effect of local composition variations on the vapour concentration via Raoult's law to the first order, i.e.", "$c_{\\alpha }=\\left(c_{\\alpha ,0}^{\\text{eq}}-c_{\\alpha }^\\infty \\right)\\tilde{c}_0 + c_{\\alpha }^\\infty + (\\partial _{y_\\text{A}}c_{\\alpha }^{\\text{eq}})\\tilde{c}_\\Delta \\,.$ The Laplace equation (REF ) splits into two Laplace equations, i.e.", "$\\tilde{\\nabla }^2 \\tilde{c}_0=0$ and $\\tilde{\\nabla }^2 \\tilde{c}_\\Delta =0$ and the boundary conditions () are transformed to $\\tilde{c}_0&=1\\text{ and }\\tilde{c}_\\Delta =y &\\text{at the liquid-gas interface and} \\\\\\tilde{c}_0&=\\tilde{c}_\\Delta =0 &\\text{far away from the droplet}\\,.$ Thereby, the evaporation rates (REF ) separate in the same way, i.e.", "$j_{\\alpha }=\\frac{D_{\\alpha }^{\\text{vap}}}{V^{1/3}}\\left[\\left(c_{\\alpha ,0}^{\\text{eq}}-c_{\\alpha }^\\infty \\right)\\tilde{j}_0 + (\\partial _{y_\\text{A}}c_{\\alpha }^{\\text{eq}})\\tilde{j}_y \\right] \\,,$ where $\\tilde{j}_0=-\\tilde{\\partial }_n\\tilde{c}_0$ only depends on the shape of the droplet, i.e.", "resembles the normalized evaporation profile of a homogeneous droplet, and $\\tilde{j}_y=-\\tilde{\\partial }_n\\tilde{c}_\\Delta $ is a linear functional of $y$ , i.e.", "the Dirichlet-to-Neumann map, accounting for deviations in the evaporation rate due to a varying interfacial composition via the composition-dependent vapour-liquid equilibrium, i.e.", "Raoult's law.", "When dropping terms of quadratic order in $y$ , the convection-diffusion equation (REF ) within the droplet becomes $\\partial _{\\tilde{t}} y_{\\text{A},0} + \\partial _{\\tilde{t}} y + \\tilde{u}\\tilde{}y = \\tilde{\\nabla }^2 y$ and the corresponding interface boundary condition (REF ) reads $-\\tilde{}{}yn= \\mbox{\\it Ev}_y\\tilde{j}_0 +\\mbox{\\it Ev}_{\\text{vap}}\\tilde{j}_y - \\mbox{\\it Ev}_{\\text{tot}} y \\tilde{j}_0$ with the non-dimensional evaporation numbers $\\mbox{\\it Ev}_y=&\\frac{1}{\\rho _0D_{0}}\\Big [(1-y_{\\text{A},0})D_{\\text{A}}^{\\text{vap}}(c_{\\text{A},0}^{\\text{eq}}-c_{\\text{A}}^\\infty )-y_{\\text{A},0}D_{\\text{B}}^{\\text{vap}}(c_{\\text{B},0}^{\\text{eq}}-c_{\\text{B}}^\\infty )\\Big ] \\\\\\mbox{\\it Ev}_{\\text{vap}}=&\\frac{1}{\\rho _0D_{0}}\\Big [(1-y_{\\text{A},0})D_{\\text{A}}^{\\text{vap}}\\partial _{y_\\text{A}}c_{\\text{A}}^{\\text{eq}} -y_{\\text{A},0}D_{\\text{B}}^{\\text{vap}}\\partial _{y_\\text{A}}c_{\\text{B}}^{\\text{eq}}\\Big ] \\\\\\mbox{\\it Ev}_{\\text{tot}}=&\\frac{1}{\\rho _0D_{0}}\\Big [D_{\\text{A}}^{\\text{vap}}(c_{\\text{A},0}^{\\text{eq}}-c_{\\text{A}}^\\infty ) +D_{\\text{B}}^{\\text{vap}}(c_{\\text{B},0}^{\\text{eq}}-c_{\\text{B}}^\\infty )\\Big ]\\,.$ The number $\\mbox{\\it Ev}_y$ quantifies the intensity of the concentration gradient induced in the liquid by preferential evaporation of one of the components, i.e.", "it compares the differences of the two diffusive vapour transports in the gas phase with the mutual diffusion in the binary liquid.", "Since the resulting composition gradient along the interface and in the bulk is the driving mechanism for Marangoni flow and natural convection, this number will become important to quantify these processes later on.", "Note that dependent on the volatilities of the components and their mass fractions in the liquid, $\\mbox{\\it Ev}_y$ may be positive or negative.", "$\\mbox{\\it Ev}_{\\text{vap}}$ is an estimate for the influence of local variations in the liquid concentration on the preferential evaporation, i.e.", "the linear feedback due to the quasi-stationary diffusion in the gas phase.", "If the composition is rather uniform in the droplet, which is usually by fast recirculating convection, the term $\\mbox{\\it Ev}_{\\text{vap}}\\tilde{j}_y$ provides only a minor contribution in (REF ), meaning that the profile of the evaporation rates is similar to the one of a pure droplet.", "Since $\\partial _{y_\\text{A}}c_{\\text{A}}^{\\text{eq}}>0$ and $\\partial _{y_\\text{A}}c_{\\text{B}}^{\\text{eq}}<0$ , i.e.", "the vapour concentration of A increases and B decreases for an increasing fraction of A in the liquid, $\\mbox{\\it Ev}_{\\text{vap}}$ is always positive.", "Large numbers of $\\mbox{\\it Ev}_{\\text{vap}}$ can actually arise towards the end of the drying of a glycerol-water droplet, as discussed later on in section .", "Finally, $\\mbox{\\it Ev}_{\\text{tot}}$ is a measure for the total evaporation speed, i.e.", "for the typical interface speed $\\tilde{u}_{\\text{I}}$ and the volume evolution.", "Note that the total evaporation speed and volume evolution is a measure for the flow towards a pinned contact line, i.e.", "the flow leading to the coffee-stain effect.", "If none of the components condensates, i.e.", "both either evaporate or are non-volatile, the modulus of $\\mbox{\\it Ev}_y$ is smaller than $\\mbox{\\it Ev}_{\\text{tot}}$ .", "Nevertheless, since the deviation from the average composition is small, i.e.", "$y\\ll 1$ , and the Marangoni convection and/or natural convection are sufficiently large, the contribution of the latter to the flow can still be dominant compared to $\\mbox{\\it Ev}_{\\text{tot}} y \\tilde{j}_0$ in (REF )." ], [ "Nondimensionalized flow", "For the Navier-Stokes equations, we employ the established Boussinesq approximation, which is valid as long as $y\\partial _{y_\\text{A}}\\rho $ is small compared to $\\rho _0$ [14].", "Due to the usually small composition gradients, this assumption is valid here.", "Therefore, except for the body force term $\\rho g e_z$ , only the zeroth order terms proportional to $\\rho _0$ are kept, whereas $y\\partial _{y_\\text{A}}\\rho $ -terms are disregarded.", "With the same argument, terms proportional to $y\\partial _{y_\\text{A}}\\mu $ and $y\\partial _{y_\\text{A}}\\sigma $ can be disregarded, whenever there is a dominant term proportional to $\\mu _0$ and $\\sigma _0$ , respectively.", "Following this argument, the Navier-Stokes equations can be written as $\\mbox{\\it Sc}^{-1}\\left(\\partial _{\\tilde{t}}\\tilde{u}+\\tilde{u}\\tilde{}\\tilde{u}\\right)&=-\\tilde{}\\tilde{p} + \\tilde{}\\left(\\tilde{}\\tilde{u} + (\\tilde{}\\tilde{u})^\\text{t} \\right) + \\mbox{\\it Ra}^* y e_z \\\\\\tilde{}\\tilde{u}&=0\\,.$ Here, the shifted nondimensionalized pressure, the Schmidt number and the incomplete Rayleigh number read $\\tilde{p}=\\frac{V^{2/3}}{D_{0}\\mu _0}\\left(p-\\rho g z\\right) \\,, \\qquad \\mbox{\\it Sc}=\\frac{\\mu _0}{D_{0}\\rho _0}\\,, \\qquad \\mbox{\\it Ra}^*=\\frac{V g\\partial _{y_\\text{A}}\\rho }{D_{0}\\mu _0}\\,.$ The Schmidt number for liquids is usually $\\mbox{\\it Sc}> {e3}$ which suggests that the lhs of (REF ) can be disregarded.", "However, since the chosen velocity and time scale in (REF ) does not necessarily coincide with the actual present scales, this argument is only valid for small Reynolds numbers.", "In small droplets with rather low volatilities and regular Marangoni flow, however, this assumption is surely met, e.g.", "$\\mbox{\\it Re} < 0.05$ in the case of the simulation in  REF .", "The incomplete Rayleigh number $\\mbox{\\it Ra}^*$ deviates from the conventional definition of the Rayleigh number just by the lack of an estimate for the composition difference, i.e.", "a term like $\\Delta y_{}$ .", "The dynamic boundary conditions at the interface, (REF ) and (REF ), read in the Boussinesq approximation $-\\tilde{p} + n\\left(\\tilde{}\\tilde{u} + (\\tilde{}\\tilde{u})^\\text{t} \\right)\\cdot n = \\frac{1}{\\mbox{\\it Ca}^*}\\left(\\tilde{\\kappa }+\\mbox{\\it Bo}\\: \\tilde{z}\\right)$ $n\\left(\\tilde{}\\tilde{u} + (\\tilde{}\\tilde{u})^\\text{t} \\right) \\cdot t = \\mbox{\\it Ma}^* \\tilde{}_{t} y t \\,.$ Here, the non-dimensional number $\\mbox{\\it Ca}^*$ , the Bond number and the incomplete Marangoni number read $\\mbox{\\it Ca}^*=\\frac{D_{0}\\mu _0}{V^{1/3}\\sigma _0}\\,, \\qquad \\mbox{\\it Bo}=\\frac{\\rho _0 g V^{2/3}}{\\sigma _0}\\,, \\qquad \\mbox{\\it Ma}^*=\\frac{V^{1/3}\\partial _{y_\\text{A}}\\sigma }{D_{0}\\mu _0}\\,.$ Note that the definition of $\\mbox{\\it Ca}^*$ does not coincide with the capillary number, i.e.", "it does not consider the actually present typical velocity scale, i.e.", "the intensity of the capillary shape relaxations during evaporation cannot be inferred from $\\mbox{\\it Ca}^*$ .", "However, both the real capillary number $\\mbox{\\it Ca}=\\mu _0U/\\sigma _0$ and $\\mbox{\\it Ca}^*$ are small in the systems considered here ($\\mbox{\\it Ca}<{1e-6}$ and $\\mbox{\\it Ca}^*<{1e-7}$ in the simulation depicted in  REF ).", "Similar to $\\mbox{\\it Ra}^*$ , the incomplete Marangoni number $\\mbox{\\it Ma}^*$ lacks in an estimate for the composition difference, i.e.", "$\\Delta y$ , as compared to the conventional definition.", "Finally, the kinematic boundary condition (REF ) becomes $\\left(\\tilde{u}-\\tilde{u_{\\text{I}}}\\right)n=\\mbox{\\it Ev}_{\\text{tot}} \\tilde{j}_0 + \\mbox{\\it Ev}_{\\text{tot,vap}} \\tilde{j}_y \\,,$ where $\\mbox{\\it Ev}_{\\text{tot,vap}}=\\frac{1}{\\rho _0D_{0}}\\Big [D_{\\text{A}}^{\\text{vap}}\\partial _{y_\\text{A}}c_{\\text{A}}^{\\text{eq}} +D_{\\text{B}}^{\\text{vap}}\\partial _{y_\\text{A}}c_{\\text{B}}^{\\text{eq}}\\Big ]$ is the analogue of $\\mbox{\\it Ev}_{\\text{vap}}$ for the total evaporation rate, i.e.", "the effect of a change in the saturation pressure due to a locally deviating composition on the total evaporation rate." ], [ "Estimation of the outwards flow", "Before focusing on natural convection and Marangoni flow in the droplet, it is beneficial to obtain an estimate for the velocity in the droplet in absence of these mechanisms, i.e.", "$\\mbox{\\it Ma}^*=\\mbox{\\it Ra}^*=0$ .", "This case, exemplified e.g.", "by a pure isothermal droplet, combined with a pinned contact line represents the purely capillary-driven outward flow, which causes the coffee-stain effect.", "For small droplets, the capillary number $\\mbox{\\it Ca}$ is small, so that the surface tension forces according to (REF ) lead to an intense relaxing traction, whenever the droplet deviates from the equilibrium shape.", "Since the Bond number $\\mbox{\\it Bo}$ is small as well for small droplets, the hydrostatic term in (REF ) can be neglected, leading to a spherical cap with a homogeneous curvature $\\tilde{\\kappa }$ as equilibrium shape.", "Hence, the shape evolution and thereby the interface velocity $\\tilde{u_{\\text{I}}}$ is solely given by the evaporation rate and the contact line kinetics, which is assumed to be pinned here.", "Since the term $\\mbox{\\it Ev}_{\\text{tot,vap}} \\tilde{j}_y$ in (REF ) is proportional to $y$ , it can be disregarded with respect to $\\mbox{\\it Ev}_{\\text{tot}} \\tilde{j}_0$ in accordance with the Boussinesq approximation.", "As a consequence, one ends up with a linear Stokes flow problem, where the entire bulk velocity is given by the instantaneous shape relaxation, which is proportional to the rate of evaporation, i.e.", "to $\\mbox{\\it Ev}_{\\text{tot}}$ .", "By integrating the evaporation rate $\\mbox{\\it Ev}_{\\text{tot}} \\tilde{j}_0$ one obtains the volume loss and thereby one can reconstruct the normal velocity of the interface $\\tilde{u}_{\\text{I}}n$ .", "The flow in the bulk $\\tilde{u}$ is subsequently given by solving the Stokes flow with the normal boundary condition $\\tilde{u}n=\\tilde{u_{\\text{I}}}n+\\mbox{\\it Ev}_{\\text{tot}}\\tilde{j}_0$ .", "In  REF , representative solutions for the bulk flow $\\tilde{u}$ with unity evaporation number, i.e.", "$\\mbox{\\it Ev}_{\\text{tot}}=1$ , are depicted.", "It is apparent that the typical bulk velocity is of the order unity, i.e.", "$\\Vert \\tilde{u}\\Vert \\sim 1$ .", "Since the flow is proportional to $\\mbox{\\it Ev}_{\\text{tot}}$ , the typical velocity corresponding to an arbitrary evaporation number $\\mbox{\\it Ev}_{\\text{tot}}$ is hence $\\Vert \\tilde{u}\\Vert \\sim \\mbox{\\it Ev}_{\\text{tot}}$ .", "This holds also for the typical interface movement, i.e.", "$\\Vert \\tilde{u}_{\\text{I}}\\Vert \\sim \\mbox{\\it Ev}_{\\text{tot}}$ ." ], [ "Quasi-stationary limit", "Knowing the fact that the capillary flow due to the volume loss is on the order of $\\mbox{\\it Ev}_{\\text{tot}}$ , we now focus on the contributions to the flow by Marangoni forces and natural convection.", "In a first step, one can consider the case where $\\mbox{\\it Ev}_{\\text{tot}}=0$ , i.e.", "no total mass transfer and hence a constant volume and shape of the droplet.", "This scenario can be realized by tuning the ambient humidities of A and B so that the evaporative mass loss of component A is balanced by the condensation of component B.", "In this case $\\mbox{\\it Ev}_{\\text{tot}}=0$ and $\\mbox{\\it Ev}_y>0$ holds.", "Again, due to the small capillary number and the small Bond number, one can assume a spherical cap shape with volume $\\tilde{V}=1$ and contact angle $\\theta $ , which are both constant now.", "Furthermore, there is no interface movement, $\\tilde{u}_{\\text{I}}n=0$ , and no total mass transfer, $\\tilde{u}n=0$ .", "By averaging (REF ) over the droplet volume $\\tilde{V}=1$ , defining the integrated evaporation rate $\\tilde{J}=\\int \\tilde{j}_0 \\mathrm {d}\\tilde{A}$ and considering only the zeroth order term in the boundary condition (REF ) in accordance with the Boussinesq approximation, one can separate the average composition $y_{\\text{A},0}$ and the deviation $y$ as follows: $\\partial _{\\tilde{t}} y_{\\text{A},0} &= -\\mbox{\\it Ev}_y \\tilde{J}_0 \\\\\\partial _{\\tilde{t}} y + \\tilde{u}\\tilde{}y &= \\tilde{\\nabla }^2 y +\\mbox{\\it Ev}_y \\tilde{J}_0 \\,.$ Here, the term $\\mbox{\\it Ev}_y \\tilde{J}_0$ assures that $y_{\\text{A},0}$ is indeed the average composition and that the average of $y$ remains zero, i.e.", "the term compensates for the imposed composition gradient at the liquid-gas interface.", "As already stated in section REF , this splitting holds only for limited time, since a variation in $y_{\\text{A},0}$ leads to a change in the liquid properties which where used for the nondimensionalization.", "Usually, however, the coupled dynamics of flow $\\tilde{u}$ and compositional differences $y$ due to Marangoni and natural convection is considerable faster than $\\mbox{\\it Ev}_y \\tilde{J}_0$ , This was already apparent from the simulations depicted in  REF and it will be validated later on in section .", "Furthermore, this observation allows to focus on stationary solutions.", "Finally, upon introducing $\\xi =\\frac{y}{\\mbox{\\it Ev}_y}\\,,$ one ends up at the following set of coupled equations: $\\tilde{u}\\tilde{}\\xi &= \\tilde{\\nabla }^2 \\xi +\\tilde{J}_0 \\\\-\\tilde{}\\tilde{p} + \\tilde{}\\left(\\tilde{}\\tilde{u} + (\\tilde{}\\tilde{u})^\\text{t} \\right) + \\mbox{\\it Ra}\\, \\xi e_z&=0 \\\\\\tilde{}\\tilde{u}&=0$ subject to the following boundary conditions $-\\tilde{}{}\\xi n&= \\tilde{j}_0 \\\\\\tilde{u}n&=0\\\\n\\left(\\tilde{}\\tilde{u} + (\\tilde{}\\tilde{u})^\\text{t} \\right) \\cdot t &= \\mbox{\\it Ma}\\: \\tilde{}_{t} \\xi t$ at the liquid-gas interface and $\\tilde{}{}\\xi n&= 0 \\\\u&=0$ at the liquid-substrate interface.", "Note that the simplified kinematic boundary condition () is now compatible with the no-slip boundary condition () at the contact line, i.e.", "a slip length is not required.", "Besides the contact angle $\\theta $ , only two parameters enter the system, namely the Marangoni number and the Rayleigh number, which read $\\mbox{\\it Ma}=\\mbox{\\it Ma}^*\\mbox{\\it Ev}_y=\\frac{V^{1/3}\\partial _{y_\\text{A}}\\sigma }{D_{0}\\mu _0}\\mbox{\\it Ev}_y\\,\\,,\\qquad \\mbox{\\it Ra}=\\mbox{\\it Ra}^*\\mbox{\\it Ev}_y=\\frac{V g\\partial _{y_\\text{A}}\\rho }{D_{0}\\mu _0}\\mbox{\\it Ev}_y\\,.$ Note that the characteristic numbers for both mechanisms are proportional to the induced composition gradient due to mass transfer, i.e.", "$\\mbox{\\it Ev}_y$ .", "Of course, in particular the tangential gradient along the interface is also strongly dependent on the contact angle $\\theta $ , since this determines the profile of the evaporation rate $\\tilde{j}_0$ .", "These equations are not only valid for the specific assumed case of $\\mbox{\\it Ev}_{\\text{tot}}=0$ , but also when the combination of Marangoni and Rayleigh flow $\\tilde{u}$ predicted by this model is considerably faster than the capillary flow, i.e.", "a flow situation with outwards flow leading to the coffee-stain effect.", "According to the estimations in section REF , this is the case if $\\tilde{u}\\gg \\mbox{\\it Ev}_{\\text{tot}}$ ." ], [ "Procedure", "Unfortunately, the analytical treatment of the model equations (REF )-() is hampered by the geometry, which demands rather complicated toroidal coordinates, and by the very strong nonlinear coupling of $\\tilde{u}$ and $\\xi $ due to the advection term.", "Therefore we investigate the system by numerical means.", "Our analysis is limited to axisymmetric solutions and we only consider the case $\\mbox{\\it Ev}_{\\text{vap}}=0$ , i.e.", "neglecting the feedback of the altered gas composition due to the liquid-vapour equilibrium on the local evaporation rate.", "Finally, we will focus on the case $\\mbox{\\it Ma}\\ge 0$ , for which evaporation leads to an overall reduction of the surface tension, as in the case of the water-glycerol depicted in  REF .", "This results in a regular flow, i.e.", "no chaotic behaviour can be found, at least not for moderate flow conditions.", "In the case of negative Marangoni numbers, chaotic flow patterns cannot be excluded due to the Marangoni instability [3], [1], [28].", "Of course, this spatio-temporal evolving type of flow cannot be captured within the assumption of a quasi-stationary process.", "One can, on the other hand, test the linear stability of the quasi-stationary solutions in the case of negative Marangoni numbers to find the transition to chaotic flow, but since also the axial symmetry is usually broken in case of negative Marangoni numbers, one also would have to generalize the entire solution procedure from axisymmetric cylindrical coordinates to the full three-dimensional problem, as done by [8].", "In order to find solutions of the system, we employed a finite element method on an axisymmetric mesh with triangular elements.", "We used linear basis functions for $\\xi $ and $\\tilde{p}$ and quadratic basis functions for $\\tilde{u}$ , i.e.", "typical Taylor-Hood elements.", "The equations have been implemented in both FEniCShttps://fenicsproject.org/ [26] and oomph-lib for mutual validation.", "The condition of zero velocity in normal direction, i.e.", "Eq.", "(), has been implemented by Lagrange multipliers.", "For an enhanced stability in the Newton method during the solution process, it has been found beneficial to replace $\\tilde{J}_0$ in (REF ) by a Lagrange multiplier which ensures that the average of $\\xi $ is zero.", "This removes the null space with respect to a constant shift in $\\xi $ and a corresponding adjustment of the pressure $\\tilde{p}$ .", "Due to the nonlinear advection term, it is in general possible that multiple solutions exist for a given parameter combination $(\\theta ,\\mbox{\\it Ma},\\mbox{\\it Ra})$ .", "For the parameter ranges considered in the following, however, we are confident that we found the generic solutions due to the following strategy: For every considered contact angle $\\theta $ , we performed adiabatic scans along $\\mbox{\\it Ra}$ in increasing and decreasing direction for fixed $\\mbox{\\it Ma}$ and vice versa.", "During that, no hysteresis, i.e.", "bistable regions, have been found.", "Furthermore, by tracing these parameter paths with continuation, we have not detected any unstable branches.", "This has been furthermore validated by investigating the eigenvalues with a shift-inverted Arnoldi methodusing Spectra https://spectralib.org.", "Finally, for each parameter combination, we performed temporal integrations of the unsteady model equations starting from a homogeneous state $\\xi =0$ .", "Since these runs converged to the same solutions as obtained by the steady parameter scans, we are sure that all solutions discussed in the following are indeed generic and stable.", "Note, however, that this is in general not true outside the considered parameter ranges." ], [ "Phase diagrams", "The phase diagrams for small and large contact angles, i.e.", "for $\\theta ={60}{}$ and $\\theta ={120}{}$ , are depicted in figures  REF and  REF , respectively.", "Here we have assumed, as in the case of the glycerol-water droplet, that the blue liquid A (e.g.", "water) is more volatile, less dense but associated with a higher surface tension than the red liquid B (e.g.", "glycerol).", "This means by definition that $\\mbox{\\it Ma}>0$ holds and that a sessile droplet is described by $\\mbox{\\it Ra}>0$ whereas a pendant droplet is given by $\\mbox{\\it Ra}<0$ .", "Depending on the Marangoni number $\\mbox{\\it Ma}$ , the Rayleigh number $\\mbox{\\it Ra}$ and the contact angle $\\theta $ , different qualitative flow scenarios can be found.", "For high Marangoni numbers and small Rayleigh numbers, the Marangoni flow dominates (Ma dominant) and vice versa (Ra dominant).", "In between, however, for sessile droplets with a contact angle below ${90}{}$ and for pendant droplets with a contact angle above ${90}{}$ , there is a region where the Marangoni effect determines the flow direction at the interface, whereas the bulk flow is driven by natural convection (Ma vs. Ra).", "In the opposite cases, i.e.", "for pendant droplets with $\\theta <{90}{}$ and sessile droplet with $\\theta >{90}{}$ , both mechanisms drive a flow in the same direction, so that one cannot directly distinguish between the two mechanisms driving the flow (Ma & Ra same dir.).", "In the limit of very strong driving of both mechanisms, however, natural convection can become so intense, that the surface tension gradient is reversed, leading to a Marangoni-induced reversal of the flow at the interface (Ra reverses Ma).", "This effect can be explained by the distortion of the internal composition field due to natural convection.", "For pendant droplets with $\\theta <{90}{}$ , the composition gradient in the bulk in normal direction is much more pronounced near the contact line as opposed to the apex.", "As a consequence, the diffusive replenishment of the blue liquid at the interface is enhanced near the contact line so that in fact more blue liquid, i.e.", "the one with higher surface tension, can be found near the contact line instead of at the apex – despite of its higher volatility and the pronounced evaporation rate at the contact line.", "The resulting Marangoni flow is therefore reversed as anticipated by considering the profile of the evaporation rate alone.", "The same explanation holds for the case $\\theta >{90}{}$ , except that one finds a more pronounced normal composition gradient near the apex as compared to the region close to the contact line, and the situation is reversed.", "All transitions between the afore-mentioned regimes are continuous.", "The drawn phase boundaries are defined by the emergence or disappearance of a second vortex.", "There is no bifurcation and/or hysteresis present at the boundaries of the regimes.", "In supplementary movie 2, a path through the parameter space is traversed and the corresponding stationary solution is shown, which illustrates the behaviour of the flow upon crossing the phase region boundaries, i.e.", "how the stationary solution gradually changes between single and two-vortex solutions.", "Finally, we also investigate the contact angle dependence of the phase diagrams by showing the corresponding regions for $\\theta ={40}{},\\,{60}{}$ and ${80}{}$ in  REF (a) and for $\\theta ={100}{},\\,{120}{}$ and ${140}{}$ in  REF (b).", "Obviously, the phase boundaries are shifted, but qualitative differences in the phase diagrams cannot be found.", "Figure: Influence of the contact angle θ\\theta on the boundaries of the phase diagram for (a) θ<90\\theta <{90}{} and (b) θ>90\\theta >{90}{}.Note again that we have assumed in the phase diagrams that the more volatile liquid (blue) is less dense in the insets in the phase diagrams.", "In the other case, the droplets depicted in the insets are required to be mirrored vertically, as the Rayleigh number is then negative for sessile droplets and positive for pendant droplets.", "Furthermore, it is noteworthy that these diagrams are for pinned droplets (CR-mode) and droplets with a constant contact angle (CA-mode), as only stationary solutions are considered anyhow.", "As long as the dominant velocity contribution is given by recirculating flow due to Marangoni and/or natural convection, any capillary flow due to shape relaxations can be disregarded.", "Finally, the diagrams can also be used for condensation instead of evaporation, as long as the $\\mbox{\\it Ma}>0$ holds.", "For condensation of component A, $\\mbox{\\it Ev}_y<0$ holds, so that $\\mbox{\\it Ma}>0$ is true if component A has the lower surface tension, i.e.", "$\\partial _{y_\\text{A}}\\sigma <0$ .", "We therefore anticipate that the diagrams can predict the flow when ethanol condenses on a pure water droplet, whereas it would fail to predict the flow when water condenses on a pure glycerol droplet ($\\mbox{\\it Ma}<0$ ).", "In fact, the latter case has been investigated experimentally, showing indeed chaotic cellular flow structures [38]." ], [ "Validation of the quasi-stationary approximation against the full numerical simulation", "Since there were a number of assumptions made in the simplification of the problem, it is necessary to validate the predicted flow by comparing it with results of the full numerical simulation, i.e.", "with the full set of equations as described in section .", "We focus on the representative simulation depicted in  REF .", "At each instant in time, we have extracted the spatially averaged water mass fraction $y_{\\text{A},0}$ , the volume $V$ to determine the spatial scale $\\@root 3 \\of {V}$ and the contact angle $\\theta $ from the simulation.", "From $y_{\\text{A},0}$ , we obtain $\\rho _0$ and $\\partial _{y_\\text{A}}\\rho $ , $\\sigma _0$ and $\\partial _{y_\\text{A}}\\sigma $ , as well as $\\mu _0$ , $D_{0}$ and $c_{\\text{A},0}^{\\text{eq}}$ from the composition-dependent properties of binary glycerol-water mixtures.", "This allows to calculate the normalized evaporation-induced composition gradient $\\mbox{\\it Ev}_y$ and the characteristic numbers $\\mbox{\\it Ra}$ and $\\mbox{\\it Ma}$ .", "On the basis of these numbers and the contact angle, we solve the simplified quasi-stationary model and re-dimensionalise the resulting velocity and composition field as well as the evaporation rate using the scales (REF ).", "This procedure allows to compare the full unsteady evolution of the droplet with the corresponding predictions at each instant by the simplified quasi-stationary model.", "Figure: Comparison of the full simulation (left) from   and the corresponding result predicted by the quasi-stationary model (right) at different times.", "The colour-code inside shows the glycerol concentration, whereas the streamlines indicate the velocity field.", "In the gas phase, the water vapour and the corresponding evaporation rate is depicted.", "(a) Initially, the full simulation has not yet attained the quasi-stationary limit, so that the intensity of the composition deviation is overpredicted in the quasi-stationary model.", "In (b-d), the quasi-stationary model predicts the result of the full simulation up to a deviation that can be barely seen by eye.", "See supplementary movie 3 for the comparison between full simulation and quasi-stationary model over the entire simulation time.The results are depicted for several instants in  REF , where the full simulation is shown on the left and the corresponding prediction of the quasi-stationary model is depicted on the right.", "Initially, i.e.", "in  REF (a), the full simulation has not attained the quasi-stationary limit.", "Hence, the quasi-stationary model slightly overpredicts the composition variations, i.e.", "it shows more glycerol (red) near the interface and more water (blue) in the bulk.", "Therefore, the flow field also slightly differs, i.e.", "the transient full simulation shows a single vortex, whereas the quasi-stationary model predicts the presence of a small counter-rotating vortex near the apex.", "Furthermore, a very gentle deviation in the spherical cap shape due to the gravitational effect in the full simulation can be seen at the apex as well (regime Ra reverses Ma).", "At later time steps, i.e.", "in  REF (b-d), however, the flow and the composition predicted by the quasi-stationary model match the results of the full simulation almost perfectly, be it in terms of the composition field, the flow pattern, the shape or the evaporation rate.", "This result substantiates the fact that the capillary outwards flow, which has been disregarded in the quasi-stationary model, can indeed be neglected as long as there is a prominent recirculating flow due to Marangoni and/or Rayleigh convection.", "Figure: Comparison of characteristic quantities of the full simulation of   and the corresponding results predicted by the quasi-stationary model.", "(a) The three input parameters for the quasi-stationary model Ra\\mbox{\\it Ra}, Ma\\mbox{\\it Ma} and θ\\theta extracted from the full simulation.", "(b) Comparison of the rms velocity.", "(c) Comparison of the maximum and minimum glycerol content inside the droplet.", "(d) Evaporation numbers extracted from the full simulation.To assess the quality of the quasi-stationary model in more detail, we have extracted some characteristic quantities of both simulations, i.e.", "from the full simulation of  REF and the corresponding quasi-stationary limit at each instant.", "In  REF (a), the time evolution of the three key parameters, namely the Rayleigh number $\\mbox{\\it Ra}$ , the Marangoni number $\\mbox{\\it Ma}$ and the contact angle $\\theta $ is shown.", "These numbers were used as input for the quasi-stationary model.", "The rms of the velocity inside the droplet is depicted in  REF (b).", "Again one can see an initial disagreement due to the fact that the full simulation has not yet attained its quasi-stationary limit.", "After that, i.e.", "after about ${50}{}$ to ${100}{}$ , the rms-velocity is well predicted until it shows again a disagreement towards the end of the drying time.", "The reason for the overpredicted velocity in the quasi-stationary model can be found in  REF (c), where the minimum and maximum glycerol concentration in the droplet according to both simulations are plotted against time.", "While it shows good agreement in the main part of the drying, the quasi-stationary model shows an enhanced maximum glycerol concentration towards the end of the drying, i.e.", "when almost only glycerol is left in the droplet.", "In fact, the glycerol concentration predicted by the quasi-stationary model even exceeds the physically realistic threshold of ${100}{}$ .", "Obviously, this overprediction of the composition differences explains the elevated prediction of the rms velocity.", "The reason of the overpredicted composition difference can finally be seen in  REF (d), where the evaporation numbers are depicted.", "When the droplet almost entirely consists of glycerol, the evaporation number $\\mbox{\\it Ev}_{\\text{vap}}$ , quantifying the reduction of the water vapour pressure for vanishing water at the interface on the evaporation dynamics (cf.", "Eq.", "()), becomes very large ($\\mbox{\\it Ev}_{\\text{vap}}\\rightarrow 18$ at $t={1000}{}$ ).", "This effect is not considered in the quasi-stationary model, since it assumes the averaged composition $y_{\\text{A},0}$ to predict the vapour-liquid equilibrium, not the local composition at the interface.", "Thereby, the amount of water vapour is strongly overestimated which results in a high evaporation rate and thereby in an unrealistically high composition difference.", "Obviously, the quasi-stationary model loses validity when $\\mbox{\\it Ev}_{\\text{vap}}$ becomes too large, meaning that the dependence of the vapour-liquid equilibrium on the local interface composition cannot be neglected any more.", "For a more detailed model, this effect can easily be incorporated into the quasi-stationary model, but it would introduce a fourth parameter besides $\\mbox{\\it Ma}$ , $\\mbox{\\it Ra}$ and $\\theta $ into the set of equations, which is beyond the scope of this article." ], [ "Conclusion", "During the evaporation of a binary droplet, multiple flow scenarios can be found, which is a result of an interplay of differences in the volatilities, mass densities and surface tensions of the two constituents.", "The difference in the volatilities induces compositional gradients in the bulk and also, due to the in general non-homogeneous evaporation rate, along the interface.", "Due to the composition-dependent mass density and surface tension, natural convection and Marangoni flow can set in, leading to a recirculating flow in the droplet that is usually much faster than the typical capillary outwards flow towards the contact line, which can be seen in pure droplets and leads to the coffee-stain effect in particle-laden droplets.", "Based on justified assumptions, we simplified the full model equations to a quasi-stationary model that only requires three parameters, namely the contact angle, the Rayleigh and the Marangoni number.", "Both, the Rayleigh and Marangoni number, linearly scale with a non-dimensional evaporation number, $\\mbox{\\it Ev}_y$ , which is a measure for the induced composition gradient by the preferential evaporation of one or the other component.", "By numerically solving for stationary solutions of the simplified model, we have explored the phase space in terms of these three quantities.", "The obtained phase diagrams allow for the prediction of the flow types in sessile and pendant binary droplets, with contact angles below and above 90.", "We found in total five different flow patterns: If one of both mechanisms, i.e.", "either natural convection or Marangoni flow, gets sufficiently strong as compared to the other one, it can dominate and control the flow direction in the entire droplet.", "This scenario can usually be seen in the case when the corresponding number, namely the Rayleigh or Marangoni number, respectively, is much larger than the other.", "In these cases, a single vortex can be seen in the droplet.", "If both mechanisms drive the flow into a different direction and are comparably strong in terms of their non-dimensional numbers, one can find two vortices, one in the bulk driven by natural convection, and a counter-rotating vortex at the interface due to Marangoni flow.", "The fourth flow type is the case, when both mechanisms act in the same direction, so that one cannot distinguish the particular cause of the driving and only a single vortex is present.", "Remarkably, however, in particular regimes in the phase space, the Marangoni flow can be reversed due to the natural convection in the bulk, leading to the fifth solution, where again two vortices can be found.", "In this situation, the bulk flow driven by natural convection deforms the internal composition field so that the diffusion dynamics in the liquid are altered, which eventually reverses the composition gradient at the interface and hence the Marangoni flow.", "To use the phase diagrams presented in this article, several requirements have to be fulfilled: First of all, the influence of thermal effects must be negligible compared to the solutal ones.", "The two liquids must be miscible and the droplet must not be too large, so that the capillary number and the Bond number are small in order to guarantee a spherical cap shape during the evaporation.", "Furthermore, the Reynolds number must be small and the spatial variations in the composition must be small enough in order to allow for a first order Taylor expansion of the composition-dependent liquid properties according to (REF ).", "Also the requirements for the Bousinessq approximation must hold.", "The Marangoni number, as defined in (REF ), must be positive, i.e.", "evaporation leads to an overall decrease of the surface tension, so that Marangoni-unstable chaotic flow can be excluded and the recirculating flow must be sufficiently faster than the movement of the interface.", "Finally, the influence of a change in the local composition on the vapour-liquid equilibrium may not be too strong, as it has been discussed on the basis on the evaporation number $\\mbox{\\it Ev}_{\\text{vap}}$ describing the feedback of local composition changes on the evaporation rate in section .", "If all these requirements are fulfilled and the composition-dependence of the required physical properties are known, the phase diagrams of this article allow for a prediction of the qualitative flow pattern in an evaporating binary droplet, probably with exception of a short initial transient phase.", "The method described in this article can be directly transferred to thermally driven Marangoni flow and natural convection in a pure droplet.", "Instead of a convection-diffusion equation for one component, one would have to consider the convection-diffusion equation for the temperature field.", "The boundary conditions will be different, e.g.", "a Dirichlet boundary condition of constant temperature at a highly conducting substrate and non-dimensional evaporative cooling instead of the number $\\mbox{\\it Ev}_y$ , but the methodological principle can remain the same.", "Also a generalization to negative Marangoni numbers could be interesting, but it would require the consideration of the problem in three dimensions.", "This would allow to predict axial symmetry breaking and also bifurcations into chaotic Marangoni flow regimes by performing a linear stability analysis of the quasi-stationary solutions.", "This work is part of an Industrial Partnership Programme (IPP) of the Netherlands Organization for Scientific Research (NWO).", "This research programme is co-financed by Canon Production Printing Holding B.V., University of Twente and Eindhoven University of Technology.", "DL gratefully acknowledges support by his ERC-Advanced Grant DDD (project number 740479)." ], [ "Declaration of Interests", "The authors report no conflict of interest.", "Since even the detailed full model is subject to some assumptions, e.g.", "the diffusion-limited vapour transport and the disregard of thermal effects, we also performed experiments on various sessile and pendant binary droplets with different volumes and contact angles.", "The details of the experimental setup are described by [23].", "Here, we are more interested in a qualitative agreement, i.e.", "whether the flow direction is dominated by natural convection or not.", "Simultaneously, we address the Grashof number (also known as Archimedes number) in the following.", "This number, defined as $\\mbox{\\it Gr}=gh^3\\rho _0(\\rho _\\text{A,pure}-\\rho _\\text{B,pure})/\\mu ^2\\,,$ with $h$ being the height of the droplet and $\\mu $ the averaged viscosity of both liquids, was used in our previous publication [23] as an indicator whether the flow in the droplet is dominated by natural convection ($\\mbox{\\it Gr}\\gg 1$ ) or not ($\\mbox{\\it Gr}\\ll 1$ ).", "Compared to the non-dimensional numbers presented in this manuscript, i.e.", "the evaporation number $\\mbox{\\it Ev}_y$ and the Rayleigh number $\\mbox{\\it Ra}$ , this number is independent of the current droplet composition.", "Instead, it just takes the pure densities of both fluids and the averaged density and viscosity into account.", "This means that the Grashof number $\\mbox{\\it Gr}$ is easily accessible, whereas the non-dimensional numbers used throughout this manuscript require the knowledge of the instantaneous average composition and the full composition-dependence of all properties, which is not always possible in an experimental setup.", "Therefore, we are interested to substantiate the argumentation by [23] that the much simpler Grashof number $\\mbox{\\it Gr}$ can be used as indicator whether to expect natural convection ($\\mbox{\\it Gr}\\gg 1$ ) or not ($\\mbox{\\it Gr}\\ll 1$ ).", "To investigate the validity, we replace the Rayleigh number by the Grashof number in the following.", "If one assumes that $\\partial _{y_\\text{A}}\\rho $ is independent of the composition, i.e.", "a linear dependence of the mass density on the mass fractions, one can obtain the Grashof number via the relation $\\mbox{\\it Gr}=\\frac{3}{\\pi }\\, \\frac{1-\\cos \\theta }{2+\\cos \\theta }\\,\\frac{\\mbox{\\it Ra}}{\\mbox{\\it Ev}_y\\mbox{\\it Sc}}\\,,$ where the factor depending on the contact angle $\\theta $ is a consequence of the different characteristic length scales, i.e.", "$\\@root 3 \\of {V}$ for $\\mbox{\\it Ra}$ and $h$ for $\\mbox{\\it Gr}$ .", "While the Schmidt number $\\mbox{\\it Sc}=\\mu /(\\rho D)$ for liquids is typically $\\mbox{\\it Sc}\\sim \\mathcal {O}(10^4-10^5)$ , for moderately volatile liquids like e.g.", "water at typical ambient conditions, $\\mbox{\\it Ev}_y\\sim \\mathcal {O}(10^{-1}-10^0)$ holds.", "In order to obtain diagrams independent of these quantities, we set the factor $\\mbox{\\it Ev}_y\\mbox{\\it Sc}=1000$ in (REF ) for the determination of the boundaries in the phase diagrams.", "Figure: Same as   and  , but expressed in terms of Gr\\mbox{\\it Gr} instead of Ra\\mbox{\\it Ra}.", "Obviously, the onset of gravity-driven flow, even in presence of rather strong Marangoni driving happens close to Gr=1\\mbox{\\it Gr}=1 (indicated by the grey line).", "Furthermore, experimental data of is also indicated.The phase diagrams rescaled to the Grashof number via this way are depicted in  REF .", "One can infer from these diagrams that even in competition with a strong Marangoni effect, the onset of gravity-driven bulk flow happens approximately at a Grashof number of $\\mbox{\\it Gr}\\sim \\mathcal {O}(1)$ for a contact angle of $\\theta ={70}{}$ in (a) and $\\theta ={100}{}$ in (b).", "Furthermore, the experimental results of [23] are indicated as dots.", "The 1,2-propanediol-water droplets with a contact angle of $\\theta ={70}{}$ discussed in [23] clearly show the effect of natural convection for an apex height of $h={800}{}$ , whereas is was not visible for $h={410}{}$ .", "This clearly coincides with the prediction of the phase diagram in  REF (a).", "The experiments on glycerol-water droplets with $\\theta ={100}{}$ , as discussed in the supplementary information of [23], reveal an absence of observable natural convection for $h={154}{}$ , whereas the presence of natural convection was found at heights $h\\ge {320}{}$ , with increasing velocity for elevated heights.", "Also this can be inferred from the $\\mbox{\\it Ma}$ -$\\mbox{\\it Gr}$ -diagram depicted in  REF (b).", "Thus we conclude that the Grashof number $\\mbox{\\it Gr}$ is indeed an indicator for the presence or absence of decisive natural convection in a binary droplet.", "The $\\mbox{\\it Ma}$ -$\\mbox{\\it Ra}$ -diagrams presented in this manuscript, however, provide a much more detailed prediction of the possible flow scenarios." ] ]

[ [ "Moir\\'e magnons in twisted bilayer magnets with collinear order" ], [ "Abstract We explore the moir\\'e magnon bands in twisted bilayer magnets with next-nearest neighboring Dzyaloshinskii-Moriya interactions, assuming that the out-of-plane collinear magnetic order is preserved under weak interlayer coupling.", "By calculating the magnonic band structures and the topological Chern numbers for four representative cases, we find that (i) the valley moir\\'e bands are extremely flat over a wide range of continuous twist angles; (ii) the topological Chern numbers of the lowest few flat bands vary significantly with the twist angle; and (iii) the lowest few topological flat bands in bilayer antiferromagnets entail nontrivial thermal spin transport in the transverse direction; These properties make twisted bilayer magnets an ideal platform to study the magnonic counterparts of moir\\'e electrons, where the statistical distinction between magnons and electrons leads to fundamentally new physical behavior." ], [ "Introduction", "The experimental discovery of two-dimensional (2D) ferromagnetism [1], [2] spurred a flurry of research, rapidly reshaping the field of 2D materials with an emerging frontier.", "Since then, 2D magnetic van der Waals materials has become a new and versatile platform to study fundamental physics in reduced dimensions [3], [4], [5], [6].", "A recent highlight in 2D materials is the observation of unconventional superconductivity [7] and Mott insulator [8] in twisted bilayer graphene where the electronic bands around the Fermi level are nearly flat, magnifying the relative strength of interaction effects.", "It has been demonstrated that in insulating magnetic materials, the spin-wave excitations (or magnons) can play the role of electrons in transporting spin angular momenta [9].", "In particular, the Dzyaloshinskii-Moriya interaction (DMI) acts as an effective spin-orbit coupling (SOC) on magnons, enabling the magnonic counterparts of pure spin phenomena usually associated with the electron spin [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].", "At this point, it is tempting to ask if twisting a bilayer magnet can generate any nontrivial magnonic bands and the consequential transport phenomena.", "If magnon bands become flat, twisted bilayer magnets (TBM) can be exploited to study the magnon-magnon and magnon-phonon interactions, the influences of which have been treated only perturbatively in traditional scenarios.", "Furthermore, if flat magnon bands are simultaneously topologically non-trivial, exotic thermomagnetic transport can arise.", "The physics behind flat magnon bands is not a trivial extension of that of electrons because the statistical distinction between magnons and electrons does not just superficially manifest in their thermal population but can also reflect profoundly in the governing equations [11].", "In this paper, we study moiré magnons of insulating TBM in honeycomb lattices, where symmetry allows for the next-nearest neighboring DMI in each layer.", "We assume that the interlayer coupling is of exchange nature and sufficiently weak compared to the intralayer exchange interactions, so the ground state remains in the collinear regime.", "In a comparative manner, we study four representative ground states of TBM [as illustrated in Fig.", "REF (c)] classified by the relative signs of intralayer and interlayer couplings.", "The moiré band structures will be obtained by plane-wave expansions [23], based on which we will further compute the associated Berry curvatures and topological Chern numbers.", "Except for the bilayer ferromagnetic (FM) state [case i in Fig.", "REF (c)], which has the same governing equation as a twisted bilayer transition metal dichalcogenides [24], [25], [26], [27], [28] or time reversal breaking layered materials [29], all TBMs involving antiferromagnetic (AFM) features [case ii, iii and iv in Fig.", "REF (c)] do not have direct correspondence to electronic systems.", "Therefore, the appearance of flat magnon bands in TBM reveals a new physical horizon different from the well-established twisted electronic systems.", "While magnons are bosonic excitations that preferably populate the high-symmetry $\\Gamma $ point, the nontrivial topological properties such as transverse spin transport originate from valley magnons around $K$ and $K^{\\prime }$ points, where the bands can reach local minima depending on the DMI.", "In particular, when two AFM layers are coupled ferromagnetically (such as MnPS$_3$ and its variances [30], [31], [32]), the lowest few bands in the valleys can be tuned into topologically flat bands at certain twist angle.", "Even though it is not clear if the flatness of a magnon band is directly linked with its topology, we make a crucial attempt to characterize this subtle relation by calculating the dependence of bandwidth and Chern numbers on the twist angle.", "It is worthwhile to distinguish the scope of our work from the findings of two recent papers [33], [34] published during the preparation of our manuscript.", "Ref.", "[Hejazi2020] studied the non-collinear ground states of TBM and the twist-induced phase transitions among those states, unravelling the strong interlayer coupling regime.", "On the other hand, this paper studies the moié magnons on top of collinear ground states, which holds in the parameter regime of weak interlayer coupling and strong anisotropy.", "Ref.", "[Ghader2019Magnon] focused on the FM ground state [corresponding to our case i in Fig.", "REF (c)] where the governing equation can be fully mapped to a Schrödinger equation describing a twisted bilayer electronic system.", "In contrast, four representative collinear ground states have been explored in this paper, providing a general and comparative picture of twisted TBM, where we especially identify the unique features originating from the AFM couplings (which can be interlayer, intralayer or both) that has no counterparts in electronic systems.", "The rest of this paper is organized as follows.", "In Sec.", ", we introduce the formalism starting with the model Hamiltonian.", "In Sec.", ", we show the moiré band structures and the Berry curvatures associated with the lowest few flat bands.", "In Sec.", ", we discuss the manipulation of the flatness and topological Chern numbers of the topological flat bands through the twist angle.", "A brief summary is given in Sec.", "." ], [ "Model and formalism", "Let us consider a magnetic insulator consisting of two honeycomb layers twisted by an angle $\\theta $ .", "In the tight-binding limit, this system can be described by the effective Hamiltonian $&\\mathcal {H}=\\sum _{n=1,2}\\mathcal {H}_n+\\mathcal {H}_T,$ with $&\\mathcal {H}_{n}=-J_{1}\\sum _{\\left<i,j\\right>}\\mathbf {S}_{n,i}\\cdot \\mathbf {S}_{n,j}+\\kappa \\sum _{i}(S_{n,i}^z)^2\\\\&\\qquad \\qquad +D_{2}\\sum _{\\left<\\left<i,j\\right>\\right>}\\epsilon _{ij}\\mathbf {\\hat{z}}\\cdot \\mathbf {S}_{n,i}\\times \\mathbf {S}_{n,j},\\\\&\\mathcal {H}_{T}=-\\sum _{i,j}J_{T}(i,j)\\mathbf {S}_{1,i}\\cdot \\mathbf {S}_{2,j}.$ Here, $\\mathcal {H}_n$ is the intralayer Hamiltonian for layer $n$ ($n=1,\\ 2$ denotes top, bottom layer) and $\\mathcal {H}_T$ is the interlayer exchange Hamiltonian with the coupling strength $J_{T}(i,j)$ depending on the distance $r_{ij}$ between site $i$ and site $j$ .", "$J_1$ is the nearest-neighbor exchange interaction ($J_1>0$ for FM and $J_1<0$ for AFM couplings), $D_2$ is the next-nearest neighboring DMI with $\\epsilon _{ij}=\\pm 1$ representing the chirality of atomic bonds, and $\\kappa <0$ is the perpendicular easy-axis anisotropy.", "On a single-layer honeycomb lattice, symmetry forbids nearest-neighboring DMI [11] unless the lattice is strained.", "By sticking to the regime of weak interlayer coupling, we also omit higher order interlayer interactions beyond $J_T$ .", "In electronic systems, the tight-binding model is problematic when the bands under consideration is topological nontrivial because constructing well-localized Wannier functions respecting all symmetries becomes impossible [35], [36], [37].", "Nevertheless, this problem does not even exist in topological magnons as long as the DMI does not destroy the collinear ground state.", "This is because the real-space basis of spin waves, unlike electronic Bloch waves subject to a Fourier transformation, is infinitely localized on the atomic sites.", "According to the signs of $J_1$ and $J_T$ , we classify bilayer magnets into four different cases labeled as i, ii, iii and iv as illustrated in Fig.", "REF (c).", "In particular, ii is representative of layered AFM such as CrI$_3$  [1] and iii is representative of intrinsic AFM with FM interlayer coupling such as MnPS$_3$  [30].", "At the same incommensurate angle, iii and iv exhibit local FM and AFM configurations, respectively, at the AA stacking regions due to the opposite sign of $J_T$ .", "In the following, the moiré magnon bands will be calculated by the plane-wave expansion under the single-layer basis, which is valid only when $J_T$ is weak so that the collinear ground state is preserved.", "Figure: (color online).", "(a) Schematics for a single-layer honeycomb lattice with c i=1,2,3 c_{i=1,2,3} and d i=1,2,3 d_{i=1,2,3} connecting the nearest and next-nearest neighbors, respectively.", "(b) Moiré pattern of a twisted bilayer magnet with alternating AAAA and ABAB stacking.", "(c) Schematic illustration of four different cases of twisted bilayer magnets classified by the signs of J 1 J_1 and J T J_T.", "(d) The moiré BZ formed by overlapping the top and bottom BZs at twist angle θ\\theta .", "(e) and (f): band structures of monolayer FM (J 1 >0J_1>0) and AFM (J 1 <0J_1<0) with different DMIs.", "The positive (negative) branch in (d) refers to the right-handed (left-handed) magnon excitations.We first solve the single-layer magnon bands.", "Neglecting magnon-magnon interactions, we adopt the linearized Holstein-Primakoff transformation [38] $S_{n,i}^{+}\\approx \\sqrt{2S}a_{n,i},\\quad S_{n,i}^{-}\\approx \\sqrt{2S}a_{n,i}^{\\dagger },\\quad S_{n,i}^{z}=S-a_{n,i}^{\\dagger }a_{n,i},$ for up spins and $S_{n,i}^{+}\\approx \\sqrt{2S}b_{n,i}^{\\dagger },\\quad S_{n,i}^{-}\\approx \\sqrt{2S}b_{n,i},\\quad S_{n,i}^{z}=b_{n,i}^{\\dagger }b_{n,i}-S,$ for down spins, where $S_{n,i}^{\\pm }=S_{n,i}^x\\pm S_{n,i}^y$ and $a_{n,i}^{\\dagger }$ ($b_{n,i}^{\\dagger }$ ) is the magnon creation operator for site $i$ on layer $n$ when there is an up (down) spin there.", "Next we take the Fourier transformation $a_{\\mathbf {k}}=\\frac{1}{\\sqrt{N}}\\sum _{\\mathbf {k}}e^{-i\\mathbf {k}\\cdot \\mathbf {r}_i}a_i$ and $b_{\\mathbf {k}}=\\frac{1}{\\sqrt{N}}\\sum _{\\mathbf {k}}e^{i\\mathbf {k}\\cdot \\mathbf {r}_i}b_i$ to transform $\\mathcal {H}_n$ into the momentum space, which in the pseudo-spin representation (i.e., $A$ and $B$ sublattices) takes the form $H_{n}\\left(\\mathbf {k}\\right)/S=\\begin{bmatrix}C+D_{2}g_{n\\mathbf {k}}&&-J_{1}f_{n\\mathbf {k}}\\\\-J_{1}f^{*}_{n\\mathbf {k}}&&C-D_{2}g_{n\\mathbf {k}}\\end{bmatrix},$ where the zero-point energy has been discarded.", "If an individual layer is AFM, the pseudospin space is furnished by the Nambu basis $\\psi _{\\mathbf {k}}=[a_{\\mathbf {k}}, b^\\dagger _{\\mathbf {k}}]^T$ such that $\\mathcal {H}_n=\\sum _{\\mathbf {k}}\\psi _{\\mathbf {k}}^\\dagger H_n\\psi _{\\mathbf {k}}$ .", "In Eq.", "(REF ), $C=3\\left|J_{1}\\right|+\\left|J_{T}\\right|-2\\kappa $ , $f_{n\\mathbf {k}}=\\sum _{i=1}^{3}\\exp \\left(i\\mathbf {k}\\cdot \\mathbf {c}_{n i}\\right)$ , $g_{n\\mathbf {k}}=2\\sum _{i=1}^{3}\\left(-1\\right)^{i-1}\\sin \\left(\\mathbf {k}\\cdot \\mathbf {d}_{n i}\\right)$ where $\\mathbf {c}_{n i}=e^{-i\\theta _{n}}\\mathbf {c}_{i}$ and $\\mathbf {d}_{n i}=e^{-i\\theta _{n}}\\mathbf {d}_{i}$ with the twist angle of the top (bottom) layer $\\theta _{1,2}=\\mp \\theta /2$ .", "The nearest neighbor links are $\\mathbf {c}_{1}=a_{0}\\begin{pmatrix}0,&&-1\\end{pmatrix}$ and $\\mathbf {c}_{2,3}=a_{0}\\begin{pmatrix}\\pm \\sqrt{3}/2,&&1/2\\end{pmatrix}$ , and the next-nearest neighbor links are $\\mathbf {d}_{1}=a_{0}\\begin{pmatrix}\\sqrt{3},&&0\\end{pmatrix}$ and $\\mathbf {d}_{2,3}=a_{0}\\begin{pmatrix}-1/2,&&\\pm \\sqrt{3}/2\\end{pmatrix}$ .", "The single-layer band structures obtained by diagonalizing Eq.", "REF are plotted in Fig.", "REF (e) and (f).", "For both FM and AFM, the lowest energy appears at the $\\Gamma $ point, which is not affected by the DMI.", "Recent studies have pointed out that even though the $D_2$ interaction originates from the intrinsic symmetry breaking of the honeycomb lattice, which is independent of the Rashba SOC, it does act on the magnons as an effective Rashba SOC thanks to the Holstein-Primakoff transformation [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].", "Next we turn to the interlayer hopping terms.", "In the continuum limit [23], the effective interlayer exchange Hamiltonian can be taken as $H_T\\left(\\mathbf {r}\\right)/S=\\sum _{i=1}^{3}T\\left(\\mathbf {q}_{i}\\right)e^{-i\\mathbf {q}_{i}\\cdot \\mathbf {r}},$ where $\\begin{split}&T\\left(\\mathbf {0}\\right)=-J_{T}\\begin{bmatrix}0&&1\\\\1&&0\\end{bmatrix},\\\\&T\\left(\\mathbf {G}_1\\right)=-J_{T}\\begin{bmatrix}0&&1\\\\e^{i2\\pi /3}&&0\\end{bmatrix},\\\\&T\\left(\\mathbf {G}_1+\\mathbf {G}_2\\right)=-J_{T}\\begin{bmatrix}0&&1\\\\e^{-i2\\pi /3}&&0\\end{bmatrix},\\end{split}$ are the interlayer hopping matrices connecting the three nearest neighbors in the momentum space labeled by the reciprocal lattice vectors $\\mathbf {G}_1=K_{\\theta }(\\frac{1}{2},\\ \\frac{\\sqrt{3}}{2})$ , $\\mathbf {G}_{2}=K_{\\theta }(-1,\\ 0)$ with $K_{\\theta }=2\\cdot 4\\pi /\\left(3a_{0}\\right)\\cdot \\sin \\left(\\theta /2\\right)$ .", "Similar to the Moiré electrons [39], magnons are essentially accumulated in the $AB$ stacking regions at small twist angles.", "So, to a good approximation, we set the interlayer hopping for the $AA$ stacking regions to be $t_{AA}=0$ while $t_{AB}=J_T$ for the $AB$ stacking regions, as shown in Fig.", "REF (b).", "We now use the plane-wave expansion to calculate the moiré magnon bands of a twisted bilayer magnet [23].", "Since the interlayer Hamiltonian $H_T(\\mathbf {r})$ is constructed in the real space above, we need to define the real-space intralayer Hamiltonian as well: $H_n(\\mathbf {r})=\\int d\\mathbf {k}H_n(\\mathbf {k})e^{-i\\mathbf {k}\\cdot \\mathbf {r}}$ .", "Then the total Hamiltonian becomes $\\mathcal {H}\\left(\\mathbf {r}\\right)=\\begin{bmatrix}H_1\\left(\\mathbf {r}\\right)&&H_{T}\\left(\\mathbf {r}\\right)\\\\H_{T}\\left(\\mathbf {r}\\right)^{\\dagger }&&H_2\\left(\\mathbf {r}\\right)\\end{bmatrix},$ from which the moiré magnon bands can be solved with a proper truncation in the Brillouin zone (BZ).", "Different from that of electrons, the eigen-equation for magnons $E\\eta \\Psi \\left(\\mathbf {r}\\right)=\\mathcal {H}\\left(\\mathbf {r}\\right)\\Psi \\left(\\mathbf {r}\\right),$ comes with an effective metric $\\eta =\\begin{bmatrix}1&&0\\\\0&&\\mathrm {sgn}\\left(J_{T}\\right)\\end{bmatrix}\\otimes \\begin{bmatrix}1&&0\\\\0&&\\mathrm {sgn}\\left(J_1\\right)\\end{bmatrix}$ that depends on the signs of both interlayer and intralayer couplings.", "The essential difference in band topology between electrons and magnons lies in this $\\eta $ matrix [11].", "If one (or both) of $J_T$ and $J_1$ becomes negative (AFM coupling), the eigenvalue problem here will not bear a correspondence in electronic systems, and intuitions acquired from electronic systems no longer apply.", "Moreover, since symmetry guarantees that the bands of twisted AFM (case ii, iii, and iv) come in pairs of right-handed chirality (positive branch) and left-handed chirality (negative branch) as those in single layer AFM, hereafter we will only show the right-handed ones for simplicity.", "To calculate the Berry curvature numerically, we follow the method in Ref.[fukui2005chern].", "Let the eigenvector in the momentum space be $\\Psi \\left(\\mathbf {k}\\right)$ , the Berry curvature can be expressed as $\\Omega \\left(\\mathbf {k}\\right)=\\frac{1}{2\\pi i}\\ln &\\left[U_x(\\mathbf {k})U_y(\\mathbf {k}+\\delta k_x)\\right.", "\\\\&\\left.\\times U_x(\\mathbf {k}+\\delta k_y)^{-1}U_y(\\mathbf {k})^{-1}\\right],$ where $U_j(\\mathbf {k})=\\frac{\\langle \\Psi (\\mathbf {k})|\\eta _{\\mathbf {k}}|\\Psi (\\mathbf {k}+k_j)\\rangle }{\\left|\\langle \\Psi (\\mathbf {k})|\\eta _{\\mathbf {k}}|\\Psi (\\mathbf {k}+k_j)\\rangle \\right|},\\quad j=x,y$ Here, the factor $\\eta _{\\mathbf {k}}=I_N\\otimes \\eta $ where $I_N$ is an $N\\times N$ identity matrix with $N$ the dimension of the Hamiltonian after truncation.", "The Chern number can then be obtained straightforwardly by integrating the Berry curvature over the first BZ $C=\\frac{1}{2\\pi }\\int {d^2k\\Omega \\left(\\mathbf {k}\\right)}.$ In our calculation, the plane-wave expansion is truncated at the crystal momentum $k_{t}=4K_{\\theta }$ if $\\theta >1^{}$ and $k_{t}=4K_{\\theta }/\\theta $ otherwise – a strategy widely adopted in the study of Moiré electrons [23].", "We adopt the scaling convention that the lattice constant $a_{0}=1$ and $\\left|J_1\\right|=1$ .", "Other parameters are taken to be $S=1$ , $\\kappa =-0.05$ , and $\\left|J_{T}\\right|=0.1$ unless otherwise specified." ], [ "$\\Gamma $ point magnons", "Magnons are bosons that preferably occupy lower-energy states.", "Therefore, at low temperatures, magnons are mostly populated in the vicinity of the $\\Gamma $ point.", "However, because of high symmetry, the moiré bands near new center of the shrunk moiré BZ–$\\gamma $ point–do not take on essential new features beyond what already exist near the single-layer $\\Gamma $ -point, thus impatient readers can skip to the next subsection directly.", "Figure: (color online).", "(a) and (b): the lowest moiré bands shrunk from the Γ\\Gamma point of the FM (case i) and AFM (case iii) single layers at θ=1 \\theta =1^{}.", "In both cases, J T =0.1J_{T}=0.1 and D 2 =0D_{2}=0.", "(c) and (d): the widths of the lowest band as a function of θ\\theta for different values of DMI.Nevertheless, for theoretical completeness, it is instructive to first study the comparably trivial $\\Gamma $ point.", "The commensurate condition requires that the $\\Gamma $ points of the top and bottom layers coincide with the new $\\gamma $ point of the moiré BZ up to integers of reciprocal lattice vector [41] as illustrated in Fig.", "REF (d).", "Consequently, twisting only introduces a phase difference between the Hamiltonians of the top and bottom layers within only the nearest-neighbor coupling in the momentum space.", "Meanwhile, the interlayer Hamiltonian reduces to $H_T\\left(\\mathbf {r}\\right)=T\\left(\\mathbf {0}\\right)=-J_{T}\\left[\\begin{matrix}1&&0\\\\0&&1\\end{matrix}\\right]$ .", "Then the total Hamiltonian becomes $\\mathcal {H}=\\left[\\begin{matrix}H_1\\left(\\mathbf {k}\\right)&&T\\left(\\mathbf {0}\\right)\\\\T\\left(\\mathbf {0}\\right)^{\\dagger }&&H_2\\left(\\mathbf {k}\\right) \\end{matrix}\\right],$ where $\\mathbf {k}$ is restricted to the first moiré BZ and the band structure can be solved by directly diagonalizing this Hamiltonian.", "Near the $\\Gamma $ point, single-layer bands shrink into the moiré BZ with minor deformation due to the weak interlayer coupling.", "Fig.", "REF (a) and (b) plot the lowest few bands for cases i and iii at twist angle $\\theta =1^{}$ , where the only visible difference is the zero energy gap at the $\\gamma $ point.", "Those bands are quadratic, topologically trivial and independent of the DMI.", "Moreover, varying the twist angle does not lead to flat bands.", "Fig.", "REF (c) and (d) plot the width of the lowest band as a function of $\\theta $ at different DMIs, where the reduction of the bandwidth with a decreasing $\\theta $ is simply due to the shrinking of the moiré BZ instead of any twist-induced qualitative change.", "To confirm that we have not missed any essential physics due to crude approximation, we have refined the above calculation by both including higher order terms in the interlayer coupling (in the $\\mathbf {k}$ space) and truncating the plane-wave expansion at larger momenta.", "However, by doing so, we do not observe any visible change with the same resolution in Fig.", "REF ." ], [ "$K$ point magnons", "Even though single-layer magnons are mostly populated near the $\\Gamma $ point, non-trivial transport properties are attributed to magnons from the $K$ and $K^{\\prime }$ -valleys [10], [11], [12], [13], [14].", "As shown in Fig.", "REF (e) and (f), a substantial DMI can even yield the energy of valley magnons comparable to that of $\\Gamma $ magnons.", "Therefore, with an increasing temperature, valley magnons become more and more important in determining spin transport.", "We expect similar outcomes for the morié magnons because the morié bands are calculated based on single-layer Hamiltonians in the presence of interlayer coupling.", "Figure: (color online).", "(a), (b), (c) and (d) are the moiré bands of valley magnons of the four cases defined in Fig. (c).", "The band Chern numbers are labeled on the top of each band.", "Note that the band Chern number is not well defined if two bands intersect with each other.", "The DMI is taken to be 0.05 in (a) and (b), and 0.35 in (c) and (d).", "The twist angle is θ=1.0 \\theta =1.0^{}.", "(e), (f), (g) and (h) are the corresponding Berry curvatures of the lowest band colored red.To study magnons folded from both the $\\Gamma $ and $K$ points uniformly, we shift the $\\gamma $ point of the shrunk moiré BZ to the origin by adding a twist-angle dependent constant $\\mathbf {k}_{0}=\\left\\lbrace 4\\pi \\cdot \\left(\\cos {\\theta /2}+\\sqrt{3}\\sin {\\theta /2}\\right)/\\left(3\\sqrt{3}a_{0}\\right),\\ 0\\right\\rbrace $ to the crystal momentum, defining $\\tilde{H}_n\\left(\\mathbf {r}\\right)=\\int d\\mathbf {k}H_n(\\mathbf {k}+\\mathbf {k}_0)e^{-i\\mathbf {k}\\cdot \\mathbf {r}}$ with $\\mathbf {k}$ restricted to the shrunk moiré BZ.", "Hence, the real-space Hamiltonian becomes $\\mathcal {H}\\left(\\mathbf {r}\\right)=\\begin{bmatrix}\\tilde{H}_1\\left(\\mathbf {r}\\right)&&H_{T}\\left(\\mathbf {r}\\right)\\\\H_{T}\\left(\\mathbf {r}\\right)^{\\dagger }&&\\tilde{H}_2\\left(\\mathbf {r}\\right)\\end{bmatrix},$ where $\\mathbf {k}$ is restricted to the shrunk moiré BZ.", "Fig.", "REF plots the moiré bands and the Berry curvatures associated with the lowest valley band colored red at $\\theta =1^{}$ for the four different cases illustrated in Fig.", "REF (c).", "In the bilayer FM case shown in Fig.", "REF (a), the bands are symmetric about the plane $E_c=3\\left|J_{1}\\right|+\\left|J_{T}\\right|-2\\kappa =3.2$ ; the Chern numbers of the bands above $E_c$ are opposite to those below $E_c$ .", "Regarding the mathematical similarity between FM magnons and Haldane's electrons on the single-layer honeycomb lattice [14], [13], [10], the eigenvalue problem in the twisted bilayer FM can be viewed as a magnonic version of twisted bilayer transition metal dichalcogenides [26], [27], where the Rashba SOC plays a crucial role in determining the band structures.", "Therefore, it is straightforward to understand why the two bands mostly close to $E_c$ are flat basing on intuitions of Moiré electrons.", "A recent study strongly suggests that $D_2$ is appreciably strong [20], which under the analogy to Haldane's electrons, corresponds to a much stronger effective Rasbha SOC acting on the magnons.", "In the other three cases plotted in Fig.", "REF (b), (c) and (d), AFM characteristics manifest in one way or another as the metric $\\eta $ appearing in Eq.", "REF involves the $-1$ factors.", "But they all share the common feature that magnon bands come in pairs of right-handed and left-handed chiralities, which mirror each other under certain symmetry transformations.", "This trait of symmetry has been discussed in detail for a single-layer honeycomb AFM [11], [12], but it becomes much less obvious when turning to the twisted bilayers.", "We believe that the reason is that the plane wave expansion is performed on the single-layer basis, which holds only for weak $J_T$ .", "Similar to the bilayer FM case, in Fig.", "REF (b), (c) and (d) the widths of the first few bands (colored) above the lowest energy point of single-layer valley magnons turn out to be negligible compared to the band gaps.", "It is noticeable that, when expressed in absolute values, they are even flatter than the electronic flat bands in twisted bilayer graphene.", "Unlike the bilayer FM case, however, they have no direct correspondence to the electronic systems because of the metric $\\eta $ , thus the flat bands in twisted AFMs are not superficial extensions of any flat-band electronic systems.", "The topological properties embedded in those flat bands are indicated by the Chern numbers marked in Fig.", "REF .", "While bands symmetrically distributed around $E_c=3.2$ in Fig.", "REF (a) have opposite Chern numbers, the effective symmetry plane in twisted bilayer AFMs is $E=0$ , which separates the right-handed sector from the left-handed sector The eigenfrequency $\\omega $ can be both positive and negative, depending on the chirality of magnons.", "But the energy is always positive.", "So here, $E$ should be understood as $\\hbar \\omega $ ..", "Accordingly, we find that the Chern numbers of the right-handed and the left-handed modes differ by a minus sign, allowing pure spin transport to be elaborated in the next section.", "Except for the case of Fig.", "REF (b)–layered AFM, the energetically relevant moiré magnon bands are either flat or topologically non-trivial, or both.", "In particular, the lowest flat band shown in Fig.", "REF (c) is simultaneously topologically non-trivial, which is amenable to thermally-induced pure spin transport.", "Even though the Chern numbers of the lowest four bands are all zero in Fig.", "REF (d) at $\\theta =1.0^{}$ , they can be tuned by varying $\\theta $ as will be shown in Fig.", "REF later.", "Moreover, the Berry curvature exhibits a six-fold (three-fold) rotational symmetry if the interlayer exchange interaction $J_T$ is FM (AFM), which can be attributed to the $\\eta $ factor unique to bosonic systems.", "In deriving the above results, we find that the plane-wave expansion is good only if the local band dispersion around the (single-layer) $K$ points is above a certain minimum.", "This sets an upper limit of DMI about $0.1$ for case i and ii and a lower limit of DMI about $0.3$ for case iii and iv, as the twist-angle varies from $0.5^{}$ to $5^{}$ ." ], [ "Manipulating Flatness and Topology By Twisting", "In this section we discuss how the width and topology of the lowest few flat bands of valley magnons (colored in Fig.", "REF ) depend on system parameters and the resulting effects in the transverse transport.", "Fig.", "REF plots the width $\\delta $ of the lowest valley magnon band as a function of the twist angle $\\theta $ and the DMI $D_2$ for the four different cases, respectively.", "When the single-layer ground state is FM (AFM), the range of $D_2$ is restricted to be 0 to $0.1$ ($0.3$ to $0.4$ ) to ensure the validity of the continuum model.", "We notice that this band is kept flat over a wide continuum of $\\theta $ rather than showing up at discrete and sharp “magic angles\".", "Of the four cases, only the bilayer FM (case i) shown in Fig.", "REF (a) is analogous to a twisted transition metal dichalcogenide [26], [27], whereas all the other cases do not have direct correspondence to any electronic systems due to the $\\eta $ matrix in Eq.", "REF .", "While we can still find the local minima of $\\delta $ as marked by the dashed line, it no longer represents where the band suddenly turns flat, which is almost invisible for any infinitesimal $D_2$ .", "$\\delta $ shows almost a monotonic dependence on $\\theta $ .", "However, the sharp pattern of discrete “magic angles\" where $\\delta $ suddenly turns flat can be retrieved in Fig.", "REF (a) (case i) when $D_2$ is exactly zero, where both the Hamiltonian and the governing equation reduce to those of a twisted bilayer graphene.", "Here, an infinitesimal $D_2$ opens a topological non-trivial gap at the $K$ and $K^{\\prime }$ points in the single-layer FM band [10], [13], which amounts to a discontinuous change of the bases in diagonalizing Eq.", "REF .", "The above finding indicates a profound fact beyond the magnonic morié systems: even in twisted bilayer electronic systems, including the Rashba SOC may remove the pattern of “magic angles\" and introduce a wide continuum of flat bands [27].", "Figure: (color online).Bandwidth δ\\delta of the lowest flat bands (see red curves in Fig. )", "as a function of the twist angle θ\\theta and the DMI D 2 D_{2} for |J T |=0.1|J_{T}|=0.1.", "The white dashed line in (a) indicates the local minima of δ\\delta .", "The white star marks where parameters in Fig.", "are chosen: θ=1 \\theta =1^{} and D 2 =0.05D_{2}=0.05 (0.350.35) for cases i and ii (iii and iv).Fig.", "REF further shows the dependence of $\\delta $ on the interlayer exchange interaction $J_{T}$ .", "For a given twist angle, the bandwidth decreases with an increasing $J_{T}$ in all cases, suggesting that the interlayer coupling enhances the lowest flat band.", "In Fig.", "REF (a), the white dashed lines mark the local minima of $\\delta $ .", "Similar to Fig.", "REF , the bandwidth $\\delta $ does not suddenly turn flat on these lines but remain essentially flat even away from these local minima, persisting over a wide range of angles.", "In addition, We notice that the bandwidth of case iv [Fig.", "REF (d)] is much smaller than other cases, which is consistent with and in fact incorporates what we have shown in Fig.", "REF (d).", "Next we turn to the topological property and the resulting transverse magnon transport.", "As has been established in single-layer magnonic systems, under a temperature gradient $\\nabla T$ , the band $m$ with a Berry curvature $\\Omega _m(\\mathbf {k})$ contributes to the total magnon flow with a magnon Hall current density as [43], [44] $\\begin{split}\\mathbf {J}_m=&\\frac{k_{B}}{h}\\hat{\\mathbf {z}}\\times \\nabla {T} \\int {\\frac{d^2k}{2\\pi }\\Omega _{m}\\left(\\mathbf {k}\\right)}\\left\\lbrace \\right.\\rho _{m}\\left(\\mathbf {k}\\right)\\ln {\\rho _{m}\\left(\\mathbf {k}\\right)}\\\\&-\\left[1+\\rho _{m}\\left(\\mathbf {k}\\right)\\right]\\ln {\\left[1+\\rho _{m}\\left(\\mathbf {k}\\right)\\right]}\\left.\\right\\rbrace ,\\end{split}$ where $k_B$ is the Boltzmann constant, $h$ is the Planck constant, and $\\rho _m(\\mathbf {k})=1/[e^{E_m(\\mathbf {k})/k_{B}T}-1]$ is the Bose-Einstein distribution function.", "If band $m$ is flat (i.e., $E_m(\\mathbf {k})$ is approximately a constant), $\\rho _m$ will be approximately a constant throughout the moiré BZ.", "As a result, $\\mathbf {J}_m$ can be simplified into $\\mathbf {J}_m\\approx \\frac{k_B}{h}\\hat{\\mathbf {z}}\\times \\nabla {T} f_mC_m$ where $C_m$ is the Chern number and $f_m=\\rho _m\\ln {\\rho _m}-(1+\\rho _m)\\ln {(1+\\rho _m)}$ is a constant depending only on temperature.", "When the lowest few bands are all flat, the total magnon Hall current density has a dominant contribution proportional to $\\sum _mf_mC_m$ .", "This simplification due to band flatness allows us to analyze the transverse transport in terms of topological Chern numbers (at least qualitatively) even though magnons are statistically distinct from electrons.", "Figure: (color online).Bandwidth δ\\delta of the lowest flat band (red curve in Fig. )", "as a function of the twist angle θ\\theta and the interlayer interaction J T J_{T} at D 2 =0.05D_2=0.05 (0.350.35) for cases i and ii (iii and iv).", "The white dashed lines in (a) denote the local minima of δ\\delta .Case i is quite straightforward.", "Even though we cannot obtain the dispersion relations and Chern numbers of all bands due to the limit of plane-wave expansion, we can claim a finite magnon Hall effect based on the acquired information.", "As discussed above, the bands in Fig.", "REF (a) are symmetrically distributed around $E_c$ and exhibit opposite Chern numbers for $E>E_c$ and $E<E_c$ .", "On the other hand, the thermal distribution $\\rho _m$ does not respect such symmetry.", "It is highly skewed around $E_c$ .", "Therefore, the magnon Hall currents arising from different bands, especially the contribution of the first few flat bands $\\sum _mf_mC_m$ , do not cancel.", "The most interesting cases are iii and iv, where the lowest few bands are topological flat bands so that they are energetically relevant to the transverse spin transport.", "In Fig.", "REF (a) and (b), we plot the Chern numbers of the lowest four topological flat bands as functions of $\\theta $ with color schemes matching those in Fig.", "REF (c) and (d).", "It is interesting to see that even though Fig.", "REF (d) (case iv) exhibits all zero Chern numbers at $\\theta =1^{}$ , changing $\\theta $ can induce nonzero Chern numbers, resulting in many topologically distinct phases within the range of $0.5^{}<\\theta <4^{}$ .", "Besides, the right-handed and left-handed magnons always have opposite Chern numbers regardless of the twisting angle, which establishes a direct connection between non-zero Chern numbers and the spin Nernst effect (SNE).", "Therefore, twisting becomes an effective operation to manipulate the band topology, hence the transverse spin transport in bilayer AFMs.", "Although this does not reflect the overall magnitude of SNE because higher bands are neglected, it reveals that the dominant contribution of SNE is indeed controllable via twisting.", "Figure: (color online).", "(a) and (b) are the Chern numbers of the lowest four bands depicted in Fig.", "(c) and (d) as functions of the twist angle with the same coloring.", "(c) and (d) are the total contribution of these four bands to the SNE coefficient at T=0.1T=0.1.", "Other parameters are taken to be the same as those in Fig. .Fig.", "REF (c) and (d) show the contributions of the lowest four topological flat bands to the SNE coefficient $\\kappa _{xy}$ , defined as $\\kappa _{xy}=\\hbar (J^{\\uparrow }-J^{\\downarrow })/(\\hat{z}\\times \\nabla {T})$ with $J^{\\uparrow }(J_{\\downarrow })$ the Hall current originated from left (right)-handed magnon band, for the cases of iii and iv in Fig.", "REF (a) and (b), respectively.", "In particular, $\\kappa _{xy}$ can even flip sign by varying $\\theta $ as shown in Fig.", "REF (c).", "While a more accurate and complete investigation of higher bands is needed to determine quantitatively the SNE in twisted bilayer magnonic systems, Fig.", "REF provides a first-step exploration into AFM moiré magnons in TBM, the dynamics of which, unlike twisted bilayer FM, have no direct correspondence to any electronic systems.", "In all four cases of TBM studied above, twisting proves to be a useful tuning knob of both the band flatness and the band topology.", "The subtle relation between the two seemingly different properties will be investigated in future studies." ], [ "Summary", "In summary, we have studied the moiré magnons in twisted bilayer magnets for four representative cases.", "By using the plane-wave expansion, we demonstrated that the moiré magnon bands shrunk from the $\\Gamma $ point of the single-layer magnet are neither flat nor topologically nontrivial.", "In striking contrast, magnon bands shrunk from the $K$ point become topological flat bands in the moiré BZ over a wide range of twist angles, which is robust against the interlayer exchange interaction as long as the collinear ground state is preserved.", "Those topological flat bands can generate transverse magnon transport that is controllable by the twist angle, opening an intriguing playground for topological Moiré magnons.", "Finally, we remark that the rapid development in 2D magnets is approaching a similar level of understanding as 2D electronic materials [26].", "In particular, monolayer magnet of long-range collinear order has been realized experimentally in varies magnetic materials.", "Considering that Moiré electronic bilayers can be fabricated by “tear-and-stack\" technique using monolayers, we expect a similar technique be devised for 2D magnets in the perceivable future." ], [ "ACKNOWLEDGMENTS", "This work is supported by the startup fund of UC-Riverside.", "The authors acknowledge useful discussions with Y. Zhang." ] ]

[ [ "Trimodal structure of Hercules stream explained by originating from bar\n resonances" ], [ "Abstract Gaia Data Release 2 revealed detailed structures of nearby stars in phase space.", "These include the Hercules stream, whose origin is still debated.", "Most of the previous numerical studies conjectured that the observed structures originate from orbits in resonance with the bar, based on static potential models for the Milky Way.", "We, in contrast, approach the problem via a self-consistent, dynamic, and morphologically well-resolved model, namely a full $N$-body simulation of the Milky Way.", "Our simulation comprises about 5.1 billion particles in the galactic stellar bulge, bar, disk, and dark-matter halo and is evolved to 10 Gyr.", "Our model's disk component is composed of 200 million particles, and its simulation snapshots are stored every 10 Myr, enabling us to resolve and classify resonant orbits of representative samples of stars.", "After choosing the Sun's position in the simulation, we compare the distribution of stars in its neighborhood with Gaia's astrometric data, thereby establishing the role of identified resonantly trapped stars in the formation of Hercules-like structures.", "From our orbital spectral-analysis we identify multiple, especially higher order resonances.", "Our results suggest that the Hercules stream is dominated by the 4:1 and 5:1 outer Lindblad and corotation resonances.", "In total, this yields a trimodal structure of the Hercules stream.", "From the relation between resonances and ridges in phase space, our model favored a slow pattern speed of the Milky-Way bar (40--45 $\\mathrm{km \\; s^{-1} \\; kpc^{-1}}$)." ], [ "Introduction", "The European Space Agency (ESA) is operating an astrometric mission Gaia which observed our Milky Way.", "Its second data release provides five astrometric parameters for 1.3 billion sources and additional line of sight velocities for 7.2 million sources.", "For stars, these astrometric data provide snapshots of their orbits in the Galactic potential and we here aim to obtain information on the dynamical structure of the Milky Way (MW) by investigating the phase-space distributions of its stars.", "In this context, velocity-space structures of the solar neighborhood have previously been repeatedly studied , , , but mainly under analytical or numerical approximations which we here drop.", "Figure: Radial and angular velocity-space distribution of stars within 0.2 kpc from the Sun, binned by 2 km s -1 ×2 km s -1 2\\;\\mathrm {km \\; s^{-1}} \\times 2\\;\\mathrm {km \\; s^{-1}}.", "From Gaia DR2 catalogue we selected stars whose relative errors in parallax are smaller than 10%.In Fig.", "REF , we present the distribution of radial velocities, $v_R$ , versus angular velocities, $v_{\\phi }$ , of stars observed with Gaia .", "In this map, we see several arch-shaped over-densities with such as the so-called `horn' near $(v_R,v_{\\phi }) \\sim (-50, 220) \\; \\mathrm {km\\;s^{-1}}$ , and the so-called `hat' near $(v_R,v_{\\phi }) \\simeq (\\pm 50, >250) \\; \\mathrm {km\\;s^{-1}}$ , .", "The most prominent structure is called the `Hercules' stream, and is located between $(v_R,v_{\\phi }) \\simeq (100, 200) \\; \\mathrm {km\\;s^{-1}}$ and $(v_R,v_{\\phi }) \\simeq (-70, 170) \\; \\mathrm {km\\;s^{-1}}$ .", "originally identified the Hercules stream as $U$ -anomaly from the Hipparcos data , , and Gaia DR2 , revealed further detailed structures within.", "Gaia DR2 showed for the first time the trimodal structure of the Hercules stream, which is composed of three sub-streams at $v_{\\phi } \\simeq 220$ , 200, and $180\\,\\mathrm {km\\;s^{-1}}$ , , , .", "Although we refer to the bottom stream in Fig.", "REF as `Hercules 3', it is identical to a moving group, HR 1614 , , .", "Soon after the first discovery of the Hercules stream, a scenario on its origin was proposed by , .", "He calculated the evolution of the stellar phase space distribution in the simple $m=2$ bar potential (where $m$ indicates the azimuthal Fourier mode), and then argued that stars trapped in the 2:1 outer Lindblad resonance (OLR) of the bar can create moving groups in the solar neighborhood.", "The exact position of the formed structure in phase space depends on the location of the OLR and the OLR position depends in turn on the pattern speed of the bar ($\\Omega _{\\mathrm {b}}$ ).", "Hence, to reproduce the observed features, this scenario requires a fast rotating bar to bring the 2:1 OLR around 8kpc, i.e.", "$\\Omega _{\\mathrm {b}}/\\Omega _0 = 1.85$ (where $\\Omega _0$ is the local circular frequency); this corresponds to $\\Omega _{\\mathrm {b}}= 53 \\pm 3 \\; \\mathrm {km \\; s^{-1} kpc^{-1}}$ , , , [3].", "Such a fast ($\\Omega _{\\mathrm {b}}\\sim 50\\; \\mathrm {km \\; s^{-1} kpc^{-1}}$ ) bar model is also favored by test particle integration in a more realistic Milky Way (MW) Galaxy potential that was constructed from an $N$ -body simulation .", "On the other hand, recent observations suggests slower pattern speeds of the bar.", "Combining data from Gaia DR2 and further surveys, both and estimated a pattern speed of $\\Omega _{\\mathrm {b}}= 41 \\pm 3 \\; \\mathrm {km \\; s^{-1} \\; kpc^{-1}}$ , and estimated $\\Omega _{\\mathrm {b}}= 37.5\\; \\mathrm {km \\; s^{-1} \\; kpc^{-1}}$ .", "Recent measurements of the bar length also support a slow bar model.", "The measured ratio of the corotaion radius ($R_{\\rm CR}$ ) to the bar length ($R_{\\rm b}$ ) $R_{\\mathrm {CR}}/R_{\\mathrm {b}} = 1.2 \\pm 0.2$ for external galaxies [1], [2].", "The bar length of $R_{\\mathrm {b}} = 5.0 \\pm 0.2$  kpc in the Milky Way , therefore, suggests the corotation radius of $R_{\\mathrm {CR}} \\simeq $ 5–7 kpc or the pattern speed of $\\Omega _{\\mathrm {b}}\\simeq $ 34–47 $\\mathrm {km \\; s^{-1} \\; kpc^{-1}}$ .", "In addition, such slow and long bar is also supported by the dynamical modelling of bulge stars and gas kinematics , .", "With such a slow pattern speed, the 2:1 OLR should be located further than 8 kpc.", "The slow and long bar model enhanced the studies of the Hercules stream's origin, and new scenarios have been suggested.", "Most of them agree with the concept that resonant orbits due to non-axisymmetric structures, such as a bar and/or spiral arms, create the moving groups in the solar neighborhood revealed by Gaia DR2.", "One of the new scenarios is the bar's corotation (CR) scenario.", "integrated orbits of test particles in a gravitational field constructed from a self-consistent $N$ -body model of the Milky Way to match observational data using the Made-to-Measure (M2M) method , and then proposed that the Hercules-like stream can be made of stars orbiting around Lagrangian points of the bar.", "These stars move outward from the bar's CR radius to visit the solar neighbourhood.", "In this study, they assumed a slow bar ($\\Omega _{\\mathrm {b}}\\sim 40 \\; \\mathrm {km \\; s^{-1} \\; kpc^{-1}}$ ).", "This scenario is also supported by analytic calculations of perturbed distribution functions in resonance regions , , [11], and full $N$ -body simulations of a MW-like galaxy .", "Another new scenario attributes the Hercules stream to higher-order resonances of the slow, long bar.", "integrated orbits of test particles in analytic bar potentials including an $m=4$ Fourier mode and suggested that the 4:1 OLR of a slowly rotating bar can lead to a bi-modal structure in the Hercules stream.", "also studied the relation between the bar's higher-order resonances and the solar neighborhood kinematics using a resonant distribution function model in the same realistic bar potential as that used in .", "They showed that CR and 6:1 OLR of the bar create a Hercules-like stream and a horn-like structure, respectively.", "Similarly, performed test particle simulations in a slow-bar potential with higher-order multipole moments combined with a spiral potential, and suggested that the Hercules stream associated with the spiral's 8:1 ILR [7].", "On the other hand, also showed that Hercules-like streams originate from orbit families associated with the 5:1 inner Lindblad resonance (ILR) in the simple $m=2$ slow-bar + spiral potentials.", "Note that most of the above studies are based on test particle simulations in `static' (or analytic) potentials.", "We can easily capture resonant orbits of stars in static potentials; however, previously performed self-consistent $N$ -body simulations of (barred) spiral galaxies have suggested that the structures in the spiral arms and bar change with time in complicated ways , , [5], [6], , , , , .", "Such transient nature may affect stellar orbits in higher-order resonances.", "Higher order resonances are usually weaker than CR or 2:1 OLR.", "In order to detect higher order resonances such as 4:1 and 6:1 OLR in self-consistent $N$ -body simulations, a high enough resolution is required.", "Recently, captured stars in the CR resonance using fully self-consistent $N$ -body simulations, but they did not report higher order resonances as the origins of the Hercules stream.", "Hence, so far no high-order resonances have been reported in self-consistent $N$ -body simulations.", "Phase space structures have been found in these self-consistent $N$ -body models, but their origin is unknown.", "performed $N$ -body simulations of disk galaxy models and found a MW-like model, of which some observed structures and kinematics match those of the MW.", "They performed simulations using a maximum of eight billion particles (more than two hundred million particles for the disk) with the dark-matter halo modeled with $N$ -body particles (live halo).", "This is an order of magnitude higher mass-resolution than previous similar studies .", "showed $U$ -$V$ maps obtained in their simulations and discussed if Hercules-like streams are found in $N$ -body simulations.", "Indeed, they found some Hercules stream-like structures in their simulations, but the origin of the structures has not been investigated.", "In this paper, we analyze the results of the $N$ -body simulations performed in , which represents an isolated Milky Way-like galaxy modeled completely as an $N$ -body system.", "In this simulation, we track and classify the stellar orbits.", "The purpose of the classification is to identify stars trapped in resonance.", "We then show that stars trapped in higher-order resonances actually exist in the live disk potential and that the trimodal structure of the Hercules stream can be explained by such higher-order resonances.", "We use one of the Milky Way $N$ -body simulations that were performed by .", "Here, we describe their model and simulation methods.", "We use model MWa of .", "This model is a Milky Way-like galaxy composed of a live stellar disk, a live classical bulge, and a live dark-matter (DM) halo, and the initial conditions were generated using GalactICS , .", "The stellar disk follows an exponential profile with a mass of $3.73 \\times 10^{10}M_{}$ , an initial scale-length ($R_{\\rm d}$ ) of $2.3$  kpc, and an initial scale-height of $0.2$ pc.", "The classical bulge follows the Hernquist profile , whose mass and scale-length are $5.42 \\times 10^9 M_{}$ and 750 pc, respectively.", "The DM halo follows the Navarro–Frenk–White (NFW) profile , whose mass and scale radius are $8.68 \\times 10^{11}M_{}$ and 10 kpc, respectively.", "A more detailed model description can be found in .", "The simulations were performed using the parallel GPU tree-code, BONSAIhttps://github.com/treecode/Bonsai [8], [9] with a GPU cluster, Piz Daint.", "The simulation was started from a disk without any structures.", "After $\\sim 2$  Gyr, a bar started to form, and continued to grow until $\\sim 5$  Gyr.", "During the evolution, the bar slowed down with oscillations up to $\\sim 8$  Gyr.", "Spiral structures also formed, and are most prominent at $\\sim 5$  Gyr.", "However, they faint at later times due to the dynamical heating of the disk.", "The simulation was continued up to 10 Gyr.", "This simulation is one of the most highly resolved $N$ -body simulations that we have performed.", "The numbers of particles of this models are 30M, 208M, and 4.9B for the bulge, disk, and halo, and the same mass resolution is used for all the three components.", "This large number of particles enables us to perform a direct comparison of simulation results with observed data.", "A further advantage of this simulation is the high time resolution of the particle data output.", "The position- and velocity-snapshots of disk particles were stored every 9.76 Myr.", "We therefore can directly recover the actual orbits of individual stars from the snapshots." ], [ "Determination of the bar's pattern speed in the simulation", "In this simulation we determine the bar's pattern speed using the Fourier decomposition as was also done in .", "We divide the galactic disk into annuli with a width of 1 kpc, and then Fourier decompose the disk's surface density in each annulus: $\\Sigma (R, \\phi ) =\\sum _{m=0}^{\\infty } A_m(R) \\exp \\lbrace im[\\phi - \\phi _m(R)]\\rbrace ,$ where $A_m(R)$ and $\\phi _m(R)$ are the $m$ -th mode's amplitude and phase angle, respectively.", "We define $\\phi _2(R)$ averaged in $R<3$  kpc as the angle of the bar in the snapshot .", "We obtained the angle of the bar of the last 64 snapshots, which corresponds to $t=9.38$ –10 Gyr.", "Finally, the bar's pattern speed, $\\Omega _{\\mathrm {b}}$ , is determined using the least squares fitting to $\\phi _2 (t) = \\Omega _{\\mathrm {b}}t + \\phi _{2, 0}$ , where $\\phi _{2, 0}$ is the angle of the bar in the first snapshot we used.", "The thus obtained pattern speed of the bar in the simulation is $\\Omega _{\\mathrm {b}}=46.12\\; \\mathrm {km\\;s^{-1}\\;kpc^{-1}} = 1.53 \\Omega _{\\mathrm {8kpc}}$ where $\\Omega _{\\mathrm {8kpc}}$ is the circular frequency at $R=8$  kpc in our MW model.", "Recent studies such as and suggested a pattern speed of the Galactic bar of $\\Omega _{\\mathrm {b}}= 41\\; \\mathrm {km\\;s^{-1}\\;kpc^{-1}} \\simeq 1.4\\Omega _0$ , where $\\Omega _0$ is the circular velocity at the solar distance.", "The bar's pattern speed in our simulation is slightly higher than this value, but slower than that favored by 's 2:1 OLR model for the Hercules stream ($\\Omega _{\\mathrm {b}}\\simeq 50\\; \\mathrm {km\\;s^{-1}\\;kpc^{-1}} \\simeq 1.8\\Omega _0$ )." ], [ "Distribution of simulated stars in velocity space", "To study the origin of the Hercules stream, we begin by locating a Sun-like position in the simulated galactic disk.", "We take the last snapshot of the simulations ($t=10$  Gyr) and plot the velocity-space distribution of particles within 0.2 kpc from the “Sun” and iterate over positions in the disk.", "The results are shown in Fig.", "REF .", "Note that we assume the position of the “Sun” in the galactic mid-plane ($z=0$ ) in the MW model.", "At certain locations, we find a velocity-space structure similar to the one observed in Gaia DR2 ( and Fig.", "REF ).", "These observational studies suggest that the Sun has a distance of $R_{}=(8.178 \\pm 13 \\pm 22)$  kpc from the Galactic center and lies at an angle of $\\phi _{}=27^{\\circ } \\pm 2^{\\circ }$ , $ 29^{\\circ }\\pm 2^{\\circ }$ , $ 24^{\\circ }$ –$27^{\\circ }$ , and $ 20^{\\circ }$ –$25^{\\circ }$ [13] with respect to the major axis of the Galactic bar.", "By analysing our simulations, we select a distance and position relative to the bar of $(R, \\phi )=(8\\, \\mathrm {kpc},20^{\\circ })$ .", "This choice is based on the similarity of the structure observed in Fig.", "REF and compared with the kaleidoscope of similar phase-space images in Fig.REF .", "The velocity-space map at $(R, \\phi )=(8\\, \\mathrm {kpc},30^{\\circ })$ also shows the Hercules-like stream, but it is most clearly identified in the map at $(R, \\phi )=(8\\, \\mathrm {kpc},20^{\\circ })$ .", "The space-coordinates framed in red in Fig.REF has the closest correspondence with the observed image.", "We then select this particular phase-space position as the one to represent the Sun in our simulations.", "Fig.", "REF is the enlarged figure of the velocity-space distribution at $(R, \\phi )=(8\\, \\mathrm {kpc},20^{\\circ })$ (same as the panel framed by a red rectangle in Fig.", "REF ).", "A Hercules-like stream is located from $(v_R, v_{\\phi } )\\simeq (80, 200 )\\, \\mathrm {km\\; s^{-1}}$ to $(v_R, v_{\\phi } )\\simeq (-60, 190 )\\, \\mathrm {km\\; s^{-1}}$ in this figure.", "In addition, a Hat-like stream and a Horn-like structure are also seen from $(v_R, v_{\\phi } ) \\simeq (50, 260 )$ to $(v_R, v_{\\phi } ) \\simeq (-50, 270 )\\, \\mathrm {km\\; s^{-1}}$ and around $(v_R, v_{\\phi } ) = (-50, 220 )\\, \\mathrm {km\\; s^{-1}}$ , respectively.", "Some other moving groups known from observations, such as the Hyades and Pleiades, cannot be clearly mapped to structures in the simulated MW model.", "The key advantage of working with an $N$ -body simulation is that resonances can be tested in live systems.", "The spiral structures and bars evolve with time in a live potential, which is not the case in static potential simulations.", "In such a time-dependent potential, stars in a resonance may not be able to stay stable in the resonance.", "In order to capture resonances in a live potential, the orbits of individual stellar orbits have to be followed in the live simulations.", "To this aim, we thus use the snapshots of our $N$ -body simulation, and trace the orbits of the stars that we are interested in.", "We especially determine the orbital frequencies of the disk particles in order to classify the particles by their orbital characteristics such as resonances.", "For this purpose, we perform a frequency analysis of stellar orbits obtained from the $N$ -body simulation with a following frequency measurement method used in .", "Here, we detail upon the classification scheme used to identify stellar properties in our simulation.", "First, we determine the radial frequency, $\\Omega _R$ , using the Discrete Fourier Transformation (DFT) for $R(i)$ where $R(i) \\; (i=1,\\ldots ,64)$ is a radial coordinate in the $i$ -th snapshot.", "We employ a zero-padding technique for Fourier transforming: 960 zero points are added at the end of the data series.", "We then sample frequency space with 512 points between $0\\; \\mathrm {km\\; s^{-1}\\; kpc^{-1}}$ and $315\\; \\mathrm {km\\; s^{-1}\\; kpc^{-1}}$ , whereby the upper bound is given by the Nyquist frequency.", "We identify a resonant $\\Omega _R$ as a frequency that causes a local maximum in the Fourier spectrum: obviously the local maximum indicates an over-abundance of stars at this frequency.", "In contrast, the associated angular frequency $\\Omega _{\\phi }$ is determined by a regression analysis instead of a Discrete Fourier Transform.", "From the snapshots, we collect per particle the series of measured angles $\\phi (i)$ as a function of time $t(i)$ , where $i$ iterates over the 64 snapshots available.", "For each particle, this results in pairs $[t(i), \\phi (i)]\\; (i=1,\\ldots ,64)$ to which we fit the function $\\phi = \\Omega _{\\phi } t +\\phi _0$ using a least squares method.", "The resulting Least-Squares Estimator yields the angular frequency $\\Omega _{\\phi }$ of the studied star.", "Figure: Orbital frequency ratios for the particles within 0.2 kpc from (R,φ)=(8 kpc ,20 ∘ )(R, \\phi )=(8\\, \\mathrm {kpc}, 20^{\\circ }).", "The four vertical solid lines indicate (Ω φ -Ω b )/Ω R =(\\Omega _{\\phi } - \\Omega _{\\mathrm {b}})/\\Omega _R = -1/2, -1/3 -1/4, and -1/5, corresponding to the 2:1, 3:1, 4:1, and 5:1 resonances, respectively.", "Pairs of dashed lines beside solid lines indicate frequency ratios of -1/2±0.01-1/2 \\pm 0.01, -1/3±0.01-1/3 \\pm 0.01, -1/4±0.01-1/4 \\pm 0.01, and -1/5±0.01-1/5 \\pm 0.01, respectively.", "For each resonance, particles whose frequency ratios are between a dashed line pair are selected as resonantly trapped particles.In Fig.", "REF , we show the distribution of the frequency ratio $(\\Omega _{\\phi } - \\Omega _{\\mathrm {b}})/\\Omega _R$ for the particles within 0.2 kpc from our “Sun's” location at $(R, \\phi )=(8\\, \\mathrm {kpc}, 20^{\\circ })$ .", "Here, we use the bar's pattern speed ($\\Omega _{\\mathrm {b}}$ ) obtained from our simulations (see Section 2.2).", "The clearly visible statistically significant peaks in the distribution correspond to stars in resonances with the bar, whereas small fluctuations simply arise from binning.", "In Fig.", "REF , we indicate the positions of multiple outer Lindblad resonances (OLR) as vertical solid lines.", "In the region that we assumed as the solar neighborhood, we find at the four OLR of 2:1, 3:1, 4:1, and 5:1.", "We assume that particles within a rage of $\\pm 0.01$ from the exact resonance frequency ratio are also in resonance; these are the two adjacent bins to the exact resonance frequency.", "We select these as resonant particles in the following analyzes.", "As an example, for particles trapped in the 2:1 resonance, we select those whose frequency ratios are in the range of $-0.51<(\\Omega _{\\phi } - \\Omega _{\\mathrm {b}})/\\Omega _R < -0.49$ which corresponds to the bins between the two vertical dashed lines next to the line of $(\\Omega _{\\phi } - \\Omega _{\\mathrm {b}})/\\Omega _R = -0.5$ .", "Figure: Examples of the orbits trapped in the bar resonances.", "The gray ellipse in the figure represents the bar orientation.", "The orbits are shown in the bar's rotating frame.", "The galaxy rotates clockwise, hence particles rotate counter-clockwise in this frame.Figure: Resonantly trapped orbits in a single frame.", "The blue dashed, green dotted, magenta dash-dotted, and orange solid lines represent the 2:1, 3:1, 4:1, and 5:1 resonance orbits, respectively.", "In this figure, the galaxy rotates clockwise.", "The star symbol presents the position of the Sun in our model, (R,φ)=(8 kpc ,20 ∘ )(R, \\phi )=(8\\, \\mathrm {kpc},20^{\\circ }).In order to confirm the validity of this selection procedure, we verify whether the selected particles are in resonant orbits.", "to verify the results of the Fourier analysis we randomly select 100 particles from the various resonant areas and plot their orbits over time.", "By visually inspecting them, as the sub-sample of 5 cases for each of the orbital resonances, presented in Fig.", "REF , we confirm that except a few odd cases, all stars are on the appropriate resonance orbit.", "We show the panels that demonstrate this procedure for 100 different plotted orbits in the online material.", "All the orbits in Fig.", "REF are shown in the rotating frame of the bar, which is represented by the gray ellipse in each panel.", "In this figure, the galaxy rotates clockwise.", "In this frame, the particles accordingly rotate counter-clockwise as their angular frequencies are slower than the bar's pattern speed.", "Stars trapped in the $m$ :1 resonances oscillate $m$ times in the radial direction while they circle the bar.", "Thus, we confirm that our method properly identifies stars in resonant orbits.", "Note that naively, one would expect that orbits of the 2:1 resonance to align with the bar or to be perpendicular to it in the case of OLR.", "In our simulation, however, we find that the orbits in the 2:1 OLR are inclined with respect to the orientation of the bar as shown in the top five panels in Fig.", "REF .", "In other regions, however, particles which are exactly aligned with or perpendicular to the bar are dominant.", "In the region which we assumed as the solar neighborhood, particles in inclined orbits are dominant.", "The reason is currently not clear, but this may be because the position we \"observed\" is not exactly on the 2:1 OLR radius.", "Other possible reasons are spiral arms and/or the bar's slow-down.", "Orbital radii decrease from lower to higher order resonances (from the top to the bottom in Fig.", "REF ).", "This is more apparent in Fig.", "REF , in which we show four resonantly trapped orbits in a single frame.", "This is simply because higher-order resonances have smaller guiding radii." ], [ "Results for Gaia DR2", "Using the orbit analysis of our simulation detailed upon in Sect.", ", we succeeded in identifying the simulation's bar pattern speed, and the resonance type of stars trapped in resonant orbits.", "By now comparing to the Gaia data, we can thus use our simulation analysis to constrain the properties of the Milky-Way galaxy.", "The thus found implications for the origin of the Hercules stream are presented in Sect.", "REF , and constraints on the Milky Way's bar pattern speed are presented in REF ." ], [ "The origin of the Hercules stream from resonances", "The Gaia DR2, the Hercules stream creates a prominent overdensity of stars in the $v_R$ versus $v_{\\phi }$ projection of the Gaia data (see Fig.", "REF ).", "The recognition of this overdensity's trimodality is recent, and requires an explanation.", "We here present the evidence that stars trapped in resonant orbits will form such a trimodal stream structure.", "This can be seen from the top panel of Fig.", "REF , where we over plot the position of particles in resonance in the velocity map (Fig.", "REF ).", "In the bottom panel of the figure, we show the velocity map colored by the fraction of particles in resonance in each bin.", "We find that between 60% and 100% of all stars in the Hercules-like stream of our simulation are trapped in the 4:1 and 5:1 OLR.", "This implies that the stars within the Hercules stream (two streams at $v_{\\phi }=220$ and 200 km s$^{-1}$ ) will also be dominated by resonantly trapped stars.", "Although the two resonantly trapped families are blurred in our velocity map of Fig.", "REF , these two streams light up and are clearly separated when classified by resonance type in Fig.", "REF , thereby providing a natural explanation for multiple streams of Gaia's observations of the Hercules stream.", "In our simulations, stars trapped in the 2:1 OLR have velocities much higher than the rotation speed at the position of the Sun.", "This is because the guiding center of the 2:1 OLR in our model is outer than 8kpc.", "These stars appear to populate the area in phase space that was characterized by the `hat' structure in the observation.", "Stars in the 3:1 OLR are distributed around the circular velocity at 8kpc.", "Among the 3:1 OLR stars, those with $v_{\\rm R}<0$ are part of the `horn' structure.", "The guiding-center radius ($R_{\\rm g}$ ) for each OLR for our simulated model can be determined as we know the structure of the simulated galactic disk.", "The radial frequency under the epicycle approximation is given by [12]: $\\kappa ^2 (R_\\mathrm {g}) ={\\left(R \\frac{\\mathrm {d}\\Omega ^2}{\\mathrm {d}R} + 4\\Omega ^2\\right)}_{R_\\mathrm {g}}.$ Here, we calculate $\\kappa $ and $\\Omega $ at each radius from the particle data of the $N$ -body simulation.", "We also determine the circular frequency, $\\Omega $ , by averaging the radial accelerations of particles in an annulus with a width of 50 pc at each radius.", "In Fig.", "REF , we show $\\Omega + \\kappa /m$ ($m=$ 2, 3, 4, and 5) and $\\Omega $ as functions of $R$ .", "The horizontal line in Fig.", "REF indicates the pattern speed of the bar.", "The radii at which the line intersects with the frequency curves correspond to the guiding-center radii for the 2:1, 3:1, 4:1, 5:1, and the corotation resonances.", "In our model of the Milky way, the Sun, at a distance of $\\sim 8$  kpc from the Galactic center, is close to the 3:1 OLR.", "It is then not surprising that this resonance is dominant in the local stellar population.", "Figure: The orbital frequencies as functions of the galactic radius, RR.", "Ω\\Omega was determined from the velocity curve and Ω+κ/m\\Omega +\\kappa /m (m=m=2, 3, 4, and 5) as expected within the epicycle approximation are plotted.", "The vertical axis is normalized by Ω 8 kpc \\Omega _{8\\mathrm {kpc}}.", "The horizontal line indicates the bar's pattern speed, and the vertical lines represent the guiding radii for the respective resonant orbits." ], [ "The Milky Way's bar pattern speed", "When projecting the Gaia DR2 data into the $R$ -$v_{\\phi }$ plane, a series of ridges appears over a wide range of $R$ , [4], .", "Although the origin of these structures is still in debate, previous studies , , , [7] have discussed that these ridges may originate from resonances, while other studies have suggested perturbations by a satellite galaxy such as the Sagittarius dwarf , or by winding transient spiral structure , .", "In this section, we study the relation between the observed ridges in Gaia DR2 and resonances in our simulated MW model: by projecting our simulated and classified stars into the $R$ -$v_{\\phi }$ plane, we can directly compare to the same projection of the Gaia DR2 data.", "If the stars classified by resonance type align with the observed ridges, then this provides compelling evidence that these ridges are indeed caused by resonantly trapped stars.", "In Fig.", "REF , we show the distributions of stars in the $R$ -$v_{\\phi }$ plane as it arises in our simulated MW model (left column) and in the Gaia DR2 (right column).", "The top panels show 2D-histograms of particles within 2 kpc from the Sun [$(R,\\phi ) = (8\\, \\mathrm {kpc}, 20^{\\circ })$ , in our model].", "This panel reveals a selection effect caused by Gaia's measurement uncertainties: due to us having selected stars with relative parallax error below 10% the right-hand panel displays systematically fewer stars in approximately concentric rings around the Sun's position at $8.2\\, \\mathrm {kpc}.$ Visually, this causes the smaller extend of the red area in the right-hand panel.", "We expected, however, that our analysis is robust with respect to this selection effect: it implies that fewer stars make it into our DR2 projection than into the projection of our simulation, but those stars that are selected are not systematically biased.", "The middle panels show the mean $v_R$ values in the respective bins in the top panels.", "In order to make this figure, we selected stars from the Gaia DR2 catalogue whose relative errors in parallaxes are less than 10%, and whose errors in radial velocities are less than $5 \\; \\mathrm {km \\; s^{-1}}$ .", "Their distances from the Galactic mid-plane are less than 0.2 kpc, and their distances from the Sun are less than 2 kpc.", "The total number of stars in this sample is 2,289,755.", "We assume that the distance of the Sun from the Galactic center is $R_0 = 8.2$  kpc, and that the distance of the Sun from the Galactic mid-plane is about $z_0 = 25$  pc, whereby the velocity of the Sun with respect to the Local Standard of Rest (LSR) is about $(U_{},V_{}) = (10, 11) \\, \\mathrm {km \\, s^{-1}}$ , and the circular velocity at $R=R_0$ of $\\Theta _0 = 238 \\; \\mathrm {km \\; s^{-1}}$ [14].", "In the middle right panel in Fig.", "REF , we can identify several streams, or ridges, in the Gaia data.", "For convenience, we refer to ridges with $v_R>0$ as `red' and $v_R<0$ as `blue'.", "From top to bottom, red ridges alternate with blue ridges.", "Amongst these, two red ridges of $v_R>0$ are located at $v_{\\phi } \\sim 200 \\; \\mathrm {km\\; s^{-1}}$ and $v_{\\phi } \\sim 180 \\; \\mathrm {km\\; s^{-1}}$ at 8 kpc, are associated with the Hercules stream.", "In the middle panel our simulation is shown on the left-hand side.", "It shows ridges similar to those observed in the Gaia data (middle panel on the right-hand side).", "In our previous discussion, we argued that the Hercules-like stream in our MW model corresponds to a ridge from $(R, v_{\\phi }) \\simeq (6\\, \\mathrm {kpc}, 250 \\,\\mathrm {km \\, s^{-1}})$ to $(R, v_{\\phi }) \\simeq (9\\, \\mathrm {kpc}, 180 \\, \\mathrm {km \\, s^{-1}})$ .", "This ridge, eventually branches into two separate structures at around 8 kpc.", "In order to study the relation between the resonances and ridges in our simulation, we overplot the positions of particles in the resonances in the bottom left panel of Fig.", "REF .", "We here randomly sample 5000 particles within 2 kpc from $(R, \\phi ) = (8\\,\\mathrm {kpc}, 20^{\\circ })$ and extract resonant particles from them as we do in Sec. 3.", "In addition, we plot curves of constant angular momentum ($L_z = R v_{\\phi }$ ) in the figure.", "These correspond to the angular momenta of the circular orbits at the resonance radii determined in Fig.", "REF .", "As discussed in studies such as and , resonant orbits approximately follow constant $L_z$ curves if their radial oscillations are small.", "The overplotting reveals that both the resonant particles and the constant $L_z$ curves follow the ridges.", "While showed that the constant $L_z$ curves are blue ridges ($v_{R}<0$ ) for 2:1, 3:1, and 4:1 resonances according to their Figure 6, our $L_z$ curves fall just between red ($v_R>0$ ) and blue ridges.", "In addition, we find that the resonant particles are always distributed below the constant $L_z$ curves in $R$ -$v_{\\phi }$ map.", "This arises due to their radial oscillations not being negligible.", "Orbits in a resonance follow a line in $L_z$ vs. $J_R$ space where $J_R$ is a radial action (see Fig.", "4 in [10]).", "As the resonant lines have negative slopes, if resonant orbits $J_R$ values are large, their $L_z$ values are smaller than those of the circular orbits ($J_R=0)$ .", "Turning to the Gaia data, the ridges seem to trace the constant angular momentum lines also in our Milky Way's data.", "The solid curves in the bottom right panel of Fig.", "REF represent the angular momenta of the resonances for the bar's pattern speed of $\\Omega _{\\mathrm {b}}= 1.4\\Omega _0$ , which is suggested by and .", "We also present the same ones but for $\\Omega _{\\mathrm {b}}= 1.55\\Omega _0$ , which is the pattern speed in our simulation.", "In order to determine the radii of the resonances and corresponding angular momenta, we assume a flat rotation curve of $v_{\\mathrm {c}}=238\\;\\mathrm {km \\; s^{-1}}$ .", "Consider that the top red ridge corresponds to the 2:1 OLR, the full curve (at $\\Omega _{\\mathrm {b}}= 1.4\\Omega _0$ ) then is close to the relation between the color and constant $L_z$ curve as shown in our simulation (the curve is located just above the red ridge).", "However, with $\\Omega _{\\mathrm {b}}= 1.4\\Omega _0=40$  km s$^{-1}$  kpc$^{-1}$ , there are no OLR around the Hercules stream, but the corotation resonance is located at the lowest value of $v_{\\phi }$ of the Hercules stream.", "Accordingly, if we assume a higher pattern speed, i.e., $\\Omega _{\\mathrm {b}}= 1.55\\Omega _0 = 45$  km s$^{-1}$  kpc$^{-1}$ , the 2:1 resonance is located slightly below the top red ridge, but the relation between the resonances and ridges for the 4:1 and 5:1 resonances looks similar to that seen in our simulations.", "In both pattern speeds, the pronounced red ridge near the bottom which we identify with the third Hercules stream, correspond to the corotation resonance.", "We find the same phenomena in our simulation, but without the stars in corotation resonance.", "The latter is a consequence if the relatively low-resolution of our simulations and the fact that in the Gaia data the corotation resonance is closer to the Sun than in our simulations.", "This is a consequence of our models not perfectly matching to the real Milky Way galaxy.", "Thus, our comparison between the Gaia data and our simulation suggest that the pattern speed of the Milky Way's bar is relatively slow, with $\\Omega _{\\mathrm {b}}= 1.4$ –$1.55\\Omega _0$ which corresponds to 40–45 $\\mathrm {km \\; s^{-1} \\; kpc^{-1}}$ )" ], [ "Summary", "We have analyzed a finely resolved $N$ -body simulation of a Milky Way-like galaxy obtained in in order to investigate the origins of the phase-space structures in the Milky-Way galaxy as observed by Gaia.", "We investigated the distribution of particles in the $v_R$ -$v_{\\phi }$ plane by iterating over multiple positions in the disk.", "This revealed a Hercules-like stream around $(R,\\phi ) = (8\\, \\mathrm {kpc}, 20^{\\circ })$ in our simulation, as is also known from the actual solar neighborhood.", "From a spectral analysis of stellar orbits, we found mainly four resonances, namely the 2:1, 3:1, 4:1, and 5:1 resonances around there.", "The observed structures in the $v_R$ -$v_{\\phi }$ plane, can be explained by stars being trapped in the 4:1 and 5:1 outer Lindblad resonances.", "In our simulations, these resonances give rise to structures similar to those observed in the actual Hercules stream.", "Our results therefore favor that the Hercules stream is composed out of stars trapped in the 4:1 and 5:1 OLR and CR, which would also explain its trimodal structure as revealed by Gaia DR2.", "In addition, the stars in the 2:1 and 3:1 OLR match to the `hat' and `horn' structures, respectively.", "We further compared the distribution of stars in the $R$ -$v_{\\phi }$ plane in our simulation with that obtained from the Gaia data.", "Particles identified to be in resonance in our simulation follow ridges in the $R$ -$v_{\\phi }$ plane.", "Similar ridges have also been found in the Gaia data and matching the observed ridges to resonances, our results suggest a relatively low pattern speed of the Milky Way's bar, namely $\\Omega _{\\mathrm {b}}= 1.4$ –$1.55\\Omega _0$ , which corresponds to 40–45 $\\mathrm {km \\; s^{-1} \\; kpc^{-1}}$ .", "This is consistent with recent studies , .", "In contrast to test particle models using static potentials, $N$ -body models have some difficulties when comparing the results with observational data due to the time-dependence of the bar and spiral arms.", "showed that $v_R$ -$v_{\\phi }$ maps obtained from $N$ -body simulations change with time.", "This is a natural consequences of dynamic spiral arms , [6].", "In fact, , discussed phase mixing by dynamic transient spiral arms, which in combination with bar resonances, creates a velocity-space distribution similar to what is seen in observations .", "Furthermore, the evolution of the bar can also affect the stellar velocity-space distribution.", "Recently, showed that stars trapped in the resonances of the bar are dragged in the phase space when the bar slows down using secular perturbation theory and test particle simulations , .", "This can also cause the formation of a Hercules-like structure.", "We will discuss the time dependence of the velocity distribution, caused by the dynamical evolution of spiral arms and bar in fully self-consistent $N$ -body simulations, in a forthcoming paper." ], [ "Acknowledgements", "We thank the anonymous referee for the useful comments.", "We thank Kohei Hattori and Anthony Brown for helpful discussions.", "This work was supported by JSPS KAKENHI Grant Nos.", "18K03711, 18H01248 and 19H01933, and the Netherlands Research School for Astronomy (NOVA).", "MF is supported by The University of Tokyo Excellent Young Researcher Program.", "Simulations are performed using GPU clusters, HA-PACS at the University of Tsukuba, Piz Daint at CSCS, Little Green Machine II (621.016.701) and the ALICE cluster at Leiden University.", "Initial development has been done using the Titan computer Oak Ridge National Laboratory.", "This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s548 and s716.", "This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No.", "DE-AC05-00OR22725 and by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 671564 (COMPAT pro ject).", "This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).", "Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.", "The data underlying this article will be shared on reasonable request to the corresponding author." ] ]

[ [ "QEBA: Query-Efficient Boundary-Based Blackbox Attack" ], [ "Abstract Machine learning (ML), especially deep neural networks (DNNs) have been widely used in various applications, including several safety-critical ones (e.g.", "autonomous driving).", "As a result, recent research about adversarial examples has raised great concerns.", "Such adversarial attacks can be achieved by adding a small magnitude of perturbation to the input to mislead model prediction.", "While several whitebox attacks have demonstrated their effectiveness, which assume that the attackers have full access to the machine learning models; blackbox attacks are more realistic in practice.", "In this paper, we propose a Query-Efficient Boundary-based blackbox Attack (QEBA) based only on model's final prediction labels.", "We theoretically show why previous boundary-based attack with gradient estimation on the whole gradient space is not efficient in terms of query numbers, and provide optimality analysis for our dimension reduction-based gradient estimation.", "On the other hand, we conducted extensive experiments on ImageNet and CelebA datasets to evaluate QEBA.", "We show that compared with the state-of-the-art blackbox attacks, QEBA is able to use a smaller number of queries to achieve a lower magnitude of perturbation with 100% attack success rate.", "We also show case studies of attacks on real-world APIs including MEGVII Face++ and Microsoft Azure." ], [ "Introduction", "Recent developments of machine learning (ML), especially deep neural networks (DNNs), have advanced a number of real-world applications, including object detection [30], drug discovery [8], and robotics [22].", "In the meantime, several safety-critical applications have also adopted ML, such as autonomous driving vehicles [7] and surgical robots [31], [32].", "However, recent research have shown that machine learning systems are vulnerable to adversarial examples, which are inputs with small magnitude of adversarial perturbations added and therefore cause arbitrarily incorrect predictions during test time [13], [40], [4], [14], [5], [6].", "Such adversarial attacks have led to great concerns when applying ML to real-world applications.", "Thus in-depth analysis of the intrinsic properties of these adversarial attacks as well as potential defense strategies are required.", "First, such attacks can be categorized into whitebox and blackbox attacks based on the attacker's knowledge about the victim ML model.", "In general, the whitebox attacks are possible by leveraging the gradient of the model — methods like fast gradient sign method (FGSM)  [14], optimization based attack [4], projected gradient descent based method (PGD) [25] have been proposed.", "However, whitebox attack is less practical, given the fact that most real-world applications will not release the actual model they are using.", "In addition, these whitebox attacks are shown to be defendable [25].", "As a result, blackbox adversarial attack have caught a lot of attention in these days.", "In blackbox attack, based on whether an attacker needs to query the victim ML model, there are query-free (e.g.", "transferability based attack) and query-based attacks.", "Though transferability based attack does not require query access to the model, it assumes the attacker has access to the large training data to train a substitute model, and there is no guarantee for the attack success rate.", "The query based attack includes score-based and boundary-based attacks.", "Score-based attack assumes the attacker has access to the class probabilities of the model, which is less practical compared with boundary-based attack which only requires the final model prediction, while both require large number of queries.", "In this paper, we propose Query-Efficient Boundary-based blackbox Attack (QEBA) based only on model's final prediction labels as a general framework to minimize the query number.", "Since the gradient estimation consumes the majority of all the queries, the main challenge of reducing the number of queries for boundary-based blackbox attack is that a high-dimensional data (e.g.", "an image) would require large number of queries to probe the decision boundary.", "As a result, we propose to search for a small representative subspace for query generation.", "In particular, queries are generated by adding perturbations to an image.", "We explore the subspace optimization methods from three novel perspectives for perturbation sampling: 1) spatial, 2) frequency, and 3) intrinsic component.", "The first one leverages spatial transformation (e.g.", "linear interpolation) so that the sampling procedure can take place in a low-dimensional space and then project back to the original space.", "The second one uses intuition from image compression literature and samples from low frequency subspace and use discrete consine transformation (DCT) [15] to project back.", "The final one performs scalable gradient matrix decomposition to select the major principle components via principle component analysis (PCA) [39] as subspace to sample from.", "In addition , we theoretically prove the optimality of them on estimating the gradient compared with estimating the gradient directly over the original space.", "To demonstrate the effectiveness of the proposed blackbox attack QEBA methods, we conduct extensive experiments on high dimensional image data including ImageNet [11] and CelebA [24].", "We perform attacks on the ResNet model [17], and show that compared with the-state-of-the-art blackbox attack methods, the different variations of QEBA can achieve lower magnitude of perturbation with smaller number of queries (attack success rate 100%).", "In order to show the real-world impact of the proposed attacks, we also perform QEBA against online commercial APIs including MEGVII Face++[26] and Microsoft Azure[28].", "Our methods can successfully attack the APIs with perturbations of reasonable magnitude.", "Towards these different subspaces, our conjecture is that the over-all performance on different subspaces depends on multiple factors including dataset size, model smoothness, adversarial attack goals etc.", "Therefore, our goal here is to make the first attempt towards providing sufficient empirical observations for these three subspaces, while further extensive studies are required to compare different factors of these subspaces, as well as identifying new types of subspaces.", "The contributions of this work are summarized as follows: 1) We propose a general Query-Efficient Boundary-based blackbox Attack QEBA to reduce the number of queries based on boundary-based attack.", "The QEBA contains three variations based on three different representative subspaces including spatial transformed subspace, low frequency subspace, and intrinsic component subspace; 2) We theoretically demonstrate that gradient estimation in the whole gradient space is inefficient in terms of query numbers, and we prove the optimality analysis for our proposed query-efficient gradient estimation methods; 3) We conduct comprehensive experiments on two high resolution image datasets: ImageNet and CelebA.", "All the different variations of QEBA outperform the state-of-the-art baseline method by a large margin; 4) We successfully attack two real-world APIs including Face++[26] and Azure[28] and showcase the effectiveness of QEBA." ], [ "Problem Definition", "Consider a $k$ -way image classification model $f({\\bf x})$ where ${\\bf x}\\in \\mathbb {R}^m$ denotes the input image with dimension $m$ , and $f({\\bf x})\\in \\mathbb {R}^k$ represents the vector of confidence scores of the image belonging to each classes.", "In boundary-based black-box attacks, the attacker can only inquire the model with queries $\\lbrace {\\bf x}_i \\rbrace $ (a series of updated images) and get the predicted labels $\\tilde{y}_i=F({\\bf x_i}) = \\operatornamewithlimits{arg\\,max}_j [f({\\bf x_i})]_j,$ where $[f]_j$ represents the score of the $j$ -th class.", "The parameters in the model $f$ and the score vector $\\bf s$ are not accessible.", "There is a target-image ${\\bf x}_{tgt}$ with a benign label $y_{ben}$ .", "Based on the malicious label $y_{mal}$ of their choice, the adversary will start from a source-image ${\\bf x}_{src}$ selected from the category with label $y_{mal}$ , and move ${\\bf x}_{src}$ towards ${\\bf x}_{tgt}$ on the pixel space while keeping $y_{mal}$ to guarantee the attack.", "An image that is on the decision boundary between the two classes (e.g.", "$y_{ben}$ and $y_{mal}$ ) and is classified as $y_{mal}$ is called boundary-image.", "The adversary's goal is to find an adversarial image(adv-image) ${\\bf x}_{adv}$ such that $F({\\bf x}_{adv}) = y_{mal}$ and $D({\\bf x}_{tgt}, {\\bf x}_{adv})$ is as small as possible, where $D$ is the distance metric (usually $L_2$ -norm or $L_\\infty $ -norm distance).", "By definition, adv-image is a boundary-image with an optimized (minimal) distance from the target-image.", "In the paper we focus on targeted attack and the approaches can extend to untargeted scenario naturally." ], [ "Query-Efficient Boundary-based blackbox Attack (QEBA)", "In this section we first introduce the pipeline of QEBA which is based on HopSkipJumpAttack (HSJA) [9].", "We then illustrate the three proposed query reduction approaches in detail.", "We provide the theoretic justification of QEBA in Section .", "The pipeline of the proposed Query-Efficient Boundary-based blackbox Attack (QEBA) is shown in Figure REF as an illustrative example.", "The goal is to produce an adv-image that looks like ${\\bf x}_{tgt}$ (cat) but is mislabeled as the malicious label (fish) by the victim model.", "First, the attack initializes the adv-image with ${\\bf x}_{src}$ .", "Then it performs an iterative algorithm consisting of three steps: estimate gradient at decision boundary which is based on the proposed representative subspace, move along estimated gradient, and project to decision boundary which aims to move towards ${\\bf x}_{tgt}$ .", "First, define the adversarial prediction score $S$ and the indicator function $\\phi $ as: $S_{{\\bf x}_{tgt}}({\\bf x}) &= [f({\\bf x})]_{y_{mal}} - \\max _{y \\ne y_{mal}} [f({\\bf x})]_y,\\\\\\phi _{{\\bf x}_{tgt}}({\\bf x}) &= \\text{sign}(S_{{\\bf x}_{tgt}}({\\bf x})) ={\\left\\lbrace \\begin{array}{ll}1 & \\text{if } S_{{\\bf x}_{tgt}}({\\bf x})\\ge 0;\\\\-1 & \\text{otherwise}.\\end{array}\\right.", "}$ We abbreviate the two functions as $S({\\bf x})$ and $\\phi ({\\bf x})$ if it does not cause confusion.", "In boundary-based attack, the attacker is only able to get the value of $\\phi $ but not $S$ .", "In the following, we first introduce the three interative steps in the attack in Section REF , then introduce three different methods for generating the optimized representative subspace in Section REF -REF ." ], [ "Estimate gradient at decision boundary", "Denote ${\\bf x}_{adv}^{(t)}$ as the adv-image generated in the $t$ -th step.", "The intuition in this step is that we can estimate the gradient of $S({\\bf x}_{adv}^{(t)})$ using only the access to $\\phi $ if ${\\bf x}_{adv}^{(t)}$ is at the decision boundary.", "This gradient can be sampled via Monte Carlo method: $\\widetilde{\\nabla S} = \\frac{1}{B} \\sum _{i=1}^B \\phi ({\\bf x}_{adv}^{(t)}+\\delta {\\bf u}_b) {\\bf u}_b$ where $\\lbrace {\\bf u}_b\\rbrace $ are $B$ randomly sampled perturbations with unit length and $\\delta $ is a small weighting constant.", "An example of this process is shown in Figure REF .", "The key point here is how to sample the perturbation ${\\bf u}_b$ 's and we propose to draw from a representative subspace in $\\mathbb {R}^n$ .", "Formally speaking, let $W=[w_1, \\ldots , w_n] \\in \\mathbb {R}^{m\\times n}$ be $n$ orthonormal basis vectors in $\\mathbb {R}^m$ , meaning $W^\\intercal W = I$ .", "Let $\\text{span}(W) \\subseteq \\mathbb {R}^m$ denote the $n$ -dimensional subspace spanned by $w_1, \\ldots , w_n$ .", "We would like to sample random perturbations from $\\text{span}(W)$ instead of from the original space $\\mathbb {R}^m$ .", "In order to do that, we sample ${\\bf v}_b \\in \\mathbb {R}^n$ from unit sphere in $\\mathbb {R}^n$ and let ${\\bf u}_b = W {\\bf v}_b$ .", "The detailed gradient estimation algorithm is shown in Alg.REF .", "Note that if we let $\\text{span}(W)=\\mathbb {R}^m$ , this step will be the same as in [9].", "However, we will sample from some representative subspace so that the gradient estimation is more efficient, and the corresponding theoretic justification is discussed in Section .", "Gradient Approximation Based QEBA [1] a data point on the decision boundary ${\\bf x} \\in \\mathbb {R}^m$ , basis of the subspace $W \\in \\mathbb {R}^{m\\times n}$ , number of random sampling $B$ , access to query the decision of victim model $\\phi $ .", "the approximated gradient $G$ sample $B$ random Gaussian vectors of the lower dimension: $V_{rnd} \\in \\mathbb {R}^{B \\times n}$ .", "project the random vectors onto the gradient basis to get the perturbation vectors: $U_{rnd} = V_{rnd} \\cdot W^\\intercal $ .", "get query points by adding perturbation vectors with the original point on the decision boundary: ${\\bf x}_q[i] = {\\bf x} + U_{rnd}[i]$ .", "Monte Carlo approximation for the gradient: $G = \\frac{1}{B}\\sum _{i=1}^{B} \\phi ({\\bf x}_q[i]) \\cdot U_{rnd}[i]$ $G$" ], [ "Move along estimated gradient", "After we have estimated the gradient of adversarial prediction score $\\nabla S$ , we will move the ${\\bf x}_{adv}^{(t)}$ towards the gradient direction: $\\hat{\\bf x}_{t+1} = {\\bf x}_{adv}^{(t)} + \\xi _t \\cdot \\frac{\\widetilde{\\nabla S}}{||\\widetilde{\\nabla S}||_2}$ where $\\xi _t$ is the step size at the $t$ -th step.", "Hence, the prediction score of the adversarial class will be increased." ], [ "Project to decision boundary", "Current $\\hat{\\bf x}_{t+1}$ is beyond the boundary, we can move the adv-image towards the target image so that it is projected back to the decision boundary: ${\\bf x}_{adv}^{(t+1)} = \\alpha _t \\cdot {\\bf x}_{tgt} + (1-\\alpha _t) \\cdot \\hat{\\bf x}_{t+1}$ where the projection is achieved by a binary search over $\\alpha _t$ .", "Note that we assume ${\\bf x}_{adv}^{(t)}$ lies on the boundary while ${\\bf x}_{src}$ does not lie on the boundary.", "Therefore, in the initialization step we need to first apply a project operation as in Eqn.", "REF to get ${\\bf x}_{adv}^{(0)}$ .", "In the following sections, we will introduce three exploration for the representative subspace optimization from spatial, frequency, and intrinsic component perspectives." ], [ "Spatial Transformed Subspace (QEBA-S)", "First we start with the spatial transformed query reduction approach.", "The intuition comes from the observation that the gradient of input image has a property of local similarity[20].", "Therefore, a large proportion of the gradients lies on the low-dimensional subspace spanned by the bilinear interpolation operation[34].", "In order to sample random perturbations for an image, we first sample a lower-dimensional random perturbation $Q$ of shape $\\lfloor \\frac{N}{r} \\rfloor \\times \\lfloor \\frac{N}{r} \\rfloor $ , where $r$ is the hyperparameter of dimension reduction factor.", "Then we use bilinear-interpolation to map it back the original image space, $X = \\text{Bil\\_Interp}(Q)$ .", "The basis of this spatial transformed subspace is the images transformed from unit perturbations in the lower space: $w^{(i,j)} = \\text{Bil\\_Interp}(e^{(i,j)}),\\quad 0\\le i,j \\le \\lfloor N/r \\rfloor $ where $e^{(i,j)}$ represents the unit vector that has 1 on the $(i,j)$ -th entry and 0 elsewhere." ], [ "Low Frequency Subspace (QEBA-F)", "In general the low frequency subspace of an image contains the most of the critical information, including the gradient information[15]; while the high frequency signals contain more noise than useful content.", "Hence, we would like to sample our perturbations from the low frequency subspace via Discrete Cosine Transformation(DCT)[1].", "Formally speaking, define the basis function of DCT as: $\\phi (i,j) = \\cos \\bigg (\\frac{(i+\\frac{1}{2})j}{N} \\pi \\bigg )$ The inverse DCT transformation is a mapping from the frequency domain to the image domain $X=\\text{IDCT}(Q)$ : $X_{i_1,i_2}=\\sum _{j_1=0}^{N-1}\\sum _{j2=0}^{N-1} N_{j_1} N_{j_2} Q_{j_1,j_2} \\phi (i_1,j_1) \\phi (i_2,j_2)$ where $N_j=\\sqrt{1/N}$ if $j=0$ and otherwise $N_j=\\sqrt{2/N}$ .", "We will use the lower $\\lfloor N/r \\rfloor $ part of the frequency domain as the subspace, i.e.", "$w^{(i,j)} = \\text{IDCT}(e^{(i,j)}),\\quad 0\\le i,j \\le \\lfloor N/r \\rfloor $ where hyperparameter $r$ is the dimension reduction factor." ], [ "Intrinsic Component Subspace (QEBA-I)", "Principal Component Analysis (PCA)[39] is a standard way to perform dimension reduction in order to search for the intrinsic components of the given instances.", "Given a set of data points in high dimensional space, PCA aims to find a lower dimensional subspace so that the projection of the data points onto the subspace is maximized.", "Therefore, it is possible to leverage PCA to optimize the subspace for model gradient matrix.", "However, in order to perform PCA we will need a set of data points.", "In our case that should be a set of gradients of $S({\\bf x})$ w.r.t.", "different $\\bf x$ .", "This is not accessible under black-box setting.", "Hence, we turn to a set of `reference models' to whose gradient we have access.", "As shown in Figure REF , we will use a reference model to calculate a set of image gradients ${\\bf g}_1, {\\bf g}_2, \\ldots , {\\bf g}_K \\in \\mathbb {R}^m$ Then we perform a PCA to extract its top-$n$ principal components - ${\\bf w}_1, \\ldots , {\\bf w}_n \\in \\mathbb {R}^m$ .", "These $w$ 's are the basis of the Intrinsic Component Subspace.", "Note that different from transferability, we do not restrict the reference models to be trained by the same training data with the original model, since we only need to search for the intrinsic components of the give dataset which is relatively stable regarding diverse models.", "In practice, the calculation of PCA may be challenging in terms of time and memory efficiency based on large high-dimensional dataset (the data dimension on ImageNet is over 150k and we need a larger number of data points, all of which are dense).", "Therefore, we leverage the randomized PCA algorithms[16] which accelerates the speed of PCA while achieving comparable performance.", "An additional challenge is that the matrix $X$ may be too large to be stored in memory.", "Therefore, we store them by different rows since each row (i.e.", "gradient of one image) is calculated independently with the others.", "The multiplication of $X$ and other matrices in memory are then implemented accordingly." ], [ "Theoretic Analysis on QEBA", "We theoretically analyze how dimension reduction helps with the gradient estimation in QEBA.", "We show that the gradient estimation bound is tighter by sampling from a representative subspace rather than the original space.", "We consider the gradient estimation as in Eqn.", "REF and let $\\rho = \\frac{||\\text{proj}_{\\text{span}(W)}(\\nabla S)||_2}{||\\nabla S||_2}$ denote the proportion of $\\nabla S$ that lies on the chosen subspace $\\text{span}(W)$ .", "Then we have the following theorem on the expectation of the cosine similarity between $\\nabla S$ and estimated $\\widetilde{\\nabla S}$ : Theorem 1 Suppose 1) $S({\\bf x})$ has $L$ -Lipschitz gradients in a neighborhood of $\\bf x$ , 2) the sampled ${\\bf v}_1, \\ldots , {\\bf v}_B$ are orthogonal to each other, and 3) $W^\\intercal W = I$ , then the expected cosine simliarity between $\\widetilde{\\nabla S}$ and $\\nabla S$ can be bounded by: $&\\bigg ( 2\\bigg (1-(\\frac{L\\delta }{2||\\nabla S||_2})^2\\bigg )^{\\frac{n-1}{2}} - 1 \\bigg )c_n\\rho \\sqrt{\\frac{B}{n}} \\\\\\le & \\mathbb {E}\\big [\\cos (\\widetilde{\\nabla S}, \\nabla S) \\big ]\\\\\\le & c_n\\rho \\sqrt{\\frac{B}{n}}$ where $c_n$ is a coefficient related with the subspace dimension $n$ and can be bounded by $c_n \\in (2/\\pi , 1)$ .", "In particular: $\\lim _{\\delta \\rightarrow 0}\\mathbb {E}\\big [\\cos (\\widetilde{\\nabla _W S}, \\nabla S) \\big ] = c_n\\rho \\sqrt{\\frac{B}{n}}.$ The theorem proof is in Appendix .", "If we sample from the entire space (i.e.", "$\\text{span}(W) = \\mathbb {R}^m$ ), the expected cosine similarity is $c_m\\sqrt{\\frac{B}{m}}$ .", "If we let $m=3\\times 224\\times 224$ and $B=100$ , the similarity is only around 0.02.", "On the other hand, if the subspace basis $w$ 's are randomly chosen, then $\\rho \\approx \\sqrt{\\frac{n}{m}}$ and the estimation quality is low.", "With larger $\\rho $ , the estimation quality will be better than sampling from the entire space.", "Therefore, we further explore three approaches to optimize the representative subspace that contains a larger portion of the gradient as discussed in Section .", "For example, in the experiments we see that when $n=m/16$ , we can reach $\\rho =0.5$ and the expected cosine similarity increase to around 0.06.", "This improves the gradient estimation quality which leads to more efficient attacks." ], [ "Experiments", "In this section, we introduce our experimental setup and quantitative results of the proposed methods QEBA-S, QEBA-F, and QEBA-I, compared with the HSJA attack[9], which is the-state-of-the-art boundary-based blackbox attack.", "Here we focus on the strongest baseline HSJA, which outperforms all of other Boundary Attack [2], Limited Attack [19] and Opt Attack [10] by a substantial margin.", "We also show two sets of qualitative results for attacking two real-world APIs with the proposed methods." ], [ "Datasets", "We evaluate the attacks on two offline models on ImageNet[11] and CelebA[24] and two online face recognition APIs Face++[26] and Azure[28].", "We use a pretrained ResNet-18 model as the target model for ImageNet and fine-tune a pretrained ResNet-18 model to classify among 100 people in CelebA.", "We randomly select 50 pairs from the ImageNet/CelebA validation set that are correctly classified by the model as the source and target images." ], [ "Attack Setup", "Following the standard setting in [9], we use $\\xi _t=||{\\bf x}_{adv}^{(t-1)}-{\\bf x}_{tgt}||_2/\\sqrt{t}$ as the size in each step towards the gradient.", "We use $\\delta _t=\\frac{1}{m} ||{\\bf x}_{adv}^{(t-1)}-{\\bf x}_{tgt}||_2$ as the perturbation size and $B=100$ queries in the Monte Carlo algorithm to estimate the gradient, where $m=3\\times 224\\times 224$ is the input dimension in each Monte Carlo step.", "We provide two evaluation metrics to evaluate the attack performance.", "The first is the average Mean Square Error (MSE) curve between the target image and the adversarial example in each step, indicating the magnitude of perturbation.", "The smaller the perturbation is, the more similar the adversarial example is with the target-image, thus providing better attack quality.", "The second is the attack success rate based on a limited number of queries, where the `success' is defined as reaching certain specific MSE threshold.", "The less queries we need in order to reach a certain perturbation threshold, the more efficient the attack method is.", "As for the dimension-reduced subspace, we use the dimension reduction factor $r=4$ in spatial transformed and low frequency subspace, which gives a 9408 dimensional subspace.", "In order to generate the Intrinsic Component Subspace, we first generate a set of image gradient vectors on the space.", "We average over the gradient of input w.r.t.", "five different pretrained substitute models - ResNet-50[17], DenseNet-121[18], VGG16[33], WideResNet[41] and GoogleNet[36].", "We use part of the ImageNet validation set (280000 images) to generate the gradient vectors.", "Finally we adopt the scalable approximate PCA algorithm[16] to extract the top 9408 major components as the intrinsic component subspace." ], [ "Commercial Online APIs", "Various companies provide commercial APIs (Application Programming Interfaces) of trained models for different tasks such as face recognition.", "Developers of downstream tasks can pay for the services and integrate the APIs into their applications.", "Note that although typical platform APIs provide the developers the confidence score of classes associated with their final predictions, the end-user using the final application would not have access to the scores in most cases.", "For example, some of Face++'s partners use the face recognition techniques for log-in authentication in mobile phones [27], where the user only knows the final decision (whether they pass the verification or not).", "We choose two representative platforms for our real-world experiments based on only the final prediction.", "The first is Face++ from MEGVII[26], and the second is Microsoft Azure[28].", "Face++ offers a `compare' API [26] with which we can send an HTTPS request with two images in the form of byte strings, and get a prediction confidence of whether the two images contain the same person.", "In all the experiments we consider a confidence greater than 50% meaning the two images are tagged as the same person.", "Azure has a slightly more complicated interface.", "To compare two images, each image first needs to pass a `detect' API call [28] to get a list of detected faces with their landmarks, features, and attributes.", "Then the features of both images are fed into a `verify' function [29] to get a final decision of whether they belong to the same person or not.", "The confidence is also given, but we do not need it for our experiments since we only leverage the binary prediction for practical purpose.", "In the experiments, we use the examples in Figure REF as source-image and target-image.", "More specifically, we use a man-woman face as the source-target pair for the `compare' API Face++, and we use a cat-woman face as the pair for the `detect' API Azure face detection.", "Figure: The source and target images for online API experiments.", "All images are resized to 3×224×2243\\times 224\\times 224.Image  is the target-image for both APIs.", "Image  is the source-image for attacking Face++ `compare' API, and the source-image for Azure `detect' API.Figure: The attack results on ImageNet and CelebA datasets." ], [ "Discretization Optimization for Attacking APIs", "The attack against online APIs suffers from the problem of `discretization'.", "That is, in the attack process we assume the pixel values to be continuous in $[0,1]$ , but we need to round it into 8-bit floating point in the uploaded RGB images when querying the online APIs.", "This would cause error in the Monte Carlo gradient estimation format in Equation REF since the real perturbation between the last boundary-image and the new query image after rounding is different from the weighted perturbation vector $\\delta {\\bf u}_b$ .", "In order to mitigate this problem, we perform discretization locally.", "Let $P_{rd}$ be a projection from a continuous image $\\bf x_c$ to a discrete image $\\bf x_d = P_{rd}(\\bf x_c)$ .", "Let $\\delta {\\bf u^{\\prime }}_b = P_{rd}(\\bf x+\\delta {\\bf u}_b) - x$ , the new gradient estimation format becomes: $\\widetilde{\\nabla f} &= \\frac{1}{B} \\sum _{i=1}^B \\phi (P_{rd}({\\bf x}+\\delta {\\bf u}_b)) {\\bf u^{\\prime }}_b.$" ], [ "Experimental Results on Offline Models", "To evaluate the effectiveness of the proposed methods, we first show the average MSE during the attack process of ImageNet and CelebA using different number of queries in Figure REF and Figure REF respectively.", "We can see that all the three proposed query efficient methods outperform HSJA significantly.", "We also show the attack success rate given different number of queries in Table REF using different MSE requirement as the threshold.", "In addition, we provide the attack success rate curve in Figure REF and REF using $10^{-3}$ as the threshold for ImageNet and $10^{-5}$ for CelebA to illustrate convergence trend for the proposed QEBA-S, QEBA-F, and QEBA-I, comparing with the baseline HJSA.", "We observe that sampling in the optimized subspaces results in a better performance than sampling from the original space.", "The spatial transforamed subspace and low-frequency subspace show a similar behaviour since both of them rely on the local continuity.", "The intrinsic component subspace does not perform better than the other two approaches, and the potential reason is that we are only using 280000 cases to find intrinsic components on the 150528-dimensional space.", "Therefore, the extracted components may not be optimal.", "We also observe that the face recognition model is much easier to attack than the ImageNet model, since the face recognition model has fewer classes (100) rather than 1000 as of ImageNet.", "A qualitative example process of attacking the ImageNet model using different subspaces is shown in Figure REF .", "In this example, the MSE (shown as $d$ in the figures) reaches below $1\\times 10^{-3}$ using around 2K queries when samlping from the subspaces, and it is already hard to tell the adversarial perturbations in the examples.", "When we further tune the adv-image using 10K queries, it reaches lower MSE." ], [ "Results of Attacking Online APIs", "The results of attacking online APIs Face++ and Azure are shown in Figure REF and Figure REF respectively.", "The labels on the y-axis indicate the methods.", "Each column represents successful attack instances with increasing number of API calls.", "As is the nature of boundary-based attack, all images are able to produce successful attack.", "The difference lies in the quality of attack instances.", "For attacks on Face++ `compare' API, the source-image is a man and the target-image is a woman as shown in Figure REF .", "Notice the man's eyes appear in a higher position in the source-image than the woman in the target-image because of the pose.", "All the instances on the first row in Figure REF based on HJSA attacks contain two pairs of eyes.", "The MSE scores ($d$ in the figures) also confirm that the distance between the attack instance and the target-image does not go down much even with more than 6000 queries.", "On the other hand, our proposed methods QEBA- can optimize attack instances with smaller magnitude of perturbation more efficiently.", "The perturbations are also smoother.", "The attack results on Azure `detect' API show similar observations.", "The source-image is a cat and the target-image is the same woman.", "Sampling from the original high-dimensional space (HJSA) gives us attack instances that presents two cat ear shapes at the back of the human face as shown in the first row in Figure REF .", "With the proposed query efficient attacks, the perturbations are smoother.", "The distance metric ($d$ ) also demonstrates the superiority of the proposed methods.", "Figure: Comparison of attacks on Face++ `compare' API.", "Goal: obtain an image that is tagged as `same person' with the source-image person 2 (Figure ) by the API when humans can clearly see person 1 here.Figure: Comparison of attacks on Azure `detect' API.", "Goal: get an image that is tagged as `no face' by the API when humans can clearly see a face there.", "The source-image is a cat as shown in Figure ." ], [ "Boundary-based Attack", "Boundary Attack [2] is one of the first work that uses final decisions of a classifier to perform blackbox attacks.", "The attack process starts from the source-image, which is classified as the adversarial malicious-class.", "Then it employs a reject sampling mechanism to find a boundary-image that still belongs to the malicious-class by performing random walk along the boundary.", "The goal is to minimize the distance between the boundary-image and the target-image.", "However, as the steps taken are randomly sampled, the convergence of this method is slow and the query number is large.", "Several techniques have been proposed to improve the performance of Boundary Attack.", "[3], [35], [15] propose to choose the random perturbation in each step more wisely instead of Gaussian perturbation, using Perlin noise, alpha distribution and DCT respectively.", "[19], [21], [23], [9] propose a similar idea - approximating the gradient around the boundary using Monte Carlo algorithm.", "There are two other blackbox attacks which are not based on the boundary.", "[10] proposes to transform the boundary-based output into a continuous metric, so that the score-based attack techniques can be adopted.", "[12] adopts evolution algorithm to achieve the decision-based attack against face recognition system." ], [ "Dimension Reduction in Score-based Attack", "Another line of work involves the dimension reduction techniques only for the score-based attacks, which requires access to the prediction of confidence for each class.", "In [15], the authors draw intuition from JPEG codec [38] image compression techniques and propose to use discrete cosine transform (DCT) for generating low frequency adversarial perturbations to assist score-based adversarial attack.", "AutoZoom [37] trains an auto-encoder offline with natural images and uses the decoder network as a dimension reduction tool.", "Constrained perturbations in the latent space of the auto-encoder are generated and passed through the decoder.", "The resulting perturbation in the image space is added to the benign one to obtain a query sample." ], [ "Conclusion", "Overall we propose QEBA, a general query-efficient boundary-based blackbox attack framework.", "We in addition explore three novel subspace optimization approaches to reduce the number of queries from spatial, frequency, and intrinsic components perspectives.", "Based on our theoretic analysis, we show the optimality of the proposed subspace based gradient estimation compared with the estimation over the original space.", "Extensive results show that the proposed QEBA significantly reduces the required number of queries and yields high quality adversarial examples against both offline and onlie real-world APIs." ], [ "Acknowledgement", "We would like to thank Prof. Yuan Qi and Prof.", "Le Song for their comments and advice in this project.", "This work is partially supported by NSF grant No.1910100." ], [ "Proof of Theorem ", "We first prove a lemma of the gradient estimation quality which samples from the entire subspace: Lemma 1 For a boundary point $x$ , suppose that $S(x)$ has $L$ -Lipschitz gradients in a neighborhood of $x$ , and that ${\\bf u}_1, \\ldots , {\\bf u}_B$ are sampled from the unit ball in $\\mathbb {R}^m$ and orthogonal to each other.", "Then the expected cosine similarity between $\\widetilde{\\nabla S}$ and $\\nabla S$ can be bounded by: $& \\bigg ( 2\\bigg (1-(\\frac{L\\delta }{2||\\nabla S||_2})^2\\bigg )^{\\frac{m-1}{2}} - 1 \\bigg )c_m\\sqrt{\\frac{B}{m}}\\\\\\le & \\mathbb {E}\\big [\\cos (\\widetilde{\\nabla S}, \\nabla S) \\big ]\\\\\\le & c_m\\sqrt{\\frac{B}{m}}$ where $c_m$ is a constant related with $m$ and can be bounded by $c_m \\in (2/\\pi , 1)$ .", "In particular, we have: $\\lim _{\\delta \\rightarrow 0}\\mathbb {E}\\big [\\cos (\\widetilde{\\nabla S}, \\nabla S) \\big ] = c_m\\sqrt{\\frac{B}{m}}.$ Let ${\\bf u}_1, \\ldots , {\\bf u}_B$ be the random orthonormal vectors sampled from $\\mathbb {R}^m$ .", "We expand the vectors to an orthonormal basis in $\\mathbb {R}^m$ : ${\\bf q}_1={\\bf u}_1, \\ldots , {\\bf q}_B={\\bf u}_B, {\\bf q}_{B+1}, \\ldots , {\\bf q}_m$ .", "Hence, the gradient direction can be written as: $\\frac{\\nabla S}{||\\nabla S||_2} = \\sum _{i=1}^m a_i{\\bf q}_i$ where $a_i=\\langle \\frac{\\nabla S}{||\\nabla S||_2}, {\\bf q}_i \\rangle $ and its distribution is equivalent to the distribution of one coordinate of an $(m-1)$ -sphere.", "Then each $a_i$ follows the probability distribution function: $p_a(x) = \\frac{(1-x^2)^{\\frac{m-3}{2}}}{\\mathcal {B}(\\frac{m-1}{2}, \\frac{1}{2})},~x\\in (-1,1)$ where $\\mathcal {B}$ is the beta function.", "According to the conclusion in the proof of Theorem 1 in [9], if we let $w=\\frac{L\\delta }{2||\\nabla S||_2}$ , then it always holds true that $\\phi ({\\bf x}+\\delta {\\bf u_i})=1$ when $a_i>w$ , -1 when $a_i<-w$ regardless of $u_i$ and the decision boundary shape.", "Hence, we can rewrite $\\phi _i$ in term of $a_i$ : $\\phi _i = \\phi ({\\bf x}+\\delta {\\bf u_i}) ={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } a_i \\in [w, 1)\\\\-1, & \\text{if } a_i \\in (-1, -w]\\\\\\text{undetermined}, & \\text{otherwise}\\end{array}\\right.", "}$ Therefore, the estimated gradient can be rewritten as: $\\widetilde{\\nabla S}=\\frac{1}{B}\\sum _{i=1}^B \\phi _i{\\bf u}_i$ Combining Eqn.", "REF and REF , we can calculate the cosine similarity: $\\mathop {\\mathbb {E}}\\big [ \\cos (\\widetilde{\\nabla S}, \\nabla S)\\big ] &= \\mathop {\\mathbb {E}}_{a_1, \\ldots ,a_B}\\frac{\\sum _{i=1}^Ba_i\\phi _i}{\\sqrt{B}}\\\\&= \\sqrt{B}\\cdot \\mathop {\\mathbb {E}}_{a_1} \\big [ a_1\\phi _1 \\big ]$ In the best case, $\\phi _1$ has the same sign with $a_1$ everywhere on $(-1,1)$ ; in the worst case, $\\phi _1$ has different sign with $a_1$ on $(-w,w)$ .", "In addition, $p_a(x)$ is symmetric on $(-1, 1)$ .", "Therefore, the expectation is bounded by: $&2\\int _{w}^{1} p_a(x)\\cdot xdx - 2\\int _0^w p_a(x)\\cdot xdx\\\\\\le & \\mathop {\\mathbb {E}}_{a_1} \\big [ a_1\\phi _1 \\big ]\\\\\\le & 2\\int _0^1 p_a(x)\\cdot xdx$ By calculating the integration, we have: $&\\bigg ( 2\\bigg (1-w^2\\bigg )^{\\frac{m-1}{2}} - 1 \\bigg ) \\cdot \\frac{2\\sqrt{B}}{\\mathcal {B}(\\frac{m-1}{2}, \\frac{1}{2})\\cdot (m-1)}\\\\\\le & \\mathop {\\mathbb {E}}\\big [ \\cos (\\widetilde{\\nabla S}, \\nabla S)\\big ]\\\\\\le & \\frac{2\\sqrt{B}}{\\mathcal {B}(\\frac{m-1}{2}, \\frac{1}{2})\\cdot (m-1)}$ The only problem is to calculate $\\mathcal {B}(\\frac{m-1}{2}, \\frac{1}{2})\\cdot (m-1)$ .", "It is easy to prove by scaling that $\\mathcal {B}(\\frac{m-1}{2}, \\frac{1}{2})\\cdot (m-1) \\in (2\\sqrt{m}, \\pi \\sqrt{m})$ .", "Hence we can get the conclusion in the theorem.", "Having Lemma REF , Theorem REF follows by noticing that $\\mathbb {E}\\big [\\cos (\\widetilde{\\nabla S}, \\nabla S) \\big ]=\\rho \\mathbb {E}\\big [\\cos (\\widetilde{\\nabla S}, \\text{proj}_{\\text{span}(W)}(\\nabla S)) \\big ]$ ." ] ]

[ [ "The darkweb: a social network anomaly" ], [ "Abstract We analyse the darkweb and find its structure is unusual.", "For example, $ \\sim 87 \\%$ of darkweb sites \\emph{never} link to another site.", "To call the darkweb a \"web\" is thus a misnomer -- it's better described as a set of largely isolated dark silos.", "As we show through a detailed comparison to the World Wide Web (www), this siloed structure is highly dissimilar to other social networks and indicates the social behavior of darkweb users is much different to that of www users.", "We show a generalized preferential attachment model can partially explain the strange topology of the darkweb, but an understanding of the anomalous behavior of its users remains out of reach.", "Our results are relevant to network scientists, social scientists, and other researchers interested in the social interactions of large numbers of agents." ], [ "Introduction", "Studies of the World Wide Web (www) have had much impact.", "An understanding of its topology have let us better understand how its information is stored and can be retrieved [11], [6].", "Insight into its paradoxical resilience [13] have allowed us to design more fault tolerant networks [12], and the universal dynamics of it growth patterns [3] have shed light on the behaviors of many others classes of networks [4].", "And perhaps most importantly, being one of the earliest studied networksalongside smallworld networks [5].", "[11], [6], the www helped give rise to the flame of attention that network science enjoys today.", "This paper is about the www's shady twin: the darkweb.", "Though the darkweb is similar to the www, both being navigated through a web browser, a key difference the two is that the identities of darkweb users are hidden – that's what makes it `dark'.", "This gives the darkweb an infamous air of mystery, which, along with the sparse academic attention it has received [17], [18], makes it ripe for analysis.", "And beyond satisfying curiosity, its reasonable to think studies of the darkweb could have as much applied impact as the studies of the www.", "Insight about the structure or dynamics of the darkweb, for instance, could potentially allow policy-makers to better police its more sinister aspects.", "There are many natural questions to ask about the darkweb.", "Does it have the same topology as the www, and therefore hold and distribute information in the same way?", "Is it resilient to attack?", "Social questions can also be posed.", "Results from psychology and game theory show people behave more socially when they are watched [7], [8] or have reputations to uphold [29].", "Do darkweb users, with their masked faces, therefore behave more selfishly than www users, whose faces are bare?", "Here we address some of these questions by performing a graph theoretical analysis of the darkweb (we define exactly what we mean by the darkweb below) and comparing it to the www.", "We find the topologies of the darkweb and the www are starkly different – in fact, the darkweb is much different to many social networks – and conjecture this is due to their users' differing social behaviour.", "We hope our work stimulates further research on the darkweb's structure as well as the social dynamics of its users." ], [ "Data Collection", "There is no single definition of the darkweb.", "It is sometimes loosely defined as “anything seedy on the Internet”, but in this work we define the darkweb as all domains underneath the “.onion” psuedo-top-level-domain[16] (which is sometimes called the onionweb).", "Specifically, we mean the subset of the web where websites are identified not by a human-readable hostname (e.g., yahoo.com) or by a IP number (e.g., 206.190.36.45), but by a randomly generated 16-character address (specifically, a hash fingerprint).", "Each website can be accessed via its hash, but it is very difficult to learn the IP number of a website from its hash – this is what makes the web `dark'; without an IP number, it is exceedingly difficult to trace the geographical origin of a communication.", "Crawling the darkweb is not much harder than crawling the regular web.", "In our case, we crawled the darkweb through the popular tor2web proxy onion.link.", "Onion.link does all of the interfacing with Tor, and one can crawl all darkweb pages without a login simply by setting a standard crawler to specifically crawl the domain *.onion.link.", "Darkweb pages are written in the same HTML language as the regular web which means we could crawl onion.link using the commercial service scrapinghub.com.", "Starting from two popular lists of darkweb sites,http://directoryvi6plzm.onion and https://ahmia.fi/onions/ we accessed each page and crawled all linked pages using breadth-first search.", "Most analyses of the www are at the page-level, where each node is an individual URL.", "One could adopt this model for the darkweb.", "It would be a natural choice for engineering motivated research question, such as studying crawlability.", "But the page-level model is not natural for socially motivated research questions, which we are interested in in this study, because the page-level graph is influenced more by the various choices of content management system, rather than by social dynamics.", "So we instead follow the modeling choice in [20] and aggregate by second-level domain (for the onionweb the second-level domain is equivalent to [20]'s “pay-level domain”).", "This means that links within a second-level domain are ignored as socially irrelevant self-connections.", "In this formulation, each vertex in the darkweb is a domain and every directed edge from $u \\rightarrow v$ means there exists a page within domain $u$ linking to a page within domain $v$ .", "The weight of the edge from $u \\rightarrow v$ is the number of pages on domain $u$ linking to pages on domain $v$ .", "The Tor Project Inc. (the creators and custodians of the darkweb) state there are $\\sim \\!\\!", "60,000$ distinct, active .onion addresses [21].", "In our analysis, however, we found merely $7,178$ active .onion domains.", "We attribute this high-discrepancy to various messaging services—particularly TorChat [22], Tor Messenger [23], and Ricochet [24].", "In all of these services, each user is identified by a unique .onion domain.", "The darkweb has a notoriously high attrition rate; its sites regularly appear and disappear.", "This complicates our analysis because it creates dead links, links to pages which have been removed.", "We do not want the dead links in our datasets so we collected responsive sites only; if we discover a link to a page on domain $v$ , but domain $v$ could not be reached after $>\\!10$ attempts across November 2016–February 2017, we delete node $v$ and all edges to node $v$ .", "Before pruning nonresponding domains, our constructed graph had 13,117 nodes and 39,283 edges.", "After pruning, it has $7,178$ nodes and $25,104$ edges (55% and 64% respectively).", "The pruned graph is the one used in the rest of this paper, which from now on we call “ the darkweb graph”.", "We note that the darkweb as defined above is different from the network described in [18].", "There, the network represents volunteer-operated nodes that could be connected in the sense that they could appear as consecutive nodes in a path through the Tor network.", "This is completely different from our network." ], [ "Graph-theoretic Results", "Table REF reports summary statistics of the darkweb and www.", "In what follows, we discuss these and other statistics.", "Table: Summarized network level properties between the www and the darkweb.", "Asterisk for the entries requiring conversion to an undirected graph." ], [ "Degree distribution", "We begin with statistics on degree distributions, reported in Figure REF .", "Panels (a) and (b) show the in and out degree distributions of the darkweb resemble power lawsUnfortunately however, we were able to confirm the power laws quantitatively.", "Following [25], which describes how to fit to power laws rigorously, we tried to fit the degree distribution data using the python package plfit, but the fitting failed on account of there not being enough data (the range of the data didn't cover enough orders of magnitude.", "), just like the www, but with one crucial difference: the location of the $y$ -intercept.", "In (a) we see $\\sim \\!", "30\\%$ of domains have exactly one incoming link ($k_{in} = 1$ ), with $62\\%$ come from one of the five largest out-degree hubs.", "In (b), we see a whopping $87\\%$ of sites do not link to any other site ($k_{out} = 0$ )!", "– these are the dark silos that we mentioned in the abstract.", "These findings tell us the darkweb is a sparse hub-and-spoke place.", "The bulk of its sites live in seclusion, not wanting to connect with the outside world.", "Panel (c) confirms this picture by showing the vast majority of pages have low pagerank (and so are isolated).", "Panel (d) shows that when a site does link to another, $32\\%$ of the time it's only a single page linking out.", "As we show in the next section, this siloed structure is not shared by many social networks.", "Figure: The distribution of the in-degree, out-degree, pagerank, and edgeweights.", "In (c) we exclude the three domains with the highest pagerank because they are such extreme outliers.", "For all plots with a log-scale axis, we follow following , to use the Fibonacci binning from ." ], [ "Bow-tie decomposition", "A useful way to describe directed graphs is via bow-tie decomposition, where the nodes are divided into six disjoint parts [28]: CORE — Also called the “Largest Strongly Connected Component”.", "It is defined as the largest subset of nodes such that there exists a directed path (in both directions, from $u \\rightarrow \\cdots \\rightarrow v$ as well as $v \\rightarrow \\cdots \\rightarrow u$ ) between every pair of nodes in the set.", "IN — The set of nodes, excluding those in the CORE, that are ancestors of a CORE node.", "OUT — The set of nodes, excluding those in the CORE, that are descendants of a CORE node.", "TUBES — The set of nodes, excluding those in the CORE, IN, and OUT, who have an ancestor in IN as well as a descendant in OUT.", "TENDRILS — Nodes that have an ancestor in IN but do not have a descendant in OUT.", "Also, nodes that have a descendant in OUT but do not have an ancestor in IN.", "DISCONNECTED — Everything else.", "Figure REF compares the bowtie decompositions of the darkweb and www.", "The www numbers are taken from [19], [20], [15].", "We chose these works because of the size of their crawls and the rigor of their analyses.", "Notice the www has each of one the 6 components of the bow-tie decomposition, whereas the darkweb only has a CORE and an OUT component.", "Moreover, the OUT component of the darkweb contains $\\sim 96\\%$ of the mass (these are the dark silos), leaving the CORE with just $\\sim 4 \\%$ .", "This is unusual for a social network; most have large COREs.", "The www's CORE has $> 50 \\%$ of the mass, while the cores of Orkut, YouTube, Flickr [30] and Twitter [31] are even larger.", "In this sense, the darkweb is a social network anomaly.", "Figure: Bow-tie decompositions of the www and darkweb.", "The figures for the www were taken from ." ], [ "Diameter analysis", "Next we examine the darkweb's internal connectivity by computing the shortest-path-length (SPL) between nodes.", "Figure REF (a) and (b) shows the SPL's for the www and darkweb.", "For all pairs of nodes $\\lbrace u,v\\rbrace $ in the darkweb, only 8.11% are connected by a directed path from $u \\rightarrow \\cdots \\rightarrow v$ or $v \\rightarrow \\cdots \\rightarrow u$ .", "This is drastically lower than the $\\sim \\!\\!43.42\\%$ found in the www [20].", "We again attribute this to the low out-degree per REF .", "Of the connected pairs, the darkweb's average shortest path length is $4.35$ compared to the $4.27$ in the world-wide-web [20].", "It's surprising to see a graph as small as the darkweb have a higher mean SPL than the entire world-wide-web, and is a testament to how sparse the darkweb graph really is.", "Figure REF (c) plots the distribution of SPLs for the 297 nodes of the CORE.", "To our surprise, the mean SPL within the CORE is $3.97$ , only $9\\%$ less than the entire darkweb.", "From this we conclude the CORE is not densely interconnected.", "Figure: Comparing shortest path lengths between the world-wide-web and the darkweb considering directed edges.", "Whereas in the www 56.65%56.65\\% of node pairs have have ∞\\infty path-length (no path connecting them), in the darkweb 91.89%91.89\\% of node-pairs have no path connecting them.", "Moreover, even within that 8.11%8.11\\% of pairs with a directed path between them, the darkweb's average SPL (μ=4.35\\mu = 4.35 ) is higher than that of the www (μ=4.27\\mu = 4.27 )." ], [ "Robustness and Fragility", "Does the peculiar structure of the darkweb make it resilient to attack?", "Figure REF shows it does not.", "As seen, the entire network (WCC) as well as the CORE quickly disintegrates under node removal.", "In Figures REF (a) and (b) we see the familiar resistance to random failure yoked with fragility to targeted attacks, in keeping with the findings of [32].", "Figures REF (b) shows that, unlike the www [20], the WCC is more susceptible to high in-degree deletions than the CORE.", "This elaborates the view from Figures REF (c) that the CORE is – in addition to not being strongly interconnected – also not any kind of high in-degree nexus.", "Figures REF (c) and (d) show the breakdown when removing central nodes.", "In (c), the CORE is largely unaffected by low centrality deletions.", "In Figure REF (c) we see that although the CORE isn't disproportionately held together by high in-degree nodes, it is dominated by very central nodes.", "Comparing Figures REF (b) and (f) we see the CORE relative to the entire network consists of more high-pagerank nodes than high in-degree nodes.", "This implies CORE nodes are not created by their high-indegree (REF ), but by their high centrality, amply corroborated by Figures REF (c) and (d).", "Likewise, Figures REF (e) recapitulates REF , that nodes with especially low in-degree or centrality are, unsurprisingly, not in the CORE.", "Figure: Deleting nodes from the darkweb graph and seeing how quickly the WCC and CORE disintegrate.", "In all plots, we shuffled the order of nodes with the same value until reaching stable statistics, e.g., in , 98%98\\% of nodes are tied for the lowest pagerank; so when removing only 10%10\\% of the nodes (e.g., ), it's ambiguous which nodes should be deleted first.", "So in our analysis we shuffled the ordering of the nodes with the same value and recomputed sizes of the WCC/CORE until the median was stable." ], [ "Reciprocal Connections", "The authors of [19] stress the importance of reciprocal connections in maintaining the www's graph properties.", "We compute two of their measures.", "First, we compute [19]'s measure $\\frac{\\langle k_{in}k_{out} \\rangle }{ \\langle k_{in} \\rangle \\langle k_{out} \\rangle } = \\frac{ \\mathbb {E}[ k_{in} k_{out} ] }{ \\mathbb {E}[ k_{in} ] \\mathbb {E}[ k_{out} ] }$ , to quantify in-degree and out-degree's deviation from independence.", "For the darkweb, we arrive at $\\frac{\\langle k_{in}k_{out} \\rangle }{\\langle k_{in} \\rangle \\langle k_{out} \\rangle }=3.70$ .", "This is in the middle of the road of prior estimates of the www, and means that the out-degree and in-degree are positively correlated.", "For greater clarity, we also plot the average out-degree as a function of the in-degree, given as, $\\langle k_{out}(k_{in}) \\rangle = \\frac{1}{N_{k_{in}}} \\sum _{i \\in \\Upsilon (k_{in})} k_{out,i} \\; ,$ which is simply “For all nodes of a given in-degree, what is the mean out-degree?”.", "The results are depicted in Figure REF .", "In short, in the darkweb there's no obvious pattern to the relationship between in-degree and out-degree.", "Figure: (a): Scatter plot of k in k_{in} and k out k_{out}.", "(b) Plot of averaged k out k_{out} versus k in k_{in}.", "(c) Comparing the rates of the darkweb sites linking to the www versus linking to other darkweb sites.", "They are essentially the same." ], [ "Network growth model", "Are standard network growth models able to capture the topology of the darkweb?", "Here we show a generalized preferential attachment model approximately can.", "In regular preferential attachment, nodes of only one type are added sequentially to a network.", "Here, we generalize this procedure to two types of nodes: pages and portals.", "Page nodes model the nodes in the darkweb which do not link to anyone (and so have $k_{in} = 0$ ).", "Portals model the rest of the nodes, which act as `portals' to the page nodes.", "The dynamics of the `page-portal' model are defined as follows.", "At $t = 0$ $N_0$ portals are arranged in a given network.", "At each time step after $t>0$ , a new page is added to the network and $m$ different portals are chosen to link to $m$ end nodes (these end nodes can be either pages or portals).", "The portal which does the linking is chosen according to preferential attachment, that is, chosen with probability $\\propto (1 + k_{out}^{\\beta }$ ), where $k_{out}$ is the out degree and $\\beta $ is a free parameter.", "The end node chosen with probability $\\propto (1 + k_{in}^{\\beta })$ .", "Notice this scheme means that only portals can link, which means page nodes forever remain with $k_{in} = 0$ , as desired.", "The model has three free parameters $m, \\beta , N_0$ .", "We chose $N_0 = 397$ so as to match the number of portals in our datasets (defined as all nodes with $k_{in} > 0$ ) and ran the model for 6242 timesteps i.e added 6242 page nodes) so that there were 7178 nodes at the end of the process which again matched the dataset.", "Figure REF shows the page-portal model with parameters $(m,\\beta ) = (4, 2)$ approximates the darkweb, the in and out degree distributions of the model approximately mimicking the data.", "We report exponents of best fit, found using the powerlaw package python.", "But keep in mind that as mentioned earlier, there are not enough orders of magnitude in the data for the estimates of the exponents to be reliable; thus the page and portal model is intended as a first step in modeling the darkweb.", "The values were $(\\alpha _{in}, \\alpha _{out})_{data} = (3.09, 2.10)$ and $(\\alpha _{in}, \\alpha _{out})_{model} = (4.3 \\pm 1.3 , 2.4 \\pm 0.2 )$ , where the model figures report means of 10 realizations and the error is given as half the range.", "The fitting was performed using the python powerlaw package.", "$\\alpha _{in}$ was much more difficult to measure than $\\alpha _{out}$ .", "As much as half of realizations led to estimates $ \\alpha _{in} > 50$ , which we interpret as the failure of the fitting to converge, and so were discarded.", "Figure: Comparison of in (k in k_{in}) and out (k out k_{out}) degree statistics of darkweb and page-portal model.", "(a) Probability density functions of k in k_{in} and k out k_{out} for page and portal model.", "(c)-(d) Histograms of above quantities." ], [ "Conclusion", "Our primary finding is that the darkweb is a social network anomaly.", "Its light CORE and massive OUT components distinguishes it from other popular social networks.", "In fact, calling it a `web' is a connectivity misnomer; it is more accurate to view it as a set of dark silos – a place of isolation, entirely distinct from the well connected world of the www and other social network.", "What causes the darkweb to be so isolated?", "We see two possible explanations: The technological explanation.", "In the darkweb, sites go up and go down all the time.", "Why bother linking if there's little chance that the destination will still exist?", "The social explanation.", "People who create sites on the darkweb are cut from a different social cloth than those who create sites on the www (or at least when using the darkweb, these people behave differently) To test the technological explanation, we performed a second crawl collecting instances of darkweb sites linking to the www and compared the relative rates of outbound linking in Figure REF (c).", "There are essentially equal rates of outbound linking to the www as well as the darkweb which tells us (i) the low outbound linking is not due to the impermanence of onion sites and (ii) if onion sites got drastically more stable, we would still see very low rates of linking.", "Taken together, these indicate the technological explanation is likely untrue.", "Thus, the social explanation is likely the cause of the darkweb's anomalous topology.", "Rigorously testing the social hypothesis is however beyond the scope of this work.", "Although, in a sense we have taken a first step in this direction by generalizing preferential attachment which itself can be viewed as model of social behavior; it is a model of trust: highly linked nodes are perceived as `trustworthy' sources of information, and so receive a disproportionate number of links; the rich get richer.", "The isolated silos of the darkweb, however, indicate trust does not play a role in the dynamics governing its evolution.", "Rather, one might say it indicates that distrust does.", "The passive pages of the page and portal model (which recall, do not link to anybody through the dynamics, and are in that sense passive) were a crude way to incorporate this effect.", "But a more principled behavioral model (i.e.", "one consistent with known results from psychology) is needed, which were were unable to develop.", "We hope psychology-fluent researchers will take up this task in future work.", "Future work could also study the temporal aspects of the darkweb.", "Is the topology we have found stationary?", "For example, in the work most closely related to ours [18], it was found that the resilience of the studied `darknet' evolved over time (as discussed in the Data Collection section, our darkweb graph is much different to the darknet in [18]).", "It would be interesting to see if the resilience of our darkweb graph behaves like this too." ], [ "Acknowledgment", "The authors would like to thank all members of the MIT Senseable City Lab consortium." ] ]

[ [ "Empirical Evaluation of Pretraining Strategies for Supervised Entity\n Linking" ], [ "Abstract In this work, we present an entity linking model which combines a Transformer architecture with large scale pretraining from Wikipedia links.", "Our model achieves the state-of-the-art on two commonly used entity linking datasets: 96.7% on CoNLL and 94.9% on TAC-KBP.", "We present detailed analyses to understand what design choices are important for entity linking, including choices of negative entity candidates, Transformer architecture, and input perturbations.", "Lastly, we present promising results on more challenging settings such as end-to-end entity linking and entity linking without in-domain training data." ], [ "Introduction", "Traditionally, entity linking approaches have relied on knowledge bases, complicated modelling and task-specific hand-engineered features to achieve high performance.", "More recently, [1], [18] and [33] show that using large-scale pretrained language models like BERT [6], pretraining on Wikipedia entity links, and fine-tuning on a specific entity linking corpus leads to state-of-the-art performance without relying on such features.", "However, [18] focused mainly on constructing general-purpose entity representations, [33] on building strong zero-shot entity linking systems, and [1] on end-to-end linking, so that the limits of pretraining for entity disambiguation have not been fully explored.", "In this paper we present a thorough study of pretraining strategies for supervised entity linking.", "We establish new upper bounds for performance on the widely studied CoNLL and TAC-KBP 2010 entity linking tasks.", "We also show that our pretraining approach yields a very competitive entity linking system without any further domain specific tuning.", "We present a detailed analysis of significant design choices including the choice of negative candidates used during training, and the document context encoded for each mention.", "We find that the optimal choice of negative candidates is dependent on whether or not the final linking system has access to an alias table.", "For a system that will use an alias table at inference, it is helpful to pretrain the models with lexically similar candidates.", "However, when no alias table is used for the downstream task, ensuring candidates are random improves the model ability to distinguish the right entity among all possible entities.", "In our studies we found two surprising results: (1) it is possible to achieve optimal entity linking results with a four layer transformer, which is one third of the size of BERT-base, and (2) given the abundance of supervision from Wikipedia links, we did not get any gains in performance from training with an auxiliary language modeling loss.", "However, we did find that the input perturbations introduced by [6] themselves increase the quality of our pretraining approach and we present an analysis of how these perturbations add robustness to the model.", "Finally, to demonstrate the generality of our model, we present results on the end-to-end entity linking task in which both mention location and identity are predicted.", "On this task, our model outperforms all but the tailored methods introduced by [16], [1].", "We argue that our model is more practical, and easier to integrate, than heavily engineered existing approaches, and we believe that downstream tasks such as information extraction and question answering can benefit from this robust standalone end-to-end entity linker (e.g.", "[7])." ], [ "Related Work", "Early entity linking systems [2], [21] focused on matching the context of the mention with that of the entity page.", "In addition to context features, systems have relied on $\\mathbb {P}(e|m)$ , the prior probability that mention $m$ refers to entity $e$ , computed from Wikipedia mention counts.", "The set of entities in a document should be globally coherent, and several approaches have introduced sophisticated global disambiguation methods [10], [3], [29] that consider all mentions in a document to make predictions.", "In contrast, we do not model document-level disambiguation explicitly.", "However, our long context windows contain several mentions which should allow such disambiguation.", "Other approaches have also sought to incorporate types and knowledge base information in their modelling, such as [26], [27].", "We pretrain our model by learning distributed representation of entities directly from Wikipedia text, similarly to [34], [35], [9], [18].", "Unlike [33], [11], our embeddings are learned directly rather than generated through entity descriptions.", "In contrast with [34], [35], we do not use additional features, such as prior probabilities or string match features.", "Our method is therefore most similar to [18] and [1] who also use a transformer.", "We differ from [18] by simultaneously considering all mentions and entities in a context and, in contrast to both, only use a four layer, randomly-initialized transformer instead of twelve layers initialized from large scale language modelling pretraining.", "In addition, we experiment with end-to-end entity linking [29], [20], [16], where instead of predicting the entity for gold spans, a system must both predict the span and its label.", "A closely related task is multilingual entity linking.", "Approaches have used multilingual embeddings to link text in several languages [30], [31].", "Zero-shot entity linking [19], [33] is another relevant task.", "In that setting, entities predicted at test time are not seen in training.", "Instead, the system relies on the entity name and description." ], [ "Task Definition", "Let $\\mathcal {E} = \\lbrace e_1 \\dots e_N\\rbrace $ be a predefined set of entities, and let $\\mathcal {V} = \\lbrace \\textsc {[mask]}, w_1 \\dots w_M\\rbrace $ be a vocabulary of words.", "A context $\\mathbf {x}= [x_0 \\dots x_t]$ is a sequence of words $x_i \\in \\mathcal {V}$ .", "A span $\\mathbf {s}= (s_{start}, s_{end})$ , is a tuple with $0 \\le s_{start}, s_{end}< t$ which defines a contiguous sequence of tokens in a given context.", "A mention label $\\mathbf {l}= (s_k, e_k)$ consists of a span $s_i$ and an entity label, $e_i \\in \\mathcal {E} \\cup \\varnothing $ .", "We use $\\overline{\\mathbf {l}}$ to denote a set of such mention labels.", "The NULL-symbol $\\varnothing $ indicates a span that is labeled as a mention, but without an entity linking label.", "Our training data, $\\mathcal {D} = \\lbrace (\\mathbf {x}_0, \\overline{\\mathbf {l}}_0) \\dots (\\mathbf {x}_N, \\overline{\\mathbf {l}}_N)\\rbrace $ , is a corpus of contexts, each paired with a set of mention labels, one for each mention in the context.", "Given an input context $\\mathbf {x}_i$ , our goal is to predict the set of entity mentions $\\overline{\\mathbf {l}}_i$ .", "In Entity Disambiguation, we are given the set of spans, and predict the entity linked by each span.", "In End-to-End Entity Linking, we must predict both the set of mention spans, and their linked entities." ], [ "Contextual Language Representation", "Our model is built using the now-standard Transformer-based architecture [32].", "The model computes a matrix representation $\\hat{\\mathbf {H}} \\in \\mathcal {R}^{t \\times d}$ of a text sequence through successive application of a Transformer block to the output of the previous layer: $\\mathbf {H}_{i} &= \\mathtt {TransformerBlock}(\\mathbf {H}_{i-1}) \\\\&= \\mathtt {MLP}(\\mathtt {MultiHeadAttention}(\\mathbf {H}_{i-1}, \\mathbf {H}_{i-1}, \\mathbf {H}_{i-1}))$ $\\mathbf {H}_0$ is a sequence of context-independent token embeddings and $\\hat{\\mathbf {H}} = \\mathbf {H}_n$ , where $n$ is the number of Transformer layers." ], [ "Entity Disambiguation", "Each entity $e \\in \\mathcal {E}$ is mapped directly onto a dedicated vector in $\\mathbb {R}^d$ via a $\\mathcal {|\\mathcal {E}|} \\times d$ dimensional embedding matrix.", "In our experiments, we have a distinct embedding for every concept that has an English Wikipedia page, resulting in approximately 5.7m entity embeddings.", "In order to perform entity linking for a particular span with word-piece token indices $(i, j)$ , we (following [17]) first obtain a representation of the span by concatenating the representation at the span start and end, and pass this through a multi-layer perceptron which projects the span representation into the same space as the entity embeddings.", "$\\hat{s}_{\\mathbf {s}_i} = \\texttt {MLP}([\\mathbf {H}_{n, s_{start}}, \\mathbf {H}_{n, s_{end}}])$ Our model scores each span-entity pair by taking the dot-product between the projected span representation and the embedding of $\\mathbf {e_c}$ .", "Thus, the conditional probability that the span $\\mathbf {s}_i$ refers to entity $e_c$ is defined as: $\\mathbb {P}(e_c | \\mathbf {s}_i) = \\frac{\\exp (\\hat{s}_{\\mathbf {s}_i} \\cdot e_c)}{\\sum _{\\mathbf {c^{\\prime }} \\in \\mathcal {E}}\\exp (\\hat{s}_{\\mathbf {s}_i} \\cdot e_{c^{\\prime }})}$ In practice, this is expensive to compute for large $|\\mathcal {E}|$ .", "Therefore, for every $\\overline{\\mathbf {l}}$ we select a set $\\mathcal {C}_{\\overline{\\mathbf {l}}}$ of $k$ candidates, which contains the entity labels for all $\\mathbf {l}\\in \\overline{\\mathbf {l}}$ as well as a set of negative candidates.", "We do not have an entity linking loss on mentions that do not have a label.", "Therefore, our per-example entity linking loss is: $l_{linking}(\\overline{\\mathbf {l}}) = \\sum _{\\mathbf {l}_i \\in \\overline{\\mathbf {l}}} \\frac{\\exp (\\hat{s}_{\\mathbf {s}_i} \\cdot e_{i}) \\mathbb {1}_{e_i != \\varnothing }}{\\sum _{c \\in \\mathcal {C}_{\\overline{\\mathbf {l}}}} \\exp (\\hat{s}_{\\mathbf {s}_i} \\cdot e_{c})}$ We will discuss further how we select $\\mathcal {C}_{\\overline{\\mathbf {l}}}$ in Section REF ." ], [ "Mention Detection", "For many entity linking tasks, the target spans are provided.", "In order to be able to do end-to-end entity linking, we additionally train our model to predict mentions, independently of entity linking.", "One way to do this would be to score every possible span-entity pair, and either use a score threshold to filter spans where no entity link achieves a sufficiently high score, or to additionally score a special NULL-link embedding.", "However, enumerating all spans for the long contexts we use in our model would be prohibitively expensive.", "We take the approach of encoding mentions as a BIO sequence, and train an MLP on the context representation to predict this sequence with a standard cross-entropy loss.", "Our final loss sums the mention detection loss and the linking loss." ], [ "Wikipedia Pretraining", "We build a training corpus of contexts paired with entity mention labels from the 2019-04-14 dump of English Wikipedia.", "We first divide each article into chunks of 1000 unicode characters, resulting in a corpus of over 17.5 million contexts with over 17 million entity mentions covering over 5.7 million entities.", "These are processed with the BERT tokenizer, limited to 256 word-piece tokens.", "In addition to the Wikipedia links, we annotate each sentence with unlinked mention spans using a state-of-the-art named entity recognizer.", "These are used as additional signal for our mention detection component." ], [ "Entity Candidates Selection", "Training the model with a full softmax over all 5.7 million entities for every mention is computationally expensive.", "A common solution is to use a noise contrastive loss [12], [22] and sample candidates according to their relative frequency, as in [18].", "In this work, we experiment with other approaches to candidate generation that might provide better negatives in training.", "In addition to negatives selected uniformly at random from the entire entity vocabulary, we define two types of hard negatives: Page candidates, which is the set of all entities linked to in the article from which the given context was taken.", "This is meant to capture semantically related concepts.", "Phrase table candidates, the set of lexically related entities for each mention candidate, obtained from the Phrase Table provided by SLING [28].", "Throughout the paper, we will use $|\\mathcal {C}_{\\overline{\\mathbf {l}}}| = 768$ .", "In our base setup, we use up to 256 page candidates, and 384 phrase table candidates, equally divided between each mention in the example.", "Any remaining room in the set of 768 is filled with random candidates sampled uniformly from the entity vocabulary (meaning a minimum of 128 random candidates per example).", "We will study the impact of different candidate selection methods in Section REF .", "In addition to those candidates, for every example in a batch, we use the candidates of other examples as additional negatives." ], [ "Input Noising", "We also add noise to the input data.", "We apply the same noise function as is used in [6]: 15% of the tokens are chosen to be modified.", "80% of those tokens are changed to the [mask] token, 10% are changed to a random token and 10% are left unmodified." ], [ "Pretraining hyperparameters", "We use Adam [15] with a learning rate of 1e-4 to optimize our model.", "We use a linear warmup schedule for the first 10% of training, decay the learning rate afterwards and use gradient clipping with a norm of 1.0.", "We train from scratch for up to a million steps and use a large batch size of 8192 for pretraining.", "We follow BERT [6] base for many of our model parameters, though we do not use large-scale language-modeling pretraining and only use four layers, as we did not find more to layers to further improve performance.", "We use the same word-piece vocabulary as the lowercase version of BERT.", "We use entity embeddings of size 256 unless mentioned otherwise.", "We weight both the entity disambiguation loss and the mention detection loss to 1.", "We use a context window of 256 tokens." ], [ "Entity Linking Datasets", "We evaluate our model on two popular entity linking benchmarks: AIDA CoNLL YAGO [13] and TAC-KBP 2010 [14].", "The first is comprehensively annotated with approximately 34,000 mentions on 1,393 newswire document on the full Wikipedia vocabulary, while the second is sparsely annotated for target entities only on a smaller entity vocabulary." ], [ "Textual Context", "Most CoNLL documents do not fit in our limit of 256 tokens.", "Therefore, we split the document into “sentences” at each newline in the document.", "We experiment with three methods to add document context to these sentences: (i) taking the sentence as-is, (ii) adding the title of the document to the sentence, (iii) adding the title of the document as well as the first two sentences to the sentence.", "Throughout our experiments we will use (iii), though we show the impact of this choice in Section REF ." ], [ "Entity Candidates Selection", "Our candidates for CoNLL come from alias tables - resources which provide a list of possible strings for a given entity.", "A key challenge with evaluating entity linking systems on the CoNLL dataset is inconsistent use of alias tables.", "[10] describe the difficulty of resolving older resources due to changes in Wikipedia links and unicode, and provide statistics for two commonly-used alias tables: The YAGO extended “means” mapping of  [13], and the “PPRforNED” mapping of  [23].", "We find that through careful resolution of unicode and Wikipedia redirects, we achieve a slightly higher conversion rate than reported by [10] (statistics provided in Table REF ).", "This leads to a higher gold recall, but also a larger number of candidate for each mention, meaning our system must distinguish between more candidates.", "We report results using both alias tables." ], [ "Finetuning", "We finetune our model – including the entity embeddings – on the CoNLL training set, using the alias table candidates for each mention.", "We used a batch size of 256, a learning rate of 1e-6, and train for 2000 steps.", "Figure: Statistics for alias table conversions, computed on the CoNLL test split.", "Gold recall is the percentage of mentions for which the gold entity is included in the candidate set.", "Average ambiguity is the total number of candidates divided by the number of mentions." ], [ "TAC-KBP 2010 is another widely used dataset for evaluating entity disambiguation systems.", "In contrast with CoNLL, the mentions are sparsely annotated among documents.", "It contains 1074 annotated entities in the training set and 1020 in the evaluation set.", "The entities for this dataset are part of the TAC Knowledge Base, containing 818,741 entities.", "Due to the reduced entity vocabulary, we can fine-tune without resorting to an alias table and we adopt this setting throughout our results.", "This is consistent with the prior state-of-the-art approach of [33].", "To select the context for a mention, we take the 256 bytes before and after the first occurrence of the mention in the document.", "We select the fine-tuning parameters on training by doing cross-validation on the training set.", "We used a batch size of 32, trained for 1,000 steps and found it was best to freeze the entity embeddings.", "Our final model is trained on all the training data with the parameters selected in cross-validation.", "However, we report the result on the evaluation (test) set number in all tables, including ablations.", "Indeed, we found that this was more reflective of task performance, as the training set is significantly easier." ], [ "End-to-end entity linking", "We also experiment with end-to-end entity linking on CoNLL (TAC-KBP is not suitable due to its sparse annotations).", "In this case, we do not use an alias table.", "In this setting, we follow the hyperparameters of Section REF .", "Instead of using candidates, we train our model to predict BIO-tagged mention boundaries and to disambiguate among all entities.", "At training and fine-tuning time, gold spans are used for the disambiguation task.", "At inference, we use the BIO-tagged predictions as our spans and predict entities for each span among all possible entities.", "We use the standard strong matching micro-F1 score." ], [ "Entity Linking", "Table REF shows that our approach outperforms all prior approaches on CoNLL and TAC-KBP 2010.", "On CoNLL, we outperform methods in both alias-table settings.", "Additionally, we note that unlike many previous systems, we do not use alias priors, knowledge-base features, or other entity features.", "Table: Test accuracy on the CoNLL and TAC-KBP entity disambiguation tasks.", "CoNLL H refers to papers using the  “means” alias table while P refers to the  table.", "*It is not clear which alias table, if any, is used by .", "†\\dagger -marked systems do not use features beyond the text and alias table." ], [ "End-to-end entity linking", "For end-to-end entity linking, we do not use the alias table.", "Instead of using candidates, we predict BIO-tagged mention boundaries and disambiguate mentions among all entities.", "At training and fine-tuning time, gold spans are used for the disambiguation task.", "At inference, we use the BIO predictions as our spans and predict entities for all these spans.", "Table REF shows our model fares well against other models, with the exception of [1] and [16].", "The former use a much larger Transformer model, and also initialize from BERT-base model, which is pretrained on a corpus of unlabeled text much larger than our training data.", "[16] relies on an alias table to generate candidate mentions at both training and inference time.", "In addition, it introduces a clever mechanism to jointly optimize and select mention boundaries and entity candidates, whereas we use a simpler pipelined approach.", "Finally, they also introduce a document-level disambiguation coherence penalty and a coreference resolution heuristic.", "We believe the use of an alias table as well as the aforementioned differences explain the gap between our method and [16], and we will look to bridge this gap in future work.", "Nevertheless, our model stands as a strong baseline of what can be achieved with simple modelling and low inference cost.", "Table: End-to-end entity linking strong matching micro-F1 score on the development and test sets of CoNLL.", "Despite the simplicity of our setup, our system is competitive with most prior work, with the noteworthy exception of" ], [ "Classifying all entities or classifying candidates", "We trained our model to distinguish the correct linked entity among candidates.", "An alternative approach is to predict among all entities.", "This is computationally more expensive as it requires doing a softmax over 5.7 million entities for every mention in the batch.", "Thus we use a batch size to 2048 and set the entity embedding dimension to 64 for both this model and the one trained with candidates.", "Table REF shows impressive accuracy without an alias table for the system classifying among all entities.", "However, it does not fare better than the model trained with candidates when using one.", "Given the considerable cost of doing the full sofmax for every mention, we use candidates for our other experiments.", "Note that our model gets 88.4% accuracy when trained only on the CoNLL data.", "Table: Entity disambiguation accuracy on the CoNLL development set for different pre-training and fine-tuning setups.", "Numbers in the first part of the table do not use an alias table, whereas the ones on the second part use 's table." ], [ "Impact of candidate selection", "In Table REF , we show the impact of different candidate selection pretraining strategies.", "On CoNLL, where we do use an alias table for evaluation, we find that our candidate selection heuristic seems to help the model in pre-training, achieving better performance than any of our ablations.", "Training with lexically related (phrase table) candidates is particularly important, as this is similar to the disambiguation task the model has to perform.", "Page candidates are semantically related but generally not lexically related and thus do not bring the same benefits.", "In fact, they might even distort the distribution of negative candidates as the model performs worse in this setting than with random candidates.", "After fine-tuning, all models fare similarly, which we believe is due to CoNLL having enough fine-tuning data so that all our models approach the performance upper-bound on this task (see Section REF ).", "For TAC-KBP 2010, where we do not use an alias table at fine-tuning or inference time, the results are markedly different.", "After pre-training, ours which has less random candidates performs worse than all other alternatives.", "This is likely because having more random candidates is closer to the full classification setup used in TAC-KBP 2010.", "Given TAC-KBP's small training set, these differences carry over in fine-tuning performance.", "Table: Impact of the candidate selection method on development performance on CoNLL and TAC-KBP 2010.", "Unsurprisingly, pretraining methods that are closer to the final task setup perform better: For CoNLL, we emphasize the importance of phrase table candidates in pre-training to emulate the use of the alias table , whereas for TAC-KBP, setups that use random candidates are more successful as they are closer to the full classification setup used in this task." ], [ "Impact of adding noise during pretraining", "Table REF shows adding noise in pretraining helps both pre-training and fine-tuning performance.", "We hypothesize that input noise implicitly trains the model to generalize to alternative aliases: for instance, given the mention “Yuri Gagarin”, the model might have to learn to recognize “Yuri [MASK]” or “[MASK] Gagarin”.", "Table: Impact of adding BERT-style input noise during pre-training.", "We report development accuracy on CoNLL using the   alias table.Encouraged by the success of [6], we experimented with also pretraining our model with a masked-language modelling objective, with the expectation that such pre-training would help our model learn better representations of language.", "We tried different architectures and loss weights (including a 4 layer transformer with both objectives at layer 4, a 12 layer with the entity linking loss at layer 4, etc.)", "but overall found this to not improve further on simply adding noise in pretraining." ], [ "Impact of context selection methods on CoNLL", "Table REF shows the impact of varying CoNLL's context type in pre-training and fine-tuning.", "We find that larger contexts, especially those that include the beginning of the document, considerably boost performance before fine-tuning.", "However, similarly to our observations in Sections REF and REF , we find that improvements are less marked after fine-tuning, likely because our performance is already very high.", "Table: Impact of additional context beyond a single sentence used for entity disambiguation performance on the CoNLL task.", "We report development accuracy on CoNLL using the  alias table." ], [ "Error analysis", "Figure REF shows three sample errors on the CoNLL development set.", "Most errors are due to varying levels of specificity in the CoNLL labels.", "Some errors are due to changes in Wikipedia.", "For instance, in text A, the Bulgaria U21 soccer team Wikipedia page was built in 2013, after CoNLL.", "Also, in text C, our model correctly disambiguates between Austin, Michaella and Richard Krajicek, which are all three tennis players (only Richard is a dutch).", "Figure: Sample of errors on the CoNLL development set for our model." ], [ "Conclusion", "In this paper we present a thorough study of pretraining strategies for supervised entity linking, achieving state-of-the-art performance on both CoNLL and TAC-KBP 2010 with a four-layer Transformer-based model.", "Given the limited headroom remaining in these datasets, and the strong impact of alias tables in simplifying the problem, we believe the creation of new datasets, and more difficult entity linking settings, such as zero-shot and low resource domains, are crucial areas for future work.", "The authors wish to thank Dan Bikel and Eunsol Choi for their helpful comments in the preparation of this paper, as well as the anonymous reviewers." ] ]

[ [ "Self-supervised Modal and View Invariant Feature Learning" ], [ "Abstract Most of the existing self-supervised feature learning methods for 3D data either learn 3D features from point cloud data or from multi-view images.", "By exploring the inherent multi-modality attributes of 3D objects, in this paper, we propose to jointly learn modal-invariant and view-invariant features from different modalities including image, point cloud, and mesh with heterogeneous networks for 3D data.", "In order to learn modal- and view-invariant features, we propose two types of constraints: cross-modal invariance constraint and cross-view invariant constraint.", "Cross-modal invariance constraint forces the network to maximum the agreement of features from different modalities for same objects, while the cross-view invariance constraint forces the network to maximum agreement of features from different views of images for same objects.", "The quality of learned features has been tested on different downstream tasks with three modalities of data including point cloud, multi-view images, and mesh.", "Furthermore, the invariance cross different modalities and views are evaluated with the cross-modal retrieval task.", "Extensive evaluation results demonstrate that the learned features are robust and have strong generalizability across different tasks." ], [ "Introduction", "Self-supervised feature learning methods learn visual features from large-scale datasets without requiring any manual annotations.", "The core of the self-supervised feature learning is to define a pretext task and the visual features are learned through the processing of accomplishing the pretext task.", "Since it can be easily scaled up to large-datasets, recently some self-supervised methods achieved comparable or even better performance on some downstream tasks than supervised learning methods [30], [17], [7], [2], [24].", "Most of the existing self-supervised feature learning methods only focus on learning features for one modality.", "As a rising trend to model 3D visual features, various methods were proposed to learn point cloud features from point cloud either by reconstructing point cloud [1], [10], [54], [57], by generating point cloud with Generative Adversarial Networks [29], [46], [51], or by accomplishing pretext tasks [16], [55].", "Only a few of them [22] explored to use cross-modality correspondence of 3D data as supervision signal for 3D self-supervised feature learning.", "Figure: Multi-modality and multi-view representations of 3D objects and pair examples.", "With cross-modal invariant and cross-view invariant constraints, modal- and view-invariant features can be obtained with heterogeneous networks for different modalities and views of data.Generally 3D data are inherently multi-modalities.", "Fig.", "REF shows different modalities of same objects in mesh, point cloud, and multi-view images.", "No matter what data format is used to represent an object, its identity remains unchanged.", "Thus, it is possible to learn features for an object that invariant to its modalities and views.", "A straightforward idea is to employ the identity invariance of different modalities and views as supervision signals to learn features from unlabeled data.", "Jing et al.", "[22] formulated the identity invariance as a classification task and jointly trained multiple networks to verify whether the inputs from different modalities belong to same object by using the cross-entropy loss.", "However, the learned features, are not modality-invariant which make it impossible to directly compare the features from different modalities of 3D data.", "Recently, contrastive learning has shown great promise and obtained promising performance for recent self-supervised feature learning methods [7], [17], [14].", "Similar to triplet loss [41], contrastive loss is to maximize the feature similarity between positive pairs and minimize the feature similarity of other negative pairs.", "The most commonly used approach to generate positive and negative training pairs is data augmentation on input data [3], [7], [17].", "However, dramatic data augmentation, such as cutout and jittering, inevitably changes the distribution of the training inputs far from the real testing data.", "To enable heterogeneous networks to learn features of different modalities and views in same universal space, in this paper, we propose to employ two constraints derived from the attributes of 3D data as supervision signal for self-supervised learning: cross-modal invariance constraint and cross-view modal invariance constraint.", "The cross-modal invariance constraint forces networks to learn identity features from different modalities of the same object, while the cross-view invariance constraint forces networks to learn an identity features for each object regardless of its view.", "When jointly learned with the two constraints, modal- and view-invariant features in the universal space can be obtained for each object.", "Inspired by the contrastive learning [7], we propose a framework to capture modal-invariant and view-invariant features with heterogeneous networks on three different modalities including mesh, point cloud, and multi-view images.", "Specifically, the features from different modalities and rendered views of 3D objects are extracted with corresponding subnetworks and combined as positive pairs (sampled with the same object identity) and negative pairs (sampled with different object identities).", "Then the feature distance of positive cross-modality pairs is minimized, and the feature distance of negative cross-modality pairs is maximized by using contrastive loss.", "The modal- and view-invariant features are obtained after optimizing networks with the two types of constraints over the positive and negative pairs.", "The main contributions of this paper are summarized as follows: We propose a novel self-supervised feature learning schema to jointly learn modal- and view-invariant features for 3D objects end-to-end without using any manual labels.", "The proposed framework maps the features of different modalities and views of 3D data into same universal space which makes cross-modal retrieval possible for 3D objects.", "To the best of our knowledge, we are the first to explore the cross-modal retrieval for 3D objects with multiple modalities in a self-supervised learning way..", "The effectiveness and generalization of the learned features are demonstrated with extensive experiments with three different modalities on five different tasks including 3D object recognition, few-shot 3D object recognition, part segmentation, in-domain 3D object retrieval, and cross-modal retrieval." ], [ "Related Work", "2D Self-supervised Feature Learning: Many methods have been proposed to learn visual features from unlabeled 2D data including videos and images.", "Based on the source of supervision signal, there are four types of self-supervised learning methods: Generation-based method, context-based method, free semantic label-based method, and cross-modal based method.", "The generation-based methods learn features by reconstructing the data including Auto-encoder, and Generative adversarial networks [13], super resolution [28], colorization [56], and video future prediction [44].", "The context-based methods learn features by using spatial context or temporal context including Jigsaw puzzle [32], geometric transformation [11], [23], clustering [4], frame order reasoning [31].", "The free semantic label-based methods learn features either by data from game engine or to distill features from other unsupervised learning features [34].", "The cross-modal-based methods learn features by the correspondence between a pair of channels of data including video-audio [27] and video-text.", "Recently, more researchers explore to apply these self-supervised leaning methods to 3D data [22], [55], [40], [16].", "3D Self-supervised Feature Learning: Several self-supervised learning learning methods have been proposed to learn 3D features for point cloud objects by reconstructing point cloud data  [1], [10], [54], [57], by generating point cloud with GANs  [29], [46], [47], [51], or by training networks to solve pre-defined pretext tasks [16], [40], [55], [22].", "Sauder et al.", "proposed to learn point cloud features by training networks to recognize the relative position of two segments of point cloud [40].", "Zhang et al.", "designed a clustering followed by contrastive as pretext task to train networks to learn point cloud features.", "Hassani et al.", "proposed to train networks with multiple pre-defined pretext tasks including clustering, prediction, and reconstruction for point cloud data [16].", "Jing et al.", "proposed to utilize cross-modality relations of point clouds and multi-view images as supervision signal to jointly learn point cloud and image features for 3D objects.", "However, the point cloud and image features by the network in [22] are not modality invariant.", "To thoroughly utilize the cross-modality inherent attributes of 3D data, here we propose to learn modal- and view-invariant features for 3D objects with three different modalities including point cloud, mesh, and images.", "Contrastive Self-supervised Learning: The basic principle of contrastive learning, such as Noise Contrastive Estimation (NCE) [15], is to learn representations by contrasting positive and negative pairs.", "By maximizing the similarity between an anchor sample and a positive sample while minimizing similarity to all other (negative) samples, contrastive learning has shown empirical success in self-supervised learning methods [3], [7], [17], [19], [20], [30], [48], [53], [14], [35] of which the core is to generate positive and negative training pairs by pretext tasks.", "Most recent work performed training pairs generation in image domain.", "Studies [3], [7], [17] applied dramatic data augmentation such as color jittering, cropping, cutout, and flipping on original images.", "PIRL [30] cropped an image into jigsaw patches, then combines the original image and the shuffled patches into a positive pair.", "[19] divided an image into a grid of overlapping patches and predicts the unseen regions by context patches with Contrastive Predictive Coding (CPC) [33].", "[14] extended the contrastive learning on videos by sampling frames from same video as multiple positive pairs.", "Contrastive Multiview Coding (CMC) [48] used transferred representations (such as Lab color space, depth, and segmentation) of a source image as paired samples.", "Very few work adopted cross-modality invariance on contrastive learning.", "[35] generated training pairs by video clips and their corresponding audios.", "The number of negative pairs affects the probability that positive sample paired with hard negative.", "Some work [30], [53], [14] introduced memory banks storing previous feature outputs to enlarge the negative pairs pool.", "Momentum Contrast (MoCo) [17] further increased the number of negative samples by a slowly updated negative feature extractor.", "In this paper, we propose to pair positive and negative object samples by cross-modality invariance in three domains (mesh, point cloud, and image) and cross-view invariance in image domain." ], [ "Method", "An overview of the proposed framework is shown in Fig.", "REF .", "The core of our method is to optimize heterogeneous networks to learn modal- and view-invariant features under cross-modality invariance constraint and cross-view invariance constraint by contrasting.", "Three heterogeneous networks are employed to extract features for three modalities of data including mesh, point cloud, and images, respectively.", "The framework contains three backbone networks (an image-extracting network $F_{img}$ , a point cloud-extracting graph network $F_{p}$ , and a mesh-extracting network $F_{m}$ ) and three corresponding projection heads ($F_{img\\_h}$ , $F_{p\\_h}$ , $F_{m\\_h}$ ) mapping features of different modalities into the universal space.", "Two types of constraints including modality-invariant constraint and view-invariant constraint are used to optimize the network by contrasting paired features of all objects in the universal space.", "The detailed formulation of our approach is explained in subsection REF , and the network architectures are described in subsection REF .", "Figure: An overview of the proposed self-supervised modal- and view-invariant feature learning framework.", "Mesh, point cloud, and multi-view image features are extracted by MeshNet, DGCNN, ResNet, and corresponding projection heads, respectively.", "With contrastive learning to minimum the feature similarity of positive pairs and maximum the feature similarity of negative pairs under modality- and view-invariant constraints, the modal- and view-invariant features can be learned with the proposed heterogeneous framework in the same universal space." ], [ "Model Parameterization", "The contrastive learning enables the networks to learn representations by maximizing the similarity between an anchor sample and a positive sample while minimizing similarity to all other (negative) samples, similar to the triplet loss [41].", "In this paper, the positive pairs are sampled from different modality and view representations with same object identity, while negative pairs with different object identities as shown in Fig REF .", "The contrastive learning implementation in this paper uses the following procedure.", "Given a batch of $k$ anchor features $\\lbrace A^{(1)}, ..., A^{(k)}\\rbrace $ and $k$ corresponding positive features $\\lbrace P^{(1)}, ..., P^{(k)}\\rbrace $ .", "The contrastive loss for this single anchor-positive batch $l_{ap}$ is defined as $l_{ap} = -\\frac{1}{k} \\sum _{i=1}^{k} \\log \\frac{e^{sim(A^{(i)}, P^{(i)})}}{\\sum _{j = 1, j \\ne i}^{k} e^{sim(A^{(i)}, A^{(j)})} + \\sum _{j=1}^{k} e^{sim(A^{(i)}, P^{(j)})}},$ where $sim(A, P)$ denotes the pairwise cosine similarity between two feature vectors as shown in Eq.", "REF .", "$sim(A^{(i)}, P^{(i)})$ calculates the feature similarity of positive pairs, while all other feature similarity calculations are for negative pairs.", "The optimization of contrastive loss pulls the positive feature pairs closer and pushes the negative feature pairs further in the universe space.", "$sim(A, P) = A^\\top P/ (\\tau \\Vert A \\Vert \\Vert P\\Vert ),$ where $\\tau $ denotes a temperature parameter.", "When considering the anchor features as the positive corresponding to original positive features, we can calculate the contrastive loss for the positive-anchor batch $l_{pa}$ as $l_{pa} = -\\frac{1}{k} \\sum _{i=1}^{k} \\log \\frac{e^{sim(P^{(i)}, A^{(i)})}}{\\sum _{j = 1, j \\ne i}^{k} e^{sim(P^{(i)}, P^{(j)})} + \\sum _{j=1}^{k} e^{sim(P^{(i)}, A^{(j)})}}.$ Therefore, the complete contrastive loss for the anchor-positive combination is $L_{AP} = l_{ap} + l_{pa}.$ Let $\\mathcal {D} = \\lbrace X^{(1)}, ..., X^{(N)}\\rbrace $ denotes training data with $N$ specific objects.", "The $i$ -th input sample $X^{(i)} = \\lbrace p^{(i)}, m^{(i)}, img_{1}^{(i)}, img_{2}^{(i)}\\rbrace $ , where $m^{(i)}$ and $p^{(i)}$ represent the 3D mesh object and the corresponding sampled point cloud, $img_{1}^{(i)}$ and $img_{2}^{(i)}$ are two images under different randomly rendered views generated from the same 3D mesh object.", "Passing $X^{(i)}$ through the framework, we can obtain corresponding features: $f_p^{(i)}$ , $f_m^{(i)}$ , $f_{i1}^{(i)}$ , and $f_{i2}^{(i)}$ .", "Given a training minibatch $\\lbrace X^{(i)}\\rbrace _{i=1}^{k}$ of $k$ samples, the positive pairs are derived from same $X^{(i)}$ , while the negative pairs are sampled between $X^{(i)}$ and all other samples.", "In our proposed self-supervised learning schema, two types of feature learning are formulated as supervision signals for contrastive learning to optimize the networks: modal-invariant feature learning and view-invariant feature learning.", "Modal-invariant Feature Learning: The object identity information of 3D objects is utilized as the modality-invariant constraint to sample training pairs from different modalities.", "In this paper, the invariance from three pairs of different data modalities are constructed as training samples including: mesh-point (mesh and point cloud), mesh-image (mesh and image), and point-image (point cloud and image).", "Taking the mesh and point cloud pair as an example, in a training minibatch of $k$ samples, a batch of $k$ mesh features $\\lbrace f_m^{(1)}, ..., f_m^{(k)}\\rbrace $ and $k$ corresponding point cloud features $\\lbrace f_p^{(1)}, ..., f_p^{(k)}\\rbrace $ are considered as the anchor and positive feature batches alternately with each other.", "Then, following the equations REF , REF , and REF , the complete cross-modal contrastive loss for the mesh and point cloud pair is indicated as $L_{MP}$ .", "The cross-modal contrastive loss for the other two cross-modality pairs (mesh-image and point-image) are calculated in the same way indicated as $L_{MI}$ and $L_{PI}$ , respectively.", "View-invariant Feature Learning: In image modality, two randomly selected views for each object are used for view-invariant contrastive learning as shown in Fig.", "REF .", "With the same loss functions REF , REF , and REF , the cross-view contrastive loss $L_{II}$ is calculated with each image feature with $\\lbrace f_{i1}^{(1)}, ..., f_{i1}^{(k)}\\rbrace $ and $\\lbrace f_{i2}^{(1)}, ..., f_{i2}^{(k)}\\rbrace $ in the training minibatch.", "When jointly trained with the two constraints, a linear weighted combination of all loss functions is employed to optimize all the networks.", "The final loss to optimize the framework is as: $\\mathcal {L} = L_{MP} + L_{MI} + L_{PI} + L_{II}.$ The details of the joint training process are illustrated in Algorithm REF .", "After the jointly training finished, three networks $F_{img}$ , $F_{p}$ , and $F_{m}$ are obtained as pre-trained models for three different modalities.", "The joint training enables the three feature extractors to map the features from different modalities and views into the same universal space.", "[!t] The proposed self-supervised feature learning algorithm by contrastive learning cross multimodality and multiviews.", "minibatch size: $k$ ; 2D image features:$f_{i}$ ; 3D point cloud features: $f_{p}$ ; 3D mesh features: $f_{m}$ ; all sampled mini-batch $\\lbrace X^{(i)}\\rbrace _{i=1}^{k}$ # feature extraction for all $i\\in \\lbrace 1, \\ldots , k\\rbrace $ do $~~~~$ $f_{m}^{(i)} = F_{m\\_h}(F_{m}(m^{(i)}))$ ; $f_{p}^{(i)} = F_{p\\_h}(F_{p}(p^{(i)}))$ ; $~~~~$ $f_{i1}^{(i)} = F_{img\\_h}(F_{img}(img_{1}^{(i)}))$ ; $f_{i2}^{(i)} = F_{img\\_h}(F_{img}(img_{2}^{(i)}))$ ; end for # loss calculation under modality-invariant constraint $L_{MP} = l_{mp} + l_{pm}$ $L_{MI} = l_{mi} + l_{im}$ $L_{PI} = l_{pi} + l_{ip}$ # loss calculation under view-invariant constraint $L_{II} = l_{i1i2} + l_{i2i1}$ # final loss $\\mathcal {L} = L_{MP} + L_{MI} + L_{PI} + L_{II}$ update networks $F_{img}$ , $F_{img\\_h}$ , $F_{p}$ , $F_{p\\_h}$ , $F_{m}$ , and $F_{m\\_h}$ to minimize $\\mathcal {L}$ return pre-trained networks $F_{img}$ , $F_{p}$ , and $F_{m}$" ], [ "Framework Architecture", "As shown in Fig.", "REF , the MeshNet [9], dynamic graph convolutional neural network (DGCNN) [50], and ResNet [18] are employed as backbone networks to extract representation features from mesh, point cloud, and rendered images, respectively.", "$F_{m\\_h}$ , $F_{p\\_h}$ , $F_{img\\_h}$ are three corresponding two-layer fully connected projection heads to map the extracted representation features into an universe space for contrastive learning.", "The architecture of backbone networks is described as follows.", "MeshNet: The backbone architecture for mesh data is MeshNet, denoted as $F_{m}$ .", "MeshNet contains three main blocks: spatial descriptor, structural descriptor, and mesh convolution block.", "The spatial descriptor applies fully-connected layers (64, 64) to extract spatial features from face’s center.", "The structural descriptor contains a face rotate convolution within fully-connected layers (32, 32) and (64, 64), and a face kernel correlation with 64 kernels.", "Two mesh convolution blocks are used to aggregate features with neighboring information which the input/output channels of spatial and structural features are configured as (64, 131, 256, 256) and (256, 256, 512, 512), respectively.", "After the two mesh convolution blocks, a fully-connected layer (1024) further fuses the neighboring features and a max-pooling layer is employed to extract 512-dimension global features from the aggregated features.", "DGCNN: The 3D point cloud feature learning network ($F_{p}$ ) employs DGCNN as the backbone model due to its capability to model local structures of each point by dynamically constructed graphs.", "There are four EdgeConv layers and the number of kernels in each layer is 64, 64, 64, and 128, and the EdgeConv layers aim to construct graphs over $k$ nearest neighbors calculated by KNN and the features for each point are calculated by an MLP over all the $k$ closest points.", "After the four EdgeConv blocks, a 512-dimension fully connected layer is used to extract per-point features for each point and then a max-pooling layer is employed to extract global features for each object.", "ResNet: ResNet18 is employed as the image feature capture network ($F_{img}$ ) for 2D images.", "It contains four convolution blocks with a number of {64, 128, 256, and 512} kernels.", "Each convolution block includes two convolution layers followed by a batch-normalization layer and a ReLU layer, except the first convolution block which consists of one convolution layer, one batch-normalization layer, and one max-pooling layer.", "A global average pooling layer, after the fourth convolution blocks, is used to obtain the global features for each image.", "Unless specifically pointed out, a 512-dimensional vector after the global average pooling layer is used for all our experiments." ], [ "Experimental Setup", "Self-supervised learning: The proposed framework is jointly trained on ModelNet40 dataset using a SGD optimizer with an initial learning rate of $0.001$ , the moment of $0.9$ , and weight decay of $0.0005$ .", "The network is optimized with a mini-batch size of 96 for $160,000$ iterations and the learning rate decrease by $90\\%$ every $40,000$ iteration.", "Data augmentation used for point cloud network includes randomly rotated between [0, $2\\pi $ ] degrees along the up-axis, randomly jittered the position of each point by Gaussian noise with zero mean and $0.02$ standard deviation.", "Data augmentation for images include randomly cropped and randomly flipped with 50% probability.", "Data augmentation for mesh includes random rotation with a degree between [0, $2\\pi $ ].", "Datasets: Two 3D object benchmarks: ModelNet40 [52] and ShapeNet [5] are used to evaluate the proposed method.", "The ModelNet40 dataset contains about $12.3k$ meshed models covering 40 object classes, while about $9.8k$ are used for training and about $2.5k$ for testing.", "The ShapeNet dataset contains 16 object categories with about $12.1k$ models for training and about $2.9k$ for testing.", "Training data generation: The point cloud data and multi-view image data are sampled and rendered from same 3D mesh objects, respectively.", "Specifically, following [37], the point cloud set is sampled from surfaces of mesh objects by Farthest Point Sampling (FPS) algorithm.", "For each object, uniform $2,048$ points are sampled and normalized into a unit sphere to keep the object shape as much as possible.", "Same as in [22], we employ Phong reflection model [36] as the rendering engine to render images from 180 virtual cameras (viewpoints) to capture perspective of mesh objects as comprehensive as possible.", "All virtual cameras are randomly placed along a sphere surface pointing toward the centroid of mesh objects, and one image is rendered form each camera.", "Note that two of the rendered images are randomly selected for each input training sample.", "Evaluation of learned 2D and 3D features: The effectiveness of the self-supervised pre-trained backbone networks $F_{img}$ , $F_{p}$ , and $F_{m}$ are validated on different downstream tasks for 3D objects including object recognition, few-shot recognition, part segmentation, in-domain and cross-modal retrieval.", "The features, from the universal space, extracted by the three backbones and corresponding projection heads ($F_{img\\_h}$ , $F_{p\\_h}$ , $F_{m\\_h}$ ) are used for the cross-modal retrieval task." ], [ "Transfer to Object Recognition Tasks", "The proposed framework can jointly learn features for data with different modalities and views.", "We validate the effectiveness of the self-supervised pre-trained $F_{img}$ , $F_{p}$ , and $F_{m}$ on three down-stream supervised tasks: image recognition, point cloud object recognition, and mesh object recognition on the ModelNet40 dataset.", "Specifically, three linear SVM classifiers are trained based on the extracted features by $F_{img}$ , $F_{p}$ , and $F_{m}$ , respectively.", "The performance of the SVMS on the testing splits of ModelNet40 dataset are reported and compared.", "Each feature for the 2D classifier is average from $v$ extracted features of $v$ random views.", "Table: The performance of object recognition tasks by using self-supervised learned models as feature extractors on the ModelNet40 dataset.", "\"Views\" indicates how many views of images are used to obtain the image features.As shown in Table REF , the self-supervised pre-trained networks $F_{img}$ , $F_{p}$ , and $F_{m}$ obtain high accuracy (almost 90%) on object recognition tasks with different modalities, showing that the discriminative semantic features are indeed learned through the self-supervised learning process.", "For the image network, when only one view is available, the performance is only 78%, and the performance is significantly boosted when more views are available.", "Given enough image views, the three networks for the three modalities obtained comparable performance showing that the proposed framework learn robust features for all the modalities." ], [ "Transfer to Few-shot Object Recognition Task", "To further evaluate the generalization ability of the learned features, we also evaluate the self-supervised pre-trained $F_{img}$ , $F_{p}$ , and $F_{m}$ on 2D/3D few-shot object recognition tasks and showing the performance in Table REF on ModelNet40 dataset.", "Similar to the settings in Section REF , three corresponding linear SVM classifiers are trained based on the features of 5, 10, and 20 labeled data for each object category, and the image features for 2D recognition task are generated by max-pooling.", "Table: The performance of few-shot object recognition tasks of the features learned by the proposed self-supervised learning method on ModelNet40 dataset.", "\"S-#\\#\" indicates the number of shots for each class.As shown in Table REF , even only a few labeled data are available for each class, the networks on mesh and point cloud obtain relative high performance for few-shot learning.", "When only one view of images available, the performance for object recognition with images are much lower than the other two modalities.", "The performance of object recognition with images is significantly boosted up when more views are available.", "Overall, the features from mesh modality are more robust than the other two modalities." ], [ "Transfer to 3D Part Segmentation Task", "For a more thorough effectiveness validation of the learned features across different tasks, we further conduct transfer learning experiments on 3D part segmentation task on the ShapeNet point cloud dataset with a few labeled data available.", "Since this data only contains labels for the point cloud data, only $F_{p}$ is evaluated on part segmentation task.", "Four fully connected layers are added on the top of $F_{p}$ , and the outputs from all the four EdgeConv blocks and the global features are used to predict the pixel-wise labels.", "We vary the amount of training data on three experimental setups: (1) with random initialization and supervised training from scratch by the same network [37], (2) updating parameters on four newly added layers with frozen $F_p$ , and (3) fine-tuning parameters with the pre-trained $F_p$ (unfrozen).", "The performance is shown in Table REF .", "Table: The performance of the three types of settings on different amount of training data from the ShapeNet dataset for object part segmentation task.As shown in Table REF , for both $F_{p}$ -Frozen and $F_{p}$ -Unfrozen, the performance of 3D part segmentation can be boosted up in overall accuracy, class mean IOU, and instance mean IOU.", "Specifically, when only 1% labeled data available, the parameter-frozen setup can significantly ($+4\\%$ on class mIOU, and $+2.9\\%$ on instance mIOU) and increases the performance than training from scratch.", "It validates that $F_{p}$ is able to learn semantic features from modality-invariant constraints and transfer them across datasets and tasks.", "As more data are available for training, the overall performance of the network keeps improving and the network significantly benefits from the learned weights.", "These results suggest that the proposed pretext task lead to learn strong features that are able to be generalized to other tasks." ], [ "Transfer to Retrieval and Cross-modal Retrieval Tasks", "Compared to all other self-supervised learning models for 3D objects, our method learns modal- and view-invariant features which makes the features of different data modalities be directly comparable.", "To evaluate the quality of the modal- and view-invariant features, we propose to evaluate them on in-domain and cross-domain retrieval tasks.", "The performance on the cross-domain retrieval task can show generalizability of the modal-invariance while the performance on the in-domain retrieval task can show generalizability of view-invariance of the learned features.", "The features for different modalities are extracted by the self-supervised pre-trained backbone networks ($F_{img\\_h}$ , $F_{p\\_h}$ , $F_{m\\_h}$ ), and then followed by L1 normalization.", "The Euclidean distance of features is employed to indicate the similarity of two features.", "All the experiments for in-domain retrieval and cross-domain retrieval tasks are performed on ModelNet40 dataset.", "Table: Performance of in-domain retrieval tasks with the learned mesh, point cloud, and image features on ModelNet40 dataset.", "Results of XMV  are reproduced.", "The networks with * ^* are pre-trained on ImageNet dataset.In-domain Retrieval: The performance of three retrieval tasks (image-to-image, point-to-point, and mesh-to-mesh) are shown in Table REF .", "For a fair comparison, the networks used in each domain are with the same architecture and trained on the same dataset, except ResNet18 [18] which is pre-trained on ImageNet dataset.", "The results from XMV [22] are tested by the released pre-trained model.", "For all the in-domain retrieval tasks, our network achieves relatively high performance and significantly outperforms the recent state-of-the-art self-supervised learning models and the ImageNet pre-trained model.", "Even when only 1 image view for each object is available, our model achieves 57.9% mAP for the in-domain retrieval showing that our model indeed learns view-invariant features.", "When 4 views of images for each object is available, the performance of in-domain retrieval task improves by $4.4$ % on ModelNet40 dataset.", "Table: Performance of cross-modal retrieval tasks with the learned images, point cloud, and mesh features on ModelNet40 dataset.Cross-modal retrieval: The jointly learned modal-invariant features in the universal feature space for three different data modalities make the cross-modal retrieval for 3D objects possible, which is, as far as we know, not explored by any other self-supervised or supervised methods.", "The cross-modal retrieval task aims to match input data from one modality to different representations from another modality.", "Here as shown in the Table REF , the feature representation abilities for different modalities are extensively evaluated by six cross-modal retrieval tasks (image-to-point, point-to-image, image-to-mesh, mesh-to-image, mesh-to-point, and point-to-mesh).", "Our models achieve relatively high performance on all the six different cross-modal combinations showing that the network indeed learns the modal-invariant features.", "In general, the retrieval accuracy between mesh and point cloud modalities is better than image-involved retrieval due to the inputs of mesh and point cloud contain overall structure information of 3D objects, while a few input views (one or two) are insufficient.", "When more views (four) are available, the image-involved retrieval accuracy is basically the same.", "The qualitative visualization results of the top-10 ranking lists for six query samples on ModelNet40 dataset are shown in Fig.", "REF .", "Only one view for each object is used as query or galleries.", "For objects with unique structures like the airplane, guitar, and car, our models achieves high precision for these classes." ], [ "Comparison with the State-of-the-art Methods", "Object recognition with 2D multi-view images.", "The performance of our self-supervised pre-trained $F_{img}$ and the state-of-the-art image-based methods on the ModelNet40 benchmark is shown in Table REF .", "Methods with multi-view inputs are compared, including hand-crafted model [39] and supervised feature learning models [8], [45], [22].", "The setups of our models are same as in subsection REF .", "Note that DeCAF [8] and MVCNN [45] demand large-scale labeled data ImageNet1K for pre-training.", "XMV [22] is pre-trained with two types of modalities (image and point cloud), and the learned features are not modality-invariant.", "When using same number of views, the performance of our model $F_{img}$ consistently outperforms the state-of-the-art self-supervised learning method [22] and obtained comparable performance with the supervised methods [8], [45].", "Table: The comparison with the state-of-the-art methods for 3D point cloud object recognition on ModelNet40 dataset.", "* indicates the results in based on mesh modality.3D object recognition with point cloud and mesh.", "Table REF compares the proposed self-supervised pre-trained modelS $F_{p}$ and $F_{m}$ against both self-supervised learning methods [1], [6], [10], [12], [16], [22], [25], [42], [51], [54], [57] and supervised learning methods [9], [10], [21], [26], [29], [38], [43], [49], [50] on the ModelNet40 benchmark.", "Our self-supervised learning approach achieves comparable performance to the supervised methods on the ModelNet40 dataset.", "The performance of our model are very close ($0.5\\%$ lower) to previous self-supervised learning methods, while the learned modal-invariant features are applicable for more downstream tasks such as retrieval task.", "Worth to note that most of the self-supervised learning only learn features for point cloud data, while our method is able to learn modal- and view-invariant features for different modalities, and it can be easily extend to other modalities." ], [ "Conclusion", "In this paper, we have proposed a novel self-supervised learning method to jointly learn features which are invariant to different modalities and views.", "Different from all the previous self-supervised learning methods, our method is able to learn features for different modalities in the same universal space which makes it possible to explore a new task, i.e., cross-modal 3D object retrieval.", "The image features, mesh features, and point cloud features learned by three different networks have been extensively tested on different tasks including 3D object recognition, few-shot learning, part segmentation, 3D objects retrieval and cross-modal retrieval, and demonstrated the strong generalizability of the learned features." ], [ "Acknowledgement", "This material is partially based upon the work supported by National Science Foundation (NSF) under award number IIS-1400802.", "Figure: Top-10 ranking lists for six query samples on ModelNet40 dataset by our models.", "The results with green boundaries belong to the same category as the query, and images with red borders do not." ] ]

[ [ "Relation Extraction with Explanation" ], [ "Abstract Recent neural models for relation extraction with distant supervision alleviate the impact of irrelevant sentences in a bag by learning importance weights for the sentences.", "Efforts thus far have focused on improving extraction accuracy but little is known about their explainability.", "In this work we annotate a test set with ground-truth sentence-level explanations to evaluate the quality of explanations afforded by the relation extraction models.", "We demonstrate that replacing the entity mentions in the sentences with their fine-grained entity types not only enhances extraction accuracy but also improves explanation.", "We also propose to automatically generate \"distractor\" sentences to augment the bags and train the model to ignore the distractors.", "Evaluations on the widely used FB-NYT dataset show that our methods achieve new state-of-the-art accuracy while improving model explainability." ], [ "Introduction", "Relation extraction with distant supervision associates a pair of entities with a bag of sentences, each containing mentions of both entities.", "The bag is tagged with relations between the pair in a Knowledge Base (KB), without explicitly indicating which sentence(s) support the relation(s).", "This method avoids the burden of manual annotations, but presents inherent ambiguity, creating challenges for learning.", "To alleviate the impact of the irrelevant sentences many approaches have been proposed including models based on attention [29], [13], [15], [17], [6], [27], [19], [3], approaches that use additional resources [26], [16] and methods that utilize supervision data [20], [2], [5].", "These studies primarily focus on improving relation extraction accuracy and little is known about whether the models are making right decision for the right reason or because of some irrelevant biases [1], [9], [7].", "This paper examines two strong baseline relation extraction models with several explanation mechanisms.", "We manually annotated a test set from the widely used FB-NYT dataset with ground truth explanations to evaluate the quality of the explanation afforded by these models.", "We also introduce two different methods for improving relation extraction.", "First, we demonstrate that replacing the entity mentions with their fine-grained entity types for sentence representation leads to improvement in both the extract accuracy and model explainability.", "Second, we augment the bags with automatically generated “distractor” sentences (i.e., sentences that contain no supporting information for the relation) and train the model to appropriately ignore the irrelevant information.", "Our evaluation on the widely used FB-NYT dataset verifies that the proposed methods achieve the new state of the art for the extraction performance along with improved model explainability." ], [ "Problem Setup", "Given entity pair $(e_i, e_j)$ , we form a bag $B_{i,j}=\\lbrace s_1, \\dots s_{N_{ij}}\\rbrace $ with $N_{ij}$ sentences that contain mentions of both entities and label it by the set of relations between $e_i$ and $e_j$ from the KB.", "Neural models for relation extraction encode each sentences into a vector representation and a bag $B_{i,j}$ is thus represented by $\\lbrace x_1, \\dots x_{N_{ij}}\\rbrace $ where $x_i \\in \\mathbb {R}^{d}$ .", "Given a set of bags and the associated labels, the training objective is to learn a model that predicts the probability $P(r=k|B_{i, j})$ that relation $k$ exists between $e_i$ and $e_j$ based on $B_{i,j}$ , where $k\\in 1\\dots K$ and $K$ is the total number of relations in the KB.", "There are zero to multiple possible relation labels for each bag.", "Importantly, only some sentences in the bag express any of the relations and the others are irrelevant (provide no information regarding the relations), but such sentences are not labeled." ], [ "Baseline Models", "We consider two baselines.", "The first is DirectSup, a recent model achieving the state-of-the-art performance by utilizing auxiliary supervision [5].", "The second baseline (CNNs+ATT) revamps the classic attention based method by [13] but adopts the same sentence encoder as DirectSup for ease of comparisons.", "In this work, we add a ReLU at the end of the sentence encoder [5] to produce positive sentence representations.", "See [5] for detailed information regarding the sentence encoder.", "DirectSup.", "Given a bag of sentences, DirectSup encodes each sentence using CNNs with different filter sizes.", "The outputs of the CNNs with different filter sizes are concatenated to produce the encoding of the sentence.", "Given a bag $B$ and the encoding of its sentences $\\lbrace x_1, x_2,...,x_{N}\\rbrace $ , DirectSup assigns an importance weight for each sentence based on the output of a binary classifier learned from an additional direct supervision data in a multi-task manner.", "Given a sentence encoding $x_n$ , the binary classifier provides a weight $\\alpha _n \\in [0, 1]$ indicating the likelihood that $x_n$ expresses some form of relations in the KB.", "As a result, for a bag $B_{i,j}$ , we have importance weights $\\lbrace \\alpha _1, \\dots , \\alpha _N\\rbrace $ .", "It then produces a single bag representation as follows: $\\bar{x} = \\mbox{Max-pool}(\\lbrace \\alpha _1 x_1, \\dots , \\alpha _n x_{N}\\rbrace )$ and the prediction for relation $k$ is given by: $P(r=k|B) = \\sigma (\\bar{x} r_k + b_k)$ where $r_k$ is an embedding of relation $k$ , $b_k$ is a bias variable and $\\sigma $ is the Sigmoid function.", "CNNs+ATT.", "This model uses the same sentence encoder as DirectSup but differs in the attention mechanism used to decide sentence importance.", "Specifically, it follows [13] and computes the importance weights of the sentences in bag $B$ with encodings $\\lbrace x_1, \\dots , x_N\\rbrace $ as follows: $\\alpha _{k,n} = \\frac{\\mbox{exp}(x_n A q_k)}{\\sum _{i=1}^{N}\\mbox{exp}(x_i A q_k)}$ where $q_k$ is a learned query vector associated with relation $k$ and $A$ is a diagonal matrix.", "Given $\\lbrace \\alpha _{k,1}, ..., \\alpha _{k,N}\\rbrace $ , we compute a bag representation specific for relation $k$ by: $\\bar{x}_k = \\sum _{n=1}^{N}{\\alpha _{k,n} x_n}$ and the prediction for relation $k$ is given by: $P(r=k|B) = \\sigma (\\bar{x}_k r_k + b_k)$ where $r_k$ is relation $k$ 's embedding and $b_k$ is the bias.", "Entity embedding.", "Prior work has demonstrated that incorporating entity embeddings into the relation extraction model leads to improved accuracy [11], [5].", "Here we also consider this strategy with the baseline models.", "Specifically, let $v_i$ and $v_j$ be the entity embedding of $e_i$ and $e_j$ , we concatenate the bag representations $\\bar{x}$ with $v_i-v_j$ and $v_i\\circ v_j$ , where $\\circ $ is element-wise product.", "We then apply a linear project layer with ReLU to produce a new bag representation for final prediction with Eq.", "REF and REF .", "For any entity $e_i$ its embedding vector $v_i$ is obtained by concatenating the average of its skip-gram [18] word embeddings and the embeddings produced by [30] (produced by using TransE on Wikipedia factual tuples).", "Training objective.", "For all the models in this work we use the binary cross entropy loss function for training: $\\begin{aligned}l = -\\sum _{B_{i,j}}\\sum _{k=1}^{K} \\mathbb {1}_{i,j,k} \\hspace{2.84526pt} \\mbox{log} \\hspace{2.84526pt} P(r=k|B_{i,j}) +\\\\(1-\\mathbb {1}_{i,j,k}) \\hspace{2.84526pt} \\mbox{log} \\hspace{2.84526pt} (1-P(r=k|B_{i,j}))\\end{aligned}$ where $\\mathbb {1}_{i,j,k}$ is an indicator function that takes value 1 if relation $k$ exists for bag $B_{i,j}$ ." ], [ "Explanation Mechanisms", "The importance weights ($\\alpha $ 's, aka attention), generated by the models can be interpreted as explanations.", "However, recent studies [8], [10], [28] have questioned the validity of attention as a faithful explanation of model's behavior.", "Thus we consider the following additional explanation mechanisms: Saliency.", "Recent works show that a model's prediction can be explained by examining the input saliency, based on the gradient of the output w.r.t.", "the inputs [25], [22], [7].", "We define the saliency of sentence $n$ for relation $k$ , denoted by $S_{x_n,k}$ , as the L1 norm of the gradient of relation $k$ logit $o_k$ with respect to $x_n$ .", "(Appendix.", "REF ).", "Gradient $\\times $ input.", "This is a commonly used measure for input attributions [24], [23].", "We will refer to this measure as $GI_{x_n,k}$ , computed as $\\sum _i x_n[i] \\times \\frac{\\partial o_k}{\\partial x_n}[i]$ .", "Leave One Out (loo).", "This measures the sensitivity of $o_k$ to the removal of a sentence.", "We refer to this measure as $loo_{x_n, k} = (o_k-o_{k, -n})$ , where $o_{k, -n}$ is the new logit of relation $k$ after removing sentence $x_n$ from its bag." ], [ "Proposed Methods", "We propose two different approaches for improving relation extraction.", "The first method we propose, introduces a subtle change to the representation of the sentences, which lead to higher performance and better explanation quality.", "We further propose to automatically generate “distractor” sentences and train the model to appropriately ignore them.", "Sentence representation.", "Each sentence in a bag contains entity mentions $m_i$ and $m_j$ for entities $e_i$ and $e_j$ respectively.", "In prior work $m_i$ and $m_j$ are kept unchanged [13], [5].", "We argue that when entity mentions are used to compute the sentence representation, they provide such rich information that the model may not need to look at the rest of the sentence to deduce a relation.", "To ensure that our predictions are supported by appropriate sentences, we need to remove this effect.", "We propose to replace the entity mentions with their Fine-Grained Entity Types (FGET) [14] to force the model to identify the relations through the sentences.", "Learning from distractors.", "Prior work studied learning from human provided rationales [12], [22], [4], [7] in order to improve model explainability.", "However, human rationales are expensive to acquire.", "In this work we propose to learn from automatically generated “distractor” sentences.", "Let $B_{i, j}$ be a positive training bag (contains at least one relation) with entities $(e_i, e_j)$ of FGET $(t_i, t_j)$ .", "Let $R_{ij} (|R_{ij}|> 1)$ be the set of annotated relations for $B_{i, j}$ .", "For each $k$ in $R_{ij}$ , we sample a “distractor” sentence $s^{\\prime }_k$ from the set of sentences in the training set such that 1) it belongs to a bag whose FGET is $(t_i, t_j)$ 2) the bag is not annotated with relation label $k$ .", "If $s^{\\prime }_k$ is not found this way, we simply choose a random sentence from a random negative bag (bag with no relation).", "Given $s^{\\prime }_k$ , we replace its entity mentions with $e_i$ and $e_j$ (or $t_i$ and $t_j$ for FGET-based sentence representation) of a sentence in $B_{i, j}$ and add it to the bag, resulting in an augmented bag $B^{\\prime }_{i,j}$ for relation $k$ .", "To learn from the augmented bags, we feed $B^{\\prime }_{i,j}$ into the model and the goal is to lower the contribution of the distractor sentence in relation to the original sentences in the bag.", "Specifically, we use $GI$ to measure the sentence-level contribution and define the distractor loss for relation $k$ as follows: $\\begin{aligned}l^{\\prime }_{d,k} = \\mbox{max}(0, \\gamma + GI_{x^{\\prime }_k, k} - \\underset{x \\in B_{i, j}}{\\mbox{max}} GI_{x, k}) \\\\ + |GI_{x^{\\prime }_k, k}|\\end{aligned}$ where $x^{\\prime }_k$ is the encoding of distractor sentence $s^{\\prime }_k$ and $\\gamma $ is a hyper-parameter for margin.", "The first term ensures that the contribution of the distractor is lower than the maximum contribution of all the sentences in the original bag and the second term reduces the absolute contribution of the distractor.", "Although we use $GI$ in Eq.REF , other explanation measures such as saliency or the positive portion of the contributions can also be applied here.", "Moreover a more advanced mechanism for generating distractors will likely lead to a higher performance.", "We hence update the loss in Eq.", "REF with: $l_m = l + \\lambda l^{\\prime }_d$ where $l^{\\prime }_d=\\sum _k l^{\\prime }_{d,k}$ and $\\lambda $ tradeoffs the regular learning loss with the distractor loss.", "Figure: PR with entity" ], [ "Experiments", "In this section, we empirically evaluate our proposed methods both in terms of their relation extraction performance and their explainability." ], [ "Dataset and Setup", "Dataset.", "Similar to our baselines and prior work, we use the modified version of FB-NYT dataset.", "The original FB-NYT dataset was built by [21] on New York Times articles which was aligned to Freebase facts.", "It later was modified by [13].", "There are 52 relations in this dataset where “place lived”, “captial”, “neighborhood of”, “natinality” and “location” are the most frequent relations.", "Tab.", "REF shows the size of the modified dataset.", "Table: FB-NYT modified dataset.Setup and Training.", "All models are implemented in PyTorch, trained with a Adam optimizer with learning rate 0.001 for a maximum of 30 epochs.", "We use 300-d skip-gram [18] word embeddings and FGET embeddings and 5-d position embedding.", "During training we freeze the word and entity embeddings.", "All reported results are averaged over three different random runs.", "We train on 90% of the training set and keep the remaining 10% for validation.", "We select $\\lambda $ from the set $\\lbrace 0.01, 0.1, 1.0, 10.0, 100.0\\rbrace $ and set $\\lambda =1.0$ based on validation AUC and the margin is fixed at $\\gamma =0.00001$ .", "Ground-truth explanations.", "There are 1950 positive bags (6444 sentences) in the test split of FB-NYT.", "For each pair of sentence-relation in a bag we annotate whether the sentence entails the relation or not.", "Based on the annotations, we extract a set called expl-eval (see Appendix REF for details) including tuples of (bag-id, relation, positive sentence in bag, negative sentence in bag).", "Each tuple provides a desired ordering of two sentences when measuring their importance to the model.", "expl-eval is then used to compute the Kendall Tau correlation between the annotation and the explanations, which measures how consistently the importance weights ranks the sentences compared to the ground truth." ], [ "Relation Extraction Performance", "Similar to prior work we use precision-recall (PR) curves to characterize the extraction performance and report the area under the PR curve (AUC) up to 0.4 recall.", "Tab.", "REF reports the AUCs of the baselines and different variants of our proposed models with (+E) and without (-E) incorporating entity embeddings.", "Specifically, we consider two different ways of incorporating the FGET representations.", "Rows 3-4 show the AUCs of the two baseline models when we replace entity mentions with their FGET (+F), whereas rows 5-6 show the AUCs when we concatenate the FGET with the entity mentions (+FE).", "From the results we can see that both baselines see clear performance gain from incorporating FGET into the representations.", "Combining FGET with entity mention (+FE) achieves higher performance than using only FGET (+F), but our hypothesis is that the former will lead to less explainable models, which we will examine in the next section.", "Finally the last three rows of the table show that adding LD to different base models can further improve the AUCs.", "Similar to prior work, we observe that incorporating entity embeddings(+E) to the model leads to substantial performance gain across the board.", "We also observe very similar performance gain when adding FGET and LD to the base models both with and without entity embeddings.", "Our best model achieved an AUC of 0.341, which improves the previous state-of-the-art by 5.7%." ], [ "Evaluation of Explanations", "We apply the explanation mechanisms described in Section  to produce sentence importance scores for the test set and compute the Kendall Tau correlations for the importance scores using expl-eval.", "For each model, to understand its behavior when it predicts correctly versus incorrectly, we consider the subset $H$ ($L$ ) of bags/relations that the model outputs high (low) probability, i.e., $p\\in [0.76,1]$ ($[0,0.25]$ ), for the correct relation.", "We report the performance on $H$ and $L$ separately in Tab.", "REF .", "Comparing correlation values for $H$ and $L$ in Tab.", "REF , we observe that when the models are making correct and confident predictions ($H$ ), the values of correlation tend to be higher.", "In contrast, when the model fails to detect the correct relation ($L$ ), we see substantially lower correlation scores.", "By replacing entity mentions with their FGET in both CNNs+ATT and DirectSup (+F), we observe substantially increased correlation scores for correct predictions (H).", "The improvement is consistent across all methods that are used to compute the importance scores.", "Recall that Tab.", "REF shows that concatenating FGET with entity mention (+FE) yields improved relation extraction performance for both CNNs+ATT and DirectSup.", "In contrast, the explanation results presented here show that this comes at the cost of explainability, as demonstrated by the substantially lower correlation scores of CNNs+ATT+FE and DirectSup+FE.", "This confirms our conjecture that removing entity mentions from the sentence representation leads to more explainable models, possibly by forcing the model to focus on the textual evidence contained in the sentence rather than the word embedding of the mentions.", "Finally, we note that adding LD further improves the correlation score on $H$ for $S$ , $GI$ and $\\alpha $ .", "This suggests that learning from distractors is a valuable strategy that not only produces better relation extraction performance, but also enhances the model explanability." ], [ "Conclusion", "In this work we provided an annotated test set with ground-truth sentence-level explanations to evaluate the explanation quality of relation extraction models with distant supervision.", "Our examination of two baselines show that a model with lower relation extraction accuracy could have higher explanation quality.", "We proposed methods to improve both the accuracy and explainability.", "Our proposed methods are based on changing the representation of the sentences and learning from distractor to teach the model to ignore irrelevant information in a bag.", "Our evaluation on the widely used FB-NYT dataset show the effectiveness of our method in achieving state-of-the art performance in both accuracy and explanation quality." ], [ "Saliency and (Gradient $\\times $ input)", "Assume that a neural model outputs a logit score $o$ which is a differentiable function and parameterized by $x \\in \\mathbb {R}^{d}$ , $\\theta $ and etc.", "The Taylor series of the given function $o$ near input $a$ is given by: $o(x) = o(a) + \\frac{\\partial o}{\\partial x}(a)(x-a) + \\frac{1}{2!}", "\\frac{{\\partial o}^{2}}{\\partial x^{2}}(a)(x-a)^{2} + \\dots $ Approximating the function $o$ as a linear function, the first order approximation of the Taylor series is given by: $o(x) \\approx \\frac{\\partial o}{\\partial x}(a)x + b$ Note that $\\frac{\\partial o}{\\partial x}(a) \\in \\mathbb {R}^{d}$ .", "Therefore for each dimension $i$ the bigger $\\frac{\\partial o}{\\partial x}(a)[i]$ , the more (positive or negative) the impact of $a[i]$ is on $o$ .", "The whole impact of $a$ on $o$ is given by $\\sum _{i}\\frac{\\partial o}{\\partial x}(a)[i]$ of its absolute value $\\sum _{i}|\\frac{\\partial o}{\\partial x}(a)[i]|$ .", "Regarding our task, the logit score of the model for a relation $k$ is $o_k$ .", "For a given sentence $x_n$ , the amount of positive or negative impact of $x_n$ on $o_k$ is approximated by $\\sum _{i}|\\frac{\\partial o_k}{\\partial x}(x_n)[i]|$ which is saliency.", "The (Gradient $\\times $ input) for a given sentence $x_n$ is equivalent to the linear approximation of $o_k$ at $x_n$ which is $\\sum _{i} x_n[i] \\times \\frac{\\partial o_k}{\\partial x}(x_n)[i]$ ." ], [ "Ground-truth explanation set.", "We annotate the positive bags of the test split of FB-NYT with ground-truth explanations.", "There are 1950 bags and 6444 sentences.", "For each pair of (sentence, relation) in a bag, the sentence is either a rationale (supportive) to the relation or it is irrelevant.", "For example: Table: NO_CAPTIONFollowing the annotation of the sentence-relation contributions which is either rationale or irrelevant, we extract a set “expl-eval” (which is going to be used to evaluate the explanation quality of the models) as follows: expl-eval = set() For each (bag-id, bag):  For each relation label (*$k$*) given to the bag:     For each pair of rationale (*$s^{+}$*) an irrelevant (*$s^{-}$*)        for (*$k$*):             expl-eval.add((bag-id, k, (*$s^{+}$*), (*$s^{-}$*)))   The size of the generated expl-eval is 1097 tuples of (bag-id, $k$ , rationale sentence, irrelevant sentence).", "Please note that the relation label $k$ is one of the ground-truth labels assigned to bag-id." ] ]

[ [ "Hyperbolic quotients of projection complexes" ], [ "Abstract This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes.", "In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups.", "In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex.", "We show that under certain conditions that the quotient complex is $\\delta$-hyperbolic.", "Additionally, under certain circumstances, we show that if the original action on the projection complex was a non-elementary WPD action, then so is the action of the quotient group on the quotient of the projection complex.", "This implies that the quotient group is acylindrically hyperbolic." ], [ "Introduction", "Projection complexes were originally defined by Bestvina–Bromberg–Fujiwara and were used to show that the mapping class group of an orientable surface has finite asymptotic dimension [3].", "The motivating idea behind these complexes is the following.", "Start with a collection of subspaces $\\lbrace Z_i\\rbrace $ contained in some metric space $X$ .", "We want these subspaces to satisfy some properties akin to negative curvature; in particular, we require that the nearest point projection from any one subset $Z_i$ to another subset $Z_j$ has uniformly bounded diameter.", "For example, one could take $X$ to be the hyperbolic plane and the collection $\\lbrace Z_i\\rbrace $ to be the orbit of a geodesic in $X$ under a discrete group of isometries of $X$ .", "The projection complex built out of this data is the graph with vertex set $\\lbrace Z_i\\rbrace $ where two vertices $Z_i$ and $Z_j$ are joined by an edge if the diameter of the union of their projections to any other $Z_k$ is small.", "A key feature of a projection complex is that, in general, it is a quasi-tree, in other words, it is quasi-isometric to a tree [3].", "Projection complexes have found several useful applications lately by many authors [10], [9], [4], [2], [11], [12], [13].", "In previous work with Margalit, we studied group actions on projection complexes [8].", "We derived a structure theorem for normal subgroups generated by elliptic elements under some hypotheses; see Section REF for the exact statement.", "We were able to apply our structure theorem to produce new examples of normal subgroups of mapping class groups of orientable surfaces that are isomorphic to right-angled Artin groups.", "In particular, we produced examples that were not free.", "In this paper, we work in the general setting of a group acting on a projection complex with the same set of hypotheses as before and study both the quotients of the projection complex by such normal subgroups and the action of the quotient group on the corresponding quotient complex.", "These appear as Theorem REF and Theorem REF respectively.", "To state these results, we describe the set up we studied before and continue to study in this paper.", "Briefly, a projection complex is a graph $\\mathcal {P}$ and a collection of functions $d_v \\colon \\,V \\setminus \\lbrace v\\rbrace \\times V \\setminus \\lbrace v\\rbrace \\rightarrow \\mathbb {R}_{\\ge 0}$ where $V$ is the set of vertices of $\\mathcal {P}$ and $v \\in V$ .", "The full definition appears in Section REF .", "Following our previous work and as explained below, our definition is a mild modification of the original definition of Bestvina–Bromberg–Fujiwara.", "Let $\\mathcal {P}$ be a projection complex, and let $G$ be a group that acts on $\\mathcal {P}$ .", "Further, for each vertex $v$ of $\\mathcal {P}$ , let $R_v$ be a subgroup of the stabilizer of $v$ in $G$ .", "Let $L > 0$ .", "We say that the family of subgroups $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ if it satisfies the following two conditions: Equivariance: If $g$ lies in $G$ and $v$ is a vertex of $\\mathcal {P}$ then $gR_vg^{-1} = R_{gv}.$ Spinning: For any distinct vertices $v$ and $w$ of $\\mathcal {P}$ and any nontrivial $h \\in R_v$ we have $d_v(w,hw) \\ge L.$ By the equivariance condition, for each vertex $v$ the subgroup $R_v$ is normal in $\\mathrm {Stab}_G(v)$ , and the subgroup $H$ of $G$ generated by the $R_v$ is normal in $G$ .", "If $\\lbrace v_i\\rbrace $ is a set of orbit representatives for the action of $G$ on the vertices of $\\mathcal {P}$ , then $H$ is the normal closure of the set $\\left\\lbrace R_{v_i}\\right\\rbrace $ .", "We can now state our theorem regarding the quotient complex.", "Theorem 1.1 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group acting on $\\mathcal {P}$ .", "There exists a constant $L_{\\rm hyp}(\\mathcal {P})$ with the following property.", "If $L \\ge L_{\\rm hyp}(\\mathcal {P})$ and if $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ then $\\mathcal {P}/\\langle {R_v}\\rangle $ is $\\delta $ –hyperbolic.", "We also examine the action of the quotient group $G/\\langle {R_v}\\rangle $ on the quotient space $\\mathcal {P}/\\langle {R_v}\\rangle $ .", "Our result on the action briefly says that certain features of the action of $G$ on $\\mathcal {P}$ persist in the quotient action.", "Before we can state our result on this action, we need to define a number of notions.", "Let $X$ be a geodesic metric space and let $G$ be a group that acts on $X$ by isometries.", "An element $f$ of $G$ is hyperbolic if $ \\lim _{n \\rightarrow \\infty } \\frac{d(x,f^nx)}{n} $ is positive for some $x \\in X$ , equivalently, for any $x \\in X$ .", "Two hyperbolic elements, $f_1$ and $f_2$ , of $G$ are independent if $d(f_1^{n_1}x,f_2^{n_2}x) \\rightarrow \\infty $ as $n_1,n_2 \\rightarrow \\pm \\infty $ for some $x \\in X$ , equivalently, for any $x \\in X$ .", "An element $f$ of $G$ is a WPD element if $f$ is hyperbolic and if for all points $x \\in X$ and for all $D \\ge 0$ , there is an $M \\ge 0$ such that the set $ \\lbrace g \\in G \\mid d(x,gx) \\le D \\mbox{ and } d(f^M x, gf^Mx ) \\le D \\rbrace $ is finite.", "We remark that it suffices to demonstrate finiteness of the above set at a single point in $X$ .", "The notion of a WPD element was introduced by Bestvina–Fujiwara as a tool for constructing quasi-morphisms [6].", "There are several known examples of WPD elements: pseudo-Anosov mapping classes acting on the corresponding curve complex [6] and fully irreducible outer automorphisms of a free group acting on the corresponding free factor complex [5] for instance.", "If the action of $G$ on $X$ is properly discontinuous, then any hyperbolic element is a WPD element.", "The action of $G$ on $X$ is a non-elementary WPD action if there exist two elements in $G$ that are WPD elements and independent.", "We remark that if $X$ is $\\delta $ –hyperbolic, $G$ is not virtually cyclic and there is one element $f$ in $G$ that is a WPD element, then for some element $g$ in $G$ , the elements $f$ and $gfg^{-1}$ are independent WPD elements.", "In fact, one can take any $g \\in G$ such that $\\langle {f}\\rangle \\cap g\\langle {f}\\rangle g^{-1}$ is finite.", "We can now state our theorem on the action of the quotient group on the quotient complex.", "Theorem 1.2 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group with a non-elementary WPD action on $\\mathcal {P}$ .", "There exists a constant $L_{\\rm WPD}(\\mathcal {P},G)$ with the following property.", "If $L \\ge L_{\\rm WPD}(\\mathcal {P},G)$ and if $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ then the action of $G/\\langle {R_v}\\rangle $ on $\\mathcal {P}/\\langle {R_v}\\rangle $ is a non-elementary WPD action.", "Precisely, if $f_1$ and $f_2$ are independent WPD elements of $G$ for its action on $\\mathcal {P}$ , then there is a constant $L_{\\rm WPD}(\\mathcal {P},f_1,f_2)$ such that their images $\\bar{f}_1$ and $\\bar{f}_2$ in $G/\\langle {R_v}\\rangle $ are independent WPD elements for the action of $G/\\langle {R_v}\\rangle $ on $\\mathcal {P}/\\langle {R_v}\\rangle $ when $L \\ge L_{\\rm WPD}(\\mathcal {P},f_1,f_2)$ and $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Whereas the constant in Theorem REF does not depend on $G$ , the constant in Theorem REF necessarily does.", "Indeed, if $G$ is equal to $\\langle {R_v}\\rangle $ then the quotient group is trivial.", "Hence we must choose $L$ after $G$ —more precisely after choosing two independent WPD elements—to ensure that the quotient is as claimed.", "There is a strengthening of the WPD condition called acylindricity that arises in several settings that we describe now.", "Let $X$ be a metric space and let $G$ be a group acting on $X$ by isometries.", "The action is acylindrical if for all $D \\ge 0$ there exist $R \\ge 0$ and $N \\ge 0$ such that for all points $x$ and $y$ in $X$ where $d(x,y) \\ge R$ , the set $ \\lbrace g \\in G \\mid d(x,gx) \\le D \\mbox{ and } d(y,gy) \\le D \\rbrace $ contains at most $N$ elements.", "A group $G$ is acylindrically hyperbolic if it admits an acylindrical action on a hyperbolic space for which there exist elements $f_1$ and $f_2$ in $G$ that are hyperbolic and independent.", "Both the mapping class group of an orientable surface [7] and the outer automorphism group of a finitely generated free group are acylindrically hyperbolic [15].", "There are several other examples and much is known about this class of groups.", "The paper by Osin contains a survey of examples and results for acylindrically hyperbolic groups [15].", "Osin derived a number of conditions that are equivalent to acylindrical hyperbolicity, one of which is that the group is not virtually cyclic and admits an action on a $\\delta $ –hyperbolic space where one element is a WPD element [15].", "Hence we obtain the following corollary of Theorems REF and  REF .", "Corollary 1.3 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group with non-elementary WPD action on $\\mathcal {P}$ .", "There exists a constant $L_{\\rm WPD}(\\mathcal {P},G)$ with the following property.", "If $L \\ge L_{\\rm WPD}(\\mathcal {P},G)$ and if $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ then $G/\\langle {R_v}\\rangle $ is acylindrically hyperbolic.", "In Section  we describe new examples of acylindrically hyperbolic groups coming from this construction.", "These groups are quotients of the mapping class group of an orientable surfaces by the normal subgroups we produced in our previous work.", "The strategy to prove Theorem REF and Theorem REF is very similar to strategy of Dahmani–Hagen–Sisto in a recent paper [11].", "In this paper, Dahmani–Hagen–Sisto consider the action of the subgroup of the mapping class group generated by $k$ th powers of Dehn twists on the curve graph, i.e., the 1–skeleton of the curve complex and they prove results similar to Theorem REF and Theorem REF .", "They make use of the fact shown by Dahmani that the curve graph has the structure of a composite projection graph [9].", "That is, there is a partition of the curve graph into finitely many pieces that behave like a projection complex, along with certain combatility conditions on how the pieces interact.", "Since we deal with a projection complex as opposed to a composite projection graph, some parts of their strategy can be simplified.", "In order to prove Theorem REF , we show that geodesic triangles in $\\mathcal {P}/\\langle {R_v}\\rangle $ lift to geodesic triangles in $\\mathcal {P}$ (Proposition REF ).", "As $\\mathcal {P}$ is a quasi-tree, it is a $\\delta $ –hyperbolic metric space and hence geodesic triangles are $\\delta _0$ –thin for some $\\delta _0$ .", "As the quotient map $p \\colon \\,\\mathcal {P}\\rightarrow \\mathcal {P}/\\langle {R_v}\\rangle $ is 1–Lipschitz, this shows that the geodesic triangles in $\\mathcal {P}/\\langle {R_v}\\rangle $ are $\\delta _0$ –thin as well.", "The proof of Theorem REF is similar except that it involves lifting geodesic quadrilaterals.", "A key fact needed here is that a geodesic in $\\mathcal {P}$ for which the projection of any two of its vertices to any other vertex in $\\mathcal {P}$ is uniformly bounded is isometrically embedded in the quotient (Lemma REF ).", "A closed path in $\\mathcal {P}/\\langle {R_v}\\rangle $ can be lifted to a path in $\\mathcal {P}$ with endpoints $x$ and $hx$ for some $h \\in \\langle {R_v}\\rangle $ .", "We describe a technique called path bending for replacing the lifted path with a new lift.", "There is a notion of complexity for an element in $\\langle {R_v}\\rangle $ .", "We show that when $x \\ne hx$ , we can bend the given lift to get a lift from $x$ to $h^{\\prime }x$ where $h^{\\prime }$ has less complexity than that of $h$ (Proposition REF ).", "This is the technique known as shortening and it plays a key role in understanding both lifts (Proposition REF ) and images (Lemma REF ).", "This technique was introduced by Dahmani–Hagen–Sisto and is also essential to their work [11]." ], [ "Outline of Paper", "Section  collects the necessary facts on projection complexes that are needed for the remainder.", "Starting in Section , we follow the strategy of Dahmani–Hagen–Sisto [11].", "In Section , we prove the main technical tool of the paper, Proposition REF .", "This is the technique known as shortening and allows us to replace a lift of a path in the quotient of the projection complex with another lift that is simpler in a precise sense.", "We apply the shortening tool in Section  to show that geodesic quadrilaterals in the quotient of the projection complex lift to geodesic quadrilaterals.", "The proof of Theorem REF appears in Section .", "In Section , we show that when vertices along a geodesic in the projection complex have bounded projections, the image of the geodesic in the quotient graph is still a geodesic.", "Using this, we can establish that certain WPD elements for the action of $G$ on $\\mathcal {P}$ have images in $G/\\langle {R_v}\\rangle $ that are still WPD elements for the action of $G/\\langle {R_v}\\rangle $ on $\\mathcal {P}/\\langle {R_v}\\rangle $ .", "In Section , we prove Theorem REF .", "Finally, in Section  we present some examples when $G$ is the mapping class group of a surface." ], [ "Acknowledgments", "We would like to thank Alessandro Sisto for suggesting that the techniques of his paper with Dahmani and Hagen could apply in our setting as well.", "We are immensely grateful to Dan Margalit for initiating our projects on windmills in projection complexes and for ideas, questions, and conversations.", "We thank the anonymous referee for reading our paper carefully and for providing useful comments.", "The first author is partially supported by the Simons Foundation Grant No. 316383.", "The second author is supported by National Science Foundation Grant No.", "DMS–1812021." ], [ "Projection complexes, Windmills and Pivot Points", "In this section we provide the definitions of projection complexes, windmills and pivot points.", "The majority of the discussion in this section appears in our previous work with Margalit [8].", "The essential material that is needed for the sequel is recorded in Lemma REF ." ], [ "Projection Complexes", "We begin with the definition of a projection complex.", "Let $\\mathbb {Y}$ be a set and let $\\theta \\ge 0$ be a constant.", "Assume that for each $y \\in \\mathbb {Y}$ there is a function $d_{y} \\colon \\,\\mathbb {Y}\\setminus \\lbrace y\\rbrace \\times \\mathbb {Y}\\setminus \\lbrace y\\rbrace \\rightarrow \\mathbb {R}_{\\ge 0}$ with the following properties.", "Symmetry: $d_{y}(x,z) = d_{y}(z,x)$ for all $x,y,z \\in \\mathbb {Y}$ Triangle inequality: $d_{y}(x,z) + d_{y}(z,w) \\ge d_{y}(x,w)$ for all $x,y,z,w \\in \\mathbb {Y}$ Inequality on triples: $\\min \\lbrace d_{y}(x,z), d_{z}(x,y) \\rbrace \\le \\theta $ for all $x,y,z \\in \\mathbb {Y}$ Finiteness: $\\#\\lbrace y \\in \\mathbb {Y}\\mid d_{y}(x,z) > \\theta \\rbrace $ is finite for all $x,z \\in \\mathbb {Y}$ These conditions are known as the projection complex axioms.", "When we say that a set $\\mathbb {Y}$ and a collection of functions $\\lbrace d_y\\rbrace _{y \\in \\mathbb {Y}}$ as above satisfy the projection complex axioms the constant $\\theta $ is implicit.", "For a given $K \\ge 0$ , we will define a graph $\\mathcal {P}_K(\\mathbb {Y})$ with vertices corresponding to the elements in $\\mathbb {Y}$ .", "The edges are defined using the notion of modified distance functions.", "Given the functions $\\lbrace d_y\\rbrace $ , Bestvina–Bromberg–Fujiwara [3] constructed another collection of functions $\\lbrace d_{y}^{\\prime }\\rbrace _{y \\in \\mathbb {Y}}$ , where each $d_y^{\\prime }$ shares the same domain and target as $d_y$ .", "Because the definition of the $d_y^{\\prime }$ is technical and because we do not use the definition in this paper, we do not state it here.", "Bestvina–Bromberg–Fujiwara [3] showed that the modified functions are coarsely equivalent to the original functions: for $x \\ne y \\ne z \\in \\mathbb {Y}$ , $d^{\\prime }_{y}(x,z) \\le d_{y}(x,z) \\le d^{\\prime }_{y}(x,z) + 2\\theta $ .", "Fix $K \\ge 0$ .", "Then two vertices $x,z$ of $\\mathcal {P}_{K}(\\mathbb {Y})$ are connected by an edge if $d^{\\prime }_{y}(x,z) \\le K$ for all $y \\in \\mathbb {Y}- \\lbrace x,z\\rbrace $ .", "Let $d$ denote the resulting path metric on $\\mathcal {P}_{K}(\\mathbb {Y})$ .", "Bestvina–Bromberg–Fujiwara showed that for $K$ large enough relative to $\\theta $ , there are constants $C_{\\rm e}$ , $C_{\\rm p}$ , and $C_{\\rm g}$ , so that the following properties hold (see [3]): Bounded edge image.", "If $x\\ne y \\ne z$ are vertices of $\\mathcal {P}_K(\\mathbb {Y})$ and $d(x,z)=1$ , then $d_{y}(x,z) \\le C_{\\rm e}$ .", "Bounded path image.", "If a path in $\\mathcal {P}_K(\\mathbb {Y})$ connects vertices $x$ to $z$ without passing through the 2–neighborhood of the vertex $y$ , then $d_y(x,z) \\le C_{\\rm p}$ .", "Bounded geodesic image.", "If a geodesic in $\\mathcal {P}_K(\\mathbb {Y})$ connects vertices $x$ to $z$ without passing through the vertex $y$ , then $d_y(x,z) \\le C_{\\rm g}$ .", "(The bounded edge image property follows from the definition of the edges of $\\mathcal {P}_K(\\mathbb {Y})$ , with $C_{\\rm e}= K+2\\theta $ .)", "If $K$ is large enough so that the graph $\\mathcal {P}_K(\\mathbb {Y})$ satisfies the bounded edge, path, and geodesic properties for some $C_{\\rm e}$ , $C_{\\rm p}$ , and $C_{\\rm g}$ , then we say that $\\mathcal {P}_K(\\mathbb {Y})$ is a projection complex.", "This is the same definition as we used in our previous work [8].", "As mentioned there, we note that our terminology is not standard; in the papers by Bestvina–Bromberg–Fujiwara [3] and Bestvina–Bromberg–Fujiwara–Sisto [4], every $\\mathcal {P}_K(\\mathbb {Y})$ is called a projection complex." ], [ "We say that a group $G$ acts on a projection complex $\\mathcal {P}_K(\\mathbb {Y})$ if $G$ acts on the set $\\mathbb {Y}$ in such a way that the associated distance functions $d_y$ are $G$ –invariant, i.e., $d_{gy}(gx,gz) = d_{y}(x,z)$ .", "We note that if the original distance functions $d_y$ are $G$ –invariant, then the modified distance functions are $G$ –invariant as well—as is evident from the definition [3]—and so the action of $G$ on $\\mathbb {Y}$ extends an action of $G$ on the graph $\\mathcal {P}_{K}(\\mathbb {Y})$ by simplicial automorphisms." ], [ "Windmills", "To understand the action of $\\langle {R_v}\\rangle $ on $\\mathcal {P}$ , in our previous work we used the notion of a windmill.", "This tool is also necessary in this current work and we review the construction now.", "Given an action of a group $G$ on a projection complex $\\mathcal {P}$ with an equivariant family of subgroups $\\lbrace R_v\\rbrace $ of $G$ , we can inductively define a sequence of subgraphs $W_i$ of $\\mathcal {P}$ , a sequence of subsets $\\mathcal {O}_i$ of the set of vertices of $\\mathcal {P}$ , and a sequence of subgroups $H_i$ of $G$ as follows.", "Let $v_0$ be some base point for $\\mathcal {P}$ .", "To begin the inductive definitions at $i=0$ , we define: $H_0 = R_{v_0}$ and $W_0 = \\mathcal {O}_0 = \\lbrace v_0\\rbrace $ .", "For $i \\ge 1$ , we denote by $N_i$ the 1–neighborhood of $W_{i-1}$ , we denote by $L_i$ the vertices of $N_i \\setminus W_{i-1}$ , and we define: $H_i = \\langle {R_v \\mid v \\in N_{i}}\\rangle $ , $W_i = H_i \\cdot N_i$ , and $\\mathcal {O}_i =$ a set of orbit representatives for the action of $H_{i-1}$ on $L_i$ .", "The set $\\lbrace (H_i,W_i,\\mathcal {O}_i)\\rbrace _{i=0}^{\\infty }$ is called a set of windmill data for the equivariant family $\\lbrace R_v\\rbrace $ .", "We observe that each $W_i$ is connected.", "The subgroup $H$ of $G$ generated by the $R_v$ is the direct limit of the $H_i$ .", "Let $\\mathcal {O}$ be the union of the sets of representatives $\\mathcal {O}_i$ .", "In our previous work with Margalit [8], we proved the existence of a constant $L(\\mathcal {P})$ such that if $L \\ge L(\\mathcal {P})$ and $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups then $ H \\cong \\raisebox {-1.5pt}{ \\underset{\\mbox{\\scriptsize {$v \\in \\mathcal {O}$}}}{\\mbox{\\Huge {$\\ast $}}}}\\, R_v.", "$ For the remainder, we will always assume that $L \\ge L(\\mathcal {P})$ whenever we are discussing an equivariant $L$ –spinning family so that this free product decomposition is valid.", "Each of the constants of the form $L_\\ast $ defined in the sequel is at least $L(\\mathcal {P})$ ." ], [ "Pivot Points", "In our previous work, we introduced the notion of the set of pivot points for an element $h$ of $H$ in order to understand the group structure of $H$  [8].", "We review this notion now and state Lemma REF which records the necessary technical facts required for the shortening argument in Section .", "The level of a nontrivial element $h \\in H$ is the minimal index $i$ such that $h \\in H_i$ .", "We define the level of the identity element to be $-1$ .", "Each $h \\in H$ with level $i$ has a syllable decomposition $h_1 \\cdots h_n$ where each syllable $h_k$ is either a nontrivial element of $H_{i-1}$ or a nontrivial element of $R_{v_k}$ with $v_k \\in \\mathcal {O}_i$ .", "Moreover no two consecutive syllables are of the first type and consecutive syllables $h_k$ and $h_{k+1}$ of the second type have distinct corresponding fixed vertices $v_k$ and $v_{k+1}$ .", "We refer to $n$ as the syllable length of $h$ .", "As long as $L \\ge L(\\mathcal {P})$ , which will be our standing assumption, this syllable decomposition is unique for an equivariant $L$ –spinning family $\\lbrace R_v \\rbrace $ .", "Let $i \\ge 1$ and fix some element $h$ of $H$ with level $i$ and with syllable decomposition $h=h_1\\cdots h_n$ .", "For $k \\in \\lbrace 1,\\dots , n\\rbrace $ with $h_k \\notin H_{i-1}$ and with corresponding fixed vertex $v_k$ we define a vertex $w_k$ of $\\mathcal {P}$ as follows: $w_k = h_1 \\cdots h_{k-1}v_k.$ Note that $v_k$ and $w_k$ are not defined for the syllables $h_k$ that lie in $H_{i-1}$ .", "Let $\\operatorname{Piv}(h)$ be the ordered list of points $w_k$ , and call these the pivot points for $h$ .", "For $h \\in H_0$ we define $\\operatorname{Piv}(h)$ to be empty.", "There are several key properties regarding windmills and pivot points that we recall now.", "Lemma 2.1 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group acting on $\\mathcal {P}$ .", "There are constants $L_0$ and $m$ with the following properties.", "Suppose $L \\ge L_0$ and suppose $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Let $H = \\langle {R_v}\\rangle $ and choose windmill data $\\lbrace (H_i,W_i,\\mathcal {O}_i)\\rbrace $ .", "If $h$ is an element of $H$ and if $w \\in \\operatorname{Piv}(h)$ , then $d_{w}(v_0,hv_0) > L/2$ .", "If $h$ is an element of $H$ and if $w$ , $w^{\\prime }$ are pivot points for $h$ with $w < w^{\\prime }$ , then $d_{w}(v_0,w^{\\prime }) > L/2 - \\theta \\mbox{ and } d_{w^{\\prime }}(v_0,w) \\le \\theta .$ For all $i \\ge 1$ , if $x \\in N_{i}$ and $v \\notin W_{i-1}$ with $v \\ne x$ , then $d_v(v_0,x) \\le m$ .", "For all $i \\ge 1$ , if $h$ has level $i$ , then no pivot point for $h$ lies in $W_{i-1}$ .", "Using the constants associated with $\\mathcal {P}$ , we set $m = 11C_{\\rm e}+ 6C_{\\rm g}+ 5C_{\\rm p}$ and $L_0= 4(m + \\theta ) + 1$ .", "We remark that $m$ is the same constant the proof of Theorem 1.6 in our prior work [8] and that $L_0\\ge L(\\mathcal {P})$ from that same theorem.", "The above listed facts follow from results and arguments appearing in the proof of Theorem 1.6 in that paper as we now explain.", "Proof of (1).", "Fix an element $h$ in $H$ with syllable decomposition $h = h_1\\cdots h_n$ .", "If the level of $h$ is less than 1, the statement is vacuous.", "Hence suppose that the level of $h$ is at least 1.", "Consider a pivot point $w = h_1 \\cdots h_{k-1}v_k$ for $h$ .", "Equation (1) in the proof of Theorem 1.6 in our prior work states that $ d_{w}(v_0,hv_0) \\ge d_{w}(v_0,h_kv_0) - 2(m+\\theta ).", "$ As $d_{w}(v_0,h_kv_0) \\ge L$ and $L/2 > 2(m+\\theta )$ , the statement holds.", "Proof of (2).", "Again, fix an element $h$ in $H$ and assume that the level of $h$ is at least 1 as the statement is vacuous otherwise.", "Let $w$ and $w^{\\prime }$ be pivot points for $h$ with $w < w^{\\prime }$ .", "Statement (B) of the inductive hypothesis in the proof of Theorem 1.6 implies that there is a geodesic from $v_0$ to $w$ avoiding $w^{\\prime }$ .", "Hence we have $d_{w^{\\prime }}(v_0,w) \\le C_{\\rm g}$ .", "Therefore, using the first item, we have $d_{w^{\\prime }}(w,hv_0) \\ge d_{w^{\\prime }}(v_0,hv_0) - d_{w^{\\prime }}(v_0,w) > L/2 - C_{\\rm g}> \\theta .$ Thus by the Inequality on triples, we find $d_{w}(w^{\\prime },hv_0) \\le \\theta $ .", "From this, using the first item again, we conclude $ d_{w}(v_0,w^{\\prime }) \\ge d_{w}(v_0,hv_0) - d_{w}(w^{\\prime },hv_0) > L/2 - \\theta .", "$ As $d_w(v_0,w^{\\prime }) > L/2-\\theta > \\theta $ , by the Inequality on triples, we have $d_{w^{\\prime }}(v_0,w) \\le \\theta $ .", "Proof of (3).", "This is statement (C) of the inductive hypothesis in the proof of Theorem 1.6.", "Proof of (4).", "Fix $i \\ge 1$ and let $h$ be an element of $H$ with level $i$ .", "The first pivot point for $h$ , $w$ , lies in $L_i$ by definition.", "As $L_i$ is disjoint from $W_{i-1}$ , the statement holds for this pivot point.", "Let $w^{\\prime }$ be another pivot point for $h$ .", "By the second item, we have $d_{w}(v_0,w^{\\prime }) > L/2 - \\theta > m$ .", "If $w^{\\prime } \\in W_{i-1} \\subset N_i$ , then as $w \\notin W_{i-1}$ , the third item would imply that $d_{w}(v_0,w^{\\prime }) \\le m$ .", "This is a contradiction, hence $w^{\\prime } \\notin W_{i-1}$ ." ], [ "Shortening via pivot points", "In this section we introduce the key technical tool: shortening.", "The precise statement is given in Proposition REF .", "This proposition will allow us to bend paths in $\\mathcal {P}$ without changing their images in the quotient $\\mathcal {P}/\\langle {R_v}\\rangle $ .", "The bent path has a lower complexity in a precise sense that we will explain.", "This will allow us to conclude that certain closed paths in $\\mathcal {P}/\\langle {R_v}\\rangle $ lift to closed paths in $\\mathcal {P}$ .", "Before we can state Proposition REF we need to alter the notions of level and pivot points so that they are better suited for conjugacy classes.", "Assume that $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Let $H = \\langle {R_v}\\rangle $ and choose windmill data $\\lbrace (H_i,W_i,\\mathcal {O}_i)\\rbrace $ ." ], [ "The complexity of an element $h \\in H$ is the ordered pair $(\\text{\\sf i}(h),\\text{\\sf n}(h))$ where $\\text{\\sf i}(h)$ is the minimal index of any $H$ –conjugate of $h$ and $\\text{\\sf n}(h)$ is the minimal syllable length of any $H$ –conjugate of $h$ that has level $\\text{\\sf i}(h)$ .", "Lexicographical order on the pair $(\\text{\\sf i}(h),\\text{\\sf n}(h))$ gives a weak order on the elements in $H$ .", "The only element with $\\text{\\sf i}(h) = -1$ is the trivial element.", "Also, we remark that if $\\text{\\sf i}(h) = 0$ , then $\\text{\\sf n}(h) = 1$ ." ], [ "Given an element $h \\in H$ with $\\text{\\sf i}(h) = i$ and $\\text{\\sf n}(h) = n$ , we can express $h$ as a reduced word $h = g (h_{1}\\cdots h_{n})g^{-1}$ where each $h_k$ is either a nontrivial element of $H_{i-1}$ or a nontrivial element of $R_{v_k}$ with $v_k \\in \\mathcal {O}_i$ and $g \\in H$ .", "If $\\text{\\sf n}(h) > 1$ , then minimality of $\\text{\\sf n}(h)$ implies that if $h_1 \\in H_{i-1}$ then $h_n \\notin H_{i-1}$ , and that if $h_1 \\in R_{v_1}$ , then $h_n \\notin R_{v_1}$ .", "The subset of $\\operatorname{Piv}(h)$ corresponding to the syllables $h_k$ that lie in $R_{v_k}$ for some $v_k \\in \\mathcal {O}_i$ are called essential pivot points.", "We denote this subset by $\\operatorname{Piv}^{*}(h)$ .", "This set is nonempty so long as $\\text{\\sf i}(h) \\ge 1$ .", "The following lemma, whose proof is an easy exercise from the definitions, justifies calling these pivot points essential.", "Lemma 3.1 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group acting on $\\mathcal {P}$ .", "Suppose $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Let $H = \\langle {R_v}\\rangle $ and choose windmill data $\\lbrace (H_i,W_i,\\mathcal {O}_i)\\rbrace $ .", "The following statements are true.", "If the syllable length of $h \\in H$ equals $\\text{\\sf n}(h)$ , then every pivot point is essential, i.e., $\\operatorname{Piv}^*(h) = \\operatorname{Piv}(h)$ .", "If $h$ and $g$ are elements of $H$ , then $\\operatorname{Piv}^{*}(ghg^{-1}) = g\\operatorname{Piv}^{*}(h)$ .", "If $h$ is an element of $H$ and $k \\ge 1$ , then $\\operatorname{Piv}^{*}(h^k) = \\bigcup _{j=0}^{k-1} h^j\\operatorname{Piv}^{*}(h)$ as ordered sets.", "Items (1) and (2) imply that if the syllable length of $h$ equals $\\text{\\sf n}(h)$ , then $\\operatorname{Piv}^*(ghg^{-1}) = g\\operatorname{Piv}(h)$ .", "We remark that items (2) and (3) of Lemma REF are false for the set of all pivot points.", "We now state and prove the shortening proposition.", "Proposition 3.2 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group acting on $\\mathcal {P}$ .", "There is a constant $L_{\\rm short}$ with the following properties.", "Suppose $L \\ge L_{\\rm short}$ and suppose $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Let $H = \\langle {R_v}\\rangle $ and choose windmill data $\\lbrace (H_i,W_i,\\mathcal {O}_i)\\rbrace $ .", "Let $x$ be a vertex in $\\mathcal {P}$ and $h \\in H$ such that $hx \\ne x$ .", "Then there exists a vertex $v$ of $\\mathcal {P}$ and element $h_{v}$ of $R_{v}$ such that either $v \\in \\lbrace x,hx\\rbrace $ or $d_{v}(x,hx) > L/10$ ; and $h_vh < h$ .", "The first item roughly translates as stating that $v$ lies on the geodesic from $x$ to $hx$ .", "Let $L_0$ and $m$ be the constants from Lemma REF .", "Set $L_{\\rm short}= \\max \\lbrace L_0,5m,14\\theta \\rbrace $ .", "Take $L \\ge L_{\\rm short}$ and suppose that $G$ is acting on $\\mathcal {P}$ and that $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Fix a vertex $x$ of $\\mathcal {P}$ and an element $h$ of $H$ such that $hx \\ne x$ .", "Let $i = \\text{\\sf i}(h)$ , $n = \\text{\\sf n}(h)$ and express $h$ as a reduced word $h = gh_{1}\\cdots h_{n}g^{-1}$ where each $h_k$ is either a nontrivial element of $H_{i-1}$ or a nontrivial element of $R_{v_k}$ with $v_k \\in \\mathcal {O}_i$ .", "First, suppose that $i = 0$ and so $h = gh_1g^{-1}$ where $h_1 \\in R_{v_0}$ .", "In this case, we take $v = gv_0$ and $h_v = gh_1^{-1}g^{-1} \\in R_v$ .", "If $v \\notin \\lbrace x, hx\\rbrace $ , then $d_v(x,hx) = d_{v_0}(g^{-1}x,h_1g^{-1}x) \\ge L > L/10$ .", "As $h_v h$ is the identity, clearly $h_v h < h$ .", "Hence for the remainder, we assume that $i$ is at least 1.", "In particular, the set $\\operatorname{Piv}^{*}(h)$ is nonempty.", "Our strategy is to find an essential pivot point $w$ for $h$ and an integer $p$ such that $v = h^{p}w$ satisfies the first item.", "Given such a pivot point $w = gh_\\sigma v_k$ , where $h_\\sigma = h_1\\cdots h_{k-1}$ , we take $h_{v} = h^{p}(gh_{\\sigma }) h_{k}^{-1}(gh_{\\sigma })^{-1}h^{-p} \\in R_{v}$ .", "Then $h_{v}h &= \\bigl (h^{p}(gh_{\\sigma }) h_{k}^{-1}(gh_{\\sigma })^{-1}h^{-p}\\bigr )h \\\\&= h^{p}\\bigl ((gh_{\\sigma })h_{k}^{-1}(gh_{\\sigma })^{-1}h\\bigr )h^{-p} \\\\&= h^{p}(gh_{1}\\cdots h_{k-1}h_{k+1} \\cdots h_{n}g^{-1})h^{-p}.$ Hence for this element we have $h_vh < h$ , which is the second item.", "If $\\lbrace x,hx\\rbrace \\cap \\operatorname{Piv}^{*}(h) \\ne \\emptyset $ , we can take $w$ to be an essential pivot point in this intersection and set $v = w$ .", "Thus we may assume that $\\lbrace x, hx \\rbrace \\cap \\operatorname{Piv}^{*}(h) = \\emptyset $ .", "There are two cases depending on whether $x \\in gW_{i}$ or $x \\notin gW_{i}$ .", "Set $\\bar{h} = h_1\\cdots h_n$ so that $h = g\\bar{h}g^{-1}$ .", "We observe that $\\bar{h}$ has level $i$ .", "For the first case, we initially assume that $x \\in gN_{i} \\subset gW_{i}$ .", "Let $w$ be a pivot point for $\\bar{h}$ , thus $gw$ is an essential pivot point for $h$ .", "By Lemma REF (1), we have that $d_{w}(v_{0},\\bar{h}v_{0}) > L/2$ .", "By Lemma REF (4), we have that $w \\notin W_{i-1}$ .", "Since $\\bar{h}^{-1}w$ is a pivot point for $\\bar{h}^{-1}$ , Lemma REF (4) also implies that $\\bar{h}^{-1}w \\notin W_{i-1}$ as well.", "Hence by Lemma REF (3) as $g^{-1}x \\in N_{i}$ and $w, \\bar{h}^{-1}w \\notin W_{i-1}$ we have that $d_{w}(g^{-1}x,v_{0}) \\le m$ and $d_{w}(\\bar{h}g^{-1}x,\\bar{h}v_{0}) = d_{\\bar{h}^{-1}w}(g^{-1}x,v_{0}) \\le m$ .", "Therefore $ d_{gw}(x,hx) = d_{w}(g^{-1}x,\\bar{h}g^{-1}x) \\ge d_{w}(v_{0},\\bar{h}v_{0}) - d_{w}(v_{0},g^{-1}x) - d_{w}(\\bar{h}v_{0},\\bar{h}g^{-1}x) > L/2 - 2m \\ge L/10.", "$ Hence we may set $v = gw$ .", "Now suppose that $x \\in gW_{i} - gN_{i}$ .", "Then there is an $h_0 \\in H_i$ such that $h_0x \\in gN_{i}$ .", "Let $h^{\\prime } = h_0 h h_0^{-1}$ and $x^{\\prime } = h_0 x$ .", "We have $h^{\\prime }x^{\\prime } \\ne x^{\\prime }$ .", "Fix some pivot point $w$ for $\\bar{h}$ and so $gw$ is an essential pivot point for $h$ .", "By Lemma REF (1), we have $h_0gw \\in \\operatorname{Piv}^{*}(h^{\\prime })$ .", "As $x,hx \\notin \\operatorname{Piv}^{*}(h)$ , we have that $h_0gw \\ne x^{\\prime },h^{\\prime }x^{\\prime }$ .", "Thus as $x^{\\prime } \\in gN_{i}$ , the above case applies and we have that $ d_{gw}(x,hx) = d_{h_0gw}(x^{\\prime },h^{\\prime }x^{\\prime }) > L/10.", "$ Hence we may set $v = gw$ .", "Lastly, we deal with the second case that $x \\notin gW_{i}$ .", "In this case, we will be considering the projection of $x$ to various points of the form $h^jw$ where $w$ is an essential pivot point for $h$ and $j$ is an integer.", "As $w$ lies in $gW_i$ by definition and $W_i$ is $H_i$ –invariant, we have that $h^jw$ lies in $gW_{i}$ .", "In particular, $x \\ne h^jw$ for any essential pivot point $w$ for $h$ and any integer and therefore projections of $x$ to such points are always defined.", "Fix any essential pivot point $w$ for $h$ .", "By Lemma REF (2) we have that $h^{j}w$ is an essential pivot point for $h^{k}$ whenever $0 \\le j < k$ and additionally, such points are ordered $h^{j_1}w < h^{j_2}w$ if $j_1 < j_2$ .", "By Lemma REF (2), we have that for $1 \\le j_{1} < j_{2}$ that $d_{h^{j_{1}}w}(w,h^{j_{2}}w) \\ge d_{h^{j_{1}}w}(v_{0},h^{j_{2}}w) - d_{h^{j_{1}}w}(v_{0},w) \\ge L/2 - 2\\theta .$ By a similar argument we have $d_{h^{j_{1}}w}(h^{j_{0}}w,h^{j_{2}}w) \\ge L/2 - 2\\theta $ for all integers $j_{0} < j_{1} < j_{2}$ .", "Claim.", "There is an integer $J$ such that $d_{h^{j}w}(h^{j-1}w,x) > \\theta $ for $j \\le J$ and $d_{h^{j}w}(h^{j-1}w,x) \\le \\theta $ for all $j > J$ .", "We first show that the set $\\lbrace j \\in \\mathbb {Z}\\mid d_{h^{j}w}(h^{j-1}w,x) \\le \\theta \\rbrace $ has the form $(J,+\\infty )$ for some $J \\in \\mathbb {Z}\\cup \\lbrace -\\infty ,+\\infty \\rbrace $ .", "To this end, we suppose that $d_{h^{j}w}(h^{j-1}w,x) \\le \\theta $ .", "If $d_{h^{j+1}w}(h^{j}w,x) > \\theta $ , then by the Inequality on triples we have $d_{h^{j}w}(h^{j+1}w,x) \\le \\theta $ .", "In this case we find that $ L/2 - 2\\theta \\le d_{h^{j}w}(h^{j-1}w,h^{j+1}w) \\le d_{h^{j}w}(h^{j-1}w,x) + d_{h^{j}w}(h^{j+1}w,x) \\le 2\\theta .", "$ This is a contradiction as $L > 8\\theta $ and therefore $d_{h^{j+1}w}(h^{j}w,x) \\le \\theta $ too.", "If $J = -\\infty $ , then $d_{h^jw}(h^{j-1}w,x) \\le \\theta $ for all integers $j$ .", "Thus for all $j \\le -1$ we find that $d_{h^{j}w}(w,x) \\ge d_{h^{j}w}(h^{j-1}w,w) - d_{h^{j}w}(h^{j-1}w,x) \\ge L/2 - 3\\theta > \\theta .$ This contradicts the Finiteness axiom.", "If $J = +\\infty $ , then by the Inequality on Triples we have $d_{h^{j}w}(h^{j+1}w,x) \\le \\theta $ for all integers $j$ .", "Thus for all $j \\ge 1$ we find that $d_{h^{j}w}(w,x) \\ge d_{h^{j}w}(h^{j+1}w,w) - d_{h^{j}w}(h^{j+1}w,x) \\ge L/2 - 3\\theta > \\theta .$ Again, this contradicts the Finiteness axiom.", "This completes the proof of the claim.", "Let $J$ be as defined in the Claim.", "To complete the proof of the proposition, there are two cases based on $d_{h^Jw}(h^{J-1}w,x)$ .", "We will show that we can take $v$ to be either $h^Jw$ or $h^{J+1}w$ .", "First, suppose that $d_{h^{J}w}(h^{J-1}w,x) \\le L/4$ .", "We have $d_{h^{J}w}(h^{J-1}w,x) > \\theta $ and by the Inequality on triples and invariance we have $d_{h^{J}w}(h^{J+1}w,hx) = d_{h^{J-1}w}(h^{J}w,x) \\le \\theta $ .", "Thus $d_{h^{J}w}(x,hx) &\\ge d_{h^{J}w}(h^{J-1}w,h^{J+1}w) - d_{h^{J}w}(h^{J-1}w,x) - d_{h^{J}w}(h^{J+1}w,hx) \\\\& \\ge L/2 - \\theta - L/4 - \\theta \\ge L/4 - 2\\theta > L/10.$ Hence we can set $v = h^{J}w$ .", "Else, we have that $d_{h^{J}w}(h^{J-1}w,x) = d_{h^{J+1}w}(h^{J}w,hx) > L/4$ .", "As $d_{h^{J+1}}(h^Jw,x) \\le \\theta $ we have $d_{h^{J+1}w}(x,hx) &\\ge d_{h^{J+1}w}(h^{J}w,hx) - d_{h^{J+1}w}(h^{J}w,x) \\\\& \\ge L/4 - \\theta > L/10.$ Hence we can set $v = h^{J+1}w$ ." ], [ "Lifting Quadrilaterals", "In this section, we apply the shortening argument of Proposition REF to show that geodesic quadrilaterals in the quotient of the projection complex $\\mathcal {P}/\\langle {R_v}\\rangle $ lift to geodesic quadrilaterals in the projection complex $\\mathcal {P}$ .", "This is stated in Proposition REF .", "As mentioned in the Introduction, the strategy to show that $\\mathcal {P}/\\langle {R_v}\\rangle $ is $\\delta $ –hyperbolic is to lift geodesic triangles in $\\mathcal {P}/\\langle {R_v}\\rangle $ to geodesic triangles in $\\mathcal {P}$ .", "As a triangle is a degenerate quadrilateral where one side has length 0, Proposition REF applies to geodesic triangles as well.", "The reason we work with quadrilaterals is to show that the action of $G/\\langle {R_v}\\rangle $ on $\\mathcal {P}/\\langle {R_v}\\rangle $ is a non-elementary WPD action, so long as the action of $G$ on $\\mathcal {P}$ is and $L$ , the spinning constant, is large enough.", "There are two items we need to discuss before stating and proving Proposition REF ." ], [ "Throughout this section we will be lifting geodesics from $\\mathcal {P}/\\langle {R_v}\\rangle $ to $\\mathcal {P}$ and modifying the lifts.", "It will be important to have a way of certifying that these lifts and their modifications are geodesics.", "This is the content of the following lemma.", "Throughout the rest of the paper, we will always assume that paths are 1–Lipschitz.", "Lemma 4.1 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group acting on $\\mathcal {P}$ .", "Suppose that $H$ is a subgroup of $G$ and let $p \\colon \\,\\mathcal {P}\\rightarrow \\mathcal {P}/H$ be the quotient map.", "The following statements are true.", "If $\\bar{\\alpha } \\colon \\,[0,n] \\rightarrow \\mathcal {P}/H$ is a path and $x$ is a point in $\\mathcal {P}$ that satisfies $p(x) = \\bar{\\alpha }(0)$ , then there exists a path $\\alpha \\colon \\,[0,n] \\rightarrow \\mathcal {P}$ such that $p \\circ \\alpha = \\bar{\\alpha }$ and $\\alpha (0) = x$ .", "If $\\alpha \\colon \\,[0,n] \\rightarrow \\mathcal {P}$ is a path and $n = d_{\\mathcal {P}/H}(p(\\alpha (0)),p(\\alpha (n)))$ , then $\\alpha $ is a geodesic.", "The first statement is obvious.", "The second statement follows as the map $p \\colon \\,\\mathcal {P}\\rightarrow \\mathcal {P}/H$ is 1–Lipschitz.", "Indeed, if $\\alpha $ is not a geodesic, then there is a geodesic $\\alpha ^{\\prime } \\colon \\,[0,n^{\\prime }] \\rightarrow \\mathcal {P}$ where $\\alpha ^{\\prime }(0) = \\alpha (0)$ , $\\alpha ^{\\prime }(n^{\\prime }) = \\alpha (n)$ and $n^{\\prime } < n$ .", "As $p$ is 1–Lipschitz, we find $ n = d_{\\mathcal {P}/H}(p(\\alpha ^{\\prime }(0)),p(\\alpha ^{\\prime }(n^{\\prime }))) \\le n^{\\prime }.", "$ This a contradiction and hence $\\alpha $ is a geodesic." ], [ "Let $v$ be a vertex in $\\mathcal {P}$ .", "Suppose $\\alpha \\colon \\,[0,n] \\rightarrow \\mathcal {P}$ is a path and that $v = \\alpha (n_0)$ for some $n_0 \\in \\lbrace 0,\\ldots ,n\\rbrace $ .", "For any $h_v \\in R_v$ we define a new path $\\alpha \\vee _v h_v \\colon \\,[0,n] \\rightarrow \\mathcal {P}$ by $\\bigl (\\alpha \\vee _v h_v\\bigr )(t) = {\\left\\lbrace \\begin{array}{ll}\\alpha (t) & \\mbox{ if } 0 \\le t \\le n_0, \\mbox{ or} \\\\h_v\\alpha (t) & \\mbox{ if } n_0 \\le t \\le n.\\end{array}\\right.", "}$ As $\\alpha (n_0) = h_v\\alpha (n_0)$ , this does define a path.", "We say that $\\alpha \\vee _v h_v$ is obtained by bending $\\alpha $ at $v$ using $h_v$.", "Writing $\\alpha $ as the concatenation of two paths $\\alpha _1$ and $\\alpha _2$ where $\\alpha _1$ ends at $v$ and $\\alpha _2$ begins at $v$ , the bent path $\\alpha \\vee _v h_v$ is the concatenation of $\\alpha _1$ and $h_v\\alpha _2$ .", "See Figure REF .", "Figure: The paths α\\alpha and α∨ v h v \\alpha \\vee _v h_vLemma 4.2 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group acting on $\\mathcal {P}$ .", "Suppose $\\lbrace R_v\\rbrace $ is an equivariant family of subgroups of $G$ .", "Let $H = \\langle {R_v}\\rangle $ and let $p \\colon \\,\\mathcal {P}\\rightarrow \\mathcal {P}/H$ be the quotient map.", "Let $\\alpha \\colon \\,[0,n] \\rightarrow \\mathcal {P}$ be a path and let $v$ be a vertex in the image of $\\alpha $ .", "Then for any $h_v \\in R_v$ the following statements are true.", "We have $p \\circ \\alpha = p \\circ \\bigl (\\alpha \\vee _v h_v \\bigr )$ .", "For any $0 \\le t_1 < t_2 \\le n$ , if $p \\circ \\alpha |[t_1,t_2]$ is a geodesic, then so is $\\bigl (\\alpha \\vee _v h_v \\bigr )|[t_1,t_2]$ .", "The first statement follows immediately from the definitions.", "The second statement follows from the first statement and Lemma REF (2).", "Let $X$ be a graph considered as a metric space where every edge has length one.", "A geodesic quadrilateral $Q$ in $X$ consists of four geodesics and four points: $\\alpha _k$ from $x_k$ to $x_{k+1 \\mod {4}}$ for $k = 0,1,2,3$ .", "We write $Q = \\cup _{k=0}^3 \\alpha _k$ .", "Proposition 4.3 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group acting on $\\mathcal {P}$ .", "For any $B \\ge 0$ , there is a constant $L_{\\rm lift}(B)$ with the following properties.", "Suppose $L \\ge L_{\\rm lift}(B)$ and suppose $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Let $H = \\langle {R_v}\\rangle $ and let $p \\colon \\,\\mathcal {P}\\rightarrow \\mathcal {P}/H$ be the quotient map.", "For each geodesic quadrilateral $\\bar{Q} = \\cup _{k=0}^3 \\bar{\\alpha }_k$ in $\\mathcal {P}/H$ there exists a geodesic quadrilateral $Q = \\cup _{k=0}^3 \\alpha _k$ in $\\mathcal {P}$ so that $p(\\alpha _k) = \\bar{\\alpha }_k$ for $k = 0,1,2,3$ .", "Additionally, $Q$ satisfies the following property.", "If there are lifts $\\tilde{\\alpha }_0$ from $\\tilde{x}_0$ to $\\tilde{x}_1$ and $\\tilde{\\alpha }_2$ from $\\tilde{x}_2$ to $\\tilde{x}_3$ of $\\bar{\\alpha }_0$ and $\\bar{\\alpha }_2$ respectively such that $d_v(\\tilde{x}_0,\\tilde{x}_1) \\le B$ and $d_v(\\tilde{x}_2,\\tilde{x}_3) \\le B$ when defined, then the geodesics $\\alpha _0$ and $\\alpha _2$ in $Q$ are $H$ –translates of $\\tilde{\\alpha }_0$ and $\\tilde{\\alpha }_2$ respectively.", "Fix $B \\ge 0$ and set $L_{\\rm lift}(B) = \\max \\lbrace L_{\\rm short},40B,40C_{\\rm g}\\rbrace $ .", "Take $L \\ge L_{\\rm lift}(B)$ and suppose that $G$ is acting on $\\mathcal {P}$ and that $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Let $\\bar{x}_{0}, \\bar{x}_{1}, \\bar{x}_{2}$ and $\\bar{x}_{3}$ be the vertices of the geodesic quadrilateral $\\bar{Q}$ in $\\mathcal {P}/H$ .", "By Lemma REF (1), for any point $x_0 \\in p^{-1}(\\bar{x}_0)$ , we can iteratively lift the geodesics $\\bar{\\alpha }_k$ to paths $\\alpha _k$ from $x_k$ to $x_{k+1}$ where $p(x_k) = \\bar{x}_k$ .", "By Lemma REF (2), the paths $\\alpha _k$ are geodesics.", "If $\\alpha _0$ and $\\alpha _2$ as in the statement of the proposition exists, then we can ensure that $\\alpha _0$ and $\\alpha _2$ are $H$ –translates of these geodesics.", "We denote the concatenation of the paths $\\alpha _k$ by $\\alpha $ and we say that $\\alpha $ is a special lift of $\\bar{Q}$ .", "For each special lift $\\alpha $ of $\\bar{Q}$ , with endpoints denoted $x_0$ and $x_4$ , there is an element $\\text{\\sf h}(\\alpha ) \\in H$ with minimal complexity such that $x_{4} = \\text{\\sf h}(\\alpha )x_{0}$ .", "Let $\\alpha $ be a special lift of $\\bar{Q}$ so that $\\text{\\sf h}(\\alpha )$ has minimal complexity among all special lifts of $\\bar{Q}$ .", "We claim that $x_0 = x_4$ , which shows that $\\alpha $ defines a geodesic quadrilateral $Q$ as in the statement of the proposition.", "Indeed, if not we will show that we can bend $\\alpha $ to a new path $\\alpha ^{\\prime }$ that is a special lift with $\\text{\\sf h}(\\alpha ^{\\prime }) < \\text{\\sf h}(\\alpha )$ .", "This contradicts the minimality of $\\text{\\sf h}(\\alpha )$ .", "To this end, suppose that $x_0 \\ne x_4 = \\text{\\sf h}(\\alpha )x_0$ .", "Apply Proposition REF to $x = x_0$ and $h = \\text{\\sf h}(\\alpha )$ and let $v$ be the corresponding vertex of $\\mathcal {P}$ and $h_v \\in R_v$ the corresponding element.", "We have that $h_v\\text{\\sf h}(\\alpha ) < \\text{\\sf h}(\\alpha )$ .", "We claim that $v$ lies in the image of $\\alpha $ .", "Indeed, if $v \\notin \\lbrace x_0,x_4\\rbrace $ , then $d_v(x_0,x_4) > L/10$ .", "If further that $v \\notin \\lbrace x_1,x_2,x_3\\rbrace $ , then by the triangle inequality, we have that $d_v(x_n,x_{n+1}) > L/40$ for some $n$ .", "As $L/40 \\ge C_{\\rm g}$ , there is a $n_0$ such that $\\alpha _n(n_0) = v$ .", "Moreover, as $L/40 \\ge B$ , if lifts $\\tilde{\\alpha }_0$ and $\\tilde{\\alpha }_2$ as in the statement of the proposition exists, we must have that $n = 1$ or $n = 3$ .", "This shows that $v$ lies in the image of $\\alpha $ .", "We consider the path $\\alpha ^{\\prime } = \\alpha \\vee _v h_v$ .", "By Lemma REF (2), $\\alpha ^{\\prime }$ consists of four geodesic segments $\\alpha ^{\\prime }_k$ for $k = 0,1,2,3$ .", "Moreover, we observe that $\\alpha ^{\\prime }$ is a special lift of $\\bar{Q}$ as if lifts $\\tilde{\\alpha }_0$ and $\\tilde{\\alpha }_2$ as in the statement of the proposition exists, then the segments $\\alpha ^{\\prime }_0$ and $\\alpha ^{\\prime }_2$ are $H$ –translates the segments $\\alpha _0$ and $\\alpha _2$ respectively.", "Letting $x^{\\prime }_4$ denote the terminal point of $\\alpha ^{\\prime }$ we find $ x^{\\prime }_4 = h_vx_4 = h_v\\text{\\sf h}(\\alpha )x_0 $ so that $\\text{\\sf h}(\\alpha ^{\\prime }) \\le h_v\\text{\\sf h}(\\alpha ) < \\text{\\sf h}(\\alpha )$ .", "This contradicts the minimality of $\\text{\\sf h}(\\alpha )$ ." ], [ "Proof of Theorem ", "In this section we prove the first of the two main results of this paper.", "Theorem REF states that if a group $G$ acts on a projection complex $\\mathcal {P}$ then there exists a constant $L_{\\rm hyp}(\\mathcal {P})$ so that if $L \\ge L_{\\rm hyp}(\\mathcal {P})$ and if $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ then $\\mathcal {P}/\\langle {R_v}\\rangle $ is $\\delta $ –hyperbolic.", "The proof proceeds by showing that geodesic triangles in $\\mathcal {P}/\\langle {R_v}\\rangle $ can be lifted to geodesic triangles in $\\mathcal {P}$ .", "Let $\\mathcal {P}$ be a projection complex and set $L_{\\rm hyp}(\\mathcal {P}) = L_{\\rm lift}(0)$ .", "Bestvina–Bromberg–Fujiwara proved the $\\mathcal {P}$ is a quasi-tree [3].", "Let $\\delta $ be such that $\\mathcal {P}$ is $\\delta $ –hyperbolic.", "Take $L \\ge L_{\\rm hyp}(\\mathcal {P})$ and suppose that $G$ is acting on $\\mathcal {P}$ and that $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family.", "Let $H = \\langle {R_v}\\rangle $ .", "Let $\\bar{\\alpha }_0$ , $\\bar{\\alpha }_1$ and $\\bar{\\alpha }_2$ be the three sides of a geodesic triangle in $\\mathcal {P}/H$ .", "We set $\\bar{\\alpha }_3$ to be the trivial path at the endpoint of $\\bar{\\alpha }_2$ .", "This gives a (degenerate) geodesic quadrilateral $\\bar{Q} = \\cup _{k=0}^3 \\bar{\\alpha }_k$ .", "By Proposition REF , there is a geodesic quadrilateral $Q = \\cup _{k=0}^3 \\alpha _k$ so that $p(\\alpha _k) = \\bar{\\alpha }_k$ for $k = 0,1,2,3$ .", "As $\\alpha _3$ is a trivial path, $Q$ is in fact a geodesic triangle in $\\mathcal {P}$ .", "As the map $p \\colon \\,\\mathcal {P}\\rightarrow \\mathcal {P}/H$ is 1–Lipschitz and as $Q$ is $\\delta $ –thin, the geodesic triangle $\\bar{Q}$ is $\\delta $ –thin as well.", "Hence $\\mathcal {P}/H$ is $\\delta $ –hyperbolic." ], [ "Bounded projections", "There are two key results in this section.", "First, we show that geodesics $\\alpha \\colon \\,[0,n] \\rightarrow \\mathcal {P}$ with bounded projections are mapped by $p$ to geodesics in $\\mathcal {P}/\\langle {R_v}\\rangle $ .", "This appears as Lemma REF .", "The proof of this lemma is very similar to the proof of Proposition REF as it involves bending and shortening.", "Secondly, we apply Lemma REF to show that given a WPD element in $G$ where some the orbit os some point has bounded projections, its image in $G/\\langle {R_v}\\rangle $ acts as a WPD element on $\\mathcal {P}/\\langle {R_v}\\rangle $ .", "This appears as Lemma REF .", "The proof of this lemma uses Proposition REF .", "Lemma 6.1 Let $\\mathcal {P}$ be a projection complex and let $G$ be a group acting on $\\mathcal {P}$ .", "For any $B \\ge 0$ , there is a constant $L_{\\rm pro}(B)$ with the following property.", "Suppose $L \\ge L_{\\rm pro}(B)$ and suppose $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Let $H = \\langle {R_v}\\rangle $ and let $p \\colon \\,\\mathcal {P}\\rightarrow \\mathcal {P}/H$ be the quotient map.", "If $\\alpha \\colon \\,[0,n] \\rightarrow \\mathcal {P}$ is a geodesic, and $d_v(\\alpha (0),\\alpha (n))) \\le B$ for all vertices $v$ of $\\mathcal {P}$ other than $\\alpha (0)$ and $\\alpha (n)$ , then $p \\circ \\alpha \\colon \\,[0,n] \\rightarrow \\mathcal {P}/H$ is a geodesic.", "Set $L_{\\rm pro}(B) = \\max \\lbrace L_{\\rm short},10B + 10C_{\\rm g}\\rbrace $ .", "Take $L \\ge L_{\\rm pro}(B)$ and suppose that $G$ is acting on $\\mathcal {P}$ and that $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family.", "Let $\\bar{\\beta } \\colon \\,[0,n^{\\prime }] \\rightarrow \\mathcal {P}/H$ be a geodesic from $p(\\alpha (0))$ to $p(\\alpha (n))$ .", "We will argue that $n = n^{\\prime }$ , showing that $p \\circ \\alpha $ is a geodesic.", "For each $H$ –translate $h\\alpha \\colon \\,[0,n] \\rightarrow \\mathcal {P}$ of $\\alpha $ , we say a lift $\\beta \\colon \\,[0,n^{\\prime }] \\rightarrow \\mathcal {P}$ of $\\bar{\\beta }$ is compatible with $h\\alpha $ if $h\\alpha (0) = \\beta (0)$ .", "In this situation, there is an element $\\text{\\sf h}(h\\alpha ,\\beta )$ with minimal complexity such that $\\beta (n^{\\prime }) = \\text{\\sf h}(h\\alpha ,\\beta )h\\alpha (n)$ .", "We replace $\\alpha $ by an $H$ –translate and let $\\beta \\colon \\,[0,n^{\\prime }] \\rightarrow \\mathcal {P}$ be a compatible lift of $\\bar{\\beta }$ so that $\\text{\\sf h}(\\alpha ,\\beta )$ minimizes complexity among all $H$ –translates of $\\alpha $ and compatible lifts.", "We claim that $\\alpha (n) = \\beta (n^{\\prime })$ , which shows that $n = n^{\\prime }$ as both $\\alpha $ and $\\beta $ are geodesics.", "Indeed, if not we will show that we can find a translate $\\alpha ^{\\prime }$ of $\\alpha $ and a compatible lift $\\beta ^{\\prime }$ with $\\text{\\sf h}(\\alpha ^{\\prime },\\beta ^{\\prime }) < \\text{\\sf h}(\\alpha ,\\beta )$ .", "The path $\\beta ^{\\prime }$ is obtained by translating or bending $\\beta $ .", "This contradicts the minimality of $\\text{\\sf h}(\\alpha ,\\beta )$ .", "To this end, suppose that $\\alpha (n) \\ne \\beta (n^{\\prime })$ .", "Apply Proposition REF to $x = \\alpha (n)$ and $h = \\text{\\sf h}(\\alpha ,\\beta )$ and let $v$ be the corresponding vertex and $h_v \\in R_v$ the corresponding element.", "We have that $h_v \\text{\\sf h}(\\alpha ,\\beta ) < \\text{\\sf h}(\\alpha ,\\beta )$ .", "There are two cases now depending on $v$ .", "If $v = \\alpha (n)$ , then for the $H$ –translate $h_v\\alpha $ and compatible lift $h_v\\beta $ , we have $h_v\\beta (n^{\\prime }) = h_v\\text{\\sf h}(\\alpha ,\\beta )\\alpha (n) = h_v\\text{\\sf h}(\\alpha ,\\beta )h_v\\alpha (n)$ so that $\\text{\\sf h}(h_v\\alpha ,h_v\\beta ) \\le h_v\\text{\\sf h}(\\alpha ,\\beta ) < \\text{\\sf h}(\\alpha ,\\beta )$ .", "This contradicts the minimality of $\\text{\\sf h}(\\alpha ,\\beta )$ .", "Else, we claim that $v$ lies in the image of $\\beta $ .", "Indeed, if $v \\ne \\beta (n)$ then $d_v(\\alpha (n),\\beta (n^{\\prime })) > L/10$ .", "If further $v \\ne \\beta (0)$ , then as $d_v(\\alpha (0),\\alpha (n)) \\le B$ and $\\alpha (0) = \\beta (0)$ , have have that $ d_v(\\beta (0),\\beta (n^{\\prime })) \\ge d_v(\\alpha (n),\\beta (n^{\\prime })) - d_v(\\alpha (0),\\alpha (n)) > L/10 - B > C_{\\rm g}.", "$ This shows that $v$ lies in the image of $\\beta $ .", "We define $\\beta ^{\\prime } = \\beta \\vee _v h_v$ .", "By Lemma REF , $\\beta ^{\\prime }$ is a compatible lift.", "Next, we find that $ \\beta ^{\\prime }(n^{\\prime }) = h_v\\beta (n^{\\prime }) = h_v\\text{\\sf h}(\\alpha ,\\beta )\\alpha (n) $ so that $\\text{\\sf h}(\\alpha ,\\beta ^{\\prime }) \\le \\text{\\sf h}(\\alpha ,\\beta ) < \\text{\\sf h}(\\alpha ,\\beta )$ .", "This contradicts the minimality of $\\text{\\sf h}(\\alpha ,\\beta )$ .", "Lemma 6.2 Let $\\mathcal {P}$ be a projection complex, let $G$ be group acting on $\\mathcal {P}$ and let $B \\ge 0$ .", "Suppose $L \\ge \\max \\lbrace L_{\\rm lift}(B),L_{\\rm pro}(B)\\rbrace $ and suppose $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ .", "Let $H = \\langle {R_v}\\rangle $ .", "If $f \\in G$ is a hyperbolic isometry of $\\mathcal {P}$ so that $d_v(x_0,f^nx_0) \\le B$ for all $n \\in \\mathbb {Z}$ when defined, then its image $\\bar{f} \\in G/H$ is a hyperbolic isometry of $\\mathcal {P}/H$ .", "Additionally, if $f$ is a WPD element, then so is $\\bar{f}$ .", "Fix $B \\ge 0$ and suppose that $G$ is acting on $\\mathcal {P}$ and that $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family where $L \\ge \\max \\lbrace L_{\\rm lift}(B),L_{\\rm pro}(B)\\rbrace $ .", "Suppose that $f \\in G$ is a hyperbolic isometry of $\\mathcal {P}$ and $x_0$ is a vertex of $\\mathcal {P}$ so that $d_v(x_0,f^nx_0) \\le B$ for all $n \\in \\mathbb {Z}$ when defined.", "Let $\\bar{x}_0 = p(x_0)$ .", "As $L \\ge L_{\\rm pro}(B)$ , by Lemma REF , we have that $d_{\\mathcal {P}/H}(\\bar{x}_0,\\bar{f}^n\\bar{x}_0) = d_{\\mathcal {P}}(x_0,f^nx_0)$ .", "Hence as $f$ is hyperbolic, $\\bar{f}$ is also hyperbolic.", "Now assume further that $f$ is a WPD element.", "Fix $D \\ge 0$ and let $M \\ge 0$ be such that the set $ \\lbrace g \\in G \\mid d_{\\mathcal {P}}(x_0,gx_0) \\le D \\mbox{ and } d_{\\mathcal {P}}(f^Mx_0,gf^Mx_0) \\le D \\rbrace $ is finite.", "Let $K$ denote the cardinality of this set.", "Suppose that $\\lbrace \\bar{g}_1, \\ldots , \\bar{g}_{K^{\\prime }} \\rbrace $ is a set of elements of $G/H$ so that $d_{\\mathcal {P}/H}(\\bar{x}_0,\\bar{g}_j\\bar{x}_0) \\le D \\mbox{ and } d_{\\mathcal {P}/H}(\\bar{f}^M\\bar{x}_0,\\bar{g}_j\\bar{f}^M\\bar{x}_0) \\le D.$ Fix elements $g_j \\in G$ whose images are the $\\bar{g}_j$ s. We consider the geodesic quadrilateral $\\bar{Q}_j = \\cup _{k=0}^3 \\bar{\\alpha }_k$ where: $\\bar{\\alpha }_0$ is a geodesic from $\\bar{x}_0$ to $\\bar{f}^M\\bar{x}_0$ , $\\bar{\\alpha }_1$ is a geodesic from $\\bar{f}^M\\bar{x}_0$ to $\\bar{g}_j\\bar{f}^M\\bar{x}_0$ , $\\bar{\\alpha }_2$ is a geodesic from $\\bar{g}_j\\bar{f}^M\\bar{x}_0$ to $\\bar{g}_j\\bar{x}_0$ , and $\\bar{\\alpha }_3$ is a geodesic from $\\bar{g}_j\\bar{x}_0$ to $\\bar{x}_0$ .", "As $L \\ge L_{\\rm lift}(B)$ for each $1 \\le j \\le K^{\\prime }$ , there is a geodesic quadrilateral $Q_j = \\cup _{k=0}^3 \\alpha _k$ so that $p(\\alpha _k) = \\bar{\\alpha }_k$ .", "Moreover, there are elements $h_0,h_2 \\in H$ such that $\\alpha _0$ is a geodesic from $h_0\\bar{x}_0$ to $h_0f^Mx_0$ and $\\alpha _2$ is a geodesic from $h_2g_jf^Mx_0$ to $h_2g_jx_0$ .", "In particular for each $1 \\le j \\le K^{\\prime }$ we find that $d_{\\mathcal {P}}(x_0,h_0^{-1}h_2g_j) \\le D \\mbox{ and } d_{\\mathcal {P}}(f^Mx_0,h_0^{-1}h_2g_jf^M x_0) \\le D. $ This shows that $K^{\\prime } \\le K$ .", "As it suffices to check finiteness at a single point, this shows that $\\bar{f}$ is a WPD element." ], [ "Proof of Theorem ", "In this section we give the proof of the second of the main results in this paper.", "Theorem REF states that if a group $G$ admits a non-elementary WPD action on a projection complex $\\mathcal {P}$ then there exists a constant $L_{\\rm WPD}(\\mathcal {P},G)$ so that if $L \\ge L_{\\rm WPD}(\\mathcal {P},G)$ and if $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ then the action of $G/\\langle {R_v}\\rangle $ on $\\mathcal {P}/\\langle {R_v}\\rangle $ is a non-elementary WPD action." ], [ "In order to apply the results of Section , we need to know that hyperbolic isometries of a projection complex have bounded projections.", "This is an application of the Finiteness axiom of a projection complex as we now show.", "Lemma 7.1 Let $\\mathcal {P}$ be a projection complex and let $f$ be a hyperbolic isometry of $\\mathcal {P}$ .", "Then for any vertex $x_0$ of $\\mathcal {P}$ , there is a constant $B_f$ such that $d_v(x_0,f^n x_0) \\le B_f$ for all $n \\in \\mathbb {Z}$ when defined.", "Let $M_1 = \\max \\lbrace d_v(x_0,f x_0) \\mid v \\notin \\lbrace x_0,f x_0\\rbrace \\rbrace $ and $M_2 = d_{x_0}(f^{-1}x_0,f x_0)$ .", "We remark that $M_1$ is finite by the Finiteness axiom.", "Set $M = \\max \\lbrace M_1,M_2 \\rbrace $ .", "Fix a geodesic $\\alpha $ from $x_0$ to $f x_0$ .", "Let $N$ be such that $d(x,f^n y) > 4$ if $x$ and $y$ lie on $\\alpha $ , and $n \\ge N$ .", "Define $B_f = NM + 2C_{\\rm p}$ .", "By equivariance, it suffices to prove the lemma for non-negative integers.", "Fix an $n \\in \\mathbb {N}$ and suppose that $v \\notin \\lbrace x_0,f^n x_0\\rbrace $ .", "If $v$ does not lie in the 2–neighborhood of the path $\\alpha \\cup f \\alpha \\cup \\cdots \\cup f^{n-1}\\alpha $ , then $d_v(x_0,f^nx_0) \\le C_{\\rm p}\\le B_f$ .", "Else, there are indices $0 \\le i_0 \\le i_1 \\le n-1$ such that $i_1 - i_0< N$ and $v$ lies in the 2–neighborhood of $f^j \\alpha $ only if $i_0 \\le j \\le i_1$ .", "Thus as $v$ does not lie in the 2–neighborhood of $\\alpha \\cup \\cdots \\cup f^{i_0-1}\\alpha $ nor in the 2–neighborhood of $f^{i_1 + 1}\\alpha \\cup \\cdots \\cup f^n\\alpha $ , we have $d_v(x_0,f^{i_0}x_0) \\le C_{\\rm p}$ and $d_v(f^{i_1 + 1}x_0,f^n x_0) \\le C_{\\rm p}$ .", "Suppose that $v \\ne f^jx_0$ for any $i_0 < j \\le i_1$ (by the definition of $i_0$ and $i_1$ these are the only possible indices).", "Then we find that $d_v(f^{i_0}x_0,f^{i_1+1}x_0) \\le \\sum _{j = i_0}^{i_1} d_v(f^jx_0,f^{j+1}x_0) = \\sum _{j = i_0}^{i_1} d_{f^{-j}v}d(x_0,f x_0) \\le NM_1 \\le NM.$ Else, we have that $v = f^{j_0}x_0$ for some $i_0 < j_0 \\le i_1$ .", "In this case we find $d_v(f^{i_0}x_0,f^{i_1+1}x_0) & \\le \\sum _{j = i_0}^{j_0-2} d_v(f^jx_0,f^{j+1}x_0) + d_{f^{j_0}x_0}(f^{j_0-1}x_0,f^{j_0+1}x_0) + \\sum _{j = j_0+1}^{i_1} d_v(f^jx_0,f^{j+1}x_0) \\\\& (j_0 - 1 - i_0)M_1 + M_2 + (i_1 - j_0)M_1 \\le (N-2)M_1 + M_2 \\le NM.$ Therefore $ d_v(x_0,f^nx_0) \\le d_v(x_0,f^{i_0}x_0) + d_v(f^{i_0}x_0,f^{i_1+1}x_0) + d_v(f^{i_1+1}x_0,f^n x_0) \\le NM + 2C_{\\rm p}= B_f.", "$ Let $\\mathcal {P}$ be a projection complex and let $G$ be a group with a non-elementary WPD action on $\\mathcal {P}$ .", "Let $f_1$ and $f_2$ be independent WPD elements in $G$ .", "Fix some point $x_0$ in $\\mathcal {P}$ and let $B_{f_1}$ and $B_{f_2}$ be the constants from Lemma REF .", "Let $B_0 = \\max \\lbrace d_v(f_1x_0,f_2x_0) \\mid v \\notin \\lbrace f_1x_0,f_2x_0 \\rbrace \\rbrace $ .", "Set $B = B_0 + B_{f_1} + B_{f_2}$ .", "Let $v$ be a vertex of $\\mathcal {P}$ and suppose that $d_v(f_1^{n_1}x_0,f_2^{n_2}x_0)$ is defined for some integers $n_1$ and $n_2$ .", "If $v \\ne x_0$ , then $ d_v(f_1^{n_1}x_0,f_2^{n_2}x_0) \\le d_v(f_1^{n_1}x_0,x_0) + d_v(x_0,f_2^{n_2}x_0) \\le B_{f_1} + B_{f_2} \\le B.", "$ Else, if $v = x_0$ , then $d_v(f_1^{n_1}x_0,f_2^{n_2}x_0) \\le d_v(f_1^{n_1}x_0,f_1x_0) + d_v(f_1x_0,f_2x_0) + d_v(f_2x_0,f_2^{n_2}x_0)\\le B_{f_1} + B_0 + B_{f_2} = B.$ Let $L_{\\rm WPD}(\\mathcal {P},G) = \\max \\lbrace L_{\\rm lift}(B),L_{\\rm pro}(B)\\rbrace $ .", "Suppose that $\\lbrace R_v\\rbrace $ is an equivariant $L$ –spinning family of subgroups of $G$ where $L \\ge L_{\\rm WPD}(\\mathcal {P},G)$ .", "Let $H = \\langle {R_v}\\rangle $ .", "By Lemma REF , the images $\\bar{f}_1$ and $\\bar{f}_2$ are WPD elements of $G/H$ acting on $\\mathcal {P}/H$ .", "Additionally, by Lemma REF , we have that $d_{\\mathcal {P}/H}(\\bar{f}_1^{n_1}\\bar{x}_0,\\bar{f}_2^{n_2}\\bar{x}_0) = d_{\\mathcal {P}}(f_1^{n_1}x_0,f_2^{n_2}x_0)$ for integers $n_2$ and $n_2$ .", "As $f_1$ and $f_2$ are independent, this shows that $\\bar{f}_1$ and $\\bar{f}_2$ are independent as well." ], [ "Examples", "In this final section we present two examples when $G$ is $\\mathrm {Mod}(S)$ , the mapping class group of an orientable surface $S$ .", "In the first example, the subgroup $H$ is the normal closure of a pseudo-Anosov mapping class; in the second example the subgroup $H$ is the normal closure of a partial pseudo-Anosov defined on an orbit-overlapping subsurface.", "The first example in fact applies more generally, whenever $G$ is a group acting on a $\\delta $ –hyperbolic metric space and $g$ is a WPD element for this action.", "The relevant background material and definitions relating to the mapping class group that appear in this section can be found in our previous paper with Margalit [8].", "Before we give the examples, we first recall the criteria of Bestvina–Bromberg–Fujiwara for showing that an element $g$ of $G$ acts on a projection complex $\\mathcal {P}$ as a WPD element." ], [ "Suppose that $\\mathcal {P}$ is a projection complex.", "Bestvina–Bromberg–Fujiwara proved the existence of a constant $C_{\\rm WPD}$ which can be used to ensure that an element acting on $\\mathcal {P}$ is a WPD element [3].", "The set-up is as follows.", "Assume that $G$ is a group that acts on $\\mathcal {P}$ and that $g$ is an element of $G$ that satisfies the following two conditions.", "There is a vertex $v$ in $\\mathcal {P}$ and $n > 0$ such that $d_v(g^{-n}v,g^n v) > C_{\\rm WPD}$ ; and there is an $m > 0$ such that the subgroup of $G$ that fixes $v, gv, \\ldots , g^m v$ is finite.", "Then $g$ is a WPD element of $G$ ." ], [ "First example", "Let $S$ be an orientable surface where $\\chi (S) < 0$ and let $f$ be a pseudo-Anosov mapping class of $S$ .", "There is a projection complex $\\mathcal {P}$ built using $f$ and its action on the curve complex $S)$ .", "We briefly recall this construction here; full details can be found in our previous paper [8].", "Fix a point $x$ in $S)$ and consider the quasi-axis bundle $\\beta = \\operatorname{EC}(f) \\cdot x$ , where $\\operatorname{EC}(f)$ is the elementary closure of $f$.", "In this context, $\\operatorname{EC}(f)$ is the stabilizer of the set of transverse measured foliations associated to $f$ considered in the space of projectivized measured foliations on $S$ .", "The vertex set of $\\mathcal {P}$ consists of the $\\mathrm {Mod}(S)$ –translates of $\\beta $ .", "Next we define the distance functions.", "Given three vertices $\\alpha _1$ , $\\alpha _2$ and $\\beta $ of $\\mathcal {P}$ , we define $d_\\beta (\\alpha _1,\\alpha _2)$ to be diameter of the union of the projections of $\\alpha _1$ and $\\alpha _2$ to $\\beta $ .", "Let $g$ be a mapping class of $S$ that does not lie in $\\operatorname{EC}(f)$ .", "We claim that $gf^n$ is a WPD element for the action on $\\mathcal {P}$ for $n$ sufficiently large.", "As $g$ does not lie in $\\operatorname{EC}(f)$ , we have that $\\beta \\notin \\lbrace g\\beta ,g^{-1}\\beta \\rbrace $ .", "Next, we have that $d_\\beta ((gf^n)^{-1}\\beta ,gf^n\\beta ) = d_\\beta (g^{-1}\\beta ,f^ng\\beta ) \\ge d_\\beta (g\\beta ,f^ng\\beta ) - d_\\beta (g\\beta ,g^{-1}\\beta )$ and thus $d_\\beta ((gf^n)^{-1}\\beta ,gf^n\\beta )$ is bounded below by $An - B$ for some constants $A,B > 0$ .", "In particular, $d_\\beta ((gf^n)^{-1}\\beta ,gf^n\\beta ) > C_{\\rm WPD}$ for sufficiently large $n$ .", "Further, as $g$ does not stabilize the measured foliations associated to $f$ , the stabilizer of $\\beta $ and $g\\beta $ is finite; see [1].", "By the Bestvina–Bromberg–Fujiwara WPD criterion we have that $f_1 = gf^n$ is a WPD element for some fixed sufficiently large enough $n$ .", "Given an element $h$ that does not lie in $\\operatorname{EC}(f_1)$ , the element $f_2 = hf_1h^{-1}$ is a WPD element and $f_1$ and $f_2$ are independent.", "Thus the action of $\\mathrm {Mod}(S)$ on $\\mathcal {P}$ is a non-elementary WPD action.", "As explained in the proof of Theorem 1.7 in our previous work, for each $L \\ge 0$ , there is an $p > 0$ such that the collection of subgroups $R_{h\\beta } = \\langle {hf^ph^{-1}}\\rangle $ is an equivariant $L$ –spinning family of subgroups.", "Therefore, by Theorem REF , the elements of $\\bar{f}_1$ and $\\bar{f}_2$ in $\\mathrm {Mod}(S)/\\left\\langle \\hspace{-1.99168pt}\\left\\langle {f^p}\\right\\rangle \\hspace{-1.99168pt}\\right\\rangle $ are independent WPD elements for its action on $\\mathcal {P}/\\left\\langle \\hspace{-1.99168pt}\\left\\langle {f^p}\\right\\rangle \\hspace{-1.99168pt}\\right\\rangle $ for a certain large enough $p$ .", "As mentioned at the beginning of this section, this above discussion works in the larger context of a group $G$ acting on a $\\delta $ –hyperbolic space using a WPD element $f$ of $G$ ." ], [ "Second example", "Again, let $S$ be an orientable surface where $\\chi (S) < 0$ .", "Let $X$ be a connected subsurface of $S$ so that for all $h \\in \\mathrm {Mod}(S)$ , either $X = hX$ or $X$ and $hX$ have nontrivial intersection (what is called an orbit-overlapping subsurface in our previous work [8]).", "There is a projection complex $\\mathcal {P}$ built using $X$ and the curve complex $S)$ .", "We briefly recall this construction here; full details can be found in our previous paper [8].", "The vertices of $\\mathcal {P}$ are the $\\mathrm {Mod}(S)$ –translates of $X$ .", "Given three vertices $Y_1$ , $Y_2$ and $X$ of $\\mathcal {P}$ , the distance $d_X(Y_1,Y_2)$ is the diameter in $X)$ of the Masur–Minsky subsurface projections of $Y_1$ and $Y_2$ to $X$  [14].", "There is a well-defined map $\\mathrm {Mod}(X) \\rightarrow \\mathrm {Mod}(S)$ ; fix an element $f$ in $\\mathrm {Mod}(S)$ that is the image of a pseudo-Anosov element on $X$ .", "Let $g$ be a mapping class of $S$ such that $\\partial X$ and $g\\partial X$ fill $S$ .", "We claim that $gf^n$ is a WPD element for the action on $\\mathcal {P}$ for sufficiently large $n$ .", "The proof is similar to the first example and left to the reader.", "Hence, as above, there are elements $f_1$ and $f_2$ in $\\mathrm {Mod}(S)$ that are independent WPD elements for the action on $\\mathcal {P}$ .", "By taking certain $p$ sufficiently large, we can ensure that the equivariant family of subgroups $R_{hX} = \\left\\langle \\hspace{-1.99168pt}\\left\\langle {hf^ph^{-1}}\\right\\rangle \\hspace{-1.99168pt}\\right\\rangle _{\\mathrm {Stab}(hX)}$ is $L$ –spinning for arbitrary $L$ .", "Hence we can ensure that the images $\\bar{f}_1$ and $\\bar{f}_2$ are independent WPD elements for the action of $\\mathrm {Mod}(S)/\\left\\langle \\hspace{-1.99168pt}\\left\\langle {f^p}\\right\\rangle \\hspace{-1.99168pt}\\right\\rangle $ on $\\mathcal {P}/\\left\\langle \\hspace{-1.99168pt}\\left\\langle {f^p}\\right\\rangle \\hspace{-1.99168pt}\\right\\rangle $ .", "Similar arguments apply to the other subgroups of $\\mathrm {Mod}(S)$ constructed in our previous work." ] ]

[ [ "A Vision to Smart Radio Environment: Surface Wave Communication\n Superhighways" ], [ "Abstract Complementary to traditional approaches that focus on transceiver design for bringing the best out of unstable, lossy fading channels, one radical development in wireless communications that has recently emerged is to pursue a smart radio environment by using software-defined materials or programmable metasurfaces for establishing favourable propagation conditions.", "This article portraits a vision of communication superhighways enabled by surface wave (SW) propagation on \"smart surfaces\" for future smart radio environments.", "The concept differs from the mainstream efforts of using passive elements on a large surface for bouncing off radio waves intelligently towards intended user terminals.", "In this vision, energy efficiency will be ultra-high, due to much less pathloss compared to free space propagation, and the fact that SW is inherently confined to the smart surface not only greatly simplifies the task of interference management, but also makes possible exceptionally localized high-speed interference-free data access.", "We shall outline the opportunities and associated challenges arisen from the SW paradigm.", "We shall also attempt to shed light on several key enabling technologies that make this realizable.", "One important technology which will be discussed is a software-controlled fluidic waveguiding architecture that permits dynamic creation of high-throughput data highways." ], [ "Introduction", "Wireless communications has come a long way, from mobile telephony in the 70s to nowadays massive, dynamic multimedia information access anywhere anytime in the era of Internet of everything (IoE).", "We have also witnessed a shift in research focus from outdoors to indoors since most data demand now tends to take place in indoor environments.", "This has motivated the emerging concept of smart radio environment which uses software-defined materials or software-controlled metasurfaces [1], [2] to engineer a radio environment suitable for wireless communications.", "Many believe this will play a role in 6G [3].", "The use of software-controlled metasurfaces for improving wireless communications is a thriving research area.", "A majority of recent efforts have been devoted to the study of adopting passive radiation elements on a programmable surface that has the ability to apply arbitrary phase shifts on the receiving radio signals and reflecting them constructively as a focussed beam towards the intended receiver.", "This approach is widely known as reconfigurable intelligent surfaces (RISs) [4].", "RIS is particularly attractive for their low power consumption and hardware cost, because of the use of relatively cheap radiating elements.", "It can be interpreted as using large surfaces present in the environment as a large aperture for collecting and transmitting radio signals for improved energy efficiency.", "Remarkably, it was reported in [5] that it is possible to achieve a 21.7 dBi antenna gain using an RIS with 256 2-bit elements at 2.3 GHz, and a 19.1 dBi antenna gain at 28.5 GHz.", "It should be noted that programmable metasurfaces can also be used to directly modulate radio-frequency (RF) carrier signals, without the need of standard components such as mixers, filters, and phase shifters, greatly simplifying the hardware complexity for wireless communications systems [6].", "The concept of smart radio environment is, however, much more than a low-cost alternative to relaying, beamforming and communication transceivers, and represents a new paradigm of engineering the radio environment through carefully designed, software-controlled metamaterials (or “meta-atoms”) that can alter their electromagnetic (EM) properties to suit the purpose of various communication applications.", "Reducing interference, enhancing security, and extending the range of communication are amongst the most obvious applications [1].", "Although the main advantages of metasurfaces are their low hardware cost and power consumption, such as in the case of RIS, utilizing programmable metasurfaces to create a smart radio environment may mean that additional signal processing and network intelligence will add to the cost and power consumption.", "This article proposes a new vision of smart radio environment that considers the use of non-radiative, trapped surface wave propagation [7], [8], as opposed to free-space propagation where radio waves are launched from the surface in [1], [2].", "The surface waves considered in this article are trapped surface waves [8] which glide at the interface of materials with different dielectric constants and the radio propagation is made to be confined to the surface.", "A unique advantage of surface wave communications (SWC) over free-space communications (FSC) is its much more favourable pathloss, which is inverse of the distance $1/d$ , instead of the inverse of the squared distance $1/d^2$ in the case of FSC.", "Also, confining the communication to the surface means that interference management becomes a lot easier since communication can be managed on a particular pathway using software-controlled waveguiding surfaces, the concept that can be enabled by a software-controlled fluidic structure [9].", "The outcome resembles a transportation network of SWC superhighways on surfaces of meta-atoms, providing various functionalities of a smart radio environment.", "The rest of this article is organized as follows.", "In Section , we provide a high-level background of SWC and highlight the unique advantages that make it particularly appealing for the smart radio environment application.", "Section presents our vision of SWC superhighways.", "Then Section describes the key enabling technologies for software-controlled SWC while Section discusses the main challenges of the proposed SWC paradigm.", "Finally, we conclude this article in Section ." ], [ "Surface Wave", "Surface wave is a non-radiating wave that propagates along the interface between two different media [7].", "The definition can be formally classified into eleven types according to their physical properties [10].", "When a radio wave is incident at a boundary from a denser medium and if the incident angle is equal to or greater than the critical angle of the media, then the radio wave will be `trapped' in the denser medium, with the evanescent fields in the rarer medium, and the wave will be confined to the surface.", "Figure REF illustrates the geometry of the directions of the waves at the interface.", "In practice, both media have finite losses, and the E-field of the surface wave will attenuate as it propagates along the interface.", "A classical result is that the power of a trapped surface wave is inversely proportional to the propagation distance, $d$ [7]: $P_{\\sf SWC}(d)\\propto \\frac{1}{d},$ which is much more desirable than the inverse squared law of what normally occurs in space wave propagation or FSC.", "Figure: Illustration of different angles and the corresponding waves.The surface resistance is associated with energy dissipation and determines the attenuation of the surface wave in the propagation direction whereas the surface reactance is associated with the energy stored in the interface, which defines the decay of the wave away from the surface in the propagation direction.", "The higher the surface reactance, the more closely the energy is stored to the surface and hence the more closely bound the wave is to the surface.", "Rectangular apertures of finite height are usually adopted as transducers to excite surface wave, with minimal space waves and reflected waves.", "The most effective surface for SWC is the one which has a purely reactive surface impedance.", "Two possible approaches to obtain high impedance surfaces are corrugated surfaces and dielectric coated conductors.", "Corrugated surfaces have purely reactive impedance that depends upon the dimensions of the grooves and humps of the surface.", "The main limitation of a corrugated structure is its directional periodic structure, which is difficult to fabricate in millimeter-wave frequency bands.", "A more viable solution is the use of dielectric coated conductors which are much easier to make.", "The dielectric layer should have a high dielectric constant and low conductivity.", "Also, the surface impedance can be further adjusted by layering several different dielectric layers on top of each other.", "In [8], a 52 GHz wideband trapped SWC system by utilizing the dielectric coated conductor approach was implemented.", "Figure REF shows the E-field distribution along a dielectric coated conductor with surface impedance $j200\\Omega $ at 60 GHz.", "Figure: Simulation results showing the E-field distribution along the direction of propagation at 60 GHz with the surface impedance of j200Ωj200\\Omega ." ], [ "The Vision: SWC Superhighways", "Less propagation loss of SWC means that it can be preferable to take a detour and travel a longer distance along walls or surfaces but still have higher received power than a direct path propagating in the free space, see Figure REF .", "SWC is thus super-energy-efficient and data can reach farther distance with the same energy consumption.", "More remarkably, confining the communication signals to the surface helps contain high-speed data streams, and simplifies interference management.", "Figure: SWC versus FSC: Longer distance beats shorter distance.Our vision of the future smart radio environment therefore resembles a transportation network of communications superhighways with exceptionally localized data access, as depicted in Figure REF .", "In this vision, dedicated pathways are created on programmable metasurfaces to carry superfast data that travel along the surfaces to reach the user equipment (UE) or arrive at a position near the UE.", "In the latter case, FSC will provide the last hop from the metasurface to the UE over a very short distance.", "The pathways are software-controlled to adapt to the radio environment and always provide the best routes requiring the least power consumption to the UEs free of interference, thanks to the extraordinary spatial selectivity of SWC.", "Figure: The vision of SWC superhighways in a smart radio environment with exceptionally localized wireless coverage (contactless or not).In this vision, one reality is that radio signals only appear where they should, and wireless coverage, contactless or not, is exceptionally localized.", "By contrast, communications relying on FSC tends to have radio waves unintentionally occupying the entire space unless multiple antennas are used to beamform the radio signals to be confined into certain space that requires advanced signal processing and resource management.", "This is the inherent problem in wireless communications.", "As a matter of fact, much processing and intelligence in wireless networks for 5G and beyond go to the management and control of radio signals for massive connectivity that allow signals to coexist without causing harmful interference to each other.", "This is a very challenging task because radio waves naturally propagate in all directions, and when radio waves hit objects along the way in the environment, the reflection and diffraction further complicate the interference pattern.", "This task will, however, be greatly simplified in SWC, since the radio waves will be kept on the surface and their presence is absolutely predictable.", "As envisaged in Figure REF , in addition to walls, surfaces like office desk can also be equipped with meta-atoms to provide zonal, targeted data access, making possible the ultimate interference-free communications in indoor environments.", "Figure: SWC inherently confines communication signals to the surfaces and causes no interference to coexisting UEs without the need of sophisticated signal processing and interference management.A natural result of this vision is that interference management becomes traffic control, and it is highly location-driven.", "In other words, the key will be to localize all the UEs relative to the infrastructure of metasurfaces in the environment, and determine the best possible routes to serve the UEs.", "Interestingly, localization in the SWC paradigm can be less complex.", "The reason is that there will be a large number of meta-atoms as anchors with known locations available in the environment that can take part in the localization task for UEs.", "In addition, the UEs are likely to be in close proximity of the anchors, and have direct line-of-sights (LoS) for ranging and localization.", "The assurance of sufficiently large number of direct LoS paths within a short distance from the UEs makes low-complexity high-resolution localization realizable.", "It is also worth pointing out that simultaneous SWC in the same pathway is possible as the metasurface can operate on a wide frequency band.", "Another feature in the SWC-based smart radio environment paradigm is the massive deployment of antennas on surfaces.", "Apart from the cases where communications takes places via direct contact with the UEs (e.g., laptops on desk), as in the conventional RIS concept, the surfaces will be equipped with antennas as widely as possible so that the UEs can be reached using short-range FSC anywhere in the environment.", "The key difference is that the meta-atoms on the surfaces now need to switch between FSC and SWC, by acting as radiating elements and the propagation medium, respectively, depending on the situations, and such switching is done seamlessly." ], [ "Enabling Technologies", "The vision of the SWC paradigm is not a dream but can be a realistic revolution being sought in 6G.", "In this section, we offer some ideas in the potential enabling technologies." ], [ "Dual-Functionality Meta-Atoms", "The SWC paradigm suggests a hybrid communication network taking full advantage of SWC and FSC.", "To realize this, each meta-atom may serve as a tuneable propagation medium or radiating element at any given time and needs to be able to switch between the two functionalities.", "SWC propagation over a metasurface as the denser medium is a natural phenomenon, and the tricky part is how the signal comes off the surface to reach the intended UE.", "In general, two `exits' are possible.", "In the first type of `exit' where the UE is in direct contact with the metasurface (for example, a laptop on a “smart” desk), it is rather straightforward to have a transducer integrated on the UE to easily capture the signal and receive the data.", "By contrast, in the case where the UE is not in contact with the surface, it requires that the metasurface have the capability to transform SWC into space wave to propagate over air to leap to the UE.", "Also, it is expected that the metasurface when acting as radiating elements, has the signal processing capability to form focused signal beams towards the intended UE and avoid interference, in the same way as in the RIS applications.", "In the leaky-wave and holographic antenna literature, periodic metallic geometries have been well studied to radiate an RF signal from a surface [11].", "In particular, a trapped surface wave can be diffracted by a periodic metallic geometry that will let part of the surface wave scatter into the free space as space wave.", "It has also proven possible to accurately control the direction of the radiation off the metasurface [12].", "Several approaches have been proposed to reconfigure the amount of radiation and the angle of departure of the space wave from a diffracted surface, and they include using active semiconductor devices and metamaterials [13].", "Furthermore, recent advances in transparent conductive sheet using graphene, carbon nanotube or metallic compounds will help the realization of meta-atoms that can be lightweight and invisible on the surfaces." ], [ "Software-Controlled Fluidic Waveguiding Structures", "The SWC vision is only realizable if we have a mechanism to create on demand dynamic pathways on surfaces for SWC.", "This is important for interference management and pathloss reduction of the data streams on the metasurface.", "Normally, an RF signal that leaves the surface wave transducer will travel along the surface wave plane, following a radial pattern with an angular coverage determined by the electrical size of aperture width of the transducer [8].", "Research on dynamic creation of pathways on the surface is very limited although high surface impedance will surely enhance surface wave propagation.", "A great deal of research efforts in the SWC area have been trying to realize reconfigurable surface impedances on a surface.", "Flexibly and dynamically creating pathways on a surface is a much more challenging task but practically achievable.", "One possible approach is to leverage a microfluidic system where a large number of micro-tubes are pre-fabricated in the surface substrate of a few millimeters thickness.", "The micro-tubes are connected to an array of software-controlled pumps which can inject conductive fluid into the tubes if required.", "If some of the pumps are activated, a selected group of the micro-tubes will be filled with conductive fluid, which then creates an integrated waveguiding structure to form a tunnel of pathway.", "The pattern which the conductive fluid-filled micro-tubes form, determines the pathway for SWC.", "In Figure REF , some preliminary results are provided to illustrate the feasibility of such concept with a surface impedance of $j96.5\\Omega $ at the operating frequency of 28 GHz.", "In this example, Galinstan that is a liquid metal alloy with high conductivity is chosen as the conductive fluid.", "It is worth noting that software-controlled fluidic structures have recently been investigated for antenna design [9].", "The knowhow in that application is anticipated to be useful in the engineering and signal processing of the microfluidic system for programmable metasurfaces.", "This architecture makes possible joint optimization of the signal and data streams, the resource allocation and management of the communication, and the propagation media that accommodate the communications.", "Intelligence in such holistic approach will be essential.", "Figure: Guided surface wave propagation over a preset pathway." ], [ "Artificial Intelligence (AI) Empowered SWC", "Recent advances have seen AI especially deep learning to be given a major role for 5G and beyond communications [3].", "There have been numerous successful examples of employing AI for wireless communications, from physical layer designs such as channel estimation and precoding, to network resource allocation such as traffic control and user pairing, to security and authentication, and to topology formation and management, to fault prediction and detection, and so on.", "Parallel can be drawn in the SWC paradigm and AI will serve as the brain to empower the superhighway network on the metasurface.", "There are several technical directions in which AI is anticipated to be key to the realization of the concept.", "One obvious avenue is the localization of UEs in the smart radio environment where connections need to be established seamlessly.", "In the SWC paradigm, the UEs' locations are most important as most communication remains on the surface but only leaps to reach the UEs in the last hop if required.", "The availability of the locations of the UEs allows the data traffic to be managed on the surface with carefully designed pathways such that the power consumption (and hence the pathloss) is minimized and the interference over the surface is also eliminated.", "Localization of the UEs is expected to be done easily from simple energy sensing, as the last hop from the surface to the UEs is likely to be short-ranged and has the direct LoS.", "Also, the large number of meta-atoms ensures more than sufficient LoS paths to exist and localize the UEs in a way similar to the traditional multilateration approach.", "The difference is that in this case, most of the paths come from the same direction with different energy levels, instead of paths coming from a variety of directions.", "Apart from this, metasurfaces are preinstalled to form a smart radio environment, meaning that a fingerprinting approach can be effectively utilized to localize the UEs.", "This can be addressed using deep learning by training an artificial neural network (ANN) over some simulated energy level data given the floor plan of the environment.", "Real-time localization of the UEs can then be easily achieved by a simple pass to the ANN after taking energy measurement from the meta-atoms.", "AI can also be useful to find the best pathways for the data to travel from the base station (which is now equipped with a surface wave launcher connecting to a metasurface network) to the data-receiving UEs.", "Such SWC map can be derived from simulated data using deep learning.", "The key would be to link the UEs' locations and data traffic demand with the physical structures and resources of the metasurfaces making up the radio environment.", "The optimal SWC superhighway network as a function of the UEs' parameters will be learned." ], [ "Fully Reconfigurable Meta-Atoms", "Metamaterials have been researched for nearly two decades, having generated many mind-blowing results including cloaking that can make objects invisible to sensors by controlling their EM radiation.", "For mobile communications, metamaterial-based antennas are also hopeful to make small-sized highly efficient wideband antennas possible.", "The notion of metasurface-empowered smart radio environments is expected to make a huge impact in 6G and has been led by the mainstream efforts of RIS where passive meta-atoms are considered.", "Despite the promising results of RIS, the fact that the meta-atom is based on the current microstrip patch antenna technology means that the bandwidth unfortunately tends to be narrow, which hinders its development for high frequency bands.", "Exploring the use of double negative (DNG) metamaterials would be key to unlock the bandwidth limitation and miniaturize the meta-atoms for increasing the aperture for performance improvement.", "In addition, the SWC paradigm requires that the meta-atoms not only act as antennas for radiation but operate as a medium for surface wave propagation.", "For the latter mission, research is required to devise a technology to adaptively control the characteristic impedance of the meta-atoms so that the surface can be optimized for adhering the radio waves to the surface for superior propagation loss and ease of interference management.", "A micro-electronics approach that can achieve fine control and resolution of the impedance of a meta-atom via a digital signal should be studied.", "Note that the current state-of-the-art meta-atom technology nevertheless comes with the limitation that the amplitude and phase of the radiation off the meta-atoms are strongly coupled.", "This will need to be tackled if metasurfaces are to be fully intelligent for advanced coding and signal processing.", "Even if this can be achieved, deciding on the appropriate surface impedance is far from trivial which couples with the action to create SWC pathways that affects the characteristic impedance of the medium above the surface.", "Depending upon the communication networking and whereabouts of the UEs, it may also be necessary for a meta-atom to act as a radiating element on one frequency but a propagation medium for SWC on another frequency at the same time.", "Such dual functionality will certainly need a more advanced meta-atom technology that can achieve independent control of the radiation characteristics over multiple frequency bands.", "All in all, programmable, reconfigurable and multi-functional meta-atom technologies will need to be sought." ], [ "Pathway Creation and Control", "Creating dynamic pathways on surfaces is the central idea of the SWC paradigm.", "In Section REF , an enabling technology which utilizes a microfluidic system of liquid metal to provide on-demand pathways is discussed.", "However, many obstacles are yet to be overcome.", "One apparent issue is the choice of the conductive fluid used for creating the pathways.", "Liquid metals such as mercury are toxic and should not be used.", "There is also a nontrivial trade-off between the response time which is related to its density and the conductivity of the fluid.", "More will need to be investigated to fully understand the practicality and feasibility of such approach in a living environment.", "Another pressing challenge is the fabrication of tube spaces in millimetre scale for creating an adaptive metasurface.", "It gets much more difficult when realizing the fact that micro-tubes do have finite thickness and will distort propagation along the fluid-made pathways.", "A thorough analysis and proper design of the architecture that integrates with the micro-fluidic system to make possible rapid distribution of conductive fluid with great precision and control will be indispensable.", "Moreover, the size of the micro-tubes and their spacing are frequency-dependent.", "How to have an implementable structure that permits flexible size and spacing to accommodate different frequency bands is extremely challenging, not to mention the difficulty of realizing different pathways in the same space at the same time for different frequencies.", "Although this issue may be mitigated by careful frequency planning and pathway optimization, much more research will need to be conducted to obtain a feasible architecture for dynamic waveguide technologies such as the fluid-based approach discussed above." ], [ "Model-Driven AI Signal Processing and Networking", "Signal processing and communication networking in such a smart radio environment depend greatly upon the floor plan of the indoor environment because metasurfaces on walls dictate the paths and positions in which data can be delivered to the UEs.", "One would expect that it is possible to use the floor plan as a model to develop and train a deep ANN that takes the UE's locations and service demands as inputs and produces as output the signal processing and networking solution.", "If such AI solution becomes available, then real-time optimization of the signal processing for SWC will be straightforward.", "Doing this, however, requires several obstacles to be tackled.", "The first hurdle is to come up with a general representation of the floor plan that contains the essential information in a concise and manageable fashion.", "This task will impact on the training efficiency of the ANN that extracts the logical features of the environment and translates them into parameters that are ready for the optimization.", "Secondly, the inputs to the ANN need to be properly defined which may be the locations of the UEs, their service demands, or even their energy signatures.", "It is in fact more challenging to define the outputs of the ANN as the variable space can be gigantic.", "This could include, from the coding rate of transmission to the transmit power, the pathway, the frequency allocation, the beamforming of meta-atoms to even security-related processing and remote charging.", "Besides, the physical constraints of the meta-atoms and the metasurface as a whole will need to be incorporated in the design.", "Even after the inputs and outputs of the ANN are defined, it is unclear how the ANN can be trained properly.", "Supervised learning requires the availability of labelled training examples whereas unsupervised learning looks for undetected patterns of datasets with no pre-existing labels.", "There is a long way to figure out how reliable datasets can be obtained to train such ANN.", "Ad hoc heuristics by using a combination of traditional techniques should present useful examples to start building a good dataset.", "Transfer learning is also expected to be useful to ensure generalizability to cope with a range of situations." ], [ "User-Centric Context-Aware Service-Based Coverage", "The concept of smart radio environment suggests strongly an integration between UEs and the environment.", "The large area of metasurfaces makes available multi-dimensional time-series of energy spectrum of the UEs that captures very rich behavioural data of the users and contextual information.", "The enriched understanding of the UEs and their needs mean that context-aware service and applications can be provided to the level that has not been achievable before.", "Note that the large number of meta-atoms brings the service of a massive number of sensors as well that enable numerous applications such as remote patient monitoring.", "Opportunities are plenty in terms of applications but it would be tricky to interpret the signals from sensors with highly uneven distributions." ], [ "Security and Anti-Wiretapping", "With increasing reliance on mobile applications on our daily life, security has become a major concern.", "In 5G and beyond systems, physical-layer security schemes appear to provide an additional layer of defence to complement traditional cryptographic approaches by exploiting the randomness of wireless channels and their unreplicability.", "The SWC network however makes the communication data more exposed since the signals glide on surfaces and are more predictable, although one merit of SWC is its high predictability to allow simple interference management.", "Wiretapping on metasurfaces is a real threat that needs to be looked at carefully and addressed.", "Anti-wiretapping techniques will be key to ensure security on metasurfaces.", "Apart from operating as radiating and propagation elements, meta-atoms should act as sensors to probe, identify and localize any suspected wiretappers or adversaries so that SWC can be rerouted.", "Together with AI, metasurfaces should possess the brain power to predict suspicious activities of malicious intrusion, and optimize SWC accordingly." ], [ "Conclusion", "Recent research has seen the notion of smart radio environment emerge as a mainstream effort to shape the environment to support communication needs by using software-controlled metasurfaces.", "Contrary to the conventional studies, this article has advocated the vision to utilize metasurface not only as a radiating platform but propagation medium taking advantage of SWC for much less pathloss and simple interference control.", "The SWC paradigm greatly reduces unnecessary radiation off the surfaces and only beams to the UEs in the last leg if needed.", "This novel SWC concept is made possible by several enabling technologies, including the one that can dynamically adapt communication pathways on a metasurface by digitally controlling a microfluidic system of liquid metal, all of which have been discussed.", "This article has also touched upon the opportunities and challenges that come with the vision.", "It is hoped that this article will serve as a catalyst to trigger further research to make the SWC vision practically feasible.", "[Figure: NO_CAPTION His current research centers around 5G and beyond mobile communications.", "He is a co-recipient of the 2013 IEEE Signal Processing Letters Best Paper Award and the 2000 IEEE VTS Japan Chapter Award at the IEEE Vehicular Technology Conference in Japan in 2000, and a few other international best paper awards.", "He is Fellow of IEEE and IET and is also on the editorial board of several international journals.", "He is the Editor-in-Chief for IEEE Wireless Communications Letters since 2020.", "[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION" ] ]

[ [ "Constraints on low-mass, relic dark matter candidates from a\n surface-operated SuperCDMS single-charge sensitive detector" ], [ "Abstract This article presents an analysis and the resulting limits on light dark matter inelastically scattering off of electrons, and on dark photon and axion-like particle absorption, using a second-generation SuperCDMS high-voltage eV-resolution detector.", "The 0.93 gram Si detector achieved a 3 eV phonon energy resolution; for a detector bias of 100 V, this corresponds to a charge resolution of 3% of a single electron-hole pair.", "The energy spectrum is reported from a blind analysis with 1.2 gram-days of exposure acquired in an above-ground laboratory.", "With charge carrier trapping and impact ionization effects incorporated into the dark matter signal models, the dark matter-electron cross section $\\bar{\\sigma}_{e}$ is constrained for dark matter masses from 0.5--$10^{4} $MeV$/c^{2}$; in the mass range from 1.2--50 eV$/c^{2}$ the dark photon kinetic mixing parameter $\\varepsilon$ and the axioelectric coupling constant $g_{ae}$ are constrained.", "The minimum 90% confidence-level upper limits within the above mentioned mass ranges are $\\bar{\\sigma}_{e}\\,=\\,8.7\\times10^{-34}$ cm$^{2}$, $\\varepsilon\\,=\\,3.3\\times10^{-14}$, and $g_{ae}\\,=\\,1.0\\times10^{-9}$." ], [ "Introduction", "During the past two decades, many significant constraints on weakly interacting massive particle (WIMP) dark matter (DM) for masses above 10 GeV$/c^{2}$ have been published (e.g.", "[1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and references therein).", "In contrast to the standard WIMP, well-motivated alternative models at masses below a few GeV$/c^{2}$ that require at least one new gauge boson to satisfy the observed relic density remain relatively unexplored [11].", "We undertook a search for such candidates with a SuperCDMS high-voltage eV-resolution (HVeV) detector [12], [13], [14].", "We constrain three DM candidates: (1) light DM $\\chi $ , referring to thermal DM particles that inelastically scatter with electrons via a new dark sector force mediator [15], [16]; (2) dark photons $V$ that kinetically mix with Standard Model (SM) photons [17], [18], [19]; and (3) axion-like particles (ALPs) that are absorbed by an atom via the axioelectric effect [20], [21], [22].", "These candidates can create electron-hole ($e^{-}h^{+}$ ) pairs in the phonon-mediated cryogenic silicon HVeV detector.", "In a prior work [12], we undertook an above-ground search with a first-generation Si HVeV detector.", "Contemporaneously, the SENSEI Collaboration reported an underground search with Skipper CCDs [23].", "Both works excluded new parameter space for light DM scattering and dark photon absorption in similar mass ranges, but did not report on the axioelectric coupling, which is most strongly constrained by astronomical observations [24], [25], [26], [27].", "In this work, we analyze a slightly larger above-ground exposure of 1.2 gram-days of a second-generation Si HVeV detector with the same dimensions but modified sensor design compared to [12], leading to an improved phonon energy resolution as good as $\\sigma _{E}=3$  eV at the single-$e^{-}h^{+}$ -pair energy (3 % charge resolution for a 100 V bias).", "Using signal models that include the contributions from charge carrier trapping and impact ionization, we report the constraints obtained from a blind analysis on $\\chi $ scattering for DM masses from 0.5–$10^{4}$  MeV$/c^{2}$ , as well as $V$ and ALP absorption for masses from 1.2–50 eV$/c^{2}$ ." ], [ "Experimental setup", "The data were acquired in a surface laboratory at Northwestern University (Evanston, IL), with the overburden and environmental radioactivity of a typical steel-concrete building.", "The detector is made of a 0.93 gram high-purity Si crystal ($1\\times 1\\times 0.4$  cm$^{3}$ ).", "We clamped the detector between two printed circuit boards for thermal sinking and electrical connections.", "To reject correlated environmental noise, we installed an anti-coincidence (veto) detector adjacent to the HVeV detector in the same light-tight copper housing mounted to the cold finger of an Adiabatic Demagnetization Refrigerator (ADR).", "More information about the detector setup and the infrared radiation shield is available in Ref. [14].", "SuperCDMS HVeV detectors measure phonons created by particle interactions in the Si crystals with two distributed channels of Quasiparticle-trap-assisted Electrothermal-feedback TESsTransition edge sensors [28].", "(QETs) [14].", "The QETs fabricated on this device have a superconducting transition temperature $T_{c}\\approx 65$  mK.", "One QET channel is a square with a sensitive area of 0.5 cm$^{2}$ , and the other is a surrounding frame of equal area.", "Both are on the detector's top surface.", "On the bottom surface, an aluminum grid with 5 % surface coverage provides a uniform electric field between the two surfaces.", "The veto detector consists of a single TES on a thin Si wafer that is identical to the TES described in Ref.", "[29] but with $T_c\\approx 52$  mK.", "We cycled the ADR daily from 4 K to the base temperature and then regulated it at 50–52 mK during data taking to obtain a 10–12 hour/day hold time [14].", "To induce Neganov-Trofimov-Luke (NTL) amplification [30], [31], the aluminum grid was biased at $V_{\\textrm {bias}} = 100$  V while the QETs and detector housing were held at ground potential.", "At the start of each daily cycle, we set the operating point of each QET to $\\sim 300~\\textrm {m}\\Omega $ (about 45 % of its normal-state resistance) and recalibrated the detector using a 635-nm laser that was fiber-coupled from room temperature.", "Each QET was read out with a DC superconducting quantum interference device (SQUID) at 1 K operated in a flux-locked feedback loop, and the signals were digitized continuously at 1.51 MHz.", "The laser intensity was adjusted to achieve a mean number of $e^{-}h^{+}$ pairs per pulse between 1 and 4, which produced enough events for calibration up to seven $e^{-}h^{+}$ pairs per event.", "We also took a dedicated laser dataset in which we varied the crystal temperature but held the QETs at their nominal operating point; we used this dataset to reconstruct the temperature dependence of the QET responsivity." ], [ "Data Collection and Event Reconstruction", "A raw exposure of 3.0 gram-days was collected over 7 days during April–May of 2019.", "By partitioning the continuous-acquisition data stream into 10-second long intervals, we performed a three-stage blind analysis.", "The first second of each interval, i.e.", "10 % of the data, was used to develop the analysis pipeline but was not included in the final spectrum.", "Data from seconds 2–3 of each interval were unblinded to verify that the analysis pipeline was indeed invariant under the presence of a larger statistical sample.", "Given that the initial unblinding satisfied this condition, we unblinded the remaining data and analyzed seconds 2–10 from each data partition, i.e.", "90 % of the data defined as the DM-search data, to extract the final results.", "To identify physics events, we triggered on pulses within the continuous-acquisition data stream offline.", "To issue triggers, we first applied a matched filter to the data stream using an exponential pulse template (20 $\\mu $ s rise time and 80 $\\mu $ s fall time) and then set a trigger threshold equivalent to $\\sim $  0.4 $e^{-}h^{+}$  pairs for event identification.", "The event trigger time is the time at which the triggered pulse is at its maximum.", "After verifying that the two QET channels on the HVeV detector have equal gain, this trigger scheme was applied to the sum of the two channels' data streams and separately to the veto detector.", "Pulse energy and time were reconstructed using an optimal filter (OF) algorithm [32], [33].", "The OF algorithm requires a pulse template and the noise power spectral density (PSD).", "We constructed the pulse template for the OF algorithm from the laser-calibration event pulses with high-frequency noise filtered out.", "We measured the noise PSD on an hourly basis to account for variations of the environmental noise, using the first 100 seconds of each one-hour data partition with triggered pulses removed.", "The pulse amplitude and time that minimize the frequency-domain $\\chi ^{2}$ were determined within a time window of [$-\\,678\\,\\mu $ s, $+\\,2034\\,\\mu $ s] centered on the trigger.", "These amplitude and time quantities of the OF algorithm were also used to compute a time-domain $\\chi ^{2}$ which was used later in the analysis.", "Temperature fluctuations at the detector and small variations in the HV bias resulted in a small variation (${<1\\,\\%}$ ) in the detector gain.", "We used the quantized $e^{-}h^{+}$ -pair peaks in the aforementioned temperature-controlled sample spectrum, as well as the daily laser-calibration spectra, to linearly correct for the temperature and voltage dependencies.", "We then corrected for nonlinearities in the detector response with a second-order polynomial.", "To calibrate the OF pulse amplitudes to energies we rescaled the $e^{-}h^{+}$ -pair peaks assuming $E_{n}= n(E_{\\gamma } + e\\cdot V_{\\textrm {bias}}), $ where $n$ denotes the $n$ -photon absorption peak, ${E_{\\gamma }=1.95}$  eV is the laser photon energy, and $e$ is the absolute value of the electron charge.", "Figure REF (top panel) shows the resulting spectra from the DM-search and laser-calibration data.", "Figure: The top panel shows the DM-search spectrum (red) in units of event rate per 3 eV bin (left y-axis) and the laser-calibration spectrum (blue) in units of events per 3 eV bin (right y-axis).", "Both spectra show the data measured with a detector bias of 100 V after applying the live-time and data-quality cuts.", "The peak seen at ∼\\sim  50 eV in the DM-search data is due to non-quantized events restricted to the outer QET channel .", "Light gray-shaded regions on the left- and right-hand sides mark the energy ranges outside the region of interest; vertical lines correspond to the phonon energy E n E_{n} of the nn-photon absorption peak (Eq. ).", "The black curve is an example of a signal produced by electron-recoiling dark matter particles with a mass of 1 GeV/c 2 /c^{2} and form factor F DM ∝1/q 2 F_{\\textrm {DM}} \\propto 1/q^{2}.", "This model assumes a Fano factor of F=0.155F=0.155, an impact ionization (II) probability of 2 %, and a charge trapping (CT) probability that varies from 0–15 % shown by the hatched region.", "The curve is scaled to the dark matter-electron cross section σ ¯ e \\bar{\\sigma }_{e} that sets the limit at the 2 nd ^{\\textrm {nd}} e - h + e^{-}h^{+}-pair peak.", "The bottom panel shows the binned efficiency data ϵ(E i )\\epsilon (E_{i}) (grey solid line), where the corresponding shaded region indicates the 1σ1\\,\\sigma statistical uncertainty in each bin.", "The red dashed curve is the efficiency curve, and the corresponding shaded region is the conservative efficiency uncertainty envelope, which accounts for the statistical and systematic uncertainties." ], [ "Data selection", "To ensure accurate event reconstruction, individual live-time intervals from the DM-search and laser-calibration data were discarded (cut) based on various criteria: (1) the ADR temperature to exclude data outside the range of the temperature calibration; (2) the pre-pulse baseline averaged over one-second intervals to reject periods of time when the detector was still recovering from a preceding high energy deposition; and (3) the trigger rate to remove bursts of non-DM triggers.", "The trigger-rate cut was not applied to the laser-calibration data.", "The data remaining after these live-time cuts define the science exposure for this analysis, and yielded a DM-search exposure of 1.2 gram-days.", "To reject poorly reconstructed events in the DM-search exposure, a set of event-by-event data-quality cuts were applied based on: (1) the difference between the OF-determined pulse time and the event trigger time to reject noise triggers and pulses affected by pile-up events; (2) the event-by-event average pre-pulse baseline to ensure the detector is at a steady working condition before an event occurs; and (3) the energy-dependent frequency- and time-domain $\\chi ^{2}$ quantities to remove pile-up events and baseline excursions that are unlikely to have been caused by DM-triggers.", "To define the cuts, we determined the nominal distributions of each parameter using the laser-calibration data and discarded events in the DM-search exposure exhibiting an excursion of $>3\\,\\sigma $ in any of these parameters.", "Lastly, we rejected events with a coincident triggered event in the veto detector.", "The cut efficiency as a function of phonon energy was determined using the laser-calibration data after applying the live-time cuts to pulses coincident with the laser trigger signal.", "Pile-up events that occurred within a laser-event trigger window were included as part of the efficiency calculation.", "The binned efficiency $\\epsilon (E_{i})$ is the fraction of events in the $i$ -th bin that pass the quality cuts.", "We expect the efficiency to be smooth; however, our measurement of $\\epsilon (E_{i})$ shown in Fig.", "REF (bottom panel) has both statistical fluctuations and systematic uncertainty.", "In order to avoid folding these effects into the final results, we fit a smooth function to $\\epsilon (E_{i})$ and assigned a conservative envelope that accounts for the statistical and systematic uncertainties.", "The systematic uncertainty is due to pile-up events that were not rejected by the live-time cuts, resulting in a misreconstruction of the energy.", "Although this envelope was propagated as part of the total experimental uncertainty in the final results, we verified that it is not the dominant source of uncertainty.", "The energy region of interest (ROI) for this analysis is 50–650 eV.", "The lower bound guarantees inclusion of the full single-$e^{-}h^{+}$ -pair peak at 100 V bias and a trigger efficiency consistent with unity.", "We set the upper bound at 650 eV to focus on the corresponding low-mass ranges where this analysis has competitive sensitivity." ], [ "DM Signal Models", "The blinded DM-search data were analyzed to set exclusion limits on light DM $\\chi $ scattering as well as dark photon $V$ and axion-like particle (ALP) absorption.", "The DM models for $\\chi $ , $V$ , and ALPs are identical to those used in Ref.", "[12] and [34].", "We set limits on the kinetic mixing parameter $\\varepsilon $ for $V$ , the axio-electric coupling $g_{ae}$ for ALPs, and the effective DM-electron cross section $\\bar{\\sigma }_{e}$ for $\\chi $ .", "In all cases we assume that the respective DM candidate constitutes all of the galactic DM with a local mass density of $\\rho _{\\textrm {DM}} = 0.3$  GeV$c^{-2}$ cm$^{-3}$ .", "The $V$ and ALP absorption rates are proportional to the photoelectric absorption cross section $\\sigma _{pe}$ of the Si detector.", "Discrepancies in the literature for $\\sigma _{pe}$ [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50] for regions within our analysis range led us to define nominal, upper, and lower photoelectric cross-section curves to accommodate the range of published values.", "The nominal $\\sigma _{pe}$ curve follows the approach taken in Ref.", "[51], with data from Ref.", "[42] for photon energies below 1 keV.", "The upper and lower $\\sigma _{pe}$ curves are derived from tracing upper and lower bounds of the published data after applying temperature corrections, along with the nominal curve data that did not have temperature corrections applied.", "The corrections account for the temperature dependence of indirect, phonon-assisted absorption that occurs at energies below the direct band gap ($\\sim $ 3 eV).", "We followed the methodology and analytical model for photon absorption found in Ref.", "[52] to extrapolate the data below 4 eV to a temperature of 50 mK.", "This analysis adopted the same ionization production model as used in Ref.", "[12] to compute the mean number of $e^{-}h^{+}$ pairs $n_{eh}$ produced for an interaction with a given recoil/absorption energy.", "For recoil/absorption energies above the Si band gap $E_{gap} = 1.2$  eV but below the average energy per $e^{-}h^{+}$ pair $\\epsilon _{eh} = 3.8$  eV, $n_{eh}=1$ ; for energies above $\\epsilon _{eh}$ , we determined $e^{-}h^{+}$ pair probabilities from binomial distributions using selected Fano factor values, $F$ .", "The total phonon energy measured for an event, $E_{ph}$ , is the recoil/absorption energy $E_{r}$ plus the energy produced by the NTL effect: $E_{ph} = E_{r} + n_{eh} \\cdot e \\cdot V_{\\textrm {bias}}$ where the ionization production model and Fano statistics determine the distribution for $n_{eh}$ .", "We combined the signal models with a charge trapping (CT) and impact ionization (II) likelihood model, which mainly contributes to the distribution of events between quantized $e^{-}h^{+}$ -pair peaks [53].", "Charge trapping occurs when an electron or hole falls into a charge vacancy in the crystal, reducing the total number of electrons or holes that traverse the entire detector and lowering the measured energy for an event.", "Impact ionization occurs when a moving charge in the crystal liberates an additional loosely bound charge, thereby increasing the measured energy for an event.", "We determined the CT and II probabilities by fitting the model used in Ref [53] to the laser-calibration data.", "The results from the fit are $11 \\pm 3\\,\\%$ and $2^{+3}_{-2}\\,\\%$ for the CT and II probabilities, respectively, and were subsequently used to generate the signal models.", "Because we were unable to determine an energy dependence of the energy resolution within the ROI for this analysis (50–650 eV), the signal models were convolved with a weighted average energy resolution: $\\sigma _{\\left< E \\right>} = 3.6$  eV.", "We determined $\\sigma _{\\left< E \\right>}$ by averaging over the resolutions of the first six, Gaussian-fitted $e^{-}h^{+}$ -pair peaks from the combined laser-calibration data weighted by the corresponding uncertainty in each peak.", "Lastly, we multiplied each signal model by the efficiency curve (bottom panel of Fig.", "REF ) as well as the exposure (1.2 gram-days).", "An example of a 1 GeV$/c^{2}$ light DM signal model is shown in the top panel of Fig.", "REF .", "Figure: 90 % C.L.", "limits on the effective dark matter-electron scattering cross section with form factor F DM =1F_{\\textrm {DM}}=1 (top) and F DM ∝1/q 2 F_{\\textrm {DM}}\\propto 1/q^{2} (bottom) and with Fano factor of 0.155 (solid-blue curve).", "The light blue band represents our estimate of the systematic uncertainty, which is dominated by varying the Fano factor assumption in the ionization model from FF = 10 -4 ^{-4} to 0.3.", "Other direct detection constraints shown include SuperCDMS HVeV R1  (red), DAMIC  (green), SENSEI  (orange), XENON10 , (teal), and XENON1T  (pink).Figure: 90 % C.L.", "limits on the dark photon (VV) kinetic mixing parameter ε\\varepsilon (top) and axioelectric coupling constant g ae g_{ae} (bottom) with Fano factor of 0.155 (solid-blue curve).", "The light blue band represents our estimate of the systematic uncertainty, which for masses ≳\\gtrsim 4×10 -3 4\\times 10^{-3} keV/c 2 /c^{2} is dominated by varying the Fano factor assumption in the ionization model from FF = 10 -4 ^{-4} to 0.3; for masses ≲\\lesssim 4×10 -3 4\\times 10^{-3} keV/c 2 /c^{2}, the uncertainty is dominated by the discrepancy in the photoelectric absorption cross section.", "Other direct detection constraints shown for VV and ALPs include SuperCDMS Soudan  (maroon), XENON10 (teal), and XENON100 (purple) ; additional constraints on VV include SuperCDMS HVeV R1  (red), DAMIC  (green), SENSEI  (orange), and anomalous energy loss mechanisms in the Sun .", "For the axioelectric coupling, the entire region shown is disfavored by the observed cooling of red giant , and white dwarf stars , ." ], [ "Limit Setting", "The Poisson exclusion limit for each DM model was calculated independently for the first six $e^{-}h^{+}$ -pair peaks using a limit setting window of $\\pm ~3\\,\\sigma _{\\left< E \\right>}$ centered on each peak.", "While taking into account the look-elsewhere effect, we selected the lowest limit amongst the individual $e^{-}h^{+}$ -pair peaks at each DM mass to obtain a final limit with a 90 % confidence level (C.L.).", "This limit calculation differs from Ref.", "[12], which determined the limits using the Optimum Interval (OI) method [59], [60].", "Due to the improved energy resolution of this analysis compared to Ref.", "[12], the OI method was found to be overly sensitive to the shape of the expected DM signals measured in the detector and thus to the effects of CT and II, leading to systematic uncertainties that are difficult to estimate.", "In contrast, the Poisson method applied to this analysis is insensitive to these systematic effects.", "A comparison of the two methods finds up to a factor of 2 stronger limits with the OI method due to the sensitivity to the model shape (the same comparison performed on the Ref.", "[12] analysis results in no such difference due to the poorer energy resolution).", "In this analysis we used the more conservative Poisson limit setting method, as it is more effective at constraining the systematic uncertainties.", "To quantify the impact of systematic uncertainties, the limits were recalculated with Gaussian distributed random variates for the energy calibration, energy resolution, CT and II fractions, and efficiency, according to their corresponding means and uncertainties.", "For the photoelectric absorption cross section, we made a random choice between the lower, upper, and nominal curves.", "At each mass, we took the average from all trials and used the $\\pm \\,1\\,\\sigma $ equivalent values from the resulting limit distribution as the limit uncertainty.", "The limits and their propagated uncertainty are calculated separately using three different values for the Fano factor: the one measured at high energy, $F=0.155$ [61], and the values of $F=10^{-4}$ and $F=0.3$ assumed to cover the systematic uncertainty of the Fano factor at these energies.", "Figures REF and REF show the limits on $\\chi $ scattering and $V$ /ALP absorption, respectively, compared to existing limits.", "The limits on $\\chi $ assume a DM form factor of either $F_{\\textrm {DM}} = 1$ or $F_{\\textrm {DM}} \\propto 1/q^{2}$  [62].", "The light blue bands representing our estimates of the systematic uncertainty envelops the $\\pm 1\\,\\sigma $ values of all three limits obtained using the different Fano factor assumptions in the ionization model.", "At most masses, the uncertainty bands are dominated by the varying Fano factor assumption; the exception is for $\\lesssim 4$  eV$/c^{2}$ in the $V$ and ALP absorption models, where the uncertainty is dominated by the discrepancy in the photoelectric absorption cross section." ], [ "Discussion and Outlook", "The limits in Figs.", "REF and REF are remarkably close to those from our previous run [12] despite the $\\sim $  2.5 times larger exposure.", "They are in fact weaker at some masses due to the higher observed event rate in the 3$^{\\textrm {rd}}$ $e^{-}h^{+}$ -pair peak coupled with the higher CT and II probabilities in this measurement, as well as the use of a more conservative limit setting method (see Ref.", "[53] for recent CT and II measurements of the detector used in Ref. [12]).", "Table REF compares the efficiency-corrected event rates for each $e^{-}h^{+}$ -pair peak within a $\\pm ~3\\,\\sigma _{E}$ window.", "The event rate observed in each peak is similar in this run compared to Ref.", "[12] despite a different detector design, cryostat, location, overburden, and shielding.", "Another 0.39 gram-days of data were taken with a bias of 60 V across the detector in order to determine if the results are voltage-dependent.", "Table REF shows that the resulting event rate for each number of $e^{-}h^{+}$ pairs is similar to the corresponding rate from the 100 V data, suggesting a voltage-independent result.", "Furthermore, the event rate above the first $e^{-}h^{+}$ -pair peak is comparable to that seen in other charge-readout experiments [12], [23], [54], [63], and adds to the growing narrative of unexplained, $\\mathcal {O}(\\textrm {Hz/kg})$ low-energy excesses measured in many sub-GeV DM searches (Refs.", "[64], [65], [66] and references therein).", "This result from our detector with unparalleled energy resolution provides a new dataset that can contribute to understanding the origin of these unknown background events.", "A third run with an identically designed detector is planned in a dilution refrigerator in a shallow underground site with 255 m.w.e.", "overburden (NEXUS Facility [11]) to probe the correlation between the unknown events and known environmental background sources.", "Table: Comparison of the efficiency-corrected event rate in each e - h + e^{-}h^{+}-pair peak between this work and Ref. .", "The event rates displayed from this analysis are calculated from the DM-search data measured with a bias voltage of 100 V, as well as from the additional dataset measured with a bias voltage of 60 V. For each number of e - h + e^{-}h^{+} pairs, the event rate is determined by counting the number of observed events within a ±3σ E \\pm ~3\\,\\sigma _{E} window centered on the peak.", "The uncertainty shown is the 3 σ\\sigma uncertainty in the number of observed events assuming Poisson statistics.Finally, due to the significant impact that charge trapping and impact ionization have on the signal reconstruction, there is an ongoing effort toward understanding these charge propagation effects and investigating factors that influence them.", "A DM model spectrum with CT and II included is shown in the top panel of Fig REF .", "The black curve shows the DM signal model for a 1 GeV$/c^{2}$ light DM particle with form factor $F_{\\textrm {DM}} \\propto 1/q^{2}$ , scaled to the limit of excluded $\\bar{\\sigma }_{e}$ , using an II probability of 2 %.", "The CT probability shown by the hatched region is varied from 0–15 %.", "For this model and limit-setting scheme, these processes do not determine the ultimate sensitivity.", "However, lower CT and II rates combined with a more robust understanding [67], [68] will allow us to use the region between the peaks in the limit-setting procedure to improve the sensitivity of future analyses, as well as to fully utilize the improved resolution of this detector for additional background discrimination." ], [ "Acknowledgements", "We would like to thank Rouven Essig and Tien-Tien Yu for helpful discussions and assistance with using QEdark [62] to generate the dark matter model used in this analysis.", "We thank Noemie Bastidon for her work in the preliminary design of our optical fiber setup and wire bonding.", "We gratefully acknowledge support from the U.S. Department of Energy (DOE) Office of High Energy Physics and from the National Science Foundation (NSF).", "This work was supported in part under NSF Grants No.", "1809730 and No.", "1707704, as well as by the Arthur B. McDonald Canadian Astroparticle Physics Research Institute, NSERC Canada, the Canada Excellence Research Chair Fund, Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No.", "420484612 and under Germany’s Excellence Strategy - EXC 2121 “Quantum Universe\" - 390833306, and the Department of Atomic Energy Government of India (DAE) under the project - Research in basic sciences (Dark matter).", "Fermilab is operated by Fermi Research Alliance, LLC, under Contract No.", "DE-AC02-37407CH11359 with the US Department of Energy.", "Pacific Northwest National Laboratory (PNNL) is operated by Battelle Memorial Institute for the DOE under Contract No. DE-AC05-76RL01830.", "SLAC is operated under Contract No.", "DEAC02-76SF00515 with the United States Department of Energy." ] ]

[ [ "Performance of Multibeam Very High Throughput Satellite Systems Based on\n FSO Feeder Links with HPA Nonlinearity" ], [ "Abstract Due to recent advances in laser satellite communications technology, free-space optical (FSO) links are presented as an ideal alternative to the conventional radio frequency (RF) feeder links of the geostationary satellite for next generation very high throughput satellite (VHTS) systems.", "In this paper, we investigate the performance of multibeam VHTS systems that account for nonlinear high power amplifiers at the transparent fixed gain satellite transponder.", "Specifically, we consider the forward link of such systems, where the RF user link is assumed to follow the shadowed Rician model and the FSO feeder link is modeled by the Gamma-Gamma distribution in the presence of beam wander and pointing errors where it operates under either the intensity modulation with direct detection or the heterodyne detection.", "Moreover, zero-forcing precoder is employed to mitigate the effect of inter-beam interference caused by the aggressive frequency reuse in the user link.", "The performance of the system under study is evaluated in terms of the outage probability, the average bit-error rate (BER), and the ergodic capacity that are derived in exact closed-forms in terms of the bivariate Meijer's G function.", "Simple asymptotic results for the outage probability and the average BER are also obtained at high signal-to-noise ratio." ], [ "Introduction", "The design of a unified platform that offers ubiquitous broadband global network coverage with very low latency communications and very high data rates in the order of Tbit/s is increasingly becoming a challenging task for 5G and beyond 5G wireless communication systems [1], [2], [3], [4], [5].", "To this end, the integration of satellite communications (SatCom), aerial networks, and terrestrial communications into a single wireless network, called space-air-ground integrated network (SAGIN), is deemed from now on crucial [6].", "More specifically, broadband multibeam SatCom systems are expected to provide seamless reliable and high data rate services at any place on the earth, particularly unserved and underserved areas [6].", "In such platform, the ground stations feed the satellite through a high capacity link, i.e.", "the feeder link, and then the satellite communicates the signal to different user terminals (UTs) via multiple beams, i.e.", "the user links [7].", "Based on the orbit type and the altitude, satellites can be classified into three categories, namely, Low Earth Orbit (LEO), Medium Earth Orbit (MEO), and Geosynchronous Orbit (GEO) where the latter provides the greatest coverage [6].", "Currently existing GEO satellite systems are based on the radio-frequency (RF) technology such as Ka-Sat with a throughput of 70 Gbit/s, Viasat 1 with 140 Gbit/s, Viasat 2 with 350 Gbit/s, and Viasat 3 that expectedly will provide throughput in the range of 1 Tbit/s by 2020 [7].", "In recent years, different constellations of satellites have been proposed to provide global broadband access to internet including the Starlink supported by SpaceX with 12000 LEO satellites [8], Oneweb with 900 LEO satellites [9], and Telesat LEO with 300 to 500 satellites [10].", "All these constellations are based on the conventional RF solutions for both feeder and user links operating at the Ku-band (12-18 GHz), the Ka-band (27-40 Ghz), and the Q/V band (40-50 GHz).", "Obviously, the bandwidth limitation remains one of the key challenges when increasing the capacity.", "For instance, around 50 ground stations are required to reach a satellite capacity of 1 Tbit/s with the traditional RF feeder links, and the number of these ground stations increases linearly with the system throughput [7].", "Another key issue with RF links is the high risk of interference with other communication systems, leading to signal interception or jamming.", "Free-space optical (FSO) technologies are being substantially considered as an attractive alternative to the existing RF feeder links for next generation very high throughput satellite systems (VHTS) [11], [12], [13], [14], [15], [16], [17], [18], [19].", "By using dense-division-multiplexing (DWDM), a fiber based technique, more than one Tbit/s can be sent by a single optical ground station (OGS) to the GEO satellite, leading to the minimization of the number of required ground stations and hence drastically reducing the ground network cost [15].", "Besides the wide available bandwidth (THz) without any restriction or regulation (license-free spectrum), FSO communications hold the advantages of immunity to interference due to the very narrow laser beams along with lesser size, weight, and power compared to their RF counterparts.", "FSO systems can be classified into two categories based on the detection type at the receiver side, namely coherent and non-coherent.", "Non-coherent systems, also known as intensity modulation with direct-detection (IM/DD), are commonly used in FSO links mainly because of their simplicity and low cost [20].", "In such systems, the receiver directly detects the intensity of the emitted light.", "With recent advances in integrated circuits as well as high-speed digital signal processing, coherent detection is becoming more attractive [21], [22], [23].", "In such systems, the incoming optical signal is mixed with a local oscillator (LO) before photo-detection, which improves the receiver sensitivity [24].", "Another interesting property of coherent detection is that amplitude, frequency, and phase modulation can be employed, which considerably increase the system spectral efficiency [23].", "Furthermore, coherent detection allows background noise rejection [25].", "Although most of laser satellite communication (laser SatCom) systems, currently under development, are operating using the IM/DD technique, coherent detection systems have also been employed as a viable alternative for certain applications [26], [27], [28].", "On the other hand, the primary concerns of the FSO feeder link are atmospheric turbulence, beam wander, and misalignment pointing errors.", "The atmospheric turbulence is caused by fluctuations in the refractive index resulting in strong intensity fluctuations, or scintillations, that may cause severe performance degradation of the FSO link.", "The scintillation index, i.e.", "normalized variance of the irradiance, is generally used to characterize these irradiance fluctuations.", "As for the beam wander, it is caused by deviations of the beam from the boresight due to the presence of turbulent eddies larger than the beam diameter.", "This beam wander effect can hence lead to strong fading of the received signal [26].", "Lastly, maintaining a constant line-of-sight (LOS) communication between the transceivers is very essential to have a 100% availability of the FSO feeder link, where the optical beam is highly directional with very narrow beam divergence.", "Due to the satellite mechanical vibration [29], [30], [31], [32], [33], the transmitted beam to the receiver satellite vibrates leading to a misalignment between the transmitter and the receiver, known also as pointing error.", "These pointing errors may lead to significant performance degradation or result in failure of the FSO link [33], [34].", "Despite all these technical challenges, the FSO feeder-link remains the most promising technological option for next generation VHTS systems[14], [7].", "For instance, in the frame of the Terabit-throughput optical satellite system technology (THRUST) project, the German Aerospace Agency DLR set the world-record in FSO communications to 1.72 Tbit/s and 13.16 Tbit/s in 2016 and 2017, respectively [35], [36].", "On the RF user link side, full frequency reuse is employed where all beams operate at the same frequency in order to enhance the bandwidth efficiency of the system.", "However, such an aggressive use of the spectrum introduces inter-beam interference, i.e.", "each UT receives interference from adjacent beams.", "An efficient interference mitigation technique consists of precoding the signals at the OGS before transmitting them to the different UTs, but requires the availability of accurate channel state information (CSI) at the OGS [37], [38], [39], [40].", "However, acquiring up to date and reliable CSI at the OGS introduces a long delay, leading to an outdated CSI.", "Subsequently, the slow fading channel in fixed satellite services (FSS), where the UTs have fixed positions inside the beams [38], [41], [42], would facilitate the CSI acquisition process as CSI needs to be updated less frequently.", "Interestingly, in [43], a novel zero-forcing (ZF) precoding scheme has been proposed for FSS multibeam SatCom systems that only exploits the UTs positions and the antenna beam radiation pattern, without requiring any CSI at the OGS.", "More precisely, the OGS can generate the deterministic multibeam matrix without requiring any feedback from the UTs [44], [45], [38], [46], [47], [43], [48].", "The vast majority of conventional high power satellite transponders employ travelling wave tube amplifier (TWTA) and solid-state power amplifier (SSPA) as onboard memoryless high power amplifiers (HPAs).", "For high output powers, TWTAs are commonly employed, in particular at higher frequency bands, because they offer higher data rates and greater bandwidth with better efficiency than SSPAs.", "For lower frequency bands and for lower transmitter power applications, SSPAs are generally preferred as they exhibit higher reliability, lower mass, and better linearity [42], [49].", "However, these HAPs models have two major nonlinear characteristics, namely, amplitude to amplitude modulation (AM/AM) and amplitude to phase modulation (AM/PM) conversions that should be taken into account as they can lead to severe performance degradation.", "Therefore, several models have been proposed to represent these nonlinear characteristics, mainly, Saleh model [50] and Rapp model [51] to characterize the nonlinear distortion due to TWTA and SSPA models, respectively.", "A few works have studied the impact of hardware impairments on the performance of satellite relay networks [52], [53], [54].", "Their results demonstrate that the impairments degrade the system performance, in particular when the impairments level is larger.", "Motivated by the DLR experimental demonstration in [36], we propose in this work to investigate the performance of VHTS FSO systems with multi-beam RF capabilities.", "As per authors' best knowledge, the first performance analysis of multibeam high throughput satellite systems with optical feeder links has been carried out in [47].", "More specifically, the FSO feeder link is considered to be operating using direct detection over the lognormal distribution whose scope is restricted to weak turbulence channel conditions, and the RF user link is modeled by the double-lognormal fading.", "Based on the precoding scheme presented in [43], the authors provided approximations for the outage probability, the average BER for MQAM and MPSK modulation schemes, and the ergodic capacity in the case of linear power amplifier (PA) at the fixed gain satellite transponder.", "However, to the best of our knowledge, there are no exact closed form expressions that capture the outage probability, the average BER for a variety of modulation schemes, and the ergodic capacity performance under both IM/DD and heterodyne detection techniques with HPA nonlinearity taken into account.", "In this context, this work presents, for the first time, a unified analytical framework for the calculation of the fundamental performance metrics of multibeam VHTS systems with HPA nonlinearity in exact closed form, applicable to both types of detection techniques.", "The FSO feeder link is modeled by the Gamma-Gamma distribution, a good model for atmospheric turbulence under both small and large scales atmospheric fluctuations [55], in the presence of beam wander and pointing errors.", "On the other hand, the RF user links are modeled as shadowed Rician channels that have been proposed in [56] for land mobile satellite channels (LMS).", "Indeed, it has been shown in [56] that the shadowed Rician model provides an excellent fit to the experimental data and has a simple mathematical form, making it attractive from a performance analysis point of view.", "Hence, the main contributions of this work are stated as follows.", "We present a detailed description of the system and channel models with a particular focus on the statistics of the FSO feeder link to stress that there is a great difference between modeling horizontal propagation paths and slant paths, where it is required to consider changes in the refractive index structure parameter along the path.", "We introduce TWTA and SSPA nonlinear amplifiers along with their impairment parameters, and utilize the Bussgang linearization theory to linearize the distortion introduced by these two HPAs.", "We first derive the end-to-end signal-to-noise-plus-distortion-ratio (SNDR) in the case of fixed gain transparent satellite transponders, considering both types of detection techniques for the FSO feeder link (i.e.", "IM/DD and heterodyne) and using the ZF precoder proposed in [43].", "Capitalizing on this result, we present closed-form expressions for the cumulative distribution function (CDF) and the probability density function (PDF) in terms of the bivariate Meijer's G function, and the moments in terms of simple functions.", "We then derive the outage probability, the average bit-error rate (BER) of a variety of modulation schemes, and the ergodic capacity, all in terms of the bivariate Meijer's G function.", "Finally, we present very tight asymptotic expressions for the outage probability and the average BER in the high signal-to-noise ratio (SNR) region in terms of simple elementary functions which are particularly useful to reveal some physical insights.", "The remainder of this paper is organized as follows.", "The system and channel models are outlined in Section II.", "We derive the statistics of the end-to-end SNDR in Section III and we present closed-form expressions for the performance metrics along with the asymptotic results at high SNR regime in Section IV.", "Numerical and simulation results are then provided in Section V followed by the conclusions in Section VI." ], [ "System and Channel Models", "We consider the forward link of a multibeam VHTS system which is defined as the end-to-end link from the OGS to the different UTs.", "More specifically, it includes the uplink of the feeder link (i.e.", "the link between the OGS and the GEO satellite), the transparent or non-regenerative GEO satellite with $N$ antenna feeds, and the downlink of the user link (i.e.", "the link between the GEO satellite and the UTs).", "In addition, we consider that the feeder link is a high capacity FSO single-input single-output (SISO) link, whereas the user link is a multiuser multiple-input single-output (MISO) Ka-band RF link as shown in Fig.", "REF .", "In this paper, we assume a high-energy FSO link whose performance is limited by shot noise as well as thermal noise.", "In this case, the noise can be modeled to high accuracy as zero mean, signal independent additive white Gaussian noise (AWGN) (a widely accepted assumption in many reported works in the literature [57], [58], [59], [60].", "Figure: Architecture of a multibeam very high throughput GEO satellite system with FSO feeder links.In this context, a single OGS simultaneously serves multiple UTs via $N$ beams, in a single feed per beam scenario.", "Moreover, on the user link side, full-frequency reuse is assumed with a cluster size $K=1$ , where all beams operate at the same frequency.", "The coverage area of the GEO satellite is filled up with seven beams arranged in a circular way, resulting in overlapping regions as detailed in Fig.", "REF .", "With beam radius $R$ , the coordinates of each beam center are determined by $C_1(0,0)$ , $C_2\\left( -\\frac{\\sqrt{3} }{2}R,\\frac{3}{2}R\\right)$ , $C_3\\left( \\frac{\\sqrt{3} }{2}R,\\frac{3}{2}R\\right)$ , $C_4\\left( \\sqrt{3}R,0\\right)$ , $C_5\\left( \\frac{\\sqrt{3}}{2}R,-\\frac{3}{2}R\\right)$ , $C_6\\left( -\\frac{\\sqrt{3}}{2}R,-\\frac{3}{2}R\\right)$ , and $C_7\\left( -\\sqrt{3}R,0\\right)$ .", "Furthermore, we focus herein on FSS systems and therefore the UTs have fixed positions inside the beams and can be ultra small aperture terminals (USATs) [42], usually equipped with single antennas, as shown in Fig.", "REF ." ], [ "FSO Feeder Link", "We assume that the data is first precoded at the OGS before being transmitted in order to mitigate inter-beam interference.", "Moreover, DWDM techniques are used to provide the aggregated throughput of multiple Tbit/s where the optical carriers, modulated using either intensity modulation or coherent modulation, are multiplexed into a single-mode fiber (SMF), amplified, and sent through the telescope of the OGS towards the GEO satellite.", "At the GEO satellite, the optical signal is captured by the telescope, demultiplexed to separate the individual DWDM channels, converted to electrical RF channels in the Ka-band, and sent to the users [15].", "Then, the received signal at the GEO satellite, ${\\bf y}_1 \\in {N\\times 1}$ , can be expressed as ${\\bf y}_1= \\eta I {\\bf x}+{\\bf n}_1,$ where $\\eta $ stands for the effective photoelectric conversion ratio, $I$ represents the received optical irradiance, ${\\bf x}\\in {N\\times 1}$ is the precoded transmit signal vector with a total power constraint of ${\\mathbb {E}}[{\\bf x}{\\bf x}^{\\mbox{\\tiny H}} ] \\le P_g$ , and ${\\bf n}_1 \\in {N\\times 1}$ refers to the additive noise vector consisting of circularly symmetric complex Gaussian entries with zero-mean and variance $\\sigma _1^2$ , i.e.", "$\\mathcal {CN}(0,\\sigma _1^2)$ .", "The irradiance $I$ includes the effect of the path loss $I_l$ , the attenuation caused by the atmospheric turbulence $I_a$ , and the attenuation due to pointing errors $I_p$ , i.e.", "$I=I_l I_a I_p$ .", "The path loss $I_l$ is deterministic and is described by the exponential Beers-Lambert Law as $I_l=\\exp (- \\sigma L)$ where $\\sigma $ represents the atmospheric attenuation coefficient and $L$ is the FSO link length [61].", "The pointing error loss due to misalignment is caused by the displacement of the laser beam along elevation and azimuth directions that are typically modeled as independent and identically distributed Gaussian random variables with zero mean value and variance $\\sigma _s^2$ .", "The resulting radial displacement at the receiver $r$ is therefore statistically characterized by a Rayleigh distribution [29], [31], [33], [61] for which the PDF of the irradiance $I_p$ is given by [61] $f_{I_p}(I_p)=\\frac{\\xi ^{2}}{A_0^{\\xi ^{2}}}\\,I_p^{\\xi ^{2}-1},\\quad 0\\le I_p \\le A_0,$ where $A_0$ is the fraction of the collected power at $r=0$ and $\\xi $ is defined as the ratio between the equivalent beam radius at the receiver and the jitter standard deviation at the receiver, and used to quantify the severity of the pointing error effect [61].", "The atmospheric turbulence $I_a$ is modeled by the Gamma-Gamma distribution whose PDF is given in [26] as $f_{I_a}(I_a)=\\frac{2 (\\alpha \\beta )^{\\frac{\\alpha +\\beta }{2}}}{\\Gamma (\\alpha )\\Gamma (\\beta )} I_a^{\\frac{\\alpha +\\beta }{2}-1}K_{\\alpha -\\beta }\\left( 2 \\sqrt{\\alpha \\beta I_a} \\right), \\quad I_a> 0$ where $\\Gamma (\\cdot )$ represents the gamma function [62], $K_{\\alpha -\\beta }(\\cdot )$ stands for the modified Bessel function of the second kind with order $\\alpha -\\beta $ , and $\\alpha $ and $\\beta $ are positive parameters which are related to the large- and small-scale irradiance fluctuations, respectively.", "Taking into account the effect of beam wander, the parameters $\\alpha $ and $\\beta $ for an untracked collimated beam are defined as [26] $\\nonumber \\alpha &=\\left[5.95 (H-h_0)^2 \\sec ^2(\\zeta )\\left(\\frac{ 2 W_0}{r_0} \\right)^{\\frac{5}{3}} \\left( \\frac{\\alpha _{\\rm {pe}}}{W} \\right)^2\\right.\\\\&\\left.+\\exp \\left( \\frac{0.49 \\sigma _{\\rm {Bu}}^2 }{\\left( 1+0.56 \\sigma _{\\rm {Bu}}^{\\frac{12}{5}} \\right)^{\\frac{7}{6}}}\\right)-1 \\right]^{-1}$ and $\\beta =\\left[\\exp \\left( \\frac{0.51 \\sigma _{\\rm {Bu}}^2 }{\\left( 1+0.69 \\sigma _{\\rm {Bu}}^{\\frac{12}{5}} \\right)^{\\frac{5}{6}}}\\right)-1 \\right]^{-1},$ where $H$ represents the altitude of the GEO satellite in m, $h_0$ is the altitude of the optical ground station in m, $\\zeta $ refers to the zenith angle, $W_0$ denotes the beam radius at the transmitter, $W$ is the beam radius at the receiver $W=W_0\\sqrt{\\Theta _0^2+\\Lambda _0^2}$ where $\\Theta _0$ and $\\Lambda _0$ are the transmitter beam parameters defined as $\\Theta _0^2=1-L/F_0$ and $\\Lambda _0=2 L/(k\\, W_0^2)$ with $F_0$ being the phase front radius of curvature at the transmitter ($F \\rightarrow \\infty $ for a collimated beam), $L=(H-h_0) \\sec (\\zeta )$ , $k= 2\\pi /\\lambda $ denotes the wavelength number, $r_0$ stands for the Fried parameter, $\\alpha _{\\text{pe}}$ is the beam wander-induced angular pointing error and $\\sigma _{\\rm {Bu}}^2$ is the Rytov variance given by [63] $\\nonumber \\sigma _{\\rm {Bu}}^2&=2.25\\,k^{\\frac{7}{6}}(H-h_0)^{\\frac{5}{6}}\\sec ^{\\frac{11}{6}}(\\zeta )\\int _{h_0}^{H}C_n^2(h)\\\\& \\times \\left( 1-\\frac{h-h_0}{H-h_0} \\right)^{\\frac{5}{6}}\\left(\\frac{h-h_0}{H-h_0} \\right)^{\\frac{5}{6}}dh.$ The Fried parameter $r_0$ is defined as [26] $r_0=\\left[0.42\\, \\sec (\\zeta )\\,k^2 \\int _{h_0}^{H}C_n^2(h)\\,dh \\right]^{-\\frac{3}{5}},$ where $C_n^2(h)$ is the refractive index structure parameter that varies as a function of the altitude $h$ based on the most widely used Hufnagel-Valley (H-V) model as [26] $\\nonumber C_n^2(h)&=0.00594 \\left( \\frac{w}{27} \\right)^2 \\left( 10^{-5}h \\right)^{10}\\exp \\left( -\\frac{h}{1000} \\right)\\\\&+2.7 \\times 10^{-16}\\exp \\left( -\\frac{h}{1500} \\right)+C_n^2(0) \\exp \\left( -\\frac{h}{100} \\right),$ where $w$ denotes the rms windspeed in m/s and $C_n^2(0)$ refers to the ground level turbulence in m$^{-2/3}$ .", "The beam wander-induced pointing error variance $\\sigma _{\\rm {pe}}^2$ is related to $\\alpha _{\\rm {pe}}$ such that $\\alpha _{\\rm {pe}}=\\sigma _{\\rm {pe}}/L$ and can be expressed as [26] $\\sigma _{\\rm {pe}}^2&=0.54\\,(H-h_0)^2 \\sec ^2(\\zeta ) \\left(\\frac{ \\lambda }{2 W_0}\\right)^2 \\left(\\frac{2 W_0}{r_0} \\right)^{\\frac{5}{3}}\\left[ 1-\\left(\\frac{C_r^2 W_0^2/r_0^2}{1+C_r^2 W_0^2/r_0^2} \\right) ^{\\frac{1}{6}}\\right],$ where $C_r$ is a scaling constant set as $2 \\pi $ [63] and $r_0$ given by (REF ).", "Based on (REF ), (REF ), and the path loss expression, the PDF of $I=I_l I_a I_p$ under the combined effect of atmospheric turbulence, beam wander, pointing errors, and path loss can be written as $f_{I}(I)=\\frac{\\xi ^2 \\alpha \\beta }{A_0 I_l\\Gamma (\\alpha )\\Gamma (\\beta )}\\, {\\rm {G}}_{1,3}^{3,0}\\left[ \\frac{\\alpha \\beta }{A_0 I_l}I \\left| \\begin{matrix} {\\xi ^2} \\\\ {\\xi ^2-1,\\alpha -1,\\beta -1} \\\\ \\end{matrix} \\right.", "\\right],$ where ${\\rm {G}}_{p,q}^{m,n}(\\cdot )$ is the Meijer's G function [62]." ], [ "Nonlinear Satellite Transponder", "Considering nonlinear HPA at the satellite transponder, the amplification process is performed in two distinct steps.", "In the first step, the RF precoded signal ${\\bf y}_1$ is amplified with a constant gain matrix ${\\bf F}=G{\\bf I}_N$ , that is ${\\bf y}_s={\\bf F} {\\bf y}_1$ , where the amplification factor $G=\\sqrt{\\frac{P_r}{P_g \\,\\mathbb {E} \\left[(\\eta I)^r \\right]+\\sigma _1^2}}$ is selected such that the total transmit power constraint at the satellite transponder is met, i.e.", "$\\mathbb {E}\\left[ \\left\\Vert {\\bf F} {\\bf y}_1\\right\\Vert ^2 \\right]\\le P_r$ , where $P_r$ is the mean signal power at the output of the gain block.", "In the second phase, the amplified version of the signal ${\\bf y}_s$ is passed through a nonlinear circuit and then the signal at the output of the memoryless nonlinear HPA can be given as [64] ${\\bf y}_{s_{\\rm {NL}}}=f_A(\\left\\Vert {\\bf y}_s \\right\\Vert )\\exp \\left( j\\left(f_P(\\left\\Vert {\\bf y}_s \\right\\Vert )+{\\rm {arg}}({\\bf y}_s)\\right)\\right),$ where $f_A(\\cdot )$ and $f_P(\\cdot )$ represent the AM/AM and AM/PM characteristic functions, respectively, and $j^2=-1$ .", "As discussed in Section I, we consider two types of nonlinear amplifiers which are widely employed in conventional high power satellite transponders, namely TWTA and SSPA.", "For the TWTA model, the AM/AM and AM/PM conversions are given as [50] $f_A(\\left\\Vert {\\bf y}_s \\right\\Vert )=A_{\\rm {sat}}^2 \\frac{\\left\\Vert {\\bf y}_s \\right\\Vert }{\\left\\Vert {\\bf y}_s \\right\\Vert ^2+A_{\\rm {sat}}^2};\\quad f_P(\\left\\Vert {\\bf y}_s \\right\\Vert )={\\rm {\\Phi _0}} \\frac{\\left\\Vert {\\bf y}_s \\right\\Vert }{\\left\\Vert {\\bf y}_s \\right\\Vert ^2+A_{\\rm {sat}}^2},$ where $A_{\\rm {sat}}$ represents the input saturation amplitude level and ${\\rm {\\Phi _0}}$ controls the maximum phase distortion introduced by TWTA.", "The AM/AM and AM/PM functions of the SSPA model can be expressed as [51] $f_A(\\left\\Vert {\\bf y}_s \\right\\Vert )= \\frac{\\left\\Vert {\\bf y}_s \\right\\Vert }{\\left[\\left(\\frac{\\left\\Vert {\\bf y}_s \\right\\Vert }{A_{\\rm {sat}}} \\right)^{2v}+1 \\right]^{\\frac{1}{2v}}};\\quad f_P(\\left\\Vert {\\bf y}_s \\right\\Vert )=0,$ where $v$ refers to the smoothness factor that controls the transition from linear to saturation region.", "It is worthy to mention that the nonlinear distortion created by both TWTA and SSPA models as shown by (REF ) makes it very hard to obtain closed-form and easy-to-use expressions for important performance metrics such as the outage probability and the average BER of the VHTS FSO system under consideration.", "However, we can linearize this distortion by means of using the Bussgang's linearization theory [65] since the input signal ${\\bf y}_s$ can be approximately modeled as Gaussian distributed.", "This is due to the fact that the precoded transmit signal ${\\bf x}$ given in (REF ) is a weighted sum of independent and identically distributed (i.i.d.)", "random variables that can be approximated by a Gaussian distribution according the central limit theorem.", "By using the Bussgang's theorem, the output of the nonlinear HPA can be expressed as ${\\bf y}_{s_{\\rm {NL}}}=K {\\bf y}_s +{\\bf n}_{\\rm {NL}},$ where $K$ is the linear scale parameter and ${\\bf n}_{\\rm {NL}} \\in {N\\times 1}$ is the nonlinear distortion term uncorrelated with ${\\bf y}_s$ and modeled as $\\mathcal {CN}(0,\\sigma _{\\rm {NL}}^2)$ .", "As discussed in Section I, we consider two types of nonlinear amplifiers which are widely employed in conventional high power satellite transponders, namely TWTA and SSPA.", "More specifically, in the case of TWTA model, the impairment parameters $K$ and $\\sigma _{\\rm {NL}}^2$ can be given under the assumption of negligible AM/PM effects by [66], [67] $\\begin{aligned}&K=\\sqrt{\\frac{A_{\\rm {sat}}^2}{4P_r}}\\left[ \\sqrt{\\frac{4A_{\\rm {sat}}^2}{P_r}} -\\sqrt{\\pi }\\exp \\left( \\frac{A_{\\rm {sat}}^2}{P_r} \\right){\\rm {erfc}}\\left( \\sqrt{\\frac{A_{\\rm {sat}}^2}{P_r}} \\right)\\left( \\frac{2A_{\\rm {sat}}^2}{P_r} -1\\right)\\right]\\\\&\\sigma _{\\rm {NL}}^2=A_{\\rm {sat}}^2 \\left[ 1+\\frac{A_{\\rm {sat}}^2}{P_r} \\exp \\left( \\frac{A_{\\rm {sat}}^2}{P_r} \\right){\\rm {Ei}}\\left( -\\frac{A_{\\rm {sat}}^2}{P_r} \\right)\\right]-K^2 P_r,\\end{aligned}$ where $A_{\\rm {sat}}$ represents the input saturation amplitude level, ${\\rm {erfc}}(\\cdot )$ is the complementary error function [62], and ${\\rm {Ei}}(\\cdot )$ is the exponential integral function [62].", "Additionally, for the SSPA model, $K$ and $\\sigma _{\\rm {NL}}^2$ can be expressed using [67] as $\\begin{aligned}&K=\\frac{A_{\\rm {sat}}^2}{P_r}\\left[ 1+\\frac{A_{\\rm {sat}}^2}{P_r} \\exp \\left( \\frac{A_{\\rm {sat}}^2}{P_r} \\right){\\rm {Ei}}\\left( -\\frac{A_{\\rm {sat}}^2}{P_r} \\right)\\right]\\\\&\\sigma _{\\rm {NL}}^2=-\\frac{A_{\\rm {sat}}^4}{P_r} \\left[ \\left( 1+\\frac{A_{\\rm {sat}}^2}{P_r} \\right) \\exp \\left( \\frac{A_{\\rm {sat}}^2}{P_r} \\right){\\rm {Ei}}\\left( -\\frac{A_{\\rm {sat}}^2}{P_r} \\right)+1\\right]-K^2 P_r.\\end{aligned}$ It is important to mention that in practice, the satellite transponders are not operated at saturation but backed off in order to reduce the nonlinearity.", "Indeed, the actual transmission power is reduced by a given amount below the HPA saturation point, which is known as the input back-off (IBO) and is defined as [66] ${\\rm {IBO}}=\\frac{A_{\\rm {sat}}^2}{P_r}.$ Moreover, it is noteworthy that $K$ and $\\sigma _{\\rm {NL}}^2$ depend on IBO for both TWTA and SSPA models, and are constants for a fixed IBO value." ], [ "RF User Link", "The received signal vector at all the UTs can be given as ${\\bf y}_2={\\bf H}{\\bf y}_{s_{\\rm {NL}}} +{\\bf n}_2=K G \\eta I {\\bf H}\\,{\\bf x}+K G {\\bf H}{\\bf n}_1+{\\bf H}{\\bf n}_{\\rm {NL}}+{\\bf n}_2,$ where the user link channel matrix ${\\bf H}\\in {N\\times N}$ represents the channel gains between the $N$ feeds and the $N$ UTs and takes into account the atmospheric fading, the beam radiation pattern, and path losses, and ${\\bf n}_2 \\in {N\\times 1}$ refers to the noise vector with elements drawn from $\\mathcal {CN}(0,1)$ .", "Using [39], [47], [43], the user link channel matrix can be expressed as ${\\bf H}={\\bf D}{\\bf B},$ where ${\\bf B}\\in {\\mathbb {R}}^{N\\times N}$ is the multibeam gain matrix that models the satellite antenna radiation pattern, the receive antenna gain, and the path loss [48], [47].", "Assuming that the receive antenna gains for all UTs are identical and equal to $G_t$ , the transmitter antenna gains for all satellite feeds are identical and equal to $G_r$ , neglecting the Earth curvature so that all UTs have a common slant range equal to the GEO satellite elevation distance, and employing the Bessel function model for a typical tapered-aperture antenna, the beam gain from the $j$ -th feed towards the $i$ -th UT can be expressed as [48], [47] $[{\\bf B}]_{ij}= \\frac{c\\sqrt{G_t G_r }}{4 \\pi f D \\sqrt{\\kappa _B T_r B_w}} \\left(\\frac{{\\rm J}_1 (u_{ij})}{2 u_{ij}}+36 \\frac{{\\rm J}_3 (u_{ij})}{u_{ij}^3}\\right),$ where $c$ is the speed of light, $f$ is the carrier frequency, $\\kappa _B$ refers to the Boltzman constant, $T_r$ denotes the receiver noise temperature, $B_w$ stands for the bandwidth of the user link, ${\\rm J}_1(\\cdot )$ and ${\\rm J}_3(\\cdot )$ are the Bessel functions of the first kind of order 1 and 3, respectively.", "In (REF ), $u_{ij}=2.07123\\sin (\\theta _{ij})/\\sin (\\theta _{3\\rm {dB}})$ is a function of the off-axis angle with respect to the beam's boresight $\\theta _{ij}=\\arctan (d_{ij}/D)$ where $d_{ij}$ represents the distance between the $i$ -th UT and the $j$ -th beam boresight (slant-range), $D$ is the distance from the UT to the satellite, and $\\theta _{3\\rm {dB}}=R/D$ refers to the beam's 3 dB angle, with $R$ denoting the beam radius.", "Since we consider that UTs have fixed positions on earth, the beam gain between the $i$ -th UT and the $j$ -th satellite feed is fixed and therefore the multibeam gain matrix ${\\bf B}$ is deterministic [46], [47], [43].", "In (REF ), ${\\bf D}\\in {N\\times N} $ is a diagonal matrix that represents the fading in the user link with the diagonal entry $d_i$ referring to the fading gain for the $i$ -th UT, $i \\in {\\mathbb {N}}$ which is assumed to follow the shadowed Rician model for LMS channels with the PDF given in [56] by $f_{\\left|d_i \\right|}(y)&=\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i}\\frac{y}{b_i}\\exp \\left( -\\frac{y^{2}}{2b_i} \\right)_{1}F_{1}\\left( m_i,1,\\frac{\\Omega _i y^{2}}{2 b_i\\left( 2 b_i m_i+\\Omega _i \\right)} \\right), y\\ge 0,$ where $\\Omega _i$ refers to the average power of the line-of-sight (LOS) component, $2b_i$ represents the average power of the multipath component, $m_i$ stands for the fading severity parameter, and $_{1}F_{1}(\\cdot ,\\cdot ,\\cdot )$ is the confluent hypergeometric function [62].", "For $m_i=0$ , (REF ) reduces to the Rayleigh PDF, while for $m_i \\rightarrow \\infty $ it simplifies to the Rice PDF." ], [ "Zero-Forcing Precoding", "The transmit precoded signal can be expressed as ${\\bf x}={\\bf T}{\\bf s},$ where ${\\bf T}\\in {N\\times N}$ is the precoding matrix and ${\\bf s}\\in {N\\times 1}$ represents the UTs data symbols at the OGS with ${\\mathbb {E}}[{\\bf s}{\\bf s}^{\\mbox{\\tiny H}} ]={\\bf I}_N$ and the transmit power constraint can be re-written as ${\\mathbb {E}}\\left[ \\left\\Vert {\\bf x}\\right\\Vert ^2\\right]={\\mathbb {E}}\\left[ \\left\\Vert {\\bf T}{\\bf s}\\right\\Vert ^2\\right]=\\operatorname{tr}\\left({\\bf T}{\\bf T}^{\\mbox{\\tiny H}} \\right)\\le P_g.$ Substituting (REF ) and (REF ) into (REF ), the received signal vector can be expressed as ${\\bf y}_2=K G \\eta I {\\bf D}{\\bf B}{\\bf T}{\\bf s}+K G {\\bf D}{\\bf B}{\\bf n}_1+{\\bf D}{\\bf B}{\\bf n}_{\\rm {NL}}+{\\bf n}_2.$ Using the ZF precoding technique presented in [43] which does not require CSI at the OGS and is only based on the deterministic multibeam matrix ${\\bf B}$ , the precoding matrix ${\\bf T}$ can be given as ${\\bf T}=\\sqrt{c_{\\rm {ZF}} }{\\bf B}^{\\mbox{\\tiny H}} \\left({\\bf B}{\\bf B}^{\\mbox{\\tiny H}} \\right)^{-1},$ where $c_{\\rm {ZF}}$ is set such that [43], [47] $c_{\\rm {ZF}}=\\frac{P_g}{\\operatorname{tr}\\left[ \\left({\\bf B}{\\bf B}^{\\mbox{\\tiny H}} \\right)^{-1} \\right]}.$ By plugging (REF ) into (REF ), the received signal at the $i$ -th UT simplifies to ${\\bf y}_{2,i}=\\sqrt{c_{\\rm {ZF}}} K G \\eta I d_i s_i+K G d_i {\\bf b}_i^{\\mbox{\\tiny T}} {\\bf n}_1+d_i {\\bf b}_i^{\\mbox{\\tiny T}} {\\bf n}_{\\rm {NL}}+{\\bf n}_{2,i}.$ Finally, the end-to-end SNDR at the $i$ -th UT can be expressed after some manipulations as $\\gamma _i=\\frac{1}{\\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2} \\frac{\\gamma _1 \\gamma _{2,i}}{\\kappa \\gamma _{2,i}+\\operatorname{tr}\\left[ \\left({\\bf B}{\\bf B}^{\\mbox{\\tiny H}} \\right)^{-1} \\right]\\overline{\\gamma }_1+\\kappa },$ where $\\gamma _1=\\frac{P_g (\\eta I)^r}{\\sigma _1^2 \\operatorname{tr}\\left[ \\left({\\bf B}{\\bf B}^{\\mbox{\\tiny H}} \\right)^{-1} \\right]}$ is the electrical SNR of the FSO feeder link operating under either IM/DD (i.e.", "$r=2$ ) or heterodyne detection (i.e.", "$r=1$ ), $\\gamma _{2,i}=\\frac{P_s \\left| d_i \\right|^2 \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2}{N }$ is the SNR of the $i$ -th UT, $\\frac{P_s}{N}$ is the average transmitted power at the satellite satisfying $\\frac{P_s}{N}=K^2 P_r+\\sigma _{\\rm {NL}}^2$ , and $\\kappa $ is the ratio between the average received SNR and the average transmitted SNDR at the relay given by [66] $\\kappa =1+\\frac{\\sigma _{\\rm {NL}}^2}{K^2 G^2 \\sigma _1^2}.$ Note that the parameter $\\kappa $ in (REF ) plays a key role in this paper as it describes the level of impairments, under both TWTA and SSPA models.", "In addition, when $\\kappa =1$ , (REF ) reduces to the end-to-end SNR in the case of linear PA at the satellite transponder as it implies that $\\sigma _{\\rm {NL}}^2=0$ .", "Considering both IM/DD and heterodyne detections, the PDF of $\\gamma _1$ can be obtained from (REF ) as $f_{\\gamma _1}(\\gamma _1)=\\frac{\\xi ^2}{r\\Gamma (\\alpha )\\Gamma (\\beta )\\gamma _1}\\, {\\rm {G}}_{1,3}^{3,0}\\left[ \\frac{\\alpha \\beta \\, \\xi ^2}{(\\xi ^2+1)}\\left( \\frac{\\gamma _1}{\\mu _r} \\right)^{\\frac{1}{r}} \\left| \\begin{matrix} {\\xi ^2+1} \\\\ {\\xi ^2,\\alpha ,\\beta } \\\\ \\end{matrix} \\right.", "\\right],$ where $\\mu _r$ is the average electrical SNR given by $\\mu _r=\\frac{P_g}{\\sigma _1^2 \\operatorname{tr}\\left[ \\left({\\bf B}{\\bf B}^{\\mbox{\\tiny H}} \\right)^{-1} \\right]}\\left(\\eta A_0 I_l\\xi ^2/(\\xi ^2+1) \\right)^r$ and can be written in terms of the average SNR of the FSO feeder link, $\\overline{\\gamma }_1$ , as $\\mu _r=\\frac{(\\xi ^2+r)\\left(\\alpha \\beta \\xi ^2 \\right)^r \\Gamma (\\alpha )\\Gamma (\\beta )}{\\xi ^2 (\\xi ^2+1)^r\\,\\Gamma (\\alpha +r)\\Gamma (\\beta )}\\overline{\\gamma }_1.$ Moreover, using (REF ), the PDF of the SNR $\\gamma _{2,i}$ can be given as $f_{\\gamma _{2,i}}(\\gamma _2)&=\\frac{m_i}{\\overline{\\gamma }_{i,2}} \\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1}\\exp \\left( -\\frac{(2 b_i m_i+\\Omega _i)\\gamma _2}{2b_i\\, \\overline{\\gamma }_{2,i}} \\right)_{1}F_{1}\\left( m_i,1,\\frac{\\Omega _i \\gamma _2}{2 b_i \\overline{\\gamma }_{2,i}} \\right),$ where $\\overline{\\gamma }_{2,i}=\\frac{P_s \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2}{N } (2 b_i m_i+\\Omega _i)$ is the average SNR at the $i$ -th RF user link.", "For integer values of the fading parameter, i.e.", "$m_i \\in \\mathbb {N}$ , the PDF expression in (REF ) can be simplified by utilizing [68] then [68] as $\\nonumber f_{\\gamma _{2,i}}(\\gamma _2)&=\\frac{m_i}{\\overline{\\gamma }_{i,2}} \\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1}\\exp \\left( -\\frac{m_i\\gamma _2}{\\overline{\\gamma }_{2,i}} \\right)\\\\&\\times \\sum _{k=0}^{m_i-1}\\frac{(-1)^k (1-m_i)_k}{k!^2} \\left(\\frac{\\Omega _i \\gamma _2}{2 b_i \\overline{\\gamma }_{i,2}} \\right)^k,$ where $(a)_k=\\Gamma (a+k)/\\Gamma (a)$ denotes the Pochhammer symbol [62]." ], [ "Statistical Analysis", "In this section, we derive new exact closed-form expressions for the end-to-end SNDR statistics for the multibeam VHTS system with FSO feeder links, accounting for nonlinearities at satellite transponder." ], [ "Cumulative Distribution Function", "A unified expression for the CDF of the overall SNDR at the $i$ -th UT considering both IM/DD and heterodyne detection techniques for the FSO feeder link in the presence of HPA nonlinearity can be derived in exact closed-form in terms of the bivariate Meijer's G function whose implementation is presented in [69], [70] as $\\nonumber F_{\\gamma _i}(x)&= 1-\\frac{\\xi ^2\\, r^{\\alpha +\\beta -2}}{\\Gamma (\\alpha )\\Gamma (\\beta ) (2 \\pi )^{r-1}}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1} \\\\\\nonumber & \\times \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k \\\\ & \\times {\\rm {G}}_{1,0:0,2:3r,r+1}^{1,0:2,0:0,3r}\\begin{bmatrix}\\begin{matrix}0\\end{matrix}\\Bigg |\\begin{matrix}-\\\\j,1\\end{matrix}\\Bigg |\\begin{matrix}\\mathcal {K}_1\\\\\\Delta (r,-\\xi ^2),0\\end{matrix}\\Bigg |\\frac{C m_i}{\\kappa \\,\\overline{\\gamma }_{2,i}},\\frac{r^{2r}(\\xi ^2+1)^r \\mu _r }{(\\alpha \\beta \\xi ^2)^r \\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2 x}\\end{bmatrix},$ where $C=\\operatorname{tr}\\left[ \\left({\\bf B}{\\bf B}^{\\mbox{\\tiny H}} \\right)^{-1} \\right]\\overline{\\gamma }_1+\\kappa $ , $\\mathcal {K}_1=\\Delta (r,1-\\xi ^2),\\Delta (r,1-\\alpha ),\\Delta (r,1-\\beta )$ , and $\\Delta (r,u)=\\frac{u}{r},\\frac{u+1}{r},\\ldots ,\\frac{u+r-1}{r}$ .", "See Appendix .", "Note that by setting $\\kappa =1$ in (REF ), we can easily obtain the CDF expression in the case of linear PA at the satellite transponder." ], [ "Probability Distribution Function", "The PDF of the end-to-end SNDR at the $i$ -th UT, can be obtained by taking the derivative of (REF ), yielding $\\nonumber f_{\\gamma _i}(x)&= \\frac{\\xi ^2\\, r^{\\alpha +\\beta -2}}{\\Gamma (\\alpha )\\Gamma (\\beta ) (2 \\pi )^{r-1}x}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1} \\\\\\nonumber & \\times \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k \\\\&\\times {\\rm {G}}_{1,0:0,2:3r,r+1}^{1,0:2,0:0,3r}\\begin{bmatrix}\\begin{matrix}0\\end{matrix}\\Bigg |\\begin{matrix}-\\\\j,1\\end{matrix}\\Bigg |\\begin{matrix}\\mathcal {K}_1\\\\\\Delta (r,-\\xi ^2),1\\end{matrix}\\Bigg |\\frac{C m_i}{\\kappa \\,\\overline{\\gamma }_{2,i}},\\frac{r^{2r}(\\xi ^2+1)^r \\mu _r }{(\\alpha \\beta \\xi ^2)^r \\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2 x}\\end{bmatrix},$" ], [ "Moments", "The $n$ -th moments of the end-to-end SNDR at the $i$ -th UT defined as $\\mathbb {E}[\\gamma _i^n]\\triangleq \\int _{0}^{\\infty }x^n\\,f_{\\gamma ,i}(x)\\,dx$ , can be given as $\\nonumber &\\mathbb {E}[\\gamma _i^n]=\\frac{\\xi ^2\\Gamma (\\alpha +r\\,n)\\Gamma (\\beta + r\\,n)}{(\\xi ^2+r\\,n)\\Gamma (\\alpha )\\Gamma (\\beta )\\Gamma (n)}\\left(\\frac{(\\xi ^2+1)^r \\mu _r}{(\\alpha \\beta \\xi ^2)^r \\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2} \\right)^n\\\\& \\times \\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1}\\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k{\\rm {G}}_{1,2}^{2,1}\\left[\\frac{C m_i}{\\kappa \\overline{\\gamma }_{2,i}}\\left| \\begin{matrix} {1-n} \\\\ {j,1} \\\\ \\end{matrix} \\right.", "\\right].$ See Appendix ." ], [ "Performance Evaluation", "This section derives new closed-form expressions for the performance metrics of the multibeam VHTS system with an FSO feeder link under the presence of satellite transponder nonlinearity.", "Additionally, this section provides tractable asymptotic expressions for the outage probability and the average BER at the high SNR regime." ], [ "Exact Analysis", "The outage probability is defined as the probability that the end-to-end SNDR falls below a predefined threshold $\\gamma _{\\rm {th}}$ and can be easily obtained at the $i$ -th UT by setting $x=\\gamma _{\\rm {th}}$ in (REF ), that is, $P_{{\\mathrm {out}},i}(\\gamma _{th})=F_{\\gamma _i}(\\gamma _{th}).$ It can be concluded from (REF ) that for low values of IBO, the term $\\kappa $ grows very large and the outage probability $P_{{\\mathrm {out}},i}(\\gamma _{th}) \\rightarrow 1$ for any $\\gamma _{th}$ , especially in the case of TWTA.", "This shows the deleterious impact of the nonlinear amplifier at the relay and demonstrates that it is necessary to increase IBO in order to reduce the distortion introduced by both TWTA and SSPA models.", "To obtain more engineering insights on the impact of the hardware impairments, we elaborate further on the asymptotic analysis at high SNR regime." ], [ "Asymptotic Analysis", "Starting from (REF ), applying [71] then [71], the outage probability at the $i$ -th UT can be given asymptotically at high SNR of the FSO link after performing some algebraic manipulations as $\\nonumber P_{{\\mathrm {out}},i}(\\gamma _{\\rm {th}})& \\underset{\\mu _r\\gg 1}{\\mathop {\\approx }}1-\\frac{\\xi ^2}{\\Gamma (\\alpha )\\Gamma (\\beta )}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1} \\\\& \\times \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k\\sum _{v=1}^{4} \\mathcal {J}_v \\, \\left(\\frac{\\gamma _{\\rm {th}}}{\\mu _r}\\right)^{\\theta _v},$ where $\\theta _v=\\left\\lbrace j, \\frac{\\xi ^2}{r} ,\\frac{\\alpha }{r},\\frac{\\beta }{r} \\right\\rbrace $ and $\\mathcal {J}_1=\\frac{\\Gamma (\\alpha -rj)\\Gamma (\\beta -rj)}{\\xi ^2-rj}\\left(\\frac{C\\, m_i\\,\\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2(\\alpha \\beta \\xi ^2)^r }{(\\xi ^2+1)^r\\, \\overline{\\gamma }_{2,i}}\\right)^j,$ $\\nonumber \\mathcal {J}_2&=\\frac{\\Gamma (\\alpha -\\xi ^2)\\Gamma (\\beta -\\xi ^2)}{r}\\left(\\frac{\\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2(\\alpha \\beta \\xi ^2)^r }{(\\xi ^2+1)^r\\, }\\right)^{\\frac{\\xi ^2}{r}}\\\\& \\times \\left(\\Gamma \\left(j-\\frac{\\xi ^{2}}{r}\\right)\\left(\\frac{C m_i }{\\kappa \\overline{\\gamma }_{2,i}}\\right)^{\\frac{\\xi ^{2}}{r}}+\\frac{{\\rm {G}}_{1,2}^{2,1}\\left[\\frac{C m_i}{\\kappa \\overline{\\gamma }_{2,i}}\\left| \\begin{matrix} {1+\\frac{\\xi ^2}{r}} \\\\ {j,1} \\\\ \\end{matrix} \\right.", "\\right]}{\\Gamma \\left( 1- \\frac{\\xi ^2}{r}\\right)}\\right),$ $\\nonumber \\mathcal {J}_3&=\\frac{\\Gamma (\\beta -\\alpha )}{r(\\xi ^2-\\alpha )}\\left(\\frac{\\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2(\\alpha \\beta \\xi ^2)^r }{(\\xi ^2+1)^r\\, }\\right)^{\\frac{\\alpha }{r}}\\\\& \\times \\left( \\Gamma \\left( j- \\frac{\\alpha }{r}\\right) \\left(\\frac{C m_i }{\\kappa \\overline{\\gamma }_{2,i}}\\right)^\\frac{\\alpha }{r}+\\frac{{\\rm {G}}_{1,2}^{2,1}\\left[\\frac{C m_i}{\\kappa \\overline{\\gamma }_{2,i}}\\left| \\begin{matrix} {1+\\frac{\\alpha }{r}} \\\\ {j,1} \\\\ \\end{matrix} \\right.", "\\right]}{\\Gamma \\left( 1- \\frac{\\alpha }{r}\\right)}\\right),$ $\\nonumber \\mathcal {J}_4&=\\frac{\\Gamma (\\alpha -\\beta )}{r(\\xi ^2-\\beta )}\\left(\\frac{\\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2(\\alpha \\beta \\xi ^2)^r }{(\\xi ^2+1)^r\\, }\\right)^{\\frac{\\beta }{r}}\\\\& \\times \\left( \\Gamma \\left( j- \\frac{\\beta }{r}\\right) \\left(\\frac{C m_i }{\\kappa \\overline{\\gamma }_{2,i}}\\right)^\\frac{\\beta }{r}+\\frac{{\\rm {G}}_{1,2}^{2,1}\\left[\\frac{C m_i}{\\kappa \\overline{\\gamma }_{2,i}}\\left| \\begin{matrix} {1+\\frac{\\beta }{r}} \\\\ {j,1} \\\\ \\end{matrix} \\right.", "\\right]}{\\Gamma \\left( 1- \\frac{\\beta }{r}\\right)}\\right).$ Note that at high SNR of the RF user link, $\\overline{\\gamma }_{2,i}$ , $\\mathcal {J}_2$ , $\\mathcal {J}_3$ , and $\\mathcal {J}_4$ can be further simplified by using [72] as $\\mathcal {J}_2=\\frac{2\\Gamma (\\alpha -\\xi ^2)\\Gamma (\\beta -\\xi ^2)\\Gamma \\left(j-\\frac{\\xi ^2}{r}\\right)}{r}\\left(\\frac{C m_i \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2(\\alpha \\beta \\xi ^2)^r }{(\\xi ^2+1)^r\\, \\overline{\\gamma }_{2,i}}\\right)^{\\frac{\\xi ^2}{r}},$ $\\mathcal {J}_3=\\frac{2\\Gamma (\\beta -\\alpha )\\Gamma (j-\\frac{\\alpha }{r})}{r(\\xi ^2-\\alpha )}\\left(\\frac{C\\, m_i\\,\\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2(\\alpha \\beta \\xi ^2)^r }{(\\xi ^2+1)^r\\, \\overline{\\gamma }_{2,i}}\\right)^{\\frac{\\alpha }{r}},$ $\\mathcal {J}_4=\\frac{2\\Gamma (\\alpha -\\beta )\\Gamma (j-\\frac{\\beta }{r})}{r(\\xi ^2-\\beta )}\\left(\\frac{C\\, m_i\\,\\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2(\\alpha \\beta \\xi ^2)^r }{(\\xi ^2+1)^r\\, \\overline{\\gamma }_{2,i}}\\right)^{\\frac{\\beta }{r}}.$ It can be inferred from (REF ) that when $\\mu _r \\rightarrow \\infty $ , the parameter $\\kappa $ grows towards infinity too and the outage probability is saturated by an irreducible floor regardless of the nonlinear HPA model at the relay.", "Indeed, the impairments become very severe at high SNR range and the outage probability does not decrease with an increase in the average electrical SNR.", "However, in the case of linear PA at the relay, the outage probability converges to zero when $\\mu _r \\rightarrow \\infty $ .", "This confirms that the hardware impairments can significantly limit the performance of VHTS systems and therefore should be considered in the design of such systems." ], [ "Exact Analysis", "A generalized expression for the average BER of the $i$ -th UT for a variety of modulation schemes can be expressed as [73] $\\overline{P}_{e,i}= \\frac{\\delta }{2\\Gamma (p)}\\sum _{u=1}^{n}q_u^p\\int _{0}^{\\infty }x^{p-1}e^{-q_u x}F_{\\gamma _i}(x)\\,dx,$ where $n$ , $\\delta $ , $p$ , and $q_u$ vary depending on the modulation technique and the type of detection (i.e IM/DD or heterodyne detection) and are listed in Table REF .", "It is important to mention here that for IM/DD technique, we investigate the average BER for on-off keying (OOK) modulation since it is the most commonly used intensity modulation technique in practical FSO systems due to its simplicity and resilience to laser nonlinearity.", "For heterodyne detection and in addition to binary modulation schemes, we analyze the average BER for multilevel phase shift keying (MPSK) and quadrature amplitude (MQAM) that are commonly deployed in coherent systems.", "By substituting (REF ) in (REF ), integrating using [62] and applying [74], a unified expression for the average BER of the $i$ -th UT for all these modulation schemes can be derived in exact closed-form in terms of the bivariate Meijer's G function as $\\nonumber \\overline{P}_{e,i}&= \\frac{\\delta \\, n}{2}-\\frac{\\delta \\,\\xi ^2\\, r^{\\alpha +\\beta -2}}{2\\Gamma (\\alpha )\\Gamma (\\beta )\\Gamma (p) (2 \\pi )^{r-1}}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1} \\\\\\nonumber & \\times \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\sum _{u=1}^{n}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k \\\\& \\times {\\rm {G}}_{1,0:0,2:3r,r+2}^{1,0:2,0:1,3r}\\begin{bmatrix}\\begin{matrix}0\\end{matrix}\\Bigg |\\begin{matrix}-\\\\j,1\\end{matrix}\\Bigg |\\begin{matrix}\\mathcal {K}_1\\\\p, \\Delta (r,-\\xi ^2),0\\end{matrix}\\Bigg |\\frac{C m_i}{\\kappa \\,\\overline{\\gamma }_{2,i}},\\frac{r^{2r}q_u(\\xi ^2+1)^r \\mu _r }{(\\alpha \\beta \\xi ^2)^r \\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2}\\end{bmatrix}.$ Table: Parameters for Different Modulation SchemesIt is worth noting that the major advantage of (REF ) is that it presents a unified BER expression in a compact form that is valid for both heterodyne and IM/DD techniques and applicable to a variety of modulation schemes.", "In addition, to reveal some useful insights, we derive an asymptotic expression for the average BER at high SNR regime as shown by (REF )." ], [ "Asymptotic Analysis", "Similar to the asymptotic outage probability analysis, a simpler closed-form expression for the average BER of the $i$ -th UT for a variety of modulation techniques can be obtained at high SNR regime by substituting the CDF expression at high SNR, obtained from (REF ), into (REF ), and utilizing [62] as $\\nonumber \\overline{P}_{e,i}& \\underset{\\mu _r\\gg 1}{\\mathop {\\approx }} \\frac{\\delta \\, n}{2}-\\frac{\\delta \\,\\xi ^2}{2\\Gamma (\\alpha )\\Gamma (\\beta )\\Gamma (p) }\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1}\\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k} \\\\& \\times \\sum _{u=1}^{n}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k \\sum _{v=1}^{4} \\mathcal {J}_v \\Gamma (p+\\theta _v)(q_u \\mu _r)^{-\\theta _v}.$ It is important to mention that the asymptotic expression of the average BER given in (REF ) is simpler and much more analytically tractable than the exact expression of the BER obtained in terms of the bivariate Meijer's G function in (REF ), which is a quite complex function and not a standard built-in function in most of the well-known mathematical software tools such as MATHEMATICA and MATLAB.", "Interestingly, (REF ) is very accurate and converges perfectly to the exact result in (REF ) at high SNR regime, which is illustrated in section V. Similar to what was concluded from the asymptotic expression of the outage probability in (REF ), it can be easily shown that a BER floor is created at high SNR range due to HPA nonlinearity, which becomes higher as IBO gets lower." ], [ "Exact Analysis", "The ergodic capacity of the $i$ -th UT of an FSO-based mutilbeam VHTS system with HPA nonlinearity can be calculated as [75],[76] [77] $\\overline{C_i}\\triangleq \\mathbb {E}[\\ln (1+\\tau \\,\\gamma _i)]=\\frac{\\tau }{\\ln (2)}\\int _{0}^{\\infty }(1+\\tau \\,x)^{-1}F_{\\gamma _i}^c(x)\\,dx,$ where $\\tau =e/(2\\pi )$ for the IM/DD technique and $\\tau =1$ for the heterodyne detection technique.", "It is worthy to mention that the expression in (REF ) is exact for $r=1$ while it is a lower-bound for $r=2$ , and can be achieved in exact closed-form in terms of the bivariate Meijer's G function by substituting (REF ) into (REF ), applying [68], and using [74] as $\\nonumber \\overline{C}_{i}& = \\frac{\\xi ^2\\, r^{\\alpha +\\beta -2}}{\\ln (2)\\Gamma (\\alpha )\\Gamma (\\beta ) (2 \\pi )^{r-1}}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1} \\\\\\nonumber & \\times \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k \\\\& \\times {\\rm {G}}_{1,0:0,2:3r+1,r+2}^{1,0:2,0:1,3r+1}\\begin{bmatrix}\\begin{matrix}0\\end{matrix}\\Bigg |\\begin{matrix}-\\\\j,1\\end{matrix}\\Bigg |\\begin{matrix}1,\\mathcal {K}_1\\\\1,\\Delta (r,-\\xi ^2),0\\end{matrix}\\Bigg |\\frac{C m_i}{\\kappa \\,\\overline{\\gamma }_{2,i}},\\frac{r^{2r}\\tau (\\xi ^2+1)^r \\mu _r }{(\\alpha \\beta \\xi ^2)^r \\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2}\\end{bmatrix},$ In the special case when $\\kappa =1$ , (50) reduces to the ergodic capacity of the $i$ -th UT of an FSO-based mutilbeam VHTS system with linear PA at the satellite transponder." ], [ "Numerical Results", "In this section, we examine the performance of a FSO-based multibeam VHTS system in the presence of atmospheric turbulence, beam wander effect, pointing errors, and HPA nonlinearities using the set of parameters listed in Table REF [47], [26].", "Table: System ParametersMonte-Carlo simulations are also included and compared with the obtained analytical results over $10^6$ realizations.", "A very good match between all the derived and the respective simulated results is observed, and hence, the accuracy of the proposed framework is verified.", "As illustrated in Fig.", "REF , we consider a FSS system with a coverage area composed of $N=7$ beams that serves multiple single antenna UTs.", "Assuming that there is only one UT per beam, each UT has a fixed position within each beam as shown by Fig.", "REF .", "Without loss of generality, we consider three types of turbulence conditions based on three values of nominal ground turbulence levels, i.e.", "$C_n^2(0)=1\\times 10^{-13} {\\rm m}^{-\\frac{2}{3}}$ , $C_n^2(0)=5\\times 10^{-13} {\\rm m}^{-\\frac{2}{3}}$ , and $C_n^2(0)=1\\times 10^{-12} {\\rm m}^{-\\frac{2}{3}}$ [63].", "Hence, from (REF ) and (REF ), the scintillation parameters ($\\alpha $ , $\\beta $ ) can be computed as (8.41, 14.67) with $\\sigma _{\\rm {pe}}=154.9$ , (2.57, 5.36) with $\\sigma _{\\rm {pe}}=141.59$ , and (1.52, 3.29) with $\\sigma _{\\rm {pe}}=133.18$ , respectively, when beam wander effects are included, whereas ($\\alpha $ , $\\beta $ )=(15.4, 14.67), (5.76, 5.36), and (3.62, 3.29), respectively, when beam wander effects are ignored.", "In addition, for the RF user link, two channel fading conditions are considered, namely, unfrequent light $\\left\\lbrace m_i, b_i, \\Omega _i \\right\\rbrace =\\left\\lbrace 19, 0.158, 1.29 \\right\\rbrace $ and frequent heavy $\\left\\lbrace m_i, b_i, \\Omega _i \\right\\rbrace =\\left\\lbrace 1, 0.063, 8.97\\times 10^{-4} \\right\\rbrace $ shadowing as provided in [56].", "Moreover, Since the noise power is normalized by $\\kappa _B B_w T_r$ in (REF ), we can assume that $\\sigma _1^2=1$ and we select $\\sigma _2^2=1$ , $I_l=1$ , and $G=1$ .", "Furthermore, we evaluate the performance relevant to the central beam which is located in the center of the coverage area and receives the maximum interference from the six adjacent beams as illustrated in Fig.", "REF .", "The outage probability versus the average electrical SNR $\\mu _r$ of the FSO feeder link under SSPA and TWTA models is plotted in Fig.", "REF for different values of $\\gamma _{\\rm {th}}$ .", "For both nonlinear HPA models, IBO is set to 25 dB.", "Results of the linear PA are also included for comparison purposes.", "It can be observed that the outage performance improves with the increase of $\\mu _r$ up to 35 dB under both TWTA and SSPA models.", "Moreover, both TWTA and SSPA as well as linear PA have the same impact on the outage probability up to 35 dB of the average electrical SNR.", "However, when $\\mu _r$ exceeds 35 dB, the nonlinearity effect of the power amplifier becomes more pronounced and the outage probability does not decrease even if $\\mu _r$ proceeds to increase.", "Indeed, as $\\mu _r$ gets larger, an outage floor is introduced regardless of the nonlinear HPA model while, it does not occur in the case of linear PA system that evidently performs better than the system with nonlinear power amplifier.", "Also, it can be noted that above 35 dB, TWTA and SSPA have different effects on the outage performance and the degradation of the outage probability caused by TWTA model is the largest.", "Furthermore, it is evident that the greater the value of the effective SNDR $\\gamma _{\\rm {th}}$ , the higher will be the outage probability of the system for both HPA models.", "The asymptotic results of the outage probability at high average electrical SNR values obtained by using (REF ) are also included in Fig.", "REF .", "Obviously, the asymptotic results of the outage probability match perfectly the analytical results in the high SNR regime.", "This justifies the accuracy and the tightness of the derived asymptotic expression in (REF ).", "Figure: OP under TWTA and SSPA models with 25 dB IBO for different values of γ th \\gamma _{\\rm {th}} when ξ=1.1\\xi =1.1 and C n 2 (0)=1×10 -12 C_n^2(0)=1 \\times 10^{-12} with beam wander effect under light shadowing conditions using IM/DD along with the asymptotic results at high SNR.Fig.", "REF illustrates the effect of beam wander associated with an untracked collimated beam on the outage performance for three different transmitter beam sizes $W_0$ corresponding to 1, 2, and 5 cm.", "We consider the SSPA model with ${\\rm {IBO}}=25$ dB.", "Based on (REF ), a value of $r_0=1.8$ cm is calculated for the Fried's atmospheric coherence width.", "The effect of the pointing error is fixed at $\\xi =1.1$ .", "As clearly seen for this figure, the outage performance under both IM/DD (i.e.", "$r=2$ ) and heterodyne (i.e.", "$r=1$ ) techniques is reduced when the transmitter beam size increases and it becomes worse when $W_0/r_0>>1$ .", "This is due to the fact that the scintillation index (SI) becomes higher as the ratio $W_0/r_0$ increases as demonstrated in [26].", "Indeed, using [26], the above mentioned transmitter beam sizes correspond to SI$=$ 0.81, 1.05, and 1.97, respectively for $C_n^2(0)=1\\times 10^{-12} {\\rm m}^{-\\frac{2}{3}}$ .", "Moreover, Fig.", "REF indicates that the heterodyne detection always performs better than the IM/DD technique for all SNR range, as expected.", "Although most of laser SatCom systems are based on the direct detection technique due to its simplicity and ease of deployment [26], coherent detection for the feeder link is preferred as it offers better spectral efficiency and higher sensitivity, compared to the IM/DD technique [28].", "Other outcomes, specifically for the high SNR asymptotic results and the outage floor due to HPA nonlinearity, can be clearly seen similar to Fig.", "REF above.", "Figure: OP under SSPA model with IBO =25{\\rm IBO}=25 dB for different values of the transmitter beam size W 0 W_0 with C n 2 (0)=1×10 -12 m -2 3 C_n^2(0)=1\\times 10^{-12} {\\rm m}^{-\\frac{2}{3}} under light shadowing conditions for ξ=1.1\\xi =1.1 and γ th =0\\gamma _{\\rm {th}}=0 dB.Fig.", "REF depicts the effect of changing the nominal ground turbulence levels on the outage performance under both IM/DD and heterodyne techniques without beam wander effect in the case when $\\gamma _{\\rm {th}}=5$ dB.", "We set the pointing error parameter to $\\xi =1.1$ and consider the TWTA model for nonlinear HPA with ${\\rm IBO}=25$ dB.", "We can observe that reducing the ground turbulence level results in better performance under both detection techniques.", "This phenomenon is due to the fact that the strength of the optical turbulence decreases as $C_n^2(0)$ becomes smaller.", "Figure: OP under TWTA model with IBO =25{\\rm IBO}=25 dB for different values of C n 2 (0)C_n^2(0) without beam wander effect under light shadowing conditions for ξ=1.1\\xi =1.1 and γ th =5\\gamma _{\\rm {th}}=5 dB with the asymptotic results at high SNR.Figure: Average BER with OOK under IM/DD for different TWTA IBOs and pointing errors with C n 2 (0)=1×10 -13 m -2 3 C_n^2(0)=1\\times 10^{-13} {\\rm m}^{-\\frac{2}{3}} under light shadowing conditions.The average BER performance with OOK modulation under TWTA model with 10 and 20 dB IBOs and different values of the pointing error parameter $\\xi $ without beam wander effect is shown in Fig.", "REF .", "Under the same conditions, results of the linear PA are also plotted in Fig.", "REF .", "Similar to the outage probability analysis, the same floor effect is observed here and as clearly seen, its level becomes higher as the value of IBO gets smaller and it vanishes when linear PA is employed.", "Moreover, the average BER performance improves as IBO increases for both values of $\\xi $ .", "This can be attributed to the fact that larger IBO values are associated with higher input power saturation levels $A_{\\rm {sat}}$ and consequently result in lower nonlinear distortion caused by HPA.", "Fig.", "REF also illustrates the effect of the pointing error on the BER performance.", "We can observe that for higher values of $\\xi $ , the effect of the pointing error is less severe and the average BER gets better, especially for higher values of IBO.", "To further illustrate the effect of IBO, the average BER with OOK modulation under different TWTA and SSPA IBO values is depicted in Fig.", "REF for $\\xi =1.1$ .", "As can be seen, the distortion caused by the amplifier's nonlinearity under both TWTA and SSPA models results in a degradation of the average BER performance, which becomes larger for lower values of IBO and a BER floor is introduced under both TWTA and SSPA especially at high SNR.", "As can be seen, the distortion caused by the amplifier's nonlinearity under both TWTA and SSPA models results in a degradation of the average BER performance, which becomes larger for lower values of IBO and a BER floor is introduced under both TWTA and SSPA especially at high SNR.", "Indeed, the effect of IBO on the average BER performance becomes more significant with the increase of the average electrical SNR.", "At low values of $\\mu _r$ , the IBO has a negligible impact on the performance and the system operates efficiently.", "This can be clearly observed from this figure as the average BER is almost the same for all values of IBO, especially for an average electrical SNR less than 15 dB.", "As $\\mu _r$ increases, the IBO parameter becomes more involved and the average BER performance improves as IBO gets larger.", "Figure: Average BER with OOK under IM/DD under TWTA and SSPA models with different values of IBO for ξ=1.1\\xi =1.1and C n 2 (0)=1×10 -12 m -2 3 C_n^2(0)=1\\times 10^{-12} {\\rm m}^{-\\frac{2}{3}} with beam wander effect under light shadowing conditions.Moreover, for all values of IBO, SSPA performs better than TWTA but its performance is still inferior to that of the linear PA. Additionally, it can be concluded that, it is necessary to use large values of IBO to obtain similar performance to the linear PA at least up to 40 dB of average electrical SNR.", "Furthermore, it can be observed that in the high SNR regime, the asymptotic expression of the average BER derived in (REF ) converges perfectly to the exact result proving the tightness of this asymptotic result.", "The BER performance for 64-QAM, 16-PSK, 16 QAM, and BPSK modulation schemes under the heterodyne detection technique and TWTA with an IBO of 25 dB, is shown in Fig.", "REF with varying effects of the pointing error ($\\xi =0.5$ and $1.1$ ).", "Clearly, we can observe that the BER performance for all modulation techniques gets better for lower effect of the pointing error (i.e.", "higher values of $\\xi $ ).", "Moreover, it can be seen from Fig.", "REF that 16-QAM outperforms 16-PSK, as expected when $M > 4$ [78] and BPSK modulation offers the best performance compared to the presented modulation techniques.", "Other outcomes, particulary for the asymptotic result at high SNR, can be noticed similar to Fig.", "REF .", "Figure: Average BER for different modulation schemes under TWTA with IBO =25{\\rm IBO}=25 dB and for varying effects of the pointing error withC n 2 (0)=1×10 -12 m -2 3 C_n^2(0)=1 \\times 10^{-12} {\\rm m}^{-\\frac{2}{3}} with beam wander effect under light shadowing conditions.Fig.", "REF presents the ergodic capacity using the IM/DD technique for negligible effect of the pointing error $\\xi =6.7$ in the presence of HPA nonlinearity.", "Both TWTA and SSPA models are considered with different values of IBO.", "As clearly seen from the figure, a capacity ceiling is created under both HPA models specially for small values of IBO.", "Moreover, it can be observed that there is an enhancement in the ergodic capacity as IBO increases and this improvement is greater when SSPA is used.", "For instance, both SSPA with IBO$=$ 25 dB and linear PA have the same impact on the ergodic capacity up to 40 dB.", "This confirms that SSPA performs better than TWTA.", "Figure: Ergodic capacity under TWTA and SSPA with different values of IBO for ξ=6.7\\xi =6.7 and C n 2 (0)=1×10 -13 m -2 3 C_n^2(0)=1\\times 10^{-13} {\\rm m}^{-\\frac{2}{3}} with beam wander effect under light shadowing conditions.Figure: Ergodic capacity under TWTA with IBO =25{\\rm IBO}=25 dB for different shadowing conditionswith C n 2 (0)=1×10 -13 C_n^2(0)=1 \\times 10^{-13} without beam wander effect under both IM/DD and heterodyne techniques.In Fig.", "REF , the ergodic capacity performance is plotted for both IM/DD and heterodyne techniques under light as well as heavy shadowing conditions.", "TWTA is considered with 25 dB IBO.", "It can be observed that as the shadowing conditions get severe, the performance under both types of detection is reduced.", "Moreover, the performance of each scheme saturates at the same level for high SNR regardless of the shadowing conditions because of the dominance of the HPA nonlinearity effect.", "In addition, as shown earlier in the outage performance analysis, the heterodyne technique performs much better than IM/DD under all shadowing conditions." ], [ "Conclusion", "In this paper, the performance of a multibeam VHTS system that uses the FSO technology in the feeder link and accounts for HPA nonlinearity has been analyzed in terms of the outage probability, the average BER, and the ergodic capacity when the FSO link operates under either IM/DD or heterodyne techniques.", "Closed-form expressions for these performance metrics are obtained in terms of the bivariate Meijer's G function considering the Gamma-Gamma distribution with beam wander and pointing error effects in the FSO feeder link, and the shadowed Rician fading channel in the RF user link.", "In addition, asymptotic results for the outage probability and the average BER in the high SNR regime are derived in terms of simple functions.", "The presented numerical results have demonstrated the notable effects of the atmospheric turbulence, the beam wander, the pointing errors, the shadowing conditions, and the HPA nonlinearity on the overall system performance.", "It has been shown that increasing the transmitted beam size or the nominal ground turbulence levels can result in severe performance degradation because of the increase in the scintillation index.", "Moreover, pointing errors can significantly degrade the performance, particularly for small values of the pointing error coefficient.", "Furthermore, the use of the heterodyne detection can considerably reduce the outage probability and the BER and increase the capacity, thereby improving the system performance.", "Our results also manifested the deleterious effects of the nonlinear distortion introduced by both TWTA and SSPA models compared to the linear PA case, especially with low IBO values, and revealed that the TWTA model leads to the greatest performance degradation." ], [ "CDF of the End-to-End SNDR", "This appendix derives closed-form expression for the CDF of the end-to-end SNDR at the $i$ -th UT $\\gamma _i$ .", "We start by deriving the CDF of $\\Lambda _i= \\frac{\\gamma _1 \\gamma _{2,i}}{\\kappa \\gamma _{2,i}+C}$ which can be written as $\\nonumber F_{\\Lambda _i}(x)&={\\rm {Pr}}\\left[ \\frac{\\gamma _1\\,\\gamma _{2,i}}{\\kappa \\gamma _{2,i}+C} \\le x \\right]\\\\\\nonumber &=1- \\int _{0}^{\\infty }\\left( 1- {\\rm {Pr}}\\left[ \\frac{\\gamma _1\\,\\gamma _{2,i}}{\\kappa \\gamma _{2,i}+C} \\le x \\,|\\,\\gamma _1 \\right]\\right)f_{\\gamma _1}(\\gamma _1)\\,d\\gamma _1\\\\&=1-\\int _{0}^{\\infty }\\overline{F}_{\\gamma _{2,i}}\\left( \\frac{C x}{z} \\right)f_{\\gamma _1}(\\kappa \\,x +z)\\,dz,$ where $C=\\operatorname{tr}\\left[ \\left({\\bf B}{\\bf B}^{\\mbox{\\tiny H}} \\right)^{-1} \\right]\\overline{\\gamma }_1+\\kappa $ and $\\overline{F}_{\\gamma _{2,i}}(\\cdot )$ stands for the complementary CDF of $\\gamma _{2,i}$ derived from (REF ) by applying [62] as $\\nonumber \\overline{F}_{\\gamma _{2,i}}(x)&= \\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1}\\exp \\left( -\\frac{m_i \\,x}{\\overline{\\gamma }_{2,i}} \\right)\\\\&\\times \\sum _{k=0}^{m_i-1}\\frac{(-1)^k (1-m_i)_k}{k!}", "\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k\\sum _{j=0}^{k}\\frac{1}{j!", "}\\left( \\frac{m_i\\, x}{\\overline{\\gamma }_{2,i}} \\right)^j.$ Substituting (REF ) and (REF ) into (REF ), transforming the $\\exp (\\cdot )$ function to its correspondent Meijer's G function by applying [68], using the definition of the Meijer's G function given in [62], and interchanging the integrals, the CDF of $\\Lambda $ becomes $\\nonumber F_{\\Lambda _i}(x)&= 1-\\frac{\\xi ^2}{r\\, \\Gamma (\\alpha )\\Gamma (\\beta )}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1}\\\\\\nonumber & \\times \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!}", "\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k\\left( \\frac{Cm_i\\, x}{\\overline{\\gamma }_{2,i}} \\right)^j \\frac{1}{(2\\pi i)^{2}}\\\\\\nonumber & \\times \\int \\limits _{\\mathcal {L}_1}\\int \\limits _{\\mathcal {L}_2}\\Gamma (-s)\\left( \\frac{C m_i x}{\\overline{\\gamma }_{2,i}} \\right)^s \\frac{\\Gamma (\\xi ^2-t)\\Gamma (\\alpha -t)\\Gamma (\\beta -t)}{\\Gamma (\\xi ^2+1-t)}\\\\& \\times \\left(\\frac{\\alpha \\beta \\xi ^2}{(\\xi ^2+1)\\mu _r^{\\frac{1}{r}}} \\right)^t\\int _{0}^{\\infty }z^{-j-s}(z+\\kappa \\, x)^{\\frac{t}{r}-1}\\,dz \\,ds\\, dt,$ where $\\mathcal {L}_1$ and $\\mathcal {L}_2$ represent the $s$ - and $t$ -plane contours, respectively.", "Utilizing [62] then [62], $\\int _{0}^{\\infty }z^{-j-s}(z+\\kappa \\, x)^{\\frac{t}{r}-1}\\,dz $ reduces to $(\\kappa \\,x)^{\\frac{t}{r}-s-j}\\Gamma (1-j-s)\\Gamma (j+s-\\frac{t}{r})/\\Gamma (1-\\frac{t}{r})$ , and (REF ) can be re-written as $\\nonumber F_{\\Lambda _i}(x)&= 1-\\frac{\\xi ^2}{\\Gamma (\\alpha )\\Gamma (\\beta )}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1} \\\\\\nonumber & \\times \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k\\frac{1}{(2\\pi i)^{2}}\\int \\limits _{\\mathcal {L}_1}\\int \\limits _{\\mathcal {L}_2}\\Gamma (s+t)\\\\\\nonumber & \\times \\Gamma (j-s) \\Gamma (1-s)\\left( \\frac{C m_i}{\\kappa \\,\\overline{\\gamma }_{2,i}} \\right)^s \\frac{\\Gamma (\\xi ^2+rt)\\Gamma (\\alpha +rt)\\Gamma (\\beta +rt)}{\\Gamma (\\xi ^2+1+rt)\\Gamma (1+t)}\\\\&\\times \\left(\\frac{(\\xi ^2+1)^r \\mu _r }{(\\alpha \\beta \\xi ^2)^r \\kappa \\,x} \\right)^t\\,ds \\,dt.$ Plugging $\\Gamma (n z)=n^{n z-\\frac{1}{2}} (2 \\pi )^{\\frac{1-n}{2}}\\prod _{k=0}^{n-1}\\Gamma \\left( z+\\frac{k}{n} \\right)$ for $n \\in \\mathbb {N}$ in (REF ) yields $\\nonumber F_{\\Lambda _i}(x)&= 1-\\frac{\\xi ^2\\, r^{\\alpha +\\beta -2}}{\\Gamma (\\alpha )\\Gamma (\\beta ) (2 \\pi )^{r-1}}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1}\\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k} \\\\\\nonumber & \\times \\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k\\frac{1}{(2\\pi i)^{2}}\\int \\limits _{\\mathcal {L}_1}\\int \\limits _{\\mathcal {L}_2}\\Gamma (s+t)\\Gamma (j-s) \\\\\\nonumber & \\times \\Gamma (1-s)\\left( \\frac{C m_i}{\\kappa \\,\\overline{\\gamma }_{2,i}} \\right)^s \\frac{\\prod _{i=0}^{r-1}\\Gamma \\left(\\frac{\\xi ^2+i}{r}+t\\right)\\prod _{i=0}^{r-1}\\Gamma \\left(\\frac{\\alpha +i}{r}+t\\right)}{\\prod _{i=0}^{r-1}\\Gamma \\left(\\frac{\\xi ^2+1+i}{r}+t\\right)\\Gamma (1+t)}\\\\& \\times \\prod _{i=0}^{r-1}\\Gamma \\left(\\frac{\\beta +i}{r}+t\\right) \\left(\\frac{r^{2r}(\\xi ^2+1)^r \\mu _r }{(\\alpha \\beta \\xi ^2)^r \\kappa \\,x} \\right)^t\\,ds \\,dt.$ With the help of [74], the CDF of $\\Lambda _i$ can be derived in terms of the bivariate Meijer's G function as $\\nonumber F_{\\Lambda _i}(x)&= 1-\\frac{\\xi ^2\\, r^{\\alpha +\\beta -2}}{\\Gamma (\\alpha )\\Gamma (\\beta ) (2 \\pi )^{r-1}}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1} \\\\\\nonumber & \\times \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k \\\\&\\times {\\rm {G}}_{1,0:0,2:3r,r+1}^{1,0:2,0:0,3r}\\begin{bmatrix}\\begin{matrix}0\\end{matrix}\\Bigg |\\begin{matrix}-\\\\j,1\\end{matrix}\\Bigg |\\begin{matrix}\\mathcal {K}_1\\\\\\Delta (r,-\\xi ^2),0\\end{matrix}\\Bigg |\\frac{C m_i}{\\kappa \\,\\overline{\\gamma }_{2,i}},\\frac{r^{2r}(\\xi ^2+1)^r \\mu _r }{(\\alpha \\beta \\xi ^2)^r \\kappa \\,x}\\end{bmatrix},$ where $\\mathcal {K}_1=\\Delta (r,1-\\xi ^2),\\Delta (r,1-\\alpha ),\\Delta (r,1-\\beta )$ and $\\Delta (r,u)=\\frac{u}{r},\\frac{u+1}{r},\\ldots ,\\frac{u+r-1}{r}$ .", "Finally, the desired CDF expression of the end-to-end SNDR at the $i$ -th UT, $\\gamma _i$ , can be easily obtained from (REF ) using a simple RV transformation as shown by (REF )." ], [ "Moments", "By applying [74], the bivariate Meijer's G function in (REF ) can be written as a definite integral involving the product of three Meijer's G functions and therefore, the moments can be expressed as $\\nonumber \\mathbb {E}[\\gamma _i^n]&=\\frac{\\xi ^2\\, r^{\\alpha +\\beta -2}}{\\Gamma (\\alpha )\\Gamma (\\beta )(2 \\pi )^{r-1}}\\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1} \\sum _{k=0}^{m_i-1}\\sum _{j=0}^{k}\\\\\\nonumber & \\times \\frac{(-1)^k (1-m_i)_k}{k!\\,j!", "}\\left(\\frac{\\Omega _i}{2 b_i m_i} \\right)^k\\int _{0}^{\\infty }\\frac{e^{-z}}{z}{\\rm {G}}_{0,2}^{2,0}\\left[\\frac{C\\,m_i z}{\\kappa \\overline{\\gamma }_{2,i}}\\left| \\begin{matrix} {-} \\\\ {j,1} \\\\ \\end{matrix} \\right.", "\\right]\\\\& \\times \\int _{0}^{\\infty } x^{n-1}{\\rm {G}}_{3r,r+1}^{0,3r}\\left[\\frac{r^{2r}(\\xi ^2+1)^r \\mu _r z}{(\\alpha \\beta \\xi ^2)^r \\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2 x} \\left| \\begin{matrix} {\\mathcal {K}_1}\\\\{\\Delta (r,-\\xi ^2),1} \\\\ \\end{matrix} \\right.", "\\right]dx\\, dz.$ By transforming the Meijer's G function to its correspondent Fox's H function with the help of [71], inverting the argument of the obtained Fox's H function via [71], applying [71], and utilizing the integral identity [79], (REF ) reduces to $\\nonumber \\mathbb {E}[\\gamma _i^n]&=\\frac{\\xi ^2\\Gamma (\\alpha +r\\,n)\\Gamma (\\beta + r\\,n)}{(\\xi ^2+r\\,n)\\Gamma (\\alpha )\\Gamma (\\beta )\\Gamma (n)}\\left(\\frac{(\\xi ^2+1)^r \\mu _r}{(\\alpha \\beta \\xi ^2)^r \\kappa \\left\\Vert {\\bf b}_i^{\\mbox{\\tiny T}} \\right\\Vert ^2 } \\right)^n\\\\&\\times \\left(\\frac{2b_i m_i}{2 b_i m_i+\\Omega _i} \\right)^{m_i-1}\\int _{0}^{\\infty }z^{n-1}e^{-z}{\\rm {G}}_{0,2}^{2,0}\\left[\\frac{Cm_i z}{\\kappa \\overline{\\gamma }_{2,i}}\\left| \\begin{matrix} {-} \\\\ {j,1} \\\\ \\end{matrix} \\right.", "\\right]dz.$ Finally, (REF ) is further simplified to (REF ) by exploiting the integral identity [62]." ] ]

[ [ "Realistic GUT Yukawa Couplings from a Random Clockwork Model" ], [ "Abstract We present realistic models of flavor in SU(5) and SO(10) grand unified theories (GUTs).", "The models are renormalizable and do not require any exotic representations in order to accommodate the necessary GUT breaking effects in the Yukawa couplings.", "They are based on a simple clockwork Lagrangian whose structure is enforced with just two (one) vectorlike U(1) symmetries in the case of SU(5) and SO(10) respectively.", "The inter-generational hierarchies arise spontaneously from products of matrices with order one random entries." ], [ "Introduction", "The puzzling flavor structure of the Standard Model (SM) has inspired a lot of model building over the past decades.", "The large ratios of masses and mixing parameters displayed in table REF , commonly referred to as flavor hierarchies, cannot be the result of a generic UV theory with order-one couplings.", "Another feature of the SM matter sector is the peculiar structure of its five gauge representations, which strongly hints at some unified gauge group with only two $SU(5)$ or one $SO(10)$ representation.", "These so-called grand-unified theories (GUTs) in turn strongly motivate the extension of the SM to its minimally supersymmetric version, the MSSM, as the latter provides an accurate unification of the gauge couplings near a scale of the order of $10^{16}$ GeV, commonly referred to as the GUT scale.", "However, the unification of couplings is much less impressive in the Yukawa sector.", "For instance, in $SU(5)$ unification, one finds the GUT-scale relation $\\mathcal {Y}_d=\\mathcal {Y}_e^T$ for the Yukawa couplings matrices of the charged-lepton and down-quark sectors.", "A quick glance at table REF (which shows the couplings at the GUT scale) reveals that this is certainly not the case.", "Even though supersymmetric threshold corrections are more important than in the gauge sector (and more dependent on the superpartner spectrum), it is very hard to attribute the large differences in the couplings to this alone.", "Typically it is possible to adjust the spectrum such that only some but not all of the Yukawa couplings become unified.", "Certain GUT breaking effects at the high scale thus seem to be unavoidable.", "Some efforts have been made to include such breaking effects, for instance via exotic Higgs representations [1], nonrenormalizable operators [2], [3], vectorlike representations [4], or combinations of these [5].", "In minimal $SO(10)$ unification, all Yukawa couplings are predicted to be exactly equal, $\\mathcal {Y}_u=\\mathcal {Y}_d=\\mathcal {Y}_e^T$ .", "The necessary GUT breaking effects now become even larger, as a comparison between the different columns of table REF shows.", "Table: Quark and Lepton data at the GUT scale in the MSSM .Here, y i ' =y i sinβy^{\\prime }_i=y_i\\sin \\beta and y i '' =y i cosβy^{\\prime \\prime }_i=y_i\\cos \\beta .The values are representative, as they depend on the supersymmetric threshold corrections, and indirectly on tanβ\\tan \\beta via the renormalization group running.The Clockwork (CW) mechanism, originally formulated in [7], [8] in the context of the Relaxion, has soon been recognized as a general framework for constructing natural hierarchies [9].", "In the flavor sector, it has been applied to explain the smallness of neutrino masses [10], [11], [12], [13], [14] as well as the charged flavor hierarchies mentioned above [15], [16], [17], [18], [19], [20].", "The CW Lagrangian is basically a one dimensional lattice (\"theory space\") of nearest-neighbor interactions enforced by some symmetry.", "The generation of hierarchies can be attributed to a controlled localization of zero modes towards the boundaries of theory space.", "However, it has also been pointed out [21], [20], [22] that when the parameters in the CW Lagrangian are chosen at random, sharp localization of the zero modes occurs in the bulk of theory space, still leading to hierarchical suppression of couplings.", "Moreover, when the couplings of the CW model become $3\\times 3$ matrices in flavor space, the three zero modes spontaneously localize at different points in the lattice, and generate inter-generational hierarchies (the vertical hierarchies in table REF ) [20].", "On a technical level, this is closely related to a peculiar property of products of $N$ random matrices, which feature a very hierarchical spectrum despite their matrix elements being all of order one [16].", "In this paper, we are going to build models of natural flavor hierarchies along the lines of Refs.", "[16], [20] in supersymmetric $SU(5)$ and $SO(10)$ unification.", "The necessary GUT breaking effects are naturally present in these models, as the symmetries of the CW Lagrangian allow for renormalizable Yukawa couplings of the vectorlike fields (CW gears) with the GUT-breaking Higgs field(s).", "The flavor hierarchies and Yukawa unification thus have a common origin in the framework of a completely renormalizable model without any exotic Higgs or matter representations.", "This paper is organized as follows.", "In section we describe our models and briefly review the mechanism of flavor hierarchies.", "In section we perform comprehensive scans over the parameter space of these models, and quantify how well the SM flavor structure is reproduced.", "Some phenomenological considerations are given in section , and in section we present our conclusions." ], [ "Charged Fermions", "We denote the usual $SU(5)$ GUT field content, consisting of 3 generations of $\\bf \\bar{5}$ and $\\bf 10$ each by $F^c$ and $A$ respectively, where here and in the following we suppress all flavour indices.", "Firstly, we add to this $N$ vectorlike copies $(F,F^c)_{i}$ , which are charged under a vectorlike $U(1)$ symmetry under which $F_i$ has charge $i$ and $F_i^c$ has charge $-i$ .", "The index $i=1\\dots N$ will be referred to as a site (in theory space).", "In addition we introduce the $U(1)$ breaking spurion $\\phi $ with charge $-1$ .", "We will denote the chiral MSSM GUT fields with an index $i=0$ , a notation consistent with their vanishing $U(1)$ charge.", "The unique renormalizable superpotential allowed by symmetries is thus of the clockwork type $W_5=\\sum _{i=1}^N \\phi (F^c_{i-1})^Ti F_{i}+(F^c_i)^T(m\\mathcal {A}_i+\\Sigma \\mathcal {B}_i)F_i\\,,$ where $m$ is a mass scale, $\\Sigma $ is the $SU(5)$ breaking adjoint field, and $\\mathcal {A}_i$ , $\\mathcal {B}_i$ and $i$ are dimensionless couplings that are $3\\times 3$ arbitrary complex matrices.Throughout this paper we adopt the convention that calligraphic capital letters denote complex $3\\times 3 $ matrices.", "The $U(1)$ symmetry is anomaly free (and hence can be gauged) once a conjugate field $\\phi ^c$ is included, we will assume $\\langle \\phi \\rangle \\gg \\langle \\phi ^c\\rangle $ such that we can simply ignore it.Including $\\phi ^c$ allows for the other nearest neighbor interactions, of the kind $\\phi ^c F_{i+1}^c F_i$ .", "This interaction was included in the model of Ref.", "[16] and leads to very similar structure for the physical Yukawa couplings.", "Secondly, in a completely analogous manner we introduce copies of the fields transforming in the antisymmetric $\\bf 10$ representation, charged under a $U(1)^{\\prime }$ symmetry, with superpotentialWe use \"strength-one\" notation for all gauge invariants, i.e.", "$F^cF=F^c_aF^a$ , $A^cA=\\frac{1}{2}A^c_{ab}A^{ab}$ , $FAA^{\\prime }=\\frac{1}{4}F^a A^{bc}A^{\\prime de}\\epsilon _{abcde}$ , $F^c\\Sigma F=F^c_a\\Sigma ^a_{\\ b}F^c$ , $A^c\\Sigma A=A^c_{ab}\\Sigma ^a_{\\ d}A^{db}$ .", "Matrix notation (such as $^T$ and $^\\dagger $ ) is then exclusively reserved for flavor space.", "$W_{10}=\\sum _{i=1}^{N^{\\prime }}\\phi ^{\\prime } (A^c_{i})^T\\, _i A_{i-1}+(A^c_i)^T(m^{\\prime }\\mathcal {A}^{\\prime }_i+\\Sigma \\,\\mathcal {B}^{\\prime }_i)A_i\\,,$ Furthermore, we denote the two Higgs doublets as $H^c$ and $H$ .", "The Yukawa superpotential reads $W_Y=H^c(A_0)^T\\, \\mathcal {Y} F_0^c + \\frac{1}{2} H(A_0)^T \\mathcal {Y}^{\\prime }A_0\\,,$ with $\\mathcal {Y}^{\\prime T}=\\mathcal {Y}^{\\prime }$ and $\\mathcal {M}_\\nu ^T=\\mathcal {M}_\\nu $ .", "In terms of MSSM fields, this gives $W_Y=H_d (Q_0)^T \\mathcal {Y}D^c_0 + H_d (E^c_0)^T \\mathcal {Y}\\, L_0 + H_u (Q_0)^T \\mathcal {Y}^{\\prime }\\,U_0^c+\\dots \\,,$ where the ellipsis denotes terms with the Higgs triplet.", "The field content of our model is summarized in table REF .", "Table: Field content and symmetries of the model.", "All matter fields and their vectorlike partners, F i F_i, F i c F_i^c, A i A_i, A i c A^c_i, and N c N^c carry an additional generation index (not shown).The complete superpotential of the charged fermions is $W=W_5+W_{10}+W_Y$ .", "Integrating out the clockwork fields with $i\\ne 0$ exactly, the superpotential becomes $W=W_Y$ , while the Kähler potential turns into $K=(F_0^c)^\\dagger \\mathcal {Z}F^c_0+ (A_0)^\\dagger \\mathcal {Z}^{\\prime } A_0\\,,$ with $\\mathcal {Z}\\equiv \\sum _{i=0}^N (\\mathcal {\\mathcal {Q}}_i\\cdots \\mathcal {Q}_1)^\\dagger (\\mathcal {Q}_i\\cdots \\mathcal {Q}_1)\\,,\\qquad \\mathcal {Q}_i\\equiv (m\\mathcal {A}_i+v_{24}Y\\mathcal {B}_i)^{-1}\\phi \\,i\\,,$ and analogously for $\\mathcal {Z}^{\\prime }$ .", "Here $Y$ is the hypercharge (with SM normalization, $Y_Q=1/6$ ).", "Notice that the flavor matrices $\\mathcal {Z}$ become hypercharge dependent and thus provide a source of GUT breaking.", "It is this effect that we want to exploit in order to separate the down quark and charged lepton Yukawa couplings.", "After canonical normalization, the physical Yukawa couplings read $\\mathcal {Y}_u^*=(\\mathcal {E}_Q)^T \\mathcal {Y}^{\\prime } \\mathcal {E}_{U^c}\\,,\\qquad \\mathcal {Y}_d^*=(\\mathcal {E}_Q)^T \\mathcal {Y}\\mathcal {E}_{D^c}\\,,\\qquad \\mathcal {Y}_e^*=(\\mathcal {E}_{L})^T\\mathcal {Y}^T \\mathcal {E}_{E^c}\\,,$ where the Hermitian matrices $\\mathcal {E}_X$ are defined as $\\mathcal {E}_{D^c}\\equiv (\\mathcal {Z}_{{2}/{3}})^{-{1}/{2}}\\,,\\qquad \\mathcal {E}_{L}\\equiv (\\mathcal {Z}_{-{1}/{2}})^{-{1}/{2}}\\,,$ and $\\mathcal {E}_{Q}\\equiv (\\mathcal {Z}^{\\prime }_{{1}/{6}})^{-1/2}\\,,\\qquad \\mathcal {E}_{U^c}\\equiv (\\mathcal {Z}^{\\prime }_{-{2}/{3}})^{-1/2}\\,,\\qquad \\mathcal {E}_{E^c}\\equiv (\\mathcal {Z}^{\\prime }_{1})^{-{1}/{2}}\\,.$ Here the subscripts on the $\\mathcal {Z}$ , $\\mathcal {Z}^{\\prime }$ refer to hypercharge.", "The eigenvalues of $\\mathcal {Z}$ ($\\mathcal {E}$ ) are always larger (smaller) than one.", "Assuming no further relation between couplings it is reasonable to expect that the matrices $\\mathcal {A}^{(\\prime )}$ , $\\mathcal {B}^{(\\prime )}$ , ${(\\prime )}$ , and $\\mathcal {Y}^{(\\prime )}$ have $\\mathcal {O}(1)$ complex entries.", "As has been shown [16], [20], the matrices $\\mathcal {E}_X$ , even though not containing any a priori large or small parameters, spontaneously develop strongly hierarchical spectra, i.e., their three eigenvalues satisfy $\\epsilon _{X^1}\\ll \\epsilon _{X^2}\\ll \\epsilon _{X^3}\\le 1\\,.$ We will refer to this kind of hierarchies as inter-generational hierarchies, to distinguish them from inter-species hierarchies between the different matter representations (e.g.", "between top and tau).", "In order to better understand this spontaneous generation of inter-generational hierarchies, let us define positive parameters $a\\equiv \\frac{m}{\\phi }\\,,\\qquad b\\equiv \\frac{v_{24}}{\\phi }\\,,$ and $a^{\\prime }\\equiv \\frac{m^{\\prime }}{\\phi ^{\\prime }}\\,,\\qquad b^{\\prime }\\equiv \\frac{v_{24}}{\\phi ^{\\prime }}\\,.$ In the extreme limit $a+b\\rightarrow \\infty $ , we have $\\mathcal {Z}\\approx 1\\,,$ and there are no inter-generational hierarchies, while in the opposite limit of $a+b\\rightarrow 0$ , we obtain a product structure $\\mathcal {Z}\\approx (\\mathcal {\\mathcal {Q}}_N\\cdots \\mathcal {Q}_1)^\\dagger (\\mathcal {Q}_N\\cdots \\mathcal {Q}_1)\\,,$ which spontaneously generates large inter-generational hierarchies between the three eigenvalues.", "This can be understood in terms of a general property of products of random $O(1)$ matrices [16].", "The parameters $a$ and $b$ thus interpolate between a hierarchical and non-hierarchical situation.", "It is noteworthy that the hierarchies in Eq.", "(REF ) become independent of $a,b$ (in the sense that only the eigenvalue's overall size but not their ratios depend on them).", "Interestingly, even in the case $a+b\\sim 1$ , More precisely $a+|Y| b\\sim 1$ .", "a strong hierarchy is still present, especially between the third and second generations (see also the discussion in [20]).", "A complementary explanation of this mechanism can be provided by the localization of the zero modes.", "As has also been shown in ref.", "[20] the zero modes of the matter fields spontaneously localize sharply around some random site in theory space, a property first pointed out in [21] in similar models.", "In this interpretation, the zero modes of the three generations are localized at different sites in theory space, which explains their hierarchical overlap with the site-zero fields.", "The parameters $a$ and $b$ set a bias for this localization, the larger $a$ and $b$ , the more the zero modes are localized towards site zero.", "Moving to a basis in which the $\\mathcal {E}_X$ are diagonal, the Yukawa couplings assume the structure $(\\mathcal {Y}_u)_{ij}\\sim \\epsilon _{Q_i}\\epsilon _{U^c_j}$ etc, familiar from Froggatt-Nielsen [23], extra-dimensional [24], [25], [26], or strongly coupled [27] models, and the hierarchies of masses and mixings follow in a way similar to those (see for instance Ref. [28]).", "The CKM angles scale as $\\theta _{ij}\\sim \\epsilon _{Q^i}/\\epsilon _{Q^j}$ ($i<j$ ) and similarly for the PMNS anlges with $Q\\rightarrow L$ .", "The charged fermion masses on the other hand scale as $\\epsilon _{Q^i}\\epsilon _{U^i}$ , $\\epsilon _{Q^i}\\epsilon _{D^i}$ , and $\\epsilon _{L^i}\\epsilon _{E^i}$ , respectively.", "Then, the CKM hierarchies roughly determine the $\\epsilon _{Q^i}$ hierarchies.", "As a general rule, this saturates the hierarchies in the down quark masses, i.e., the required hierarchies in the $\\epsilon _{D^i}$ are rather mild, while the hierarchies in the up quark masses require further suppression from the $\\epsilon _{U^i}$ .", "In the lepton sector, clearly one must avoid a large hierarchy in the $\\epsilon _{L^i}$ in order to keep the PMNS angles large.", "The charged lepton hierarchies then come mostly from the $\\epsilon _{E^i}$ parameters.", "From these rough considerations alone, it is already quite apparent that the structure is rather $SU(5)$ compatible (large hierarchies in the $\\bf 10$ sector and mild hierarchies in the $\\bf \\bar{5}$ sector).", "In section we will indeed show that the $SU(5)$ model works remarkably well.", "However, it turns out that our ideas are even partially successful in $SO(10)$ unified models, and it is worthwhile then to develop an $SO(10)$ version of the model, which will be done in section ." ], [ "Neutrinos", "To generate neutrino masses we will employ the seesaw mechanism [29], [30], [31], [32], [33], [34].", "We will assume no clockwork fields for the neutrinos, such that we have the simple superpotential $W_\\nu =H_u (N^c)^T \\mathcal {Y}^{\\prime \\prime } L_0 +\\frac{1}{2}(N^c)^T \\mathcal {M}N^c+\\dots \\,,$ where we have already discarded the Higgs triplets.", "Integrating out the right handed (RH) neutrinos as well as the clockwork leptons gives the Weinberg operator $W_{\\nu }=-\\frac{1}{2}(H_u L)^T (\\mathcal {E}_L^T\\mathcal {Y}^{\\prime \\prime T}\\mathcal {M}^{-1}\\mathcal {Y}^{\\prime \\prime }\\mathcal {E}_L)\\,(H_uL)\\,.$ We parametrize $\\mathcal {M}=m_R \\, , where $ mR$ is a scale and $ is a further dimensionless order one complex (symmetric) matrix." ], [ "An $SO(10)$ extension.", "It is possible to extend the previous model to $SO(10)$ .", "The SM fields (including the right handed neutrino) unify into a spinorial $\\bf 16$ representation, denoted by $S$ .", "To break $SO(10)$ down to the SM one needs more than one irreducible representation.", "We will choose a $\\bf 45$ and a vectorlike ($\\bf 126+\\overline{126}$ ).", "These are the lowest-dimensional representations that satisfy the following three criteria: $(i)$ breaking of $SO(10)\\rightarrow $ SM , $(ii)$ possibility to write Yukawa couplings in the clockwork Lagrangian with $\\bf 16$ , $\\bf \\overline{16}$ and a GUT breaking Higgs field, $(iii)$ possibility to write Majorana neutrino masses [32], [33].Criterion $(i)$ allows also for other low-dimensional representations, for instance $\\bf 54$ instead of $\\bf 45$ and $\\bf 16$ instead of $\\bf 126$ .", "However criterion $(ii)$ selects $\\bf 45$ over $\\bf 54$ , and criterion $(iii)$ selects $\\bf 126$ over $\\bf 16$ ." ], [ "Charged Fermions", "We are then lead to consider a single $U(1)$ symmetry (the combination $Q-Q^{\\prime }$ of the symmetries in the $SU(5)$ model which commutes with $SO(10)$ ).", "The field content is given in table REF .", "The model is defined by $W_{16}=\\sum _{i=1}^{N}\\phi \\, (S^c_{i})^T i\\, S_{i-1}+(S^c_i)^T(m\\mathcal {A}_i+\\Sigma \\,\\mathcal {B}_i)\\, S_i\\,,$ along with the unique Yukawa coupling $W_Y=\\frac{1}{2}H(S_0)^T\\mathcal {Y}S_0\\,.$ Table: Breaking patterns of 45+126+126 ¯\\bf 45 +126+\\overline{126}.", "Note that ℍ 126 =SU(5)\\mathbb {H}_{126}=SU(5) always.", "The Yukawa relations marked with ** are ruled out.The VEV of the $\\bf 45$ is a generic linear combination of hypercharge and $B-L$ generators, which breaks $SO(10)$ down to $\\mathbb {H}_{45}=$ SM$\\times (B-L)$ .", "At the same time the VEV of the $\\bf 126$ representation breaks $SO(10)$ to $\\mathbb {H}_{126}=SU(5)$ , leaving as the unbroken subgroup $\\mathbb {H}_{45}\\cap \\mathbb {H}_{126}=\\rm SM$ .", "In certain particular directions of $\\langle \\Sigma \\rangle $ , the group $\\mathbb {H}_{45}$ can be enhanced.", "These directions are summarized in table REF .", "Since only the 45 VEV participates in the GUT breaking of the Yukawa couplings, an enhanced $\\mathbb {H}_{45}$ may lead to some relations between Yukawa couplings even when $\\mathbb {H}_{45}\\cap \\mathbb {H}_{126}=\\rm SM$ .", "This is the case for all but the last row in table REF .It is worthwhile to point out that the first row in table REF corresponds to the Dimopoulos-Wilczek mechanism which is therefore impossible to realized within our model.", "Let us write the VEV of $\\Sigma $ as $\\langle \\Sigma \\rangle = v_{45} [\\sin \\alpha \\, Y+\\cos \\alpha \\, (B-L)]\\,.$ The model then has one discrete and 4 continuous non-stochastic parameters: $N$ , $a\\equiv m/\\phi $ , $b\\equiv v_{45}/\\phi $ , $\\tan \\alpha $ , and $\\tan \\beta $ .A further parameter, the seesaw scale, only affects the neutrino sector According to table REF , the values $\\tan \\alpha =0,\\ -\\frac{4}{5},\\ -2$ are ruled out by either $\\mathcal {Y}_u=\\mathcal {Y}_d$ or $\\mathcal {Y}_d=\\mathcal {Y}_e$ ." ], [ "Neutrinos", "With the symmetry assignments as in table REF , there is a unique Majorana mass term for the RH neutrinos $W_{R}=\\frac{1}{2}v_{126}N_0^{cT} N̥_0^{c}\\,,$ where $ is another complex order one symmetric, dimensionless matrix.All other Majorana mass terms are forbidden by the nonzero $ U(1)$ charges.", "Considering the non-hierarchical nature of neutrino masses, this is actually very welcome, since this implies that the hierarchical factors $ E$ drop out of the Weinberg operator.We could, for instance, first integrate out the clockwork gauge-singlets, and the RH neutrino fields $ Nc0$ would obtain the usual hierarchical wave function renormalization factor.", "But the normalization of the RH neutrino^{\\prime }s kinetic term is irrelevant, as they are integrated out in the see-saw mechanism.$" ], [ "Simulations", "In this section we would like to find a sets of parameters such that the mechanism leads to a successful generation of the SM flavor structure.", "We distinguish two kind of model parameters.", "The first kind quantify some underlying physics assumption, such as the scales of symmetry breaking, or the number of vectorlike fields.", "The following parameters are of this type: $N,\\ N^{\\prime },\\ a,\\ b,\\ a^{\\prime },\\ b^{\\prime },\\ \\tan \\beta ,\\ m_R\\qquad SU(5) {\\rm \\ model,}$ $N,\\ a,\\ b,\\ \\tan \\alpha ,\\ \\tan \\beta ,\\ m_R\\qquad SO(10) {\\rm \\ model.", "}$ We will refer to these parameters as non-stochastic or deterministic.", "The remaining parameters are the coupling matrices $\\mathcal {A}^{(\\prime )}_i$ , $\\mathcal {B}^{(\\prime )}_i$ , ${(\\prime )}_i$ , $\\mathcal {Y}^{(\\prime )}$ and $.", "In the absence of additional structure, such as symmetries that would constrain the form of these matrices, it is natural to assume that they have order-one complex entries.Choosing them randomly from some suitable prior distribution defines an ensemble of models (with the same physics assumptions).", "We can then compute the distributions of physical observables (masses and mixings), for each choice of the deterministic parameters.In order to model the property \"order one\" for the matrix elements, we will chose flat uniform priors with $ |Re(Ai)kl| 1$ and $ |Im(Ai)kl| 1$ etc.", "Of course, the \"posterior distributions\" depend to some extent on the choice of priors.\\footnote {We have checked though that the posterior that for instance the \\chi ^2 values reported in tables \\ref {tab:simu1} and \\ref {tab:simu2} are virtually identical if we substitute the uniform priors with Gaussians.", "}$ In order to quantify the success of the mechanism, we proceed as follows.", "Let us define the variables $x_i\\equiv \\log _{10} O_i\\,,$ where the observables $O_i$ run over the nine physical Yukawa couplings, the three CKM mixing angles $\\theta _{ij}$ , as well as the three quantities $\\sin ^2\\theta _{ij}$ of the PMNS matrix.", "For the time being we will focus our attention to these 15 observables (all taken at the GUT scale).", "We will comment on the remaining observables (neutrino mass squared differences and CP violating phases) below.", "It turns out that the logarithms of the observables roughly follow a multi-dimensional Gaussian distribution.", "It is therefore useful to approximate this distribution by a Gaussian with mean and covariance taken from the exact (simulated) distribution.", "This defines a $\\chi ^2$ function $\\chi (x_i)=(x_i-\\bar{x}_i)(x_j-\\bar{x}_j)C^{-1}_{ij}\\,,$ where the means $\\bar{x}_i$ and covariances $C_{ij}$ depend on the deterministic parameters.", "This $\\chi ^2$ function can be used to quantify how the experimental point $x_i^{\\rm exp}$ compares to the typical models in the ensemble.", "We use as experimental input the values given in table REF .", "This should be seen as a representative case only, as the precise values depend on the supersymmetric threshold corrections.", "Furthermore, we can ignore the experimental uncertainties which are completely negligible with respect to the width of the theoretical distribution.", "Instead of $\\chi ^2(x_i^{\\rm exp})$ , an equivalent but more meaningful quantity to look at is the associated \"$p$  -value\", the proportion of models that have a larger $\\chi ^2$ than the experimental point, that is, which are less likely.", "In our context, a $p$  -value $\\sim 1$ indicates that the experimental value roughly coincides with the mean of the theoretical distribution, implying that the ensemble typically features a SM-like flavor structure.", "One can then optimize the deterministic parameters in order to yield larger $p$  -values, that is, the SM point belongs to the most likely models of the ensemble (it sits near the mean of the distribution).", "We should make a disclaimer though to avoid misconceptions.", "We are not performing a usual $\\chi ^2$ fit of a model to experimental data.", "Rather, we are optimizing the deterministic parameters such that the theoretical distributions of models have the SM point as a typical outcome.", "To quantify this statements we use $\\chi _{\\rm models}^2(x_{i}^{\\rm exp})$ (and the associated $p$  -values) as a measure.", "We will take a look at the $SU(5)$ and $SO(10)$ cases separately." ], [ "$SU(5)$", "We will consider two benchmark values, $\\tan \\beta =40$ and $\\tan \\beta =10$ .", "For each pair of $N,N^{\\prime }$ , we can then optimize the continuous parameters $a$ , $b$ , $a^{\\prime }$ and $b^{\\prime }$ .", "However, roughly speaking, the optimal values of $a,b$ ($a^{\\prime },b^{\\prime }$ ) only depend on $N$ ($N^{\\prime }$ ) and not $N^{\\prime }$ ($N$ ).", "Therefore, we can group the parameters as shown in table REF and REF .", "A more refined tuning of the continuous parameters can lead to slightly smaller $\\chi ^2$ for some values of $(N,N^{\\prime })$ , but we don't believe that this adds anything to the general conclusions.", "We also give in figure REF the distributions for the case $N=1$ , $N^{\\prime }=5$ .", "Several features are worth pointing out.", "The $p$  -values in tables REF and REF can be quite close to one, especially in the $\\tan \\beta =40$ case, indicating that our mechanism results in very SM-like masses and mixings.", "As evident from Fig.", "REF , only weak correlations between the lepton and down sectors (for instance between mu and strange masses) persist, due to the GUT breaking effects built into the clockwork Lagrangian.", "One easily realizes large differences such as $m_\\mu /m_s\\sim 5$ .", "The breaking of the degeneracy between $\\mathcal {Y}_d$ and $\\mathcal {Y}_e$ comes mostly from the $\\bf \\bar{5}$ and not from the $\\bf 10$ sector (the fit prefers $a<b$ and $a^{\\prime }>b^{\\prime }$ ).", "The reason is that the hierarchies must mainly come from the $\\bf 10$ sector, as explained at the end of section REF , but the hypercharges predict larger hierarchies in $\\mathcal {E}_{E^c}$ than in $\\mathcal {E}_Q$ , which goes in the wrong direction.", "Therefore, the GUT breaking terms proportional to $b^{\\prime }$ cannot be too large, and $N\\ne 0$ (even if not strictly needed to generate the hierarchies) is crucial to get $\\mathcal {Y}_d\\ne \\mathcal {Y}_e^T$ .", "For $N=0$ , we could find no choices of parameters with $\\chi ^2<40$ ($p>10^{-4}$ ).", "At larger $N$ ($N^{\\prime }$ ) a too strong inter-generational hierarchy can be mitigated by larger values of $a,b$ ($a^{\\prime },b^{\\prime })$ , see eq.", "(REF ).", "On the other hand for $N^{\\prime }<2$ , the inter-generational hierarchies are too small, and this cannot be compensated for by going to very small $a^{\\prime },b^{\\prime }$ , which in this regime have a universal effect on all three generations (see eq.", "(REF ) and the discussion there).", "There exist some deviations from the Gaussian approximation for the leptonic observables $\\sin ^2\\theta _{12}$ and $\\sin ^2\\theta _{23}$ , due to the fact that their distributions peak near the upper limit $\\sin ^2\\theta _{ij}= 1$ .", "However, the true probability density at the experimental point is larger than the the one given by the Gaussian approximation, hence our estimate for the global $\\chi ^2$ is conservative.", "Table: Simulation for the SU(5)SU(5) model for the scenario with tanβ=40\\tan \\beta =40.", "We display the χ 2 \\chi ^2 of the physical couplings, corresponding to 15 degrees of freedom.", "The parenthesis give the pp -value, the proportion of models with lower probability density (larger χ 2 \\chi ^2) than the SM.Table: Simulation for the SU(5)SU(5) model for the scenario with tanβ=10\\tan \\beta =10.", "We display the χ 2 \\chi ^2 of the physical couplings, corresponding to 15 degrees of freedom.", "The parenthesis give the pp -value, the proportion of models with lower probability density (larger χ 2 \\chi ^2) than the SM.Figure: The distribution of masses and mixings (parameters x i x_i) in the SU(5)SU(5) model, for the case N=1N=1, N ' =5N^{\\prime }=5 and tanβ=40\\tan \\beta =40.", "In the lower triangle, we display scatter plots of the two-dimensional marginal distributions, with the solid (dashed) contour representing one (two) sigma, and the red dots the experimental value.", "In the diagonal, we show the one dimensional marginal distributions over a 3 sigma range with the red lines indicating the experimental value.", "The numbers on the bottom of each entry are the mean and standard deviation of the theoretical distributions.", "In the upper triangle, we show the correlation coefficients.Before turning to the $SO(10)$ case, let us comment on the remaining observables, namely the mass-squared differences of the neutrinos and the CP phases in the CKM and PMNS matrices.", "Inverted neutrino mass ordering would imply almost complete degeneracy between the heavier two neutrinos, which is difficult to realize in our models.", "Hence normal mass ordering is strongly preferred.", "In this case, neutrino masses can always be fit rather well.", "We have checked that enlarging the set of observables by $\\Delta m_{21}^2$ and $\\Delta m_{32}^2$ adds very little to the global $\\chi ^2$ (after adjusting the see-saw scale $m_R$ ), approximately $\\sim 0.25$ for $N=1$ , and $\\sim 0.5$ for $N=3$ .", "However, since there are now two more degrees of freedom, the $p$  -values are actually slightly higher than those reported in tables REF and REF .", "As for the phases $\\delta _{\\rm CKM}$ and $\\delta _{\\rm PMNS}$ , their distributions are pretty much flat over the entire range $[0,2\\pi )$ .", "Since the Gaussian approximation certainly fails for these variables, it is not very meaningful to include them in the analysis above.", "On the other hand the flatness of the distribution tells us that the experimental values are neither preferred nor disfavored in our class of models." ], [ "$SO(10)$", "The $SO(10)$ model depends on $\\tan \\alpha $ , defined in eq.", "(REF ), which parametrizes the relative direction of $SO(10)$ breaking by the $\\bf 45$ representation.", "In order to assess the favoured values of $\\tan \\alpha $ , let us define the quantity $b_{X}\\equiv b\\left|Y\\sin \\alpha + (B-L)\\,\\cos \\alpha \\right|\\,,$ where $X=Q,L,U^c,D^c,E^c$ .", "The quantity $a+b_X$ controls the average localization of the zero modes of the field $X$ : the larger $a+b_X$ , the more it is localized towards site zero, while smaller values repel the zero modes from site zero and reduce the couplings to the Higgs.", "Ideally, we would like $b_L$ to be large, as it reduces the hierarchy in the PMNS angles.", "One has that $b_L$ is the largest amongst the $b_X$ in the regime $-0.8< \\tan \\alpha < 0$ .", "At the same time we need to avoid to be too close to the end points of this interval, as they correspond to points where some of the Yukawa couplings matrices exactly unify, see table REF .", "A value that works well in this regime is $\\tan \\alpha \\sim -0.6$ , which we will use as our main benchmark.At this point, one has $b_X=b\\times \\lbrace 0.6,\\ 0.46,\\ 0.34,\\ 0.2,\\ 0.05\\rbrace $ for $X=L,D^c,E^c,Q,U^c$ respectively.", "Since the $a$ and $b$ parameters affect all fields simultaneously, it is no longer possible (contrary to the $SU(5)$ case) to suppress the down type and charged lepton masses by reducing $a,b$ without for instance also reducing the top mass.", "We must instead resort to large values of $\\tan \\beta $ .", "We find that a reasonable value is $\\tan \\beta \\sim 50$ .", "Table: Simulation for the SO(10)SO(10) model, with tanα=-0.6\\tan \\alpha =-0.6 and tanβ=50\\tan \\beta =50.", "The χ 2 \\chi ^2 corresponds to 15 degrees of freedom.", "The row marked pp -value gives the proportion of models with lower probability density (larger χ 2 \\chi ^2) than the standard model.We show in table REF the results of optimizing the remaining parameters $a$ and $b$ , and in figure REF we show the distributions in a representative case.", "We can make the following observations.", "While certainly less impressive than the $SU(5)$ model, the $p$  -values are nevertheless surprisingly good, considering the necessity of rather large $SO(10)$ breaking effects.", "For $N\\ge 5$ , the experimental point $x_i^{\\rm \\exp }$ is less than one sigma away from the mean of the distribution.", "As in the case of $SU(5)$ , at large values of $N$ , potentially too large inter-generational hierarchies are partially erased by increasing $a$ or $b$ , which explains why the $\\chi ^2$ values do not increase again at larger $N$ .", "However, they also do not improve any further for $N>7$ .", "Neutrinos with normal mass ordering can again be fit easily in our model.", "The typical increase in $\\chi ^2$ is of the order of $\\sim 0.3$ , which for two additional degrees of freedom slightly increases the $p$  -values.", "Figure: The distribution of masses and mixings in the SO(10)SO(10) model (parameters x i x_i), for the case N=6N=6.", "In the lower triangle, we display scatter plots of the two-dimensional marginal distributions, with the solid (dashed) contour representing one (two) sigma, and the red dots the experimental value.", "In the diagonal, we show the one dimensional marginal distributions over a 3 sigma range with the red lines indicating the experimental value.", "The numbers on the bottom of each entry are the mean and standard deviation.", "In the upper triangle, we show the correlation coefficients." ], [ "Phenomenology", "There are two principle impacts on low energy observables that could be used to constrain this class of models.", "Firstly, as any supersymmetric GUT model, exchange of the triplet Higgs can mediate proton decay via dimension-five operators.", "In minimal $SU(5)$ GUTs, precision gauge coupling unification requires a triplet Higgs mass that allows for proton decay at a rate incompatible with data [35].", "In our model, there are additional vectorlike matter fields with associated threshold corrections.", "Notice that only mass ratios enter in the generation of the Yukawa couplings, and the overall mass scale of these new particles is a free parameter.", "For each model in our scan, one can in principle adjust this mass scale and the triplet mass to achieve precise gauge coupling unification, and calculate the associated proton lifetime.", "Such an analysis is however not independent of the solution to the doublet triplet splitting problem, and the two issues should be dealt with together.", "For instance, the missing partner mechanism [36], [37], [38] typically require extended Higgs sectors.", "Even though we have presented our mechanism for a minimal Higgs sector, we expect it to work similarly well for extended Higgs sectors, as long as we can write some Yukawa interactions of the CW matter fields with the GUT-breaking Higgs fields.", "On the other hand it would also be interesting to try and take advantage of the CW mechanism itself to split the doublet and triplet masses.", "A fully realistic model in this regard, including doublet-triplet splitting and an analysis of proton decay, is left to future work.", "Secondly, we should comment about low energy flavor violating signatures of these models.", "The wave function renormalization factors, see e.g. eq.", "(REF ), will also strongly reduce flavor violation in the soft masses [39], [40].", "The effect is virtually identical to strongly coupled [27] or extra-dimensional [41], [42] models of supersymmetric flavor, though quite different from models with horizontal symmetries, which are more constrained than the former [40].", "One of the most constraining observables is the decay of $\\mu \\rightarrow e\\gamma $ .", "If the charged lepton hierarchy is mostly coming from the $\\mathcal {E}_e$ (as in our $SU(5)$ scenario), one has [40] $\\left( \\frac{A_0}{100\\ \\rm GeV}\\right)\\left(\\frac{400\\ \\rm GeV}{\\tilde{m}_\\ell }\\right)^4<0.4\\,,$ where $A_0$ is the trilinear soft term and $\\tilde{m}_\\ell $ the slepton mass (similar bounds have been derived in ref. [27]).", "When $\\mathcal {E}_\\ell $ is somewhat hierarchical (as in our $SO(10)$ model), the bounds become weaker.", "In the quark sector, the strongest bounds come from the neutron EDM which require squark masses $\\tilde{m}_q\\gtrsim 1$ TeV [40].", "Since the sfermion masses are dominated by the gaugino loops, squark and slepton masses are related as $\\tilde{m}_q\\sim 5 \\tilde{m}_\\ell $ , and the leptonic bounds are more constraining." ], [ "Conclusions", "We have presented a renormalizable GUT model of flavor which accounts very well for the observed hierarchies of masses and mixings in the charged fermion sector.", "The model features one and two spontaneously broken $U(1)$ symmetries in the case of $SU(5)$ and $SO(10)$ respectively.", "Contrary to Froggatt-Nielsen type models, the MSSM chiral matter fields are uncharged under this symmetry.", "We have taken advantage of the GUT breaking terms present in the most general renormalizable CW Lagrangian in order to lift the degeneracy of the down and lepton Yukawa couplings.", "Inter-generational hierarchies result spontaneously from products of $\\mathcal {O}(1)$ matrices, while inter-species hierarchies can either arise from a CW-like suppression or from large $\\tan \\beta $ .", "In $SU(5)$ , for a GUT breaking scale slightly smaller than the $U(1)$ breaking scales we obtain distributions of models that feature the SM point amongst the $\\sim 5\\%$ most likely models, i.e., very close to the mean value of the distribution.", "This requires roughly about one vectorlike copy of the $\\bf \\bar{5}$ matter fields, and $\\gtrsim 5$ copies of the $\\bf 10$ .", "For the best $SO(10)$ case, the SM fits slightly worse in the distributions, belonging only to about the $60\\%$ most likely models, approximately one sigma away from the mean of the distribution.", "A good fit requires about $N\\gtrsim 5$ vectorlike copies of the entire MSSM matter content.", "The fact that the probability density is unsuppressed at the SM point can also be interpreted in terms of fine-tuning.", "Accidental cancellations occur only very rarely at random.", "We could exclude such points from our distributions, but this would at most affect the far tails of the distributions, implying that there are many non-fine tuned models which reproduce the experimental data of table REF precisely." ] ]

[ [ "New insight into the origin of the GeV flare in the binary system PSR\n B1259-63 from the 2017 periastron passage" ], [ "Abstract PSR B1259-63 is a gamma-ray binary system hosting a radio pulsar orbiting around a O9.5Ve star, LS 2883, with a period of ~3.4 years.", "The interaction of the pulsar wind with the LS 2883 outflow leads to unpulsed broad band emission in the radio, X-rays, GeV and TeV domains.", "While the radio, X-ray and TeV light curves show rather similar behaviour, the GeV light curve appears very different with a huge outburst about a month after a periastron.", "The energy release during this outburst seems to significantly exceed the spin down luminosity of the pulsar and the GeV light curve and energy release varies from one orbit to the next.", "In this paper we present for the first time the results of optical observations of the system in 2017, and also reanalyze the available X-ray and GeV data.", "We present a new model in which the GeV data are explained as a combination of the bremsstrahlung and inverse Compton emission from the unshocked and weakly shocked electrons of the pulsar wind.", "The X-ray and TeV emission is produced by synchrotron and inverse Compton emission of energetic electrons accelerated on a strong shock arising due to stellar/pulsar winds collision.", "The brightness of the GeV flare is explained in our model as a beaming effect of the energy released in a cone oriented, during the time of flare, in the direction of the observer." ], [ "Introduction", "Gamma-ray binary systems are composed of a compact object, either a black hole or a neutron star, orbiting a massive O or B type star.", "They are distinguished from X-ray binaries of a similar nature by non-thermal emission that peaks at energies $\\gtrsim 1$  MeV [18].", "Of all $\\gamma $ -ray binaries the only systems where the nature of the compact object is known are PSR B1259-63 and PSR J2032+4127, both of which are radio pulsars.", "In PSR B1259-63 the pulsar has a spin period of 47.76 ms and is orbiting a O9.5Ve star (LS 2883) with a period of $\\sim 1236.7$  days in a highly eccentric orbit ($e\\sim 0.87$ ) [26], [34], [38].", "Based on the parallax data in the Gaia DR2 Archive [21] the distance to the system is $2.39 \\pm 0.19$  kpc, which is consistent with the value of $2.6^{+0.4}_{-0.3}$ kpc is reported by [33].", "The optical spectrum of the companion shows evidence of an equatorial disc, which is thought to be inclined with respect to the orbital plane by $\\sim 10-40^\\circ $ [31], which causes the pulsar to cross the disc twice during the periastron passage.", "The interaction of the pulsar wind with the companion's outflow leads to the generation of the unpulsed non-thermal emission in radio, X-ray, GeV and TeV energies.", "X-ray emission is observed throughout the orbit but the unpulsed radio, GeV and TeV radiation occurs only within a few months before and after the periastron [27], [16], [25], [23].", "The unpulsed radio and X-ray emission exhibits a similar two peak light curve with the peaks occurring during the time when the pulsar crosses the disc of the companion.", "Current H.E.S.S.", "observations indicate that TeV emission can have a similar behaviour [23], but more sensitive observations are needed to confirm this.", "Hopefully CTA will address this issue in the very near future [15].", "The GeV emission, however, shows a very different behaviour and is characterised by a strong flare, which started about $\\sim 30$  days after the 2010 and 2014 periastron passages [1], [10] and has no obvious flaring counterparts at other wavelength.", "The only visible effect coinciding in time with a GeV flare is a rapid decrease of the H $\\alpha $ equivalent width [16], usually interpreted as a measure of the companion's star disc.", "The destruction of the disc is also evident in Chandra observations of the source far away from the periastron [35], [37], [36], [24].", "These data demonstrates the presence of X-ray emitting clumps moving away from the binary with speeds of about 0.1 of the speed of light.", "The clumps are being ejected at least once per binary period, 3.4 years, presumably around binary periastra and are probably associated with the destruction of the companion's disc.", "The most recent periastron of PSR B1259-63 (September 22, 2017; $t_p = \\mbox{ MJD~} 58018.1$ ) presents another opportunity to examine the nature and mechanics of the GeV flare using the available broadband observations.", "The GeV behaviour of PSR B1259-63 turned out to be very different from previous periastra.", "The flare only started 40 days after the periastron and reveals variability on hour timescales [25], [39], [11].", "In this paper we present for the first time the results of the optical observations of the system in 2017, discuss the available multiwavelength data (optical, X-ray and GeV) and propose a new model to explain the origin of the GeV flare and how its observed luminosity apparently exceeds the spin-down luminosity.", "In section  of this paper we describe the details of the data analysis, in section  we present our model, and give our conclusions in section .", "Optical observations during the 2017 periastron passage were limited because of the position of the source relative to the Sun, and PSR B1259-63/LS 2883 was only visible for a short period just after sunset, allowing limited observations before periastron.", "The system was observed with the SAAO 1.9-m telescope between 2017 August 24 and 2017 September 04 ($\\tau \\sim -28$  d to $\\tau \\sim -17$  d; where $\\tau $ is the time from periastron) using the SpUpNIC grating spectrograph [17].", "The spectroscopic observations with SpUpNIC were performed using a 1200 lines mm$^{-1}$ grating, with a spectral resolution of $\\sim 1$  Å, covering a wavelength range of $\\sim 6150 - 7150$  Å.", "Each night, multiple exposures of the target were taken with a typical exposure time of 60 seconds, while the source was high enough to be observed.", "Arc observations of a CuNe arc lamp were taken before and after every science exposure.", "Dome flats where taken each day using the same configuration and a spectroscopic standard (CD-32 9927) was observed at the beginning of each night.", "The data reduction and wavelength calibration was perform following the standard iraf/noao procedures.", "Each night's observations were combined and normalized.", "As previously reported, the spectrum shows a strong H $\\alpha $ emission line that remains single peaked through all observations.", "The double peaked He i ($\\lambda $ 6678) line is also present in the observations and the variation of the ratio of the peaks of the violet to red (V/R) components of the line was measured.", "Further photometric observations in the H $\\alpha $ filter were undertaken using the Watcher Robotic Telescope [19] from 2017 September 04 to 2017 September 21 ($\\tau \\sim -17$   to $\\sim -0.4$  d).", "Multiple 30 second exposure in the H $\\alpha $ filter were taken per night.", "All images on the same night were combined to increase the signal to noise and differential photometry, using stars on the same field of view, was performed.", "The results are shown in Fig.", "REF (and also bottom panel of Fig.", "REF ) and compared to the observations around the 2014 periastron passage [13], [42].", "The top panel shows that the H $\\alpha $ equivalent width follows the same trend as the previous periastron passage, with the line strength increasing towards the point of the first disc crossing.", "While the spectroscopic observations could not be taken beyond this, the photometric H $\\alpha $ observations (middle panel, shown in arbitrary flux units) show that the line continues to follow the same trend; the line strength decreases after the first disc crossing, then continues to grow towards periastron.", "This confirms the strong interaction between the pulsar and the circumstellar disc around the periastron.", "This is also shown by the V/R variation (bottom panel) which follows a similar trend, though at a lower scale, with the V component increasing towards the first disc crossing.", "The optical spectroscopic results are given in Table.", "REF .", "Figure: Optical observations before the 2017 periastron passage.", "Top: H α\\alpha equivalent width.", "Middle: H α\\alpha photometry (in arbitrary units) Bottom: V/R line ratio from the He I profile.", "The results are compared to the 2014 periastron passage.Table: The H α\\alpha equivalent width and V/R ratio of the double peaked He i line measured around the first disc crossing phase before the 2017 periastron passage." ], [ "X-ray data", "A full overview of the X-ray flux and spectral slope for the observations around 2004, 2007, 2010, 2014 and 2017 years are presented in the panels (b) and (c) of Fig.", "REF .", "Historical data in this figure are taken from [16]" ], [ "The 2017 periastron passage of PSR B1259-63 was closely monitored by the Swift satellite [22].", "We have analysed all available data taken from March, 26th, 2017 to January, 6th, 2018.", "The data were reprocessed and analysed as suggested by the Swift/XRT teamSee e.g.", "the Swift/XRT User's Guide with the xrtpipeline v.0.13.5 and heasoft v.6.27 software package.", "The spectral analysis of Swift/XRT spectra was performed with XSPEC v.12.11.0.", "The spectrum was extracted from a circle of radius $36^{\\prime \\prime }$ around the position of PSR B1259-63 and the background estimated from a co-centred annulus with inner/outer radii of $60^{\\prime \\prime }/300^{\\prime \\prime }$ .", "During the spectral analysis we noticed that the quality of the Swift/XRT data does not allow the hydrogen column density to be firmly determined in each individual observation and thus we chose to fix it to the mean value.", "To do this we have fitted all the data with an absorbed power law model with a common value of column density in all the observations.", "The resulting value of $N_H=0.55\\times 10^{22}$ cm$^{-2}$ is in good agreement with previous observations [16].", "These data were also presented in the paper of [39], but in that work the value of the column density was fixed in a rather model-dependent way." ], [ "We accompanied our analysis with the analysis of historic, publicly available Chandra data taken during the 2014 and 2017 periastron passages (February to June 2014, ObsIds: 16563, 16583, 16624, 16625 and July 2017, ObsIds: 19281,20116).", "We analyzed these data using the most recent CIAO v.4.12 software and CALDB 4.9.0.", "The data were reprocessed with the chandra_repro utility.", "The source and background spectra, with corresponding RMFs and ARFs, were extracted with the specextract tool.", "We note that two Chandra observations of PSR B1259-63 in June 2014 (ObsIds: 16624, 16625) were performed in asic-cc (continuous clocking) mode, in which only 1-dimensional spatial information is available.", "In this case, to extract the source and background spectra, we used box-shaped regionsSee caveats of asic-cc mode data analysis.", "centred on PSR B1259-63 and on a nearby source-free region, respectively.", "For the rest of observations we utilized standard circular regions.", "Figure: Evolution of multiwavelength PSR B1259-63 spectral characteristics over the different periastron passages.", "Panel a: Fermi/LAT flux measurements in the E >> 100 MeV energy range with a weekly bin size.", "Flux is given in 10 -6 ^{-6} cm -2 ^{-2} s -1 ^{-1}.", "Panel b: 1-10 keV X-ray flux in units of 10 -11 ^{-11} erg cm -2 ^{-2} s -1 ^{-1}.", "Panel c: X-ray slope.", "Panel d: H α\\alpha equivalent width." ], [ "The analysis of Fermi/LAT data was performed using Fermitools version 1.2.23 (released 11th February 2020).", "For the analysis of the 2017 periastron passage and the combined periastron data the analysis was carried out using the latest Pass 8 reprocessed data (P8R3) from the SOURCE event class.", "All gamma-ray photons used for this analysis were within the energy range 0.1 – 100 GeV and within a circular region of $15^\\circ $ around the ROI centred on PSR B1259-63.", "The selected maximum zenith angle was $90^\\circ $ .", "The spatial-spectral model built in order to perform the likelihood analysis included the Galactic and isotropic diffuse emission components and the known gamma-ray sources within $20^\\circ $ of the ROI centre from the 4FGL catalogue [2].", "A likelihood analysis was applied to each observation data set twice to achieve the best model for that time span and energy range.", "The first likelihood run frees the normalization of every source within $15^\\circ $ of the ROI centre and the index of PSR B1259-63.", "The second run fixes the normalization of all sources outside $5^\\circ $ of PSR B1259-63 to the value calculated from the first likelihood fit.", "The output model of this second likelihood analysis was used for the lightcurve and spectrum generation.", "The gamma-ray flux, light-curves and spectral results for PSR B1259-63 presented here were calculated using a binned likelihood fit using the BinnedAnalysis module from the FermiTools Python packages.", "PSR B1259-63 was modelled using a single powerlaw with the normalization left free and the index was fixed to the value from the preliminary analysis.", "When generating the light curves and spectra, any free sources (except PSR B1259-63) with a TS $<$ 1 was removed from the fitted model.", "If any bin had a poor detection of PSR B1259-63 (TS $<$ 1 or Flux $<$ Flux Error) the calculated flux was replaced with a 95% confidence upper limit on the photon flux above 100 MeV using the IntegralUpperLimits functions from gtlike.", "The spectral model output of the poor detection fit was used to calculate the upper limit values.", "Data covering a time frame from 50 days before the periastron to 100 days after periastron were used to produce lightcurves of the 2017 periastron passage.", "The results of the lightcurve analysis were used to define the time periods for further spectral analysis.", "As seen in the top panel of Fig.", "REF , the weekly binned GeV light curves of the 2010, 2014 and 2017 periastra show differences in the shape of the post-periastron flaring period.", "The 2017 GeV light curve, with daily binning, is shown in Fig.", "REF .", "This light curve demonstrates strong day to day flux variability.", "The different colour highlights on this figure are used to show the data sets used for spectral analysis (see also Table REF ).", "The results of this light curve analysis are in line with previous analyses of the 2017 periastron passage by [39] and [25] where discrepancies are likely caused by the use of different catalogues and updated software.", "For the spectral analysis we split the available data into several time periods (see Table REF ).", "The period before the GeV flare (pre-flare), was divided into two: from 20 days before until periastron, and from periastron until 20 days after (prfl1 and prfl2 data sets correspondingly).", "For these periods data from the 2010, 2014 and 2017 periastra were included to improve the statistics.", "For the flare analysis we use only data from the 2017 periastron to examine the spectra of the average flare period and the daily short flares that can be seen in the Fig.", "REF .", "The pre-flare spectra before and after the periastron are compared in Fig.", "REF and, along with the flare spectra, are also shown in Fig.", "REF .", "All time periods defined in Table REF were also analysed in the $0.1 - 2.0$  GeV energy range.", "We used a super exponential cut-off power law (PLSuperExpCutoff) model to match the shape of the GeV peak spectra; see Table REF for the best-fit parameters.", "Spectral fitting was done twice, the first time with all the parameters free, and second time with $\\gamma 2$ fixed to the value of the first fit to better constrain the other model parameters.", "Figure: Daily-binned light curve of the 2017 periastron passage.", "The highlights indicate the time periods that were used for spectral analysis and modelling; the details are given in Table .", "The different periods are shown as: prfl1 = Red, prfl2 = Magenta, avfl = Yellow, pkfl = Green.Table: Details of the time periods used for spectral analysis during the 2017 periastron.", "t p =MJD58018.1t_p=\\mbox{MJD } 58018.1 corresponds to the time of 2017 periastron passage.Table: Spectral model parameters from the binned likelihood analysis of different phases of the 2017 periastron from 0.1 - 2.0 GeV.", "The model used is the PLSuperExpCutoff where dN/dE=N 0 (E/E c ) γ1 exp(-(E/E 0 ) γ2 )dN/dE = N_0 (E/E_c)^{\\gamma 1} exp(-(E/E_{0})^{\\gamma 2}).The spectral shape of the GeV emission turned out to be very different before and after the periastron, see Figure REF .", "Blue and red points correspond to the spectrum of the source averaged over 20 days before and after the periastron correspondingly.", "The spectrum before periastron (prfl1) is well described with a simple power law.", "The GeV emission after the periastron (prfl2) has a higher flux in the 0.1 – 2 GeV energy range and is characterised by a strong cut-off, see Table REF .", "This noticeable difference in spectral characteristics between different time periods of the preflare implies that around periastron there is a change of the spectrum of electron population responsible for the GeV production.", "While there is a notable distinction in the characteristics between the time periods of the preflare, such a difference was not observed in the analysis of the flare periods.", "The spectra of both the average flare period (avfl) and the peaks of the flare (pkfl) are quite similar (see Figure REF and Table REF ), and differ mainly by the normalisation.", "The energy release in PSR B1259-63 is known to be extremely efficient, for example around periastron more than 10% of the spin-down luminosity is released in the $1 - 10$  keV energy band alone [14].", "The energy release during the GeV flare is even larger, reaching values higher than 50% of the spin-down luminosity in 2010 and 2014 years, without considering possible beaming effects [10].", "The GeV flare in 2017 differs a lot from the previous ones, as it starts about 10 days later and consists of a number of short individual flares, more intensive than previously observed [39], [25].", "The brightness of these flares allows their structure to be reconstructed on timescales shorter than 1 day, and even to find 15 minute long sub-flares.", "The energy release during these sub-flares reached a value exceeding the spin-down luminosity by a factor of 30, with no clear counterparts at other wavelength [25].", "This makes it clear that the GeV flare is produced as a separate and highly anisotropic component.", "In what follows we propose a new model, which suggests the presence of two populations of relativistic electrons: (i): electrons of the unshocked and weakly shocked pulsar wind and (ii): strongly shocked electrons.", "The spectrum of unshocked electrons was selected to be a power law with the slope $-2$ in energy range $E_e=0.6-1$  GeV.", "A small fraction of electrons are additionally accelerated at the strong shock near the apex to $E_e\\sim 500$  TeV energies with the similar slope $\\Gamma _e=-2$ on a characteristic timescale (see Fig.", "REF ) $t_{\\rm acc} \\approx 0.1\\left(E_e/1\\,\\mbox{TeV}\\right)\\eta (B_0/1\\,\\mbox{G})^{-1}\\quad \\mbox{s}$ where $B_0$ is a magnetic field in the region and $\\eta \\ge 1$ is the acceleration efficiency [28].", "The rest of the electrons flying into the shock direction will be reverted to flow along the shock at the surface of stellar-pulsar wind interaction cone far from the apex, and could be additionally mildly accelerated on a weak shock, see Fig.", "REF for a sketch of the model.", "This leads to a power law tail in the spectrum of diverted electrons with a slope $\\sim -3$ which continues above 1 GeV to at least $E_e\\sim 5$  GeV.", "This slope is characteristic of particles acceleration on weak shocks [7], [8].", "Hereafter we will refer to these diverted electrons as weakly shocked electrons.", "The spectra of both populations will be additionally modified by radiative (IC, synchrotron or bremsstrahlung) and non-radiative (adiabatic or escape) losses operating in the system.", "In our calculations the resulting electron spectrum was determined by numerically calculating the radiative losses of a continuously injected spectrum of electrons, until a steady solution has been obtained.", "The time that electrons spend in the emitting region, $R/c$ , is about a few thousand seconds, and is long enough to substantially modify the injected spectrum due to synchrotron losses (in our calculations we took t$_{esc}$ =4000 s; see Figure REF and Table REF ).", "We would like to note that the losses substantially modify the injected spectrum and thus properly accounting for such losses is important for modelling the PSR B1259-63 system.", "Additionally one has to account for the radiation efficiency of the considered mechanisms.", "This is rather low for the majority of the considered processes and subsequently an increase of the total energy of the pulsar wind is required (in comparison to 100% efficiency case).", "Figure: GeV emission of PSR B1259-63 during the periods twenty days before (blue points, prfl1) and after (red points, prfl2) the periastron.", "Blue and red curves show best fit models in 100 MeV – 2 GeV energy range, see Table .", "Shaded regions show 1 σ\\sigma confidence range for fitted models.According to our model, similar to previous works [13], [16], [12], the X-ray and TeV components are explained as synchrotron and IC emission of the strongly shocked electrons.", "The GeV component in our model is a separate component due to the combination of the IC and bremsstrahlung emission of the less energetic, unshocked electrons interacting with the clumps of the matter from the Be-star wind/disk which penetrated through the shock.", "The modelled SEDs along with the observed keV-TeV data are shown in Figure REF .", "In this figure the synchrotron and IC emission of the strongly shocked electrons are shown with solid and dashed magenta curves correspondingly.", "The contributions of IC and bremsstrahlung emission of the unshocked and weakly shocked electrons are indicated with dashed and dashed-dot curves correspondingly, where the colours refer to the prfl1 (blue) and prfl2 (red) periods.", "The TeV points shown are taken from [3], and the coloured region at TeV energies represent the range of multi-years H.E.S.S.", "measurements reported in Fig.", "2 of [23].", "The shaded region at X-ray energies represents the range of fluxes observed by Swift in 2017 before (left panel), and after (right panel) the GeV flare.", "The synchrotron, bremsstrahlung and IC emission was calculated with the naima v.0.8.3 package [44], which uses the approximations for the IC, synchrotron and bremsstrahlung emission from  [4], [5], [30], [6].", "Table: Details of the models.", "D is a distance from the Be star to the emission region.", "Effective luminosity L of the pulsar wind electrons (without considering beaming effects) is measured in units of spin-down luminosity L sd =8.2×10 35 _{sd}=8.2\\times 10^{35} erg/sFigure: A sketch of the geometry of the proposed model (not to the scale) for the period of GeV flare.", "Red sphere presents the Be star with the disk shown with yellow semi-transparent circle.", "Stellar/pulsar winds interaction cone is shown with cyan.", "PSR B1259-63 pulsar orbit is illustrated with black line.", "The unshocked pulsar electrons (magenta) are strongly accelerated in the region close to the cone's tip (blue region; “strongly shocked electrons”) and weakly accelerated at the rest of the cone surface(cyan region; “weakly shocked electrons”).", "The flight directions of these electrons are shown with cyan/magenta arrows.", "The clumps of stellar wind are shown with yellow spheres.", "See text for the detailed model description.Figure: Comparison of the cooling times to acceleration and escape time for various radiation processes in the case of avfl model, see text for details.To explain the observed luminosity one has to consider that for relativistic electrons both IC and bremsstrahlung radiation is strongly peaked in the forward direction and most of the energy radiated as a photon moving in the same direction as the initial electron.", "To explain the observed excess of the energy released during the short flares an initially isotropic pulsar wind should be reversed after the shock and confined within a cone, pointing in the direction of the observer, similar to the geometry described in [29].", "The difference in true anomaly between days 40 and 80 after the periastron (period including all short flares) is about 20$^\\circ $ .", "This angle is comfortably smaller than the apex angle of the shock $4\\pi /30$ needed to explain the observed luminosity during the short flares in the case of 100% efficiency.", "The cone of such a size is a result of the interaction of the isotropic pulsar wind with the isotropic Be star wind if the winds ram pressure ratio is $\\frac{L_{sd}}{\\dot{M}V_wc}=0.05$ [29].", "In Figure REF we show Fermi/LAT spectra averaged over the total flare (green points), and over the peak of the flares only (red points).", "These spectra can be explained as a combination of IC and bremstrahlung emission on the clumps with densities of about $2\\times 10^{10}$ cm$^{-3}$ and less then $1\\times 10^{9}$ cm$^{-3}$ correspondingly, see Table REF .", "Please note that in the Figure REF we show only the dominant component (IC for the average flare and bremssrahlung for the peaks) in order to make the figure more readable.", "Also please note that the split of the IC and bremsstrahlung is very model dependent and requires detailed hydrodynamic simulations beyond the scope of this paper.", "To explain the observed luminosity of the 15 minutes long flare one needs to assume a 1000 second long interaction with a clump of material with a density of $4\\times 10^{11}$ cm$^{-3}$ .", "The required densities of the clumps are higher than the average density of an undisturbed disc.", "At the same time the required averaged density both around the periastron and during the period of GeV flare is consistent with the unperturbed, smoothly decreasing disc density model in [41], which gives a density at the base of disc of $n_e \\approx 6\\times 10^{13}$  cm$^{-3}$ .", "At the binary separation distance at 40 days from periastron the disc density will have decreased to $\\sim 10^8$  cm$^{-3}$ (within the disc).", "This, combined with the observed H $\\alpha $ variation (Fig.", "REF ), clearly indicates that the disc must be strongly clumped and disrupted near periastron.", "The difference in the 2017 Fermi/LAT light curve (rapid flares) from the 2010 and 2014 periastra, also suggests a more complicated disc behaviour, that was unfortunately not observable during the 2017 periastron [20].", "Figure: Left: Broad band spectrum emission of PSR B1259-63 during the 20 days before (blue points) and after (red points) the periastron period.", "Green X-ray points are NuSTAR observations of the 2014 periastron from Right: Broad band spectrum emission of PSR B1259-63 during the GeV flare.", "Blue points represent the flare averaged over the whole duration, and red points correspond to the sum of peak periods.The TeV points shown are taken from ,and the shaded regions at TeV energies represent the range of multi-years H.E.S.S.", "measurements reported in Fig.", "2 of .The shaded regions at X-ray energies represent the range of fluxes observed by SWIFT in 2017 before (left panel), and after (right panel) the GeV flare.", "In both panels dashed lines show an IC component, solid magenta line corresponds to synchrotron emission of strongly shocked electrons, dash-dotted line shows the bremsstrahlung component and black solid line corresponds to overall model emission.Recently a very hard TeV spectrum with the slope reaching values of $\\Gamma \\sim 2.5$ at certain orbital phases was reported in [23], see shaded region in Fig.", "REF .", "Assuming that this emission is produced by IC in a strong Klein-Nishina regime the spectral slope of the corresponding electrons can be estimated to be $\\Gamma _e\\sim 2.5$ , similar to the value used in the modelling presented here.", "In the case that the electrons propagate through regions with a non-zero magnetic field their spectrum, at TeV energies, undergoes severe cooling due to synchrotron losses , see e.g.", "Fig.", "REF .", "This leads to the formation of a break in the electron spectrum with a typical softening after the break of $\\Delta \\Gamma _e=1$ .", "Thus, to match the observed HESS spectrum, an initially extremely hard spectrum of electrons with a slope $\\Gamma _{e,0}=1.5$ with a break to $\\Gamma _e=2.5$ at TeV energies has to be considered.", "The initial slope $\\Gamma _{e,0}=1.5$ corresponds in the proposed model to the spectrum of the shocked electrons.", "This slope is substantially different from a “standard” $\\Gamma _e=2$ slope of Fermi-mechanism accelerated electrons.", "We would like to note, however, that very hard slopes up to $\\Gamma \\sim 1$ (at least at relatively broad energy range close to the spectral cut-off) were reported for diffusive shock acceleration on a multiple shocks, see e.g.", "[32], [9], [43].", "Such shocks can potentially form in PSR B1259-63 in a pulsar wind/Be star disc interaction region, assuming that the disc hosts multiple clumps.", "GeV emission around the periastron period can be explained as IC emission of the unshocked electrons (see Figure REF ).", "In this Figure we also show NuSTAR data points taken around 2014 periastron [16].", "Within our model softening of the X-ray slope around the periastron can be attributed to additional cooling losses due to the higher value of magnetic field and the increased photon energy density near the periastron.", "To explain higher intensity and much sharper cutoff of the GeV flux after the periastron, one needs to assume that the acceleration becomes more efficient and the slope of the electrons is equal to -2.5 above 1 GeV.", "The parameters of all the models are summarised in Table REF ." ], [ "Conclusions", "In this paper we present optical observations (spectroscopy and H $\\alpha $ photometry) and discuss spectral characteristics of the GeV emission both around the periastron and during the flare.", "We propose a new model to explain the origin of the short bright GeV flares observed during 2017 periastron passage by Fermi.", "We show that: 1) The optical observations show that the behaviour of the disc around the first disc crossing is similar to the previous periastron, clearly indicating the pulsar significantly disrupts the disc.", "2) The observed X-ray and TeV emission both around the periastron and during GeV flare can be explained as a synchrotron and IC emission of the strongly shocked electrons of the pulsar wind.", "3) The GeV component is a combination of the IC emission of unshocked/ weakly shocked electrons and bremsstrahlung emission.", "4) The luminosity of the GeV flares can be understood if it is assumed that the initially isotropic pulsar wind after the shock is reversed and confined within a cone looking, during the flare, in the direction of the observer.", "5) The observed softening of the spectrum close to the periastron corresponds to the shift of the break in the electrons spectrum due to the cooling losses.", "The position of the break is determined by the strength of the magnetic field in the emitting region, which is higher around the periastron.", "We foresee that the break position can be located at higher energies further from the periastron, which can lead to the detectable break in X-ray/TeV spectra.", "A hint of such a break was detected by Suzaku in 2007 [40]." ], [ "Acknowledgements", "Authors thank Prof. F. Aharonian for fruitful discussions.", "This paper uses observations made at the South African Astronomical Observatory (SAAO).", "This paper uses observations obtained at the Boyden Observatory, University of the Free State, South Africa.", "Watcher data made available through support from Science Foundation Ireland grant 07/RFP/PHYF295.", "The authors acknowledge support by the state of Baden-Württemberg through bwHPC.", "This work was supported by DFG through the grant MA 7807/2-1.", "The authors wish to acknowledge the DJEI/DES/SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support.", "We would also like to acknowledge networking support by the COST Actions CA16214 and CA16104.", "DM acknowledges support from the Irish Research Council through grant GOIPG/2014/453.", "Optical data underlying this article were provided by permission of SALT/Watcher collaborations.", "Data will be shared on request to the corresponding author with the permission of SALT/Watcher collaborations.", "Other data used in the article will be shared on reasonable request to the corresponding author." ] ]

[ [ "Homological Percolation: The Formation of Giant k-Cycles" ], [ "Abstract In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation.", "Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components).", "Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit.", "We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation.", "Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing." ], [ "Introduction", "Percolation theory focuses on the formation of large-scale structures, and originally introduced as a model for propagation of liquid in porous media.", "The first percolation model, introduced by Broadbent and Hammersley [11], is known today as the bond-percolation model where bonds (connection between sites) can be either open or closed independently at random.", "Since then, percolation theory has become one of the dominant areas in mathematics and statistical physics, see [14] for a survey on the field.", "In this paper we focus on a continuum percolation model that was introduced first by Gilbert [21] as a model for ad-hoc wireless networks.", "In continuum percolation, geometric objects (grains) are deployed at random in space, and we consider the structure formed by their union (see [29]).", "We introduce a new higher-dimensional generalization of percolation phenomena using the language of algebraic topology.", "In order to do so, we will be considering percolation in a finite (yet large) medium, where structures such as “giant\" connected components, one-arm events, and crossing components, may appear.", "Our main observation is that these formations are mostly topological in nature, i.e.", "they are concerned with qualitative aspects of connectivity rather than quantitative measures of geometry.", "In algebraic topology, connected components are considered “0-dimensional cycles\" (or rather equivalence classes of cycles), forming the first class in a sequence known as the homology groups $\\left\\lbrace \\mathrm {H}_k\\right\\rbrace _{k\\ge 0}$ (see Section REF ).", "For example, elements in $\\mathrm {H}_1$ (1-cycles) can be thought of as loops surrounding holes, elements in $\\mathrm {H}_2$ (2-cycles) can be thought of as surfaces enclosing cavities, and there is a general notion for $k$ -dimensional cycles.", "Our goal is to introduce a notion of “giant $k$ -dimensional cycles,\" and explore the probability of these structures to appear.", "We focus on the continuum percolation model where the grains are balls of a fixed (nonrandom) radius.", "Suppose that we have a homogenous Poisson process $\\mathcal {P}_n$ with rate $n$ generated over a space $\\mathcal {S}$ , and consider $\\mathcal {O}_r$ to be the union of balls of radius $r$ around $\\mathcal {P}_n$ .", "We define as giant $k$ -cycles in $\\mathcal {O}_r$ any $k$ -cycle in $\\mathcal {O}_r$ that is also a $k$ -cycle of $\\mathcal {S}$ , i.e.", "elements in the image of the map $\\mathrm {H}_k(\\mathcal {O}_r)\\rightarrow \\mathrm {H}_k(\\mathcal {S})$ (see Section for formal definitions).", "Note, that taking $\\mathcal {S}$ to be a box (or any compact and convex space) will be pointless, since a box has $\\mathrm {H}_k=0$ (i.e.", "no $k$ -cycles) for all $k>0$ .", "Instead, we focus on the $d$ -dimensional flat torus $\\mathbb {T}^d$ , i.e.", "the unit box $[0,1]^d$ with a periodic boundary.", "In this case, it is known that $\\operatorname{rank}(\\mathrm {H}_k(\\mathbb {T}^d)) = \\binom{d}{k}$ , and therefore we expect to observe giant $k$ -cycles emerging in $\\mathcal {O}_r$ for all $1\\le k\\le d$ .", "Our main result is the following.", "We define $E_k$ to be the event that there exists a giant $k$ -cycle in $\\mathcal {O}_r$ (i.e.", "$\\operatorname{Im}(\\mathrm {H}_k(\\mathcal {O}_r)\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)) \\ne 0$ ), and $A_k$ to be the event that $\\mathcal {O}_r$ contains all giant $k$ -cycles (i.e.", "$\\operatorname{Im}(\\mathrm {H}_k(\\mathcal {O}_r)\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)) = \\mathrm {H}_k(\\mathbb {T}^d)$ ).", "Similarly to the study of random geometric graphs [32], we study the limit when $n\\rightarrow \\infty $ and $r = r(n)$ satisfies $nr^d = \\lambda $ (i.e.", "$r = (\\lambda /n)^{1/d}$ ), known as the thermodynamic limit.", "Our results show that there exist two sequences of threshold values $\\lambda _{0,1}\\le \\cdots \\le \\lambda _{0,d-1}$ , and $\\lambda _{1,1}\\le \\cdots \\le \\lambda _{1,d-1}$ , such that: (a) If $\\lambda < \\lambda _{0,k}$ then $\\mathbb {P}\\left(E_k\\right)\\rightarrow 0$ exponentially fast.", "(b) If $\\lambda > \\lambda _{1,k}$ then $\\mathbb {P}\\left(A_k\\right)\\rightarrow 1$ exponentially fast.", "Clearly, $\\lambda _{0,k}\\le \\lambda _{1,k}$ and we conjecture that these values are equal.", "We will prove equality for $k=1$ , while for $k>1$ this remains an open problem." ], [ "Related work.", "A few higher-dimensional percolation notions have been studied in the past.", "In [4], [23] the model of random plaquettes was studied, as a generalization for bond percolation models.", "The main idea here is to consider $\\mathbb {Z}^d$ , and instead of setting edges to be open/close, we do the same for the $k$ -dimensional cubical faces ($2\\le k \\le d$ ).", "The study in [4] focuses on 2-dimensional plaquettes, and asks whether an arbitrarily large loop of edges in $\\mathbb {Z}^d$ is covered by a 2-dimensional surface of random plaquettes.", "Note, that loops that are not covered by any surface are exactly what we refer to as (nontrivial) 1-cycles.", "The results in [23] consider $(d-1)$ -dimensional plaquettes, and address the formation of unoccupied spheres around the origin, which are related to entanglement.", "These spheres can also be thought of as $(d-1)$ -cycles in homology.", "Using a similar cubical model, [25] studied percolation in a graph generated by neighboring $(d-1)$ -cycles.", "Another model is a generalization of the Erdős-Rényi $G(n,p)$ random graph.", "Instead of a graph, we consider a simplicial complex (an object consisting of vertices, edges, triangles, tetrahedra and higher dimensional simplexes).", "The random $k$ -complex $X_k(n,p)$ is generated by taking $n$ vertices, including all possible simplexes of dimensions $0,\\ldots , k-1$ , and setting the state of the $k$ -simplexes independently as open with probability $p$ , and closed otherwise.", "In the $G(n,p)$ graph a giant component consisting of $\\Theta (n)$ vertices is known to emerge when $p=1/n$ [19].", "The shadow of a graph is the set of all edges not in the graph, whose addition to the graph generates a new cycle.", "When the giant component emerges, we observe that the shadow is giant, i.e.", "contains a fraction of the edges.", "Similarly, in [28] it was shown that when $p=c/n$ for a known $c>0$ , a giant $k$ -shadow emerges in $X_k(n,p)$ where here the shadow refers to all the $k$ -faces not in the complex, whose addition generates a new $k$ -cycle.", "The phenomena above and the formation of giant $k$ -cycles introduced in this paper are not obviously related.", "In particular, they are not directly comparable, as the plaquette model is infinite, the random $k$ -complex has no underlying topology, and the torus we study here is both finite and has nontrivial homology.", "Nontheless, all these results describe different aspects of percolative behavior in higher dimensions.", "It is an interesting question whether any of these notions coincide for any particular model.", "Finally, we note that the topological study of $\\mathcal {O}_r$ here is equivalent (via the Nerve Lemma [10]) to the study of the random Čech complex [27].", "Recall that our results describe the appearance of the giant $k$ -cycles of the torus in the random process $\\mathcal {O}_r$ .", "A different transition related to $\\mathcal {O}_r$ (via the Čech complex), which can be referred to as homological connectivity [6], describes the stage where the $k$ -th homology of $\\mathcal {O}_r$ not only contains all the $k$ -cycles of the torus ($\\operatorname{Im}(\\mathrm {H}_k(\\mathcal {O}_r)\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)) = \\mathrm {H}_k(\\mathbb {T}^d)$ ) but is completely identical (isomorphic) to it (i.e.", "$\\mathrm {H}_k(\\mathcal {O}_r) \\cong \\mathrm {H}_k(\\mathbb {T}^d)$ ).", "For the flat torus, it was shown in [6] that a sharp phase transition for the $k$ -th homological connectivity occurs when $nr^d = \\frac{1}{\\omega _d}(\\log n + (k-1)\\operatorname{\\log \\log }n)$ , where $\\omega _d$ is the volume of a unit ball in $\\mathbb {R}^d$ .", "Note that this regime is much denser than the thermodynamic limit we consider here ($nr^d = \\text{const}$ ).", "For the top-dimensional homology ($\\mathrm {H}_d$ ), the homological percolation and homological connectivity phase transitions are the same, and occur when $nr^d = \\frac{1}{\\omega _d}(\\log n + (d-1)\\operatorname{\\log \\log }n)$ , which is also the coverage threshold.", "Hence, we do not consider the case of $k=d$ in this paper.", "With respect to the thermodynamic limit, we also note that several limit theorems have been proved in the past, for examples – the Betti numbers [38], the Euler characteristic [36], and the topological type distribution [5]." ], [ "Applied topology.", "While the study in this paper is mainly motivated and inspired by percolation theory, and the main goal is to seek higher dimensional analogue to percolation phenomena, we also want to highlight another interesting application of the results.", "The field of applied topology (or topological data analysis) promotes the use of mathematical topology in data and network analysis [12], [18], [20] One of the most powerful tools developed in this field is persistent homology [17], [39].", "Briefly, it is an algebraic tool that can be used to detect $k$ -cycles that appear at different scales in observed data.", "For example, given a point-cloud $\\mathcal {X}$ , we can consider the filtration generated by the union of balls of varying radii $\\left\\lbrace B_r(\\mathcal {X})\\right\\rbrace _{r=0}^\\infty $ .", "As we increase the radius, $k$ -cycles can be formed (born), and later filled in (die).", "The $k$ -th persistent homology, denoted $\\operatorname{PH}_k(\\mathcal {X})$ is the collection of all such $k$ -cycles, where each cycle $\\gamma $ is assigned with an interval $[\\operatorname{birth}(\\gamma ), \\operatorname{death}(\\gamma ))$ representing the range of scales (radii) in which the feature was observed, see Figure REF .", "Figure: Persistent homology for a random point cloud.", "(a) A random sample generated in an annulus whose inner radius is 0.50.5 and outer radius is 1.", "(b) We consider the persistent homology PH 1 \\operatorname{PH}_1 (i.e.", "holes) generated by drawing balls of radius rr around the points, and increasing rr.", "Each bar corresponds to a 1-cycle, and its endpoints are the birth and death times (radii) of that cycle.", "Note that there is a single giant 1-cycle here (representing the hole of the annulus), and its death time is roughly 0.5 (same as the inner radius).One of the key problems in the field is to decide, among all the features in $\\operatorname{PH}_k$ , which ones represent statistically significant phenomena that one should look into (or the “signal\" underlying the data), and which are merely artifacts of our finite sampling or other sources of randomness that should be ignored (“noise\").", "Over the years, several ideas have been proposed (see the survey [37]), but none has grown into a robust statistical framework, so the problem is still very much open.", "The probabilistic analysis of persistent homology is highly challenging due to the potentially global and dependent nature of the algebraic-topological transformation.", "Nevertheless, over the past decade, some significant progress has been achieved [1], [26], [31].", "Considering individual cycles in $\\operatorname{PH}_k$ , the following theoretical result is the only one available to date.", "For each $k$ -cycle $\\gamma \\in \\operatorname{PH}_k$ we can associate a measure of topological persistence by taking $\\pi (\\gamma ) := \\operatorname{death}(\\gamma )/\\operatorname{birth}(\\gamma )$ .", "Suppose that the data $\\mathcal {X}$ are sampled over a space with a trivial homology (e.g.", "a box, ball, etc.).", "In this case, all the cycles in $\\operatorname{PH}_k$ should be considered as noise, since the signal is trivial.", "We define the extremal noise persistence as $\\Pi _k(n) := \\max _{\\gamma \\in \\operatorname{PH}_k(\\mathcal {X}_n)} \\pi (\\gamma )$ .", "The following result was proved in [7].", "Theorem 1.1 ([7]) If $\\mathcal {X}= \\mathcal {X}_n$ is a homogeneous Poisson process with rate $n$ , then there exist $A,B > 0$ , such that with high probability we have $A\\left(\\frac{\\log n}{\\operatorname{\\log \\log }n}\\right)^{1/k} \\le \\Pi _k(n) \\le B\\left(\\frac{\\log n}{\\operatorname{\\log \\log }n}\\right)^{1/k}.$ In other words, this result provides us with the asymptotic rate of the most persistent noisy cycle.", "While [7] proved this result in a box, we note that this result will hold for any smooth compact manifold, as long $\\Pi _k(n)$ is taken over the noisy cycles (i.e.", "ignoring precisely the giant cycles we study in this paper).", "With this result in hand, an obvious question is then – how does the scaling in Theorem REF compares to the persistence of the signal (giant) cycles?", "Notice that the death of a signal cycle (in the limit) is non-random, and depends on the geometry of the underlying space only.", "For example, sampling from an annulus, then the death time of the giant 1-cycle is the inner radius (asymptotically), see Figure REF .", "Therefore, in order to estimate the persistence ratio of the signal cycles, we need to evaluate their birth time.", "The results in this paper provide the correct scaling for these birth times, and by that can be used to highlight the asymptotic differences between signal and noise in geometric models (see Section )." ], [ "Homology", "Homology is an algebraic invariant which characterizes spaces and functions using groups and homomorphisms.", "In a nutshell, if $X$ is a topological space, we have a sequence of groups $\\left\\lbrace \\mathrm {H}_k(X)\\right\\rbrace _{k\\ge 0}$ , where loosely speaking, the generators of $\\mathrm {H}_0(X)$ correspond to the connected components of $X$ , the generators of $\\mathrm {H}_1(X)$ correspond to closed loops surrounding holes in $X$ , $\\mathrm {H}_2(X)$ corresponds to surfaces enclosing voids in $X$ , and in general the elements of $\\mathrm {H}_k(X)$ are considered nontrivial $k$ -dimensional cycles.", "We refer the reader to  [24] for more precise definitions, as for the most part we will not require it.", "We assume homology is computed using field coefficients, denoted by $\\mathbb {F}$ , and then the homology groups are simply vector spaces and the dimension of these vector spaces are known as the Betti numbers, denoted $\\beta _k(X)$ .", "In this paper, we limit ourselves to the case of the $d$ -dimensional torus.", "The homology groups in this case are $\\mathrm {H}_k \\cong \\mathbb {F}^{\\binom{d}{k}}$ , where $\\mathbb {F}$ is the field of coefficients we use.", "See Figure REF (a) for the case $d=2$ .", "More concretely, we will study the flat torus $\\mathbb {T}^d = \\mathbb {R}^d / \\mathbb {Z}^d$ , which is a topological torus with a locally flat metric.", "A useful way to think of $\\mathbb {T}^d$ is using the unit box $Q^d = [0,1]^d$ with a periodic boundary, i.e.", "$\\mathbb {T}^d = Q^d / \\lbrace 0\\sim 1\\rbrace $ .", "In this case, we can view the cycles in $\\mathrm {H}_k(\\mathbb {T}^d)$ as follows.", "Let $\\gamma _{k,1} := \\left([0,1]^k\\times \\left\\lbrace 0\\right\\rbrace ^{d-k}\\right) / \\left\\lbrace 0\\sim 1\\right\\rbrace $ , i.e.", "we take a $k$ -dimensional face of the $Q^d$ , with the periodic boundary of the torus.", "Then each $\\gamma _{k,1}$ introduces a $k$ -dimensional cycle in $\\mathbb {T}^d$ .", "For each $k$ , we can similarly generate a basis for $\\mathrm {H}_k(\\mathbb {T}^d)$ $\\left\\lbrace \\gamma _{k,1},\\gamma _{k,2},\\ldots ,\\gamma _{k,\\binom{d}{k}}\\right\\rbrace $ by taking $k$ -faces of $Q^d$ in all possible $\\binom{d}{k}$ directions (i.e.", "that are not parallel).", "We call these cycles the “essential cycles\" of the torus $\\mathbb {T}^d$ .", "See Figure REF (b).", "The careful reader should note that by cycle, we are referring to a cycle representative of a non-trivial homology class.", "In addition to describing the properties of a single space $X$ , homology groups can also be used to study functions between spaces.", "For a given function $f:X\\rightarrow Y$ we have a collection of induced maps $f_*:\\mathrm {H}_k(X)\\rightarrow \\mathrm {H}_k(Y)$ , that describe what happens to every $k$ -cycle in $X$ after applying $f$ .", "As $\\mathrm {H}_k(X),\\mathrm {H}_k(Y)$ are vector spaces, $f_*$ are linear transformations.", "Figure: The homology of the torus.", "(a) The 2d2d torus as a manifold.", "There is a single connected component – H 0 ≅𝔽\\mathrm {H}_0\\cong \\mathbb {F}, two independent 1-cycles (dashed line) – H 1 ≅𝔽 2 \\mathrm {H}_1\\cong \\mathbb {F}^2, and a single “air pocket\" – H 2 ≅𝔽\\mathrm {H}_2 \\cong \\mathbb {F}.", "(b) The 3d3d flat torus 𝕋 3 =[0,1] 3 /0∼1\\mathbb {T}^3 = [0,1]^3/\\left\\lbrace 0\\sim 1\\right\\rbrace , where H 1 ≅𝔽 3 \\mathrm {H}_1\\cong \\mathbb {F}^3 and H 2 ≅𝔽 3 \\mathrm {H}_2\\cong \\mathbb {F}^3.", "On the first row we mark the essential 1-cycles, and on the second row the essential 2-cycles.", "The columns are ordered so that the 2-cycle at the bottom is the dual (via Lemma ) of the 1-cycle above." ], [ "Continuum percolation", "Percolation theory focuses primarily on the formation of infinite components in random media.", "In continuum percolation (see [29]), the medium is generated by geometric objects (grains) placed at random in space.", "In its simplest form, we have a homogeneous Poisson process in $\\mathbb {R}^d$ with rate $\\lambda $ , denoted $\\mathcal {P}_\\lambda $ , and the grains are fixed-size balls.", "We define the occupancy and vacancy processes as $\\mathcal {O}:= \\bigcup _{p\\in \\mathcal {P}_\\lambda } B_1(p)\\quad \\text{and}\\quad \\mathcal {V}:= \\mathbb {R}^d \\backslash \\mathcal {O},$ where $B_r(p)$ is the ball of radius $r$ around $p$ .", "The fundamental results in percolation theory are concerned with probability to form an infinite component.", "To this end, we define the event when the origin is part of an infinite component in $\\mathcal {O}$ as $I_0$ , where `infinite' could refer to either the diameter, volume or the number of points (cf. [29]).", "Similarly, we define $\\bar{I}_0$ for the vacancy $\\mathcal {V}$ .", "Next, we define the percolation probabilities $\\theta (\\lambda ) := \\mathbb {P}\\left(I_0\\right),\\quad \\text{and}\\quad \\bar{\\theta }(\\lambda ) := \\mathbb {P}\\left(\\bar{I}_0\\right),$ and the percolation thresholds $\\lambda _c := \\inf \\left\\lbrace \\lambda : \\theta (\\lambda ) >0 \\right\\rbrace ,\\quad \\text{and}\\quad \\bar{\\lambda }_c:= \\sup \\left\\lbrace \\lambda : \\bar{\\theta }(\\lambda ) > 0\\right\\rbrace .$ A fundamental result in continuum percolation then states that for all $d\\ge 2$ we have $0 < \\lambda _c \\le \\bar{\\lambda }_c < \\infty ,$ with equality for $d=2$ [34], and a strict inequality for $d>2$ [35].", "The critical values $\\lambda _c,\\bar{\\lambda }_c$ were shown to control various phenomena related to the connected components of the occupancy/vacancy processes.", "Of a particular interest to us will be those related to crossing paths in a finite box.", "Define $W_n := [-\\frac{n}{2}, \\frac{n}{2}]^d$ , and let $\\mathcal {O}(n),\\mathcal {V}(n)$ be the occupancy and vacancy processes generated by the points in $\\mathcal {P}_\\lambda \\cap W_n$ .", "For every $n$ , these processes are finite, and we can ask whether either of them contains a path that crosses the box from one side to the other.", "This question is interesting mainly in the limit as $n\\rightarrow \\infty $ .", "In the theory of random geometric graphs [32], an alternative and nearly equivalent model is studied, which will be useful for us in this paper.", "Instead of taking the growing box $W_n$ and a fixed radius $r=1$ , we take the fixed unit box $Q^d = [0,1]^d$ , consider the homogeneous Poisson process of rate $n$ in this box $\\mathcal {P}_n$ , and study the processes $ \\mathcal {O}_r := \\bigcup _{p\\in \\mathcal {P}_n} B_r(p),\\quad \\text{and}\\quad \\mathcal {V}_r := Q\\backslash \\mathcal {O}_r.", "$ To make the models equivalent we set $n_\\lambda = (n/\\lambda )^{1/d}$ , and note that the limiting behavior of the processes $\\mathcal {O}(n),\\mathcal {V}(n)$ (in $W_n$ ) is same as $\\mathcal {O}(n_\\lambda ),\\mathcal {V}(n_\\lambda )$ (in $W_{n_\\lambda }$ , for any fixed $\\lambda >0$ ).", "In addition, by a scaling argument – taking $\\mathcal {P}_\\lambda $ in $W_{n_\\lambda }$ with balls of radius 1 is equivalent to taking $\\mathcal {P}_n$ in $Q^d$ with balls of radius $r=(\\lambda /n)^{1/d}$ .", "In other words, the processes $\\mathcal {O}(n_\\lambda )$ and $n_\\lambda \\mathcal {O}_r$ have the same distribution (up to translation).", "To conclude, we will consider the model in (REF ), under the condition $ nr^d = \\lambda , $ for a fixed $\\lambda \\in (0,\\infty )$ .", "Note that this implies that $r = (\\lambda /n)^{1/d} \\rightarrow 0$ as $n\\rightarrow \\infty $ .", "To prove our main result, we will need the following statements that are adapted from the continuum percolation literature.", "For any two sets $A,B\\subset Q$ we denote by $A \\stackrel{\\mathcal {O}_r}{\\longleftrightarrow }B$ the event that there exists a path in the occupancy process that connects a point in $A$ to a point in $B$ .", "Similarly, we define $A \\stackrel{\\mathcal {V}_r}{\\longleftrightarrow }B$ for the vacancy processes.", "The first statements we need are about the exponential decay of the one-armed probabilities.", "Proposition 2.1 Let $c$ be the center point of the cube $Q$ , and $\\partial B_{R}(c)$ be the boundary of the ball of radius $R<1/2$ centered at $c$ .", "If $\\lambda < \\lambda _c$ , there exists $C_1>0$ (possibly depends on $\\lambda $ ) such that $\\mathbb {P}\\left(c \\stackrel{\\mathcal {O}_r}{\\longleftrightarrow } \\partial B_{R}(c)\\right) \\le e^{-C_1 R n^{1/d}}.$ If $\\lambda > \\bar{\\lambda }_c$ , there exists $C_2>0$ (possibly depends on $\\lambda $ ) such that $\\mathbb {P}\\left(c \\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c)\\right) \\le e^{-C_2 R n^{1/d}}.$ This is merely a scaled version of Theorem 2 and 4 in [16].", "In [16] it is proved that if $\\lambda <\\lambda _c$ then for any $\\tilde{R}>0$ we have $\\mathbb {P}\\left(0\\stackrel{\\mathcal {O}}{\\longleftrightarrow }\\partial B_{\\tilde{R}}(0)\\right) \\le e^{-c_\\lambda \\tilde{R}},$ for some $c_\\lambda >0$ .", "Scaling by $r$ and shifting by $c$ , we have $\\mathbb {P}\\left(c\\stackrel{\\mathcal {O}_r}{\\longleftrightarrow }\\partial B_{\\tilde{R}r}(c)\\right) \\le e^{-c_\\lambda \\tilde{R}}.$ Finally, since $r = (\\lambda /n)^{1/d}$ we have $\\mathbb {P}\\left(c\\stackrel{\\mathcal {O}_r}{\\longleftrightarrow }\\partial B_{R}(c)\\right) \\le e^{-c_\\lambda R/r} = e^{-C_1 R n^{1/d}}.$ Similarly, we can prove the statement for $\\mathcal {V}_r$ .", "The next statement we need is about the crossing paths and uniqueness of the giant component.", "Proposition 2.2 Suppose that $d\\ge 2$ and $\\lambda > \\lambda _c$ .", "Take any $D \\le 1$ .", "Denote by $E$ the events that: There exists a unique component of $\\mathcal {O}_r$ that crosses the box $Q$ in all directions.", "The diameter of all other components in $\\mathcal {O}_r$ is at most $D$ .", "Then there exists $C_3>0$ (possibly depends on $\\lambda $ ), so that $\\mathbb {P}\\left(E\\right) \\ge 1-e^{-C_3 D n^{1/d}}.$ We use a scaled version of Proposition 2 in [33].", "For the cube $W_n$ it is proved in [33] that when $\\lambda > \\lambda _c$ , for any $\\log n \\ll \\phi _n \\le n$ the probability that there exists a unique giant component in $\\mathcal {O}(n)$ , crossing the box $W_n$ in all directions, and that all other components have diameter less than $\\phi _n$ is bounded from below by $1-e^{-C_3\\phi _n}$ for some $C_3>0$ .", "Scaling from $\\mathcal {O}(n)$ to $\\mathcal {O}_r$ , then implies that for $n^{-1/d}\\log n \\ll D \\le 1$ we have $\\mathbb {P}\\left(E\\right) \\ge 1-e^{-C_3 D n^{1/d}}$ .", "Remark 2.3 While Theorem REF is stated for a cube, the proof in [33] applies for a box of any fixed-size dimensions." ], [ "Main result", "Throughout this paper we let $d\\ge 2$ be fixed and consider the flat torus $\\mathbb {T}^d = \\mathbb {R}^d / \\mathbb {Z}^d$ (see Section REF ).", "Let $\\left\\lbrace X_1,X_2,\\ldots \\right\\rbrace $ be iid random variables uniformly distributed in $\\mathbb {T}^d$ , let $N\\sim \\mathrm {Poisson}\\left({n}\\right)$ be another independent variable, and define $\\mathcal {P}_n := \\left\\lbrace X_1,\\ldots , X_N\\right\\rbrace $ .", "Then $\\mathcal {P}_n$ is a homogeneous Poisson process on $\\mathbb {T}^d$ with rate $n$ , and define the occupancy and vacancy processes as above by $\\mathcal {O}_r := \\bigcup _{p\\in \\mathcal {P}_n} B_r(p),\\quad \\text{and}\\quad \\mathcal {V}_r := \\mathbb {T}^d \\backslash \\mathcal {O}_r,$ where we use balls with respect to the toroidal metric (which is locally flat).", "In order to define the giant $k$ -cycles, we consider the inclusion maps $i:\\mathcal {O}_r \\rightarrow \\mathbb {T}^d$ and $\\bar{i}: \\mathcal {V}_r\\rightarrow \\mathbb {T}^d$ and consider their induced maps (homomorphisms) in homology, $ i_{k}: \\mathrm {H}_k(\\mathcal {O}_r)\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d),\\quad \\text{and}\\quad \\bar{i}_{k}: \\mathrm {H}_k(\\mathcal {V}_r)\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d).$ Loosely speaking, the image of $ i_{k}$ (resp.", "$\\bar{i}_{k}$ ) corresponds to the $k$ -cycles of the tours that have a representative element in $\\mathcal {O}_r$ (resp.", "$\\mathcal {V}_r$ ).", "We define the $k$ -th homological percolation events as $E_k := \\lbrace \\operatorname{Im}( i_{k}) \\ne 0\\rbrace ,\\quad \\text{and}\\quad A_k := \\lbrace \\operatorname{Im}( i_{k}) = \\mathrm {H}_k(\\mathbb {T}^d)\\rbrace ,$ and similarly for vacancy we define $\\bar{E}_k, \\bar{A}_k$ .", "The event $E_k$ asserts that at least one of the $k$ -cycles of the torus is represented in $\\mathcal {O}_r$ , while $A_k$ asserts that all of them are.", "In Figure REF , we observe 1-cycles that realize the event $A_1$ in the 2-dimensional torus and the 3-dimensional torus.", "In Figure REF we show the 2-cycles that realize $A_2$ for the 3-dimensional torus.", "One important remark is that the examples illustrate that the giant cycles need not be simple, e.g.", "top-to-bottom or left-to-right.", "Figure: The formation of giant 1-cycles in the flat torus.", "(a) We plot realizations of the 1-cycles generated by random balls in 𝕋 2 \\mathbb {T}^2 (box with periodic boundary), where we have two giant cycles.", "The first cycle consists of the green+red paths, and the second cycle consists of the blue+red paths.", "(b) We plot the 1-cycles generated in 𝕋 3 \\mathbb {T}^3, where we have three of them.", "To simplify the picture we do not show the balls here, only the paths (red,gree,blue) that correspond to the =1-cycles (note that the cycles may overlap).Figure: The formation of giant 2-cycles in the flat torus.", "Here we take 𝕋 3 \\mathbb {T}^3 ([0,1] 3 [0,1]^3 with periodic boundary) and draw the giant 2-cycles formed by the union of balls over a random sample.", "To simplify the picture we only show a triangulated version (the nerve) of the balls generating the 2-cycles.", "Each of the 2d surfaces presented is a 2-cycles, meaning that it encloses a cavity in the structure.We can now state the main result of the paper, which considers the limiting probability of the events $E_k,A_k$ as $n\\rightarrow \\infty $ .", "Theorem 3.1 Let $d\\ge 2$ , and let $nr^d = \\lambda $ .", "Then there exist two sequences $ 0<\\lambda _{0,1} \\le \\lambda _{0,2} \\le \\cdots \\le \\lambda _{0,d-1} < \\infty ,\\quad \\text{and}\\quad 0<\\lambda _{1,1} \\le \\lambda _{1,2} \\le \\cdots \\le \\lambda _{1,d-1} < \\infty , $ with $\\lambda _{0,k} \\le \\lambda _{1,k}$ , such that the following holds.", "If $\\lambda < \\lambda _{0,k}$ then $ \\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log \\mathbb {P}\\left(A_k\\right) \\le \\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log \\mathbb {P}\\left(E_k\\right) < 0, $ and if $\\lambda > \\lambda _{1,k}$ then $ \\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log {(1-\\mathbb {P}\\left(E_k\\right))} \\le \\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log {(1-\\mathbb {P}\\left(A_k\\right))} < 0.", "$ Further, we have that $\\lambda _{0,1} = \\lambda _{1,1} = \\lambda _{c}$ , and $\\lambda _{1,d-1} \\le \\bar{\\lambda }_{c}$ , where $\\lambda _c,\\bar{\\lambda }_c$ are the critical values for continuum percolation discussed in Section .", "In other words, the theorem implies that there exist $C_{0,k},C_{1,k}>0$ (possibly depending on $\\lambda $ ) such that for $\\lambda < \\lambda _{0,k}$ and for large enough $n$ we have $\\mathbb {P}\\left(A_k\\right)\\le \\mathbb {P}\\left(E_k\\right) \\le e^{-C_{0,k}n^{1/d}},$ and for $\\lambda > \\lambda _{1,k}$ , for large enough $n$ , $\\mathbb {P}\\left(E_k\\right) \\ge \\mathbb {P}\\left(A_k\\right) \\ge 1-e^{-C_{1,k} n^{1/d}}.$ This, in particular, implies that $\\lim _{n\\rightarrow \\infty }\\mathbb {P}\\left(E_k\\right) = \\lim _{n\\rightarrow \\infty }\\mathbb {P}\\left(A_k\\right) = {\\left\\lbrace \\begin{array}{ll} 1 & \\lambda > \\lambda _{1,k},\\\\0 & \\lambda < \\lambda _{0,k}.", "\\end{array}\\right.", "}$ The main conclusion from Theorem REF is that the giant cycles of all dimensions $0< k<d$ appear within the thermodynamic limit (i.e.", "$nr^d = \\text{const}$ ).", "This observation is not so obvious, since forming giant $k$ -cycles require $\\mathcal {O}_r$ to cover large $k$ -dimensional surfaces, while the process itself is still relatively sparse.", "For example, homological connectivity – the phase when $\\mathrm {H}_k(\\mathcal {O}_r) \\cong \\mathrm {H}_k(\\mathbb {T}^d)$ , occurs at a much later stage, when $nr^d \\sim \\log n$ [6].", "Note that $k=d$ is excluded from the theorem.", "The $d$ -cycle of the torus can only appear in $\\mathcal {O}_r$ upon coverage, which also occurs when $nr^d \\sim \\log n$ [6].", "Another conclusion from the theorem is that the appearance of the giant $k$ -cycles occurs in an orderly fashion, increasing in $k$ .", "In addition, all the cycles are formed in the interval $[\\lambda _{c},\\bar{\\lambda }_{c}]$ .", "Further, once the giant component in $\\mathcal {O}_r$ appears (at $\\lambda _{c}$ ) it already includes (w.h.p.)", "all the giant 1-cycles, and hence $\\lambda _{0,1} = \\lambda _{1,1} = \\lambda _{c}$ .", "This behavior will be made clearer in the proof.", "Note that Theorem REF provides a sharp phase transition only for the case of $k=1$ , and the inequalities between the thresholds are not strict.", "However, since sharpness is a key property in most percolation models [3], [30], [15], we believe that a stronger statement is true here as well.", "The proof of this statement will remain as future work.", "Conjecture 3.2 For all $0<k<d$ we have $\\lambda _{0,k} = \\lambda _{1,k} := \\lambda _k$ , and in addition $\\lambda _c = \\lambda _1 < \\lambda _2 < \\cdots < \\lambda _{d-1} = \\bar{\\lambda }_c.$" ], [ "Proofs", "In this section we prove Theorem REF .", "We start by defining $ \\begin{split} \\lambda _{0,k} &= \\sup \\left\\lbrace \\lambda : \\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log \\mathbb {P}\\left(E_k\\right) < 0\\right\\rbrace , \\\\\\lambda _{1,k} &= \\inf \\left\\lbrace \\lambda : \\limsup _{n\\rightarrow \\infty } n^{-1/d} \\log (1-\\mathbb {P}\\left(A_k\\right))<0\\right\\rbrace .", "\\end{split} $ In case the first set is empty, we set $\\lambda _{0,k }= -\\infty $ , and in case the second set is empty we set $\\lambda _{1,k} = \\infty $ (we will show later that neither set is empty).", "Note that by definition, (REF ) and (REF ) hold.", "From the definitions we also have $\\lambda _{0,k}\\le \\lambda _{1,k}$ for all $k$ , since if $\\mathbb {P}\\left(A_k\\right)\\rightarrow 1$ then surely $\\mathbb {P}\\left(E_k\\right) \\lnot \\rightarrow 0$ .", "Thus, in order to prove Theorem REF we have to show that all thresholds are in $(0,\\infty )$ and are increasing in $k$ as in (REF ).", "We will break the proof of Theorem REF into three parts.", "We start by proving that $\\lambda _{0,1} = \\lambda _{1,1} = \\lambda _c > 0$ .", "Next, we prove that $\\lambda _{1,d+1}\\le \\bar{\\lambda }_c < \\infty $ .", "Finally, we prove that for all $1\\le k \\le d-2$ we have $\\lambda _{0,k} \\le \\lambda _{0,k+1}$ , and $\\lambda _{1,k} \\le \\lambda _{1,k+1}$ .", "That will conclude the proof." ], [ "Giant 1-cycles", "Our goal in this section is to prove the following lemma.", "Lemma 4.1 The thresholds for the giant 1-cycles satisfy $\\lambda _{0,1} = \\lambda _{1,1} = \\lambda _c$ .", "Suppose first that $\\lambda < \\lambda _{c}$ .", "Recall that we can consider the torus $\\mathbb {T}^d$ as the quotient $Q^d/\\left\\lbrace 0\\sim 1\\right\\rbrace $ , and take a discretization of $Q^d$ by the grid $\\varepsilon r\\cdot \\mathbb {Z}^d$ , where $\\varepsilon $ is chosen small enough that any ball of radius $r$ intersects at least one grid point (i.e.", "$\\varepsilon <1/\\sqrt{d}$ ).", "Suppose that $\\operatorname{Im}{i_1}\\ne 0$ , i.e.", "there exists a non-trivial 1-cycle in $\\mathrm {H}_1(\\mathcal {O}_r)$ that is mapped to a non-trivial 1-cycle in $\\mathrm {H}_1(\\mathbb {T}^d)$ .", "Denote by $\\gamma $ one of the (possibly many) combinations of balls in $\\mathcal {O}_r$ that realizes this cycle.", "Since $\\gamma $ contains at least one ball, it must intersect with at least one of the grid points, denoted $x_0$ .", "In addition, fixing $R < 1/2$ , then the ball $B_R(x_0) \\subset \\mathbb {T}^d$ is contractible, and therefore its homology is trivial.", "Thus, any cycle supported on a component that is contained in $B_R(x_0)$ will be mapped to a trivial cycle in $\\mathbb {T}^d$ (see Figure REF ).", "We therefore conclude that $\\gamma $ must intersect with the boundary $\\partial B_R(x_0)$ .", "In other words, there must be a path in $\\mathcal {O}_r$ connecting $x_0$ to $\\partial B_R(x_0)$ .", "By Proposition REF , and the translation invariance of the torus, this occurs with probability at most $e^{-C_1 R n^{1/d}}$ .", "Since there are $M = (\\varepsilon r)^{-d} = O(n)$ many grid points, taking a union bound we conclude that $\\mathbb {P}\\left(E_1\\right) \\le M e^{-C_1 R n^{1/d}} = O\\left(n e^{-C_1 R n^{1/d}}\\right).$ Thus, we conclude that $\\lambda \\le \\lambda _{0,1}$ .", "Since we assumed $\\lambda < \\lambda _c$ , we have $\\lambda _{0,1}\\ge \\lambda _c$ .", "Figure: (a) The path presented here generates a giant 1-cycle in 𝒪 r \\mathcal {O}_r.", "The point xx is on this cycle, and we can see that there is a path from xx to ∂B R (x)\\partial B_R(x).", "(b) Here we have a 1-cycle in 𝒪 r \\mathcal {O}_r that is fully contained in B R (x)B_R(x).", "Indeed, this not a giant 1-cycle, since in 𝕋 d \\mathbb {T}^d this loop does not surround a hole (i.e.", "it is a boundary).Next, suppose that $\\lambda > \\lambda _{c}$ .", "Note that we can also think of the torus $\\mathbb {T}^d$ as $\\mathbb {T}^d = ([0,4/3]\\times [0,1]^{d-1}) / \\mathbb {Z}^d$ .", "With this in mind, we define the boxes $R_i = \\left(\\left[i/3, (i+2)/3\\right]\\times [0,1]^{d-1}\\right) / \\mathbb {Z}^d \\subset \\mathbb {T}^d,\\quad i=0,1,2,$ as well as their intersections $R_{i,j} := R_i\\cap R_j$ .", "For each of the boxes $R_i$ , we can define a Poisson process $\\mathcal {P}_n^{(i)} = \\mathcal {P}_n\\cap R_i$ .", "Next, we define the occupancy process $\\mathcal {O}_r^{(i)}$ as the union of $r$ -balls around $\\mathcal {P}_n^{(i)}$ in $R_i$ with its Euclidean (rather than toroidal) metric.", "Denote by $B_i$ the event that The process $\\mathcal {O}_r^{(i)}$ contains a path crossing $R_i$ along its short (2/3) side.", "There is a unique component in $\\mathcal {O}_r^{(i)}$ whose diameter is larger than $1/6$ .", "According to Theorem REF , we have that $\\mathbb {P}\\left(B_i\\right) \\ge 1-e^{-\\frac{1}{6}C_3n^{1/d}}$ .", "Using a union bound we then have that $\\mathbb {P}\\left(B_1\\cap B_2 \\cap B_3\\right) \\ge 1-3e^{-\\frac{1}{6}C_3 n^{1/d}}.$ Under the event $B = B_1\\cap B_2\\cap B_3$ , we denote by $L_i$ the largest component in $\\mathcal {O}_r^{(i)}$ , so that it contains a crossing path on the short side, denoted $\\pi _i$ .", "Note that each $\\pi _i$ also contains a path crossing $R_{i,j}$ ($j\\ne i$ ) along the shorter side, denoted $\\pi _{i,j}$ .", "While $\\pi _{i,j}$ is not necessarily contained in $\\mathcal {O}_r^{(j)}$ , it is true that the diameter of $\\pi _{i,j}\\cap \\mathcal {O}_r^{(j)}$ is at least $1/3-r > 1/6$ .", "Therefore, we conclude that $\\pi _{i,j}\\cap \\mathcal {O}_r^{(j)}\\subset L_j$ , implying that there is a path in $\\mathcal {O}_r^{(j)}$ connecting $\\pi _{i,j}$ and $\\pi _j$ .", "To conclude, under the event $B$ we have the following sequence of connected paths, $\\pi _1 \\stackrel{\\mathcal {O}_r^{(1)}}{\\longrightarrow }\\pi _{1,2} \\stackrel{\\mathcal {O}_r^{(2)}}{\\longrightarrow }\\pi _2 \\stackrel{\\mathcal {O}_r^{(2)}}{\\longrightarrow }\\pi _{2,3} \\stackrel{\\mathcal {O}_r^{(3)}}{\\longrightarrow }\\pi _3 \\stackrel{\\mathcal {O}_r^{(3)}}{\\longrightarrow }\\pi _{3,1}\\stackrel{\\mathcal {O}_r^{(1)}}{\\longrightarrow }\\pi _1.$ In other words, we showed that under $B$ we can find a path in $\\mathcal {O}_r$ that loops along one of the sides of the torus.", "Such a loop will generate an element in $\\mathrm {H}_1(\\mathcal {O}_r)$ that is homologous to the essential 1-cycle of the torus $\\gamma _{1,1}$ (see Section REF ).", "Repeating the same arguments in all $d$ -directions, and using a union bound will imply that $\\mathbb {P}\\left(A_k\\right) \\ge 1-3d e^{-\\frac{1}{6}C_3 n^{1/d}}.$ Thus, we must have $\\lambda \\ge \\lambda _{1,1}$ , and since $\\lambda >\\lambda _c$ we conclude that $\\lambda _{1,1}\\le \\lambda _c$ .", "Finally, we showed that $\\lambda _{1,1}\\le \\lambda _c \\le \\lambda _{0,1}$ .", "On the other hand, (REF ) implies that $\\lambda _{0,1}\\le \\lambda _{1,1}$ .", "Thus, we conclude that $\\lambda _{0,1}=\\lambda _{1,1} = \\lambda _c$ , concluding the proof.", "Figure: Considering the torus 𝕋 d \\mathbb {T}^d as the quotient [0,4/3]×[0,1] d-1 /ℤ d [0,4/3]\\times [0,1]^{d-1}/\\mathbb {Z}^d.", "We then split the torus into the boxes R 1 ,R 2 ,R 3 R_1,R_2,R_3 and their intersection.", "Notice that the top and bottom rectangle are identical (R 1,3 R_{1,3}).", "Using the gluing arguments in the proof, and connecting the dots from 1 to 6, we get a loop that generates the top-bottom 1-cycle in the picture.Note that as mentioned in the proof, the paths π i,j ∩𝒪 r (j) \\pi _{i,j}\\cap \\mathcal {O}_r^{(j)} are not necessarily crossing for R i,j R_{i,j}, as can be seen in the figure.Observation 4.2 The proof that a giant cycle or equivalently a non-contractible loop exists (in the case of the torus) follows from the uniqueness of the crossing component.", "This uniqueness also implies that a cycle cannot “wind around\" the torus multiple times before forming a loop, as this would imply multiple crossing components in all of the boxes." ], [ "Duality", "The proofs for $k>1$ will require the following duality between the occupancy and vacancy processes.", "Lemma 4.3 Recall that $ i_{k}:\\mathrm {H}_k(\\mathcal {O}_r) \\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)$ and $\\bar{i}_{k}: \\mathrm {H}_k(\\mathcal {V}_r) \\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)$ are the maps (group homomorphisms) induced by the inclusion map.", "Then, $\\beta _k(\\mathbb {T}^d) := \\operatorname{rank}(\\mathrm {H}_k(\\mathbb {T}^d)) = \\operatorname{rank}( i_{k}) + \\operatorname{rank}(\\bar{i}_{d-k}).$ Note that since we are using field coefficients, the homology groups are vector spaces, and we can simply replace $\\operatorname{rank}$ with $\\dim $ .", "Recall the definitions of the events $A_k,E_k,\\bar{A}_k,\\bar{E}_k$ .", "The following corollary will be very useful for us.", "Corollary 4.4 The event $A_k$ occurs if and only if $\\bar{E}_{d-k}$ does not.", "In other words, $A_k$ and $\\bar{E}_{d-k}$ are complementing events.", "The event $A_k$ occurs if and only if $\\operatorname{rank}(i_k) = \\beta _k(\\mathbb {T}^d)$ .", "Using Lemma REF , this holds if and only if $\\operatorname{rank}(\\bar{i}_{d-k}) =0$ .", "Finally, by definition $\\operatorname{rank}(\\bar{i}_{d-k}) =0$ if and only if $\\bar{E}_{d-k}$ does not hold.", "This completes the proof.", "The proof of Lemma REF requires more familiarity with algebraic topology than the rest of the paper, but is not required in order to understand the rest of the paper.", "We use a form of Alexander duality, which relates the homology of a suitably well-behaved subset of a space with the cohomology of its complement (see [24]).", "Lemma 4.5 ([24] Thm 3.44) Let $M$ be a closed orientable $d$ -manifold, and let $K\\subset M$ be compact and locally contractible.", "Then, $\\mathrm {H}_{k}(M, M-K) \\cong \\mathrm {H}^{d-k}(K)$ Before continuing we make a few remarks.", "First, the locally contractible condition follows in our case as the number of balls intersecting any point is finite almost surely.", "We also note that since we are considering (co)homology over a field, homology and cohomology are dual vector spaces, so their ranks/dimensions are the same.", "Finally, we note for the reader that although Alexander duality is most commonly stated for the case $M=\\mathbb {S}^d$ , it remains true for any compact manifold.", "[Proof of Lemma REF ] Take $M=\\mathbb {T}^d$ and $K=\\mathcal {O}_r$ (so that $M-K = \\mathcal {V}_r$ ) in Lemma REF , and consider the following diagram, $@C+1pc{\\mathrm {H}_{d-k}(\\mathcal {V}_r)[r]^{{\\bar{i}_{d-k}}}& \\mathrm {H}_{d-k}(\\mathbb {T}^d)[d]^{\\cong }[r]^{j}& \\mathrm {H}_{d-k}(\\mathbb {T}^d,\\mathcal {V}_r)[d]^{\\cong }\\\\& \\mathrm {H}^{k}(\\mathbb {T}^d)[r]^{i^k}& \\mathrm {H}^{k}(\\mathcal {O}_r)}$ The first row in this diagram is a part of the long exact sequence for relative homology.", "The left vertical map is the isomorphism given by Poincaré duality, and the second vertical map is the isomorphism provided by Lemma REF .", "The fact that this diagram commutes arises as part of the proof of Lemma REF (see [24]).", "By the rank-nullity theorem, we have $ \\operatorname{rank}(\\mathrm {H}_{d-k}(\\mathbb {T}^d)) = \\operatorname{rank}(\\ker (j)) + \\operatorname{rank}(\\operatorname{Im}(j)).", "$ Since the top row is exact we have that $\\ker (j) = \\operatorname{Im}(\\bar{i}_{d-k})$ , implying that $\\operatorname{rank}(\\ker (j)) = \\operatorname{rank}(\\operatorname{Im}(\\bar{i}_{d-k}))$ .", "In addition, since we are assuming field coefficients, and using Poincaré duality, we have that $\\operatorname{rank}(\\mathrm {H}_{d-k}(\\mathbb {T}^d)) = \\operatorname{rank}(\\mathrm {H}^k(\\mathbb {T}^d)) = \\operatorname{rank}(\\mathrm {H}_k(\\mathbb {T}^d)) = \\beta _k(\\mathbb {T}^d).$ Finally, since the square in the diagram commutes, and both vertical maps are isomorphisms, we have that $\\operatorname{rank}(\\operatorname{Im}(j)) = \\operatorname{rank}(\\operatorname{Im}(i^k))$ .", "Since we assume field coefficients, the rank of the vector space and its dual are the same [13], and therefore $\\operatorname{rank}(\\operatorname{Im}(i^k)) = \\operatorname{rank}(\\operatorname{Im}(i_k))$ .", "Putting all these arguments into (REF ) completes the proof." ], [ "Giant $(d-1)$ -cycles", "The duality in Lemma REF and Corollary REF imply that if we can prove a phase transition for $\\mathrm {H}_1(\\mathcal {V}_r)$ , it will imply a phase transition for $\\mathrm {H}_{d-1}(\\mathcal {O}_r)$ as they are complementing.", "We note that dualities of a similar spirit have been used for bond percolation in $\\mathbb {R}^2$  [22], as well as implicitly in “blocking surface\" arguments [15], [2].", "Our proof of Lemma REF shows that the transition for $\\mathrm {H}_1$ is equivalent to the transition for the giant component.", "While uniqueness is known for the giant component in $\\mathcal {O}_r$  [33], to the best of our knowledge, to date no proof exists for uniqueness of the giant component in $\\mathcal {V}_r$ (i.e.", "the equivalent of Proposition REF for the vacancy).", "While we expect such statement to be true, there are numerous technical obstacles when dealing with the vacancy process, primarily due to its more complicated geometry (see Figure REF ).", "Thus, for the time being we make the following weaker statement.", "Figure: An approximation of the vacancy (shown in blue) at thresholds small enough (λ<λ ¯ c \\lambda < \\bar{\\lambda }_c) so that A ¯ 1 \\bar{A}_1 has occurred.", "(a) In 𝕋 2 \\mathbb {T}^2, we have A ¯ 1 =E 1 c \\bar{A}_1 = E_1^c, and therefore we observe no 1-cycles in the occupancy (white).", "(b) In 𝕋 3 \\mathbb {T}^3, we have A ¯ 1 =E 2 c \\bar{A}_1 = E_2^c, and thus the occupancy contains no 2-cycles.", "As can be seen, the vacancy has a much more challenging geoemtry as components can be arbitrarily small (whereas in the occupancy, the volume of a component is lower bounded by the volume of a ball, i.e.", "Ω(r d )\\Omega (r^d)).Lemma 4.6 The thresholds for the giant ${d-1}$ -cycles satisfy $\\lambda _{0,d-1} \\le \\lambda _{1,d-1} \\le \\bar{\\lambda }_c < \\infty ,$ where $\\bar{\\lambda }_c$ is the percolation threshold for the vacancy process in $\\mathbb {R}^d$ .", "Before proving the lemma, we require one intermediate technical result.", "Lemma 4.7 Let $c$ be the center point of $Q^d$ , and let $B_{R}(c)$ be a ball centered at the origin of radius $R<1/2$ , and be $Q_{\\varepsilon r}(c)$ be a box of side-length $\\varepsilon r$ centered at $c$ .", "If $\\lambda > \\bar{\\lambda }_c$ then there exists $C_4>0$ such that $\\mathbb {P}\\left(Q_{\\varepsilon r}(c) \\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c) \\right)\\le e^{-C_4 R n^{1/d}}.$ From Proposition REF we have when $\\lambda > \\bar{\\lambda }_c$ we have $ \\mathbb {P}\\left(c\\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c)\\right) \\le e^{-C_2 R n^{1/d}}.", "$ Next, $ \\mathbb {P}\\left(Q_{\\varepsilon r}(c) \\subset \\mathcal {V}_r\\right) \\ge \\mathbb {P}\\left( B_{r(1+\\sqrt{d}\\varepsilon /2)} \\cap \\mathcal {P}_n = \\emptyset \\right) = e^{-\\lambda \\omega _d(1+\\sqrt{d}\\varepsilon /2)^d} := C $ Note that if we have that both $Q_{\\varepsilon r}(c) \\subset \\mathcal {V}_r$ and $Q_{\\varepsilon r}(c) \\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c)$ , then necessarily $c\\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c)$ .", "Thus, $\\mathbb {P}\\left(Q_{\\varepsilon r}(c) \\subset \\mathcal {V}_r \\text{ and } Q_{\\varepsilon r}(c) \\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c)\\right) \\le \\mathbb {P}\\left(c\\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c)\\right) \\le e^{-C_2R n^{1/d}}.$ Since both events on the LHS are decreasing, we can use the FKG inequality (see, e.g.", "[29]) together with (REF ) and have $C\\cdot \\mathbb {P}\\left(Q_{\\varepsilon r}(c) \\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c)\\right)\\le e^{-C_2 R n^{1/d}}.$ Thus, we can find $C_4>0$ such that $\\mathbb {P}\\left(Q_{\\varepsilon r}(c) \\stackrel{\\mathcal {V}_r}{\\longleftrightarrow } \\partial B_{R}(c)\\right)\\le e^{-C_4 R n^{1/d}}$ , completing the proof.", "[Proof of Lemma REF ] Suppose that $\\lambda > \\bar{\\lambda }_c$ .", "The proof is mostly similar to the proof of Lemma REF .", "The main difference here is that there is no discretization $Q^d \\cap ( \\varepsilon r\\cdot \\mathbb {Z}^d)$ that guarantees that a component in $\\mathcal {V}_r$ will intersect any of the grid points.", "Instead, for every $x$ in the grid we take $Q_{\\varepsilon r}(x)$ to be the box of side-length $\\varepsilon r$ centered at $x$ .", "Since the union of these boxes covers $Q^d$ , we have that every component in $\\mathcal {V}_r$ must intersect at least one of these boxes.", "As in the proof of Lemma REF we argue that if $\\gamma \\subset \\mathcal {V}_r$ is a realization of a non-trivial 1-cycle in $\\mathrm {H}_1(\\mathcal {V}_r)$ that is mapped to a non-trivial cycle in $\\mathrm {H}_1(\\mathbb {T}^d)$ , then $\\gamma $ cannot be contained in a ball of radius $R<1/2$ .", "For any point $x\\in \\mathbb {T}^d$ , using the translation-invariance of the torus, and Lemma REF , we have that $\\mathbb {P}\\left(Q_{\\varepsilon r}(x) \\stackrel{\\mathcal {V}_r}{\\longrightarrow } \\partial B_R(x)\\right) \\le e^{-C_4 R n^{1/d}}.$ Since we have $M= (\\varepsilon r)^{-d} = O(n)$ boxes, using a union bound, we have $\\mathbb {P}\\left(\\bar{E}_1\\right) \\le Me^{-C_4 R n^{1/d}} = O\\left(ne^{-C_4 R n^{1/d}}\\right).$ Using Corollary REF , we have that $A_{d-1} = {\\bar{E}_1}^c$ .", "Thus, we have $\\mathbb {P}\\left(A_{d-1}\\right) \\ge 1-Me^{-C_4 R n^{1/d}},$ implying that $\\lambda > \\lambda _{1,d-1}$ .", "Therefore, we conclude that $\\lambda _{1,d-1}\\le \\bar{\\lambda }_c$ , completing the proof." ], [ "Giant $k$ -cycles, {{formula:fd710f87-3cef-466b-baf3-b49ef9f06c5e}}", "In this section we will prove that the appearance of all giant $k$ -cycles ($1<k<d-1$ ) occurs between $\\lambda _c$ and $\\bar{\\lambda }_c$ , and in an increasing order, as staged in Theorem REF .", "The following lemma is the main result of this section.", "Lemma 4.8 For every $1\\le k \\le d-2$ and $\\theta =0,1$ , we have $\\lambda _{\\theta ,k} \\le \\lambda _{\\theta ,k+1}$ .", "To prove Lemma REF , we will use the following statement, which is a consequence of the duality in Lemma REF .", "Lemma 4.9 The events $A_k,E_k$ satisfy $A_1 \\supset A_2 \\supset \\cdots \\supset A_{d-1}\\quad \\text{and}\\quad E_1 \\supset E_2 \\supset \\cdots \\supset E_{d-1},$ and the same holds for $\\bar{A}_k,\\bar{E}_k$ .", "For $i=1,\\ldots ,d$ define $\\mathbb {T}^d_i:= (\\mathbb {R}^{i-1}\\times \\left\\lbrace 0\\right\\rbrace \\times \\mathbb {R}^{d-i})/\\mathbb {Z}^d$ .", "In other words, $\\mathbb {T}^d_i$ are $(d-1)$ -dimensional flat tori embedded in $\\mathbb {T}^d$ .", "Let $\\mathcal {O}_r^{(i)} := \\mathcal {O}_r\\cap \\mathbb {T}_i^d$ , and $\\mathcal {V}_r^{(i)} := \\mathcal {V}_r\\cap \\mathbb {T}_i^d$ be the induced (or projected) processes.", "Similarly to the events $E_k,A_k$ we can define $E^{(i)}_k,A_k^{(i)},\\bar{E}^{(i)}_k,\\bar{A}_k^{(i)}$ ($1\\le k \\le d-2$ ) with respect to the processes $\\mathcal {O}_r^{(i)},\\mathcal {V}_r^{(i)}$ , and the $(d-1)$ -torus $\\mathbb {T}_i^d$ .", "Fix $1\\le k\\le d-2$ , and suppose that $A_k^{(i)}$ occurs, then $ \\operatorname{Im}(\\mathrm {H}_k(\\mathcal {O}_r^{(i)})\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d_i)) = \\mathrm {H}_k(\\mathbb {T}^d_i).", "$ We now require two topological facts: The inclusion $\\mathbb {T}_i^d\\hookrightarrow \\mathbb {T}^d$ induces an injective map in homology, i.e.", "the map $\\mathrm {H}_k(\\mathbb {T}_i^d)\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)$ is injective; For $k<d$ , the $k$ -dimensional classes in $\\mathbb {T}^d$ are spanned by the $k$ -dimensional classes in the $d$ subtorii, $\\mathbb {T}^d_i$ , i.e.", "$\\sum _{i=1}^d \\mathrm {H}_k(\\mathbb {T}^d_i) = \\mathrm {H}_k(\\mathbb {T}^d)$ , where the summation represents the sum of the vector spaces as subspaces of $\\mathrm {H}_k(\\mathbb {T}^d)$ .", "These two results are well-known.", "However, for completeness we include proofs in Appendix .", "The fact that $\\mathrm {H}_k(\\mathbb {T}_i^d)\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)$ is injective, implies that $\\operatorname{Im}(\\mathrm {H}_k(\\mathcal {O}_r^{(i)})\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d_i)) \\cong \\operatorname{Im}(\\mathrm {H}_k(\\mathcal {O}_r^{(i)})\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)).$ If $A_k^{(i)}$ occur for all $i=1,\\ldots ,d$ we have $\\mathrm {H}_k(\\mathbb {T}^d) = \\sum _{i=1}^d \\mathrm {H}_k(\\mathbb {T}^d_i) \\cong \\sum _{i=1}^d \\operatorname{Im}(\\mathrm {H}_k(\\mathcal {O}_r^{(i)})\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d))\\subset \\operatorname{Im}(\\mathrm {H}_k(\\mathcal {O}_r)\\rightarrow \\mathrm {H}_k(\\mathbb {T}^d)) \\subset \\mathrm {H}_k(\\mathbb {T}^d),$ implying that the last relation is an equality, so that $A_k$ holds as well.", "In other words, we showed that $ A_{k}^{(1)} \\cap \\cdots \\cap A_k^{(d)}\\subset A_k, $ and similarly we can show that $ E_{k}^{(1)} \\cup \\cdots \\cup E_k^{(d)}\\subset E_k.", "$ The same inclusions will apply for $\\bar{A}_k, \\bar{E}_k$ .", "Next, from Corollary REF we have that $A_k = (\\bar{E}_{d-k})^c$ , and since $\\mathbb {T}^d_i$ is a $(d-1)$ -torus, we also have that $ A_k^{(i)} = (\\bar{E}_{d-1-k}^{(i)})^c$ .", "Putting all these connections together we have that $\\begin{split} A_k &\\supset \\left(A_k^{(1)}\\cap \\cdots \\cap A_k^{(d)}\\right)= \\left((\\bar{E}_{d-1-k}^{(1)})^c \\cap \\cdots \\cap (\\bar{E}_{d-1-k}^{(d)})^c \\right)\\supset (\\bar{E}_{d-1-k})^c = A_{k+1}, \\end{split} $ and $\\begin{split} E_k &\\supset \\left(E_k^{(1)}\\cup \\cdots \\cup E_k^{(d)}\\right)= \\left((\\bar{A}_{d-1-k}^{(1)})^c \\cup \\cdots \\cup (\\bar{A}_{d-1-k}^{(d)})^c \\right)\\supset (\\bar{A}_{d-1-k})^c = E_{k+1}.", "\\end{split} $ Similarly, we can prove that $\\bar{A}_k \\supset \\bar{A}_{k+1}$ and $\\bar{E}_k \\supset \\bar{E}_{k+1}$ , concluding the proof.", "[Proof of Lemma REF ] For any $\\lambda < \\lambda _{0,k}$ we have $\\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log \\mathbb {P}\\left(E_k\\right) < 0$ .", "From Lemma REF we have that $E_{k+1}\\subset E_k$ , and therefore we also have $\\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log \\mathbb {P}\\left(E_{k+1}\\right) < 0$ , implying that $\\lambda < \\lambda _{0,k+1}$ .", "Thefeore, we conclude that $\\lambda _{0,k}\\le \\lambda _{0,k+1}$ .", "Similarly, for all $\\lambda > \\lambda _{1,k+1}$ we have $\\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log (1- \\mathbb {P}\\left(A_{k+1}\\right))< 0$ , and from Lemma REF we have $\\limsup _{n\\rightarrow \\infty } n^{-1/d}\\log (1- \\mathbb {P}\\left(A_{k}\\right))< 0$ , implying that $\\lambda > \\lambda _{1,k}$ .", "Therefore, $\\lambda _{1,k} \\le \\lambda _{1,k+1}$ .", "This completes the proof." ], [ "Discussion", "In this paper we have defined a notion of giant $k$ -dimensional cycles that emerge in a continuum percolation model on the torus.", "We have shown the existence of thresholds for the appearance of these cycles and that these thresholds are in the thermodynamic limit.", "In this section we provide some insights and directions for future work.", "The main open problem remains proving Conjecture REF , i.e.", "that all transitions are sharp, and that the ordering between the thresholds is strict.", "As we stated earlier, to prove sharpness for $k=d-1$ all that is required is a uniqueness statement for the giant vacancy component.", "For the intermediate dimensions, it is less clear how to prove sharpness.", "A desirable extension would be to state and prove analogous results for general manifolds as well as the appearance of the fundamental group.", "In the case of the torus, the sharp threshold for the fundamental group follows directly from the homological statements in this paper.", "The main challenges in manifolds will be that we must deal with (a) curvature, and (b) the existence and representability of giant cycles.", "In principle, we do not expect curvature to change these statements, however it adds significant technical complications (see [8]).", "The lack of a product structure in general manifolds makes relating giant cycles of different dimensions more difficult as well.", "However, we note that these do not apply to the fundamental group.", "In this paper we used balls with a fixed radius $r$ .", "The most common model studied in continuum percolation is where the grains are balls with random radii.", "Most of the statements in this paper can be translated to the random-radii case (assuming bounded moments).", "However, the equality $\\lambda _{0,1} = \\lambda _{1,1} = \\lambda _c$ requires a quantitative uniqueness of the crossing component, i.e.", "bounds on the second largest component, which to the best of our knowledge, has not been proved for the general case.", "In a recent paper [9], we experimentally studied the homological percolation thresholds in various models including continuum percolation, site percolation, and Gaussian random fields.", "We compared these thresholds to the zeros of the expected Euler characteristic curve (as a function of $\\lambda $ ), which has an explicit expression.", "The simulation results in [9] show that the percolation thresholds always appear very near the zeros of the expected EC.", "This is a somewhat surprising result, as the EC is a quantitive descriptor (counting cycles), while the percolation thresholds describe a qualitative phenomenon (the emergence of giant cycles).", "It remains an open question as to the nature of this observed correlation, and whether the zeros of the EC curve (which can be evaluated analytically) can potentially be used to approximate or at least bound the percolation thresholds.", "The definitions we used here for giant cycles, can be applied in the context on various other percolation models such as bond and site percolation.", "In principle, the general behavior should follow similarly to the one we observed here, while the proof might require a slightly different approach.", "In particular, the duality statement we have here, does not apply directly to other models.", "In the applied topology aspect of this work, recall that we wish to distinguish between the signal and noise cycles in persistent homology.", "For every signal (giant) cycle $\\gamma _{\\text{signal}}$ , we have $\\operatorname{death}(\\gamma _{\\text{signal}}) = \\text{const}$ , while the results in this paper imply that $\\operatorname{birth}(\\gamma ) = \\text{const}\\cdot n^{-1/d}$ (since the giant cycles are formed when $nr^d = \\lambda $ ).", "Therefore, the persistence value for the giant cycles, satisfies $\\pi (\\gamma _{\\text{signal}}) := \\frac{\\operatorname{death}(\\gamma _{\\text{signal}})}{\\operatorname{birth}(\\gamma _{\\text{signal}})} = \\Theta (n^{1/d}).$ In [7] it was shown that the persistence of all the noise cycles satisfies $\\pi (\\gamma _{\\text{noise}}) = O\\left(\\left(\\frac{\\log n}{\\operatorname{\\log \\log }n}\\right)^{1/k}\\right).$ In other words, the results in this paper indicate that asymptotically the persistence of the signal and the noise cycles differ by orders of magnitudes.", "From the applied topology perspective, this is an optimistic statement, since it means that given a large sample, we could use persistence to distinguish between signal and noise." ], [ "A topological supplement for the proof of Lemma ", "Recall that $\\mathbb {T}^d$ is the flat $d$ -torus and $\\mathbb {T}_i^d$ is the $(d-1)$ -torus defined by the $i$ -th flat.", "First, we show that the inclusion map $\\mathbb {T}_i^d\\hookrightarrow \\mathbb {T}^d$ , induces an injective map on homology $\\mathrm {H}_k(\\mathbb {T}_i^d)\\hookrightarrow \\mathrm {H}_k(\\mathbb {T}^d),$ Express the $d$ -torus as the $d$ -fold cartesian product of circles $\\mathbb {T}^d = \\mathbb {S}^1\\times \\ldots \\times \\mathbb {S}^1$ .", "Hence, we can rewrite $\\mathbb {T}^d$ as the cartesian product $\\mathbb {T}^d = \\mathbb {T}_i^{d}\\times \\mathbb {S}^1$ .", "Taking homology, we have the following isomorphism via the Künneth formula, $ \\bigoplus \\limits _{k+\\ell =m} \\mathrm {H}_k(\\mathbb {T}_i^{d})\\otimes \\mathrm {H}_\\ell (\\mathbb {S}^1) \\cong \\mathrm {H}_k(\\mathbb {T}^{d})$ Restricting to $\\ell =0$ yields the required injective map.", "Note that this follows from the fact that there is no torsion since we are working over field coefficients.", "Next we show that for $k<d$ $ \\sum \\limits _{i=1}^d \\mathrm {H}_k(\\mathbb {T}_i^d) = \\mathrm {H}_k(\\mathbb {T}^d).$ where $\\mathrm {H}_k(\\mathbb {T}_i^d)$ are taken as vector subspaces of the vector space $\\mathrm {H}_k(\\mathbb {T}^d)$ .", "This is well defined since the maps are injective by the argument above.", "Hence, we can take the sum of the individual vector spaces as subspaces, denoted by $ \\sum \\limits _{i=1}^d \\mathrm {H}_k(\\mathbb {T}_i^d)$ .", "For each $i$ , $\\mathrm {H}_k(\\mathbb {T}_i^d)\\subseteq \\mathrm {H}_k(\\mathbb {T}^d)$ , so it follows that $\\sum \\limits _{i=1}^d \\mathrm {H}_k(\\mathbb {T}_i^d)\\subseteq \\mathrm {H}_k(\\mathbb {T}^d).$ In the other direction, on can again use the representation of $\\mathbb {T}^d$ as the Cartesian product of circles.", "Applying the Künneth formula $d$ times, we obtain $\\mathrm {H}_k(\\mathbb {T}^d) = \\sum \\limits _{\\sum _{i} k(j) = k } \\mathrm {H}_{k(1)}(\\mathbb {S}^1) \\otimes \\cdots \\otimes \\mathrm {H}_{k(d)}(\\mathbb {S}^1) $ As the homology of $\\mathbb {S}^1$ is only non-zero for $k=0,1$ , a $k$ -cycle in $\\mathbb {T}^d$ can be represented by taking the $\\mathrm {H}_1(\\mathbb {S}^1)$ in some $k$ coordinates and $\\mathrm {H}_0(\\mathbb {S}^1)$ in the others.", "This is an element of any $\\mathrm {H}_k(\\mathbb {T}^d_i)$ where $k(i)=0$ and since $k<d$ , there must be at least one.", "The result follows." ] ]

[ [ "On-Shell Perspectives on the Massless Limit of Massive Supergravity" ], [ "Abstract Massive gravity exhibits a famous discontinuity in its 2-point linearized amplitude for t-channel scattering of gravitational sources, in the $m \\to 0$ limit.", "In essence, the source of this vDVZ discontinuity is in the failure of the zero-helicity mode to decouple in this limit.", "In [1], we showed how this result could be understood in the context of modern on-shell methods and, in particular, the BCFW construction.", "In this article, we provide a similar on-shell perspective to the equally interesting but lesser known discontinuity first discovered by Deser, Kay and Stelle in massive supergravity." ], [ "Introduction", "Endowing the graviton with a small but nonzero mass is an appealing idea for many reasons, the most significant of which is a credible explanation for the observed late-time acceleration of the Universe without the need to invoke exotic forms of matter and energy.", "However, since nothing in life is free, this approach is not without its own pathologies.", "Among these are a ghost mode and a, less scary but perhaps more famous, discontinuity in the 2-point function of the theory in the massless limit.", "The former was resolved in the recent seminal work of de Rham et.al.", "[2] while the latter boils down to a noncommutativity of limits.", "For this, there are two possibilities: turning off interactions does not necessarily commute with the massless limit.", "In other words, in order to resolve the vDVZ discontinuity, it is necessary to go beyond the linearised Fierz-Pauli action, leading to the famed Vainshtein screening mechanism of [3], or the massless limit does not commute with the limit of vanishing cosmological constant.", "Either case will break Birkhoff's theorem resulting in a vDVZ-like discontinuity.", "To clarify the situation surrounding this discontinuity, in [1] we attempted to re-phrase the discontinuity in the language of scattering amplitudes, largely because these modern on-shell methods are unencumbered by much of the baggage of the usual Lagrangian formulation of the problem.", "Recently, there have been several advances in our understanding of gravity by harnessing the on-shell scattering amplitudes paradigm [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], including approaches based on the double copy [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67] (see [68] for a recent review), the classical double copy [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96], [97], [98], [14], [99], [100], [101], [102], [103], [104], [105], [106] and those probing the on-shell structure of massive gravity [107], [108].", "In [1], we utilised this modern approach to show how the vDVZ discontinuity manifests at the level of the scattering amplitudes but did not go quite far enough in disambiguating between the two sources above, mostly because computing massive on-shell amplitudes in gravity at higher (than linear) order was beyond the scope of that article.", "At the same time we were reminded that this massive to massless discontinuity of the spin-2 graviton is also shared by a spin-$\\frac{3}{2}$ Rarita-Schwinger field coupled to a conserved vector-spinor current $j^{\\mu }$ .", "This is not an unexpected result since, when the spin-$\\frac{3}{2}$ field is coupled to the current, the supermatter interactions resulting from single-fermion interactions have precisely the form required by supersymmetry to complement single-graviton exchange between stress tensor sources [109].", "Demonstrating this result using standard methods is a nontrivial exercise in supergravity manipulations [110].", "Furthermore, we are motivated by the fact that, while all evidence so far suggest a massless graviton, the same is not true for the Rarita-Schwinger field.", "First, we would expect to have observed the gravitino were it massless.", "Secondly, various supersymmetry breaking mechanisms (such as the super-Higgs effect, or gravitationally induced SUSY breaking) have been shown to endow the Rarita-Schwinger particle with a non-zero mass [111], [112], [113].", "In this article, we approach the spin-$\\frac{3}{2}$ discontinuity from the point of view of on-shell massive amplitudes following the analysis in [1].", "There it was shown that such a discontinuity can be observed in a purely on-shell manner by directly constructing the massive scattering amplitudes of the theory, taking their massless limit and comparing them with the constructed massless amplitudes.", "To be concrete, we consider scattering amplitudes in $\\mathcal {N} = 1$ 4D supergravity, whose gauge multiplet consists entirely of a (spin-2) graviton and one Majorana (spin-$\\frac{3}{2}$ ) spinor gravitino.", "This can be coupled to matter multiplets that preserve the supersymmetry, specifically an $\\mathcal {N} = 1$ vector $(1,\\frac{1}{2})$ multiplet consisting of a gauge-boson and gaugino, and an $\\mathcal {N} = 1$ chiral $(\\frac{1}{2},0)$ multiplet consisting of a spin-$\\frac{1}{2}$ fermion and a complex scalar.", "Giving the graviton a non-zero mass, increases the (on-shell) degrees of freedom from two to five, introducing two vectors and one scalar.", "On the other hand, giving the gravitino mass results in two additional fermionic modes, corresponding to four on-shell degrees of freedom that are grouped by spin as $\\lbrace \\pm \\frac{3}{2}, \\pm \\frac{1}{2}\\rbrace $ .", "In the case of the massive spin-2 action, the vector modes completely decouple, while the scalar modes couple to the trace of the matter stress energy tensor.", "In the massless limit then, any coupled matter that has a trace-free stress energy sector won't suffer a discontinuity, while any matter stress energy tensor with a non-zero trace will suffer one.", "From the on-shell scattering amplitude perspective, this same phenomena can be observed by computing the scattering of scalars via a massive graviton and comparing the massless limit of this amplitude with the same amplitude constructed from a massless graviton.", "The supersymmetric analog of this statement is that the fermionic modes of the gravitino couple to the Dirac gamma-trace of the current $\\gamma _\\mu j^\\mu $ so in this case, we compare to amplitudes with non-zero gamma-trace." ], [ "The Supersymmetric vDVZ Discontinuity", "To begin, we should first clarify what is the analogue of the vDVZ discontinuity in the supersymmetric context.", "Towards this end, we will utilise the Stückelberg formalism, appropriately adapted.", "Consider then, the free massive Rarita-Schwinger action $S &= -\\int d^4x~e\\left(\\frac{1}{2}\\overline{\\Psi }_\\mu ^\\alpha \\gamma ^{\\mu \\rho \\nu }\\partial _\\rho \\Psi _{\\nu \\alpha } - \\frac{m}{2}\\overline{\\Psi }_\\mu ^\\alpha \\gamma ^{\\mu \\nu }\\Psi _{\\nu \\alpha } + \\overline{\\Psi }_\\mu ^\\alpha j^\\mu _\\alpha \\right) \\equiv -\\int d^4x~e~\\mathcal {L},$ where $\\alpha $ is a spinor index, $\\mu ,\\nu ,\\rho $ are Lorentz indices and $e$ is, as usual, the determinant of the frame field $e^{a}_{\\mu }(x)$ .", "Our $\\gamma $ -conventions read $\\gamma ^{\\mu \\rho \\nu } \\equiv i\\epsilon ^{\\mu \\sigma \\rho \\nu }\\gamma _5\\gamma _\\sigma , ~~~~~\\gamma ^{\\mu \\nu } \\equiv \\frac{i}{2}[\\gamma ^\\mu ,\\gamma ^\\nu ].$ Without the mass term, this action is invariant under $\\Psi ^{\\alpha }_\\mu \\longrightarrow \\Psi ^{\\alpha }_\\mu + \\partial _\\mu \\chi ^\\alpha ,~~~~~\\overline{\\Psi }^{\\alpha }_\\mu \\longrightarrow \\overline{\\Psi }^{\\alpha }_\\mu + \\overline{\\chi }^\\alpha \\overset{\\leftarrow }{\\partial }_\\mu ,$ for some spinor $\\chi ^\\alpha $ .", "This symmetry is broken by the mass term but can be restored if we introduce the supercovariant derivative $D_\\mu = \\partial _\\mu + \\frac{1}{2} m\\gamma _\\mu ,$ in terms of which the Lagrangian becomes $\\mathcal {L} = \\frac{1}{2}\\overline{\\Psi }_\\mu ^\\alpha \\gamma ^{\\mu \\rho \\nu }D_\\rho \\Psi _{\\nu \\alpha } + \\overline{\\Psi }_\\mu ^\\alpha j^\\mu _\\alpha ,$ after judicious use of the identity $\\epsilon ^{\\mu \\nu \\rho \\sigma }\\gamma _5\\gamma _\\nu = 2\\gamma ^\\mu \\gamma ^{\\rho \\sigma }$ .", "We can now introduce $\\chi $ as a (spinorial) Stückelberg field via the transformationThe factor of $\\frac{1}{\\sqrt{3}m}$ ensures a canonical fermionic kinetic term.", "$\\Psi _\\mu ^\\alpha \\longrightarrow \\Psi _\\mu ^\\alpha + \\frac{1}{\\sqrt{3}m}D_\\mu \\chi ^\\alpha .$ Under this transformation, the Lagrangian density varies as $\\delta \\mathcal {L} = -\\frac{\\sqrt{3}m}{2}\\left(\\overline{{\\Psi }}^\\alpha \\chi _\\alpha + \\overline{\\chi }^\\alpha {\\Psi }_\\alpha \\right) - \\overline{\\chi }^\\alpha ({\\partial } + m)\\chi _\\alpha + \\frac{1}{\\sqrt{6}}\\overline{\\chi }^\\alpha {j}_\\alpha \\,.$ Subsequently, taking the massless limit does not lead to the original massless Lagrangian since, $\\mathcal {L}^{massive}\\bigg |_{m\\longrightarrow 0} = \\mathcal {L}^{massless} + \\frac{1}{\\sqrt{6}}\\overline{\\chi }^\\alpha {j}_\\alpha .$ It is this that we identify as the SUSY equivalent of the vDVZ discontinuity, since any matter with a $\\gamma $ -traceless current will couple differently than matter with a non-vanishing $\\gamma $ -trace.", "As alluded to in the introduction, we wish to couple to two chiral multiplets: a scalar multiplet ($\\frac{1}{2},0$ ) and a vector multiplet $(1,\\frac{1}{2})$ .", "The corresponding vector-spinor currents are, by Noethers theorem, $j_{\\alpha }^\\mu [\\Phi ,\\psi ] &= \\left[i\\gamma ^\\nu \\partial _\\nu (\\phi _1 -i\\gamma _5\\phi _2) -m(\\phi _1 + i\\gamma _5\\phi _2)\\right]\\gamma ^\\mu \\psi _\\alpha \\\\j_{\\alpha }^\\mu [A,\\lambda ] &= \\gamma ^\\rho \\gamma ^\\nu \\gamma ^\\mu \\lambda _\\alpha F_{\\rho \\nu },$ where $\\Phi = \\phi _1 + i\\phi _2$ is a complex scalar, $\\psi _\\alpha $ a Majorana fermion (both with mass $m_\\Phi $ ), $\\lambda _\\alpha $ a massless photino and $F_{\\rho \\nu }$ the Maxwell tensor for photon $A_\\nu $ .", "These are both conserved, i.e.", "that $\\partial _\\mu j_{\\alpha }^\\mu [\\Phi ,\\psi ] = \\partial _\\mu j_{\\alpha }^\\mu [A,\\lambda ] = 0$ , however only one has a non-zero Dirac trace, e.g.", "${j}_{\\alpha }[\\Phi ,\\psi ] \\ne 0,~~~~~{j}_{\\alpha }[A,\\lambda ] = 0.$ While this formulation is clearly off-shell, it will inform the on-shell investigation to which we now turn." ], [ "$\\mathcal {N} = 1$ Supersymmetric Discontinuity", "Let's start by thinking about the supermultiplets.", "To ensure that we preserve the correct symmetries we have to let all the particles in the multiplet have the same mass.", "To this end we will draw on the vector- and scalar-multiplet currents as they are stated in the previous section.", "In the interest of self-containment, many of the techniques and conventions used throughout this chapter will be set up in this first section." ], [ "Bold Notation and the Stückelberg Decomposition", "As described in [6], we can significantly reduce the notational overhead that comes with incorporating $SU(2)$ little group indices in favour of bold-facing the massive particle spinors.", "Given our treatment of the action in section , it will be useful to perform a similar analysis at the level of the on-shell three-particle amplitudes.", "To this end, consider the coupling of a massless photon and massless fermion to a massive gravitino.", "The associated amplitude is given by $\\mathcal {M}_3^{\\lbrace IJK\\rbrace }[1^{-1/2},2^{-1},\\textbf {3}^{3/2}] = \\frac{\\kappa }{m}\\mathinner {\\langle {1\\textbf {3}}\\rangle }\\mathinner {\\langle {2\\textbf {3}}\\rangle }^2.$ Simply unbolding this expression does not yield the correct result, since taking the massless limit tells us that the spin $-3/2$ mode of the gravitino diverges.", "Instead, we need to write it in a form where we can unbold, i.e.", "$\\begin{split}\\mathcal {M}_3^{\\lbrace IJK\\rbrace }[1^{-1/2},2^{-1},\\textbf {3}^{3/2}]&=\\frac{\\kappa }{m^2} \\mathinner {\\langle {1}|}p_2\\vert \\textbf {3} ]\\langle 2\\textbf {3} \\rangle ^2\\\\&=-\\kappa \\frac{\\langle 12 \\rangle \\langle 2\\textbf {3} \\rangle ^2[ \\textbf {3}2 ]}{\\langle 2\\textbf {3}^{I} \\rangle [ \\textbf {3}_{I}2 ]}.\\end{split}$ Now, when we unbold, we find that the longitudinal spin $1/2$ mode of the gravitino is non-vanishing in the massless limit, since $\\mathcal {M}_3^{\\lbrace IJK\\rbrace }[1^{-1/2},2^{-1},\\textbf {3}^{3/2}]\\bigg |_{m\\longrightarrow 0} = -\\frac{\\kappa }{3}\\mathinner {\\langle {12}\\rangle }\\mathinner {\\langle {23}\\rangle },$ and with the inclusion of the relevant symmetry factor.", "This is consistent with the field theory treatment.", "On the other hand, we find that the $3/2$ mode does indeed survive, but for a different helicity choice of the fermion and photon $\\mathcal {M}_3^{\\lbrace IJK\\rbrace }[1^{+1/2},2^{-1},\\textbf {3}^{3/2}] = \\kappa \\frac{\\langle 2\\textbf {3} \\rangle ^3}{\\langle 12 \\rangle } ~~\\longrightarrow ~~ \\kappa \\frac{\\langle 23 \\rangle ^3}{\\langle 12 \\rangle }.$ This implies that the act of expanding three-paticle amplitudes into their various helicity components is the on-shell avatar of the Stückelberg decomposition, reflecting our result in section .", "This example manifests a subtlety in this approach; for the `unbolding' of massive amplitudes to be physically meaningful, we will often have to make sure that we have teased out as much of the explicit mass dependence as possible, a point that will be important for what follows.", "Another, perhaps more easily digested, example is the three-particle amplitude of a massive vector and two massless scalars.", "This amplitude is treated in detail in [6] by making appropriate choices for the massive spinor indices to reveal the underlying helicities in the massless limit.", "The amplitude can be written in the generic form $\\mathcal {A}^{\\lbrace IJ\\rbrace }[1^0,2^0,\\textbf {3}^1]\\propto \\frac{\\mathinner {\\langle {\\textbf {3}}|}p_1p_2\\mathinner {|{\\textbf {3}}\\rangle }}{m^2} + \\frac{\\mathinner {\\langle {\\textbf {3}}|}p_1\\vert \\textbf {3} ]}{m} + \\frac{[ \\textbf {3} \\vert p_1p_2\\vert \\textbf {3} ]}{m^2}.$ Naively unbolding any of these terms to take the massless limit appears to be divergent.", "However, using the fact that $m^2 = \\langle 12 \\rangle [ 12 ]$ it can be rewritten as $\\mathcal {A}^{\\lbrace IJ\\rbrace }[1^0,2^0,\\textbf {3}^1]\\propto \\frac{\\langle \\textbf {3}1 \\rangle \\langle 2\\textbf {3} \\rangle }{\\langle 12 \\rangle } + m\\frac{\\langle \\textbf {3}1 \\rangle [ 1\\textbf {3} ]}{\\langle 1\\textbf {3}^I \\rangle [ \\textbf {3}_I 1 ]} + \\frac{[ \\textbf {3}1 ][ 2\\textbf {3} ]}{[ 12 ]}.$ Now that the mass factors in the denominators have been taken care of we can simply unbold and retain the term with the appropriate helicity.", "The first term is clearly the $h_3=-1$ helicity mode, the last term the $h_3=+1$ helicity mode and the scalar mode vanishes in the massless limit as expected.", "We see that we can simply extract the appropriate modes of the vector in the massless limit by finding the explicit mass dependence of a general form of the amplitude and then simply unbolding." ], [ "Vector Mulitplet", "We begin by computing the amplitudes involving an $\\mathcal {N} = 1$ massless vector multiplet $(1,\\frac{1}{2})$ .", "This produces a three-particle vertex consisting of a gauge boson (a photon), a fermion (the photino) and a gravitino interacting with a coupling $\\kappa $ .", "We'll require the propagating gravitino to be massive in order to study the effects of taking the massless limit.", "The associated four-particle diagram is given by Fig.", "REF .", "Figure: Vector multiplet four-particleWe need only construct the left hand three-particle amplitude for all the possible helicity configurations.", "This is because the right hand three-particle amplitudes can be obtained by complex conjugation of eqs.", "(REF ) and (), and making the replacements $p_1\\longrightarrow p_3, ~p_2\\longrightarrow p_4$ .", "Throughout this article we will use the formalism developed in Ref.", "[6], including bold notation to highlight when a spinor represents a massive particle, suppressing the $SU(2)$ little group indices unless required for clarity.", "Using this formalism, we find that the possible three-particle amplitudes are $\\mathcal {M}_3^{\\lbrace JKL \\rbrace }[1^{+1/2},2^{-1},\\textbf {p}^{3/2}]&=\\frac{\\kappa }{m^2} \\langle 2\\textbf {p} \\rangle ^3 [ 12 ]=\\kappa \\frac{\\langle 2\\textbf {p} \\rangle ^3}{\\langle 12 \\rangle },\\\\\\mathcal {M}_3^{\\lbrace JKL \\rbrace }[1^{-1/2},2^{-1},\\textbf {p}^{3/2}]&=\\frac{\\kappa }{m} \\langle 1\\textbf {p} \\rangle \\langle 2\\textbf {p} \\rangle ^2.$ From this it is now simple to compute the two possible four-particle amplitudes by contracting the massive indices of the internal particle $\\begin{split}\\mathcal {M}_4[1^{+1/2},2^{-1},3^{-1/2},4^{+1}]&=\\mathcal {M}^{\\lbrace IJK\\rbrace }[1^{+1/2},2^{-1},\\textbf {p}^{3/2}]\\frac{1}{p^2+m^2} \\tilde{\\mathcal {M}}_{\\lbrace IJK\\rbrace }[3^{-1/2},4^{+1},-\\textbf {p}^{3/2}]\\\\&= -\\frac{\\kappa ^2}{t+m^2} \\frac{[ 14 ]}{[ 34 ]}\\mathinner {\\langle {2}|}p_1\\vert 4 ]^2 \\\\&= \\frac{[ 14 ]}{[ 34 ]}\\mathcal {M}_4[1^{0},2^{-1},3^0,4^{+1}],\\end{split}$ where $\\mathcal {M}_4[1^{0},2^{-1},3^0,4^{+1}]$ is the massive graviton mediated scalar-photon amplitude [1].", "This explicitly shows that the supersymmetric Ward identity is satisfied, as expected.", "This amplitude only has an explicit mass dependence in the propagator, and we can therefore easily take the massless limit.", "We want to compare the massless limit of the amplitude with the four-particle amplitude with an initially massless gravitino, which must be constructed from the three-particle amplitude $\\mathcal {M}_3[1^{+1/2},2^{-1},p^{-3/2}] = \\kappa \\frac{\\langle 2p \\rangle ^3}{\\langle 12 \\rangle }.$ Using this, we find that the massless four-particle amplitude is $\\mathcal {M}_4[1^{+1/2},2^{-1},3^{-1/2},4^{+1}]\\bigg |_{m=0}&= \\sum _{\\pm } \\mathcal {M}_3[1^{+1/2},2^{-1},p^{\\pm 3/2}] \\frac{1}{p^2} \\mathcal {M}_3[3^{-1/2},4^{+1},-p^{\\mp 3/2}]\\nonumber \\\\&=-\\frac{\\kappa ^2}{t} \\frac{[ 14 ]}{[ 34 ]}\\mathinner {\\langle {2}|}p_1\\vert 4 ]^2,$ in agreement with eq.", "(REF ).", "Explicitly, $\\mathcal {M}_4[1^{+1/2},2^{-1},3^{-1/2},4^{+1}]\\bigg |_{m\\longrightarrow 0} = \\mathcal {M}_4[1^{+1/2},2^{-1},3^{-1/2},4^{+1}]\\bigg |_{m=0}.$ If our only concern was reproducing the field-theory result in section we could well stop at this point, noting that this is the only chiral amplitude possible for this field configuration.", "However, we do not wish to be led by the field theory construction, and so with no a priori reason to discard the other helicity possibility, we have include it.", "To construct the non-chiral four-particle amplitude, notice that the three-particle amplitude in eq.", "() is symmetric in two of its indices and, summing over the internal $SU(2)$ indices, allows us to write $\\mathcal {M}_3^{\\lbrace JKL\\rbrace }\\mathcal {M}_{3\\lbrace JKL\\rbrace } = \\frac{1}{6}\\mathcal {M}_3^{JKL}\\mathcal {M}_{3\\lbrace JKL\\rbrace } = \\frac{1}{3}\\mathcal {M}_3^{JKL}(\\mathcal {M}_{3JKL}+\\mathcal {M}_{3KJL}+\\mathcal {M}_{3LKJ}),$ which in turn permits the four-particle amplitude to be written as $\\begin{split}\\mathcal {M}_4[1^{-1/2},2^{-1},3^{+1/2},4^{+1}]&=\\mathcal {M}_3^{\\lbrace IJK\\rbrace }[1^{-1/2},2^{-1},\\textbf {p}^{3/2}]\\frac{1}{p^2+m^2} \\tilde{\\mathcal {M}}_{3\\lbrace IJK\\rbrace }(3^{+1/2},4^{+1},-\\textbf {p}^{3/2})\\\\&= \\frac{\\kappa ^2}{3 m^2(p^2+m^2)}( \\mathinner {\\langle {1}|}p\\vert 3 ]\\mathinner {\\langle {2}|}p\\vert 4 ]^2+2\\mathinner {\\langle {1}|}p\\vert 4 ]\\mathinner {\\langle {2}|}p\\vert 3 ]\\mathinner {\\langle {2}|}p\\vert 4 ])\\\\&= -\\frac{\\kappa ^2}{3(p^2+m^2)}\\langle 12 \\rangle [ 34 ]\\mathinner {\\langle {2}|}p\\vert 4 ] + \\mathcal {O}(m^2)\\,,\\end{split}$ where, in the last line we have used the Schouten identityTo see that the second piece of the amplitude is $\\propto m^4$ , multiply by $[ 12 ]^2/[ 12 ]^2$ and use conservation of momentum.. Now we can easily take the massless limit to find $\\begin{split}\\mathcal {M}_4[1^{-1/2},2^{-1},3^{+1/2},4^{+1}]\\bigg |_{m\\longrightarrow 0}&=-\\frac{\\kappa ^2}{3t} \\langle 12 \\rangle [ 34 ]\\mathinner {\\langle {2}|}p\\vert 4 ] \\\\&= \\frac{1}{3}\\mathcal {M}_4[1^{-1/2},2^{-1},3^{+1/2},4^{+1}]\\bigg |_{m=0}\\end{split}$ and where $\\mathcal {M}_4[1^{-1/2},2^{-1},3^{+1/2},4^{+1}]\\bigg |_{m=0} = \\mathcal {M}_3[1^{-1/2},2^{-1},p^{+1/2}]\\mathcal {M}_3[3^{+1/2},4^{+1},-p^{-1/2}]/t\\,.$ From this it is clear that this helicity structure produces a contribution due to the spin-$\\frac{1}{2}$ mode of the gravitino but with an overall factor of a $1/3$ when compared with its massless counterpart." ], [ "Scalar Mulitplet", "For the scalar multiplet, we need to consider a massive gravitino mediated interaction of a massive fermion and a massive scalar.", "For simplicity, we will take the masses of the latter two to be the same.", "The corresponding four-particle diagram is given in Fig.", "REF .", "Figure: Scalar multiplet four-particleFollowing Ref.", "[6] the all-massive three-particle amplitude can be constructed as $\\begin{split}\\mathcal {M}_3^{I\\lbrace JKL\\rbrace }[\\textbf {1}^{1/2},\\textbf {2}^0,\\textbf {p}^{3/2}]&=g_1 \\langle \\textbf {1}\\textbf {p} \\rangle \\langle \\textbf {p}\\textbf {p} \\rangle \\\\&+g_2 (\\langle \\textbf {1}\\textbf {p} \\rangle \\mathinner {\\langle {\\textbf {p}}|}p_1p\\mathinner {|{\\textbf {p}}\\rangle }+\\langle \\textbf {p}\\textbf {p} \\rangle (\\mathinner {\\langle {\\textbf {1}}|}p_1p\\mathinner {|{\\textbf {p}}\\rangle }+\\mathinner {\\langle {\\textbf {p}}|}p_1p\\mathinner {|{\\textbf {1}}\\rangle }))\\\\&+g_3 (\\mathinner {\\langle {\\textbf {1}}|}p_1p\\mathinner {|{\\textbf {p}}\\rangle }+\\mathinner {\\langle {\\textbf {p}}|}p_1p\\mathinner {|{\\textbf {1}}\\rangle })\\mathinner {\\langle {\\textbf {p}}|}p_1p\\mathinner {|{\\textbf {p}}\\rangle }),\\end{split}$ where we have chosen to expand in a basis $(\\mathcal {O}^{ab} = p_1^{\\lbrace a\\dot{b}}p_{\\dot{b}}^{b\\rbrace }, \\epsilon ^{ab})$ and the coupling functionsThe `coupling functions' are functions of particle mass and the gravitational coupling $\\kappa $ only, with the mass dependence needing to be fixed by e.g.", "demanding the correct high-energy behaviour $g_i$ have an undetermined mass dependence that will be fixed shortly.", "To simplify this, note that since the external $SU(2)$ indices of the gravitino have to be symmetrised, any term that has a factor $\\langle \\textbf {p}^I\\textbf {p}^J \\rangle =m\\epsilon ^{IJ}$ (where $IJ$ are free indices) will vanish once symmetrised and can be ignored.", "Taking the external particle masses to be $m_1$ and the internal gravitino to have mass $m$ , reduces the above amplitude to $ \\begin{split}\\mathcal {M}_3^{I\\lbrace JKL\\rbrace }(\\textbf {1}^{1/2},\\textbf {2}^0,\\textbf {p}^{3/2})&=m(-g_2+g_3m^2)\\langle \\textbf {1}\\textbf {p} \\rangle \\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ]+2m^2m_1g_3[ \\textbf {1}\\textbf {p} ]\\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ].\\end{split}$ In getting to this point we have made liberal use of the identities in appendix .", "In order to fix the coupling functions $g_i$ , we can compare these amplitudes with the ones derived in limit where one particle becomes massless.", "Taking the external mass to zero then, we find $\\mathcal {M}_3^{I\\lbrace JKL\\rbrace }[\\textbf {1}^{1/2},\\textbf {2}^0,\\textbf {p}^{3/2}]\\bigg |_{m_1\\longrightarrow 0}={\\left\\lbrace \\begin{array}{ll}\\frac{\\kappa }{m} \\langle 1\\textbf {p} \\rangle \\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ] ,~~~h_1=-1/2\\\\\\frac{\\kappa }{m} [ 1\\textbf {p} ]\\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ] ,~~~h_1=+1/2,\\end{array}\\right.", "}$ which implies that the following limits must hold $\\lim _{m_1 \\longrightarrow 0} m(-g_2+g_3m^2) = \\frac{\\kappa }{m},~~~~~\\lim _{m_1\\longrightarrow 0}2m^2m_1g_3 = \\frac{\\kappa }{m}.$ Next, to isolate any $m_1$ dependence, take $m\\longrightarrow 0$ so that $\\mathcal {M}_3^{I\\lbrace JKL\\rbrace }[\\textbf {1}^{1/2},\\textbf {2}^0,\\textbf {p}^{3/2}]\\bigg \\vert _{m\\longrightarrow 0}\\simeq {\\left\\lbrace \\begin{array}{ll}\\kappa m_1 \\langle \\textbf {1}p \\rangle ,~~~h_p=-1/2,\\\\\\kappa \\frac{\\langle \\textbf {1}p \\rangle \\mathinner {\\langle {p}|}p_1\\vert \\xi ]}{[ p\\xi ]} ,~~~h_p=-3/2,\\\\\\kappa m_1 [ \\textbf {1}p ] ,~~~h_p=+1/2,\\\\\\kappa \\frac{[ \\textbf {1}p ]\\mathinner {\\langle {\\xi }|}p_1\\vert p ]}{\\langle \\xi p \\rangle } ,~~~h_p=+3/2\\end{array}\\right.", "}$ Now we can, for example, look at the $-3/2$ mode of the gravitino, choose $\\xi = \\eta $ and compare with the $I=J=K=1$ amplitude, $\\mathcal {M}_3^{I\\lbrace 111\\rbrace }[\\textbf {1}^{1/2},\\textbf {2}^0,\\textbf {p}^{3/2}] = \\kappa \\frac{\\langle \\textbf {1}p \\rangle \\mathinner {\\langle {p}|}p_1\\vert \\eta ]}{m} + \\mathcal {O}(m).$ Recognising that in the massless limit $m \\longrightarrow [p\\eta ]$ , implies that the limits in eq.", "(REF ) must hold.", "In other words, the amplitude has no explicit $m_1$ -dependence (although it will have implicit dependence on $m_{1}$ to recover e.g.", "the $h_p = \\pm 1/2$ amplitudes).", "Putting this together then, the three-particle amplitude must take the form $ \\begin{split}\\mathcal {M}_3^{I\\lbrace JKL\\rbrace }\\left[\\textbf {1}^{1/2},\\textbf {2}^0,\\textbf {p}^{3/2}\\right]&=\\frac{\\kappa }{m}(\\langle \\textbf {1}\\textbf {p} \\rangle +[ \\textbf {1}\\textbf {p} ])\\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ].\\end{split}$ With the three-particle amplitude in hand, we can now construct the four-particle amplitude.", "Given its lack of index symmetry however, the four-particle amplitude will have six distinct tensor structures and is particularly unwieldy.", "We will spare the reader the gory details.", "Suffice it to say that, with judicious use of the identities in appendix , it can be written as $\\mathcal {M}_4^{I_1I_3}\\left[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}\\right]&=&\\frac{\\kappa ^2}{6\\textbf {t}m^2}(\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ] - \\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ])(-6m^2(2p_1\\cdot p_3) +2 (2p\\cdot p_1)^2\\nonumber \\\\&+&4m^2m_1^2) -4mm_1^2(\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ] + \\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ])(2p\\cdot p_1)\\\\&+&4m(\\langle \\textbf {13} \\rangle + [ \\textbf {13} ])(2p\\cdot p_1)^2) +\\mathcal {O}(m).\\nonumber $ Now, using the fact that $2p\\cdot p_1=-2p\\cdot p_3= -m^2$ , we take the massless limit to get, $\\begin{split}\\mathcal {M}_4^{I_1I_3}\\left[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}\\right]\\bigg \\vert _{m\\longrightarrow 0}&=-\\frac{\\kappa ^2}{t}(\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ]-\\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ])(2p_1 \\cdot p_3 -\\frac{2}{3} (m_1^2))\\,.\\end{split}$ This in turn needs to be compared to the four-particle amplitude with a massless gravitino exchange, $\\mathcal {M}^{3/2}_4{}^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m=0}\\!\\!\\!\\!&=&\\sum _{\\pm 3/2}\\mathcal {M}_3^{I_1\\lbrace JKL\\rbrace }(\\textbf {1}^{1/2},\\textbf {2}^0,p^{\\pm 3/2})\\frac{1}{t}\\mathcal {M}_3^{I_3}{}_{\\lbrace JKL\\rbrace }(\\textbf {3}^{1/2},\\textbf {4}^0,-p^{\\mp 3/2})\\nonumber \\\\&=&\\frac{\\kappa ^2}{t}(\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ]\\frac{\\mathinner {\\langle {\\zeta }|}p_3 p p_1\\vert \\xi ]}{\\mathinner {\\langle {\\zeta }|}p\\vert \\xi ]}-\\mathrm {C.C.", "}),$ where $\\zeta $ and $\\xi $ are reference spinors.", "To facilitate this comparison, we choose $\\zeta =\\xi $ and utilise conservation of momentum and the Schouten identity to write the amplitude as $\\mathcal {M}^{3/2}_4{}^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m=0}&=&-\\frac{\\kappa ^2}{t}(\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ]-\\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ])(2p_1\\cdot p_3)\\,.$ To make a concrete comparison of (REF ) with its massless propagator counterparts we also need to compute the four particle amplitude with a massless fermion exchange.", "Using the three particle amplitudes given in [6] this is straightforwardly computed as $\\mathcal {M}^{1/2}_4{}^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m=0}&=&\\sum _{\\pm 1/2}\\mathcal {M}_3^{I_1\\lbrace JKL\\rbrace }(\\textbf {1}^{1/2},\\textbf {2}^0,p^{\\pm 1/2})\\frac{1}{t}\\mathcal {M}_3^{I_3}{}_{\\lbrace JKL\\rbrace }(\\textbf {3}^{1/2},\\textbf {4}^0,-p^{\\mp 1/2})\\nonumber \\\\&=&\\frac{\\kappa ^2}{t}(\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ]-\\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ])(m_1^2).$ We are now in a position to write the amplitude (REF ) in terms of its massless counterparts (REF ) and (REF ) as $\\mathcal {M}_4^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m\\longrightarrow 0} \\!\\!\\!\\!\\!\\!= \\mathcal {M}^{3/2}_4{}^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m=0} + \\frac{2}{3}\\mathcal {M}^{1/2}_4{}^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m=0}.$ From this expression, it is evident that a discontinuity should be expected in the scalar sector, in precisely the same form as from the field theory analysis.", "Furthermore, taking the external particle masses to zero and making a specific helicity choice shows that the four-particle amplitude satisfies the expected Ward identity $\\mathcal {M}_4[1^{-1/2},2^{0},3^{+1/2},4^{0}] = \\frac{\\mathinner {\\langle {12}\\rangle }}{\\mathinner {\\langle {32}\\rangle }}\\mathcal {M}_4[1^{0},2^{0},3^{0},4^{0}],$ where $\\mathcal {M}_4[1^{0},2^{0},3^{0},4^{0}]$ is the massless limit of eq.", "3.18 in Ref [1]." ], [ "Supersymmetry Breaking and the Discontinuity", "An interesting question that arises from our analysis above is whether this discontinuity persists below the supersymmetry breaking scale.", "Indeed, from the field theory perspective, none of the arguments for the existence of the discontinuity depend in any substantial way on whether the supersymmetry is broken or not, just that a massive spin-$\\frac{3}{2}$ propagator couples to a current with $j\\cdot \\gamma = 0$ which is then compared to one for which $j\\cdot \\gamma \\ne 0$ .", "Our on-shell analysis however provides a simple test of this hypothesis by considering multiplets with distinct masses, e.g.", "a massive scalar and fermion with masses $m_s \\ne m_f$ .", "The natural multiplets to consider in this case are, the vector multiplet with a massless photon and a fermion of mass $m_f$ and the scalar multiplet with a massive scalar and massive fermion with respective masses $m_s \\ne m_f$ .", "However there is a subtlety that requires some discussion.", "In the previous case with unbroken supersymmetry, the vector multiplet amplitudes only contain a single mass with which to constrain the coupling function, and while the scalar multiplet contains two distinct masses, there are well defined massless amplitudes to compare to (after taking appropriate massless limits), thereby allowing us to derive the correct mass structure.", "Breaking supersymmetry, on the other hand, results in even more masses that could form part of the coupling function.", "In this case, little group scaling and dimensional analysis alone are insufficient to constrain the masses.", "For example, in the scalar multiplet where only the fermion is massive, according to the one-massive formula given in Ref.", "[6], the amplitude $\\mathcal {M}_3^{I}[\\textbf {1}^{1/2},2^{0},p^{\\pm 3/2}] = 0$ .", "This is due to the requirement that $S+h_1-h_2$ and $S+h_2-h_1$ must both be positive, which is clearly not the case if $S = 1/2, h_1 = 0$ and $h_2 = \\pm 3/2$ .", "On the other hand, the fully massless amplitude does exist and is given by $\\mathcal {M}_3[1^{-1/2},2^{0},p^{-3/2}] = \\kappa \\frac{\\langle 1p \\rangle ^2\\langle 2p \\rangle }{\\langle 12 \\rangle }\\,.$ This is obviously not a limit of the one-massive case above, a fact that is also true if only the scalar is left massive.", "It seems then that the only consistent way to fix the mass dependence of the coupling function $g_i$ is to demand that every amplitude has a well defined massless limit and discarding it if it doesn't.", "To constrain such coupling functions, let's consider the two-massive-one-massless three-particle amplitude, taking care to retain the gravitino as one of the massive particles.", "We will compare this to the amplitude with only the gravitino massive $\\mathcal {M}_3^{\\lbrace IJK\\rbrace }[1^{-1/2},2^{0},\\textbf {p}^{3/2}] = \\frac{\\kappa }{m} \\langle 1\\textbf {p} \\rangle \\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ]$ since this does recover the massless amplitude in eq.", "(REF ).", "Demanding that we recover this amplitude from the two-massive amplitudes $\\begin{split}\\mathcal {M}_3^{I\\lbrace JKL\\rbrace }[\\textbf {1}^{1/2},2^{0},\\textbf {p}^{3/2}] &= \\frac{\\kappa }{m} (\\langle \\textbf {1}\\textbf {p} \\rangle \\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ] +[ \\textbf {1}\\textbf {p} ]\\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ])\\\\\\mathcal {M}_3^{\\lbrace JKL\\rbrace }[1^{-1/2},\\textbf {2}^{0},\\textbf {p}^{3/2}] &= \\frac{\\kappa }{m} \\langle 1\\textbf {p} \\rangle \\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ],\\end{split}$ constrains the three-massive particle amplitude to the form, $\\mathcal {M}_3^{I\\lbrace JKL\\rbrace }[\\textbf {1}^{1/2},\\textbf {2}^0,\\textbf {p}^{3/2}]=\\frac{\\kappa }{m}(\\langle \\textbf {1}\\textbf {p} \\rangle +[ \\textbf {1}\\textbf {p} ])\\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ].$ The construction of the four-particle amplitude now mirrors that outlined in the previous section exactly, with the result that $\\mathcal {M}_4^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]&=&\\frac{\\kappa ^2}{6tm^2}((\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ] - \\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ])(-6m^2(2p_1\\cdot p_3) +2 (2p\\cdot p_1)^2\\nonumber \\\\&+&4m^2m_f^2) -4mm_f^2(\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ] + \\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ])(2p\\cdot p_1)\\\\&+&4m(\\langle \\textbf {13} \\rangle + [ \\textbf {13} ])(2p\\cdot p_1)^2) +\\mathcal {O}(m).\\nonumber $ The difference between this expression and (REF ) is that $2p\\cdot p_1 = m^2 +m_f^2 -m_s^2 = -2p\\cdot p_3$ .", "Correspondingly, the amplitude (REF ) will have terms of order $\\mathcal {O}(m^{-1})$ and therefore has no consistent massless limit.", "This appears to be an artefact of constructing massive three-particle amplitudes in this formalism, and specifically fixing the mass structure of the arbitrary coupling functions.", "If this inconsistency were somehow resolved, we would expect the mass factor in the denominator to have an additive form, $m+m_f-m_s$ , which would allow the massless limit to be taken in the four-particle amplitude.", "That said, it is not clear to us how such a denominator could arise from a local quantum field theory with polynomial interactions.", "For example, an action containing a term of the form $\\sim \\phi {\\Psi }^\\alpha \\psi _\\alpha $ will have at most a single mass in the denominator coming from the Rarita-Schwinger polarization.", "In this sense, a mass dependence of $\\mathcal {O}(m^{-1})$ is to be expected and therefore the formalism does seemingly produce the correct amplitude, but one which apparently doesn't have well defined massless limits.", "It is entirely plausible that this problem is cured by a high energy mechanism, e.g.", "the Higgs, which does in some circumstances restore the high energy limit via the goldstone equivalence theorem.", "The gravitino-goldstino equivalence theorem shows that spontaneously broken symmetries involving longitudinally polarized gravitinos can have a well defined high energy behaviour via the goldstone equivalence [114].", "However, it is not clear to us that this will solve all of the issues raised in this section, and is at any rate beyond the scope of this article and so we leave this investigation to the future.", "Amplitudes for the vector multiplet exhibit a similar inconsistency in that the amplitude with a single massive photino does not reduce to the required massless limit.", "Specifically, $\\mathcal {M}_3^{I}[\\textbf {1}^{1/2},2^{-1},p^{-3/2}] = \\frac{\\kappa }{m_f}\\mathinner {\\langle {\\textbf {1}p}\\rangle }\\mathinner {\\langle {p2}\\rangle }^2$ cannot reproduce the amplitude $\\mathcal {M}_3[1^{-1/2},2^{-1},p^{-3/2}] = 0$ in the limit that $m_f\\rightarrow 0$ .", "Again we turn to the case where the gravitino is the only massive particle to fix the coupling function.", "The corresponding amplitude, $ M^{I \\lbrace JKL \\rbrace }[\\textbf {1}^{ 1/2},2^{-1},\\textbf {p}^{3/2}]&=\\frac{\\kappa }{m} ([ \\textbf {1}\\textbf {p} ]\\langle \\textbf {p}2 \\rangle ^2 + \\langle \\textbf {1}\\textbf {p} \\rangle \\langle \\textbf {p}2 \\rangle ^2),$ correctly reproduces all of the expected three-particle amplitudes in the relevant limits.", "From this, and with some straightforward but tedious simplification, we find the four-particle amplitude $\\mathcal {M}_4^{I_1I_3}[\\textbf {1}^{1/2},2^{-1},\\textbf {3}^{1/2},4^{-1}]&=&\\frac{\\kappa ^2}{3tm^2}(\\langle \\textbf {1}2 \\rangle [ \\textbf {3}4 ](-m^2-m_f^2-3u)+\\langle 2\\textbf {3} \\rangle [ \\textbf {1}4 ](3m_f^2+3m^2)\\nonumber \\\\&+&(\\langle \\textbf {13} \\rangle +[ \\textbf {13} ])\\mathinner {\\langle {2}|}p\\vert 4 ](3m_f-m))\\mathinner {\\langle {2}|}p\\vert 4 ].$ Again there is no consistent massless limit to be taken and a similar argument as in the scalar multiplet follows.", "Clearly, this issue requires further analysis." ], [ "Discussion", "Using the recently developed formalism for massive scattering amplitudes, we have shown in this article that the chiral scalar multiplet and the non-chiral vector multiplet of massive supergravity both produce discontinuous scattering amplitudes in the $m\\rightarrow 0$ limit, reproducing on-shell the field theory result of [110].", "That said, scattering amplitudes themselves are not observables.", "Indeed, the primary significance of the original vDVZ discontinuity of massive gravity lies in the emperical fact that the light bending angle in massive gravity deviates from observation by a factor of $\\frac{3}{4}$ , provided the amplitudes are normalised to recover Newtonian gravity.", "This argument is obviously less relevant in the context of supergravity.", "However, our results establish the discontinuity at the amplitude level as an on-shell avatar of the Stückelberg decomposition.", "Let's unpack this a little, beginning with our results from the supersymmetric case.", "In all the diagrams in this article, our conventions are that time moves upwards and external particles are considered outgoing.", "This results in a decided difference in the two amplitudes computed in the supersymmetric vector multiplet.", "The first that we computed in (REF ) describes the chiral multiplet.", "This amplitude does not have a discontinuity in the massless limit in that $\\mathcal {M}_4[1^{+1/2},2^{-1},3^{-1/2},4^{+1}]\\bigg |_{m\\longrightarrow 0}=\\mathcal {M}^{3/2}_4[1^{+1/2},2^{-1},3^{-1/2},4^{+1}]\\bigg |_{m=0}$ .", "The next one that we computed, (REF ) describes a non-chiral multiplet.", "This amplitude does indeed exhibit a discontinuity in the massless limit, $\\mathcal {M}_4[1^{-1/2},2^{-1},3^{+1/2},4^{+1}]\\bigg |_{m\\longrightarrow 0}=\\frac{1}{3}\\mathcal {M}^{1/2}_4[1^{-1/2},2^{-1},3^{+1/2},4^{+1}]\\bigg |_{m=0}$The superscript on the amplitude here denotes the spin of the massless propagator..", "Guided by the fact that the field theory is chiral, we need only consider that piece.", "It is worth noting that if our only source of information came from on-shell amplitude methods we would necessarily have to include the non-chiral part, resulting in an ambiguity in the realization of discontinuity.", "An interesting problem for the future would be to understand how to project out the chiral part of the amplitude.", "Moving on to the scalar multiplet, we find that $\\mathcal {M}_4^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m\\longrightarrow 0}&=&\\mathcal {M}^{3/2}_4{}^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m=0}\\\\&+& \\frac{2}{3}\\mathcal {M}^{1/2}_4{}^{I_1I_3}[\\textbf {1}^{1/2},\\textbf {2}^{0},\\textbf {3}^{1/2},\\textbf {4}^{0}]\\bigg \\vert _{m=0},$ manifesting the discontinuity anticipated from the field theory analysis.", "Next, we attempted to see the discontinuity with supersymmetry broken, endowing the external particle species with distinct masses.", "Sadly, we were unable to express the relevant four-particle amplitudes in a form in which the massless limit can be cleanly taken.", "This is due to how the structure of the coupling functions in the three-particle amplitudes is determined.", "Specifically, the amplitudes can be organised such that they reveal a mass dependence of the order $\\sim 1/m$ and is not finite in the massless limit while, for all of the massless limits to be well defined, we require a mass dependence of the form $1/(m+m_f-m_s)$ .", "We remain unsatisfied with this puzzle but leave its resolution for future work.", "Another noteworthy point in the non-supersymmetric case is that the vector multiplet amplitude (REF ) contains terms that correspond to both the chiral and non-chiral pieces as a result of the now massive fermion.", "The on-shell technology developed in [6] for massive particle scattering is both conceptually and computationally powerful, but like any new technology, it requires extensive beta-testing to iron out any bugs.", "This article details one such test.", "We set out to give an on-shell derivation of the spin-$\\frac{3}{2}$ analogue of the famous vDVZ discontinuity of massive gravity, expecting the calculation to be clean and unambiguous.", "We were met with several subtlties, some of which we were able to resolve and some, like how to treat symmetry breaking and chirality, we remain puzzled by.", "Clearly though, there is still much work that remains to be done." ], [ "Acknowledgements", "We would like to thank the anonymous referee of one of our previous papers for pointing us to this problem.", "JM is supported by the NRF of South Africa under grant CSUR 114599.", "DB and NM is supported by funding from the DST/NRF SARChI in Physical Cosmology.", "Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard." ], [ "Conventions and Notation", "In the interests of self-containment, we collect here some of our conventions for computing massive amplitudes that we use in the main text.", "Our convention for massive spinors can be summarized as: If no up-down massive spinor index pair is explicit on adjacent massive spinors they are considered to be contracted.", "For spinor indices we use the lower case Latin alphabet and for little-group indices we use the upper case Latin alphabet.", "Epsilon conventions and helicity basis: $\\epsilon _{ab} = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix}\\,,\\quad \\epsilon ^{ab} = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix}\\,,\\quad \\zeta ^{-I}= \\begin{bmatrix} 1\\\\0\\end{bmatrix}\\,,\\quad \\zeta ^{+I}= \\begin{bmatrix} 0\\\\1\\end{bmatrix}$ Mandelstam variables: $s=-(p_1 + p_4)^2\\,,\\quad u=-(p_1 + p_3)^2\\,,\\quad t=-(p_1 + p_2)^2$ Massless spinor relations: $p^{a\\dot{b}} = -\\mathinner {|{p}\\rangle }^{a}[ p \\vert ^{\\dot{b}}\\,,\\quad \\mathinner {|{-p}\\rangle } =-\\mathinner {|{p}\\rangle }\\,,\\quad \\vert -p ] =\\vert p ]$ Massive spinor relations: $p^{a\\dot{b}} &=& \\mathinner {|{\\textbf {p}}\\rangle }^{aI}{}_{I}[ \\textbf {p} \\vert ^{\\dot{b}} = -\\mathinner {|{p}\\rangle }^a[ p \\vert ^{\\dot{b}}-\\mathinner {|{\\eta _p}\\rangle }^a[ \\eta _p \\vert ^{\\dot{b}}\\,,\\quad p^2=\\mathrm {det}(p^{a\\dot{b}}) = -m_p^2\\,,\\\\\\langle p\\eta _p \\rangle &=& [ p\\eta _p ] = m_p \\quad \\mathinner {\\langle {\\textbf {p}}|}_a^Ip^{a\\dot{b}} = m [ \\textbf {p} \\vert ^{\\dot{b}I}\\,,\\quad [ \\textbf {p} \\vert ^{\\dot{a}I}p_{\\dot{a}b} = -m \\mathinner {\\langle {\\textbf {p}}|}_{b}^{ I}\\,,\\\\\\langle i|pp|j \\rangle &=& -m_p^2 \\langle ij \\rangle \\,,\\quad {}^I\\langle \\textbf {p}\\textbf {p} \\rangle ^J = m \\epsilon ^{IJ}\\quad {}_I[ \\textbf {p}\\textbf {p} ]_J = -m \\epsilon _{IJ}\\,,\\\\\\langle i\\textbf {p} \\rangle ^I{}_I\\langle \\textbf {p}j \\rangle &=& m\\langle ij \\rangle \\,,\\quad [ i\\textbf {p} ]^I{}_I[ \\textbf {p}j ] = m[ ij ]\\,,\\quad \\mathinner {|{\\textbf {p}}\\rangle }^{I}{}_{J}[ \\textbf {p} \\vert = -\\mathinner {|{\\textbf {p}}\\rangle }_{I}{}^{J}[ \\textbf {p} \\vert $" ], [ "Some Miscellaneous Identities", " Simplification of the general scalar multiplet amplitude where we symmetrize over the massive relevant massive indices $\\begin{split}\\langle \\textbf {p}^{\\lbrace I}\\textbf {p}^{J\\rbrace } \\rangle &=m\\epsilon ^{\\lbrace IJ\\rbrace }=0,\\\\\\mathinner {\\langle {\\textbf {1}}|}p_1p\\mathinner {|{\\textbf {p}}\\rangle }&= -m_f m [ \\textbf {1}\\textbf {p} ],\\\\\\mathinner {\\langle {\\textbf {p}}|}p_1p\\mathinner {|{\\textbf {1}}\\rangle }&=m m_f [ \\textbf {1}\\textbf {p} ] - (m^2+m_f^2-m_s^2)\\langle \\textbf {1}\\textbf {p} \\rangle ,\\\\\\mathinner {\\langle {\\textbf {p}}|}p_1p\\mathinner {|{\\textbf {p}}\\rangle }&=-m \\mathinner {\\langle {\\textbf {p}}|}p_1\\vert \\textbf {p} ].\\end{split}$ Here are some identities commonly used in the scalar multiplet amplitudes.", "We keep the masses of the three particles distinct for clarity.", "For the terms in the all massive three-particle amplitude that are not listed here, simply exchange $1\\rightleftarrows 3$ or where relevant complex conjugate.", "$\\begin{split}\\mathinner {\\langle {\\textbf {p}^I}|}p_1\\vert \\textbf {p}^J ][ \\textbf {p}_J \\vert p_3\\vert \\textbf {p}_I ]&= -m^2(2p_1\\cdot p_3)\\\\\\mathinner {\\langle {\\textbf {p}^I}|}p_1pp_3\\vert \\textbf {p}_I ]&= m^2(2p_1\\cdot p_3)+(2p\\cdot p_1)(2p\\cdot p_3)\\\\\\mathinner {\\langle {\\textbf {1}}|}pp_3pp_1p\\vert \\textbf {3} ]&= m^2\\mathinner {\\langle {\\textbf {1}}|}pp_3p_1\\vert \\textbf {3} ] + m^2 m_f \\langle \\textbf {1}\\textbf {3} \\rangle (2p\\cdot p_1) +\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ](2p\\cdot p_1)(2p\\cdot p_3)\\\\\\mathinner {\\langle {\\textbf {1}}|}pp_3pp_1\\mathinner {|{\\textbf {3}}\\rangle }&=m^2\\mathinner {\\langle {\\textbf {1}}|}p_3p_1\\mathinner {|{\\textbf {3}}\\rangle } - m_f \\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ](2p\\cdot p_3) + \\langle \\textbf {13} \\rangle (2p\\cdot p_3)^2 \\\\ \\mathinner {\\langle {\\textbf {1}}|}pp_3p_1p\\mathinner {|{\\textbf {3}}\\rangle }&=m^2\\mathinner {\\langle {\\textbf {1}}|}p_3p_1\\mathinner {|{\\textbf {3}}\\rangle } - m_f\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ](2p\\cdot p_1) + \\langle \\textbf {13} \\rangle (2p\\cdot p_1)^2 - m_f \\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ](2p.p_1)\\\\ \\mathinner {\\langle {\\textbf {1}}|}p_3pp_1p\\mathinner {|{\\textbf {3}}\\rangle }&=-m^2\\mathinner {\\langle {\\textbf {1}}|}p_3p_1\\mathinner {|{\\textbf {3}}\\rangle } +m_f\\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ](2p\\cdot p_1) - \\langle \\textbf {13} \\rangle (2p\\cdot p_1)^2\\\\ \\mathinner {\\langle {\\textbf {1}}|}p_3p_1p\\vert \\textbf {3} ]&=-m_f^2 \\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ] - m_f [ \\textbf {13} ](2p\\cdot p_3) + \\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ](2p_1\\cdot p_3)\\\\\\mathinner {\\langle {\\textbf {1}}|}p_3pp_1\\vert \\textbf {3} ]&=m_f^2\\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ] - m_f \\langle \\textbf {13} \\rangle (2p\\cdot p_1) - \\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ](2p_1\\cdot p_3) + m_f [ \\textbf {13} ](2p\\cdot p_3)\\\\\\mathinner {\\langle {\\textbf {1}}|}pp_3p_1\\vert \\textbf {3} ]&=-m_f^2\\mathinner {\\langle {\\textbf {3}}|}p\\vert \\textbf {1} ] + m_f \\langle \\textbf {13} \\rangle (2p\\cdot p_1) + \\mathinner {\\langle {\\textbf {1}}|}p\\vert \\textbf {3} ](2p_1\\cdot p_3)\\end{split}$" ] ]

[ [ "Contradistinguisher: A Vapnik's Imperative to Unsupervised Domain\n Adaptation" ], [ "Abstract A complex combination of simultaneous supervised-unsupervised learning is believed to be the key to humans performing tasks seamlessly across multiple domains or tasks.", "This phenomenon of cross-domain learning has been very well studied in domain adaptation literature.", "Recent domain adaptation works rely on an indirect way of first aligning the source and target domain distributions and then train a classifier on the labeled source domain to classify the target domain.", "However, this approach has the main drawback that obtaining a near-perfect alignment of the domains in itself might be difficult/impossible (e.g., language domains).", "To address this, we follow Vapnik's imperative of statistical learning that states any desired problem should be solved in the most direct way rather than solving a more general intermediate task and propose a direct approach to domain adaptation that does not require domain alignment.", "We propose a model referred Contradistinguisher that learns contrastive features and whose objective is to jointly learn to contradistinguish the unlabeled target domain in an unsupervised way and classify in a supervised way on the source domain.", "We achieve the state-of-the-art on Office-31 and VisDA-2017 datasets in both single-source and multi-source settings.", "We also notice that the contradistinguish loss improves the model performance by increasing the shape bias." ], [ "Introduction", "The recent success of deep neural networks for supervised learning tasks in several areas like computer vision, speech, and natural language processing can be attributed to the models trained on large amounts of labeled data.", "However, acquiring massive amounts of labeled data in some domains can be very expensive or not possible at all.", "Additionally, the amount of time required for labeling the data to use existing deep learning techniques can be very high initially for a new domain.", "This is known as cold-start.", "On the contrary, cost-effective unlabeled data can be easily obtained in large amounts for most new domains.", "So, one can aim to transfer the knowledge from a labeled source domain to perform tasks on an unlabeled target domain.", "To study this, under the purview of transductive transfer learning, several approaches like domain adaptation, sample selection bias, co-variance shift have been explored in recent times.", "Existing domain adaptation approaches mostly rely on domain alignment, i.e., align both domains so that they are superimposed and indistinguishable in the latent space.", "This domain alignment can be achieved in three main ways: [(a)] discrepancy-based methods [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], reconstruction-based methods [19], [20] and adversarial adaptation methods [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40].", "These domain alignment strategies of indirectly addressing the task of unlabeled target domain classification have three main drawbacks.", "[(i)] (i) The sub-task of obtaining a perfect alignment of the domain in itself might be impossible or very difficult due to large domain shifts (e.g., language domains).", "(ii) The use of multiple classifiers and/or GANs to align the distributions unnecessarily increases the complexity of the neural networks leading to over-fitting in many cases.", "(iii) Due to distribution alignment, the domain-specific information is lost as the domains get morphed.", "A particular case where the domain alignment and the classifier trained on the source domain might fail is that the target domain is more suited to classification tasks than the source domain with lower classification performance.", "In this case, it is advised to perform the classification directly on the unlabeled target domain in an unsupervised manner as domain alignment onto a less suited source domain only leads to loss of information.", "It is reasonable to assume that for the main objective of unlabeled target domain classification, one can use all the information in the target domain and optionally incorporate any useful information from the labeled source domain and not the other way around.", "These drawbacks push us to challenge the idea of solving domain adaptation problems without solving the general problem of domain alignment.", "In this work, we study unsupervised domain adaptation by learning contrastive features in the unlabeled target domain in a fully unsupervised manner with the help of a classifier simultaneously trained on the labeled source domain.", "More importantly, we derive our motivation from the Vapnik's imperative that motivated the statistical learning theory [41], [42].", "“When solving a given problem, try to avoid solving a more general problem as an intermediate step.” [42] In the context of domain adaptation, the desired problem is classification on the unlabeled target domain, and domain alignment followed by most standard methods is the general intermediate.", "Considering the various drawback of the domain alignment approach, in this paper, we propose a method for domain adaptation that does not require domain alignment and approach the problem directly.", "This work extends our earlier conference paper [43] in the following way.", "[(i)] We provide additional experimental results on more complex domain adaptation dataset Office-31 [44] which includes images from three different sources, AMAZON ($\\mathcal {A}$ ), DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ) categorized into three domains respectively with only a few labeled high-resolution images.", "We provide additional experimental results on the benchmark VisDA-2017 [45] dataset for unsupervised domain adaptation that includes synthetic and real-world images from 12 different classes.", "We provide several ablation studies and demonstrations that will provide insights into the working of our proposed method CUDA [43].", "Additionally, we observed that the proposed contradistinguish loss helps to learn high-level features related to the shapes of the objects in the image.", "We extend our algorithm to the case of multi-source domain adaptation and establish benchmark results on Office-31 [44] dataset and blackDigits datasets.", "Figure: Demonstration of difference in domain alignment and proposed method CUDA on the 2-dimensional blobs synthetic toy-dataset for domain distributions from popular scikit-learnscikit{ - }learn .We provide two example settings of different data distributions for domains D0D0 and D1D1 in both the directions of domain adaptation D0↔D1D0{ \\leftrightarrow }D1 separated by the vertical lines.In each setting, the top row corresponds to the domain alignment approach.", "The bottom row corresponds to the proposed method CUDA compared to their respective domain alignment in the top row in both D0↔D1D0{ \\leftrightarrow }D1 domain adaptation tasks.", "The yellow dotted lines indicate the domain alignment process to superimpose the target domain onto the source domain, thereby morphing both the domains.similarly, the two sub-columns indicate the experiments with swapped source and target domains.Unlike the domain alignment approach, where the classifier is learnt only on source domain, CUDA demonstrates the contradistinguisher jointly learnt to classify on both the domains.As seen above, swapping domains affect the classifier learnt in domain alignment because the classifier depends on the source domain.", "However, because of joint learning on both the domains simultaneously, contradistinguisher shows almost the same decision boundary irrespective of the source domain, i.e., irrespective of the direction of the domain adaptation, i.e., D0→D1D0{ \\rightarrow }D1 or D1→D0D1{ \\rightarrow }D0.", "(Best viewed in color.", ")A summary of our contributions in this paper is as follows.", "We propose a simple method Contradistinguisher for Unsupervised Domain Adaptation (CUDA) that directly addresses the problem of domain adaptation by learning a single classifier, which we refer to as Contradistinguisher, jointly in an unsupervised manner over the unlabeled target domain and in a supervised manner over the labeled source domain.", "Hence, overcoming the drawbacks of distribution alignment-based techniques.", "We formulate a `contradistinguish loss' to directly utilize unlabeled target domain and address the classification task using unsupervised feature learning.", "Note that a similar approach called DisCoder [47] was used for a much simpler task of semi-supervised feature learning on a single domain with no domain distribution shift.", "We extend our experiments to more complex domain adaptation datasets Office-31 [44] and VisDA-2017 [45].", "From our experiments, we show that by jointly training contradistinguisher on the source domain and the target domain distributions, we can achieve above/on-par results over several recently proposed domain adaptation methods.", "We also observed an improvement in the classification performance on VisDA-2017 [45] over the vanilla CUDA with the data augmentation.", "We further demonstrate our proposed method's simplicity and effectiveness by easily extending single-source domain adaptation to a more complex and general multi-source domain adaptation.", "We demonstrate the effectiveness of the multi-source domain adaptation extension by performing experiments on Office-31 [44] dataset and blackDigits datasets ( USPS ($us$) [48], MNIST ($mn$) [49], SVHN ($sv$) [50], MNIST-M ($mm$) [21], SYNNUMBERS ($sn$) [21]) in a multi-source setting.", "Apart from these real-world benchmark datasets, we also validate the proposed method using the synthetically created toy-datasets (Fig.", "REF ).", "From our toy-dataset experiments, we provide two main insights.", "[(i)] (6) CUDA does indeed address the classification directly on the target domain in a fully unsupervised way without the domain alignment.", "(7) Since the classification is done directly on the unlabeled target domain in a fully unsupervised manner, the target domain classification performance is not limited by the source domain classification performance, i.e., the irrespective of the domain is used as the labeled source domain and the unlabeled target domain, the performance is the respective domains are similar.", "In other words, swapping of the domains or the direction of the domain adaptation has little effect on the classification performance on each individual domain.", "The rest of this paper is structured as follows.", "Section discusses related works in domain adaptation.", "In Section , we elaborate on the problem formulation, neural network architecture used by us, loss functions, model training, and inference algorithms of our proposed method.", "Section deals with the discussion of the experimental setup, results and analysis on visual datasets.", "Finally, in Section , we conclude by highlighting the key contributions of CUDA.", "Related Work As mentioned earlier, almost all domain adaptation approaches rely on domain alignment techniques.", "Here we briefly outline three main techniques of domain alignment.", "[(a)] Discrepancy-based methods: Deep Adaptation Network (DAN) [1] proposes mean-embedding matching of multi-layer representations across domain by minimizing Maximum Mean Discrepancy (MMD) [51], [52], [53] in a reproducing kernel Hilbert space (RKHS).", "Residual Transfer Network (RTN) [2] introduces separate source and target domain classifiers differing by a small residual function along with fusing the features of multiple layers in a reproducing kernel Hilbert space (RKHS) to match the domain distributions.", "Joint Adaptation Network (JAN) [3] proposes to optimize Joint Maximum Mean Discrepancy (JMMD), which measures the Hilbert-Schmidt norm between kernel mean embedding of empirical joint distributions of source and target domain.", "Associative Domain Adaptation (ADA) [4] learns statistically domain invariant embeddings by associating the embeddings of the final fully-connected layer before applying softmax as an alternative to MMD loss.Maximum Classifier Discrepancy (MCD) [5] aligns source and target distributions by maximizing the discrepancy between two separate classifiers.", "Self Ensembling (SE) [6] uses mean teacher variant [54] of temporal ensembling [55] with heavy reliance on data augmentation to minimize the discrepancy between student and teacher network predictions.", "Variational Fair Autoencoder (VFAE) [7] uses Variational Autoencoder (VAE) [56] with MMD to obtain domain invariant features.", "Central Moment Discrepancy (CMD) [8] proposes to match higher-order moments of source and target domain distributions.", "Rozantsev et al.", "[9] propose to explicitly model the domain shift using two-stream architecture, one for each domain along with MMD to align the source and target representations.", "A more recent approach multi-domain Domain Adaptation layer (mDA-layer) [10], [11] proposes a novel idea of replacing standard Batch-Norm layers [57] with specialized Domain Alignment layers [12], [13] thereby reducing the domain shift by discovering and handling multiple latent domains.", "Geodesic Flow Subspaces (GFS/SGF) [14] performs domain adaptation by first generating two subspaces of the source and the target domains by performing PCA, followed by learning a finite number of the interpolated subspaces between source and target subspaces based on the geometric properties of the Grassmann manifold.", "In the presence of multi-source domains, this method is very effective as this identifies the optimal subspace for domain adaptation.sFRAME (sparse Filters, Random fields and Maximum Entropy) [15] models are defined as Markov random field model that model data distributions based on maximum entropy distribution to fit the observed data by identifying the patterns in the observed data.", "Transferrable Prototypical Networks (TPN) [16] propose to identify prototypes for each class in source and target domains that are close in the embedding space and minimize the distance between these prototypes for domain adaptation.", "Contrastive Adaptation Network (CAN) [17] uses MMD-loss for the feature encodings along with the heuristic clustering schema to selectively pick a subset of the high confidence image samples from the target domain.", "These samples are then utilized in training the classifier on the target domain instead of the entire target domain.", "blackMoment Matching for Multi-Source Domain Adaptation (M3SDA) [18] proposes to dynamically align multiple labeled source domains and the unlabeled target domain by matching the moments of the feature distributions.", "Reconstruction-based methods: Deep Reconstruction-Classification Networks (DRCN) [19] and Domain Separation Networks (DSN) [20] approaches learn shared encodings of source and target domains using reconstruction networks.", "Adversarial adaptation methods: Reverse Gradient (RevGrad) [21] or Domain Adversarial Neural Network (DANN) [22] uses domain discriminator to learn domain invariant representations of both the domains.", "Coupled Generative Adversarial Network (CoGAN) [23] uses Generative Adversarial Network (GAN) [58] to obtain domain invariant features used for classification.", "Adversarial Discriminative Domain Adaptation (ADDA) [24] uses GANs along with weight sharing to learn domain invariant features.", "Generate to Adapt (GTA) [25] learns to generate an equivalent image in the other domain for a given image, thereby learning common domain invariant embeddings.", "Cross-Domain Representation Disentangler (CDRD) [26] learns cross-domain disentangled features for domain adaptation.", "Symmetric Bi-Directional Adaptive GAN (SBADA-GAN) [27] aims to learn symmetric bidirectional mappings among the domains by trying to mimic a target image given a source image.", "Cycle-Consistent Adversarial Domain Adaptation (CyCADA) [28] adapts representations at both the pixel-level and feature-level over the domains.", "Moving Semantic Transfer Network (MSTN) [29] proposes a moving semantic transfer network that learns semantic representations for the unlabeled target samples by aligning labeled source centroids and pseudo-labeled target centroids.", "Conditional Domain Adversarial Network (CDAN) [30] conditions the adversarial adaptation models on discriminative information conveyed in the classifier predictions.", "Decision-boundary Iterative Refinement Training with a Teacher (DIRT-T) [32] and Augmented Cyclic Adversarial Learning (ACAL) [33] learn by using a domain discriminator along with data augmentation for domain adaptation.", "Deep Cocktail Network (DCTN) [35] proposes a k-way domain discriminator and category classifier for digit classification and real-world object recognition in a multi-source domain adaptation setting.", "Batch Spectral Penalization (BSP) [36] investigates the transferability and the discriminability of the features learnt using the standard adversarial domain adaptation techniques.", "Also, BSP proposes an additional batch spectral loss as it is observed that the transferable features learnt using adversarial domain adaptation result in the loss of the discriminability of the classifier.", "Transferable Normalization (TransNorm) [37] proposes a further improvement in transferability by replacing the normal batch-normalization layer with separate normalization layers for source and target domain input batches.", "Adversarial Tight Match (ATM) [38] proposes to combine the adversarial training with discrepancy metric by introducing a novel discrepancy metric Maximum Density Divergence (MDD) to minimize inter-domain divergence and maximize the intra-class density.", "Certainty based Attention for Domain Adaptation (CADA) [39] propose to identify features that increase the certainty of the domain discriminator prediction to improve the classifier.", "Progressive Feature Alignment Network (PFAN) [40] proposes to align the discriminative features across domains progressively and effectively, via exploiting the intra-class variation in the target domain.", "Apart from these approaches, a slightly different method that has been recently proposed is called Tri-Training.", "Tri-Training algorithms use three classifiers trained on the labeled source domain and refine them for the unlabeled target domain.", "To be precise, in each round of tri-training, a target sample is pseudo-labeled if the other two classifiers agree on the labeling, under certain conditions such as confidence thresholding.", "Asymmetric Tri-Training (ATT) [59] uses three classifiers to bootstrap high confidence target domain samples by confidence thresholding.", "This way of bootstrapping works only if the source classifier has very high accuracy.", "In the case of low source classifier accuracy, target samples are never obtained to bootstrap, resulting in a bad model.", "Multi-Task Tri-training (MT-Tri) [60] explores the tri-training technique on the language domain adaptation tasks in a multi-task setting.", "All the domain adaptation approaches mentioned earlier have a common unifying theme: they attempt to morph the target and source distributions so as to make them indistinguishable.", "However, aligning domains is a complex task than the classification task.", "In this paper, we propose a completely different approach: instead of focusing on aligning the source and target distributions, we learn a single classifier referred to as Contradistinguisher, jointly on both the domain distributions using contradistinguish loss for the unlabeled target domain data and supervised loss for the labeled source domain data.", "Proposed Method: CUDA A domain $\\mathcal {D}_d$ is specified by its input feature space $\\mathcal {X}_d$ , the label space $\\mathcal {Y}_d$ and the joint probability distribution $p(\\mathbf {x}_d, \\mathbf {y}_d)$ , where $\\mathbf {x}_d{ \\in }\\mathcal {X}_d$ and $\\mathbf {y}_d{ \\in }\\mathcal {Y}_d$ .", "Let $\\left|\\mathcal {Y}_d \\right|{ = }K$ be the number of class labels such that $\\mathbf {y}_d{ \\in }\\lbrace 0, \\ldots , K{ - }1\\rbrace $ for any instance $\\mathbf {x}_d$ .", "Domain adaptation, in particular, consists of two domains $\\mathcal {D}_{s}$ and $\\mathcal {D}_{t}$ that are referred as the source and target domains respectively.", "We define $(\\mathbf {x}_s,\\mathbf {y}_s)$ as the random variables that denote the source domain input features and the corresponding source domain label.", "Similarly, we define $(\\mathbf {x}_t,\\mathbf {y}_t)$ as the random variables that denote the target domain input features and the corresponding target domain label.", "A common assumption in domain adaptation is that the input feature space as well as the label space remains unchanged across the source and the target domain, i.e., $\\mathcal {X}_s{ = }\\mathcal {X}_t{ = }\\mathcal {X}_d$ and $\\mathcal {Y}_s{ = }\\mathcal {Y}_t{ = }\\mathcal {Y}_d$ .", "Hence, the only difference between the source and target domain is input-label space distributions, i.e., $p(\\mathbf {x}_s,\\mathbf {y}_s){ \\ne }p(\\mathbf {x}_t,\\mathbf {y}_t)$ .", "This is referred to as domain shift in the domain adaptation literature.", "In particular, in an unsupervised domain adaptation, the training data consists of the labeled source domain instances ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ corresponding to the random variables $(\\mathbf {x}_s,\\mathbf {y}_s)$ and the unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variables $(\\mathbf {x}_t)$ .", "Observe that in the unsupervised domain adaptation setting, the target domain labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $(\\mathbf {y}_t)$ are unobserved/missing.", "Given a labeled data in the source domain, it is straightforward to learn a classifier by maximizing the conditional probability $p(\\mathbf {y}_s|\\mathbf {x}_s)$ over the labeled samples.", "However, the task at hand is to learn a classifier on the unlabeled target domain by transferring the knowledge from the labeled source domain to the unlabeled target domain.", "Overview Figure: Architecture of the proposed method CUDA with Contradistinguisher (Encoder and Classifier).", "Three optimization objectives with their respective inputs involved in training of CUDA: (i) Source supervised (), (ii) Target unsupervised () and Adversarial regularization ().The outline of the proposed method CUDA that involves contradistinguisher and the respective losses involved in training are depicted in Fig.", "REF .", "The objective of contradistinguisher is to find a clustering scheme using the most contrastive features on unlabeled target in such a way that it also satisfies the target domain prior over the labels, i.e., target domain prior enforcing.", "We achieve this by jointly training on labeled source samples in a supervised manner and unlabeled target samples in an unsupervised end-to-end manner by using a contradistinguish loss same as [47].", "This fine-tunes the classifier learnt from the source domain also to the target domain, as demonstrated in Fig.", "REF .", "The crux of our approach is the contradistinguish loss (REF ) which is discussed in detail in Section REF .", "Hence, the apt name contradistinguisher for our neural network architecture.", "Note that the objective of contradistinguisher is not the same as a classifier, i.e., distinguishing is not the same as classifying.", "Suppose there are two contrastive entities $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ , where $C_1, C_2$ are two classes.", "The aim of a classifier is to classify $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ , where to train a classifier one requires labeled data.", "On the contrary, the job of contradistinguisher is to just identify $e_1{ \\ne }e_2$ , i.e., contradistinguisher can classify $e_1{ \\in }C_1$ (or $C_2$ ) and $e_2{ \\in }C_2$ (or $C_1$ ) indifferently.", "To train contradistinguisher, we do not need any class information but only need unlabeled entities $e_1$ and $e_2$ .", "Using unlabeled target data, contradistinguisher is able to find a clustering scheme by distinguishing the unlabeled target domain samples in an unsupervised way.", "However, since the final task is classification, one would require a selective incorporation of the pre-existing informative knowledge required for the task of classification.", "This knowledge of assigning the label to the clusters is obtained by jointly training, thus classifying $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ .", "Supervised Source Classification For the labeled source domain instances ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ corresponding to the random variables $(\\mathbf {x}_s,\\mathbf {y}_s)$ of the labeled source domain, we define the conditional-likelihood of observing $\\mathbf {y}_s$ given $\\mathbf {x}_s$ as, $p_{\\theta }(\\mathbf {y}_{s}|\\textbf {x}_{s})$ , where $\\theta $ denotes the parameters of contradistinguisher.", "We estimate $\\theta $ by maximizing the conditional log-likelihood of observing the labels given the labeled source domain samples.", "Therefore, the source domain supervised objective to maximize is given as $\\mathcal {L}_{s}(\\theta )&{}={}&\\sum _{i=1}^{n_s}\\log (p_{\\theta }(\\mathbf {y}^{i}_{s}|\\textbf {x}^{i}_{s}))\\hspace{5.0pt}.$ Alternatively, one can minimize the cross-entropy loss, as used in practical implementation, instead of maximizing (REF ), i.e., $\\mathcal {L}_{ce}(\\theta )&{}={}&-\\sum _{i=1}^{n_s}\\sum _{k=0}^{K-1}{\\mathbb {1}[\\mathbf {y}^{i}_s{ = }k]}\\log (\\hat{\\mathbf {y}}^{ik}_{s})\\hspace{5.0pt},$ where $\\hat{\\mathbf {y}}^{ik}_{s}$ is the softmax output of contradistinguisher that represents the probability of class $k$ for the given sample $\\mathbf {x}^{i}_s$ .", "Unsupervised Target Classification For the unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $\\mathbf {x}_t$ of the unlabeled target domain, the corresponding labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $\\mathbf {x}_t$ are unknown/missing.", "Hence, a naive way of predicting the target labels is to directly use the classifier trained only with a supervised loss given in (REF ).", "While this approach may perform reasonably well in certain cases, it fails to deliver state-of-the-art performance.", "This may be attributed to the following reason: the support for the distribution $p_{\\theta }$ is defined only over the source domain instances $\\mathbf {x}_s$ and not the target domain instances $\\mathbf {x}_t$ .", "Hence, we model a non-trivial joint distribution $\\hat{q}_\\theta (\\textbf {x}_t,\\mathbf {y}_t)$ parameterized by the same $\\theta $ over target domain with only the target domain instances as the support as, $\\hat{q}_{\\theta }(\\textbf {x}_t,\\mathbf {y}_t)&{}={}&\\frac{p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t)}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t^{\\ell })}\\hspace{5.0pt}.$ However (REF ) is not a joint distribution yet because $\\sum _{\\ell =1}^{n_t}\\hat{q}_{\\theta }(\\textbf {x}^{\\ell }_{t}, \\mathbf {y}_{t}){ \\ne }p(\\mathbf {y}_t)$ , i.e., marginalizing over all ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ does not yield the target prior distribution, i.e., $p(\\mathbf {y}_t)$ .", "We modify (REF ) so as to include the marginalization condition.", "Hence, we refer to this as target domain prior enforcing.", "$q_{\\theta }(\\textbf {x}_t,\\mathbf {y}_t)&{}={}&\\frac{p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t) {p(\\mathbf {y}_t)}}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t^{\\ell })}\\hspace{5.0pt},$ where $p(\\mathbf {y}_t)$ denotes the target domain prior probability over the labels.", "Note that $q_\\theta (\\mathbf {x}_t, \\mathbf {y}_t)$ defines a non-trivial approximate of joint distribution over the target domain as a function of $p_\\theta $ learnt over source domain.", "The resultant unsupervised maximization objective for the target domain is given by maximizing the log-probability of the joint distribution $q_\\theta (\\mathbf {x}_t, \\mathbf {y}_t)$ which is $\\mathcal {L}_{t}(\\theta , {\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1})&{}={}&\\sum _{j=1}^{n_t}\\log (q_{\\theta }(\\textbf {x}^{j}_{t}, \\mathbf {y}^{j}_{t}))\\hspace{5.0pt},$ Next, we discuss how the objective given in (REF ) is solved, and the reason why (REF ) is referred to as contradistinguish loss.", "Since the target labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ are unknown, one needs to maximize (REF ) over the parameters $\\theta $ as well as the unknown target labels $\\mathbf {y}_t$ .", "As there are two unknown variables for maximization, we follow a two-step approach to maximize (REF ) as analogous to Expectation-Maximization (EM) algorithm [61].", "The two optimization steps are as follows.", "[(i)] Pseudo-label selection: We maximize (REF ) only with respect to the label $\\mathbf {y}_t$ for every $\\mathbf {x}_t$ by fixing $\\theta $ as $\\hat{\\mathbf {y}}^{j}_t&{}={}&\\operatornamewithlimits{arg\\,max}_{\\mathbf {y}^{j} \\in \\mathcal {Y}_t} \\frac{p_{\\theta }(\\mathbf {y}^{j}|\\textbf {x}^{j}_t) {p(\\mathbf {y}_t)}}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}^{\\ell }| \\textbf {x}_t^{\\ell })}\\hspace{5.0pt},$ Pseudo-labeling approach under semi-supervised representation learning setting has been well studied in [62] and shown equivalent to entropy regularization [63].", "As previously mentioned, the pseudo-label selection is analogous to E-step in the EM algorithm.", "Moreover, we derive the motivation from [47] that also uses pseudo-labeling in the context of semi-supervised representation learning.", "However, the proposed method addresses a more complex problem of domain adaptation in the presence of a domain shift.", "The pseudo-labeling essentially tries to cluster by assigning labels using source domain features of the classifier trained on the source domain.", "This is effectively is similar to the E-step in EM algorithm in spirit.", "Maximization: By fixing the pseudo-labels ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{n_t}_{j=1}$ from (REF ), we train contradistinguisher to maximize (REF ) with respect to the parameter $\\theta $ .", "$\\mathcal {L}_t(\\theta )&{}={}&\\sum _{j=1}^{n_t}\\log (p_{\\theta }(\\hat{\\mathbf {y}}^{j}_t|\\textbf {x}^{j}_t)) + \\sum _{j=1}^{n_t}\\log ({p(\\mathbf {y}_t)})\\nonumber \\\\&&{-}\\:\\sum _{j=1}^{n_t}\\log (\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\hat{\\mathbf {y}}^{j}_t| \\textbf {x}_t^{\\ell }))\\hspace{5.0pt}.$ Since the pseudo-labels from (REF ) are used for the maximization, this constrains the model to learn the features to further improve the current pseudo-labeling for the next iteration.", "This step is similar to the M-step in the EM algorithm in spirit.", "The first term, i.e., log-probability for a label $\\hat{\\mathbf {y}}^{j}_t$ given $\\mathbf {x}^{j}_t$ forces contradistinguisher to choose features to classify $\\mathbf {x}^{j}_t$ to $\\hat{\\mathbf {y}}^{j}_t$ .", "The second term is a constant, hence it has no effect on the optimization with respect to $\\theta $ .", "The third term is the negative of log-probability for the pseudo-label $\\hat{\\mathbf {y}}^{j}_t$ given all the samples $\\mathbf {x}^{\\ell }_t$ in the entire domain.", "Maximization of this term forces contradistinguisher to choose features to not classify all the other $\\mathbf {x}^{\\ell { \\ne }j}_t$ to selected pseudo-label $\\hat{\\mathbf {y}}^{j}_t$ except the given sample $\\mathbf {x}^{j}_t$ .", "This forces contradistinguisher to extract the most unique features of a given sample $\\mathbf {x}^{j}_t$ against all the other samples $\\mathbf {x}^{\\ell { \\ne }j}_t$ , i.e., most unique contrastive feature of the selected sample $\\mathbf {x}^{j}_t$ over all the other samples $\\mathbf {x}^{\\ell { \\ne }j}_t$ to distinguish a given sample from all others.", "The first and third term together in (REF ) enforce that contradistinguisher learns the most contradistinguishing features among the samples $\\mathbf {x}_t{ \\in }\\mathcal {X}_t$ , thus performing unlabeled target domain classification in a fully unsupervised way.", "We refer to the unsupervised target domain objective (REF ) as contradistinguish loss because of this contradistinguishing feature learning.", "Ideally, one would like to compute the third term in (REF ) using the complete target training data for each input sample.", "Since it is expensive to compute the third term over the entire $\\mathbf {x}_t$ for each individual sample during training, one evaluates the third term in (REF ) over a mini-batch.", "In our experiments, we have observed that the mini-batch strategy does not cause any problem during training as far as it includes at least one sample from each class, which is a fair assumption for a reasonably large mini-batch size of 128.", "For numerical stability, we use $\\log \\sum \\exp $ trick to optimize third term in (REF ).", "Adversarial Regularization To prevent contradistinguisher from over-fitting to the chosen pseudo labels during the training, we use adversarial regularization.", "In particular, we train contradistinguisher to be confused about the set of fake negative samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{n_f}_{j=1}$ by maximizing the conditional log-probability over the given fake sample such that the sample belongs to all $K(\\left|\\mathcal {Y}_d \\right|)$ classes simultaneously.", "The adversarial regularization objective is to multi-label the fake sample (e.g., a noisy image that looks like a cat and a dog) equally to all $K$ classes as labeling to any unique class introduces more noise in pseudo labels.", "This strategy is similar to entropy regularization [63] in the sense that instead of minimizing the entropy for the real target samples, we maximize the conditional log-probability over the fake negative samples.", "Therefore, we add the following maximization objective to the total contradistinguisher objective as a regularizer.", "$\\mathcal {L}_{adv}(\\theta )&{}={}&\\sum _{j=1}^{n_f}\\log (p_{\\theta }(\\hat{\\mathbf {y}}_t^{j}|\\hat{\\mathbf {x}}^{j}_{t}))\\hspace{5.0pt},$ for all $\\hat{\\mathbf {y}}^{j}_t{ \\in }\\mathcal {Y}_t$ .", "As maximization of (REF ) is analogous to minimizing the binary cross-entropy loss (REF ) of a multi-class multi-label classification task, in our practical implementation, we minimize (REF ) for assigning labels to all the classes for every sample.", "$\\mathcal {L}_{bce}(\\theta )&{}={}&-\\sum _{j=1}^{n_f}\\sum _{k=0}^{K-1}\\log (\\hat{\\mathbf {y}}^{jk}_{t})\\hspace{5.0pt},$ where $\\hat{\\mathbf {y}}^{jk}_{t}$ is the softmax output of contradistinguisher which represents the probability of class $k$ for the given sample $\\hat{\\mathbf {x}}^{j}_t$ .", "The fake negative samples $\\hat{\\mathbf {x}}_t$ can be directly sampled from, say, a Gaussian distribution in the input feature space $\\mathcal {X}_t$ with the mean and standard deviation of the samples $\\mathbf {x}_t{ \\in }\\mathcal {X}_t$ .", "For the language domain, fake samples are generated randomly, as mentioned above, because the input feature is the form of embeddings extracted from denoising auto-encoder with bag-of-words as the input.", "In case of visual datasets, as the feature space is high dimensional, the fake images $\\hat{\\mathbf {x}}_t$ are generated using a generator network $G_{\\phi }$ with parameter $\\phi $ that takes Gaussian noise vector $\\eta _t$ as input to produce a fake sample $\\hat{\\mathbf {x}}_t$ , i.e., $\\hat{\\mathbf {x}}_t = G_{\\phi }(\\eta _t)$ .", "Generator $G_{\\phi }$ is trained by minimizing kernel-MMD loss [64], i.e., a modified version of MMD loss between the encoder output $\\rho _{enc}(\\hat{\\mathbf {x}}_t)$ and $\\rho _{enc}(\\mathbf {x}_t)$ of $n_f$ fake images $\\hat{\\mathbf {x}}_t$ and $n_t$ real target domain images $\\mathbf {x}_t$ respectively.", "$\\mathcal {L}_{gen}(\\phi )&{}={}&\\frac{1}{n_f^{2}}\\sum _{i=1}^{n_f}\\sum _{j=1}^{n_f} k(\\rho _{enc}(\\hat{\\mathbf {x}}_t^{i}),\\rho _{enc}(\\hat{\\mathbf {x}}_t^{j}))\\nonumber \\\\&&{+}\\:\\frac{1}{n_t^{2}}\\sum _{i=1}^{n_t}\\sum _{j=1}^{n_t} k(\\rho _{enc}(\\mathbf {x}_t^{i}),\\rho _{enc}(\\mathbf {x}_t^{j}))\\nonumber \\\\&&{-}\\:\\frac{2}{n_t n_f}\\sum _{i=1}^{n_f}\\sum _{j=1}^{n_t} k(\\rho _{enc}(\\hat{\\mathbf {x}}_t^{i}),\\rho _{enc}(\\mathbf {x}_t^{j})),$ where $k(x,x^{\\prime }) = e^{-\\gamma \\left\\Vert x-x^{\\prime }\\right\\Vert ^{2}}$ is the Gaussian kernel.", "Note that the generator's objective is not to generate realistic images but to generate fake noisy images with mixed image attributes from the target domain.", "This reduces the effort of training powerful generators, which is the focus in adversarial based domain adaptation approaches [25], [26], [27], [28], [29] used for domain alignment.", "Algorithm REF and REF list steps involved in CUDA training and inference, respectively.", "[t!]", "$b{ = }batch\\_size$ , $epochs{ = }max\\_epoch$ , $n_{batch}{ = } number\\ of\\ batches$ $\\theta $ /* parameter of contradistinguisher */${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ , ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ target domain prior $p(\\mathbf {y}_t)$ is known use $p(\\mathbf {y}_t)$ for the contradistinguish loss (REF ) /* target domain prior enforcing */ compute $p(\\mathbf {y}_t)$ assuming $p(\\mathbf {y}_t) = p(\\mathbf {y}_s)$ /* fair assumption as most datasets are well balanced */ $epoch = 1$ to $epochs$ $batch = 1$ to $n_{batch}$ sample a mini-batch ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{b}_{i=1}$ , ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{b}_{j=1}$ compute $\\mathcal {L}_{s}(\\theta )$  (REF ) using ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{b}_{i=1}$ /* source supervised loss */ compute ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{b}_{j=1}$  (REF ) using ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{b}_{j=1}$ /* pseudo label selection step */ compute $\\mathcal {L}_{t}(\\theta )$  (REF ) fixing ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{b}_{j=1}$ /* maximization step */ /* steps REF and REF together optimize unsupervised contradistinguish loss (REF ) */ adversarial regularization is enabled Generator $G_\\phi $ is used get fake samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ from Gaussian noise vectors ${\\lbrace \\eta ^{j}_t\\rbrace }^{b}_{j=1}$ using $G_\\phi $ , compute $\\mathcal {L}_{gen}(\\phi )$ (REF ) /* generator training */ get fake samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ by random sampling in the input feature space $\\mathcal {X}_t$ compute $\\mathcal {L}_{adv}(\\theta )$  (REF ) using ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ /* fake samples are assigned to all classes equally */ combine losses in steps REF ,REF ,REF and REF to compute gradients using back-propagation update $\\theta $ using gradient descent /* and $\\phi $ if $G_\\phi $ is used */ CUDA Training [t!]", "${\\lbrace \\mathbf {x}^{i}_{test}\\rbrace }^{n_{test}}_{i=1}$ /* input test samples */${\\lbrace \\hat{\\mathbf {y}}^{i}_{test}\\rbrace }^{n_{test}}_{i=1}$ /* predicted labels */ $i = 1$ to $n_{test}$ predict label as $\\hat{\\mathbf {y}}^{i}_{test}{}={}\\operatornamewithlimits{arg\\,max}_{\\mathbf {y}\\in \\mathcal {Y}_t} p_{\\theta }(\\mathbf {y}|\\textbf {x}^{i}_{test})$ CUDA Inference Extension to Multi-Source Domain Adaptation Here, we argue that our proposed method CUDA has an implicit advantage in dealing with multi-source domain adaption problems over the techniques based on domain alignment.", "In a multi-source adaption setting, domain alignment methods need to consider the domain-shift between a source and a target domain and consider the domain-shift between the multiple source domains.", "Therefore, domain alignment methods are required to solve the even more complex intermediate problem of aligning multiple source and target domain distributions in addition to the complex intermediate problem of source and target domain alignment to deal with the multi-source domain adaptation problems.", "However, as the proposed method does not depend on the domain alignment, the extension to multi-source in the proposed method is very simple.", "As our main focus is to perform unsupervised learning directly on the target domain, the model obtained using is better generalized to the target domain and reduces overfitting on the source, which usually results in a negative transfer.", "We believe that this is one of the main advantages of addressing the domain adaptation by performing the primary task of target domain classification rather than the intermediate task of domain alignment.", "We propose a simple extension our proposed method to perform multi-source domain adaptation in the following manner.", "Let us suppose we are given with $R$ source domains ${\\lbrace s_1,\\ldots ,s_R\\rbrace }$ , consisting of labeled training data $({\\lbrace (\\mathbf {x}^{i}_{s_1}, \\mathbf {y}^{i}_{s_1})\\rbrace }^{n_{s_1}}_{i=1}, \\ldots , {\\lbrace (\\mathbf {x}^{i}_{s_R}, \\mathbf {y}^{i}_{s_R})\\rbrace }^{n_{s_R}}_{i=1})$ and unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ .", "We compute the source supervised loss for the $r^{th}$ source domain using (REF ), i.e., $\\mathcal {L}_{s_r}(\\theta )$  (REF ) with ${\\lbrace (\\mathbf {x}^{i}_{s_r}, \\mathbf {y}^{i}_{s_r})\\rbrace }^{n_{s_r}}_{i=1}$ training data.", "We further compute the total multi-source supervised loss as $\\mathcal {L}_{s_{total}}(\\theta ) = \\sum _{r=1}^{R}\\mathcal {L}_{s_r}(\\theta ).$ We replace $\\mathcal {L}_{s}(\\theta )$  (REF ) in the total optimization objective with $\\mathcal {L}_{s_{total}}(\\theta )$  (REF ) in step REF of Algorithm REF .", "It should be noted that the unsupervised loss for the target domain is still unmodified irrespective of the number of source domains.", "We experimentally demonstrate the efficacy of the proposed multi-source domain adaptation extension on Office-31 [44] and Digits datasets [48], [49], [50], [21].", "Experiments For our domain adaptation experiments, we consider both synthetic and real-world datasets.", "Under synthetic datasets, we experiment using 2-dimensional blobs with different source and target domain probability distributions to demonstrate the effectiveness of the proposed method under different domain shifts.", "Under real-world datasets, we consider only the complex, high-resolution Office-31 [44] and VisDA-2017 [45] object classification datasets for our experiment as the low-resolution datasets are already addressed in our conference paper CUDA: Contradistinguisher for Unsupervised Domain Adaptation (CUDA) [43].", "We have published our python code for all the experiments at https://github.com/sobalgi/cuda.", "black Table: Details of visual domain adaptation datasets.Table REF provides details on the visual datasets used in our experiments.", "We also experiment and report the results of our ablation study carried out with different combinations of the three optimization objectives with their respective domains as the inputs involved in CUDA training: [(i)] source supervised loss: $ss$ described in (REF ), source/target unsupervised loss: $su/tu$ described in (REF ), source/target adversarial regularization loss: $sa/ta$ described in (REF ).", "$ss$ indicates the minimum target domain test accuracy that can be attained with a chosen contradistinguisher neural network by training only using the labeled source domain.", "Any improvement over $ss$ using CUDA (i.e., combination of $su/tu/sa/ta$ ) indicates effectiveness of CUDA as the chosen contradistinguisher neural network is fixed.", "Experiments on Synthetic Toy-dataset Figure: Contour plots show the probability contours along with clear decision boundaries on different toy-dataset settings trained using CUDA.", "(source domain: ×\\times , target domain: ++, class 0: blue, class 1: red.)", "(Best viewed in color.", ")We validate our proposed method by performing experiments on synthetically created simple datasets that model different source and target domain distributions in a 2-dimensional input feature space using different blobs of source-target domain orientations and offsets (i.e., domain shift).", "We create blobs for source and target domains with 4000 samples using standard $scikit{ - }learn$  [46] as indicated in Fig.", "REF and REF .", "We further evenly split these 4000 data-points into equal train and test sets.", "Each of the splits consists of the same number of samples corresponding to both the class labels.", "The main motivation of the experiments on toy-dataset is to understand and visualize the behavior of the proposed method under some typical domain distribution scenarios and analyze the performance of  CUDA.", "$Blobs$ toy-dataset plots in Fig.", "REF shows clear comparisons of the classifier decision boundaries learnt using CUDA over domain alignment approaches.", "The top row in Fig.", "REF corresponds to the domain alignment classifier trained only on the labeled source domain, i.e., $ss$ .", "However, the bottom row in Fig.", "REF corresponds to contradistinguisher trained using the proposed method CUDA with labeled source and unlabeled target domain, i.e., $ss{ + }tu{ + }ta$ .", "Fig.", "REF demonstrates the classifier learnt using CUDA on the synthetic datasets with different complex shapes and orientations of the source and target domain distributions for the input data.", "Fig.", "REF and REF indicate the simplest form of the domain adaptation tasks with similar orientations in source and target domain distributions.It is important to note that the prior enforcing used in pseudo-label selection is the reason such fine classifier boundaries are observed, especially in Fig.", "REF ,REF and REF -REF .", "Fig.", "REF and REF represent more complex configurations of source and target domain distributions that indicate the hyperbolic decision boundaries jointly learnt on both the domains simultaneously using a single classifier without explicit domain alignment.", "Similarly, Fig.", "REF represents a complex configuration of source and target domain distributions that indicates an elliptical decision boundary.", "These simulated experiments points to some significant inner workings of our approach CUDA.", "These are the two main takeaways from the toy-dataset experiments.", "[(i)] The non-necessity of the domain alignment in the form of distribution distance metric minimization or data augmentation.", "In the case of these toy-datasets, it is not possible to perform any form of data augmentation, unlike some of the visual domain adaptation tasks, because the data is directly available in the form of encoded features that cannot be easily data augmented through standard heuristics.", "In such a case, it is necessary to realize a generic approach applicable to multiple modalities of the input, e.g., similar to the toy-dataset in language domain adaptation tasks.", "The features are presented in the form of word2vec/doc2vec, and no data augmentation is possible.", "Fig.", "REF and REF provide some interesting observations.", "Here, we can observe that the classes in the source domain are overlapping, resulting in less than 100% classification on the source domain, which in turn results in less than 100% classification on the target domain when considering domain alignment approaches.", "However, CUDA does not try to morph the target domain onto the source domain by directly classifying on the target domain resulting in a perfect classification.", "Since the classification is done directly on the unlabeled target domain in a fully unsupervised manner, the target domain classification performance is not limited by the source domain classification performance, i.e., the irrespective of the domain is used as the labeled source domain and the unlabeled target domain, the performance is the respective domains are similar.", "In other words, swapping of the domains or the direction of the domain adaptation has little effect on the classification performance on each individual domain.", "Experiments on Real-world Datasets In our previous work [43], we have demonstrated the effectiveness of CUDA in real-world domain adaptation on low-resolution visual datasets and language datasets.", "In contrast to low-resolution visual datasets, we consider the complex, high-resolution Office-31 [44] and VisDA-2017 [45] object classification datasets for domain adaptation.", "In addition to the single-source domain adaptation experiments, we also extend CUDA to Office-31 [44] and Digits datasets [48], [49], [50], [21].", "Office-31 Dataset Figure: Illustrations of samples from all the three domains of high resolution Office-31  dataset with one instance per each class from every domain (column {1,4,7,10}: 𝒜\\mathcal {A}, {2,5,8,11}: 𝒟\\mathcal {D}, {3,6,9,12}: 𝒲\\mathcal {W}).", "(Best viewed in color.", ")Figure: Illustrations of samples from all the three data-splits of VisDA-2017  dataset with one instance per each class from every domain ({row 1}: 𝒱 syn \\mathcal {V}_{syn} source domain synthetic images (training set), {row 2}: 𝒱 real \\mathcal {V}_{real} target domain real-world images (validation set), {row 3}: 𝒱 real \\mathcal {V}_{real} target domain real-world images (testing set)).", "It should be noted that unlike the Office-31 dataset and other standard benchmark domain adaptation datasets discussed in , most of the real-world images in the target domain of the VisDA-2017 dataset contains multiple true labels, which are only annotated with only one of the multiple labels.", "(Best viewed in color.", ")In high-resolution visual datasets, we consider Office-31 [44] dataset for our experiments.", "Unlike low-resolution visual datasets, here, we have only a few hundreds of training samples that make this an even more challenging task.", "Office objects: Office-31 [44] dataset consists of high resolution images of objects belonging to 31 classes obtained from three different domains AMAZON ($\\mathcal {A}$ ), DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ).", "Fig.", "REF shows illustrations of the images from all the three above mentioned domains of the Office-31 [44] dataset.", "AMAZON ($\\mathcal {A}$ ) domain consists of synthetic images with clear white background.", "DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ) domains consist of real-world images with noisy background and surroundings.", "We consider all possible six combinatorial tasks of domain adaptation involving all the three domains, i.e., $\\mathcal {A}{ \\leftrightarrow }\\mathcal {D}$ , $\\mathcal {A}{ \\leftrightarrow }\\mathcal {W}$ and $\\mathcal {D}{ \\leftrightarrow }\\mathcal {W}$ .", "Compared to low-resolution visual datasets, Office-31 [44] dataset domain adaptation tasks have increased complexity due to the small number of training images.", "Unlike low-resolution visual datasets, the high-resolution Office-31 [44] dataset does not have separate pre-defined train and test splits.", "Since we do not use any labels from the target domain during training, we report ten-crop test accuracy on the target domain by summing the softmax values of all the ten crops of the image and assign the label with maximum aggregate softmax value for the given image as in CDAN [30] in Table REF .", "To further alleviate the lack of a large number of training samples, pre-trained networks such as ResNet-50 [65] and ResNet-152 [65] were used to extract 2048 dimensional features from high-resolution images similar to CDAN [30].", "Since the images are not well centered and have a high resolution, we use the standard ten-crop of the image to extract features from the same images during training and testing, also similar to CDAN [30].", "The use of pre-trained models leads to two choices of training, [(i)] Fine-tune the pre-trained model used as feature extractor along with the final classifier layer: This requires careful selection of several hyper-parameters such as learning rate, learning rate decay, batch size, etc., to fine-tune the network to the current dataset while preserving the ability of the pre-trained network.", "We observed that fine-tuning also depends on the loss function used for training [66], which in our case, the use of contradistinguish loss greatly affected the changes in the pre-trained model as it is trained only using cross-entropy loss.", "Fine-tuning is also computationally expensive and time-consuming as each iteration requires computing gradients of all the pre-trained model parameters.", "Fix the pre-trained model and only train the final classifier layer: Alternative to fine-tuning is to fix the pre-trained model and use it only as a feature extractor.", "This approach has multiple practical benefits such as, [(a)] The computational time and cost of fine-tuning the parameters of the pre-trained model are alleviated.", "Since the feature extractor is fixed, it requires only once to extract and store the features locally instead of extracting the same features every iteration.", "Hence reducing the training time and the GPU memory as it is only required to train the final classifier.", "VisDA-2017 Dataset The VisDA-2017 dataset consists of two domains, (i) synthetic and (ii) real, with three predefined data splits.", "Fig.", "REF indicates the samples from all the 12 classes of the three data splits.", "The three predefined data splits in the VisDA-2017 dataset are as follows.", "[(i)] Training set: This split includes 152,397 labeled synthetic images obtained using 2D renderings of 3D models from different angles and different lighting conditions.", "This split is considered as a labeled source domain for training.", "Validation set: This split includes 55,388 real-world images obtained from a curated subset of MS COCO [67] dataset.", "This split is considered an unlabeled target domain training set, and this is used during the training without labels.", "Testing set: This split includes 72,372 real-world images obtained from YouTube Bounding Boxes [68] dataset.", "This split is considered as the target domain testing set used for evaluation and to report the results.", "Analysis of Experimental Results on Real-world Datasets Figure: Row 1 and 2: t-SNE  plots for embeddings from the output of contradistinguisher with samples from Office-31  dataset as input corresponding to the highest mean accuracy setting ss+tu+su+tass{ + }tu{ + }su{ + }ta indicated in Table  for single-source domain adaptation using ResNet-152  as the fixed encoder.Row 3: t-SNE  plots for embeddings from the output of contradistinguisher corresponding to the samples from Office-31  dataset in high-resolution visual tasks after applying softmax trained with CUDA with ResNet-50  as the encoder in a multi-source domain adaptation setting as indicated in Table .We can observe the clear class-wise clustering among all the 31 classes in the Office-31  datasets.We achieve high accuracies in spite of having only a few hundred training samples in each domain.", "(Best viewed in color.", ")Table: Target domain accuracy (%) on high resolution Office-31  dataset containing three domains.CUDA corresponds to our best results obtained with the best hyper-parameter settings.ssss: source supervised (), tutu: target unsupervised (), susu: source unsupervised (), sasa: source adversarial regularization () and tata: target adversarial regularization () represents different training configurations.Table: Target domain accuracy (%) on high resolution Office-31  dataset under multi-source domain adaptation setting by combining two domains into a single source domain and the remaining domain as the target domain with ResNet-50  as the encoder.CUDA corresponds to our best results obtained with the best hyper-parameter settings.ssss: source supervised (), tutu: target unsupervised (), susu: source unsupervised (), sasa: source adversarial regularization () and tata: target adversarial regularization () represents different training configurations.Table: Target domain accuracy reported on the test set (%) on all 5 combinations of Digits datasets under multi-source domain adaptation setting.Figure: The t-SNE plots of the unseen test set samples corresponding to the CUDA result in Table .", "The t-SNE plots show clear clustering of all the 10 classes in Digits datasets distinctively.", "(Best viewed in color.", ")Figure: The t-SNE plots of CUDA/CUDA * ^{*} shows the clear clustering of all the twelve classes of VisDA-2017 distinctively compared to the t-SNE plots of BSP/CAN.The t-SNE plots of CUDA/CUDA * ^{*} represent some important visual semantics of the image embeddings obtained from contradistinguisher in the following manner.", "(i) The vehicular classes such as `bus', `car', `train', and `truck' can be seen clustered closely as semantically these classes are similar to each other (region bounded in red).", "(ii) The two-wheeler classes such as `bicycle' and `motorcycle' are clustered closely as these are semantically similar to each other compared to vehicular classes that are clustered exactly opposite (region bounded in green).", "(iii) Irrespective of the approach used, there is always confusion between `knife' and `skateboard' classes.", "This confusion between `knife' and `skateboard' classes represented in the confusion matrices, which is also clearly seen in the t-SNE plots as well, can be attributed to the nature of images of these classes in the dataset on close observation (region bounded in blue).", "(iv) The remaining classes such as `aeroplane', `horse', `person' and `plant' can be seen clustered independently and distinctively as these classes have almost no visual semantic similarities to one another.", "(Best viewed in color.", ")Table: Results on VisDA-2017 dataset reproduced from the current state-of-the-art method BSP, CAN and our proposed method CUDA reported on both the validation set and test set.", "We report all the evaluation metrics such as precision, recall, and accuracy, unlike BSP/CAN, where the recall scores are mistakenly reported as accuracy.", "CUDA * ^{*} represents the results reproduced using vanilla CUDA with the data augmentation and target domain clustering similar to CAN for a fair comparison of the effect of the CUDA over CAN.The results reported for BSP, CAN, and CUDA/CUDA * ^{*} are from our own best reproduction from the original source code.Table: Total classification accuracy (%) on VisDA-2017 dataset reported on both the validation set and test set.", "The results from JAN, GTA, CDAN and TransNorm are reported from TransNorm .The results reported for BSP, CAN and CUDA/CUDA * ^{*} are from our own best reproduction from the original source code.CUDA * ^{*} represents the results reproduced using vanilla CUDA with the data augmentation and target domain clustering similar to CAN for a fair comparison of the effect of the CUDA over CAN.Figure: We indicate few samples that are misclassified by the contradistinguisher in the following subcaption format `original_label|predicted_label'.In most cases, the original ground truth labels are dubious, and the predicted labels make more sense realistically.Subplots (6), (7), (11), (12), (14), (18), (23) and (25) shows that the object is identified based on the shape and not if the object is present only in the foreground.", "This indicates that the contradistinguisher makes the predictions based on the clearly visible shapes and not the presence of the object in the foreground/background.The visualization of the features responsible for the respective predicted outcome indicates the shape bias as mostly the features are detected as edges corresponding to the shape of the object in the image.", "This shows the importance of shape bias to achieve high performance in transfer learning and domain adaptation tasks.Office-31 Single-Source Domain Adaptation Results We report the standard ten-crop accuracy on the target domain images as reported by several state-of-the-art domain adaptation methods [30], [25], [3].", "Since there are no explicit test split specified in the dataset and no labels are used from the target domain during training, it is common to report ten-crop accuracy considering the whole target domain.", "In Table REF , we report accuracies obtained by fine-tuning ResNet-50 [65] using the learning rate schedule followed in CDAN [30] and also without fine-tuning ResNet-50 [65].", "Apart from fixed ResNet-50 [65], we also report accuracies with fixed ResNet-152 [65] in Table REF for comparison.", "Fig.", "REF -REF indicate the t-SNE [69] plots of the softmax output after aggregating the ten-crop of each image corresponding to training configuration $ss{ + }tu{ + }su{ + }ta$ reported in Table REF .", "Fig.", "REF reports the t-SNE [69] plots of the training setting using ResNet-152 [65] encoder with the highest mean accuracy of all the six domain adaptation tasks.", "We clearly observe that CUDA outperforms several state-of-the-art methods that also use ResNet-50 [65] and even further surpasses by using ResNet-152 [65] encoder with CUDA.", "Among the three domains in Office-31 [44] dataset, $\\mathcal {A}$ can be considered as a well-curated synthetic dataset with clear background and $\\lbrace \\mathcal {D},\\mathcal {W}\\rbrace $ as an uncurated real-world dataset with noisy background and surroundings.", "We report the six domain adaptation tasks in the order of their complexity from low to high as, [(i)] Fig.", "REF and REF indicate highest accuracies because of similar real-world to real-world domain adaptation task, Fig.", "REF and REF indicate moderately high accuracies because of synthetic to real-world domain adaptation task and Fig.", "REF and REF indicate the lowest accuracies among all the six tasks because of real-world to synthetic domain adaptation task.", "Comparing CUDA with $ss$ in Tables REF and REF , we can see significant improvements in the target domain test accuracies due to the use of contradistinguish loss (REF ) demonstrating the effectiveness of contradistinguisher.", "As our method is mainly dependent on the contradistinguish loss (REF ), we observed further improved results by experimenting with better neural networks, e.g., using ResNet-152 over ResNet-50 along with our contradistinguish loss (REF ).", "From our ablations study in Table REF , we observe the effect of selection of ImageNet pre-trained ResNet-50 and ResNet-152 models on the domain adaptation with similar implications with the work [70].", "In general, one can always obtain better results irrespective of the approach by using better/deeper pre-trained models and/or data augmentation.", "However, since our main aim is to isolate, observe and benchmark only the true effect of different benchmark approaches, in our experiments, we maintain all the other parameters such as pre-trained neural network/data augmentation similar except for the core idea of the approaches and report our results both on Office-31 and VisDA-2017 datasets.", "Multi-Source Domain Adaptation Results We also extend the experiments to multi-source domain adaptation on the Office-31 [44] and Digits datasets [48], [49], [50], [21].", "In Table REF , we can clearly observe that in $\\mathcal {A}{ + }\\mathcal {D}{ \\rightarrow }\\mathcal {W}$ task, multi-source domain adaptation provides better results than their respective best single source domain adaptation experiments.", "However in case of $\\mathcal {D}{ + }\\mathcal {W}{ \\rightarrow }\\mathcal {A}$ and $\\mathcal {W}{ + }\\mathcal {A}{ \\rightarrow }\\mathcal {D}$ , the multi-source domain adaptation improves over $ss$ , it underperforms compared to best single source domain adaptation task.", "This can be attributed to the fact that the model tends to overfit on the source domains resulting in a negative transfer.This negative transfer behavior is also prevalent in other multi-source domain adaptation approaches since all the other multi-source domain adaptation methods also underperform compared to their best single source domain adaptation results, as reported in Table REF .", "Fig.", "REF -REF indicates t-SNE [69] plots for embeddings from the output of contradistinguisher corresponding to the samples from Office-31 [44] dataset after applying softmax trained with CUDA with ResNet-50 [65] as the encoder in a multi-source domain adaptation setting.", "We can observe the best results when the target domain is one of the real-world domain, i.e., $\\mathcal {D}$ and $\\mathcal {W}$ .", "It was consistently observed that domain adaptation tasks with synthetic domain $\\mathcal {A}$ as the target domain to be the most complex tasks of all the domain adaptation tasks across all the domain adaptation methods.", "Similarly, Table REF presents the results of multi-source domain adaptation on Digits datasets against benchmark approaches.", "In Fig.", "REF , we see the t-SNE plots on the test set depicting clear class-wise clustering that indicates the efficacy of CUDA single-source to multi-source extension.", "VisDA-2017 Single-Source Domain Adaptation Results For experiments on the VisDA-2017 dataset, we consider the most recent state-of-the-art benchmark domain adaptation approaches BSP [36] and CAN [17].", "Like BSP and CAN, we use the same neural network architecture with Imagenet pre-trained ResNet-101 with contradistinguish loss for training.", "BSP and CAN report the evaluation metric of accuracy in their papers.", "However, on reproducing the results with BSPhttps://github.com/thuml/Batch-Spectral-Penalization and CANhttps://github.com/kgl-prml/Contrastive-Adaptation-Network-for-Unsupervised-Domain-Adaptation to set the baseline for comparison, we noticed that the results reported had the following inconsistencies.", "[(i)] The results reported in the paper as accuracy in actual were the class-wise recall scores.", "The most standard procedure in machine learning is to report experimental performances on the test split, which is unseen during the training.", "However, in BSP and CAN, the results are reported on the validation set, which is used as the unlabeled target domain training set.", "Since the VisDA-2017 dataset has pre-defined splits for evaluation, reporting the results on the validation set used during the training does not indicate these models' generalizing capability, which is the most important aspect one would base an evaluation.", "We correct the above misreporting by reproducing the results from BSP and CAN and report all the relevant metrics for both validation and test splits of the VisDA-2017 dataset.", "Apart from these, we also validate our results on the official challenge evaluation portalhttps://competitions.codalab.org/competitions/17052#results by submitting the results of our approach CUDA on the VisDA-2017 dataset.", "In Fig.", "REF , we report the t-SNE plots reproduced from the current state-of-the-art unsupervised domain adaptation approach BSP [36] and CAN [17], in comparison with our approach CUDA/CUDA$^*$ on both the pre-defined validation split (seen target domain training set) and testing split (unseen target domain testing set) of the VisDA-2017 dataset.", "The results reported as CUDA corresponds to the CUDA experiments without any data augmentation using the BSP source code as the baseline to keep all the parameters the same for a fair comparison with BSP.", "Similarly, the results reported as CUDA$^*$ corresponds to the CUDA experiments with data augmentation and clustering of high confidence target domain samples using the CAN source code as the baseline to keep all the parameters same for a fair comparison with CAN.", "We can see the classwise clusters in BSP/CUDA are narrower compared to CAN/CUDA$^*$ , which are broader due to the use of data augmentation.", "Data augmentation helps modify/broaden the data distribution aiding in an improvement over the vanilla approaches without data augmentation.", "We further validate the results best results from our approach, i.e., CUDA$^*$ by submitting to the official challenge evaluation leaderboard.", "In Table REF we compare the per-class precision, recall, and accuracies on both the pre-defined validation set and test set of the VisDA-2017 against the results reproduced from the BSP/CAN.", "In Table REF we compare the total classification accuracies on both the pre-defined validation set and test set of the VisDA-2017 dataset against different benchmark methods.", "The results in Tables REF and REF indicate the superior performance of the proposed method CUDA/CUDA$^*$ over the current state-of-the-art domain alignment approaches BSP/CAN on both the pre-defined validation set and a test set of VisDA-2017 dataset.", "Even though the validation set and test set belong to the real-world domain, there is an inherent domain shift between them as both the data splits are collected from two different datasets, i.e., MS COCO [67] and YouTube Bounding Boxes ,[68] respectively.", "Results from CUDA indicate a better generalization to real-world domain as the scores in validation and test sets are closer compared to other approaches on the VisDA-2017 dataset.", "We can also observe that the t-SNE plot of CUDA in Fig.", "REF and REF clearly shows the visual semantics captured between the classes of images in the VisDA-2017 dataset.", "Apart from setting CUDA as the solid baseline for VisDA-2017, we further put a conscious effort to carefully investigate the reasons for the misclassification using the contradistinguisher to check if we can further improve the results.", "In most misclassified cases, we have observed that the labels predicted by CUDA appeared to be correct in comparison to the ground truth label of the dataset.", "In Fig.", "REF , we present some of these instances where the predicted label is more close to the real label than the ground truth.", "We can explain this misclassification as a limitation of the VisDA-2017 dataset in the following way.", "The misclassification observed in Fig.", "REF is due to the fact that the images in the VisDA-2017 dataset consist of objects belonging to more than one of the twelve classes, i.e., the images in the dataset consists of multiple labels for a single image, but the dataset only records one of these several true labels.", "As the assumed task for domain adaptation is single-label multi-class We see this as a limitation of the VisDA-2017 dataset compared other benchmark domain adaptation datasets such as Office-31 or the low-resolution visual datasets demonstrated in our conference paper [43].", "It is necessary that the datasets be consistent in the sense that each image has a unique label corresponding to it so that during the evaluation, there is no ambiguity between the original label and the predicted label from the trained model.", "The presence of this ambiguity in the dataset classification would then lead to observing the true evaluation metrices resulting in improper benchmarking for any given approach.", "However, in the case of the VisDA-2017 dataset, the predicted label from the model cannot be considered as a wrong label as it contains the object of the predicted label in the image.", "We believe that this is one of the reasons for the overall low performance apart from the complexity of the VisDA-2017 dataset compared to other visual datasets.", "It should also be observed that CUDA identifies the most distinguishing/prominent and assigns the label irrespective of the position (foreground/background) of the object as indicated in some of the subplots in Fig.", "REF .", "These misclassified cases indicates one of the strong drawback/limitation of VisDA-2017 dataset compared to other visual datasets, i.e., VisDA-2017 dataset has image samples with multiple true labels instead of a unique label for each image sample.", "Since the images might contain multiple true classes for an image, ideally all these true labels are to be associated with the image in the dataset to rightly evaluate any trained model for its efficacy.", "Because we perform single-label multi-class classification, predicting any one of the true labels of the image should be considered as right for the evaluation metric.", "However, this is not the case as the dataset does not record all the true labels for the images.", "So, if one plans to rightly use this dataset, all the true labels are to be annotated for each of the images in the dataset or use other benchmark datasets such as DomainNet/LSDAC (Large Scale Domain Adaptation Challenge) dataset [18] that alleviates the problem of multi-labels of VisDA-2017 dataset as DomainNet dataset only consist of single true label per each image in the dataset, resulting in correct evaluation without the issue of misclassification we have indicated above for the VisDA-2017 dataset.", "Apart from analyzing the limitation of the VisDA-2017 dataset, we also analyze the nature of the feature representations learnt by contradistinguisher.", "In order to visualize the features that prompted the predicted label, we use Captumhttps://captum.ai/tutorials/Resnet_TorchVision_Interpret, an open-source, extensible library for model interpretability built on PyTorchhttps://pytorch.org/ [71].", "We use gradient-based attribution to compute the integrated gradients for a given image using the predicted label.", "We obtain the high-level features or the saliency maps [72] for the given image.", "In terms of high-level features in a given image, one can imagine features such as shape, color, texture, size, etc.", "to be the features that help in predicting the classifier outcome.", "Out of all these features, the most natural and basic feature influencing the outcome is observed to be the shapes of the objects.", "Extensive research materials in psychology such as [73], [74], [75], [76], [77] have indicated that human babies and adults tend to utilize shapes than color/material/texture to assign a word label to the given object.", "This particular phenomenon is widely termed as `shape bias' in the literature.", "However, recently, it was shown that the ImageNet pre-trained models possess a texture bias over shape bias [78].", "To improve the shape bias, [78] propose a new modified dataset called `Stylized-ImageNet' to overcome the texture bias.", "By increasing the shape bias, [78] demonstrated improved performance and robustness.", "Since we use the ImageNet pre-trained ResNet-101 as a feature extractor, it is necessary to understand the nature of extracted features from the input images.", "Unlike `Stylized-ImageNet', in domain adaptation tasks, one cannot always expect to get such a curated dataset with ground truth labels on the target domain for each task.", "Instead, it might be easy and desirable to change the loss function that enhances the shape features with the same training dataset.Surprisingly, in our observations, we find that the features learnt by the classifier indicate the high-level features or the saliency maps [72] representing the shape of the object in the image.", "The contradistinguish loss is formulated and optimized in such a way that the features extracted are most unique and contrastive for a given image in comparison to other images in the dataset.", "This consequently is observed as the features corresponding to shapes in the form of silhouette in the feature visualizations in Fig.", "REF as each object posses a unique shape as it's most contrasdistinguishing character, i.e., the character which is most discriminative and unique to the given image.", "Concluding Remarks In this paper, we have proposed a direct approach to solve the problem of unsupervised domain adaptation that is different from the standard distribution alignment approaches.", "In our approach, we jointly learn a Contradistinguisher on the source and target domain distribution in the same input-label feature space using contradistinguish loss for unsupervised target domain to identify contrastive features.", "We have shown that contrastive learning overcomes the need and drawbacks of domain alignment, especially in tasks where domain shift is very high (e.g., language domains) and data augmentation techniques cannot be applied.", "Due to the inclusion of prior enforcing in the contradistinguish loss, the proposed unsupervised domain adaptation method CUDA could incorporate any known target domain prior to overcoming the drawbacks of skewness in the target domain, thereby resulting in a skew-robust model.", "We validated the efficacy of CUDA by experimenting on the synthetically created toy-dataset.", "We further demonstrated the simplicity and effectiveness of our proposed method by performing multi-source domain adaptation on Office-31 and Digits datasets to consistently outperform other multi-source domain adaptation approaches.", "We have also tested the proposed method CUDA on the recent benchmark visual domain adaptation datasets such Office-31 and VisDA-2017 classification datasets and demonstrated above/on-par results with the state-of-the-art approaches.", "We further analyzed the nature of the feature representation learnt using contradistinguish loss to identify the features related to the shapes that influence the predicted outcome.", "As the features related to shapes are learnt, we observed that it helps improving the performance and robustness of the trained model as the model is not biased to colors/textures in the images.", "We concluded that learning and improving shape bias is one of the keys to achieve ideal transfer learning and domain adaptation.", "Acknowledgments The authors would like to thank the Ministry of Human Resource Development (MHRD), Government of India, for their generous funding towards this work through the UAY Project: IISc 001.", "The authors thank Tejas Duseja for helping the authors with setting up some experiments.", "The authors would also like to thank anonymous reviewers for providing their valuable feedback that helped in improving the manuscript.", "[Figure: NO_CAPTION [Figure: NO_CAPTION" ], [ "Related Work", "As mentioned earlier, almost all domain adaptation approaches rely on domain alignment techniques.", "Here we briefly outline three main techniques of domain alignment.", "[(a)] Discrepancy-based methods: Deep Adaptation Network (DAN) [1] proposes mean-embedding matching of multi-layer representations across domain by minimizing Maximum Mean Discrepancy (MMD) [51], [52], [53] in a reproducing kernel Hilbert space (RKHS).", "Residual Transfer Network (RTN) [2] introduces separate source and target domain classifiers differing by a small residual function along with fusing the features of multiple layers in a reproducing kernel Hilbert space (RKHS) to match the domain distributions.", "Joint Adaptation Network (JAN) [3] proposes to optimize Joint Maximum Mean Discrepancy (JMMD), which measures the Hilbert-Schmidt norm between kernel mean embedding of empirical joint distributions of source and target domain.", "Associative Domain Adaptation (ADA) [4] learns statistically domain invariant embeddings by associating the embeddings of the final fully-connected layer before applying softmax as an alternative to MMD loss.Maximum Classifier Discrepancy (MCD) [5] aligns source and target distributions by maximizing the discrepancy between two separate classifiers.", "Self Ensembling (SE) [6] uses mean teacher variant [54] of temporal ensembling [55] with heavy reliance on data augmentation to minimize the discrepancy between student and teacher network predictions.", "Variational Fair Autoencoder (VFAE) [7] uses Variational Autoencoder (VAE) [56] with MMD to obtain domain invariant features.", "Central Moment Discrepancy (CMD) [8] proposes to match higher-order moments of source and target domain distributions.", "Rozantsev et al.", "[9] propose to explicitly model the domain shift using two-stream architecture, one for each domain along with MMD to align the source and target representations.", "A more recent approach multi-domain Domain Adaptation layer (mDA-layer) [10], [11] proposes a novel idea of replacing standard Batch-Norm layers [57] with specialized Domain Alignment layers [12], [13] thereby reducing the domain shift by discovering and handling multiple latent domains.", "Geodesic Flow Subspaces (GFS/SGF) [14] performs domain adaptation by first generating two subspaces of the source and the target domains by performing PCA, followed by learning a finite number of the interpolated subspaces between source and target subspaces based on the geometric properties of the Grassmann manifold.", "In the presence of multi-source domains, this method is very effective as this identifies the optimal subspace for domain adaptation.sFRAME (sparse Filters, Random fields and Maximum Entropy) [15] models are defined as Markov random field model that model data distributions based on maximum entropy distribution to fit the observed data by identifying the patterns in the observed data.", "Transferrable Prototypical Networks (TPN) [16] propose to identify prototypes for each class in source and target domains that are close in the embedding space and minimize the distance between these prototypes for domain adaptation.", "Contrastive Adaptation Network (CAN) [17] uses MMD-loss for the feature encodings along with the heuristic clustering schema to selectively pick a subset of the high confidence image samples from the target domain.", "These samples are then utilized in training the classifier on the target domain instead of the entire target domain.", "blackMoment Matching for Multi-Source Domain Adaptation (M3SDA) [18] proposes to dynamically align multiple labeled source domains and the unlabeled target domain by matching the moments of the feature distributions.", "Reconstruction-based methods: Deep Reconstruction-Classification Networks (DRCN) [19] and Domain Separation Networks (DSN) [20] approaches learn shared encodings of source and target domains using reconstruction networks.", "Adversarial adaptation methods: Reverse Gradient (RevGrad) [21] or Domain Adversarial Neural Network (DANN) [22] uses domain discriminator to learn domain invariant representations of both the domains.", "Coupled Generative Adversarial Network (CoGAN) [23] uses Generative Adversarial Network (GAN) [58] to obtain domain invariant features used for classification.", "Adversarial Discriminative Domain Adaptation (ADDA) [24] uses GANs along with weight sharing to learn domain invariant features.", "Generate to Adapt (GTA) [25] learns to generate an equivalent image in the other domain for a given image, thereby learning common domain invariant embeddings.", "Cross-Domain Representation Disentangler (CDRD) [26] learns cross-domain disentangled features for domain adaptation.", "Symmetric Bi-Directional Adaptive GAN (SBADA-GAN) [27] aims to learn symmetric bidirectional mappings among the domains by trying to mimic a target image given a source image.", "Cycle-Consistent Adversarial Domain Adaptation (CyCADA) [28] adapts representations at both the pixel-level and feature-level over the domains.", "Moving Semantic Transfer Network (MSTN) [29] proposes a moving semantic transfer network that learns semantic representations for the unlabeled target samples by aligning labeled source centroids and pseudo-labeled target centroids.", "Conditional Domain Adversarial Network (CDAN) [30] conditions the adversarial adaptation models on discriminative information conveyed in the classifier predictions.", "Decision-boundary Iterative Refinement Training with a Teacher (DIRT-T) [32] and Augmented Cyclic Adversarial Learning (ACAL) [33] learn by using a domain discriminator along with data augmentation for domain adaptation.", "Deep Cocktail Network (DCTN) [35] proposes a k-way domain discriminator and category classifier for digit classification and real-world object recognition in a multi-source domain adaptation setting.", "Batch Spectral Penalization (BSP) [36] investigates the transferability and the discriminability of the features learnt using the standard adversarial domain adaptation techniques.", "Also, BSP proposes an additional batch spectral loss as it is observed that the transferable features learnt using adversarial domain adaptation result in the loss of the discriminability of the classifier.", "Transferable Normalization (TransNorm) [37] proposes a further improvement in transferability by replacing the normal batch-normalization layer with separate normalization layers for source and target domain input batches.", "Adversarial Tight Match (ATM) [38] proposes to combine the adversarial training with discrepancy metric by introducing a novel discrepancy metric Maximum Density Divergence (MDD) to minimize inter-domain divergence and maximize the intra-class density.", "Certainty based Attention for Domain Adaptation (CADA) [39] propose to identify features that increase the certainty of the domain discriminator prediction to improve the classifier.", "Progressive Feature Alignment Network (PFAN) [40] proposes to align the discriminative features across domains progressively and effectively, via exploiting the intra-class variation in the target domain.", "Apart from these approaches, a slightly different method that has been recently proposed is called Tri-Training.", "Tri-Training algorithms use three classifiers trained on the labeled source domain and refine them for the unlabeled target domain.", "To be precise, in each round of tri-training, a target sample is pseudo-labeled if the other two classifiers agree on the labeling, under certain conditions such as confidence thresholding.", "Asymmetric Tri-Training (ATT) [59] uses three classifiers to bootstrap high confidence target domain samples by confidence thresholding.", "This way of bootstrapping works only if the source classifier has very high accuracy.", "In the case of low source classifier accuracy, target samples are never obtained to bootstrap, resulting in a bad model.", "Multi-Task Tri-training (MT-Tri) [60] explores the tri-training technique on the language domain adaptation tasks in a multi-task setting.", "All the domain adaptation approaches mentioned earlier have a common unifying theme: they attempt to morph the target and source distributions so as to make them indistinguishable.", "However, aligning domains is a complex task than the classification task.", "In this paper, we propose a completely different approach: instead of focusing on aligning the source and target distributions, we learn a single classifier referred to as Contradistinguisher, jointly on both the domain distributions using contradistinguish loss for the unlabeled target domain data and supervised loss for the labeled source domain data.", "Proposed Method: CUDA A domain $\\mathcal {D}_d$ is specified by its input feature space $\\mathcal {X}_d$ , the label space $\\mathcal {Y}_d$ and the joint probability distribution $p(\\mathbf {x}_d, \\mathbf {y}_d)$ , where $\\mathbf {x}_d{ \\in }\\mathcal {X}_d$ and $\\mathbf {y}_d{ \\in }\\mathcal {Y}_d$ .", "Let $\\left|\\mathcal {Y}_d \\right|{ = }K$ be the number of class labels such that $\\mathbf {y}_d{ \\in }\\lbrace 0, \\ldots , K{ - }1\\rbrace $ for any instance $\\mathbf {x}_d$ .", "Domain adaptation, in particular, consists of two domains $\\mathcal {D}_{s}$ and $\\mathcal {D}_{t}$ that are referred as the source and target domains respectively.", "We define $(\\mathbf {x}_s,\\mathbf {y}_s)$ as the random variables that denote the source domain input features and the corresponding source domain label.", "Similarly, we define $(\\mathbf {x}_t,\\mathbf {y}_t)$ as the random variables that denote the target domain input features and the corresponding target domain label.", "A common assumption in domain adaptation is that the input feature space as well as the label space remains unchanged across the source and the target domain, i.e., $\\mathcal {X}_s{ = }\\mathcal {X}_t{ = }\\mathcal {X}_d$ and $\\mathcal {Y}_s{ = }\\mathcal {Y}_t{ = }\\mathcal {Y}_d$ .", "Hence, the only difference between the source and target domain is input-label space distributions, i.e., $p(\\mathbf {x}_s,\\mathbf {y}_s){ \\ne }p(\\mathbf {x}_t,\\mathbf {y}_t)$ .", "This is referred to as domain shift in the domain adaptation literature.", "In particular, in an unsupervised domain adaptation, the training data consists of the labeled source domain instances ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ corresponding to the random variables $(\\mathbf {x}_s,\\mathbf {y}_s)$ and the unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variables $(\\mathbf {x}_t)$ .", "Observe that in the unsupervised domain adaptation setting, the target domain labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $(\\mathbf {y}_t)$ are unobserved/missing.", "Given a labeled data in the source domain, it is straightforward to learn a classifier by maximizing the conditional probability $p(\\mathbf {y}_s|\\mathbf {x}_s)$ over the labeled samples.", "However, the task at hand is to learn a classifier on the unlabeled target domain by transferring the knowledge from the labeled source domain to the unlabeled target domain.", "Overview Figure: Architecture of the proposed method CUDA with Contradistinguisher (Encoder and Classifier).", "Three optimization objectives with their respective inputs involved in training of CUDA: (i) Source supervised (), (ii) Target unsupervised () and Adversarial regularization ().The outline of the proposed method CUDA that involves contradistinguisher and the respective losses involved in training are depicted in Fig.", "REF .", "The objective of contradistinguisher is to find a clustering scheme using the most contrastive features on unlabeled target in such a way that it also satisfies the target domain prior over the labels, i.e., target domain prior enforcing.", "We achieve this by jointly training on labeled source samples in a supervised manner and unlabeled target samples in an unsupervised end-to-end manner by using a contradistinguish loss same as [47].", "This fine-tunes the classifier learnt from the source domain also to the target domain, as demonstrated in Fig.", "REF .", "The crux of our approach is the contradistinguish loss (REF ) which is discussed in detail in Section REF .", "Hence, the apt name contradistinguisher for our neural network architecture.", "Note that the objective of contradistinguisher is not the same as a classifier, i.e., distinguishing is not the same as classifying.", "Suppose there are two contrastive entities $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ , where $C_1, C_2$ are two classes.", "The aim of a classifier is to classify $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ , where to train a classifier one requires labeled data.", "On the contrary, the job of contradistinguisher is to just identify $e_1{ \\ne }e_2$ , i.e., contradistinguisher can classify $e_1{ \\in }C_1$ (or $C_2$ ) and $e_2{ \\in }C_2$ (or $C_1$ ) indifferently.", "To train contradistinguisher, we do not need any class information but only need unlabeled entities $e_1$ and $e_2$ .", "Using unlabeled target data, contradistinguisher is able to find a clustering scheme by distinguishing the unlabeled target domain samples in an unsupervised way.", "However, since the final task is classification, one would require a selective incorporation of the pre-existing informative knowledge required for the task of classification.", "This knowledge of assigning the label to the clusters is obtained by jointly training, thus classifying $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ .", "Supervised Source Classification For the labeled source domain instances ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ corresponding to the random variables $(\\mathbf {x}_s,\\mathbf {y}_s)$ of the labeled source domain, we define the conditional-likelihood of observing $\\mathbf {y}_s$ given $\\mathbf {x}_s$ as, $p_{\\theta }(\\mathbf {y}_{s}|\\textbf {x}_{s})$ , where $\\theta $ denotes the parameters of contradistinguisher.", "We estimate $\\theta $ by maximizing the conditional log-likelihood of observing the labels given the labeled source domain samples.", "Therefore, the source domain supervised objective to maximize is given as $\\mathcal {L}_{s}(\\theta )&{}={}&\\sum _{i=1}^{n_s}\\log (p_{\\theta }(\\mathbf {y}^{i}_{s}|\\textbf {x}^{i}_{s}))\\hspace{5.0pt}.$ Alternatively, one can minimize the cross-entropy loss, as used in practical implementation, instead of maximizing (REF ), i.e., $\\mathcal {L}_{ce}(\\theta )&{}={}&-\\sum _{i=1}^{n_s}\\sum _{k=0}^{K-1}{\\mathbb {1}[\\mathbf {y}^{i}_s{ = }k]}\\log (\\hat{\\mathbf {y}}^{ik}_{s})\\hspace{5.0pt},$ where $\\hat{\\mathbf {y}}^{ik}_{s}$ is the softmax output of contradistinguisher that represents the probability of class $k$ for the given sample $\\mathbf {x}^{i}_s$ .", "Unsupervised Target Classification For the unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $\\mathbf {x}_t$ of the unlabeled target domain, the corresponding labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $\\mathbf {x}_t$ are unknown/missing.", "Hence, a naive way of predicting the target labels is to directly use the classifier trained only with a supervised loss given in (REF ).", "While this approach may perform reasonably well in certain cases, it fails to deliver state-of-the-art performance.", "This may be attributed to the following reason: the support for the distribution $p_{\\theta }$ is defined only over the source domain instances $\\mathbf {x}_s$ and not the target domain instances $\\mathbf {x}_t$ .", "Hence, we model a non-trivial joint distribution $\\hat{q}_\\theta (\\textbf {x}_t,\\mathbf {y}_t)$ parameterized by the same $\\theta $ over target domain with only the target domain instances as the support as, $\\hat{q}_{\\theta }(\\textbf {x}_t,\\mathbf {y}_t)&{}={}&\\frac{p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t)}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t^{\\ell })}\\hspace{5.0pt}.$ However (REF ) is not a joint distribution yet because $\\sum _{\\ell =1}^{n_t}\\hat{q}_{\\theta }(\\textbf {x}^{\\ell }_{t}, \\mathbf {y}_{t}){ \\ne }p(\\mathbf {y}_t)$ , i.e., marginalizing over all ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ does not yield the target prior distribution, i.e., $p(\\mathbf {y}_t)$ .", "We modify (REF ) so as to include the marginalization condition.", "Hence, we refer to this as target domain prior enforcing.", "$q_{\\theta }(\\textbf {x}_t,\\mathbf {y}_t)&{}={}&\\frac{p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t) {p(\\mathbf {y}_t)}}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t^{\\ell })}\\hspace{5.0pt},$ where $p(\\mathbf {y}_t)$ denotes the target domain prior probability over the labels.", "Note that $q_\\theta (\\mathbf {x}_t, \\mathbf {y}_t)$ defines a non-trivial approximate of joint distribution over the target domain as a function of $p_\\theta $ learnt over source domain.", "The resultant unsupervised maximization objective for the target domain is given by maximizing the log-probability of the joint distribution $q_\\theta (\\mathbf {x}_t, \\mathbf {y}_t)$ which is $\\mathcal {L}_{t}(\\theta , {\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1})&{}={}&\\sum _{j=1}^{n_t}\\log (q_{\\theta }(\\textbf {x}^{j}_{t}, \\mathbf {y}^{j}_{t}))\\hspace{5.0pt},$ Next, we discuss how the objective given in (REF ) is solved, and the reason why (REF ) is referred to as contradistinguish loss.", "Since the target labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ are unknown, one needs to maximize (REF ) over the parameters $\\theta $ as well as the unknown target labels $\\mathbf {y}_t$ .", "As there are two unknown variables for maximization, we follow a two-step approach to maximize (REF ) as analogous to Expectation-Maximization (EM) algorithm [61].", "The two optimization steps are as follows.", "[(i)] Pseudo-label selection: We maximize (REF ) only with respect to the label $\\mathbf {y}_t$ for every $\\mathbf {x}_t$ by fixing $\\theta $ as $\\hat{\\mathbf {y}}^{j}_t&{}={}&\\operatornamewithlimits{arg\\,max}_{\\mathbf {y}^{j} \\in \\mathcal {Y}_t} \\frac{p_{\\theta }(\\mathbf {y}^{j}|\\textbf {x}^{j}_t) {p(\\mathbf {y}_t)}}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}^{\\ell }| \\textbf {x}_t^{\\ell })}\\hspace{5.0pt},$ Pseudo-labeling approach under semi-supervised representation learning setting has been well studied in [62] and shown equivalent to entropy regularization [63].", "As previously mentioned, the pseudo-label selection is analogous to E-step in the EM algorithm.", "Moreover, we derive the motivation from [47] that also uses pseudo-labeling in the context of semi-supervised representation learning.", "However, the proposed method addresses a more complex problem of domain adaptation in the presence of a domain shift.", "The pseudo-labeling essentially tries to cluster by assigning labels using source domain features of the classifier trained on the source domain.", "This is effectively is similar to the E-step in EM algorithm in spirit.", "Maximization: By fixing the pseudo-labels ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{n_t}_{j=1}$ from (REF ), we train contradistinguisher to maximize (REF ) with respect to the parameter $\\theta $ .", "$\\mathcal {L}_t(\\theta )&{}={}&\\sum _{j=1}^{n_t}\\log (p_{\\theta }(\\hat{\\mathbf {y}}^{j}_t|\\textbf {x}^{j}_t)) + \\sum _{j=1}^{n_t}\\log ({p(\\mathbf {y}_t)})\\nonumber \\\\&&{-}\\:\\sum _{j=1}^{n_t}\\log (\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\hat{\\mathbf {y}}^{j}_t| \\textbf {x}_t^{\\ell }))\\hspace{5.0pt}.$ Since the pseudo-labels from (REF ) are used for the maximization, this constrains the model to learn the features to further improve the current pseudo-labeling for the next iteration.", "This step is similar to the M-step in the EM algorithm in spirit.", "The first term, i.e., log-probability for a label $\\hat{\\mathbf {y}}^{j}_t$ given $\\mathbf {x}^{j}_t$ forces contradistinguisher to choose features to classify $\\mathbf {x}^{j}_t$ to $\\hat{\\mathbf {y}}^{j}_t$ .", "The second term is a constant, hence it has no effect on the optimization with respect to $\\theta $ .", "The third term is the negative of log-probability for the pseudo-label $\\hat{\\mathbf {y}}^{j}_t$ given all the samples $\\mathbf {x}^{\\ell }_t$ in the entire domain.", "Maximization of this term forces contradistinguisher to choose features to not classify all the other $\\mathbf {x}^{\\ell { \\ne }j}_t$ to selected pseudo-label $\\hat{\\mathbf {y}}^{j}_t$ except the given sample $\\mathbf {x}^{j}_t$ .", "This forces contradistinguisher to extract the most unique features of a given sample $\\mathbf {x}^{j}_t$ against all the other samples $\\mathbf {x}^{\\ell { \\ne }j}_t$ , i.e., most unique contrastive feature of the selected sample $\\mathbf {x}^{j}_t$ over all the other samples $\\mathbf {x}^{\\ell { \\ne }j}_t$ to distinguish a given sample from all others.", "The first and third term together in (REF ) enforce that contradistinguisher learns the most contradistinguishing features among the samples $\\mathbf {x}_t{ \\in }\\mathcal {X}_t$ , thus performing unlabeled target domain classification in a fully unsupervised way.", "We refer to the unsupervised target domain objective (REF ) as contradistinguish loss because of this contradistinguishing feature learning.", "Ideally, one would like to compute the third term in (REF ) using the complete target training data for each input sample.", "Since it is expensive to compute the third term over the entire $\\mathbf {x}_t$ for each individual sample during training, one evaluates the third term in (REF ) over a mini-batch.", "In our experiments, we have observed that the mini-batch strategy does not cause any problem during training as far as it includes at least one sample from each class, which is a fair assumption for a reasonably large mini-batch size of 128.", "For numerical stability, we use $\\log \\sum \\exp $ trick to optimize third term in (REF ).", "Adversarial Regularization To prevent contradistinguisher from over-fitting to the chosen pseudo labels during the training, we use adversarial regularization.", "In particular, we train contradistinguisher to be confused about the set of fake negative samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{n_f}_{j=1}$ by maximizing the conditional log-probability over the given fake sample such that the sample belongs to all $K(\\left|\\mathcal {Y}_d \\right|)$ classes simultaneously.", "The adversarial regularization objective is to multi-label the fake sample (e.g., a noisy image that looks like a cat and a dog) equally to all $K$ classes as labeling to any unique class introduces more noise in pseudo labels.", "This strategy is similar to entropy regularization [63] in the sense that instead of minimizing the entropy for the real target samples, we maximize the conditional log-probability over the fake negative samples.", "Therefore, we add the following maximization objective to the total contradistinguisher objective as a regularizer.", "$\\mathcal {L}_{adv}(\\theta )&{}={}&\\sum _{j=1}^{n_f}\\log (p_{\\theta }(\\hat{\\mathbf {y}}_t^{j}|\\hat{\\mathbf {x}}^{j}_{t}))\\hspace{5.0pt},$ for all $\\hat{\\mathbf {y}}^{j}_t{ \\in }\\mathcal {Y}_t$ .", "As maximization of (REF ) is analogous to minimizing the binary cross-entropy loss (REF ) of a multi-class multi-label classification task, in our practical implementation, we minimize (REF ) for assigning labels to all the classes for every sample.", "$\\mathcal {L}_{bce}(\\theta )&{}={}&-\\sum _{j=1}^{n_f}\\sum _{k=0}^{K-1}\\log (\\hat{\\mathbf {y}}^{jk}_{t})\\hspace{5.0pt},$ where $\\hat{\\mathbf {y}}^{jk}_{t}$ is the softmax output of contradistinguisher which represents the probability of class $k$ for the given sample $\\hat{\\mathbf {x}}^{j}_t$ .", "The fake negative samples $\\hat{\\mathbf {x}}_t$ can be directly sampled from, say, a Gaussian distribution in the input feature space $\\mathcal {X}_t$ with the mean and standard deviation of the samples $\\mathbf {x}_t{ \\in }\\mathcal {X}_t$ .", "For the language domain, fake samples are generated randomly, as mentioned above, because the input feature is the form of embeddings extracted from denoising auto-encoder with bag-of-words as the input.", "In case of visual datasets, as the feature space is high dimensional, the fake images $\\hat{\\mathbf {x}}_t$ are generated using a generator network $G_{\\phi }$ with parameter $\\phi $ that takes Gaussian noise vector $\\eta _t$ as input to produce a fake sample $\\hat{\\mathbf {x}}_t$ , i.e., $\\hat{\\mathbf {x}}_t = G_{\\phi }(\\eta _t)$ .", "Generator $G_{\\phi }$ is trained by minimizing kernel-MMD loss [64], i.e., a modified version of MMD loss between the encoder output $\\rho _{enc}(\\hat{\\mathbf {x}}_t)$ and $\\rho _{enc}(\\mathbf {x}_t)$ of $n_f$ fake images $\\hat{\\mathbf {x}}_t$ and $n_t$ real target domain images $\\mathbf {x}_t$ respectively.", "$\\mathcal {L}_{gen}(\\phi )&{}={}&\\frac{1}{n_f^{2}}\\sum _{i=1}^{n_f}\\sum _{j=1}^{n_f} k(\\rho _{enc}(\\hat{\\mathbf {x}}_t^{i}),\\rho _{enc}(\\hat{\\mathbf {x}}_t^{j}))\\nonumber \\\\&&{+}\\:\\frac{1}{n_t^{2}}\\sum _{i=1}^{n_t}\\sum _{j=1}^{n_t} k(\\rho _{enc}(\\mathbf {x}_t^{i}),\\rho _{enc}(\\mathbf {x}_t^{j}))\\nonumber \\\\&&{-}\\:\\frac{2}{n_t n_f}\\sum _{i=1}^{n_f}\\sum _{j=1}^{n_t} k(\\rho _{enc}(\\hat{\\mathbf {x}}_t^{i}),\\rho _{enc}(\\mathbf {x}_t^{j})),$ where $k(x,x^{\\prime }) = e^{-\\gamma \\left\\Vert x-x^{\\prime }\\right\\Vert ^{2}}$ is the Gaussian kernel.", "Note that the generator's objective is not to generate realistic images but to generate fake noisy images with mixed image attributes from the target domain.", "This reduces the effort of training powerful generators, which is the focus in adversarial based domain adaptation approaches [25], [26], [27], [28], [29] used for domain alignment.", "Algorithm REF and REF list steps involved in CUDA training and inference, respectively.", "[t!]", "$b{ = }batch\\_size$ , $epochs{ = }max\\_epoch$ , $n_{batch}{ = } number\\ of\\ batches$ $\\theta $ /* parameter of contradistinguisher */${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ , ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ target domain prior $p(\\mathbf {y}_t)$ is known use $p(\\mathbf {y}_t)$ for the contradistinguish loss (REF ) /* target domain prior enforcing */ compute $p(\\mathbf {y}_t)$ assuming $p(\\mathbf {y}_t) = p(\\mathbf {y}_s)$ /* fair assumption as most datasets are well balanced */ $epoch = 1$ to $epochs$ $batch = 1$ to $n_{batch}$ sample a mini-batch ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{b}_{i=1}$ , ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{b}_{j=1}$ compute $\\mathcal {L}_{s}(\\theta )$  (REF ) using ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{b}_{i=1}$ /* source supervised loss */ compute ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{b}_{j=1}$  (REF ) using ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{b}_{j=1}$ /* pseudo label selection step */ compute $\\mathcal {L}_{t}(\\theta )$  (REF ) fixing ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{b}_{j=1}$ /* maximization step */ /* steps REF and REF together optimize unsupervised contradistinguish loss (REF ) */ adversarial regularization is enabled Generator $G_\\phi $ is used get fake samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ from Gaussian noise vectors ${\\lbrace \\eta ^{j}_t\\rbrace }^{b}_{j=1}$ using $G_\\phi $ , compute $\\mathcal {L}_{gen}(\\phi )$ (REF ) /* generator training */ get fake samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ by random sampling in the input feature space $\\mathcal {X}_t$ compute $\\mathcal {L}_{adv}(\\theta )$  (REF ) using ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ /* fake samples are assigned to all classes equally */ combine losses in steps REF ,REF ,REF and REF to compute gradients using back-propagation update $\\theta $ using gradient descent /* and $\\phi $ if $G_\\phi $ is used */ CUDA Training [t!]", "${\\lbrace \\mathbf {x}^{i}_{test}\\rbrace }^{n_{test}}_{i=1}$ /* input test samples */${\\lbrace \\hat{\\mathbf {y}}^{i}_{test}\\rbrace }^{n_{test}}_{i=1}$ /* predicted labels */ $i = 1$ to $n_{test}$ predict label as $\\hat{\\mathbf {y}}^{i}_{test}{}={}\\operatornamewithlimits{arg\\,max}_{\\mathbf {y}\\in \\mathcal {Y}_t} p_{\\theta }(\\mathbf {y}|\\textbf {x}^{i}_{test})$ CUDA Inference Extension to Multi-Source Domain Adaptation Here, we argue that our proposed method CUDA has an implicit advantage in dealing with multi-source domain adaption problems over the techniques based on domain alignment.", "In a multi-source adaption setting, domain alignment methods need to consider the domain-shift between a source and a target domain and consider the domain-shift between the multiple source domains.", "Therefore, domain alignment methods are required to solve the even more complex intermediate problem of aligning multiple source and target domain distributions in addition to the complex intermediate problem of source and target domain alignment to deal with the multi-source domain adaptation problems.", "However, as the proposed method does not depend on the domain alignment, the extension to multi-source in the proposed method is very simple.", "As our main focus is to perform unsupervised learning directly on the target domain, the model obtained using is better generalized to the target domain and reduces overfitting on the source, which usually results in a negative transfer.", "We believe that this is one of the main advantages of addressing the domain adaptation by performing the primary task of target domain classification rather than the intermediate task of domain alignment.", "We propose a simple extension our proposed method to perform multi-source domain adaptation in the following manner.", "Let us suppose we are given with $R$ source domains ${\\lbrace s_1,\\ldots ,s_R\\rbrace }$ , consisting of labeled training data $({\\lbrace (\\mathbf {x}^{i}_{s_1}, \\mathbf {y}^{i}_{s_1})\\rbrace }^{n_{s_1}}_{i=1}, \\ldots , {\\lbrace (\\mathbf {x}^{i}_{s_R}, \\mathbf {y}^{i}_{s_R})\\rbrace }^{n_{s_R}}_{i=1})$ and unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ .", "We compute the source supervised loss for the $r^{th}$ source domain using (REF ), i.e., $\\mathcal {L}_{s_r}(\\theta )$  (REF ) with ${\\lbrace (\\mathbf {x}^{i}_{s_r}, \\mathbf {y}^{i}_{s_r})\\rbrace }^{n_{s_r}}_{i=1}$ training data.", "We further compute the total multi-source supervised loss as $\\mathcal {L}_{s_{total}}(\\theta ) = \\sum _{r=1}^{R}\\mathcal {L}_{s_r}(\\theta ).$ We replace $\\mathcal {L}_{s}(\\theta )$  (REF ) in the total optimization objective with $\\mathcal {L}_{s_{total}}(\\theta )$  (REF ) in step REF of Algorithm REF .", "It should be noted that the unsupervised loss for the target domain is still unmodified irrespective of the number of source domains.", "We experimentally demonstrate the efficacy of the proposed multi-source domain adaptation extension on Office-31 [44] and Digits datasets [48], [49], [50], [21].", "Experiments For our domain adaptation experiments, we consider both synthetic and real-world datasets.", "Under synthetic datasets, we experiment using 2-dimensional blobs with different source and target domain probability distributions to demonstrate the effectiveness of the proposed method under different domain shifts.", "Under real-world datasets, we consider only the complex, high-resolution Office-31 [44] and VisDA-2017 [45] object classification datasets for our experiment as the low-resolution datasets are already addressed in our conference paper CUDA: Contradistinguisher for Unsupervised Domain Adaptation (CUDA) [43].", "We have published our python code for all the experiments at https://github.com/sobalgi/cuda.", "black Table: Details of visual domain adaptation datasets.Table REF provides details on the visual datasets used in our experiments.", "We also experiment and report the results of our ablation study carried out with different combinations of the three optimization objectives with their respective domains as the inputs involved in CUDA training: [(i)] source supervised loss: $ss$ described in (REF ), source/target unsupervised loss: $su/tu$ described in (REF ), source/target adversarial regularization loss: $sa/ta$ described in (REF ).", "$ss$ indicates the minimum target domain test accuracy that can be attained with a chosen contradistinguisher neural network by training only using the labeled source domain.", "Any improvement over $ss$ using CUDA (i.e., combination of $su/tu/sa/ta$ ) indicates effectiveness of CUDA as the chosen contradistinguisher neural network is fixed.", "Experiments on Synthetic Toy-dataset Figure: Contour plots show the probability contours along with clear decision boundaries on different toy-dataset settings trained using CUDA.", "(source domain: ×\\times , target domain: ++, class 0: blue, class 1: red.)", "(Best viewed in color.", ")We validate our proposed method by performing experiments on synthetically created simple datasets that model different source and target domain distributions in a 2-dimensional input feature space using different blobs of source-target domain orientations and offsets (i.e., domain shift).", "We create blobs for source and target domains with 4000 samples using standard $scikit{ - }learn$  [46] as indicated in Fig.", "REF and REF .", "We further evenly split these 4000 data-points into equal train and test sets.", "Each of the splits consists of the same number of samples corresponding to both the class labels.", "The main motivation of the experiments on toy-dataset is to understand and visualize the behavior of the proposed method under some typical domain distribution scenarios and analyze the performance of  CUDA.", "$Blobs$ toy-dataset plots in Fig.", "REF shows clear comparisons of the classifier decision boundaries learnt using CUDA over domain alignment approaches.", "The top row in Fig.", "REF corresponds to the domain alignment classifier trained only on the labeled source domain, i.e., $ss$ .", "However, the bottom row in Fig.", "REF corresponds to contradistinguisher trained using the proposed method CUDA with labeled source and unlabeled target domain, i.e., $ss{ + }tu{ + }ta$ .", "Fig.", "REF demonstrates the classifier learnt using CUDA on the synthetic datasets with different complex shapes and orientations of the source and target domain distributions for the input data.", "Fig.", "REF and REF indicate the simplest form of the domain adaptation tasks with similar orientations in source and target domain distributions.It is important to note that the prior enforcing used in pseudo-label selection is the reason such fine classifier boundaries are observed, especially in Fig.", "REF ,REF and REF -REF .", "Fig.", "REF and REF represent more complex configurations of source and target domain distributions that indicate the hyperbolic decision boundaries jointly learnt on both the domains simultaneously using a single classifier without explicit domain alignment.", "Similarly, Fig.", "REF represents a complex configuration of source and target domain distributions that indicates an elliptical decision boundary.", "These simulated experiments points to some significant inner workings of our approach CUDA.", "These are the two main takeaways from the toy-dataset experiments.", "[(i)] The non-necessity of the domain alignment in the form of distribution distance metric minimization or data augmentation.", "In the case of these toy-datasets, it is not possible to perform any form of data augmentation, unlike some of the visual domain adaptation tasks, because the data is directly available in the form of encoded features that cannot be easily data augmented through standard heuristics.", "In such a case, it is necessary to realize a generic approach applicable to multiple modalities of the input, e.g., similar to the toy-dataset in language domain adaptation tasks.", "The features are presented in the form of word2vec/doc2vec, and no data augmentation is possible.", "Fig.", "REF and REF provide some interesting observations.", "Here, we can observe that the classes in the source domain are overlapping, resulting in less than 100% classification on the source domain, which in turn results in less than 100% classification on the target domain when considering domain alignment approaches.", "However, CUDA does not try to morph the target domain onto the source domain by directly classifying on the target domain resulting in a perfect classification.", "Since the classification is done directly on the unlabeled target domain in a fully unsupervised manner, the target domain classification performance is not limited by the source domain classification performance, i.e., the irrespective of the domain is used as the labeled source domain and the unlabeled target domain, the performance is the respective domains are similar.", "In other words, swapping of the domains or the direction of the domain adaptation has little effect on the classification performance on each individual domain.", "Experiments on Real-world Datasets In our previous work [43], we have demonstrated the effectiveness of CUDA in real-world domain adaptation on low-resolution visual datasets and language datasets.", "In contrast to low-resolution visual datasets, we consider the complex, high-resolution Office-31 [44] and VisDA-2017 [45] object classification datasets for domain adaptation.", "In addition to the single-source domain adaptation experiments, we also extend CUDA to Office-31 [44] and Digits datasets [48], [49], [50], [21].", "Office-31 Dataset Figure: Illustrations of samples from all the three domains of high resolution Office-31  dataset with one instance per each class from every domain (column {1,4,7,10}: 𝒜\\mathcal {A}, {2,5,8,11}: 𝒟\\mathcal {D}, {3,6,9,12}: 𝒲\\mathcal {W}).", "(Best viewed in color.", ")Figure: Illustrations of samples from all the three data-splits of VisDA-2017  dataset with one instance per each class from every domain ({row 1}: 𝒱 syn \\mathcal {V}_{syn} source domain synthetic images (training set), {row 2}: 𝒱 real \\mathcal {V}_{real} target domain real-world images (validation set), {row 3}: 𝒱 real \\mathcal {V}_{real} target domain real-world images (testing set)).", "It should be noted that unlike the Office-31 dataset and other standard benchmark domain adaptation datasets discussed in , most of the real-world images in the target domain of the VisDA-2017 dataset contains multiple true labels, which are only annotated with only one of the multiple labels.", "(Best viewed in color.", ")In high-resolution visual datasets, we consider Office-31 [44] dataset for our experiments.", "Unlike low-resolution visual datasets, here, we have only a few hundreds of training samples that make this an even more challenging task.", "Office objects: Office-31 [44] dataset consists of high resolution images of objects belonging to 31 classes obtained from three different domains AMAZON ($\\mathcal {A}$ ), DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ).", "Fig.", "REF shows illustrations of the images from all the three above mentioned domains of the Office-31 [44] dataset.", "AMAZON ($\\mathcal {A}$ ) domain consists of synthetic images with clear white background.", "DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ) domains consist of real-world images with noisy background and surroundings.", "We consider all possible six combinatorial tasks of domain adaptation involving all the three domains, i.e., $\\mathcal {A}{ \\leftrightarrow }\\mathcal {D}$ , $\\mathcal {A}{ \\leftrightarrow }\\mathcal {W}$ and $\\mathcal {D}{ \\leftrightarrow }\\mathcal {W}$ .", "Compared to low-resolution visual datasets, Office-31 [44] dataset domain adaptation tasks have increased complexity due to the small number of training images.", "Unlike low-resolution visual datasets, the high-resolution Office-31 [44] dataset does not have separate pre-defined train and test splits.", "Since we do not use any labels from the target domain during training, we report ten-crop test accuracy on the target domain by summing the softmax values of all the ten crops of the image and assign the label with maximum aggregate softmax value for the given image as in CDAN [30] in Table REF .", "To further alleviate the lack of a large number of training samples, pre-trained networks such as ResNet-50 [65] and ResNet-152 [65] were used to extract 2048 dimensional features from high-resolution images similar to CDAN [30].", "Since the images are not well centered and have a high resolution, we use the standard ten-crop of the image to extract features from the same images during training and testing, also similar to CDAN [30].", "The use of pre-trained models leads to two choices of training, [(i)] Fine-tune the pre-trained model used as feature extractor along with the final classifier layer: This requires careful selection of several hyper-parameters such as learning rate, learning rate decay, batch size, etc., to fine-tune the network to the current dataset while preserving the ability of the pre-trained network.", "We observed that fine-tuning also depends on the loss function used for training [66], which in our case, the use of contradistinguish loss greatly affected the changes in the pre-trained model as it is trained only using cross-entropy loss.", "Fine-tuning is also computationally expensive and time-consuming as each iteration requires computing gradients of all the pre-trained model parameters.", "Fix the pre-trained model and only train the final classifier layer: Alternative to fine-tuning is to fix the pre-trained model and use it only as a feature extractor.", "This approach has multiple practical benefits such as, [(a)] The computational time and cost of fine-tuning the parameters of the pre-trained model are alleviated.", "Since the feature extractor is fixed, it requires only once to extract and store the features locally instead of extracting the same features every iteration.", "Hence reducing the training time and the GPU memory as it is only required to train the final classifier.", "VisDA-2017 Dataset The VisDA-2017 dataset consists of two domains, (i) synthetic and (ii) real, with three predefined data splits.", "Fig.", "REF indicates the samples from all the 12 classes of the three data splits.", "The three predefined data splits in the VisDA-2017 dataset are as follows.", "[(i)] Training set: This split includes 152,397 labeled synthetic images obtained using 2D renderings of 3D models from different angles and different lighting conditions.", "This split is considered as a labeled source domain for training.", "Validation set: This split includes 55,388 real-world images obtained from a curated subset of MS COCO [67] dataset.", "This split is considered an unlabeled target domain training set, and this is used during the training without labels.", "Testing set: This split includes 72,372 real-world images obtained from YouTube Bounding Boxes [68] dataset.", "This split is considered as the target domain testing set used for evaluation and to report the results.", "Analysis of Experimental Results on Real-world Datasets Figure: Row 1 and 2: t-SNE  plots for embeddings from the output of contradistinguisher with samples from Office-31  dataset as input corresponding to the highest mean accuracy setting ss+tu+su+tass{ + }tu{ + }su{ + }ta indicated in Table  for single-source domain adaptation using ResNet-152  as the fixed encoder.Row 3: t-SNE  plots for embeddings from the output of contradistinguisher corresponding to the samples from Office-31  dataset in high-resolution visual tasks after applying softmax trained with CUDA with ResNet-50  as the encoder in a multi-source domain adaptation setting as indicated in Table .We can observe the clear class-wise clustering among all the 31 classes in the Office-31  datasets.We achieve high accuracies in spite of having only a few hundred training samples in each domain.", "(Best viewed in color.", ")Table: Target domain accuracy (%) on high resolution Office-31  dataset containing three domains.CUDA corresponds to our best results obtained with the best hyper-parameter settings.ssss: source supervised (), tutu: target unsupervised (), susu: source unsupervised (), sasa: source adversarial regularization () and tata: target adversarial regularization () represents different training configurations.Table: Target domain accuracy (%) on high resolution Office-31  dataset under multi-source domain adaptation setting by combining two domains into a single source domain and the remaining domain as the target domain with ResNet-50  as the encoder.CUDA corresponds to our best results obtained with the best hyper-parameter settings.ssss: source supervised (), tutu: target unsupervised (), susu: source unsupervised (), sasa: source adversarial regularization () and tata: target adversarial regularization () represents different training configurations.Table: Target domain accuracy reported on the test set (%) on all 5 combinations of Digits datasets under multi-source domain adaptation setting.Figure: The t-SNE plots of the unseen test set samples corresponding to the CUDA result in Table .", "The t-SNE plots show clear clustering of all the 10 classes in Digits datasets distinctively.", "(Best viewed in color.", ")Figure: The t-SNE plots of CUDA/CUDA * ^{*} shows the clear clustering of all the twelve classes of VisDA-2017 distinctively compared to the t-SNE plots of BSP/CAN.The t-SNE plots of CUDA/CUDA * ^{*} represent some important visual semantics of the image embeddings obtained from contradistinguisher in the following manner.", "(i) The vehicular classes such as `bus', `car', `train', and `truck' can be seen clustered closely as semantically these classes are similar to each other (region bounded in red).", "(ii) The two-wheeler classes such as `bicycle' and `motorcycle' are clustered closely as these are semantically similar to each other compared to vehicular classes that are clustered exactly opposite (region bounded in green).", "(iii) Irrespective of the approach used, there is always confusion between `knife' and `skateboard' classes.", "This confusion between `knife' and `skateboard' classes represented in the confusion matrices, which is also clearly seen in the t-SNE plots as well, can be attributed to the nature of images of these classes in the dataset on close observation (region bounded in blue).", "(iv) The remaining classes such as `aeroplane', `horse', `person' and `plant' can be seen clustered independently and distinctively as these classes have almost no visual semantic similarities to one another.", "(Best viewed in color.", ")Table: Results on VisDA-2017 dataset reproduced from the current state-of-the-art method BSP, CAN and our proposed method CUDA reported on both the validation set and test set.", "We report all the evaluation metrics such as precision, recall, and accuracy, unlike BSP/CAN, where the recall scores are mistakenly reported as accuracy.", "CUDA * ^{*} represents the results reproduced using vanilla CUDA with the data augmentation and target domain clustering similar to CAN for a fair comparison of the effect of the CUDA over CAN.The results reported for BSP, CAN, and CUDA/CUDA * ^{*} are from our own best reproduction from the original source code.Table: Total classification accuracy (%) on VisDA-2017 dataset reported on both the validation set and test set.", "The results from JAN, GTA, CDAN and TransNorm are reported from TransNorm .The results reported for BSP, CAN and CUDA/CUDA * ^{*} are from our own best reproduction from the original source code.CUDA * ^{*} represents the results reproduced using vanilla CUDA with the data augmentation and target domain clustering similar to CAN for a fair comparison of the effect of the CUDA over CAN.Figure: We indicate few samples that are misclassified by the contradistinguisher in the following subcaption format `original_label|predicted_label'.In most cases, the original ground truth labels are dubious, and the predicted labels make more sense realistically.Subplots (6), (7), (11), (12), (14), (18), (23) and (25) shows that the object is identified based on the shape and not if the object is present only in the foreground.", "This indicates that the contradistinguisher makes the predictions based on the clearly visible shapes and not the presence of the object in the foreground/background.The visualization of the features responsible for the respective predicted outcome indicates the shape bias as mostly the features are detected as edges corresponding to the shape of the object in the image.", "This shows the importance of shape bias to achieve high performance in transfer learning and domain adaptation tasks.Office-31 Single-Source Domain Adaptation Results We report the standard ten-crop accuracy on the target domain images as reported by several state-of-the-art domain adaptation methods [30], [25], [3].", "Since there are no explicit test split specified in the dataset and no labels are used from the target domain during training, it is common to report ten-crop accuracy considering the whole target domain.", "In Table REF , we report accuracies obtained by fine-tuning ResNet-50 [65] using the learning rate schedule followed in CDAN [30] and also without fine-tuning ResNet-50 [65].", "Apart from fixed ResNet-50 [65], we also report accuracies with fixed ResNet-152 [65] in Table REF for comparison.", "Fig.", "REF -REF indicate the t-SNE [69] plots of the softmax output after aggregating the ten-crop of each image corresponding to training configuration $ss{ + }tu{ + }su{ + }ta$ reported in Table REF .", "Fig.", "REF reports the t-SNE [69] plots of the training setting using ResNet-152 [65] encoder with the highest mean accuracy of all the six domain adaptation tasks.", "We clearly observe that CUDA outperforms several state-of-the-art methods that also use ResNet-50 [65] and even further surpasses by using ResNet-152 [65] encoder with CUDA.", "Among the three domains in Office-31 [44] dataset, $\\mathcal {A}$ can be considered as a well-curated synthetic dataset with clear background and $\\lbrace \\mathcal {D},\\mathcal {W}\\rbrace $ as an uncurated real-world dataset with noisy background and surroundings.", "We report the six domain adaptation tasks in the order of their complexity from low to high as, [(i)] Fig.", "REF and REF indicate highest accuracies because of similar real-world to real-world domain adaptation task, Fig.", "REF and REF indicate moderately high accuracies because of synthetic to real-world domain adaptation task and Fig.", "REF and REF indicate the lowest accuracies among all the six tasks because of real-world to synthetic domain adaptation task.", "Comparing CUDA with $ss$ in Tables REF and REF , we can see significant improvements in the target domain test accuracies due to the use of contradistinguish loss (REF ) demonstrating the effectiveness of contradistinguisher.", "As our method is mainly dependent on the contradistinguish loss (REF ), we observed further improved results by experimenting with better neural networks, e.g., using ResNet-152 over ResNet-50 along with our contradistinguish loss (REF ).", "From our ablations study in Table REF , we observe the effect of selection of ImageNet pre-trained ResNet-50 and ResNet-152 models on the domain adaptation with similar implications with the work [70].", "In general, one can always obtain better results irrespective of the approach by using better/deeper pre-trained models and/or data augmentation.", "However, since our main aim is to isolate, observe and benchmark only the true effect of different benchmark approaches, in our experiments, we maintain all the other parameters such as pre-trained neural network/data augmentation similar except for the core idea of the approaches and report our results both on Office-31 and VisDA-2017 datasets.", "Multi-Source Domain Adaptation Results We also extend the experiments to multi-source domain adaptation on the Office-31 [44] and Digits datasets [48], [49], [50], [21].", "In Table REF , we can clearly observe that in $\\mathcal {A}{ + }\\mathcal {D}{ \\rightarrow }\\mathcal {W}$ task, multi-source domain adaptation provides better results than their respective best single source domain adaptation experiments.", "However in case of $\\mathcal {D}{ + }\\mathcal {W}{ \\rightarrow }\\mathcal {A}$ and $\\mathcal {W}{ + }\\mathcal {A}{ \\rightarrow }\\mathcal {D}$ , the multi-source domain adaptation improves over $ss$ , it underperforms compared to best single source domain adaptation task.", "This can be attributed to the fact that the model tends to overfit on the source domains resulting in a negative transfer.This negative transfer behavior is also prevalent in other multi-source domain adaptation approaches since all the other multi-source domain adaptation methods also underperform compared to their best single source domain adaptation results, as reported in Table REF .", "Fig.", "REF -REF indicates t-SNE [69] plots for embeddings from the output of contradistinguisher corresponding to the samples from Office-31 [44] dataset after applying softmax trained with CUDA with ResNet-50 [65] as the encoder in a multi-source domain adaptation setting.", "We can observe the best results when the target domain is one of the real-world domain, i.e., $\\mathcal {D}$ and $\\mathcal {W}$ .", "It was consistently observed that domain adaptation tasks with synthetic domain $\\mathcal {A}$ as the target domain to be the most complex tasks of all the domain adaptation tasks across all the domain adaptation methods.", "Similarly, Table REF presents the results of multi-source domain adaptation on Digits datasets against benchmark approaches.", "In Fig.", "REF , we see the t-SNE plots on the test set depicting clear class-wise clustering that indicates the efficacy of CUDA single-source to multi-source extension.", "VisDA-2017 Single-Source Domain Adaptation Results For experiments on the VisDA-2017 dataset, we consider the most recent state-of-the-art benchmark domain adaptation approaches BSP [36] and CAN [17].", "Like BSP and CAN, we use the same neural network architecture with Imagenet pre-trained ResNet-101 with contradistinguish loss for training.", "BSP and CAN report the evaluation metric of accuracy in their papers.", "However, on reproducing the results with BSPhttps://github.com/thuml/Batch-Spectral-Penalization and CANhttps://github.com/kgl-prml/Contrastive-Adaptation-Network-for-Unsupervised-Domain-Adaptation to set the baseline for comparison, we noticed that the results reported had the following inconsistencies.", "[(i)] The results reported in the paper as accuracy in actual were the class-wise recall scores.", "The most standard procedure in machine learning is to report experimental performances on the test split, which is unseen during the training.", "However, in BSP and CAN, the results are reported on the validation set, which is used as the unlabeled target domain training set.", "Since the VisDA-2017 dataset has pre-defined splits for evaluation, reporting the results on the validation set used during the training does not indicate these models' generalizing capability, which is the most important aspect one would base an evaluation.", "We correct the above misreporting by reproducing the results from BSP and CAN and report all the relevant metrics for both validation and test splits of the VisDA-2017 dataset.", "Apart from these, we also validate our results on the official challenge evaluation portalhttps://competitions.codalab.org/competitions/17052#results by submitting the results of our approach CUDA on the VisDA-2017 dataset.", "In Fig.", "REF , we report the t-SNE plots reproduced from the current state-of-the-art unsupervised domain adaptation approach BSP [36] and CAN [17], in comparison with our approach CUDA/CUDA$^*$ on both the pre-defined validation split (seen target domain training set) and testing split (unseen target domain testing set) of the VisDA-2017 dataset.", "The results reported as CUDA corresponds to the CUDA experiments without any data augmentation using the BSP source code as the baseline to keep all the parameters the same for a fair comparison with BSP.", "Similarly, the results reported as CUDA$^*$ corresponds to the CUDA experiments with data augmentation and clustering of high confidence target domain samples using the CAN source code as the baseline to keep all the parameters same for a fair comparison with CAN.", "We can see the classwise clusters in BSP/CUDA are narrower compared to CAN/CUDA$^*$ , which are broader due to the use of data augmentation.", "Data augmentation helps modify/broaden the data distribution aiding in an improvement over the vanilla approaches without data augmentation.", "We further validate the results best results from our approach, i.e., CUDA$^*$ by submitting to the official challenge evaluation leaderboard.", "In Table REF we compare the per-class precision, recall, and accuracies on both the pre-defined validation set and test set of the VisDA-2017 against the results reproduced from the BSP/CAN.", "In Table REF we compare the total classification accuracies on both the pre-defined validation set and test set of the VisDA-2017 dataset against different benchmark methods.", "The results in Tables REF and REF indicate the superior performance of the proposed method CUDA/CUDA$^*$ over the current state-of-the-art domain alignment approaches BSP/CAN on both the pre-defined validation set and a test set of VisDA-2017 dataset.", "Even though the validation set and test set belong to the real-world domain, there is an inherent domain shift between them as both the data splits are collected from two different datasets, i.e., MS COCO [67] and YouTube Bounding Boxes ,[68] respectively.", "Results from CUDA indicate a better generalization to real-world domain as the scores in validation and test sets are closer compared to other approaches on the VisDA-2017 dataset.", "We can also observe that the t-SNE plot of CUDA in Fig.", "REF and REF clearly shows the visual semantics captured between the classes of images in the VisDA-2017 dataset.", "Apart from setting CUDA as the solid baseline for VisDA-2017, we further put a conscious effort to carefully investigate the reasons for the misclassification using the contradistinguisher to check if we can further improve the results.", "In most misclassified cases, we have observed that the labels predicted by CUDA appeared to be correct in comparison to the ground truth label of the dataset.", "In Fig.", "REF , we present some of these instances where the predicted label is more close to the real label than the ground truth.", "We can explain this misclassification as a limitation of the VisDA-2017 dataset in the following way.", "The misclassification observed in Fig.", "REF is due to the fact that the images in the VisDA-2017 dataset consist of objects belonging to more than one of the twelve classes, i.e., the images in the dataset consists of multiple labels for a single image, but the dataset only records one of these several true labels.", "As the assumed task for domain adaptation is single-label multi-class We see this as a limitation of the VisDA-2017 dataset compared other benchmark domain adaptation datasets such as Office-31 or the low-resolution visual datasets demonstrated in our conference paper [43].", "It is necessary that the datasets be consistent in the sense that each image has a unique label corresponding to it so that during the evaluation, there is no ambiguity between the original label and the predicted label from the trained model.", "The presence of this ambiguity in the dataset classification would then lead to observing the true evaluation metrices resulting in improper benchmarking for any given approach.", "However, in the case of the VisDA-2017 dataset, the predicted label from the model cannot be considered as a wrong label as it contains the object of the predicted label in the image.", "We believe that this is one of the reasons for the overall low performance apart from the complexity of the VisDA-2017 dataset compared to other visual datasets.", "It should also be observed that CUDA identifies the most distinguishing/prominent and assigns the label irrespective of the position (foreground/background) of the object as indicated in some of the subplots in Fig.", "REF .", "These misclassified cases indicates one of the strong drawback/limitation of VisDA-2017 dataset compared to other visual datasets, i.e., VisDA-2017 dataset has image samples with multiple true labels instead of a unique label for each image sample.", "Since the images might contain multiple true classes for an image, ideally all these true labels are to be associated with the image in the dataset to rightly evaluate any trained model for its efficacy.", "Because we perform single-label multi-class classification, predicting any one of the true labels of the image should be considered as right for the evaluation metric.", "However, this is not the case as the dataset does not record all the true labels for the images.", "So, if one plans to rightly use this dataset, all the true labels are to be annotated for each of the images in the dataset or use other benchmark datasets such as DomainNet/LSDAC (Large Scale Domain Adaptation Challenge) dataset [18] that alleviates the problem of multi-labels of VisDA-2017 dataset as DomainNet dataset only consist of single true label per each image in the dataset, resulting in correct evaluation without the issue of misclassification we have indicated above for the VisDA-2017 dataset.", "Apart from analyzing the limitation of the VisDA-2017 dataset, we also analyze the nature of the feature representations learnt by contradistinguisher.", "In order to visualize the features that prompted the predicted label, we use Captumhttps://captum.ai/tutorials/Resnet_TorchVision_Interpret, an open-source, extensible library for model interpretability built on PyTorchhttps://pytorch.org/ [71].", "We use gradient-based attribution to compute the integrated gradients for a given image using the predicted label.", "We obtain the high-level features or the saliency maps [72] for the given image.", "In terms of high-level features in a given image, one can imagine features such as shape, color, texture, size, etc.", "to be the features that help in predicting the classifier outcome.", "Out of all these features, the most natural and basic feature influencing the outcome is observed to be the shapes of the objects.", "Extensive research materials in psychology such as [73], [74], [75], [76], [77] have indicated that human babies and adults tend to utilize shapes than color/material/texture to assign a word label to the given object.", "This particular phenomenon is widely termed as `shape bias' in the literature.", "However, recently, it was shown that the ImageNet pre-trained models possess a texture bias over shape bias [78].", "To improve the shape bias, [78] propose a new modified dataset called `Stylized-ImageNet' to overcome the texture bias.", "By increasing the shape bias, [78] demonstrated improved performance and robustness.", "Since we use the ImageNet pre-trained ResNet-101 as a feature extractor, it is necessary to understand the nature of extracted features from the input images.", "Unlike `Stylized-ImageNet', in domain adaptation tasks, one cannot always expect to get such a curated dataset with ground truth labels on the target domain for each task.", "Instead, it might be easy and desirable to change the loss function that enhances the shape features with the same training dataset.Surprisingly, in our observations, we find that the features learnt by the classifier indicate the high-level features or the saliency maps [72] representing the shape of the object in the image.", "The contradistinguish loss is formulated and optimized in such a way that the features extracted are most unique and contrastive for a given image in comparison to other images in the dataset.", "This consequently is observed as the features corresponding to shapes in the form of silhouette in the feature visualizations in Fig.", "REF as each object posses a unique shape as it's most contrasdistinguishing character, i.e., the character which is most discriminative and unique to the given image.", "Concluding Remarks In this paper, we have proposed a direct approach to solve the problem of unsupervised domain adaptation that is different from the standard distribution alignment approaches.", "In our approach, we jointly learn a Contradistinguisher on the source and target domain distribution in the same input-label feature space using contradistinguish loss for unsupervised target domain to identify contrastive features.", "We have shown that contrastive learning overcomes the need and drawbacks of domain alignment, especially in tasks where domain shift is very high (e.g., language domains) and data augmentation techniques cannot be applied.", "Due to the inclusion of prior enforcing in the contradistinguish loss, the proposed unsupervised domain adaptation method CUDA could incorporate any known target domain prior to overcoming the drawbacks of skewness in the target domain, thereby resulting in a skew-robust model.", "We validated the efficacy of CUDA by experimenting on the synthetically created toy-dataset.", "We further demonstrated the simplicity and effectiveness of our proposed method by performing multi-source domain adaptation on Office-31 and Digits datasets to consistently outperform other multi-source domain adaptation approaches.", "We have also tested the proposed method CUDA on the recent benchmark visual domain adaptation datasets such Office-31 and VisDA-2017 classification datasets and demonstrated above/on-par results with the state-of-the-art approaches.", "We further analyzed the nature of the feature representation learnt using contradistinguish loss to identify the features related to the shapes that influence the predicted outcome.", "As the features related to shapes are learnt, we observed that it helps improving the performance and robustness of the trained model as the model is not biased to colors/textures in the images.", "We concluded that learning and improving shape bias is one of the keys to achieve ideal transfer learning and domain adaptation.", "Acknowledgments The authors would like to thank the Ministry of Human Resource Development (MHRD), Government of India, for their generous funding towards this work through the UAY Project: IISc 001.", "The authors thank Tejas Duseja for helping the authors with setting up some experiments.", "The authors would also like to thank anonymous reviewers for providing their valuable feedback that helped in improving the manuscript.", "[Figure: NO_CAPTION [Figure: NO_CAPTION" ], [ "Proposed Method: CUDA", "A domain $\\mathcal {D}_d$ is specified by its input feature space $\\mathcal {X}_d$ , the label space $\\mathcal {Y}_d$ and the joint probability distribution $p(\\mathbf {x}_d, \\mathbf {y}_d)$ , where $\\mathbf {x}_d{ \\in }\\mathcal {X}_d$ and $\\mathbf {y}_d{ \\in }\\mathcal {Y}_d$ .", "Let $\\left|\\mathcal {Y}_d \\right|{ = }K$ be the number of class labels such that $\\mathbf {y}_d{ \\in }\\lbrace 0, \\ldots , K{ - }1\\rbrace $ for any instance $\\mathbf {x}_d$ .", "Domain adaptation, in particular, consists of two domains $\\mathcal {D}_{s}$ and $\\mathcal {D}_{t}$ that are referred as the source and target domains respectively.", "We define $(\\mathbf {x}_s,\\mathbf {y}_s)$ as the random variables that denote the source domain input features and the corresponding source domain label.", "Similarly, we define $(\\mathbf {x}_t,\\mathbf {y}_t)$ as the random variables that denote the target domain input features and the corresponding target domain label.", "A common assumption in domain adaptation is that the input feature space as well as the label space remains unchanged across the source and the target domain, i.e., $\\mathcal {X}_s{ = }\\mathcal {X}_t{ = }\\mathcal {X}_d$ and $\\mathcal {Y}_s{ = }\\mathcal {Y}_t{ = }\\mathcal {Y}_d$ .", "Hence, the only difference between the source and target domain is input-label space distributions, i.e., $p(\\mathbf {x}_s,\\mathbf {y}_s){ \\ne }p(\\mathbf {x}_t,\\mathbf {y}_t)$ .", "This is referred to as domain shift in the domain adaptation literature.", "In particular, in an unsupervised domain adaptation, the training data consists of the labeled source domain instances ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ corresponding to the random variables $(\\mathbf {x}_s,\\mathbf {y}_s)$ and the unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variables $(\\mathbf {x}_t)$ .", "Observe that in the unsupervised domain adaptation setting, the target domain labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $(\\mathbf {y}_t)$ are unobserved/missing.", "Given a labeled data in the source domain, it is straightforward to learn a classifier by maximizing the conditional probability $p(\\mathbf {y}_s|\\mathbf {x}_s)$ over the labeled samples.", "However, the task at hand is to learn a classifier on the unlabeled target domain by transferring the knowledge from the labeled source domain to the unlabeled target domain." ], [ "Overview", "The outline of the proposed method CUDA that involves contradistinguisher and the respective losses involved in training are depicted in Fig.", "REF .", "The objective of contradistinguisher is to find a clustering scheme using the most contrastive features on unlabeled target in such a way that it also satisfies the target domain prior over the labels, i.e., target domain prior enforcing.", "We achieve this by jointly training on labeled source samples in a supervised manner and unlabeled target samples in an unsupervised end-to-end manner by using a contradistinguish loss same as [47].", "This fine-tunes the classifier learnt from the source domain also to the target domain, as demonstrated in Fig.", "REF .", "The crux of our approach is the contradistinguish loss (REF ) which is discussed in detail in Section REF .", "Hence, the apt name contradistinguisher for our neural network architecture.", "Note that the objective of contradistinguisher is not the same as a classifier, i.e., distinguishing is not the same as classifying.", "Suppose there are two contrastive entities $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ , where $C_1, C_2$ are two classes.", "The aim of a classifier is to classify $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ , where to train a classifier one requires labeled data.", "On the contrary, the job of contradistinguisher is to just identify $e_1{ \\ne }e_2$ , i.e., contradistinguisher can classify $e_1{ \\in }C_1$ (or $C_2$ ) and $e_2{ \\in }C_2$ (or $C_1$ ) indifferently.", "To train contradistinguisher, we do not need any class information but only need unlabeled entities $e_1$ and $e_2$ .", "Using unlabeled target data, contradistinguisher is able to find a clustering scheme by distinguishing the unlabeled target domain samples in an unsupervised way.", "However, since the final task is classification, one would require a selective incorporation of the pre-existing informative knowledge required for the task of classification.", "This knowledge of assigning the label to the clusters is obtained by jointly training, thus classifying $e_1{ \\in }C_1$ and $e_2{ \\in }C_2$ ." ], [ "Supervised Source Classification", "For the labeled source domain instances ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ corresponding to the random variables $(\\mathbf {x}_s,\\mathbf {y}_s)$ of the labeled source domain, we define the conditional-likelihood of observing $\\mathbf {y}_s$ given $\\mathbf {x}_s$ as, $p_{\\theta }(\\mathbf {y}_{s}|\\textbf {x}_{s})$ , where $\\theta $ denotes the parameters of contradistinguisher.", "We estimate $\\theta $ by maximizing the conditional log-likelihood of observing the labels given the labeled source domain samples.", "Therefore, the source domain supervised objective to maximize is given as $\\mathcal {L}_{s}(\\theta )&{}={}&\\sum _{i=1}^{n_s}\\log (p_{\\theta }(\\mathbf {y}^{i}_{s}|\\textbf {x}^{i}_{s}))\\hspace{5.0pt}.$ Alternatively, one can minimize the cross-entropy loss, as used in practical implementation, instead of maximizing (REF ), i.e., $\\mathcal {L}_{ce}(\\theta )&{}={}&-\\sum _{i=1}^{n_s}\\sum _{k=0}^{K-1}{\\mathbb {1}[\\mathbf {y}^{i}_s{ = }k]}\\log (\\hat{\\mathbf {y}}^{ik}_{s})\\hspace{5.0pt},$ where $\\hat{\\mathbf {y}}^{ik}_{s}$ is the softmax output of contradistinguisher that represents the probability of class $k$ for the given sample $\\mathbf {x}^{i}_s$ ." ], [ "Unsupervised Target Classification", "For the unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $\\mathbf {x}_t$ of the unlabeled target domain, the corresponding labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ corresponding to the random variable $\\mathbf {x}_t$ are unknown/missing.", "Hence, a naive way of predicting the target labels is to directly use the classifier trained only with a supervised loss given in (REF ).", "While this approach may perform reasonably well in certain cases, it fails to deliver state-of-the-art performance.", "This may be attributed to the following reason: the support for the distribution $p_{\\theta }$ is defined only over the source domain instances $\\mathbf {x}_s$ and not the target domain instances $\\mathbf {x}_t$ .", "Hence, we model a non-trivial joint distribution $\\hat{q}_\\theta (\\textbf {x}_t,\\mathbf {y}_t)$ parameterized by the same $\\theta $ over target domain with only the target domain instances as the support as, $\\hat{q}_{\\theta }(\\textbf {x}_t,\\mathbf {y}_t)&{}={}&\\frac{p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t)}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t^{\\ell })}\\hspace{5.0pt}.$ However (REF ) is not a joint distribution yet because $\\sum _{\\ell =1}^{n_t}\\hat{q}_{\\theta }(\\textbf {x}^{\\ell }_{t}, \\mathbf {y}_{t}){ \\ne }p(\\mathbf {y}_t)$ , i.e., marginalizing over all ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ does not yield the target prior distribution, i.e., $p(\\mathbf {y}_t)$ .", "We modify (REF ) so as to include the marginalization condition.", "Hence, we refer to this as target domain prior enforcing.", "$q_{\\theta }(\\textbf {x}_t,\\mathbf {y}_t)&{}={}&\\frac{p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t) {p(\\mathbf {y}_t)}}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}_t|\\textbf {x}_t^{\\ell })}\\hspace{5.0pt},$ where $p(\\mathbf {y}_t)$ denotes the target domain prior probability over the labels.", "Note that $q_\\theta (\\mathbf {x}_t, \\mathbf {y}_t)$ defines a non-trivial approximate of joint distribution over the target domain as a function of $p_\\theta $ learnt over source domain.", "The resultant unsupervised maximization objective for the target domain is given by maximizing the log-probability of the joint distribution $q_\\theta (\\mathbf {x}_t, \\mathbf {y}_t)$ which is $\\mathcal {L}_{t}(\\theta , {\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1})&{}={}&\\sum _{j=1}^{n_t}\\log (q_{\\theta }(\\textbf {x}^{j}_{t}, \\mathbf {y}^{j}_{t}))\\hspace{5.0pt},$ Next, we discuss how the objective given in (REF ) is solved, and the reason why (REF ) is referred to as contradistinguish loss.", "Since the target labels ${\\lbrace \\mathbf {y}^{j}_t\\rbrace }^{n_t}_{j=1}$ are unknown, one needs to maximize (REF ) over the parameters $\\theta $ as well as the unknown target labels $\\mathbf {y}_t$ .", "As there are two unknown variables for maximization, we follow a two-step approach to maximize (REF ) as analogous to Expectation-Maximization (EM) algorithm [61].", "The two optimization steps are as follows.", "[(i)] Pseudo-label selection: We maximize (REF ) only with respect to the label $\\mathbf {y}_t$ for every $\\mathbf {x}_t$ by fixing $\\theta $ as $\\hat{\\mathbf {y}}^{j}_t&{}={}&\\operatornamewithlimits{arg\\,max}_{\\mathbf {y}^{j} \\in \\mathcal {Y}_t} \\frac{p_{\\theta }(\\mathbf {y}^{j}|\\textbf {x}^{j}_t) {p(\\mathbf {y}_t)}}{\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\mathbf {y}^{\\ell }| \\textbf {x}_t^{\\ell })}\\hspace{5.0pt},$ Pseudo-labeling approach under semi-supervised representation learning setting has been well studied in [62] and shown equivalent to entropy regularization [63].", "As previously mentioned, the pseudo-label selection is analogous to E-step in the EM algorithm.", "Moreover, we derive the motivation from [47] that also uses pseudo-labeling in the context of semi-supervised representation learning.", "However, the proposed method addresses a more complex problem of domain adaptation in the presence of a domain shift.", "The pseudo-labeling essentially tries to cluster by assigning labels using source domain features of the classifier trained on the source domain.", "This is effectively is similar to the E-step in EM algorithm in spirit.", "Maximization: By fixing the pseudo-labels ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{n_t}_{j=1}$ from (REF ), we train contradistinguisher to maximize (REF ) with respect to the parameter $\\theta $ .", "$\\mathcal {L}_t(\\theta )&{}={}&\\sum _{j=1}^{n_t}\\log (p_{\\theta }(\\hat{\\mathbf {y}}^{j}_t|\\textbf {x}^{j}_t)) + \\sum _{j=1}^{n_t}\\log ({p(\\mathbf {y}_t)})\\nonumber \\\\&&{-}\\:\\sum _{j=1}^{n_t}\\log (\\sum _{\\ell =1}^{n_t}p_{\\theta }(\\hat{\\mathbf {y}}^{j}_t| \\textbf {x}_t^{\\ell }))\\hspace{5.0pt}.$ Since the pseudo-labels from (REF ) are used for the maximization, this constrains the model to learn the features to further improve the current pseudo-labeling for the next iteration.", "This step is similar to the M-step in the EM algorithm in spirit.", "The first term, i.e., log-probability for a label $\\hat{\\mathbf {y}}^{j}_t$ given $\\mathbf {x}^{j}_t$ forces contradistinguisher to choose features to classify $\\mathbf {x}^{j}_t$ to $\\hat{\\mathbf {y}}^{j}_t$ .", "The second term is a constant, hence it has no effect on the optimization with respect to $\\theta $ .", "The third term is the negative of log-probability for the pseudo-label $\\hat{\\mathbf {y}}^{j}_t$ given all the samples $\\mathbf {x}^{\\ell }_t$ in the entire domain.", "Maximization of this term forces contradistinguisher to choose features to not classify all the other $\\mathbf {x}^{\\ell { \\ne }j}_t$ to selected pseudo-label $\\hat{\\mathbf {y}}^{j}_t$ except the given sample $\\mathbf {x}^{j}_t$ .", "This forces contradistinguisher to extract the most unique features of a given sample $\\mathbf {x}^{j}_t$ against all the other samples $\\mathbf {x}^{\\ell { \\ne }j}_t$ , i.e., most unique contrastive feature of the selected sample $\\mathbf {x}^{j}_t$ over all the other samples $\\mathbf {x}^{\\ell { \\ne }j}_t$ to distinguish a given sample from all others.", "The first and third term together in (REF ) enforce that contradistinguisher learns the most contradistinguishing features among the samples $\\mathbf {x}_t{ \\in }\\mathcal {X}_t$ , thus performing unlabeled target domain classification in a fully unsupervised way.", "We refer to the unsupervised target domain objective (REF ) as contradistinguish loss because of this contradistinguishing feature learning.", "Ideally, one would like to compute the third term in (REF ) using the complete target training data for each input sample.", "Since it is expensive to compute the third term over the entire $\\mathbf {x}_t$ for each individual sample during training, one evaluates the third term in (REF ) over a mini-batch.", "In our experiments, we have observed that the mini-batch strategy does not cause any problem during training as far as it includes at least one sample from each class, which is a fair assumption for a reasonably large mini-batch size of 128.", "For numerical stability, we use $\\log \\sum \\exp $ trick to optimize third term in (REF ).", "Adversarial Regularization To prevent contradistinguisher from over-fitting to the chosen pseudo labels during the training, we use adversarial regularization.", "In particular, we train contradistinguisher to be confused about the set of fake negative samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{n_f}_{j=1}$ by maximizing the conditional log-probability over the given fake sample such that the sample belongs to all $K(\\left|\\mathcal {Y}_d \\right|)$ classes simultaneously.", "The adversarial regularization objective is to multi-label the fake sample (e.g., a noisy image that looks like a cat and a dog) equally to all $K$ classes as labeling to any unique class introduces more noise in pseudo labels.", "This strategy is similar to entropy regularization [63] in the sense that instead of minimizing the entropy for the real target samples, we maximize the conditional log-probability over the fake negative samples.", "Therefore, we add the following maximization objective to the total contradistinguisher objective as a regularizer.", "$\\mathcal {L}_{adv}(\\theta )&{}={}&\\sum _{j=1}^{n_f}\\log (p_{\\theta }(\\hat{\\mathbf {y}}_t^{j}|\\hat{\\mathbf {x}}^{j}_{t}))\\hspace{5.0pt},$ for all $\\hat{\\mathbf {y}}^{j}_t{ \\in }\\mathcal {Y}_t$ .", "As maximization of (REF ) is analogous to minimizing the binary cross-entropy loss (REF ) of a multi-class multi-label classification task, in our practical implementation, we minimize (REF ) for assigning labels to all the classes for every sample.", "$\\mathcal {L}_{bce}(\\theta )&{}={}&-\\sum _{j=1}^{n_f}\\sum _{k=0}^{K-1}\\log (\\hat{\\mathbf {y}}^{jk}_{t})\\hspace{5.0pt},$ where $\\hat{\\mathbf {y}}^{jk}_{t}$ is the softmax output of contradistinguisher which represents the probability of class $k$ for the given sample $\\hat{\\mathbf {x}}^{j}_t$ .", "The fake negative samples $\\hat{\\mathbf {x}}_t$ can be directly sampled from, say, a Gaussian distribution in the input feature space $\\mathcal {X}_t$ with the mean and standard deviation of the samples $\\mathbf {x}_t{ \\in }\\mathcal {X}_t$ .", "For the language domain, fake samples are generated randomly, as mentioned above, because the input feature is the form of embeddings extracted from denoising auto-encoder with bag-of-words as the input.", "In case of visual datasets, as the feature space is high dimensional, the fake images $\\hat{\\mathbf {x}}_t$ are generated using a generator network $G_{\\phi }$ with parameter $\\phi $ that takes Gaussian noise vector $\\eta _t$ as input to produce a fake sample $\\hat{\\mathbf {x}}_t$ , i.e., $\\hat{\\mathbf {x}}_t = G_{\\phi }(\\eta _t)$ .", "Generator $G_{\\phi }$ is trained by minimizing kernel-MMD loss [64], i.e., a modified version of MMD loss between the encoder output $\\rho _{enc}(\\hat{\\mathbf {x}}_t)$ and $\\rho _{enc}(\\mathbf {x}_t)$ of $n_f$ fake images $\\hat{\\mathbf {x}}_t$ and $n_t$ real target domain images $\\mathbf {x}_t$ respectively.", "$\\mathcal {L}_{gen}(\\phi )&{}={}&\\frac{1}{n_f^{2}}\\sum _{i=1}^{n_f}\\sum _{j=1}^{n_f} k(\\rho _{enc}(\\hat{\\mathbf {x}}_t^{i}),\\rho _{enc}(\\hat{\\mathbf {x}}_t^{j}))\\nonumber \\\\&&{+}\\:\\frac{1}{n_t^{2}}\\sum _{i=1}^{n_t}\\sum _{j=1}^{n_t} k(\\rho _{enc}(\\mathbf {x}_t^{i}),\\rho _{enc}(\\mathbf {x}_t^{j}))\\nonumber \\\\&&{-}\\:\\frac{2}{n_t n_f}\\sum _{i=1}^{n_f}\\sum _{j=1}^{n_t} k(\\rho _{enc}(\\hat{\\mathbf {x}}_t^{i}),\\rho _{enc}(\\mathbf {x}_t^{j})),$ where $k(x,x^{\\prime }) = e^{-\\gamma \\left\\Vert x-x^{\\prime }\\right\\Vert ^{2}}$ is the Gaussian kernel.", "Note that the generator's objective is not to generate realistic images but to generate fake noisy images with mixed image attributes from the target domain.", "This reduces the effort of training powerful generators, which is the focus in adversarial based domain adaptation approaches [25], [26], [27], [28], [29] used for domain alignment.", "Algorithm REF and REF list steps involved in CUDA training and inference, respectively.", "[t!]", "$b{ = }batch\\_size$ , $epochs{ = }max\\_epoch$ , $n_{batch}{ = } number\\ of\\ batches$ $\\theta $ /* parameter of contradistinguisher */${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{n_s}_{i=1}$ , ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ target domain prior $p(\\mathbf {y}_t)$ is known use $p(\\mathbf {y}_t)$ for the contradistinguish loss (REF ) /* target domain prior enforcing */ compute $p(\\mathbf {y}_t)$ assuming $p(\\mathbf {y}_t) = p(\\mathbf {y}_s)$ /* fair assumption as most datasets are well balanced */ $epoch = 1$ to $epochs$ $batch = 1$ to $n_{batch}$ sample a mini-batch ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{b}_{i=1}$ , ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{b}_{j=1}$ compute $\\mathcal {L}_{s}(\\theta )$  (REF ) using ${\\lbrace (\\mathbf {x}^{i}_s, \\mathbf {y}^{i}_s)\\rbrace }^{b}_{i=1}$ /* source supervised loss */ compute ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{b}_{j=1}$  (REF ) using ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{b}_{j=1}$ /* pseudo label selection step */ compute $\\mathcal {L}_{t}(\\theta )$  (REF ) fixing ${\\lbrace \\hat{\\mathbf {y}}^{j}_t\\rbrace }^{b}_{j=1}$ /* maximization step */ /* steps REF and REF together optimize unsupervised contradistinguish loss (REF ) */ adversarial regularization is enabled Generator $G_\\phi $ is used get fake samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ from Gaussian noise vectors ${\\lbrace \\eta ^{j}_t\\rbrace }^{b}_{j=1}$ using $G_\\phi $ , compute $\\mathcal {L}_{gen}(\\phi )$ (REF ) /* generator training */ get fake samples ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ by random sampling in the input feature space $\\mathcal {X}_t$ compute $\\mathcal {L}_{adv}(\\theta )$  (REF ) using ${\\lbrace \\hat{\\mathbf {x}}^{j}_t\\rbrace }^{b}_{j=1}$ /* fake samples are assigned to all classes equally */ combine losses in steps REF ,REF ,REF and REF to compute gradients using back-propagation update $\\theta $ using gradient descent /* and $\\phi $ if $G_\\phi $ is used */ CUDA Training [t!]", "${\\lbrace \\mathbf {x}^{i}_{test}\\rbrace }^{n_{test}}_{i=1}$ /* input test samples */${\\lbrace \\hat{\\mathbf {y}}^{i}_{test}\\rbrace }^{n_{test}}_{i=1}$ /* predicted labels */ $i = 1$ to $n_{test}$ predict label as $\\hat{\\mathbf {y}}^{i}_{test}{}={}\\operatornamewithlimits{arg\\,max}_{\\mathbf {y}\\in \\mathcal {Y}_t} p_{\\theta }(\\mathbf {y}|\\textbf {x}^{i}_{test})$ CUDA Inference Extension to Multi-Source Domain Adaptation Here, we argue that our proposed method CUDA has an implicit advantage in dealing with multi-source domain adaption problems over the techniques based on domain alignment.", "In a multi-source adaption setting, domain alignment methods need to consider the domain-shift between a source and a target domain and consider the domain-shift between the multiple source domains.", "Therefore, domain alignment methods are required to solve the even more complex intermediate problem of aligning multiple source and target domain distributions in addition to the complex intermediate problem of source and target domain alignment to deal with the multi-source domain adaptation problems.", "However, as the proposed method does not depend on the domain alignment, the extension to multi-source in the proposed method is very simple.", "As our main focus is to perform unsupervised learning directly on the target domain, the model obtained using is better generalized to the target domain and reduces overfitting on the source, which usually results in a negative transfer.", "We believe that this is one of the main advantages of addressing the domain adaptation by performing the primary task of target domain classification rather than the intermediate task of domain alignment.", "We propose a simple extension our proposed method to perform multi-source domain adaptation in the following manner.", "Let us suppose we are given with $R$ source domains ${\\lbrace s_1,\\ldots ,s_R\\rbrace }$ , consisting of labeled training data $({\\lbrace (\\mathbf {x}^{i}_{s_1}, \\mathbf {y}^{i}_{s_1})\\rbrace }^{n_{s_1}}_{i=1}, \\ldots , {\\lbrace (\\mathbf {x}^{i}_{s_R}, \\mathbf {y}^{i}_{s_R})\\rbrace }^{n_{s_R}}_{i=1})$ and unlabeled target domain instances ${\\lbrace \\mathbf {x}^{j}_t\\rbrace }^{n_t}_{j=1}$ .", "We compute the source supervised loss for the $r^{th}$ source domain using (REF ), i.e., $\\mathcal {L}_{s_r}(\\theta )$  (REF ) with ${\\lbrace (\\mathbf {x}^{i}_{s_r}, \\mathbf {y}^{i}_{s_r})\\rbrace }^{n_{s_r}}_{i=1}$ training data.", "We further compute the total multi-source supervised loss as $\\mathcal {L}_{s_{total}}(\\theta ) = \\sum _{r=1}^{R}\\mathcal {L}_{s_r}(\\theta ).$ We replace $\\mathcal {L}_{s}(\\theta )$  (REF ) in the total optimization objective with $\\mathcal {L}_{s_{total}}(\\theta )$  (REF ) in step REF of Algorithm REF .", "It should be noted that the unsupervised loss for the target domain is still unmodified irrespective of the number of source domains.", "We experimentally demonstrate the efficacy of the proposed multi-source domain adaptation extension on Office-31 [44] and Digits datasets [48], [49], [50], [21].", "Experiments For our domain adaptation experiments, we consider both synthetic and real-world datasets.", "Under synthetic datasets, we experiment using 2-dimensional blobs with different source and target domain probability distributions to demonstrate the effectiveness of the proposed method under different domain shifts.", "Under real-world datasets, we consider only the complex, high-resolution Office-31 [44] and VisDA-2017 [45] object classification datasets for our experiment as the low-resolution datasets are already addressed in our conference paper CUDA: Contradistinguisher for Unsupervised Domain Adaptation (CUDA) [43].", "We have published our python code for all the experiments at https://github.com/sobalgi/cuda.", "black Table: Details of visual domain adaptation datasets.Table REF provides details on the visual datasets used in our experiments.", "We also experiment and report the results of our ablation study carried out with different combinations of the three optimization objectives with their respective domains as the inputs involved in CUDA training: [(i)] source supervised loss: $ss$ described in (REF ), source/target unsupervised loss: $su/tu$ described in (REF ), source/target adversarial regularization loss: $sa/ta$ described in (REF ).", "$ss$ indicates the minimum target domain test accuracy that can be attained with a chosen contradistinguisher neural network by training only using the labeled source domain.", "Any improvement over $ss$ using CUDA (i.e., combination of $su/tu/sa/ta$ ) indicates effectiveness of CUDA as the chosen contradistinguisher neural network is fixed.", "Experiments on Synthetic Toy-dataset Figure: Contour plots show the probability contours along with clear decision boundaries on different toy-dataset settings trained using CUDA.", "(source domain: ×\\times , target domain: ++, class 0: blue, class 1: red.)", "(Best viewed in color.", ")We validate our proposed method by performing experiments on synthetically created simple datasets that model different source and target domain distributions in a 2-dimensional input feature space using different blobs of source-target domain orientations and offsets (i.e., domain shift).", "We create blobs for source and target domains with 4000 samples using standard $scikit{ - }learn$  [46] as indicated in Fig.", "REF and REF .", "We further evenly split these 4000 data-points into equal train and test sets.", "Each of the splits consists of the same number of samples corresponding to both the class labels.", "The main motivation of the experiments on toy-dataset is to understand and visualize the behavior of the proposed method under some typical domain distribution scenarios and analyze the performance of  CUDA.", "$Blobs$ toy-dataset plots in Fig.", "REF shows clear comparisons of the classifier decision boundaries learnt using CUDA over domain alignment approaches.", "The top row in Fig.", "REF corresponds to the domain alignment classifier trained only on the labeled source domain, i.e., $ss$ .", "However, the bottom row in Fig.", "REF corresponds to contradistinguisher trained using the proposed method CUDA with labeled source and unlabeled target domain, i.e., $ss{ + }tu{ + }ta$ .", "Fig.", "REF demonstrates the classifier learnt using CUDA on the synthetic datasets with different complex shapes and orientations of the source and target domain distributions for the input data.", "Fig.", "REF and REF indicate the simplest form of the domain adaptation tasks with similar orientations in source and target domain distributions.It is important to note that the prior enforcing used in pseudo-label selection is the reason such fine classifier boundaries are observed, especially in Fig.", "REF ,REF and REF -REF .", "Fig.", "REF and REF represent more complex configurations of source and target domain distributions that indicate the hyperbolic decision boundaries jointly learnt on both the domains simultaneously using a single classifier without explicit domain alignment.", "Similarly, Fig.", "REF represents a complex configuration of source and target domain distributions that indicates an elliptical decision boundary.", "These simulated experiments points to some significant inner workings of our approach CUDA.", "These are the two main takeaways from the toy-dataset experiments.", "[(i)] The non-necessity of the domain alignment in the form of distribution distance metric minimization or data augmentation.", "In the case of these toy-datasets, it is not possible to perform any form of data augmentation, unlike some of the visual domain adaptation tasks, because the data is directly available in the form of encoded features that cannot be easily data augmented through standard heuristics.", "In such a case, it is necessary to realize a generic approach applicable to multiple modalities of the input, e.g., similar to the toy-dataset in language domain adaptation tasks.", "The features are presented in the form of word2vec/doc2vec, and no data augmentation is possible.", "Fig.", "REF and REF provide some interesting observations.", "Here, we can observe that the classes in the source domain are overlapping, resulting in less than 100% classification on the source domain, which in turn results in less than 100% classification on the target domain when considering domain alignment approaches.", "However, CUDA does not try to morph the target domain onto the source domain by directly classifying on the target domain resulting in a perfect classification.", "Since the classification is done directly on the unlabeled target domain in a fully unsupervised manner, the target domain classification performance is not limited by the source domain classification performance, i.e., the irrespective of the domain is used as the labeled source domain and the unlabeled target domain, the performance is the respective domains are similar.", "In other words, swapping of the domains or the direction of the domain adaptation has little effect on the classification performance on each individual domain.", "Experiments on Real-world Datasets In our previous work [43], we have demonstrated the effectiveness of CUDA in real-world domain adaptation on low-resolution visual datasets and language datasets.", "In contrast to low-resolution visual datasets, we consider the complex, high-resolution Office-31 [44] and VisDA-2017 [45] object classification datasets for domain adaptation.", "In addition to the single-source domain adaptation experiments, we also extend CUDA to Office-31 [44] and Digits datasets [48], [49], [50], [21].", "Office-31 Dataset Figure: Illustrations of samples from all the three domains of high resolution Office-31  dataset with one instance per each class from every domain (column {1,4,7,10}: 𝒜\\mathcal {A}, {2,5,8,11}: 𝒟\\mathcal {D}, {3,6,9,12}: 𝒲\\mathcal {W}).", "(Best viewed in color.", ")Figure: Illustrations of samples from all the three data-splits of VisDA-2017  dataset with one instance per each class from every domain ({row 1}: 𝒱 syn \\mathcal {V}_{syn} source domain synthetic images (training set), {row 2}: 𝒱 real \\mathcal {V}_{real} target domain real-world images (validation set), {row 3}: 𝒱 real \\mathcal {V}_{real} target domain real-world images (testing set)).", "It should be noted that unlike the Office-31 dataset and other standard benchmark domain adaptation datasets discussed in , most of the real-world images in the target domain of the VisDA-2017 dataset contains multiple true labels, which are only annotated with only one of the multiple labels.", "(Best viewed in color.", ")In high-resolution visual datasets, we consider Office-31 [44] dataset for our experiments.", "Unlike low-resolution visual datasets, here, we have only a few hundreds of training samples that make this an even more challenging task.", "Office objects: Office-31 [44] dataset consists of high resolution images of objects belonging to 31 classes obtained from three different domains AMAZON ($\\mathcal {A}$ ), DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ).", "Fig.", "REF shows illustrations of the images from all the three above mentioned domains of the Office-31 [44] dataset.", "AMAZON ($\\mathcal {A}$ ) domain consists of synthetic images with clear white background.", "DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ) domains consist of real-world images with noisy background and surroundings.", "We consider all possible six combinatorial tasks of domain adaptation involving all the three domains, i.e., $\\mathcal {A}{ \\leftrightarrow }\\mathcal {D}$ , $\\mathcal {A}{ \\leftrightarrow }\\mathcal {W}$ and $\\mathcal {D}{ \\leftrightarrow }\\mathcal {W}$ .", "Compared to low-resolution visual datasets, Office-31 [44] dataset domain adaptation tasks have increased complexity due to the small number of training images.", "Unlike low-resolution visual datasets, the high-resolution Office-31 [44] dataset does not have separate pre-defined train and test splits.", "Since we do not use any labels from the target domain during training, we report ten-crop test accuracy on the target domain by summing the softmax values of all the ten crops of the image and assign the label with maximum aggregate softmax value for the given image as in CDAN [30] in Table REF .", "To further alleviate the lack of a large number of training samples, pre-trained networks such as ResNet-50 [65] and ResNet-152 [65] were used to extract 2048 dimensional features from high-resolution images similar to CDAN [30].", "Since the images are not well centered and have a high resolution, we use the standard ten-crop of the image to extract features from the same images during training and testing, also similar to CDAN [30].", "The use of pre-trained models leads to two choices of training, [(i)] Fine-tune the pre-trained model used as feature extractor along with the final classifier layer: This requires careful selection of several hyper-parameters such as learning rate, learning rate decay, batch size, etc., to fine-tune the network to the current dataset while preserving the ability of the pre-trained network.", "We observed that fine-tuning also depends on the loss function used for training [66], which in our case, the use of contradistinguish loss greatly affected the changes in the pre-trained model as it is trained only using cross-entropy loss.", "Fine-tuning is also computationally expensive and time-consuming as each iteration requires computing gradients of all the pre-trained model parameters.", "Fix the pre-trained model and only train the final classifier layer: Alternative to fine-tuning is to fix the pre-trained model and use it only as a feature extractor.", "This approach has multiple practical benefits such as, [(a)] The computational time and cost of fine-tuning the parameters of the pre-trained model are alleviated.", "Since the feature extractor is fixed, it requires only once to extract and store the features locally instead of extracting the same features every iteration.", "Hence reducing the training time and the GPU memory as it is only required to train the final classifier.", "VisDA-2017 Dataset The VisDA-2017 dataset consists of two domains, (i) synthetic and (ii) real, with three predefined data splits.", "Fig.", "REF indicates the samples from all the 12 classes of the three data splits.", "The three predefined data splits in the VisDA-2017 dataset are as follows.", "[(i)] Training set: This split includes 152,397 labeled synthetic images obtained using 2D renderings of 3D models from different angles and different lighting conditions.", "This split is considered as a labeled source domain for training.", "Validation set: This split includes 55,388 real-world images obtained from a curated subset of MS COCO [67] dataset.", "This split is considered an unlabeled target domain training set, and this is used during the training without labels.", "Testing set: This split includes 72,372 real-world images obtained from YouTube Bounding Boxes [68] dataset.", "This split is considered as the target domain testing set used for evaluation and to report the results.", "Analysis of Experimental Results on Real-world Datasets Figure: Row 1 and 2: t-SNE  plots for embeddings from the output of contradistinguisher with samples from Office-31  dataset as input corresponding to the highest mean accuracy setting ss+tu+su+tass{ + }tu{ + }su{ + }ta indicated in Table  for single-source domain adaptation using ResNet-152  as the fixed encoder.Row 3: t-SNE  plots for embeddings from the output of contradistinguisher corresponding to the samples from Office-31  dataset in high-resolution visual tasks after applying softmax trained with CUDA with ResNet-50  as the encoder in a multi-source domain adaptation setting as indicated in Table .We can observe the clear class-wise clustering among all the 31 classes in the Office-31  datasets.We achieve high accuracies in spite of having only a few hundred training samples in each domain.", "(Best viewed in color.", ")Table: Target domain accuracy (%) on high resolution Office-31  dataset containing three domains.CUDA corresponds to our best results obtained with the best hyper-parameter settings.ssss: source supervised (), tutu: target unsupervised (), susu: source unsupervised (), sasa: source adversarial regularization () and tata: target adversarial regularization () represents different training configurations.Table: Target domain accuracy (%) on high resolution Office-31  dataset under multi-source domain adaptation setting by combining two domains into a single source domain and the remaining domain as the target domain with ResNet-50  as the encoder.CUDA corresponds to our best results obtained with the best hyper-parameter settings.ssss: source supervised (), tutu: target unsupervised (), susu: source unsupervised (), sasa: source adversarial regularization () and tata: target adversarial regularization () represents different training configurations.Table: Target domain accuracy reported on the test set (%) on all 5 combinations of Digits datasets under multi-source domain adaptation setting.Figure: The t-SNE plots of the unseen test set samples corresponding to the CUDA result in Table .", "The t-SNE plots show clear clustering of all the 10 classes in Digits datasets distinctively.", "(Best viewed in color.", ")Figure: The t-SNE plots of CUDA/CUDA * ^{*} shows the clear clustering of all the twelve classes of VisDA-2017 distinctively compared to the t-SNE plots of BSP/CAN.The t-SNE plots of CUDA/CUDA * ^{*} represent some important visual semantics of the image embeddings obtained from contradistinguisher in the following manner.", "(i) The vehicular classes such as `bus', `car', `train', and `truck' can be seen clustered closely as semantically these classes are similar to each other (region bounded in red).", "(ii) The two-wheeler classes such as `bicycle' and `motorcycle' are clustered closely as these are semantically similar to each other compared to vehicular classes that are clustered exactly opposite (region bounded in green).", "(iii) Irrespective of the approach used, there is always confusion between `knife' and `skateboard' classes.", "This confusion between `knife' and `skateboard' classes represented in the confusion matrices, which is also clearly seen in the t-SNE plots as well, can be attributed to the nature of images of these classes in the dataset on close observation (region bounded in blue).", "(iv) The remaining classes such as `aeroplane', `horse', `person' and `plant' can be seen clustered independently and distinctively as these classes have almost no visual semantic similarities to one another.", "(Best viewed in color.", ")Table: Results on VisDA-2017 dataset reproduced from the current state-of-the-art method BSP, CAN and our proposed method CUDA reported on both the validation set and test set.", "We report all the evaluation metrics such as precision, recall, and accuracy, unlike BSP/CAN, where the recall scores are mistakenly reported as accuracy.", "CUDA * ^{*} represents the results reproduced using vanilla CUDA with the data augmentation and target domain clustering similar to CAN for a fair comparison of the effect of the CUDA over CAN.The results reported for BSP, CAN, and CUDA/CUDA * ^{*} are from our own best reproduction from the original source code.Table: Total classification accuracy (%) on VisDA-2017 dataset reported on both the validation set and test set.", "The results from JAN, GTA, CDAN and TransNorm are reported from TransNorm .The results reported for BSP, CAN and CUDA/CUDA * ^{*} are from our own best reproduction from the original source code.CUDA * ^{*} represents the results reproduced using vanilla CUDA with the data augmentation and target domain clustering similar to CAN for a fair comparison of the effect of the CUDA over CAN.Figure: We indicate few samples that are misclassified by the contradistinguisher in the following subcaption format `original_label|predicted_label'.In most cases, the original ground truth labels are dubious, and the predicted labels make more sense realistically.Subplots (6), (7), (11), (12), (14), (18), (23) and (25) shows that the object is identified based on the shape and not if the object is present only in the foreground.", "This indicates that the contradistinguisher makes the predictions based on the clearly visible shapes and not the presence of the object in the foreground/background.The visualization of the features responsible for the respective predicted outcome indicates the shape bias as mostly the features are detected as edges corresponding to the shape of the object in the image.", "This shows the importance of shape bias to achieve high performance in transfer learning and domain adaptation tasks.Office-31 Single-Source Domain Adaptation Results We report the standard ten-crop accuracy on the target domain images as reported by several state-of-the-art domain adaptation methods [30], [25], [3].", "Since there are no explicit test split specified in the dataset and no labels are used from the target domain during training, it is common to report ten-crop accuracy considering the whole target domain.", "In Table REF , we report accuracies obtained by fine-tuning ResNet-50 [65] using the learning rate schedule followed in CDAN [30] and also without fine-tuning ResNet-50 [65].", "Apart from fixed ResNet-50 [65], we also report accuracies with fixed ResNet-152 [65] in Table REF for comparison.", "Fig.", "REF -REF indicate the t-SNE [69] plots of the softmax output after aggregating the ten-crop of each image corresponding to training configuration $ss{ + }tu{ + }su{ + }ta$ reported in Table REF .", "Fig.", "REF reports the t-SNE [69] plots of the training setting using ResNet-152 [65] encoder with the highest mean accuracy of all the six domain adaptation tasks.", "We clearly observe that CUDA outperforms several state-of-the-art methods that also use ResNet-50 [65] and even further surpasses by using ResNet-152 [65] encoder with CUDA.", "Among the three domains in Office-31 [44] dataset, $\\mathcal {A}$ can be considered as a well-curated synthetic dataset with clear background and $\\lbrace \\mathcal {D},\\mathcal {W}\\rbrace $ as an uncurated real-world dataset with noisy background and surroundings.", "We report the six domain adaptation tasks in the order of their complexity from low to high as, [(i)] Fig.", "REF and REF indicate highest accuracies because of similar real-world to real-world domain adaptation task, Fig.", "REF and REF indicate moderately high accuracies because of synthetic to real-world domain adaptation task and Fig.", "REF and REF indicate the lowest accuracies among all the six tasks because of real-world to synthetic domain adaptation task.", "Comparing CUDA with $ss$ in Tables REF and REF , we can see significant improvements in the target domain test accuracies due to the use of contradistinguish loss (REF ) demonstrating the effectiveness of contradistinguisher.", "As our method is mainly dependent on the contradistinguish loss (REF ), we observed further improved results by experimenting with better neural networks, e.g., using ResNet-152 over ResNet-50 along with our contradistinguish loss (REF ).", "From our ablations study in Table REF , we observe the effect of selection of ImageNet pre-trained ResNet-50 and ResNet-152 models on the domain adaptation with similar implications with the work [70].", "In general, one can always obtain better results irrespective of the approach by using better/deeper pre-trained models and/or data augmentation.", "However, since our main aim is to isolate, observe and benchmark only the true effect of different benchmark approaches, in our experiments, we maintain all the other parameters such as pre-trained neural network/data augmentation similar except for the core idea of the approaches and report our results both on Office-31 and VisDA-2017 datasets.", "Multi-Source Domain Adaptation Results We also extend the experiments to multi-source domain adaptation on the Office-31 [44] and Digits datasets [48], [49], [50], [21].", "In Table REF , we can clearly observe that in $\\mathcal {A}{ + }\\mathcal {D}{ \\rightarrow }\\mathcal {W}$ task, multi-source domain adaptation provides better results than their respective best single source domain adaptation experiments.", "However in case of $\\mathcal {D}{ + }\\mathcal {W}{ \\rightarrow }\\mathcal {A}$ and $\\mathcal {W}{ + }\\mathcal {A}{ \\rightarrow }\\mathcal {D}$ , the multi-source domain adaptation improves over $ss$ , it underperforms compared to best single source domain adaptation task.", "This can be attributed to the fact that the model tends to overfit on the source domains resulting in a negative transfer.This negative transfer behavior is also prevalent in other multi-source domain adaptation approaches since all the other multi-source domain adaptation methods also underperform compared to their best single source domain adaptation results, as reported in Table REF .", "Fig.", "REF -REF indicates t-SNE [69] plots for embeddings from the output of contradistinguisher corresponding to the samples from Office-31 [44] dataset after applying softmax trained with CUDA with ResNet-50 [65] as the encoder in a multi-source domain adaptation setting.", "We can observe the best results when the target domain is one of the real-world domain, i.e., $\\mathcal {D}$ and $\\mathcal {W}$ .", "It was consistently observed that domain adaptation tasks with synthetic domain $\\mathcal {A}$ as the target domain to be the most complex tasks of all the domain adaptation tasks across all the domain adaptation methods.", "Similarly, Table REF presents the results of multi-source domain adaptation on Digits datasets against benchmark approaches.", "In Fig.", "REF , we see the t-SNE plots on the test set depicting clear class-wise clustering that indicates the efficacy of CUDA single-source to multi-source extension.", "VisDA-2017 Single-Source Domain Adaptation Results For experiments on the VisDA-2017 dataset, we consider the most recent state-of-the-art benchmark domain adaptation approaches BSP [36] and CAN [17].", "Like BSP and CAN, we use the same neural network architecture with Imagenet pre-trained ResNet-101 with contradistinguish loss for training.", "BSP and CAN report the evaluation metric of accuracy in their papers.", "However, on reproducing the results with BSPhttps://github.com/thuml/Batch-Spectral-Penalization and CANhttps://github.com/kgl-prml/Contrastive-Adaptation-Network-for-Unsupervised-Domain-Adaptation to set the baseline for comparison, we noticed that the results reported had the following inconsistencies.", "[(i)] The results reported in the paper as accuracy in actual were the class-wise recall scores.", "The most standard procedure in machine learning is to report experimental performances on the test split, which is unseen during the training.", "However, in BSP and CAN, the results are reported on the validation set, which is used as the unlabeled target domain training set.", "Since the VisDA-2017 dataset has pre-defined splits for evaluation, reporting the results on the validation set used during the training does not indicate these models' generalizing capability, which is the most important aspect one would base an evaluation.", "We correct the above misreporting by reproducing the results from BSP and CAN and report all the relevant metrics for both validation and test splits of the VisDA-2017 dataset.", "Apart from these, we also validate our results on the official challenge evaluation portalhttps://competitions.codalab.org/competitions/17052#results by submitting the results of our approach CUDA on the VisDA-2017 dataset.", "In Fig.", "REF , we report the t-SNE plots reproduced from the current state-of-the-art unsupervised domain adaptation approach BSP [36] and CAN [17], in comparison with our approach CUDA/CUDA$^*$ on both the pre-defined validation split (seen target domain training set) and testing split (unseen target domain testing set) of the VisDA-2017 dataset.", "The results reported as CUDA corresponds to the CUDA experiments without any data augmentation using the BSP source code as the baseline to keep all the parameters the same for a fair comparison with BSP.", "Similarly, the results reported as CUDA$^*$ corresponds to the CUDA experiments with data augmentation and clustering of high confidence target domain samples using the CAN source code as the baseline to keep all the parameters same for a fair comparison with CAN.", "We can see the classwise clusters in BSP/CUDA are narrower compared to CAN/CUDA$^*$ , which are broader due to the use of data augmentation.", "Data augmentation helps modify/broaden the data distribution aiding in an improvement over the vanilla approaches without data augmentation.", "We further validate the results best results from our approach, i.e., CUDA$^*$ by submitting to the official challenge evaluation leaderboard.", "In Table REF we compare the per-class precision, recall, and accuracies on both the pre-defined validation set and test set of the VisDA-2017 against the results reproduced from the BSP/CAN.", "In Table REF we compare the total classification accuracies on both the pre-defined validation set and test set of the VisDA-2017 dataset against different benchmark methods.", "The results in Tables REF and REF indicate the superior performance of the proposed method CUDA/CUDA$^*$ over the current state-of-the-art domain alignment approaches BSP/CAN on both the pre-defined validation set and a test set of VisDA-2017 dataset.", "Even though the validation set and test set belong to the real-world domain, there is an inherent domain shift between them as both the data splits are collected from two different datasets, i.e., MS COCO [67] and YouTube Bounding Boxes ,[68] respectively.", "Results from CUDA indicate a better generalization to real-world domain as the scores in validation and test sets are closer compared to other approaches on the VisDA-2017 dataset.", "We can also observe that the t-SNE plot of CUDA in Fig.", "REF and REF clearly shows the visual semantics captured between the classes of images in the VisDA-2017 dataset.", "Apart from setting CUDA as the solid baseline for VisDA-2017, we further put a conscious effort to carefully investigate the reasons for the misclassification using the contradistinguisher to check if we can further improve the results.", "In most misclassified cases, we have observed that the labels predicted by CUDA appeared to be correct in comparison to the ground truth label of the dataset.", "In Fig.", "REF , we present some of these instances where the predicted label is more close to the real label than the ground truth.", "We can explain this misclassification as a limitation of the VisDA-2017 dataset in the following way.", "The misclassification observed in Fig.", "REF is due to the fact that the images in the VisDA-2017 dataset consist of objects belonging to more than one of the twelve classes, i.e., the images in the dataset consists of multiple labels for a single image, but the dataset only records one of these several true labels.", "As the assumed task for domain adaptation is single-label multi-class We see this as a limitation of the VisDA-2017 dataset compared other benchmark domain adaptation datasets such as Office-31 or the low-resolution visual datasets demonstrated in our conference paper [43].", "It is necessary that the datasets be consistent in the sense that each image has a unique label corresponding to it so that during the evaluation, there is no ambiguity between the original label and the predicted label from the trained model.", "The presence of this ambiguity in the dataset classification would then lead to observing the true evaluation metrices resulting in improper benchmarking for any given approach.", "However, in the case of the VisDA-2017 dataset, the predicted label from the model cannot be considered as a wrong label as it contains the object of the predicted label in the image.", "We believe that this is one of the reasons for the overall low performance apart from the complexity of the VisDA-2017 dataset compared to other visual datasets.", "It should also be observed that CUDA identifies the most distinguishing/prominent and assigns the label irrespective of the position (foreground/background) of the object as indicated in some of the subplots in Fig.", "REF .", "These misclassified cases indicates one of the strong drawback/limitation of VisDA-2017 dataset compared to other visual datasets, i.e., VisDA-2017 dataset has image samples with multiple true labels instead of a unique label for each image sample.", "Since the images might contain multiple true classes for an image, ideally all these true labels are to be associated with the image in the dataset to rightly evaluate any trained model for its efficacy.", "Because we perform single-label multi-class classification, predicting any one of the true labels of the image should be considered as right for the evaluation metric.", "However, this is not the case as the dataset does not record all the true labels for the images.", "So, if one plans to rightly use this dataset, all the true labels are to be annotated for each of the images in the dataset or use other benchmark datasets such as DomainNet/LSDAC (Large Scale Domain Adaptation Challenge) dataset [18] that alleviates the problem of multi-labels of VisDA-2017 dataset as DomainNet dataset only consist of single true label per each image in the dataset, resulting in correct evaluation without the issue of misclassification we have indicated above for the VisDA-2017 dataset.", "Apart from analyzing the limitation of the VisDA-2017 dataset, we also analyze the nature of the feature representations learnt by contradistinguisher.", "In order to visualize the features that prompted the predicted label, we use Captumhttps://captum.ai/tutorials/Resnet_TorchVision_Interpret, an open-source, extensible library for model interpretability built on PyTorchhttps://pytorch.org/ [71].", "We use gradient-based attribution to compute the integrated gradients for a given image using the predicted label.", "We obtain the high-level features or the saliency maps [72] for the given image.", "In terms of high-level features in a given image, one can imagine features such as shape, color, texture, size, etc.", "to be the features that help in predicting the classifier outcome.", "Out of all these features, the most natural and basic feature influencing the outcome is observed to be the shapes of the objects.", "Extensive research materials in psychology such as [73], [74], [75], [76], [77] have indicated that human babies and adults tend to utilize shapes than color/material/texture to assign a word label to the given object.", "This particular phenomenon is widely termed as `shape bias' in the literature.", "However, recently, it was shown that the ImageNet pre-trained models possess a texture bias over shape bias [78].", "To improve the shape bias, [78] propose a new modified dataset called `Stylized-ImageNet' to overcome the texture bias.", "By increasing the shape bias, [78] demonstrated improved performance and robustness.", "Since we use the ImageNet pre-trained ResNet-101 as a feature extractor, it is necessary to understand the nature of extracted features from the input images.", "Unlike `Stylized-ImageNet', in domain adaptation tasks, one cannot always expect to get such a curated dataset with ground truth labels on the target domain for each task.", "Instead, it might be easy and desirable to change the loss function that enhances the shape features with the same training dataset.Surprisingly, in our observations, we find that the features learnt by the classifier indicate the high-level features or the saliency maps [72] representing the shape of the object in the image.", "The contradistinguish loss is formulated and optimized in such a way that the features extracted are most unique and contrastive for a given image in comparison to other images in the dataset.", "This consequently is observed as the features corresponding to shapes in the form of silhouette in the feature visualizations in Fig.", "REF as each object posses a unique shape as it's most contrasdistinguishing character, i.e., the character which is most discriminative and unique to the given image.", "Concluding Remarks In this paper, we have proposed a direct approach to solve the problem of unsupervised domain adaptation that is different from the standard distribution alignment approaches.", "In our approach, we jointly learn a Contradistinguisher on the source and target domain distribution in the same input-label feature space using contradistinguish loss for unsupervised target domain to identify contrastive features.", "We have shown that contrastive learning overcomes the need and drawbacks of domain alignment, especially in tasks where domain shift is very high (e.g., language domains) and data augmentation techniques cannot be applied.", "Due to the inclusion of prior enforcing in the contradistinguish loss, the proposed unsupervised domain adaptation method CUDA could incorporate any known target domain prior to overcoming the drawbacks of skewness in the target domain, thereby resulting in a skew-robust model.", "We validated the efficacy of CUDA by experimenting on the synthetically created toy-dataset.", "We further demonstrated the simplicity and effectiveness of our proposed method by performing multi-source domain adaptation on Office-31 and Digits datasets to consistently outperform other multi-source domain adaptation approaches.", "We have also tested the proposed method CUDA on the recent benchmark visual domain adaptation datasets such Office-31 and VisDA-2017 classification datasets and demonstrated above/on-par results with the state-of-the-art approaches.", "We further analyzed the nature of the feature representation learnt using contradistinguish loss to identify the features related to the shapes that influence the predicted outcome.", "As the features related to shapes are learnt, we observed that it helps improving the performance and robustness of the trained model as the model is not biased to colors/textures in the images.", "We concluded that learning and improving shape bias is one of the keys to achieve ideal transfer learning and domain adaptation.", "Acknowledgments The authors would like to thank the Ministry of Human Resource Development (MHRD), Government of India, for their generous funding towards this work through the UAY Project: IISc 001.", "The authors thank Tejas Duseja for helping the authors with setting up some experiments.", "The authors would also like to thank anonymous reviewers for providing their valuable feedback that helped in improving the manuscript.", "[Figure: NO_CAPTION [Figure: NO_CAPTION" ], [ "Experiments", "For our domain adaptation experiments, we consider both synthetic and real-world datasets.", "Under synthetic datasets, we experiment using 2-dimensional blobs with different source and target domain probability distributions to demonstrate the effectiveness of the proposed method under different domain shifts.", "Under real-world datasets, we consider only the complex, high-resolution Office-31 [44] and VisDA-2017 [45] object classification datasets for our experiment as the low-resolution datasets are already addressed in our conference paper CUDA: Contradistinguisher for Unsupervised Domain Adaptation (CUDA) [43].", "We have published our python code for all the experiments at https://github.com/sobalgi/cuda.", "black Table: Details of visual domain adaptation datasets.Table REF provides details on the visual datasets used in our experiments.", "We also experiment and report the results of our ablation study carried out with different combinations of the three optimization objectives with their respective domains as the inputs involved in CUDA training: [(i)] source supervised loss: $ss$ described in (REF ), source/target unsupervised loss: $su/tu$ described in (REF ), source/target adversarial regularization loss: $sa/ta$ described in (REF ).", "$ss$ indicates the minimum target domain test accuracy that can be attained with a chosen contradistinguisher neural network by training only using the labeled source domain.", "Any improvement over $ss$ using CUDA (i.e., combination of $su/tu/sa/ta$ ) indicates effectiveness of CUDA as the chosen contradistinguisher neural network is fixed.", "Experiments on Synthetic Toy-dataset Figure: Contour plots show the probability contours along with clear decision boundaries on different toy-dataset settings trained using CUDA.", "(source domain: ×\\times , target domain: ++, class 0: blue, class 1: red.)", "(Best viewed in color.", ")We validate our proposed method by performing experiments on synthetically created simple datasets that model different source and target domain distributions in a 2-dimensional input feature space using different blobs of source-target domain orientations and offsets (i.e., domain shift).", "We create blobs for source and target domains with 4000 samples using standard $scikit{ - }learn$  [46] as indicated in Fig.", "REF and REF .", "We further evenly split these 4000 data-points into equal train and test sets.", "Each of the splits consists of the same number of samples corresponding to both the class labels.", "The main motivation of the experiments on toy-dataset is to understand and visualize the behavior of the proposed method under some typical domain distribution scenarios and analyze the performance of  CUDA.", "$Blobs$ toy-dataset plots in Fig.", "REF shows clear comparisons of the classifier decision boundaries learnt using CUDA over domain alignment approaches.", "The top row in Fig.", "REF corresponds to the domain alignment classifier trained only on the labeled source domain, i.e., $ss$ .", "However, the bottom row in Fig.", "REF corresponds to contradistinguisher trained using the proposed method CUDA with labeled source and unlabeled target domain, i.e., $ss{ + }tu{ + }ta$ .", "Fig.", "REF demonstrates the classifier learnt using CUDA on the synthetic datasets with different complex shapes and orientations of the source and target domain distributions for the input data.", "Fig.", "REF and REF indicate the simplest form of the domain adaptation tasks with similar orientations in source and target domain distributions.It is important to note that the prior enforcing used in pseudo-label selection is the reason such fine classifier boundaries are observed, especially in Fig.", "REF ,REF and REF -REF .", "Fig.", "REF and REF represent more complex configurations of source and target domain distributions that indicate the hyperbolic decision boundaries jointly learnt on both the domains simultaneously using a single classifier without explicit domain alignment.", "Similarly, Fig.", "REF represents a complex configuration of source and target domain distributions that indicates an elliptical decision boundary.", "These simulated experiments points to some significant inner workings of our approach CUDA.", "These are the two main takeaways from the toy-dataset experiments.", "[(i)] The non-necessity of the domain alignment in the form of distribution distance metric minimization or data augmentation.", "In the case of these toy-datasets, it is not possible to perform any form of data augmentation, unlike some of the visual domain adaptation tasks, because the data is directly available in the form of encoded features that cannot be easily data augmented through standard heuristics.", "In such a case, it is necessary to realize a generic approach applicable to multiple modalities of the input, e.g., similar to the toy-dataset in language domain adaptation tasks.", "The features are presented in the form of word2vec/doc2vec, and no data augmentation is possible.", "Fig.", "REF and REF provide some interesting observations.", "Here, we can observe that the classes in the source domain are overlapping, resulting in less than 100% classification on the source domain, which in turn results in less than 100% classification on the target domain when considering domain alignment approaches.", "However, CUDA does not try to morph the target domain onto the source domain by directly classifying on the target domain resulting in a perfect classification.", "Since the classification is done directly on the unlabeled target domain in a fully unsupervised manner, the target domain classification performance is not limited by the source domain classification performance, i.e., the irrespective of the domain is used as the labeled source domain and the unlabeled target domain, the performance is the respective domains are similar.", "In other words, swapping of the domains or the direction of the domain adaptation has little effect on the classification performance on each individual domain.", "Experiments on Real-world Datasets In our previous work [43], we have demonstrated the effectiveness of CUDA in real-world domain adaptation on low-resolution visual datasets and language datasets.", "In contrast to low-resolution visual datasets, we consider the complex, high-resolution Office-31 [44] and VisDA-2017 [45] object classification datasets for domain adaptation.", "In addition to the single-source domain adaptation experiments, we also extend CUDA to Office-31 [44] and Digits datasets [48], [49], [50], [21].", "Office-31 Dataset Figure: Illustrations of samples from all the three domains of high resolution Office-31  dataset with one instance per each class from every domain (column {1,4,7,10}: 𝒜\\mathcal {A}, {2,5,8,11}: 𝒟\\mathcal {D}, {3,6,9,12}: 𝒲\\mathcal {W}).", "(Best viewed in color.", ")Figure: Illustrations of samples from all the three data-splits of VisDA-2017  dataset with one instance per each class from every domain ({row 1}: 𝒱 syn \\mathcal {V}_{syn} source domain synthetic images (training set), {row 2}: 𝒱 real \\mathcal {V}_{real} target domain real-world images (validation set), {row 3}: 𝒱 real \\mathcal {V}_{real} target domain real-world images (testing set)).", "It should be noted that unlike the Office-31 dataset and other standard benchmark domain adaptation datasets discussed in , most of the real-world images in the target domain of the VisDA-2017 dataset contains multiple true labels, which are only annotated with only one of the multiple labels.", "(Best viewed in color.", ")In high-resolution visual datasets, we consider Office-31 [44] dataset for our experiments.", "Unlike low-resolution visual datasets, here, we have only a few hundreds of training samples that make this an even more challenging task.", "Office objects: Office-31 [44] dataset consists of high resolution images of objects belonging to 31 classes obtained from three different domains AMAZON ($\\mathcal {A}$ ), DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ).", "Fig.", "REF shows illustrations of the images from all the three above mentioned domains of the Office-31 [44] dataset.", "AMAZON ($\\mathcal {A}$ ) domain consists of synthetic images with clear white background.", "DSLR ($\\mathcal {D}$ ) and WEBCAM ($\\mathcal {W}$ ) domains consist of real-world images with noisy background and surroundings.", "We consider all possible six combinatorial tasks of domain adaptation involving all the three domains, i.e., $\\mathcal {A}{ \\leftrightarrow }\\mathcal {D}$ , $\\mathcal {A}{ \\leftrightarrow }\\mathcal {W}$ and $\\mathcal {D}{ \\leftrightarrow }\\mathcal {W}$ .", "Compared to low-resolution visual datasets, Office-31 [44] dataset domain adaptation tasks have increased complexity due to the small number of training images.", "Unlike low-resolution visual datasets, the high-resolution Office-31 [44] dataset does not have separate pre-defined train and test splits.", "Since we do not use any labels from the target domain during training, we report ten-crop test accuracy on the target domain by summing the softmax values of all the ten crops of the image and assign the label with maximum aggregate softmax value for the given image as in CDAN [30] in Table REF .", "To further alleviate the lack of a large number of training samples, pre-trained networks such as ResNet-50 [65] and ResNet-152 [65] were used to extract 2048 dimensional features from high-resolution images similar to CDAN [30].", "Since the images are not well centered and have a high resolution, we use the standard ten-crop of the image to extract features from the same images during training and testing, also similar to CDAN [30].", "The use of pre-trained models leads to two choices of training, [(i)] Fine-tune the pre-trained model used as feature extractor along with the final classifier layer: This requires careful selection of several hyper-parameters such as learning rate, learning rate decay, batch size, etc., to fine-tune the network to the current dataset while preserving the ability of the pre-trained network.", "We observed that fine-tuning also depends on the loss function used for training [66], which in our case, the use of contradistinguish loss greatly affected the changes in the pre-trained model as it is trained only using cross-entropy loss.", "Fine-tuning is also computationally expensive and time-consuming as each iteration requires computing gradients of all the pre-trained model parameters.", "Fix the pre-trained model and only train the final classifier layer: Alternative to fine-tuning is to fix the pre-trained model and use it only as a feature extractor.", "This approach has multiple practical benefits such as, [(a)] The computational time and cost of fine-tuning the parameters of the pre-trained model are alleviated.", "Since the feature extractor is fixed, it requires only once to extract and store the features locally instead of extracting the same features every iteration.", "Hence reducing the training time and the GPU memory as it is only required to train the final classifier.", "VisDA-2017 Dataset The VisDA-2017 dataset consists of two domains, (i) synthetic and (ii) real, with three predefined data splits.", "Fig.", "REF indicates the samples from all the 12 classes of the three data splits.", "The three predefined data splits in the VisDA-2017 dataset are as follows.", "[(i)] Training set: This split includes 152,397 labeled synthetic images obtained using 2D renderings of 3D models from different angles and different lighting conditions.", "This split is considered as a labeled source domain for training.", "Validation set: This split includes 55,388 real-world images obtained from a curated subset of MS COCO [67] dataset.", "This split is considered an unlabeled target domain training set, and this is used during the training without labels.", "Testing set: This split includes 72,372 real-world images obtained from YouTube Bounding Boxes [68] dataset.", "This split is considered as the target domain testing set used for evaluation and to report the results.", "Analysis of Experimental Results on Real-world Datasets Figure: Row 1 and 2: t-SNE  plots for embeddings from the output of contradistinguisher with samples from Office-31  dataset as input corresponding to the highest mean accuracy setting ss+tu+su+tass{ + }tu{ + }su{ + }ta indicated in Table  for single-source domain adaptation using ResNet-152  as the fixed encoder.Row 3: t-SNE  plots for embeddings from the output of contradistinguisher corresponding to the samples from Office-31  dataset in high-resolution visual tasks after applying softmax trained with CUDA with ResNet-50  as the encoder in a multi-source domain adaptation setting as indicated in Table .We can observe the clear class-wise clustering among all the 31 classes in the Office-31  datasets.We achieve high accuracies in spite of having only a few hundred training samples in each domain.", "(Best viewed in color.", ")Table: Target domain accuracy (%) on high resolution Office-31  dataset containing three domains.CUDA corresponds to our best results obtained with the best hyper-parameter settings.ssss: source supervised (), tutu: target unsupervised (), susu: source unsupervised (), sasa: source adversarial regularization () and tata: target adversarial regularization () represents different training configurations.Table: Target domain accuracy (%) on high resolution Office-31  dataset under multi-source domain adaptation setting by combining two domains into a single source domain and the remaining domain as the target domain with ResNet-50  as the encoder.CUDA corresponds to our best results obtained with the best hyper-parameter settings.ssss: source supervised (), tutu: target unsupervised (), susu: source unsupervised (), sasa: source adversarial regularization () and tata: target adversarial regularization () represents different training configurations.Table: Target domain accuracy reported on the test set (%) on all 5 combinations of Digits datasets under multi-source domain adaptation setting.Figure: The t-SNE plots of the unseen test set samples corresponding to the CUDA result in Table .", "The t-SNE plots show clear clustering of all the 10 classes in Digits datasets distinctively.", "(Best viewed in color.", ")Figure: The t-SNE plots of CUDA/CUDA * ^{*} shows the clear clustering of all the twelve classes of VisDA-2017 distinctively compared to the t-SNE plots of BSP/CAN.The t-SNE plots of CUDA/CUDA * ^{*} represent some important visual semantics of the image embeddings obtained from contradistinguisher in the following manner.", "(i) The vehicular classes such as `bus', `car', `train', and `truck' can be seen clustered closely as semantically these classes are similar to each other (region bounded in red).", "(ii) The two-wheeler classes such as `bicycle' and `motorcycle' are clustered closely as these are semantically similar to each other compared to vehicular classes that are clustered exactly opposite (region bounded in green).", "(iii) Irrespective of the approach used, there is always confusion between `knife' and `skateboard' classes.", "This confusion between `knife' and `skateboard' classes represented in the confusion matrices, which is also clearly seen in the t-SNE plots as well, can be attributed to the nature of images of these classes in the dataset on close observation (region bounded in blue).", "(iv) The remaining classes such as `aeroplane', `horse', `person' and `plant' can be seen clustered independently and distinctively as these classes have almost no visual semantic similarities to one another.", "(Best viewed in color.", ")Table: Results on VisDA-2017 dataset reproduced from the current state-of-the-art method BSP, CAN and our proposed method CUDA reported on both the validation set and test set.", "We report all the evaluation metrics such as precision, recall, and accuracy, unlike BSP/CAN, where the recall scores are mistakenly reported as accuracy.", "CUDA * ^{*} represents the results reproduced using vanilla CUDA with the data augmentation and target domain clustering similar to CAN for a fair comparison of the effect of the CUDA over CAN.The results reported for BSP, CAN, and CUDA/CUDA * ^{*} are from our own best reproduction from the original source code.Table: Total classification accuracy (%) on VisDA-2017 dataset reported on both the validation set and test set.", "The results from JAN, GTA, CDAN and TransNorm are reported from TransNorm .The results reported for BSP, CAN and CUDA/CUDA * ^{*} are from our own best reproduction from the original source code.CUDA * ^{*} represents the results reproduced using vanilla CUDA with the data augmentation and target domain clustering similar to CAN for a fair comparison of the effect of the CUDA over CAN.Figure: We indicate few samples that are misclassified by the contradistinguisher in the following subcaption format `original_label|predicted_label'.In most cases, the original ground truth labels are dubious, and the predicted labels make more sense realistically.Subplots (6), (7), (11), (12), (14), (18), (23) and (25) shows that the object is identified based on the shape and not if the object is present only in the foreground.", "This indicates that the contradistinguisher makes the predictions based on the clearly visible shapes and not the presence of the object in the foreground/background.The visualization of the features responsible for the respective predicted outcome indicates the shape bias as mostly the features are detected as edges corresponding to the shape of the object in the image.", "This shows the importance of shape bias to achieve high performance in transfer learning and domain adaptation tasks.Office-31 Single-Source Domain Adaptation Results We report the standard ten-crop accuracy on the target domain images as reported by several state-of-the-art domain adaptation methods [30], [25], [3].", "Since there are no explicit test split specified in the dataset and no labels are used from the target domain during training, it is common to report ten-crop accuracy considering the whole target domain.", "In Table REF , we report accuracies obtained by fine-tuning ResNet-50 [65] using the learning rate schedule followed in CDAN [30] and also without fine-tuning ResNet-50 [65].", "Apart from fixed ResNet-50 [65], we also report accuracies with fixed ResNet-152 [65] in Table REF for comparison.", "Fig.", "REF -REF indicate the t-SNE [69] plots of the softmax output after aggregating the ten-crop of each image corresponding to training configuration $ss{ + }tu{ + }su{ + }ta$ reported in Table REF .", "Fig.", "REF reports the t-SNE [69] plots of the training setting using ResNet-152 [65] encoder with the highest mean accuracy of all the six domain adaptation tasks.", "We clearly observe that CUDA outperforms several state-of-the-art methods that also use ResNet-50 [65] and even further surpasses by using ResNet-152 [65] encoder with CUDA.", "Among the three domains in Office-31 [44] dataset, $\\mathcal {A}$ can be considered as a well-curated synthetic dataset with clear background and $\\lbrace \\mathcal {D},\\mathcal {W}\\rbrace $ as an uncurated real-world dataset with noisy background and surroundings.", "We report the six domain adaptation tasks in the order of their complexity from low to high as, [(i)] Fig.", "REF and REF indicate highest accuracies because of similar real-world to real-world domain adaptation task, Fig.", "REF and REF indicate moderately high accuracies because of synthetic to real-world domain adaptation task and Fig.", "REF and REF indicate the lowest accuracies among all the six tasks because of real-world to synthetic domain adaptation task.", "Comparing CUDA with $ss$ in Tables REF and REF , we can see significant improvements in the target domain test accuracies due to the use of contradistinguish loss (REF ) demonstrating the effectiveness of contradistinguisher.", "As our method is mainly dependent on the contradistinguish loss (REF ), we observed further improved results by experimenting with better neural networks, e.g., using ResNet-152 over ResNet-50 along with our contradistinguish loss (REF ).", "From our ablations study in Table REF , we observe the effect of selection of ImageNet pre-trained ResNet-50 and ResNet-152 models on the domain adaptation with similar implications with the work [70].", "In general, one can always obtain better results irrespective of the approach by using better/deeper pre-trained models and/or data augmentation.", "However, since our main aim is to isolate, observe and benchmark only the true effect of different benchmark approaches, in our experiments, we maintain all the other parameters such as pre-trained neural network/data augmentation similar except for the core idea of the approaches and report our results both on Office-31 and VisDA-2017 datasets.", "Multi-Source Domain Adaptation Results We also extend the experiments to multi-source domain adaptation on the Office-31 [44] and Digits datasets [48], [49], [50], [21].", "In Table REF , we can clearly observe that in $\\mathcal {A}{ + }\\mathcal {D}{ \\rightarrow }\\mathcal {W}$ task, multi-source domain adaptation provides better results than their respective best single source domain adaptation experiments.", "However in case of $\\mathcal {D}{ + }\\mathcal {W}{ \\rightarrow }\\mathcal {A}$ and $\\mathcal {W}{ + }\\mathcal {A}{ \\rightarrow }\\mathcal {D}$ , the multi-source domain adaptation improves over $ss$ , it underperforms compared to best single source domain adaptation task.", "This can be attributed to the fact that the model tends to overfit on the source domains resulting in a negative transfer.This negative transfer behavior is also prevalent in other multi-source domain adaptation approaches since all the other multi-source domain adaptation methods also underperform compared to their best single source domain adaptation results, as reported in Table REF .", "Fig.", "REF -REF indicates t-SNE [69] plots for embeddings from the output of contradistinguisher corresponding to the samples from Office-31 [44] dataset after applying softmax trained with CUDA with ResNet-50 [65] as the encoder in a multi-source domain adaptation setting.", "We can observe the best results when the target domain is one of the real-world domain, i.e., $\\mathcal {D}$ and $\\mathcal {W}$ .", "It was consistently observed that domain adaptation tasks with synthetic domain $\\mathcal {A}$ as the target domain to be the most complex tasks of all the domain adaptation tasks across all the domain adaptation methods.", "Similarly, Table REF presents the results of multi-source domain adaptation on Digits datasets against benchmark approaches.", "In Fig.", "REF , we see the t-SNE plots on the test set depicting clear class-wise clustering that indicates the efficacy of CUDA single-source to multi-source extension.", "VisDA-2017 Single-Source Domain Adaptation Results For experiments on the VisDA-2017 dataset, we consider the most recent state-of-the-art benchmark domain adaptation approaches BSP [36] and CAN [17].", "Like BSP and CAN, we use the same neural network architecture with Imagenet pre-trained ResNet-101 with contradistinguish loss for training.", "BSP and CAN report the evaluation metric of accuracy in their papers.", "However, on reproducing the results with BSPhttps://github.com/thuml/Batch-Spectral-Penalization and CANhttps://github.com/kgl-prml/Contrastive-Adaptation-Network-for-Unsupervised-Domain-Adaptation to set the baseline for comparison, we noticed that the results reported had the following inconsistencies.", "[(i)] The results reported in the paper as accuracy in actual were the class-wise recall scores.", "The most standard procedure in machine learning is to report experimental performances on the test split, which is unseen during the training.", "However, in BSP and CAN, the results are reported on the validation set, which is used as the unlabeled target domain training set.", "Since the VisDA-2017 dataset has pre-defined splits for evaluation, reporting the results on the validation set used during the training does not indicate these models' generalizing capability, which is the most important aspect one would base an evaluation.", "We correct the above misreporting by reproducing the results from BSP and CAN and report all the relevant metrics for both validation and test splits of the VisDA-2017 dataset.", "Apart from these, we also validate our results on the official challenge evaluation portalhttps://competitions.codalab.org/competitions/17052#results by submitting the results of our approach CUDA on the VisDA-2017 dataset.", "In Fig.", "REF , we report the t-SNE plots reproduced from the current state-of-the-art unsupervised domain adaptation approach BSP [36] and CAN [17], in comparison with our approach CUDA/CUDA$^*$ on both the pre-defined validation split (seen target domain training set) and testing split (unseen target domain testing set) of the VisDA-2017 dataset.", "The results reported as CUDA corresponds to the CUDA experiments without any data augmentation using the BSP source code as the baseline to keep all the parameters the same for a fair comparison with BSP.", "Similarly, the results reported as CUDA$^*$ corresponds to the CUDA experiments with data augmentation and clustering of high confidence target domain samples using the CAN source code as the baseline to keep all the parameters same for a fair comparison with CAN.", "We can see the classwise clusters in BSP/CUDA are narrower compared to CAN/CUDA$^*$ , which are broader due to the use of data augmentation.", "Data augmentation helps modify/broaden the data distribution aiding in an improvement over the vanilla approaches without data augmentation.", "We further validate the results best results from our approach, i.e., CUDA$^*$ by submitting to the official challenge evaluation leaderboard.", "In Table REF we compare the per-class precision, recall, and accuracies on both the pre-defined validation set and test set of the VisDA-2017 against the results reproduced from the BSP/CAN.", "In Table REF we compare the total classification accuracies on both the pre-defined validation set and test set of the VisDA-2017 dataset against different benchmark methods.", "The results in Tables REF and REF indicate the superior performance of the proposed method CUDA/CUDA$^*$ over the current state-of-the-art domain alignment approaches BSP/CAN on both the pre-defined validation set and a test set of VisDA-2017 dataset.", "Even though the validation set and test set belong to the real-world domain, there is an inherent domain shift between them as both the data splits are collected from two different datasets, i.e., MS COCO [67] and YouTube Bounding Boxes ,[68] respectively.", "Results from CUDA indicate a better generalization to real-world domain as the scores in validation and test sets are closer compared to other approaches on the VisDA-2017 dataset.", "We can also observe that the t-SNE plot of CUDA in Fig.", "REF and REF clearly shows the visual semantics captured between the classes of images in the VisDA-2017 dataset.", "Apart from setting CUDA as the solid baseline for VisDA-2017, we further put a conscious effort to carefully investigate the reasons for the misclassification using the contradistinguisher to check if we can further improve the results.", "In most misclassified cases, we have observed that the labels predicted by CUDA appeared to be correct in comparison to the ground truth label of the dataset.", "In Fig.", "REF , we present some of these instances where the predicted label is more close to the real label than the ground truth.", "We can explain this misclassification as a limitation of the VisDA-2017 dataset in the following way.", "The misclassification observed in Fig.", "REF is due to the fact that the images in the VisDA-2017 dataset consist of objects belonging to more than one of the twelve classes, i.e., the images in the dataset consists of multiple labels for a single image, but the dataset only records one of these several true labels.", "As the assumed task for domain adaptation is single-label multi-class We see this as a limitation of the VisDA-2017 dataset compared other benchmark domain adaptation datasets such as Office-31 or the low-resolution visual datasets demonstrated in our conference paper [43].", "It is necessary that the datasets be consistent in the sense that each image has a unique label corresponding to it so that during the evaluation, there is no ambiguity between the original label and the predicted label from the trained model.", "The presence of this ambiguity in the dataset classification would then lead to observing the true evaluation metrices resulting in improper benchmarking for any given approach.", "However, in the case of the VisDA-2017 dataset, the predicted label from the model cannot be considered as a wrong label as it contains the object of the predicted label in the image.", "We believe that this is one of the reasons for the overall low performance apart from the complexity of the VisDA-2017 dataset compared to other visual datasets.", "It should also be observed that CUDA identifies the most distinguishing/prominent and assigns the label irrespective of the position (foreground/background) of the object as indicated in some of the subplots in Fig.", "REF .", "These misclassified cases indicates one of the strong drawback/limitation of VisDA-2017 dataset compared to other visual datasets, i.e., VisDA-2017 dataset has image samples with multiple true labels instead of a unique label for each image sample.", "Since the images might contain multiple true classes for an image, ideally all these true labels are to be associated with the image in the dataset to rightly evaluate any trained model for its efficacy.", "Because we perform single-label multi-class classification, predicting any one of the true labels of the image should be considered as right for the evaluation metric.", "However, this is not the case as the dataset does not record all the true labels for the images.", "So, if one plans to rightly use this dataset, all the true labels are to be annotated for each of the images in the dataset or use other benchmark datasets such as DomainNet/LSDAC (Large Scale Domain Adaptation Challenge) dataset [18] that alleviates the problem of multi-labels of VisDA-2017 dataset as DomainNet dataset only consist of single true label per each image in the dataset, resulting in correct evaluation without the issue of misclassification we have indicated above for the VisDA-2017 dataset.", "Apart from analyzing the limitation of the VisDA-2017 dataset, we also analyze the nature of the feature representations learnt by contradistinguisher.", "In order to visualize the features that prompted the predicted label, we use Captumhttps://captum.ai/tutorials/Resnet_TorchVision_Interpret, an open-source, extensible library for model interpretability built on PyTorchhttps://pytorch.org/ [71].", "We use gradient-based attribution to compute the integrated gradients for a given image using the predicted label.", "We obtain the high-level features or the saliency maps [72] for the given image.", "In terms of high-level features in a given image, one can imagine features such as shape, color, texture, size, etc.", "to be the features that help in predicting the classifier outcome.", "Out of all these features, the most natural and basic feature influencing the outcome is observed to be the shapes of the objects.", "Extensive research materials in psychology such as [73], [74], [75], [76], [77] have indicated that human babies and adults tend to utilize shapes than color/material/texture to assign a word label to the given object.", "This particular phenomenon is widely termed as `shape bias' in the literature.", "However, recently, it was shown that the ImageNet pre-trained models possess a texture bias over shape bias [78].", "To improve the shape bias, [78] propose a new modified dataset called `Stylized-ImageNet' to overcome the texture bias.", "By increasing the shape bias, [78] demonstrated improved performance and robustness.", "Since we use the ImageNet pre-trained ResNet-101 as a feature extractor, it is necessary to understand the nature of extracted features from the input images.", "Unlike `Stylized-ImageNet', in domain adaptation tasks, one cannot always expect to get such a curated dataset with ground truth labels on the target domain for each task.", "Instead, it might be easy and desirable to change the loss function that enhances the shape features with the same training dataset.Surprisingly, in our observations, we find that the features learnt by the classifier indicate the high-level features or the saliency maps [72] representing the shape of the object in the image.", "The contradistinguish loss is formulated and optimized in such a way that the features extracted are most unique and contrastive for a given image in comparison to other images in the dataset.", "This consequently is observed as the features corresponding to shapes in the form of silhouette in the feature visualizations in Fig.", "REF as each object posses a unique shape as it's most contrasdistinguishing character, i.e., the character which is most discriminative and unique to the given image.", "Concluding Remarks In this paper, we have proposed a direct approach to solve the problem of unsupervised domain adaptation that is different from the standard distribution alignment approaches.", "In our approach, we jointly learn a Contradistinguisher on the source and target domain distribution in the same input-label feature space using contradistinguish loss for unsupervised target domain to identify contrastive features.", "We have shown that contrastive learning overcomes the need and drawbacks of domain alignment, especially in tasks where domain shift is very high (e.g., language domains) and data augmentation techniques cannot be applied.", "Due to the inclusion of prior enforcing in the contradistinguish loss, the proposed unsupervised domain adaptation method CUDA could incorporate any known target domain prior to overcoming the drawbacks of skewness in the target domain, thereby resulting in a skew-robust model.", "We validated the efficacy of CUDA by experimenting on the synthetically created toy-dataset.", "We further demonstrated the simplicity and effectiveness of our proposed method by performing multi-source domain adaptation on Office-31 and Digits datasets to consistently outperform other multi-source domain adaptation approaches.", "We have also tested the proposed method CUDA on the recent benchmark visual domain adaptation datasets such Office-31 and VisDA-2017 classification datasets and demonstrated above/on-par results with the state-of-the-art approaches.", "We further analyzed the nature of the feature representation learnt using contradistinguish loss to identify the features related to the shapes that influence the predicted outcome.", "As the features related to shapes are learnt, we observed that it helps improving the performance and robustness of the trained model as the model is not biased to colors/textures in the images.", "We concluded that learning and improving shape bias is one of the keys to achieve ideal transfer learning and domain adaptation.", "Acknowledgments The authors would like to thank the Ministry of Human Resource Development (MHRD), Government of India, for their generous funding towards this work through the UAY Project: IISc 001.", "The authors thank Tejas Duseja for helping the authors with setting up some experiments.", "The authors would also like to thank anonymous reviewers for providing their valuable feedback that helped in improving the manuscript.", "[Figure: NO_CAPTION [Figure: NO_CAPTION" ], [ "Concluding Remarks", "In this paper, we have proposed a direct approach to solve the problem of unsupervised domain adaptation that is different from the standard distribution alignment approaches.", "In our approach, we jointly learn a Contradistinguisher on the source and target domain distribution in the same input-label feature space using contradistinguish loss for unsupervised target domain to identify contrastive features.", "We have shown that contrastive learning overcomes the need and drawbacks of domain alignment, especially in tasks where domain shift is very high (e.g., language domains) and data augmentation techniques cannot be applied.", "Due to the inclusion of prior enforcing in the contradistinguish loss, the proposed unsupervised domain adaptation method CUDA could incorporate any known target domain prior to overcoming the drawbacks of skewness in the target domain, thereby resulting in a skew-robust model.", "We validated the efficacy of CUDA by experimenting on the synthetically created toy-dataset.", "We further demonstrated the simplicity and effectiveness of our proposed method by performing multi-source domain adaptation on Office-31 and Digits datasets to consistently outperform other multi-source domain adaptation approaches.", "We have also tested the proposed method CUDA on the recent benchmark visual domain adaptation datasets such Office-31 and VisDA-2017 classification datasets and demonstrated above/on-par results with the state-of-the-art approaches.", "We further analyzed the nature of the feature representation learnt using contradistinguish loss to identify the features related to the shapes that influence the predicted outcome.", "As the features related to shapes are learnt, we observed that it helps improving the performance and robustness of the trained model as the model is not biased to colors/textures in the images.", "We concluded that learning and improving shape bias is one of the keys to achieve ideal transfer learning and domain adaptation." ], [ "Acknowledgments", "The authors would like to thank the Ministry of Human Resource Development (MHRD), Government of India, for their generous funding towards this work through the UAY Project: IISc 001.", "The authors thank Tejas Duseja for helping the authors with setting up some experiments.", "The authors would also like to thank anonymous reviewers for providing their valuable feedback that helped in improving the manuscript.", "[Figure: NO_CAPTION [Figure: NO_CAPTION" ] ]

[ [ "High-resolution observations of the solar photosphere, chromosphere and\n transition region. A database of coordinated IRIS and SST observations" ], [ "Abstract NASA's Interface Region Imaging Spectrograph (IRIS) provides high resolution observations of the solar atmosphere through UV spectroscopy and imaging.", "Since the launch of IRIS in June 2013, we have conducted systematic observation campaigns in coordination with the Swedish 1-m Solar Telescope (SST) on La Palma.", "The SST provides complementary high-resolution observations of the photosphere and chromosphere.", "The SST observations include spectro-polarimetric imaging in photospheric Fe I lines and spectrally-resolved imaging in the chromospheric Ca II 8542 A, H-alpha, and Ca II K lines.", "We present a database of co-aligned IRIS and SST datasets that is open for analysis to the scientific community.", "The database covers a variety of targets including active regions, sunspots, plage, quiet Sun, and coronal holes." ], [ "Introduction", "The solar atmosphere is a very dynamic region, where fundamental physical processes take place on small spatial scales and short dynamical time scales, often leading to rapid changes in the thermodynamic state of the plasma.", "Resolving these processes in observations requires high resolution in the combined spatial, temporal, and spectral domains.", "Furthermore, the combination of multiple spectral diagnostics, preferably with sensitivity to line formation conditions that cover a large range in temperatures, densities, and magnetic field topologies, are of fundamental importance for advancing our understanding of the solar atmosphere.", "The simultaneous acquisition of vastly different spectral diagnostics is possible through coordinated observations between space-borne and ground-based observing facilities.", "Telescopes in space provide unique access to the short wavelength regime with seeing-free diagnostics of the chromosphere, transition region and corona.", "Ground-based telescopes allow for high resolution in photospheric and chromospheric diagnostics, as well as high-sensitivity polarimetric measurements of the magnetic field with instrumentation that can be more complex than in space, and which is not limited by data transfer rates.", "Coordinated observations, therefore, strongly enhance the potential to unravel connections in the solar atmosphere that span from the photosphere, through the chromosphere and transition region to the corona.", "The Interface Region Imaging Spectrograph [8], a NASA Small Explorer (SMEX) satellite, was launched on 2013-Jun-27.", "It combines high resolution in the spatial (03–04), temporal (down to 1 s), and spectral domains (velocity determination down to 1 km$\\;$ s$^{-1}$ ).", "Spectral diagnostics include the Mgii h & k resonance lines (chromosphere), the Cii lines at 1335 Å (upper chromosphere and transition region), and the Siiv lines at 1400 Å (transition region).", "Furthermore, the (weaker) Oiv lines around 1400 Å as well as the Fexii 1349 Å and Fexxi 1354 Å lines provide diagnostics on the corona and high-energy flares.", "Slit-jaw imaging in the Mgii k core, Mgii h wing, Cii, and Siiv lines provides valuable context information.", "The IRIS satellite offers considerable flexibility in its observing configuration, and, for example, allows for a wide variety in area coverage (i.e., field-of-view (FOV) size), temporal cadence, and choice of spectral diagnostics.", "Target selection is organized through a system with relatively short communication lines and allows for effective coordination with ground-based telescopes and other observing facilities.", "This has opened up possibilities to expand on IRIS's rich arsenal of spectral diagnostics, for example by adding photospheric and chromospheric spectropolarimetry and high-resolution imaging in various spectral lines at and around the area covered by the IRIS spectrograph slit.", "Shortly after IRIS was launched, scientists from the University of Oslo and from the Lockheed Martin Solar and Astrophysics Laboratory (LMSAL) started organizing coordinated observing campaigns with the Swedish 1-m Solar Telescope [34] on La Palma.", "Every year, four campaigns – typically two weeks each – are conducted during the SST observing season (April – October).", "The SST is capable of providing high-quality time series of spectrally resolved photospheric and chromospheric diagnostics that under excellent seeing conditions reach the diffraction limit of $<$ 01 over the full arcmin$^2$ FOV.", "Furthermore, the versatile CRISP instrument can provide spectropolarimetric data that enable the measurement of the magnetic field topology.", "In addition, the tunable filter system CHROMIS, installed in 2016, can simultaneously provide narrowband filtergrams at several wavelengths in the core of the Caii K line.", "Data from the coordinated campaigns have been used to study a variety of topics, including: the disk counterparts of spicules [7], [29], [32], [18], [1], chromospheric bright grains in the internetwork [19] and active region plage [39], penumbral microjets in sunspots [43], [9], the atmospheric stratification in plage [3] and sunspots [2], the relation between Ellerman Bombs and ultraviolet (UV) bursts [44], [10], [30], [42], [23], Ellerman bombs in the quiet Sun [31], magnetic flux emergence from the photosphere to the transition region [22], surges [20], and the chromospheric counterparts of transition-region unresolved fine structure loops [26].", "In this paper, we describe the public release of co-aligned IRIS and SST data.", "At first, the public release is limited to data products that share the same plate scale as IRIS (017 per pixel) for easier data analysis.", "This pixel scale implies that the spatial resolution of the SST data is degraded.", "The release of the corresponding full spatial resolution SST data is planned for future data releases.", "The IRIS telescope design and instrumentation are described in [8].", "The IRIS satellite acquires spectra in three spectral regions: in the far UV from 1332 to 1358 Å (FUV1), in the far UV from 1389 to 1407 Å (FUV2), and in the near UV from 2783 to 2834 Å (NUV).", "The FUV1 region is dominated by the Cii lines at 1334 and 1335 Å that are formed in the upper chromosphere [27], [28], the FUV2 region is dominated by the Siiv lines at 1394 and 1403 Å that are formed in the transition region.", "The NUV region is dominated by the chromospheric Mgii h and k lines [12], [13], and further hosts the upper photospheric and lower chromospheric Mgii triplet lines [24] and a large number of (upper) photospheric blends in the strong Mgii wings [25].", "The 033 wide spectrograph slit has a length of 175 and can be displaced with respect to the solar surface to build up a raster that samples an area up to 130$\\times $ 175.", "There are several choices of step sizes between consecutive slit positions: dense sampling with 035 steps, sparse sampling with 1 steps, or coarse sampling with 2 steps.", "Alternatively, the spectrograph can record data in a sit-and-stare mode, where the slit does not move and stays at a fixed location (with or without tracking for solar rotation).", "The IRIS satellite can take slit-jaw images (SJIs) with different filters to provide context around the spectrograph slit.", "The four science SJI channels are: SJI 2796, centered on Mgii k (4 Å bandpass); SJI 2832, centered at 2830 Å in the Mgii h wing (4 Å bandpass); SJI 1330, centered at 1340 Å and dominated by the Cii lines (55 Å bandpass); and SJI 1400, centered at 1390 Å and dominated by the Siiv lines (55 Å bandpass).", "Slit-jaw images from different channels are recorded sequentially and have the same exposure time as the spectrograms recorded with the spectrograph.", "Various choices can be made to reduce the data volume in order to fit within the daily limits of data transfer from the spacecraft to ground stations.", "For each spectral line of interest, the wavelength range can be selected to limit the data transferred, or the spatial extent of the raster can be limited by transferring only data from a reduced part along the slit.", "Other measures to reduce data transfer are compression, data binning (spatially and/or spectrally), and omitting one or several SJI channels (most frequently SJI 2832 is omitted, although this is often done to improve the cadence of the other SJI channels).", "Taken together, the various possible choices in raster step size, number of slit positions, slit length, SJI channel selection, exposure time, spatial and spectral binning, compression, and spectral line selection (line lists) constitute a considerable number of possible observing programs.", "These programs are identified by a unique number, the OBS number [8].", "The OBSID, together with the observing date and start time, constitute a unique identifier for each dataset (see the first three columns in Table )." ], [ "SST", "The SST telescope design and its main optical elements are described in [34].", "A description of upgrades of optical components and instrumentation, as well as a thorough evaluation of optical performance is provided by [36].", "An adaptive optics system is fully integrated in the optical system [35] and was upgraded with an 85-electrode deformable mirror operating at 2 kHz in 2013.", "A dichroic beam splitter divides the beam on the optical table into a red ($>500$  nm) and a blue beam.", "Both beams are equipped with tunable filter instruments: the CRISP imaging spectropolarimeter [37] on the red beam, and the CHROMIS imaging spectrometer on the blue beam.", "Both CRISP and CHROMIS are dual Fabry–Pérot filtergraph systems based on the design by [33] and are capable of fast wavelength sampling of spectral lines.", "Before the installation of CHROMIS in September 2016, the blue beam was equipped with a number of interference filters, including a full width at half maximum (FWHM) of 10 Å wide filter for photospheric imaging at 3954 Å, and an FWHM=1 Å wide filter centered on the Caii H line core at $\\lambda =3968$  Å [16].", "The CRISP instrument has a pair of liquid crystals that together with a polarising beam splitter allow measurements of circular and linear polarisation in for example the photospheric Fei 6173 Å, Fei 6301 Å, and Fei 6302 Å lines, and the chromospheric Caii 8542 Å line.", "The CRISP instrument has a plate scale of 0058 per pixel and the SST diffraction limit ($\\lambda /D$ ) is 014 at the wavelength of H$\\alpha $ (with the telescope aperture diameter $D$ =0.97 m).", "The transmission profile of CRISP has FWHM=60 mÅ at the wavelength of H$\\alpha $.", "The CHROMIS instrument has a plate scale of 0038 per pixel and the SST diffraction limit is 008 at the wavelength of Caii K. The transmission profile of CHROMIS has FWHM$\\approx $ 120 mÅ at the wavelength of Caii K. The FOV of CRISP and CHROMIS is approximately 1$\\times $ 1.", "We note that sunlight collected by the SST is split by a dichroic beam splitter such that CRISP and CHROMIS can operate independently and in parallel, without reducing the efficiency of either instrument.", "Image restoration by means of the multi-object multi-frame blind deconvolution [15], [40] method is applied to all data to enhance image quality over the full FOV.", "The MOMFBD restoration is integrated in the CRISP and CHROMIS data processing pipelines [6], [17].", "These pipelines include the method described by [11] for consistency between sequentially recorded liquid crystal states and wavelengths, with destretching performed as in [38].", "The CRISP and CHROMIS instruments include auxiliary wideband (WB) systems which are essential as anchor channels in MOMFBD restoration.", "Furthermore, they provide photospheric reference channels that facilitate precise co-alignment between CRISP and CHROMIS data (or blue beam filter data before 2016), or co-alignment with data from IRIS and the Solar Dynamic Observatory [14]." ], [ "SST observing programs", "The SST observing programs vary from campaign to campaign, and often during campaigns as well, depending on the target and science goals.", "Common to all datasets in the database is the inclusion of at least one chromospheric line, H$\\alpha $ or Caii 8542 Å, and often both.", "In order to keep the temporal cadence below 20 s, the Caii 8542 Å observations were most often carried out in non-polarimetric mode.", "During the 2013 and 2014 observing seasons, photospheric spectropolarimetry was limited to one single position in the blue wing of the Fei 6302 Å line.", "The Stokes V maps serve as effective locators of the strongest magnetic field regions and polarity indicators.", "An example of such a blue wing Fei 6302 Å Stokes V map can be seen in Fig.", "REF for the 2014-Sep-09 and 2014-Sep-15 datasets, as well as in Fig.", "REF .", "During later campaigns, spectral sampling of photospheric Fei lines was extended.", "These observations were subjected to a fast and robust pixel-to-pixel Milne-Eddington (ME) inversion procedure.", "The parallel C++ implementationhttps://github.com/jaimedelacruz/pyMilne [4] is based upon the analytical intensity derivatives described by [21] and an efficient Levenberg-Marquardt algorithm that is described in [5].", "Example line of sight (LOS) magnetic field strength ($B_\\textrm {LOS}$ ) maps from Fei 6173 Å inversions are shown in Fig.", "REF for observing date 2015-Jun-26 and in Fig.", "REF for 2015-Sep-17.", "The database contains maps of $B_\\textrm {LOS}$ , plane of the sky magnetic field strength $B_\\textrm {perp}$ , and LOS velocity from these ME inversions.", "For datasets for which spectropolarimetric Caii 8542 Å observations were taken, we include magnetograms that were constructed by summing Stokes V data from the blue wing of the Caii line, and subtracting the corresponding sum from the red wing.", "These serve as photospheric magnetic field maps in a similar way as the Fei 6302 Å Stokes V maps.", "Examples can be found in Fig.", "REF for observing dates 2016-Apr-29 and 2016-Sep-03." ], [ "IRIS and SST co-alignment", "For the co-alignment of the IRIS and SST data, we employ cross-correlation of image pairs that are morphologically as similar as possible.", "Most often, the SJI 2796 and Caii 8542 Å wing (at 0.8 – 1.2 Å offset from line core) or Caii K wing show similar enough scenes to give satisfying results.", "This is particularly true for more quiet regions with the characteristic mesh-like pattern from acoustic shocks and the surrounding network of high-contrast bright regions.", "For active regions with enhanced flaring activity or large sunspots, the SJI 2796 and Caii 8542 Å wing pair can have more dissimilar scenes and therefore the co-alignment can be less reliable.", "The combination SJI 2832 with CRISP WB or H$\\alpha $ far wing gives excellent co-alignment results since both channels show pure photospheric scenes.", "However, SJI 2832 is not always selected for the IRIS observing programs to limit the data rate and improve the cadence of the other SJI channels.", "Before cross-correlation, the plate scales between image pairs are matched.", "Offsets are then determined by cross-correlation over a subfield of the common FOV of image pairs that are closest in time.", "Examples of such subfields are outlined by white rectangles in Figs.", "REF –REF .", "The raw offsets are then smoothed with a temporal window to account for jitter due to noise.", "The offsets that are applied to the data are interpolated to the relevant time grid of the particular diagnostic.", "The precision of the alignment is limited by a number of factors.", "Formation height differences between the diagnostics used for cross-correlation may introduce a systematic offset that is difficult to account for.", "This is probably of limited concern for cross-correlation between photospheric diagnostics involving SJI 2832, but it is more uncertain between SJI 2796 and the Caii 8542 wing.", "The systematic offset may be higher for oblique observing angles towards the limb and may also depend on the type of target (for example, active regions with flaring activity that appears less prominent in the Caii 8542 wing than in the Mgii k core).", "Furthermore, varying seeing conditions at the SST inevitably lead to image distortions that cannot be fully accounted for in post-processing.", "We estimate that the error in the co-alignment can be as good as or better than one IRIS pixel (017, in the case of SST data taken under excellent conditions and closely matching diagnostic pairs in the cross-correlation).", "However, we also see local offsets due to image warping that can be as large as $\\sim $ 2 IRIS pixels.", "These local offsets vary in magnitude proportionally with the seeing conditions.", "For the current release of data to the database, the IRIS data is kept as reference.", "This means that the SST data is down-scaled to the IRIS plate scale (for CRISP with a factor 2.9, for CHROMIS with a factor 4.4), rotated and clipped to match the IRIS FOV and orientation, and clipped in time to match the IRIS observation duration.", "We have also applied the reverse approach, keeping at least the SST data at its superior spatial resolution for analysis in earlier publications.", "These types of data products are considered for future data releases but we note that the quality control is a laborious effort, partly due to the alignment uncertainties outlined above.", "For future studies one can consider the use of the more highly resolved SST data to uncover possible fine structure below the IRIS resolution, which could be of importance for the interpretation of the data.", "Such analysis could be performed by comparing individual spectra from both datasets, or by using a newly developed spatially coupled inversion method that allows for the combining of datasets acquired at different spatial resolution [4].", "Each dataset would set constraints in the reconstructed model down to the smallest spatial scales that are present in the data without affecting the information provided by the other datasets that are included in the inversion." ], [ "Data in the database", "Table  gives an overview of various parameters that characterize datasets in the public database.", "There is a variety of targets, including the quiet Sun, coronal holes, enhanced networks, active regions with and without sunspots, and plages.", "The data can be accessed through the database, available through the public web portal at the IRIS web pages at LMSALhttps://iris.lmsal.com/search/.", "The data will also be available through the Hinode Science Data Centre Europe hosted at the University of Oslohttp://sdc.uio.no/sdc/.", "The data products that are publicly released are FITS files in so-called IRIS level 3 format.", "Level 3 data are data cubes that are a recast of the standard IRIS level 2 data files.", "Level 2 data are the science-ready data files that have been processed to include corrections for dark current and flat field, geometric distortions and wavelength calibration [45].", "The level 3 data are four-dimensional data cubes with $(x,y,\\lambda ,t)$ -axes: the spatial $x$ -axis along the raster slit positions, the spatial $y$ -axis along the spectrograph slit, the $\\lambda $ -axis along the wavelength dimension, and $t$ the temporal axis.", "The level 3 data cubes in the database can be readily accessed with CRISPEX [41], [17], a graphical user interface written in the Interactive Data Language (IDL).", "It allows for side-by-side browsing and basic time-series analysis of the IRIS rasters, SJIs and CRISP or CHROMIS data.", "The CRISPEX interface is distributed as part of the IRIS package in SolarSoft IDL and can also be downloaded https://github.com/grviss/crispex separately; however, it requires SolarSoft for full functionality when inspecting the IRIS-SST data.", "Tutorials for its use are available online.REF ,https://iris.lmsal.com/tutorials.html Overview of the datasets available in the database.", "Table: NO_CAPTION aObserving date in format year, month, day.", "bStarting time (UT) of observations in format hour, min, s. cThe OBSID number encodes the IRIS observing configuration in a unique number [8].", "The combination <Date>_<Time>_<OBSID> constitutes a unique identifier to the dataset.", "dThe IRIS spectrograph slit covers a region on the Sun through a raster of $n$ slit positions, with a separation of type: dense (035), sparse (1), or coarse (2).", "Raster type s&s is the “sit-and-stare” mode for which the slit remains fixed at one location.", "The area covered is shown in the FOV column, and the temporal cadence in the Cad.", "column.", "eTarget: AR: active region, QS: quiet Sun, CH: coronal hole, S: sunspot.", "fPointing coordinates at the beginning of the time series, the target is followed by tracking solar rotation.", "g$\\mu = \\cos \\theta $ with $\\theta $ the observing angle.", "hDuration of overlap of SST observations with IRIS in format hh:mm:ss.", "iSpectral lines observed with SST.", "CRISP is operated independently from the instruments on the blue beam; with fixed interference filters (Caii H) or CHROMIS (Caii K).", "The instruments have their own cadences and overlap times and are separated in rows in the table.", "jCadence of the SST observations.", "kReferences to publications based on these data sets: 1: [7], 2: [29], 3: [19], 4: [3] 5: [43], 6: [44], 7: [31] 8: [39] 9: [32], 10: [18], 11: [20], 12: [30], 13: [42], 14: [2], 15: [23] 16: [9].", "Figure: Sample images from four different datasets.", "Each row shows four different diagnostics.", "The first two images on each row show the channel pairs that were used for IRIS and SST co-alignment and the area outlined by the white rectangle marks the region used for cross-correlation to determine offsets.", "The dashed red line in the second image marks the location of the IRIS slit in the SJI image to the left.", "The dashed purple lines in the third image mark the area covered by the IRIS raster.", "The SST images are down-scaled to the IRIS plate scale.Figure: Sample images from four different datasets.", "The format is the same as Fig.", ".Figure: Sample images from four different datasets.", "The format is the same as Fig.", ".Figure: Sample images from four different datasets with large IRIS rasters.", "Each row shows five different diagnostics.", "The first two images in each row show the channel pairs that were used for IRIS and SST co-alignment and the area outlined by the white rectangle marks the region used for cross-correlation to determine offsets.", "The dashed red line in the second image marks the location of the IRIS slit in the SJI image to the left.", "The dashed purple lines mark the area covered by the IRIS raster.", "The three right-most images show spectroheliograms constructed from the raster data cubes: at the nominal line core wavelengths of the Mgii k and Hα\\alpha line cores, and a blue wing Fei 6302 Å Stokes V map at -0.048-0.048 Å.", "The SST images are down-scaled to the IRIS plate scale.This paper is dedicated to Ted Tarbell who passed away in April 2019.", "Ted was the leader of the LMSAL SVST/SST campaigns since the 1980s through the 2000s and participated with great enthusiasm in the LMSAL campaigns in 2013 and 2014.", "Ted was a great friend and an inspiring mentor to his junior colleagues.", "The Swedish 1-m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.", "The Institute for Solar Physics is supported by a grant for research infrastructures of national importance from the Swedish Research Council (registration number 2017-00625).", "IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre.", "We thank the following people for their assistance at the SST: Jack Carlyle, Tiffany Chamandy, Henrik Eklund, Thomas Golding, Chris Hoffmann, Charalambos Kanella, Ingrid Marie Kjelseth, and Bhavna Rathore.", "We further acknowledge excellent support at the SST by Pit Sütterlin.", "We are also grateful to the IRIS planners for the IRIS-SST coordination.", "This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 824135.", "This research is supported by the Research Council of Norway, project number 250810, and through its Centres of Excellence scheme, project number 262622.", "BDP and colleagues at LMSAL and BAERi acknowledge support from NASA contract NNG09FA40C (IRIS).", "JdlCR is supported by grants from the Swedish Research Council (2015-03994), the Swedish National Space Agency (128/15) and the Swedish Civil Contingencies Agency (MSB).", "This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (SUNMAG, grant agreement 759548).", "VMJH and SJ receive funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.", "682462).", "JMS is supported by NASA grants NNX17AD33G, 80NSSC18K1285, and NSF grant AST1714955.", "GV is supported by a grant from the Swedish Civil Contingencies Agency (MSB).", "CF acknowledges funding from CNES.", "We made much use of NASA's Astrophysics Data System Bibliographic Services." ] ]

[ [ "Restoring the night sky darkness at Observatorio del Teide: First\n application of the model Illumina version 2" ], [ "Abstract The propagation of artificial light into real environments is complex.", "To perform its numerical modelling with accuracy one must consider hyperspectral properties of the lighting devices and their geographic positions, the hyperspectral properties of the ground reflectance, the size and distribution of small-scale obstacles, the blocking effect of topography, the lamps angular photometry and the atmospheric transfer function (aerosols and molecules).", "A detailed radiative transfer model can be used to evaluate how a particular change in the lighting infrastructure may affect the sky radiance.", "In this paper, we use the new version (v2) of the Illumina model to evaluate a night sky restoration plan for the Teide Observatory located on the island of Tenerife, Spain.", "In the past decades, the sky darkness was severely degraded by growing light pollution on the Tenerife Island.", "In this work, we use the contribution maps giving the effect of each pixel of the territory to the artificial sky radiance.", "We exploit the hyperspectral capabilities of Illumina v2 and show how the contribution maps can be integrated over regions or municipalities according to the Johnson-Cousins photometric bands spectral sensitivities.", "The sky brightness reductions per municipality after a complete shutdown and a conversion to Light-Emitting Diodes are calculated in the Johnson-Cousins B, V, R bands.", "We found that the conversion of the lighting infrastructure of Tenerife with LED (1800K and 2700K), according to the conversion strategy in force, would result in a zenith V band sky brightness reduction of about 0.3 mag arcsec-2." ], [ "INTRODUCTION", "The propagation of light in the nocturnal environment involves multiple physical interactions [3].", "In order to model that propagation with reasonable level of accuracy, one must include the information about the optical properties of the artificial light sources (spectral power distribution, angular emission) and their positions (latitude, longitude, elevation and height above ground).", "Other parameters such as the presence of blocking obstacles like trees and buildings, the spectral ground reflectance and the optical transfer function of the atmosphere (including aerosols) significantly influence light propagation [2], [37], [19], [39], [40], [41], [7], [32], [42], [43].", "The use of numerical models to study light pollution dates back to the 1980s with the Garstang model [21].", "Numerical models allows a full control of the environmental parameters and provide the possibility to identify the origin of the light detected at a particular location in any viewing angle.", "The Garstang model contained many simplifying assumptions, mainly motivated by the low power of computers available at that time.", "Since then, many models have been developed with increased complexity to better describe the light pollution propagation in the nocturnal environment [6], [2], [30], [33], [15], [8], [33], [3], [20], [5].", "The Teide Observatory or Observatorio del Teide (OT) was founded in 1964.", "It is located in the Canary island of Tenerife at 2390 m of altitude.", "It is operated by the Instituto de Astrofísica de Canarias (IAC).", "It hosts many international telescopes and is the reference in solar astronomy.", "It benefits of good seeing conditions and good image quality [48], [49], [36], [35], [50], [38].", "Its artificial skyglow increased with the development of the touristic industry and with the general population increase.", "Its research capacities in the visible have been considerably reduced accordingly.", "For that reason, subsequent major optical telescopes were built at Roque de los Muchachos Observatory (ORM) on the nearby island of La Palma.", "In order to protect ORM sky from being altered too by light pollution, la ley del cielo [28], [22], [51], a national light pollution abatement, was voted by the Spanish government.", "This law comprises strict regulations to the lighting practices on the Island of La Palma but also some important restrictions for the Island of Tenerife.", "In 1997, the Spanish Government has subsidized the cost of a programme of street lighting replacement on La Palma island to minimize light pollution [16].", "The areas of Tenerife island facing ORM, hereafter called the protected area, experience more restrictive lighting rules than the rest of the island (hereafter called unprotected area).", "Thanks to that law, in the protected area, any lighting replacement has to be done using Phosphor Converted Amber light (PCamber) Light-Emitting Diodes (LED) with a reduction of output flux of 20% (i.e., output flux is 0.8 of initial value).", "In the unprotected area, lighting replacement has to be done using 2700 Kelvin LEDs (LED2700K) with an output flux reduction of 70% (i.e., output flux is 0.3 of its initial value).", "The smaller flux reduction for the protected area is explained by the fact that for this area, in the past, the allowed output flux was more restrictive than in the unprotected area.", "Basically, the protected area luminous flux was already reduced.", "At the end, when all light fixtures of the island will be converted to LEDs, both areas will have similar lighting levels.", "For the whole island of Tenerife, there is an additional flux reduction after midnight (output flux after midnight is 0.65 of the output flux before midnight).", "The aim of this paper is to show up to what extent darkness of the sky around zenith at OT can be improved on the basis of its artificial sky radiance reduction.", "To achieve that, we first model the multispectral artificial sky radiance toward zenith and at 30 degrees from zenith for the present situation.", "The crucial step to reach this first milestone was to define the lighting infrastructure and the obstacles properties all over the modelling domain.", "The modelled present artificial radiance is compared with All Sky Transmission Monitor (ASTMON, [1]) Sky Brightness (SB) measurements in the B V and R Johnson-Cousins (JC) photometric bands for instruments installed at OT and ORM.", "Such a comparison is required in order to get a relevant estimate of the natural SB.", "This natural component comes from many sources like the starlight, the sky glow, the zodiacal light and so on.", "The natural SB and corresponding natural radiance are used to transform the calculation of the artificial sky radiance into the total SB (artificial + natural).", "In addition to modelling the present situation, two other modelled scenarios were performed to determine the effect of a full replacement of the light fixtures by: 1- PCamber, and 2- LED2700K.", "For these last two scenarios, we maintained the output flux equal to its present values.", "The results are weighted by the output flux reduction rules identified for the protected and unprotected areas." ], [ "METHODOLOGY", "A simulation of the OT and ORM skies have been done by [4] using version 0 of Illumina (v0).", "It was a comparison experiment with the MSNsRAu model [30].", "Among many differences with the version 2 (v2) used in the present paper, Illumina v0 was monochromatic and was using the Defense Meteorological Satellite Program - Operational Linescan System (DMSP-OLS, [27], [17]) satellite data with much lower resolution and bad radiometric accuracy compared to Visible Infrared Imaging Radiometer Suite Day Night Band (VIIRS-DNB, [18]) used in v2.", "It also had a crude correction for subgrid obstacles.", "There are not many methods to evaluate to what extent the sky quality of OT can be improved on the basis of its artificial sky radiance reduction.", "One of them can be that proposed by [10] where they compute the relative contribution of an area to the sky brightness of another.", "The present method is similar in the sense that we also integrate the contribution over municipalities, but since we are only concerned about a precise observing location we can model exactly the contributions to the sky radiance of the location using a complete radiative transfer model such as Illumina instead of relying on the use of uniform point-spread functions.", "In this paper we are using the new version (v2) of the radiative transfer model Illumina to simulate the sky radiance in several wavelengths (spectral bins).", "Prior to the numerical calculations, it is important to define, as accurately as possible, the light fixture inventory, a list of the properties of the light sources (spectral power distribution and angular emission functions) and obstacles of the domain.", "This is probably the most difficult part of the work.", "As a result of the numerical calculations, we can exploit the modelled sky radiance in every spectral bin and combine them to create the artificial sky spectrum.", "We can also exploit the contribution maps.", "Such maps give the geographical distribution of the origin of the modelled sky radiance.", "There is one contribution map per spectral bin, per viewing angle and per lighting scenario.", "We integrated the contribution maps using the three JC bands (B, V, R) in order to compare them to the observed SB.", "The natural radiance in each JC band needs to be determined in order to calculate the total SB and radiance (natural + artificial).", "Contribution maps are also integrated over geographical limits of Tenerife municipalities and over the protected / unprotected area of the Tenerife Island.", "Such method is also applied to lighting conversion plans which allow the evaluation of the expected radiance reductions and SB decrease associated with the conversion or shutdown of each municipality or area." ], [ "Illumina v2 model", "Illumina is a heterogeneous radiative transfer model dedicated to the simulation of the artificial sky radiance in any wavelength [6], [2], [3], [5].", "The model is calculating the following physical interactions: 1- the aerosol (scattering and absorption) and molecular extinction (scattering only); 2- the 1st and 2nd order of scattering; 3- the ground reflection (lambertian); 4- the lamp flux; 5- the lamp angular emission function (horizontally averaged); 6- the topography; 7- the subgrid obstacles blocking (trees and buildings when the horizontal and vertical resolution cannot resolve them); 8- the reflection by overhead clouds.", "Illumina cannot yet calculate the molecular absorption.", "For that reason the use of Illumina must be restricted to the atmospheric windows but especially to the visible range.", "Since we used the newly released version of the model, it is worth highlighting the changes compared to the previous version (v1).", "The basic novelties of the model comprise An improvement of the calculation of the scattering probability and extinction.", "The probability of scattering is obtained from: $p=1-\\exp \\left( \\frac{\\ln (T_\\infty ) \\exp (-{z}/{H}) \\mathrm {d}l }{H} \\right)$ Where $T_\\infty $ is the vertical transmittance of the aerosols or molecules for the entire atmospheric vertical column.", "$H$ is the scale height ($H=8$ km for molecules and 2 km for aerosols), $z$ is the altitude above ground and $\\mathrm {d}l$ is the length of the scattering voxel.", "Similarly the transmittance of a light path is given by equations REF and REF for a horizontal and an oblique light beam respectively: $T=\\exp \\left( \\frac{\\ln (T_\\infty ) \\exp (-{z}/{H}) d }{H} \\right)$ $T=\\exp \\left( \\frac{\\ln (T_\\infty )}{\\cos (\\theta _z)} \\left[\\mathrm {e}^{-z_a/H} - \\mathrm {e}^{-z_b/H} \\right] \\right)$ Where $d$ is the horizontal distance of the light path, $z_a$ and $z_b$ are the bottom and top heights and $\\theta _z$ is the zenith angle.", "$T_\\infty $ for molecules ($T_{m\\infty }$ ) is obtained using the extinction coefficient given by [29] in their equation 18 (see equation REF below).", "For aerosols, $T_{a\\infty }$ is given by equation REF .", "The improvement of the accuracy in determining the solid angles.", "Especially when the scattering medium is located near the source, the first order scattering point, or the observer.", "In v1, the 3D space was divided into a fixed and coarse mesh grid, while in v2, we are defining small voxels on the fly.", "No vertical mesh grid is used anymore.", "The cloud base height can be set by the user, and a correction for the cloud fraction was added on the basis of [44] observations.", "We do not use this feature in this work since we are only concerned about clear skies.", "The addition of the direct radiance calculation.", "In v1, only the sky radiance was calculated.", "No direct sight to the light fixtures was allowed.", "This feature is not used in this work since we are focusing on the sky radiance.", "The direct radiance data are more suited for health and ecosystem studies.", "The Moderate Resolution Imaging Spectroradiometer (MODIS) reflectance product used in v1 is replaced by a weighted combination of surface reflectances to be defined by the user.", "This change has been implemented because 1) the low resolution of the MODIS data (500 m) that include many types of surface at the street-level scale so that the reflectance was not only representative of the ground below the light fixtures but rather of an average of surfaces, some lighted, some not; 2) the coarse resolution of VIIRS-DNB of 750 m do not allow the precise localization of the source and this is not enough accurate to identify the right reflectance to use even if we use high resolution reflectance data like the one from the Land Satellite (LANDSAT, [34]); 3- satellite-based evaluation of the reflectance can be biased by obstacles that can hide the lighted surfaces and then introduce significant mismatch between the detected reflectance and the one of the surfaces underlying the lamp fixture.", "In v2 the reflectance is constant for all the modelling domain but has to be representative of the ground underlying the lighting devices.", "The ASTER spectral library [9] is routinely used for that purpose.", "The introduction of a multiscale grid that can allow a finer description of the environment near the observer.", "With this new feature, there is virtually no limit to the spatial resolution.", "In v1, the spatial resolution was fixed to 1 km.", "With v2, one can use very high resolution lidar data and then resolve the 3D buildings and trees effect on the light propagation.", "In v1, only a subgrid statistical obstacle correction was possible.", "Note that such statistical subgrid correction is still available in v2, depending on the resolution used in the multiscale definition of the modelling domain.", "The point source inventory can be directly used in the model to improve satellite-derived inventory.", "In v1, only satellite-derived inventory was possible.", "We do not use this feature in the present work.", "As for Illumina v1, Illumina v2 requires an as accurate as possible definition of a set of input data: angular emission function of the lamps*; spectra of the lamps*; lamp flux; lamp height relative to the ground; obstacles properties (height, distance, filling factor)*; underlying ground spectral reflectance; topography; minimum ground surface atmospheric pressure; relative humidity; $\\tau _a$ , Angström coefficient ($\\alpha $ ) and the aerosol model Most of them are currently quite easy to define except the ones marked with an asterisk.", "Their determination requires the collaboration with a local expert that has good knowledge of the lighting infrastructure.", "We hope that with the rapid evolution of remote sensing techniques, having a local expert will not be required in a near future." ], [ "Modelling experiments", "The aim of that work is to the evaluate the current level of light pollution and its possible change upon conversion of the lighting infrastructure with less polluting devices and better lighting practices.", "In that scope, it is very important to correctly define the geographical domain, the lighting infrastructure, and environmental properties over that domain.", "In order to accurately model the contribution of the different municipalities of Tenerife Island on the sky radiance at OT we defined a finer resolution inventory for Tenerife while keeping a coarse definition for the other islands.", "It is well known, and since a long time, that the effect of light pollution is decreasing rapidly with distance [13], [46], [12].", "This fact stresses the importance of a better definition of the sources close to the observer.", "The experiment use 14 layers ranging from 20 m of resolution in the first central layer to a resolution of $\\approx $ 1493 m for the 14th layer.", "The resolution scale factor between two consecutive layers is 1.393.", "Given that, the resolution of the second layer is $\\approx $ 28 m, the third $\\approx $ 39 m and so on.", "The dimensions of each layer were 255x255 pixels.", "Figures REF and REF show the various circular zones that were defined to characterize the different lighting and environment properties.", "On a given circular zone, we assume that the spectra, the angular emission functions, the lamp height and the obstacles properties are uniform.", "The light flux inside a given zone can vary given that it is derived using the VIIRS-DNB satellite monthly data (April 2019 in this study).", "Topography also varies inside a zone.", "The method used to convert VIIRS-DNB into flux is explained in [5].", "The complete set of data used for the inventory is given in Table .", "Figure: Circular zones used to define the properties of lighting devices and of the obstacles over the whole modelling domain.", "In the setting of the properties, the smaller zones overwrite the larger if ever there is an intersection between them.Figure: Enlargement of figure showing circular zones used to define the properties of lighting devices and of the obstacles over the Tenerife Island.The modelling domain can be seen in Figure REF .", "This figure corresponds to the original VIIRS-DNB upward radiance data.", "The overall modelling domain covers $\\approx $ 380 km E-W by $\\approx $ 380 km N-S.", "The domain is centred on the observer position at OT (28.301197$^\\circ $ N, 16.510761$^\\circ $ W).", "We assume the observer to be 5 m above ground.", "A zoomed view on Tenerife Island is provided on Figure REF .", "On that figure we filtered the original VIIRS-DNB radiances with a threshold of 0.8 nW/sr/cm$^2$ .", "Such a threshold allowed to remove the background light over the ocean surface along with over the unlighted dense forest of the islands.", "The calculations are made for 14 25 nm-wide spectral bins covering the spectral range of 380 nm to 730 nm.", "The sea-level air pressure is set to 101.3 kPa with an air relative humidity of 70%.", "The sky is defined as cloudless.", "The maximum distance to calculate the effect of reflection on the ground is set to 9.99 m and the ground reflectance is defined by a weighted spectrum obtained assuming 90% of asphalt and 10% of grass.", "Reflectance spectra are taken from the ASTER spectral library [9].", "We are using a $\\tau _a$ at 500 nm of 0.04 and an angstrom coefficient of 1.1.", "Both values corresponding to the average of clear sky conditions for that period (April 2019) according to the Izaña sunphotometer of the Aerosol Robotic Network (AERONET, [25]).", "This Izaña sunphotometer is located only about one kilometer away from OT and is at almost the same altitude.", "We used the maritime aerosol model as defined by [45].", "We modelled three cases: 1- the present situation, 2- a complete conversion of the lamps to LED2700K, and 3- a complete conversion to PCamber.", "For both conversion scenarios, we first assumed that the output luminous flux was kept identical as it is in the present situation.", "At the end, to estimate the effect of real conversions, we weigh these results by their output luminous flux reductions according to the legal prescriptions described in the introduction (-20% in the protected area and -70% in the unprotected).", "In addition, we assume that replacement in the protected area is done using PCamber while LED2700K to be used in the unprotected area.", "Figure: Original VIIRS-DNB radiances over the modelling domain.Figure: Filtered VIIRS-DNB data over the Tenerife Island." ], [ "Integration over geographical limits and JC bands", "There is one artificial radiance contribution map for each viewing angle and spectral bin.", "Since we are focused on the analysis of the sky brightness in the JC bands (B, V, R), we need to integrate the spectral information over each band.", "The process consists of doing the product of the spectral sensitivity of the JC band integrated over the spectral bin by the artificial radiance of that bin and integrate this quantity over the spectral range for each pixel of the modelling domain.", "The result is the JC band artificial radiance contribution map.", "Knowing the geographical limits of the municipalities and protected/unprotected areas, it is then possible to add up radiances of all pixel falling inside the limits of each municipality or area.", "Again we obtain an artificial radiance but it corresponds to the artificial radiance in JC bands for each municipality or area.", "The limits of the various municipalities and areas of Tenerife Island are illustrated in Figure REF .", "The star on that figure illustrates the position of the OT.", "Figure: Limits of the various municipalities of Tenerife and protected (grey) / unprotected area.", "The observatory is marked by a star.", "Green color represent natural parks and reserves." ], [ "Atmospheric and obstacle correction to the VIIRS-DNB inversion", "At the moment of writing this paper, Illumina do not correct for the VIIRS-DNB signal reduction caused by the atmospheric extinction and obstacles blocking.", "These corrections will soon be incorporated into the model.", "Their effect on each pixel of the modelling domain should be different because the obstacles and angular emission of light vary from one pixel to the other.", "In this study, we have made an approximate correction on the modelling output results instead of the modelling inputs.", "For the atmospheric correction, the correction is the same everywhere.", "To compensate for the molecular transmittance ($T_m$ ) and the aerosol transmittance ($T_a$ ), we use the following expression: $F_T = \\frac{1}{T_{a\\infty } T_{m\\infty }}$ The molecular transmittance is calculated with equation REF derived from [29].", "$T_{m\\infty } = \\exp \\left( \\frac{-1}{ \\lambda ^4 \\left( 115.6406-\\frac{1.335}{\\lambda ^2} \\right)} \\right)$ $\\lambda $ is in units of $\\mu $ m. The aerosol transmittance is calculated using $T_{a\\infty } = e^{-\\tau _a}$ Where $\\tau _a$ is calculated at any wavelength ($\\tau _a(\\lambda )$ ) from $\\tau _a$ at 500 nm ($\\tau _a (0.5 \\mu \\mathrm {m})$ ) and with the Angstrom exponent $\\alpha $ .", "$\\tau _a(\\lambda ) = \\tau _a (0.5 \\mu \\mathrm {m}) \\left( \\frac{\\lambda }{0.5} \\right)^{-\\alpha }$ $F_T$ can be easily estimated for the effective wavelength ($\\lambda _e$ ) of the B, V, and R bands given in Table REF .", "The buildings and trees are unresolved obstacles in the model but we include their statistical effects.", "However they are only considered to solve the radiative transfer but not to produce the input data.", "Obstacles blocking is determined by the average horizontal distance between the lamp and the obstacle ($d_o$ ), the average lamp height ($h_l$ ) and the average obstacle height ($h_o$ ) along with the obstacle filling factor ($f_o$ ).", "$f_o$ accounts for the fact that not all the light is intercepted by the obstacles, a part of it can pass through because there can be some space between the buildings and trees.", "These parameters were defined while building the inventory (see Table ).", "In Table , Obst.", "Distance is equal to $2 \\times d_o$ .", "The VIIRS-DNB radiance monthly product is an average of radiances corresponding to a variety of zenith angles from 0 to $70^\\circ $ .", "But many of these angles are partly blocked by the subgrid obstacles.", "This blocking effect impacts in different ways the VIIRS-DNB radiance monthly product.", "There could be two components to the upward radiance: 1- the direct light and 2- the light reflected by the ground and obstacles surfaces.", "In most cases, the 2nd component is the dominant one.", "This is because most light fixtures do not emit significantly at $\\theta _z < 70^\\circ $ .", "If we define the obstacle correction factor as $F_{o}$ , we can write the corrected radiance ${R_a}$ as a function of the uncorrected artificial radiance (${R_a}^*$ ) as follows: ${R_a} \\approx {R_a}^* F_T F_{o}$ If we assume that the street surface is the most lighted surface and then neglect the reflected light from the obstacles walls, we can define the limit zenith angle that allows reflected light to reach the satellite.", "This angle is given by: $\\theta _{lim} = \\arctan \\left(\\frac{d_o}{h_o}\\right)$ The obstacles correction factor can be calculated by a weighting function of the solid angles.", "$F_{o} \\approx \\frac{\\int _{0}^{70^\\circ } \\sin \\theta _z \\mathrm {d}\\theta _z}{\\int _{0}^{\\theta _{lim}} \\sin \\theta _z \\mathrm {d}\\theta _z + (1-f_o) \\int _{\\theta _{lim}}^{70^\\circ } \\sin \\theta _z \\mathrm {d}\\theta _z }$ $F_{o} \\approx \\frac{ 1 - \\cos 70^\\circ }{1- f_o \\cos \\theta _{lim} + (f_o-1) \\cos 70^\\circ }$ For typical Tenerife values of $h_o \\approx $ 9 m, $d_o \\approx 4$ m (i.e., about 8 m in diagonal between facing buildings), and $f_o \\approx $ 0.9.", "This leads to $\\theta _{lim} \\approx $ 24$^\\circ $ and then $F_o \\approx 4.6$ .", "As said, the obstacle correction varies from one pixel to another.", "For that reason the above correction is very approximate.", "We know that the obstacle correction is independent of the wavelength.", "It should be the same for the three Jonhson-Cousins bands.", "In this paper we decided to use sky brightness data acquired with ASTMON cameras during April 2019, both in OT and ORM, to determine if our $F_o$ estimate fit the observations and ultimately find a better value to use.", "We use the difference in the sky brightness in the V and B bands between the two sites.", "More specifically we use the 75 percentile (P75) values (see table REF ).", "P75 values were estimated to provide the best proxy of the darkest conditions during April 2019.", "Using 99 percentile (P99) data should normally be better but, for the month of April 2019, there was some contamination in the P99 band data that disappeared in the 75 percentile data.", "We assume that the sky brightness at ORM in its best atmospheric conditions, is very close to the natural sky brightness.", "This assumption should be true within 0.03 mag arcsec-2 according to [11].", "This assumption do not apply to the R band.", "We exclude the R band because that, on La Palma Island, many of the light fixtures are either Low-pressure sodium or monochromatic amber LEDs.", "These artificial lights only emit in the R band.", "For that reason, we cannot assume that the R band sky brightness at ORM is as representative of the natural sky brightness as the B and V bands.", "Table REF show the P99 zenith sky brightness recorded at ORM for the years 2018-2019.", "These measurements are brighter than the natural sky brightness $S_{bg}$ estimates made by [11] in the B and V (-0.07 mag arcsec-2 in B, -0.09 mag arcsec-2 in V).", "But we recall that [11] suggested to add 0.03 mag arcsec-2 in all bands to determine the natural level.", "This is what we have done in B and V. Considering that, the new 2018-2019 measurements are consistent with the 1998 measurements in B and V bands.", "In the R band, the P99 measurement is darker than the [11] estimate (+0.15 mag arcsec-2).", "The ORM sky brightness decrease in the R band since 1998 is probably due to the significant change in the lighting systems on La Palma.", "In 1998, there was a lot of Low Pressure Sodium lamps that are emitting in the R band.", "Then we must admit that the natural SB evaluation made by [11] in the R band was overestimated of at least 0.15 mag arcsec-2.", "For that reason we will use the 2018-2019 P99 data as the best estimate of $S_{bg}$ in the R band while keeping the [11] values for B and V bands.", "Table: Percentile 75 ASTMON sky brightness measurements and differences in April 2019 at OT and ORM.", "The Δ S \\Delta _S values of that table are used to determine precisely the correction factor F o F_o.Table: Percentile 99 ASTMON sky brightness measurements for 2018-2019 at ORM.The obstacles correction factor was empirically verified so that the total modelled zenith sky brightness reduction upon a complete shutdown of the lights (see Table REF ) fit with the measured SB differences in B and V bands between OT and ORM (see Table REF ).", "This exercise led us to a value of $F_o=$ 5.05 instead of 4.6 (i.e., 10% larger).", "With that empirical value, we obtain a fit of B and V bands reductions that is within 0.02 mag arcsec-2.", "Part of this correction can come from the fact that VIIRS-DNB data are not acquired at the same moment than SB measurements.", "But it is most probably coming from the use of only one set of obstacles values to estimate $F_o$ that may not correctly represent the influence of all the pixels of the domain to the modelled sky brightness.", "This is why we should implement this directly to the input data in the future.", "The obstacle model used only assumes a single layer of uniform obstacles.", "But we assume that a more complete description such as the one made by [31] could not provide a significant improvement to the correction of the VIIRS-DNB signal because of the relatively low zenith angles considered ($\\theta _z \\le 70^\\circ $ ).", "Table: Correction factors to the modelled radiance assuming h o h_o=9, f o f_o=0.9, and d o d_o=6." ], [ "Conversion from radiance to sky brightness", "Illumina calculates only the artificial sky radiance.", "In order to determine the total SB equivalent in units of mag arcsec-2 it is mandatory to get a good estimate of the natural component of the SB for the site and period.", "The natural SB is highly variable with time, altitude, season and observing direction.", "It is composed of light from multiple sources such as the zodiacal light, the starlight, the sky glow and the Milky Way [11].", "In this study we are using natural sky brightness estimates made by [11] to determine the background SB and radiance.", "This natural sky brightness excludes starlight.", "For that reason, the ASTMON measurements are shifted compared to the background SB, but the shift is the same for both OT and ORM when considering the same period.", "This is why we used the SB differences between the two sites instead of absolute values to calibrate the model results as explained in Section REF .", "For a given JC band, let’s call the radiance responsible for the natural contribution without starlight, the background radiance ($R_{bg}$ ).", "We also define $R_a$ as the artificial component of the radiance, and $R$ as the total sky radiance excluding starlight.", "The total radiance is defined as $R=R_a+R_{bg}$ .", "According to the definition of the magnitude, we can write: $R_{bg} = R_0 10^{-0.4 S_{bg}}$ The zero point radiances $R_0$ are obtained from [14] and given in Table REF .", "They were derived with the relative absolute energy distribution of [23] standards and the absolute flux calibration for $\\alpha $ Lyrae given by [24].", "Values of $R_{bg}$ and corresponding $S_{bg}$ measurements ([11] in B & V bands and ASTMON in the R band) are given in table REF for OT in April 2019.", "Table: Zero point radiances for the JC photometric system derived from .Table: Natural sky brightness, background radiances, artificial radiances and total radiances toward zenith at OT in the B V R bands.", "B and V were determined using while R was determined with P99 measurements of table .Once $R_{bg}$ is known, the SB from any modelled artificial radiance integrated over the JC band (${R_a}$ ) can be determined from the definition of the magnitude.", "$S = -2.5 \\log \\left( \\frac{{R_a} + R_{bg}}{R_0} \\right)$ $R_{a}$ have to be determined by integrating the modelled artificial radiances for the spectral bins over the JC band.", "In a similar way, we can express the variation in SB ($\\Delta {{S}}$ ) after a shutdown or after a reduction of the radiance of a given area ($\\Delta {{R_a}}$ ).", "$\\Delta {S}= -2.5 \\log \\left( \\frac{{R_a} - \\Delta {{R_a}} + R_{bg}}{{R_a} + R_{bg}} \\right)$ Equation REF is used to calculate the values of Tables REF and REF .", "For the case of a lamp conversion, $\\Delta {{R_a}}$ is the difference in artificial radiance contribution of an area (present situation minus converted).", "For a complete shutdown of an area, $\\Delta {{R_a}}$ is simply the present artificial radiance contribution of the area." ], [ "RESULTS AND ANALYSIS", "Figure REF shows the relative importance of the different part of the modelling domain in terms of their contribution to the zenith artificial radiance in the V band.", "On that figure, it is noticeable that most of the zenith artificial radiance at OT is coming from the island of Tenerife itself.", "The second contributor is Gran Canaria, and the third is the small island of La Gomera.", "The contribution of La Palma island is negligible.", "We did not calculate the effect of El Hierro but it is for sure a lot smaller.", "The low contribution of La Palma is easy to explain because of its modest lighting infrastructure combined to its large distance from Tenerife.", "Figure REF gives more details about the contribution of different parts of the Island of Tenerife.", "In this figure, we can clearly perceive that the contribution is a complex combination of the distance to OT and the installed lamp fluxes.", "It can be noticed for example that Santa Cruz de Tenerife is not a huge contributor even if it emits a large amount of light as seen from Figure REF .", "A more detailed view of that information is given in Table REF .", "In this table, the percentage of the total artificial radiance is given for each municipality and protected/unprotected areas.", "This table shows that toward zenith in the V band and for the present situation, 97% of the observed radiance comes from Tenerife Island (only 3% comes from the other islands).", "The most contributing municipality is La Orotava with around 17% followed by Güímar with around 11%.", "The protected area contributes to about 43% while it is about 54% for the unprotected area.", "The capital Santa Cruz is not the most important contributor with about 7%.", "Another interesting result is that Los Cristianos and Playa de Las Américas (Arona), highest density tourists areas, contributes with about 2%.", "Some of these contributions may appear low and counter intuitive.", "This is because that our feeling of light pollution levels on site is driven by the observation of light domes toward the main sources.", "Here we show that their contributions are relatively low when looking toward zenith.", "In the advent of a complete conversion to LED, these numbers change a bit.", "The contribution of other islands becomes relatively more important (between 6% and 11%) but not in absolute values since in this conversion scenario only Tenerife is converted.", "After conversion, the relative contribution of Güímar is reduced while Los Realejos increase significantly to become the second contributing municipality.", "This is because that a part of Los Realejos is already converted to PCamber in the present situation, so that its absolute contribution is less reduced after the conversion in comparison to other municipalities.", "Figure: Contribution of the different part of the Tenerife Island to the V band artificial zenith sky radiance at OT.Table: Fraction of the zenith artificial sky radiance to the total by municipality or area.", "Converted case stand for the change of all lighting devices of the protected area to PCamber with 20% output flux reduction and the change of all lighting devices of the unprotected area to LED2700K with 70% output flux reduction.", "The 5 most contributing municipalities to the present V band radiance are in bold.", "We also indicate their decreasing order of importance in the V band before their names.Figure: Colour index of the artificial sky brightness of the different parts of the Canary Islands to the artificial zenith sky radiance at OT.Figures REF and REF show the $B-V$ colour index in magnitude calculated using the artificial radiances only.", "These figures are for the present situation.", "It is clear on Figure REF that La Palma does not have much blue light ($B-V \\approx 4$ ), but there are also some low blue content spots on the island of Tenerife (Figure REF ), Los Realejos being one of them.", "These low blue radiance spots are places mostly already converted to PCamber.", "Figure: Colour index of the artificial sky brightness of the different parts of the Tenerife Island to the artificial zenith sky radiance at OT.Table REF shows the total artificial radiances in B, V and R at zenith angles $\\theta _z=0^\\circ $ and $\\theta _z=30^\\circ $ for the present situation and for the conversion scenario.", "The zenith artificial radiance after conversion is $\\approx $ 33% of its present value in the B band while it is around 52% in the two other filters.", "This result shows clearly that the blue content of the sky brightness is more efficiently reduced after conversion but the reduction is also significant in the V and R bands (about a factor of 2).", "These reductions are the result of a combination of the change in colour of the lamp spectra and a general reduction of the upward emitted light since the LED fixture used for the conversion have an Upward Light Output Ratio (ULOR) of 0.", "Table: Comparison of the artificial radiances after midnight for the present situation and the full conversion.One interesting aspect of modelling the sky brightness is that we can a test unlimited number of changes in the lighting infrastructure and environmental variables to determine their effects on the sky brightness.", "Table REF shows the SB change associated with the shutdown of each municipality.", "This table shows that a complete shutdown of Tenerife Island would improve zenith OT SB by 0.155 mag arcsec-2 in the B band, 0.427 mag arcsec-2 in the V band and 0.258 mag arcsec-2 in the R band.", "These numbers are the maximal SB reductions availables but implies a complete shutdown of the light fixtures.", "That is certainly not realistic in the real world.", "With such a SB reduction, OT sky would be as dark as ORM sky.", "The five most contributing municipalities to the zenith SB in V band are in order of decreasing importance: La Orotava, Güímar, Los Realejos, Santa Cruz de Tenerife and Granadilla.", "The SB reductions at $\\theta _z=30^\\circ $ are even larger (e.g., 0.461 mag arcsec-2 in the V band).", "Table: Reduction of the sky brightness after midnight in the B V R JC bands if a municipality or area is totally shutdown.", "The 5 most contributing municipalities to the present V band SB are in bold.", "We also indicate their decreasing order of importance in the V band before their names.A more realistic scenario is presented in Tables REF and REF .", "Table REF shows the expected SB reduction after a conversion of a municipality or area to the relevant LED technology.", "As shown in the table, no gain in zenith SB may be achieved with the lighting conversion of other islands to PCamber.", "Furthermore, only a tiny reduction ($\\le 0.002$ mag arcsec-2) can be obtained at $30^\\circ $ zenith angle from the conversions of the other islands.", "The five municipalities conversion that may deliver the maximum zenith V band SB reduction are, in order of decreasing effect, 1- Güímar (pop.", "20 190), 2- La Orotava (pop.", "42 029), 3- Santa Cruz de Tenerife (pop.", "207 312), 4- Granadilla (pop.", "50 146), 5- Arico (pop.", "7 988).", "The available SB reduction is relatively similar from one to the other municipality with 0.036 mag arcsec-2 for Güímar and 0.025 mag arcsec-2 for Arico.", "Some of these 5 municipalities are clearly less populated and thus involve less light points to be converted.", "These municipalities should be prioritized to get the maximum SB reduction with minimal investment.", "The ratio of total sky radiance reduction per inhabitant for each municipality is shown in Table REF .", "If we consider the maximum zenith sky radiance reduction for minimal investment in V band, Fasnia should certainly be the best starting point followed by Arico, Arafo, Güímar and San Miguel.", "These five municipalities may reduce the total V band sky radiance by 9.2% with only 6.25 % of the Tenerife population, compared to the complete conversion o the island that should result in a radiance reduction of 24.1%.", "In terms of SB, the complete conversion of the Tenerife Island should improve the zenith V band SB by 0.299 mag arcsec-2.", "This is actually the best SB reduction in the V band that can be achieved with the lighting conversion rules currently in place.", "The conversion of the five municipalities listed above should deliver a SB reduction of $\\approx $ 0.1 mag arcsec-2 in the V band.", "Table: Reduction of the sky brightness after midnight in the B V R JC bands if a municipality or area is converted to LED.", "PCamber with 20% output flux reduction in the protected area and LED2700K with 70% output flux reduction in the unprotected area.", "Output flux assumed to remain constant for other islands.", "The 5 municipalities with the largest reduction of the V band SB after a conversion to LED are in bold.", "We also indicate their decreasing order of importance in the V band before their names.", "Note that the ordre is different from tables and .Table: Relative reduction of the total zenith sky radiances (artificial + natural) in the B V R bands and equivalent reduction per inhabitant .", "The municipalities in bold are the 5 most efficient to reduce the zenith V band sky brightness when considering the investment per inhabitant.", "We estimate that the replacement program should focus first on that list of municipalities." ], [ "Conclusions", "In this paper we show how a radiative transfer code dedicated to the modelling of the sky radiance can be used to plan an efficient light conversion to restore the night sky brightness to ist natural value.", "The methodology presented is applied to the case of Observatorio del Teide, Tenerife.", "We showed how the determination of the sky radiance for the present situation and for some LED conversion plans can be combined to optimize the night sky darkness restoration.", "The integration of the results according to the Johnson-Cousins bands spectral responses and for defined territories (like municipality limits) render it possible to identify the most urgent municipalities to convert in order to get the maximum decrease of the sky brightness with the minimum financial and human resources.", "We demonstrated that just the completion of the undergoing lighting infrastructure conversion plan of the Tenerife Island should translate into a V band sky brightness reduction of $\\approx $ 0.3 mag arcsec-2.", "Such improvement would not be enough to recover a sky darkness typical of astronominal sites like Observatorio del Roque de los Muchachos, however it is not that far from it.", "We would need a zenith V band sky brightness reduction of $\\approx $ 0.44 mag arcsec-2 to be reaching the same sky darkness.", "A reduction of $\\approx $ 0.3 mag arcsec-2 means a reduction of the total V band sky radiance by 24% and a reduction of the artificial V band sky radiance by 48%.", "This is actually a dramatic reduction of the light pollution.", "A complete shutdown of the Tenerife light would improve the V band zenith sky brightness by $\\approx $ 0.43 mag arcsec-2.", "The capital Santa Cruz de Tenerife and the busy tourist places of Los Cristianos and Playa de Las Américas (Arona) are producing respectively 7% and 2% of the artificial sky radiance toward zenith.", "Their contribution would be dominant when looking closer to horizon in their respective directions but it was not evaluated in this study.", "Actually, our results show clearly that nearby sources are the main contributors to the sky brightness toward zenith.", "We can expect that, for some specific research applications needing to observe closer to horizon, it can represents an important problem and then any further touristic development should consider strong mitigation measures to restrict their light pollution emissions.", "We also showed that the sky radiance reduction per inhabitant can be an efficient proxy to optimize sky brightness reductions with limited resources.", "Applying the conversion plan for the 5 municipalities showing the highest ratio of radiance reduction per inhabitant (see table REF ) allows the reduction of the sky brightness by $\\approx $ 0.1 mag arcsec-2 (i.e., a total V band sky radiance reduction of $\\approx $ 9% compared to the maximum available of $\\approx $ 24%).", "But these municipalities represent only about 6% of the Tenerife population.", "This study showed how the atmospheric correction and, most importantly, the obstacles blocking correction play significant role in determining the lamp fluxes from the VIIRS-DNB radiances.", "For that reason, we will emphasize the addition of this feature to the Illumina model." ], [ "Acknowledgements", "We applied the sequence-determines-credit approach [47] for the sequence of authors.", "An important part of that research funded by M. Aubé's Fonds de recherche du Québec – Nature et technologies grant.", "Most computations were carried out on the Mammouth Parallel II cluster managed by Calcul Québec and Compute Canada.", "The operation of these supercomputers is funded by the Canada Foundation for Innovation (CFI), NanoQuébec, Réseau de Médecine Génétique Appliquée, and the Fonds de recherche du Québec – Nature et technologies (FRQNT).", "Thanks to Julio A. Castro Almazán for his help in processing the ASTMON data.", "This study was carried out in part at IAC, we want to thank that institution and its deputy director for the financial support and warm welcome." ], [ "Light fixture and obstacles inventory for the present situation", "lccccccl Light fixture and obstacles inventory 1lLatitude 1cLongitude 1cRadius 1cObst.", "height 1cObst.", "Distance 1cObst.", "Filling factor 1cLamp height 1lLamp spectra and ULOR degree degree km m m - m 8c 10 – continued from previous page 1lLatitude 1cLongitude 1cRadius 1cObst.", "height 1cObst.", "Distance 1cObst.", "Filling factor 1cLamp height 1lLamp spectra and ULOR degree degree km m m - m 28.30094 -16.510816 49.97 5 7 0.2 6 100_HPS_0 27.93533 -15.598841 35.15 15 15 0.9 6 90_HPS_2 10_L40_0 28.66761 -17.848418 25.76 15 12 0.95 6 15_PCA_0 10_HPS_0 75_LPS_0 28.46752 -16.592312 19.5 6 10 0.2 5 100_HPS_2 28.24940 -16.814967 19.27 8 35 0.1 9 100_HPS_0 28.63933 -16.359244 17.11 8 35 0.1 9 100_HPS_0 28.12460 -17.230469 14.59 7 15 0.8 6 90_HPS_2 10_L40_0 28.33264 -16.396496 5.07 5 100 0.1 9 100_HPS_0 28.04566 -16.575587 2.44 13 100 0.25 9 100_HPS_0 28.51875 -16.387880 2.08 6 30 0.4 7 99_HPS_0 1_PCA_0 28.12210 -16.734982 1.97 12 18 0.7 9 100_HPS_0 28.37376 -16.851106 1.95 10 10 0.9 6 100_HPS_0 28.08387 -16.732077 1.79 16 28 0.6 8 10_PCA_0 2_L27_0 78_HPS_0 28.01340 -16.649815 1.73 14 12 0.9 6 80_HPS_0 15_MV3_0 5_PCA_0 28.51767 -16.300309 1.723 6 25 0.3 6 100_HPS_0 28.44829 -16.458228 1.71 8 30 0.5 7 50_PCA_0 50_HPS_1 28.23371 -16.841736 1.69 16 22 0.7 7 97_PCA_0 3_L27_0 28.46672 -16.256862 1.68 24 30 0.9 9 90_HPS_0 5_L40_0 5_MH_1 28.42892 -16.493302 1.67 24 30 0.7 6 100_HPS_0 28.10089 -16.755249 1.67 12 30 0.6 9 100_HPS_0 28.37183 -16.815418 1.62 8 10 0.9 4 100_HPS_3 28.44793 -16.305677 1.59 9 11 0.9 5 95_HPS_0 5_MH_1 28.46175 -16.285635 1.56 32 16 0.75 6 95_HPS_0 5_MH_1 28.39869 -16.572359 1.49 5 30 0.8 4 100_HPS_0 28.41183 -16.545088 1.49 20 15 0.8 7 60_PCA_0 21_L27_0 19_HPS_0 28.36733 -16.714263 1.47 24 7 0.9 4 100_HPS_0 28.48344 -16.416816 1.45 8 30 0.5 3 80_PCA_0 20_HPS_0 28.07720 -16.557724 1.41 8 10 0.9 7 100_HPS_0 28.46759 -16.379026 1.38 8 40 0.4 9 100_HPS_0 28.33341 -16.370738 1.38 7 50 0.85 10 90_HPS_0 2_L40_0 8_MH_5 28.37304 -16.785446 1.37 8 25 0.4 8 100_HPS_0 28.02369 -16.615692 1.33 10 26 0.7 5 100_HPS_0 28.05784 -16.731080 1.32 14 40 0.75 9 10_PCA_0 2_L27_0 78_HPS_0 28.36792 -16.760711 1.32 7 6 0.9 6 50_HPS_0 50_HPS_1 28.48450 -16.341955 1.31 5 60 0.5 9 100_HPS_0 28.31567 -16.411176 1.31 10 16 0.85 6 95_HPS_0 3_HPS_5 2_MH_20 28.12714 -16.774962 1.3 5 25 0.75 4 25_HPS_3 5_HPS_50 70_HPS_0 28.47380 -16.304124 1.28 12 10 0.9 9 95_HPS_0 5_MH_1 28.37888 -16.686270 1.28 9 15 0.7 5 100_HPS_0 28.48549 -16.391648 1.18 9 17 0.5 5 100_PCA_0 28.16630 -16.501999 1.16 5 8 0.8 7 90_HPS_0 10_HPS_1 28.44652 -16.264041 1.16 5 18 0.8 7 95_HPS_2 2.5_MH_2 2.5_FLU_2 28.05080 -16.711846 1.15 16 60 0.6 9 10_PCA_0 2_L27_0 78_HPS_0 28.08682 -16.500563 1.14 8 65 0.3 9 5_MH_10 95_HPS_0 28.52364 -16.344300 1.14 7 12 0.7 7 100_HPS_0 28.35083 -16.703138 1.12 7 10 0.5 7 50_HPS_3 50_HPS_0 28.05334 -16.616540 1.1 8 35 0.6 9 5_MH_10 95_HPS_0 28.56943 -16.324811 1.1 8 6 0.6 7 95_HPS_0 3_PCA_0 2_L27_0 28.39099 -16.523835 1.09 9 12 0.9 5 100_HPS_0 28.48621 -16.318925 1.08 18 12 0.9 9 75_HPS_0 10_L40_0 10_L30_0 28.12130 -16.576724 1.08 10 20 0.9 6 3_HPS_38 97_HPS_0 28.21147 -16.778658 1.07 8 5 0.9 4 50_HPS_0 50_HPS_3 28.50047 -16.317123 1.05 7 12 0.6 5 90_HPS_0 10_L40_0 28.36897 -16.368080 1.03 8 30 0.8 6 100_HPS_0 28.46480 -16.403630 1.03 9 15 0.7 6 100_HPS_0 28.49336 -16.201951 0.98 6 35 0.4 8 95_HPS_0 2.5_MH_2 2.5_FLU_2 28.04623 -16.537383 0.97 20 15 0.8 7 100_HPS_0 28.07029 -16.655582 0.97 8 15 0.75 9 100_HPS_0 28.12996 -16.754614 0.96 13 10 0.8 9 100_HPS_0 28.02337 -16.698863 0.96 6 18 0.8 6 100_HPS_0 28.44967 -16.368691 0.94 8 10 0.6 7 100_HPS_0 28.35613 -16.780046 0.94 8 20 0.8 6 30_HPS_1 70_HPS_1 28.14254 -16.756250 0.93 5 12 0.4 6 100_HPS_0 28.05023 -16.678293 0.92 16 12 0.7 8 10_MH_5 90_HPS_0 28.53401 -16.361629 0.91 9 8 0.8 7 99_HPS_0 1_PCA_0 28.48467 -16.225677 0.91 6 100 0.1 7 30_HPS_0 70_L40_0 28.20192 -16.827965 0.9 15 25 0.8 9 100_HPS_0 28.49411 -16.372853 0.89 8 25 0.5 7 100_HPS_0 28.07346 -16.671971 0.89 8 10 0.8 7 100_HPS_0 28.08931 -16.658905 0.87 9 12 0.75 5 100_HPS_0 28.37785 -16.552871 0.85 9 10 0.9 5 100_HPS_0 28.09282 -16.631381 0.84 5 8 0.5 5 90_HPS_1.5 10_HPS_0 28.46777 -16.447119 0.84 8 25 0.5 9 100_HPS_0 28.29346 -16.377225 0.82 16 30 0.7 8 100_HPS_0 28.55389 -16.344736 0.82 16 20 0.6 6 95_HPS_0 3_PCA_0 2_L27_0 28.37744 -16.639092 0.82 4 18 0.6 6 100_HPS_0 28.03476 -16.637584 0.82 9 12 0.9 6 100_HPS_0 28.47452 -16.436175 0.82 7 8 0.5 7 100_HPS_0 28.02908 -16.605236 0.82 16 50 0.4 4 70_HPS_38 30_HPS_0 28.12925 -16.529170 0.82 8 9 0.5 5 100_HPS_0 28.15614 -16.635884 0.81 7 15 0.9 7 5_HPS_38 95_HPS_0 28.50974 -16.193821 0.81 12 12 0.7 6 100_HPS_0 28.38366 -16.611910 0.8 12 25 0.8 7 50_HPS_3 50_HPS_0 28.35657 -16.734756 0.8 7 10 0.7 7 100_HPS_3 28.09959 -16.680719 0.8 8 9 0.9 6 10_HPS_0 90_HPS_38 28.38511 -16.584514 0.8 24 10 0.9 9 100_PCA_0 28.37792 -16.570143 0.8 7 18 0.6 6 100_PCA_0 28.45926 -16.420817 0.78 8 12 0.6 9 100_HPS_0 28.35389 -16.373236 0.78 16 10 0.8 5 95_HPS_0 5_MH_0 28.09810 -16.617865 0.78 6 6 0.7 5 90_HPS_38 10_HPS_0 28.37363 -16.652382 0.78 6 10 0.9 5 40_HPS_0 60_HPS_2 28.18275 -16.480165 0.77 8 12 0.7 6 100_HPS_0 28.38805 -16.506205 0.75 10 9 0.9 6 100_HPS_1 28.25800 -16.426538 0.74 8 100 0.05 6 80_HPS_3 20_HPS_20 28.41205 -16.504643 0.74 10 28 0.5 5 60_HPS_0 40_PCA_0 28.36623 -16.499300 0.74 4 15 0.5 8 100_HPS_0 28.40159 -16.509613 0.73 4 18 0.9 5 100_HPS_0 28.38382 -16.596720 0.73 8 9 0.5 4 100_HPS_0 28.15916 -16.766020 0.73 10 7 0.6 6 97_HPS_0 3_HPS_1 28.08029 -16.680308 0.7 14 10 0.7 6 100_HPS_0 28.33809 -16.419592 0.7 7 7 0.9 5 95_HPS_0 5_HPS_38 28.04859 -16.660106 0.69 8 16 0.7 6 100_HPS_0 28.33082 -16.399790 0.69 6 8 0.9 5 100_HPS_0 28.26909 -16.819540 0.69 10 18 0.5 7 97_PCA_0 3_L27_0 28.10252 -16.587477 0.65 8 10 0.7 6 100_HPS_0 28.01262 -16.668922 0.65 12 8 0.9 8 75_HPS_0 25_HPS_3 28.07459 -16.694939 0.64 6 12 0.6 4 90_HPS_1.5 10_HPS_0 28.15323 -16.728272 0.63 6 12 0.75 6 100_HPS_0 28.28033 -16.409189 0.62 8 9 0.5 6 100_HPS_0 28.11100 -16.595715 0.62 10 20 0.7 7 40_HPS_38 10_HPS_1.5 50_HPS_3 28.37534 -16.584072 0.62 9 7 0.6 8 100_PCA_0 28.37693 -16.512386 0.61 8 13 0.7 8 100_HPS_0 28.34693 -16.722251 0.61 6 20 0.7 6 100_HPS_1 28.32740 -16.804861 0.61 8 4 0.5 7 97_PCA_0 3_L27_0 28.23876 -16.796690 0.6 8 14 0.75 8 100_HPS_3 28.39785 -16.554391 0.6 8 15 0.7 7 100_HPS_3 28.44270 -16.282202 0.6 7 30 0.95 7 90_HPS_0 10_MH_5 28.39578 -16.544884 0.59 10 65 0.6 9 80_HPS_0 20_MH_15 28.29615 -16.815608 0.59 8 27 0.4 7 97_PCA_0 3_L27_0 28.43321 -16.321034 0.59 8 17 0.8 6 100_HPS_0 28.18253 -16.818130 0.58 14 10 0.8 6 95_HPS_0 5_L27_0 28.51595 -16.360996 0.58 8 40 0.1 4 100_HPS_0 28.39829 -16.357672 0.57 4 10 0.3 6 100_HPS_1 28.42350 -16.318772 0.57 12 17 0.6 7 95_HPS_2 2.5_MH_2 2.5_FLU_2 28.16574 -16.430795 0.57 6 22 0.7 6 50_HPS_0 50_HPS_50 28.04286 -16.614751 0.56 6 25 0.8 6 80_MH_50 20_HPS_0 28.42400 -16.299263 0.56 15 15 0.8 7 80_HPS_0 20_L40_10 28.05432 -16.525011 0.55 8 25 0.8 4 100_HPS_0 28.40166 -16.321832 0.54 5 7 0.7 7 95_HPS_0 5_MH_5 28.41454 -16.317553 0.54 6 17 0.5 10 100_HPS_3 28.37970 -16.360827 0.53 30 20 0.6 10 95_HPS_0 5_MH_0 28.23671 -16.440496 0.53 16 20 0.6 6 100_HPS_0 28.43874 -16.369566 0.53 7 25 0.4 9 100_HPS_0 28.42918 -16.307490 0.52 7 13 0.9 7 95_HPS_0 5_MH_0 28.49136 -16.214020 0.52 10 30 0.4 7 80_HPS_0 20_MH_5 28.06746 -16.625952 0.52 5 22 0.8 8 40_HPS_38 10_HPS_1.5 50_HPS_3 28.18666 -16.765573 0.51 12 9 0.5 8 30_HPS_3 70_HPS_0 28.33915 -16.790778 0.51 8 7 0.5 6 50_HPS_1 50_HPS_38 28.38759 -16.561770 0.51 9 15 0.7 8 100_PCA_0 28.32649 -16.787447 0.51 8 12 0.5 6 50_HPS_38 50_HPS_0 28.14390 -16.443432 0.5 6 8 0.9 5 100_HPS_38 28.48085 -16.245588 0.5 30 30 0.6 7 90_HPS_0 10_MH_3 28.23301 -16.456764 0.5 8 10 0.4 6 100_HPS_0 28.06971 -16.510903 0.5 7 100 0.05 9 100_L40_100 28.12227 -16.462628 0.5 9 6 0.9 6 100_HPS_1 28.33926 -16.756488 0.48 8 15 0.1 7 100_HPS_0 28.22711 -16.782947 0.48 4 8 0.8 7 95_HPS_1 5_HPS_0 28.40917 -16.333454 0.48 5 10 0.7 10 100_HPS_0 28.49304 -16.354686 0.47 6 12 0.6 8 95_HPS_0 5_PCA_0 28.39138 -16.670902 0.47 6 15 0.8 6 100_HPS_0 28.56031 -16.218691 0.47 4 7 0.8 6 100_HPS_0 28.40934 -16.311675 0.46 10 15 0.5 9 100_HPS_0 28.26815 -16.806913 0.46 9 7 0.6 7 97_PCA_0 3_L27_0 28.39667 -16.347677 0.46 4 8 0.9 4 75_HPS_1 25_HPS_0 28.34160 -16.731272 0.44 4 6 0.7 7 80_HPS_1 20_HPS_0 28.10510 -16.559746 0.43 4 25 0.4 6 99_HPS_0 1_HPS_38 28.52626 -16.154935 0.43 10 11 0.5 7 100_HPS_0 28.18068 -16.792192 0.41 12 12 0.8 9 100_HPS_0 28.39212 -16.657583 0.41 6 20 0.6 4 100_HPS_0 28.09897 -16.481605 0.41 6 6 0.9 6 100_HPS_1 28.16870 -16.733814 0.4 5 30 0.2 6 99_HPS_0 1_FLU_38 28.39388 -16.591268 0.4 9 15 0.9 5 90_PCA_0 10_HPS_0 28.14233 -16.522501 0.4 8 10 0.5 5 100_HPS_0 28.41704 -16.304883 0.4 5 10 0.8 4 100_HPS_0 28.19417 -16.424306 0.4 9 40 0.3 7 100_HPS_0 28.27356 -16.384774 0.4 8 16 0.75 6 100_HPS_0 28.38185 -16.373466 0.4 10 7 0.7 9 90_HPS_2 10_HPS_0 28.38211 -16.534929 0.4 8 22 0.8 4 100_HPS_0 28.36673 -16.528122 0.38 8 12 0.6 5 100_PCA_0 28.38796 -16.553718 0.37 9 10 0.8 5 100_HPS_0 28.14599 -16.792426 0.37 48 40 0.5 7 100_HPS_2 28.39359 -16.649390 0.37 8 13 0.7 5 100_HPS_0 28.40723 -16.325813 0.36 8 20 0.7 4 100_HPS_0 28.28347 -16.801950 0.36 4 8 0.7 3 97_PCA_0 3_L27_0 28.48806 -16.238554 0.35 12 6 0.8 6 100_HPS_0 28.15812 -16.746611 0.35 7 20 0.2 7 99_HPS_0 1_HPS_1 28.17986 -16.802851 0.34 12 9 0.75 6 100_HPS_0 28.40318 -16.332080 0.33 8 10 0.8 6 95_HPS_15 5_MH_15 28.43660 -16.288420 0.32 7 60 0.2 7 80_HPS_0 20_MH_5 28.22189 -16.413374 0.31 8 12 0.6 7 100_HPS_0 28.29604 -16.565715 0.29 8 45 0.2 4 100_FLU_15 28.57158 -16.197841 0.29 12 4 0.95 6 100_HPS_0 28.28710 -16.813983 0.29 6 9 0.5 7 97_PCA_0 3_L27_0 28.45347 -16.342539 0.29 6 17 0.95 7 100_MH_5 28.39628 -16.641086 0.28 6 8 0.9 6 95_HPS_0 5_PCA_0 28.24050 -16.403156 0.25 12 10 0.9 7 100_HPS_0 28.10965 -16.470852 0.25 7 13 0.7 7 100_HPS_3 28.14601 -16.547024 0.24 4 10 0.6 6 95_HPS_0 5_HPS_3 28.26879 -16.426897 0.24 6 9 0.7 8 100_HPS_0 28.10215 -16.476654 0.23 8 10 0.9 7 100_HPS_2 28.10540 -16.473615 0.21 10 9 0.9 6 100_HPS_1 28.24013 -16.411299 0.21 5 5 0.85 6 100_MH_3 28.24899 -16.398963 0.19 12 4 0.7 7 100_HPS_0 28.26028 -16.392639 0.13 8 10 0.6 5 100_HPS_0 28.30583 -16.564861 0.06 6 100 0.4 6 100_MV3_15" ] ]

[ [ "Theory of magnetotransport in shaped topological insulator nanowires" ], [ "Abstract We show that shaped topological insulator (TI) nanowires, i.e.", "such that their cross-section radius varies along the wire length, can be tuned into a number of different transport regimes when immersed in a homogeneous coaxial magnetic field.", "This is in contrast with widely studied tubular nanowires with constant cross-section, and is due to magnetic confinement of Dirac surface carriers.", "In flat 2D systems such a confinement requires non-homogeneous magnetic fields, while for shaped nanowires of standard size homogeneous fields of the order of $B\\sim\\,1$T are sufficient.", "We put recent work [Kozlovsky et al., Phys.", "Rev.", "Lett.", "124, 126804 (2020)] into broader context and extend it to deal with axially symmetric wire geometries with arbitrary radial profile.", "A dumbbell-shaped TI nanowire is used as a paradigmatic example for transport through a constriction and shown to be tunable into five different transport regimes: (i) conductance steps, (ii) resonant transmission, (iii) current suppression, (iv) Coulomb blockade, and (v) transport through a triple quantum dot.", "Switching between regimes is achieved by modulating the strength of a coaxial magnetic field and does not require strict axial symmetry of the wire cross-section.", "As such, it should be observable in TI nanowires fabricated with available experimental techniques." ], [ "Introduction", "Topological materials have been a central topic in solid state research for roughly two decades.", "Many distinct topological phases are currently known [2], that of (strong) topological insulators (TIs) being a most prominent one [3], [4].", "The low-energy electronic structure of a flat TI surface is characterized by a single Dirac cone [5].", "This case, possibly the simplest and most widely studied one, is already enough to produce a number of notable transport phenomena See e.g. Refs.", "[48], [49], [50] or Refs.", "[51], for reviews.. Geometrically more complex than a flat surface, TI nanowires (TINWs) have also been intensively studied [7], [8], [9], [10], [11].", "One notable reason for this is that their high surface-to-volume ratio enhances the visibility of surface transport features.", "Moreover, transport takes place on a surface which is closed along the transversal direction, enclosing the (nominally) insulating three-dimensional (3D) TI bulk.", "This leads to the interplay between the spin Berry phase of surface states and an Aharonov-Bohm phase acquired in the presence of a coaxial magnetic field [1], [12], [13], [14], [15], [16], [17], [18], [19].", "Insight into the physics of 3DTI surfaces of more complex geometry (beyond flat or cylindrical) can be obtained from an effective surface Dirac theory derived either from the 3D bulk Hamiltonian of the paradigmatic bismuth-based TIs [20], [21], or from a field theoretic approach [22].", "In this paper we apply such a theory to axially symmetric TINWs whose radius varies arbitrarily along their length, which we dub from now on simply shaped TINWs.", "Our goal is a systematic study of their magnetotransport properties, thereby extending previous work [1] on truncated TI nanocones (TINCs), see Fig.", "REF (a).", "The latter were shown to offer rich magnetotransport signatures, ranging from conductance quantization to resonant transmission through Dirac Landau levels and Coulomb blockade-type transport.", "We will discuss how further regimes become available in shaped TINWs of experimentally realistic sizes.", "Note that such TINWs are structurally shaped on mesoscopic scales, in stark contrast to the overall cylindrical but (randomly) rippled TINWs considered in Ref. [xypakis2017].", "Figure: Examples for shaped TINWs subject to a coaxial magnetic field.", "The bulk is assumed to be perfectly insulating, while the electronic structure on the metallic surface (blue) is modeled by the Hamiltonian ().", "(a) The region between z 0 z_0 and z 1 z_1 (which correspond to radii R 0 R_0 and R 1 R_1) will be referred to as a TI nanocone (TINC).", "Defining δ≡arctan𝒮\\delta \\equiv \\arctan \\mathcal {S}, where 𝒮≡(R 1 -R 0 )/(z 1 -z 0 )\\mathcal {S}\\equiv (R_1-R_0)/(z_1-z_0) is the slope of the TINC, we get the magnetic field component B ⊥ =BsinδB_\\perp =B\\sin \\delta piercing the surface.", "It is negative for R 1 <R 0 R_1<R_0.", "(b) The region between z 0 z_0 and z 3 z_3 will be referred to as a TI dumbbell.", "For arguments concerning transport, we will consider the cylindrical leads to either side of the junctions to be metallic.", "(c) Smoothed TI dumbbell.When a shaped TINW is subjected to an arbitrary magnetic field, the latter can be decomposed into two components, one perpendicular to the nanowire axis, the other coaxial.", "It is instructive to study each component separately.", "If the component perpendicular to the TINW axis is strong, i.e.", "the associated magnetic length $l_B$ is much smaller than the nanowire diameter, the nanowire conductance will be dominated by chiral side (hinge) states [1], [23], [24], [25], [26], [27].", "These states are largely independent of the TINW shape.", "This geometry-insensitive configuration is not interesting for our purposes, ergo we focus here on purely coaxial magnetic fields, $\\mathbf {B}=B\\hat{z}$ , with $\\hat{z}$ the coaxial unit vector.", "In such a configuration the TINW transport properties depend strongly on its radial profile.", "Indeed, in a shaped TINW, the magnetic flux through the nanowire cross section is a function of $z$ , and so is the out-of-surface component $B_\\perp =B_\\perp (z)$ experienced by the surface electrons, see Fig.", "REF .", "If the phase coherence length is sufficiently long, the $z$ -dependent cross section leads to a $z$ -dependent Aharonov-Bohm phase.", "We will show that the latter, together with quantum confinement due to the finite circumference, generates a mass-like $z$ -dependent potential.", "This potential can be used to qualitatively predict the transport characteristics of any shaped TINW, and was recently discussed for the simplest case of a single TINC [1].", "In order to understand magnetotransport in shaped nanowires, an essential step will be to consider such TINCs as building blocks of more complex geometries, see Fig.", "REF .", "The paper consists of three main parts, Secs.", "-, plus a concluding one, Sec. .", "In Sec.", "we derive an effective 1D Dirac equation that can be used to describe the electronic surface structure of shaped TINWs.", "Such a Dirac equation can be solved analytically for simple cases and numerically for more complex TINW geometries.", "The reader not interested in the technical aspect of the derivation can skip directly to Sec.", "REF , which discusses the ensuing physical picture – namely, that transport properties are determined by an effective mass-like potential entering the 1D Dirac equation.", "Sec.", "reviews pedagogically the physics of TINCs in a coaxial magnetic field.", "This is instructive, as any shaped nanowire can be constructed from a succession of infinitesimal conical segments with constant slope.", "We will show that provided $B_\\perp $ is strong enough, such that $l_B$ is small compared to the length of the TINC, the conductance is determined by resonant transmission through Dirac Landau levels (LLs) that form on the wire's surface.", "In Sec.", "we introduce the dumbbell-shaped TINW of Fig.", "REF (b), representing the important case of a nanowire constriction and simply referred to from now on as TI dumbbell.", "In magnetic fields of intermediate strength ($B\\sim 1\\,$ T) such an object can be tuned into three fundamentally different transport regimes by varying the field magnitude.", "First, if the lead Fermi level $E_F$ matches the energy of a LL from the side TINCs, the latter become transparent and resonant transmission into and out of the central region takes place.", "If on the other hand $E_F$ lies in between LLs, the latter become opaque and act as barriers, suppressing overall transmission ($G\\approx 0$ ).", "Finally, between these two special cases, the conductance of the individual TINCs fulfills the condition $0<G\\ll e^2/h$ , a prerequisite for Coulomb blockade physics.", "We show that the Coulomb blockade regime should be accessible in the TI dumbbell, with single-particle energy levels of the confined Dirac electrons modulating the periodicity of Coulomb blockade oscillations.", "Most notably, the dependence of the transport regimes on $E_F$ allows to switch on and off Coulomb blockade physics by a simple tuning of the magnetic field strength, which shifts the LL ladder.", "We also treat the more realistic scenario of smoothly shaped TINWs, whose geometry is comparable to experimentally realized nanowires [28], see Fig.", "REF (c).", "For such geometries, yet another transport regime emerges for high magnetic fields from the interplay between the magnetic length and the length scale of the smoothing.", "Sec.", "concludes and sums up our findings.", "A series of technical details are discussed in the Appendices -.", "We are interested in topological surface transport [13], [14], [15], so the starting point is the surface Dirac Hamiltonian $H$ .", "In experimental samples, the bulk is usually not perfectly insulating, but several techniques, e.g.", "gating or compensation, can be used to suppress its transport contribution, see for example Refs.", "[29], [30], [31].", "We thus neglect bulk contributions throughout.", "The model Hamiltonian $H$ satisfies the time-independent surface Dirac equation $H\\psi =\\epsilon \\psi $ and can be derived starting from either of the two approaches mentioned above: microscopic or field theoretic.", "Both derivations can be consulted in detail in Ref. [xypakis2017].", "In order to introduce our notation, the derivation of $H$ is sketched very briefly in Appendix .", "One finds $\\begin{aligned}H=v_F&\\left[\\frac{1}{\\sqrt{1+R^{\\prime 2}}}\\left(p_z-\\frac{i\\hbar }{2}\\frac{R^{\\prime }}{R}\\right)\\sigma _z\\right.\\\\&\\left.+\\left(p_\\varphi +\\frac{\\hbar }{R}\\frac{\\Phi }{\\Phi _0}\\right)\\sigma _y\\right],\\end{aligned}$ where $R\\equiv R(z)$ is the radius of the shaped TINW as a function of the coaxial coordinate $z$ , $R^{\\prime } \\equiv \\mathrm {d}R/\\mathrm {d}z$ , $\\Phi \\equiv \\pi BR^2$ the magnetic flux enclosed by the wire, $\\Phi _0\\equiv h/e$ the flux quantum, and $\\sigma _{y,z}$ are Pauli matrices.", "The momentum operators are defined as $p_z\\equiv -i\\hbar \\partial _z$ , and $p_\\varphi \\equiv -i\\hbar R^{-1}\\partial _\\varphi $ .", "Note the different origin of the second and fourth terms: The shift in coaxial momentum is due to the spin connection [32], [33], while the shift in azimuthal momentum is due to the magnetic field.", "Note that the spin connection term in Eq.", "(REF ) can be gauged away by the local transformation $\\begin{aligned}\\psi &\\rightarrow \\tilde{\\psi }=\\sqrt{R}\\psi ,\\\\H&\\rightarrow \\tilde{H}=\\sqrt{R}H\\frac{1}{\\sqrt{R}},\\end{aligned}$ such that $\\tilde{H}=v_F\\left[\\frac{1}{\\sqrt{1+R^{\\prime 2}}}p_z\\sigma _z+\\left(p_\\varphi +\\frac{\\hbar }{R}\\frac{\\Phi }{\\Phi _0}\\right)\\sigma _y\\right].$ In terms of the arc length coordinate $s$ running along the TINW surface, such that $ds^2=dz^2+dR^2$ , Eq.", "(REF ) becomes $\\begin{aligned}\\tilde{H}=v_F\\left[p_s\\sigma _z+\\left(p_\\varphi +\\frac{\\hbar }{R}\\frac{\\Phi }{\\Phi _0}\\right)\\sigma _y\\right],\\end{aligned}$ where $p_s\\equiv -i\\hbar \\partial _s$ .", "This last form of the Hamiltonian is particularly well suited for simulating transport through shaped TINWs with a numerical tight-binding approach, see Ref.", "[Kozlovsky2020] or Appendix  for details.", "For our analytics and general discussions we will, however, express everything in terms of the coaxial coordinate $z$ throughout the paper.", "That is, we solve the eigenvalue problem Eq.", "(REF ) using the Hamiltonian (REF ).", "Exploiting rotational symmetry, the solution to Eq.", "(REF ) can be written as $\\psi =e^{-i(l+1/2)\\varphi }\\chi _{nl}(z),$ where $\\chi _{nl}(z)$ is a two-spinor and $l\\in \\mathbb {Z}$ denotes the orbital angular momentum quantum number.", "The latter can only assume discrete values due to the azimuthal size confinement.", "From now on $\\chi _{nl}\\equiv \\chi _{nl}(z)$ for brevity.", "The shift of $1/2$ in the angular momentum quantization represents the presence of a spin Berry phase of $\\pi $ , which is a distinct feature of 3D TINWs [7], [14], [15], [9].", "The spin Berry phase ensures antiperiodic boundary conditions in the azimuthal direction.", "The meaning of the quantum number $n\\in \\mathbb {N}$ will become clear in Sec. .", "(Essentially, for a given angular momentum $l$ , it labels a series of bound, quasi-bound and/or scattering states depending on the character of the corresponding effective potential landscape, cf.", "Eq.", "(REF ) below.)", "With the ansatz (REF ), we obtain the 1D Dirac equation $\\left[\\frac{v_F}{\\sqrt{1+R^{\\prime 2}}}\\left(p_z-\\frac{i\\hbar }{2}\\frac{R^{\\prime }}{R}\\right)\\sigma _z+V_l\\sigma _y\\right]\\chi _{nl}=\\epsilon _{nl}\\chi _{nl}.$ Here, the angular momentum term $V_l\\equiv \\hbar v_F k_l\\equiv -\\frac{\\hbar v_F}{R}\\left(l+\\frac{1}{2}-\\frac{\\Phi }{\\Phi _0}\\right)$ induces a position- and magnetic field-dependent energy gap.", "For given $B$ , this leads to a mass-like potential landscape along the wire that, unlike an electrostatic potential, does not admit Klein tunneling [34].", "Consequently, the sign of $V_l$ is unimportant: a state with angular momentum quantum number $l$ sees the effective potential $|V_l|$ .", "The role of $|V_l|$ will be demystified in Sec.", "REF .", "Here we just add two remarks.", "(i) It is enlightening to consider the limit of a cylindrical TINW.", "In this case, $|V_l|$ is constant along the wire and simply equal to the energy minimum of the corresponding subband $\\epsilon _{l}(k_z)$ .", "The quantum number $n$ then takes the form of a continuous coaxial wave number: $n\\rightarrow k_z$ .", "(ii) In Sec.", "REF , we will discuss in detail the effective potential for a TINC.", "In the course of this, we will point out (and elaborate on it in Appendix ) that mass potential landscapes analogous to $|V_l|$ appear in any system where Dirac carriers feel an effectively inhomogeneous magnetic field, for example when magnetic step barriers are formed in graphene [35], [36]." ], [ "Solution of the surface Dirac equation", "Equation (REF ) can be decoupled into two second-order partial differential equations.", "We define $\\gamma \\equiv 1/\\sqrt{1+R^{\\prime 2}}$ for compact notation, insert the two-spinor $\\smash{\\chi _{nl}=(\\chi _{nl}^{(1)},\\chi _{nl}^{(2)})^T}$ and define $\\smash{\\chi _{nl}^\\pm \\equiv \\chi _{nl}^{(1)}\\pm \\chi _{nl}^{(2)}}$ , such that the Dirac equation (REF ) becomes $\\mathcal {O}_l^\\pm \\chi _{nl}^\\pm =\\epsilon _{nl}^2\\chi _{nl}^\\pm ,$ with $\\mathcal {O}_l^\\pm \\equiv -(\\hbar v_F\\gamma )^2\\left[\\partial _z^2-R^{\\prime }\\left(\\gamma ^2R^{\\prime \\prime }-\\frac{1}{R}\\right)\\partial _z+\\mathcal {P}^\\pm _l\\right].$ The magnetic field and angular momentum dependence enter only into the last term, $\\begin{aligned}\\mathcal {P}^\\pm _l&\\equiv -\\frac{1}{\\gamma }\\left(\\frac{k_l^2}{\\gamma }\\mp k_l^{\\prime }\\right)+\\frac{1}{2R}\\left(\\gamma ^2R^{\\prime \\prime }-\\frac{R^{\\prime 2}}{2R}\\right).\\end{aligned}$ Let us check the limit of a cylindrical TINW: $R^{\\prime }=k_l^{\\prime }=0$ , $\\gamma =1$ , $\\smash{\\chi _{nl}^\\pm \\rightarrow e^{ik_zz}(\\chi _l^{(1)}\\pm \\chi _l^{(2)})}$ and $\\epsilon _{nl}\\rightarrow \\epsilon _{l}(k_z)$ , where $\\smash{\\chi _l^{(1,2)}}$ are independent of $z$ .", "This yields the energy dispersion $\\epsilon _{l}(k_z)=\\pm \\hbar v_F\\sqrt{k_z^2+k_l^2}$ , as expected [17], [19].", "For an arbitrary geometry $R(z)$ , magnetic field $B$ and angular momentum quantum number $l$ , one can numerically solve Eq.", "(REF ).", "The numerical implementation of $R(z)$ is described in Appendix ." ], [ "Effective mass-potential – physical picture", "The purpose of this Subsection is to convey a better understanding of the role of the effective mass-potential $|V_l|$ , Eq.", "(REF ).", "We therefore visualize its effect on the transport properties of a shaped TINW in a coaxial magnetic field by presenting a concrete example, namely a smoothed TI dumbbell [cf.", "Fig.", "REF (c), the corresponding radial profile being defined in Appendix ] with parameters given in Fig.", "REF .", "Note that, in Section , we will discuss the magnetotransport properties of such TI dumbbells in great detail, for all magnetic fields from $B=0$ to several Tesla, and the choice of parameters will be very relevant; in particular, we will mostly focus on the regime of about one Tesla, where Coulomb blockade physics arises.", "In the present Subsection, in contrast, our aim is purely pedagogical, so the dimensions of the junction and $B$ -value are not overly important.", "In particular, the low value $B=200\\,$ mT chosen here is pedagogically useful (not too many $l$ -modes present in Fig.", "REF ), but not a very interesting choice from the point of view of Section .", "In order to analyze the pristine effect of $|V_l|$ on the conductance of the dumbbell, we take a clean system (no disorder) with simple cylindrical TINW leads attached.", "The conductance simulations are performed with kwant [37].", "For details on the numerics, such as the non-uniform lattice we use, we refer to Appendix .", "Note that we fix the Fermi velocity to $v_F=5\\cdot 10^5\\,$ m$\\,$ s$^{-1}$ for all our numerical calculations, which is a typical value for bulk TIs [5].", "The effective potential $|V_l(z)|$ for the set of parameters given in the caption of Fig.", "REF , and the corresponding conductance $G$ as a function of the Fermi level $E_F$ , are displayed in Fig.", "REF .", "The transport characteristics of the smoothed TI dumbbell can be qualitatively understood in terms of $|V_l|$ in the following way: Transport is blocked below $E_F\\approx 4.7\\,$ meV (below the red dashed line) because all open lead modes [i.e.", "modes with $\\epsilon _l(k_z=0)<4.7\\,$ meV in the cylindrical leads, see $l = 3$ (brown) and $l = 2$ (purple)] do not have enough energy to overcome the central barriers.", "For $E_F \\gtrsim 4.7\\,$ meV, the $l = 1$ mode can pass through the TINW, since the local maximum of $|V_{l=1}|$ (red) at the wire center is only slightly above 2$\\,$ meV.", "Hence we observe a conductance step at $E_F\\approx 4.7\\,$ meV.", "The situation is slightly different for the $l = 2$ mode (purple).", "It is already present in the leads at $E_F\\approx 2.6\\,$ meV, but the potential has its maximum in the center at $|V_{l=2}|\\approx 6.3\\,$ meV.", "Comparing the corresponding conductance step with the one at $E_F\\approx 4.7\\,$ meV, it is apparent that its slope is lower.", "The reason for this is that the electronic mode can tunnel through the $l = 2$ barrier at energies below $6.3\\,$ meV, leading to a finite conductance contribution at lower energies.", "The same behavior can be observed for the $l=3$ mode.", "This exemplary discussion shows that plotting $|V_l|$ for all values of $l$ relevant at low energy is an efficient way to predict qualitative features of the conductance $G(E_F)$ in shaped TINWs, for any given $B$ .", "Note that due to rotational symmetry, coupling between different $l$ -modes is absent and the crossings in Fig.", "REF are real crossings, not avoided crossings.", "Thus an electron cannot traverse the wire by changing its orbital angular momentum quantum number.", "Rotational symmetry is broken for instance by disorder, which allows coupling between $l$ -modes.", "We will see the effect of $l$ -mode coupling later on.", "The TI nanocone (TINC) depicted in Fig.", "REF (a) represents the elementary building block of shaped TINWs, so we shall study in detail its transport characteristics.", "The latter were recently pointed out in Ref.", "[Kozlovsky2020], and we provide here a more pedagogical and complete treatment of the subject, focusing exclusively on coaxial magnetic fields." ], [ "Effective mass-potential of a TI nanocone", "For a given TINC geometry, Fig.", "REF shows how the potential $|V_l|$ evolves as a function of the magnetic field.", "For $B=0$ , $|V_l|\\propto 1/R$ due to size confinement only, and modes with angular momentum quantum number $l$ and $-l-1$ are degenerate.", "Upon turning on the magnetic field the degeneracy is lifted: modes with $l<0$ move upward in energy, while $l\\ge 0$ modes move downward.", "The potential for a given mode possesses a root inside the TINC as soon as the critical radius $\\tilde{R}_l(B)\\equiv \\sqrt{\\frac{\\Phi _0}{\\pi B}\\left(l+\\frac{1}{2}\\right)}$ fulfills the condition $\\tilde{R}_l(B)\\le R_0$ .", "This root is located at $\\tilde{z}_l(B)\\equiv \\frac{1}{\\mathcal {S}}\\left[\\tilde{R}_l(B)-R_0\\right],$ since $R=R_0+\\mathcal {S}z$ with the slope $\\mathcal {S}\\equiv \\frac{R_1-R_0}{z_1-z_0}<0$ .", "The higher the magnetic field, the larger the number of effective potentials $|V_{l\\ge 0}|$ developing a wedge-shaped well, whose minimum migrates from $\\tilde{z}_l=z_0$ to $\\tilde{z}_l=z_1$ and vanishes as soon as $\\tilde{R}_l(B)<R_1$ .", "We will see in Sec.", "REF that this can be understood as the formation of Landau levels As can be seen from Fig.", "REF , the twofold angular momentum degeneracy gets lifted for $B\\ne 0$ and is never restored for higher $B$ , due to the out-of-surface component.", "This is in marked contrast to cylindrical TINWs..", "Note the interesting analogy to graphene in an inhomogeneous magnetic field: Consider the step-like profile of $B_\\perp $ , see inset to Fig.", "REF (a).", "Due to the Dirac character of the surface electrons, we expect similar physics as for a magnetic step barrier in graphene, studied in Ref.", "[35].", "In the same way, more complicated profiles of $B_\\perp $ , see for example Fig.", "REF (b), are analogous to more complicated magnetic barriers in graphene, see Ref.", "[36].", "Indeed, a shaped TINW with profile $B_\\perp (z)$ is, in many respects, similar to graphene subject to an equivalent inhomogeneous magnetic field.", "In such 2D Dirac systems, the role of the (angular momentum-based) effective potential (REF ) is taken by a transverse momentum-based effective potential, as we point out in Appendix ; this potential develops wedges similar to those in Fig.", "REF for suitable system parameters.", "In view of this analogy (for more details see Appendix ), one expects that a TINC may act as a strong magnetic barrier – indeed, we will see that it does – and that Dirac electrons can be confined in between two TINCs.", "One should, however, remember that a magnetic step barrier in (2D) graphene requires an inhomogenoeus magnetic field, while the (3D) TINC allows for similar physics by just using a homogenoeus magnetic field." ], [ "Electronic structure of a TI nanocone", "We now discuss solutions of the 1D effective Dirac equation (REF ) for the TINC geometry with leads as depicted in Fig.", "REF (a).", "Note that $|V_l|$ is constant in the leads with values $|V_l(z_0)|$ ($|V_l(z_1)|$ ) in the left (right) lead.", "There are different regimes separated by the energy thresholds $\\epsilon _l^\\text{min}\\equiv \\text{min}(|V_l(z_0)|,|V_l(z_1)|)$ and $\\epsilon _l^\\text{max}\\equiv \\text{max}(|V_l(z_0)|,|V_l(z_1)|)$ , which is explained in the following.", "For energies $\\epsilon _{nl}>\\epsilon _l^\\text{max}$ , a solution $\\chi _{nl}$ of Eq.", "(REF ) is fully extended across the TINC and extends into both leads, and $n$ is a continuous index; in the limit of zero slope, this index is simply $n=k_z$ .", "For $\\epsilon _l^\\text{min}\\le \\epsilon _{nl}\\le \\epsilon _l^\\text{max}$ , any solution $\\chi _{nl}$ extends into one of the leads, while decaying exponentially on the other side, and again $n$ is continuous.", "For intermediate/high magnetic fields potential wedges are present, see, for instance, the last panel in Fig.", "REF , and $\\epsilon _l^\\text{min}$ represents the depth of such a wedge.", "Thus, the possibility for bound states within the effective potential with energies $\\epsilon _{nl}<\\epsilon _l^\\text{min}$ arises.", "As pointed out in Ref.", "[1], the energies of these bound states are given by the Dirac LL energies $E_n=\\text{sgn}(n)\\frac{\\hbar v_F}{l_B}\\sqrt{2|n|},\\hspace{28.45274pt}n\\in \\mathbb {Z},$ where $l_B\\equiv \\sqrt{\\hbar /(e|B_\\perp |)}$ is the magnetic length and $B_\\perp $ is the magnetic field component perpendicular to the surface of the TINC, cf. Fig.", "REF (a).", "This outcome is corroborated in the following with numerical results and an intuitive physical picture.", "Figure: (a) Effective potentials |V l ||V_l| for cone parameters z 0 =0z_0=0, z 1 =594.7z_1=594.7\\,nm, R 0 =2R 1 =156.6R_0=2R_1=156.6\\,nm, and B=2B=2\\,T, which yields l B ≈50l_B\\approx 50 nm, L cone =600L_{\\rm cone}=600 nm.", "Note that |V l ||V_l| are plotted for every fifth ll-value only.", "Additionally, the eigenstate probability distributions [for the solutions of Eq.", "()] |χ n=0,l=15 (z)| 2 |\\chi _{n=0,l=15}(z)|^2 (blue solid line), |χ n=1,l=15 (z)| 2 |\\chi _{n=1,l=15}(z)|^2 (blue dashed line), |χ n=0,l=25 (z)| 2 |\\chi _{n=0,l=25}(z)|^2 (green solid line) and |χ n=1,l=25 (z)| 2 |\\chi _{n=1,l=25}(z)|^2 (green dashed line) are shown (using arbitrary units).", "The bound states are identified as QH states (see main text).", "(b) Bar plot for the bound state energies ϵ nl \\epsilon _{nl}.", "The width of the bars is 0.40.4\\,meV.", "Dashed vertical lines give the analytical values of the Dirac LL energies with the perpendicular magnetic field component |B ⊥ |=Bsin(β/2)|B_\\perp |=B\\sin (\\beta /2).The inset shows a schematic side view of the TINC, with opening angle β\\beta .In Fig.", "REF , we choose parameters $\\lbrace R_0,R_1,z_0,z_1,B\\rbrace $ with $l_B \\ll L_{\\rm cone} \\equiv (z_1-z_0)/\\cos (\\beta /2)$ ($l_B\\approx 50$  nm, $L_{\\rm cone}=600$  nm), which highlights the formation of LLs.", "Here, $L_{\\rm cone}$ is the arc length along the TINC and $\\beta $ the opening angle.", "Figure REF (a) shows the effective potentials $|V_l|$ for the corresponding TINC together with the probability distributions of the eigenstates $|\\chi _{nl}(z)|^2$ for the two wedges $l=15$ and $l=25$ , where the solid (dashed) line corresponds to the $n=0$ ($n=1$ ) state.", "Note that, for clarity of the Figure, $|V_l|$ is not plotted for all $l$ -values (unlike in Fig.", "REF ).", "Figure REF (b) shows a bar plot which counts eigenenergies $\\epsilon _{nl}$ in an energy window of $0.4\\,$ meV.", "Here, all energies $\\epsilon _{nl}$ were used for which the states $\\chi _{nl}$ are bound states on the TINC, i.e.", "reside between $z_0$ and $z_1$ .", "As expected, we observe a large degeneracy in $l$ at the Dirac LL energies $E_n$ (up to numerical precision) marked with vertical dashed lines.", "Moreover, the eigenstate probability distributions $|\\chi _{nl}(z)|^2$ show one maximum for $n=0$ and two maxima for $n=1$ , as expected from quantum Hall (QH) states which derive from a harmonic oscillator equation (for which a state with index $n$ has $n+1$ maxima).", "Hence, we can indeed identify the bound states of the effective potential wedges $|V_l|$ as QH states, and all QH states (labeled by $n,l$ ) for a given $n$ together form the $n$ -th LL.", "The intuitive physical picture is the following: The perpendicular magnetic field component is constant throughout the cone, which means that the 2D Dirac electrons on the surface are subject to a homogeneous magnetic field and thus form LLs.", "The only condition which needs to be fulfilled is that the magnetic length is small compared to the length of the cone, such that the QH states (in classical terms cyclotron orbits) fit onto the TINC.", "This is equivalent to the condition that effective potential wedges form within the TINC.", "These arguments are also reflected in the degeneracies of the LLs in Fig.", "REF (b), which is given by the height of the bars.", "Quantum Hall states with larger $n$ extend more in space, and thus less QH states fit onto the cone.", "Consequently, the degeneracy decreases with increasing $n$ ." ], [ "Transport through a TI nanocone", "The setup we consider for transport is a TINC connected to cylindrical, highly-doped TI leads in a coaxial magnetic field, see Fig.", "REF (a).", "Its transport properties are determined by the states available at a given energy, i.e.", "those discussed in Sec.", "REF : At high energies ($\\epsilon _{nl}>\\epsilon _l^\\text{max}, n$ continuous), states are fully extended across the TINC and hybridize with both leads.", "At intermediate energies ($\\epsilon _l^\\text{min}\\le \\epsilon _{nl}\\le \\epsilon _l^\\text{max}, n$ continuous), states couple strongly to one of the leads and weakly if at all to the other.", "If $\\epsilon _{nl}<\\epsilon _l^\\text{min}$ , quasi-bound states centered at $\\tilde{z}_l(B)$ exist.", "For $\\epsilon _{nl}\\ll \\epsilon _l^\\text{min}$ their energy coincides with LL states $\\epsilon _{nl}=E_n$ .", "Closer to the potential threshold, $\\epsilon _{nl}\\lesssim \\epsilon _l^\\text{min}$ , the tail of the wave function enters the leads, and $\\epsilon _{nl}\\lesssim E_n$ .", "The considerations above, together with knowledge from Sec.", "REF , allow us to make qualitative predictions for the conductance $G$ as a function of the lead Fermi level $E_F$ and the coaxial magnetic field $B$ .", "These predictions will be confirmed by numerical transport simulations later on.", "Low magnetic field – Inspecting the effective potential in Fig.", "REF , one expects $G(E_F)$ to be characterized by steps centered around energies $\\epsilon _l^\\text{max}$ , since the effective potential of the left (right) lead is given by $|V_l(z_0)|$ ($|V_l(z_1)|$ ).", "Intermediate/high magnetic field – In the situation shown in Fig.", "REF for $B=0.5\\,$ T, only the potential wedges belonging to $l=7,8$ feature thresholds above the first LL.", "Consequently, the first LL can only form in the central part of the TINC, and higher LLs are absent.", "Thus we choose the TINC parameters from Fig.", "REF , where LLs consisting of many QH states form and their role in transport is enhanced.", "Figure: Transport through a TINC in intermediate/high magnetic field for parameters as in Fig. .", "(a) Effective potentials |V l ||V_l| on the TINC.", "Note that |V l ||V_l| are plotted for every third ll-value only.", "LL energies are indicated by orange dashed lines.", "The shaded stripes denote typical broadening of the LLs, as extracted from the numerical results in panel (b).", "(b) Logarithmic plot of the disorder-averaged conductance around the first and second LL, as a function of lead Fermi level E F E_F and calculated using Kwant.", "Vertical lines have the same meaning as in panel (a).", "The labels (I, II, III) serve to explain the transport regimes of the TI dumbbell (see Sec.", "and Fig.", ").The effects on transport of such strong LL quantization are presented in Fig.", "REF , showing $|V_l|$ and $G(E_F)$ for a TINC geomentry defined in the caption and a coaxial magnetic field of $B=2\\,$ T. Here disorder is added, which couples different $l$ -modes (i.e.", "QH states).", "This causes a broadening of the LLs, sketched in Fig.", "REF (a) as gray shaded areas around their central (ideal) energies (horizontal orange lines).", "Independently of the value of $E_F$ , a lead electron can enter the outer TINC region via states of type [B].", "However, $E_F$ determines whether it can enter the central region: Off-resonance – If $E_F$ is far away from a LL energy, an electron in the outer TINC region (where no LL forms), cannot find states for elastic transport further into the TINC, hence the conductance is suppressed.", "On-resonance – If $E_F$ lies within a disorder-broadened LL, characterized by a certain width $\\Delta E_n$ , an electron in the outer TINC region can, via disorder-induced scattering, be transferred to a state of type [C].", "From there, it can travel elastically through the central TINC region (via disorder-coupled QH states), such that the conductance is finite.", "For visualization, consider a lead electron from the left at $E_F<E_2-\\Delta E_2/2$ [black dashed line in Fig.", "REF (a)].", "It can enter the TINC via states of type [B], with $31\\lesssim l\\lesssim 40$ .", "Then, however, elastic transport is obstructed, because potential wedges in the center of the TINC only host states of type [C], with $\\epsilon _{nl}\\approx E_n$ .", "In contrast, if $E_2-\\Delta E_2/2\\lesssim E_F\\lesssim E_2+\\Delta E_2/2$ [black dotted line in Fig.", "REF (a)], a lead electron from the left, after accessing the TINC via modes $31\\lesssim l\\lesssim 40$ , can elastically tunnel through the core region of the TINC, via states of type [C] which exist for $13\\lesssim l\\lesssim 30$ , and exit the TINC on the other side.", "We conclude that, at low energies, the TINC is transparent for $E_F\\approx E_n$ , while it is opaque for $E_F$ in between two consecutive LLs.", "Fig.", "REF (b) shows the TINC conductance around the first and second LLs, calculated using the kwant [37] software.", "The resonant conductance peaks, already numerically obtained in Ref.", "[Kozlovsky2020], are explained in an intuitive way by the microscopic picture outlined above.", "Highly-doped leads were used for the calculations, together with Gaussian-correlated disorder, $\\mathinner {\\langle {V(\\mathbf {r})V(\\mathbf {r}^{\\prime })}\\rangle }=K\\hbar v_Fe^{-|\\mathbf {r}-\\mathbf {r}^{\\prime }|^2/2\\xi ^2}/(2\\pi \\xi ^2)$ , with the (dimensionless) disorder strength $K$ and the correlation length $\\xi $ .", "For numerical results presented throughout this paper, averages were taken over 600 disorder configurations with $K=0.1$ .", "(For more details on the methodology of the numerics, see Ref.", "[Kozlovsky2020] or Appendix ).", "Importantly, Fig.", "REF (b) is also the starting point for describing transport in a TI dumbbell.", "Figure: Radius of a smoothed TINC as given by Eq.", "() for three values of σ\\sigma , with parameters chosen such that the (perfectly sharp) TINC geometry used in Figs.", "& is recovered in the limit σ→∞\\sigma \\rightarrow \\infty ." ], [ "Smoothed TI nanocone", "Consider now a more realistically shaped TINC with smooth connections to the leads, cf.", "Fig.", "REF (c).", "The parametrization of the corresponding radius $R(z)$ , cf.", "Fig.", "REF , is given by Eq.", "(REF ).", "The smoothing strength is determined by the parameter $\\sigma $ , where a small (large) $\\sigma $ corresponds to strong (weak) smoothing.", "In the low magnetic field regime there is no qualitative difference with the ideal TINC, and conductance steps centered around energies $\\epsilon _l^\\text{max}$ are expected.", "In the remainder of the present section, we focus on intermediate and high fields, starting as usual by solving Eq.", "(REF ).", "In Fig.", "REF , we show the effect of smoothing for the same system parameters as in Fig.", "REF , meaning that the TINC from Fig.", "REF is recovered in the limit $\\sigma \\rightarrow \\infty $ .", "When the TINC is smoothed (by decreasing $\\sigma $ ), the value of $|B_\\perp |$ gets lowered, the effect being considerably stronger in the vicinity of the leads than in the middle of the TINC, cf. Fig.", "REF .", "In the language of the effective potential, this means that a given wedge $|V_l|$ shifts and gets distorted (mostly its lead-facing branch gets lowered), see Fig.", "REF (a).", "This effect is stronger for wedges close to the leads.", "Therefore, the smoothing can have two different effects on a given LL bound state.", "(i) The bound state disappears.", "This is relevant for states close to the leads.", "(ii) The bound state survives but gets lowered in energy because of reduced $B_\\perp $ (increasing magnetic length).", "Thus, upon smoothing, the $l$ -degeneracy of the LLs, present for the perfect TINC in Sec.", ", is lifted, see Fig.", "REF (a).", "Note that the zeroth LL stays $l$ -degenerate, since it is not affected by the smoothing.", "For all states belonging to class (ii), we define the decrease in energy $\\Delta \\epsilon _{nl}(\\sigma )\\equiv \\epsilon _{nl}(\\sigma \\rightarrow \\infty )-\\epsilon _{nl}(\\sigma )>0$ , where $\\epsilon _{nl}(\\sigma \\rightarrow \\infty )=E_n$ .", "It is plotted for LL indices $n=1$ to $n=3$ in Fig.", "REF (b), for a selected number of angular momenta.", "Figure: (a) Effective potentials for a smoothed TINC, with the same parameters as in Fig.", ", but lowering the value of σ\\sigma in Eq.", "() to σ=10μ\\sigma =10\\,\\mu m -1 ^{-1}.", "Orange dashed lines represent LL energies E n E_n in the limit σ→∞\\sigma \\rightarrow \\infty of a perfect TINC, while solid orange lines are bound state energies ϵ nl \\epsilon _{nl}.", "(c) Difference in energy Δϵ nl (σ)\\Delta \\epsilon _{nl}(\\sigma ) between LLs of the perfect TINC [dashed orange lines in panel (a)] and bound state energies ϵ nl \\epsilon _{nl} of the potential well |V l ||V_l| [solid orange lines in panel (a)].Consider now the transport characteristics of the smoothed TINC.", "An electron from the lead at given $E_F$ can typically proceed a bit further into the TINC than in the limit $\\sigma \\rightarrow \\infty $ .", "This is due to the lowering of the lead-facing effective potential branches, see Fig.", "REF (a).", "However, the energies of states in the TINC center stay practically the same as in the limit $\\sigma \\rightarrow \\infty $ .", "Thus, transport through the TINC is still suppressed for $E_F$ placed in between LLs.", "Pursuing this line of thought, we can predict another interesting transport regime in smoothed TINCs.", "If the decrease in energy of bound states relatively close to the leads [for example the state $(n,l)=(2,14)$ in Fig.", "REF (a)] becomes large enough ($\\Delta \\epsilon _{nl}(\\sigma )>\\Delta E_n/2$ ), these energies exit the disorder-broadened transport channel.", "Such states are no longer available for elastic transport, even though the Fermi level is tuned \"on resonance”, i.e.", "$E_F\\approx E_n$ .", "This phenomenon can be achieved in our setup using relatively strong magnetic fields $B\\gtrsim 5\\,$ T. Consequently, magnetic barriers arise close to the TINC ends, such that a single (smoothed) TINC becomes a quantum dot-like object.", "If a gate electrode is attached to the TINC, this may lead to Coulomb blockade-type physics [1].", "In summary, the qualitative form of the conductance $G(E_F)$ shown in Fig.", "REF (b) is robust against variations of the geometry, as long as the local QH state energies lie within the disorder broadening.", "For a smoothing [as defined by Eq.", "(REF )] strength for which the QH states close to the leads are moved beyond the disorder broadening, magnetic barriers appear and we expect transport to be dominated by Coulomb blockade-like physics, as discussed in Ref.", "[1].", "As an example of shaped TINWs beyond the relatively simple TINC, we now consider a TI dumbbell [see Fig.", "REF (b)], which is representative for a mesoscopic TI nanowire constriction and hosts a rich variety of magnetotransport regimes.", "For simplicity we take the TI dumbbell to be composed of two symmetrically arranged TINCs ($R_2=R_1$ , $R_3=R_0$ ), cf. Fig.", "REF (b), each with the same parameters as used in Figs.", "REF and REF .", "The length of the intermediate cylindrical part is chosen as $L\\equiv z_2-z_1=(z_1-z_0)/2$ ." ], [ "Low magnetic field", "The effective potentials feature an $l$ -degeneracy for $B=0$ [see Fig.", "REF (a) and recall Fig.", "REF ], leading to a conductance profile as shown in Fig.", "REF (b).", "This $l$ -degeneracy gets lifted for $B\\ne 0$ .", "When this happens, the precise form of the conductance $G(E_F)$ depends on the particular value of $B$ , but its qualitative structure (smoothed steps originating from mode opening) stays the same as long as $B_\\perp $ is too weak for LLs to form on the two TINCs.", "(Although the dumbbell considered there is of slightly different dimensions than the one discussed in the present Section, this degeneracy lifting, as well as the qualitatively unchanged conductance profile, can be observed explicitly in Fig.", "REF , which is plotted in the low $B$ regime.)", "Due to the mirror symmetry with respect to the plane $z=(z_3-z_0)/2$ , these steps are located at the same values $E_F=\\epsilon _l^\\text{max}$ that one would expect for a single TINC.", "Figure: Transport through a TI dumbbell for B=0B=0.", "Here we choose z 0 =0z_0=0, z 1 =594.7z_1=594.7\\,nm, z 2 =3z 1 /2z_2=3z_1/2, z 3 =5z 1 /2z_3=5z_1/2, and R 0 =R 3 =2R 1 =2R 2 =156.6R_0=R_3=2R_1=2R_2=156.6\\,nm.", "(a) Effective potentials |V l ||V_l| and (b) disorder-averaged conductance as a function of the lead Fermi level E F E_F, calculated using kwant." ], [ "Intermediate/high magnetic field", "In this regime, a simple tuning of the magnitude of $B$ allows access to three scenarios: (I) current suppression, (II) Coulomb blockade, and (III) resonant transmission.", "Given our choice of parameters, the effective potential landscape on the dumbbell's left side is the one shown in Fig.", "REF (a), followed by a constant $|V_l|$ in the cylindrical center and the mirrored version of Fig.", "REF (a) on the right, see Fig.", "REF (a).", "Figure: (a) Effective potentials for a TI dumbbell for B=2B=2\\,T and parameters as in Fig. .", "Note that |V l ||V_l| are plotted for every third ll-value only, except in the important central region, which is the focus of this Figure.", "(b) Zoom into the central region.", "Energies of eigenstates residing on the central cylinder (on the TINCs)are marked by blue (orange) lines.Potential wells appear in the central region, which may host bound states whose coupling to the leads depends on the transparency of the TINCs.", "This transparency is the same for both TINCs, and governed by the conductance $G(E_F)$ shown in Fig.", "REF (b), because the ladder of LLs depends only on the absolute value of $B_\\perp $ .", "Figure REF (b) depicts the energies [obtained from solving Eq.", "(REF ) in the presence of cylindrical TI leads] corresponding to QH states on the TINCs as orange lines, and those corresponding to states confined between the two TINCs as blue lines.", "As a guide to the eye, the extent of the lines in the $z$ -direction is chosen such that their ends touch the potential well they belong to (this encodes the $l$ -value of a given energy level).", "Within each given $l$ -potential well, the level spacing (distance between blue lines) can be controlled by the longitudinal confinement and is proportional to $1/L$ .", "However, it is in general not possible to define a constant level spacing characterizing the whole central region, as states belonging to different $l$ -potentials may cluster.", "Using Fig.", "REF (b), three different transport regimes can be identified, depending on the lead Fermi level position relative to the LL energies.", "If $E_F$ is placed such that $G\\approx 0$ ($G$ being the conductance through a single TINC), both TINCs act as strong barriers, and transport through the dumbbell is suppressed.", "This is indicated by the dashed-dotted line in Fig.", "REF .", "If $E_F$ is placed such that the TINC conductances are $0<G\\ll e^2/h$ , the central cylindrical region can be viewed as a quantum dot weakly coupled to the leads.", "As we discuss below, this leads to conductance oscillations of Coulomb blockade type once the central region is gated.", "This is indicated by the dashed line in Fig.", "REF .", "Both TINCs are transparent ($G\\sim e^2/h$ ) for a lead Fermi level fulfilling $E_F\\approx E_n$ .", "If $E_F$ is in addition in resonance with a (disorder broadened) bound state of the central cylinder (blue line), we expect a finite conductance.", "Otherwise the transmission is suppressed.", "This is indicated by the dotted line in Fig.", "REF .", "In the remainder of this Section, we focus on case (II)." ], [ "Coulomb blockade in the TI dumbbell", "If we assume a gate electrode to be applied to the central cylindrical region, a decisive quantitiy is the charging energy $E_C=e^2/C$ , where the capacitance $C$ depends on the experimental setup (the geometry and the materials, including the dielectrics).", "Some typical values for $C$ are provided by the literature.", "For experiments on strained HgTe TINWs [19] of dimensions comparable to our situation, a numerical solution of the Poisson equation in the presence of SiO2/Al2O3 dielectrics and a gold top gate yields an effective capacitance per surface area $C_\\text{eff} \\approx 4\\cdot 10^{-4}\\,\\text{F}\\,\\text{m}^{-2}$ .", "A different experiment, studying TI quantum dots based on Bi2Se3 TINWs on a SiO2/Si substrate [39], found $C=2\\cdot 10^{-17}\\,$ F for a surface area of $8.6\\cdot 10^{-14}\\,$ m$^2$ , corresponding to $C_\\text{eff}=2.3\\cdot 10^{-4}\\,\\text{F}\\,\\text{m}^{-2}$ .", "We can thus estimate the charging energy for the TI dumbbell by $E_C=e^2/(2\\pi R_1LC_\\text{eff})$ .", "For our example, we choose a value of $E_C=1.5\\,$ meV, which corresponds to $C_\\text{eff}=7.3\\cdot 10^{-4}\\,\\text{F}\\,\\text{m}^{-2}$ , the same order of magnitude as in the above experiments.", "This choice results in $E_C \\approx \\delta \\epsilon $ ; here, $\\delta \\epsilon $ is the typical level spacing in the central region, defined as follows: Each $\\epsilon _{nl}$ quasi-bound state in the central region [blue lines in Fig.", "REF (b); $n$ counts the quasi-bound states within a given well $|V_l|$ ] has a broadening $\\Gamma _{nl}$ determined by its coupling to the leads.", "Blue lines clustering such that their energy distance satisfies $|\\epsilon _{nl}-\\epsilon _{n^{\\prime }l^{\\prime }}|< \\frac{1}{2}{\\rm max}\\left\\lbrace \\Gamma _{nl},\\Gamma _{n^{\\prime }l^{\\prime }}\\right\\rbrace $ cannot be resolved, yielding effectively a single central region level (which can be multiply occupied).", "The spacing $\\delta \\epsilon $ is taken among such central region levels, be they single $\\epsilon _{nl}$ levels or clusters as just defined, and should be viewed as an order-of-magnitude estimate obtained by inspecting Fig.", "REF (b).", "In the presence of a gate electrode, we expect Coulomb blockade oscillations [40] in the current-gate voltage characteristics.", "The particular Coulomb blockade regime is determined by the ratio of the three energy scales $\\delta \\epsilon $ , $k_BT$ and $\\Gamma _{nl}$ .", "An order of magnitude estimate for $\\Gamma _{nl} = \\hbar / \\tau _{nl}$ follows from the dwell time $\\tau _{nl}$ in the central region.", "The two side TINCs can be treated as “black boxes” with a transmission given by the conductance from Fig.", "REF (b), such that the system is mapped to a quasi-1D double-barrier structure of inner length $L$ .", "Estimating $\\tau _{nl}$ requires two ingredients: (i) The average distance (in the $z$ -direction) covered by an $\\epsilon _{nl}$ quasi-bound state is $d=L\\frac{1+{\\cal R}_2}{1-{\\cal R}_1 {\\cal R}_2}$ , with ${\\cal R}_i, i=1,2$ the reflection probabilities at the individual barriers.", "In our symmetric setup we have ${\\cal R}_1={\\cal R}_2\\equiv {\\cal R}$ , and $d=L/{\\cal T}$ , with ${\\cal T}=1-{\\cal R}\\ll 1$ .", "(ii) The average velocity (in the $z$ -direction) can be estimated as $\\hbar v_{nl}\\sim \\partial \\epsilon _l(k_{z,n})/\\partial k_{z,n}$ where $\\epsilon _l(k_{z,n})=\\hbar v_F[(l+1/2-\\Phi /\\Phi _0)^2/R^2+k_{z,n}^2]^{1/2}$ is the band structure of a cylindrical TINW, with $k_{z,n}=(\\pi /L)n$ the wavevector values corresponding to a given quasi-bound state $n$ .", "It follows $\\Gamma _{nl}=\\frac{\\hbar }{\\tau _{nl}}\\sim \\frac{\\hbar v_{nl}}{d}\\sim \\frac{\\cal T}{L}\\frac{(\\hbar v_F)^2 k_{z,n}}{\\epsilon _l(k_{z,n})}.$ Thus, using a typical value ${\\cal T}=5\\cdot 10^{-3}$ [cf. Fig.", "REF (b)], one has $\\Gamma _{nl}\\sim 5\\,\\mu $ eV.", "While this implies that a few of the $\\epsilon _{nl}$ levels from Fig.", "REF (b) form clusters, spacing among the latter is such that at temperatures around $T\\approx 0.5\\,$ K various central region levels (single $\\epsilon _{nl}$ or clusters thereof) are resolved, i.e.", "the condition $\\Gamma _{nl}\\ll k_BT\\ll \\delta \\epsilon $ is fulfilled.", "In this regime, only a single level contributes to transport significantly, and the conductance is given by [40] $G=\\frac{e^2}{h}\\Gamma _{nl}\\frac{f(\\Delta _{nl})[1-f(\\Delta _{nl})]}{2k_BT}.$ The quantity $\\Delta _{nl}\\equiv \\epsilon _{nl}-E_F+(N_{nl}-1/2)E_C-\\alpha eV_g$ entering the Fermi-Dirac distribution $f(x)=1/(1+e^{x/k_BT})$ leads to a conductance peak whenever $\\Delta _{nl}=0$ .", "Here $N_{nl}$ is the number of electrons on the quantum dot, i.e.", "the number of levels with $\\epsilon _{n^{\\prime }l^{\\prime }}<\\epsilon _{nl}$ ; The proportionality constant $\\alpha $ between gate voltage $V_g$ and the associated electrostatic energy is, like $C$ , a function of the capacitance matrix [40] and needs to be determined experimentally [41].", "The resulting Coulomb blockade oscillations for the dumbbell from Fig.", "REF are shown in Fig.", "REF , taking $E_F=11.3\\,$ meV [dashed black line in Figs.", "REF and REF ], $E_C=1.5\\,$ meV, $T=0.5\\,$ K and ${\\cal T}=5\\cdot 10^{-3}$ .", "The fluctuations of conductance peak positions reflect the level spacings of the confined states living in the central cylindrical region.", "Note that our discussion of Coulomb blockade, and more generally of all the transport regimes considered above, aims at identifying intrinsic geometry-induced features.", "Therefore, the role of additional system-specific properties, e.g.", "voltage ripples [42] that might affect the Coulomb blockade oscillations, are not considered in our theoretical model (REF ).", "Figure: Coulomb blockade oscillations for a (surface) quantum dot formed due to magnetic confinement in a TI dumbbell with the same parameters as in Fig.", ".", "The conductance is calculated using Eq.", "(), at a temperature T=0.5T=0.5\\,K, lead Fermi level E F =11.3E_F=11.3\\,meV and charging energy E C =1.5E_C=1.5\\,meV." ], [ "Smoothed TI dumbbell", "The knowledge about a single smoothed TINC, discussed in Sec.", "REF , is straightforwardly generalized to more complex shaped TINWs.", "In particular, the smoothed TI dumbbell from Fig.", "REF (c) (plotted for $\\sigma =15\\,\\mu $ m$^{-1}$ ) still exhibits qualitatively the same transport behavior as the model junction discussed in the course of the present Section.", "As for the smoothed TINC, a new transport regime emerges at rather high magnetic fields.", "In this case, the smoothed dumbbell is still a double-barrier structure off resonance.", "However, it is a quadruple-barrier structure on resonance, i.e.", "a triple quantum dot, since each TINC features a pair of potential barriers.", "Table: Summary of low-energy magnetotransport regimes in simple examples of smoothly shaped TI nanowires, with comparison to the cylindrical limit.", "Each entry describes the shape of G(E F )G(E_F) for fixed BB with its physical origin in brackets.", "If the transport regime for a given geometry and BB-field depends on the Fermi level E F E_F, possible different regimes are separated by a slash.", "Abbreviations: osc.", "–– Van Hove singularity-induced conductance oscillation on top of an increasing conductance background; SBO –– subband opening as E F E_F is increased; quan. cond. pl.", "–– quantized conductance plateau; CSS –– chiral side surface states; cond.", "steps –– conductance steps; MO –– mode opening; res. trans.", "–– resonant transmission; LL –– Landau levels; CB –– Coulomb blockade; curr. supp.", "–– current suppression; triple QD –– transport through a triple quantum dot." ], [ "Summary & Conclusions", "We showed that numerous transport regimes are accessible when a shaped TI nanowire (TINW), i.e.", "an axially symmetric TINW with varying cross section along its length, is immersed in a homogeneous magnetic field.", "The results are summarized in Tab.", "REF , which is briefly outlined in the following.", "Consider first a strong ($l_B \\ll $ wire width) perpendicular magnetic field ($\\mathbf {B}\\perp \\hat{z}$ ), such that the top and bottom TINW surfaces are in the quantum Hall regime.", "Here, transport is dominated by chiral side surface states which do not “feel” the geometry of the nanowire.", "Thus, shaped TINWs behave qualitatively the same as cylindrical TINWs.", "A characteristic transport feature in this regime is a quantized conductance plateau in a magnetic field-dependent energy window [1], [23], [24], [25], [26], [27].", "In contrast, for weak perpendicular magnetic fields, states wrap around the wire and transport is thus geometry-dependent.", "For cylindrical nanowires, subbands open as $E_F$ is increased, which leads to Van Hove singularity-induced oscillations on top of an increasing conductance background.", "For shaped TINWs in a weak perpendicular magnetic field, quantum confinement-induced potentials need to be overcome.", "The corresponding transport signatures are steps in the conductance due to mode opening.", "In a coaxial magnetic field ($\\mathbf {B}\\parallel \\hat{z}$ ), the focus of our work, states wrap around the circumference and enclose the magnetic flux – hence, transport is highly sensitive to both magnetic field and geometry.", "In the presence of rotational symmetry, the angular momentum $l$ is a good quantum number and the effective mass-type potential $|V_l|$ is a useful tool to predict the transport behavior of any shaped TINW.", "For cylindrical nanowires, $G(E_F)$ is determined by Van Hove singularity-induced oscillations on top of an increasing conductance background, independently of the magnetic field strength – a result of translational invariance along the wire.", "Smoothed TI nanocones (TINCs) in contrast, first introduced in Ref.", "[1] and discussed at length in Sec.", ", can be tuned between three different regimes by a simple variation of the coaxial magnetic field strength: conductance steps due to mode opening (a mode with angular momentum quantum number $l$ opens as soon as $E_F > |V_l|$ ), resonant transmission, and Coulomb blockade-like transport through LLs.", "Using the single TINC as a building block, more complex-shaped TINWs can be composed.", "We focused on the paradigmatic example of a TINW constriction, dubbed TI dumbbell.", "In this system, a triple quantum dot structure can form for magnetic fields beyond approximately 3-4 Teslas (for our choice of parameters), adding one more fundamentally different transport regime to those available in a TINC.", "Most notably however, in the intermediate magnetic field regime ($B\\approx 1-2\\,$ T), one can switch on and off Coulomb blockade oscillations at will, simply upon altering the $B$ -value: If $E_F$ is far from the disorder-broadened LL energies, the TINCs on the sides of the dumbbell act as infinitely strong barriers and transport is suppressed.", "On the other hand, the same TINCs act as finite tunnel barriers if $E_F$ is in the vicinity of (but not exactly in resonance with) the LL energies, so that conductance oscillations of Coulomb blockade type should be visible if a gate electrode is applied to the central region.", "Concerning experimental realization, we point out that TI nanowires with non-uniform cross section have been fabricated, see e.g. Refs.", "[28], [19].", "All-round (homogeneous) gating, as assumed in the discussion of Coulomb blockade, is also currently possible [43], [44].", "The realization of a finely shaped TI tube thus appears challenging but within current experimental capabilities.", "Moreover, shaped TINWs represent a substantial practical advantage from the experimental point of view: The Coulomb blockade regime in the TI dumbbell arises from magnetic confinement of Dirac electrons.", "To the contrary of proposals for 2D geometries, where non-homogeneous magnetic fields are necessary to confine Dirac fermions [35], [36], in our shaped nanowires a homogeneous magnetic field is enough (see also Appendix ).", "This fact is the decisive ingredient behind the relatively simple on/off-switch mechanism between different magnetotransport regimes.", "Finally, let us stress that magnetic confinement is not restricted to the axially symmetric TI dumbbell explicitly considered.", "Essentially, one can use TINCs – tunable into barriers due to Landau quantization – and cylindrical TINWs – where free motion follows from $B_\\perp =0$ – as elemental building blocks and connect them in series to build arbitrary magnetic barrier profiles in homogeneous magnetic fields.", "Furthermore, strict axial symmetry is not required: a TINC with a somewhat distorted cross-section acts as a magnetic barrier as long as the local $l_B$ is smaller than its geometrical size, ensuring the formation of QH states throughout its (distorted) perimeter.", "We thus expect our results to be valid guidelines for the analysis of magnetotransport in a wide range of TINWs of any shape.", "Acknowledgements: We thank Andrea Donarini, Milena Grifoni, Dominik Hahn, Gilles Montambaux and Max Nitsch for useful discussions.", "CG thanks the STherQO members, and in particular Guillaume Weick, for useful comments.", "This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project-ID 314695032—SFB 1277 (subproject A07), and within Priority Programme SPP 1666 \"Topological Insulators\" (project Ri681-12/2).", "Support by the Elitenetzwerk Bayern Doktorandenkolleg \"Topological Insulators\" as well as the École Doctorale Physique en Île de France (EDPIF) is also acknowledged." ], [ "Dirac surface Hamiltonian for a shaped TI nanowire", "The derivation of the Hamiltonian (REF ) using a fiel theoretic approach is sketched here very briefly; the details can be found in Ref. [xypakis2017].", "Fermions on a (2+1)-dimensional curved manifold fulfill the covariant Dirac equation $\\gamma ^\\mu D_\\mu \\Psi =0,$ where $D_\\mu \\equiv \\partial _\\mu +\\Gamma _\\mu $ is the covariant derivative and $\\gamma ^\\mu \\equiv V_a^\\mu \\xi ^a$ are covariant Dirac matrices.", "The additional term $\\Gamma _\\mu $ is known as the spin connection [32], [33].", "The $\\xi ^a$ are local Dirac matrices satisfying the Clifford algebra $\\lbrace \\xi ^a,\\xi ^b\\rbrace =2\\eta ^{ab}$ , where $\\eta ^{ab}$ is the Minkowski metric, and $V_a^\\mu $ are the inverse vielbeins.", "The metric for the shaped TINW is given as [22] $dl^2=-dt^2+(1+R^{\\prime 2})dz^2+R^2d\\varphi ^2$ , where $\\varphi $ is the azimuthal angle and $R\\equiv R(z)$ is the radius as a function of the coaxial coordinate $z$ ; note that we work in natural units.", "We choose the following set of Dirac matrices (different from the choice in Ref.", "[22]): $\\xi ^0=i\\sigma _x,\\hspace{14.22636pt}\\xi ^1=\\sigma _y,\\hspace{14.22636pt}\\xi ^2=-\\sigma _z,$ such that $\\gamma ^0=i\\sigma _x,\\hspace{14.22636pt}\\gamma ^1=\\frac{1}{\\sqrt{1+R^{\\prime 2}}}\\sigma _y,\\hspace{14.22636pt}\\gamma ^2=-\\frac{1}{R}\\sigma _z.$ For the spin connection one finds $\\Gamma _t=\\Gamma _z=0$ and $\\Gamma _\\varphi =\\frac{i}{2}\\frac{R^{\\prime }}{\\sqrt{1+R^{\\prime 2}}}\\sigma _x.$ Then the Dirac equation (REF ) becomes $\\begin{aligned}i\\sigma _x\\partial _t\\Psi =&\\left[-\\frac{1}{\\sqrt{1+R^{\\prime 2}}}\\left(\\partial _z+\\frac{R^{\\prime }}{2R}\\right)\\sigma _y+\\frac{1}{R}\\partial _\\varphi \\sigma _z\\right]\\Psi .\\end{aligned}$ Restoring the fundamental constants and left-multiplying by $\\sigma _x$ such that a Hamiltonian can be defined by $H\\Psi =i\\hbar \\partial _t\\Psi $ , one arrives at the surface Dirac Hamiltonian for a shaped TINW: $H=v_F\\left[\\frac{1}{\\sqrt{1+R^{\\prime 2}}}\\left(p_z-\\frac{i\\hbar }{2}\\frac{R^{\\prime }}{R}\\right)\\sigma _z+p_\\varphi \\sigma _y\\right],$ where $p_z\\equiv -i\\hbar \\partial _z$ and $p_\\varphi \\equiv -i\\hbar R^{-1}\\partial _\\varphi $ .", "The term $\\propto R^{\\prime }/R$ represents the nontrivial spin connection.", "In the presence of a homogeneous coaxial magnetic field $\\mathbf {B}=B\\hat{z}$ , the vector potential in the symmetric gauge is given by $\\mathbf {A}=A_\\varphi \\,\\hat{\\varphi }=Br/2\\,\\hat{\\varphi }$ .", "We replace $p_\\varphi \\rightarrow p_\\varphi +eA_\\varphi $ , where $e>0$ , to obtain $\\begin{aligned}H=v_F&\\left[\\frac{1}{\\sqrt{1+R^{\\prime 2}}}\\left(p_z-\\frac{i\\hbar }{2}\\frac{R^{\\prime }}{R}\\right)\\sigma _z\\right.\\\\&\\left.+\\left(p_\\varphi +\\frac{\\hbar }{R}\\frac{\\Phi }{\\Phi _0}\\right)\\sigma _y\\right].\\end{aligned}$ Here, $\\Phi \\equiv \\pi BR^2$ is the magnetic flux piercing the wire and $\\Phi _0\\equiv h/e$ is the magnetic flux quantum.", "This is the Hamiltonian (REF ) provided in the main part of this paper." ], [ "Conductance simulations with ", "Non-uniform lattice – The effective tight-binding Hamiltonian used to compute transport throughout this work with the software package kwant is obtained by discretizing Eq.", "(REF ).", "In the following, we use the short-hand notation $\\Psi (s_i, \\varphi _j) \\equiv \\Psi _{i,j}$ for the two-component spinor wave function $\\Psi $ on the numerical grid defined by the grid points $(i,j)$ (where $i,j$ are integers and $s$ is the arclength along the wire).", "Using this notation, a discretization of the transversal wave number operator $\\hat{k}_\\varphi =-\\mathrm {i}\\partial _\\varphi /R(s)$ with the standard symmetric finite difference method yields $\\hat{k}_\\varphi (s) \\Psi _{i,j} \\rightarrow &- \\frac{\\mathrm {i}}{R(s_i)} \\frac{1}{2 \\Delta \\varphi } \\left(\\Psi _{i,j+1} - \\Psi _{i,j-1} \\right) \\\\\\equiv &- \\frac{\\mathrm {i}}{2a_\\varphi (s_i)} \\left(\\Psi _{i,j+1} - \\Psi _{i,j-1} \\right),$ where the angle $\\Delta \\varphi $ is determined by the number of grid points in the transversal direction $N_\\varphi $ , namely $\\Delta \\varphi = 2\\pi /N_\\varphi $ .", "In Eq.", "(REF ), we introduce the $s$ -dependent transversal grid constant $a_\\varphi (s) \\equiv R(s) \\Delta \\varphi $ to highlight that effectively the transversal grid spacing is changing such that $N_\\varphi a_\\varphi (s) = 2 \\pi R(s)$ .", "With the standard discretization of the longitudinal wave number operator $\\hat{k}_s \\Psi _{i,j} = -\\mathrm {i}\\left(\\Psi _{i+1,j} - \\Psi _{i-1,j} \\right)/(2 a_s)$ , where $a_s$ is the grid spacing in the longitudinal direction, we arrive at the tight-binding Hamiltonian $\\begin{split}H_\\text{TB} = - \\frac{\\mathrm {i}\\hbar v_F}{2} \\sum _{i,j} \\left( \\frac{1}{a_s} \\sigma _z \\mathinner {|{i,j}\\rangle }\\mathinner {\\langle {i+1,j}|}\\right.", "\\\\+ \\left.", "\\frac{1}{a_\\varphi (s_i)} \\sigma _y \\mathinner {|{i,j}\\rangle }\\mathinner {\\langle {i,j+1}|} \\right) + \\text{h.c.}\\end{split}$ The coaxial magnetic field is implemented using the usual Peierls substitution.", "Modeling disorder on curved surfaces – For creating correlated disorder we use the so-called Fourier filtering method (FFM), which is discussed in detail for instance in Ref.", "[45].", "For shaped TINWs, we construct a disorder landscape with the desired correlation length in a 3D box, in which the TINW is embedded.", "The values for the disorder potential $V_\\text{dis}(\\mathbf {r})$ are then evaluated within the box on the surface of the TINW and added as an onsite potential to the tight-binding Hamiltonian (REF )." ], [ "Effective mass potential for graphene subject to a magnetic step barrier", "In view of the step-like profile of $B_\\perp $ for a single TINC, see Fig.", "REF (a), we here provide the connection to the related and well-known problem of a magnetic step barrier in (single-valley) graphene [35].", "More generally, the form of Eq.", "(REF ) that we found for a shaped TINW is very similar to the effective Schrödinger equation found in graphene subject to various magnetic field profiles [35], [36], [46], [47].", "In this Appendix, we show that the results of Ref.", "[35] can be reinterpreted in the language of an effective mass potential, in full analogy to the effective potential introduced in Eq.", "(REF ).", "Consider an infinite graphene sheet, subject to a magnetic step barrier that is translationally invariant in the $y$ -direction and nonzero only in the region $-d\\le x\\le d$ , such that $B(x,y)=B_0\\Theta (d^2-x^2)$ [35].", "Assume inter-valley scattering to be absent.", "Choosing the gauge $\\mathbf {A}(x,y)=A(x)\\hat{y}$ , where $A(x)=B_0{\\left\\lbrace \\begin{array}{ll}-d, & x<-d \\,\\, (\\text{region I})\\\\x, & |x|\\le d \\,\\, (\\text{region II})\\\\d, & x>d \\,\\, (\\text{region III})\\end{array}\\right.", "}$ the Dirac equation becomes $v_F\\left\\lbrace p_x\\sigma _x+\\left[\\hbar k_y+eA(x)\\right]\\sigma _y\\right\\rbrace \\psi (x)=\\epsilon \\psi (x),$ where we exploited the fact that transverse momentum $\\hbar k_y$ is a good quantum number.", "Equation (REF ) is easily decoupled to give $\\mathcal {O}_y^\\pm \\psi _\\pm =\\epsilon ^2\\psi _\\pm ,$ where the Dirac spinor $\\psi =(\\psi _+,\\psi _-)^T$ and $\\begin{aligned}\\mathcal {O}_y^\\pm &=-(\\hbar v_F)^2(\\partial _x^2+\\mathcal {P}_y^\\pm ),\\\\\\mathcal {P}_y^\\pm &=\\mp \\frac{e}{\\hbar }A^{\\prime }-\\left(k_y+\\frac{e}{\\hbar }A\\right)^2.\\end{aligned}$ The analogy to Eq.", "(REF ) is evident, discrete angular momentum being replaced by continuous transverse momentum, and the coaxial coordinate replaced by $x$ .", "Comparing Eq.", "(REF ) to Eq.", "(REF ), we observe that transverse momentum acts as a mass potential: $V(x)\\equiv \\hbar v_Fk(x)=\\hbar v_F(k_y+eA(x)/\\hbar )$ .", "With the gauge (REF ) one has [35] $k(x)=\\tilde{\\epsilon }\\sin \\phi +{\\left\\lbrace \\begin{array}{ll}0, & x<-d \\,\\, (\\text{region I})\\\\(d+x)/l_B^2, & |x|\\le d \\,\\, (\\text{region II})\\\\2d/l_B^2, & x>d \\,\\, (\\text{region III})\\end{array}\\right.", "}$ where $\\tilde{\\epsilon }\\equiv \\epsilon /(\\hbar v_F)$ , $l_B\\equiv \\sqrt{\\hbar /(eB_0)}$ is the magnetic length and $\\phi $ is the kinematic incidence angle.", "Now, if $\\sin \\phi \\ge 0$ , it is clear that $k(x)\\ge 0$ , i.e.", ", $V(x)$ cannot become negative.", "However, if $\\sin \\phi <0$ , we have two possibilities: (i) $\\tilde{\\epsilon }|\\sin \\phi |\\ge 2d/l_B^2$ , then $k(x)\\le 0$ always.", "(ii) $\\tilde{\\epsilon }|\\sin \\phi |<2d/l_B^2$ , then $k(x){\\left\\lbrace \\begin{array}{ll}<0, & x<-d \\,\\, (\\text{region I})\\\\\\le 0, & -d\\le x\\le x_0 \\,\\, (\\text{region II})\\\\>0, & x_0<x\\le d \\,\\, (\\text{region II})\\\\>0.", "& x>d \\,\\, (\\text{region III})\\\\\\end{array}\\right.", "}$ Here, $x_0\\equiv \\tilde{\\epsilon }|\\sin \\phi |l_B^2-d<d$ denotes the root of the effective potential, analogous to $\\tilde{z}_l$ , cf. Eq.", "(REF ).", "In full analogy to Section REF , a root in $V(x)$ corresponds to a minimum and a surrounding potential wedge in $|V(x)|$ .", "Consequently, bound states within the effective potential $|V(x)|$ may exist if the two necessary criteria $\\sin \\phi <0$ and $|\\sin \\phi |<2d/(\\tilde{\\epsilon }l_B^2)$ are fulfilled.", "This is visualized in Fig.", "REF .", "If bound states exist, they correspond to Landau levels (LLs) [36].", "This duality of Landau level formation and bound states in the effective potential at LL energies is in complete analogy to what we find in Section REF .", "Figure: Effective potential landscape |k(x)||k(x)| seen by a state incident on a magnetic step barrier in graphene, for different values of transverse momentum (dd and l B l_B are fixed), and as calculated from Eq. ().", "The linear form of the vector potential leads to perfectly triangular potential wells.Moreover, the picture of an effective potential landscape $|V(x)|$ can explain intuitively the perfect reflection criterion found in Ref.", "[35], Eq.", "(12): $\\tilde{\\epsilon }\\le d/l_B^2.$ When considering Fig.", "REF and varying the parameter $\\tilde{\\epsilon }\\sin \\phi $ arbitrarily, it is clear that the minimal energy threshold $k_\\text{max}\\equiv \\text{max}(|k(-d)|,|k(d)|)$ an incoming state can see (the analog of $\\epsilon _l^\\text{max}$ in Section REF ) is $k_\\text{max}=d/l_B^2$ .", "Hence, no transmission can occur in principle if $\\tilde{\\epsilon }\\le k_\\text{max}$ .", "This is precisely the criterion (REF ).", "Obviously, the discussion conducted in this Appendix for a simple magnetic step barrier can be extended to situations where more complicated inhomogeneous magnetic fields profiles are applied to graphene.", "For example, for each of the magnetic field profiles studied in Ref.", "[36], we can construct the corresponding shaped TINW by choice of the profile of $B_\\perp (z)$ , cf.", "the insets of Fig.", "REF ." ], [ "Parametrization of shaped TI nanowires", "We use the NDEigensystem routine in Wolfram Mathematica to solve Eq.", "(REF ) numerically.", "The TI nanocone is parametrized with $\\begin{aligned}R_\\sigma ^\\text{TINC}(z)&\\equiv R_0+(R_1-R_0)\\Theta _\\sigma (z-z_1)\\\\&+\\mathcal {S}(z-z_0)[\\Theta _\\sigma (z-z_0)-\\Theta _\\sigma (z-z_1)],\\end{aligned}$ where $\\Theta _\\sigma (z-z^{\\prime })\\equiv \\frac{1}{2}+\\frac{1}{\\pi }\\arctan [\\sigma (z-z^{\\prime })]$ is a smoothed Heaviside function with a step at $z=z^{\\prime }$ , such that $\\Theta _{\\sigma \\rightarrow \\infty }(z-z^{\\prime })=\\Theta (z-z^{\\prime })$ .", "With this, one can construct any shaped TINW at will.", "For example, for the TI dumbbell, see Fig.", "REF , where we assume $z_3-z_2=z_1-z_0$ and $R_2=R_1$ , $R_3=R_0$ for simplicity, we have $R_\\sigma ^\\text{TIDB}(z)\\equiv \\, R_\\sigma ^\\text{TINC}(z)+R_\\sigma ^\\text{TINC}(-z+z_1+z_2)-R_1.$ Changing the value of $\\sigma $ allows to interpolate between the ideal TI dumbbell [Fig.", "REF (b)] and the smoothed version shown in Fig.", "REF (c)." ] ]

[ [ "MARVEL analysis of the measured high-resolution rovibronic spectra of\n the calcium monohydroxide radical (CaOH)" ], [ "Abstract The calcium monohydroxide radical (CaOH) is an important astrophysical molecule relevant to cool stars and rocky exoplanets, amongst other astronomical environments.", "Here, we present a consistent set of highly accurate rovibronic (rotation-vibration-electronic) energy levels for the five lowest electronic states ($\\tilde{X}\\,^2\\Sigma^+$, $\\tilde{A}\\,^2\\Pi$, $\\tilde{B}\\,^2\\Sigma^+$, $\\tilde{C}\\,^2\\Delta$, $\\tilde{D}\\,^2\\Sigma^+$) of CaOH.", "A comprehensive analysis of the published spectroscopic literature on this system has allowed 1955 energy levels to be determined from 3204 rovibronic experimental transitions, all with unique quantum number labelling and measurement uncertainties.", "The dataset covers rotational excitation up to $J=62.5$ for molecular states below 29\\,000~cm$^{-1}$.", "The analysis was performed using the MARVEL algorithm, which is a robust procedure based on the theory of spectroscopic networks.", "The dataset provided will significantly aid future interstellar, circumstellar and atmospheric detections of CaOH, as well as assisting in the design of efficient laser cooling schemes in ultracold molecule research and precision tests of fundamental physics." ], [ "Introduction", "The calcium monohydroxide radical ($^{40}$ Ca$^{16}$ O$^{1}$ H) is a linear triatomic molecule of increasing astronomical interest due to its expected presence in the atmospheres of hot rocky super-Earth exoplanets [7], [59].", "This class of exoplanets are very close to their host star and tidally-locked, with their dayside exposed to extremely high temperatures, e.g.", "2000–4000 K. The material present on the surface of the planet, including rock-forming elements such as silicon, magnesium, iron, calcium, and so on, will vaporise to some extent and produce an atmosphere strongly dependant on planetary composition [55], [23].", "Investigating the spectroscopy of hot rocky super-Earths requires accurate spectroscopic data on simple molecules composed of rock-forming elements, like calcium monohydroxide.", "However, data for CaOH is not necessarily available or easily accessible.", "For example, a recent systematic study modelling M-dwarf photospheres by [53] noted missing opacity from the benchmark BT-Settl model due to three molecules: NaH, AlH and CaOH.", "The ExoMol project has since computed line lists for NaH [54] and AlH [72] meaning that CaOH, notably its band around 18 000 cm$^{-1}$ , remains as the only identified missing source of opacity in these objects.", "Given the high cosmic abundance of calcium with respect to molecular hydrogen, it is reasonable to expect calcium-bearing molecules such as CaOH in other interstellar and circumstellar environments.", "For example, a possible formation mechanism in the interstellar medium (ISM) is through the reaction of Ca$^+$ ions with small oxide interstellar grains to release gas-phase CaOH [21].", "[69] predicted that CaOH would be the most abundant calcium-bearing molecule in oxygen-rich late-type stars at temperatures of $T=1000$ –2000 K. While the $\\tilde{B}\\,^2\\Sigma ^+$ –$\\tilde{X}\\,^2\\Sigma ^+$ electronic band of CaOH was tentatively assigned in the spectra of late-type M-dwarf stars [51].", "A large number of experimental studies have measured the rovibronic (rotation-vibration-electronic) spectrum of CaOH, however, there is no centralised source containing this information, aside from the CDMS database [48], [47], [22] but this only covers the microwave region (0–34 cm$^{-1}$ ).", "In this work, we present a dataset of highly accurate molecular rovibronic transitions and energy levels for the CaOH molecule, obtained by evaluating all available spectroscopic data on CaOH from the published literature using the MARVEL (Measured Active Rotational-Vibrational Energy Levels) algorithm [26], [17], [25], [68].", "This procedure takes a set of assigned transition frequencies with measurement uncertainties and converts it into a consistent set of empirical energy levels, each with their own measurement uncertainty and unique quantum numbers and state labels.", "A MARVEL dataset for CaOH will considerably aid future astronomical detection of this molecule, particularly because of its large wavelength and rotational excitation coverage.", "Furthermore, it will benefit the calculation of a molecular line list for CaOH, which is currently being undertaken by the ExoMol project [58], [65].", "MARVEL datasets of empirical energy levels can greatly improve the accuracy of computed molecular opacities, as was done for the recent MARVEL titanium oxide dataset [45] and TiO line list [44], whose detection in exoplanet atmospheres had been hampered by the inaccuracy of line positions in the available line lists [32].", "We mention that the alkaline earth monohydroxide radicals, including CaOH [41], [3], are relevant in studies of ultracold molecules and precision tests of fundamental physics due to their favourable energy level structure.", "A list of highly accurate energy levels across multiple electronic states can be useful in this field, especially for the design of efficient laser cooling schemes which requires knowledge of molecular rovibronic structure to a high degree of accuracy.", "The MARVEL procedure [26], [17], [25], [68] is based on the theory of spectroscopic networks [18], [25], [24], [2] and offers an elegant way to construct and represent complex networks such as those contained in a molecule's spectroscopy.", "Energy levels are represented as nodes with the allowed transitions linking them and the corresponding transition intensities acting as weights.", "Provided with a dataset of assigned transitions with measurement uncertainties, MARVEL will produce a consistent set of uniquely labelled empirical-quality energy levels, with the uncertainties propagated from the input transitions to the output energies.", "From a user-perspective, transitions included in the MARVEL dataset must have a measurement uncertainty and every energy level has to be uniquely labelled, typically by a set of quantum numbers as discussed below.", "The chosen set of quantum numbers must be consistent across the whole dataset but need not be physically meaningful.", "However, a sensible choice will benefit comparisons with other data and allow the final dataset to be readily utilized in future studies.", "MARVEL is publicly available through a user-friendly web interface at http://kkrk.chem.elte.hu/marvelonline and numerous MARVEL studies have been performed on astronomically important diatomic and small polyatomic molecules: NH$_3$  [1], C$_2$  [27], TiO [45], C$_2$ H$_2$  [12], H$_2$ S [13], ZrO [46], NH [19], SO$_2$  [67], H$_3^+$  [29] and isotopologues [28], and H$_2{}^{16}$ O and its isotopologues [68], [63], [60], [61], [62], [64]." ], [ "Electronic structure and spectroscopy of CaOH", "The calcium monohydroxide radical is an open-shell system with a relatively complex electronic structure.", "To date, only the lowest-lying eight electronic states up to the $\\tilde{G}\\,^2\\Pi $ state [30] at approximately 32 633 cm$^{-1}$ are known.", "In this work we consider the lowest five electronic states ($\\tilde{X}\\,^2\\Sigma ^+$ , $\\tilde{A}\\,^2\\Pi $ , $\\tilde{B}\\,^2\\Sigma ^+$ , $\\tilde{C}\\,^2\\Delta $ , $\\tilde{D}\\,^2\\Sigma ^+$ ) and the transitions linking them, as shown in Figure REF .", "This choice was governed by the availability of laboratory rovibronic transtion data in the literature.", "The wavenumber regions considered for each state were: 0–2599 cm$^{-1}$ ($\\tilde{X}\\,^2\\Sigma ^+$ ); 15 966–17 677 cm$^{-1}$ ($\\tilde{A}\\,^2\\Pi $ ); 18 023–18 849 cm$^{-1}$ ($\\tilde{B}\\,^2\\Sigma ^+$ ); 22 197–23 457 cm$^{-1}$ ($\\tilde{C}\\,^2\\Delta $ ); 28 157–28 898 cm$^{-1}$ ($\\tilde{D}\\,^2\\Sigma ^+$ ).", "Figure: The rovibronic states and transitions of CaOH considered in this work.Interestingly, the spectrum of CaOH is affected by the Renner-Teller effect, see e.g.", "[37] for a recent review, which is caused by the interaction of electronic orbital and vibrational angular momenta in linear molecules.", "This effect manifests itself when the molecule bends by lifting the degeneracy of the electronic states, for example, the first excited $\\tilde{A}\\,^2\\Pi $ state of CaOH is split into two components $A^{\\prime }$ and $A^{\\prime \\prime }$ at bent configurations.", "The Renner-Teller effect complicates spectral analysis due to the increased energy level congestion in band systems but its treatment is necessary for a correct description and several experimental studies have considered Renner splittings in CaOH [43], [42], [36], [14], [31]." ], [ "Vibronic coupling, symmetry and quantum numbers", "MARVEL requires that each transition is between energy levels with unique state labels and quantum numbers that are consistent across the entire dataset.", "For CaOH, different authors have used different combinations of quantum numbers, as listed in Table REF .", "In this work we have selected eight quantum numbers, shown in Table REF , that form a consistent set and allow each state to be uniquely labelled.", "The electronic state and the vibronic state are labelled along with the rotational angular momentum quantum number $J$ , the rotationless parity $e$ /$f$ , and the quantum labels $F_1$ and $F_2$ denoting spin components $J = N + 1/2$ and $J = N - 1/2$ , respectively.", "Normal mode notation $(v_1,v_2^{L},v_3)$ is used for the vibrational states, where $v_1, v_2, v_3$ represent the symmetric stretch, bending, and asymmetric stretch modes.", "The quantum number $L$ will be used to refer to the absolute value of the vibrational angular momentum quantum number $l$ associated with the $\\nu _2$ bending mode, $L=|l|$ , where the vibrational quantum number $l$ takes the following values: $|l| = v_2, v_2-2, v_2-4, \\ldots , 0\\,({\\rm or}\\,1).$ For example, the $v_2 = 1$ state has two degenerate components $l = \\pm 1$ .", "The $v_2 = 2$ state assumes three states of $l = \\pm 2$ and $l=0$ .", "The $v_2 = 3$ state splits into four components $l = \\pm 3$ and $l=\\pm 1$ and so on.", "The typical designation of the linear bending quantum mode therefore includes $|l|$ : $(v_1, v_2^{|l|},v_3)$ .", "For $v_2 = 0$ and often for $v_2 = 1$ the superscript $|l|$ can be omitted.", "Coupling with other degrees of freedom (such as molecular rotation) lifts the degeneracy of the $l\\ne 0 $ states.Here we use the linear triatomic version of the bending quantum number $v_2$ , which is related to the bent molecule case as $v_2 \\equiv v_2^{\\rm linear} = 2 v_{2}^{\\rm bent} + L$ , where $L$ is constrained to $L = K_a \\pm \\Lambda $ [11], and $K_a = |k_a|$ is the rotational quantum number associated with the projection of the total rotational angular momentum on the $a$ axis (or $z$ axis in our case).", "Table: State labels and quantum numbers used by different authors for energy levels of CaOH.Table: State labels and quantum numbers of CaOH adopted for the MARVEL dataset.When the molecule is linear, the ${\\mathcal {C}}_{\\infty {\\rm v}}$ (M) molecular symmetry group is used to classify the rotation, vibration and electronic degrees of freedom.", "The lowest five electronic states are $\\tilde{X}\\,^2\\Sigma ^+$ , $\\tilde{A}\\,^2\\Pi $ , $\\tilde{B}\\,^2\\Sigma ^+$ , $\\tilde{C}\\,^2\\Delta $ , $\\tilde{D}\\,^2\\Sigma ^+$ with the projections of the electronic angular momentum on the $z$ axis $\\Lambda = 0$ ($\\tilde{X}$ , $\\tilde{B}$ , $\\tilde{D}$ ), $\\pm 1$ ($\\tilde{A}$ ) and $\\pm 2$ ($\\tilde{C}$ ).", "The symmetry of the vibrational motion is controlled by the vibrational angular momentum quantum number $L$ in the same way that the symmetry of the electronic state is defined by $\\Lambda $ : $\\Sigma $ , $\\Pi $ , $\\Delta $ , ..., for $l =0,\\pm 1,\\pm 2,\\ldots ,$ respectively.", "The electronic angular momentum can be coupled to the vibrational angular momentum giving the vibronic angular momentum with projections $\\Lambda +l$ , which are also described using the notation $\\Sigma ^{\\pm }, \\Pi , \\Delta , \\Phi , \\ldots $ .", "These vibronic symmetries are then used to label the final vibronic states while the electronic symmetries are omitted.", "Consider, for instance, the vibrational state $(0,1^1,0)$ in the first excited electronic state $\\tilde{A}\\,^2\\Pi $ .", "Coupling its vibrational angular momentum with $l=\\pm 1$ ($\\Pi $ ) and the electronic angular momentum with $\\Lambda = \\pm 1$ leads to the three vibronic components $|\\Lambda +l| = 0,0,2$ , i.e.", "$\\Sigma ^{+}$ , $\\Sigma ^{-}$ and $\\Delta $ , with a doubly degenerate state $\\Delta $ .", "These vibronic states are then assigned $\\tilde{A} (0,1^1,0)$ $^2\\Sigma ^{\\pm }$ and $\\tilde{A}(0,1^1,0)$ $^2\\Delta $ .", "The bending motion of the molecule ($v_2>0$ ) lifts the electronic degeneracy of $\\Lambda \\ne 0$ states ($\\tilde{A}\\,^2\\Pi $ and $\\tilde{C}\\,^2\\Delta $ ) via the Renner-Teller interaction ($\\Lambda $ doubling), see, e.g., [52], [33], [34], [35].", "There are two alternative notations for these components used in the literature.", "Experimental studies of CaOH tend to use $\\mu $ and $\\kappa $ for the two vibronic sub-bands associated with the Renner-Teller splitting, while a more general consideration denotes the two Renner-Teller components $A^{\\prime }$ and $A^{\\prime \\prime }$ of the $C_s$ symmetry group to classify the molecular states, as well as the different degrees of freedom of CaOH.", "According to this convention, the $\\kappa $ levels are always higher in energy than the $\\mu $ levels of the same $J$ [35].", "In the above example of $(0,1^1,0)$ in the excited electronic state $\\tilde{A}\\,^2\\Pi $ , the final vibronic notations inherit the symmetry of a linear molecule and are given by $\\tilde{A}(0,1^1,0)$ $\\mu /\\kappa $ $^2\\Sigma ^{\\pm }$ and $\\tilde{A}(0,1^1,0)$ $^2\\Delta $ ." ], [ "Symmetry considerations", "When coupling different angular momenta of CaOH, it is common to first couple the electronic and vibrational angular momenta towards the vibronic momentum with projections $\\Lambda +l$ (also assigned $^2\\Sigma ^{\\pm }, \\Pi , \\Delta , \\Phi , \\ldots $ ) and then with the unpaired spin ($S=1/2$ ) towards the total angular momentum with the projection $\\Omega = \\Lambda +l+\\Sigma $ , where $\\Sigma $ is the projection of the electron spin angular momentum on the $z$ axisNote that this label $\\Sigma $ is different from the label $\\Sigma $ used to assign the $\\Lambda =0$ state.. For CaOH, it is common to use the $F_1$ and $F_2$ quantum labels to distinguish the two spin components with $|\\Lambda +l|\\pm 1/2$ .", "For example, in the $^2\\Sigma $ states the rotational angular momentum $\\bf {N}$ is coupled to spin as $\\bf {J} = \\bf {N} + \\bf {S} $ and thus their projections on the $z$ axis (at linear configurations) are related by $J=N\\pm 1/2$ , where $N$ is the rotational angular momentum.", "Each rotational state of $^2\\Sigma $ consists of two spin sub-components, assigned $F_1$ and $F_2$ .", "$\\Omega $ is sometimes used as an alternative to $F_1$ and $F_2$ to identify vibronic sub-components, e.g.", "as $^2\\Pi _{1/2}$ and $^2\\Pi _{3/2}$ , $^2\\Delta _{3/2}$ and $^2\\Delta _{5/2}$ etc.", "Most of the sources presented in this work, however, use the $F_1$ and $F_2$ notation.", "The symmetry of the rovibronic state is a product of the symmetries of the electronic, vibrational, spin and rotational parts and can also be associated with the corresponding angular momenta $\\bf {G}$ , $\\bf {L}$ , $\\bf {S}$ and $\\bf {N}$ .", "However, the final rovibronic state can only be $\\Sigma ^+$ (parity is $+$ ) or $\\Sigma ^-$ (parity is $-$ ) of the molecular symmetry group ${\\mathcal {C}}_{\\infty {\\rm v}}$ (M) [70].", "The parity of $^2\\Sigma ^+$ or $^2\\Sigma ^-$ states are $(-1)^{N}$ and $(-1)^{N+1}$ , respectively.", "The state parity always changes for the electric dipole transitions $+ \\leftrightarrow -$ , which can be used to reconstruct the parity of the upper/lower state if only one of them is known.", "The rotationless $e/f$ parity is related to the $+/-$ parity as follows: levels with parity $+(-1)^{J-0.5} $ are $e$ levels, and levels with parity $-(-1)^{J-0.5} $ are $f$ levels [10].", "Some authors use the Mulliken' labels $c, d$ [49] to distinguish states of different parities, for example in the case of $v_2=3$  [74] with $(0,3^{1c},0)$ , $(0,3^{1d},0)$ , $(0,3^{3c},0)$ and $(0,3^{3d},0)$ .", "The electric dipole selection rules are $J^{\\prime } &= J^{\\prime \\prime } \\pm 1; \\\\e &\\leftrightarrow e \\quad {\\rm and} \\quad f \\leftrightarrow f \\quad {\\Delta J = \\pm 1}; \\\\e &\\leftrightarrow f \\quad \\Delta J = 0; \\\\\\Sigma ^+ &\\leftrightarrow \\Sigma ^- \\quad (\\rm total~symmetry).$" ], [ "Examples of coupling vibration, spin and electronic angular momenta", "For non-$\\Sigma $ states, the interaction with the electron spin can involve the vibrational and electronic angular momentum.", "For example, the vibrational excitation $(0,1^1,0)$ of the electronic state $\\tilde{X}\\,^2\\Sigma ^+$ gives rise to the total vibronic angular momentum of the symmetry $^2\\Pi $ ($l=\\pm 1$ ), referenced to as $\\tilde{X}(0,1^1,0)\\,^2\\Pi $ .", "The resulting rovibronic energy pattern forms a quartet, which can be designated using the four combinations of $e,f$ and $F_1,F_2$ .", "Including the electronic angular momentum complicates this picture even more.", "For example, in order to couple all angular momenta in the vibronic state $\\tilde{A}(0,1^1,0)\\,^2\\Pi $ , in this work we first couple the electronic and vibrational momenta with projections $l=\\pm 1$ and $\\Lambda =\\pm 1$ , respectively, towards the vibronic angular momentum with $l+\\Lambda = 0^{+/-}, \\pm 2$ .", "The zero vibronic components $0^{+/-}$ have two symmetries $^2\\Sigma ^+$ and $^2\\Sigma ^-$ with $|\\Omega | = 1/2$ .", "In this case $F_1$ and $F_2$ are redundant and can be chosen to match $e$ and $f$ states.", "The state with $l+\\Lambda = \\pm 2$ has the symmetry $^2\\Delta $ with $|\\Omega |$ taking values $3/2$ and $5/2$ , and therefore can also be designated $(0,1^1,0){}^2\\Delta _{3/2}$ ($F_1$ ) and $(0,1^1,0){}^2\\Delta _{5/2}$ ($F_2$ ), respectively, where the labels $F_1$ and $F_2$ are assigned to the $|\\Omega | = 1/2$ and $3/2$ states based on the order of the corresponding energy values.", "Consider another example of the $\\tilde{A}(0,2,0)\\,^2\\Pi $ state ($\\Lambda =\\pm 2$ , $l=0,\\pm 2$ ) with the vibronic angular momenta $\\Lambda +l = \\pm 1$ and $\\pm 3$ , which represent the vibronic states $^2\\Pi $ and $^2\\Phi $ , respectively.", "The $\\Lambda = \\pm 2$ degeneracies are lifted due to the interaction with the bending mode (Renner-Teller) leading to the $\\mu $ and $\\kappa $ sub-components (or $A^{\\prime }$ and $A^{\\prime \\prime }$ ).", "The interaction with spin introduces the spin splitting with the final values of the projections of the angular momenta $1/2$ and $3/2$ ($^2\\Pi $ ) or $5/2$ and $7/2$ ($^2\\Phi $ ), i.e.", "to $^2\\Pi _{1/2}$ , $^2\\Pi _{3/2}$ , $^2\\Phi _{5/2}$ and $^2\\Phi _{7/2}$ .", "However, if $l=0$ but $\\Lambda \\ne 0$ , such as e.g.", "for the $\\tilde{A}(0,0,0)\\,^2\\Pi $ state, the final projections of the total angular momentum $\\Omega = \\Lambda + \\Sigma $ are $\\pm 1/2$ and $\\pm 3/2$ , and these states can be assigned $^2\\Pi _{1/2}$ and $^2\\Pi _{3/2}$ , respectively." ], [ "Experimental data sources", "Spectroscopic data was extracted from thirteen published sources [20], [74], [56], [43], [16], [42], [15], [73], [36], [14], [8], [9], [31].", "These data are summarised in Table REF .", "Only data from [20] was provided in digital format while all other literature sources had to be processed using digitisation software.", "A unique reference label was assigned to each extracted transition, which is a requirement for the MARVEL input file.", "The reference indicates the data source, Table (or page) and line number that the transition originates from.", "The data source tag, for example 06DiShWa, is based on the notation employed by the IUPAC task group on water [61], [63].", "Aside from the numerous experimental measurements of CaOH spectra, there have been several studies investigating the electronic structure of CaOH using quantum chemical methods [6], [4], [5], [50], [38], [39], [66], [57] but these have largely focused on molecular structures and properties rather than rovibronic spectroscopy.", "Table: Experimental sources of CaOH spectra and their coverage." ], [ "Comments on literature sources", "83HiQiHa: [31] contains transitions from the $\\tilde{A}(0,0,0)\\,^2\\Pi $ –$\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ band.", "Most of these data are included in the more recent study by [20] but some of the line positions show deviations up to 0.3 cm$^{-1}$ .", "For example, the transition $\\tilde{A}(0,0,0)\\,^2\\Pi $ , $J=36.5$ $F=2$ , $e$ $\\leftarrow $ $\\tilde{X}(0,0,0)\\,^2\\Pi $ , $J=36.5$ $F=2$ , $f$ appears in [31] as 16 029.986 cm$^{-1}$ and in [20] as 16 030.285 cm$^{-1}$ .", "We believe that the recent data [20] is more reliable.", "In these instances we have removed the lines by [31] from our MARVEL analysis by changing the sign of the corresponding frequency value (MARVEL convention).", "84BeKi: [9] contains transitions from the $\\tilde{B}(0,0,0)\\,^2\\Sigma ^+$ –$\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ and $\\tilde{B}(1,0,0)\\,^2\\Sigma ^+$ –$\\tilde{X}(1,0,0)\\,^2\\Sigma ^+$ band.", "85BeBr: [8] contains transitions from the $\\tilde{A}(0,0,0)\\,^2\\Pi $ –$\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ band.", "91CoLiPr: [14] contains transitions from the $\\tilde{A}(1,0,0)\\,^2\\Pi $ –$\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ band.", "92LiCo: [42] reported transitions from the $\\tilde{A}(0,2^0,0)\\,^2\\Pi $ –$\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ band system.", "92CoLiPr: [15] studied the $\\tilde{A}\\,^2\\Pi $ –$\\tilde{X}\\,^2\\Sigma ^+$ system, covering the $\\tilde{A}(1,0,0)\\,^2\\Pi $ –$\\tilde{X}(1,0,0)\\,^2\\Sigma ^{+}$ , $\\tilde{A}(1,0,0)\\,^2\\Pi $ –$\\tilde{X}(2,0,0)\\,^2\\Sigma ^{+}$ , $\\tilde{A}(1,0,0)\\,^2\\Pi $ –$\\tilde{X}(3,0,0)\\,^2\\Sigma ^{+}$ , $\\tilde{A}(1,0,0)\\,^2\\Pi $ –$\\tilde{X}(4,0,0)\\,^2\\Sigma ^{+}$ , $\\tilde{A}(1,0,0)\\,^2\\Pi $ –$\\tilde{X}(0,2^0,0)\\,^2\\Sigma ^+$ and $\\tilde{A}(1,0,0)\\,^2\\Pi $ –$\\tilde{X}(0,2^2,0)\\,^2\\Delta $ vibronic bands.", "92ZiBaAn: [73] investigated eleven pure rotational transitions in the $\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ ground state.", "The total angular momentum quantum number $F$ used to account for hyperfine structure was not included in our dataset.", "Instead, the uncertainty of these transitions was increased to match the hyperfine splittings.", "The dataset was converted to cm$^{-1}$ from the original units of MHz.", "92JaBe: [36] studied the $\\tilde{C}\\,^2\\Delta $ –$\\tilde{X}\\,^2\\Sigma ^+$ system covering the $\\tilde{C}(0,1^1,0)\\,^2\\Pi $ –$\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ band.", "93ScFlSt: [56] investigated the hyperfine structure of the three lowest pure rotational transitions of the $\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ state.", "The total angular momentum quantum number $F$ used to account for hyperfine structure was not included in our dataset.", "Instead, the uncertainties for these transitions were increased to 0.0001 cm$^{-1}$ to account for the fact that we have neglected hyperfine effects.", "94CoLiPr: [16] measured transitions in the $\\tilde{A}\\,^2\\Pi $ –$\\tilde{X}\\,^2\\Sigma ^+$ band system: $\\tilde{A}(0,1^1,0)\\,^2\\Sigma ^+$ –$\\tilde{X}(0,1^1,0)\\,^2\\Pi $ and $\\tilde{A}(0,1^1,0)\\,^2\\Sigma ^-$ –$\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ bands.", "The upper doubling states are assigned $\\kappa \\,^2\\Sigma ^-$ or $\\mu \\,^2\\Sigma ^+$ .", "95LiCo: [43] studied the $v_2 = 1$ bending vibrational levels of the $\\tilde{A}\\,^2\\Pi $ and $\\tilde{X}\\,^2\\Sigma ^+$ states, covering transitions of the following bands: $\\tilde{A}(0,1^1,0)\\kappa \\,^2\\Sigma ^-$ , $^2\\Delta , \\mu \\,^2\\Sigma ^+$ $\\leftarrow $ $\\tilde{X}(0,1^1,0)\\,^2\\Pi $ , $(0,0,0)\\,^2\\Sigma ^+$ .", "It should be noted that $\\tilde{A}(0,1^1,0)$ –$\\tilde{X}(0,0,0)$ is electric dipole forbidden.", "Even though the $\\tilde{A}(0,1,0)\\mu {}^2\\Sigma ^+$ and $\\tilde{A}(0,1^1,0)\\kappa {}^2\\Sigma ^-$ states can be correlated to $A^{\\prime }$ and $A^{\\prime \\prime }$ , we retained the experimental labels $\\mu $ and $\\kappa $ .", "The $e/f$ parity of all states was reconstructed from the $+/-$ parities $p$ , which in turn were obtained from the parity $p^{\\prime \\prime }$ of the lower vibronic state $\\Sigma ^+$ as $(-1)^{N}$ .", "For non-$\\Sigma $ lower vibronic states ($\\tilde{X}\\,^2\\Pi $ ) we used the upper state parity instead if the vibronic symmetries were either $\\Sigma ^+$ ($\\mu $ ) or $\\Sigma ^-$ ($\\kappa $ ): the parities $p^{\\prime }$ are $(-1)^{N^{\\prime }}$ or $-(-1)^{N^{\\prime }}$ , respectively.", "96ZiFlAn: [74] has pure rotational and rovibrational transitions in the $\\tilde{X}\\,^2\\Sigma ^+$ ground electronic state: $(0,0,0)$ , $(0,1^{1c},0)$ , $(0,1^{1d},0)$ , $(0,2^{0},0)$ , $(0,2^{2c},0)$ , and $(0,2^{2d},0)$ where the superscript $c/d$ represents $l$ -type doubling effects.", "The $(0,1^{1},0)$ and $(0,2^{2},0)$ vibronic states of $\\tilde{X}\\,^2\\Sigma ^+$ have the vibrational symmetries $^2\\Pi $ and $^2\\Delta $ , respectively.", "The correlation between $c/d$ and $e/f$ does not appear to follow the rules given by [10].", "In order to reconstruct this correlation we used the $(0,1^{1},0)$ spectroscopic constants from [74] in the program PGOPHER [71], assuming the vibronic symmetry $^2\\Pi $ , and computed the corresponding transitions allowing us to correlate the $e/f$ and $c/d$ parities as follows: for $c$ , $F_1=e$ and $F_2=f$ ; for $d$ , $F_1=f$ and $F_2=e$ , but only for the $(0,0,0)$ , $(0,1^{1c},0)$ , $(0,1^{1d},0)$ states.", "The CaOH spectroscopic constants of $\\tilde{X}(0,2^{2},0)\\,^2\\Delta $ by [74] only work with PGOPHER if assuming the symmetry $^2\\Pi $ instead of $^2\\Delta $ , which in this case worked with the same conversion rules as for the $(0,1^1,0)$ transitions.", "Apparently this is due to the effective rotational Hamiltonian model used in the fit in the original work.", "The dataset was converted to cm$^{-1}$ from the original units of MHz.", "06DiShWa: [20] covers transitions from the $\\tilde{A}(0,0,0)\\,^2\\Pi $ –$\\ \\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ and $\\tilde{D}(0,0,0)\\,^2\\Sigma ^+$ –$\\tilde{A}(0,0,0)\\,^2\\Pi $ bands.", "In their analysis, [20] used the $\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ –$\\tilde{X}(0,0,0)\\,^2\\Sigma ^+$ transitions from the work by [73].", "A number of the $\\tilde{A}$ –$\\tilde{X}$ transitions were measured in the work of [31]." ], [ "Results and Discussion", "Our final MARVEL transition file (input, see Table REF ) consists of 3204 experimental (excluding the 16 transitions from [31] deemed less reliable) and 20 pseudo-experimental (PGOPHER) transitions and has the following structure: Table: NO_CAPTIONwhere $\\tilde{\\nu }$ is the transition wavenumber (cm$^{-1}$ ), unc is the experimental uncertainty (cm$^{-1}$ ), QN$^{\\prime }$ and QN$^{\\prime \\prime }$ are the quantum numbers of the upper and lower states, respectively, and `source' is the abbreviation of the literature source concatenated with a counting number $i$ of the data from this source.", "source$_i$ is a unique ID of the transitions in the MARVEL dataset.", "The pseudo-experimental transition wavenumbers were reconstructed with the PGOPHER program using the spectroscopic constants from [43] and [74] to support the corresponding datasets compensating for missing lines.", "Their uncertainties were set to 1 cm$^{-1}$ in order not to interfere with the true experimental values.", "The QN set comprises 8 quantum numbers/labels selected to represent a general rovibronic state of CaOH as in Tables REF and REF : Vibronic state label, $e/f$ parity, $J$ , $v_1$ , $v_2$ , $L$ , $v_3$ , and $F = F_1, F_2$ .", "The MARVEL energy file (output) was processed via the online MARVEL app using the Cholesky (analytic) approach with a 0.05 cm$^{-1}$ threshold on the uncertainty of the “very bad” lines, which produced a MARVEL energy file containing 1955 states.", "The MARVEL dataset covers rotational excitation up to $J=62.5$ for molecular states below 29 000 cm$^{-1}$ .", "The MARVEL output and structure of the MARVEL energy levels is illustrated in Table REF and also plotted in Fig REF as a function of $J$ .", "Figure: The MARVEL energy term values for CaOH shown for different electronic bands.Table: Extract from the MARVEL transition file.", "The quantum numbers/labels are described in Table .The MARVEL frequency wavenumber ν ˜\\tilde{\\nu } and uncertainties are in cm -1 ^{-1}.Table: Extract from the MARVEL energy file.", "The quantum numbers/labels are described in Table , which are followed by the MARVEL energy term value (cm -1 ^{-1}), uncertainty (cm -1 ^{-1}) and the number of transitions supporting the state in question.Figure REF offers a visual representation of the CaOH MARVEL network, where the upper state energies are connected with the lower state energies via circles.", "The size of a circle is $\\log (n+1)$ , where $n$ is the number of transitions supporting the corresponding upper state.", "The vertical bars along the horizontal-axis show the lower state energies, while the horizontal bars along the vertical-axis give the upper state energies.", "The value of $n$ ranges from 1 (dark blue) to 38 (red).", "Figure: The MARVEL network for CaOH: the upper state energies are plotted against corresponding lower state energies.", "The vertical bars along the horizontal-axis represent the lower state energies, while the horizontal bars along the vertical-axis give the upper state energies.", "Each circle represents a particular transition, with the size proportional to the log of the number of transitions nn plus 1, going to the upper state.", "The value ranges from 1 (dark blue) to 38 (red)." ], [ "Conclusions", "We have comprehensively evaluated the published spectroscopic literature on CaOH and extracted all meaningful molecular rovibronic transition data.", "These data were analysed using the robust MARVEL algorithm which converts assigned transitions into a consistent set of uniquely labelled empirical energy levels with measurement uncertainties.", "The dataset covers rotational excitation up to $J=62.5$ for 1955 molecular states below 29 000 cm$^{-1}$ .", "The MARVEL input and output files for CaOH are provided as supplementary material and can be readily updated to include new experimental rovibronic measurements.", "While we have analyzed data from five electronic states and a large range of rotational levels, the experimental data has only limited coverage of vibrationally excited states.", "There is no empirical information on the $\\nu _3$ stretching vibrational mode and only limited information on vibrational excitation of the other modes in electronically excited states.", "This means that any line list constructed for this molecule will have to rely on ab initio predictions for the missing quantities.", "With the renewed interest in CaOH we expect the new MARVEL dataset to help future detection of this molecule in astronomical environments.", "The most immediate benefit will be in the calculation of a comprehensive CaOH molecular line list as part of the ExoMol project [58], [65].", "A list of highly accurate empirical energy levels is necessary to refine the theoretical spectroscopic model of a molecule to achieve orders-of-magnitude improvements in the accuracy of the predicted line positions.", "Accurate molecular opacities of CaOH are essential for detecting this molecule in hot rocky super-Earth exoplanets and this work brings us closer to this goal.", "As mentioned before, a detailed knowledge of the energy level structure in CaOH will help the design of efficient laser cooling schemes in ultracold molecule research and precision tests of fundamental physics.", "The alkaline earth monohydroxide radicals are attractive molecules in this pursuit due to their favourable energy level structure, as demonstrated by SrOH, which was the first untrapped polyatomic molecule to be laser-cooled [40].", "This work was supported by the STFC Project No.", "ST/R000476/1.", "We thank Peter Bernath for collecting experimental data and many helpful discussions.", "Yixin Wang's visit was supported by Physics Boling Class in Nankai University.", "MARVEL, PGOPHER" ] ]

[ [ "Error analysis of Nitsche's and discontinuous Galerkin methods of a\n reduced Landau-de Gennes problem" ], [ "Abstract We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau-de Gennes framework.", "The main results are (i) a priori error estimates for the energy norm, within the Nitsche's and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient {\\it a posteriori} analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases.", "Numerical examples that validate the theoretical results, are presented separately." ], [ "Introduction", "This paper focuses on the numerical analysis of a system of two second order semi-linear elliptic partial differential equations, with a lower order cubic non-linearity, defined on bounded two-dimensional domains with Lipschitz boundaries and inhomogeneous boundary conditions.", "Such systems arise naturally in different contexts for two-dimensional systems, our primary motivation being two-dimensional liquid crystal systems [11].", "Liquid crystals are intermediate phases of matter between the conventional solid and liquid states of matter with versatile properties of both phases.", "Nematic liquid crystals are amongst the most commonly used liquid crystals, for which the constituent rod-like molecules translate freely but exhibit locally preferred directions of orientational ordering, and these locally distinguished directions are referred to as nematic directors [11].", "Consequently, nematics are directional or anisotropic materials with direction-dependent physical properties.", "The Landau-de Gennes (LdG) theory is perhaps the most celebrated continuum theory for nematic liquid crystals, that describes the nematic state by the LdG order parameter - $\\textit {Q}$ -tensor order parameter: a symmetric traceless $3\\times 3$ matrix that contains information about the nematic directors and the degree of orientational ordering, within the matrix eigenvectors and eigenvalues, respectively [26].", "Reduced two-dimensional LdG approaches have been rigorously derived for two-dimensional domains [13], [35], for certain model situations.", "In the reduced case, the order parameter is a symmetric, traceless $2\\times 2$ matrix, with simply two independent components, $u$ and $v$ .", "More precisely, if $\\mathbf {n} = \\left( \\cos \\theta , \\sin \\theta \\right)$ is the nematic director in the plane, where $\\theta \\in \\left[0, 2\\pi \\right)$ and $s$ is a scalar order parameter that measures the degree of orientational order, then $u = s \\cos 2 \\theta $ and v =$s \\sin 2 \\theta $ .", "Define the two-dimensional vector, $\\Psi :=\\left(u, v \\right)$ on an open, bounded domain, $\\Omega \\subset \\mathbb {R}^2$ , with a polygonal boundary, $\\partial \\Omega $ .", "Then for Dirichlet boundary conditions and in the absence of external fields, the dimensionless reduced LdG free energy [22] is $ \\mathcal {E}(\\Psi _\\epsilon ) =\\int _\\Omega (\\vert \\nabla \\Psi _\\epsilon \\vert ^2 +\\epsilon ^{-2}(\\vert \\Psi _\\epsilon \\vert ^2 - 1)^2) \\,{\\rm dx},$ where $\\Psi _\\epsilon = \\mathbf {g} $ on $\\partial \\Omega $ , and $\\epsilon $ is a material-dependent parameter that depends on the elastic constant, domain size and temperature.", "Informally speaking, the limit $\\epsilon \\rightarrow 0$ corresponds to macroscopic domains with size much greater than material-dependent characteristic nematic correlation lengths [25].", "The experimentally observable and physically relevant states are local or global minimizers of the reduced energy, which are weak solutions, $\\Psi _\\epsilon \\in \\mathbf {H}^1(\\Omega ):=H^1(\\Omega ) \\times H^1(\\Omega ) $ , of the corresponding Euler-Lagrange equations: $ -\\Delta \\Psi _\\epsilon =2\\epsilon ^{-2}(1-\\vert \\Psi _\\epsilon \\vert ^2)\\Psi _\\epsilon \\text{ in } \\Omega \\,\\, \\text{ and }\\,\\, \\Psi _\\epsilon = \\mathbf {g} \\text{ on } \\partial \\Omega .$ In what follows, we work with fixed but small values of $\\epsilon $ , which describe large domains [16].", "The non-linear system (REF ) is a Poisson-type equation for a two-dimensional vector with a non-linear (cubic) lower order term.", "In fact, these equations are the celebrated Ginzburg-Landau partial differential equations with a rescaled $\\epsilon $ , which have been extensively studied in [3], [28].", "In this paper, we apply Nitsche's finite-element approximation method to this system (REF ) and our main contribution is to relax regularity assumptions on $\\Psi _\\epsilon $ , as will be explained below.", "The Nitsche's method is well-studied in the literature; see [15], [17], [23], [27] for applications of Nitsche's method to Poisson's equation with different boundary conditions, a priori and a posteriori error analysis for such problems and applications of medius analysis.", "Further results for the a posteriori analysis of the Poisson problem are also given in [5] with a saturation assumption, that can be stated as the approximate solution of the problem in a finer mesh constitutes a better approximation to the exact solution than the approximate solution on a coarser mesh, in the energy norm.", "In [2] and [20], the authors derive a posteriori error bounds for the finite-element analysis of the Poisson's equation with $C^0$ - Dirichlet boundary conditions, without the saturation assumption.", "Further, a priori and a posteriori error analysis of dGFEM (Discontinuous Galerkin Finite Element Methods) for the von Kármán equations are studied in [8].", "The references are not exhaustive and the techniques in these papers are adapted to deal with the novel aspects of our problem, as will be outlined below.", "The specific model (REF ) has been applied with success to the planar bistable nematic device reported in [33], where the authors study nematic liquid crystals-filled shallow square wells, with experimentally imposed tangent boundary conditions.", "They report the existence of six distinct stable solutions: two diagonal and four rotated solutions.", "The nematic director is aligned along the square diagonals in the diagonal solutions, and rotates by $\\pi $ radians for the rotated solutions.", "In [22], the authors study the numerical convergence of the diagonal and rotated solutions, in a conforming finite-element set-up, as a function of $\\epsilon $ .", "In [24], the authors carry out a rigorous a priori error analysis for the dGFEM approximation of 2-regular solutions of (REF ) in convex polygonal domains.", "Optimal linear (resp.", "quadratic) order of convergence in energy (resp.", "$\\mathbf {L}^2$ ) norm for solutions, $\\Psi _\\epsilon \\in 2(\\Omega )$ , accompanied by an analysis of the $h-\\epsilon $ dependency, where $h$ is the mesh size or the discretization parameter, along with some numerical experiments are discussed.", "However, there are no a posteriori error estimates in [24].", "This paper builds on the results in [24] with several non-trivial generalisations and extensions.", "Define the admissible space $\\mathcal {{X}}=\\left\\lbrace \\mathbf {w} \\in \\mathbf {H}^1(\\Omega ) :\\mathbf {w}= \\mathbf {g} \\,\\, \\text{on} \\,\\, \\partial \\Omega \\right\\rbrace $ and we restrict attention to solutions $\\Psi _{\\epsilon } \\!", "\\in \\!", "\\mathcal {{X}}\\cap \\mathbf {H}^{1+\\alpha }(\\Omega )$ for $0<\\alpha \\le 1$ , where $\\alpha $ is the index of elliptic regularity in this manuscript.", "The main contributions of this article are summarised below: an a priori finite element error analysis for (REF ), using the Nitsche's method to incorporate the non-homogeneous boundary conditions, along with a proof of $h^\\alpha $ (resp.", "$h^{2\\alpha }$ )-convergence of the energy (resp.", "$\\mathbf {L}^2$ ) norm where $h$ is the discretization parameter; a reliable and efficient residual type a posteriori error estimate for (REF ) with the assumption that the boundary function, $\\mathbf {g} \\in {\\frac{1}{2}}(\\partial \\Omega )$ belongs to $\\mathbf {C}^0(\\overline{\\partial \\Omega })$ ; a priori and a posteriori error estimates for dGFEM under the mild regularity assumption, $\\Psi _{\\epsilon } \\in \\mathcal {{X}}\\cap \\mathbf {H}^{1+\\alpha }(\\Omega )$ for $0<\\alpha \\le 1$ ; numerical experiments for uniform as well as adaptive refinement that validate the theoretical estimates above.", "There are new technical challenges in this manuscript, compared to [24].", "The analysis in [24] is restricted to solutions in $\\mathbf {H}^2(\\Omega )$ , or equivalently $\\alpha =1$ .", "The first challenge with a less regular solution, $\\Psi _{\\epsilon }$ as above, is to handle the normal derivatives $\\nabla \\Psi _{\\epsilon } \\nu \\notin \\mathbf {L}^2(E)$ across the element boundaries.", "In this paper, the medius analysis [15] that combines ideas of both a posteriori and a priori analysis is employed to overcome this.", "New local efficiency results are proved to establish the stability of a perturbed bilinear form.", "For the a posteriori analysis, a lifting operator $\\Psi _\\mathbf {g}$ is used such that $\\Psi _\\mathbf {g} = \\mathbf {g}$ on $\\partial \\Omega $ , and this technique requires the additional continuity constraints on $\\mathbf {g}$ .", "These continuity constraints might be relaxed by using saturation techniques, and will be pursued in future work.", "The generalisations to dGFEM with $\\alpha <1$ , involve additional jump and average terms, due to the lack of inter-element continuity in the dGFEM discrete space.", "For $\\alpha =1$ , there are $\\epsilon $ -independent estimates for the $H^2$ -norm of solutions, $\\Psi _\\epsilon $ (see[4]) which allows for a $h-\\epsilon $ dependency study in [24]; this is not easily possible for $\\alpha <1$ and hence, the value of $\\epsilon $ is fixed in this manuscript, unlike the study in [24].", "The reduced regularity assumption for the exact solution is relevant for non-convex polygons, domains with re-entrant corners or slit edges, with $\\alpha <1$ [14].", "Further, a posteriori error estimators provide a systematic way of controlling errors for adaptive mesh refinements [1], [34], as is illustrated by means of several numerical experiments in Section .", "The numerical estimates confirm the a priori and a posteriori estimates and establishes the advantages of adaptive mesh refinements in terms of computational cost and rates of convergence, captured by informative convergence plots for the estimators in Section .", "The standard notation for Sobolev spaces $H^s(\\Omega )$ $(\\text{resp.}", "\\,W^{s,p}(\\Omega ))$ with $s,p$ positive real numbers, equipped with the usual norm ${\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert }_s$ $(\\text{resp. }", "{\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert }_{s,p} )$ is used throughout the paper and the space $\\mathbf {H}^s(\\Omega )$ (resp.", "$\\mathbf {L}^p(\\Omega )$ ) is defined to be the product space $H^s(\\Omega ) \\times H^s(\\Omega )$ $(\\text{resp. }", "L^p(\\Omega ) \\times L^p(\\Omega ))$ equipped with the corresponding norm ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_s$ ($\\text{resp.", "}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{s,p}$ ) defined by ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_s\\!=({\\vert \\hspace{-1.0625pt}\\vert \\varphi _1\\vert \\hspace{-1.0625pt}\\vert }_s^2+{\\vert \\hspace{-1.0625pt}\\vert \\varphi _2\\vert \\hspace{-1.0625pt}\\vert }_s^2)^{\\frac{1}{2}}$ for all $ \\Phi \\!=\\!", "(\\varphi _1, \\varphi _2) \\in \\!\\mathbf {H}^s(\\Omega ) $ $ ( \\text{resp.", "}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{s,p}\\!\\!=({\\vert \\hspace{-1.0625pt}\\vert \\varphi _1\\vert \\hspace{-1.0625pt}\\vert }_{s,p}^2+{\\vert \\hspace{-1.0625pt}\\vert \\varphi _2\\vert \\hspace{-1.0625pt}\\vert }_{s,p}^2)^{\\frac{1}{2}}$ for all $\\Phi \\!=\\!", "(\\varphi _1, \\varphi _2) \\!\\in \\!\\mathbf {W}^{s,p}(\\Omega )).$ The norm on $\\mathbf {L}^2(\\Omega )$ space is simply ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0\\!=({\\vert \\hspace{-1.0625pt}\\vert \\varphi _1\\vert \\hspace{-1.0625pt}\\vert }_0^2+{\\vert \\hspace{-1.0625pt}\\vert \\varphi _2\\vert \\hspace{-1.0625pt}\\vert }_0^2)^{\\frac{1}{2}}$ for all $ \\Phi \\!=\\!", "(\\varphi _1, \\varphi _2) \\in \\!\\mathbf {L}^2(\\Omega ).", "$ Set $V:=H_0^1(\\Omega )= \\big \\lbrace \\varphi \\in L^2(\\Omega ): \\frac{\\partial \\varphi }{\\partial x}, \\frac{\\partial \\varphi }{\\partial y}\\in L^2(\\Omega ) ,$ $ \\varphi |_{\\partial {\\Omega }}=0 \\big \\rbrace $ and $ = \\,\\!\\mathbf {H}_0^1(\\Omega )=H_0^1(\\Omega ) \\times H_0^1(\\Omega )$ .", "Throughout the manuscript, $C_s$ denotes a generic constant.", "The paper is organized as follows.", "In the next section, the weak formulation and the Nitsche's method are introduced.", "Section REF is devoted to the main results for both a priori and a posteriori error analysis for Nitsche's method.", "Section contains some auxiliary results followed by the rigorous a priori error estimates for Nitsche's method.", "In Section , a reliable and efficient a posteriori error analysis for Nitsche's method is presented.", "Section  focuses on the the generalisations to dGFEM and is followed by numerical experiments that confirm the theoretical findings in Section .", "Section  concludes with some brief perspectives.", "The proofs of the local efficiency results are given in the Appendix." ], [ "Preliminaries and main results", "The weak formulation of the non-linear system (REF ), the Nitsche's method and the main results in the Nitsche framework are given in this section." ], [ "Weak formulation", "The weak formulation of (REF ) seeks $\\Psi _\\epsilon \\in \\mathcal {{X}}$ such that for all $\\Phi \\in ,{\\begin{@align}{1}{-1} N(\\Psi _\\epsilon ;\\Phi ):=A(\\Psi _\\epsilon ,\\Phi )+B(\\Psi _\\epsilon ,\\Psi _\\epsilon ,\\Psi _\\epsilon ,\\Phi )+C(\\Psi _\\epsilon ,\\Phi )=0.", "\\,\\,\\,\\,\\end{@align}}Here for all $ =(1,2), = (1, 2), =(1,2), =(1,2) X:=H1() ,${\\begin{@align*}{1}{-1}&A(\\Theta ,\\Phi ):=a(\\theta _1,\\varphi _1)+ a(\\theta _2,\\varphi _2),\\,\\,C(\\Theta ,\\varphi ):=c(\\theta _1,\\varphi _1)+ c(\\theta _2,\\varphi _2),\\\\&B(\\Xi ,\\eta ,\\Theta ,\\Phi ):=\\frac{2}{3\\epsilon ^{2}}\\int _\\Omega \\left((\\Xi \\cdot \\eta )(\\Theta \\cdot \\Phi )+2(\\Xi \\cdot \\Theta )(\\eta \\cdot \\Phi )\\right) \\,{\\rm dx}=\\frac{1}{3}(3b(\\xi _1,\\eta _1,\\theta _1,\\varphi _1)+3b(\\xi _2,\\eta _2,\\theta _2,\\varphi _2) \\\\&\\qquad \\qquad \\qquad \\qquad + 2b(\\xi _2,\\eta _1,\\theta _2,\\varphi _1)+ 2 b(\\xi _1,\\eta _2,\\theta _1,\\varphi _2)+ b(\\xi _2,\\eta _2,\\theta _1,\\varphi _1)+b(\\xi _1,\\eta _1,\\theta _2,\\varphi _2)),\\\\&\\text{and for } \\xi ,\\eta ,\\theta ,\\varphi \\in H^1(\\Omega ),\\: a(\\theta ,\\varphi ):=\\int _{\\Omega } \\nabla \\theta \\cdot \\nabla \\varphi \\,{\\rm dx}, \\: b(\\xi ,\\eta ,\\theta ,\\varphi ):= 2\\epsilon ^{-2}\\int _\\Omega \\xi \\eta \\theta \\varphi \\,{\\rm dx}\\nonumber \\\\& \\text{ and } c(\\theta ,\\varphi ):=-2\\epsilon ^{-2} \\int _\\Omega \\theta \\varphi \\,{\\rm dx}.", "\\end{@align*}}See \\cite {MultistabilityApalachong, DGFEM} for a proof of existence of minimizers of the Landau-de Gennes energy functional (\\ref {Landau-de Gennes energy functional}) that are solutions to (\\ref {continuous nonlinear}).The analysis of this article is applicable to the cases where the exact solution $$ of (\\ref {continuous nonlinear}) belongs to $ X 1+(), 0 < <1,$ for example in non-convex polygons.When $$ is a convex polygon, $ =1$; that is, the solution of (\\ref {continuous nonlinear}) belongs to $ X 2()$.The regular solutions (also referred to as non-singular solutions in literature; see \\cite {Gaurang_apriori_aposteriori} and the references therein) $$ of (\\ref {continuous nonlinear strong form}) for \\textbf {a fixed $$} are approximated.This implies that the linearized operator $ DN(), $ is invertible in the Banach space and the following inf-sup conditions \\cite {Ern} hold:{\\begin{@align}{1}{-1}0< \\beta := \\inf _{\\begin{array}{c}\\Theta \\in {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1\\end{array}} \\sup _{\\begin{array}{c}\\Phi \\in {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1\\end{array}}\\langle DN(\\Psi _\\epsilon )\\Theta , \\Phi \\rangle , \\text{ and } 0< \\beta = \\inf _{\\begin{array}{c}\\Phi \\in {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1\\end{array}} \\sup _{\\begin{array}{c}\\Theta \\in {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1\\end{array}}\\langle DN(\\Psi _\\epsilon )\\Theta , \\Phi \\rangle ,\\end{@align}}where $ DN(), :=A(,)+3B(,,,)+C(,)$ and the inf-sup constant $$ depends on $$.", "Here and throughout the paper, $ , $ denotes the duality pairing between $ *$ and $ .", "The parameter $\\epsilon $ in $\\Psi _\\epsilon $ is suppressed for notational brevity and is chosen fixed in the sequel." ], [ "Nitsche's method", "Consider a shape-regular triangulation $\\mathcal {T}$ of $\\Omega $ into triangles [10].", "Define the mesh discretization parameter $h= \\max _{T \\in \\mathcal {T}} h_T,$ where $h_T= diam(T) $ .", "Denote $\\mathcal {E}_h^{i} $ ( resp.", "$ \\mathcal {E}_h^{\\partial }$ ) to be the interior (resp.", "boundary) edges of $\\mathcal {T}$ and let $\\mathcal {E}:=\\mathcal {E}_h^{i} \\cup \\mathcal {E}_h^{\\partial }$ .", "The length of an edge $E$ is denoted by $h_E.$ Define the finite element subspace of $\\mathbf {X}$ by $\\mathbf {X}_h:=X_h \\times X_h$ with $X_h:= \\lbrace v \\in H^1(\\Omega )|\\,\\, v|_T \\in P_1(T) \\text{ for all } T \\in \\mathcal {T} \\rbrace $ and let the discrete norm be defined by $\\displaystyle {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_h:=\\int _{\\Omega } \\vert \\nabla v\\vert ^2 \\,{\\rm dx}+ \\sum _{ E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E} \\int _{E} v^2 \\,{\\rm ds}\\text{ for all } v \\in X_h.$ Here $\\sigma >0$ is the penalty parameter and $P_1(T)$ denotes affine polynomials defined on $T$ .", "The space $\\mathbf {X}_h $ is equipped with the product norm ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}:= \\left( {\\vert \\hspace{-1.0625pt}\\vert \\varphi _{1}\\vert \\hspace{-1.0625pt}\\vert }_h^2 + {\\vert \\hspace{-1.0625pt}\\vert \\varphi _{2}\\vert \\hspace{-1.0625pt}\\vert }_h^2\\right)^{1/2} $ for all $\\Phi _{h}=(\\varphi _{1},\\varphi _{2}) \\in \\mathbf {X}_h$ .", "Define ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0, E}^2:= {\\vert \\hspace{-1.0625pt}\\vert \\varphi _{1}\\vert \\hspace{-1.0625pt}\\vert }_{0, E}^2 + {\\vert \\hspace{-1.0625pt}\\vert \\varphi _{2}\\vert \\hspace{-1.0625pt}\\vert }_{0, E}^2 $ and ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0, T}^2:= {\\vert \\hspace{-1.0625pt}\\vert \\varphi _{1}\\vert \\hspace{-1.0625pt}\\vert }_{0, T}^2 + {\\vert \\hspace{-1.0625pt}\\vert \\varphi _{2}\\vert \\hspace{-1.0625pt}\\vert }_{0, T}^2 $ for $\\Phi _{h} \\in \\mathbf {X}_h$ such that for $v \\in X_h,$ ${\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{0, E}^2:= \\int _{ E} v^2 \\,{\\rm ds}$ and ${\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{0, T}^2:= \\int _{ T} v^2 \\,{\\rm dx}$ , respectively.", "Define $H^1(\\mathcal {T}):= \\lbrace v \\in L^2(\\Omega )| \\, v \\in H^1(T) \\text{ for all } T \\in \\mathcal {T}\\rbrace $ and $\\mathbf {H}^1(\\mathcal {T}):=H^1(\\mathcal {T}) \\times H^1(\\mathcal {T}).", "$ For an interior edge $E$ shared by the triangles $T^+$ and $T^-$ , define the jump and average of $\\varphi \\in H^1(\\mathcal {T})$ across $E$ as $[\\varphi ]_E:= \\varphi |_{T^+} - \\varphi |_{T^-}$ and $\\lbrace \\varphi \\rbrace _E:=\\frac{1}{2}(\\varphi |_{T^+} + \\varphi |_{T^-} )$ , respectively, and for an boundary edge $E $ of the triangle $T$ , $[\\varphi ]_E:= \\varphi |_{T} $ and $\\lbrace \\varphi \\rbrace _E:=\\varphi |_{T}$ , respectively.", "For a vector function, jump and average are defined component-wise.", "For $\\theta , \\varphi \\in H^1(\\Omega )$ , $\\mathbf {g}=(g_1, g_2)$ and the penalty parameter $\\sigma >0$ , let $&a_h(\\theta ,\\varphi ):= \\int _\\Omega \\nabla \\theta \\cdot \\nabla \\varphi \\,{\\rm dx}- \\langle \\frac{\\partial \\theta }{\\partial \\nu }, \\varphi \\rangle _{\\partial \\Omega } - \\langle \\theta , \\frac{\\partial \\varphi }{\\partial \\nu } \\rangle _{\\partial \\Omega } + \\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E} \\langle \\theta , \\varphi \\rangle _E,\\\\ & \\text{ and }\\,\\,l^i_h(\\varphi ):=- \\langle g_i, \\frac{\\partial \\varphi }{\\partial \\nu } \\rangle _{\\partial \\Omega } + \\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E} \\langle g_i, \\varphi \\rangle _E \\text{ for } 1\\le i \\le 2,$ where $\\langle \\cdot , \\cdot \\rangle _{\\partial \\Omega }$ denotes the duality pairing between ${H}^{-\\frac{1}{2}}(\\partial \\Omega )$ and ${H}^{\\frac{1}{2}}(\\partial \\Omega )$ and $\\nu $ denotes the outward unit normal associated to $\\partial \\Omega $ .", "In the sequel, $\\langle \\cdot , \\cdot \\rangle _E$ is the duality pairing between ${H}^{-\\frac{1}{2}}(E)$ and ${H}^{\\frac{1}{2}}(E)$ for $E \\in \\mathcal {E}.$ For $ \\Theta =(\\theta _1,\\theta _2),\\, \\Phi =(\\varphi _1,\\varphi _2)\\in \\mathbf {X}$ , let $A_{h}(\\Theta ,\\Phi ) :=a_{h}(\\theta _1,\\varphi _1)+a_{h}(\\theta _2,\\varphi _2),$ and $L_h(\\Phi _{h})=l^1_h(\\varphi _1)+l^2_h(\\varphi _2)$ .", "The Nitsche's method corresponding to (REF ) seeks $ \\Psi _{h}\\!\\in \\!\\mathbf {X}_h$ , such that for all $ \\Phi _{h} \\in \\mathbf {X}_h,$ $N_h(\\Psi _h;\\Phi _h):=A_{h}(\\Psi _{h},\\Phi _{h})+B(\\Psi _{h},\\Psi _{h},\\Psi _{h},\\Phi _{h})+C(\\Psi _{h},\\Phi _{h})-L_h(\\Phi _{h})=0.$ Remark 2.1 The restrictions of the bilinear and quadrilinear forms $C(\\cdot , \\cdot ),$ $B(\\cdot , \\cdot ,\\cdot , \\cdot )$ to $T \\in \\mathcal {T}$ are denoted as $C_T(\\cdot , \\cdot ), $ $B_T(\\cdot , \\cdot ,\\cdot , \\cdot )$ , respectively.", "Define the bilinear form $A_T(\\Theta , \\Phi ):= \\int _T \\nabla \\Theta \\cdot \\nabla \\Phi \\,{\\rm dx}$ for all $\\Theta , \\Phi \\in \\mathbf {X}$ .", "For $\\Phi _h=(\\varphi _1,\\varphi _2) \\in \\mathbf {X}_{h}$ , let $\\nabla \\Phi _h\\nu _E: =(\\frac{\\partial \\varphi _1}{\\partial \\nu _E}, \\frac{\\partial \\varphi _2}{\\partial \\nu _E})$ on an edge $E$ with outward unit normal $\\nu _E$ to $E$ ." ], [ "Main results", "The main results in this manuscript for Nitsche's method are stated in this sub-section.", "Theorems REF and REF establish a priori error estimates in energy and $\\mathbf {L}^2(\\Omega )$ norms, and a posteriori error estimates for Nitsche's method, respectively, when the exact solution $\\Psi $ of () has the regularity $\\mathcal {{X}} \\cap {1+\\alpha }(\\Omega ) , \\; 0< \\alpha \\le 1$ .", "Throughout the sequel, $0 < \\alpha \\le 1$ denotes the index of elliptic regularity.", "The results are extended for dGFEM and are presented in Section .", "Theorem 2.2 (A priori error estimate) Let $\\Psi $ be a regular solution of ().", "For a sufficiently large penalty parameter $\\sigma >0$ , and a sufficiently small discretization parameter $h$ , there exists a unique solution $\\Psi _{h}$ to the discrete problem (REF ) that approximates $\\Psi $ such that $(i)\\,\\,\\, {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h \\lesssim h^{\\alpha },\\quad (ii)\\,\\,\\, {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 \\lesssim h^{2\\alpha },$ where $0 < \\alpha \\le 1$ denotes the index of elliptic regularity.", "As per standard convention $a \\lesssim b$ $\\iff $ $a \\le Cb$ where the constant $C$ is independent of the discretization parameter $h$ .", "A reliable and efficient a posteriori error estimate for (REF ) is the second main result of the paper.", "For each element $T \\!\\in \\!", "\\mathcal {T}$ and edge $E \\!\\in \\!", "\\mathcal {E}$ , define the volume and edge contributions to the estimators by $&\\vartheta _T^2: = h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert 2\\epsilon ^{-2} (\\vert \\Psi _h\\vert ^2-1)\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T}, \\,\\,\\,(\\vartheta _E^i)^2 := h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\nabla \\Psi _h \\nu _E]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\text{ for all } E \\in \\mathcal {E}_h^{i}, \\\\& \\text{and } (\\vartheta _E^{\\partial })^2:= \\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{h} - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\text{ for all } E \\in \\mathcal {E}_h^{\\partial }.", "$ Define the estimator $ \\displaystyle \\vartheta ^2:= \\sum _{ T \\in \\mathcal {T}} \\vartheta _T^2 +\\sum _{ E \\in \\mathcal {E}_h^{i}} (\\vartheta _E^i)^2 + \\sum _{ E \\in \\mathcal {E}_h^{\\partial } }(\\vartheta _E^{\\partial })^2.$ Theorem 2.3 (A posteriori error estimate) Let $\\Psi $ be a regular solution of () and $\\Psi _{h}$ solve (REF ).", "For a sufficiently large penalty parameter $\\sigma >0$ , and a sufficiently small discretization parameter $h$ , there exist $h$ -independent positive constants $C_{\\text{rel}}$ and $C_{\\text{eff}}$ such that $C_{\\text{eff}} \\vartheta \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Psi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le C_{\\text{rel}} \\big ( \\vartheta + h.o.t \\big ),$ where $h.o.t$ expresses one or several terms of higher order (as will be explained in Section )." ], [ "A priori error estimate", "This section is devoted to the proof of Theorem REF .", "Some auxiliary results are presented first.", "This is followed by a discrete inf-sup condition and the construction of a non-linear map for the application of Brouwer's fixed point theorem.", "The energy and $\\mathbf {L}^2$ - norm estimates follow as a consequence of the fixed point and duality arguments." ], [ "Auxiliary results", "Lemma 3.1 (Poincaré type inequalities)[21] Let $\\Omega $ be a bounded open subset of $\\mathbb {R}^2$ with Lipschitz continuous boundary $\\partial \\Omega $ .", "For $\\varphi \\in H^1_0(\\Omega )$ , there exists a positive constant $\\alpha _0 =\\alpha _0(\\Omega )$ such that $\\alpha _0 {\\vert \\hspace{-1.0625pt}\\vert \\varphi \\vert \\hspace{-1.0625pt}\\vert }_{0} \\le {\\vert \\hspace{-1.0625pt}\\vert \\nabla \\varphi \\vert \\hspace{-1.0625pt}\\vert }_{0} .$ For $\\varphi \\in H^1(\\mathcal {T})$ , there exists a constant $C_P>0$ independent of $h $ and $\\varphi $ such that for $1 \\le r < \\infty $ , ${\\vert \\hspace{-1.0625pt}\\vert \\varphi \\vert \\hspace{-1.0625pt}\\vert }_{L^r(\\Omega )} \\le C_P {\\vert \\hspace{-1.0625pt}\\vert \\varphi \\vert \\hspace{-1.0625pt}\\vert }_{h}.$ Lemma 3.2 (Trace inequalities) [12] $\\displaystyle (i) \\; \\text{For }{\\rm v}\\!\\in \\!", "H^1(T), T \\!\\in \\!", "\\mathcal {T}, {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_{0,\\partial T}^2 \\lesssim (h_T^{-1} {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2 + {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}{\\vert \\hspace{-1.0625pt}\\vert \\nabla {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}).$ $\\displaystyle (ii) \\;\\text{For all }\\Phi _{h}\\!\\in \\!", "\\mathbf {X}_{h},\\sum _{E \\in \\mathcal {E}_h^{\\partial }} h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\Phi _{h}\\nu _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0}^2.", "$ Lemma 3.3 (Interpolation estimate)[10] For ${\\rm v} \\!\\in \\!", "H^{1+\\alpha }(\\Omega ) \\text{ with } \\alpha \\!\\in \\!", "(0,1]$ , there exists ${\\rm {I}}_{h}{\\rm v} \\!\\in \\!", "X_h$ such that ${\\vert \\hspace{-1.0625pt}\\vert {\\rm v}-{\\rm {I}}_{h}{\\rm v} \\vert \\hspace{-1.0625pt}\\vert }_{0} +h {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}-{\\rm {I}}_{h}{\\rm v} \\vert \\hspace{-1.0625pt}\\vert }_{1} \\le C_I h^{1+\\alpha } \\vert {\\rm v}\\vert _{H^{1+ \\alpha }(\\Omega )},$ where $C_I$ is a positive constant independent of $h$ .", "Remark 3.4 Trace inequality stated in Lemma REF $(i)$ yields $ {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}-{\\rm {I}}_{h}{\\rm v} \\vert \\hspace{-1.0625pt}\\vert }_{h} \\le C_I h ^{\\alpha } \\vert {\\rm v}\\vert _{H^{1+\\alpha }(\\Omega )}$ for a positive constant $C_I$ independent of $h$ .", "Lemma 3.5 (Extension operator) [6], [18], [23] Define the operator $\\Pi _h: X_h \\rightarrow V_h:=X_h \\cap H^1_0(\\Omega ) $ using nodal values of freedom: $\\displaystyle \\Pi _h{\\rm v}(n) = {\\left\\lbrace \\begin{array}{ll}0 \\quad \\quad \\text{ for a node } n \\text{ on } \\partial \\Omega \\\\{\\rm v}(n) \\quad \\quad \\text{ for a node } n \\text{ on } \\Omega \\setminus \\partial \\Omega \\end{array}\\right.", "}$ .", "For all ${\\rm v}\\in X_h$ , it holds that $\\big (\\sum _{T \\in \\mathcal {T}} h_T^{-2} {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}-\\Pi _h {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T} + \\sum _{ E \\in \\mathcal {E}_h^{i}} h_E^{-1} {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}-\\Pi _h {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,E}\\big )^{\\frac{1}{2}} \\le C_{e_1} {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_h , \\\\{\\vert \\hspace{-1.0625pt}\\vert {\\rm v}-\\Pi _h {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_h \\le C_{e_2}(\\sum _{ E \\in \\mathcal {E}^{\\partial }_h}h_E^{-1}\\int _{ E}{\\rm v}^2{ \\rm ds} )^{\\frac{1}{2}} \\le C_{e_2} {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_h, \\quad {\\vert \\hspace{-1.0625pt}\\vert \\Pi _h {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_h \\le C_{e_3} {\\vert \\hspace{-1.0625pt}\\vert {\\rm v}\\vert \\hspace{-1.0625pt}\\vert }_h ,$ where the constants $C_{e_1}$ , $C_{e_2}$ and $C_{e_3} $ are independent of $h$ .", "The next lemma states boundedness and coercivity results for $A(\\cdot ,\\cdot )$ (resp.", "$A_h(\\cdot ,\\cdot )$ ), boundedness results for $B(\\cdot ,\\cdot ,\\cdot ,\\cdot )$ and $C(\\cdot ,\\cdot )$ .", "These results are a consequence of Hölder's inequality, Lemma REF , and the Sobolev embedding results [10] $H^{1}(\\Omega ) \\hookrightarrow L^{4}(\\Omega )$ and $H^{1+\\alpha }(\\Omega ) \\hookrightarrow L^{\\infty }(\\Omega )$ for $\\Omega \\subset \\mathbb {R}^2$ and $\\alpha >0$ .", "For detailed proofs, see [24].", "Lemma 3.6 (Properties of bilinear and quadrilinear forms)[24] The following properties hold.", "For all $\\Theta $ , $\\Phi \\!\\in \\!", ", $ A(,) 1 1, and A(,) 12.", "$\\item For the choice of a sufficiently large parameter $$, there exists a positive constant $ Cell > 0$ such that $ for all h, h Xh$, $ Ah(h,h)h h h h, and Ah(h, h) Cell h h2.$\\item For all $$, $ X$,$ C(,)-2 0 0$ and$ C(,)-2 h h.$\\item For $ ,,, X$, (resp.", "$ , H1+()$ with $ 0<1$, $$, $ X$),{\\begin{@align*}{1}{-1}& \\int _{\\Omega }(\\Xi \\cdot \\eta )(\\Theta \\cdot \\Phi ) {\\,{\\rm dx}} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 \\text{ and }\\int _\\Omega (\\Xi \\cdot \\eta )(\\Theta \\cdot \\Phi ) \\,{\\rm dx}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}, \\\\& B(\\Xi ,\\eta ,\\Theta ,\\Phi )\\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1\\text{ and }B(\\Xi ,\\eta ,\\Theta ,\\Phi ) \\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}.", "\\\\&(\\text{resp.", "}B(\\Xi ,\\eta ,\\Theta ,\\Phi )\\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0).\\end{@align*}}\\item For $ X$ and for all $ h, h Xh$ (resp.", "$ 1+() with 0<1$),{\\begin{@align*}{1}{-1}&B(\\eta _h,\\eta _h, \\eta _h, \\Phi _{h})- B(\\eta ,\\eta , \\eta , \\Phi _{h}) \\lesssim \\epsilon ^{-2}({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _h-\\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_h ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1) + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _h-\\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1}^2) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h \\\\& (\\text{resp.", "}B({\\rm I}_{h}\\eta ,{\\rm I}_{h}\\eta , {\\rm I}_{h}\\eta , \\Phi _{h})- B(\\eta ,\\eta , \\eta , \\Phi _{h}) \\lesssim \\epsilon ^{-2} h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^3{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}, \\\\&B(\\eta ,\\eta , \\Theta _{h}, \\Phi _{h}) - B({\\rm I}_{h}\\eta ,{\\rm I}_{h}\\eta , \\Theta _{h}, \\Phi _{h}) \\lesssim \\epsilon ^{-2} h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}), \\end{@align*}}$ where the hidden constant in $\"\\lesssim \"$ depends on the constants from $C_P, C_S$ and $\\alpha _0$ , and are independent of $h.$ We now state the well-posedness and regularity of solutions of a second-order linear system of equations () and a perturbation result () that is important to prove the discrete inf-sup condition in the next section.", "The proof follows analogous to the proof in Theorem 4.7 of [24] and is skipped.", "Lemma 3.7 (Linearized systems) Let $\\Psi $ be a regular solution of ().", "For a given $\\Theta _{h} \\!\\in \\!", "\\mathbf {X}_h$ with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ , there exist $\\xi $ and $\\eta \\!\\in \\!", "\\mathbf {H}^{1+\\alpha }(\\Omega )\\cap that solve the linear systems{\\begin{@align}{1}{-1}A(\\xi ,\\Phi )=3B(\\Psi ,\\Psi ,\\Theta _{h},\\Phi )+C(\\Theta _{h},\\Phi ) \\,\\, \\text{ for all } \\Phi \\in \\,\\,\\text{and}\\\\A(\\eta ,\\Phi )=3B(\\Psi ,\\Psi , \\Pi _h\\Theta _{h},\\Phi )+C(\\Pi _h\\Theta _{h},\\Phi ) \\,\\, \\text{ for all } \\Phi \\in \\,\\,\\,\\,\\,\\,\\,\\end{@align}} such that{\\begin{@align}{1}{-1}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha } \\lesssim \\epsilon ^{-2}(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)\\quad \\text{ and } \\quad {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\eta -\\xi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0}\\lesssim \\epsilon ^{-2}h(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2),\\end{@align}}where the constant hidden in $ \"\" $ depends on $ CS$, $ 0$ and $ Ce1$.$ The next three lemmas concern local efficiency type estimates that yield lower bounds for the errors, and are necessary for the medius analysis.", "The proofs follow from standard bubble function techniques extended to the non-linear system considered in this paper and is sketched in the Appendix.", "Lemma 3.8 (Local efficiency I) Let $\\Psi \\!\\in \\!", "\\mathcal {{X}}$ be a regular solution of ().", "For $\\Phi _h \\!\\in \\!", "\\mathbf {X}_h$ , define $ \\eta _T := (2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h})|_T$ , where $T \\!\\in \\!", "\\mathcal {T}$ and $ \\eta _E := [\\nabla \\Phi _h \\nu _E]_E$ , where $E$ is an edge of $T$ .", "Then the following estimates hold.", "$(i) \\sum _{ T \\in \\mathcal {T}} h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2 + \\sum _{ E \\in \\mathcal {E}_h^{i}} h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{h} (1+ \\epsilon ^{-2} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1}+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1}) + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1}^2 +1))^2.$ $(ii)$ For $\\Psi \\!\\in \\!", "\\mathcal {{X}}\\cap {1+\\alpha }(\\Omega )$ , $0 < \\alpha \\le 1$ , $\\Phi _h:= {\\rm I}_h \\Psi $ in the definitions of $ \\eta _T$ and $ \\eta _E$ , $\\sum _{ T \\in \\mathcal {T}} h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2 + \\sum _{ E \\in \\mathcal {E}_h^{i}} h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\lesssim h^{2\\alpha } (1+ \\epsilon ^{-2}h^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2))^2{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2.$ The next lemma is a local efficiency type result for () that helps to prove the discrete inf-sup condition for a linear problem in the next section.", "Lemma 3.9 (Local efficiency II) Let $\\xi $ be the solution of () with interpolant ${\\rm {I}}_h \\xi \\!\\in \\!", "h:= \\mathbf {X}_h\\cap 1_0(\\Omega )$ .", "If the exact solution $\\Psi \\!\\in \\!", "\\mathcal {{X}}\\cap {1+\\alpha }(\\Omega )$ , $0 < \\alpha \\le 1$ , then $\\sum _{ T \\in \\mathcal {T}} h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2 + \\sum _{ E \\in \\mathcal {E}_h^{i}} h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2\\lesssim \\epsilon ^{-4}h^{2\\alpha }(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)^2,$ where $ \\eta _T:=(2 \\epsilon ^{-2}(\\vert {\\rm {I}}_h \\Psi \\vert ^2 \\Theta _{h} + 2 ({\\rm {I}}_h \\Psi \\cdot \\Theta _{h}) {\\rm {I}}_h \\Psi -\\Theta _{h}))|_T$ is defined on a triangle $T \\!\\in \\!", "\\mathcal {T}$ , $ {\\rm {I}}_h \\Psi \\!\\in \\!", "\\mathbf {X}_h$ is the interpolant of $\\Psi $ and $ \\eta _E= [\\nabla ({\\rm {I}}_h\\xi ) \\nu _E]_E$ on the edge $E$ of $T$ and $\\Theta _h \\in \\mathbf {X}_h$ .", "For $G \\!\\in \\!", "\\mathbf {L}^2(\\Omega )$ , the well-posed dual problem admits a unique $\\chi \\!\\in \\!", "\\cite {DGFEM} such that{\\begin{@align}{1}{-1} \\langle DN(\\Psi )\\Phi , \\chi \\rangle = (G, \\Phi ) \\,\\,\\quad \\text{for all } \\Phi \\in \\end{@align}}that satisfies{\\begin{@align}{1}{-1}\\quad {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\chi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }\\lesssim (1+ \\epsilon ^{-2}(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert G\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0,\\end{@align}}where $ 0 < 1$ denotes the index of elliptic regularity.$ A local efficiency type result for () is needed to establish $\\mathbf {L}^2 $ - norm error estimates and is stated below.", "Lemma 3.10 (Local efficiency III) Let $\\Psi $ be a regular solution of () and ${\\rm I}_h\\Psi \\!\\in \\!", "\\mathbf {X}_h$ be its interpolant.", "For a given $G \\!\\in \\!", "\\mathbf {L}^2(\\Omega )$ , let $\\chi $ solve () and let its interpolant be ${\\rm {I}}_{h}\\chi \\!\\in \\!", "h$ .", "Then, the following result holds.", "$\\sum _{ T \\in \\mathcal {T}} h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2 + \\sum _{ E \\in \\mathcal {E}_h^{i}} h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2\\lesssim h^{2\\alpha }(1+ \\epsilon ^{-2}(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{1+\\alpha }))^4 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert G\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0^2+ (Osc(G))^2,$ where $ \\eta _T :=( G -2\\epsilon ^{-2} (\\vert {\\rm {I}}_{h}\\Psi \\vert ^2 {\\rm {I}}_{h}\\chi +2 ({\\rm I}_h\\Psi \\cdot {\\rm I}_h{ \\chi }){\\rm {I}}_{h}\\Psi - {\\rm {I}}_{h}\\chi ))|_T$ is defined on a triangle $T \\in \\mathcal {T} $ , $ \\eta _E:=[ \\nabla ({\\rm {I}}_{h}\\chi ) \\nu _E ]_E $ on edge $E$ of $T$ and $\\displaystyle Osc(G)=\\big (\\sum _{ T \\in \\mathcal {T}}h_T^2 (\\inf _{G_h \\in P_1(T)} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert G- G_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2)\\big )^{\\frac{1}{2}}.$ Remark 3.11 In this article, we consider the case when exact solution belongs to $\\mathcal {{X}} \\cap {1+\\alpha }(\\Omega ), \\; 0< \\alpha \\le 1.$ Hence globally continuous piece-wise affine polynomials in $X_h$ lead to optimal order estimates.", "However, if the solution belongs to $s(\\Omega )$ for $\\frac{3}{2} < s \\le p+1, $ $p \\in \\mathbb {N},$ then choose $X_h= \\lbrace v_h \\in {C}^0(\\overline{\\Omega }), v_h|_T \\in P_p(T), \\text{ for all } T \\in \\mathcal {T}\\rbrace $ [17].", "In this case, the local efficiency terms $ \\eta _T:=-\\Delta \\Phi _h+ (2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h})|_T$ in Lemma REF and $\\eta _T$ will include $\\Delta (\\textrm {I}_h \\xi )$ (resp.", "$\\Delta (\\textrm {I}_h \\chi )$ ) in Lemma REF (resp.", "Lemma REF ).", "Proof of a priori estimates This subsection focuses on the proof of a priori error estimates in Theorem REF .", "The key idea is to establish a discrete inf-sup condition that corresponds to a perturbed bilinear form defined for all $\\Theta _{h}, \\Phi _{h} \\!\\in \\!", "\\mathbf {X}_h$ as $\\langle DN_h(\\textrm {I}_h\\Psi )\\Theta _{h}, \\Phi _{h} \\rangle := A_{h}(\\Theta _{h},\\Phi _{h})+3B(\\textrm {I}_h\\Psi ,\\textrm {I}_h\\Psi ,\\Theta _{h},\\Phi _{h})+C(\\Theta _{h},\\Phi _{h}).$ in Theorem REF when the exact solution $\\Psi $ of () belongs to $\\mathcal {{X}} \\cap {1+\\alpha }(\\Omega )$ with $0< \\alpha \\le 1$ .", "The proofs in [24] assume that the exact solution belongs to $ \\mathcal {{X}} \\cap 2(\\Omega )$ .", "The non-trivial modification of the proof techniques appeal to a clever re-grouping of the terms that involve the boundary terms and an application of Lemma REF .", "Moreover, in [24], the stability of the perturbed bilinear form $\\langle DN_h({{\\rm I}_{h}\\Psi })\\cdot ,\\cdot \\rangle $ is established by first proving the stability of $\\langle DN_h({\\Psi })\\cdot ,\\cdot \\rangle $ (see [24]).", "In this article, we provide an alternate simplified proof that directly establishes the stability of the perturbed bilinear form using Lemma REF .", "Lemma 3.12 Let $\\Psi $ be a regular solution of () and ${\\rm I}_{h}\\Psi $ be its interpolant.", "For $\\Theta _{h} \\!\\in \\!", "\\mathbf {X}_h$ with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ , and the interpolant ${\\rm {I}}_h \\xi \\in {h}$ of the solution $\\xi $ of (), it holds that ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h} + {\\rm {I}}_h\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\lesssim \\langle DN_h({\\rm I}_{h}\\Psi )\\Theta _{h},\\Phi _{h} \\rangle + \\epsilon ^{-2}h^\\alpha (1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{1+\\alpha }).$ Since $\\Theta _{h} + \\textrm {I}_h \\xi \\!\\in \\!", "\\mathbf {X}_h,$ the discrete coercivity condition in Lemma REF $(ii)$ implies that there exists $ \\Phi _{h}\\!\\in \\!", "\\mathbf {X}_h$ with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ such that $\\displaystyle {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h} + \\textrm {I}_h\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\lesssim A_h(\\Theta _{h} + \\textrm {I}_h\\xi ,\\Phi _{h}).", "$ This inequality with (), (REF ) and a regrouping of terms yields ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h} + \\textrm {I}_h\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} &\\lesssim \\langle DN_h({\\rm I}_{h}\\Psi )\\Theta _{h},\\Phi _{h} \\rangle +(A_h(\\textrm {I}_h\\xi ,\\Phi _{h})-A(\\xi ,\\Pi _h\\Phi _{h}))+(3B({\\rm I}_{h}\\Psi ,{\\rm I}_{h}\\Psi ,\\Theta _{h},\\Pi _h\\Phi _{h}-\\Phi _{h}) \\nonumber \\\\& \\quad + C(\\Theta _{h},\\Pi _h\\Phi _{h}-\\Phi _{h})) +3(B(\\Psi ,\\Psi ,\\Theta _{h},\\Pi _h\\Phi _{h})- B({\\rm I}_{h}\\Psi ,{\\rm I}_{h}\\Psi ,\\Theta _{h},\\Pi _h\\Phi _{h}) ).", "$ The definition of $A_h(\\cdot ,\\cdot )$ and $\\textrm {I}_h\\xi =0$ on $\\partial \\Omega $ lead to $ A_h(\\textrm {I}_h\\xi ,\\Phi _{h})-A(\\xi ,\\Pi _h\\Phi _{h})& = (A(\\textrm {I}_h\\xi ,\\Phi _{h}-\\Pi _h\\Phi _{h})- \\langle \\nabla (\\textrm {I}_h\\xi ) \\nu , \\Phi _{h} \\rangle _{\\partial \\Omega }) +A(\\textrm {I}_h\\xi -\\xi ,\\Pi _h\\Phi _{h}) .$ An integration by parts element-wise, and $\\Delta (\\textrm {I}_h\\xi )=0$ in $T$ , $\\Pi _h\\Phi _{h} =0 $ on $\\partial \\Omega $ , and $ [\\Phi _{h}-\\Pi _h \\Phi _{h} ]_E =0 $ for all $E\\in \\mathcal {E}^i_h$ lead to an estimate for the first term in the right-hand side of (REF ) as $A(\\textrm {I}_h\\xi ,\\Phi _{h}-\\Pi _h\\Phi _{h})- \\langle \\nabla (\\textrm {I}_h\\xi ) \\nu , \\Phi _{h}-\\Pi _h\\Phi _{h} \\rangle _{\\partial \\Omega }= \\sum _{E \\in \\mathcal {E}_h^i} \\langle [\\nabla (\\textrm {I}_h\\xi ) \\nu _E ]_E, \\Phi _{h}-\\Pi _h \\Phi _{h} \\rangle _{E}.$ Note that the above term can be combined with the third term on the right-hand side of (REF ) to rewrite the expression with the help of local term $ \\eta _T=(2 \\epsilon ^{-2}(\\vert {\\rm {I}}_h \\Psi \\vert ^2 \\Theta _{h} + 2 ({\\rm {I}}_h \\Psi \\cdot \\Theta _{h}) {\\rm {I}}_h \\Psi -\\Theta _{h}))|_T$ on a triangle $T$ and $ \\eta _E= [\\nabla ({\\rm {I}}_h\\xi ) \\nu _E]_E$ on the edge $E$ can be rewritten as $& A(\\textrm {I}_h\\xi ,\\Phi _{h}-\\Pi _h\\Phi _{h})- \\langle \\nabla (\\textrm {I}_h\\xi ) \\nu , \\Phi _{h}-\\Pi _h\\Phi _{h} \\rangle _{\\partial \\Omega }+ (3B({\\rm I}_{h}\\Psi ,{\\rm I}_{h}\\Psi ,\\Theta _{h},\\Pi _h\\Phi _{h}-\\Phi _{h}) +C(\\Theta _{h},\\Pi _h\\Phi _{h}-\\Phi _{h})) \\\\ & = -\\sum _{T \\in \\mathcal {T}}\\int _T \\eta _T \\cdot (\\Phi _{h}-\\Pi _h\\Phi _{h})\\,{\\rm dx}+\\sum _{E \\in \\mathcal {E}^i_h} \\langle \\eta _E, \\Phi _{h}-\\Pi _h \\Phi _{h} \\rangle _{E}.$ A Cauchy-Schwarz inequality, Lemma REF and the inequality (REF ) applied to the right-hand side of the last equality yield $\\sum _{T \\in \\mathcal {T}}\\int _T \\eta _T \\cdot (\\Pi _h\\Phi _{h}-\\Phi _{h})\\,{\\rm dx}+\\sum _{E \\in \\mathcal {E}^i_h} \\langle \\eta _E, \\Phi _{h}-\\Pi _h \\Phi _{h} \\rangle _{E}& \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} (\\sum _{T \\in \\mathcal {T}} h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2+\\sum _{E \\in \\mathcal {E}^i_h} h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2)^{\\frac{1}{2}}\\nonumber \\\\&\\lesssim \\epsilon ^{-2}h^\\alpha (1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{1+\\alpha }).$ Next we proceed to estimate the terms that remain on the right-hand side of (REF ) and (REF ).", "Lemma REF $(i)$ , Lemma REF , (), ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ and () lead to $A(\\textrm {I}_h\\xi -\\xi ,\\Pi _h\\Phi _{h}) \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\textrm {I}_h\\xi -\\xi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\lesssim h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\lesssim \\epsilon ^{-2}h^\\alpha (1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{1+\\alpha }).$ Lemma REF $(v)$ , (), ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ and ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ lead to $3(B(\\Psi ,\\Psi ,\\Theta _{h},\\Pi _h\\Phi _{h}) - B({\\rm I}_{h}\\Psi ,{\\rm I}_{h}\\Psi ,\\Theta _{h},\\Pi _h\\Phi _{h}) )\\lesssim \\epsilon ^{-2}h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{1+\\alpha }.$ A substitution of (REF )- (REF ) in (REF ) concludes the proof of Lemma REF .", "Theorem 3.13 (Stability of perturbed bilinear form).", "Let $\\Psi $ be a regular solution of () and ${{\\rm I}_{h}}\\Psi $ be its interpolant.", "For a sufficiently large $\\sigma $ , and a sufficiently small discretization parameter $h$ , there exists a constant $\\beta _0 $ such that the perturbed bilinear form in (REF ) satisfies the following discrete inf-sup condition: $0< \\beta _0 \\le \\inf _{\\begin{array}{c}\\Theta _{h} \\in \\mathbf {X}_h \\\\ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1\\end{array}} \\sup _{\\begin{array}{c}\\Phi _{h} \\in \\mathbf {X}_h \\\\ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1\\end{array}}\\langle DN_h({{\\rm I}_{h}\\Psi })\\Theta _{h},\\Phi _{h} \\rangle .$ The inf-sup condition in (), () and Lemma REF $(i)$ imply that there exists $\\Phi \\!\\in \\!", "with $ 1 =1$ such that$$ \\displaystyle \\beta {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Pi _h \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1\\le \\langle DN(\\Psi )\\Pi _h \\Theta _{h},\\Phi \\rangle =A(\\Pi _h \\Theta _{h} + \\eta ,\\Phi ) \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Pi _h \\Theta _{h} + \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}.", "$$$ Recall that $\\xi $ is the solution of () and ${\\rm {I}}_h \\xi $ is its interpolant.", "A triangle inequality followed by an application of the last displayed inequality yields $1={\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} &\\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}-\\Pi _h \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Pi _h \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}-\\Pi _h \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Pi _h \\Theta _{h} + \\eta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\\\& \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}-\\Pi _h \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h} + \\textrm {I}_h\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} +{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_h\\xi -\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla ({\\xi }-{\\eta })\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0,$ where ${\\xi }-{\\eta } =0$ on $\\mathcal {E}^{\\partial }_h$ is used in the last term.", "Since $\\xi =0$ on $\\mathcal {E}^{\\partial }_h$ , () and a triangle inequality yield ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}-\\Pi _h \\Theta _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le C_{e_2}(\\sum _{ E \\in \\mathcal {E}^{\\partial }_h}h_E^{-1}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}+\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0, E} )^{\\frac{1}{2}}\\le C_{e_2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}+\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le C_{e_2} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h}+\\textrm {I}_h\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\xi -\\textrm {I}_h\\xi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}).$ Use this in (REF ) and apply Lemmas REF , REF and () to obtain $\\displaystyle 1 \\le C_1 (\\langle DN_h({\\rm I}_{h}\\Psi )\\Theta _{h},\\Phi _{h} \\rangle + \\epsilon ^{-2}h^\\alpha (1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{1+\\alpha })),$ where the constant $C_1$ depends on $\\alpha _0, C_S, C_I, C_{e_1} $ , $C_{e_2},$ $C_{e_3}$ and is independent of $h$ .", "Therefore, for a given $\\epsilon $ , the discrete inf-sup condition holds with $\\beta _0= \\frac{1}{C_1}$ for $h < h_0:=\\left(\\frac{\\epsilon ^2}{2C_1(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{1+\\alpha })} \\right)^{\\frac{1}{\\alpha }} $ .", "Remark 3.14 In [24], under the assumption that exact solution has 2 regularity, the discrete inf-sup condition is established for a choice of $h=O(\\epsilon ^2)$ .", "Though $h-\\epsilon $ dependency is not the focus of this paper, for the case $\\alpha =1$ , where it is well-known[3] that ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{2}$ is bounded independent of $\\epsilon ,$ $h-\\epsilon $ dependency results can be derived analogous to [24].", "The proof of the energy norm error estimate in Theorem REF utilizes the methodology of [24].", "However, Lemma REF establishes the estimate that requires non-trivial modifications of the techniques used in [24] to prove energy norm error estimates.", "Lemma 3.15 (An intermediate estimate) Let $\\Psi $ be a regular solution of () and ${\\rm I}_h\\Psi \\in \\mathbf {X}_{h}$ be it's interpolant.", "Then, for any $\\Phi _h \\in \\mathbf {X}_h$ with $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1,$ it holds that $A_{h}({\\rm {I}}_{h}\\Psi , \\Phi _{h} ) + B({\\rm {I}}_{h}\\Psi ,{\\rm {I}}_{h}\\Psi ,{\\rm {I}}_{h}\\Psi , \\Phi _{h} )+ C({\\rm {I}}_{h}\\Psi , \\Phi _{h} ) -L_{h}(\\Phi _{h})\\lesssim h^{\\alpha } (1+ \\epsilon ^{-2}h^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ Add and subtract $(A_{h}(\\textrm {I}_{h}\\Psi , \\Pi _h\\Phi _{h} )-L_{h}( \\Pi _h\\Phi _{h})) $ to rewrite the left-hand side of the above displayed inequality as $A_{h}(\\textrm {I}_{h}\\Psi , \\Phi _{h} ) + B&(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Phi _{h} )+ C(\\textrm {I}_{h}\\Psi , \\Phi _{h} )-L_{h}(\\Phi _{h})=(A_{h}(\\textrm {I}_{h}\\Psi , \\Phi _{h}- \\Pi _h\\Phi _{h} )-L_{h}(\\Phi _{h}- \\Pi _h\\Phi _{h}))\\\\& +(C(\\textrm {I}_{h}\\Psi , \\Phi _{h} ) +B(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Phi _{h} )) +(A_{h}(\\textrm {I}_{h}\\Psi , \\Pi _h\\Phi _{h} )-L_{h}( \\Pi _h\\Phi _{h})).$ The definition of $A_h(\\cdot ,\\cdot )$ and $L_h(\\cdot )$ , followed by an integration by parts element-wise for the term $A(\\cdot ,\\cdot )$ , $\\Delta (\\textrm {I}_{h}\\Psi ) = 0$ and $[\\Phi _{h}- \\Pi _h\\Phi _{h}]_E=0$ for $E \\in \\mathcal {E}^i_h$ show that $& A_{h}(\\textrm {I}_{h}\\Psi , \\Phi _{h}- \\Pi _h\\Phi _{h} ) -L_{h}(\\Phi _{h}- \\Pi _h\\Phi _{h})= \\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle [\\nabla (\\textrm {I}_{h}\\Psi )\\nu _E ]_E,\\Phi _{h}- \\Pi _h\\Phi _{h} \\rangle _{E} \\nonumber \\\\& \\qquad + \\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E}\\langle \\textrm {I}_{h}\\Psi - \\mathbf {g}, \\Phi _{h}- \\Pi _h\\Phi _{h} \\rangle _E+\\langle \\mathbf {g}-\\textrm {I}_{h}\\Psi , \\nabla (\\Phi _{h}- \\Pi _h\\Phi _{h})\\nu \\rangle _{\\partial \\Omega }.$ Set $ \\eta _T=( 2\\epsilon ^{-2}(\\vert \\textrm {I}_{h}\\Psi \\vert ^2 -1)\\textrm {I}_{h}\\Psi )|_T \\text{ on a triangle } T \\text{ and } \\eta _E = [\\nabla (\\textrm {I}_{h}\\Psi )\\nu _E]_E \\text{ on the edge } E$ and observe that $\\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle [\\nabla (\\textrm {I}_{h}\\Psi )\\nu _E ]_E,\\Phi _{h}&- \\Pi _h\\Phi _{h} \\rangle _{E} +C(\\textrm {I}_{h}\\Psi , \\Phi _{h} )+B(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Phi _{h} )=(\\sum _{T \\in \\mathcal {T}}\\int _T \\eta _T \\cdot (\\Phi _{h}- \\Pi _h\\Phi _{h})\\,{\\rm dx}\\\\& + \\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle \\eta _E ,\\Phi _{h}- \\Pi _h\\Phi _{h} \\rangle _{E}) +(B(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Pi _h\\Phi _{h} ) +C({\\rm I}_{h}\\Psi , \\Pi _h\\Phi _{h})).$ The definition of $A_h(\\cdot ,\\cdot ) $ , the consistency of the exact solution $\\Psi $ given by $ N_h(\\Psi , \\Pi _h\\Phi _{h})= L_{h}( \\Pi _h\\Phi _{h}) $ and $\\Pi _h\\Phi _{h} = 0 $ on ${\\partial \\Omega }$ yield, $ A_{h}(\\textrm {I}_{h}\\Psi , \\Pi _h\\Phi _{h} )-L_{h}( \\Pi _h\\Phi _{h})& = A(\\textrm {I}_{h}\\Psi -\\Psi , \\Pi _h\\Phi _{h} )+\\langle \\mathbf {g}-\\textrm {I}_{h}\\Psi , \\nabla (\\Pi _h\\Phi _{h})\\nu \\rangle _{\\partial \\Omega } \\nonumber \\\\& \\quad - (B(\\Psi ,\\Psi ,\\Psi , \\Pi _h\\Phi _{h}) + C(\\Psi , \\Pi _h\\Phi _{h} )).$ An application of (REF )-(REF ) in (REF ), a cancellation of a boundary term and a suitable re-arrangement of terms leads to $& A_{h}({\\rm {I}}_{h}\\Psi , \\Phi _{h} ) + B({\\rm {I}}_{h}\\Psi ,{\\rm {I}}_{h}\\Psi ,{\\rm {I}}_{h}\\Psi , \\Phi _{h} )+ C({\\rm {I}}_{h}\\Psi , \\Phi _{h} ) -L_{h}(\\Phi _{h}) =\\sum _{T \\in \\mathcal {T}}\\int _T \\eta _T \\cdot (\\Phi _{h}- \\Pi _h\\Phi _{h})\\,{\\rm dx}\\nonumber \\\\& + \\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle \\eta _E ,\\Phi _{h}- \\Pi _h\\Phi _{h} \\rangle _{E} + (A(\\textrm {I}_{h}\\Psi -\\Psi , \\Pi _h\\Phi _{h} )+ C({\\rm I}_{h}\\Psi -\\Psi , \\Pi _h\\Phi _{h})) + \\langle \\mathbf {g}-\\textrm {I}_{h}\\Psi , \\nabla \\Phi _{h} \\nu \\rangle _{\\partial \\Omega }\\nonumber \\\\& +\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E}\\langle \\textrm {I}_{h}\\Psi - \\mathbf {g}, \\Phi _{h}- \\Pi _h\\Phi _{h} \\rangle _E+ (B(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Pi _h\\Phi _{h} )-B(\\Psi ,\\Psi ,\\Psi , \\Pi _h\\Phi _{h})) .", "$ Now we estimate the terms on the right-hand side of (REF ).", "A Cauchy-Schwarz inequality, Lemma REF $(ii)$ and (REF ) with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ leads to $&\\sum _{T \\in \\mathcal {T}}\\int _T \\eta _T \\cdot (\\Phi _{h}- \\Pi _h\\Phi _{h})\\,{\\rm dx}+ \\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle \\eta _E ,(\\Phi _{h}- \\Pi _h\\Phi _{h}) \\rangle _{E} \\lesssim h^{\\alpha } (1+ \\epsilon ^{-2}h^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ A use of Lemma REF , (REF )$(ii)$ (resp.", "$(iii)$ ), Remark REF and () yields $A(\\textrm {I}_{h}\\Psi -\\Psi , \\Pi _h\\Phi _{h} )+C({\\rm I}_{h}\\Psi -\\Psi , \\Pi _h\\Phi _{h}) &\\lesssim ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h}\\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}+ \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h}\\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\\\&\\lesssim h^{\\alpha }(1+\\epsilon ^{-2} h^{\\alpha }){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha } .$ The next two estimates are obtained using Cauchy-Schwarz inequality, the definition of $\\Vert .\\Vert _h$ , Remark REF , () and Lemma REF $(ii)$ .", "$&\\langle \\mathbf {g}-\\textrm {I}_{h}\\Psi , \\nabla \\Phi _{h} \\nu \\rangle _{\\partial \\Omega } \\le \\big (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h}\\Psi - \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}} \\big (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{h_E}{\\sigma } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\Phi _{h}\\nu _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}}\\lesssim h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha } ,\\\\&\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E}\\langle \\textrm {I}_{h}\\Psi - \\mathbf {g}, \\Phi _{h}- \\Pi _h\\Phi _{h} \\rangle _E\\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h}\\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}- \\Pi _h\\Phi _{h} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\lesssim h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ Lemma REF $(v)$ , () and ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ yield $&B(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Pi _h\\Phi _{h} )-B(\\Psi ,\\Psi ,\\Psi , \\Pi _h\\Phi _{h}) \\lesssim \\epsilon ^{-2} h^{2\\alpha }{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^3.$ A combination of the estimates in (REF )- (REF ) completes the proof of Lemma REF .", "A use of Lemma REF and the methodology of [24] leads to the proof of Theorem REF .", "An outline is sketched for completeness.", "[Proof of energy norm estimate in Theorem REF] For $\\Phi _{h} \\!\\in \\!", "\\mathbf {X}_h$ , let the non-linear map $\\mu _{h}:\\mathbf {X}_h \\rightarrow \\mathbf {X}_h$ be defined by $\\langle DN_h({{\\rm I}_{h}\\Psi }) \\mu _{h}(\\Theta _{h}),\\Phi _{h} \\rangle = 3B(\\textrm {I}_{h}\\Psi , \\textrm {I}_{h}\\Psi ,\\Theta _{h},\\Phi _{h}) - B(\\Theta _{h},\\Theta _{h}, \\Theta _{h},\\Phi _{h}) +L_h(\\Phi _{h}),$ and let $\\mathbb {B}_R(\\textrm {I}_{h}\\Psi ):= \\lbrace \\Phi _h \\!\\in \\!", "\\mathbf {X}_h: {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_h\\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le R\\rbrace .$ Theorem REF helps to establish that $\\mu _{h}$ is well-defined and any fixed point of $\\mu _{h}$ is a solution of the discrete non-linear problem (REF ).", "Moreover, for a sufficiently large choice of the penalization parameter $\\sigma $ , and a sufficiently small choice of discretization parameter $h$ , there exists a positive constant $R(h)$ such that $\\mu _{h}$ maps the closed convex ball $\\mathbb {B}_{R(h)}({\\rm I}_{h}\\Psi )$ to itself; that is, ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h} - {\\rm I}_{h}\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le R(h) \\Rightarrow {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mu _{h}(\\Theta _{h}) - {\\rm I}_{h}\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le R(h) \\text{ for all } \\Theta _{h} \\in \\mathbf {X}_{h} .", "$ The definition of $\\langle DN_h({\\rm I}_{h}\\Psi ) \\cdot ,\\cdot \\rangle $ from (REF ), (REF ), followed by simple algebra and a re-arrangement of terms leads to $& \\langle DN_h({\\rm I}_{h}\\Psi ) ({\\rm I}_{h}\\Psi -\\mu _{h}(\\Theta _{h})) ,\\Phi _{h} \\rangle =(A_{h}(\\textrm {I}_{h}\\Psi , \\Phi _{h} ) + B(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Phi _{h} )+ C(\\textrm {I}_{h}\\Psi , \\Phi _{h} )-L_{h}(\\Phi _{h})) \\nonumber \\\\& \\quad \\quad +(2B(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Phi _{h} ) -3B(\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\Psi , \\Theta _{h},\\Phi _{h}) + B(\\Theta _{h},\\Theta _{h}, \\Theta _{h},\\Phi _{h}) )=:T^{\\prime }_1 +T^{\\prime }_2.$ The term $T^{\\prime }_1$ is estimated using Lemma REF .", "Set $\\mathbf {\\tilde{e}}:=\\Theta _{h} -\\textrm {I}_{h}\\Psi $ .", "The definition of $B(\\cdot ,\\cdot ,\\cdot ,\\cdot )$ , the Cauchy-Schwarz inequality and some straight forward algebraic manipulations lead to $T^{\\prime }_2 \\lesssim 2\\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {\\tilde{e}} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}^2 ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {\\tilde{e}} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}.$ Discrete inf-sup condition in Lemma REF yields that there exists a $\\Phi _{h}\\!\\in \\!", "\\mathbf {X}_h$ with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ such that $\\beta _0{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\rm I}_{h}\\Psi -\\mu _{h}(\\Theta _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le \\langle DN_h({\\rm I}_{h}\\Psi ) ({\\rm I}_{h}\\Psi -\\mu _{h}(\\Theta _{h})) ,\\Phi _{h} \\rangle .$ Since $\\Theta _{h} \\!\\in \\!", "\\mathbb {B}_R(\\textrm {I}_{h}\\Psi ), $ ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\tilde{\\mathbf {e}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}= {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{h} -\\textrm {I}_{h}\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\le R(h)$ .", "For a fixed value of $\\epsilon $ , a use of Lemma REF and (REF ) in (REF ), and (REF ) with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}=1$ leads to ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\rm I}_{h}\\Psi -\\mu _{h}(\\Theta _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le C_2(h^{\\alpha }(1+h^{\\alpha })+ R(h)^2 ( R(h)+1)),$ where $C_2$ is a constant independent of $h.$ For a choice of $R(h):= 2C_2h^{\\alpha }$ and $h< h_2:= \\min (h_0, h_1)$ with $h_1^{\\alpha }< \\frac{1}{1+ 4C_2^2 ( 2C_2 h_0^{\\alpha } +1)} $ , a simple algebraic calculation leads to ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\rm I}_{h}\\Psi -\\mu _{h}(\\Theta _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le 2C_2h^{\\alpha }= R(h).$ Analogous ideas as [24] establishes that $\\text{ for all } \\Theta _1, \\Theta _2 \\!\\in \\!", "\\mathbb {B}_{R(h)}({\\rm I}_{h}\\Psi )$ , ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mu _{h}(\\Theta _1)-\\mu _{h}(\\Theta _2)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\lesssim h^{\\alpha }(h^{\\alpha }+1){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _1-\\Theta _2\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}.$ Thus the map $\\mu _h$ is well-defined, continuous and maps a closed convex subset $\\mathbb {B}_R(\\textrm {I}_{h}\\Psi )$ of a Hilbert space $\\mathbf {X}_{h}$ to itself.", "Therefore, Brouwer's fixed point theorem and contraction result stated above establishes the existence and uniqueness of the fixed point, say $\\Psi _{h}$ in the ball $\\mathbb {B}_R(\\textrm {I}_{h}\\Psi )$ .", "A triangle inequality, ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h}\\Psi - \\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\lesssim h^{\\alpha }$ and Remark REF yield the a priori error estimate in energy norm.", "Remark 3.16 The proof of the energy norm estimate relies on the techniques of medius analysis [15] to deal with the milder regularity of the exact solution.", "This involves a different strategy for the proof using the local efficiency results when compared to [24], where $2(\\Omega )$ regularity is assumed for the exact solution.", "Remark 3.17 For $\\alpha =1$ , that is, $\\Psi \\!\\in \\!", "{2}(\\Omega ),$ it is well-known [3] that ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_2$ is bounded independent of $\\epsilon $ .", "In this case, $\\displaystyle {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\rm I}_{h}\\Psi -\\mu _{h}(\\Theta _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\le C_3(h (1+ \\epsilon ^{-2}h)+\\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {\\tilde{e}} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}^2 ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {\\tilde{e}} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}+1)), $ where the constant $C_3$ is independent of $h$ and $\\epsilon $ .", "For a sufficiently small choice of the discretization parameter chosen as $h =O(\\epsilon ^{2+\\tau }), \\tau > 0$ and $R(h)= 2C_3 h $ , $\\mu _{h}$ maps the ball $\\mathbb {B}_R({\\rm I}_h \\Psi )$ to itself, and it is a contraction map on $\\mathbb {B}_R({\\rm I}_h \\Psi )$ .", "The modification of the proof in above theorem follows analogous to Theorem $5.1$ in [24] and yields $h$ -$\\epsilon $ dependent estimates for this case.", "Next, the $L^2$ norm error estimate is derived using the Aubin-Nitsche [10] duality technique.", "The proof relies on energy norm error bounds that has been established for a fixed $\\epsilon $ .", "However, when $\\Psi \\in 2(\\Omega )$ , the proof can be modified as in Theorem $3.5$ in [24] to obtain $h-\\epsilon $ dependent estimates.", "[Proof of ${\\bf L}^2$ estimate in Theorem REF] Set ${\\varphi }_h=\\textrm {I}_{h}\\Psi - \\Psi _h$ and choose $G= {\\varphi }_h$ , $\\Phi = \\Pi _h {\\varphi }_h $ in the continuous dual linear problem () to deduce $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0^2= ({\\varphi }_h, {\\varphi }_h)= ({\\varphi }_h, {\\varphi }_h- \\Pi _h {\\varphi }_h) + \\langle DN(\\Psi ) \\Pi _h {\\varphi }_h , \\chi \\rangle .$ Let $\\textrm {I}_{h}\\chi \\!\\in \\!", "h \\subset 1_0(\\Omega )$ denotes the interpolant of $\\chi $ .", "A use of $\\textrm {I}_h\\chi =0$ on $\\partial \\Omega $ implies $\\langle DN(\\Psi ) {\\varphi }_h , \\textrm {I}_{h}\\chi \\rangle = \\langle DN_h(\\Psi ) {\\varphi }_h, \\textrm {I}_{h}\\chi \\rangle + \\langle {\\varphi }_h, \\nabla ( \\textrm {I}_{h} \\chi ) \\nu \\rangle _{\\partial \\Omega }.$ Add and subtract $\\langle DN(\\Psi ) ({\\varphi }_h-\\Pi _h {\\varphi }_h ), \\textrm {I}_{h}\\chi \\rangle $ in the right-hand side of (REF ), use the definition of $\\langle DN(\\Psi )\\cdot , \\cdot \\rangle $ and the last displayed identity with $\\Pi _h{\\varphi }_h=0$ on $\\partial \\Omega $ , and re-arrange the terms to obtain $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0^2 &=({\\varphi }_h, {\\varphi }_h- \\Pi _h {\\varphi }_h)+ (-A( \\textrm {I}_{h}\\chi ,{\\varphi }_h-\\Pi _h{\\varphi }_h)+ \\langle \\nabla ( \\textrm {I}_{h} \\chi ) \\nu ,{\\varphi }_h -\\Pi _h{\\varphi }_h \\rangle _{\\partial \\Omega }) +(C(\\textrm {I}_{h}\\chi ,\\Pi _h{\\varphi }_h-{\\varphi }_h )\\\\& \\quad +3B(\\Psi ,\\Psi , \\textrm {I}_{h}\\chi , \\Pi _h{\\varphi }_h-{\\varphi }_h ) )+\\langle DN(\\Psi ) \\Pi _h {\\varphi }_h , \\chi -\\textrm {I}_{h}\\chi \\rangle +\\langle DN_h(\\Psi ) {\\varphi }_h, \\textrm {I}_{h}\\chi \\rangle \\\\& =:\\mathrm {T}_1+\\mathrm {T}_2+ \\mathrm {T}_3+\\mathrm {T}_4+\\mathrm {T}_5.$ A use of Hölder's inequality, (REF ) and the estimate ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}= {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h} \\Psi -\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\lesssim h^\\alpha $ from the proof of Theorem REF leads to $\\displaystyle \\mathrm {T}_1=({\\varphi }_h, {\\varphi }_h- \\Pi _h {\\varphi }_h) \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h- \\Pi _h {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 \\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0.", "$ Apply integration by parts element-wise for the term $A({\\varphi }_h-\\Pi _h{\\varphi }_h,\\textrm {I}_{h}{\\chi } )$ in the expression of ${\\mathrm {T}}_2$ , use $\\Delta ( \\textrm {I}_{h}\\chi )=0$ and recall the definitions of the local term $ \\eta _E=[\\nabla ( \\textrm {I}_{h} \\chi ) \\nu _E]_E $ on $E$ from Lemma REF with $G= {\\varphi }_h$ and $Osc({\\varphi }_h)=0$ .", "This with a Cauchy-Schwarz inequality, Lemma REF , (REF ) and the estimate ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h} \\lesssim h^\\alpha $ leads to $ {\\mathrm {T}}_2= -A( \\textrm {I}_{h}\\chi ,{\\varphi }_h-\\Pi _h{\\varphi }_h)+ \\langle \\nabla ( \\textrm {I}_{h} \\chi ) \\nu ,{\\varphi }_h -\\Pi _h{\\varphi }_h \\rangle _{\\partial \\Omega }= \\sum _{ E \\in \\mathcal {E}_h^{i}} \\langle \\eta _E, \\Pi _h{\\varphi }_h -{\\varphi }_h \\rangle _E\\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0}.$ Lemma REF , (), () and ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\lesssim h^\\alpha $ yield $ {\\mathrm {T}}_3= 3B(\\Psi ,\\Psi , \\textrm {I}_{h}\\chi , \\Pi _h{\\varphi }_h-{\\varphi }_h )+C(\\textrm {I}_{h}\\chi ,\\Pi _h{\\varphi }_h-{\\varphi }_h )\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h}\\chi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Pi _h{\\varphi }_h -{\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0\\lesssim h^{2\\alpha }{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0.$ The boundedness and interpolation estimates in Lemmas REF , REF , () and ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\lesssim h^\\alpha ,$ and (), leads to a bound for the fourth term of (REF ) as ${\\mathrm {T} }_4 &= A( \\Pi _h{\\varphi }_h , \\chi -\\textrm {I}_{h}\\chi )+3B(\\Psi ,\\Psi , \\Pi _h{\\varphi }_h , \\chi -\\textrm {I}_{h}\\chi )+ C( \\Pi _h{\\varphi }_h , \\chi -\\textrm {I}_{h}\\chi )\\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\chi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }\\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0.$ The discrete nonlinear problem (REF ) plus the consistency of the exact solution $\\Psi $ yield $\\displaystyle N_h(\\Psi , \\textrm {I}_{h}\\chi )= L_{h}( \\textrm {I}_{h}\\chi ) = N_h(\\Psi _h, \\textrm {I}_{h}\\chi ).$ Recall that ${\\varphi }_h=\\textrm {I}_{h}\\Psi - \\Psi _h$ and re-write the last term in (REF ) using the above displayed identity, and the definitions of $DN_h$ and $N_h$ as ${\\mathrm {T}}_5 =& \\langle DN_h(\\Psi ) {\\varphi }_h, \\textrm {I}_{h}\\chi \\rangle + N_h(\\Psi _{h}, \\textrm {I}_{h}\\chi )- N_h(\\Psi , \\textrm {I}_{h}\\chi )= A_h(\\textrm {I}_{h}\\Psi - \\Psi , \\textrm {I}_{h}\\chi )+(C(\\textrm {I}_{h}\\Psi - \\Psi , \\textrm {I}_{h}\\chi )\\\\& +3B(\\Psi ,\\Psi ,\\textrm {I}_{h}\\Psi - \\Psi ,\\textrm {I}_{h}\\chi ))+(2 B(\\Psi ,\\Psi ,\\Psi ,\\textrm {I}_{h}\\chi ) -3 B(\\Psi ,\\Psi , \\Psi _h,\\textrm {I}_{h}\\chi )+ B(\\Psi _h,\\Psi _h,\\Psi _h,\\textrm {I}_{h}\\chi )).$ An integration by parts element-wise for the term $A(\\Psi -\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h} \\chi )$ , $\\Delta \\textrm {I}_{h} \\chi =0,$ a Cauchy-Schwarz inequality, Lemma REF with $G= {\\varphi }_h$ , $Osc({\\varphi }_h)=0$ and Lemma REF lead to an estimate for the first term on the right-hand side of ${\\mathrm {T} }_5$ above as $\\sum _{ E \\in \\mathcal {E}_h^{i}} \\langle \\eta _E,\\textrm {I}_{h}\\Psi -\\Psi \\rangle _E\\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 .$ Here, $ \\eta _E$ is the local term as defined in the above estimates.", "Lemma REF , Remark REF and () leads to an estimate for the second term in the expression on the right-hand side for ${\\mathrm {T}}_5$ above as $C(\\textrm {I}_{h}\\Psi - \\Psi , \\textrm {I}_{h}\\chi )+ 3B(\\Psi ,\\Psi , \\Psi -\\textrm {I}_{h}\\Psi ,\\textrm {I}_{h}\\chi )\\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\chi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha } \\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 .$ Proceed as in the estimate for $T_2^{\\prime }$ (also see [24]), and use Remark REF and () to estimate the third term in the expression on the right-hand side for ${\\mathrm {T}}_5$ as $&2 B(\\Psi ,\\Psi ,\\Psi ,\\textrm {I}_{h}\\chi ) -3 B(\\Psi ,\\Psi , \\Psi _h,\\textrm {I}_{h}\\chi )+ B(\\Psi _h,\\Psi _h,\\Psi _h,\\textrm {I}_{h}\\chi ) \\\\&\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}^2({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h}\\chi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}\\lesssim h^{2\\alpha }{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\chi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }\\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0.$ A combination of the estimates in (REF )- (REF ) yields ${\\mathrm {T}}_5\\lesssim h^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0.$ Substitute the estimates derived for ${\\mathrm {T}}_1$ to ${\\mathrm {T}}_5$ in (REF ) and cancel the term ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\varphi }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0$ to obtain $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h} \\Psi - \\Psi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 \\lesssim h^{2\\alpha }.", "$ This estimate, a triangle inequality and Lemma REF yield ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Psi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0\\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\textrm {I}_{h} \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{h} \\Psi - \\Psi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0 \\lesssim h^{2\\alpha }$ and this concludes the proof.", "A posteriori error estimate In this section, we present some auxiliary results followed by the a posteriori error analysis for the Nitsche's method.", "Note that, to derive the a posteriori estimates, it is assumed that $\\mathbf {g}$ (the inhomogeneous Dirichlet boundary condition) belongs to $ {\\frac{1}{2}}(\\partial \\Omega )\\cap \\mathbf {C}^0(\\overline{\\partial \\Omega }).$ The approximation properties of the Scott-Zhang interpolation operator [31] are introduced first.", "Lemma 4.1 (Scott-Zhang interpolation )[31] For $l, m \\in \\mathbb {N}$ with $1 \\le l < \\infty $ , there exists an interpolation operator ${\\rm I}^{SZ}_h: H^l_0(\\Omega ) \\rightarrow V_h:= X_h \\cap H^1_0(\\Omega ) $ that satisfies the stability and aproximation properties given by: (a) for all $0\\le m \\le \\min (1,l)$ , ${\\vert \\hspace{-1.0625pt}\\vert {\\rm I}^{SZ}_h v \\vert \\hspace{-1.0625pt}\\vert }_{m, \\Omega } \\le C_{SZ}{\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{l,\\Omega } \\,\\text{ for all } v \\in H^{l}_0(\\Omega ) ,$ (b) provided $l \\le 2$ , for all $0 \\le m \\le l$ , ${\\vert \\hspace{-1.0625pt}\\vert v- {\\rm I}^{SZ}_h v \\vert \\hspace{-1.0625pt}\\vert }_{m, T} \\le C_{SZ}h_T^{l-m}\\vert v\\vert _{l, \\omega _T} \\text{ for all } v \\in H_0^{l}(\\omega _T) \\text{ and } T \\in \\mathcal {T},$ where the constant $C_{SZ}>0$ is independent of $h$ , and $\\omega _T$ is the set of all triangles in $\\mathcal {T}$ that share at least one vertex with $T$ .", "Lemma 4.2 [19] Let $\\Psi _{\\mathbf {g}} \\in \\mathcal {{X}}$ solve $\\displaystyle \\int _{\\Omega } \\nabla \\Psi _{\\mathbf {g}} \\cdot \\nabla \\Phi \\, {\\rm dx} = \\sum _{ T \\in \\mathcal {T}} \\int _T \\nabla \\Psi _{h} \\cdot \\nabla \\Phi \\, {\\rm dx} \\text{ for all } \\Phi \\in $ where $\\Psi _h$ is the solution of (REF ).", "Then there exists a constant $C>0$ , depending only on the minimum angle of $\\mathcal {T} $ such that $\\displaystyle \\sum _{ T \\in \\mathcal {T}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Psi _{\\mathbf {g}} - \\Psi _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0, T}^2 \\le C (\\vartheta _{hot}^{\\partial })^2, $ where $\\displaystyle (\\vartheta _{hot}^{\\partial })^2:= \\sum _{ E \\in \\mathcal {E}_h^{\\partial } }h_E^{-1} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {g} - \\mathbf {g}_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2+ h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\mathbf {g} - \\mathbf {g}_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2$ , $ \\mathbf {g}_h$ being the standard Lagrange interpolant [10] of $\\mathbf {g}$ from $\\mathbf {P}_2(\\mathcal {E}^{\\partial }_h) \\cap \\mathbf {C}^0(\\overline{\\partial \\Omega } ).$ Remark 4.3 Note that the benchmark liquid crystal example: Example REF in [22] has Lipschitz continuous boundary conditions.", "Hence the a posteriori error analysis of this paper is applicable to this example and the results are illustrated in Section .", "The proof of Theorem REF , stated in Subsection REF , is presented in this section.", "An abstract estimate for the case of non-homogeneous boundary conditions and quartic nonlinearity is derived modifying the methodology in [9], [34] first and this result is crucial to prove Theorem REF .", "Theorem 4.4 (An abstract estimate) Let $\\Psi $ be a regular solution to () and $\\Psi _{\\mathbf {g}} \\in \\mathcal {{X}}$ .", "Then, $DN$ is locally Lipschitz continuous at $\\Psi $ , that is given $R_0>0$ , $DN$ restricted to $B(\\Psi ,R_0)$ is Lipschitz continuous.", "Moreover, (a) $\\displaystyle \\gamma := \\sup _{\\eta \\in B(\\Psi , R_0)} \\dfrac{{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN(\\eta ) - DN(\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}}{{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta - \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}} < \\infty ,$ and (b) there exists a constant $R>0$ such that for all ${\\eta }_{h} \\in B(\\Psi , R)$ , $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*} +(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h,$ where the constant in $\"\\lesssim \"$ depends on $\\gamma $ , continuous inf-sup constant $\\beta $ and Poincaré constant $C_P$ , and the nonlinear (resp.", "linearized) operator $N(\\cdot )$ (resp.", "$DN(\\cdot )$ ) is defined in () (resp.", "()).", "In the first step, it is established that $DN$ is locally Lipschitz continuous at $\\Psi $ and $\\gamma <\\infty $ .", "Let $R_0>0$ be given and $\\eta \\!\\in \\!", "B(\\Psi , R_0).$ For $\\Theta \\!\\in \\!\\mathbf {X}$ and $\\Phi \\!\\in \\!", "$ the definition of $DN(\\cdot )$ , $B(\\cdot , \\cdot , \\cdot , \\cdot )$ , a re-grouping of terms and Lemma REF $(iv)$ leads to $ &\\langle DN(\\eta ) \\Theta , \\Phi \\rangle - \\langle DN(\\Psi ) \\Theta , \\Phi \\rangle = 3B(\\eta ,\\eta , \\Theta , \\Phi ) - 3 B(\\Psi ,\\Psi , \\Theta , \\Phi ) \\\\& = 2\\epsilon ^{-2} \\int _{\\Omega } ((\\eta -\\Psi )\\cdot ( \\eta +\\Psi )(\\Theta \\cdot \\Phi )+ 2 (\\eta -\\Psi ) \\cdot \\Theta (\\eta \\cdot \\Phi ) + 2 (\\Psi \\cdot \\Theta ) (\\eta -\\Psi ) \\cdot \\Phi ) \\,{\\rm dx}\\\\&\\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1}(R_0 + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1.$ The above displayed inequality with definition of ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN(\\eta ) - DN(\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}$ leads to the Lipschitz continuity.", "This and Lemma REF concludes the proof of the first step.", "Step two establishes (REF ).", "The continuous formulation () and a Taylor expansion lead to $0=N(\\Psi ;\\Phi )= N({\\eta }_h; \\Phi ) + \\bigg \\langle \\int _{0}^{1}DN(\\Psi + t( {\\eta }_h-\\Psi ))(\\Psi - {\\eta }_h)\\,{\\rm dt}, \\Phi \\bigg \\rangle .", "$ Introduce $\\pm \\langle DN(\\Psi )(\\Psi - {\\eta }_h), \\Phi \\rangle $ in the above displayed expression and rearrange the terms to obtain $\\langle DN(\\Psi )(\\Psi - {\\eta }_h), \\Phi \\rangle = -N({\\eta }_h; \\Phi ) - \\bigg \\langle \\int _{0}^{1}(DN(\\Psi + t({\\eta }_h-\\Psi )) -DN(\\Psi ) )(\\Psi - {\\eta }_h )\\,{\\rm dt}, \\Phi \\bigg \\rangle .$ Rewrite $\\Psi - {\\eta }_h$ as $ (\\Psi - \\Psi _{\\mathbf {g}})+ (\\Psi _{\\mathbf {g}} - {\\eta }_h )$ in the left-hand side of the above term, use linearity of $\\langle DN(\\Psi )\\cdot , \\cdot \\rangle $ , introduce $\\pm \\langle DN({\\eta }_h)(\\Psi _{\\mathbf {g}} - {\\eta }_h), \\Phi \\rangle $ in the first step; and bound in the second step below to obtain $&\\langle DN(\\Psi )(\\Psi - \\Psi _{\\mathbf {g}}), \\Phi \\rangle = - N({\\eta }_h;\\Phi ) +\\langle (DN({\\eta }_h)-DN(\\Psi ))(\\Psi _{\\mathbf {g}} - {\\eta }_h), \\Phi \\rangle - \\langle DN({\\eta }_h)(\\Psi _{\\mathbf {g}} - {\\eta }_h), \\Phi \\rangle \\\\&\\quad - \\bigg \\langle \\int _{0}^{1}( DN(\\Psi + t( {\\eta }_h-\\Psi )) - DN(\\Psi ) )(\\Psi - {\\eta }_h)\\,{\\rm dt}, \\Phi \\bigg \\rangle \\\\&\\lesssim \\big ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*} + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_h)-DN(\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 \\\\& \\quad + \\int _{0}^{1} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert ( DN(\\Psi + t( {\\eta }_h-\\Psi ))-DN(\\Psi ) )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 \\,{\\rm dt}\\big ){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1.$ Since $\\Psi _{\\mathbf {g}} \\!\\in \\!", "\\mathcal {{X}}$ , $\\Psi - \\Psi _{\\mathbf {g}}\\!\\in \\!", "$ For $\\delta >0 $ small enough, the continuous inf-sup condition () implies that there exists $\\Phi \\!\\in \\!", "with $ 1 = 1$ such that $ (- ) - g 1 DN()(- g), .$A triangle inequality, $ - g=0$ on $$ and the last displayed inequality yield{\\begin{@align*}{1}{-1} (\\beta - \\delta ) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h \\le (\\beta - \\delta )({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Psi _{\\mathbf {g}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h) \\lesssim \\langle DN(\\Psi )(\\Psi - \\Psi _{\\mathbf {g}}), \\Phi \\rangle + (\\beta - \\delta ){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h.\\end{@align*}}Take $ 0$ to obtain $ - h h DN()(- g), + g - h h. $A combination of (\\ref {2.4}), the last displayed inequality and the definition of $$ plus Lemma \\ref {Poincare type inequality} for $ 1=1$ leads to{\\begin{@align*}{1}{-1}C_4 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*} + (1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h +{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_h,\\end{@align*}} where the constant $ C4$ depends on $$, $$ and $ CP.", "$For a choice of $ R := {R0, C4/2}$, use $ - h h < C4/2$ and $ - h 2h <C4/2 - h h $ in the second and third terms, respectively, in the right-hand side of the above inequality to obtain $ C4/2 - h h N(h) * + (1+C4/2+DN(h) L(X, *) )g - h h,$and this leads to the desired conclusion.$ Next, the main result of this section is proved in the following text.", "[Proof of Theorem REF] Theorem REF guarantees the existence of $R>0$ such that (REF ) holds for a choice of $ \\eta _h=\\Psi _h.$ Choose $\\Psi _{\\mathbf {g}}$ as in Lemma REF .", "A posteriori reliability (resp.", "efficiency) estimate provides an upper bound (resp.", "lower bound) on the discretization error, up to a constant.", "To establish the reliability, Theorem REF is utilized and the term $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N(\\Psi _h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*}$ is estimated first.", "Since $ is a Hilbert space, there exists a $ with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1= 1$ such that ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N(\\Psi _h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*}= N(\\Psi _h; \\Phi )= N(\\Psi _h; \\Phi -{\\textrm {I}^{SZ}_h} \\Phi )+N(\\Psi _h; \\textrm {I}^{SZ}_h\\Phi ),$ where $\\textrm {I}^{SZ}_h:h$ is the Scott-Zhang interpolation in Lemma REF .", "The second term in (REF ) can be rewritten using (REF ) with test function $\\textrm {I}^{SZ}_h\\Phi $ (that vanishes on $\\partial \\Omega $ ) as $N(\\Psi _h; \\textrm {I}^{SZ}_h\\Phi ) = \\langle \\Psi _h- \\mathbf {g},\\nabla (\\textrm {I}^{SZ}_h\\Phi ) \\nu \\rangle _{\\partial \\Omega } &\\lesssim \\big (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _h - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}} \\big (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{h_E}{\\sigma } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\textrm {I}^{SZ}_h\\Phi ) \\nu _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}} \\\\& \\lesssim \\big (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _h - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 = (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} (\\vartheta _E^{\\partial })^2 )^{{1}/{2}},$ where for ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1$ , a Cauchy-Schwarz inequality, Lemmas REF ($ii$ ) and REF are utilized in the second and third steps.", "Apply integration by parts element-wise for $A(\\Psi _h, \\Phi -\\textrm {I}^{SZ}_h\\Phi ) $ in the expression of $N(\\Psi _h; \\Phi -\\textrm {I}^{SZ}_h \\Phi )$ , use $[\\Phi -\\textrm {I}^{SZ}_h \\Phi ]_E=0$ on $E \\!\\in \\!", "\\mathcal {E}_h^i$ , $\\Phi -\\textrm {I}^{SZ}_h \\Phi = 0$ on $\\partial \\Omega $ , $\\Delta \\Psi _{h}=0$ and recall the definition of the local terms $ \\eta _T := (2\\epsilon ^{-2}(\\vert \\Psi _{h}\\vert ^2 -1)\\Psi _{h})|_T$ defined on a triangle $T\\!\\in \\!", "\\mathcal {T}$ and $ \\eta _E := [\\nabla \\Psi _h \\nu _E]_E$ on the edge $E$ of $T$ .", "For ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1$ , the above arguments, Cauchy-Schwarz inequality and Lemma REF lead to $ & N(\\Psi _h; \\Phi -\\textrm {I}^{SZ}_h\\Phi ) = A(\\Psi _h, \\Phi -\\textrm {I}^{SZ}_h\\Phi )+ B(\\Psi _h,\\Psi _h,\\Psi _h, \\Phi -\\textrm {I}^{SZ}_h \\Phi )+ C(\\Psi _h, \\Phi -\\textrm {I}^{SZ}_h \\Phi )\\\\&= \\sum _{T \\in \\mathcal {T}} \\int _T \\eta _T \\cdot (\\Phi -\\textrm {I}^{SZ}_h \\Phi ) \\,{\\rm dx}+\\sum _{E \\in \\mathcal {E}_h^i} \\langle \\eta _E, \\Phi -\\textrm {I}^{SZ}_h \\Phi \\rangle _E\\lesssim \\big (\\sum _{T \\in \\mathcal {T}} \\vartheta _T^2+\\sum _{E \\in \\mathcal {E}_h^i} (\\vartheta _E^{i})^2 \\big )^{\\frac{1}{2}} \\\\&\\quad \\times \\big (\\sum _{T \\in \\mathcal {T}} h_T^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi -\\textrm {I}^{SZ}_h \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T}+\\sum _{E \\in \\mathcal {E}_h^i} h_E^{-1} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi -\\textrm {I}^{SZ}_h \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}}\\lesssim \\big (\\sum _{T \\in \\mathcal {T}} \\vartheta _T^2+\\sum _{E \\in \\mathcal {E}_h^i} (\\vartheta _E^{i})^2 \\big )^{\\frac{1}{2}}, $ where $\\vartheta _T^2 = h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert 2\\epsilon ^{-2} (\\vert \\Psi _h\\vert ^2-1)\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T},$ and $(\\vartheta _E^i)^2 = h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\nabla \\Psi _h \\nu _E]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\text{ for all } E \\in \\mathcal {E}_h^{i}$ .", "A use of (REF ), (REF ) in (REF ) leads to the estimate of ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N(\\Psi _h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*}.$ The definition of ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}$ and Lemma REF yield ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - \\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_h = \\sum _{ T \\in \\mathcal {T}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Psi _{\\mathbf {g}} - \\Psi _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0, T}^2 + \\sum _{ E \\in \\mathcal {E}_h^{\\partial } } \\frac{\\sigma }{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{h}- \\mathbf {g} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0, E} \\\\ \\lesssim (\\vartheta _{hot}^{\\partial })^2+\\sum _{ E \\in \\mathcal {E}_h^{\\partial } }(\\vartheta _E^{\\partial })^2,$ where $(\\vartheta _{hot}^{\\partial })^2:= \\sum _{ E \\in \\mathcal {E}_h^{\\partial } }h_E^{-1} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {g} - \\mathbf {g}_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 + h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\mathbf {g} - \\mathbf {g}_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2$ (see Lemma REF ).", "This leads to the bound for the second term in (REF ) by higher order terms (h.o.t.)", "[7] that consist of (i) the errors arising due to the polynomial approximation of the boundary data $\\mathbf {g}$ that depends on the given data smoothness and (ii) the terms ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN(\\Psi _h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}}- \\Psi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}$ .", "To establish the efficiency estimate, set $\\Phi _h = \\Psi _h$ and $ \\eta _T = (2\\epsilon ^{-2}(\\vert \\Psi _{h}\\vert ^2 -1)\\Psi _{h})|_T $ on a triangle $T$ and $ \\eta _E=[\\nabla \\Psi _h \\nu _E]_E$ on a edge $E$ in Lemma REF .", "A use of the local efficiency estimates in Lemma REF (i) and $\\displaystyle \\sum _{ E \\in \\mathcal {E}^{\\partial }_h} ( \\vartheta _E^{\\partial })^2=\\sum _{ E \\in \\mathcal {E}^{\\partial }_h} \\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{h} - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Psi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}^2$ establishes the lower bound in Theorem REF .", "Remark 4.5 For $X_h= \\lbrace v_h \\in {C}^0(\\overline{\\Omega }), v_h|_T \\in P_p(T), \\text{ for all } T \\in \\mathcal {T}\\rbrace $ , that is, if we use higher order polynomials for the approximation, then $\\vartheta _T^2 = h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert -\\Delta \\Psi _h+2\\epsilon ^{-2} (\\vert \\Psi _h\\vert ^2-1)\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T}$ in (REF ).", "Extension to discontinuous Galerkin FEM In this section, we extend the results in Section REF to dGFEM.", "The discrete space for dGFEM consists of piecewise linear polynomials defined by $\\displaystyle X_{{\\rm dG}}:=\\lbrace v \\in L^2(\\Omega ): v|_T \\in P_1(T) \\text{ for all } T \\in \\mathcal {T}\\rbrace ,$ and the mesh dependent norm $ \\displaystyle {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{{\\rm dG}}:=\\sum _{ T \\in \\mathcal {T}} \\int _T \\vert \\nabla v\\vert ^2 \\,{\\rm dx}+ \\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E} \\int _{E} [v]_E^2 \\,{\\rm ds},$ where $\\sigma _{\\rm dG}> 0$ is the penalty parameter.", "Let $\\mathbf {X}_{{\\rm dG}}\\!", ": =X_{{\\rm dG}} \\times X_{{\\rm dG}} $ be equipped with the product norm defined by ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}^2= {\\vert \\hspace{-1.0625pt}\\vert \\varphi _1\\vert \\hspace{-1.0625pt}\\vert }^2_{\\rm dG}+ {\\vert \\hspace{-1.0625pt}\\vert \\varphi _2\\vert \\hspace{-1.0625pt}\\vert }^2_{\\rm dG}$ for all $\\Phi _{\\rm dG}=(\\varphi _1,\\varphi _2) \\in \\mathbf {X}_{{\\rm dG}}$ .", "The dGFEM formulation corresponding to () seeks $ \\Psi _{{\\rm dG}}\\!\\in \\!\\mathbf {X}_{{\\rm dG}}$ such that for all $ \\Phi _{{\\rm dG}} \\!\\in \\!", "\\mathbf {X}_{{\\rm dG}},$ $N_{{\\rm dG}}(\\Psi _{\\rm dG};\\Phi _{\\rm dG}):=A_{{\\rm dG}}(\\Psi _{{\\rm dG}},\\Phi _{{\\rm dG}})+B(\\Psi _{{\\rm dG}},\\Psi _{{\\rm dG}},\\Psi _{{\\rm dG}},\\Phi _{{\\rm dG}})+C(\\Psi _{{\\rm dG}},\\Phi _{{\\rm dG}})-L_{\\rm dG}(\\Phi _{{\\rm dG}})=0,$ where for $ \\Theta =(\\theta _1,\\theta _2) ,\\, \\Phi =(\\varphi _1,\\varphi _2)\\in 1(\\mathcal {T}) $ , $A_{{\\rm dG}}(\\Theta ,\\Phi ) :=a_{{\\rm dG}}(\\theta _1,\\varphi _1)+a_{{\\rm dG}}(\\theta _2,\\varphi _2), $ $L_{\\rm dG}(\\Phi _{{\\rm dG}})=l^1_{\\rm dG}(\\varphi _1)+l^2_{\\rm dG}(\\varphi _2)$ , and for $\\theta , \\varphi \\in H^1(\\mathcal {T})$ , and for $-1 \\le \\lambda \\le 1,$ $&a_{{\\rm dG}}(\\theta ,\\varphi ):=\\sum _{ T \\in \\mathcal {T}} \\int _T \\nabla \\theta \\cdot \\nabla \\varphi \\,{\\rm dx}- \\sum _{E \\in \\mathcal {E}} \\langle \\lbrace \\frac{\\partial \\theta }{\\partial \\nu _E}\\rbrace _E, [\\varphi ]_E \\rangle _E-\\lambda \\sum _{E \\in \\mathcal {E}} \\langle \\lbrace \\frac{\\partial \\varphi }{\\partial \\nu _E}\\rbrace _E, [\\theta ]_E \\rangle _E+ \\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E} \\langle [\\theta ]_E, [\\varphi ]_E \\rangle _E\\\\&\\text{ and }\\,\\,l^i_{\\rm dG}(\\varphi ):=- \\sum _{E \\in \\mathcal {E}_h^{\\partial }}\\langle g_i, \\frac{\\partial \\varphi }{\\partial \\nu _E} \\rangle _{E} + \\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma _{\\rm dG}}{h_E} \\langle g_i, \\varphi \\rangle _E \\text{ for } 1\\le i \\le 2.$ The operators $B(\\cdot ,\\cdot ,\\cdot ,\\cdot )$ and $ C(\\cdot ,\\cdot )$ are as defined in Section REF .", "The proofs of results in this section follow on similar lines to the results established in Sections and for the Nitsche's method.", "Hence the main resuts and the auxiliary results needed to establish them are stated and parts of proofs where ideas differ are highlighted.", "Lemma 5.1 (Boundedness and coercivity of $A_{\\rm dG}$ )[29] For the choice of a sufficiently large parameter $\\sigma _{\\rm dG}$ , there exists a positive constant $\\alpha _2 > 0$ such that for $\\Theta _{{\\rm dG}}, \\Phi _{{\\rm dG}} \\!\\in \\!", "\\mathbf {X}_{\\rm dG}$ , $A_{{\\rm dG}}(\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}})\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}, \\text{ and } A_{{\\rm dG}}(\\Phi _{{\\rm dG}}, \\Phi _{{\\rm dG}}) \\ge \\alpha _2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}^2, $ where the hidden constant in $\"\\lesssim \"$ is independent of $h.$ Lemma 5.2 (Interpolation estimate)[30] For ${\\rm v} \\in H^s(\\Omega ) \\text{ with } s \\ge 1$ , there exists ${\\rm {I}_{\\rm dG}v} \\in X_{\\rm dG}$ such that for any $T \\in \\mathcal {T},$ $\\displaystyle {\\vert \\hspace{-1.0625pt}\\vert \\rm v-\\rm {I}_{\\rm dG}v \\vert \\hspace{-1.0625pt}\\vert }_{H^l(T)} \\le C_I h_T ^{s-l} {\\vert \\hspace{-1.0625pt}\\vert \\rm v\\vert \\hspace{-1.0625pt}\\vert }_{H^s(T)}$ for $l=0,1$ where $C_I$ denotes a generic interpolation constant independent of $h$ .. Lemma 5.3 (Enrichment operator).", "[6], [18] There exists an enrichment operator ${\\rm E}_h : X_{\\rm dG} \\rightarrow V_h\\subset H_0^1(\\Omega )$ , where $V_h $ is the Lagrange $P_1$ conforming finite element space associated with the triangulation $\\mathcal {T}$ that satisfies the following properties.", "For any $\\varphi _{\\rm dG} \\in X_{\\rm dG}$ , $\\displaystyle (a) \\sum _{ T \\in \\mathcal {T}} h_T^{-2}{\\vert \\hspace{-1.0625pt}\\vert {\\rm E}_h\\varphi _{\\rm dG} -\\varphi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2 + {\\vert \\hspace{-1.0625pt}\\vert {\\rm E}_h\\varphi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert }_{1}^2 \\le C_{en_1} {\\vert \\hspace{-1.0625pt}\\vert \\varphi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG}^2,$ and $\\displaystyle (b)\\, {\\vert \\hspace{-1.0625pt}\\vert {\\rm E}_h\\varphi _{\\rm dG} - \\varphi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG}^2 \\le C_{en_2}(\\sum _{ E \\in \\mathcal {E}}\\int _{ E}\\frac{1}{h_E}[\\varphi _{\\rm dG}]_E^2 {\\rm ds} ) ,$ where $C_{en_1}$ and $C_{en_2}$ are positive constants independent of $h$ .", "Remark 5.4 (Modified local efficiency results) Similar local efficiency results in Lemmas REF -REF hold for dGFEM with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}$ is replaced by ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}$ and the interpolation operator ${\\rm I}_h$ replaced by ${\\rm I}_{\\rm dG}.$ The discrete inf-sup condition corresponding to the perturbed bilinear form $\\langle DN_{{\\rm dG}}(\\textrm {I}_{{\\rm dG}}\\Psi )\\Theta _{{\\rm dG}}, \\Phi _{{\\rm dG}} \\rangle := A_{{\\rm dG}}(\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}})+3B(\\textrm {I}_{{\\rm dG}}\\Psi ,\\textrm {I}_{{\\rm dG}}\\Psi ,\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}})+C(\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}})$ is stated first.", "This is crucial in establishing the error estimates.", "Lemma 5.5 (Stability of perturbed bilinear form).", "Let $\\Psi \\!", "\\in \\!", "\\mathcal {{X}} \\cap \\mathbf {H}^{1+\\alpha }(\\Omega ), \\, 0<\\alpha \\le 1,$ be a regular solution of () and ${{\\rm I}_{\\rm dG}}\\Psi $ be its interpolant.", "For a sufficiently large $\\sigma _{\\rm dG}$ and a sufficiently small discretization parameter $h$ , there exists a constant $\\beta _1 $ such that $\\displaystyle 0< \\beta _1 \\le \\inf _{\\begin{array}{c}\\Theta _{{\\rm dG}} \\in \\mathbf {X}_{{\\rm dG}} \\\\ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}=1\\end{array}} \\sup _{\\begin{array}{c}\\Phi _{{\\rm dG}} \\in \\mathbf {X}_{{\\rm dG}} \\\\ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}=1\\end{array}}\\langle DN_{{\\rm dG}}({{\\rm I}_{{\\rm dG}}\\Psi })\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}} \\rangle .$ The proof follows along similar lines as the proofs of Lemma REF and Theorem REF , except for the additional terms $ \\displaystyle \\sum _{E \\in \\mathcal {E}^i_h}\\langle \\lbrace \\nabla (\\textrm {I}_{{\\rm dG}}\\xi ) \\nu _E\\rbrace _E, [\\Phi _{{\\rm dG}}]_E \\rangle _{E} , \\; \\displaystyle \\sum _{E \\in \\mathcal {E}}\\langle \\lbrace \\nabla \\Phi _{{\\rm dG}} \\nu _E\\rbrace _E, [\\textrm {I}_{{\\rm dG}}\\xi ]_E \\rangle _{E} , \\; \\sum _{E \\in \\mathcal {E}}\\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\xi ]_E , \\;[\\Phi _{{\\rm dG}}]_E \\rangle _{E} $ that appear in $A_{{\\rm dG}}(\\textrm {I}_{{\\rm dG}}\\xi ,\\Phi _{{\\rm dG}})-A(\\xi ,\\textrm {E}_h\\Phi _{{\\rm dG}})$ (see (REF )).", "Since $[{\\rm E}_h\\Phi _{{\\rm dG}}]_{E}=0$ and $[\\xi ]_E=0$ for all $E \\in \\mathcal {E}_h^i $ , the above displayed terms are equal to $\\displaystyle \\sum _{E \\in \\mathcal {E}^i_h}\\langle \\lbrace \\nabla (\\textrm {I}_{{\\rm dG}}\\xi ) \\nu _E\\rbrace _E, [\\Phi _{{\\rm dG}} -\\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _{E}, \\sum _{E \\in \\mathcal {E}}\\langle \\lbrace \\nabla \\Phi _{{\\rm dG}} \\nu _E\\rbrace _E, [\\textrm {I}_{{\\rm dG}}\\xi -\\xi ]_E \\rangle _{E} , \\sum _{E \\in \\mathcal {E}}\\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\xi -\\xi ]_E , [\\Phi _{{\\rm dG}} -\\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _{E},$ respectively.", "A Cauchy-Schwarz inequality and Lemmas REF , REF yield estimate for the above terms.", "The rest of the details are skipped for brevity.", "Lemma 5.6 Let $\\Psi $ be a regular solution of () and ${\\rm I}_{\\rm dG}\\Psi \\in \\mathbf {X}_{{\\rm dG}}$ be its interpolant.", "Then, for any $\\Phi _{\\rm dG}\\in \\mathbf {X}_{\\rm dG}$ with $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}=1,$ it holds that $&A_{{\\rm dG}}({\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} ) + B({\\rm {I}}_{{\\rm dG}}\\Psi ,{\\rm {I}}_{{\\rm dG}}\\Psi ,{\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} )+ C({\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} ) -L_{{\\rm dG}}(\\Phi _{{\\rm dG}})\\\\& \\qquad \\lesssim h^{\\alpha } (1+ \\epsilon ^{-2}h^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ The proof of Lemma 3.15 is modified and the steps that are different are detailed.", "The definitions of $A_{\\rm dG}(\\cdot , \\cdot ) $ (with an integration by parts) and $L_{\\rm dG}( \\cdot ) $ will lead to the inter-element jump and average terms in the identities corresponding to (REF ) and (REF ).", "Utilize $[\\Psi ]_E=0$ for all $E \\in \\mathcal {E}^i$ to rewrite these identities as follows.", "$A_{{\\rm dG}}(\\textrm {I}_{{\\rm dG}}&\\Psi , \\Phi _{{\\rm dG}}- \\textrm {E}_{h}\\Phi _{{\\rm dG}} ) -L_{{\\rm dG}}(\\Phi _{{\\rm dG}}- {\\rm E}_h\\Phi _{{\\rm dG}})= \\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle [\\nabla (\\textrm {I}_{{\\rm dG}}\\Psi )\\nu _E ]_E,\\lbrace \\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}\\rbrace _E \\rangle _{E} \\\\&+ \\lambda \\sum _{ E \\in \\mathcal {E}}\\langle [ \\Psi -\\textrm {I}_{{\\rm dG}}\\Psi ]_E, \\lbrace \\nabla (\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}})\\nu _E \\rbrace _E \\rangle _{E} + \\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\Psi - \\Psi ]_E, [\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _E,\\\\& \\hspace{-31.2982pt} A_{{\\rm dG}}(\\textrm {I}_{{\\rm dG}}\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} )-L_{{\\rm dG}}( \\textrm {E}_h\\Phi _{{\\rm dG}})= A(\\textrm {I}_{{\\rm dG}}\\Psi -\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} )+\\lambda \\sum _{ E \\in \\mathcal {E}}\\langle [{\\Psi }-\\textrm {I}_{{\\rm dG}}\\Psi ]_E, \\lbrace \\nabla (\\textrm {E}_h\\Phi _{{\\rm dG}})\\nu _E\\rbrace _E \\rangle _{E} \\nonumber \\\\& \\quad \\quad \\quad \\hspace{85.35826pt} - (B(\\Psi ,\\Psi ,\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}}) + C(\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} )).$ The inclusion of jump and average terms in the above displayed identities will modify (REF ) as $& A_{{\\rm dG}}({\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} ) + B({\\rm {I}}_{{\\rm dG}}\\Psi ,{\\rm {I}}_{{\\rm dG}}\\Psi ,{\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} )+ C({\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} ) -L_{{\\rm dG}}(\\Phi _{{\\rm dG}}) \\nonumber =\\sum _{T \\in \\mathcal {T}}\\int _T \\eta _T \\cdot (\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}})\\,{\\rm dx}\\\\& \\quad + \\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle \\eta _E ,\\lbrace \\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}\\rbrace _E \\rangle _{E} + (A(\\textrm {I}_{{\\rm dG}}\\Psi -\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} ) + C({\\rm I}_{{\\rm dG}}\\Psi -\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}})) \\nonumber \\\\& \\quad + (B(\\textrm {I}_{{\\rm dG}}\\Psi ,\\textrm {I}_{{\\rm dG}}\\Psi ,\\textrm {I}_{{\\rm dG}}\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} )-B(\\Psi ,\\Psi ,\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}})) +\\lambda \\sum _{ E \\in \\mathcal {E}}\\langle [{\\Psi }-\\textrm {I}_{{\\rm dG}}\\Psi ]_E, \\lbrace \\nabla \\Phi _{{\\rm dG}} \\nu _E \\rbrace _E \\rangle _{E} \\\\&\\quad +\\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\Psi - \\Psi ]_E, [\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _E :=T_1 + \\cdots +T_6, $ where $ \\eta _T:=( 2\\epsilon ^{-2}(\\vert \\textrm {I}_{{\\rm dG}}\\Psi \\vert ^2 -1)\\textrm {I}_{{\\rm dG}}\\Psi )|_T \\text{ on } T \\text{ and } \\eta _E: = [\\nabla (\\textrm {I}_{{\\rm dG}}\\Psi )\\nu _E]_E \\text{ on } E.$ The terms $T_1$ to $ T_4 $ are estimated in similar lines to the corresponding terms in Lemma REF .", "Apply Cauchy-Schwarz inequality, Lemma REF , Lemmas REF , REF and $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}=1$ to $T_5$ and $T_6$ .", "$&T_5:=\\sum _{E \\in \\mathcal {E}}\\langle [\\Psi -\\textrm {I}_{{\\rm dG}}\\Psi ]_E, \\lbrace \\nabla \\Phi _{{\\rm dG}} \\nu _E \\rbrace _E \\rangle _{E} \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{{\\rm dG}}\\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}\\lesssim h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha },\\\\&T_6:=\\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\Psi - \\Psi ]_E, [\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _E \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{{\\rm dG}}\\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}\\lesssim h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ A combination of the estimates lead to the desired result.", "The next abstract estimate is analogous to Theorem REF in Section and is useful to establish a reliable and efficient a posteriori error estimate for dGFEM.", "Lemma 5.7 Let $\\Psi $ be a regular solution to () and $\\Psi _{\\mathbf {g}} \\!", "\\in \\!", "\\mathcal {{X}}$ .", "Then, $DN$ is locally Lipschitz continuous at $\\Psi $ , that is given $R_0>0$ , $DN$ restricted to $B(\\Psi ,R_0)$ is Lipschitz continuous.", "Moreover, (a) $\\displaystyle \\gamma := \\sup _{\\eta \\in B(\\Psi , R_0)} \\dfrac{{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN(\\eta ) - DN(\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}}{{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta - \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG}} < \\infty ,$ and (b) there exists a constant $R>0$ such that for all ${\\eta }_{\\rm dG} \\in B(\\Psi , R)$ , $\\displaystyle {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N({\\eta }_{\\rm dG})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*} +(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_{\\rm dG})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG},$ where the constant in $\"\\lesssim \"$ depends on $\\gamma $ , continuous inf-sup constant $\\beta $ and Poincaré constant, and the nonlinear (resp.", "linearized) operator $N(\\cdot )$ (resp.", "$DN(\\cdot )$ ) is defined in () (resp.", "()).", "For each element $T$ and edge $E$ , the volume and edge contributions to the estimators for dGFEM are $&\\vartheta _T^2: = h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert -\\Delta \\Psi _{\\rm dG} + 2\\epsilon ^{-2} (\\vert \\Psi _{\\rm dG}\\vert ^2-1)\\Psi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T}, \\,\\,\\, (\\vartheta _E^{\\partial })^2:= \\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\rm dG} - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\text{ for all } E \\in \\mathcal {E}_h^{\\partial },\\\\& \\text{and }(\\vartheta _E^i)^2 := h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\nabla \\Psi _{\\rm dG} \\nu _E]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 +\\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\Psi _{\\rm dG}]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\text{ for all } E \\in \\mathcal {E}_h^{i} .$ Define the estimator $ \\displaystyle \\vartheta ^2:= \\sum _{ T \\in \\mathcal {T}} \\vartheta _T^2 +\\sum _{ E \\in \\mathcal {E}_h^{i}} (\\vartheta _E^i)^2 + \\sum _{ E \\in \\mathcal {E}_h^{\\partial } }(\\vartheta _E^{\\partial })^2.$ The main result of this section is presented now.", "Theorem 5.8 (A priori and a posteriori error estimates) Let $\\Psi $ be a regular solution of ().", "For a sufficiently large penalty parameter $\\sigma _{\\rm dG}>0$ and a sufficiently small discretization parameter $h$ , there exists a unique solution $\\Psi _{\\rm dG}$ to the discrete problem (REF ) that approximates $\\Psi $ such that (Energy norm estimate) $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Psi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG} \\lesssim h^{\\alpha }$ , where $\\Psi \\!", "\\in \\!", "\\mathcal {{X}}\\cap \\mathbf {H}^{1+\\alpha }(\\Omega ) ,$ $0 < \\alpha \\le 1$ is the index of elliptic regularity, (A posteriori estimates) There exist $h$ -independent positive constants ${\\rm C}_{\\text{rel}}$ and ${\\rm C}_{\\text{eff}}$ such that ${\\rm C}_{\\text{eff}} \\vartheta \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Psi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG} \\le {\\rm C}_{\\text{rel}} \\big ( \\vartheta + h.o.t \\big ),$ where $h.o.t$ expresses terms of higher order.", "The basic ideas of proofs of both a priori and a posteriori error estimates follow from Theorems REF and REF .", "The modifications for the case of dGFEM are sketched for the sake of clarity.", "1.", "(Energy norm estimate): The energy norm error estimate in a priori error analysis is proved using Brouwer's fixed point theorem.", "The non-linear map [24] $\\mu _{\\rm dG}: \\mathbf {X}_{{\\rm dG}} \\rightarrow \\mathbf {X}_{{\\rm dG}}$ is defined in this case as $\\langle DN_{\\rm dG}({{\\rm I}_{\\rm dG}\\Psi }) \\mu _{{\\rm dG}}(\\Theta _{\\rm dG}),\\Phi _{\\rm dG} \\rangle = 3B(\\textrm {I}_{\\rm dG}\\Psi , \\textrm {I}_{\\rm dG}\\Psi ,\\Theta _{\\rm dG},\\Phi _{\\rm dG}) - B(\\Theta _{\\rm dG},\\Theta _{\\rm dG}, \\Theta _{\\rm dG},\\Phi _{\\rm dG}) +L_h(\\Phi _{\\rm dG}).$ The proof now follows analogous to Theorem REF , using Lemmas REF and REF .", "2.", "(A posteriori estimate): Lemma REF and techniques used in proof of Theorem REF lead to a posteriori estimates.", "The jump and average terms of $A_{\\rm dG}(\\cdot ,\\cdot )$ in the expansion of $N_{\\rm dG}(\\Psi _{\\rm dG}; \\textrm {I}^{SZ}_h\\Phi )$ in (REF ) will modify (REF ) to $N(\\Psi _{\\rm dG}; \\textrm {I}^{SZ}_h\\Phi )=\\lambda \\sum _{ E \\in \\mathcal {E}_h^{i}} \\langle [\\Psi _{\\rm dG}]_E,\\lbrace \\nabla (\\textrm {I}^{SZ}_h\\Phi ) \\nu _E\\rbrace _E \\rangle _E+ \\lambda \\sum _{ E \\in \\mathcal {E}_h^{\\partial } } \\langle \\Psi _{\\rm dG}- \\mathbf {g},\\nabla (\\textrm {I}^{SZ}_h\\Phi ) \\nu _E \\rangle _{E}.$ The Cauchy-Schwarz inequality, Lemmas REF ($ii$ ) and REF plus ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1$ yield $N(\\Psi _{\\rm dG}; \\textrm {I}^{SZ}_h\\Phi ) \\lesssim \\big (\\sum _{E \\in \\mathcal {E}_h^{i}} \\frac{\\sigma _{\\rm dG}}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\Psi _{\\rm dG}]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2+\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma _{\\rm dG}}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\rm dG}- \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0^2 \\big )^{\\frac{1}{2}} $ and leads to interior edge estimator term $ \\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\Psi _{\\rm dG}]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2$ .", "Moreover, a use of $[\\Psi ]_E=0$ for all $E \\in \\mathcal {E}_h^{i}$ shows $ \\sum _{ E \\in \\mathcal {E}_h^{i}}\\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\Psi _{\\rm dG}]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\rm dG}- \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}$ and establishes the efficiency bound.", "The remaining part of the proof uses ideas similar to the proof of Theorem REF .", "Numerical results In this section, we present some numerical experiments that confirm the theoretical results obtained in Sections -, and illustrate the practical performances of the error indicators in adaptive mesh refinement for both Nitsche's method and dGFEM.", "Preliminaries The uniform refinement process divides each triangle in the triangulation of $\\bar{\\Omega }$ into four similar triangles for subsequent mesh refinements using red refinement.", "Let $e(n)$ and $h(n)$ (resp.", "$e({n-1})$ and $h({n-1})$ ) denote the error and the discretization parameter at the $n$ -th (resp.", "$n-1$ -th) level of uniform refinements, respectively.", "The convergence rate at $n$ -th level is defined by $\\displaystyle \\alpha _n:=log(e(n)/e({n-1}))/log(h(n)/h({n-1}))$ .", "The penalty parameters $\\sigma =\\sigma _{\\rm dG}=10$ is chosen for the numerical experiments.", "Numerical experiments are performed for different values of $\\epsilon $ to illustrate the efficacy of the methods.", "Newton's method is employed to compute the approximated solutions of the discrete nonlinear problem (REF ).", "The Newton's iterates for Nitsche's method (see [24] for Newton's iterates in dGFEM) are given by $\\Psi ^n_{h}, \\,\\, n= 1,2, \\ldots $ $A_h(\\Psi ^n_{h}, \\Phi _h) + 3B(\\Psi ^{n-1}_{h},\\Psi ^{n-1}_{h},\\Psi ^n_{h},\\Phi _{h}) +C(\\Psi ^n_{h}, \\Phi _h)= 2 B(\\Psi ^{n-1}_{h},\\Psi ^{n-1}_{h},\\Psi ^{n-1}_{h},\\Phi _{h}) +L_h(\\Phi _{h}).$ The tolerance in the Newton's method is chosen as $10^{-8} $ in the numerical experiments unless mentioned otherwise.", "Remark 6.1 It can be established that the Newton iterates in (REF ) converge quadratically to the discrete solution [24].", "Figure: triangulation2Initial triangulation 𝒯 0 \\mathcal {T}_0 of L-shape domain in Example and triangulation3 its uniform refinement.", "Slitdomain Slit domain in Example Figure: Convergence history (a priori estimates) for Nitsche's method and dGFEM for AprioriresultLslapedomain Example , with ϵ=0.4\\epsilon =0.4, and AprioriresultSlitdomain Example , with ϵ=0.6\\epsilon =0.6.", "Example on a $L$ -shaped domain Consider (REF ) on a non-convex L-shape domain $\\Omega = (-1, 1 )\\times (-1,1) \\setminus [0,1] \\times [-1,0]$ .", "For the manufactured solution $ u= r^{{2}/{3}}\\sin ({2 \\theta }/{3}),$ $v= r^{1/2}\\sin (\\theta /2)$ , where $(r, \\theta )$ denote the system of polar coordinates, compute the corresponding right-hand side $\\mathbf {f}$ and the non-homogeneous Dirichlet boundary condition $\\mathbf {g}$ .", "In this case, the exact solutions $\\Psi \\in \\mathbf {H}^{1+1/2-\\kappa }(\\kappa \\!> \\!", "0)$ [14], and the theoretically expected rate of error reduction is $O(h^{1/2})$ and $O(h)$ in the energy norm and $\\mathbf {L}^2$ norm (see Theorems REF and REF ).", "Figure REF and REF display the initial triangulation and its uniform refinement.", "The initial guess for the Nitsche's method (resp.", "dGFEM) is chosen as $\\Psi _h^0 \\in \\mathbf {X}_{h}$ (resp.", "$\\Psi _{\\rm dG}^0 \\in \\mathbf {X}_{{\\rm dG}}$ ), where $\\Psi _h^0$ solves $A_h(\\Psi _h^0, \\Phi _h)= L_h(\\Phi _h) \\text{ for all } \\Phi _h \\in \\mathbf {X}_{h}$ (resp.", "$A_{\\rm dG}(\\Psi _{\\rm dG}^0, \\Phi _{\\rm dG})= L_{\\rm dG}(\\Phi _{\\rm dG}) \\text{ for all } \\Phi _{\\rm dG}\\in \\mathbf {X}_{{\\rm dG}}$ ), and the linear form $L_h(\\cdot )$ (resp.", "$L_{\\rm dG}(\\cdot )$ ) is modified to incorporate the information on $\\mathbf {f}$ .", "The approximations to the discrete solution to (REF ) are obtained using the Newton's method defined in (REF ).", "Figure REF presents the convergence rates in energy norm and $\\mathbf {L}^2$ norm, with $\\epsilon =0.4$ , for both Nitsche's method and dGFEM.", "Benchmark example on unit square domain Consider () on a convex domain $\\Omega =(0,1)\\times (0,1)$ with the Dirichlet boundary condition [22] given by $\\displaystyle \\mathbf {g}={\\left\\lbrace \\begin{array}{ll}(\\textit {T}_{d}(x),0) & \\text{on}\\,\\,\\,\\, y=0 \\,\\,\\,\\,\\text{and} \\,\\,\\,\\, y=1 \\\\(- \\textit {T}_d(y),0) & \\text{on}\\,\\,\\,\\, x=0 \\,\\,\\,\\, \\text{and} \\,\\,\\,\\, x=1\\end{array}\\right.", "}$ , where the parameter $d=3 \\epsilon $ with $\\epsilon = 0.02$ and the trapezoidal shape function $\\textit {T}_d:[0,1]\\rightarrow {\\mathbb {R}}$ is defined by $\\displaystyle \\textit {T}_d(t)={\\left\\lbrace \\begin{array}{ll}t/d, & 0 \\le t \\le d \\\\1, & d \\le t \\le 1- d \\\\(1-t)/d, & 1- d \\le t \\le 1\\end{array}\\right.", "}$ .", "See [22], [24] for details of construction of a suitable initial guess of Newton’s iterates in this example.", "Table: Numerical energy, errors and convergence rates for D1 and R1 solutions, respectively in energy and 𝐋 2 \\mathbf {L}^2 norms for ϵ=0.02\\epsilon =0.02.Table REF presents the computed energy, error in energy and $\\mathbf {L}^2$ norms for numerical approximation of the diagonal D1 and rotated R1 solutions, respectively obtained using the Nitsche's method in (REF ).", "The orders of convergence agrees with the theoretical orders of convergence obtained in [22], [24], [33].", "For the corresponding results for dGFEM, see [24].", "Example on a slit domain Let $\\Omega $ be the slit domain $\\lbrace (x,y) \\in \\mathbb {R}^2: \\vert x\\vert +\\vert y\\vert <1 \\rbrace \\setminus ([0,1] \\times \\lbrace 0\\rbrace )$ (see Figure REF ).", "Select the non-homogeneous Dirichlet boundary data $\\mathbf {g}$ and the right-hand side $\\mathbf {f}$ so that the manufactured solution is given by $u(r, \\theta )=v(r, \\theta )= r^{1/2}\\sin (\\theta /2)- (1/2)(r\\sin (\\theta ))^2$ .", "Figure REF presents the convergence history in energy norm, $\\mathbf {L}^2$ norm, with the parameter value $\\epsilon =0.6$ , for Nitsche's method and dGFEM.", "The rates are approximately $0.5004$ (resp.", "$0.9846$ ) in energy norm ($\\mathbf {L}^2$ norm).", "Adaptive mesh-refinement For the adaptive refinement, the order of convergence of error and estimators are related to total number of unknowns ($\\text{Ndof}(l)$ ).", "Let $e(l)$ and $\\text{Ndof}(l)$ be the error and total number of unknowns at the $l-$ th level refinement, respectively.", "The convergence rates are calculated as $\\text{Order}_e(l):= \\frac{\\log (e(l-1)/e(l))}{\\log (\\text{Ndof}(l)/\\text{Ndof}(l-1))} \\qquad \\text{and} \\qquad \\text{Order}_{\\vartheta }(l):= \\frac{\\log (\\vartheta (l-1)/\\vartheta (l))}{\\log (\\text{Ndof}(l)/\\text{Ndof}(l-1))}.$ Given an initial triangulation $\\mathcal {T}_0,$ run the steps SOLVE, ESTIMATE, MARK and REFINE successively for different levels $l=0,1,2,\\ldots $ SOLVE Compute the solution $\\Psi _l:=\\Psi _h$ (resp.", "$\\Psi _l:=\\Psi _{\\rm dG}$ ) of the discrete problem (REF ) (resp.", "REF ) for the triangulation $\\mathcal {T}_l$ .", "ESTIMATE Calculate the error indicator $\\displaystyle _{T,l}:=\\big (\\vartheta _T^2 + \\sum _{ E \\in \\partial T \\cap \\mathcal {E}_h^{i}} (\\vartheta ^i_E)^2 + \\sum _{E \\in \\partial T \\cap \\mathcal {E}_h^{\\partial }}(\\vartheta _E^{\\partial })^2 \\big )^\\frac{1}{2} $ for each element $T \\in \\mathcal {T}_l.$ Recall the volume and edge estimators for Nitsche's method (resp.", "dGFEM) given by (REF )-() (resp.", "(REF )-()).", "MARK For next refinement, choose the elements $T \\in \\mathcal {T}_l$ using Dörfler marking such that $ 0.3\\sum _{ T \\in {\\mathcal {T}_l}}_{T, l}^2 \\\\ \\le \\sum _{ T \\in \\tilde{\\mathcal {T}}}_{T, l}^2$ and collect those elements to construct a subset $\\tilde{\\mathcal {T}} \\subset \\mathcal {T}_l$ .", "REFINE Compute the closure of $\\tilde{\\mathcal {T}}$ and use newest vertex bisection [32] refinement strategy to construct the new triangulation $\\mathcal {T}_{l+1}$ .", "Consider (REF ) in L-shaped domain (Figure REF ) with the manufactured solution presented in Example REF and apply the adaptive refinement algorithm.", "The estimator is modified as $\\vartheta _T^2:= h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {f}- 2\\epsilon ^{-2}(\\vert \\Psi _h\\vert ^2- 1 )\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2$ (resp.", "$\\vartheta _T^2:= h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {f}- 2\\epsilon ^{-2}(\\vert \\Psi _{\\rm dG}\\vert ^2- 1 )\\Psi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2$ ) for Nitsche's method (resp.", "dGFEM) and this takes into account the effect of the non-zero right-hand side $\\mathbf {f}$ calculated using the manufactured solution.", "Figures REF and REF (resp.", "Figures REF and REF ) plot the discrete solutions, $u_h$ and $v_h$ of the Nitsche's method (resp.", "dGFEM), respectively, with the parameter $\\epsilon =0.4$ , and display the adaptive refinement near the vicinity of the re-entrant corner of the L-shaped domain.", "Table REF displays the computational error, estimator and convergence rates for uniform and adaptive mesh refinement for $\\epsilon =0.4$ .", "It is observed from Table REF that we have a suboptimal empirical convergence rate (calculated with respect to Ndof) of $0.25$ for uniform mesh refinement and an improved optimal empirical convergence rate of $0.5$ for adaptive mesh-refinement.", "Further, in the adaptive refinement process, the number of mesh points required to achieve convergence is significantly reduced compared to uniform meshes and the convergence is faster than the uniform refinement process.", "Figure REF displays the convergence behavior of the error and estimator along with the efficiency constant $C_{eff}$ plot, as a function of the total number of degrees of freedoms for $\\epsilon =0.2, 0.8$ .", "Here, $C_{eff}$ is the ratio between computed estimators and errors, which remains constant after the first few refinement levels.", "Figure REF displays the discrete solutions (diagonal D1 and rotated R1) and the adaptive mesh refinements in the square domain $\\Omega = (0,1) \\times (0,1)$ , for Example REF .", "Here, we observe adaptive mesh refinements near the defect points [22] of the domain (four corner points).", "Note that the estimator tends to zero as the number of degrees of freedom (Ndof) increases.", "Figure REF (resp.", "Figure REF ) is the estimator vs Ndof plot for various values of $\\epsilon $ for the diagonal, D1 solution obtained using Nitsche's method (resp.", "dGFEM).", "The tolerance used for Newton's method convergence is $10^{-6}$ and it is observed that for a fixed value of the discretization parameter $h$ , the number of Newton iterations required for the convergence increases as the value of $\\epsilon $ decreases.", "Observe that the rate of decay of the estimators is slower for smaller values of $\\epsilon $ .", "Remark 6.2 The $h$ -$\\epsilon $ dependency, discussed in [24] has been reflected for adaptive refinement in this article, in terms of Ndof-$\\epsilon $ dependency.", "It is observed in [24] that errors are sensitive to the choice of discretization parameter as $\\epsilon $ decreases.", "Figure REF (resp.", "Figure REF ) display the discrete solution corresponding to the Example REF and adaptive mesh-refinements, near the singularity at the origin for the parameter value $\\epsilon =0.6$ (resp.", "$\\epsilon =1$ ), for Nitsche's method (resp.", "dGFEM).", "Figure REF (resp.", "Figure REF ) shows the convergence history of errors in energy norm and estimators, for both uniform and adaptive refinements, for Nitsche's method (resp.", "dGFEM).", "A sub-optimal empirical convergence rate $1/3$ for uniform refinement, and an improved empirical convergence rate $0.5 $ , for adaptive mesh refinement, are obtained as a function of degrees of freedom for both Nitsche's method and dGFEM.", "Figure: Adaptive mesh refinements: Exactsolutionuh u h u_h, Exactsolutionvh v h v_h for Nitsche's method and, ExactsolutionuhdG u h u_h, ExactsolutionvhdG v h v_h for dGFEM for Example with ϵ=0.4\\epsilon =0.4.Table: Numerical errors, estimators and experimental convergence rates for uniform and adaptive mesh refinement for ϵ=0.4.\\epsilon =0.4.Figure: Ndof versus ee, ϑ\\vartheta and C eff C_{eff} for L-shape domain in Example .Figure: Adaptive mesh refinements: uD1 u h u_h, vD1 v h v_h for D1 solution.", "Adaptive mesh refinements: uR1 u h u_h, vR1 v h v_{h} for R1 solution of Example .Figure: Ndof vs estimators plot for various values of ϵ\\epsilon in square domain Example for Estimators Nitsche's method and EstimatorsdG dGFEM.Figure: Adaptive mesh refinements: SlitdomaintriangulationNitsche u h u_h for Nitsche's method with ϵ=0.6\\epsilon =0.6 .", "SlitdomaintriangulationdG u h u_h for dGFEM with ϵ\\epsilon =1.1.Figure: Ndof versus e,e, ϑ\\vartheta and C eff C_{eff} for SlitdomainErrorestimatorNitscheNitsche's method with ϵ=0.6\\epsilon =0.6 and SlitdomainErrorestimator dGFEM with ϵ=1\\epsilon =1.", "Conclusions This manuscript focuses on a priori and a posteriori error analysis for solutions with milder regularity than 2, and such solutions of lesser regularity are relevant, for example, in polygonal domains or domains with re-entrant corners that have boundary conditions of lesser regularity.", "We use Nitsche's method for our analysis; the a priori error analysis relies on medius analysis and these techniques are extended to dGFEM.", "In [24], $h$ -$\\epsilon $ dependent error estimates for $2(\\Omega )$ regular solutions are obtained, and this follows from an $\\epsilon $ independent bound for the exact solution ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_2$ , as established in [3].", "It is not clear if such estimates are feasible for exact solutions with milder regularity, ${1+\\alpha }(\\Omega ),$ $0 < \\alpha \\le 1$ , since we do not have $\\epsilon $ -independent bounds for ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }$ at hand.", "It may be possible to obtain such bounds for certain model problems, which would allow $h-\\epsilon $ dependent estimates.", "The methods in this paper will extend to modelling problems with weak anchoring or surface energies, which would translate to a Robin-type boundary condition; some dynamical models e.g.", "Allen-Cahn type evolution equations, stochastic versions of the Ginzburg-Landau system (REF ); modelling problems for composite material, such as ferronematics, which have both nematic and polar order etc.", "The overarching aim is to propose optimal estimates for the discretization parameter and number of degrees of freedom, for systems of second-order elliptic partial differential equations with lower order polynomial non-linearities, as a function of the model parameters e.g.", "$\\epsilon $ , and use these estimates for powerful new computational algorithms.", "Acknowledgements R.M.", "gratefully acknowledges support from institute Ph.D. fellowship and N.N.", "gratefully acknowledges the support by DST SERB MATRICS grant MTR/2017/000 199.", "A.M acknowledges support from the DST-UKIERI and British Council funded project on \"Theoretical and Experimental Studies of Suspensions of Magnetic Nanoparticles, their Applications and Generalizations\" and support from IIT Bombay, and a Visiting Professorship from the University of Bath.", "Appendix This section discusses the proofs of the local efficiency results in Lemmas REF -REF .", "The local cut off functions play an important role to establish the local efficiency results.", "Consider the interior bubble function [1], [34] $\\widehat{b}_T = 27\\widehat{\\lambda }_1 \\widehat{\\lambda }_2 \\widehat{\\lambda }_3 $ supported on a reference triangle $\\widehat{T}$ with the barycentric coordinate functions $\\widehat{\\lambda }_1 , \\widehat{\\lambda }_2, \\widehat{\\lambda }_3$ .", "For $T \\in \\mathcal {T}, $ let $\\mathcal {F}_T: \\widehat{T} \\rightarrow T $ be a continuous, affine and invertible transformation.", "Define the bubble function on the element $T$ by $b_T = \\widehat{b}_T \\circ \\mathcal {F}_T^{-1} $ .", "Three edge bubble functions on the reference triangle $\\widehat{T}$ are given by $\\widehat{b}_1= 4 \\widehat{\\lambda }_2 \\widehat{\\lambda }_3$ , $\\widehat{b}_2= 4 \\widehat{\\lambda }_1 \\widehat{\\lambda }_3$ and $\\widehat{b}_3= 4 \\widehat{\\lambda }_1 \\widehat{\\lambda }_2$ .", "On the edge $E$ of any triangle $T \\in \\mathcal {T}$ , define the edge bubble function to be $b_E: = \\widehat{b}_E \\circ \\mathcal {F}_T^{-1}$ , where $\\widehat{b}_E$ is the corresponding edge bubble function on $\\widehat{T}$ .", "Here, $b_E$ is supported on the pair of triangles sharing the edge $E.$ Lemma 8.1 [1], [34] Let $\\widehat{P} \\subset H^1(\\widehat{T})$ be a finite dimensional subspace on the reference triangle $\\widehat{T}$ and consider $P= \\lbrace \\widehat{v} \\circ \\mathcal {F}_T^{-1}:\\widehat{v} \\in \\widehat{P} \\rbrace $ to be the finite dimensional space of functions defined on $T$ .", "Then the following inverse estimates hold for all $v \\in P$ , ${\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{L^2(T)} \\lesssim \\int _T b_T v^2 \\,{\\rm dx}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{L^2(T)}, \\quad {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{L^2(T)} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert b_Tv\\vert \\hspace{-1.0625pt}\\vert }_{L^2(T)} + h_T{\\vert \\hspace{-1.0625pt}\\vert \\nabla (b_Tv)\\vert \\hspace{-1.0625pt}\\vert }_{L^2(T)} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{L^2(T)}.", "$ Let $E \\subset \\partial T$ be an edge and $b_E$ be the corresponding edge bubble function supported on the patch of triangles $\\omega _E$ sharing the edge $E$ .", "Let $P(E)$ be the finite dimensional space of functions defined on $E$ obtained by mapping $\\widehat{P}(\\widehat{E})\\subset H^1(\\widehat{E}).$ Then for all $v \\in P(E)$ , ${\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{L^2(E)} \\lesssim \\int _E b_E v^2 \\,{\\rm dx}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{L^2(E)}, \\quad \\,\\,\\, h_E^{ -\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert b_Ev\\vert \\hspace{-1.0625pt}\\vert }_{L^2(\\omega _E)}+ h_E^{ \\frac{1}{2}}{\\vert \\hspace{-1.0625pt}\\vert \\nabla (b_Ev)\\vert \\hspace{-1.0625pt}\\vert }_{L^2(\\omega _E)} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{L^2(E)}, $ where the hidden constants in $\"\\lesssim \"$ are independent of $h_T$ and $h_E$ .", "[Proof of Lemma REF] $(i)$ Let $T \\in \\mathcal {T}$ be arbitrary and $b_T$ be the interior bubble function supported on the triangle $T$ .", "Choose $\\displaystyle \\rho _T:=\\left\\lbrace \\begin{array}{l}\\big (-\\Delta \\Phi _{h}+2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h}\\big )b_T \\quad \\text{in } T\\\\0 \\quad \\text{in } \\Omega \\setminus T\\end{array}\\right.$ , utilize (REF ), () with $\\Phi :=\\rho _T$ and apply an integration by parts for the first term (which is a zero term) on the right-hand side below to obtain $& {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2\\lesssim \\int _T \\big (-\\Delta \\Phi _{h}+2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h}\\big ) \\cdot \\rho _T \\,{\\rm dx}\\\\&= A_T(\\Phi _{h}-\\Psi , \\rho _T ) +(B_T(\\Phi _{h},\\Phi _{h},\\Phi _{h},\\rho _T )-B_T(\\Psi ,\\Psi ,\\Psi ,\\rho _T ))+C_T(\\Phi _{h}-\\Psi ,\\rho _T ) .$ Together with Hölder's inequality, Lemma REF and (REF ), the terms on the right-hand side of (REF ) are estimated as $A_T(\\Phi _{h}-\\Psi , \\rho _T )\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Phi _{h}-\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\rho _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T} h_T^{-1}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}.\\\\C_T(\\Phi _{h}-\\Psi ,\\rho _T )\\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h-\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}\\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T},$ $B_T(\\Phi _{h},\\Phi _{h},\\Phi _{h},\\rho _T )-B_T(\\Psi ,\\Psi ,\\Psi ,\\rho _T )\\lesssim &\\epsilon ^{-2}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T}({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T}+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T})\\\\& + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T}^2) h_T^{-1}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}.$ A combination of the above three displayed estimates in (REF ) plus Lemma REF establishes $h_T{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h, T} (1+ \\epsilon ^{-2} (1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}^2 +{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h, T} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T}) )).$ To find the estimate corresponding to $ \\eta _E$ , consider the edge bubble function $b_E$ supported on the patch of triangles $\\omega _E$ sharing the edge $E$ .", "Define $\\displaystyle \\rho _E:=\\left\\lbrace \\begin{array}{l}[\\nabla \\Phi _h \\nu ]b_E \\quad \\text{in } \\omega _E\\\\0 \\quad \\text{in } \\Omega \\setminus \\omega _E\\end{array}\\right.$ and use (REF ), $[\\rho _E] = 0$ for $E \\in \\mathcal {E}_h^i $ and an integration by parts to obtain ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 & \\lesssim \\int _{E}[\\nabla \\Phi _{h} \\nu ] \\cdot \\rho _E\\,{\\rm ds}= \\int _{E } [\\nabla \\Phi _{h} \\nu ] \\cdot \\lbrace \\ \\rho _E \\rbrace \\,{\\rm ds}+\\int _{E} \\lbrace \\nabla \\Phi _{h} \\nu \\rbrace \\cdot [\\rho _E] \\,{\\rm ds}\\\\& =\\sum _{T \\in \\omega _E} \\int _T (\\Delta \\Phi _{h} \\cdot \\rho _E + \\nabla \\Phi _{h} \\cdot \\nabla \\rho _E) \\,{\\rm dx}.$ Add and subtract $\\sum _{T \\in \\omega _E} \\int _T2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h} \\cdot \\rho _E \\,{\\rm dx}$ in the right-hand side of (REF ) to rewrite the expression with the help of $ \\eta _T= \\Delta \\Phi _{h}-2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h}$ (with a $-\\Delta \\Phi _{h}=0$ added).", "The expression () with $\\Phi = \\rho _E$ , a re-grouping of terms and Hölder's inequality lead to ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 &\\lesssim (\\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2)^{\\frac{1}{2}}(\\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _E \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2)^{\\frac{1}{2}} + \\sum _{T \\in \\omega _E} (A_T(\\Phi _{h}-\\Psi , \\rho _E) +C_T(\\Phi _{h}-\\Psi , \\rho _E)\\\\&\\quad + (B_T( \\Phi _{h}, \\Phi _{h}, \\Phi _{h},\\rho _E)-B_T( \\Psi , \\Psi , \\Psi ,\\rho _E))) .$ A combination of Hölder's inequality, Lemma REF and (REF ) yields $\\sum _{T \\in \\omega _E} A_T(\\Phi _{h}- \\Psi , \\rho _E) \\lesssim \\sum _{T \\in \\omega _E}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Psi -\\Phi _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\rho _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}\\lesssim h_E^{-\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Psi -\\Phi _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,\\omega _E},\\\\\\sum _{T \\in \\omega _E} C_T(\\Phi _{h}- \\Psi , \\rho _E)\\lesssim \\epsilon ^{-2}\\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} \\lesssim \\epsilon ^{-2}h_E^{\\frac{1}{2}}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,\\omega _E},$ $\\sum _{T \\in \\omega _E} (B_T( \\Phi _{h}, \\Phi _{h}, \\Phi _{h},\\rho _E)-B_T( \\Psi , \\Psi , \\Psi ,\\rho _E) ) \\lesssim & \\epsilon ^{-2} h_E^{-\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}\\sum _{T \\in \\omega _E}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T} \\\\&+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T})+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}^2).$ The estimate of ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}$ in (REF ) and (REF ) together with the above three displayed estimates in (REF ) lead to $h_E^{\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E} \\lesssim \\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h, T} (1+ \\epsilon ^{-2} (1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}^2 +{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h, T} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1}+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}) )).$ A combination of (REF ) and REF completes the proof of $(i)$ in Lemma REF .", "$(ii)$ For $\\Phi _{h}={ \\rm I}_h \\Psi $ in (REF ), Lemma REF $(v)$ and (REF ) yield $B_T({\\rm I}_{h} \\Psi ,{\\rm I}_{h} \\Psi , {\\rm I}_{h} \\Psi , \\rho _T)- B_T( \\Psi , \\Psi , \\Psi , \\rho _T) &\\lesssim \\epsilon ^{-2}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha ,T}^3 (h_T^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\rho _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}+ h_T^{1+\\alpha }{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T})\\\\& \\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha ,T}^3(h_T^{2\\alpha } + h_T^{2+\\alpha } )h_T^{-1}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} .$ Substitute (REF ), (), (REF ) in (REF ) and utilize Lemma REF to arrive at $h_T{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}& \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla ({\\rm I}_{h} \\Psi -\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}+ \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\rm I}_{h} \\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}+ \\epsilon ^{-2} h_T^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^3\\\\& \\lesssim h_T^{\\alpha } (1+ \\epsilon ^{-2}h_T^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ A choice of $\\Phi _h = {\\rm I}_h \\Psi $ in (REF ), Lemma REF $(v)$ and (REF ) yield $&\\sum _{T \\in \\omega _E}( B_T({\\rm I}_{h} \\Psi ,{\\rm I}_{h} \\Psi , {\\rm I}_{h} \\Psi , \\rho _E)- B_T( \\Psi , \\Psi , \\Psi , \\rho _E)) \\lesssim \\epsilon ^{-2}\\sum _{T \\in \\omega _E}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha , T}^3 (h_T^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\rho _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}+ h_T^{1+\\alpha }{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T})\\\\&\\qquad \\lesssim \\epsilon ^{-2} h_E^{-\\frac{1}{2}}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E} \\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha ,T}^3(h_T^{2\\alpha } +h_E h_T^{2+\\alpha } ) .$ Substitute (REF ), (), (REF ) in (REF ) and employ Lemma REF to obtain $ h_E^{\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}\\lesssim \\sum _{T \\in \\omega _E} h_T^{\\alpha } (1+ \\epsilon ^{-2}h_T^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ A combination of (REF ) and (REF ) concludes the proof of $(ii)$ in Lemma REF .", "The proof of Lemma REF (resp.", "REF ) follows analgous to the proof of Lemma REF with the choice of $\\rho _T:=\\left\\lbrace \\begin{array}{ll@{:}}(\\Delta (\\textrm {I}_h \\xi ) +2 \\epsilon ^{-2}(\\vert \\textrm {I}_h \\Psi \\vert ^2 \\Theta _{h} + 2 (\\textrm {I}_h \\Psi \\cdot \\Theta _{h}) \\textrm {I}_h \\Psi -\\Theta _{h}) )b_T \\text{ in } T\\\\0 \\quad \\text{in } \\Omega \\setminus T\\end{array}\\right.\\text{ and }\\rho _E:=\\left\\lbrace \\begin{array}{l}[\\nabla (\\textrm {I}_h\\xi ) \\nu ] b_E \\text{ in } \\omega _E\\\\0 \\quad \\text{in } \\Omega \\setminus \\omega _E\\end{array}\\right..$ $ \\bigg (\\text{resp.", "}\\rho _T:=\\left\\lbrace \\begin{array}{l}\\big (G_h +\\Delta ( \\textrm {I}_{h}\\chi ) -2\\epsilon ^{-2} (\\vert \\textrm {I}_{h}\\Psi \\vert ^2 \\textrm {I}_{h}\\chi +2 (\\textrm {I}_{h} \\Psi \\cdot \\textrm {I}_{h}\\chi )\\textrm {I}_{h} \\Psi - \\textrm {I}_{h}\\chi )\\big )b_T \\quad \\text{in } T\\\\0 \\quad \\text{in } \\Omega \\setminus T\\end{array}\\right.", "$ $\\text{ and } \\rho _E:=\\left\\lbrace \\begin{array}{l}[ \\nabla (\\textrm {I}_h\\chi ) \\nu ]b_E \\quad \\text{in } \\omega _E\\\\0 \\quad \\text{in } \\Omega \\setminus \\omega _E\\end{array}\\right.", "\\bigg ).$" ], [ "A posteriori error estimate", "In this section, we present some auxiliary results followed by the a posteriori error analysis for the Nitsche's method.", "Note that, to derive the a posteriori estimates, it is assumed that $\\mathbf {g}$ (the inhomogeneous Dirichlet boundary condition) belongs to $ {\\frac{1}{2}}(\\partial \\Omega )\\cap \\mathbf {C}^0(\\overline{\\partial \\Omega }).$ The approximation properties of the Scott-Zhang interpolation operator [31] are introduced first.", "Lemma 4.1 (Scott-Zhang interpolation )[31] For $l, m \\in \\mathbb {N}$ with $1 \\le l < \\infty $ , there exists an interpolation operator ${\\rm I}^{SZ}_h: H^l_0(\\Omega ) \\rightarrow V_h:= X_h \\cap H^1_0(\\Omega ) $ that satisfies the stability and aproximation properties given by: (a) for all $0\\le m \\le \\min (1,l)$ , ${\\vert \\hspace{-1.0625pt}\\vert {\\rm I}^{SZ}_h v \\vert \\hspace{-1.0625pt}\\vert }_{m, \\Omega } \\le C_{SZ}{\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{l,\\Omega } \\,\\text{ for all } v \\in H^{l}_0(\\Omega ) ,$ (b) provided $l \\le 2$ , for all $0 \\le m \\le l$ , ${\\vert \\hspace{-1.0625pt}\\vert v- {\\rm I}^{SZ}_h v \\vert \\hspace{-1.0625pt}\\vert }_{m, T} \\le C_{SZ}h_T^{l-m}\\vert v\\vert _{l, \\omega _T} \\text{ for all } v \\in H_0^{l}(\\omega _T) \\text{ and } T \\in \\mathcal {T},$ where the constant $C_{SZ}>0$ is independent of $h$ , and $\\omega _T$ is the set of all triangles in $\\mathcal {T}$ that share at least one vertex with $T$ .", "Lemma 4.2 [19] Let $\\Psi _{\\mathbf {g}} \\in \\mathcal {{X}}$ solve $\\displaystyle \\int _{\\Omega } \\nabla \\Psi _{\\mathbf {g}} \\cdot \\nabla \\Phi \\, {\\rm dx} = \\sum _{ T \\in \\mathcal {T}} \\int _T \\nabla \\Psi _{h} \\cdot \\nabla \\Phi \\, {\\rm dx} \\text{ for all } \\Phi \\in $ where $\\Psi _h$ is the solution of (REF ).", "Then there exists a constant $C>0$ , depending only on the minimum angle of $\\mathcal {T} $ such that $\\displaystyle \\sum _{ T \\in \\mathcal {T}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Psi _{\\mathbf {g}} - \\Psi _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0, T}^2 \\le C (\\vartheta _{hot}^{\\partial })^2, $ where $\\displaystyle (\\vartheta _{hot}^{\\partial })^2:= \\sum _{ E \\in \\mathcal {E}_h^{\\partial } }h_E^{-1} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {g} - \\mathbf {g}_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2+ h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\mathbf {g} - \\mathbf {g}_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2$ , $ \\mathbf {g}_h$ being the standard Lagrange interpolant [10] of $\\mathbf {g}$ from $\\mathbf {P}_2(\\mathcal {E}^{\\partial }_h) \\cap \\mathbf {C}^0(\\overline{\\partial \\Omega } ).$ Remark 4.3 Note that the benchmark liquid crystal example: Example REF in [22] has Lipschitz continuous boundary conditions.", "Hence the a posteriori error analysis of this paper is applicable to this example and the results are illustrated in Section .", "The proof of Theorem REF , stated in Subsection REF , is presented in this section.", "An abstract estimate for the case of non-homogeneous boundary conditions and quartic nonlinearity is derived modifying the methodology in [9], [34] first and this result is crucial to prove Theorem REF .", "Theorem 4.4 (An abstract estimate) Let $\\Psi $ be a regular solution to () and $\\Psi _{\\mathbf {g}} \\in \\mathcal {{X}}$ .", "Then, $DN$ is locally Lipschitz continuous at $\\Psi $ , that is given $R_0>0$ , $DN$ restricted to $B(\\Psi ,R_0)$ is Lipschitz continuous.", "Moreover, (a) $\\displaystyle \\gamma := \\sup _{\\eta \\in B(\\Psi , R_0)} \\dfrac{{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN(\\eta ) - DN(\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}}{{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta - \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}} < \\infty ,$ and (b) there exists a constant $R>0$ such that for all ${\\eta }_{h} \\in B(\\Psi , R)$ , $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*} +(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h,$ where the constant in $\"\\lesssim \"$ depends on $\\gamma $ , continuous inf-sup constant $\\beta $ and Poincaré constant $C_P$ , and the nonlinear (resp.", "linearized) operator $N(\\cdot )$ (resp.", "$DN(\\cdot )$ ) is defined in () (resp.", "()).", "In the first step, it is established that $DN$ is locally Lipschitz continuous at $\\Psi $ and $\\gamma <\\infty $ .", "Let $R_0>0$ be given and $\\eta \\!\\in \\!", "B(\\Psi , R_0).$ For $\\Theta \\!\\in \\!\\mathbf {X}$ and $\\Phi \\!\\in \\!", "$ the definition of $DN(\\cdot )$ , $B(\\cdot , \\cdot , \\cdot , \\cdot )$ , a re-grouping of terms and Lemma REF $(iv)$ leads to $ &\\langle DN(\\eta ) \\Theta , \\Phi \\rangle - \\langle DN(\\Psi ) \\Theta , \\Phi \\rangle = 3B(\\eta ,\\eta , \\Theta , \\Phi ) - 3 B(\\Psi ,\\Psi , \\Theta , \\Phi ) \\\\& = 2\\epsilon ^{-2} \\int _{\\Omega } ((\\eta -\\Psi )\\cdot ( \\eta +\\Psi )(\\Theta \\cdot \\Phi )+ 2 (\\eta -\\Psi ) \\cdot \\Theta (\\eta \\cdot \\Phi ) + 2 (\\Psi \\cdot \\Theta ) (\\eta -\\Psi ) \\cdot \\Phi ) \\,{\\rm dx}\\\\&\\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1}(R_0 + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1.$ The above displayed inequality with definition of ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN(\\eta ) - DN(\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}$ leads to the Lipschitz continuity.", "This and Lemma REF concludes the proof of the first step.", "Step two establishes (REF ).", "The continuous formulation () and a Taylor expansion lead to $0=N(\\Psi ;\\Phi )= N({\\eta }_h; \\Phi ) + \\bigg \\langle \\int _{0}^{1}DN(\\Psi + t( {\\eta }_h-\\Psi ))(\\Psi - {\\eta }_h)\\,{\\rm dt}, \\Phi \\bigg \\rangle .", "$ Introduce $\\pm \\langle DN(\\Psi )(\\Psi - {\\eta }_h), \\Phi \\rangle $ in the above displayed expression and rearrange the terms to obtain $\\langle DN(\\Psi )(\\Psi - {\\eta }_h), \\Phi \\rangle = -N({\\eta }_h; \\Phi ) - \\bigg \\langle \\int _{0}^{1}(DN(\\Psi + t({\\eta }_h-\\Psi )) -DN(\\Psi ) )(\\Psi - {\\eta }_h )\\,{\\rm dt}, \\Phi \\bigg \\rangle .$ Rewrite $\\Psi - {\\eta }_h$ as $ (\\Psi - \\Psi _{\\mathbf {g}})+ (\\Psi _{\\mathbf {g}} - {\\eta }_h )$ in the left-hand side of the above term, use linearity of $\\langle DN(\\Psi )\\cdot , \\cdot \\rangle $ , introduce $\\pm \\langle DN({\\eta }_h)(\\Psi _{\\mathbf {g}} - {\\eta }_h), \\Phi \\rangle $ in the first step; and bound in the second step below to obtain $&\\langle DN(\\Psi )(\\Psi - \\Psi _{\\mathbf {g}}), \\Phi \\rangle = - N({\\eta }_h;\\Phi ) +\\langle (DN({\\eta }_h)-DN(\\Psi ))(\\Psi _{\\mathbf {g}} - {\\eta }_h), \\Phi \\rangle - \\langle DN({\\eta }_h)(\\Psi _{\\mathbf {g}} - {\\eta }_h), \\Phi \\rangle \\\\&\\quad - \\bigg \\langle \\int _{0}^{1}( DN(\\Psi + t( {\\eta }_h-\\Psi )) - DN(\\Psi ) )(\\Psi - {\\eta }_h)\\,{\\rm dt}, \\Phi \\bigg \\rangle \\\\&\\lesssim \\big ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*} + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_h)-DN(\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 \\\\& \\quad + \\int _{0}^{1} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert ( DN(\\Psi + t( {\\eta }_h-\\Psi ))-DN(\\Psi ) )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 \\,{\\rm dt}\\big ){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1.$ Since $\\Psi _{\\mathbf {g}} \\!\\in \\!", "\\mathcal {{X}}$ , $\\Psi - \\Psi _{\\mathbf {g}}\\!\\in \\!", "$ For $\\delta >0 $ small enough, the continuous inf-sup condition () implies that there exists $\\Phi \\!\\in \\!", "with $ 1 = 1$ such that $ (- ) - g 1 DN()(- g), .$A triangle inequality, $ - g=0$ on $$ and the last displayed inequality yield{\\begin{@align*}{1}{-1} (\\beta - \\delta ) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h \\le (\\beta - \\delta )({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Psi _{\\mathbf {g}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h) \\lesssim \\langle DN(\\Psi )(\\Psi - \\Psi _{\\mathbf {g}}), \\Phi \\rangle + (\\beta - \\delta ){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h.\\end{@align*}}Take $ 0$ to obtain $ - h h DN()(- g), + g - h h. $A combination of (\\ref {2.4}), the last displayed inequality and the definition of $$ plus Lemma \\ref {Poincare type inequality} for $ 1=1$ leads to{\\begin{@align*}{1}{-1}C_4 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*} + (1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h +{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_h + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_h,\\end{@align*}} where the constant $ C4$ depends on $$, $$ and $ CP.", "$For a choice of $ R := {R0, C4/2}$, use $ - h h < C4/2$ and $ - h 2h <C4/2 - h h $ in the second and third terms, respectively, in the right-hand side of the above inequality to obtain $ C4/2 - h h N(h) * + (1+C4/2+DN(h) L(X, *) )g - h h,$and this leads to the desired conclusion.$ Next, the main result of this section is proved in the following text.", "[Proof of Theorem REF] Theorem REF guarantees the existence of $R>0$ such that (REF ) holds for a choice of $ \\eta _h=\\Psi _h.$ Choose $\\Psi _{\\mathbf {g}}$ as in Lemma REF .", "A posteriori reliability (resp.", "efficiency) estimate provides an upper bound (resp.", "lower bound) on the discretization error, up to a constant.", "To establish the reliability, Theorem REF is utilized and the term $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N(\\Psi _h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*}$ is estimated first.", "Since $ is a Hilbert space, there exists a $ with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1= 1$ such that ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N(\\Psi _h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*}= N(\\Psi _h; \\Phi )= N(\\Psi _h; \\Phi -{\\textrm {I}^{SZ}_h} \\Phi )+N(\\Psi _h; \\textrm {I}^{SZ}_h\\Phi ),$ where $\\textrm {I}^{SZ}_h:h$ is the Scott-Zhang interpolation in Lemma REF .", "The second term in (REF ) can be rewritten using (REF ) with test function $\\textrm {I}^{SZ}_h\\Phi $ (that vanishes on $\\partial \\Omega $ ) as $N(\\Psi _h; \\textrm {I}^{SZ}_h\\Phi ) = \\langle \\Psi _h- \\mathbf {g},\\nabla (\\textrm {I}^{SZ}_h\\Phi ) \\nu \\rangle _{\\partial \\Omega } &\\lesssim \\big (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _h - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}} \\big (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{h_E}{\\sigma } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\textrm {I}^{SZ}_h\\Phi ) \\nu _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}} \\\\& \\lesssim \\big (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma }{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _h - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1 = (\\sum _{E \\in \\mathcal {E}_h^{\\partial }} (\\vartheta _E^{\\partial })^2 )^{{1}/{2}},$ where for ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1$ , a Cauchy-Schwarz inequality, Lemmas REF ($ii$ ) and REF are utilized in the second and third steps.", "Apply integration by parts element-wise for $A(\\Psi _h, \\Phi -\\textrm {I}^{SZ}_h\\Phi ) $ in the expression of $N(\\Psi _h; \\Phi -\\textrm {I}^{SZ}_h \\Phi )$ , use $[\\Phi -\\textrm {I}^{SZ}_h \\Phi ]_E=0$ on $E \\!\\in \\!", "\\mathcal {E}_h^i$ , $\\Phi -\\textrm {I}^{SZ}_h \\Phi = 0$ on $\\partial \\Omega $ , $\\Delta \\Psi _{h}=0$ and recall the definition of the local terms $ \\eta _T := (2\\epsilon ^{-2}(\\vert \\Psi _{h}\\vert ^2 -1)\\Psi _{h})|_T$ defined on a triangle $T\\!\\in \\!", "\\mathcal {T}$ and $ \\eta _E := [\\nabla \\Psi _h \\nu _E]_E$ on the edge $E$ of $T$ .", "For ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1$ , the above arguments, Cauchy-Schwarz inequality and Lemma REF lead to $ & N(\\Psi _h; \\Phi -\\textrm {I}^{SZ}_h\\Phi ) = A(\\Psi _h, \\Phi -\\textrm {I}^{SZ}_h\\Phi )+ B(\\Psi _h,\\Psi _h,\\Psi _h, \\Phi -\\textrm {I}^{SZ}_h \\Phi )+ C(\\Psi _h, \\Phi -\\textrm {I}^{SZ}_h \\Phi )\\\\&= \\sum _{T \\in \\mathcal {T}} \\int _T \\eta _T \\cdot (\\Phi -\\textrm {I}^{SZ}_h \\Phi ) \\,{\\rm dx}+\\sum _{E \\in \\mathcal {E}_h^i} \\langle \\eta _E, \\Phi -\\textrm {I}^{SZ}_h \\Phi \\rangle _E\\lesssim \\big (\\sum _{T \\in \\mathcal {T}} \\vartheta _T^2+\\sum _{E \\in \\mathcal {E}_h^i} (\\vartheta _E^{i})^2 \\big )^{\\frac{1}{2}} \\\\&\\quad \\times \\big (\\sum _{T \\in \\mathcal {T}} h_T^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi -\\textrm {I}^{SZ}_h \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T}+\\sum _{E \\in \\mathcal {E}_h^i} h_E^{-1} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi -\\textrm {I}^{SZ}_h \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\big )^{\\frac{1}{2}}\\lesssim \\big (\\sum _{T \\in \\mathcal {T}} \\vartheta _T^2+\\sum _{E \\in \\mathcal {E}_h^i} (\\vartheta _E^{i})^2 \\big )^{\\frac{1}{2}}, $ where $\\vartheta _T^2 = h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert 2\\epsilon ^{-2} (\\vert \\Psi _h\\vert ^2-1)\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T},$ and $(\\vartheta _E^i)^2 = h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\nabla \\Psi _h \\nu _E]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\text{ for all } E \\in \\mathcal {E}_h^{i}$ .", "A use of (REF ), (REF ) in (REF ) leads to the estimate of ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N(\\Psi _h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*}.$ The definition of ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}$ and Lemma REF yield ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - \\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_h = \\sum _{ T \\in \\mathcal {T}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Psi _{\\mathbf {g}} - \\Psi _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0, T}^2 + \\sum _{ E \\in \\mathcal {E}_h^{\\partial } } \\frac{\\sigma }{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{h}- \\mathbf {g} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0, E} \\\\ \\lesssim (\\vartheta _{hot}^{\\partial })^2+\\sum _{ E \\in \\mathcal {E}_h^{\\partial } }(\\vartheta _E^{\\partial })^2,$ where $(\\vartheta _{hot}^{\\partial })^2:= \\sum _{ E \\in \\mathcal {E}_h^{\\partial } }h_E^{-1} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {g} - \\mathbf {g}_h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 + h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\mathbf {g} - \\mathbf {g}_h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2$ (see Lemma REF ).", "This leads to the bound for the second term in (REF ) by higher order terms (h.o.t.)", "[7] that consist of (i) the errors arising due to the polynomial approximation of the boundary data $\\mathbf {g}$ that depends on the given data smoothness and (ii) the terms ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN(\\Psi _h)\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}}- \\Psi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}$ .", "To establish the efficiency estimate, set $\\Phi _h = \\Psi _h$ and $ \\eta _T = (2\\epsilon ^{-2}(\\vert \\Psi _{h}\\vert ^2 -1)\\Psi _{h})|_T $ on a triangle $T$ and $ \\eta _E=[\\nabla \\Psi _h \\nu _E]_E$ on a edge $E$ in Lemma REF .", "A use of the local efficiency estimates in Lemma REF (i) and $\\displaystyle \\sum _{ E \\in \\mathcal {E}^{\\partial }_h} ( \\vartheta _E^{\\partial })^2=\\sum _{ E \\in \\mathcal {E}^{\\partial }_h} \\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{h} - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Psi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}^2$ establishes the lower bound in Theorem REF .", "Remark 4.5 For $X_h= \\lbrace v_h \\in {C}^0(\\overline{\\Omega }), v_h|_T \\in P_p(T), \\text{ for all } T \\in \\mathcal {T}\\rbrace $ , that is, if we use higher order polynomials for the approximation, then $\\vartheta _T^2 = h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert -\\Delta \\Psi _h+2\\epsilon ^{-2} (\\vert \\Psi _h\\vert ^2-1)\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T}$ in (REF )." ], [ "Extension to discontinuous Galerkin FEM", "In this section, we extend the results in Section REF to dGFEM.", "The discrete space for dGFEM consists of piecewise linear polynomials defined by $\\displaystyle X_{{\\rm dG}}:=\\lbrace v \\in L^2(\\Omega ): v|_T \\in P_1(T) \\text{ for all } T \\in \\mathcal {T}\\rbrace ,$ and the mesh dependent norm $ \\displaystyle {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{{\\rm dG}}:=\\sum _{ T \\in \\mathcal {T}} \\int _T \\vert \\nabla v\\vert ^2 \\,{\\rm dx}+ \\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E} \\int _{E} [v]_E^2 \\,{\\rm ds},$ where $\\sigma _{\\rm dG}> 0$ is the penalty parameter.", "Let $\\mathbf {X}_{{\\rm dG}}\\!", ": =X_{{\\rm dG}} \\times X_{{\\rm dG}} $ be equipped with the product norm defined by ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}^2= {\\vert \\hspace{-1.0625pt}\\vert \\varphi _1\\vert \\hspace{-1.0625pt}\\vert }^2_{\\rm dG}+ {\\vert \\hspace{-1.0625pt}\\vert \\varphi _2\\vert \\hspace{-1.0625pt}\\vert }^2_{\\rm dG}$ for all $\\Phi _{\\rm dG}=(\\varphi _1,\\varphi _2) \\in \\mathbf {X}_{{\\rm dG}}$ .", "The dGFEM formulation corresponding to () seeks $ \\Psi _{{\\rm dG}}\\!\\in \\!\\mathbf {X}_{{\\rm dG}}$ such that for all $ \\Phi _{{\\rm dG}} \\!\\in \\!", "\\mathbf {X}_{{\\rm dG}},$ $N_{{\\rm dG}}(\\Psi _{\\rm dG};\\Phi _{\\rm dG}):=A_{{\\rm dG}}(\\Psi _{{\\rm dG}},\\Phi _{{\\rm dG}})+B(\\Psi _{{\\rm dG}},\\Psi _{{\\rm dG}},\\Psi _{{\\rm dG}},\\Phi _{{\\rm dG}})+C(\\Psi _{{\\rm dG}},\\Phi _{{\\rm dG}})-L_{\\rm dG}(\\Phi _{{\\rm dG}})=0,$ where for $ \\Theta =(\\theta _1,\\theta _2) ,\\, \\Phi =(\\varphi _1,\\varphi _2)\\in 1(\\mathcal {T}) $ , $A_{{\\rm dG}}(\\Theta ,\\Phi ) :=a_{{\\rm dG}}(\\theta _1,\\varphi _1)+a_{{\\rm dG}}(\\theta _2,\\varphi _2), $ $L_{\\rm dG}(\\Phi _{{\\rm dG}})=l^1_{\\rm dG}(\\varphi _1)+l^2_{\\rm dG}(\\varphi _2)$ , and for $\\theta , \\varphi \\in H^1(\\mathcal {T})$ , and for $-1 \\le \\lambda \\le 1,$ $&a_{{\\rm dG}}(\\theta ,\\varphi ):=\\sum _{ T \\in \\mathcal {T}} \\int _T \\nabla \\theta \\cdot \\nabla \\varphi \\,{\\rm dx}- \\sum _{E \\in \\mathcal {E}} \\langle \\lbrace \\frac{\\partial \\theta }{\\partial \\nu _E}\\rbrace _E, [\\varphi ]_E \\rangle _E-\\lambda \\sum _{E \\in \\mathcal {E}} \\langle \\lbrace \\frac{\\partial \\varphi }{\\partial \\nu _E}\\rbrace _E, [\\theta ]_E \\rangle _E+ \\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E} \\langle [\\theta ]_E, [\\varphi ]_E \\rangle _E\\\\&\\text{ and }\\,\\,l^i_{\\rm dG}(\\varphi ):=- \\sum _{E \\in \\mathcal {E}_h^{\\partial }}\\langle g_i, \\frac{\\partial \\varphi }{\\partial \\nu _E} \\rangle _{E} + \\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma _{\\rm dG}}{h_E} \\langle g_i, \\varphi \\rangle _E \\text{ for } 1\\le i \\le 2.$ The operators $B(\\cdot ,\\cdot ,\\cdot ,\\cdot )$ and $ C(\\cdot ,\\cdot )$ are as defined in Section REF .", "The proofs of results in this section follow on similar lines to the results established in Sections and for the Nitsche's method.", "Hence the main resuts and the auxiliary results needed to establish them are stated and parts of proofs where ideas differ are highlighted.", "Lemma 5.1 (Boundedness and coercivity of $A_{\\rm dG}$ )[29] For the choice of a sufficiently large parameter $\\sigma _{\\rm dG}$ , there exists a positive constant $\\alpha _2 > 0$ such that for $\\Theta _{{\\rm dG}}, \\Phi _{{\\rm dG}} \\!\\in \\!", "\\mathbf {X}_{\\rm dG}$ , $A_{{\\rm dG}}(\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}})\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}, \\text{ and } A_{{\\rm dG}}(\\Phi _{{\\rm dG}}, \\Phi _{{\\rm dG}}) \\ge \\alpha _2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}^2, $ where the hidden constant in $\"\\lesssim \"$ is independent of $h.$ Lemma 5.2 (Interpolation estimate)[30] For ${\\rm v} \\in H^s(\\Omega ) \\text{ with } s \\ge 1$ , there exists ${\\rm {I}_{\\rm dG}v} \\in X_{\\rm dG}$ such that for any $T \\in \\mathcal {T},$ $\\displaystyle {\\vert \\hspace{-1.0625pt}\\vert \\rm v-\\rm {I}_{\\rm dG}v \\vert \\hspace{-1.0625pt}\\vert }_{H^l(T)} \\le C_I h_T ^{s-l} {\\vert \\hspace{-1.0625pt}\\vert \\rm v\\vert \\hspace{-1.0625pt}\\vert }_{H^s(T)}$ for $l=0,1$ where $C_I$ denotes a generic interpolation constant independent of $h$ .. Lemma 5.3 (Enrichment operator).", "[6], [18] There exists an enrichment operator ${\\rm E}_h : X_{\\rm dG} \\rightarrow V_h\\subset H_0^1(\\Omega )$ , where $V_h $ is the Lagrange $P_1$ conforming finite element space associated with the triangulation $\\mathcal {T}$ that satisfies the following properties.", "For any $\\varphi _{\\rm dG} \\in X_{\\rm dG}$ , $\\displaystyle (a) \\sum _{ T \\in \\mathcal {T}} h_T^{-2}{\\vert \\hspace{-1.0625pt}\\vert {\\rm E}_h\\varphi _{\\rm dG} -\\varphi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2 + {\\vert \\hspace{-1.0625pt}\\vert {\\rm E}_h\\varphi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert }_{1}^2 \\le C_{en_1} {\\vert \\hspace{-1.0625pt}\\vert \\varphi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG}^2,$ and $\\displaystyle (b)\\, {\\vert \\hspace{-1.0625pt}\\vert {\\rm E}_h\\varphi _{\\rm dG} - \\varphi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG}^2 \\le C_{en_2}(\\sum _{ E \\in \\mathcal {E}}\\int _{ E}\\frac{1}{h_E}[\\varphi _{\\rm dG}]_E^2 {\\rm ds} ) ,$ where $C_{en_1}$ and $C_{en_2}$ are positive constants independent of $h$ .", "Remark 5.4 (Modified local efficiency results) Similar local efficiency results in Lemmas REF -REF hold for dGFEM with ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h}$ is replaced by ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\cdot \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}$ and the interpolation operator ${\\rm I}_h$ replaced by ${\\rm I}_{\\rm dG}.$ The discrete inf-sup condition corresponding to the perturbed bilinear form $\\langle DN_{{\\rm dG}}(\\textrm {I}_{{\\rm dG}}\\Psi )\\Theta _{{\\rm dG}}, \\Phi _{{\\rm dG}} \\rangle := A_{{\\rm dG}}(\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}})+3B(\\textrm {I}_{{\\rm dG}}\\Psi ,\\textrm {I}_{{\\rm dG}}\\Psi ,\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}})+C(\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}})$ is stated first.", "This is crucial in establishing the error estimates.", "Lemma 5.5 (Stability of perturbed bilinear form).", "Let $\\Psi \\!", "\\in \\!", "\\mathcal {{X}} \\cap \\mathbf {H}^{1+\\alpha }(\\Omega ), \\, 0<\\alpha \\le 1,$ be a regular solution of () and ${{\\rm I}_{\\rm dG}}\\Psi $ be its interpolant.", "For a sufficiently large $\\sigma _{\\rm dG}$ and a sufficiently small discretization parameter $h$ , there exists a constant $\\beta _1 $ such that $\\displaystyle 0< \\beta _1 \\le \\inf _{\\begin{array}{c}\\Theta _{{\\rm dG}} \\in \\mathbf {X}_{{\\rm dG}} \\\\ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Theta _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}=1\\end{array}} \\sup _{\\begin{array}{c}\\Phi _{{\\rm dG}} \\in \\mathbf {X}_{{\\rm dG}} \\\\ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}=1\\end{array}}\\langle DN_{{\\rm dG}}({{\\rm I}_{{\\rm dG}}\\Psi })\\Theta _{{\\rm dG}},\\Phi _{{\\rm dG}} \\rangle .$ The proof follows along similar lines as the proofs of Lemma REF and Theorem REF , except for the additional terms $ \\displaystyle \\sum _{E \\in \\mathcal {E}^i_h}\\langle \\lbrace \\nabla (\\textrm {I}_{{\\rm dG}}\\xi ) \\nu _E\\rbrace _E, [\\Phi _{{\\rm dG}}]_E \\rangle _{E} , \\; \\displaystyle \\sum _{E \\in \\mathcal {E}}\\langle \\lbrace \\nabla \\Phi _{{\\rm dG}} \\nu _E\\rbrace _E, [\\textrm {I}_{{\\rm dG}}\\xi ]_E \\rangle _{E} , \\; \\sum _{E \\in \\mathcal {E}}\\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\xi ]_E , \\;[\\Phi _{{\\rm dG}}]_E \\rangle _{E} $ that appear in $A_{{\\rm dG}}(\\textrm {I}_{{\\rm dG}}\\xi ,\\Phi _{{\\rm dG}})-A(\\xi ,\\textrm {E}_h\\Phi _{{\\rm dG}})$ (see (REF )).", "Since $[{\\rm E}_h\\Phi _{{\\rm dG}}]_{E}=0$ and $[\\xi ]_E=0$ for all $E \\in \\mathcal {E}_h^i $ , the above displayed terms are equal to $\\displaystyle \\sum _{E \\in \\mathcal {E}^i_h}\\langle \\lbrace \\nabla (\\textrm {I}_{{\\rm dG}}\\xi ) \\nu _E\\rbrace _E, [\\Phi _{{\\rm dG}} -\\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _{E}, \\sum _{E \\in \\mathcal {E}}\\langle \\lbrace \\nabla \\Phi _{{\\rm dG}} \\nu _E\\rbrace _E, [\\textrm {I}_{{\\rm dG}}\\xi -\\xi ]_E \\rangle _{E} , \\sum _{E \\in \\mathcal {E}}\\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\xi -\\xi ]_E , [\\Phi _{{\\rm dG}} -\\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _{E},$ respectively.", "A Cauchy-Schwarz inequality and Lemmas REF , REF yield estimate for the above terms.", "The rest of the details are skipped for brevity.", "Lemma 5.6 Let $\\Psi $ be a regular solution of () and ${\\rm I}_{\\rm dG}\\Psi \\in \\mathbf {X}_{{\\rm dG}}$ be its interpolant.", "Then, for any $\\Phi _{\\rm dG}\\in \\mathbf {X}_{\\rm dG}$ with $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}=1,$ it holds that $&A_{{\\rm dG}}({\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} ) + B({\\rm {I}}_{{\\rm dG}}\\Psi ,{\\rm {I}}_{{\\rm dG}}\\Psi ,{\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} )+ C({\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} ) -L_{{\\rm dG}}(\\Phi _{{\\rm dG}})\\\\& \\qquad \\lesssim h^{\\alpha } (1+ \\epsilon ^{-2}h^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ The proof of Lemma 3.15 is modified and the steps that are different are detailed.", "The definitions of $A_{\\rm dG}(\\cdot , \\cdot ) $ (with an integration by parts) and $L_{\\rm dG}( \\cdot ) $ will lead to the inter-element jump and average terms in the identities corresponding to (REF ) and (REF ).", "Utilize $[\\Psi ]_E=0$ for all $E \\in \\mathcal {E}^i$ to rewrite these identities as follows.", "$A_{{\\rm dG}}(\\textrm {I}_{{\\rm dG}}&\\Psi , \\Phi _{{\\rm dG}}- \\textrm {E}_{h}\\Phi _{{\\rm dG}} ) -L_{{\\rm dG}}(\\Phi _{{\\rm dG}}- {\\rm E}_h\\Phi _{{\\rm dG}})= \\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle [\\nabla (\\textrm {I}_{{\\rm dG}}\\Psi )\\nu _E ]_E,\\lbrace \\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}\\rbrace _E \\rangle _{E} \\\\&+ \\lambda \\sum _{ E \\in \\mathcal {E}}\\langle [ \\Psi -\\textrm {I}_{{\\rm dG}}\\Psi ]_E, \\lbrace \\nabla (\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}})\\nu _E \\rbrace _E \\rangle _{E} + \\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\Psi - \\Psi ]_E, [\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _E,\\\\& \\hspace{-31.2982pt} A_{{\\rm dG}}(\\textrm {I}_{{\\rm dG}}\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} )-L_{{\\rm dG}}( \\textrm {E}_h\\Phi _{{\\rm dG}})= A(\\textrm {I}_{{\\rm dG}}\\Psi -\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} )+\\lambda \\sum _{ E \\in \\mathcal {E}}\\langle [{\\Psi }-\\textrm {I}_{{\\rm dG}}\\Psi ]_E, \\lbrace \\nabla (\\textrm {E}_h\\Phi _{{\\rm dG}})\\nu _E\\rbrace _E \\rangle _{E} \\nonumber \\\\& \\quad \\quad \\quad \\hspace{85.35826pt} - (B(\\Psi ,\\Psi ,\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}}) + C(\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} )).$ The inclusion of jump and average terms in the above displayed identities will modify (REF ) as $& A_{{\\rm dG}}({\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} ) + B({\\rm {I}}_{{\\rm dG}}\\Psi ,{\\rm {I}}_{{\\rm dG}}\\Psi ,{\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} )+ C({\\rm {I}}_{{\\rm dG}}\\Psi , \\Phi _{{\\rm dG}} ) -L_{{\\rm dG}}(\\Phi _{{\\rm dG}}) \\nonumber =\\sum _{T \\in \\mathcal {T}}\\int _T \\eta _T \\cdot (\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}})\\,{\\rm dx}\\\\& \\quad + \\sum _{ E \\in \\mathcal {E}_h^{i}}\\langle \\eta _E ,\\lbrace \\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}\\rbrace _E \\rangle _{E} + (A(\\textrm {I}_{{\\rm dG}}\\Psi -\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} ) + C({\\rm I}_{{\\rm dG}}\\Psi -\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}})) \\nonumber \\\\& \\quad + (B(\\textrm {I}_{{\\rm dG}}\\Psi ,\\textrm {I}_{{\\rm dG}}\\Psi ,\\textrm {I}_{{\\rm dG}}\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}} )-B(\\Psi ,\\Psi ,\\Psi , \\textrm {E}_h\\Phi _{{\\rm dG}})) +\\lambda \\sum _{ E \\in \\mathcal {E}}\\langle [{\\Psi }-\\textrm {I}_{{\\rm dG}}\\Psi ]_E, \\lbrace \\nabla \\Phi _{{\\rm dG}} \\nu _E \\rbrace _E \\rangle _{E} \\\\&\\quad +\\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\Psi - \\Psi ]_E, [\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _E :=T_1 + \\cdots +T_6, $ where $ \\eta _T:=( 2\\epsilon ^{-2}(\\vert \\textrm {I}_{{\\rm dG}}\\Psi \\vert ^2 -1)\\textrm {I}_{{\\rm dG}}\\Psi )|_T \\text{ on } T \\text{ and } \\eta _E: = [\\nabla (\\textrm {I}_{{\\rm dG}}\\Psi )\\nu _E]_E \\text{ on } E.$ The terms $T_1$ to $ T_4 $ are estimated in similar lines to the corresponding terms in Lemma REF .", "Apply Cauchy-Schwarz inequality, Lemma REF , Lemmas REF , REF and $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}=1$ to $T_5$ and $T_6$ .", "$&T_5:=\\sum _{E \\in \\mathcal {E}}\\langle [\\Psi -\\textrm {I}_{{\\rm dG}}\\Psi ]_E, \\lbrace \\nabla \\Phi _{{\\rm dG}} \\nu _E \\rbrace _E \\rangle _{E} \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{{\\rm dG}}\\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}\\lesssim h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha },\\\\&T_6:=\\sum _{E \\in \\mathcal {E}} \\frac{\\sigma _{\\rm dG}}{h_E}\\langle [\\textrm {I}_{{\\rm dG}}\\Psi - \\Psi ]_E, [\\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}}]_E \\rangle _E \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\textrm {I}_{{\\rm dG}}\\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _{{\\rm dG}}- \\textrm {E}_h\\Phi _{{\\rm dG}} \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}\\lesssim h^\\alpha {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ A combination of the estimates lead to the desired result.", "The next abstract estimate is analogous to Theorem REF in Section and is useful to establish a reliable and efficient a posteriori error estimate for dGFEM.", "Lemma 5.7 Let $\\Psi $ be a regular solution to () and $\\Psi _{\\mathbf {g}} \\!", "\\in \\!", "\\mathcal {{X}}$ .", "Then, $DN$ is locally Lipschitz continuous at $\\Psi $ , that is given $R_0>0$ , $DN$ restricted to $B(\\Psi ,R_0)$ is Lipschitz continuous.", "Moreover, (a) $\\displaystyle \\gamma := \\sup _{\\eta \\in B(\\Psi , R_0)} \\dfrac{{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN(\\eta ) - DN(\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}}{{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta - \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG}} < \\infty ,$ and (b) there exists a constant $R>0$ such that for all ${\\eta }_{\\rm dG} \\in B(\\Psi , R)$ , $\\displaystyle {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - {\\eta }_{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert N({\\eta }_{\\rm dG})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{*} +(1+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert DN({\\eta }_{\\rm dG})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\mathcal {L}(\\mathbf {X}, *)}){\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\mathbf {g}} - {\\eta }_{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG},$ where the constant in $\"\\lesssim \"$ depends on $\\gamma $ , continuous inf-sup constant $\\beta $ and Poincaré constant, and the nonlinear (resp.", "linearized) operator $N(\\cdot )$ (resp.", "$DN(\\cdot )$ ) is defined in () (resp.", "()).", "For each element $T$ and edge $E$ , the volume and edge contributions to the estimators for dGFEM are $&\\vartheta _T^2: = h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert -\\Delta \\Psi _{\\rm dG} + 2\\epsilon ^{-2} (\\vert \\Psi _{\\rm dG}\\vert ^2-1)\\Psi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }^2_{0,T}, \\,\\,\\, (\\vartheta _E^{\\partial })^2:= \\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\rm dG} - \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\text{ for all } E \\in \\mathcal {E}_h^{\\partial },\\\\& \\text{and }(\\vartheta _E^i)^2 := h_E {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\nabla \\Psi _{\\rm dG} \\nu _E]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 +\\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\Psi _{\\rm dG}]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\text{ for all } E \\in \\mathcal {E}_h^{i} .$ Define the estimator $ \\displaystyle \\vartheta ^2:= \\sum _{ T \\in \\mathcal {T}} \\vartheta _T^2 +\\sum _{ E \\in \\mathcal {E}_h^{i}} (\\vartheta _E^i)^2 + \\sum _{ E \\in \\mathcal {E}_h^{\\partial } }(\\vartheta _E^{\\partial })^2.$ The main result of this section is presented now.", "Theorem 5.8 (A priori and a posteriori error estimates) Let $\\Psi $ be a regular solution of ().", "For a sufficiently large penalty parameter $\\sigma _{\\rm dG}>0$ and a sufficiently small discretization parameter $h$ , there exists a unique solution $\\Psi _{\\rm dG}$ to the discrete problem (REF ) that approximates $\\Psi $ such that (Energy norm estimate) $ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Psi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG} \\lesssim h^{\\alpha }$ , where $\\Psi \\!", "\\in \\!", "\\mathcal {{X}}\\cap \\mathbf {H}^{1+\\alpha }(\\Omega ) ,$ $0 < \\alpha \\le 1$ is the index of elliptic regularity, (A posteriori estimates) There exist $h$ -independent positive constants ${\\rm C}_{\\text{rel}}$ and ${\\rm C}_{\\text{eff}}$ such that ${\\rm C}_{\\text{eff}} \\vartheta \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Psi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{\\rm dG} \\le {\\rm C}_{\\text{rel}} \\big ( \\vartheta + h.o.t \\big ),$ where $h.o.t$ expresses terms of higher order.", "The basic ideas of proofs of both a priori and a posteriori error estimates follow from Theorems REF and REF .", "The modifications for the case of dGFEM are sketched for the sake of clarity.", "1.", "(Energy norm estimate): The energy norm error estimate in a priori error analysis is proved using Brouwer's fixed point theorem.", "The non-linear map [24] $\\mu _{\\rm dG}: \\mathbf {X}_{{\\rm dG}} \\rightarrow \\mathbf {X}_{{\\rm dG}}$ is defined in this case as $\\langle DN_{\\rm dG}({{\\rm I}_{\\rm dG}\\Psi }) \\mu _{{\\rm dG}}(\\Theta _{\\rm dG}),\\Phi _{\\rm dG} \\rangle = 3B(\\textrm {I}_{\\rm dG}\\Psi , \\textrm {I}_{\\rm dG}\\Psi ,\\Theta _{\\rm dG},\\Phi _{\\rm dG}) - B(\\Theta _{\\rm dG},\\Theta _{\\rm dG}, \\Theta _{\\rm dG},\\Phi _{\\rm dG}) +L_h(\\Phi _{\\rm dG}).$ The proof now follows analogous to Theorem REF , using Lemmas REF and REF .", "2.", "(A posteriori estimate): Lemma REF and techniques used in proof of Theorem REF lead to a posteriori estimates.", "The jump and average terms of $A_{\\rm dG}(\\cdot ,\\cdot )$ in the expansion of $N_{\\rm dG}(\\Psi _{\\rm dG}; \\textrm {I}^{SZ}_h\\Phi )$ in (REF ) will modify (REF ) to $N(\\Psi _{\\rm dG}; \\textrm {I}^{SZ}_h\\Phi )=\\lambda \\sum _{ E \\in \\mathcal {E}_h^{i}} \\langle [\\Psi _{\\rm dG}]_E,\\lbrace \\nabla (\\textrm {I}^{SZ}_h\\Phi ) \\nu _E\\rbrace _E \\rangle _E+ \\lambda \\sum _{ E \\in \\mathcal {E}_h^{\\partial } } \\langle \\Psi _{\\rm dG}- \\mathbf {g},\\nabla (\\textrm {I}^{SZ}_h\\Phi ) \\nu _E \\rangle _{E}.$ The Cauchy-Schwarz inequality, Lemmas REF ($ii$ ) and REF plus ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_1=1$ yield $N(\\Psi _{\\rm dG}; \\textrm {I}^{SZ}_h\\Phi ) \\lesssim \\big (\\sum _{E \\in \\mathcal {E}_h^{i}} \\frac{\\sigma _{\\rm dG}}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\Psi _{\\rm dG}]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2+\\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{\\sigma _{\\rm dG}}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\rm dG}- \\mathbf {g}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_0^2 \\big )^{\\frac{1}{2}} $ and leads to interior edge estimator term $ \\sum _{E \\in \\mathcal {E}_h^{\\partial }} \\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\Psi _{\\rm dG}]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2$ .", "Moreover, a use of $[\\Psi ]_E=0$ for all $E \\in \\mathcal {E}_h^{i}$ shows $ \\sum _{ E \\in \\mathcal {E}_h^{i}}\\frac{1}{h_E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert [\\Psi _{\\rm dG}]_E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 \\le {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi _{\\rm dG}- \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{{\\rm dG}}$ and establishes the efficiency bound.", "The remaining part of the proof uses ideas similar to the proof of Theorem REF ." ], [ "Numerical results", "In this section, we present some numerical experiments that confirm the theoretical results obtained in Sections -, and illustrate the practical performances of the error indicators in adaptive mesh refinement for both Nitsche's method and dGFEM." ], [ " Preliminaries", " The uniform refinement process divides each triangle in the triangulation of $\\bar{\\Omega }$ into four similar triangles for subsequent mesh refinements using red refinement.", "Let $e(n)$ and $h(n)$ (resp.", "$e({n-1})$ and $h({n-1})$ ) denote the error and the discretization parameter at the $n$ -th (resp.", "$n-1$ -th) level of uniform refinements, respectively.", "The convergence rate at $n$ -th level is defined by $\\displaystyle \\alpha _n:=log(e(n)/e({n-1}))/log(h(n)/h({n-1}))$ .", "The penalty parameters $\\sigma =\\sigma _{\\rm dG}=10$ is chosen for the numerical experiments.", "Numerical experiments are performed for different values of $\\epsilon $ to illustrate the efficacy of the methods.", "Newton's method is employed to compute the approximated solutions of the discrete nonlinear problem (REF ).", "The Newton's iterates for Nitsche's method (see [24] for Newton's iterates in dGFEM) are given by $\\Psi ^n_{h}, \\,\\, n= 1,2, \\ldots $ $A_h(\\Psi ^n_{h}, \\Phi _h) + 3B(\\Psi ^{n-1}_{h},\\Psi ^{n-1}_{h},\\Psi ^n_{h},\\Phi _{h}) +C(\\Psi ^n_{h}, \\Phi _h)= 2 B(\\Psi ^{n-1}_{h},\\Psi ^{n-1}_{h},\\Psi ^{n-1}_{h},\\Phi _{h}) +L_h(\\Phi _{h}).$ The tolerance in the Newton's method is chosen as $10^{-8} $ in the numerical experiments unless mentioned otherwise.", "Remark 6.1 It can be established that the Newton iterates in (REF ) converge quadratically to the discrete solution [24].", "Figure: triangulation2Initial triangulation 𝒯 0 \\mathcal {T}_0 of L-shape domain in Example and triangulation3 its uniform refinement.", "Slitdomain Slit domain in Example Figure: Convergence history (a priori estimates) for Nitsche's method and dGFEM for AprioriresultLslapedomain Example , with ϵ=0.4\\epsilon =0.4, and AprioriresultSlitdomain Example , with ϵ=0.6\\epsilon =0.6." ], [ " Example on a $L$ -shaped domain", "Consider (REF ) on a non-convex L-shape domain $\\Omega = (-1, 1 )\\times (-1,1) \\setminus [0,1] \\times [-1,0]$ .", "For the manufactured solution $ u= r^{{2}/{3}}\\sin ({2 \\theta }/{3}),$ $v= r^{1/2}\\sin (\\theta /2)$ , where $(r, \\theta )$ denote the system of polar coordinates, compute the corresponding right-hand side $\\mathbf {f}$ and the non-homogeneous Dirichlet boundary condition $\\mathbf {g}$ .", "In this case, the exact solutions $\\Psi \\in \\mathbf {H}^{1+1/2-\\kappa }(\\kappa \\!> \\!", "0)$ [14], and the theoretically expected rate of error reduction is $O(h^{1/2})$ and $O(h)$ in the energy norm and $\\mathbf {L}^2$ norm (see Theorems REF and REF ).", "Figure REF and REF display the initial triangulation and its uniform refinement.", "The initial guess for the Nitsche's method (resp.", "dGFEM) is chosen as $\\Psi _h^0 \\in \\mathbf {X}_{h}$ (resp.", "$\\Psi _{\\rm dG}^0 \\in \\mathbf {X}_{{\\rm dG}}$ ), where $\\Psi _h^0$ solves $A_h(\\Psi _h^0, \\Phi _h)= L_h(\\Phi _h) \\text{ for all } \\Phi _h \\in \\mathbf {X}_{h}$ (resp.", "$A_{\\rm dG}(\\Psi _{\\rm dG}^0, \\Phi _{\\rm dG})= L_{\\rm dG}(\\Phi _{\\rm dG}) \\text{ for all } \\Phi _{\\rm dG}\\in \\mathbf {X}_{{\\rm dG}}$ ), and the linear form $L_h(\\cdot )$ (resp.", "$L_{\\rm dG}(\\cdot )$ ) is modified to incorporate the information on $\\mathbf {f}$ .", "The approximations to the discrete solution to (REF ) are obtained using the Newton's method defined in (REF ).", "Figure REF presents the convergence rates in energy norm and $\\mathbf {L}^2$ norm, with $\\epsilon =0.4$ , for both Nitsche's method and dGFEM." ], [ " Benchmark example on unit square domain ", "Consider () on a convex domain $\\Omega =(0,1)\\times (0,1)$ with the Dirichlet boundary condition [22] given by $\\displaystyle \\mathbf {g}={\\left\\lbrace \\begin{array}{ll}(\\textit {T}_{d}(x),0) & \\text{on}\\,\\,\\,\\, y=0 \\,\\,\\,\\,\\text{and} \\,\\,\\,\\, y=1 \\\\(- \\textit {T}_d(y),0) & \\text{on}\\,\\,\\,\\, x=0 \\,\\,\\,\\, \\text{and} \\,\\,\\,\\, x=1\\end{array}\\right.", "}$ , where the parameter $d=3 \\epsilon $ with $\\epsilon = 0.02$ and the trapezoidal shape function $\\textit {T}_d:[0,1]\\rightarrow {\\mathbb {R}}$ is defined by $\\displaystyle \\textit {T}_d(t)={\\left\\lbrace \\begin{array}{ll}t/d, & 0 \\le t \\le d \\\\1, & d \\le t \\le 1- d \\\\(1-t)/d, & 1- d \\le t \\le 1\\end{array}\\right.", "}$ .", "See [22], [24] for details of construction of a suitable initial guess of Newton’s iterates in this example.", "Table: Numerical energy, errors and convergence rates for D1 and R1 solutions, respectively in energy and 𝐋 2 \\mathbf {L}^2 norms for ϵ=0.02\\epsilon =0.02.Table REF presents the computed energy, error in energy and $\\mathbf {L}^2$ norms for numerical approximation of the diagonal D1 and rotated R1 solutions, respectively obtained using the Nitsche's method in (REF ).", "The orders of convergence agrees with the theoretical orders of convergence obtained in [22], [24], [33].", "For the corresponding results for dGFEM, see [24]." ], [ " Example on a slit domain", "Let $\\Omega $ be the slit domain $\\lbrace (x,y) \\in \\mathbb {R}^2: \\vert x\\vert +\\vert y\\vert <1 \\rbrace \\setminus ([0,1] \\times \\lbrace 0\\rbrace )$ (see Figure REF ).", "Select the non-homogeneous Dirichlet boundary data $\\mathbf {g}$ and the right-hand side $\\mathbf {f}$ so that the manufactured solution is given by $u(r, \\theta )=v(r, \\theta )= r^{1/2}\\sin (\\theta /2)- (1/2)(r\\sin (\\theta ))^2$ .", "Figure REF presents the convergence history in energy norm, $\\mathbf {L}^2$ norm, with the parameter value $\\epsilon =0.6$ , for Nitsche's method and dGFEM.", "The rates are approximately $0.5004$ (resp.", "$0.9846$ ) in energy norm ($\\mathbf {L}^2$ norm)." ], [ " Adaptive mesh-refinement", " For the adaptive refinement, the order of convergence of error and estimators are related to total number of unknowns ($\\text{Ndof}(l)$ ).", "Let $e(l)$ and $\\text{Ndof}(l)$ be the error and total number of unknowns at the $l-$ th level refinement, respectively.", "The convergence rates are calculated as $\\text{Order}_e(l):= \\frac{\\log (e(l-1)/e(l))}{\\log (\\text{Ndof}(l)/\\text{Ndof}(l-1))} \\qquad \\text{and} \\qquad \\text{Order}_{\\vartheta }(l):= \\frac{\\log (\\vartheta (l-1)/\\vartheta (l))}{\\log (\\text{Ndof}(l)/\\text{Ndof}(l-1))}.$ Given an initial triangulation $\\mathcal {T}_0,$ run the steps SOLVE, ESTIMATE, MARK and REFINE successively for different levels $l=0,1,2,\\ldots $ SOLVE Compute the solution $\\Psi _l:=\\Psi _h$ (resp.", "$\\Psi _l:=\\Psi _{\\rm dG}$ ) of the discrete problem (REF ) (resp.", "REF ) for the triangulation $\\mathcal {T}_l$ .", "ESTIMATE Calculate the error indicator $\\displaystyle _{T,l}:=\\big (\\vartheta _T^2 + \\sum _{ E \\in \\partial T \\cap \\mathcal {E}_h^{i}} (\\vartheta ^i_E)^2 + \\sum _{E \\in \\partial T \\cap \\mathcal {E}_h^{\\partial }}(\\vartheta _E^{\\partial })^2 \\big )^\\frac{1}{2} $ for each element $T \\in \\mathcal {T}_l.$ Recall the volume and edge estimators for Nitsche's method (resp.", "dGFEM) given by (REF )-() (resp.", "(REF )-()).", "MARK For next refinement, choose the elements $T \\in \\mathcal {T}_l$ using Dörfler marking such that $ 0.3\\sum _{ T \\in {\\mathcal {T}_l}}_{T, l}^2 \\\\ \\le \\sum _{ T \\in \\tilde{\\mathcal {T}}}_{T, l}^2$ and collect those elements to construct a subset $\\tilde{\\mathcal {T}} \\subset \\mathcal {T}_l$ .", "REFINE Compute the closure of $\\tilde{\\mathcal {T}}$ and use newest vertex bisection [32] refinement strategy to construct the new triangulation $\\mathcal {T}_{l+1}$ .", "Consider (REF ) in L-shaped domain (Figure REF ) with the manufactured solution presented in Example REF and apply the adaptive refinement algorithm.", "The estimator is modified as $\\vartheta _T^2:= h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {f}- 2\\epsilon ^{-2}(\\vert \\Psi _h\\vert ^2- 1 )\\Psi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2$ (resp.", "$\\vartheta _T^2:= h_T^2 {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\mathbf {f}- 2\\epsilon ^{-2}(\\vert \\Psi _{\\rm dG}\\vert ^2- 1 )\\Psi _{\\rm dG}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2$ ) for Nitsche's method (resp.", "dGFEM) and this takes into account the effect of the non-zero right-hand side $\\mathbf {f}$ calculated using the manufactured solution.", "Figures REF and REF (resp.", "Figures REF and REF ) plot the discrete solutions, $u_h$ and $v_h$ of the Nitsche's method (resp.", "dGFEM), respectively, with the parameter $\\epsilon =0.4$ , and display the adaptive refinement near the vicinity of the re-entrant corner of the L-shaped domain.", "Table REF displays the computational error, estimator and convergence rates for uniform and adaptive mesh refinement for $\\epsilon =0.4$ .", "It is observed from Table REF that we have a suboptimal empirical convergence rate (calculated with respect to Ndof) of $0.25$ for uniform mesh refinement and an improved optimal empirical convergence rate of $0.5$ for adaptive mesh-refinement.", "Further, in the adaptive refinement process, the number of mesh points required to achieve convergence is significantly reduced compared to uniform meshes and the convergence is faster than the uniform refinement process.", "Figure REF displays the convergence behavior of the error and estimator along with the efficiency constant $C_{eff}$ plot, as a function of the total number of degrees of freedoms for $\\epsilon =0.2, 0.8$ .", "Here, $C_{eff}$ is the ratio between computed estimators and errors, which remains constant after the first few refinement levels.", "Figure REF displays the discrete solutions (diagonal D1 and rotated R1) and the adaptive mesh refinements in the square domain $\\Omega = (0,1) \\times (0,1)$ , for Example REF .", "Here, we observe adaptive mesh refinements near the defect points [22] of the domain (four corner points).", "Note that the estimator tends to zero as the number of degrees of freedom (Ndof) increases.", "Figure REF (resp.", "Figure REF ) is the estimator vs Ndof plot for various values of $\\epsilon $ for the diagonal, D1 solution obtained using Nitsche's method (resp.", "dGFEM).", "The tolerance used for Newton's method convergence is $10^{-6}$ and it is observed that for a fixed value of the discretization parameter $h$ , the number of Newton iterations required for the convergence increases as the value of $\\epsilon $ decreases.", "Observe that the rate of decay of the estimators is slower for smaller values of $\\epsilon $ .", "Remark 6.2 The $h$ -$\\epsilon $ dependency, discussed in [24] has been reflected for adaptive refinement in this article, in terms of Ndof-$\\epsilon $ dependency.", "It is observed in [24] that errors are sensitive to the choice of discretization parameter as $\\epsilon $ decreases.", "Figure REF (resp.", "Figure REF ) display the discrete solution corresponding to the Example REF and adaptive mesh-refinements, near the singularity at the origin for the parameter value $\\epsilon =0.6$ (resp.", "$\\epsilon =1$ ), for Nitsche's method (resp.", "dGFEM).", "Figure REF (resp.", "Figure REF ) shows the convergence history of errors in energy norm and estimators, for both uniform and adaptive refinements, for Nitsche's method (resp.", "dGFEM).", "A sub-optimal empirical convergence rate $1/3$ for uniform refinement, and an improved empirical convergence rate $0.5 $ , for adaptive mesh refinement, are obtained as a function of degrees of freedom for both Nitsche's method and dGFEM.", "Figure: Adaptive mesh refinements: Exactsolutionuh u h u_h, Exactsolutionvh v h v_h for Nitsche's method and, ExactsolutionuhdG u h u_h, ExactsolutionvhdG v h v_h for dGFEM for Example with ϵ=0.4\\epsilon =0.4.Table: Numerical errors, estimators and experimental convergence rates for uniform and adaptive mesh refinement for ϵ=0.4.\\epsilon =0.4.Figure: Ndof versus ee, ϑ\\vartheta and C eff C_{eff} for L-shape domain in Example .Figure: Adaptive mesh refinements: uD1 u h u_h, vD1 v h v_h for D1 solution.", "Adaptive mesh refinements: uR1 u h u_h, vR1 v h v_{h} for R1 solution of Example .Figure: Ndof vs estimators plot for various values of ϵ\\epsilon in square domain Example for Estimators Nitsche's method and EstimatorsdG dGFEM.Figure: Adaptive mesh refinements: SlitdomaintriangulationNitsche u h u_h for Nitsche's method with ϵ=0.6\\epsilon =0.6 .", "SlitdomaintriangulationdG u h u_h for dGFEM with ϵ\\epsilon =1.1.Figure: Ndof versus e,e, ϑ\\vartheta and C eff C_{eff} for SlitdomainErrorestimatorNitscheNitsche's method with ϵ=0.6\\epsilon =0.6 and SlitdomainErrorestimator dGFEM with ϵ=1\\epsilon =1." ], [ "Conclusions", "This manuscript focuses on a priori and a posteriori error analysis for solutions with milder regularity than 2, and such solutions of lesser regularity are relevant, for example, in polygonal domains or domains with re-entrant corners that have boundary conditions of lesser regularity.", "We use Nitsche's method for our analysis; the a priori error analysis relies on medius analysis and these techniques are extended to dGFEM.", "In [24], $h$ -$\\epsilon $ dependent error estimates for $2(\\Omega )$ regular solutions are obtained, and this follows from an $\\epsilon $ independent bound for the exact solution ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_2$ , as established in [3].", "It is not clear if such estimates are feasible for exact solutions with milder regularity, ${1+\\alpha }(\\Omega ),$ $0 < \\alpha \\le 1$ , since we do not have $\\epsilon $ -independent bounds for ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }$ at hand.", "It may be possible to obtain such bounds for certain model problems, which would allow $h-\\epsilon $ dependent estimates.", "The methods in this paper will extend to modelling problems with weak anchoring or surface energies, which would translate to a Robin-type boundary condition; some dynamical models e.g.", "Allen-Cahn type evolution equations, stochastic versions of the Ginzburg-Landau system (REF ); modelling problems for composite material, such as ferronematics, which have both nematic and polar order etc.", "The overarching aim is to propose optimal estimates for the discretization parameter and number of degrees of freedom, for systems of second-order elliptic partial differential equations with lower order polynomial non-linearities, as a function of the model parameters e.g.", "$\\epsilon $ , and use these estimates for powerful new computational algorithms." ], [ "Acknowledgements", "R.M.", "gratefully acknowledges support from institute Ph.D. fellowship and N.N.", "gratefully acknowledges the support by DST SERB MATRICS grant MTR/2017/000 199.", "A.M acknowledges support from the DST-UKIERI and British Council funded project on \"Theoretical and Experimental Studies of Suspensions of Magnetic Nanoparticles, their Applications and Generalizations\" and support from IIT Bombay, and a Visiting Professorship from the University of Bath." ], [ "Appendix", "This section discusses the proofs of the local efficiency results in Lemmas REF -REF .", "The local cut off functions play an important role to establish the local efficiency results.", "Consider the interior bubble function [1], [34] $\\widehat{b}_T = 27\\widehat{\\lambda }_1 \\widehat{\\lambda }_2 \\widehat{\\lambda }_3 $ supported on a reference triangle $\\widehat{T}$ with the barycentric coordinate functions $\\widehat{\\lambda }_1 , \\widehat{\\lambda }_2, \\widehat{\\lambda }_3$ .", "For $T \\in \\mathcal {T}, $ let $\\mathcal {F}_T: \\widehat{T} \\rightarrow T $ be a continuous, affine and invertible transformation.", "Define the bubble function on the element $T$ by $b_T = \\widehat{b}_T \\circ \\mathcal {F}_T^{-1} $ .", "Three edge bubble functions on the reference triangle $\\widehat{T}$ are given by $\\widehat{b}_1= 4 \\widehat{\\lambda }_2 \\widehat{\\lambda }_3$ , $\\widehat{b}_2= 4 \\widehat{\\lambda }_1 \\widehat{\\lambda }_3$ and $\\widehat{b}_3= 4 \\widehat{\\lambda }_1 \\widehat{\\lambda }_2$ .", "On the edge $E$ of any triangle $T \\in \\mathcal {T}$ , define the edge bubble function to be $b_E: = \\widehat{b}_E \\circ \\mathcal {F}_T^{-1}$ , where $\\widehat{b}_E$ is the corresponding edge bubble function on $\\widehat{T}$ .", "Here, $b_E$ is supported on the pair of triangles sharing the edge $E.$ Lemma 8.1 [1], [34] Let $\\widehat{P} \\subset H^1(\\widehat{T})$ be a finite dimensional subspace on the reference triangle $\\widehat{T}$ and consider $P= \\lbrace \\widehat{v} \\circ \\mathcal {F}_T^{-1}:\\widehat{v} \\in \\widehat{P} \\rbrace $ to be the finite dimensional space of functions defined on $T$ .", "Then the following inverse estimates hold for all $v \\in P$ , ${\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{L^2(T)} \\lesssim \\int _T b_T v^2 \\,{\\rm dx}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{L^2(T)}, \\quad {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{L^2(T)} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert b_Tv\\vert \\hspace{-1.0625pt}\\vert }_{L^2(T)} + h_T{\\vert \\hspace{-1.0625pt}\\vert \\nabla (b_Tv)\\vert \\hspace{-1.0625pt}\\vert }_{L^2(T)} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{L^2(T)}.", "$ Let $E \\subset \\partial T$ be an edge and $b_E$ be the corresponding edge bubble function supported on the patch of triangles $\\omega _E$ sharing the edge $E$ .", "Let $P(E)$ be the finite dimensional space of functions defined on $E$ obtained by mapping $\\widehat{P}(\\widehat{E})\\subset H^1(\\widehat{E}).$ Then for all $v \\in P(E)$ , ${\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{L^2(E)} \\lesssim \\int _E b_E v^2 \\,{\\rm dx}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }^2_{L^2(E)}, \\quad \\,\\,\\, h_E^{ -\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert b_Ev\\vert \\hspace{-1.0625pt}\\vert }_{L^2(\\omega _E)}+ h_E^{ \\frac{1}{2}}{\\vert \\hspace{-1.0625pt}\\vert \\nabla (b_Ev)\\vert \\hspace{-1.0625pt}\\vert }_{L^2(\\omega _E)} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert v\\vert \\hspace{-1.0625pt}\\vert }_{L^2(E)}, $ where the hidden constants in $\"\\lesssim \"$ are independent of $h_T$ and $h_E$ .", "[Proof of Lemma REF] $(i)$ Let $T \\in \\mathcal {T}$ be arbitrary and $b_T$ be the interior bubble function supported on the triangle $T$ .", "Choose $\\displaystyle \\rho _T:=\\left\\lbrace \\begin{array}{l}\\big (-\\Delta \\Phi _{h}+2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h}\\big )b_T \\quad \\text{in } T\\\\0 \\quad \\text{in } \\Omega \\setminus T\\end{array}\\right.$ , utilize (REF ), () with $\\Phi :=\\rho _T$ and apply an integration by parts for the first term (which is a zero term) on the right-hand side below to obtain $& {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2\\lesssim \\int _T \\big (-\\Delta \\Phi _{h}+2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h}\\big ) \\cdot \\rho _T \\,{\\rm dx}\\\\&= A_T(\\Phi _{h}-\\Psi , \\rho _T ) +(B_T(\\Phi _{h},\\Phi _{h},\\Phi _{h},\\rho _T )-B_T(\\Psi ,\\Psi ,\\Psi ,\\rho _T ))+C_T(\\Phi _{h}-\\Psi ,\\rho _T ) .$ Together with Hölder's inequality, Lemma REF and (REF ), the terms on the right-hand side of (REF ) are estimated as $A_T(\\Phi _{h}-\\Psi , \\rho _T )\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Phi _{h}-\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\rho _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}\\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T} h_T^{-1}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}.\\\\C_T(\\Phi _{h}-\\Psi ,\\rho _T )\\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h-\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}\\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T},$ $B_T(\\Phi _{h},\\Phi _{h},\\Phi _{h},\\rho _T )-B_T(\\Psi ,\\Psi ,\\Psi ,\\rho _T )\\lesssim &\\epsilon ^{-2}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T}({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T}+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T})\\\\& + {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T}^2) h_T^{-1}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}.$ A combination of the above three displayed estimates in (REF ) plus Lemma REF establishes $h_T{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h, T} (1+ \\epsilon ^{-2} (1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}^2 +{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h, T} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1,T}) )).$ To find the estimate corresponding to $ \\eta _E$ , consider the edge bubble function $b_E$ supported on the patch of triangles $\\omega _E$ sharing the edge $E$ .", "Define $\\displaystyle \\rho _E:=\\left\\lbrace \\begin{array}{l}[\\nabla \\Phi _h \\nu ]b_E \\quad \\text{in } \\omega _E\\\\0 \\quad \\text{in } \\Omega \\setminus \\omega _E\\end{array}\\right.$ and use (REF ), $[\\rho _E] = 0$ for $E \\in \\mathcal {E}_h^i $ and an integration by parts to obtain ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 & \\lesssim \\int _{E}[\\nabla \\Phi _{h} \\nu ] \\cdot \\rho _E\\,{\\rm ds}= \\int _{E } [\\nabla \\Phi _{h} \\nu ] \\cdot \\lbrace \\ \\rho _E \\rbrace \\,{\\rm ds}+\\int _{E} \\lbrace \\nabla \\Phi _{h} \\nu \\rbrace \\cdot [\\rho _E] \\,{\\rm ds}\\\\& =\\sum _{T \\in \\omega _E} \\int _T (\\Delta \\Phi _{h} \\cdot \\rho _E + \\nabla \\Phi _{h} \\cdot \\nabla \\rho _E) \\,{\\rm dx}.$ Add and subtract $\\sum _{T \\in \\omega _E} \\int _T2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h} \\cdot \\rho _E \\,{\\rm dx}$ in the right-hand side of (REF ) to rewrite the expression with the help of $ \\eta _T= \\Delta \\Phi _{h}-2\\epsilon ^{-2}(\\vert \\Phi _{h}\\vert ^2 -1)\\Phi _{h}$ (with a $-\\Delta \\Phi _{h}=0$ added).", "The expression () with $\\Phi = \\rho _E$ , a re-grouping of terms and Hölder's inequality lead to ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}^2 &\\lesssim (\\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2)^{\\frac{1}{2}}(\\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _E \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}^2)^{\\frac{1}{2}} + \\sum _{T \\in \\omega _E} (A_T(\\Phi _{h}-\\Psi , \\rho _E) +C_T(\\Phi _{h}-\\Psi , \\rho _E)\\\\&\\quad + (B_T( \\Phi _{h}, \\Phi _{h}, \\Phi _{h},\\rho _E)-B_T( \\Psi , \\Psi , \\Psi ,\\rho _E))) .$ A combination of Hölder's inequality, Lemma REF and (REF ) yields $\\sum _{T \\in \\omega _E} A_T(\\Phi _{h}- \\Psi , \\rho _E) \\lesssim \\sum _{T \\in \\omega _E}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Psi -\\Phi _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\rho _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}\\lesssim h_E^{-\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla (\\Psi -\\Phi _{h})\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,\\omega _E},\\\\\\sum _{T \\in \\omega _E} C_T(\\Phi _{h}- \\Psi , \\rho _E)\\lesssim \\epsilon ^{-2}\\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} \\lesssim \\epsilon ^{-2}h_E^{\\frac{1}{2}}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,\\omega _E},$ $\\sum _{T \\in \\omega _E} (B_T( \\Phi _{h}, \\Phi _{h}, \\Phi _{h},\\rho _E)-B_T( \\Psi , \\Psi , \\Psi ,\\rho _E) ) \\lesssim & \\epsilon ^{-2} h_E^{-\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}\\sum _{T \\in \\omega _E}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T} \\\\&+{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T})+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}^2).$ The estimate of ${\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}$ in (REF ) and (REF ) together with the above three displayed estimates in (REF ) lead to $h_E^{\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E} \\lesssim \\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi - \\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h, T} (1+ \\epsilon ^{-2} (1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}^2 +{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi -\\Phi _{h}\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{h, T} ({\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Phi _h\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1}+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1, T}) )).$ A combination of (REF ) and REF completes the proof of $(i)$ in Lemma REF .", "$(ii)$ For $\\Phi _{h}={ \\rm I}_h \\Psi $ in (REF ), Lemma REF $(v)$ and (REF ) yield $B_T({\\rm I}_{h} \\Psi ,{\\rm I}_{h} \\Psi , {\\rm I}_{h} \\Psi , \\rho _T)- B_T( \\Psi , \\Psi , \\Psi , \\rho _T) &\\lesssim \\epsilon ^{-2}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha ,T}^3 (h_T^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\rho _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}+ h_T^{1+\\alpha }{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T})\\\\& \\lesssim \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha ,T}^3(h_T^{2\\alpha } + h_T^{2+\\alpha } )h_T^{-1}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T} .$ Substitute (REF ), (), (REF ) in (REF ) and utilize Lemma REF to arrive at $h_T{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _T\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}& \\lesssim {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla ({\\rm I}_{h} \\Psi -\\Psi )\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}+ \\epsilon ^{-2} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert {\\rm I}_{h} \\Psi -\\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}+ \\epsilon ^{-2} h_T^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^3\\\\& \\lesssim h_T^{\\alpha } (1+ \\epsilon ^{-2}h_T^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ A choice of $\\Phi _h = {\\rm I}_h \\Psi $ in (REF ), Lemma REF $(v)$ and (REF ) yield $&\\sum _{T \\in \\omega _E}( B_T({\\rm I}_{h} \\Psi ,{\\rm I}_{h} \\Psi , {\\rm I}_{h} \\Psi , \\rho _E)- B_T( \\Psi , \\Psi , \\Psi , \\rho _E)) \\lesssim \\epsilon ^{-2}\\sum _{T \\in \\omega _E}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha , T}^3 (h_T^{2\\alpha } {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\nabla \\rho _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T}+ h_T^{1+\\alpha }{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\rho _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,T})\\\\&\\qquad \\lesssim \\epsilon ^{-2} h_E^{-\\frac{1}{2}}{\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E} \\sum _{T \\in \\omega _E} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha ,T}^3(h_T^{2\\alpha } +h_E h_T^{2+\\alpha } ) .$ Substitute (REF ), (), (REF ) in (REF ) and employ Lemma REF to obtain $ h_E^{\\frac{1}{2}} {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\eta _E\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{0,E}\\lesssim \\sum _{T \\in \\omega _E} h_T^{\\alpha } (1+ \\epsilon ^{-2}h_T^{\\alpha }(1+ {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }^2)) {\\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert \\Psi \\vert \\hspace{-1.0625pt}\\vert \\hspace{-1.0625pt}\\vert }_{1+\\alpha }.$ A combination of (REF ) and (REF ) concludes the proof of $(ii)$ in Lemma REF .", "The proof of Lemma REF (resp.", "REF ) follows analgous to the proof of Lemma REF with the choice of $\\rho _T:=\\left\\lbrace \\begin{array}{ll@{:}}(\\Delta (\\textrm {I}_h \\xi ) +2 \\epsilon ^{-2}(\\vert \\textrm {I}_h \\Psi \\vert ^2 \\Theta _{h} + 2 (\\textrm {I}_h \\Psi \\cdot \\Theta _{h}) \\textrm {I}_h \\Psi -\\Theta _{h}) )b_T \\text{ in } T\\\\0 \\quad \\text{in } \\Omega \\setminus T\\end{array}\\right.\\text{ and }\\rho _E:=\\left\\lbrace \\begin{array}{l}[\\nabla (\\textrm {I}_h\\xi ) \\nu ] b_E \\text{ in } \\omega _E\\\\0 \\quad \\text{in } \\Omega \\setminus \\omega _E\\end{array}\\right..$ $ \\bigg (\\text{resp.", "}\\rho _T:=\\left\\lbrace \\begin{array}{l}\\big (G_h +\\Delta ( \\textrm {I}_{h}\\chi ) -2\\epsilon ^{-2} (\\vert \\textrm {I}_{h}\\Psi \\vert ^2 \\textrm {I}_{h}\\chi +2 (\\textrm {I}_{h} \\Psi \\cdot \\textrm {I}_{h}\\chi )\\textrm {I}_{h} \\Psi - \\textrm {I}_{h}\\chi )\\big )b_T \\quad \\text{in } T\\\\0 \\quad \\text{in } \\Omega \\setminus T\\end{array}\\right.", "$ $\\text{ and } \\rho _E:=\\left\\lbrace \\begin{array}{l}[ \\nabla (\\textrm {I}_h\\chi ) \\nu ]b_E \\quad \\text{in } \\omega _E\\\\0 \\quad \\text{in } \\Omega \\setminus \\omega _E\\end{array}\\right.", "\\bigg ).$" ] ]

[ [ "Magnetic properties of quasi-one-dimensional lanthanide calcium\n oxyborates Ca$_4$LnO(BO$_3$)$_3$" ], [ "Abstract This study examines the lanthanide calcium oxyborates Ca$_4$LnO(BO$_3$)$_3$ (Ln = La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Y, Er, Yb).", "The reported monoclinic structure (space group $Cm$) was confirmed using powder X-ray diffraction.", "The magnetic Ln$^{3+}$ ions are situated in well-separated chains parallel to the c axis in a quasi-one-dimensional array.", "Here we report the first bulk magnetic characterisation of Ca$_4$LnO(BO$_3$)$_3$ using magnetic susceptibility $\\chi$(T) and isothermal magnetisation M(H) measurements at T $\\geq$ 2 K. With the sole exception of Ca$_4$TbO(BO$_3$)$_3$, which displays a transition at T = 3.6 K, no magnetic transitions occur above 2 K, and Curie-Weiss analysis indicates antiferromagnetic nearest-neighbour interactions for all samples.", "Calculation of the magnetic entropy change $\\Delta S_m$ indicates that Ca$_4$GdO(BO$_3$)$_3$ and Ca$_4$HoO(BO$_3$)$_3$ are viable magnetocaloric materials at liquid helium temperatures in the high-field and low-field regimes respectively." ], [ "Introduction", "An ideal one-dimensional (1D) system – a single chain of magnetic ions – would never display long-range order [1], [2].", "Such systems have been predicted to host exotic magnetic behaviour such as spinons [3], but are impossible to realise in the solid state.", "Certain crystal structures may, however, be quasi-1D if the spin chains are kept well separated by nonmagnetic atoms, but in most cases there is weak coupling between the chains, leading to 3D ordering at low temperatures.", "Extensive work has been carried out on quasi-1D $S=\\frac{1}{2}$ systems containing first-row transition metals, but much less is known about lanthanide systems [4], [5], [6], [7], [8].", "These have very different magnetic properties from the 3$d$ transition metals as a result of strong spin-orbit coupling, and the interplay between superexchange ($J$ ), dipolar ($D$ ) and crystal electric field (CEF) effects means that they are likely to require different models for the magnetism.", "Examples of quasi-1D lanthanide systems include the hydroxycarbonates LnOHCO$_3$ and formates Ln(HCOO)$_3$ .", "Both series of compounds have been shown to display unusual magnetic properties and to be viable magnetocaloric materials at temperatures below 10 K, although in some cases inter-chain coupling produces three-dimensional ordering at temperatures below 2 K [9], [10], [11].", "Low-dimensional magnetic systems are of interest for solid-state magnetic refrigeration as a more sustainable alternative to liquid helium for cooling to low temperatures.", "This technology relies on the magnetocaloric effect (MCE) arising from adiabatic demagnetisation of the sample: this causes the temperature to drop as magnetic domains de-align from the field direction, enabling large amounts of magnetic entropy to be extracted.", "The lower limit for magnetic cooling is set by the long-range magnetic ordering temperature of the sample; thus, low-dimensional materials are useful as they usually display suppression of this ordering temperature.", "Lanthanide compounds have been widely studied for this purpose, particularly those containing Gd$^{3+}$ , which has a large spin compared to the transition metals, and usually[12] no crystal field interactions compared with the other lanthanides, since $L=0$ [13], [14], [10], [9].", "For Gd$^{3+}$ (e.g.", "Gd$_3$ Ga$_5$ O$_{12}$ , `GGG') the MCE is maximised in high fields ($\\mu _0H > 5$  T), where Heisenberg systems have been shown to be optimised, but obtaining such fields still requires liquid helium to cool the superconducting magnets.", "At low fields $\\mu _0H \\le 2$  T a permanent magnet can be used to provide the external field; systems with significant single-ion anisotropy (e.g.", "Dy$_3$ Ga$_5$ O$_{12}$ ) tend to perform better in this regime [14], [15].", "The lanthanide calcium oxyborates with general formula Ca$_4$LnO(BO$_3$ )$_3$ (Ln = Y, La–Lu) have previously been investigated as nonlinear optic materials [16], [17].", "The synthesis of these compounds in powder form may be carried out through straightforward solid-state or sol-gel procedures [18], [19].", "The lanthanide ions in the unit cell are arranged in chains parallel to the $c$ -axis (Fig.", "REF ) with ions separated by $c \\approx $ 3.6 Å along the chains.", "These chains are well separated in the $ab$ plane by 8–9 Å, leading to a quasi-1D Ln$^{3+}$ array.", "Figure: Top: Monoclinic crystal structure of Ca 4 _4LnO(BO 3 _3) 3 _3 (Ln = Y, La–Lu).", "O atoms in red; CaO 6 _6 and LnO 6 _6 distorted octahedra in blue and purple respectively; trigonal planar (BO 3 _3) 3- ^{3-} groups in green.", "Bottom: Connectivity of Ln 3+ ^{3+} ions in Ca 4 _4LnO(BO 3 _3) 3 _3: intra-chain (solid lines, 3.6–3.8 Å) and inter-chain (dashed lines, 8.1–8.3 Å, and dotted lines, 9.0–9.2 Å).In this article we report the solid-state synthesis of twelve compounds with the formula Ca$_4$LnO(BO$_3$ )$_3$ from across the lanthanide series, followed by structural characterisation using powder X-ray diffraction (PXRD).", "Furthermore we report the bulk magnetic characterisation of these compounds using magnetic susceptibility and isothermal magnetisation measurements at $T \\ge 2$  K. Their potential application in magnetic refrigeration applications is discussed with the aid of magnetic entropy calculations." ], [ "Experimental", "Polycrystalline samples of Ca$_4$LnO(BO$_3$ )$_3$ (Ln = La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Y, Er, Yb) were synthesised according to a ceramic procedure, adapted from Ref.", "Crossno1997, from CaCO3 (99.99 %), H3BO3 (99.999 %) and Ln$_2$ O$_3$ (Ln = La, Nd, Sm, Eu, Gd, Dy, Ho, Y, Er, Yb), Pr6O11 or Tb4O7 (all lanthanide oxides $\\ge $  99.99 %).", "Lanthanide oxides were pre-dried at 800 C overnight prior to weighing out.", "Stoichiometric amounts of the reagents (except in the case of Ln = Yb: see below) were ground with a pestle and mortar and placed in an alumina crucible.", "The powder was first heated in air at 900 C for 4 h in order to effect decomposition of the boric acid and calcium carbonate.", "The sample was subsequently cooled, reground and reheated to 1200 C for several days, with intermediate regrinding every 24 hours, until the percentages of impurity phases no longer changed.", "Room temperature PXRD patterns were collected on a Bruker D8 diffractometer (Cu K$\\alpha $ , $\\lambda = 1.541$ Å) in the range $10 \\le 2\\theta () \\le 90$ with a step size of 0.01 and measurement time 1 second per step.", "Rietveld refinement[21] was carried out using the program Topas.", "[22] Magnetic susceptibility and isothermal magnetisation were measured on a Quantum Design 9 T Physical Properties Measurement System using the ACMS-II option in the temperature and field ranges $2 \\le T(K) \\le 300$ and $0 \\le \\mu _0H(T) \\le 9$ respectively.", "In a low field of 500 Oe, the $M(H)$ curve is linear for all $T$ and the susceptibility can therefore be approximated by $\\chi (T)=M/H$ ." ], [ "Crystal Structure", "From PXRD and Rietveld refinement all samples were found to adopt the previously reported monoclinic Cm structure (Fig.", "REF ); a representative X-ray refinement is given in Fig.", "REF [18], [23], [20].", "The unit cell contains two independent Ca$^{2+}$ sites, both six-coordinate but distorted from perfectly octahedral geometry.", "Additionally the crystal structure contains distorted LnO$_6$ octahedra, which share edges along the `chains', and trigonal planar BO$_3^{3-}$ groups, which are tilted at different angles from the $c$ -axis.", "A previous study found that the tilting of each of these borate groups changed in a regular way as the size of the lanthanide ions was varied [20], but such analysis is beyond the scope of this work, which is limited by the powder samples and solely X-ray diffraction measurements.", "In some cases small amounts, $\\le 5$ wt %, of non-magnetic impurities (Ca3(BO3)2, H3BO3) remained after multiple heating steps: for details, see Table REF .", "For Ca$_4$ YbO(BO$_3$ )$_3$ , 50 % excess H3BO3 was required to ensure complete reaction of Yb2O3, leading to a higher proportion of Ca3(BO3)2 in the final sample.", "There is also a larger proportion of Ca3(BO3)2 in Ca$_4$ LaO(BO$_3$ )$_3$ than in the other samples; this is attributed to either incomplete drying or reabsorption of water into La2O3 before weighing out.", "The series Ca$_4$LnO(BO$_3$ )$_3$ obeys Vegard's Law for variation of lattice parameters (Fig.", "REF ).", "The insensitivity of X-ray diffraction to oxygen and boron in the presence of heavy elements (Ca, Ln) required the atomic positions of O and B to be fixed at previously reported values [24], while the atomic coordinates of Ca were refined (Table REF ).", "Considering ionic radii, the trivalent lanthanide ions range in size from 103.2 pm (La) down to 86.8 pm (Yb) while 6-coordinate Ca$^{2+}$ is 100 pm [25]; the extent of cation mixing might then be expected to increase with increasing Ln$^{3+}$ radius.", "The presence, if any, of Ca$^{2+}$ /Ln$^{3+}$ site disorder was tested by setting a suitable mixed Ca$^{2+}$ and Ln$^{3+}$ occupancy in each of the three metal sites and refining with the overall ratio fixed at 4:1.", "No significant site disorder was observed for any compound, however, in agreement with a previous single-crystal study of Ca$_4$ LaO(BO$_3$ )$_3$ [16].", "The fractional occupancies were therefore fixed at 1 for each site.", "Figure: Room temperature PXRD pattern for Ca 4 _4DyO(BO 3 _3) 3 _3: Red dots – experimental data; black line – calculated intensities; green line – difference pattern; blue tick marks – Bragg reflection positions.Figure: Unit cell volume of all Ca 4 _4LnO(BO 3 _3) 3 _3 samples as a function of lanthanide ionic radius.", "Error bars (from individual refinements) are smaller than the datapoints.", "The dashed line is given as a guide to the eye.Table: Refined crystal structure parameters for Ca 4 _4LnO(BO 3 _3) 3 _3 samples, from room-temperature PXRD refinements in space group Cm.", "Due to the low scattering power of B and O compared with Ca and Ln, the boron and oxygen atomic positions were kept fixed at the following general sites as given for Ca 4 _4GdO(BO 3 _3) 3 _3 : B1 (2aa) = (0.3764, 0, 0.7011); B2 (4bb) = (0.9491, 0.1947, 0.0798); O1 (2aa) = (0.8252, 0, 0.4175); O2 (4bb) = (0.4614, 0.9257, 0.7492); O3 (2aa) = (0.2032, 0, 0.6043); O4 (4bb) = (0.0859, 0.1434, 0.0766); O5 (4bb) = (0.9675, 0.2695, 0.2746).", "The lanthanide ion is at site 2aa = (0, 0, 0).", "The values of thermal parameters for all atoms were kept fixed at B iso B_{iso} = 1 Å 2 ^2." ], [ "Bulk magnetic properties", "The zero-field-cooled magnetic susceptibility curves, collected on warming at $H$  = 500 Oe, are shown in Fig.", "REF for Ln = Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er and Yb.", "The samples containing Sm and Eu do not obey the Curie-Weiss law ($\\chi = \\frac{C}{T-\\theta _{CW}}$ ), but display van Vleck paramagnetism due to the mixing of ground states with low-lying excited states [26].", "The broad feature at $T > $ 6 K for Ca$_4$ PrO(BO$_3$ )$_3$ is attributed to a singlet ground state with van Vleck paramagnetism, in accordance with previous reports of Pr compounds [27], [28], [29].", "A broad magnetic transition characteristic of low-dimensional ordering is visible at $T$ = 3.6 K for Ln = Tb, while for all other Ca$_4$LnO(BO$_3$ )$_3$ no transition occurs at $T \\ge $ 2 K. Linear Curie-Weiss fitting was carried out in both high-temperature (50–150 K) and low-temperature regimes.", "The susceptibility of lanthanide compounds is strongly dependent on crystal electric field effects and the low-temperature fitting ranges were therefore varied depending on the lanthanide ion in question [30], [31].", "The resultant magnetic parameters are given in Table REF .", "All Curie-Weiss temperatures $\\theta _{CW}$ are negative, indicating antiferromagnetic interactions.", "The effective magnetic moment per Ln$^{3+}$ ion was calculated from the Curie constant $C$ for each compound and the high $T$ values agree well with the theoretical free-ion value $g_J\\sqrt{J(J+1)}$ within the bounds of experimental error.", "The Curie constants and thus the effective magnetic moments are broadly consistent between the two fitting regimes.", "Figure: Left: Magnetic susceptibility as a function of temperature for the Ca 4 _4LnO(BO 3 _3) 3 _3 samples with Ln = Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er and Yb.", "Right: Reciprocal magnetic susceptibility, χ -1 \\chi ^{-1}.Table: Bulk magnetic properties of Ca 4 _4LnO(BO 3 _3) 3 _3, Ln = Pr, Nd, Gd–Er, Yb.Isothermal magnetisation data for the Ca$_4$LnO(BO$_3$ )$_3$ compounds are shown in Fig.", "REF .", "We conclude that Ca$_4$ PrO(BO$_3$ )$_3$ has a singlet ground state, as observed in other Pr compounds [27], [32], [28], [29].", "The magnetisation saturates at 2 K and 9 T in all other compounds.", "For an isotropic (Heisenberg) spin system the magnetisation is expected to saturate at a value of $g_JJ$ , whereas easy-axis (Ising) systems tend to saturate at half this value, although other contributions may increase $M_{sat}$ above $g_JJ/2$ [33], [10].", "In particular, the exact saturation value for a particular lanthanide ion varies depending on the local point group symmetry (i.e.", "CEF) of the atomic site; further experiments such as inelastic neutron scattering are required in order to confirm the anisotropy.", "The data for Ca$_4$LnO(BO$_3$ )$_3$ indicate that Ca$_4$ GdO(BO$_3$ )$_3$ is likely a Heisenberg spin system, with the remaining compounds showing substantial local single-ion anisotropy.", "These results are consistent with other lanthanide compounds such as the Ln$_3$ Sb$_3$ Zn$_2$ O$_{14}$ kagome lattices [32], titanate pyrochlores [33], [34], gallium and aluminium garnets [35], [36], hydroxycarbonates [10] and metaborates [29].", "Figure: Magnetic susceptibility as a function of applied field for the Ca 4 _4LnO(BO 3 _3) 3 _3 samples with Ln = Pr, Nd, Gd, Tb, Dy, Ho, Er and Yb." ], [ "Discussion", "The trivalent Ln$^{3+}$ ions have highly localised 4$f$ orbitals, meaning that the superexchange between ions – which depends on the orbital overlap – is smaller than for the first-row transition metals.", "We therefore expect superexchange ($J_{nn}$ , along the 1D spin chains) to be of a similar magnitude to, or much larger than the dipolar interactions ($D$ ) depending on the electronic configuration of the lanthanide ion in question, and hence its magnetic moment.", "The superexchange can be estimated in the mean-field, isotropic approximation using $J_{nn} = \\frac{3k_B\\theta _{CW}}{2nS(S+1)}$ where $S$ in the denominator is the total spin quantum number and $n$ is the number of nearest-neighbour spins (here $n$ = 2) [37].", "For systems containing lanthanide ions the spin-orbit coupling cannot be neglected and the quantum number $J=|L \\pm S|$ is usually substituted for $S$ in equation REF , depending on whether the shell is more or less than half-filled.", "The mean-field approximation is in general a poor model for lanthanide systems due to the strong single-ion anisotropy which is often observed.", "However, in the absence of inelastic neutron spectroscopy (INS) data the mean-field approximation does allow us to infer some information regarding the magnetic interactions.", "In the case of a two-level spin system, it is more appropriate to take $S_{eff} = \\frac{1}{2}$ .", "This is true for Nd$^{3+}$ , Dy$^{3+}$ , Er$^{3+}$ and Yb$^{3+}$ , which are Kramers ions with an odd number of $f$ -electrons and therefore symmetry-constrained to have an $S_{eff} = \\frac{1}{2}$ doublet ground state at low temperatures.", "Examples may be found in references Ashtar2019,Gao2018,Xing2019,Li2015.", "For non-Kramers ions (Tb$^{3+}$ , Ho$^{3+}$ ) this constraint does not apply if the local point group symmetry of the ion is lower than cubic [41], as is the case here (Ln$^{3+}$ ions in orthorhombic sites).", "Tb$^{3+}$ and Ho$^{3+}$ have been reported to have $S_{eff} = \\frac{1}{2}$ ground states due to mixing of two low-lying singlet states in some compounds[42], [43] but this is not the case for Ho3Mg2Sb3O14 and Ho3Ga5O12 [28], [44].", "In the absence of INS data we cannot confirm the spin anisotropy for the Ca$_4$LnO(BO$_3$ )$_3$ compounds with Ln = Nd, Tb, Dy, Ho, Er and Yb, and will therefore calculate the superexchange using both $J=L \\pm S$ and $S_{eff} = \\frac{1}{2}$ .", "The dipolar interaction $D$ may be estimated using $D = \\frac{-\\mu _0\\mu ^2_{eff}}{4\\pi r^3}$ where $r$ is the distance between adjacent Ln$^{3+}$ ions in the same or neighbouring chains [45].", "This is a general expression for $D$ which does not take into account any single-ion anisotropy.", "Furthermore, a true calculation for $D$ would be long-range and cover many spins.", "As with equation REF , however, we here use the general expression to provide a ballpark estimate for $D$ under the stated approximations, since INS data are not yet available.", "The sizes of the dipolar and nearest-neighbour exchange interactions were estimated using the magnetic susceptibility data (low-temperature fitting) and the resulting parameters are given in Table REF .", "It has been proposed that the quasi-one-dimensional nature of the magnetism in these compounds is related to the relative sizes of the exchange and dipolar interactions [29].", "Regardless of the size of $J$ , we expect to see quasi-1D magnetic behaviour with signatures of low-dimensional ordering at low temperatures (as may be the case for Ca$_4$ TbO(BO$_3$ )$_3$ ) because $D_{intra-chain}$ is an order of magnitude larger than $D_{inter-chain}$ .", "At this stage we refrain from drawing solid conclusions about the true one-dimensional nature of these compounds, due to the absence of measurements at $T < 2$  K and the potential inadequacy of the mean-field approximations used to derive the interaction energies.", "Table: Dipolar (DD) and nearest-neighbour exchange (J nn J_{nn}) interactions for Ca 4 _4LnO(BO 3 _3) 3 _3, Ln = Pr, Nd, Gd, Tb, Dy, Ho, Er and Yb.", "The negative signs indicate antiferromagnetic interactions.The potential for these compounds to act as magnetocaloric materials was quantified by calculating the change in magnetic entropy per mole, $\\Delta S_m$ , according to the Maxwell thermodynamic relation [46]: $\\Delta S_m = \\int _{H_0}^{H_1} \\left( \\frac{\\partial M}{\\partial T}\\right)_H dH$ Magnetocaloric data for selected Ca$_4$LnO(BO$_3$ )$_3$ samples as a function of applied field are shown in Fig.", "REF .", "Ca$_4$ GdO(BO$_3$ )$_3$ provides the optimal MCE in fields $5 < \\mu _0H$ (T) $ < 9$ , while Ca$_4$ HoO(BO$_3$ )$_3$ is the best magnetocaloric material in this family at fields below 5 T. Selected data for the Ca$_4$LnO(BO$_3$ )$_3$ compounds are compared with the standard magnetocaloric materials Gd3Ga5O12 (GGG) and Dy3Ga5O12 (DGG) at low and high fields in Table REF [14], [36].", "Here we define a low field as $\\mu _0H \\le $  2 T, which is the largest field attainable with a permanent magnet as opposed to a superconducting one.", "We find that at $\\mu _0H = 9$  T, Ca$_4$ GdO(BO$_3$ )$_3$ is competitive with GGG in terms of MCE per Gd$^{3+}$ ion, but in terms of MCE per kilogram it performs poorly due to the four heavy Ca$^{2+}$ ions per formula unit.", "At a low field of 2 T, Ca$_4$ HoO(BO$_3$ )$_3$ has a significantly higher MCE per mole of lanthanide ion than DGG, and a comparable gravimetric MCE.", "The straightforward, scaleable solid-state synthesis of Ca$_4$LnO(BO$_3$ )$_3$ , combined with very low ($T < 4$  K) magnetic ordering temperatures, makes these materials attractive candidates for magnetic refrigeration applications.", "Tuning of the MCE at different temperatures or fields may be possible through partial chemical substitution of the lanthanide ions [15].", "We note particularly that this structure type is highly flexible in allowing substitution of a range of lanthanide ions of different sizes, without any cation mixing that would destroy the quasi-1D magnetic structure.", "Partial substitution of Sr$^{2+}$ for Ca$^{2+}$ has also been reported, further increasing the family of related compounds [20].", "Figure: Molar (top) and gravimetric (bottom) magnetocaloric data for Ca 4 _4LnO(BO 3 _3) 3 _3 (Ln = Nd, Gd, Tb, Dy, Ho, Er and Yb; filled symbols) at TT = 2 K as a function of applied field, compared with Gd3Ga5O12 (open circles) and Dy3Ga5O12 (open triangles) .Table: Comparison of ΔS m \\Delta S_m (TT = 2 K) in molar, gravimetric and volumetric units for selected Ca 4 _4LnO(BO 3 _3) 3 _3 compounds and the standard magnetocaloric materials , ." ], [ "Conclusions", "Twelve compounds in the series Ca$_4$LnO(BO$_3$ )$_3$ (Ln = Y, La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Yb) have been synthesised using a straightforward and repeatable solid-state procedure.", "The reported structure has been confirmed using X-ray diffraction.", "Bulk magnetic characterisation indicates that the quasi-one-dimensional nature of these materials leads to suppression of the magnetic ordering temperatures, which additionally makes these materials good candidates for magnetic refrigeration applications at liquid helium temperatures.", "These results contribute to the rapidly growing set of lanthanide–alkaline earth borates with low-dimensional structures and novel magnetic properties, such as the Sr$_6$LnFe(BO$_3$ )$_6$ , Ba$_3$Ln(BO$_3$ )$_3$ and ABaLn(BO$_3$ )$_2$ (A = Na$^+$ , K$^+$ , Rb$^+$ ) structural families [48], [38], [49], [31], [50], [51].", "Furthermore, the Ca$_4$LnO(BO$_3$ )$_3$ structure type is compositionally flexible, which should allow for the realisation of novel low-dimensional magnetic lattices through chemical substitution.", "We acknowledge funding from the EPSRC for a PhD studentship and the use of the Advanced Materials Characterisation Suite (EPSRC Strategic Equipment Grant EP/M000524/1).", "NDK thanks J. Paddison, P. Mukherjee and J. Tuffnell for useful discussions and C. Liu for assistance in collecting the magnetic data.", "Data files relating to this article are available at https://doi.org/10.17863/CAM.52921." ] ]

[ [ "The Effect of Substellar Continent Size on Ocean Dynamics of Proxima\n Centauri b" ], [ "Abstract The potential habitability of tidally locked planets orbiting M-dwarf stars has been widely investigated in recent work, typically with a non-dynamic ocean and without continents.", "On Earth, ocean dynamics are a primary means of heat and nutrient distribution.", "Continents are a critical source of nutrients, strongly influence ocean dynamics, and participate in climate regulation.", "In this work, we investigate how the size of a substellar land mass affects the oceans ability to transport heat and upwell nutrients on the tidally locked planet Proxima Centauri b using the ROCKE-3D coupled ocean-atmosphere General Circulation Model (GCM).", "We find that dayside ice-free ocean and nutrient delivery to the mixed layer via upwelling are maintained across all continent sizes.", "We also find that Proxima Centauri bs climate is more sensitive to differences among atmospheric GCMs than to the inclusion of ocean dynamics in ROCKE-3D.", "Finally, we find that Proxima Centauri b transitions from a lobster state where ocean heat transport distributes heat away from the substellar point to an eyeball state where heat transport is restricted and surface temperature decreases symmetrically from the substellar point when the continent size exceeds about 20 percent of the surface area.", "Our work suggests that both a dynamic ocean and continents are unlikely to decrease the habitability prospects of nearby tidally locked targets like Proxima Centauri b that could be investigated with future observations by the James Webb Space Telescope (JWST)." ], [ "Introduction", "M-dwarf stars comprise nearly 80% of all stars in the galaxy and about one out of six M-dwarfs have an Earth-sized planet within their Habitable Zone [7].", "The Habitable Zone (HZ) is defined as the range of distances from the star in which a planet can have liquid water at its surface [17].", "M-dwarfs stars are smaller and dimmer than the Sun, and consequently their HZs are much closer to the star, which makes a HZ planet likely to become tidally locked in a 1:1 spin orbit state in which one side of the planet is in constant daytime while the other is in perpetual night.", "Many planets in the HZ of M-dwarfs have already been observed [1], [10], [25].", "The discovery of Proxima Centauri b was especially exciting, because it orbits an M-dwarf star only 4.2 lightyears away and is very likely to be tidally locked [1].", "Its proximity to Earth will make it a prime target for future observations with the James Webb Space Telescope (JWST).", "These exciting discoveries have spurred theoretical research into the habitability of tidally locked planets like Proxima Centauri b.", "These habitability studies have mostly used Global Climate Models (GCMs) that assume either no continents [13], [9], [4], [42], a static “slab” ocean [15], [23], [44], [43], [3], or both [29], [39], [36], [2], [12], [18], [32].", "These studies tend to agree that without a dynamic ocean, the climate of a tidally locked planet will settle into an “eyeball” state [31] characterized by roughly radially symmetric surface temperatures that decrease away from the substellar point.", "On Earth, ocean dynamics play a vital role in heat and nutrient distribution.", "The importance of ocean circulation on Earth has inspired a few recent studies of ocean circulation on extrasolar planets [5], [13], [46], [4], [14], [30], [42].", "These have established that on tidally locked plants, ocean dynamics play a key role in heat redistribution from the dayside to the nightside.", "Moreover, a dynamic ocean introduces asymmetries in the surface temperature distribution that have been described as a “lobster” state because of the way the ocean transports heat from the substellar point towards the nightside of the planet [13].", "This greatly expands the area of open ocean and increases global average temperatures.", "Additionally, ocean dynamics are an important control on the replenishment of essential nutrients to the mixed layer that are lost when biomass sinks to the bottom of the ocean (the so-called biological pump, [27]).", "The mixed layer is the area of the surface ocean that has nearly uniform properties because it is well mixed.", "The depth of the mixed layer varies spatially and is determined by the vertical potential density gradient, which is influenced by the sea surface temperature and salinity, as well as wind strength near the surface.", "Upwelling into the mixed layer is generally discouraged by stable density stratification, with warm sunlit surface water sitting atop cold deep water.", "Upwelling occurs in regions where ocean currents diverge such that conservation of mass requires deep water to move upward into the mixed layer.", "Ocean dynamics therefore exert direct control on the distribution of nutrients in the surface ocean, and therefore photosynthetic life on Earth.", "In an exoplanet context, [30] explored oceanographic constraints on life, such as areas of upwelling.", "They argued that planets with higher rates of upwelling may be more favorable for photosynthetic life.", "Continental configuration is an important control on both ocean currents and climate.", "The importance of continents for ocean circulation and climate can be seen in idealized GCM simulations where only slivers of continents can disrupt ocean flows and lead to major reorganizations in climate [8], [33], [26].", "Moreover, the carbonate-silicate cycle requires precipitation on exposed continents to regulate atmospheric CO$_2$ [37].", "Continental weathering also delivers nutrients to the ocean that are essential for oceanic life, potentially making continents necessary not only for long-term climate stability, but also for life.", "On tidally locked planets, we expect continents to impose similar constraints on ocean currents and climate.", "In the context of tidally locked planets, [41] suggested that true polar wander will align topographical anomalies with the planet-star axis, meaning that continents will tend to congregate at the substellar or anti-stellar point [22].", "[24] used the Met Office Unified Model (UM) to explore how substellar continents affect the climate of Proxima Centauri b with no ocean dynamics.", "They found that the habitability of Proxima Centauri b is largely insensitive to the size of the substellar continent.", "For continents ranging from 4% to 39% of the total surface area of the planet, a patch of above-freezing temperatures persisted at the substellar point and temperatures elsewhere never fell below 125 K, avoiding condensation of CO$_2$ onto the surface.", "Increasing continent size also increased temperature contrasts between the day and the nightside and tended to cool the planet on a global average.", "In this work, we use the ROCKE-3D coupled ocean-atmosphere General Circulation Model (GCM) to investigate how substellar continent size on Proxima Centauri b affects two vital controls on habitability: ocean heat transport and nutrient upwelling.", "In Section , we describe our model setup with ROCKE-3D, including assumptions of planetary parameters of Proxima Centauri b.", "In Section 3, we investigate the effect of substellar continent size on ocean heat transport and nutrient upwelling on tidally locked planets.", "We discuss the effect of ocean dynamics on the habitability of Proxima Centauri b and the potential observability of biosignatures in Section and conclude in Section 5." ], [ "Methods", "We use ROCKE-3D [40], a coupled ocean-atmosphere 3D GCM adapted from the Goddard Institute for Space Studies (GISS) ModelE2.", "We use a 4x 5 latitude-longitude resolution with 40 vertical layers in the atmosphere and a topmost layer pressure of 0.1 mb.", "ROCKE-3D uses the radiative transfer model SOCRATES, described in detail by [38].", "SOCRATES was developed for the UM and was also used by [24].", "The stellar spectrum for Proxima Centauri was prepared by [28], with $T_{eff} = 3042$ K and $R = 9.8\\times 10^{7}$ m. We explore two ocean regimes to isolate the effect of ocean dynamics.", "The first is a slab ocean with constant 24 m mixed layer depth and lateral ocean heat transport set to zero (hereafter, no OHT), following [24].", "The second is a fully coupled dynamic ocean with a 5 layer resolution and a prescribed depth of 150 m. We use a relatively shallow ocean to reduce computational time.", "In both cases, sea surface salinity is 34.7 psu.", "Ocean, land, sea ice, and snow albedo prescriptions are described in detail by [40].", "We ensure that our simulations reach convergence in global mean top-of-atmosphere net radiative flux.", "Following [24], we model our planet after Proxima Centauri b with an assumed surface gravity, $g = 10.98$ m/s$^2$ , and planetary radius, $r_p = 7127$ km.", "The planet receives an incident stellar flux of 881.7 W/m$^2$ .", "We assume that Proxima Centauri b is tidally locked with an orbital period of 11.19 Earth days, and has zero eccentricity and obliquity.", "We assume a surface pressure of 1 bar and an N$_2$ -dominated atmosphere with 280 ppm of CO$_2$ .", "The substellar point is located at the equator at 0 longitude.", "Continents are centered at the substellar point, and have sizes ranging from 0% (aqua planet) to 39% of the total surface area of the planet.", "Following [30], we calculate upwelling as the vertical velocity at the base of the mixed layer.", "The mixed layer is the area of the ocean that has nearly uniform properties and is in equilibrium with the surface.", "ROCKE-3D defines the mixed layer depth (MLD) as the depth at which the potential density is 0.03 kg/m$^3$ higher than the surface [21].", "We further classify upwelling as coastal upwelling if any of the adjacent cells are land.", "We are especially interested in dayside upwelling since photosynthetic life requires access to sunlight.", "Figure: Continents at the substellar point inhibit ocean heat transport, limiting the effect of ocean dynamics.", "As continent size increases, the planet transitions from a “lobster state\" to an “eyeball state\", characteristic of a tidally locked planet with no ocean dynamics.", "The center of each map is the substellar point, and white boxes represent continents.", "The colorbars correspond to the entire row of plots.", "Arrows in the bottom row are normalized to 1600 Wm -2 ^{-2} and are determined by the direction and magnitude of N-S and E-W heat flux.", "The colorbar in the bottom row represents the magnitude of ocean heat transport." ], [ "Temperature Offset Between ROCKE-3D and UM", "We begin by comparing simulations in ROCKE-3D with a slab ocean (no OHT) to similar simulations in UM by [24].", "Figure REF shows how surface temperatures are affected by substellar continent size in both the dynamic and no OHT cases.", "The no OHT ROCKE-3D simulations have global mean surface temperatures 9-21 K higher than the UM simulations (Fig.", "1(a)).", "This difference grows with increasing continent sizes, which is likely due to differences in prescribed land albedo.", "The UM used by [24] prescribes an ice-free ocean albedo of 0.07 and a land albedo of 0.4.", "The prescribed albedos in ROCKE-3D are systematically lower, with an ice-free ocean albedo of 0.03 and a bare soil albedo of 0.2.", "For simulations with substellar continents, the large difference in land albedo means that less incident solar radiation is reflected, which leads to higher average temperatures.", "For the aqua planet simulation, the relatively small ice-free ocean albedo differences alone cannot account for the temperature differences between ROCKE-3D and UM.", "Additionally, ROCKE-3D and UM use the same radiative transfer model, SOCRATES, so radiative transfer calculations cannot account for the difference in temperature.", "We suspect that differences in the cloud and convection schemes, which can change the global-mean surface temperature by tens of Kelvin [35], are responsible for the differences between ROCKE-3D and UM when there are no continents.", "Figure: Areas of upwelling are correlated with areas of divergent ocean currents, especially off the coast of continents.", "Upwelling on the dayside is maintained for every continent size considered in this work.", "The vectors in the first and second row are determined from the direction and magnitude of N-S and E-W surface winds and ocean currents.", "The colorbars in the first two rows represent the magnitude of surface winds and ocean currents.", "Positive values of upwelling corresponds to upward movements of water at the base of the mixed layer while negative values correspond to downward movements." ], [ "Effect of Ocean Dynamics on Temperature", "Next, we consider simulations with a fully dynamic ocean to investigate whether the inclusion of a dynamic ocean affects the habitability of tidally locked planets.", "From Figure REF , we see that the effect of including a dynamic ocean diminishes as substellar continent size increases.", "This is because a primary effect of ocean dynamics on a tidally locked planet is to transport heat away from the substellar point to the nightside.", "When a substellar continent of increasing size blocks heat redistribution, this effect weakens.", "Moreover, it is important to note that our results are much more sensitive to differences between ROCKE-3D and UM than they are to the inclusion of ocean dynamics.", "There is a small difference between the surface albedo in the dynamic ocean and no-OHT cases (Figure REF (e)).", "This is due to differences in sea ice coverage, which is impacted by ocean heat transport." ], [ "Effect of a Substellar Continent on Ocean Dynamics", "Previous work has shown that the insolation pattern on a tidally locked planet creates an “eyeball” state in which surface temperatures decrease roughly symmetrically from the substellar point [31].", "In the first row of Figure REF , we show sea surface temperatures in the no OHT case.", "Heat is concentrated near the substellar point, creating an “eyeball\" pattern.", "When ocean dynamics are included, the ocean transports heat from the substellar point, which expands the area of deglaciated ocean, forming a “lobster state” [13].", "The 0% land-coverage case in the second row of Figure REF demonstrates how a dynamic ocean transports heat towards the nightside such that the sea surface temperature pattern has “lobster claws” of warm ocean away from the substellar point.", "In general, global-mean temperature is higher and the day/night temperature contrast is smaller in the lobster state due to the transport of heat away from the substellar point.", "Adding a continent at the substellar point disrupts the ocean's ability to transport heat.", "Figure REF demonstrates the planet’s transition from the lobster state with efficient ocean heat transport to the eyeball state with diminished ocean heat transport as continent size grows.", "Once the continent covers about 20% of the surface of the planet, dynamic ocean simulations yield results fairly similar to corresponding no OHT simulations.", "The bottom row of Figure REF demonstrates the effect of continent size on ocean heat transport in the dynamic ocean simulations.", "On the aqua planet (0% continent size), OHT efficiently transports heat from the substellar point towards the nightside, creating the lobster pattern in the sea surface temperature map.", "However, even the addition of a small, 4% continent dramatically disrupts OHT.", "Though the continent never entirely shuts off OHT, a 22% continent restricts OHT enough that the planet settles into an eyeball state similar to the no OHT case." ], [ "Effect of a Substellar Continent on Nutrient Fluxes", "Figure REF shows how divergent ocean currents determine areas of upwelling.", "In the third row of Figure REF , upwelling occurs on the dayside for every continent size considered in this work.", "Upwelling corresponds to areas of divergent surface ocean currents shown in the second row of Figure REF .", "This is because the conservation of mass requires an upwelling of water in areas of diverging surface currents.", "This upwelling may supply nutrients to the shallow ocean where photosynthesis is viable.", "Figure REF shows the change in upwelling and average mixed layer depth on the dayside of the planet as substellar continent size increases.", "Here, upwelling is defined as the area-weighted sum of vertical velocity at the base of the mixed layer.", "All values are normalized to the Earth-like baseline from [30].", "Both total and dayside upwelling decrease with increasing continent size due to a general decrease in ocean area (Figure REF (a)).", "Dayside upwelling constitutes the majority of total upwelling, which is favorable for photosynthetic life.", "The total (Figure REF (a)) and average (Figure REF (b)) dayside upwelling are significantly larger than the corresponding upwelling in an Earth-like simulation for all continent sizes.", "The mixed layer depth (MLD) is the depth of the surface ocean that is homogenized by surface winds.", "On Earth, the MLD is sensitive to sea surface temperatures, which affect ocean stratification, and shallows in the summer when sea surface temperatures rise.", "In our simulations, the MLD is deepest in the aqua planet case and rapidly shallows with increasing substellar continent size, corresponding to changes in sea surface temperature and therefore ocean density stratification.", "As before, we can attribute this behavior to the transition from lobster to eyeball seen in Figure REF .", "When ocean heat transport is very efficient, heat does not concentrate as strongly on the dayside, resulting in a deeper mixed layer.", "However, as ocean heat transport is inhibited by a continent, heat is trapped on the dayside, shallowing the mixed layer there.", "With continents larger than about 20%, the behavior of the average MLD on the dayside is dominated by the effect of decreasing ocean area on the dayside.", "As the continent grows, it does so at the expense of the warmest portions of the ocean with the shallowest MLD, so the average MLD is deeper.", "Ocean dynamics play a key role in heat transport and in previous work have been shown to impact the total habitable area on tidally locked planets.", "In this work, we showed that the habitability of Proxima Centauri b with substellar continents is not very sensitive to the inclusion of a dynamic ocean.", "Differences in surface albedo, cloud, and convective parameterizations between ROCKE-3D and UM have a much larger effect on climate than ocean dynamics.", "Differences across climate models are a huge uncertainty.", "This is consistent with the relatively large effect that seemingly small differences in modeling assumptions can have when GCMs are applied to exoplanets [47], [45].", "In every model scenario we considered, the dayside of the planet had liquid water, satisfying the most basic definition of habitability.", "Next, we investigated other controls on photosynthetic life, like nutrient availability and access to sunlight.", "The vast majority of oceanic life on Earth resides in the mixed layer.", "On Earth, the light penetration depth (the depth at which solar radiation can penetrate the water) is of the order of 100 m. Photosynthetic life is more productive when the mixed layer is shallow compared to the light penetration depth.", "Since tidally locked planets orbit red-dwarf stars, the incident solar radiation is shifted towards the infrared, which has a shallower light penetration depth on the order of 10 m [16].", "Therefore, we can expect that life on tidally locked planets would favor areas of the ocean with very shallow mixed layers.", "We found that the MLD initially shallows with increasing continent size due to a decrease in the efficiency of OHT.", "Additionally, continents provide a source of coastal upwelling, which delivers nutrients to the mixed layer.", "Continents are also a source of nutrients like phosphorus from chemical weathering that are vital for photosynthesis [11].", "However, if a continent is large enough to cover more than about half of the dayside, this begins to deepen the average mixed layer depth on the dayside by covering ocean that would have been extremely warm.", "This would force life to exist closer to the nightside of the planet, which could restrict photosynthetic life.", "Photosynthetic life requires both light and nutrients.", "We showed that a majority of upwelling, which provides nutrients, occurs on the dayside, where there is light.", "Photosynthesis is thus viable on synchronously rotating planets, including planets with substellar land mass.", "A high concentration of photosynthetic life can affect the spectral appearance of a planet, and thus might be remotely detectable [30], [34].", "Moreover, photosynthetic life, which translates stellar energy into chemical energy in the form of thermodynamic disequilibrium may be uniquely detectable with JWST, even if it does not produce O$_2$ [19], [20].", "In this work we prescribed a 150 m ocean to reduce computational time.", "There is evidence that the depth of an ocean can influence surface temperatures [6], [42].", "Additionally, the MLD in simulations with small continents nearly reached 150 m in some areas, so the shallowness of the ocean used in this work may impart some uncertainty to our results.", "However, day side mixed layers are typically 40-100 m deep, rarely approaching the depth of the ocean except when large continents force photosynthetic marine life to the margins of the day side (Figure REF ).", "Future work could examine how increasing ocean depth affects our results, but we expect that our interpretations regarding the suitability of tidally locked planets for photosynthetic life are robust against this uncertainty." ], [ "Conclusions", "In this work, we used ROCKE-3D to investigate how substellar continents on Proxima Centauri b affect two vital ocean processes: ocean heat transport and nutrient upwelling.", "Our main conclusions are as follows: The habitability of Proxima Centauri b is largely independent of ocean dynamics for the planetary parameters we considered.", "In both the dynamic and slab ocean scenario, liquid water is found on the dayside regardless of substellar continent size.", "Surface temperature results are more sensitive to differences in the surface albedo, convection, and cloud schemes between ROCKE-3D and the Met Office Unified Model than the inclusion of ocean dynamics.", "Substellar continents reduce the efficiency of ocean heat transport.", "Our simulations suggest that Proxima Centauri b should transition from a lobster to an eyeball state as substellar continent size increases and inhibits ocean heat transport.", "Proxima Centauri b experiences widespread dayside upwelling across all continent sizes.", "This is vital for returning nutrients to the surface ocean that are lost when biomass sinks to the deep ocean.", "The average mixed layer depth on the dayside is lowest for continents between 16-26% of the total surface area of the planet.", "Shallow mixed layers are preferable for photosynthetic life on planets around M-dwarfs due to shallower light penetration depths of their oceans.", "For all of the continent sizes we considered, dayside surface ocean conditions are conducive to photosynthetic marine life.", "This is an important result given the prevailing view that continental weathering is essential for not only longterm climate stability, but also for life.", "This work suggests that tidally locked planets with surface oceans can maintain habitable conditions for life regardless of substellar continent size, which may further the case for extraterrestrial life on synchronously rotating worlds.", "This work was completed with resources provided by the University of Chicago Research Computing Center.", "This work was partially supported by the NASA Astrobiology Program grant No.", "80NSSC18K0829 and benefited from participation in the NASA Nexus for Exoplanet Systems Science research coordination network.", "S.L.O.", "acknowledges support from the T.C.", "Chamberlin Post-doctoral Fellowship in the Department of Geophysical Sciences at the University of Chicago.", "T.D.K.", "acknowledges funding from the 51 Pegasi b Fellowship in Planetary Astronomy sponsored by the Heising-Simons Foundation.", "We thank the referee for insightful comments that improved the manuscript." ] ]

[ [ "Sound Regular Corecursion in coFJ" ], [ "Abstract The aim of the paper is to provide solid foundations for a programming paradigm natively supporting the creation and manipulation of cyclic data structures.", "To this end, we describe coFJ, a Java-like calculus where objects can be infinite and methods are equipped with a codefinition (an alternative body).", "We provide an abstract semantics of the calculus based on the framework of inference systems with corules.", "In coFJ with this semantics, FJ recursive methods on finite objects can be extended to infinite objects as well, and behave as desired by the programmer, by specifying a codefinition.", "We also describe an operational semantics which can be directly implemented in a programming language, and prove the soundness of such semantics with respect to the abstract one." ], [ "Introduction", "Applications often deal with data structures which are conceptually infinite, such as streams or infinite trees.", "Thus, a major problem for programming languages is how to finitely represent something which is infinite, and, even harder, how to correctly manipulate such finite representations to reflect the expected behaviour on the infinite structure.", "A well-established solution is lazy evaluation, as, e.g., in Haskell.", "In this approach, the conceptually infinite structure is represented as the result of a function call, which is evaluated only as much as needed.", "Focusing on the paradigmatic example of streams (infinite lists) of integers, we can define two_one = 2:1:two_one, or even represent the list of natural numbers as from 0, where from n = n:from(n+1).", "In this way, functions which only need to inspect a finite portion of the structure, e.g., getting the $i$ -th element, can be correctly implemented.", "On the other hand, functions which need to inspect the whole structure, e.g., min getting the minimal element, or allPos checking that all elements are positive, have an undefined result (that is, non-termination, operationally).", "More recently, a different, in a sense complementaryAs we will discuss further in the Conclusion., approach has been considered , , , which focuses on cyclic structures (e.g., cyclic lists, trees and graphs).", "They can be regarded as a particular case of infinite structures: abstractly, they correspond to regular terms (or trees), that is, finitely branching trees whose depth can be infinite, but contain only a finite set of subtrees.", "For instance, the list two_one is regular, whereas the list of natural numbers is not.", "Typically, cyclic data structures are handled by programming languages by relying on imperative features or ad hoc data structures for bookkeeping.", "For instance, we can build a cyclic object by assigning to a field of an object a reference to the object itself, or we can visit a graph by marking already encountered nodes.", "In this approach , , , instead, the programming language natively supports regular structures, as outlined below: Data constructors are enriched by allowing equations, e.g., $\\textit {x}= 2 : 1 : \\textit {x}$ .", "Functions are regularly corecursive, that is, execution keeps track of pending function calls, so that, when the same call is encountered the second time, this is detected, avoiding non-termination as with ordinary recursion.", "For instance, when calling min on the list $\\textit {x}= 2 : 1 : \\textit {x}$ , after an intermediate call on the list $y = 1 : 2 : y$ , the same call is encountered.", "Regular corecursion originates from co-SLD resolution , , , where already encountered goals (up to unification), called coinductive hypotheses, are considered successful.", "However, co-SLD resolution is not flexible enough to to correctly express certain predicates on regular terms; for instance, in the min example, the intuitively correct corecursive definition is not sound, because the predicate succeeds for all lower bounds of $l$ , as shown in the following.", "When moving from goals to functions calls, the same problem manifests more urgently because a result should always be provided for already encountered calls.", "To solve this issue, the mechanism of flexible regular corecursion can be adopted to allow the programmer to correctly specify the behaviour of recursive functions on cyclic structures.", "For instance, for function min, the programmer specifies that the head of the list should be returned when detecting a cyclic call; in this way, on the list $\\textit {x}= 2 : 1 : \\textit {x}$ , the result of the cyclic call is 2, so that the result of the original call is 1, as expected.", "Flexible regular corecursion as outlined above has been proposed in the object-oriented , functional , and logic paradigms (see Section  for more details).", "However, none of these proposals provides formal arguments for the correctness of the given operational semantics, by proving that it is sound with respect to some model of the behaviour of functions (or predicates) on infinite structures.", "The aim of this paper is to bridge this gap, by providing solid foundations for a programming paradigm natively supporting cyclic data structures.", "This is achieved thanks to the recently introduced framework of inference systems with corules , , allowing definitions which are neither inductive, nor purely coinductive.", "We present the approach in the context of Java-like languages, namely on an extension of Featherweight Java ($\\textsc {FJ} $ ) called $\\textsc {coFJ} $ , outlined as follows: $\\textsc {FJ} $ objects are smoothly generalized from finite to infinite by interpreting their definition coinductively, and methods are equipped with a codefinition (an alternative body).", "We provide an abstract big-step semantics for $\\textsc {coFJ} $ by an inference system with corules.", "In $\\textsc {coFJ} $ with this semantics, $\\textsc {FJ} $ recursive methods on finite objects can be extended to infinite objects as well, and behave as desired by the programmer, by specifying a codefinition.", "For instance, if the codefinitions for min and allPos are specified to return the head, and true, respectively, then min returns 1 on $\\textit {x}= 2 : 1 : \\textit {x}$ , and 0 on the list of the natural numbers, whereas allPos returns true on both lists.", "Then, we provide an operational (hence, executable) semantics where infinite objects are restricted to regular ones and methods are regularly corecursive, and we show that such operational semantics is sound with respect to the abstract one.", "At https://person.dibris.unige.it/zucca-elena/coFJ_implementation.zip we provide a prototype implementation of $\\textsc {coFJ} $ , briefly described in the Conclusion.", "A preliminary version of the operational semantics, with no soundness proof with respect to a formal model, has been given in .", "Section  is a quick introduction to inference systems with corules.", "Section  describes $\\textsc {FJ} $ and informally introduces our approach.", "In Section  we define $\\textsc {coFJ} $ and its abstract semantics, in Section  the operational semantics, in Section  we show some advanced examples, and in Section  we prove soundness.", "Finally, we discuss related work and draw conclusions in Section  and Section , respectively." ], [ "Inference systems with corules", "First we recall standard notions on inference systems , .", "Assuming a universe ${\\cal U}$ of judgments, an inference system ${\\cal I}$ is a set of (inference) rules, which are pairs $\\displaystyle \\frac{\\textit {Pr}}{\\textit {c}}$ , with $\\textit {Pr}\\subseteq {\\cal U}$ the set of premises, and $\\textit {c}\\in {\\cal U}$ the consequence (a.k.a.", "conclusion).", "A rule with an empty set of premises is an axiom.", "A proof tree (a.k.a.", "derivation) for a judgment $\\mathit {j}$ is a tree whose nodes are (labeled with) judgments, $\\mathit {j}$ is the root, and there is a node $c$ with children $\\textit {Pr}$ only if there is a rule $\\displaystyle \\frac{\\textit {Pr}}{\\textit {c}}$ .", "The inductive and the coinductive interpretation of ${\\cal I}$ , denoted $\\textit {Ind}({\\cal I})$ and $\\textit {CoInd}({\\cal I})$ , are the sets of judgments with, respectively, a finiteUnder the common assumption that sets of premises are finite, otherwise we should say well-founded., and a possibly infinite proof tree.", "In set-theoretic terms, let ${\\textit {F}_{{\\cal I}}}:\\wp ({\\cal U})\\rightarrow \\wp ({\\cal U})$ be defined by ${\\textit {F}_{{\\cal I}}}(S)=\\lbrace \\textit {c}\\mid \\textit {Pr}\\subseteq S, \\displaystyle \\frac{\\textit {Pr}}{\\textit {c}}\\in {\\cal I}\\rbrace $, and say that a set $S$ is closed if ${\\textit {F}_{{\\cal I}}}(S)\\subseteq S$ , consistent if $S\\subseteq {\\textit {F}_{{\\cal I}}}(S)$ .", "Then, it can be proved that $\\textit {Ind}({\\cal I})$ is the smallest closed set, and $\\textit {CoInd}({\\cal I})$ is the largest consistent set.", "We write ${\\cal I} \\!\\vdash \\!", "{\\mathit {j}} $ when $\\mathit {j}$ has a finite derivation in ${\\cal I}$ , that is, $\\mathit {j}\\in \\textit {Ind}({\\cal I})$ .", "An inference system with corules, or generalized inference system, is a pair $({{{\\cal I}},{{\\cal I}^{co}}})$ where ${\\cal I}$ and ${\\cal I}^{co}$ are inference systems, whose elements are called rules and corules, respectively.", "Corules can only be used in a special way, as defined below.", "For a subset $S$ of the universe, let ${{\\cal I}_{{\\sqcap }S}}$ denote the inference system obtained from ${\\cal I}$ by keeping only rules with consequence in $S$ .", "Let $({{{\\cal I}},{{\\cal I}^{co}}})$ be a generalized inference system.", "Then, its interpretation $\\textit {Gen}({\\cal I},{\\cal I}^{co})$ is defined by $\\textit {Gen}({\\cal I},{\\cal I}^{co})=\\textit {CoInd}({{\\cal I}_{{\\sqcap }\\textit {Ind}({\\cal I}\\cup {\\cal I}^{co})}})$ .", "In proof-theoretic terms, $\\textit {Gen}({\\cal I},{\\cal I}^{co})$ is the set of judgments that have a possibly infinite proof tree in ${\\cal I}$ , where all nodes have a finite proof tree in ${\\cal I}\\cup {\\cal I}^{co}$ , that is, the (standard) inference system consisting of rules and corules.", "We write $ ({{{\\cal I}},{{\\cal I}^{co}}}) \\!\\vdash \\!", "{\\mathit {j}} $ when $\\mathit {j}$ is derivable in $({{{\\cal I}},{{\\cal I}^{co}}})$ , that is, $\\mathit {j}\\in \\textit {Gen}({\\cal I},{\\cal I}^{co})$ .", "Note that $ ({{{\\cal I}},{\\emptyset }}) \\!\\vdash \\!", "{\\mathit {j}} $ is the same as ${\\cal I} \\!\\vdash \\!", "{\\mathit {j}} $ .", "We illustrate these notions by a simple example.", "As usual, sets of rules are expressed by meta-rules with side conditions, and analogously sets of corules are expressed by meta-corules with side conditions.", "(Meta-)corules will be written with thicker lines, to be distinguished from (meta-)rules.", "The following inference system defines the minimum element of a list, where $[\\textit {x}]$ is the list consisting of only $\\textit {x}$ , and $\\textit {x}:u$ the list with head $\\textit {x}$ and tail $u$ .", "$\\displaystyle \\frac{}{\\textit {min}({[\\textit {x}]},\\textit {x})}\\hspace{15.0pt}\\displaystyle \\frac{\\textit {min}(u,y)}{\\textit {min}(x{:}u,z)}z=\\min (x,y)$ The inductive interpretation gives the correct result only on finite lists, since for infinite lists an infinite proof is clearly needed.", "However, the coinductive one fails to be a function.", "For instance, for $L$ the infinite list $2:1:2:1:2:1:\\ldots $ , any judgment $\\textit {min}(L,x)$ with $x\\le 1$ can be derived, as shown below.", "$\\displaystyle \\frac{\\displaystyle \\frac{\\displaystyle \\frac{\\ldots }{\\textit {min}(L,1)}}{\\textit {min}(1{:}L,1)}}{\\textit {min}(2{:}1{:}L,1)}\\hspace{15.0pt}\\displaystyle \\frac{\\displaystyle \\frac{\\displaystyle \\frac{\\ldots }{\\textit {min}(L,0)}}{\\textit {min}(1{:}L,0)}}{\\textit {min}(2{:}1{:}L,0)}$ By adding a corule (in this case a coaxiom), wrong results are “filtered out”: $\\displaystyle \\frac{}{\\textit {min}(x{:}\\epsilon ,x)}\\hspace{15.0pt}\\displaystyle \\frac{\\textit {min}(u,y)}{\\textit {min}(x{:}u,z)}z=\\min (x,y)\\hspace{15.0pt}\\genfrac{}{}{1.7pt}0{}{\\textit {min}(x{:}u,x)}$ Indeed, the judgment $\\textit {min}(2{:}1{:}L,1)$ has the infinite proof tree shown above, and each node has a finite proof tree in the inference system extended by the corule: $\\displaystyle \\frac{\\displaystyle \\frac{\\displaystyle \\frac{\\ldots }{\\textit {min}(L,1)}}{\\textit {min}(1{:}L,1)}}{\\textit {min}(2{:}1{:}L,1)}\\hspace{15.0pt}\\displaystyle \\frac{\\genfrac{}{}{1.7pt}0{}{\\textit {min}(1{:}L,1)}}{\\textit {min}(2{:}1{:}L,1)}$ The judgment $\\textit {min}(2{:}1{:}L,0)$ , instead, has the infinite proof tree shown above, but has no finite proof tree in the inference system extended by the corule.", "Indeed, since 0 does not belong to the list, the corule can never be applied.", "On the other hand, the judgment $\\textit {min}(L,2)$ has a finite proof tree with the corule, but cannot be derived since they it has no infinite proof tree.", "We refer to , , , for other examples.", "As final remark, note that requiring the existence of a finite proof tree with corules only for the root is not enough.", "For regular proof trees, the requirement to have such a proof tree for each node can be simplified in two ways: either requiring a sufficiently large finite proof-with-corules for the root, that is, a finite proof tree for the root which includes all the nodes of the regular proof tree or requiring a finite proof-with-corules for one node taken from each infinite path.", "Let $({{{\\cal I}},{{\\cal I}^{co}}})$ be a generalized inference system.", "The bounded coinduction principle , a generalization of the standard coinduction principle, can be used to prove completeness of $({{{\\cal I}},{{\\cal I}^{co}}})$ w.r.t.", "a set $\\textit {S}$ (for “specification”) of valid judgments.", "[Bounded coinduction]If the following two conditions hold: $\\textit {S}\\subseteq \\textit {Ind}({\\cal I}\\cup {\\cal I}^{co})$ , that is, each valid judgment has a finite proof tree in ${\\cal I}\\cup {\\cal I}^{co}$ ; $\\textit {S}\\!\\subseteq \\!", "{\\textit {F}_{{\\cal I}}}(\\textit {S})$ , that is, each valid judgment is the consequence of a rule in ${\\cal I}$ with premises in $\\textit {S}$ then $\\textit {S}\\subseteq \\textit {Gen}({\\cal I},{\\cal I}^{co})$ ." ], [ "From $\\textsc {FJ} $ to {{formula:2e929a5c-13f9-4a9a-93c0-361ef88ac942}}", "We recall $\\textsc {FJ} $ , and informally explain its extension with infinite objects and codefinitions.", "Featherweight Java The standard syntax and semantics in big-step style of $\\textsc {FJ} $ are shown in Figure REF .", "We omit cast since this feature does not add significant issues.", "We adopt a big-step, rather than a small-step style as in the original $\\textsc {FJ} $ definition, since in this way the semantics is directly defined by an inference system, denoted ${\\cal I}_\\textsc {FJ} $ in the following, which will be equipped with corules to support infinite objects.", "We write $\\overline{\\textit {cd}}$ as metavariable for $\\textit {cd}_1 \\ldots \\textit {cd}_n$ , $n\\ge 0$ , and analogously for other sequences.", "We sometimes use the wildcard $\\_$ when the corresponding metavariable is not relevant.", "A sequence of class declarations $\\overline{\\textit {cd}}$ is called a class table.", "Each class has a canonical constructor whose parameters match the fields of the class, the inherited ones first.", "We assume standard $\\textsc {FJ} $ constraints, e.g., no field hiding and no method overloading.", "The only variables occurring in method bodies are parameters (including $\\texttt {this})$ .", "Values are objects, that is, constructor invocations where arguments are values in turn.", "The judgment $\\textit {e}\\!\\Downarrow \\!\\textit {v}$ is implicitly parameterized on a fixed class table.", "In the rules we use standard $\\textsc {FJ} $ auxiliary functions, omitting their formal definition.", "Notably, $\\mathit {fields}($ returns the sequence $1\\ldots n$ of the field namesWe omit types since not relevant here.", "We discuss about type systems for $\\textsc {coFJ} $ in the conclusion.", "of the class, in declaration order with the inherited first, and $\\mathit {mbody}(\\textit {m})$ , for method $\\textit {m}$ of the class, the pair of the sequence of parameters and the definition.", "Substitution $\\textit {e} [\\overline{\\textit {e}}/\\overline{\\textit {x}}]$ , for $\\overline{\\textit {e}}$ and $\\overline{\\textit {x}}$ of the same length, is defined in the customary manner.", "Finally, for $\\overline{\\textit {e}}=\\textit {e}_1\\ldots \\textit {e}_n$ and $\\overline{\\textit {v}}=\\textit {v}_1\\ldots \\textit {v}_n$ , $\\overline{\\textit {e}}\\!\\Downarrow \\!\\overline{\\textit {v}}$ is an abbreviation for $\\textit {e}_1\\!\\Downarrow \\!\\textit {v}_1\\ldots \\textit {e}_n\\!\\Downarrow \\!\\textit {v}_n$ .", "Figure: FJ\\textsc {FJ} syntax and big-step rulesRule (FJ-field) models field access.", "If the selected field is actually a field of the receiver's class, then the corresponding value is returned as result.", "Rule (FJ-new) models object creation: if the argument expressions $\\overline{\\textit {e}}$ evaluate to values $\\overline{\\textit {v}}$ , then the result is an object of class C. Rule (FJ-invk) models method invocation.", "The receiver and argument expressions are evaluated first.", "Then, method look-up is performed, starting from the receiver's class, by the auxiliary function $\\mathit {mbody}$ .", "Lastly, the definition $\\textit {e}$ of the method, where $\\texttt {this}$ is replaced by the receiver, and the parameters by the arguments, is evaluated, and its result is returned.", "Infinite objects and codefinitions We take as running example the following $\\textsc {FJ} $ implementation of lists of integers, equipped with some typical methods: isEmpty tests the emptiness, incr returns the list where all elements have been incremented by one, allPos checks whether all elements are positive, member checks whether the argument is in the list, and min returns the minimal element.", "class List extends Object {   bool isEmpty() {true}   List incr() {new EmptyList()}   bool allPos() {true}   bool member(int x) {false} } class EmptyList extends List { } class NonEmptyList extends List {   int head; List tail;   bool isEmpty() {false}   List incr() {new NonEmptyList(this.head+1,this.tail.incr())}   bool allPos() {if (this.head<=0) false else this.tail.allPos()}   bool member(int x) {if (this.head==x) true else this.tail.member(x)}   int min() {     if (this.tail.isEmpty()) this.head     else Math.min(this.tail.min(),this.head)   } } We used some additional standard constructs, such as conditional and primitive types bool and int with their operations; to avoid to use abstract methods, List provides the default implementation on empty lists, overridden in NonEmptyList, except for method min which is only defined on non empty lists.", "In $\\textsc {FJ} $ we can represent finite lists.", "For instance, the object new NonEmptyList(2, new NonEmptyList(1, new EmptyList())) which we will abbreviate $[2,1]$ , represents a list of two elements, and it is easy to see that all the above method definitions provide the expected meaning on finite lists.", "On the other hand, since the syntactic definition for objects is interpreted, like the others, inductively, in $\\textsc {FJ} $ objects are finite, hence we cannot represent, e.g., the infinite list of natural numbers $[0,1,2,3, \\ldots ]$ , abbreviated $[0..]$ , or the infinite list $[2,1,2,1,2,1,\\ldots ]$ , abbreviated $[2,1]^\\omega $ .", "To move from finite to infinite objects, it is enough to interpret the syntactic definition for values coinductively, so to obtain infinite terms as well.", "However, to make the extension significant, we should be able to generate such infinite objects as results of expressions, and to appropriately handle them by methods.", "To generate infinite objects, e.g., the infinite lists mentioned above, a natural approach is to consider method definitions as corecursive, that is, to take the coinductive interpretation of the inference system in Figure REF .", "Consider the following class: class ListFactory extends Object {   NonEmptyList from(int x) {new NonEmptyList(x, this.from(x+1)}   NonEmptyList two_one() {new NonEmptyList(2, this.one_two())}   NonEmptyList one_two() {new NonEmptyList(1, this.two_one())} } With the standard $\\textsc {FJ} $ semantics, given by the inductive interpretation of the inference system in Figure REF , the method invocation new ListFactory().from(0) (abbreviated $\\texttt {from}_0$ in the following) has no result, since there is no finite proof tree for a judgment of shape $\\texttt {from}_0\\!\\Downarrow \\!\\_$ .", "Taking the coinductive interpretation, instead, such call returns as result the infinite list of natural numbers $[0..]$ , since there is an infinite proof tree for the judgment $\\texttt {from}_0\\!\\Downarrow \\![0..", "]$ .", "Analogously, the method invocation $\\texttt {new ListFactory().two\\_one()}$ returns $[2,1]^\\omega $ .", "Moreover, the method invocations $[0..].\\texttt {incr}()$ and $[2,1]^\\omega .\\texttt {incr}($ ) correctly return as result the infinite lists $[1..]$ and $[3,2]^\\omega $ , respectively.", "However, in many cases to consider method definitions as corecursive is not satisfactory, since it leads to non-determinism, as shown for inference systems in Section .", "For instance, for the method invocation $[0..].\\texttt {allPos}()$ both judgments $[0..].\\texttt {allPos}()\\!\\Downarrow \\!\\texttt {true}$ and $[0..].\\texttt {allPos}()\\!\\Downarrow \\!\\texttt {false}$ are derivable, and analogously for $[2,1]^\\omega .\\texttt {allPos}()$ .", "In general, both results can be obtained for any infinite list of all positive numbers.", "A similar behavior is exhibited by method member: given an infinite list $L$ which does not contain $\\textit {x}$ , both judgments $L.\\texttt {member}(\\textit {x})\\!\\Downarrow \\!\\texttt {true}$ and $L.\\texttt {member}(\\textit {x})\\!\\Downarrow \\!\\texttt {false}$ are derivable.", "Finally, for the method invocation $[2,1]^\\omega .\\texttt {min}()$ , any judgment $[2,1]^\\omega .\\texttt {min}()\\!\\Downarrow \\!\\textit {x}$ with $\\textit {x}\\le 1$ can be derived.", "To solve this problem, $\\textsc {coFJ} $ allows the programmer to control the semantics of corecursive methods by adding a codefinitionThe term “codefinition” is meant to suggest “alternative definition used to handle corecursion”., that is, an alternative method body playing a special role.", "Depending on the codefinition, the purely coinductive interpretation is refined, by filtering out some judgments.", "In the example, to achieve the expected meaning, the programmer should provide the following codefinitions.", "class ListFactory extends Object {   NonEmptyList from(int x) {     new NonEmptyList(x, this.from(x+1)} corec {any}   NonEmptyList one_two() {     new NonEmptyList(1, this.two_one())} corec {any}   NonEmptyList two_one() {     new NonEmptyList(2, this.one_two())} corec {any} } class NonEmptyList extends List {   int head; List tail;   bool isEmpty() {false}   List incr() {     new NonEmptyList(this.head+1,this.tail.incr())} corec {any}   bool allPos() {     if (this.head <= 0) false else this.tail.allPos()} corec {true}   bool member(int x) {     if (this.head == x) true else this.tail.member(x)} corec {false}   int min() {     if (this.tail.isEmpty()) this.head     else Math.min(this.tail.min(),this.head)   } corec {this.head} } For the three methods of ListFactory and for the method incr the codefinition is any.", "This corresponds to keeping the coinductive interpretation as it is, as is appropriate in these cases since it provides only the expected result.", "In the other three methods, instead, the effect of the codefinition is to filter the results obtained by the coinductive interpretation.", "The way this is achieved is explained in the following section.", "Finally, for method isEmpty no codefinition is added, since the inductive behaviour works on infinite lists as well." ], [ "$\\textsc {coFJ} $ and its abstract semantics", "We formally define $\\textsc {coFJ} $ , illustrate how the previous examples get the expected semantics, and show that, despite its non-determinism, $\\textsc {coFJ} $ is a conservative extension of $\\textsc {FJ} $ .", "Formal definition of $\\textsc {coFJ} $ The $\\textsc {coFJ} $ syntax is given in Figure REF .", "Figure: coFJ\\textsc {coFJ} syntax and abstract semanticsAs the reader can note, the only difference is that method declarations include now, besides a definition $\\textit {e}$ , an optional codefinition $\\textit {e}^{\\prime }$ , as denoted by the square brackets in the production.", "Furthermore, besides $\\texttt {this}$ , there is another special variable $\\texttt {any}$ , which can only occur in codefinitions.", "The codefinition will be used to provide an abstract semantics through an inference system with corules, where the role of $\\texttt {any}$ is to be a placeholder for an arbitrary value.", "For simplicity, we require the codefinition $\\textit {e}^{\\prime }$ to be statically restricted to avoid recursive (even indirect) calls to the same method (we omit the standard formalization).", "Note that $\\textsc {FJ} $ is a (proper) subset of $\\textsc {coFJ} $ : indeed, an $\\textsc {FJ} $ class table is a $\\textsc {coFJ} $ class table with no codefinitions.", "The syntactic definition for values is the same as before, but is now interpreted coinductively, as indicated by the symbol $::=_\\textsc {co}$ .", "In this way, infinite objects are supported.", "By replacing method parameters by arguments, we obtain runtime expressions admitting infinite objects as subterms.", "The sets ${\\cal V}$ and ${\\cal E}$ of $\\textsc {FJ} $ objects and expressions are subsets of ${\\cal V}\\textsuperscript {a}$ and ${\\cal E}\\textsuperscript {a}$ , respectively.", "The judgment $\\textit {e}\\!\\Downarrow \\!\\textit {v}$ , with $\\textit {e}\\in {\\cal E}\\textsuperscript {a}$ and $\\textit {v}\\in {\\cal V}\\textsuperscript {a}$ , is defined by an inference system with corules $({{{\\cal I}_{\\scriptscriptstyle \\textsc {FJ}}},{{\\cal I}^{co}_{\\scriptscriptstyle \\textsc {FJ}}}})$ where the rules ${\\cal I}_{\\scriptscriptstyle \\textsc {FJ}}$ are thoseTo be precise, meta-rules are the same, with meta-variables $\\textit {e}$ and $\\textit {v}$ ranging on ${\\cal E}\\textsuperscript {a}$ , and ${\\cal V}\\textsuperscript {a}$ , respectively.", "However, we could have taken this larger universe in $\\textsc {FJ} $ as well without affecting the defined relation.", "of $\\textsc {FJ} $ , as in Figure REF , and the corules ${\\cal I}^{co}_{\\scriptscriptstyle \\textsc {FJ}}$ are instances of two metacorules.", "Corule (abs-co-val) is needed to obtain a value for infinite objects, as shown below.", "Corule (abs-co-invk) is analogous to the standard rule for method invocation, but uses the codefinition, and the variable $\\texttt {any}$ can be non-deterministically substituted with an arbitrary value.", "The auxiliary function co-mbody is defined analogously to mbody, but it returns the codefinition.", "Note that, even when $\\mathit {mbody}(\\textit {m})$ is defined, $\\textit {co-mbody}(\\textit {m})$ can be undefined since no codefinition has been specified.", "This can be done to force a purely inductive behaviour for the method.", "Examples As an example, we illustrate in Figure REF the role of the two corules for the call new ListFactory().from(0).", "For brevity, we write abbreviated class names.", "Furthermore, $\\texttt {from}_n$ stands for the call new ListFactory().from(n) and $[n..]$ for the infinite object new NonEmptyList(n,new NonEmptyList(n+1,...))).", "Figure: Infinite (top) and finite (bottom) proof trees for 𝚏𝚛𝚘𝚖 n ⇓[n..]\\texttt {from}_n\\!\\Downarrow \\![n..", "]In the top part of Figure REF , we show the infinite proof tree $T_n$ which can be constructed, for any natural number $n$ , for the judgment $\\texttt {from}_n\\!\\Downarrow \\![n..", "]$ without the use of corules.", "We use standard rules (n-val) and (+) to deal with integer constants and addition.", "To derive the judgment in the inference system with corules, each node in this infinite tree should have a finite proof tree with the corules.", "Notably, this should hold for nodes of shape $\\texttt {from}_n\\!\\Downarrow \\![n..", "]$ , and indeed the finite proof tree for such nodes is shown in the bottom part of the figure.", "Note that, in this example, the result for the call $\\texttt {from}_n$ is uniquely determined by the rules, hence the role of the corules is just to “validate” this result.", "To this end, the codefinition of the method from is the special variable any, which, when evaluating the codefinition, can be replaced by any value, hence, in particular, by the correct result $[n..]$ .", "Corule (abs-co-val) is needed to obtain a finite proof tree for the infinite objects of shape $[n..]$ .", "Analogous infinite and finite proof trees can be constructed for the judgments $\\texttt {new ListFactory().two\\_one()}\\!\\Downarrow \\!", "[2,1]^\\omega $ , $[0..].\\texttt {incr}()\\!\\Downarrow \\![1..", "]$ and $[2,1]^\\omega .\\texttt {incr}(\\!", ")\\Downarrow \\!", "[3,2]^\\omega $ .", "For the method call $[0..].\\texttt {allPos}()$ , instead, both judgments $[0..].\\texttt {allPos}()\\!\\Downarrow \\!\\texttt {true}$ and $[0..].\\texttt {allPos}()\\!\\Downarrow \\!\\texttt {false}$ have an infinite proof tree.", "However, no finite proof tree using the codefinition can be constructed for the latter, whereas this is trivially possible for the former.", "Analogously, given an infinite list $L$ which does not contain $\\textit {x}$ , only the judgment $L.\\texttt {member}(\\textit {x})\\!\\Downarrow \\!\\texttt {false}$ has a finite proof tree using the codefinition.", "Finally, for the method invocation $[2,1]^\\omega .\\texttt {min}()$ , for any $\\textit {v}\\le 1$ there is an infinite proof tree built without corules for the judgment $\\mathtt {[2,1]^\\omega .min()}\\!\\Downarrow \\!\\textit {v}$ as shown in Figure REF .", "Figure: Infinite proof tree for [2,1] ω .𝚖𝚒𝚗()⇓𝑣\\mathtt {[2,1]^\\omega .min()}\\!\\Downarrow \\!\\textit {v} with 𝑣≤1\\textit {v}\\le 1 (main tree at the top left corner)However, only the judgment $[2,1]^\\omega .\\texttt {min}()\\!\\Downarrow \\!1$ has a finite proof tree using the codefinition (Figure REF ).", "For space reasons in both figures ellipses are used to omit the less interesting parts of the proof trees; we use the standard rule (if-f) for conditional, and the predefined function Math.min on integers.", "Figure: Finite proof tree with codefinition for [2,1] ω .𝚖𝚒𝚗()⇓1\\mathtt {[2,1]^\\omega .min()}\\!\\Downarrow \\!\\mathtt {1} (T 0 T_0 as in Figure )Non-determinism and conservativity The $\\textsc {coFJ} $ abstract semantics is inherently non-deterministic.", "Indeed, depending on the codefinition, the non-determinism of the coinductive interpretation may be kept.", "For instance, consider the following method declaration: class C {   C m() { this.m() } corec { any } } Method m() recursively calls itself.", "In the abstract semantics, the judgment $\\texttt {new}\\ \\texttt {C} ().\\texttt {m}()\\!\\Downarrow \\!\\textit {v}$ can be derived for any value $\\textit {v}$ .", "In the operational semantics defined in Section , such method call evaluates to $({{\\textit {x}},{\\textit {x}:\\textit {x}}})$ , that is, the representation of undetermined.", "However, determinism of $\\textsc {FJ} $ evaluation is preserved.", "Indeed, $\\textsc {coFJ} $ abstract semantics is a conservative extension of $\\textsc {FJ} $ semantics, as formally stated below.", "[Conservativity] If ${\\cal I}_{\\scriptscriptstyle \\textsc {FJ}} \\!\\vdash \\!", "{\\textit {e}\\!\\Downarrow \\!\\textit {v}} $ , then $ ({{{\\cal I}_{\\scriptscriptstyle \\textsc {FJ}}},{{\\cal I}^{co}_{\\scriptscriptstyle \\textsc {FJ}}}}) \\!\\vdash \\!", "{\\textit {e}\\!\\Downarrow \\!\\textit {v}^{\\prime }} $ iff $\\textit {v}= \\textit {v}^{\\prime }$ .", "Both directions can be easily proved by induction on the definition of ${\\cal I}_{\\scriptscriptstyle \\textsc {FJ}} \\!\\vdash \\!", "{\\textit {e}\\!\\Downarrow \\!\\textit {v}} $ .", "For the left-to-right direction, the fact that each syntactic category has a unique applicable meta-rule is crucial.", "This theorem states that, whichever the codefinitions chosen, $\\textsc {coFJ} $ does not change the semantics of expressions evaluating to some value in $\\textsc {FJ} $ .", "That is, $\\textsc {coFJ} $ abstract semantics allows derivation of new values only for expressions whose semantics is undefined in standard $\\textsc {FJ} $ , as in the examples shown above.", "Note also that, if no codefinition is specified, then the $\\textsc {coFJ} $ abstract semantics coincides with the $\\textsc {FJ} $ one, because corule (abs-co-invk) cannot be applied, hence no infinite proof trees can be built for the evaluation of $\\textsc {FJ} $ expressions." ], [ "Operational semantics", "We informally introduce the operational semantics of $\\textsc {coFJ} $ , provide its formal definition, and prove that it is deterministic and conservative.", "Outline In contrast to the abstract semantics of the previous section, the aim is to define a semantics which leads to an interpreter for the calculus.", "To obtain this, there are two issues to be considered: infinite (regular) objects should be represented in a finite way; infinite (regular) proof trees should be replaced by finite proof trees.", "In the following we explain how these issues are handled in the $\\textsc {coFJ} $ operational semantics.", "To obtain (1), we use an approach based on capsules , which are essentially expressions supporting cyclic references.", "In our context, capsules are pairs $({{\\textit {e}},{\\sigma }})$ where $\\textit {e}$ is an $\\textsc {FJ} $ expression and $\\sigma $ is an environment, that is, a finite mapping from variables into $\\textsc {FJ} $ expressions.", "Moreover, the following capsule property is satisfied: writing $\\textit {FV}(\\textit {e})$ for the set of free variables in $\\textit {e}$ , $\\textit {FV}(\\textit {e})\\subseteq \\mathit {dom}(\\sigma )$ and, for all $\\textit {x}\\in \\mathit {dom}(\\sigma )$ , $\\textit {FV}(\\sigma (\\textit {x}))\\subseteq \\mathit {dom}(\\sigma )$ .", "An $\\textsc {FJ} $ source expression $\\textit {e}$ is represented by the capsule $({{\\textit {e}},{\\emptyset }})$ , where $\\emptyset $ denotes the empty environment.", "In particular, values are pairs $({{{\\mathrm {v}}},{\\sigma }})$ where ${\\mathrm {v}}$ is an open $\\textsc {FJ} $ object, that is, an object possibly containing variables.", "In this way, cyclic objects can be obtained: for instance, $({{\\textit {x}},{\\textit {x}:\\texttt {new}\\ \\texttt {NEL} (2,\\texttt {new}\\ \\texttt {NEL} (1,\\textit {x}))}})$ represents the infinite regular list $[2,1]^\\omega $ considered before.", "To obtain (2), methods are regularly corecursive.", "This means that execution keeps track of the pending method calls, so that, when a call is encountered the second time, this is detectedThe semantics detects an already encountered call by relying on capsule equivalence (Figure )., avoiding non-termination as it would happen with ordinary recursion.", "Regular corecursion in $\\textsc {coFJ} $ is flexible, since the behaviour of the method when a cycle is detected is specified by the codefinition.", "Consider, for instance, the method call $\\texttt {new ListFactory().two\\_one()}$ ; thanks to regular corecursion, the result is the cyclic object $({{\\textit {x}},{\\textit {x}:\\texttt {new}\\ \\texttt {NEL} (2,\\texttt {new}\\ \\texttt {NEL} (1,\\textit {x}))}})$ .", "Indeed, the operational semantics associates a fresh variable, say, $\\textit {x}$ , to the initial call, so that, when the same call is encountered the second time, the association $\\textit {x}:{\\textit {x}}$ is added in the environment, and the codefinition is evaluated where any is replaced by $\\textit {x}$ .", "Hence, $({{\\textit {x}},{\\textit {x}:\\textit {x}}})$ is returned as result, so that the result of the original call is $({{\\textit {x}},{\\textit {x}:\\texttt {new}\\ \\texttt {NEL} (2,\\texttt {new}\\ \\texttt {NEL} (1,\\textit {x}))}})$ .", "The call new ListFactory().from(0), instead, does not terminate in the operational semantics, since no call is encountered more than once (the resulting infinite object is non-regular).", "Consider now the call $[2,1]^\\omega .\\texttt {allPos}()$ .", "In this case, when the call is encountered the second time, after an intermediate call $[1,2]^\\omega .\\texttt {allPos}()$ , the result of the evaluation of the codefinition is true, so that the result of the original call is true as well.To be rigorous, a capsule of shape $({{\\texttt {true}},{\\_}})$ .", "If the codefinition were any, then the result would be $({{\\textit {x}},{\\textit {x}:\\textit {x}}})$ , that is, undetermined.", "Note that, if the list is finite, then no regular corecursion is involved, since the same call cannot occur more than once; the same holds if the list is cyclic, but contains a non-positive element, hence the method invocation returns false.", "The only case requiring regular corecursion is when the method is invoked on a cyclic list with all positive elements, as $[2,1]^\\omega $ .", "In the case of $[2,1]^\\omega .\\texttt {min}()$ , when the call is encountered the second time the result of the evaluation of the codefinition is 2, so that the result of the intermediate call $[1,2]^\\omega .\\texttt {min}()$ is 1, and this is also the result of the original call.", "Formal definition To formally express the approach described above, the judgment of the operational semantics has shape $\\textit {e},\\sigma ,\\tau \\!\\Downarrow \\!", "{\\mathrm {v}},{\\sigma ^{\\prime }}$ where: $({{\\textit {e}},{\\sigma }})$ is the capsule to be evaluated; $\\tau $ is a call trace, used to keep track of already encountered calls, that is, an injective map from calls ${\\mathrm {v}}_0.\\textit {m}(\\overline{{\\mathrm {v}}})$ to (possibly tagged) variables, and $({{{\\mathrm {v}}},{{\\sigma ^{\\prime }}}})$ is the capsule result.", "Variables in the codomain of the call trace have a tag $\\textsf {ck}$ during the checking step for the corresponding call, as detailed below.", "The pair $({{\\textit {e}},{\\sigma }})$ and $({{{\\mathrm {v}}},{{\\sigma ^{\\prime }}}})$ are assumed to satisfy the capsule property.", "The semantic rules are given in Figure REF .", "We denote by ${\\sigma }\\lbrace \\textit {x}:\\!", "{\\mathrm {v}}^{}\\rbrace $ the environment which gives ${\\mathrm {v}}$ on $\\textit {x}$ , and is equal to $\\sigma $ elsewhere, and analogously for other maps.", "Furthermore, we use the following notations, formally defined in Figure .", "$\\mathit {unfold}({\\mathrm {v}},\\sigma )$ is the unfolding of ${\\mathrm {v}}$ in $\\sigma $ , that is, the corresponding object, if any.", "$\\sigma _1{\\sqcup } \\sigma _2$ is the union of environments, defined if they agree on the common domain.", "$ ({{{\\mathrm {v}}},{\\sigma }}) {\\approx _{}} ({{{\\mathrm {v}}^{\\prime }},{\\sigma ^{\\prime }}})$ is the equivalence of capsules.", "As will be formalized in the first part of Section , equivalent capsules denote the same sets of abstract objects.", "This equivalence is extended by congruence to expressions, in particular to calls ${\\mathrm {v}}_0.\\textit {m}(\\overline{{\\mathrm {v}}})$ .", "$\\tau _{\\approx \\sigma }$ is obtained by extending $\\tau $ up to equivalence in $\\sigma $ .", "That is, detection of already encountered calls is performed up-to equivalence in the current environment.", "Figure: Proofs" ] ]

[ [ "Quantifying the effects of quarantine using an IBM SEIR model on\n scalefree networks" ], [ "Abstract The COVID-19 pandemic led several countries to resort to social distancing, the only known way to slow down the spread of the virus and keep the health system under control.", "Here we use an individual based model (IBM) to study how the duration, start date and intensity of quarantine affect the height and position of the peak of the infection curve.", "We show that stochastic effects, inherent to the model dynamics, lead to variable outcomes for the same set of parameters, making it crucial to compute the probability of each result.", "To simplify the analysis we divide the outcomes in only two categories, that we call {best and worst scenarios.", "Although long and intense quarantine is the best way to end the epidemic, it is very hard to implement in practice.", "Here we show that relatively short and intense quarantine periods can also be very effective in flattening the infection curve and even killing the virus, but the likelihood of such outcomes are low.", "Long quarantines of relatively low intensity, on the other hand, can delay the infection peak and reduce its size considerably with more than 50% probability, being a more effective policy than complete lockdown for short periods." ], [ "Introduction", "The novel Coronavirus [1] pandemic has changed the lives of millions of people around the world.", "The lack of effective medications or a vaccine,[2], [3], [4], [5] has made social distancing the only reliable way to slow down the virus transmission and prevent the collapse of the health system.", "[6], [7], [8], [9] Quarantine measures are, however, difficult to implement and have enormous economic consequences.", "This is leading many communities, from cities to entire countries, to end quarantine even before the infection curve has reached its peak.", "[10], [11], [12] Understanding how quarantine duration, effectiveness and starting time affect the infection curve is, therefore, key to guide public policies.", "Several approaches have been recently proposed to model the COVID-19 epidemic,[13] including muti-layer networks,[14] the Richards growth model,[15] and many others.", "[16], [17], [18], [19] Since our interest here is to quantify the effectiveness of a non-pharmacological intervention, we opted for a SEIR (susceptible, exposed, infected and recovered) individual based model (IBM) and studied how different types of quarantine change the infection dynamics.", "Individuals are modeled as nodes of a scale-free (Barabási-Albert) network [20] that can only infect their connected neighbors.", "Because the dynamics is stochastic, independent simulations with the same set of parameters can lead to quite different outcomes.", "Here we group the outcomes in only two categories, that we call best and worst scenarios.", "Stochasticity, a reality of the Sars-Cov-2 infection, is not captured by the mean field approximation of the SEIR model,[21] where outcomes depend deterministically on the model parameters.", "We find three types of quarantine that can be effective against the epidemic: (i) relatively long (10 weeks) and intense (more than $80\\%$ isolation); (ii) short (8 weeks) and of intermediate intensity (around $70\\%$ isolation) and; (iii) long (12 weeks or more) with relatively low intensity ($40\\%$ isolation).", "The first type, which completely ends the epidemic, is clearly the best but also the most difficult to achieve.", "The second type is feasible, but we find that most of the times (in most of the simulations) they result in worst case scenarios.", "The third type emerges as the most practical and easy to apply.", "It is not so effective as the previous types, but does decrease the infection peak by half.", "Also, it falls into the best case scenario more than $50\\%$ of the times and even in the worst scenarios the infection peak decreases." ], [ "The Model", "We model the spread of the virus using an extension of the SEIR model (susceptible, exposed, infected and recovered (or removed)).", "Exposed individuals simulate the incubation period of the disease, when infected subjects cannot yet pass on the virus.", "The mean field version of SEIR model is described by the equations $\\begin{array}{ll}\\dot{S} & = -\\beta SI/N \\\\\\dot{E} & = \\beta SI/N -\\sigma E \\\\\\dot{I} & = \\sigma E -\\gamma I \\\\\\dot{R} & = \\gamma I\\end{array}$ where $N$ is the population size, $\\beta $ is the infection rate, $\\sigma $ the rate at which exposed become infected and $\\gamma $ the recovery rate.", "The basic reproductive number, $R_0 = \\beta /\\gamma $ , gives the number of secondary infections generated by the first infectious individual over the full course of the epidemic in a fully susceptible population.", "Figure: Illustration of virus spread on the network: (a) initially only one individual is infected (red) and all the others susceptible (green); (b) neighbors of infected might get the virus (yellow); (c) one neighbor did get the virus and become exposed (orange); (d) neighbors of first infected individual might still get the virus, but the orange node is still in incubation time; (e) another node gets the virus from the first infected becoming exposed and the old exposed becomes infected; (f) more nodes might get the virus (yellow) and; (g) some do become exposed while the first infected becomes recovered.In order to take into account heterogeneity in the number of contacts we use instead an individual based model where the population is represented by a Barabási-Albert network [20], [22], [23] with $N$ of nodes and an average degree $D$ .", "As in a deterministic SEIR model, the nodes can be classified as susceptible, exposed, infected and recovered.", "Susceptible individuals can become exposed if connected to an infected one; exposed individual $i$ becomes infected after a period $t_i$ of virus incubation; infected individuals can recover, and once recovered it is considered immune and therefore cannot be infected again.", "At the beginning of the simulation, only one node, chosen at random, is infected while all the remaining population is susceptible.", "Every susceptible node connected to the infected individual becomes exposed with the transmission probability $p$ whereas the infected node might itself recover with probability $r$ .", "The probability $p$ can be calculated as $p=R_0/(\\tau _{symp}D)$ , where $\\tau _{symp}$ is the average time duration of symptoms.", "We assume that the symptoms last for a time $\\tau $ distributed according to an exponential distribution $\\mathcal {F}(\\tau )=\\lambda e^{-\\lambda \\tau }$ .", "Once a node is exposed, it stays exposed for an incubation time $t_i$ , chosen according to a given distribution $\\mathcal {P}(t_i)$ .", "[24] After this period it becomes infected and is able to infect other nodes.", "It follows that $r\\approx \\lambda =1/\\tau _{symp}$ , for small $\\lambda $ .", "For $\\mathcal {P}(t_i)$ we have used a $\\Gamma (\\alpha ,\\beta )$ distribution, as in [25].", "Fig.", "REF illustrates the network of contacts, states of individuals (S, I, E, or R) and the infection dynamics.", "For the simulations, we fixed the number of individuals $N=2000$ , the average degree $D\\approx 98$ , mean incubation time $6.5$ days with standard deviation $2.6$ days ($\\alpha =6.25$ and $\\beta \\approx 0.96$ ), $R_0=2.4$ and $\\tau _{symp}=14$ days.", "[26], [27], [28] We run the simulations until the epidemic ends and no new infection is possible.", "We model quarantine periods by reducing the transmission probability $p$ by the factor $(1-Q)$ , where $Q$ represents the intensity of quarantine, varying from 0 (no quarantine) to 1 (full quarantine).", "Duration and starting time are specified by period $\\left[t_s,t_s + t_d\\right]$ .", "We shall see that all three quarantine parameters (starting time $t_s$ , duration $t_d$ and intensity $Q$ ) have significant effects on the dynamics, particularly on the infection peak height and delay." ], [ "Results and Discussion", "Unlike the mean field SEIR model, Eqs.", "(REF ), the present IBM version on networks is probabilistic and different outcomes are obtained every time the model is ran with the same set of parameters.", "To obtain statistically significant data (while keeping simulation time reasonable) we have ran the model 25 times for different quarantine duration and intensities, beginning $t_s=20$ , 30, and 40 days after the first infected node appears (at the beginning of the simulation).", "The results were divided in two different scenarios, the best and the worst cases.", "For each set of parameters, the best scenario consists of simulations where the infection peak is lower than the average peak of the full set of simulations, whereas the worst scenario contains the set with higher than average peaks.", "This approach is important because in many cases the epidemic response to the quarantine is not satisfactory, and this might be solely due to stochastic effects, a common feature of real systems.", "As an example, Fig.", "REF shows the evolution curves of infected plus exposed individuals for all 25 replicas for $Q=0.9$ and $t_d=10$ weeks.", "Since independent populations, represented by different Barabási-Albert networks generated with the same specifications, under the same quarantine parameters might respond drastically different to quarantine, we also need to know the probability of each outcome.", "Figure: Evolution of number of infected plus exposed individuals for Q=0.9Q=0.9, t s =30t_s=30 days and t d =10t_d=10 weeks for 25 replicas of the simulation.", "The blue dashed line shows the average height of the highest peak of each curve.", "Red and green dashed lines show the average peak of worst (8 replicas) and best (17 replicas) scenarios respectively, i. e., the average peak of the curves in which there is a second peak, after the quarantine, and the average peak of those in which there is not a second peak.", "The average of all curves (black thick line) is not representative of any actual curve.", "The gray shaded area indicates the quarantine period.Figures REF , REF and REF show results for average peak height, time of infection peak and fraction of recovered individuals at the end of the epidemic (i.e., all individuals that had contact with the virus, as we do not take mortality into account).", "The results in each case are separated into best and worst scenarios and we compute the probability that a best scenario will happen.", "For example, a specific set of parameters might result in ending the epidemic, but its probability of occurrence can be too low, excluding it as a recommended policy.", "All results are displayed as heat-maps.", "Figure: Peak height with respect to the average `no quarantine' result, starting 20, 30 or 40 days after the first infection (left, middle and right columns respectively).", "Plots in the first and second rows show the best and worst scenarios.", "The third row shows the probability that a simulation results in a best scenario.", "Quarantine duration is measured in weeks (from 1 to 15) and quarantine intensity goes from 0 (no quarantine) to 1 (full individual lock-down, p=0p=0).", "Green, red and purple ellipses highlight parameter regions of interest.", "White vertical and horizontal reference lines mark Q=70%Q=70\\% and t d =8t_d=8 weeks.Fig.", "REF shows how peak height varies with quarantine duration, intensity and start date.", "This information is complemented by Fig.", "REF , that shows how peak center changes with quarantine parameters, and Fig.", "REF , displaying the proportion of recovered individuals at the end of the epidemic.", "The purple ellipse in Fig.", "REF marks the parameter region where quarantine is very intense and lasts for more than 8 weeks, an ideal situation that works around $90\\%$ of the times but is very hard to enforce in practice.", "In this case the epidemic stops quickly (blue areas in Fig.", "REF ) and less than $10\\%$ of the population is infected (green areas in Fig.", "REF ).", "The red ellipse in Fig.", "REF shows a transition zone where the best scenario corresponds to substantial curve flattening.", "The center of the red ellipse is at $Q \\approx 0.5$ for $t_s=20$ but shifts to $Q \\approx 0.9$ for $t_s=40$ , showing the importance of starting quarantine early.", "For all values of $t_s$ the red ellipse is centered at $t_d \\approx 6$ weeks, which is a relatively short duration.", "Peak center, however, is not delayed in the best case scenarios.", "Importantly, best case scenarios are very unlikely in this region, occurring with probability around $20\\%$ .", "Finally, the region surrounded by the green ellipse in Fig.", "REF corresponds to long but moderate intensity quarantines.", "For the three values of $t_s$ considered peak height was reduced by about $50\\%$ in the best case scenarios, which happens about $50\\%$ of the times.", "Peak center was not significantly delayed in the best scenarios, but was pushed forward in the worst scenarios, where peak height was reduced to about $70\\%$ with respect to non-quarantine height.", "Interestingly, in both scenarios about $70\\%$ of the population was infected at the end of the simulation, showing that herd immunity was achieved (corresponding to the pink areas in Fig REF ).", "Figure: Peak center (in days after the first infection) starting 20, 30 or 40 days after the first infection (left, middle and right columns respectively) for different quarantine intensities for best and worst scenarios.Quarantine can also be implemented in the mean field model, Eqs.", "(REF ).", "[27] This is accomplished by integrating the dynamical equations with the infection rate $\\beta _0$ for $t \\in [0,t_s]$ , with the reduced value $\\beta _Q = (1-Q) \\beta _0$ during quarantine period $t_s < t < ts+ t_d$ and again with $\\beta _0$ for $t>t_s+t_d$ .", "Fig.", "REF shows how results of mean field model differ from the IBM simulations.", "Panel (a) shows the dynamics without quarantine according to the mean field (thick lines) and 25 simulations with the IBM.", "Panel (b) shows the effects of quarantine on the mean field model for $Q=0.35$ , $t_d=10$ weeks and several starting times $t_s$ .", "According to the mean field model quarantine is effective only if started later, otherwise the infection curve peaks at high values when the quarantine is over.", "The right panels compare IBM simulations (c) and mean field results (d) for $t_s=30$ days and $t_d=15$ weeks for several quarantine intensities $Q$ .", "The mean field infection curves always grow to high values when quarantine is over, whereas the IBM simulations show many examples of low peak values and total epidemic control, with $I+E$ going to zero after the quarantine period.", "This highlights the importance of heterogeneous social interactions represented by the Barabási-Albert network and stochastic dynamics in epidemiological modeling.", "Figure: Proportion of recovered individuals at the end of the epidemic for quarantine starting 20, 30 or 40 days after the first infection (left, middle and right columns respectively) for different quarantine intensities for best and worst scenarios.Figure: (a) dynamics without quarantine computed with mean field equations (thick lines) and IBM simulations (thin lines); (b) mean field results with Q=0.35Q=0.35, t d =10t_d=10 weeks and several starting dates t s t_s; (c) 25 IBM simulations and (d) mean field dynamics for t s =30t_s=30 days, t d =15t_d=15 weeks and several intensities QQ.", "For the mean field equations, we set N=2000N=2000, β=R 0 γ\\beta =R_0\\gamma , γ=1/14\\gamma =1/14, σ=1/〈t i 〉\\sigma =1/\\langle t_i\\rangle , and starting with one infected individual." ], [ "Conclusions", "In this paper we considered the effects of quarantine duration, starting date and intensity in the outcome of epidemic spreading in a population presenting heterogeneous degrees of connections.", "The model is stochastic and curves representing numbers of infected individuals vary considerably from one simulation to the other even when all model parameters are fixed.", "In order to distinguish between different outcomes we have divided them into two groups with the best and worst results based on the height of the infection peak (below or above the average height, respectively).", "We have further divided the results into four qualitative classes delimited by the three ellipses in Fig.", "REF plus the rest of the diagram.", "Besides the obvious region indicated by the purple ellipse where quarantine is very intense and long, we found that short but not so intense quarantine (red ellipse) does not work, since the probability of an outcome in the best scenario is very low.", "Instead, long but average intensity quarantine is both likely to work and flattens the infection curve by around $50\\%$ , being the best alternative given the current assumptions.", "Indeed, the infection peak is considerably delayed in the region of the green ellipse when it falls into the worst scenario, confirming it as the best bet for preventing the health system breakdown (Fig.", "REF ).", "The proportion of the population that had contact with the virus at the end of the epidemic (number of recovered individuals, Fig.", "REF ) leads to more than $60\\%$ of the population, very close to achieving herd immunity.", "Comparing to the other regions, this seems to be the best option to control the epidemics under the model assumptions.", "We note, however, that the model does not account for deaths.", "If achieving herd immunity implies high mortality, the best option would be long and intense quarantine (purple ellipses in Fig.", "REF ), the only way to avoid large number of infections and, therefore.", "high mortality.", "We found that differences between mean field and stochastic models are very significant with respect to the effects of quarantine.", "In many cases as the former cannot control the epidemic, as the infection peak grows again once the quarantine period is over, whereas the latter can end the epidemic in the best case scenarios.", "We recall that we used uniform decrease in infection rate as a proxy for quarantine.", "This is a simplified approach and other methods could be implemented to verify the robustness of the results.", "Also, different network topologies might affect the spread of the epidemics.", "Random uniform (Erdos-Renyi) [22] networks should produce results similar to mean field simulations, but small-world [29], [22] or other topologies could speed up or slow down the spread dynamics.", "Our model is particularly suited to study spread between connected cities, that can be represented by modules of a larger network.", "We have also kept information about the virus DNA and its mutations, allowing us to reconstruct the phylogeny and classify its strains as it propagates.", "These results will be published in a forthcoming article.", "We thank Flavia D. Marquitti for a critical reading of this manuscript.", "This work was supported by the São Paulo Research Foundation (FAPESP), grants 2019/13341-7 (VMM), 2019/20271 and 2016/01343-7 (MAMA), and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), grant 301082/2019-7 (MAMA)." ] ]

[ [ "Image Restoration from Parametric Transformations using Generative\n Models" ], [ "Abstract When images are statistically described by a generative model we can use this information to develop optimum techniques for various image restoration problems as inpainting, super-resolution, image coloring, generative model inversion, etc.", "With the help of the generative model it is possible to formulate, in a natural way, these restoration problems as Statistical estimation problems.", "Our approach, by combining maximum a-posteriori probability with maximum likelihood estimation, is capable of restoring images that are distorted by transformations even when the latter contain unknown parameters.", "The resulting optimization is completely defined with no parameters requiring tuning.", "This must be compared with the current state of the art which requires exact knowledge of the transformations and contains regularizer terms with weights that must be properly defined.", "Finally, we must mention that we extend our method to accommodate mixtures of multiple images where each image is described by its own generative model and we are able of successfully separating each participating image from a single mixture." ], [ "Introduction", "As a first step towards the presentation of our methodology for image restoration let us introduce a simple mathematical problem.", "Assume that we are observing a vector $Y$ which is a transformed and noisy version of a hidden vector $X$ .", "We are interested in estimating $X$ from the observation $Y$ when we have available a generative model that captures the statistical behavior of $X$ .", "A number of well known computer vision problems can be formulated as the estimation problem we just described: Inpainting, which consists in reconstructing an image $X$ when we have available its partially erased version $Y$ ; Super-resolution, where from a lower resolution image $Y$ we recover a higher resolution version $X$ ; Image coloring, where from a gray-color image $Y$ we recover the full-color (RGB) version $X$ ; Image separation when we separate images from a linear (or nonlinear) combination and finally, Generative model inversion, where from an image $Y$ we identify the input to the generative model that generates an output which is as close as possible to the available image $Y$ ; are just a few of the image restoration problems of interest.", "Regarding image inpainting, classical techniques can be found in [1], [6], [15], [22], [39]; for super-resolution an overview of classical methods exists in [2], [16], [17], [19], [31] and for coloring in [21], [38].", "With the advent of generative networks [18], a new class of tools has emerged which is based on generative models.", "For inpainting such methods are considered in [40], [45], [46], with the super-resolution and coloring problem also being mentioned in the same articles.", "The inversion of a generative model enjoys exactly the same formulation and suitable solutions can be found in [13], [27].", "Finally, for the image separation problem recent efforts based on generative modeling can be found in [3], [25], [41], [43].", "Early efforts based on generative modeling [13], [25], [27], [40], [45], [46] were using the generative model partially (only the generator function).", "Only recently [3], [5], [41], [43], [48] we see techniques that employ the complete model (generator function and input density) improving the performance of the corresponding methods.", "Even in these recent efforts we observe the existence of weighting parameters that need to be tuned.", "The tuning process is carried out by applying the corresponding method multiple times with different parameter values and adopting the one delivering the best result.", "Since our own methodology relies on the well established Statistical estimation theory, we will be able to identify in a well defined way all parameters entering into the problem.", "In fact the method we are going to develop will be able to treat cases where the transformation responsible for the image distortion is not exactly known as required by all existing methods." ], [ "Statistical estimation theory", "Let us recall some very basic elements from Statistical estimation theory.", "Consider two random vectors $Z,Y$ where $Z$ is hidden while $Y$ is observed.", "Using $Y$ , we would like to estimate the hidden vector $Z$ when $Z,Y$ are statistically related, with their relationship captured by the joint probability density function (pdf) $Y,Z)$ .", "The existence of a joint density $Y,Z)$ allows for the applications of the well known Bayesian estimation theory [32] for the estimation of $Z$ from $Y$ .", "According to this theory, any deterministic function $\\hat{Z}(Y)$ of $Y$ can play the role of an estimator for $Z$ .", "Bayesian estimation provides a solid mathematical mechanism for evaluating these candidate estimators and identifying the one that will play the role of the final optimum estimator.", "Following this theory, we first need to provide a cost function $\\hat{Z},Z)$ that places a cost on each combination $\\lbrace \\hat{Z},Z\\rbrace $ of estimate and true value.", "Then, the performance criterion of an estimator is defined as the average cost $\\mathsf {E}[\\hat{Z}(Y),Z)]$ , where expectation is with respect to $Z$ and $Y$ .", "An estimator $\\hat{Z}(Y)$ will be regarded as optimum if it minimizes the average cost among all deterministic functions of $Y$ .", "In the existing theory, one can find several meaningful cost functions along with their corresponding optimum estimators.", "From the available possibilities we distinguish two candidates that are of interest to our analysis.", "Specifically, we focus on the minimum mean square error (MMSE) and the maximum a-posteriori probability (MAP) estimators [32] which we present next.", "MMSE: The cost function for this estimator is $\\hat{Z},Z)=\\Vert \\hat{Z}-Z\\Vert ^2$ .", "It is then well known [32] that the optimum estimator is defined as the conditional mean $\\hat{Z}=\\mathsf {E}[Z|Y]=\\int Z Z|Y)\\,dZ=\\int Z \\frac{Y,Z)}{Y)}\\,dZ=\\frac{\\int ZY,Z)\\,dZ}{\\int Y,Z)\\,dZ}.$ where $Z|Y)$ denotes the conditional probability density of $Z$ given $Y$ .", "MAP: Here the cost function is somewhat peculiar and defined as $\\hat{Z},Z)=\\left\\lbrace \\begin{array}{cl}1,&\\text{if}~\\Vert \\hat{Z}-Z\\Vert \\ge \\delta \\\\0,&\\text{otherwise},\\end{array}\\right.$ where $0<\\delta \\ll 1$ denotes a very small quantity (tending to 0).", "This criterion is known [32] to lead to the well known MAP estimator which is defined as $\\hat{Z}=\\text{arg}\\max _ZZ|Y)=\\text{arg}\\max _Z\\frac{Y,Z)}{Y)}=\\text{arg}\\max _ZY,Z),$ corresponding to the most likely $Z$ given the observations $Y$ .", "There are of course other popular alternatives as, for example, the minimum mean absolute error (MMAE) criterion which leads to the conditional median estimator [32].", "However, in this work we analyze only the two estimators depicted in (REF ) and (REF ) and in the simulations we basically use the MAP estimator that presents clear computational advantages." ], [ "Including unknown parameters", "The previous classical results are based on the assumption that there is available (known) a joint density $Y,Z)$ that captures the statistical relationship between $Z$ and $Y$ .", "In practice, however, the joint density may also contain a number of parameters that we express with the help of a vector $\\gamma $ .", "In other words, the joint pdf of $Y,Z$ is of the form $Y,Z|\\gamma )$ for some vector $\\gamma $ .", "The Bayesian approach treats parameters as random as well, consequently $Y,Z|\\gamma )$ is regarded as the pdf of $Y,Z$ conditioned on $\\gamma $ .", "Since $\\gamma $ is also random, its statistical behavior is expressed by a pdf $\\mathsf {p}(\\gamma )$ .", "This implies that the joint density of all three random vectors has the form $Y,Z,\\gamma )=Y,Z|\\gamma )\\mathsf {p}(\\gamma )$ .", "As before, we assume that we observe $Y$ and interested in estimating $Z$ .", "The question now is how should we treat $\\gamma $ .", "There are two possibilities: Marginalization of $\\gamma $: We can compute $Y,Z)=\\int Y,Z|\\gamma )\\mathsf {p}(\\gamma )\\,d\\gamma $ and then use (REF ) or (REF ).", "Estimation of $\\gamma $ : We can apply (REF ) or (REF ) with $\\hat{Z}$ replaced by $\\lbrace \\hat{Z},\\hat{\\gamma }\\rbrace $ , thus considering $\\gamma $ as part of the quantities to be estimated.", "Since our focus is only on $\\hat{Z}$ this implies that finding $\\hat{\\gamma }$ is simply an intermediate step towards our desired goal.", "In the applications of interest, as we will see, we have available the density $Y,Z|\\gamma )$ but not $\\mathsf {p}(\\gamma )$ .", "We can overcome this lack of knowledge by following a worst-case scenario, namely assume that $\\mathsf {p}(\\gamma )$ is the most uninformative prior.", "If $\\gamma \\in {C}$ , where ${C}$ is some known set, then this corresponds to selecting $\\mathsf {p}(\\gamma )$ to be the uniform over ${C}$ , provided that the Lebesgue measure $\\mu ({C})$ is finite.", "If $\\mu ({C})=\\infty $ then we can adopt for $\\mathsf {p}(\\gamma )$ the improper uniform [36].", "We can easily verify that in both cases we obtain the same results if we consider from the start that $\\mathsf {p}(\\gamma )=1$ for all $\\gamma \\in {C}$ .", "This is exactly what we are going to use in our subsequent analysis.", "MMSE: Here, marginalization and estimation of $\\gamma $ result in exactly the same estimate for $Z$ which, under the improper uniform, takes the form $\\bar{(Y,Z)=\\int \\limits _{\\gamma \\in {C}}Y,Z|\\gamma )\\,d\\gamma ,~~\\hat{Z}=\\frac{\\int Z\\bar{(Y,Z)\\,dZ}{\\int \\bar{(Y,Z)\\,dZ}.", "}\\textbf {MAP:} For this estimator the two approaches differ.", "In the case of the marginalization approach we obtain\\begin{equation}\\hat{Z}=\\text{arg}\\max _{Z}\\bar{(Z,Y),}with \\bar{(Z,Y) defined in (\\ref {eq:1.4-2}), while the estimation approach yields\\begin{equation}\\tilde{(Z,Y)=\\max _{\\gamma \\in {C}}Y,Z|\\gamma ),~~\\hat{Z}=\\text{arg}\\max _{Z}\\tilde{(Z,Y).", "}We observe that (\\ref {eq:1.5-2}) is equivalent to first performing a \\textit {maximum likelihood} estimation \\cite [pp.", "319–351]{veeravalli} of \\gamma followed by a MAP estimation of Z.", "}After this very brief presentation of the necessary results from Statistical estimation theory, we are now ready to apply these ideas to image restoration problems.\\end{equation}}\\end{equation}\\section {Image restoration and generative model}Let us focus on the general problem of interest and include the generative model into our formulation.", "Suppose X is a random vector described by the generative model X=\\mathsf {G}(Z) with the input Z being distributed according to the density Z).", "Both functions \\lbrace \\mathsf {G}(Z),Z)\\rbrace that comprise the generative model are assumed \\textit {known}.", "The generator \\mathsf {G}(Z) can be a neural network (deep or shallow), trained with the help of adversarial \\cite {arjovsky,basioti1,binkowski,goodfellow,nowozin} or non-adversarial techniques \\cite {basioti2,dziugaite}.", "If a non-adversarial training method is adopted then this clearly suggests that a discriminator function, which plays an important role in the techniques proposed in \\cite {daras,siavelis,yeh1,yeh2}, does not exist.", "For this reason our goal, similarly to \\cite {asim,bora,whang}, is to propose estimation techniques that do not rely on discriminator functions.", "}{E}very time a realization of X occurs, we assume that it undergoes a transformation and its noisy version is observed as a data vector Y.", "More specifically, the observation vector Y and the original vector X are related through the following equation\\begin{equation}Y=X,\\alpha )+W.\\end{equation}X,\\alpha ) is a deterministic transformation with known functional form that can possibly contain \\textit {unknown} parameters \\alpha \\in {A} with {A} a known set.\\footnote {When parameter vector \\alpha exists we assume that it may change with \\textit {each realization} of X.", "For this reason \\alpha \\textit {cannot} be tuned before hand using training data and make X,\\alpha ) completely known.", "}W is a random vector independent from X that expresses additive noise and/or \\textit {modeling errors}.", "For W we assume that it is distributed according to the density \\mathsf {g}(W|\\beta ) which has a known functional form and possibly \\textit {unknown} parameters \\beta \\in {B}, with {B} a known set.", "The problem we would like to solve is the recovery of the vector X from the observation vector Y.", "To achieve this estimate we intend to exploit the generative model in the following way: Instead of finding directly the estimate \\hat{X} as in \\cite {asim,whang},we propose to obtain the estimate \\hat{Z} of the input to the generator and then estimate X as \\hat{X}=\\mathsf {G}(\\hat{Z}).", "This simple variation allows for the adoption of \\textit {any form} for the generator function without special preference to invertible ones and, as we will see, for efficient handling of unknown parameters.", "}To apply the estimation theory presented in the previous section we first need to find the joint density of Y,Z.", "It is easy to see that the parameter vector \\gamma of the general theory corresponds to the combination \\lbrace \\alpha ,\\beta \\rbrace and\\begin{equation}Y,Z|\\alpha ,\\beta )=\\mathsf {g}\\big (Y-\\mathsf {G}(Z),\\alpha )|\\beta \\big )\\,Z).\\end{equation}In (\\ref {eq:2.3}) Z) is completely known since it is the pdf of the input to the generative model, while \\mathsf {g}(W|\\beta ) is known up to possibly some unknown parameter vector \\beta .", "If we apply the estimators of the previous section then for the MMSE in (\\ref {eq:1.4-2}) we must define\\begin{equation}\\bar{\\mathsf {g}}(W)=\\int \\limits _{\\beta \\in {B}}\\mathsf {g}(W|\\beta )\\,d\\beta ,~~~~\\bar{\\bar{\\mathsf {g}}}(Y,Z)=\\int \\limits _{\\alpha \\in {A}}\\bar{\\mathsf {g}}\\big (Y-\\mathsf {G}(Z),\\alpha )\\big )\\,d\\alpha \\end{equation}which yields the estimate\\begin{equation}\\hat{Z}=\\frac{\\int Z\\bar{\\bar{\\mathsf {g}}}(Y,Z)Z)\\,dZ}{\\int \\bar{\\bar{\\mathsf {g}}}(Y,Z)Z)\\,dZ}=\\frac{\\mathsf {E}_{Z}[Z\\bar{\\bar{\\mathsf {g}}}(Y,Z)]}{\\mathsf {E}_{Z}[\\bar{\\bar{\\mathsf {g}}}(Y,Z)]}.\\end{equation}The last ratio contains expectations with respect to Z which is distributed according to Z).", "Similarly, for the MAP estimator in (\\ref {eq:1.5-1}) and (\\ref {eq:1.5-2}) we can write for the marginalization approach that\\begin{equation}\\hat{Z}=\\text{arg}\\max _Z\\bar{\\bar{\\mathsf {g}}}(Y,Z),\\end{equation}with \\bar{\\bar{\\mathsf {g}}}(Y,Z) defined in (\\ref {eq:2.4.5}), while for the estimation approach we have\\begin{equation}\\tilde{\\mathsf {g}}(W)=\\max _{\\beta \\in {B}} \\mathsf {g}(W|\\beta ),~~\\tilde{\\tilde{\\mathsf {g}}}(Y,Z)=\\max _{\\alpha \\in {A}} \\tilde{\\mathsf {g}}\\big (Y-\\mathsf {G}(Z),\\alpha )\\big ),~~\\hat{Z}=\\text{arg}\\max _Z\\tilde{\\tilde{\\mathsf {g}}}(Y,Z)Z).\\end{equation}When the transformation X) does not contain any unknown parameters, the previous expressions simplify to\\begin{equation}\\hat{Z}=\\frac{\\int Z\\bar{\\mathsf {g}}\\big (Y-\\mathsf {G}(Z))\\big )Z)\\,dZ}{\\int \\bar{\\mathsf {g}}\\big (Y-\\mathsf {G}(Z))\\big )Z)\\,dZ}=\\frac{\\mathsf {E}_{Z}\\big [Z\\bar{\\mathsf {g}}\\big (Y-\\mathsf {G}(Z))\\big ]}{\\mathsf {E}_{Z}\\big [\\bar{\\mathsf {g}}\\big (Y-\\mathsf {G}(Z))\\big )\\big ]},\\end{equation}for the MMSE, while for the MAP estimation we have\\begin{equation}\\hat{Z}=\\text{arg}\\max _Z\\bar{\\mathsf {g}}\\big (Y-\\mathsf {G}(Z))\\big )Z),~\\text{or}~\\hat{Z}=\\text{arg}\\max _Z\\tilde{\\mathsf {g}}\\big (Y-\\mathsf {G}(Z))\\big )Z),\\end{equation}with the first corresponding to marginalization and the second to maximum likelihood estimation of \\beta .Our approach, because it is based on the classical Statistical estimation theory, enjoys a number of interesting properties: 1)~As in \\cite {asim,whang}, it uses the \\textit {complete} generator model in order to perform the estimation and does not require any discriminator function.", "2)~The final optimization problem does not contain terms in the form of regularizers that include \\textit {unknown weights} that require tuning.", "3)~We can treat transformations and noise pdfs that contain unknown parameters which are being properly identified using maximum likelihood estimation.", "The last two properties are unique to our proposed approach and are not present in any other existing generative image restoration methodology.$" ], [ "Gaussian noise and Gaussian input", "Let us now specify in more detail our mathematical model.", "For the additive noise vector $W$ appearing in the data model in (), we assume that it has Gaussian elements that are independent and identically distributed with mean zero and variance $\\beta ^2$ .", "We adopt the Gaussian model only for simplicity.", "It is possible to resort to more general noise models as for example the Student's $t$ -distribution which was successfully employed in classical (non generative) image restoration techniques [12].", "Unfortunately Student's $t$ distribution does not offer closed form expressions for its parameters as the Gaussian case.", "And this is something we would like to have in order to be able to compare our resulting cost function with the costs employed in the existing literature.", "Limiting ourselves to Gaussian noise for $W$ where each element has zero mean and variance $\\beta ^2$ , yields $\\mathsf {g}(W|\\beta )\\!=\\!\\frac{e^{-\\frac{1}{2\\beta ^2}\\Vert W\\Vert ^2}}{\\sqrt{(2\\pi )^N\\beta ^{2N}}},~~\\bar{\\mathsf {g}}(W)\\!=\\!\\int _0^\\infty \\mathsf {g}(W|\\beta )\\,d\\beta \\!=\\!\\frac{C}{\\Vert W\\Vert ^{N+1}},~~\\tilde{\\mathsf {g}}(W)\\!=\\!\\max _{\\beta \\ge 0}\\mathsf {g}(W|\\beta )\\!=\\!\\frac{C^{\\prime }}{\\Vert W\\Vert ^N},$ where $C,C^{\\prime }$ constants and $N$ is the length of the vector $W$ .", "If we assume that the input density $Z)$ is also Gaussian $\\mathcal {N}(0,I)$ , where $I$ the identity matrix, then for known transformations $X)$ the MMSE estimate in () becomes $\\hat{Z}=\\frac{\\mathsf {E}_{Z}[Z\\Vert Y-\\mathsf {G}(Z))\\Vert ^{-(N+1)}]}{\\mathsf {E}_{Z}[\\Vert Y-\\mathsf {G}(Z))\\Vert ^{-(N+1)}]}\\approx \\frac{\\sum _{i=1}^LZ_i\\Vert Y-\\mathsf {G}(Z_i))\\Vert ^{-(N+1)}]}{\\sum _{i=1}^L\\Vert Y-\\mathsf {G}(Z_i))\\Vert ^{-(N+1)}]}.$ We note that we generate realizations $\\lbrace Z_1,\\ldots ,Z_L\\rbrace $ of $Z)$ and by evoking the the Law of Large Numbers we approximate the MMSE estimate.", "For the MAP estimates in () we have $\\hat{Z}=\\text{arg}\\min _Z\\lbrace M\\log \\Vert Y-\\mathsf {G}(Z))\\Vert ^2+\\Vert Z\\Vert ^2\\rbrace ,~\\text{where}~M=N+1~\\text{or}~M=N,$ with $M=N+1$ corresponding to the marginalization and $M=N$ to the estimation of $\\beta $ .", "The two expressions are clearly very similar especially in the case where $N\\gg 1$ .", "We can now compare our optimizations in (REF ) with $\\hat{Z}=\\text{arg}\\min _Z\\lbrace \\Vert Y-\\mathsf {G}(Z))\\Vert ^2+\\lambda \\Vert Z\\Vert \\rbrace ~\\text{and}~\\hat{Z}=\\text{arg}\\min _Z\\lbrace \\Vert Y-\\mathsf {G}(Z))\\Vert ^2+\\lambda \\Vert Z\\Vert ^2\\rbrace ,$ where the first is proposed in [5] and the second in [10], [48].", "Both approaches in (REF ) contain an unknown weight $\\lambda $ which in order to be tuned properly we need to solve the corresponding optimization problem several times for different values of this parameter and select the one delivering the best results.", "In our approach in (REF ) such parameter is clearly not present.", "Another notable difference is how the error distance $\\Vert Y-\\mathsf {G}(Z))\\Vert ^2$ is combined with the input power $\\Vert Z\\Vert ^2$ .", "In our method we use the logarithm of the distance while in [5], [10], [48] it is the distance itself combined with $\\Vert Z\\Vert $ or $\\Vert Z\\Vert ^2$ .", "We would like to emphasize that the cost function of our approach is not selected in some arbitrary way but it is the outcome of a theoretical analysis which is based on the Statistical estimation theory.", "Even though the MMSE estimator in (REF ) does not involve any additional optimization when the transformation $X)$ has no parameters, it can be used only when the length $N$ of $W$ is small.", "Indeed, for large $N$ the expression $\\Vert Y-\\mathsf {G}(Z_i))\\Vert ^{-(N+1)}$ may very easily take extremely small or extremely large values which will cause computational problems due to finite precision.", "Unless $N$ is of the order of a few tens this method should be avoided.", "This observation clearly applies to images where $N$ can be several thousands.", "From now on we focus on MAP estimation and since the difference between the two versions in (REF ) is minor when $N$ is large, we adopt the second version where we estimate $\\beta $ instead of marginalizing it." ], [ "Parametric transformations", "Let us now consider the more challenging problem of a transformation $X,\\alpha )$ containing unknown parameters $\\alpha $ .", "Following our general theory developed for the case of noise and generator input being Gaussian, the MAP estimator with maximum likelihood estimation of the parameters is equivalent to $\\hat{Z}=\\text{arg}\\min _Z\\big \\lbrace N\\log \\big (\\min _{\\alpha }\\Vert Y-\\mathsf {G}(Z),\\alpha )\\Vert ^2\\big )+\\Vert Z\\Vert ^2\\big \\rbrace .$ Additionally, if the transformation is linearIn most restoration problems the transformation enters as a matrix multiplied element-by-element with the ideal image which is also expressed as a matrix.", "If we reshape the image into a vector then this multiplication becomes a classical linear matrix/vector multiplication which is what we adopt in our analysis., that is, $X,\\alpha )=\\alpha )X$ , which is the case in most restoration problems, then the optimization problem in (REF ) becomes $\\hat{Z}=\\text{arg}\\min _Z\\big \\lbrace N\\log \\big (\\min _{\\alpha }\\Vert Y-\\alpha )\\mathsf {G}(Z)\\Vert ^2\\big )+\\Vert Z\\Vert ^2\\big \\rbrace ,$ where $\\alpha )$ is a matrix parametrized with $\\alpha $ .", "Finally we can further advance our analysis if we assume that $\\alpha )$ is linear in its parameters namely it can be decomposed as $\\alpha )=\\alpha _11+\\cdots +\\alpha _MM$ where $1,\\ldots ,T_M$ are known matrices and $\\alpha =[\\alpha _1,\\ldots ,\\alpha _M]^{\\intercal }$ is the unknown parameter vector.", "As an example consider the coloring problem where we have ${\\rm R},{\\rm G},{\\rm B}$ with each matrix isolating the corresponding RGB component from the ideal colored image and the scalar quantities $\\lbrace \\alpha _{\\rm R},\\alpha _{\\rm G},\\alpha _{\\rm B}\\rbrace $ denoting the percentage by which each component contributes to the final gray level.", "If these percentages are known before hand then the resulting transformation $\\alpha _{\\rm R}{\\rm R}+\\alpha _{\\rm G}{\\rm G}+\\alpha _{\\rm B}{\\rm B}$ is also known and the coloring problem can be treated by existing techniques.", "If, however, $\\lbrace \\alpha _{\\rm R},\\alpha _{\\rm G},\\alpha _{\\rm B}\\rbrace $ are unknown, then we need to estimate these parameters in parallel with $Z$ , by solving (REF ).", "Regarding the minimization with respect to $\\alpha $ we can either combine it with the minimization with respect to $Z$ and use, for example, a gradient descent for the pair $\\lbrace Z,\\alpha \\rbrace $ or, in the linear case we can find the analytic solution of the minimization with respect to $\\alpha $ , substitute it, and then minimize only over $Z$ .", "The first idea is straightforward and requires no further explanation.", "For the second if we define the matrix ${T}=[1\\mathsf {G}(Z),\\ldots ,M\\mathsf {G}(Z)]$ then from (REF ) and (REF ) we conclude that $\\min _{\\alpha }\\Vert Y-\\alpha )\\mathsf {G}(Z)\\Vert ^2=\\min _{\\alpha }\\Vert Y-{T}\\alpha \\Vert ^2=\\Vert Y\\Vert ^2-Y^{\\intercal }{T}({T}^{\\intercal }{T})^{-1}{T}^{\\intercal }Y,$ with the last outcome following from the Orthogonality Principle [20] and expressing the projection error of $Y$ onto the space generated by the columns of ${T}$ .", "This result, when substituted in (REF ), yields $\\hat{Z}=\\text{arg}\\min _Z\\left\\lbrace N\\log \\left(\\Vert Y\\Vert ^2-Y^{\\intercal }{T}({T}^{\\intercal }{T})^{-1}{T}^{\\intercal }Y\\right)+\\Vert Z\\Vert ^2\\right\\rbrace ,$ where the only unknown is $Z$ and, we recall from its definition, that ${T}$ is also a function of $Z$ ." ], [ "The image separation problem", "In [3], [41], [43] existing single image methods are extended to combinations of multiple images where each image is described by a separate generative model.", "These extensions experience the same drawbacks as the original methods, namely, 1) they contain multiple regularizer terms with unknown weights that need to be properly determined, 2) the proposed criteria are not the outcome of some rigorous mathematical analysis, and 3) the corresponding methods in [3], [41], [43] cannot accommodate combinations involving unknown parameters.", "We can overcome the previous weaknesses by generalizing our methodology to cover multiple images as well.", "For simplicity we only consider the two image case with the analysis of any number of images being very similar.", "Suppose that we have two images $X_1,X_2$ each satisfying a generative model $X_i=\\mathsf {G}_i(Z_i)$ with input density $Z_i\\sim i(Z),~i=1,2$ .", "If the data model follows $Y=\\alpha _1X_1+\\alpha _2X_2+W$ where the additive noise/modeling error $W$ has density $\\mathsf {g}(W|\\beta )$ with parameters $\\beta $ then we can combine all parts and produce the joint probability density $Y,Z_1,Z_2|\\alpha _1,\\alpha _2,\\beta )=\\mathsf {g}\\big (Y-\\alpha _1\\mathsf {G}_1(Z_1)-\\alpha _2\\mathsf {G}_2(Z_2)|\\beta \\big )\\,1(Z_1)\\,2(Z_2).$ In (REF ) we made the assumption that $Z_1,Z_2$ are statistically independent which produced the product of the two input densities.", "Following the general theory and limiting ourselves to the MAP estimator we need to solve the optimization problem $\\max _{Z_1,Z_2}\\max _{\\alpha _1,\\alpha _2,\\beta }\\!Y,Z_1,Z_2|\\alpha _1,\\alpha _2,\\beta )\\!=\\!\\max _{Z_1,Z_2}\\!\\big \\lbrace \\!\\max _{\\alpha _1,\\alpha _2,\\beta }\\mathsf {g}\\big (Y-\\alpha _1\\mathsf {G}_1(Z_1)-\\alpha _2\\mathsf {G}_2(Z_2)|\\beta \\big )\\big \\rbrace 1(Z_1)2(Z_2).$ If, as before, $\\mathsf {g}(W|\\beta )$ is Gaussian with mean 0 and covariance $\\beta ^2I$ and both input vectors are independent Gaussian with mean 0 and unit covariance matrix then from (REF ), after maximizing over $\\beta $ , we conclude that $\\lbrace \\hat{Z}_1,\\hat{Z}_2\\rbrace =\\text{arg}\\,\\min _{Z_1,Z_2}\\big \\lbrace N\\log \\big (\\min _{\\alpha _1,\\alpha _2}\\Vert Y-\\alpha _1\\mathsf {G}_1(Z_1)-\\alpha _2\\mathsf {G}_2(Z_2)\\Vert ^2\\big )+\\Vert Z_1\\Vert ^2+\\Vert Z_2\\Vert ^2\\big \\rbrace .$ We can either apply gradient descent on the combination $\\lbrace Z_1,Z_2,\\alpha _1,\\alpha _2\\rbrace $ or solve analytically for $\\lbrace \\alpha _1,\\alpha _2\\rbrace $ , substitute, and then minimize over $\\lbrace Z_1,Z_2\\rbrace $ .", "Regarding the latter we have from the Orthogonality Principle [20] $\\min _{\\alpha _1,\\alpha _2}\\Vert Y-\\alpha _1\\mathsf {G}_1(Z_1)-\\alpha _2\\mathsf {G}_2(Z_2)\\Vert ^2=\\Vert Y\\Vert ^2-Y^{\\intercal }{G}({G}^{\\intercal }{G})^{-1}{G}^{\\intercal }Y$ where ${G}=[\\mathsf {G}_1(Z_1), \\mathsf {G}_2(Z_2)]$ .", "Substituting in (REF ) gives rise to $\\lbrace \\hat{Z}_1,\\hat{Z}_2\\rbrace =\\text{arg}\\,\\min _{Z_1,Z_2}\\left\\lbrace N\\log \\left(\\Vert Y\\Vert ^2-Y^{\\intercal }{G}({G}^{\\intercal }{G})^{-1}{G}^{\\intercal }Y\\right)+\\Vert Z_1\\Vert ^2+\\Vert Z_2\\Vert ^2\\right\\rbrace ,$ and where we observe that ${G}$ is a function of $\\lbrace Z_1,Z_2\\rbrace $ .", "We can also accommodate the case where transformations are applied to each individual image suggesting that each image can undergo a different deformation before the final mixture.", "This corresponds to replacing each component $\\alpha _i\\mathsf {G}_i(Z_i)$ in (REF ) with $i(\\mathsf {G}_i(Z_i),\\alpha _i)$ where each transformation $i(\\cdot )$ can have its own unknown local parameters $\\alpha _i$ .", "Obtaining the necessary equations for this more general setup presents no particular difficulty.", "Unlike the classical source separation problem [11], [24] where we need as many (linear) combinations of the sources as the number of sources we are seeking, here separation can be achieved from a single mixture.", "Of course this is possible because we have available a statistical description of the sources in the form of generative models.", "We recall that in classical source separation such description is not present and separation is achieved by processing directly the available combinations." ], [ "Datasets and Pretrained GAN models", "For our experiments, we use the CelebA [28] and the Caltech Birds [47] datasets.", "The first dataset contains 202,599 RGB images that are cropped and resized to 64x64x3 and then separated into two sets 202,499 for training and 100 for testing.", "For the Birds dataset, we train two models, one with the original images and the second with segmented images, namely, with removed background using the included segmentation masks.", "In both cases, the images are resized to 64x64x3 while we kept 10,609 images for training and 1179 for testing.", "The training data are used to design Wasserstein GANs [4] with the progressive method developed in [23].", "For all cases we use the following configuration, Generator: input 512 Gaussians $\\mathcal {N}(0,1)$ and five layers.", "Each layer consists of two convolutions with two kernels $3\\times 3$ except the first layer that has one $4\\times 4$ and one $3\\times 3$ kernel and the last that has two $3\\times 3$ and one $1\\times 1$ kernel resulting in an output of $64\\times 64\\times 3$ .", "After each convolution, a leaky ReLU is applied except for the last $1\\times 1$ convolution where no nonlinear function is used.", "The intermediate layers also involve an upsampling operation.", "Discriminator: Input $64\\times 64\\times 3$ with six layers in total.", "The first five layers have two convolutions with two $3\\times 3$ kernels except for the first layer which has an additional $1\\times 1$ layer and the last layer which has a $3\\times 3$ and a $4\\times 4$ kernel.", "After each convolution, we apply a leaky ReLU except for the last $4\\times 4$ kernel where no nonlinearity is used.", "In the intermediate layers, we apply downsampling except for the last layer.", "Finally, we employ a fully connected part that provides the scalar output of the discriminator.", "In all competing methods, we apply the momentum gradient descent [37] with normalized gradients where the momentum hyperparameter is set to 0.999, the learning rate to 0.001 and the algorithm runs for 200,000 iterations." ], [ "Deblurring", "Perhaps the most common deformation is due to a linear filter convolved with an image.", "In particular when the filter is one-dimensional and applied to the image row-by-row then this can model a horizontal motion blur.", "This idea can be clearly extended to cover blurring in any direction but for simplicity we consider the case of horizontal blurring.", "We use a finite impulse response filter of length 5 with coefficients $\\alpha _1=1.0187;\\alpha _2=-0.5933;\\alpha _3=-0.3501;\\alpha _4=0.4635;\\alpha _5=-0.24$ that were randomly generated.", "The goal is from the deformed image to restore the original.", "We compare the methods of [45] and [48] (which in this example coincides with [10]) against our method.", "The techniques in [45], [48] require exact knowledge of the filter coefficients.", "They also require fine-tuning of the weight $\\lambda $ appearing in (REF ).", "This is achieved by solving multiple instances of optimization problems with various weight values and selecting the one delivering the smallest reconstruction error.", "In the case of [45] this turns out to be 0.6 while in [48] 0.2.", "We should emphasize that these values are filter, transformation and data dependent meaning that if the filter coefficients or the transformation or the class of data changes we need to repeat the tuning procedure.", "What is even more crucial is the fact that tuning requires exact knowledge of the filter.", "Consequently, if the filter contains unknown parameters, the tuning process is impossible.", "Since our method has no unknown weights it can be applied directly without the need of any fine-tuning phase.", "We distinguish two versions in our approach.", "In the first we assume that we know the filter coefficients exactly in order to make fair comparisons with [45], [48].", "In the second version we assume that the filter coefficients are unknown which implies that we simultaneously estimate the filter coefficients and restore the original image by solving REF .", "Unlike our proposed technique, existing methods do not perform this combined optimization and are therefore unable to restore the original image when the transformation contains unknown parameters.", "Figure: Deblurring.", "Row a) Original; b) Blurred; c) , known parameters; d) , known parameters; e) Proposed, known parameters; f) Proposed, unknown parameters.", "Table: Reconstruction errors and PSNRs for Deblurring.As in [13], [45] we ran our simulations for every testing image three times, with different initializations and retain the solution with the smallest reconstruction error.", "In Figure REF we present the corresponding results for the problem of horizontal blurring.", "Row a) depicts the original faces; in row b) we see their blurred version when we apply the selected filter; rows c), d), e) present the restoration provided by [45], [48] and our first version respectively when the filter is known; finally row f) is the restoration results of our second version with the filter coefficients being unknown and estimated for each image at the same time with the restoration process.", "Table REF contains the corresponding restoration errorsSmaller error does not necessarily imply that the method provides visually better results.", "If this were the case then the preferable optimization would have been to minimize the restoration error.", "However it is well known that this criterion does not lead to satisfactory restorations (see for example video [35]).", "(per pixel average squared error between original and restored) and the Peak Signal to Noise Ratio (PSNR) of each method.", "Visibly it is difficult to distinguish differences between the various techniques.", "This fact is also captured in Table REF where the restoration errors and the PSNRs are comparable.", "The same observation applies even in the case of our second version that estimates the unknown coefficients.", "We must however emphasize once more that our versions compared to the existing techniques enjoy certain unique properties: 1) There are no weights that need fine-tuning; 2) The restoration quality is comparable even when the transformations have unknown parameters and 3) Our criterion is not an ad-hoc selection but the outcome of a rigorous mathematical analysis." ], [ "Colorization", "In the second set of experiments we recover an RGB image from one of its chromatic components.", "As such we select the green channel.", "This information is passed onto the methods of [45], [48] and to our first version.", "In our second version we assume that we do not know which channel generates the observed gray-level data and we attempt to find the right channel at the same time with the restoration process.", "We recall that the channel decomposition is a linear transformation that can be implemented with three matrices ${\\rm R},{\\rm G},{\\rm B}$ .", "The fact that the unknown parameter is now discrete does not pose any special difficulty in the optimization in (REF ), which must be modified as follows $\\hat{Z}=\\text{arg}\\min _Z\\big \\lbrace N\\log \\big (\\min _{i={\\rm R,G,B}}\\Vert Y-i\\mathsf {G}(Z)\\Vert ^2\\big )+\\Vert Z\\Vert ^2\\big \\rbrace ,$ For the solution of (REF ), (REF ) and (REF ) we employ the same algorithm and hyperparameters as in our first set of experiments, except of course the weight $\\lambda $ in [45] and [48] which we have to retune.", "This results in 0.1 for [45] and 0.5 for [48].", "In Figure REF , row a) depicts the original RGB images; we see the green channel in gray-level in row b); rows c), d), e) have the restoration results of [45], [48] and our proposed first version respectively Figure: Row a) Original; b) Transformed; c) , known parameters; d) , known parameters; e) Proposed, known parameters; f) Proposed, unknown parameters.", "Table: Reconstruction errors and PSNRs for Colorization.when the channel is known; finally row f) contains the results of our second version when the channel is unknown and must be discovered by solving (REF ).", "We also see in Table REF the corresponding restoration errors and PSNRs.", "Again we realize that our proposed methodology provides comparable restoration quality as the existing methods when the transformation is known without the need to fine-tune any weights.", "Additionally, it also delivers similar quality even if the transformation contains unknown parameters which may take discrete or continuous values." ], [ "Image Separation", "For the image separation problem we start by forming mixtures that are combinations of images from CelebA and segmented Caltech Birds.", "As before we implement the momentum gradient descent [37] with normalized gradients.", "The momentum hyperparameter is set to 0.999, the learning rate to 0.1 and we run the algorithm for 200,000 iterations.", "When the mixture coefficients $\\alpha _1, \\alpha _2$ are known we compare our method with the method developed in [41] which consists in solving the following optimization problem: $\\lbrace \\hat{Z}_1, \\hat{Z}_2\\rbrace = \\text{arg}\\min _{Z_1,Z_2}\\big \\lbrace \\Vert Y-\\alpha _1G_1(Z_1) - \\alpha _2 G_2(Z_2) \\Vert ^2+\\lambda _1\\Vert Z_1\\Vert ^2+\\lambda _2\\Vert Z_2\\Vert ^2\\big \\rbrace .$ Weights $\\lambda _1, \\lambda _2$ , as in the single image case, need to be fine-tuned.", "As before we select the values offering the best performance in terms of reconstruction error and PSNR.", "For the case where $\\alpha _1,\\alpha _2$ are unknown, we select values satisfying the constraint $\\alpha _1+\\alpha _2=1$ .", "Even though this is not necessary, we pass this information to our second version that estimates the two parameters in order to observe how it performs when there are also constraints present.", "Specifically we solve the problem $&\\lbrace \\hat{Z}_1,\\hat{Z}_2\\rbrace =\\text{arg}\\,\\min _{Z_1,Z_2}\\big \\lbrace N\\log \\big (\\min _{\\alpha _1,\\alpha _2}\\Vert Y-\\alpha _1\\mathsf {G}_1(Z_1)-\\alpha _2\\mathsf {G}_2(Z_2)\\Vert ^2\\big )+\\Vert Z_1\\Vert ^2+\\Vert Z_2\\Vert ^2\\big \\rbrace \\\\&\\text{for\\ } \\alpha _1+\\alpha _2=1.$ In the first set of experiments we select $\\alpha _1=\\alpha _2=0.5$ .", "For [41], fine-tuning its parameters results in $\\lambda _1=\\lambda _2=0.3$ .", "Figure REF contains separation examples using the segmented version of the Caltech Birds.", "Specifically in rows a), b) we have the original images, in row c) their mixture, in d), e) the separated images by the method in [41] when the mixture coefficients are known, in f), g) the corresponding results of our first version with the mixture coefficients being known and in h), i) the reconstructed images by our second version when the mixture coefficients are unknown.", "Table REF depicts the corresponding reconstruction errors and PSNRs per dataset.", "Figure: Rows a,b) Originals; c) Mixture; d),e) , known coefficients; f),g)  Proposed, known coefficients; h),i) Proposed, unknown coefficients.Table: Reconstruction errors and PSNRs for the segmented Caltech birds (1) and CelebA (2) datasets when α 1 =0.6,α 2 =0.4\\alpha _1 = 0.6, \\alpha _2 = 0.4.We also experimented with unequal mixing parameters and used $\\alpha _1=0.6,\\alpha _2=0.4$ .", "As we mentioned before, every time the transformation changes the weights $\\lambda _1,\\lambda _2$ defining the optimization problem in (REF ), for the method in [41], need to be retuned.", "This time tuning came up with the values $\\lambda _1=\\lambda _2 = 0.1$ .", "In Table REF we show the reconstruction errors and PSNRs achieved by [41] and our first version where $\\alpha _1,\\alpha _2$ are known and also our second version that treats $\\alpha _1,\\alpha _2$ as unknown.", "Finally, in our last set of experiments we separate CelebA faces from the original Caltech Birds.", "We use again $\\alpha _1=\\alpha _2=0.5$ .", "Since the datasets have changed, the weights $\\lambda _1, \\lambda _2$ in [41] need to be retuned.", "This time the best values we obtain are $\\lambda _1=0.5, \\lambda _2=0.4$ .", "Due to the more complicated nature of the images in the two datasets, for convergence we increased the number of iterations to 400,000.", "Figure: Rows a,b) Originals; c) Mixture; d),e) , known coefficients; f),g)  Proposed, known coefficients; h),i) Proposed, unknown coefficients.", "Table: Reconstruction errors and PSNRs for the Caltech birds (1) and CelebA (2) datasets when α 1 =α 2 =0.5\\alpha _1 = \\alpha _2 = 0.5.In Figure REF the rows a), b) have the original images, row c) their mixture, rows d) ,e) the separated images by [41] for known mixture coefficients, rows f), g) depict the results of our first version where mixture coefficients are known and rows h), i) the corresponding reconstruction by our second version when the coefficients are unknown.", "In Table REF we can see the corresponding reconstruction errors and PSNRs per dataset." ], [ "Conclusion", "We introduced a general image restoration methodology which is based on generative model description of the class (or classes) of images to be restored.", "Our processing technique relies on the Statistical estimation theory and is capable of restoring images through a mathematically well-defined optimization problem that does not require any tuning of weights of regularizer terms.", "The most important advantage of our method consists in its ability to restore and/or separate images even when the transformations responsible for the deformation contain unknown parameters.", "Experiments using popular dataset show that our technique, when applied to transformations with unknown parameters, it is capable of delivering similar restoration quality as the existing state of the art that needs exact knowledge of the transformations and tuning of weights for regularizer terms." ], [ "Acknowledgment", "This work was supported by the US National Science Foundation under Grant CIF 1513373, through Rutgers University." ] ]

[ [ "Exploiting Non-Linear Redundancy for Neural Model Compression" ], [ "Abstract Deploying deep learning models, comprising of non-linear combination of millions, even billions, of parameters is challenging given the memory, power and compute constraints of the real world.", "This situation has led to research into model compression techniques most of which rely on suboptimal heuristics and do not consider the parameter redundancies due to linear dependence between neuron activations in overparametrized networks.", "In this paper, we propose a novel model compression approach based on exploitation of linear dependence, that compresses networks by elimination of entire neurons and redistribution of their activations over other neurons in a manner that is provably lossless while training.", "We combine this approach with an annealing algorithm that may be applied during training, or even on a trained model, and demonstrate, using popular datasets, that our method results in a reduction of up to 99\\% in overall network size with small loss in performance.", "Furthermore, we provide theoretical results showing that in overparametrized, locally linear (ReLU) neural networks where redundant features exist, and with correct hyperparameter selection, our method is indeed able to capture and suppress those dependencies." ], [ "Introduction", "Modern Deep Neural Networks (DNN) have pushed the state-of-the-art in many computer vision tasks including image recognition, object detection, etc.", "Underlying the success of DNNs, are the millions of constituent parameters and their non-linear combinations that allow DNNs to accurately model a wide variety functions over their input features.", "However, running inference over these massive models imposes exorbitant memory and computational costs that make deploying them at scale in the real-world with more stringent latency, compute and energy constraints, a challenging problem.", "This problem is exacerbated if later models build on existing models, which are themselves very large.", "Often, the intermediate outputs of a pretrained image recognition model are used to encode visual information for another downstream task, such as visual question answering, image retrieval, etc.", "Since these intermediate outputs can be very high-dimensional, using them as features can introduce a large number of parameters to downstream models.", "Therefore, methods for compressing the neural models can have far reaching effects vis-a-vis their deployability and scalability.", "As deep learning models make their way from research labs to real world environments, the task of making them more resource efficient has received a great deal of attention.", "This has given rise to a large body of work focused on compressing and/or accelerating DNNs.", "One of the most common techniques for compressing neural models is parameter pruning, i.e.", "pruning away model parameters or neurons based on some metric.", "Such methods have many limitations ; one of them is that the heuristic used attempts to identify \"weak\" elements (with small magnitude or derivative, for instance), but parameters can be unnecessary in more ways than being small.", "Consider two neurons taken individually that both have large amplitude, but happen to yield identical outputs ; one of them can be pruned without loss of information, but most current pruning methods could not figure it out.", "In this paper we propose a novel neural model compression technique that exploits the dependencies in the non-linear activations of the units in each layer to identify redundant units and prune them.", "Our technique is based on the observation model optimization can potentially converge to a point at which the outputs of several units in one layer become highly correlated, even linearly dependent, with each other, and thus, by removing one of them and adjusting the outgoing weights of the other units we can obtain a smaller model with identical outputs.", "We identify redundant units by measuring the degree to which they can be predicted as a linear combination of other units in the same layer.", "Specifically, we learn a transformation matrix, $A$ , that best approximates an identity mapping for the activations of the units in this layer, while constraining the diagonal of $A$ to be zero.", "We select the units with the lowest prediction error to remove, and adjust the outgoing weights of the remaining units using the values in the corresponding columns $A$ such that input to the next layer remains the same (or is minimally perturbed).", "Once we have removed all the predictable units, removing any additional units will cause a reduction in model performance.", "We then fine-tune the model to recover the lost accuracy.", "In order to facilitate tuning, we use distillation [7] to bring the compressed model's output distribution close to the uncompressed model's output distribution raised to a high temperature.", "We demonstrate the efficacy of our technique on two popular image recognition models, AlexNet[10] and VGG[20], and three popular benchmark datasets, CIFAR-10[9] and Caltech-256[4].", "We demonstrate, theoretically and empirically, that under our proposed weight readjustment scheme, the inputs to the subsequent layer are only minimally perturbed while redundant units are present.", "Our technique can reduce the parameters of VGG and AlexNet by more than 99% on CIFAR10 and by more than 80% on Caltech-256.", "Finally, we inspect the intermediate representations of the compressed models and show that the data remains cleanly separable post-compression, which suggests that the intermediate representations continue to capture rich information about the data which may be useful for transfer learning tasks." ], [ "Related Work", "Most existing techniques for reducing the size of neural models can be grouped into three high-level categories, namely low-rank factorization, knowledge distillation and, parameter pruning.", "We argue that all of them have significant shortcomings.", "There are also methods that reduce computation while not really affecting the number of parameters in the network, such as quantization [5][17]; however these are somewhat orthogonal to the scope of this paper." ], [ "Pruning", "A common pruning approach to model compression attempts to eliminate individual parameters in the network [21], [22], [12].", "Many of them do so by enforcing a sparsity constraint, such as $L1$ regularization, to push some parameters to 0, or $L2$ regularization to simply keep weights small and then prune the small ones [6].", "Those methods can achieve reasonable performance (up to 35x compression in [5]).", "One of theses methods' main limitations however, is that their outputs take the form of sparse weight matrices, and benefiting of those in terms of computation time is not always easy in practice.", "A different family of methods overcome that shortcoming by pruning out entire neurons and/or convolution filters [11], [13], [24], [16], [18].", "Those methods identify weak neurons using heuristics such as activation value [8] or absolute sum of weights [11].", "Fine-tuning may be used afterwards [13].", "Both of these methods treat each unit independently, in that they prune the unit if that unit has little effect on the downstream computations.", "However, these techniques would not prune a unit whose output is significantly impacts downstream computation but is largely predicted as a linear combination of the outputs of the other units in the same layer.", "Such units can be pruned, and their weights distributed, to achieve lossless compression." ], [ "Matrix Factorization", "Factorization-based approaches [3], [14], [23] factor the weight matrices of the neural network into multiple low-rank components which can be then used to approximate the output of the original weight matrix.", "Those techniques are seemingly more similar to our approach : low-rank matrices eliminate some redundancies by projecting on a subspace of smaller dimension.", "The key difference is that those methods work at the weight matrix level, while we find redundancies within post-activation representations.", "The non-linearity of those activations is likely to create new redundancies, that escape factorization methods but that our approach can capture.", "However, the case of a linear activation is similar to a factorization process, and we will to some extent use it in to better understand the process of weight readjustment and how it affects the error surface." ], [ "Distillation", "Model compression using knowledge distillation [7] involves training a more compact student model to predict the outputs of a larger teacher model, instead of the original targets [15], [2].", "Distillation, or annealing, provides an interesting angle to the compression problem, but in itself suffers from several shortcomings.", "First, it requires the student model to be predefined, which is a rather tedious task.", "By default this model would need to be trained from scratch.", "Finally, this process relies solely on the extra supervision of the teacher model to overcome the challenges of training a less overparametrized model, with complex error surfaces ; this seems sub-optimal.", "On the contrary, distillation can become very useful as a complementary compression tool.", "Assuming a compression method induces some drop in performance in the smaller model, a short amount of fine-tuning may boost its performance, and using knowledge distillation from the original model at that step can speed up that process.", "We make use of distillation in such a manner, in an iterative fashion, and discuss it in REF" ], [ "Notations and task definition", "Throughout the paper, we consider a feed-forward neural network $F = \\phi _N\\circ L_N\\circ \\phi _{N-1}\\circ L_{N-1}\\circ ...\\circ \\phi _1\\circ L_1$ where $L_k$ is the k-th dense or convolutional (which is dense with shared parameters) layer and $\\phi _k$ is the following activation, in the largest sense.", "For example $\\phi _k$ may involve a pooling, or a softmax operator in the case of $\\phi _N$ .", "The weight matrix of $L_k$ is $W_k$ , of size $(n_k^o,n_{k+1}^{i^2})$ .", "Depending on the nature of $\\phi _k$ , $n_k^i$ and $n_k^o$ may be different ; however, to alleviate notations later on, we will consider that $n_k^i=n_k^o =n_k$ which does not induce a loss of generality in our algorithms.", "For a given input vector $X$ , sampled from data distribution $\\mathcal {D}$ we define the intermediate representations $Z_k$ (activations) and $Y_k$ (pre-activation), such that $Z_0=X\\;\\; ; \\;\\;Y_k = W_k.Z_{k-1}\\;\\; ; \\;\\;Z_k = \\phi _k(Y_k)$ Our goal is to eliminate redundancies within the activations of a given layer.", "To do so, we consider for each activation $Z_k[i]$ the task of predicting it as a linear combination of the neighbouring activations $Z_k[j],j\\ne i$ .", "Solving that task, evaluated with the $L_2$ norm, amounts to solving the following problem : $\\min _{A_k\\in \\mathcal {M}_n(\\mathbf {R})} \\mathbf {E}_{x\\sim D}[\\left\\Vert Z_k-A_kZ_k\\right\\Vert _2^2] \\;\\;\\;s.t.\\; diag(A_k)=0$" ], [ "Expression of the $A_k$ matrix", "Let us find the expression of redundancy matrix $A_k$ .", "We start by simplifying the problem's formulation.", "Rewriting the objective $\\min _{A_k\\in \\mathcal {M}_n(\\mathbf {R})} \\sum _{i=1}^{n_k}\\mathbf {E}_{x\\sim D}[(Z_k[i]-A_k[i].Z_k)^2]$ ($M[i]$ being the ith row vector in matrix $M$ ), elements of $A$ of different rows are clearly uncoupled both in the objective and the constraint.", "We can therefore solve that problem row-wise.", "Writing $U = I_{n_k}-A_k$ , we must solve $n_k$ problems $P_l$ of the form $\\min _{u\\in \\mathbf {R}^n_k}\\mathbf {E}_{x\\sim D}[(u^TZ_k)^2]\\;\\;\\;s.t\\; u_l=1$ .", "Define $g(u)=\\mathbf {E}_{x\\sim D}[(u^TZ_k)^2]= \\mathbf {E}_{x\\sim D}[(u^TZ_k).", "(u^TZ_k)^T]= \\mathbf {E}_{x\\sim D}[u^TZ_kZ_k^Tu]= u^TSu$ where $S = \\mathbf {E}_{x\\sim D}[Z_kZ_k^T]$ is $Z_k$ 's correlation matrix, which is positive semidefinite.", "Let's consider only the non-degenerate case where it is positive definite.", "$g_l$ is convex (of hessian $\\frac{1}{2}S$ , so the zeros of the gradients indicate exactly its minima).", "Specifically, within the hyperplane of admissible points $H={u\\in \\mathbf {R}^n_k,u_l=1}$ , $g_l$ is minimal if and only if : $\\nabla _ug = 2Su \\in H^\\bot = \\mathbf {R}e_l$ , where $e_l$ is the lth vector of the canonical base.", "Or in other words, $u$ is a minimum iff it is a multiple of $S^{-1}e_i = S^{-1}[l]^T$ , the lth column of $S^{-1}$ .", "Therefore the minimum is $u^* = \\frac{1}{S^{-1}[l][l]}S^{-1}[l]^T$ .", "In full matrix form, we conclude that the solution $A_k$ to the equation is $A_k = S^{-1}D$ , with $S = \\mathbf {E}_{x\\sim D}[Z_kZ_k^T]$ and $D = diag(S^{ -1})^{-1}$ In practice, due to the presence of a matrix inversion, it is simpler and faster to obtain $A_k$ using gradient descent, given inputs sampled in the training set." ], [ "Weight readjustment", "The previous weight matrix, and the residual error provides information regarding which activation is most predictable and should be removed.", "We then wish to adjust the remaining weight matrix of the following layer to account for this redundancy elimination.", "Assume only one activation $l$ was removed.", "We consider the compressed vectors $Z_k^l$ (of size $n_k-1$ ) where activation l was removed.", "We can infer a transformation matrix $T_k^l$ from $Z_k^l$ to $Z_k$ , with minimal error, using the lth row of $A_k$ : $Z_k \\approx T_k^lZ_k^l$ , where : $\\forall {i<l}, T_k^l[i][j] = \\delta _{ij}\\;\\;\\forall {i>l}, T_k^l[i][j] = \\delta _{i(j+1)}$ $\\forall {j<l}, T_k^l[l][j] = A_k[l][j]\\;\\;\\forall {j\\ge l}, T_k^l[l][j] = A_k[l][j+1]$ i.e.", "$ T_k^l$ is the identity matrix $I_{n_k-1}$ where $A_k[l]$ has been inserted as the lth row, minus the 0 coefficient $A_k[l][l]$ .", "Therefore, a natural adjustment for the weight matrix is $W_{k+1}^l=W_{k+1}T_k^l$ , such that $W_{k+1}^lZ_k^l \\approx W_{k+1}Z_k$ .", "If we remove more than one coefficient at once, the expression of the transformation matrix is less straightforward.", "Assume for example that activations $l$ and $j$ are removed ; Obtaining the approximate expression of $Z_k[l]$ obtained from $A_k$ , using only remaining activations, leads to the following derivations : $Z_k[l] = \\sum _{i=1}^{n_k}A_k[l,i]Z_k[i] = \\sum _{i=1\\\\i\\ne l,m}^{n_k}A_k[l,i]Z_k[i] + A_k[l,j]Z_k[j]$ $Z_k[l] =A_k[l,j]A_k[j,l]Z_k[l] \\sum _{i=1\\\\i\\ne l,j}^{n_k}A_k[l,i]Z_k[i]+ A_k[l,j]A_k[j,i]Z_k[i]$ $Z_k[l] =\\frac{\\sum _{i=1\\\\i\\ne l,j}^{n_k}A_k[l,i]Z_k[i]+ A_k[l,j]A_k[j,i]Z_k[i]}{1-A_k[l,j]A_k[j,l]}$ More generally, assume a set $J = \\lbrace j_1<...<j_m\\rbrace \\subset [1..n_k]$ of activations is eliminated.", "We note its complementary set $H=[1..n_k]\\setminus J = \\lbrace h_1<...<h_{n_k-m}\\rbrace $ .", "We define $A_k^{J+}$ , the $(m,m)$ matrix defined by $A_k^{J+}[i][p]=A_k[j_i][j_p]$ , and $A_k^{J-}$ , the $(m,n_k-m)$ matrix such that $A_k^{J-}[i][p]=A_k[j_i][h_p]$ .", "Then we can write, where $Z_k^{J}$ contains only the activations in $J$ and $Z_k^{H}$ only those not in $J$ : $Z_k^{J} = A_k^{J+}Z_k^{J}+A_k^{J-}Z_k^{H}$ i.e.", "$Z_k^{J} = (I_m-A_k^{J+})^{-1}A_k^{J-}Z_k^{H} = U_k^JZ_k^{H}$ provided that $(I_m-A_k^{J+})$ is invertible.", "From there we can easily obtain the equivalent of the transformation matrix in the single activation case :$Z = T_k^JZ_k^{H}$ , where $T_k^J$ is $U_k^J$ completed with ones on the diagonal for the additional rows." ], [ "Stability results on compression", "In practice, we wish to compress a network at some point during training, followed by further fine-tuning.", "We may expect that if we are close enough to a good local minimum in a convex region of the error surface, we may capture that minimum's redundancies and eliminate them.", "One question however arises : after compression, will we still be close enough to the minimum that the fine-tuning will converge, or may it diverge?", "The following stability result gives partial answers to that question.", "Let the kth activation in the neural network be linear, so that we can write $F(X) = f_\\theta (W_{k+1}.W_k.g_\\phi (X))$ for $X$ in the training set $\\mathcal {X}$ .", "Let $p^* = (\\theta ^*,\\phi ^*,W_{k+1}^*,W_k^*)$ a local minimum for the loss function $L$ in the parameter space $\\mathcal {P}$ , and assume an exact redundancy of the final activation of the kth layer : $W_{k+1}^{*}.W_k^{*}=W_{k+1}^{*\\prime }.W_k^{*\\prime }$ , where $W_k^{*\\prime }$ is $W_k^{*}$ minus its final row, and $W_{k+1}^{*\\prime }$ is the contracted weight matrix as computed in REF with matrix $A_k$ .", "The compression-readjustment operation projects $p*$ onto $p*^{\\prime } = (\\theta ^*,\\phi ^*,W_{k+1}^{*\\prime },W_k^{*\\prime })$ Assume there is a ${L_2}$ -ball $B\\subset \\mathcal {P}$ of radius $R$ centered $p*$ , on which $L$ is convex.", "There is an ellipsoid $E$ centered on $p*$ of equation $\\left\\Vert \\theta -\\theta ^*\\right\\Vert _2^2+\\left\\Vert \\phi -\\phi ^*\\right\\Vert _2^2+\\left\\Vert W_{k+1}-W_{k+1}^*\\right\\Vert _2^2+\\sum _{i=1}^{n_k-1}\\sum _{p=1}^{n_k}(1+A_k[n_k][i]^2)V[i][p]^2 \\le R$ onto a convex region of the compressed parameter space.", "We delay the full proof to an appendix.", "We can however discuss the implications of the theorem.", "We note first that the theorem by itself assumes linear activations.", "Assume our training brought us near a (global or local) minimum displaying some redundancies that we manage to eliminate.", "We would like regularities of the error space in that region, such as convexity, to be preserved after compression.", "That the point before compression was in a convex region around the minimum is not enough to have convexity in the compressed space; however, our theorem shows that there is a slightly different region, determined by the radius of the convex region around the minimum, that does ensure post-compression convexity.", "That region is obtained by flattening the ball onto the subspace of $W_{k+1}$ .", "Besides, the subspace corresponding to the compressed coefficient can be ignored.", "While the above theorem applies only to the linear activation case, we argue that the results extends naturally to locally linear or nearly linear activations.", "Consider for example a ReLU activation: around any parameter point, there is a ball on which the network is identical to one with linear activation; we can apply theorem 1 on that restricted area of the space." ], [ "An empirical justification for readjustment", "The proposed pruning approach is based on the hypothesis that elimination of linearly dependent (or almost dependent) neurons (or filters) within any layer with appropriate weight adjustment will result in minimal or even zero changes to the representations seen by subsequent layers in the network.", "In other words, compressing the $k$ th layer of a network as proposed should not significantly change the pre-activation values observed by layer $k+1$ .", "We evaluate this hypothesis in this section.", "In Fig.", "REF , we plot pre-activation norm differences on an AlexNet network for an intermediate convolutional layer, computed on a random input sample, after compressing the previous layer.", "The norm difference is computed as $\\frac{\\left\\Vert Z_{17}^{(i)} - Z_{17}^{(0)}\\right\\Vert _2}{\\left\\Vert Z_{17}^{(0)}\\right\\Vert _2}$ , where $Z_{17}^{(i)}$ represents the activations after the $i^{th}$ shrinking step.", "We compare the results obtained from just trimming dependent neurons without subsequent adjustment of weights, to those obtained after weight readjustment.", "As expected, trimming neurons modifies $Z$ , but subsequent weight readjustment largely eliminates the changes from trimming – after 15 compression steps we have only a $2\\%$ norm change, confirming the intuition behind our method." ], [ "When Activations Are Not Dependent", "The above analysis shows that if there is perfect linear dependence in the neuron activations, i.e.", "$\\left\\Vert Z_k-A_kZ_k\\right\\Vert _2^2=0$ , then we can achieve lossless compression, however, in many cases this condition may not hold.", "In such situations, the parameters of the pruned model, even after readjustment, may end up in a suboptimal region of the loss surface.", "This is because readjustment weights in $A$ are imperfect and error prone, and therefore will move the model parameters to a different, potentially suboptimal, point on the error surface.", "Since, reducing the size of the model makes the error surface less smooth [1], even if the operating point of the smaller model is close to the operating point of the larger model, it may have a much higher loss.", "To keep the model parameters from deviating too far from the optima during compression we employ a modified version of Annealed Model Contraction (AMC) [19], which attempts to keep the model in the optimal region by cycles between pruning and fine-tuning phases.", "Below we provide a description of AMC, and our modifications to it." ], [ "Annealed Model Contraction", "AMC is an iterative method that greedily prunes the model layer-by-layer.", "As formulated in [19], AMC starts from the first (or the last) layer of the model and proceeds to maximally shrink the current layer before moving on to the next.", "While compressing a layer, AMC alternates between pruning and fine-tuning.", "Pruning is performed by reinitializing the layer with $\\gamma \\%$ fewer neurons/filters and the whole network is then fine-tuned end-to-end.", "During fine-tuning, knowledge distillation [7] is used to facilitate training and the following loss is minimized $\\mathcal {L} = (1-\\lambda )H\\left(\\text{softmax}\\left(\\frac{\\mathbf {z}}{T}\\right),\\text{softmax}\\left(\\frac{\\mathbf {v}}{T}\\right)\\right) + \\lambda H\\left(\\mathbf {y_{true}}, \\text{softmax}\\left(\\frac{\\mathbf {v}}{1}\\right)\\right)$ Where $\\mathbf {z}$ and $\\mathbf {v}$ are the logits returned by the teacher and student models, respectively, $T$ is a hyperparameter referred to as the temperature of the distribution, and $\\lambda $ controls the contribution of the loss against the target label to the the overall loss.", "AMC continues to prune a layer as long as the pruned model's accuracy remains within a threshold, $\\epsilon $ , of the uncompressed model's accuracy.", "Once the current layer can not be pruned any further, AMC proceeds to shrink the next layer in the model.", "AMC can be applied to both, dense and convolutional layers.", "In the case of the former, it prunes neurons while in the latter it prunes convolutional filters." ], [ "Annealed Model Contraction with Lossless Redundancy Elimination", "$\\text{RemoveAndAdjust}(A, W, j)$ : adjust the weight matrix after the removal of the $j^{th}$ neuron from the previous layer using the method in REF FLREShrinkLREShrink FnFunction: $F$ , $l$ , $\\gamma $ $Z \\leftarrow F_{1:l}(\\mathcal {X})$ compute the activations of the $l^{th}$ layer.", "$A \\leftarrow \\min _A \\left\\Vert ZA - Z\\right\\Vert ^2 \\text{s.t } \\text{diag}(A)=\\mathbf {0}$ $\\mathcal {E} \\leftarrow \\arg \\text{sort}(\\left\\Vert ZA - Z\\right\\Vert ^2)[:\\lfloor \\gamma * \\text{sizeof}(F[l])\\rfloor ]$ $\\bar{W}^{(l+1)}\\leftarrow [W^{(l+1)}; b^{(l+1)}]$ Concatenate the weights and bias.", "$j\\in \\mathcal {E}$ $W^{(l)} \\leftarrow W^{(l)}_{-j.", "}$ drop the $j^{th}$ row of $W^{(l)}$ $W^{(l+1)} \\leftarrow \\text{RemoveAndAdjust}(A, \\bar{W}^{(l+1)}, j)$ $Acc\\leftarrow \\text{evaluate}(F_t)$ $F_s^{\\prime }[i_B]\\leftarrow \\text{LREShrink}(F_s, i_B, \\gamma )$ $Acc^{\\prime }\\leftarrow \\text{evaluate}(F_s^{\\prime })$ $Acc-Acc^{\\prime }\\le \\epsilon $ $F_s\\leftarrow F_s^{\\prime }$ $F_s^{\\prime }[i_B]\\leftarrow \\text{LREShrink}(F_s^{\\prime },i_B, \\gamma )$ $Acc^{\\prime }\\leftarrow \\text{evaluate}(F_s^{\\prime })$ $Acc-Acc^{\\prime }>\\epsilon $ $F_s^{\\prime }\\leftarrow \\text{distill}(F_s^{\\prime })$ $Acc^{\\prime }\\leftarrow \\text{evaluate}(F_s^{\\prime })$ LRE-AMC Algorithm While effective, AMC has the shortcoming that it takes an ad-hoc approach to parameter pruning.", "AMC removes neurons from a layer by reinitializing the layer with fewer neurons.", "The new initialization is random and therefore can land the model arbitrarily far away from the optimal point.", "On the other hand, the Lossless Redundancy Elimination (LRE) formalism presented in Section provides a method of setting the parameters of the pruned layer that guarantees (under some constraints) that the model remains near the optimal operating point.", "However, LRE only considers the activations and weights between two layer, and thus does not account for the effects of pruning on the operating point of the whole model.", "Therefore, we propose to combine LRE and AMC in a novel model compression algorithm, which we call LRE-AMC, that compensates for the inadequacies of both LRE and AMC.", "LRE-AMC (Algorithm REF ) differs from vanilla AMC in two significant ways.", "First, instead of pruning neurons by reinitializing the layer, LRE-AMC uses the LRE formalism to select neurons/filters (in the following we will use the term units to refer to neurons and filters) to prune away based on the degree of linear dependency between their activations.", "Thus, LRE-AMC retains units that have linearly independent activation and thus have learned to encode unique aspects of the data, whereas these units have to be relearned under AMC.", "Second, LRE-AMC breaks the pruning process into two phases.", "In the first phase LRE is used to remove the selected units one-by-one and adjust the weight matrix such that the outputs of the layer are minimally perturbed.", "After each pruning stage we measure the performance of the model on a held-out set and continue pruning without fine-tuning until the performance of the model remains within a threshold, $\\epsilon $ , of the original.", "When the performance drops below the threshold $\\epsilon $ , we start phase two in which we use distillation to fine-tune the model to bring the model's performance to within $\\epsilon $ of the pre-compression performance." ], [ "Datasets", "We evaluate our proposed method three datasets of real-world images, namely CIFAR-10 and Caltech-256.", "CIFAR10 contains 50,000 training images and 10,000 testing images.", "Each images has a size of $32\\times 32$ pixels and is assigned one out of 10 class labels.", "Caltech-256 contains 30,607 real-world images, of different sizes, spanning 257 classes.", "Following the protocol from [20], we construct a balanced training set for Caltech 256 with 60 images per class.", "For both, Caltech256 and CIFAR10, we used 20% of the training images for validation during training.", "We apply data augmentation to increase the size of the training data and improve generalization.", "Specifically, we augment CIFAR10 with random affine transformations, horizontal flips and grayscaling.", "Meanwhile, we augment Caltech-256 by taking a random $256\\times 256$ crop at a random scale between 0.8 and 1.0, and applying rotation, color jitter and horizontal flipping before resizing the image to $224\\times 224$ .", "The pixel values of images from both datasets are normalized by mean [0.485, 0.456, 0.406] and standard deviation [0.229, 0.224, 0.225]." ], [ "Experimental Setup", "We implemented AMC and LRE-AMC using Pytorch and Python 3.", "We use AlexNet and VGG16 as our base models which we will compress.", "Since the receptive field of the first convolutional layer in AlexNet is too large $32\\times 32$ images, we reduced it to $3\\times 3$ when training on CIFAR-10.", "When training on Caltech256, we initialized the models with weights from models pretrained on ImageNet and tuned only the final classification layer.", "The accuracy and number of parameters of the base models are presented in Table REF .", "Table: The accuracy of the baseline models on CIFAR10 and Caltech256 and the number of parameters that they contain.Since AMC does not define an order in which the layers must be shrunk, we must define one ourselves.", "We experiment with two orderings, namely, top down (TD) and round robin (RR).", "In TD we start from the penultimate layer, maximally shrink it and move down the network.", "In round robin (RR) we again start from the penultimate layer, but instead of maximally shrinking it we shrink each layer by at most a factor $\\gamma $ and then move to the next layer.", "We also introduce additional constraints in LREShrink (Algorithm REF ) to prevent the removal of neurons with independent observations and to stop removing neurons when $A$ becomes too error prone.", "Specifically, we do not apply the update if the average norm of the rows in the update is larger than the average norm of rows in the weight matrix i.e.", "${\\frac{1}{n^o_k}\\sum _i\\left\\Vert \\hat{W_{k+1}[i]} - W_{k+1}[i]\\right\\Vert _2 > \\frac{1}{n^o_l}\\sum _i\\left\\Vert W_{k+1}\\right\\Vert _2}$ or $\\mathbb {E}[|A_{.j}|\\mathbb {E}\\left[Z^l_{.j}]\\right] > 1$ .", "To measure the effect of adjusting the network parameters using LRE, run experiments in which we do not adjust the network parameters using the LRE formalism presented in REF .", "Instead, we prune the neurons with linearly dependent activations by simply dropping the corresponding columns from the weight matrix, and keeping the other columns as is.", "Unless otherwise specified, we use the following hyperparameter settings.", "For experiments with AlexNet we use a learning rate of $10^{-4}$ and set $T=4$ in equation REF .", "For experiments with VGG16 we use a learning rate of $5\\times 10^{-5}$ and set $T=5$ .", "For both the models we set $\\lambda =0.75$ in equation REF and $\\gamma =0.75$ .", "During the fine-tuning phase, we tune the model for up to 50 epochs.", "We stop with the accuracy comes within $\\epsilon =0.05$ of the precompression accuracy.", "If the accuracy on the held-out set does not improve for 3 epochs we reduce the learning rate by 50%.", "We stop tuning if the learning rate drops below $10^{-6}$ ." ], [ "Results", "We present the percentage reduction in the number of model parameters, and the consequent loss in accuracy in Table REF .", "The “wAdj” and “noAdj” settings correspond to the setting in which LRE is used and the setting in which LRE is not used.", "Under both these settings we demonstrate that our technique is able to decimate the number of parameters of AlexNet and VGG16, by pruning as much as 99% of the model parameters." ], [ "Top Down Shrinking", "We find that when we shrink the layers in top-down order we find that adjusting the model weights with LRE results in a significant reduction in model parameters.", "Adjusting the weights of AlexNet using LRE allows us to remove almost 30% more parameters on CIFAR10 and 47% more parameters on Caltech256, compared to when we did not adjust the weights.", "Furthermore, we observe that adjusting the weights allows us to prune additional neurons/filters from both, the dense and the convolution layers.", "This is an impressive result, not only because LRE-AMC able to reduce the number number of parameters in the network drastically but also because it yields better compression on the more difficult dataset.", "When we ran the same experiment with VGG16 we found that adjusting the weights using LRE results in slightly lower compression on CIFAR10 than however LRE is able to prune an additional 20% of the model parameters, most of which are pruned from the dense layers.", "Table: The percentage reduction in the number of total parameters (-Δ A -\\Delta _A), dense layer parameters (-Δ D -\\Delta _D), convolutional layer parameters (-Δ C -\\Delta _C), and classification accuracy (-Δ Acc -\\Delta _{Acc})." ], [ "Round Robin Shrinking", "When we shrink the layers in a round robin fashion we find that we can achieve greater compression of the convolutional layers.", "Since the convolution layers scan the input, computing their activations involves a lot of floating-point operations (FLOPs).", "Reducing the number of convolutional filters greatly reduces the FLOPs of the model.", "Interestingly, performing round robin shrinking has a more significant impact on the total number of model parameter in AlexNet when the weights are not adjusted using LRE.", "In fact, under round robin shrinking not adjusting the weights yields slightly better compression both in terms of reduction in the number of model parameters and the accuracy degredation.", "We also observe that under round robin shrinking, we achieve lower compression in terms of dense layer parameters on Caltech256 but we are able to prune away many more parameters from the convolutional layers.", "This seems to suggest that round robin shrinking would be ideal when minimizing FLOPs is more important than reducing memory consumption, while top-down shrinking should be preferred when memory consumption is to be optimized." ], [ "Accuracy Error Tradeoff", "In this section we present experimental results that describe the compression-performance trade-off of our approach.", "As mentioned, we have used a tolerance of $\\epsilon =0.05$ to limit the deterioration of accuracy during and after compression.", "In Figure REF we plot the decrease in accuracy against the percentage of the parameters pruned for top-down and round robin shrinking of AlexNet on Caltech-256 at different values of $\\epsilon $ .", "Figure REF exhibits the expected trend, in that, as we decrease $\\epsilon $ both, the decrease in accuracy and the fraction of removed parameters decrease.", "We see that the the parameter reduction falls much faster as we decrease $\\epsilon $ , indicating that that under the top-down shrinking scheme additional accuracy comes at a steep cost in compression performance.", "On the other hand, Figure REF exhibits a very different trend.", "As we decrease $\\epsilon $ from 0.05 to 0.03 the compression improves, however, it deteriorates as when $\\epsilon =0.01$ and improves again when $\\epsilon =0.0$ .", "Even though compression suffers when we set $\\epsilon =0.0$ , the deterioration is modest compared to the top-down shrinking.", "We do not have a reliable explanation for this phenomenon, because the repetitive nature of the round-robin shrinking approach makes its analysis complicated.", "Figure: The change in the compression percentage of AlexNet as the accuracy tolerance is reduced from 5% to 0% (a) under top-down shrinking and (b) round robin shrinking on Caltech-256.", "In both settings the weights are adjusted using LRE.It is entirely possible that removing neurons/filters in a certain order can lead to greater compression than removing neurons in some other order.", "The complexity arises if the optimal order spans across layers, something which the LRE framework does not account for.", "Though we do not prove it conclusively, the round robin shrinking approach seems to maintain compression even under very stringent accuracy constraints, and, therefore, shows promise as an effective model compression approach that could benefit from further study." ], [ "Representation learning", "As we discussed earlier, one of the main use of a compressed representation on a task such as image recognition should be to provide useful pretrained representations for potential downstream tasks.", "When using performance-guided pruning, it is possible to degrade the learned representations while resorting on the final layers to use artifacts to maintain good performance on the recognition task.", "To make sure that our method isn't such a case, we provide some insight on the final convolutional layer of our VGG16 network trained on CIFAR through low-dimensional visualizations.", "In REF we plot t-SNE visualizations to observe separability.", "We can observe the compressed representations images are almost as separable as they are in the original network.", "Since t-SNE distorts the point clusters, we also plot PCA representations to assess the shape of the clusters.", "We can see that weight readjustment plays a significant role here : without it, clusters are more scattered and less convex, while with it they are more similar to the original network.", "This vouches for redundancy elimination and weight readjustment as compression methods that respect the semantics of the data, and that arguably are compatible with transfer learning between vision tasks." ], [ "Conclusion", "We have presented a novel neural model compression technique called LRE-AMC, which eliminates units (neurons or convolutional filters) whose activations are linearly dependent on the activations of other neurons in the same layer.", "Since entire units are pruned away, and not just individual parameters, the weight matrices in compressed model are smaller, which reduces the model's memory requirements and makes computation more efficient.", "We demonstrate the efficacy of LRE-AMC by applying it to AlexNet and VGG16 and show that we can remove more than 99% of the units in both these models while suffering only a 5-6% loss in accuracy on CIFAR-10.", "We have also applied LRE-AMC to the more difficult Caltech-256 dataset and achieved more than 80% compression.", "Furthermore, we show that after compression the data remains separable in the model's intermediate layers, suggesting that the intermediate representation could carry sufficient information for transfer learning tasks.", "For future work we will explore methods of incorporating information from the derivatives of the subsequent layers to better estimate the effect of removing a unit on the overall output of the network, and prune neurons that minimally impact the output.", "We expect that this modification will result in smaller models and greater accuracy." ] ]

[ [ "Non-Uniqueness of Bubbling for Wave Maps" ], [ "Abstract We consider wave maps from $\\mathbb R^{2+1}$ to a $C^\\infty$-smooth Riemannian manifold, $\\mathcal N$.", "Such maps can exhibit energy concentration, and at points of concentration, it is known that the map (suitably rescaled and translated) converges weakly to a harmonic map, known as a bubble.", "We give an example of a wave map which exhibits a type of non-uniqueness of bubbling.", "In particular, we exhibit a continuum of different bubbles at the origin, each of which arise as the weak limit along a different sequence of times approaching the blow-up time.", "This is the first known example of non-uniqueness of bubbling for dispersive equations.", "Our construction is inspired by the work of Peter Topping [Topping 2004], who demonstrated a similar phenomena can occur in the setting of harmonic map heat flow, and our mechanism of non-uniqueness is the same 'winding' behavior exhibited in that work." ], [ "Introduction", "We consider wave maps from $(1+2)$ -dimensional Minkowski space into a compact Riemannian manifold $({\\mathcal {N}}, g)$ , which are defined formally as critical points of the Lagrangian $\\mathcal {L}(U, \\partial U) = \\frac{1}{2} \\int _{{\\mathbb {R}}^{1 + 2}} \\eta ^{\\alpha \\beta }\\, \\langle \\partial _\\alpha U , \\partial _\\beta U \\rangle _g dt dx,$ where $\\eta $ is the Minkowski metric and g is the Riemannian manifold on ${\\mathcal {N}}$ .", "In local coordinates on ${\\mathcal {N}}$ , wave maps are solutions $u$ to the system $\\left\\lbrace \\begin{aligned}&\\Box u^i = \\Gamma _{k\\ell }^i(u) \\nabla u^k \\cdot \\nabla u^\\ell , \\\\&(u, \\partial _t u ) \\big |_{t=0} = (u_0, u_1),\\end{aligned} \\right.$ where $\\Gamma _{k\\ell }^i$ are the Christoffel symbols for ${\\mathcal {N}}$ .", "There is a conserved energy associated to (REF ) given by $\\mathcal {E}(u, \\partial _t u )(t) = \\int _{\\mathbb {R}^2} |\\nabla u |_g^2+|\\partial _t u |_g^2 \\equiv \\mathcal {E}(u_0, u_1 ),$ whenever the right-hand side is finite.", "The coercivity of this energy implies that that the $\\dot{H}^1$ -norm of solutions remains bounded for all time.", "However, it is still possible for energy to concentrate producing “solitons\" or “bubbles\" in the suitably rescaled weak limit.", "Bubbling is a well-studied phenomena in nonlinear evolution equations, particularly in the parabolic setting, dating back to the work of Struwe [36], which built on the work of Sacks and Uhlenbeck [41].", "We will not attempt to do justice to the vast literature here, particularly pertaining to parabolic flows, but we refer to [12] and references therein for an overview on the history.", "The purpose of this paper is to provide the first example, in the context of (REF ) where this weak limit is non-unique, see Theorem REF below.", "In particular, we construct a solution where, after rescaling around a particular point in space-time, different solitons are obtained by considering weak limits along different sequences of times.", "To more precisely state our results, let us recall some background on wave maps.", "We consider (REF ) for smooth, finite energy initial data belonging to a certain symmetry class and satisfying certain quantitative higher regularity bounds; we will make these assumptions precise below.", "For such initial data, classical energy methods show that (REF ) admits a unique smooth solution for small times, see e.g.", "[33].", "When the domain is $\\mathbb {R}^{2+1}$ , the energy (REF ) is scale invariant, thus solutions to (REF ) can exhibit energy concentration even if the initial data is smooth or highly symmetric, see, e.g.", "[23], [30], [31].", "This concentration can be ruled in certain cases under additional assumptions, say on the smallness of the initial energy, see [44], [45], [20], or by restricting the topology of the target, see, e.g.", "[38], [22], [43].", "For general targets, Sterbenz and Tataru [35] showed that inside any light cone a dichotomy holds: either energy does not concentrate too much, in which case solution can be smoothly extended to the whole space with quantitative control, or the energy concentrates and the solution blows up via bubbling of harmonic maps.", "To state their result (in the infinite time setting), we follow the notation in [35], and fix $C_{[t_0, t_1]} = \\lbrace t_0 \\le t \\le t_1, r \\le t\\rbrace $ for the truncated light cone and $S_{t} = \\lbrace (x,t)\\mid |x| \\le t \\rbrace $ for the time sections of the light cone.", "We set $\\mathcal {E}_{S_t} = \\frac{1}{2} \\int _{S_t} |\\nabla u |_g^2 + |\\partial _t u|_g^2.$ Theorem 1.1 ([35]) Let $u: C_{[1,\\infty )} \\rightarrow \\mathcal {N}$ be a $C^\\infty $ wave map such that $\\lim _{t \\rightarrow \\infty } \\mathcal {E}_{S_t}[u] < \\infty .$ Then exactly one of the following possibilities must hold: (a) There exists a sequence of points $(t_n, x_n) \\in C_{[1,\\infty )}$ and scales $r_n$ with $t_n \\rightarrow \\infty , \\quad \\limsup _{n \\rightarrow \\infty } \\frac{|x_n|}{t_n} < 1, \\quad \\lim _{n \\rightarrow \\infty } \\frac{r_n}{t_n} = 0.$ so that the rescaled sequence of maps $u^{(n)}(t,x) = u(t_n + r_n t, x_n + r_n x)$ converges strongly in $H^1_{loc}$ to a Lorentz transform of an entire (time-independent) harmonic map $u^{(\\infty )} : \\mathbb {R}^2 \\rightarrow \\mathcal {N}$ of nontrivial energy: $u^{(\\infty )} : \\mathbb {R}^2 \\rightarrow \\mathcal {N}, \\quad 0 < \\Vert u^{(\\infty )} \\Vert _{\\dot{H}^1({\\mathbb {R}}^2)} \\le \\lim _{t \\rightarrow \\infty } \\mathcal {E}_{S_t}[u].$ (b) For each $\\varepsilon > 0$ , there exists $t_0 > 1$ and a wave map extension $u : \\mathbb {R}^2 \\times [t_0, \\infty ) \\rightarrow \\mathcal {N}$ with bounded energy: $\\mathcal {E}(u) \\le (1 + \\varepsilon ^8) \\lim _{t \\rightarrow \\infty } \\mathcal {E}_{S_t}[u],$ which satisfies $\\sup _{t \\in [t_0, \\infty )} \\sup _{k \\in \\mathbb {Z}} ( \\Vert P_k u(t) \\Vert _{L^\\infty _x} + 2^{-k} \\Vert P_k \\partial _t u(t) \\Vert _{L^\\infty _x}) \\le \\varepsilon ,$ where $P_k$ are the Littlewood-Paley projections to frequency $2^k$ .", "An analogous dichotomy holds for finite time blowup, see [35].", "Remark 1.2 In the previous theorem and throughout, we use $\\dot{H}^s({\\mathbb {R}}^n)$ to denote the usual homogeneous Sobolev space, with norm defined as $\\Vert u\\Vert _{\\dot{H}^s({\\mathbb {R}}^n)} = \\Vert (-\\Delta )^{s/2} u\\Vert _{L^2({\\mathbb {R}}^n)}.$ Remark 1.3 It was observed in [35] that part (b) in Theorem REF implies that a certain controlling norm for $u$ is finite, and in [34] it was proved that this implies that $u$ converges to a linear wave after applying a suitable gauge transformation.", "We note that this is called scattering in the terminology of [35], but is different from the usual use of that term in nonlinear dispersive equations, see [24] for some discussion.", "We adopt this terminology for brevity when referring to the behavior in part (b) of this theorem.", "Instead of working with the modified definition of scattering in [34], we instead adopt a more direct approach to demonstrate that our flow exhibits energy concentration via degree considerations.", "We believe this approach illuminates certain features of scattering in the current setting.", "This argument is inspired by arguments in [7], [40], [4], [24], see Section for more details.", "The first case in Theorem REF describes singularity formation via “bubbling” (a.k.a.", "energy concentration) to non-constant harmonic maps.", "Note that the bubbling phenomenon described in Theorem REF leaves open the question of whether the convergence along a discrete sequence of times and scales can be strengthened to convergence as $t\\rightarrow \\infty $ after rescaling, translating and applying Lorentz transformations by some continuous functions, $r(t)$ , $x(t)$ and $\\gamma (t)$ .", "To reduce the number of parameters, and to provide an example of non-uniqueness in what we speculate is the simplest possible setting, we will impose an additional symmetry on our solution, which will allow us to ignore translations and Lorentz transformations: Definition 1.4 We say that a wave map $u(x,t): {\\mathbb {R}}^{1+2} \\rightarrow \\mathcal {M}$ is quasi-equivariant if there exists a smooth one-parameter family of isometries, $\\Phi _s \\in \\mathrm {Isom}(\\mathcal {M})$ .", "with $\\Phi _0 = \\mathrm {Id}_{\\mathcal {M}}$ , such that (using polar coordinates) one has $u(r, \\theta + s, t) = \\Psi _s \\circ u(r, \\theta , t)$ for all $r > 0, s, \\theta \\in \\mathbb {S}^1$ and $t\\in [0, T_{\\max })$ .", "Remark 1.5 Many of Topping's constructions for harmonic map heat flow (see, e.g.", "[48], [46]) exhibit quasi-equivariant symmetries.", "While, Topping does not give this symmetry a name, we introduce the term “quasi-equivariance\" to stress its relation to the well-studied notion of equivariant wave maps.", "In the language of Definition REF , if $\\mathcal {M}$ is a surface of rotation and $\\Phi _s$ is a rotation of $\\mathcal {M}$ by $s$ radians around the same axis, then $u$ is equivariant.", "We can now define of uniqueness of a bubbling singularities for quasi-equivariant wave maps: Definition 1.6 We say that a quasi-equivariant wave map, $u:\\mathbb {R}^{1+2}\\rightarrow \\mathcal {M}$ , has a unique bubble at a point $0\\in \\mathbb {R}^2$ and time $T_{max} > 0$ if there exists a continuous function, $r(t): [0, T_{\\max })\\rightarrow (0, + \\infty )$ , with $r(t) = {\\left\\lbrace \\begin{array}{ll} o(T_{max} - t) & T_{max} < \\infty \\\\o(t) &T_{max} =\\infty \\end{array}\\right.", "}$ such that $u(t, r(t)x) \\rightarrow \\omega (x)$ in $C_{\\mathrm {loc}}^0(\\mathbb {R}^2\\backslash \\lbrace 0\\rbrace ; \\mathcal {M})$ for some non-trivial harmonic map $\\omega : \\mathbb {R}^{2}\\rightarrow \\mathcal {M}$ .", "In contrast to part $(a)$ of Sterbenz-Tataru's Theorem REF , which yields convergence to a harmonic map along a sequence of times, the previous definition describes convergence (after application of the relevant symmetries in the quasi-equivariant class) to a unique harmonic map along all times.", "In general, it is difficult to prove that uniqueness in the sense of Definition REF fails.", "However, our Theorem REF shows that if a singularity is “winding\" (see Definition REF ), then the bubble is not unique.", "Using this approach, our main theorem gives the first example of a bubbling solution to a non-linear wave equation for which uniqueness in the sense of Definition REF is known not to hold.", "Theorem 1.7 (Main Theorem) There exists a compact smooth Riemannian manifold $(\\mathcal {N}, g)$ given by $\\mathcal {N} = \\mathbb {T}^2 \\times _f \\mathcal {S}^2$ for a certain $C^\\infty ({\\mathbb {T}}^2)$ warping function, $f$ , and $C^\\infty $ -smooth, finite energy, quasi-equivariant initial data $(u_0, u_1)$ , which satisfy $\\Vert (u_0, u_1) \\Vert _{\\dot{H}^3 \\times \\dot{H}^2} < \\infty , \\qquad \\mathcal {E}(u_0, u_1) < \\mathcal {E}_{\\mathrm {quasi}}(\\mathcal {N}) + \\varepsilon _1,$ such that the corresponding solution $(u, u_t)$ to (REF ) has a bubbling singularity as $t \\rightarrow T_{\\max }$ which fails to be unique in the sense of Definition REF .", "Above, $\\varepsilon _1 >0$ is a constant which depends only on $\\mathcal {N}$ , and $\\mathcal {E}_{\\mathrm {quasi}}(\\mathcal {N})$ denotes the smallest energy of a non-trivial quasi-equivariant harmonic map, $\\omega : \\mathbb {S}^2 \\rightarrow \\mathcal {N}$ .", "The energy assumption above is reminiscent of work which considers wave maps with energies slightly above the “ground state\" (that is, the lowest energy of a non-trivial harmonic map into the target manifold).", "We do not show that $\\mathcal {E}_{\\mathrm {quasi}}(\\mathcal {N})$ is the energy of the ground state, but it plays that role within our considered symmetry class and we will sometimes abuse terminology and refer to it as the energy of the ground state (see Lemma REF and the remark after).", "Remark 1.8 A few clarifying remarks on Theorem REF : We note that blow-up for radially symmetric wave maps into general targets has been ruled out under a number of different weak assumptions [39], [26], [2].", "Thus it is perhaps unreasonable to expect that Theorem REF could hold under stronger symmetry assumptions on the initial data.", "The fact that $f\\in C^\\infty $ , as opposed to analytic, allows $f(p)-f(q)$ to vanish to arbitrarily high order as $p\\rightarrow q$ .", "Much previous work on wave maps has assumed non-degeneracy conditions (e.g.", "conditions (A1)-(A3) in Section 3 of [18]) which rule out this behavior.", "In addition to $C^\\infty $ smoothness and finite energy, we require some additional quantitative regularity of the initial data.", "While $\\dot{H}^{1+ \\varepsilon } \\times \\dot{H}^{\\varepsilon }$ regularity (for any $\\varepsilon > 0$ ) would suffice, we assume $\\dot{H}^3 \\times \\dot{H}^2$ -regularity of the initial data since the proof of local wellposedness proceeds by classical energy methods and is simpler than the techniques required to get the almost critical result.", "This additional regularity is used in Lemma REF .", "The proof of Theorem REF actually establishes the existence of an infinite family of quasi-equivariant initial data $(u_0, u_1)$ for which the corresponding solution has non-unique bubbling, as opposed to a specific construction of the initial data.", "Remark 1.9 We do not, currently have more precise control on the rates $\\lbrace r_n\\rbrace $ for the bubbling in Theorem REF than what is provided in [35].", "In the case of equivariant wave maps into $\\mathbb {S}^2$ , it is remarked in [4] that various possibilities exist, and we refer to [16], [27] for the construction of an infinite-time blow-up in this setting.", "In [48], Topping establishes a lower bound on the rate of blow-up in the context of harmonic map heat flow.", "A key ingredient in Topping's proof of this lower bound is a quantitative “neck estimate”, see [48], due to Qing and Tian [28], see also Lemma 2.9 in [47].", "While neck estimates have been recently made available for wave maps into spheres in the work of Grinis [14], those estimates are non-quantitative, and thus it remains unclear how to use them to gain control on the rate of bubbling for wave maps, even if we were to adapt them to the case of general targets." ], [ "Soliton Resolution Conjecture and Prior Work", "One motivation for establishing Theorem REF stems from the large amount of recent activity in establishing the soliton resolution conjecture for nonlinear wave and wave map equations.", "The soliton resolution conjecture posits that solutions for a broad class of nonlinear dispersive equations should decompose asymptotically as a sum of “bubbles\" and radiation.", "In the setting of (quasi-)equivariant wave maps, $\\psi : \\mathbb {R}^{2+1} \\rightarrow \\mathcal {M}$ , the conjecture states that if $\\psi $ develops at least one bubble at $t = \\infty $ , then there exists a collection of finite energy harmonic maps, $\\lbrace Q_j\\rbrace _{j=1}^J: \\mathbb {R}^2 \\rightarrow \\mathcal {M}$ , continuous scaling parameters, $0 < \\lambda _1(t) \\ll \\lambda _2(t) \\ll \\ldots \\ll \\lambda _J(t) \\ll t$ , and a finite energy linear wave, $\\phi _L$ , such that $\\psi (-, t) = \\phi _L(-, t) + \\sum _{j=1}^J Q_j\\left(\\frac{-}{\\lambda _j(t)}\\right) + \\varepsilon (t),$ where $\\varepsilon (t) \\rightarrow 0$ in the appropriate function space as $t\\rightarrow \\infty $ (see the introduction of [6], [14] for a more detailed discussion of the soliton resolution conjecture for wave maps into the sphere).", "In particular, (REF ) implies uniqueness in the sense of Definition REF for each of the bubbles, $Q_j$ .", "We note that such a description goes beyond Theorem REF which, together with uniqueness in the sense of Definition REF , only describes the dynamics of energy concentration at one scale.", "We refer to the examples of [17], [19] which demonstrate that such multi-scale concentration is in fact possible.", "Much progress has been made on the soliton resolution conjecture a variety of settings.", "Thus far, the full conjecture has been proved in the work of Duyckaerts, Kenig and Merle [10] and [11] for the radial, focusing energy-critical nonlinear wave equation in odd space dimensions, and wave maps into $\\mathbb {S}^2$ under various symmetry and energy assumptions, see, for instance, [3], [4], [9], [19] and references therein.", "Nonetheless, in many cases the conjecture in its full strength remains open.", "Often, one can show that an asymptotic decomposition in the vein of (REF ) holds along a sequence of times $t_n \\rightarrow T_{\\max }$ , see [6], [18], [8].", "In these instances, the difficulty becomes proving that the decomposition is independent of the sequence $\\lbrace t_n\\rbrace $ .", "Our main theorem, Theorem REF , demonstrates that in some settings, it may be impossible to move beyond the soliton resolution along a sequence of times to the full conjecture.", "We hope that our theorem may shed some light on the specific difficulties in proving continuous time soliton resolution.", "Indeed, for quasi-equivariant non-linear wave equations there are two main ways in which the decomposition could depend on the sequence of times.", "The first is that the bubbles can “switch places\", i.e.", "$Q_j$ may develop at a smaller scale than $Q_k$ along one sequence of times but at a larger scale along another sequence of times.", "The second is that the bubbles, $Q_j$ , themselves could depend on the sequence of times considered.", "In both cases, a careful understanding of the potential bubbles, i.e.", "non-trivial harmonic maps at specified energy levels, and possible interactions between bubbles separated in scale or in space has appeared to be a crucial ingredient in proofs which ultimately establish uniqueness for such problems (see, e.g.", "[19], [11] respectively).", "A complete understanding of the possible bubbles and their interactions is often achieved by imposing symmetry assumptions; for example, there is one (up to rescaling) radially symmetric stationary solution to the non-linear wave equation considered in [10].", "In other situations, energy constraints can rule out multi-soliton configurations, see, for instance, [3], [4] which establishes continuous-time soliton resolution for equivariant wave maps into $\\mathbb {S}^2$ with energy less than three times the energy of the lowest energy harmonic map into the sphere, and [9] for the result without the equivariance assumption when the energy is restricted to just above the energy of the ground state.", "In contrast to these cases, the target manifold in Theorem REF admits a continuum of quasi-equivariant harmonic maps $\\omega : \\mathbb {R}^2 \\rightarrow \\mathcal {N}$ at the lowest admissible non-zero energy level, and the richness of this family plays an essential role in the proof of non-uniqueness for Theorem REF .", "Finally, we note one further aspect of our setting which contributes to non-uniqueness, specifically compared to the setting considered in [9].", "While in both cases wave maps with energy just above the ground state are considered, in [9] the authors exploit the fact that for wave maps into spheres, the energy is coercive near the traveling waves, which traps the wave map in increasingly small neighborhoods of the traveling wave, yielding uniqueness.", "In contrast, while the second component of our target manifold is the sphere, the first component has an infinite length geodesic which, using standard coordinates $\\mathbb {T}^2 = (w, z)$ , wraps around the torus infinitely many times as it approaches the circle $\\lbrace w = 0\\rbrace $ .", "This winding behavior allows the first coordinate of the wave map to exit a small neighborhood of a soliton infinitely many times, even though the behavior of the second component of the map will be well controlled (i.e.", "the second component may be almost constant)." ], [ "Comparison with non-uniqueness in elliptic and parabolic problems", "Our construction is heavily inspired by Topping's construction of a harmonic map heat flow which develops a non-unique bubble in finite time [48].", "Our target manifold, including the warping function, is essentially equivalent to Topping's (more on this in Section ).", "We also use the same mechanism as Topping to ensure non-uniqueness, what he calls “winding\".", "Note that we record only the definition of winding as it applies in the quasi-equivariant setting, and a more general definition would take into account translational and Lorentz symmetries.", "Definition 1.10 ([48]) A quasi-equivariant wave map $u: \\mathbb {R}^{1+2} \\rightarrow \\mathcal {M}$ has a winding singularity at time $T_{\\max }$ and the origin $ 0 \\in \\mathbb {R}^2$ if there exists sequences $\\lbrace r_n\\rbrace $ , and $\\lbrace t_n\\rbrace $ , satisfying $t_n \\uparrow T_{\\max },\\qquad r_n = {\\left\\lbrace \\begin{array}{ll} o(T_{max} - t_n) & T_{max} < \\infty \\\\o(t_n) & T_{max} = \\infty \\end{array}\\right.", "}$ such that $u(t_n, r_nx) \\rightarrow \\omega (x)$ in $C^0_{\\mathrm {loc}}({\\mathbb {R}}^2\\backslash \\lbrace x_0\\rbrace ; \\mathcal {M})$ , where $\\omega $ is a non-constant harmonic map, the lifts $\\hat{u}(t_n,r_nx)$ have no convergent subsequence in $C^0_{\\mathrm {loc}}({\\mathbb {R}}^2\\backslash \\lbrace x_0\\rbrace ; \\widehat{\\mathcal {M}})$ .", "Despite the similarities in set-up, the execution of our proof differs substantially from Topping's.", "This is due to fundamental differences between the parabolic and Hamiltonian settings, even at the level of ODEs, which we had to overcome.", "To elucidate these issues, we elaborate here on two examples.", "For analytic functions, $f$ , bounded gradient flows, $\\dot{x}(t) = -\\nabla f(x(t))$ , have a unique limit as $t\\rightarrow \\infty $ , due to the Łojasiewicz inequalities [25].", "It is a beautiful observation of L. Simon [32] that this fact about ODEs can be applied to study the long time behavior of the gradient flows of many elliptic functionals which arise naturally in geometry.", "These “Łojasiewicz-Simon\" inequalities have found subsequent use in a huge range of geometric and variational problems.", "For example, to prove uniqueness of tangent objects for variational problems (e.g.", "minimal surfaces, [32] and mean curvature flow [5]) and to show the uniqueness of long time limits of geometric flows (e.g.", "Yamabe flows [1], harmonic map heat flow [13]).", "However, this phenomena does not hold in the Hamiltonian setting.", "Indeed, if $f(x,y) = \\frac{1}{2}(x^2 +y^2)$ then the equation $\\ddot{x}(t) = -\\nabla f(x(t))$ becomes $\\ddot{x}(t) = -x(t)$ .", "One solution to this ODE is the bounded flow $x(t) = (\\cos (t), \\sin (t))$ , which clearly does not have a unique limit as $x(t) \\rightarrow \\infty $ .", "When $f$ is not analytic, but is $C^\\infty $ , then the classic “goat tracks\" example $ f(r,\\theta ) = {\\left\\lbrace \\begin{array}{ll} 1&\\quad r \\le 1,\\\\1 + e^{-\\frac{1}{r-1}}\\left(\\sin (\\frac{1}{r-1} + \\theta ) + 2\\right)&\\quad r > 1,\\end{array}\\right.", "}$ generates a gradient flow $\\dot{x}(t) = -\\nabla f(x(t))$ which is bounded but doesn't have a unique limit as $t\\rightarrow \\infty $ , in fact, every point on the circle $r = 1$ is a limit point.", "This example is at the heart of Topping's construction [48] (see also the examples of non-uniqueness for singularities of harmonic maps, [49], and the long term behavior of harmonic map heat flow, [46]).", "In contrast, the corresponding flow given by $\\ddot{x}(t) = -\\nabla f(x(t))$ cannot exhibit the same asymptotic behavior as the gradient flow does.", "The Hamiltonian flow stays bounded as long as $|x(0)|$ is close to one and $\\dot{x}(0)$ is small enough, however, by working in polar coordinates, one can see that if $|x(t)| \\rightarrow 1$ as $t\\rightarrow \\infty $ it must also be the case that $|\\dot{x}(t)| \\rightarrow 0$ as $t\\rightarrow \\infty $ .", "This would violate the conservation of the energy, $|\\dot{x}|^2(t) + f(x(t))$ (provided $\\dot{x}(0) \\ne 0$ ).", "As these examples show, one needs caution when using long-term behavior of non-linear parabolic flows to provide insight into the Hamiltonian setting.", "On a practical level, while energy conservation provides some control for Hamiltonian flows, it is not a substitute for the maximum principle and energy dissipation, which hold in the parabolic setting.", "For example, as in [48] we show that the image of the flow is contained in a geodesic in the target manifold, see Lemma REF .", "However, we are faced with the additional difficulty of showing that the flow cannot leave this geodesic before the blow-up time.", "We note that this issue is not present in the parabolic setting, where stationary solutions act as a barrier to constrain the flow.", "Additionally, energy dissipation allows Topping to determine that his flow blows up in finite time via topological considerations and Lemaire's theorem.", "On the other hand, we must leave open the possibility that the winding singularity may occur at either finite or infinite time (we speculate either situation can occur).", "Nonetheless, we believe that our proof exhibits, morally, a phenomenon exploited in work of Grinis [14]: that nonlinear wave equations start to exhibit elliptic behavior in the (strict) interior of a light-cone in which energy is concentrating." ], [ "Structure of the Paper", "Here we briefly outline the structure of the paper.", "In Section we record some preliminaries about wave maps and harmonic maps into general Riemannian manifolds.", "In Section we construct the target manifold $\\mathcal {N}$ from Theorem REF .", "This follows much as in [48], with additional complications caused by the fact that we are unable to use the maximum principle to constrain the image of the flow.", "We overcome this difficulty by working with a compact target manifold and carefully defining our metric twist globally on ${\\mathbb {T}}^2$ .", "In Section we establish some preliminary results on the Hamiltonian flow.", "In subsection REF we also rule out “scattering\" to a solution of the underlying linear wave equation.", "In [48] the analogous parabolic phenomena, relaxation to a stationary solution, can be quickly ruled out using topology.", "The wave map setting requires a more involved estimate of the energy flux through the wall of the light cone.", "In Section we study the harmonic maps into $\\mathcal {N}$ which can arise as bubbles in the flow.", "Here the analysis is complicated by the fact that the metric twist had to be defined globally on ${\\mathbb {T}}^2$ .", "In particular, we use energy arguments to rule out bubbles which wrap “the wrong way\" around the ${\\mathbb {T}}^2$ component, see Lemma REF .", "Finally, in Section we establish properties of the singularity, in particular winding, proving Theorem REF ." ], [ "Acknowledgements", "This work was done while ME was visiting the University of Chicago for the AY 2019-2020; he thanks the department and especially Carlos Kenig for their hospitality.", "He also thanks Carlos Kenig for suggesting he investigate the phenomena of bubbling for harmonic map heat flow.", "DM learned about Topping's work on non-uniqueness for the harmonic map heat flow bubbling, and the open question of non-uniqueness for wave maps bubbling, in a topics course given by Carlos Kenig at the University of Chicago.", "Both ME and DM benefited from several discussions with Andrew Lawrie, including discussions about Topping's work on harmonic map heat flow and about scattering for wave maps.", "The authors thank both Carlos Kenig and Andrew Lawrie for helpful comments on an earlier version of this manuscript." ], [ "Preliminaries for Wave Maps and Harmonic Maps", "In this section we collect some basic facts about the regularity of (quasi-equivariant) wave maps and harmonic maps.", "As mentioned in the introduction, we will consider wave maps $u: \\mathbb {R}^{1+2} \\rightarrow \\mathbb {T}^2 \\times _f \\mathbb {S}^2$ for a certain $C^\\infty ({\\mathbb {T}}^2)$ warping function, $f$ .", "Using polar coordinates in the domain of $u$ , we suppose that the initial condition has the form $u_0(r,\\theta ) = (0, y_0, \\alpha _0(r), \\theta ),\\qquad u_1(r,\\theta ) = (0, y_1(r), \\alpha _1(r), 0).$ We will rely on the following local existence and persistence of regularity result, which follows via energy estimates.", "Note too that the optimal local theory is known, see for instance [21].", "Proposition 2.1 (Classical local existence and persistence of regularity) Let $s_0 > 1$ and let $(u_0, u_1) \\in \\dot{H}^{s_0 + 1} \\times \\dot{H}^{s_0}$ .", "There exists a $T_{max} \\equiv T_{max}(u) > 0$ such that for every $T < T_{max}$ , there exists a unique solution $u:[0,T] \\times \\mathbb {R}^{1+2} \\rightarrow \\mathcal {N}$ of (REF ) such that $\\sup _{0 < t \\le T} \\Vert u(t) \\Vert _{\\dot{H}^{s_0 + 1}} < \\infty .$ Moreover, if $(u_0, u_1) \\in H^{s + 1} \\times H^{s}$ for any $s > s_0$ , then $\\sup _{0 < t \\le T} \\Vert u(t) \\Vert _{ \\dot{H}^{s + 1}} < \\infty .$ The uniqueness conclusion of Proposition REF implies that the wave map retains rotational and quasi-equivarient symmetry.", "Therefore, the solution will have the form $u(t, r,\\theta ) = (X(t,r), Y(t,r) , \\alpha (t,r), \\theta ).$ In the coordinates of (REF ), the energy has the form $\\mathcal {E}(t) = 2\\pi \\int _0^\\infty \\left(\\frac{1}{2}|\\nabla (X,Y)|^2 + \\frac{1}{2}|\\partial _t (X,Y)|^2 + f(X,Y)e(\\alpha )\\right)rdr,$ where $e(\\alpha )$ is the “spherical\" part of the energy $e(\\alpha ) := \\frac{1}{2} \\left[ \\left( \\frac{\\partial \\alpha }{\\partial r}\\right)^2 +\\left( \\frac{\\partial \\alpha }{\\partial t}\\right)^2+ \\frac{\\sin ^2 \\alpha }{ r^2} \\right].$ If the initial conditions (REF ) have finite energy, $\\mathcal {E}(u_0,u_1) < \\infty $ , then $\\mathcal {E}(t) \\equiv \\mathcal {E}(u_0, u_1)$ for all $t < T_{\\max }$ .", "Furthermore, if $u$ bubbles to $\\omega $ at $T_{\\max }$ in the sense of Theorem REF or the analogous finite time result, then as stated in case (a) of that theorem, $\\int |D\\omega |^2 \\le \\mathcal {E}(u_0, u_1)$ .", "In order to satisfy $\\int _0^\\infty f(X,Y) e(\\alpha ) rdr < \\infty $ for all $t$ it must be the case that $\\alpha (t, 0) = m \\pi $ and $\\alpha (t, \\infty ) = n \\pi $ for all $t \\in I_{max}(u)$ for some integers $n, m$ .", "That the integers must be constant for all $t$ follows from the continuity in time of the flow.", "We will take $m=0$ in the sequel, and we define the degree of the wave map to be the integer $n$ ." ], [ "Regularity for Weakly Harmonic Maps from ${\\mathbb {R}}^2$", "Let $(\\mathcal {N}, g)$ be a closed smooth Riemannian manifold.", "Throughout we will assume that $\\mathcal {N}$ is smoothly embedded into $\\mathbb {R}^n$ .", "Weakly harmonic maps $\\omega : {\\mathbb {R}}^2 \\rightarrow \\mathcal {M} \\subset \\mathbb {R}^n$ are critical points of the energy $E(u) \\equiv \\int _{{\\mathbb {R}}^2} |Du|^2_g$ under perturbations of the form $u_\\varepsilon (x) = \\pi (u + \\varepsilon \\varphi (x))$ where $\\pi $ is the nearest point projection of $\\mathbb {R}^n$ onto $\\mathcal {N}$ , which is well defined and smooth in a small neighborhood of $\\mathcal {N}$ , and $\\varphi \\in C^1_c({\\mathbb {R}}^2; {\\mathbb {R}}^n)$ .", "Equivalently, these maps (weakly) satisfy the semi-linear PDE: $-\\Delta \\omega + A(\\omega )(\\nabla \\omega , \\nabla \\omega ) = 0$ where $A(\\omega )$ is the second fundamental form of $\\mathcal {N}$ .", "In general, weakly harmonic maps need not be continuous (see, e.g.", "[29]), but when the domain is two-dimensional all weakly harmonic maps are $C^\\infty $ by the fundamental result of [15].", "As such, we will refer to weakly harmonic maps from $\\mathbb {R}^2$ as simply harmonic maps.", "Finally, if $\\omega : {\\mathbb {R}}^2 \\rightarrow {\\mathcal {N}}$ is harmonic, then (composing with a stereographic projection) we get a harmonic map $\\widetilde{\\omega }: \\mathbb {S}^2\\backslash \\lbrace \\infty \\rbrace \\rightarrow {\\mathcal {N}}$ , which we can smoothly extend to all of $\\mathbb {S}^2$ by the work of Sachs-Uhlenbeck [41].", "We will often abuse notation and identify the harmonic maps $\\omega , \\widetilde{\\omega }$ .", "Our first theorem quantifies the regularity of two-dimensional harmonic maps (the precise statement non-minimizing maps on ${\\mathbb {R}}^2$ is hard to track down.", "However, one can argue as in [42] or consider the stationary case of the parabolic regularity proven in [37]): Theorem 2.2 Let $\\omega : \\mathbb {R}^2 \\rightarrow \\mathcal {N}$ be a harmonic map.", "There exists an $\\bar{\\varepsilon } > 0$ small, and depending on $\\mathcal {N}$ , such that if $ \\int _{B_R(x_0)} |D\\omega |^2 \\le \\bar{\\varepsilon },$ then, $ R^2\\sup _{B_{R/2}(x_0)}|D\\omega |^2 \\le C\\int _{B_R(x_0)} |D\\omega |^2 dx,$ where $C > 0$ depends on $\\mathcal {N}$ but not on $x_0, R$ or $\\omega $ .", "Equivalently, given the small energy condition (REF ) $ \\sup _{x,y\\in B_{R/2}(x_0)} d_{\\mathcal {N}}(\\omega (x),\\omega (y)) \\le C \\left(\\int _{B_R(x_0)} |D\\omega |^2 dx\\right)^{1/2},$ where $d_{\\mathcal {N}}(p,q)$ is the geodesic distance between $p,q \\in \\mathcal {N}$ and $C > 0$ is as above.", "There are two standard corollaries of Theorem REF which will be important to us.", "The first states that there is a least energy non-trivial harmonic map into any target.", "Corollary 2.3 For any weakly harmonic map $\\omega : {\\mathbb {R}}^2 \\rightarrow \\mathcal {N}$ , one of following two hold: $\\mathcal {E}(\\omega ) \\equiv \\int _{{\\mathbb {R}}^2}|D\\omega |^2 = 0$ and $\\omega $ is almost everywhere equal to a constant $p \\in \\mathcal {N}$ .", "$\\mathcal {E}(\\omega ) \\ge \\bar{\\varepsilon } > 0$ , where $\\bar{\\varepsilon }$ is the constant from Theorem REF The second corollary states that the shortest path between two points in the image of a harmonic map cannot have infinite length: Lemma 2.4 Let $\\omega : {\\mathbb {R}}^2 \\rightarrow \\mathcal {N}$ be a weakly harmonic map.", "There cannot be two points $p_1, p_2 \\in \\mathrm {Im}\\,\\omega $ such that the shortest path between the two points in $\\mathcal {N}$ has infinite length.", "Let $x,y \\in {\\mathbb {R}}^2$ and let $L_{x,y}$ be the line segment connecting the two.", "Let $E$ be the energy of the harmonic map.", "Since $\\omega \\in C^\\infty ({\\mathbb {R}}^2)$ , for all $z \\in L_{x,y}$ there exists an $r_z > 0$ such that $\\int _{B_{r_z}(z)}|D\\omega |^2 < \\bar{\\varepsilon }.$ By compactness there are finitely many $z_i \\in L_{x,y}$ such that $L_{x,y}\\subset \\bigcup _{i=1}^M B_{r_{z_i}}(z_i).$ Using the oscillation estimate (REF ), we have that $d_{\\mathcal {N}}(\\omega (x), \\omega (y)) \\le CM\\sqrt{\\bar{\\varepsilon }} < \\infty $ .", "We end this section with some elementary facts about harmonic maps $\\omega : \\mathbb {S}^2 \\rightarrow \\mathbb {S}^2$ .", "Lemma 2.5 Let $\\omega $ be a harmonic map $\\mathbb {S}^2 \\rightarrow \\mathbb {S}^2$ .", "Let $\\mathcal {E}_{\\mathbb {S}^2}$ denote the lowest energy level of a non-trivial harmonic map between spheres, which is guaranteed to exist by Corollary REF .", "There is a unique (up to a conformal transformation of $\\mathbb {S}^2$ ) equivariant harmonic map $\\omega (r,\\theta ) = (\\alpha (r), \\theta )$ such that $\\mathcal {E}(\\omega ) = \\mathcal {E}_{\\mathbb {S}^2}$ .", "Furthermore, if $\\omega $ is equivariant and $\\mathcal {E}(\\omega ) > \\mathcal {E}_{\\mathbb {S}^2}$ then it must be that $\\mathcal {E}(\\omega ) \\ge 2 \\mathcal {E}_{\\mathbb {S}^2}$ ." ], [ "Construction of the target", "We now construct the target manifold ${\\mathcal {N}}$ using a modification of Topping's construction in [48].", "Recall that we will construct ${\\mathcal {N}}$ as a twisted product of the torus $\\mathbb {T}^2$ with $S^2$ , i.e.", "${\\mathcal {N}}= \\mathbb {T}^2 \\times _f S^2,$ where $f \\in C^\\infty ({\\mathbb {T}}^2)$ denotes the warping function.", "We introduce coordinates $(w,z)$ on ${\\mathbb {T}}^2$ where $w,z\\in [0,1] \\cong \\mathbb {S}^1$ .", "We define the curve $\\gamma (s)= \\left(\\frac{1}{\\pi }\\cot ^{-1}(s),\\, s \\mod {1}\\right), \\quad s \\in (-\\infty , \\infty ) ,$ where inverse cotangent is defined so that $\\cot ^{-1}(s): (-\\infty , \\infty ) \\rightarrow [0,\\pi ]$ .", "Observe that $\\cot ^{-1}$ is $C^\\infty $ with uniform control on all derivatives in any $[-K, K] \\subset (-\\infty , \\infty )$ .", "We want to define a metric on ${\\mathbb {T}}^2$ such that $\\gamma $ is a geodesic.", "We claim that $h(w,z) = \\begin{pmatrix} \\pi ^2 & \\pi \\sin (\\pi w)^2\\\\\\pi \\sin (\\pi w)^2 & 1+\\sin (\\pi w)^4\\end{pmatrix}$ gives such a metric.", "To see that this is the case, and to simplify our analysis of the wave maps equation, it will be useful to make the following coordinate change: $\\Phi : (w,z) \\mapsto (x,y)= (\\cot (\\pi w)-z,z),\\qquad D\\Phi = \\begin{pmatrix} \\frac{-\\pi }{\\sin ^2(\\pi w)} & -1\\\\0 & 1\\end{pmatrix}$ We note that $D\\Phi $ is well defined and invertible away from $w = 0 \\equiv 1$ , and we will show in Lemma REF that when $t < T_{\\max }$ the flow stays away from this curve, hence the change of variables remains valid up until the first blow-up time.", "We further note that $\\Phi $ respects the symmetry of ${\\mathbb {T}}^2$ given by $w\\cong w+1$ and that the image of $\\Phi $ will have the symmetry $(x,y)\\cong (x+1, y-1)$ , and thus will be a cylinder.", "We shouldn't expect the image of $\\Phi $ to be a topological torus as $\\Phi $ is not well defined at $w= 0, 1$ .", "In the $(x,y)$ coordinates $\\gamma (s) = (0, s), \\quad s\\in (-\\infty , \\infty )$ and the pushforward of the metric, $h$ , is given by $\\textbf {h}(x,y) = \\begin{pmatrix} \\frac{1}{(1+(x+y)^2)^2} & 0\\\\0 & 1\\end{pmatrix}.$ It is now straightforward to compute that $\\gamma $ is a geodesic.", "As a check, we observe that $(x,y) \\cong (x+1, y-1)$ is an isometry of $\\mathrm {Im}\\, \\Phi $ equipped with the metric $\\textbf {h}$ .", "We now turn to the construction of the warping function $f(w,z)$ .", "Here we face a technical difficulty not present in [48] which is our inability to constrain the flow using the maximum principle.", "Thus we need to carefully define the warping function on all of $\\mathcal {N}$ and not just in a neighborhood of $w = 0$ .", "We do this by means of a smooth cutoff function, the existence of which is guaranteed by the following lemma.", "Lemma 3.1 There exists a $C^\\infty $ bump function $\\chi : {\\mathbb {T}}^2 \\rightarrow [0,1]$ with the following properties: ${\\left\\lbrace \\begin{array}{ll} \\chi \\equiv 1 & w\\in [0, 1/4]\\cup [3/4, 1] \\\\\\chi \\equiv 0 & w \\in [7/16, 9/16]\\\\\\nabla \\chi |_{\\gamma } \\parallel \\dot{\\gamma } & w\\in (1/4, 3/4).\\end{array}\\right.", "}$ We provide only a sketch of this proof.", "To ease notation, let $X = \\lbrace (w,z) \\mid w\\in (1/4, 3/4)\\rbrace \\subset {\\mathbb {T}}^2$ .", "To define the cut-off function $\\chi $ , we smoothly connect a cut-off defined in a neighborhood of the geodesic $\\gamma $ , with another depending only on $w$ outside this neighborhood.", "The latter construction is simpler, so we only describe the former.", "Let $\\gamma _\\delta = \\lbrace p \\in {\\mathbb {T}}^2\\mid \\mathrm {dist}(p, \\gamma ) < \\delta \\rbrace $ .", "Inside of $X$ , $\\gamma $ is parameterized by a $C^\\infty $ function with uniform bounds on all the derivatives, therefore, there exists a $\\delta _0 > 0$ such that we can smoothly parameterize $\\gamma _{\\delta _0}$ by $(t, n)$ where the $t$ coordinates are parallel to $\\gamma $ and the $n$ coordinates are normal to $\\gamma $ .", "In $X \\cap \\gamma _{\\delta _0/2}$ , we define $\\chi $ to depend only on $t$ (working in $t, n$ coordinates) and to $C^\\infty $ interpolate between 0 and 1 so that the first two conditions of (REF ) hold.", "With $\\chi $ defined we can now define the metric twist $\\tilde{f}$ in a neighborhood of $w = 0 \\cong 1$ : $\\tilde{f}(w,z) \\equiv e^{-2\\pi \\cot (\\pi w)}\\left(\\sin 2\\pi \\bigl (\\cot (\\pi w)-z-1/8\\bigr )+\\sqrt{2}\\right) + 1,$ and then globally define the twist by $f(w,z) = {\\left\\lbrace \\begin{array}{ll}1& \\:\\: w = 0 \\\\\\tilde{f}(w,z)& \\:\\: 0 < w \\le 1/4 \\\\\\chi (w,z)\\tilde{f}(w,z)+ (1-\\chi (w,z))M& \\:\\: 1/4 < w \\le 1/2 \\\\\\chi (w,z)\\tilde{f}(1-w,1-z) + (1-\\chi (w,z))M &\\:\\: 1/2 < w \\le 3/4\\\\\\tilde{f}(1-w, 1-z)&\\:\\: 3/2 < w < 1\\end{array}\\right.", "}$ where $M > 1$ is a constant to be chosen later.", "Note that $f$ is $C^\\infty $ away from $w= 0,1$ (as it is the sum of products of $C^\\infty $ functions).", "At $w = 0$ we observe that $\\tilde{f} - 1$ vanishes to infinite order and similarly $\\tilde{f}(1-w, 1-z)$ at $w = 1$ .", "Thus $f\\in C^\\infty ({\\mathbb {T}}^2)$ , and furthermore, $f$ is invariant under the isometries of the space, i.e.", "$(w,z) \\cong (w+n, z+m)$ for $(n,m)\\in \\mathbb {Z}\\times \\mathbb {Z}$ .", "In $(x,y)$ coordinates, we have $\\tilde{f}(x,y) = e^{-2\\pi (x+y)}\\left(\\sin 2\\pi \\bigl (x-1/8\\bigr )+\\sqrt{2}\\right) + 1.$ It is a straightforward calculation to see the following properties of $f$ : (i) $\\partial _x \\tilde{f}(0,y) = 0$ (ii) $\\partial _y \\tilde{f}(0, y) < 0$ .", "Combined with what we know about $\\chi $ , this implies that $\\nabla f|_\\gamma $ is parallel to $\\gamma $ in all of ${\\mathbb {T}}^2\\backslash \\lbrace w= 0\\rbrace $ , allowing for the fact that $\\nabla f$ is zero away from the support of $\\chi $ .", "We can define $f$ globally in $(x,y)$ coordinates by $f(x,y) = {\\left\\lbrace \\begin{array}{ll}\\tilde{f}(-x, -y)&\\:\\: x+y < -\\cot ^{-1}(\\pi /4)\\\\\\chi (x,y)\\tilde{f}(-x,-y) + (1-\\chi (x,y))M &\\:\\: x+y \\in [-\\cot ^{-1}(\\pi /4) , 0) \\\\\\chi (x,y)\\tilde{f}(x,y)+ (1-\\chi (x,y))M& \\:\\: x+y \\in [0, \\cot ^{-1}(\\pi /4)) \\\\\\tilde{f}(x,y)& \\:\\: x+y > \\cot ^{-1}(\\pi /4)\\end{array}\\right.", "}$ Furthermore, by picking $M > 2\\sup _{{\\mathbb {T}}^2}\\tilde{f}$ we can guarantee that $\\mathrm {sgn}(y) \\partial _y f(0,y) \\le 0\\textup { with equality if and only if }f \\equiv M$ (i.e.", "$y \\notin \\mathrm {supp} \\chi $ ).", "This can be seen through a chain rule computation and the fact that $\\partial _y \\tilde{f}(0, y) < 0$ .", "We end this section by summarizing the properties of $f$ and $\\mathcal {N}$ which are important to us.", "Lemma 3.2 (Properties of the target Manifold) The function $f$ , manifold $\\mathcal {N} = {\\mathbb {T}}^2\\times _f {\\mathbb {S}}^2$ and curve $\\gamma $ , described above, have the following properties: $f \\in C^\\infty ({\\mathbb {T}}^2)$ and $f \\ge 1$ always with $f = 1$ iff $w = 0$ .", "The curve $\\gamma (s): (-\\infty , \\infty ) \\rightarrow {\\mathbb {T}}^2$ is a geodesic with the following properties: For any $s\\in \\mathbb {R}$ , $\\ell (\\gamma ((-\\infty ,s)) = \\infty = \\ell (\\gamma ((s, \\infty )))$ .", "$\\mathrm {sgn}(s)\\frac{d}{ds}f(\\gamma (s)) > 0$ except in a neighborhood of 0, in which $f \\equiv M \\gg 1$ $\\lbrace w = 0\\rbrace \\subset \\overline{\\gamma ((-\\infty ,\\infty ))}$ but $\\lbrace w = 0\\rbrace \\cap \\gamma (-\\infty , \\infty ) = \\emptyset $ .", "For any $s \\in {\\mathbb {R}}$ , $\\nabla f(\\gamma (s)) \\parallel \\gamma ^{\\prime }(s)$ .", "Let us quickly comment on some of these conditions: Remark 3.3 It is not so important that $f, \\mathcal {N}$ satisfy the conditions REF , REF precisely.", "The arguments here work for any $f$ which is globally bounded away from zero and which achieves its minimum on a topological circle that is in the closure of (but does not intersect) a geodesic $\\gamma $ .", "These facts will help us control the image of possible bubbles; cf.", "Lemma REF .", "Second, conditions REF ,REF REF imply that the gradient flow generated by $f$ starting at a point along the curve $\\gamma $ will be bounded but will not have a unique limit as $t\\rightarrow \\infty $ .", "Rather, each point in $\\lbrace w = 0\\rbrace $ will be an accumulation point of the flow.", "As we mentioned in Section REF , this property would not possible if $f$ were analytic." ], [ "Analysis of the Hamiltonian Flow", "We now turn to the setting of wave maps into $\\mathcal {N}$ .", "Our first result of this section establishes that for wave maps with initial conditions of the form (REF ), the flow stays inside $\\mathrm {Im} \\gamma \\times _f \\mathbb {S}^2$ for all $t < T_{max}(u)$ .", "Throughout this section, we will denote by $P_1$ and $P_2$ the projection of ${\\mathcal {N}}$ onto its two-dimensional components: $P_1 : {\\mathcal {N}}\\rightarrow {\\mathbb {T}}^2, \\quad P_2 : {\\mathcal {N}}\\rightarrow {\\mathbb {S}}.$ Lemma 4.1 Let $u: \\mathbb {R}^{2+1} \\rightarrow {\\mathcal {N}}$ be a wave map with initial conditions (REF ), which also satisfy $\\Vert (u_0, u_1)\\Vert _{\\dot{H}^3\\times \\dot{H}^2} < \\infty $ .", "Then for all $t < T_{max}(u)$ , using the notation of (REF ), $X(t,\\cdot ) \\equiv 0$ , and $- \\infty < Y(t,\\cdot ) < \\infty .$ We prove at as long as the initial conditions lay in $\\gamma \\times T\\gamma $ we have $X(t,r) \\equiv 0$ .", "This follows from the fact that $\\gamma \\times \\mathbb {S}^2$ lies totally geodesically within ${\\mathcal {N}}$ (cf.", "[48] Section 3), and then the classical fact that wave maps with initial conditions in totally geodesic submanifolds stay in that submanifold.", "Let us briefly sketch how this works: $\\gamma $ is a geodesic in ${\\mathbb {T}}^2$ which implies that the image of $(\\Delta _{{\\mathbb {T}}^2})|_\\gamma $ is contained in $T\\gamma \\subset T{\\mathbb {T}}^2$ .", "The first component of the wave map satisfies the equation $\\partial ^2_{tt} P_1\\circ u = -\\Delta _{{\\mathbb {T}}^2} P_1\\circ u - \\nabla f(P_1\\circ u) e(\\alpha ).$ Since $\\nabla f(P_1\\circ u)$ and $\\Delta _{{\\mathbb {T}}^2}P_1\\circ u$ lie tangent to $\\gamma $ , the flow stays in $\\gamma $ for the whole time of existence.", "Finally, to obtain boundedness of $Y$ , we note that by assumption $(0,y_0) \\in \\gamma $ , so, to fix notation, suppose that $s_0 \\in \\mathbb {R}$ is such that $\\gamma (s_0) = (0,y_0)$ .", "Note that $\\ell \\bigl (\\gamma \\bigl ((s_0, \\infty )\\bigr )\\bigr ) = \\infty = \\ell \\bigl (\\gamma \\bigl ((-\\infty , s_0)\\bigr )\\bigr ), \\qquad \\forall s_0 \\in (-\\infty , \\infty ),$ so we will conclude by establishing that the image of a finite energy wave map with $\\dot{H}^{3} \\times \\dot{H}^2$ bounds cannot contain an infinite length path, which will establish that $|Y(\\cdot , t)| < \\infty $ for all $t < T_{\\max }$ proving (REF ).", "This is where we rely on the persistence of regularity result from Proposition REF .", "We note again that $\\dot{H}^{1+} \\times \\dot{H}^{0+}$ bounds would suffice for this argument.", "For any $r_0 > 0$ and $r_1, r_2 > r_0$ , letting $b = (0, Y(t, r_2))$ and $a = (0, Y(t, r_1))$ , we have $\\ell (a,b) \\le \\int _{r_1}^{r_2} |DY | dr \\le \\mathcal {E}(u) \\frac{1}{r_0},$ Hence for any given time, the only point in the domain at which $Y(t, r)$ can be infinite is $r = 0$ .", "However, for $p > 1$ and for any $r_1 \\le 1$ , letting $c =Y(t, r_1)$ and $d = Y(t, 0)$ , we further have that $\\ell (c,d) \\le \\left( \\int _{0}^{r_1} |DY |^{p} r dr \\right)^{1/p} & \\le \\left( \\int _{0}^{\\infty } |DY |^{2} r dr \\right)^{1-\\theta } \\left( \\int _{0}^{\\infty } |D^3Y |^{2} r dr \\right)^{\\theta }\\\\& \\le C\\bigl (t, \\Vert (u_0, u_1)\\Vert _{\\dot{H}^3 \\times \\dot{H}^2}, \\mathcal {E}(u_0, u_1)\\bigr )$ where the second inequality holds for some $0< \\theta < 1$ by Sobolev embedding.", "Hence $Y(t, 0)$ cannot pass through $\\lbrace \\pm \\infty \\rbrace $ ." ], [ "Energy Concentration at $T_{\\max } = \\infty $", "In this subsection we assume that $T_{\\max } = +\\infty $ .", "We want to show that scattering (i.e.", "outcome (b) in Theorem REF ) cannot occur unless the wave map is degree zero.", "We begin by rewriting the system of equations for $Y$ and $\\alpha $ , which, in light of Lemma REF , is given by $\\begin{split}\\frac{\\partial ^2 Y}{\\partial t^2} &= \\frac{\\partial ^2 Y}{\\partial r^2} + \\frac{1}{r} \\frac{\\partial Y}{\\partial r} - \\frac{\\partial f}{\\partial y}(0, Y) e(\\alpha ), \\,\\,\\,\\,\\qquad \\qquad \\qquad \\qquad \\qquad (Y, Y_t)\\big |_{t=0} = (y_0, y_1), \\\\\\frac{\\partial ^2 \\alpha }{\\partial t^2} &= \\frac{\\partial ^2 \\alpha }{\\partial r^2} + \\frac{1}{r} \\frac{\\partial \\alpha }{\\partial r} -\\frac{\\sin (2\\alpha )}{2 r^2} + \\frac{1}{f(0,Y)} \\frac{\\partial f}{\\partial y}(0, Y) \\frac{\\partial Y}{\\partial r} \\frac{\\partial \\alpha }{\\partial r} , \\qquad (\\alpha , \\alpha _t)\\big |_{t=0} = (\\alpha _0, \\alpha _1).\\end{split}$ We plan on showing that energy concentrates inside the light cone $|x| < t$ as $t\\rightarrow \\infty $ (i.e.", "that outcome (a) of Theorem REF holds).", "We start by observing that the norms of derivatives of quasi-equivariant functions have radial symmetry: Proposition 4.2 Let $u: \\mathbb {R}^{1+2} \\rightarrow (\\mathcal {M},g)$ be a quasi-equivariant wave map, then $|\\nabla _x u|_g, |\\nabla _t u|_g$ and $\\left\\langle \\nabla _x u, \\nabla _t u\\right\\rangle _g$ are all radially symmetric functions.", "Recall that $u(r, \\theta +s,t) = \\Phi _s \\circ u(r, \\theta , t)$ for $\\Phi _s \\in \\mathrm {Isom}(\\mathcal {M})$ .", "In particular this implies that $D\\Phi _s$ satisfies $\\left\\langle D\\Phi _s v, D\\Phi _s w\\right\\rangle _{g(\\Phi _s(p))} = \\left\\langle v, w\\right\\rangle _{g(p)}, \\qquad \\forall p \\in \\mathcal {M}, v,w \\in T_p \\mathcal {M}.$ We can then compute, $\\left\\langle \\nabla _x u(r,\\theta +s, t), \\nabla _t u(r, \\theta +s, t)\\right\\rangle _{g} =& \\left\\langle D\\Phi _s\\nabla _x u(r, \\theta , t), D\\Phi _s \\nabla _t u(r, \\theta , t)\\right\\rangle _g \\\\=& \\left\\langle \\nabla _x u(r, \\theta , t), \\nabla _t u(r, \\theta , t)\\right\\rangle _g.$ The same argument applies for $|\\nabla _x u|_g, |\\nabla _t u|_g$ .", "We also record a standard Hölder regularity estimate for quasi-equivariant wave maps.", "Note that we abuse notation and use $|\\cdot |$ to denote the distance within the manifold.", "Lemma 4.3 Let $u:\\mathbb {R}^{1+2}\\rightarrow \\mathcal {M}$ be a quasi-equivariant finite energy wave map into a smooth manifolds $\\mathcal {M}$ and $\\lbrace \\Phi _s\\rbrace _{s\\in \\mathbb {S}^1}$ the associated smoothly parameterized one parameter family of isometries (see Definition REF ).", "Then, for any $r_0 > 0$ there exists $C \\equiv C(\\mathcal {E}(u_0, u_1), r_0, \\sup _{s \\in {\\mathbb {S}}^1}\\Vert \\partial _s \\Phi _s\\Vert _{C^\\infty (T\\mathcal {N})}) > 0$ such that for any $t \\in \\mathbb {R}, \\theta _1, \\theta _2 \\in {\\mathbb {S}}^1$ and any $r,s \\in \\mathbb {R}$ with $r_0 < r,s$ we have $|u(t, r, \\theta _1) - u(t, s, \\theta _2)| \\le C \\bigl ( |r-s|^{1/2} + |\\theta _1-\\theta _2| \\bigr ).$ First, fix $r > 0$ .", "For any $\\phi , \\theta \\in {\\mathbb {S}}^1, t\\in {\\mathbb {R}}$ we have $|u(t, r,\\theta ) - u(t, r, \\phi )| = |u(t,r, \\theta ) - \\mathrm {\\Phi }_{\\phi -\\theta }(u(t, r, \\theta ))| < \\Vert \\mathrm {Id}-\\mathrm {\\Phi }_{\\phi -\\theta }\\Vert _{C^\\infty (\\mathcal {N})} < C|\\phi -\\theta |.$ Note, a similar argument shows that for $r,s, t > 0$ $|u(t,r,\\theta ) - u(t,s,\\theta )|$ is independent of $\\theta $ .", "Fix $\\theta \\in {\\mathbb {S}}^1$ , $s > r> r_0 > 0$ .", "Then, by Hölder's inequality, $|u(t,r,\\theta ) - u(t,s,\\theta )|^2 &= \\frac{1}{2\\pi }\\int _{0}^{2\\pi }\\left( \\int _{r}^s \\partial _r u(t, r^{\\prime },\\theta ) dr^{\\prime } \\right)^2d\\theta \\\\&\\le \\left(\\frac{1}{2\\pi }\\int _0^{2\\pi } \\int _{r}^s |\\partial _r u(t, r^{\\prime },\\theta )|^2 r^{\\prime } dr^{\\prime }d\\theta \\right) \\frac{|r-s|}{r_0} .$ Putting (REF ) and (REF ) together, the conclusion follows.", "For $0 < T < \\infty $ and define, for any $A \\ge 0$ $\\mathrm {Flux}(u, T, A) := \\int _{r = T-A} |\\nabla _x u(r, \\theta , T) + \\nabla _t u(r,\\theta ,T)|_g^2 d\\sigma (\\theta ).$ As suggested by its name, the flux measures the energy entering the (translated inwards by $A$ ) light cone at time $T$ .", "We can see this in the following energy identity: $\\int _{|x| < T_1-A} |\\nabla _{x,t}u(x,T_1)|_g^2 dx - \\int _{|x| < T_2-A}|\\nabla _{x,t} u(x,T_2)|_g^2 dx = \\int _{T_2}^{T_1} \\mathrm {Flux}(u, t, A)dt.$ It follows from (REF ) that $T\\mapsto \\int _{|x| < T-A} |\\nabla _{x,t}u(x,T)|_g^2 dx$ is monotone increasing for any $A \\ge 0$ .", "Since the integral is monotone increasing in $T$ and bounded by $E(u_0, u_1)$ , it follows that $\\lim _{T\\rightarrow \\infty } \\int _{r < T-A} |\\nabla _{x,t}u(x,T)|_g^2 dx$ exists and, therefore, $\\lim _{T\\uparrow \\infty } \\mathrm {Flux}(u,T, A) = 0.$ Our next proposition shows that the energy in a linear neighborhood of the boundary of the light cone still goes to zero in infinite time; this observation was first made in the setting of radial wave maps, c.f.", "[7], see also Proposition 2.1 in [4] and Lemma 4.1 in [40].", "Proposition 4.4 Let $u$ be finite energy quasi-equivariant wave map, with $T_{\\max } = +\\infty $ .", "Then for any $\\lambda \\in (0,1)$ we have $\\int _{\\lambda T < |x| < T-A} |\\nabla _x u(x,T)|_g^2 + |\\nabla _t u(x,T)|^2_g dx \\rightarrow 0 \\qquad \\textup {as T, A \\rightarrow \\infty for A \\le (1-\\lambda )T}.$ We follow the notation from [40], see also [33].", "Let $e := \\frac{1}{2}\\left(|\\nabla _x u|_g^2 + |\\nabla _t u|_g^2\\right), \\:\\: m := \\left\\langle \\nabla _x u, \\nabla _t u\\right\\rangle , \\:\\: L := \\frac{1}{2}\\left(|\\nabla _x u|_g^2 - |\\nabla _t u|_g^2\\right).$ From these we have the algebraic relations, $\\begin{aligned} \\partial _t(re) - \\partial _r(rm) =& 0\\\\\\partial _t(rm) - \\partial _r(re) =& L.\\end{aligned}$ Note that these identities follow from the wave map equation (REF ) and Proposition REF , which, in this context, implies that $\\partial _re = \\partial _xe$ and similarly with $m$ .", "It will be convenient to reparameterize $(t,r)$ space by the coordinates $\\xi = t-r$ and $\\eta = t+r$ .", "We also introduce the quantities $\\mathcal {A}^2 = r(e+m)$ and $\\mathcal {B}^2 = r(e-m)$ .", "Then $\\partial _\\xi \\mathcal {A}^2 = L = -\\partial _\\eta \\mathcal {B}^2$ .", "From this equation and the algebraic observation that $8r^2(e^2 -m^2) \\ge L^2$ , we see that $\\begin{aligned}|\\partial _\\xi \\mathcal {A}^2| \\le & \\frac{C}{r}\\mathcal {B}\\\\|\\partial _\\eta \\mathcal {B}^2|\\le & \\frac{C}{r}\\mathcal {A}.\\end{aligned}$ Let $Q$ denote the quadrilateral in $(\\xi , \\eta )$ space with vertices $((1-\\lambda )T,(1+\\lambda )T), \\quad (A, 2T-A), \\quad (A, 2s-(1-\\lambda )T), \\quad ((1-\\lambda )T, 2s-(1-\\lambda )T)$ where $s \\gg T > 0$ , see Figure REF (the ordering above is the order of the vertices in the Figure, starting from the lower left and moving counter-clockwise).", "Figure: The quadrilateral QQBy (REF ) the vector $(re, -rm)$ is divergence free, so we can conclude $\\begin{aligned} 0=&\\int _{\\partial Q} (re, -rm)\\cdot \\hat{n} \\\\=& \\underbrace{-\\int _{\\lambda T}^{T-A} erdr}_{I} - \\underbrace{\\int _{2T-A}^{2s-(1-\\lambda )T} \\mathcal {A}^2(A, \\eta ^{\\prime })d\\eta ^{\\prime }}_{II}\\\\+& \\underbrace{\\int _A^{(1-\\lambda )T} \\mathcal {B}^2(\\xi ^{\\prime },2s- (1-\\lambda )T)d\\xi ^{\\prime }}_{III} - \\underbrace{\\int _{(1+\\lambda )T}^{2s-(1-\\lambda )T} \\mathcal {A}^2((1-\\lambda )T, \\eta ^{\\prime })d\\eta ^{\\prime }}_{IV}.\\end{aligned}$ The terms $I-IV$ correspond to integrating along the sides of the quadrilateral $Q$ in Figure REF , beginning at $I$ and moving counter-clockwise.", "In (REF ), integral $I$ is exactly the one we want to show goes to zero as $T, A$ go to infinity, and hence we have $|I| \\le |II| + |III| + |IV|.$ We will handle each term on the right separately.", "Note that $II \\le \\int _{T}^\\infty \\mathrm {Flux}(u, t, A)dt,$ and hence letting $T\\uparrow \\infty $ , (REF ) implies that $II$ goes to zero.", "For the next terms, we introduce some notation: let $\\mathcal {E}_{\\lambda }(\\xi ) = \\int _{\\frac{1+\\lambda }{1-\\lambda }\\xi }^\\infty \\mathcal {A}^2(\\xi , \\eta ^{\\prime })d\\eta ^{\\prime }.$ Figure: The shaded region is created by taking a lightcone with vertex (0,a)(0, a), removing the lightcone with vertex (0,b)(0,b) and intersecting that region with the backwards lightcone with vertex (0,η 0 )(0,\\eta _0)Now we handle term III.", "Integrating between two light cones, i.e.", "the shaded region in Figure REF , we see that $\\int _{a}^b \\mathcal {B}^2(\\xi ^{\\prime }, \\eta _0)d\\xi ^{\\prime } = \\int _a^{\\eta _0} \\mathcal {A}^2(a, \\eta ^{\\prime })d\\eta ^{\\prime } - \\int _{b}^{\\eta _0} \\mathcal {A}^2(b, \\eta ^{\\prime })d\\eta ^{\\prime }.$ Define $\\mathcal {F}(a,b) =\\lim _{\\eta \\rightarrow \\infty }\\int _a^b \\mathcal {B}^2(\\xi ^{\\prime }, \\eta )d\\xi ^{\\prime },$ and note that the limit in this definition exists since $\\mathcal {F}(a,b) = \\mathcal {E}_0(a) - \\mathcal {E}_0(b).$ The identity (REF ) also implies that $\\mathcal {E}_0(-)$ is a decreasing function since $\\mathcal {F}(a,b) \\ge 0$ .", "Therefore, since $\\mathcal {F}(a,b) \\le \\mathcal {E}_0(a) < \\mathcal {E}(u_0, u_1)$ , we can define $\\mathcal {F}(a) \\equiv \\lim _{b\\rightarrow \\infty } \\mathcal {F}(a,b) .$ Now, noting that $III \\le \\mathcal {F}(A),$ we need to establish the limit of the right-hand side is zero.", "But since we know that $\\mathcal {E}_0(-)$ is decreasing and non-negative, $\\lim _{b\\rightarrow \\infty } \\mathcal {E}_0(b)$ exists, and, consequently, $\\lim _{a\\rightarrow \\infty } \\mathcal {F}(a) =0$ , which concludes the proof for term $III$ .", "Finally, it is clear that $IV \\le \\mathcal {E}_\\lambda ((1-\\lambda )T)$ , and hence we are done if we can show that $\\lim _{T\\uparrow \\infty } \\mathcal {E}_{\\lambda }((1-\\lambda )T) = 0.$ Here we argue exactly as in [7].", "While that lemma is written in the setting of radially symmetric wave maps (with arbitrary target), we note that all the relevant quantities in our setting are radial by Proposition REF .", "Therefore, (REF ) holds and the proof is complete.", "We can then quickly conclude that the kinetic energy of a quasi-equivariant wave map vanishes inside of the light cone (this corresponds to Corollary 2.2 in [4]): Corollary 4.5 Let $u$ be a finite energy quasi-equivariant wave map with $T_{\\max } = \\infty $ .", "Then $ \\lim _{A \\rightarrow \\infty }\\limsup _{T\\rightarrow \\infty } \\frac{1}{T} \\int _A^T \\int _0^{t-A} |\\partial _t u|^2(r,t)rdrdt = 0.$ From (REF ), it follows that $\\partial _t(r^2m) = \\partial _r(r^2e) - r|\\partial _t u|^2_g.$ The proof then follows as in [4].", "Now we narrow our focus to solutions to the system (REF ).", "We aim to prove the following: Proposition 4.6 For wave maps, $u$ , given by (REF ), scattering cannot occur when $\\alpha $ is not of degree zero.", "In particular, if $T_{\\max }(u) = + \\infty $ then there exists $t_n \\uparrow \\infty $ and $\\lambda (t_n) \\ll t_n$ such that $u(t_n, \\lambda (t_n)r)\\rightarrow \\omega $ in $H^1_{\\mathrm {loc}}$ , where $\\omega :{\\mathbb {R}}^{2} \\rightarrow \\mathcal {N}$ is a non-trivial harmonic map.", "Our arguments will follow very closely those in [4], where an analogous proposition is proven for equivariant wave maps into spheres, see [4].", "The $\\alpha $ component of our wave map is not equivariant wave map itself (it satisfies a different equation), however, we will be able to use, essentially unchanged, any arguments in [4] which are purely energy theoretic, since $E(\\alpha ) \\le E(u)$ .", "The first step is to show that the $\\alpha $ component converges to a constant when $r \\ge \\lambda t$ .", "Here we can argue exactly as in [4] with our Proposition REF and Corollary REF taking the place of [4].", "Corollary 4.7 (Cf.", "[4]) Let $\\lambda > 0$ and $u = (Y,\\alpha )$ be a finite energy wave map which solves the system (REF ).", "Then $\\alpha (\\infty ,t) := \\lim _{r\\rightarrow \\infty } \\alpha (r, t)$ exists and, furthermore, $\\lim _{t\\uparrow \\infty } \\Vert \\alpha (r,t) - \\alpha (\\infty ,t)\\Vert _{L^\\infty (r \\ge \\lambda t)} = 0.$ From Corollary REF we can construct our sequence $\\lambda (t_n)$ : Lemma 4.8 Let $u = (Y,\\alpha )$ be a finite energy wave map which solves (REF ).", "Further assume that $\\alpha $ is not of zero degree.", "For each $t > T_0$ , let $\\lambda (t)$ be such that $ 1 \\le 2\\pi \\int _{0}^{2\\lambda (t)} e(r,t) rdr \\le 2,$ then $\\lambda (t) \\ll t$ .", "We first note that if $\\alpha $ has degree not equal to zero then $E(u) > \\mathcal {E}_{\\mathbb {S}^2} = 2\\pi $ .", "So, by continuity of the energy, there exist $\\lambda (t)$ which satisfy (REF ).", "Imagine that there are $t_n \\uparrow \\infty $ and a $\\lambda > 0$ such that $\\lambda (t_n) > \\lambda t_n$ .", "It would then follow that $\\begin{aligned}\\int _0^{2\\lambda (t_n)} e(r,t_n)rdr \\ge & \\int _0^{\\lambda t_n} e(r,t_n)rdr \\ge \\int _0^{\\lambda t_n} \\left((\\partial _r \\alpha )^2 + \\frac{\\sin ^2(\\alpha )}{2r^2}\\right)rdr\\\\ \\ge & \\int _0^{\\lambda t_n} \\left|\\partial _r \\alpha \\right|\\left|\\sin (\\alpha )\\right| dr \\ge \\int _0^{\\alpha (\\lambda t_n)} \\sin (\\rho ) d\\rho \\stackrel{\\mathrm {Corollary}\\; \\ref {c:constantoutsideofcone}}{\\rightarrow } \\int _0^{\\pi } \\sin (\\rho )d\\rho = 2.\\end{aligned}$ This contradicts the defintion of $\\lambda (t)$ (i.e.", "(REF )).", "As such $\\lambda (t_n) \\ll \\lambda t_n$ for any $t_n \\uparrow \\infty $ and $\\lambda > 0$ .", "We can now finish just as in the proof of [4], we sketch the argument below: Let $\\lambda (t)$ be defined as in Lemma REF and let $A(t)$ be such that $A(t) \\rightarrow \\infty $ as $t\\rightarrow \\infty $ but also $\\lambda (t) \\le A(t) \\ll t$ .", "Arguing from Corollary REF (see [4]) there exists $t_n \\uparrow \\infty $ and $\\lambda _n := \\lambda (t_n), A_n := A(t_n)$ such that $\\lim _{n\\rightarrow \\infty } \\frac{1}{\\lambda _n}\\int _{t_n}^{t_n + \\lambda _n} \\int _0^{t-A_n}|\\partial _t u|^2 rdrdt = 0.$ Rescaling so that $u_n(x,t) = u(\\lambda _n x, \\lambda _n t +t_n)$ gives that $\\int _0^1 \\int _0^{r_n}|\\partial _t u_n|_g^2rdrdt \\rightarrow 0,$ where $r_n = (t_n-A_n)/\\lambda _n \\rightarrow \\infty $ .", "This scaling also preserves the energy.", "Thus the $u_n \\rightharpoonup u_\\infty $ in $\\dot{H}^1_{\\mathrm {loc}}$ .", "Furthermore, $u_\\infty $ is time independent by (REF ) and therefore is a finite energy harmonic map, which can be assumed to be smooth and globally defined by the work of [15] and [41].", "We now want to show that this convergence is strong, which also implies that $u_\\infty $ is non-trivial by (REF ).", "Away from 0, we may assume the sequence converges locally uniformly by local uniform Hölder continuity of Lemma REF .", "Near 0 we use (REF ) to see that $u_n(B_1(0))$ is contained in a coordinate chart in $\\mathcal {N}$ (and the chart can be taken uniformly for $n$ large enough).", "Then one can argue exactly as in [38] (see in particular (3.17) onwards) to conclude that the convergence is strong near 0." ], [ "An Energy Gap for Bubbles arising in the flow", "In this section we study the space of harmonic maps which can possible arise as bubbles in the flow (REF ).", "We begin with the following consequence of Lemma REF .", "Proposition 5.1 Let $u: \\mathbb {R}^{1+2} \\rightarrow \\mathcal {M}$ be a quasi-equivariant wave-map to the smooth manifold $\\mathcal {M}$ .", "If $u^{(n)}$ (defined above) converges in $H^1_{loc}$ to a non-trivial harmonic map, $\\omega : {\\mathbb {S}}^2 \\rightarrow \\mathcal {M}$ then, up to passing to a susbsequence, $u^{(n)}(t, x) \\rightarrow \\omega : \\mathbb {R}^2 \\rightarrow \\mathcal {M},$ in $C^0_{loc}(\\mathbb {R}^2 \\setminus \\lbrace 0\\rbrace ,\\, \\mathcal {M})$ as $n \\rightarrow \\infty $ .", "By Lemma REF , the sequence $u^{(n)} = u(t_n, r_n x)$ is uniformly Hölder continuous on any compact region $K \\subset \\subset \\mathbb {R}^2 \\setminus \\lbrace 0\\rbrace $ .", "By Arzelá-Ascoli, to extract a uniformly convergent subsequence, hence by a diagonal argument and passing to a further subsequence, we can product a subsequence which converges uniformly on any compact subset of $\\mathbb {R}^2 \\setminus \\lbrace 0\\rbrace $ .", "By uniqueness of limits, we have convergence to $\\omega $ .", "From this convergence, we can conclude first that the image of the bubble is topologically connected to the image of the rest of the flow.", "Proposition 5.2 (Connectedness of the flow) Let $u: \\mathbb {R}^{1+2} \\rightarrow \\mathcal {M}$ be a quasi-equivariant wave map into a smooth manifold $\\mathcal {M}$ , such that bubbling (as above) occurs at time $T_{\\max }(u)$ and that the rescaled wave maps converge to the non-trivial harmonic map $\\omega : \\mathbb {S}^2 \\rightarrow \\mathcal {M}$ .", "Then $\\bigcup _{t < T_{max}(u)} u(t, \\mathbb {R}^2) \\cup \\omega (\\mathbb {S}^2)$ is a connected subset of $\\mathcal {M}$ .", "The fact that union of the images of the wave map components $\\mathcal {U} := \\bigcup _{t < T_{max}(u)} u(t, \\mathbb {R}^2)$ and the image of the harmonic map $\\omega (\\mathbb {S}^2)$ are each connected follows from the continuity of the involved maps (in both space and time in the case of the wave maps).", "To see that the union of these two components is connected, we note that by Proposition REF , the set $\\omega (\\mathbb {R}^2 \\setminus \\lbrace 0\\rbrace )$ contains limit points of $\\mathcal {U}$ , which implies the result.", "The following immediate corollary of Propositions REF and REF constrains the form and image of any bubble arising in the flow (REF ): Corollary 5.3 Any harmonic map $\\omega :{\\mathbb {S}}^2\\rightarrow {\\mathcal {N}}$ arising as a non-trivial bubble in the flow (REF ) must be of the form $\\omega (r,\\theta ) = (0, Y(r), \\alpha (r), \\theta ),$ and satisfy $\\omega ({\\mathbb {S}}^2) \\subset \\overline{\\gamma }\\times _f{\\mathbb {S}}^2$ .", "We are now ready to state and prove the main result of this section: a classification of the bubbles arising in the flow (REF ) at low energies.", "Before we do, let us recall from Lemma REF that $\\mathcal {E}_{\\mathbb {S}^2}$ is the smallest possible energy of a non-trivial harmonic map $\\omega : {\\mathbb {S}}^2 \\rightarrow {\\mathbb {S}}^2$ .", "Also recall, from above, that $P_1: {\\mathcal {N}}\\rightarrow {\\mathbb {T}}^2$ is the projection map onto the first coordinate and $P_2: {\\mathcal {N}}\\rightarrow {\\mathbb {S}}^2$ is the projection map onto the second coordinate.", "Lemma 5.4 Let $\\omega : {\\mathbb {S}}^2 \\rightarrow {\\mathcal {N}}$ be a non-trivial harmonic map arising as a bubble in the flow (REF ).", "Let $\\varepsilon _0 = \\min \\lbrace \\bar{\\varepsilon }, \\cot (7\\pi /16)/2\\rbrace > 0$ , where $\\bar{\\varepsilon } > 0$ is the constant given by Theorem REF .", "Then, either $\\mathcal {E}(\\omega ) = \\mathcal {E}_{{\\mathbb {S}}^2}$ , or $\\mathcal {E}(\\omega ) > \\mathcal {E}_{{\\mathbb {S}}^2}+ \\varepsilon _0/2$ .", "In the former case, $P_1\\omega $ is a constant map into the set $\\lbrace w = 0\\rbrace $ and $P_2\\omega $ is a degree one harmonic map from ${\\mathbb {S}}^2 \\rightarrow {\\mathbb {S}}^2$ .", "In this proof we will use $Y, \\alpha $ interchangeably with $P_1\\omega , P_2\\omega $ respectively.", "By Lemma REF , $\\mathrm {Im}\\,P_1\\omega $ is either a subset of $\\gamma $ or of $\\lbrace w = 0\\rbrace \\equiv \\overline{\\gamma }\\,\\setminus \\,\\gamma $ .", "In the latter case, $f$ is constant on $\\lbrace w = 0\\rbrace $ so $P_2\\omega $ is a harmonic map between spheres and $P_1\\omega $ is a harmonic map from $\\mathbb {S}^2 \\rightarrow \\mathbb {S}^1$ .", "The maximum principle implies that $P_1\\omega $ is a constant, so $P_2\\omega $ must be non-trivial, as $\\omega $ is non-trivial.", "Therefore, $P_2\\omega $ is either a degree one harmonic map or $\\mathcal {E}(P_2\\omega ) \\ge 2 \\mathcal {E}_{{\\mathbb {S}}^2}$ , and the conclusion of the lemma holds in both cases.", "Thus, it suffices to rule out the presence of harmonic maps where $\\mathrm {Im}\\, P_1\\omega \\subset \\gamma $ satisfying our energy constraint.", "We first claim that $P_2\\omega $ must be non-trivial.", "Indeed, if $P_2\\omega $ is trivial, then $P_1\\omega $ is a bounded harmonic map from ${\\mathbb {R}}^2$ to $\\gamma (-\\infty , \\infty ) \\cong {\\mathbb {R}}$ .", "Such a map must be trivial which contradicts the non-triviality of $\\omega $ .", "Moreover, since $P_2\\omega $ is non-trivial and equivariant, the maximum principle implies that the image of $P_2\\omega $ cannot be contained in a hemisphere, and thus $\\mathcal {E}(P_2\\omega ) \\ge \\mathcal {E}_{{\\mathbb {S}}^2}$ .", "Let $S = \\lbrace r \\in {\\mathbb {R}}^2\\mid f(0, Y(r)) < 2\\rbrace ,$ and assume that $\\mathcal {E}(\\omega ) < \\mathcal {E}_{{\\mathbb {S}}^2} + \\varepsilon _0/2$ .", "Since $\\int _{\\mathbb {R}^2\\backslash S}f(0,Y) e(\\alpha )rdr > 2\\int _{\\mathbb {R}^2\\backslash S} e(\\alpha )rdr,$ it must be the case that $\\int _{S} e(\\alpha )rdr > \\mathcal {E}_{\\mathbb {S}^2} - \\varepsilon _0/2,$ and therefore $\\int _{{\\mathbb {R}}^2\\backslash S} |D\\omega |_g^2 < \\varepsilon _0.$ Thus the hypothesis of Theorem REF are fulfilled inside of any ball $B \\subset {\\mathbb {R}}^2 \\backslash S$ .", "$P_1 \\omega = Y$ is radial, so the oscillation of $Y$ inside any component of ${\\mathbb {R}}^2\\backslash S$ is bounded by $2\\varepsilon _0$ .", "Note that $M > 2$ so $f(0,Y(r)) < 2$ implies that $|Y| > \\cot (7\\pi /16)$ (see (REF ), (REF )).", "As $2\\varepsilon _0 < \\cot (7\\pi /16)$ , we conclude that $Y$ does not change sign on $\\mathbb {R}^2$ .", "Composing with the diffeomorphism $\\Psi _s: (x,y,\\alpha ,\\theta ) \\mapsto (x, y+s, \\alpha ,\\theta )$ we see that $\\frac{d}{ds}\\mathcal {E}(\\Psi _s\\circ \\omega )|_{s=0} = \\int _{{\\mathbb {R}}^2} \\partial _yf(0,Y) e(\\alpha )rdr \\ne 0,$ where the last “non-equality\" follows from (REF ) and the fact that $Y$ does not change sign on ${\\mathbb {R}}^2$ (and, consequently, $\\partial _y f(0,Y)$ has a sign).", "That the deformation in (REF ) changes the energy to first order contradicts the fact that $\\omega $ is harmonic.", "Thus there are no harmonic maps $\\omega $ with $P_1 \\omega \\subset \\gamma $ satisfying our energy constraint, and our proof is complete.", "Remark 5.5 It is not hard to show that $\\mathcal {E}_{\\mathbb {S}^2}$ is exactly the energy of the lowest energy non-trivial quasi-equivariant harmonic map, $\\omega :\\mathbb {S}^2\\rightarrow \\mathcal {N}$ , what we call $\\mathcal {E}_{\\mathrm {quasi}}(\\mathcal {N})$ in Theorem REF ." ], [ "Properties of the singularity", "We now turn to the proof of our main theorem, which we recall here: Theorem (Main theorem) There exists a compact smooth Riemannian manifold $(\\mathcal {N}, g)$ given by $\\mathcal {N} = \\mathbb {T}^2 \\times _f \\mathcal {S}^2$ for a certain $C^\\infty ({\\mathbb {T}}^2)$ warping function, $f$ , and $C^\\infty $ -smooth, finite energy, quasi-equivariant initial data $(u_0, u_1)$ , which satisfy $\\Vert (u_0, u_1) \\Vert _{\\dot{H}^3 \\times \\dot{H}^2} < \\infty , \\qquad \\mathcal {E}(u_0, u_1) < \\mathcal {E}_{\\mathrm {quasi}}(\\mathcal {N}) + \\varepsilon _1,$ such that the corresponding solution $(u, u_t)$ to (REF ) has a bubbling singularity as $t \\rightarrow T_{\\max }$ which fails to be unique in the sense of Definition REF .", "Above, $\\varepsilon _1 >0$ is a constant which depends only on $\\mathcal {N}$ , and $\\mathcal {E}_{\\mathrm {quasi}}(\\mathcal {N})$ denotes the smallest energy of a non-trivial quasi-equivariant harmonic map, $\\omega : \\mathbb {S}^2 \\rightarrow \\mathcal {N}$ .", "Before proceeding with the proof of the main result we summarize what we know so far about solutions to (REF ), or more precisely to solutions of (REF ).", "In light of Sterbenz and Tataru's dichotomy, Theorem REF , at every finite time $t_0$ , the solution either concentrates energy at a point and bubbles off (a Lorentz transform of) a finite energy harmonic map or it can be continued smoothly past $t_0$ .", "If the solution exists as $t\\rightarrow \\infty $ then it either scatters in the limit or bubbles off (a Lorentz transform of) a finite energy harmonic map.", "If the “spherical part\" of the initial data has degree one, Proposition REF implies that blow-up in the form of bubbling off a harmonic map must occur (either in finite time or as $t\\rightarrow \\infty $ ).", "Now note that quasi-equivariance implies that there is no Lorentz transform symmetry in the solutions, and hence the convergence of the bubbling sequence (as described in part (a) of Theorem REF ) must be to an entire, non-constant harmonic map.", "Moreover, singularity formation at any point in the domain requires the bubbling of at least as much energy as the lowest energy non-trivial harmonic map.", "Hence by additivity of the energy, bubbling can occur only at finitely many points, and so in the quasi-equivariant setting the only possible blow-up point in the domain is the origin $x = 0$ .", "Finally, if the initial conditions (REF ) have energy less than $\\mathcal {E}_{\\mathbb {S}^2} + \\varepsilon _0/2$ , then it must be the case that the bubble also has energy less than this.", "By Lemma REF , this implies that the harmonic map, $\\omega $ , decomposes into a constant map into $\\lbrace w=0\\rbrace \\subset {\\mathbb {T}}^2$ and a degree one equivariant harmonic map between spheres.", "Summarizing the previous discussion, we have, thus far, established that: Theorem 6.1 Consider the wave map equation (REF ) with smooth, quasi-equivariant symmetric, finite energy initial data of the form (REF ).", "Further assume that $\\alpha _0$ (the “spherical part\" of the initial conditions) is a degree one map between 2-spheres.", "Then there exists a unique solution to (REF ) on the time interval $[0, T_{\\max })$ , and a sequence of times $t_n\\uparrow T_{\\max }$ and scales $r_n$ with $r_n = {\\left\\lbrace \\begin{array}{ll} o(T_{max} - t_n) & T_{max} < \\infty \\\\o(t_n) & T_{max} = \\infty \\end{array}\\right.", "}$ so that the rescaled sequence of maps $u^{(n)}(t, x) = u(t_n, r_n x)$ converges strongly in $H^1_{loc}$ to an entire Harmonic map $\\omega : \\mathbb {R}^2 \\rightarrow \\mathcal {N}$ of nontrivial energy.", "Furthermore, if $\\mathcal {E}(u_0, u_1) < \\mathcal {E}_{\\mathbb {S}^2} + \\varepsilon _0/2,$ then $\\omega = (P_1\\omega , P_2\\omega )$ satisfies that $P_1\\omega = (0, z_0)$ for some $z_0 \\in [0,1]$ and $P_2\\omega $ is a degree one equivariant harmonic maps between 2-spheres.", "We note that it is easy to produce initial conditions which satisfy the hypothesis of Theorem REF .", "Using the notation of (REF ), one such example is letting $(\\alpha _0(r), \\theta )$ be a degree one equivariant harmonic map, $\\alpha _1(r) \\equiv 0 \\equiv Y_1(r)$ and $Y_0(r) \\equiv c$ where $c$ is a constant large enough such that $f(0,c) < 1 + \\varepsilon _0/(2\\mathcal {E}_{\\mathbb {S}^2})$ ." ], [ "Winding Singularities", "In this subsection we prove that the singularity guaranteed by Theorem REF is winding in the sense of [48], and in fact enjoys a stronger form of winding which we introduce below, which we call “strongly winding”.", "We will then prove that a strongly winding singularity implies non-uniqueness in the sense of Definition REF .", "Our stronger notion of winding requires winding along all sequences, as opposed to just one.", "We find this definition to be slightly easier to work with, and, in the situation where only one bubble develops at a given point and time, equivalent to the standard definition of winding.", "Before we introduce the following stronger notion of winding recall that if $\\mathcal {M}$ is a Riemannian manifold, then $\\widehat{\\mathcal {M}}$ its its universal cover and, given a function $u : \\mathbb {R}^{1+d} \\rightarrow \\mathcal {M}$ , we let $\\hat{u}: \\mathbb {R}^{1+d} \\rightarrow \\widehat{\\mathcal {M}}$ be its lift to the universal cover.", "Definition 6.2 (Strongly winding singularity) A quasi-equivariant wave map $u: \\mathbb {R}^{1+2} \\rightarrow \\mathcal {M}$ has a strongly winding singularity at time $T_{\\max }$ and the origin $ 0 \\in \\mathbb {R}^2$ if for any sequences $\\lbrace r_n\\rbrace $ , and $\\lbrace t_n\\rbrace $ , satisfying $t_n \\uparrow T_{\\max },\\qquad r_n = {\\left\\lbrace \\begin{array}{ll} o(T_{max} - t_n) & T_{max} < \\infty \\\\o(t_n) & T_{max} = \\infty \\end{array}\\right.", "}$ such that $u(t_n, r_nx) \\rightarrow \\omega (x)$ in $C^0_{\\mathrm {loc}}({\\mathbb {R}}^2\\backslash \\lbrace 0\\rbrace ; \\mathcal {M})$ , where $\\omega $ is a non-constant harmonic map, the lifts $\\hat{u}(t_n,r_nx)$ have no convergent subsequence in $C^0_{\\mathrm {loc}}({\\mathbb {R}}^2\\backslash \\lbrace 0\\rbrace ; \\widehat{\\mathcal {M}})$ .", "For a solution $u$ with winding singularity, we say that $u$ has a winding singularity with respect to sequences $\\lbrace t_n\\rbrace $ and $\\lbrace r_n\\rbrace $, when $\\lbrace t_n\\rbrace $ and $\\lbrace r_n\\rbrace $ are the sequences guaranteed by Definition REF .", "The following proposition provides an equivalent definition of (strongly) winding.", "Proposition 6.3 If $u:\\mathbb {R}^{1+d} \\rightarrow \\mathcal {M}$ has a singularity at $(T_{\\max }, 0)$ , then the singularity is winding with respect to sequences $\\lbrace t_n\\rbrace $ and $\\lbrace r_n\\rbrace $ if and only if there exists a compact $K \\subset \\subset {\\mathbb {R}}^d \\backslash \\lbrace 0\\rbrace $ such that every subsequence $\\lbrace \\hat{u}(t_{n_j}, r_{n_j}x)\\big |_K\\rbrace $ is unbounded.", "This proposition follows readily from the equicontinuity of the lifts and compactness, but we include the details for completeness.", "Suppose first that the singularity is winding in the sense of Definition REF .", "Since the sequence $\\lbrace u^{(n)}\\rbrace $ is equicontinuous on every compact set, the family given by the lifts $\\hat{u}^{(n)}:= \\hat{u}(t_n, r_nx)$ is still equicontinuous on every compact set.", "Indeed, fix a compact set $K \\subset \\subset {\\mathbb {R}}^d \\backslash \\lbrace 0\\rbrace $ , $x \\in K$ and $\\varepsilon > 0$ , and let $\\delta > 0$ be as in the definition of equicontinuity of the family $\\lbrace u^{(n)}\\rbrace $ .", "Making $\\delta $ smaller if necessary, we can restrict to a sufficiently small neighborhood so that the covering map trivializes.", "This yields the equicontinuity of the lifts.", "Hence, the only way the lifted sequence can fail to be precompact is if there exists a compact set on which the family is unbounded.", "Conversely, if there exists such a compact subset $K$ , then unboundedness implies that there is no convergent subsequence, and hence the singularity is winding.", "Remark 6.4 The previous proposition adapts readily to strongly winding singularities.", "We are now prepared to establish the following main proposition about the nature of the singularity.", "Proposition 6.5 The flow described (REF ) with initial conditions as in Theorem REF has a strongly winding singularity at $(T_{max}, 0)$ .", "We know the flow needs to converge to a bubble as described in Lemma REF .", "Let $\\gamma $ be the geodesic described in Section .", "By the path lifting property, $\\gamma $ may be lifted to a unique path $\\hat{\\gamma }$ in the universal cover, which is necessarily an unbounded curve since $\\gamma $ is not null-homotopic in ${\\mathcal {N}}$ and $\\pi _1({\\mathcal {N}}) = {\\mathbb {Z}}^2$ .", "Let now $\\lbrace t_n\\rbrace $ and $\\lbrace r_n\\rbrace $ be any sequences along which $u^{(n)}(t, x) = u(t_n, r_n x)$ converge to $\\omega $ , see Theorem REF .", "Lemma REF tells us that $P_1\\omega \\subset \\lbrace w=0\\rbrace $ which implies that for any $t_0 \\gg 1$ and any compact set $K$ , there exists an $N = N(t_0, K)$ such that if $n \\ge N$ , then $u^{n}(K) \\subset \\gamma ((-\\infty , -t_0] \\cup [t_0, \\infty ))$ .", "Lifting to the universal cover, the unboundedness of $\\hat{\\gamma }$ implies that the sequence $\\hat{u}^{(n)}|_{K}:= \\hat{u}(t_n, r_nx)|_{K}$ must be unbounded, and hence by Proposition REF , the singularity is winding.", "Our final result shows that a strongly winding singularity cannot give rise to a unique bubble in the sense of Definition REF .", "Theorem 6.6 If $u$ has a strongly winding singularity at $x_0$ and time $T_{\\max }$ , then it does not have a unique bubble at that point and time.", "Suppose for contradiction that $u$ has a winding singularity at the origin at time $T_{max}$ , and a unique bubble $\\omega $ to which it converges under rescaling by $r(t)$ .", "and translation by $x(t)$ .", "We will show that for every compact $K \\subset \\subset {\\mathbb {R}}^d \\backslash \\lbrace 0\\rbrace $ , there exists a subsequence $\\lbrace \\hat{u}(t_{n_j}, r_{n_j}x)\\big |_K\\rbrace $ which is bounded, contradicting Proposition REF .", "Fix a compact set $K \\subset \\subset {\\mathbb {R}}^d \\backslash \\lbrace 0\\rbrace $ and note that $\\widehat{\\omega }(K)$ is compact.", "Define the $\\varepsilon $ -neighbourhood of a set $A$ by $N_\\varepsilon (A) := \\lbrace x\\mid \\mathrm {dist}(x,A) < \\varepsilon \\rbrace .$ Fix $\\varepsilon = 1$ , since $u(t, r(t)- ) \\rightarrow \\omega $ in $C^0_{\\mathrm {loc}}$ , there exists $t_0$ , such that for all $t > t_0$ we have $u(t, r(t)K) \\subset N_1(\\omega (K)).$ But we then have that $\\lbrace \\hat{u}(t_n, r(t_n)K)\\rbrace _{\\lbrace n\\mid t_n > t_0\\rbrace } \\subset N_1(\\widehat{\\omega }(K))$ , and, in particular, the sequence is bounded, which contradicts Proposition REF , and concludes the proof." ] ]

[ [ "Codebook-Based Beam Tracking for Conformal ArrayEnabled UAV MmWave\n Networks" ], [ "Abstract Millimeter wave (mmWave) communications can potentially meet the high data-rate requirements of unmanned aerial vehicle (UAV) networks.", "However, as the prerequisite of mmWave communications, the narrow directional beam tracking is very challenging because of the three-dimensional (3D) mobility and attitude variation of UAVs.", "Aiming to address the beam tracking difficulties, we propose to integrate the conformal array (CA) with the surface of each UAV, which enables the full spatial coverage and the agile beam tracking in highly dynamic UAV mmWave networks.", "More specifically, the key contributions of our work are three-fold.", "1) A new mmWave beam tracking framework is established for the CA-enabled UAV mmWave network.", "2) A specialized hierarchical codebook is constructed to drive the directional radiating element (DRE)-covered cylindrical conformal array (CCA), which contains both the angular beam pattern and the subarray pattern to fully utilize the potential of the CA.", "3) A codebook-based multiuser beam tracking scheme is proposed, where the Gaussian process machine learning enabled UAV position/attitude predication is developed to improve the beam tracking efficiency in conjunction with the tracking-error aware adaptive beamwidth control.", "Simulation results validate the effectiveness of the proposed codebook-based beam tracking scheme in the CA-enabled UAV mmWave network, and demonstrate the advantages of CA over the conventional planner array in terms of spectrum efficiency and outage probability in the highly dynamic scenarios." ], [ "Introduction", "Unmanned aerial vehicles (UAVs) have many critical applications in civilian and military areas [1], [2], such as fire fighting, plant protection, remote monitoring, etc.", "Therein, a UAV acts as an Internet of Things (IoT) node or a peripheral node to assist IoT.", "For instance, it is very attractive to deploy UAVs to assist IoT with satisfying the requirements of high-rate and low-latency data transmission [3], [4].", "Moreover, UAVs equipped with IoT sensors can be regarded as part of IoT to perform various missions flexibly [5].", "In many mission-driven scenarios, e.g., gathering and relaying IoT data, cooperative radio transmission, environment sensing, traffic flow monitoring, etc., multiple UAVs are often teamed to collaboratively accomplish the designated missions, in which real-time sensing/monitoring information or high-definition video transmission are usually necessitated [6].", "Therefore, the high-rate data transmission among UAVs is of great importance to the development of IoT with large-scale video data or high-definition image data transmission requirements.", "In addition, UAVs can also be used as flying base-stations or mobile relay backhaul nodes to provide on-the-fly high-capacity communication links for the emergency coverage of IoT devices [7], [8].", "In such mission-driven UAV networks, high-data-rate inter-UAV communications play a pivotal role.", "MmWave band has abundant spectrum resource, and is considered as a potential avenue to support high-throughput data transmission for UAV networks [9], [10], [7].", "If the Line-of-Sight (LoS) propagation is available, mmWave communication can achieve kilometer-level communication range and gigabits-persecond data rate [11], which can support UAV networks in many scenarios [6], [7], [9].", "However, there are critical challenges to achieve reliable mmWave communications for UAV networks.", "Specifically, a UAV maintains three-dimensional or full-spatial mobility with very high dynamic, and thus the angle-of-arrival (AOA) of communication signal always varies over time in all directions.", "To this end, a powerful antenna array is important to offer full-spatial coverage capability and facilitate the mmWave link maintenance for UAV networks.", "The uniform linear array (ULA) and uniform planar array (UPA) are widely adopted in the existing studies on mmWave communication and networking [12], [13], [14], [15].", "However, their coverage capabilities are often confined within a two-dimensional space and a half three-dimensional space.", "Therefore, conventional ULA and UPA can only attain a limited coverage within a fraction of the full space, causing high communication outage probability in highly dynamic UAV mmWave networks.", "When considering UAV communications with UPA or ULA, a UAV is typically modeled as a point in space without considering its size and shape.", "Actually, the size and shape can be utilized to support more powerful and effective antenna array.", "Inspired by this basic consideration, the conformal array (CA) [16] is introduced to UAV communications.", "A CA is usually in a shape of cylindrical or spherical conforming to a predefined surface, e.g., a part of an airplane or UAV, and can reap full spatial coverage with proper array designs.", "Compared with surface-mounted multiple UPAs, a CA, conforming to the surface of a UAV, can compact the UAV design, reduce the extra drag and fuel consumption, and also facilitate an array of a larger size [16].", "Furthermore, directional radiating elements (DREs) are commonly integrated with antenna array to enhance the beamforming ability [16], [17], [18].", "In such a case, the coverage capability of CA is far stronger than that of UPA and ULA via proper array designs, due to the exploitation of size and shape.", "Specifically, a CA can enable the potential to enlarge (roll up) the surface of antenna array.", "This advantage not only achieves a larger array gain to combat path-loss but also sustains full-spatial transmitting/receiving to facilitate fast beam tracking for mobile UAV mmWave networks [19].", "Note that in mission-driven UAV networks, agile and robust beam tracking is very challenging yet critical for inter-UAV mmWave communications [10], because UAV position and attitude may vary very fast.", "By carefully exploiting the CA's full spatial transmission/reception property, the stringent constraints on beam tracking for highly dynamic moving UAVs can be relieved considerably.", "So far, however, the CA-enabled UAV mmWave network is almost untouched in the literature.", "Regarding the mmWave CA, there are only a few recent works on the radiation patterns and beam scanning characteristics [20] and the performance evaluation of CA-based beamforming for static mmWave cellular networks [21].", "These works validate the potential advantage of CA in the static mmWave networks, which are not applicable to mobile UAV mmWave networks.", "For both static and mobile mmWave networks, codebook design is of vital importance to empower the feasible beam tracking and drive the mmWave antenna array for reliable communications [22], [23].", "Recently, ULA/UPA-oriented codebook designs have been proposed for mmWave networks, which include the codebook-based beam tracking and channel estimation methods.", "For example, considering the ULA with omnidirectional radiating elements (REs), the hierarchical-codebook-based subarray and antenna deactivating strategies are proposed to achieve efficient beam training for single-user scenarios [12], [24].", "The multiuser downlink beam training algorithms regarding the ULA are proposed with the multi-resolution codebook designs for partially-connected [25] and fully-connected [15] hybrid structures, respectively.", "However, extending the aforementioned works to the CA is not straightforward.", "The reasons are as follows: When the commonly-adopted DRE is integrated with CA, the limited radiation range of DREs is no longer the same and each is affected by the DRE's location on CA, as the DRE-covered array plane is rolled up.", "The determined radiation direction of CA is only within a part of DREs' radiation range.", "This observation indicates that only a part of the DREs or some specific subarrays need to be activated with reference to the AOA or angle of departure (AOD) of transceivers.", "Therefore, the dynamic subarray localization and activation are very coupled and critical for the efficient utilization of the DRE-covered CA.", "Note that conventional ULA/UPA-oriented codebook designs mainly focus on the beam direction/width controlling via the random-like subarray activation/deactivation without specific subarray localization.", "In contrast, the codebook design for DRE-covered CA should emphasize the location of the activated subarray to achieve the promise of full-spatial coverage of the CA in UAV networks.", "Nevertheless, such work is still missing now in the literature.", "These points mentioned above motivate us to study a new beam tracking framework with the well-tailored codebook for CA-enabled UAV mmWave networks.", "In this paper, we consider a dynamic mission-driven UAV network with UAV-to-UAV mmWave communications, wherein multiple transmitting UAVs (t-UAVs) simultaneously transmit to a receiving UAV (r-UAV).", "In such a scenario, we focus on inter-UAV communications in UAV networks, and the UAV-to-ground communications are not involved.", "In particular, each UAV is equipped with a cylindrical conformal array (CCA), and a novel-codebook-based mmWave beam tracking scheme is proposed for such a highly dynamic UAV network.", "More specifically, the codebook consists of the codewords corresponding to various subarray patterns and beam patterns.", "Based on the joint UAV position-attitude prediction, an efficient codeword selection scheme is further developed with tracking error (TE) awareness, which achieves fast subarray activation/partition and array weighting vector selection.", "It is verified that our proposed scheme achieves a higher spectrum efficiency, lower outage probability and stronger robustness for inter-UAV mmWave communications.", "In summary, the key contributions of this paper are listed as follows.", "The first study on the beam tracking framework for CA-enabled UAV mmWave networks.", "We propose an overall beam tracking framework to exemplify the idea of the DRE-covered CCA integrated with UAVs, and reveal that CA can offer full-spatial coverage and facilitate beam tracking, thus enabling high-throughput inter-UAV data transmission for mission-driven UAV networking.", "To the best of our knowledge, this is the first work on the beam tracking framework for CA-enabled UAV mmWave networks.", "The specialized codebook design of the DRE-covered CCA for multi-UAV mobile mmWave communications.", "Under the guidance of the proposed framework, a novel hierarchical codebook is designed to encompass both the subarray patterns and beam patterns.", "The newly proposed CA codebook can fully exploit the potentials of the DRE-covered CCA to offer full spatial coverage.", "Moreover, the corresponding codeword selection scheme is also carefully designed to facilitate fast multi-UAV beam tracking/communication in the considered CA-enabled UAV mmWave network.", "The CCA codebook-based multi-UAV beam tracking scheme with TE awareness.", "Based on the designed codebook, by exploiting the Gaussian process (GP) tool, both the position and attitude of UAVs can be fast tracked for fast multiuser beam tracking along with dynamic TE estimation.", "Moreover, the estimated TE is leveraged to direct the selection of a proper codeword, which inspires an optimized subarray activation and beamwidth control towards the better performance with TE awareness.", "Note that there exist some mobile mmWave beam tracking schemes exploiting the position or motion state information (MSI) based on conventional ULA/UPA recently.", "For example, the beam tracking is achieved by directly predicting the AOD/AOA through the improved Kalman filtering [26], however, the work of [26] only targets at low-mobility scenarios.", "For vehicle networks, the position-assisted beam tracking methods are proposed by [27] and [28].", "Nevertheless, the impact of the attitude changes of vehicles on the beam tracking is not involved.", "The research work on the beam tracking for UAVs with mmWave communications is still rare.", "The authors in [29] and [30] consider the UAV-to-ground and UAV-to-satellite mmWave communications, respectively, with fast beam tracking by estimating the position and attitude of UAVs.", "However, previous schemes cannot be readily extended to UAV-to-UAV mmWave communications, where both transmitter and receiver are fast moving with quick attitude variations.", "Recently, we propose a position-attitude-prediction-based beam tracking scheme for the UAV-to-UAV mmWave communication with conventional UPA [31], which is an initial attempt to address the beam tracking challenge in UAV mmWave networks by adopting the GP tool.", "In a nutshell, all the aforementioned work [26], [27], [28], [29], [30], [31] is based on conventional ULA/UPA, and there is no existing work on the beam tracking solution for CA-enabled UAV mmWave networks.", "Moreover, the aforementioned work [26], [27], [28], [29], [30], [31] does not consider the TE-aware robust design, which might be much beneficial for highly dynamic UAV networks.", "Figure: NO_CAPTIONThe rest of this paper is as follows.", "In Section , the system model is introduced.", "In Section , the CCA codebook design and the codebook-based joint subarray partition and AWV selection algorithms are proposed.", "Next, the TE-aware codebook-based beam tracking with 3D beamwidth control is further proposed in Section .", "Simulation results are given in Section , and finally Section  concludes this paper." ], [ "System Model", "A CCA-enabled UAV mmWave network is considered in this paper.", "Here, we first establish the DRE-covered CCA model in Section REF .", "Then the system setup of the considered UAV mmWave network is described in Section REF .", "Finally, the beam tracking problem for the CA-enabled UAV mmWave network is modeled in Section REF ." ], [ "Conformal Array Model", "The CCA is the conformal array with a cylindrical shape that can conform some parts of UAV's configuration such as the fuselage.", "A CCA contains $M\\times N$ elements, ${{M}}\\gg 1,{{N}}\\gg 1$ , which are placed along the cylindrical shape $\\Gamma $ with $N$ and $M$ elements on the $xy$ -plane and the $z$ -axis, respectively, as shown in Fig.", "REF .", "The elements of CCA on the $xy$ -plane form a circular array and the elements of CCA on the $z$ -axis form a uniform line array.", "According to [21], [32], the steering vector is given by (REF ), where $\\alpha $ , $\\beta $ and $R_\\text{cyl}$ are the azimuth angle, the elevation angle and the radius of the cylinder, respectively.", "The inter-element distance on the $z$ -axis $d_\\text{cyl}=\\frac{\\lambda _c}{2}$ , where $\\lambda _c$ is the carrier wavelength.", "The inter-element distance on the $xy$ -plane is also set as $d_{xy}=\\frac{\\lambda _c}{2}$ .", "$\\phi _n=\\phi _\\text{c}(n)=\\frac{(2n-1-N)\\Delta \\phi _\\text{c}}{2}$ is the angular position of the $n$ -th element on the $xy$ -plane and $\\Delta \\phi _\\text{c}=\\frac{2\\pi }{N}$ is the corresponding inter-element distance in the angular domain.", "It is assumed that the DREs are modeled as ideal directional elements with the angular domain radiation coverage $\\Delta \\alpha $ and $\\Delta \\beta $  [17].", "More specifically, as shown by Fig.", "REF , each DRE has the azimuth angle coverage, $[{{\\alpha }_{n,\\min }},{{\\alpha }_{n,\\max }}]$ with $\\Delta \\alpha ={{\\alpha }_{n,\\max }}-{{\\alpha }_{n,\\min }}$ .", "The elevation angle covered by a DRE is denoted by $[{{\\beta }_{m,\\min }},{{\\beta }_{m,\\max }}]$ with $\\Delta \\beta ={{\\beta }_{m,\\max }}-{{\\beta }_{m,\\min }}$ .", "In the CCA coordinate frame shown in Fig.", "REF , the azimuth angle coverage of the $n$ -th element on the $xy$ -plane is given by $\\begin{aligned}&{{\\alpha }_{n,\\min }}=\\phi _n-\\frac{\\Delta \\alpha }{2}+2l\\pi , l\\in \\mathbb {Z},\\\\&{{\\alpha }_{n,\\max }}=\\phi _n+\\frac{\\Delta \\alpha }{2}+2l\\pi , l\\in \\mathbb {Z},\\end{aligned}$ and the elevation angle coverage of the $m$ -th element on $z$ -axis is given by $\\begin{aligned}&{{\\beta }_{m,\\min }}=-\\frac{\\Delta \\beta }{2}+2l\\pi , l\\in \\mathbb {Z},\\\\&{{\\beta }_{m,\\max }}=\\frac{\\Delta \\beta }{2}+2l\\pi , l\\in \\mathbb {Z}.\\end{aligned}$ For the CCA-enabled UAV mmWave network, the array size is usually large and the corresponding inter-element distance $\\Delta \\phi $ is small.", "Therefore, it is assumed that $\\Delta \\alpha $ and $\\Delta \\beta $ satisfy $\\Delta \\phi _{\\text{c}}\\le \\Delta \\alpha $ and $\\Delta \\beta =\\pi $ to ensure that the DRE-covered CCA covers the full angular domain.", "The analog precoding architecture adopted for DRE-covered CCA is shown in Fig.", "REF  [13], which tunes the partially-connected precoding architecture by adapting the connection between the RF chains and the antenna elements to the channel variation and forming dynamic subarrays.", "For a fixed time slot, the precoding architecture is the conventional partially-connected precoding.", "For different time slots, the connection changes mainly depend on the variations of the AOA/AOD caused by the movement of UAVs.", "Figure: The DRE-covered CCA in 3D view and top-view.Figure: The analog RF precoder structure with dynamic subarrays." ], [ "System Setup", "A mission-driven UAV network consisting of a leading UAV (LUAV) and several following UAVs (FUAVs) is considered.", "The FUAVs follow the LUAV to perform tasks and transmit data to it, and the LUAV has strong ability to communicate with and forward the aggregated data to the remote ground station.", "Since real-time and high data rate information transmission is usually required by these tasks, mmWave communication is adopted in the UAV network to meet the demands.", "As shown in Fig.", "REF , the considered UAV mmWave network is composed of $K$ t-UAVs (FUAVs) and one r-UAV (LUAV), each t-UAV and the r-UAV are equipped with a CCA with ${{M_\\text{t}}}\\times {{N_\\text{t}}}$ and ${{M_\\text{r}}}\\times {{N_\\text{r}}}$ DREs, respectively.", "It is assumed that only analog beamforming is considered and only one radio frequency (RF) chain is needed for t-UAVs.", "The r-UAV is equipped with $N_\\text{RF}$ RF chains with $N_\\text{RF}>K$ .", "Denote $H_k\\in \\mathbb {C}^{M_{\\text{t}}N_{\\text{t}}\\times {M_{\\text{r}}N_{\\text{r}}}}, k=1,\\ldots ,K,$ as the multiple-input multiple-output (MIMO) channel between the $k$ -th t-UAV and r-UAV.", "For an $M_\\text{t}N_\\text{t}\\times 1$ unit-norm analog transmit beamforming vector, i.e., the transmit antenna weight vector (AWV) $f_k$ , and an $M_\\text{r}N_\\text{r}\\times 1$ unit-norm analog combing vector, i.e., the receive AWV, $w_k$ , the received signal of the $k$ -th t-UAV at r-UAV is expressed as $r_k=\\sqrt{p_k}w_{{k}}^{H}{{H}_{{k}}}{{f}_{{k}}}s_k+w_{{k}}^{H}\\sum _{i\\ne k}{\\sqrt{p_i}H_if_is_i}+w_{{k}}^{H}{{n}_{{}}},$ where $s_k$ is the transmitted signal of the $k$ -th t-UAV with $\\mathbb {E}\\lbrace |s_k|^2\\rbrace =1$ , $p_k$ is the transmit power of the $k$ -th t-UAV, and $n$ is the $M_\\text{r}N_\\text{r}\\times 1$ noise vector, whose elements are independent and identically distributed, and obey $\\mathcal {CN}(0,\\sigma _n^2)$ , where $\\sigma _n^2$ is the noise variance.", "The rich scattering environment rarely appears in the UAV mmWave communication since there are few scatterers in the air.", "At the appropriate altitude, the airspace is relatively open, and the blockage hardly happens.", "Hence, the LoS propagation is dominated in the inter-UAV communication for our considered UAV mmWave networks [6].", "Therefore, the channel matrix $H$ from the t-UAV to r-UAV with DRE-covered CCA is given by [30], [17] $\\begin{aligned}H_k(t)\\!&=\\!\\frac{h_0}{D_k^{-\\gamma }(t)}(\\Lambda _{k}(\\alpha ^{\\text{r}}_{k}(t),\\beta ^{\\text{r}}_{k}(t))\\circ A(\\alpha ^{\\text{r}}_{k}(t),\\beta _{k}^{\\text{r}}(t)))\\\\&(\\Lambda _{k}(\\alpha _{k}^{\\text{t}}(t),\\beta _{k}^{\\text{t}}(t))\\circ A(\\alpha _{k}^{\\text{t}}(t),\\beta _{k}^{\\text{t}}(t)))^H,\\end{aligned}$ where $\\gamma $ denotes the path-loss exponent, $D_k$ denotes the distance between the $k$ -th t-UAV and r-UAV, $h_0$ is the complex channel gain and $\\circ $ represents Hadamard product.", "$A(\\alpha _{k}^{\\text{r}},\\beta _{k}^{\\text{r}})$ and $A(\\alpha _{k}^{\\text{t}},\\beta _{k}^{\\text{t}})$ are the normalized transmitting and receiving array response vectors for the $k$ -th t-UAV at the azimuth (elevation) angles of arrival (AOAs) and departure (AODs) $\\alpha _{k}^{\\text{r}}(\\beta _{k}^{\\text{r}})$ , ${{\\alpha }_{k}^{\\text{t}}}({{\\beta }_{k}^{\\text{t}}})$ , respectively.", "$\\Lambda _{k}(\\alpha _{k}^{\\text{r}},\\beta _{k}^{\\text{r}})$ and $\\Lambda _{k}(\\alpha _{k}^{\\text{t}},\\beta _{k}^{\\text{t}})$ are the antenna element gains of the CCA at the azimuth (elevation) AOAs and AODs, respectively.", "The DRE is considered as the ideal sectored element [17] and hence the element gain is given by ${{\\left[ {{\\Lambda }_{k}}\\left( {{\\alpha }^{\\text{t}}_{k}},{{\\beta }_{k}^{\\text{t}}} \\right) \\right]}_{(m,n)}}=\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}&1,\\ \\ \\ \\forall {{\\alpha }_{k}^{\\text{t}}}\\in [{{\\alpha }_{m,\\min }},{{\\alpha }_{m,\\max }}],\\\\&\\ \\ \\ \\ \\ \\forall {{\\beta }_{k}^{\\text{t}}}\\in [{{\\beta }_{n,\\min }},{{\\beta }_{n,\\max }}], \\\\&0,\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{otherwise.", "}\\end{IEEEeqnarraybox}\\right.$ Figure: The considered CC-enabled UAV mmWave network consists of a r-UAV and multiple t-UAVs.", "UAV position-attitude prediction is performed to obtain the future motion state information (MSI) before next information feedback.", "The CCA and the beam are shown in detail in the CCA view." ], [ "Beam Tracking Problem Formulation with CCA", "For the LOS channel, the AOAs and AODs in (REF ) are mainly determined by the position and attitude of the t-UAVs and r-UAV.", "Given the received signal in (REF ), the signal-to-interference-plus-noise ratio (SINR) When the transmit signals of t-UAVs have similar AOAs at the r-UAV, the combiners may not separate them correctly and the inter-UAV interference exists in the UAV networks.", "Due to the constraint on the minimum distance between UAVs, the inter-UAV interference is not the main factor affecting the sum SE and can be neglected, which will be shown in our simulations.", "of the $k$ -th t-UAV at slot $t$ is given by $\\gamma _k(t)\\!=\\!\\frac{\\left|\\sqrt{p_{k}(t)}w_{k}(t)^HH_{k}(t)f_{k}(t)\\right|^2}{\\sum \\limits _{i\\ne k}\\left|\\sqrt{p_{i}(t)}w_{k}(t)^HH_{i}(t)f_{i}(t)\\right|^2\\!+\\!n_{\\sigma }(t)},$ where $n_{\\sigma }(t)=\\sigma ^2w_{k}(t)^Hw_{k}(t)$ .", "The sum spectral efficiency (SE) of the mmWave UAV network at slot $t$ is given by $R(t)=\\sum _{k\\in \\mathcal {K}}\\log (1+\\gamma _k(t)),$ Aiming to maximize the SE at slot $t$ , the optimal beamforming and combining vector $f_k(t)$ and $w_k(t)$ , which are mainly determined by the AOAs $\\alpha _{k}^{\\text{r}}(t)(\\beta _{k}^{\\text{r}}(t))$ and AODs ${{\\alpha }_{k}^{\\text{t}}}(t)({{\\beta }_{k}^{\\text{t}}}(t))$ at slot $t$ , shall be carefully designed by efficiently solving the following problem $\\begin{aligned} & \\underset{f_{{k}},\\,w_{{k}}}{\\text{max}} & & R(t)\\\\& \\text{subject to} & & \\left\\Vert f_{{k}}\\right\\Vert =1,\\\\& & & \\left\\Vert w_{{k}}\\right\\Vert =1.\\end{aligned}$ Note that directly solving the above beam tracking problem is very challenging, especially in the considered highly dynamic UAV mmWave network.", "Therefore, developing new and efficient beam tracking solution for the CA-enabled UAV mmWave network is the major focus of our work.", "Recall that several efficient codebook-based beam training and tracking schemes have been proposed for conventional mmWave network with uniform ULA and UPA [22], [23].", "These prior works inspire us to propose a specialized new codebook design and the corresponding codeword selection/processing strategy that can drive the CCA to achieve fast beam tracking in the highly dynamic UAV mmWave network.", "To this end, the properties of the CCA should be exploited in the design of the codebook, which are briefly discussed as follows.", "Activated Subarray with Limited DREs: As shown in Fig.", "REF , given a certain azimuth angle, there are limited DREs that can be activated.", "Due to the directivity, the DREs of the CCA subarray at different positions are anisotropic, and this phenomenon is different from the UPA.", "If an inappropriate subarray is activated, the beam angle may go beyond the radiation range of certain subarray elements, degrading the beam gain and SE.", "Multiuser-resultant Receiver Subarray Partition: As shown in Fig.", "REF , the r-UAV needs to activate multiple subarrays to serve multiple t-UAVs at the same time.", "Assuming that an element can not be contained in different subarrays, then the problem of activated CCA subarray partition rises at the r-UAV side for the fast multi-UAV beam tracking.", "The dynamic CCA subarray partition can be considered as the dynamic antenna resource allocation for multiple t-UAVs, which has strong impact on the sum SE of the UAV mmWave network.", "From the aforementioned two properties of the CCA, we know that the optimal beamforming and combining vector $f_k(t)$ and $w_k(t)$ are relevant to the activated subarray and subarray partition.", "To this end, we can continue to reformulate the beam tracking problem by taking the subarray activation/partition into account, which will facilitate the specialized codebook design in the next section.", "Let us denote $\\mathcal {S}^{\\text{t}}_{k}$ as the activated subarray at the $k$ -th t-UAV with $|\\mathcal {S}_{k}^{\\text{t}}|=M_{\\text{act},t,k}N_{\\text{act},t,k}\\le M_{t}N_{t}$ , $\\mathcal {S}^{\\text{r}}_{k}$ the activated subarray for the $k$ -th t-UAV at the r-UAV, and the array partition at the r-UAV as $\\mathcal {S}^{\\text{r}}=\\mathop {\\cup }\\limits _{{k\\in \\mathcal {K}}}{\\mathcal {S}^{\\text{r}}_k}$ where $|\\mathcal {S}_{k}^{\\text{r}}|>0$ , ${\\mathcal {S}_{k}^{\\text{r}}}\\mathop {\\cap }{\\mathcal {S}_{j}^{\\text{r}}}=\\emptyset ,k\\ne j$ and $|\\mathcal {S}^{\\text{r}}|=M_{\\text{act},r}N_{\\text{act},r}\\le M_{r}N_{r}$ .", "The SE maximization problem for the codebook-based beam tracking with CCA can be formulated as $\\begin{aligned} & \\underset{f_{{k}},{\\mathcal {S}_{k}^{\\text{t}}},w_{{k}},{\\mathcal {S}_{k}^{\\text{r}}}}{\\text{max}} & & R(t)\\\\& \\text{subject to} & & f_{{k}}\\left(\\mathcal {S}_{k}^{\\text{t}}\\right)\\;\\in \\mathcal {F},\\\\& & & w_{{k}}\\left(\\mathcal {S}_{k}^{\\text{r}}\\right)\\;\\in \\mathcal {W},\\\\& & & \\mathcal {S}^{\\text{r}}=\\mathop {\\cup }\\limits _{{k\\in \\mathcal {K}}}{\\mathcal {S}_{k}^{\\text{r}}},\\\\& & & {\\mathcal {S}_{k}^{\\text{r}}}\\mathop {\\cap }{\\mathcal {S}_{j}^{\\text{r}}}=\\emptyset ,\\,k\\ne j.\\end{aligned}$ $\\mathcal {F}$ and $\\mathcal {W}$ are the sets of all analog beamforming vectors and combing vectors satisfying the hardware constraints, respectively.", "In fact, solving the above problem (REF ) requires the new codebook design and codeword selection/processing strategy.", "Noting the interdependent relationship between the beamformer/combiner (or AWV) and the activated subarray or subarray partition, a well-structured codebook should be designed to facilitate the fast localization of the activated subarray and flexible beam control.", "For this goal, the CCA codebook design and the codebook-based joint Subarray Partition and Array-weighting-vector Selection (SPAS) algorithm will be first proposed in the next section.", "In addition, the AOAs and AODs should be tracked in the highly dynamic UAV mmWave network.", "To this end, in Section  we will further propose a novel predictive AOA/AOD tracking scheme in conjunction with tracking error treatment to address the high mobility challenge, then we integrate these operations into the codebook-based SPAS to achieve reliable beam-tracking for the considered UAV mmWave network.", "In this section, we characterize the CCA from several relevent aspects in REF and design a specialized hierarchical codebook for the DRE-covered CCA in REF , wherein the subarray activation/partitioning patterns (in terms of subarray location and size) are carefully integrated with the angular domain beam patterns (in terms of beam angles and widths).", "Then, we present the basic framework of the codebook-based SPAS for the beam tracking in the considered UAV mmWave network for t-UAVs and the r-UAV in REF and REF , respectively." ], [ "CCA Characterization", "In this subsection, we characterize the CCA from serval relevant aspects to facilitate the codebook design.", "The first characteristic is the maximum codebook resolution which is related to the maximum size of the activated subarray of the DRE-covered CCA.", "More specifically, due to the directivity of the antenna element, only a subset of the elements satisfying $\\alpha _0+2l\\pi \\in [\\alpha _{n,\\min },\\alpha _{n,\\max }]$ can be activated at a certain beam angle $\\alpha _0$ , as shown in Fig.", "REF .", "Hence, the maximum number of elements which can be activated, i.e., the maximum resolution, is given by the following theorem.", "Theorem 1 For an $M \\times N$ -element DRE-covered CCA, the maximum number of the activated elements on the $xy$ -plane with a given azimuth angle $\\alpha _0$ is given by $N_{\\text{act,max}}$ .", "$\\begin{aligned}&N_{\\text{act,max}}=|n_2-n_1|,\\\\&n_1=\\biggl \\lceil {\\frac{\\alpha _0+2l\\pi -\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}}\\biggr \\rceil ,\\\\&n_2=\\biggl \\lceil {\\frac{\\alpha _0+2l\\pi +\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}}\\biggr \\rceil ,\\end{aligned}$ and the maximum number of the activated elements in the $z$ -axis is $M$ .", "Please refer to Appendix .", "According to Theorem REF , we know that all elements along the $z$ -axis can be activated at a certain elevation angle while only a part of the elements on the $xy$ -plane can be activated at a certain azimuth angle." ], [ "Multi-resolution and Beamwidth", "Given the maximum resolution of the codebook, we continue to discuss the characteristic of the multi-resolution and the beamwidth with the CCA codebook.", "For the multi-resolution codebook, the variable resolution is tuned by the beamwidth, which is determined by the number of the activated elements [12].", "Note that the beam coverage and the corresponding beamwidth are determined by both the element radiation pattern and array radiation pattern for the DRE-covered CCA.", "In particular, the beam coverage in the azimuth (elevation) plane of the activated $M_{\\text{act}} \\times N_{\\text{act}}$ subarray is $\\mathcal {CV}_{\\text{a(e)}}=\\mathcal {CV}_{\\text{a(e),array}}\\cap \\mathcal {CV}_{\\text{a(e),element}},$ where $\\mathcal {CV}_{\\text{a(e),array}}$ is the coverage of the antenna array with omnidirectional elements [12], named as the array coverage, and $\\mathcal {CV}_{\\text{a(e),element}}=\\lbrace \\alpha (\\beta )|\\lambda _s(\\alpha ,\\beta )>0\\rbrace $ is the coverage of the $M_{\\text{act}}N_{\\text{act}}$ antenna elements, named as the element coverage.", "$\\lambda _s(\\alpha ,\\beta )$ is the sum antenna elements gain at the angle $(\\alpha ,\\beta )$ that is given by $\\lambda _s(\\alpha ,\\beta )=\\sum _{m,n}\\left[{{\\Lambda }}\\left( {{\\alpha }},{\\beta } \\right) \\right]_{(m,n)}$ .", "The array coverage and the element coverage will be further explained in Section REF to analyze the coverage performance of the DRE covered CCA codebook.", "The corresponding beamwidth of the azimuth (elevation) plane of the DRE-covered CCA $BW_{\\text{a(e)}}$ is represented as $BW_{\\text{a(e)}}=\\min \\lbrace BW_{\\text{a(e),array}},BW_{\\text{a(e),element}}\\rbrace ,$ where $BW_{\\text{a(e),array}}$ is the beamwidth of the antenna array with omnidirectional elements, named as the array beamwidth, which is usually defined by the number of elements in the codebook design [12].", "In the designed codebook, $BW_{\\text{a(e),array}}$ is set as $BW_{\\text{a,array}}=\\frac{2\\pi }{N_{\\text{act}}}$ and $BW_{\\text{a,array}}=\\frac{2\\pi }{M_{\\text{act}}}$ .", "The element beamwidth $BW_{\\text{a(e),element}}$ is the width of the element coverage $\\mathcal {CV}_{\\text{a(e),element}}$ .", "Note that for DRE-covered CCA, we still need the following theorem to determine $BW_{\\text{a(e),element}}$ so as to give a full description of the beamwidth in (REF ).", "Theorem 2 When $\\Delta \\phi _{\\text{c}}\\le \\Delta \\alpha $ , the DRE coverage of $M \\times N$ CCA on the azimuth plane is $BW_{\\text{a,element}}=\\Delta \\alpha +(N-1)\\Delta \\phi +2l\\pi ,l\\in \\mathbb {Z},$ and the DRE coverage on the elevation plane is $BW_{\\text{e,element}}=\\Delta \\beta .$ Please refer to Appendix ." ], [ "Localized Subarray Activation", "According to Theorem REF , only a subarray of CCA can be activated at a certain beam angle.", "Next, the relationship between the subarray and the beam angles is studied.", "The number and position of the activated elements determine the subarray.", "Assuming that the elements in the activated subarray are adjacent to each other and the activated subarray can be expanded as a rectangle, the activated subarray is denoted by the set $\\mathcal {S}(M_{\\text{act}},N_{\\text{act}},p_c)$ , where $M_{\\text{act}}$ and $N_{\\text{act}}$ are the numbers of elements on the $z$ -axis and $xy$ -plane, respectively, and $p_c=(m_c,n_c)$ is the position of the subarray center element, which is related to the azimuth angle.", "Then we have the following theorem to localize the activated subarray.", "Theorem 3 For the $M \\times N$ -element CCA, given the beam angle $(\\alpha ,\\beta )$ and the size of subarray $(M_{\\text{act}},N_{\\text{act}})$ , when the position of the center element of the subarray $p_c$ satisfies $\\begin{aligned}&n_{c}=\\biggl \\lceil {\\frac{\\alpha +2l\\pi }{\\Delta \\phi }+(N+1)/2}\\biggr \\rceil , l\\in \\mathbb {Z},\\\\&m_{c}=\\biggl \\lfloor {\\frac{M}{2}}\\biggr \\rfloor ,\\end{aligned}$ all elements of the subarray can be activated.", "Please refer to Appendix .", "Theorem 3 provides the feasible position of the activated subarray's center element, which is related to the azimuth angle $\\alpha $ .", "This property indicates that when using the DRE-covered CCA, the activated subarray should be localized with reference to the azimuth angle.", "Figure: The proposed 3D CCA hierarchical codebook.", "The hierarchical codebook contains multiple layers with different beamwidth.", "The (i,j)(i,j)-th code of the (m,n)(m,n)-th layer contains the AWV vv and the corresponding subarray 𝒮\\mathcal {S}.", "The beam coverages of v(m s ,n s ,i,j,𝒮)v(m_s,n_s,i,j,\\mathcal {S}) in azimuth angle and elevation angle are [(i-1)BW a ,iBW a ][(i-1){BW}_{\\text{a}},i{BW}_{\\text{a}}] and [(j-1)BW e ,jBW e ][(j-1){BW}_{\\text{e}},j{BW}_{\\text{e}}], respectively." ], [ "CCA Codebook Design", "After the discussion on the characteristics of CCA, in this subsection, we continue to explain the specialized codebook design for the DRE-covered CCA.", "Revisiting Theorem REF and Theorem REF , the size and position of the activated CCA subarray are related to the azimuth angle; meanwhile, the beamwidth is determined by the size of the activated subarray according to Theorem REF .", "Therefore, the conventional codebook only consisting of different beamwidth and beam angles is not able to reveal the relationship among the beam angle, beamwidth and the corresponding supporting subarray for the DRE-covered CCA.", "In order to solve the beam tracking problem in (REF ), the subarray activation/partition and AWV selection needs to be jointly optimized at the same time.", "To this end, a new specialized hierarchical codebook $\\mathcal {V}$ should be designed to facilitate efficient beam tracking, wherein the codeword $v$ should contain both the angular-domain beam pattern information $(\\alpha _i,\\beta _i)$ and the corresponding subarray patten information $\\mathcal {S}$ .", "As shown in Fig.", "REF , we propose a hierarchical codebook $\\mathcal {V}$ that consists of multiple layers, and each layer is determined by the size of an activated subarray $(M_{\\text{act}}=m_s,N_{\\text{act}}=n_s)$ .", "In a certain layer indexed by the subarray size $(m_s,n_s)$ , the $(i,j)$ -th codeword of AWV $v(i,j,\\mathcal {S})$ is determined by the supporting subarray $\\mathcal {S}$ , and the quantized azimuth angle $\\alpha _i=i\\frac{BW_{\\text{a}}}{2}$ and the quantized elevation angle $\\beta _j=j\\frac{BW_{\\text{e}}}{2}$ , where $i\\in \\mathcal {I}_{n_s}=\\left[1,I=\\lceil \\frac{2\\pi }{BW_{\\text{a}}}\\rceil \\right],j\\in \\mathcal {J}_{m_s}=\\left[1,J=\\lceil \\frac{\\pi }{BW_{\\text{e}}}\\rceil \\right]$ .", "In particular, the supporting subarray $\\mathcal {S}(m_s,n_s,p_c(i))$ has a size of $(m_s,n_s)$ , and its center element location $p_c(i)=(m_c,n_c(i))$ is relevant to the azimuth angle, i.e., the index $i$ of the azimuth angle.", "Since the beamwidth is mainly effected by the size of the subarray according to Theorem REF , the codewords of AWV in the same layer has the same beamwidth.", "Moreover, given the size of supporting array $(m_s,n_s)$ and the beam angle $(\\alpha _i,\\beta _j)$ , the center element position of the supporting subarray $p_c(i)=(m_c,n_c(i))$ is given by (REF ) according to Theorem REF .", "Remark 1 The conventional UPA/ULA codebook design mainly controls the beamwidth by the subarray activation/deactivation with different numbers of elements.", "In contrast, the codebook for DRE-covered CCA focuses on both the number of subarray elements and the specific subarray localization.", "The number of subarray elements determines the beamwidth and the localization determines the beam angle that can be achieved by the subarray.", "Along with the overall codebook structure, we continue to reveal the inner contents of the codeword $v(i,j,\\mathcal {S})$ in the $(m_s,n_s)$ -th layer.", "Note that $v(i,j,\\mathcal {S})$ has a size of $MN$ and its $(m,n)$ -th entry is given by $[v(i,j,\\mathcal {S})]_{(m,n)}=1_{\\lbrace m,n,i,j\\rbrace }[A(\\alpha _i,\\beta _j)]_{(m,n)},$ where $A(\\alpha _i,\\beta _j)$ is given in (REF ) and $1_{\\lbrace m,n,i,j\\rbrace }$ is the indicator function $1_{\\lbrace m,n,i,j\\rbrace }=\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}& 1,(m,n)\\in \\mathcal {S}(m_s,n_s,p_c(i)), \\\\& 0,\\text{else}.\\end{IEEEeqnarraybox} \\right.$ According to (REF ), the codeword $v(i,j,\\mathcal {S})$ includes both the beam pattern information and the subarray pattern information.", "The beam pattern information mainly includes the beam angle $(\\alpha _i,\\beta _j)$ and the beam width determined by the size of $\\mathcal {S}$; the subarray pattern information includes the subarray location and size determined by $\\mathcal {S}$ .", "Finally, we elaborate the coverage property of our codebook.", "The coverage of the $(i,j)$ -th codeword in the $(m_s,n_s)$ -th layer is given by $\\mathcal {CV}_{\\text{a(e)}}(i,j,\\mathcal {S})=\\mathcal {CV}_{\\text{a(e),array}}(i,j,\\mathcal {S})\\cap \\mathcal {CV}_{\\text{a(e),element}}(i,j,\\mathcal {S}).$ Then the coverage property is characterized by the following theorem.", "Theorem 4 When the $(i,j)$ -th codeword in each layer satisfies $i\\in \\mathcal {I}_{n_s}=\\left[1,I=\\lceil \\frac{2\\pi }{BW_{\\text{a}}}\\rceil \\right]$ and $j\\in \\mathcal {J}_{m_s}=\\left[1,J=\\lceil \\frac{\\pi }{BW_{\\text{e}}}\\rceil \\right]$ , the union of the beam coverage of all codewords in each layer covers the whole azimuth and elevation angular domain, i.e., $\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}&\\underset{i\\in \\mathcal {I}}{\\cup }\\mathcal {CV}_{\\text{a}}(i,j,\\mathcal {S})=[\\alpha _{0},\\alpha _{0}+2\\pi ],\\\\&\\underset{j\\in \\mathcal {J}}{\\cup }\\mathcal {CV}_{\\text{e}}(i,j,\\mathcal {S})=[\\beta _{0},\\beta _{0}+2\\pi ],\\end{IEEEeqnarraybox} \\right.$ where $\\alpha _{0}$ and $\\beta _{0}$ is an arbitrary angle.", "Please refer to Appendix .", "In Fig.", "REF , we show the polar plots of the $(4,32)$ -th layer of the codebook coverage on the azimuth plane with $BW_{\\text{a,array}}=\\frac{\\pi }{16}$ , $\\Delta \\alpha =\\frac{2\\pi }{3}$ and $\\Delta \\beta =\\pi $ .", "It is observed that the AWVs of the same layer in the designed CCA codebook have the omnidirectional coverage.", "Figure: Polar plots of the (4,32)(4,32)-th layer of the codebook coverage on the azimuth plane with BW a,array =π 16BW_{\\text{a,array}}=\\frac{\\pi }{16}, Δα=2π 3\\Delta \\alpha =\\frac{2\\pi }{3} and Δβ=π\\Delta \\beta =\\pi ." ], [ "CCA Codebook Based Joint Subarray Partition and AWV Selection Algorithm for t-UAVs", "Based on the designed CCA codebook, the joint subarray partition and AWV selection (SPAS) algorithm is developed in this section to solve the beam tracking problem in (REF ).", "As mentioned above, it is assumed that the inter-UAV interference approximately does not exist in the considered UAV mmWave networks and the sum SE in problem (REF ) is rewritten as $R(t)=\\sum _{k\\in \\mathcal {K}}\\log (1+\\text{SNR}_k(t)),$ where $\\text{SNR}_k(t)$ is given by $\\text{SNR}_k(t)\\!=\\!\\frac{\\left|\\sqrt{p_{k}(t)}w_{k}(t)^HH_{k}(t)f_{k}(t)\\right|^2}{\\!n_{\\sigma }(t)}.$ Given the transmit power, the AOAs $\\left\\lbrace (\\alpha _{k}^{\\text{r}}(t),\\beta _{k}^{\\text{r}}(t)) \\right\\rbrace $ and AODs $\\left\\lbrace (\\alpha _{k}^{\\text{t}}(t),\\beta _{k}^{\\text{t}}(t))\\right\\rbrace $ at slot $t$ , the beam tracking problem in (REF ) can be translated to maximize the beam gain of the beamforming vector $f_{{k}}({\\mathcal {S}_{k}^{\\text{t}}})$ and the combining vector $w_{{k}}({\\mathcal {S}_{k}^{\\text{r}}})$ , respectively, without considering the interference between the t-UAVs.", "Then the problem can be solved by the SPAS algorithm for t-UAV and r-UAV, respectively.", "To start with, the beam gain of a subarray-dependent $MN\\times 1$ AWV $v\\left(\\mathcal {S}_{\\mathrm {act}}\\right)$ along the angle $(\\alpha ,\\beta )$ is expressed as $G(v,\\alpha ,\\beta )=\\sqrt{N_{\\text{act}}M_{\\text{act}}}a(\\alpha ,\\beta , \\mathcal {S}_{\\mathrm {act}})^Hv\\left(\\mathcal {S}_{\\mathrm {act}}\\right),$ where $N_{\\text{act}}M_{\\text{act}}$ is the size of the activated subarray $\\mathcal {S}_{\\mathrm {act}}$ to support $v$ and $a(\\alpha ,\\beta , \\mathcal {S}_{\\mathrm {act}})$ is the corresponding array steering vector along the angle $(\\alpha ,\\beta )$ .", "The $(m,n)$ -th entry of $a(\\alpha ,\\beta , \\mathcal {S}_{\\mathrm {act}})$ is expressed as $[a(\\alpha ,\\beta , \\mathcal {S}_{\\mathrm {act}})]_{(m,n)}=1_{\\lbrace \\mathcal {S}_{\\mathrm {act}}\\rbrace }[A(\\alpha ,\\beta )]_{(m,n)},$ where $1_{{\\lbrace \\mathcal {S}_{\\mathrm {act}}\\rbrace }}$ is the indicator function $1_{\\lbrace \\mathcal {S}_{\\mathrm {act}}\\rbrace }=\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}& 1,(m,n)\\in \\mathcal {S}_{\\mathrm {act}}, \\\\& 0,\\mathrm {else}.\\end{IEEEeqnarraybox} \\right.$ In the considered UAV mmWave network, the $k$ -th t-UAV transmits to only one r-UAV.", "Hence, the beam tracking problem for t-UAVs in (REF ) with our proposed codebook can be rewritten as $\\begin{aligned} & \\underset{f_{{k}},{\\mathcal {S}_{k}^{\\text{t}}}}{\\text{max}} & & G(f_{k}\\left(\\mathcal {S}_{k}^{\\text{t}}\\right),\\alpha _{t,k}(t),\\beta _{t,k}(t))\\\\& \\text{subject to} & & f_{{k}}\\left(\\mathcal {S}_{k}^{\\text{t}}\\right)=v(i,j,\\mathcal {S})\\;\\in \\mathcal {V}_{k}.\\end{aligned}$ The t-UAV needs to select an appropriate codeword $v(i,j,\\mathcal {S})$ from our proposed codebook $\\mathcal {V}_k$ to solve the subarray partition and AWV selection problem in (REF ).", "Note that after the codeword $v(i,j,\\mathcal {S})$ is selected, the beam pattern and the subarray pattern are determined.", "Given AODs, the maximum size of the activated subarray should be selected and the quantization error between the AODs and the beam angles in the codeword should be minimized to maximize the beam gain of the beamforming vector of the $k$ -th t-UAV.", "Therefore, the optimal codeword $v\\left(i_{k}^{*},j_{k}^{*},\\mathcal {S}\\left(m_{s,k}^{*},n_{s,k}^{*},p_{c,k}\\left(i_{k}^{*}\\right)\\right)\\right)$ to solve the problem in (REF ) at slot $t$ is given by $\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}m_{s,k}^{*}&=M_{\\text{act,max}},\\\\n_{s,k}^{*}&=N_{\\text{act,max}},\\\\i_k^{*}&=\\mathop {\\text{arg min}}\\limits _{i\\in \\mathcal {I}_{n_s}} |\\alpha _{t,k}(t)-\\alpha _i|,\\\\j_k^{*}&=\\mathop {\\text{arg min}}\\limits _{j\\in \\mathcal {J}_{m_s}} |\\beta _{t,k}(t)-\\beta _j|,\\\\m_{c,k}^{*}&=\\biggl \\lfloor {\\frac{M}{2}}\\biggr \\rfloor ,\\\\n_{c,k}^{*}&=\\biggl \\lceil {\\frac{\\alpha _{i_{k}^{*}}+2l\\pi }{\\Delta \\phi }+(N+1)/2}\\biggr \\rceil , l\\in \\mathbb {Z},\\end{IEEEeqnarraybox} \\right.$ and the corresponding supporting subarray $\\mathcal {S}(m_{s,k}^{*},n_{s,k}^{*},p_{c,k}^{*})$ is selected as the activated subarray at the same time, here $p^{*}_{c,k}=(m_{c,k}^{*}, n_{c,k}^{*})$ , which is relevant to $i_k^{*}$ .", "As shown in Fig.", "REF (a), the t-UAV subarray pattern is a part of cylinder with the size of $M_{\\text{act}}=M_{\\text{act,max}}$ and $N_{\\text{act}}=N_{\\text{act,max}}$ .", "As $\\alpha _i$ and $\\beta _j$ is the quantization of azimuth angle and elevation angle, respectively, the indexes of the optimal codewords $i_k^{*}$ and $j_k^{*}$ in the given layer of the codebook according to (REF ) are given by $i_k^{*}=\\biggl \\lceil {\\frac{\\alpha _{t,k}(t)}{BW_{a}}}\\biggr \\rceil $ , and $j_k^{*}=\\biggl \\lceil {\\frac{\\beta _{t,k}(t)}{BW_{e}}}\\biggr \\rceil $ .", "Figure: The subarray patterns on the cylinder and the corresponding expanded cylinder.", "(a) The t-UAV subarray partition pattern.", "(b) The r-UAV subarray partition pattern with conflict.", "(c) The r-UAV subarray partition pattern without conflict.", "(d) The t-UAV subarray partition pattern with beamwidth selection." ], [ "CCA Codebook Based Joint Subarray Partition and AWV Selection Algorithm for r-UAV", "In the considered UAV mmWave network, the r-UAV needs to activate multiple subarrays and select multiple combining vectors to serve multiple t-UAVs at the same time.", "Hence, the beam gain of the combining vector maximization problem for r-UAV with our proposed codebook can be rewritten as $\\begin{aligned} & \\underset{w_{{k}},{\\mathcal {S}_{k}^{\\text{r}}}}{\\text{max}} & &G(w_k\\left(\\mathcal {S}_{k}^{\\text{r}}\\right),\\alpha _{r,k}(t),\\beta _{r,k}(t))\\\\& \\text{subject to} & & w_{{k}}\\left(\\mathcal {S}_{k}^{\\text{r}}\\right)=v(i,j,\\mathcal {S})\\;\\in \\mathcal {V}_k, \\\\& & & \\mathcal {S}^{\\text{r}}=\\mathop {\\cup }\\limits _{{k\\in \\mathcal {K}}}{\\mathcal {S}_{k}^{\\text{r}}},\\\\& & & {\\mathcal {S}_{k}^{\\text{r}}}\\mathop {\\cap }{\\mathcal {S}_{j}^{\\text{r}}}=\\emptyset ,\\,k\\ne j.\\end{aligned}$ The r-UAV needs to select multiple appropriate AWVs $v(m_{s,k},n_{s,k},i_k,j_k,\\mathcal {S}_k),k\\in \\mathcal {K}$ from our proposed codebook $\\mathcal {V}$ to solve the subarray partition and AWVs selection problem.", "If an element is contained in different subarrays, there is a conflict between the subarrays.", "To solve the problem in (REF ), the joint SPAS problem without considering the conflict is discussed first and the conflict avoidance will be discussed later.", "Given AOAs, the maximum size of the activated subarray should be selected and the quantization error between the AOAs and the beam angles in the codeword should be minimized to maximize the beam gain of the combining vector for the $k$ -th t-UAV.", "Similarly with (REF ), the optimal codewords $v\\left(i_{k}^{*},j_{k}^{*},\\mathcal {S}\\left(m_{s,k}^{*},n_{s,k}^{*},p_{c,k}\\left(i_{k}^{*}\\right)\\right)\\right)$ without considering the conflict constraint is expressed as $\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}m_{s,k}^{*}&=M_{\\text{act,max}},\\\\n_{s,k}^{*}&=N_{\\text{act,max}},\\\\i_{k}^{*}&=\\mathop {\\text{arg min}}\\limits _{i\\in \\mathcal {I}_n} |\\alpha _{r,k}(t)-\\alpha _i|,\\\\j_{k}^{*}&=\\mathop {\\text{arg min}}\\limits _{j \\in \\mathcal {J}_m} |\\beta _{r,k}(t)-\\beta _j|,\\\\m_{c,k}^{*}&=\\biggl \\lfloor {\\frac{M}{2}}\\biggr \\rfloor ,\\\\n_{c,k}^{*}&=\\biggl \\lceil {\\frac{\\alpha _{r,k}(t)+2l\\pi }{\\Delta \\phi }+(N+1)/2}\\biggr \\rceil , l\\in \\mathbb {Z},\\end{IEEEeqnarraybox} \\right.$ and the activated subarray is obtained by the corresponding supporting subarray $\\mathcal {S}(m_{s,k}^{*},n_{s,k}^{*},p_{c,k}^{*})$ , $p_{c,k}^{*}=(m_{c,k}^{*},n_{c,k}^{*})$ .", "Compared to (REF ), $M_{\\text{act,max}}$ and $N_{\\text{act,max}}$ in (REF ) are determined by the array size of r-UAV, $i_{k}^{*}$ , $j_{k}^{*}$ and $n_{c,k}^{*}$ are determined by the AOAs $\\left(\\alpha _{r,k}(t),\\beta _{r,k}(t)\\right)$ instead of AODs.", "The indexes of the optimal codeword $i_{k}^{*}$ and $j_{k}^{*}$ can be found in the way similar to (REF ).", "If there is no conflict between the activated subarrays, the subarray partition pattern is shown in Fig.", "REF (c).", "However, it is possible that the optimal codewords for different t-UAVs need to activate the same antenna elements, which causes a conflict between the corresponding subarrays.", "Hence, the activated subarrays $\\mathcal {S}_k^{\\text{r}}$ should be partitioned based on the optimal codewords to avoid the conflict.", "To this end, the criterion to detect the conflict is discussed at first.", "More specifically, if and only if $\\begin{aligned}d(n_{c,k},n_{c,q})<\\frac{n_{s,k}+n_{s,q}}{2},\\\\|m_{c,k}-m_{c,q}|<\\frac{m_{s,k}+m_{s,q}}{2},\\end{aligned}$ there is a conflict between the subarrays of $M\\times N$ -element CCA $\\mathcal {S}_k(m_{s,k},n_{s,k}, p_{c,k})$ and $\\mathcal {S}_q (m_{s,q},n_{s,q},p_{c,q})$ , $(q\\ne k)$ .", "$d(n_{c,k},n_{c,q})$ is the distance between the center elements of the two subarrays on the $xy$ -plane, which is given by $d(n_{c,k},n_{c,q})=\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}N-|n_{c,k}-n_{c,q}|,|n_{c,k}-n_{c,q}|>N/2\\\\|n_{c,k}-n_{c,q}|,\\text{else},\\end{IEEEeqnarraybox} \\right.$ If (REF ) holds, there are not enough DREs between them to be activated separately and the conflict appears.", "Then the conflict between the activated subarrays of the selected optimal codewords is detected according to (REF ).", "Denote $K\\times K$ matrix $C_{\\text{sa}}$ as the conflict matrix, whose $(k,q)$ -th element represents the conflict information between the subarray $\\mathcal {S}_k$ and $\\mathcal {S}_q$ .", "If there is a conflict between the subarrays $\\mathcal {S}_k$ and $\\mathcal {S}_q$ , $C_{\\text{sa},(k,q)}=1$ , $q\\ne k$ ; otherwise, $C_{\\text{sa},(k,q)}=0$ .", "In addition, $C_{\\text{sa},(k,k)}=0$ .", "In addition, if $C_{\\text{sa},(k,q)}=1$ and $C_{\\text{sa},(p,q)}=1$ , $C_{\\text{sa},(k,p)}=1$ .", "At last, the combining vector $w_{{k}}=v_k$ and the corresponding subarray $\\mathcal {S}^{r}_k=\\mathcal {S}_k$ needs to be updated to avoid the conflict.", "The conflict set of the $k$ -th subarray is defined as $\\mathcal {C}_{\\text{sa},k}=k\\cup \\lbrace q|C_{\\text{sa},(k,q)}=1\\rbrace $ .", "$r_q$ is the index of the sorted elements of the conflict set $q$ .", "The size of the activated subarray to serve the $q$ -th UAV in the conflict set is updated by $\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}N_{\\text{act},q}&=n_{s,q},\\\\M_{\\text{act},q}&=\\biggl \\lfloor \\frac{M}{|\\mathcal {C}_{\\text{sa},q}|}\\biggr \\rfloor .\\end{IEEEeqnarraybox} \\right.$ The combining vector to serve the $q$ -th t-UAV in the conflict set is updated by $[w_q]_{(m,n)}=\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}&[w_q]_{(m,n)}, m \\in [(r_q-1)M_{\\text{act},k}+1,r_qM_{\\text{act},k}],\\\\&0, \\text{else}.\\end{IEEEeqnarraybox} \\right.$ The corresponding activated subarray is updated by $\\left\\lbrace \\begin{IEEEeqnarraybox}[][c]{ll}n_{c,q}&=n_{c,q}^{*},\\\\m_{c,q}&=\\biggl \\lceil \\frac{(2r_q-1)M_{\\text{act}}+1}{2}\\biggr \\rceil .\\end{IEEEeqnarraybox} \\right.$ The conflict matrix $C_{\\text{sa}}$ is calculated again with the new activated subarrays.", "The updating procedure is finished if $C_{\\text{sa}}=0$ .", "Meanwhile, the subarray partition and AWV selection problem for the r-UAV is sloved.", "The corresponding subarray partition pattern is shown in Fig.", "REF (b).", "To summarize, the proposed CCA codebook based SPAS algorithm for r-UAV is given in Algorithm REF .", "Complexity Analysis: For each slot, the complexity of obtaining the optimal codewords is $\\mathcal {O}(K)$ ; the complexity of calculating the conflict matrix and the conflict set is $\\mathcal {O}(K^3)$ ; the combining vector and the activated subarray update takes $\\mathcal {O}(M_{\\text{act}}N_{\\text{act}})$ operations.", "In the worst case, there are conflicts among all the optimal activated subarrays, which causes the updating procedure's complexity of $\\mathcal {O}(K^3+KM_{\\text{act}}N_{\\text{act}})$ .", "Finally, the complexity of Algorithm 1 is on the order of $\\mathcal {O}(K^3+KM_{\\text{act}}N_{\\text{act}})$ .", "In the ideal case, there is no conflict among all the subarrays, and the updating procedure is not performed.", "In this case, the complexity of Algorithm 1 is $\\mathcal {O}(K^3)$ .", "[h] CCA Codebook Based Joint Subarray Partition and Code Selection Algorithm for r-UAV [1] Input: $(\\alpha _{r,k}(t),\\beta _{r,k}(t))$ ,$\\mathcal {V}$ , $\\mathcal {K}$ Obtain the optimal codewords for $k\\in \\mathcal {K}$ according to (REF ).", "Calculate the conflict matrix $C_{\\text{sa}}$ according to (REF ).", "$C_{\\text{sa}}\\ne 0, k \\in \\mathcal {K}$ Calculate the conflict set of $k$ -th subarray $\\mathcal {C}_{\\text{sa},k}$ .", "$q \\in \\mathcal {C}_{\\text{sa},k}$ Update the combining vector $w_q$ and the activated subarray $\\mathcal {S}^{r}_q$ according to (REF ), (REF ) and (REF ).", "Output: $v_k,\\mathcal {S}^{r}_k$ .", "Tracking-Error-Aware Beam Tracking by Exploiting 3D Beamwidth Selection The CCA codebook based SPAS algorithm is proposed in the previous section to solve the joint CCA subarray partition and AWV selection problem.", "In this section, the TE-aware beam tracking problem is addressed based on the CCA codebook based SPAS algorithm.", "Tracking the AOAs and AODs is essential for beam tracking, which is closely connected with the position and attitude of the t-UAVs and r-UAV.", "The position and attitude compose the UAV's motion state information (MSI).", "In this section, the MSI prediction based AOAs and AODs estimation scheme and the protocol for beam tracking are introduced in Section REF .", "Then the TE estimation algorithm which exploits the MSI prediction error is proposed in Section REF .", "The TE-aware CCA codebook based 3D beamwidth selection algorithm is developed based on the TE estimation to achieve effective beam tracking in Section REF .", "UAV Motion State Information The AOAs and AODs of the LOS channel in (REF ) are mainly determined by the position and attitude of the t-UAVs and r-UAV.", "The t-UAV and r-UAV motion state information (MSI) mainly consists of the t-UAV and r-UAV's position and attitude, denoted as ${{X}_{t(r)}}=\\left( {{x}_{t(r)}},{{y}_{t(r)}},{{z}_{t(r)}} \\right)$ and ${{\\Theta }_{t(r)}}=\\left( {{\\psi }_{t(r)}},{{\\theta }_{t(r)}},{{\\phi }_{t(r)}} \\right)$ , respectively.", "The velocity and acceleration vectors are given by $v_{t(r)}(t)=\\frac{X_{t(r)}(t)-X_{t(r)}(t-1)}{\\delta {t}}$ and $a_{t(r)}(t)=\\frac{v_{t(r)}(t)-v_{t(r)}(t-1)}{\\delta {t}}$ , where $\\delta {t}$ is the time slot duration.", "The position and attitude of CCA related to the UAV is denoted as $X_{\\text{CCA}}$ and $\\Theta _{\\text{CCA}}$ .", "The UAVs' trajectory on the $xy$ -plane is assumed to follow the Smooth-Turn mobility model [33] that can capture the mobility of UAVs in the scenarios like patrolling.", "In this model, the UAV circles around a certain point on the horizontal plane (xy-plane) for an exponentially distributed duration until the UAV selects a new center point with the turning radius whose reciprocal obeys the normal distribution $\\mathcal {N}(0,\\sigma ^2_r)$ .", "According to [33], $\\sigma ^2_r$ plays an important role in the degree of randomness.", "The UAVs are in the state of uniform linear motion in the vertical direction with different velocity $v_{t(r),z}$ , where $v_{t(r),z}$ obeys the uniform distribution $v_{t(r),z}\\sim \\mathcal {U}(v_{t(r),z,\\text{min}},v_{t(r)z,\\text{max}})$ .", "Moreover, aiming to maintain the communication link with the r-UAV, the t-UAVs keep their positions in a limited region at arbitrary time where the distance between the t-UAV and the r-UAV is less than $D_{\\text{r,max}}$ .", "The distance between UAVs is also limited no less than $D_{\\text{r,min}}$ to ensure the flight safety.", "The relationship between the position and attitude (equations (8)-(10) in [34]) is used to determine the UAVs' attitude.", "Thanks to the integrated sensors, such as inertial measurement unit (IMU) and global position system (GPS), the UAV is able to derive its own MSI.", "However, the r-UAV also needs the MSI of all t-UAVs and each t-UAV needs the r-UAV's MSI for beam tracking, which is challenging for the r-UAV/t-UAVs.", "The GP-based MSI prediction is proposed to solve the problem in [31].", "Specifically, the r-UAV/t-UAV's historical MSI is first exchanged with the t-UAV/r-UAV over a lower-frequency band and then the t-UAV will predict the future MSI of the r-UAV based on the historical MSI by using the GP-based MSI prediction model.", "However, the MSI prediction error causes the beam tracking error, which has a negative effect on the sum SE of UAV mmWave network and is not addressed by [31].", "In this paper, a new TE-aware transmission protocol is proposed to solve the problem as shown in Fig.", "REF .", "Figure: Our frame structure design for high-mobility UAV mmWave networks.", "Therein, exchanging slot contains MSI exchanging, codeword selection, prediction, error bounding, and data transmission, while tracking slot contains beamwidth selection, TE-aware codeword selection, and data transmission.A conceptual frame structure is designed which contains two types of time slots.", "One is the exchanging slot (e-slot) and the other is the tracking slot (t-slot).", "Let us first focus on the e-slot.", "It is assumed that UAVs exchange MSI every $T$ t-slots, i.e., in an e-slot, to save resource for payload transmission.", "In the MSI exchanging period of the e-slot $t$ , the r-UAV exchanges its historical MSI with each t-UAV and the t-UAV only exchanges its historical MSI with r-UAV over the low-rate control links that work in the lower-frequency band [35].", "Then t-UAVs and r-UAV perform codeword selection.", "Employing the GP-based MSI prediction algorithm proposed in [31], each t-UAV predicts the MSI of r-UAV and r-UAV predicts the MSI of all t-UAVs in the next $T$ t-slots.", "In the tracking error bounding period, the UAVs estimate the TE of AOAs and AODs based on the GP prediction error.", "Compared to e-slot, t-slot does not have the MSI exchanging, prediction and error bounding, but has the TE-aware codeword selection.", "Specifically, in t-slot the t-UAVs and r-UAV achieve the adaptive beamwidth control against AODs/AOAs prediction errors by employing the TE-aware codeword selection.", "Compared to the motion-aware protocol in [31], the new TE-aware protocol can be applied to the UAV mmWave network with higher mobility including random trajectories and high velocity.", "Since the new TE-aware protocol contains the error bounding and TE-aware codeword selection periods, it is able to deal with the beam tracking error caused by high mobility of UAVs.", "Next, we will detail how to bound the TE and how to select the proper codeword with suitable beamwidth against the TE in the following subsections.", "Tracking Error Bounding The tracking error of beam angles has a negative influence on the beam gain obtained by CCA.", "The proposed tracking error bounding algorithm uses the position/attitiude prediction error of the GP-based MSI prediction to obtain the beam angle tracking error, wherein the geometry relationship between UAVs and the Monte-Carlo method is utilized.", "First, the algorithm used to predict MSI in the next $T$ t-slots after MSI exchanging is introduced.", "Due to the movement inertia, the MSI between adjacent slots is correlated with each other.", "Hence, the historical MSI can be used to predict the future MSI.", "According to the GP-based MSI prediction algorithm, the predicted position and attitude are estimated by the mean of the predictive distribution of the outputs (the future MSI) on the specific test dataset.", "The predictive distribution of the output (the future MSI) is given by ${\\left\\lbrace \\begin{array}{ll}& ({{Y}^{*}}|{{X}^{*}},\\mathcal {D})\\sim \\mathcal {N}(m(Y^{*}),K(Y^{*}))\\\\m(Y^{*}) & =K({{X}^{*}},X)K{{(X,X)}^{-1}}Y,\\\\K(Y^{*}) & =-K({{X}^{*}},X)K{{(X,X)}^{-1}}K(X,{{X}^{*}})\\\\& \\,\\,\\,\\,\\,+K({{X}^{*}},{{X}^{*}})\\end{array}\\right.", "}$ where $Y^{*}$ is the output of the test dataset, $X^{*}$ is the input of the test dataset, $\\mathcal {D}=\\lbrace X,Y\\rbrace $ is the training dataset, $m(Y^{*})$ is a mean matrix of the test data whose $i$ -th element is the mean function $m(y^{*}_i)$ , and $K(X,X)$ is a covariance matrix whose element is the kernel function $K(i,j)=k(x_i,x_j)$ .", "The input matrix of the training data is $X=[{x}_{1},\\ldots ,{x}_{K}]^T$ , where ${x}_{k}$ is the input of the $k$ -th training data.", "The output matrix of the training data is $Y=[{y}_{1},\\ldots ,{y}_{K}]^T$ , where ${y}_{k}$ is the output of the $k$ -th training data.", "The input and output of the test dataset $X^{*}$ and $Y^{*}$ can be defined similarly.", "Next, the specific training dataset $(X,Y)$ and the test dataset $(X^{*},Y^{*})$ for MSI prediction and tracking error bounding are introduced.", "We focus on the error bounding for the x-coordinate of UAV position and the yaw angle of UAV attitude to explain the general idea, and other positions and attitudes can be developed similarly.", "Let us first define the inputs and outputs of the $k$ -th training data for the x-coordinate of r-UAV prediction as column vectors $x_k=\\lbrace x_r(t-T),\\ldots ,x_r(t-1)\\rbrace $ and $y_k=\\lbrace x_{r}(t),\\ldots ,x_{r}(t+T_f)\\rbrace $ , respectively, where $T_f$ represents the number of slots predicted ahead of the reference time $t$ .", "Define the inputs and outputs of the $k$ -th training data for UAV yaw angle prediction as $x_k=\\lbrace \\phi _r(t-T),\\ldots ,\\phi _r(t-1),v_{r}(t),a_{r}(t)\\rbrace $ and $y_k=\\lbrace \\phi _r(t),\\ldots ,\\phi _r(t+T_f)\\rbrace $ , and the inputs and outputs of the $q$ -th test data can be defined in a similar way.", "Then we continue to bound the tracking error range.", "Note that the covariance represents the statistical uncertainty of the prediction.", "According to the properties of Gaussian distribution, the $99\\%$ confidence region can be bounded by adding and subtracting three times of the standard deviation to/from the mean value.", "That is to say, the GP-based MSI prediction algorithm can bound the prediction error within a certain range of a confidence value.", "More specifically, the covariance of the output of the $q$ -th test data is written by $k(y^{*}_{p})=\\lbrace k({y^{*}(t)}),\\ldots ,k({y^{*}(t+T_f)})\\rbrace $ .", "Then the predicted x-coordinate of r-UAV in the slot $t$ is $\\hat{x}_{r}(t)=m({x}_{r}(t))$ .", "The real position on the x-coordinate obeys the Gaussian distribution with the mean $m({x}_{r}(t))$ and the covariance $k(x_r(t))$ .", "It belongs to the range $[m({x}_{r}(t))-3\\sqrt{k(x_{r}(t))},m({x}_{r}(t))+3\\sqrt{k(x_{r}(t))}]$ which is called the error range of x-coordinate with $99\\%$ confidence.", "The distribution of the yaw angle can be given in a similar way.", "The predictive MSI $(X_r(t),\\Theta _r(t))$ and the corresponding error range is obtained after GP-based prediction.", "Meanwhile, the UAV's real position is assumed to follow the Gaussian distribution with the mean $m(X_r(t))=[m(x_r(t)),m(y_r(t)),m(z_r(t))]$ and covariance $K(X_r(t))=[k(x_r(t)),k(y_r(t)),k(z_r(t))]$ for the prediction problem, the real UAV's attitude is also assumed to obey the Gaussian distribution with the mean $m(\\Theta _r(t))=[m(\\psi _r(t)),m(\\theta _r(t)),m(\\phi _r(t))]$ and covariance $K(\\Theta _r(t))=[k(\\psi _r(t)),k(\\theta _r(t)),k(\\phi _r(t))]$ .", "The Monte-Carlo method is used to estimate the beam angle error.", "To achieve this goal, each UAV generates two random variables $\\overline{X}_{r}(t)\\sim \\mathcal {N}(m(X_r(t)),K(X_r(t)))$ and $\\overline{\\Theta }_{r}(t)\\sim \\mathcal {N}(m(\\Theta _r(t)),K(\\Theta _r(t)))$ with reference to the GP-predicted distributions.", "The corresponding beam angles $\\overline{\\alpha }(t),\\overline{\\beta }(t)$ are calculated by the geometry based beam tracking algorithm proposed in [31].", "Let us focus on the transmitting beam angle of a t-UAV to explain the basic idea, and other beam angles can be calculated similarly.", "First, we establish different coordinate frames, including the global coordinate frame (g-frame), the t-UAV coordinate frame (a-frame), the r-UAV coordinate frame (b-frame), and the antenna array coordinate frame (c-frame).", "Then, the coordinate frame transformation can be obtained by the predicted MSI of the r-UAV and the MSI of t-UAVs given by sensors, such as IMU, GPS, and so on.", "Finally, the transmitting beam angle of the t-UAV can be calculated in c-frame by using the geometric relationship between the CCA of the r-UAV and the CCA of t-UAVs.", "Repeat the above process and a lot of samples of beam angles will be generated.", "Then the empirical probability distribution functions (EDFs) of beam angles and the mean value $\\hat{\\alpha }_{\\text{mean}}(t)$ and $\\hat{\\beta }_{\\text{mean}}(t)$ are obtained based on the statistics over the set of samples $\\overline{\\alpha }(t)$ and $\\overline{\\beta }(t)$ .", "Subsequently the error range of the azimuth angle $[\\hat{\\alpha }_{\\text{min}},\\hat{\\alpha }_{\\text{max}}]$ can be derived with reference to the following probability $\\text{Pr}(\\overline{\\alpha }(t)\\in [\\hat{\\alpha }_{\\text{min}}(t),\\hat{\\alpha }_{\\text{max}}(t)])=\\text{P}_{\\alpha },$ and $[\\hat{\\alpha }_{\\text{min}},\\hat{\\alpha }_{\\text{max}}]$ is uniquely determined by assuming the range is centered by the mean value of $\\overline{\\alpha }(t)$ , i.e., $(\\hat{\\alpha }_{\\text{min}}+\\hat{\\alpha }_{\\text{max}})/2=\\hat{\\alpha }_{\\text{mean}}$ .", "In a similar way, the error range of the elevation angle $[\\hat{\\beta }_{\\text{min}},\\hat{\\beta }_{\\text{max}}]$ can be derived with reference to the following probability $\\text{Pr}(\\overline{\\beta }(t)\\in [\\hat{\\beta }_{\\text{min}}(t),\\hat{\\beta }_{\\text{max}}(t)])=\\text{P}_{\\beta }.$ In summary, the error bounding algorithm is shown in Algorithm REF , and the corresponding complexity analysis is given below.", "Complexity Analysis: For each slot, the complexity of the prediction with trained GP model is $\\mathcal {O}(N_K)$ , where $N_K$ is the dimension of kernel matrix $N_K$ .", "Generating $\\overline{X}_{r}(t)$ , $\\overline{\\Theta }_{r}$ and computing the corresponding beam angles takes $\\mathcal {O}(I_{\\text{max}})$ operations.", "Computing the EDF of the beam angles and obtaining their error ranges takes $\\mathcal {O}(I_{\\text{max}})$ operations.", "Consequently, the complexity of Algorithm REF at each slot is $\\mathcal {O}(N_K+I_{\\text{max}})$ .", "[t] UAV Position-Attitude Prediction Error Bounding Algorithm [1] Optimize the hyperparameters in the GP model using the training data.", "$t_0=1:T:T_{\\text{max}}$ Given MSI before $t$ , predict $X_r(t)$ , $X_t(t)$ , ${\\Theta }_{r}(t)$ and ${\\Theta }_{t}(t)$ , derive the covariation $K(X_r(t))$ and $K(\\Theta _r(t))$ , $t\\in [t_0,t_0+T]$ .", "each $t\\in [t_0,t_0+T]$ $i=1:I_{\\text{max}}$ Generate $\\overline{X}_{r}(t)\\sim \\mathcal {N}(m(X_r(t)),K(X_r(t)))$ , $\\overline{\\Theta }_{r}(t)\\sim \\mathcal {N}(m(\\Theta _r(t)),K(\\Theta _r(t)))$ .", "Compute $\\overline{\\alpha }_i(t),\\overline{\\beta }_i(t)$ using the geometry based beam tracking algorithm in [31].", "Compute the PDF of the beam angles based on $\\overline{\\alpha }_i(t),\\overline{\\beta }_i(t), 1\\le i\\le I_{\\max }$ .", "$[\\hat{\\alpha }_{\\text{min}},\\hat{\\alpha }_{\\text{max}}]$ and $[\\hat{\\beta }_{\\text{min}},\\hat{\\beta }_{\\text{max}}]$ according to (REF ) and (REF ).", "Tracking-Error-Aware Codeword Selection with 3D Beamwidth Control At the r-UAV side, in the presence of beam tracking error for a t-UAV, the optimal beamwidth is not always the narrowest one that makes full use of all the possible DREs.", "A wider beamwidth (a smaller subarray size) should be selected to improve the beam gain with the increase of TE.", "However, too wide beamwith (too small subarray size) can cause the beam gain reduction due to the decrease in beam directivity.", "Therefore, the appropriate beamwidth (subarray size/layer in the codebook) should be selected.", "To this end, the TE-aware codeword selection using the multi-resolution CCA codebook is proposed to achieve adaptive 3D beamwidth control for more robust beam tracking at the r-UAV side.", "Without loss of generality, let us focus on the TE-aware codeword selection for the $k$ -th t-UAV at the r-UAV side.", "The beam gain is selected as the optimization objective, and the problem of beamwidth control is translated to choose the appropriate subarray size, which corresponds to the appropriate layer in the codebook, to optimize the beam gain.", "A two-step scheme is proposed to find the suboptimal layer index in the codebook and the codewords can be selected at the same time.", "Specifically, to start with, r-UAV can get the estimation of $(\\hat{\\alpha }_{k},\\hat{\\beta }_{k})$ according to (REF ) and obtain the subarray-dependent $MN\\times 1$ combining vector $w_{k}(\\hat{\\alpha }_{k},\\hat{\\beta }_{k},\\mathcal {S}_{k}^{\\mathrm {r}})$ .", "Then the corresponding beam gain along an arbitrary angle $(\\alpha ,\\beta )$ is expressed as $G_{k}(\\alpha ,\\beta ,w_{k})=\\sqrt{N_{\\text{act}}M_{\\text{act}}}a_{k}(\\alpha ,\\beta )^{H}w_{k}\\left(\\hat{\\alpha }_{k},\\hat{\\beta }_{k},\\mathcal {S}_{k}^{\\mathrm {r}}\\right),$ where $N_{\\text{act,}k}M_{\\text{act},k}$ is the size of the activated subarray $\\mathcal {S}_{k}^{\\mathrm {r}}$ to support $v$ , and $a_{k}(\\alpha ,\\beta )$ is the array steering vector along the angle $(\\alpha ,\\beta )$ (cf.(1)).", "When the estimated elevation angle $\\hat{\\beta }_{k}$ is fixed, the minimum beam gain achieved by CCA on the azimuth plane with the azimuth angle error range $[\\alpha _{k,\\mathrm {min}},\\alpha _{k,\\mathrm {max}}]$ is defined as $G_{k}(w_{k})_{\\text{a,min}}=\\min \\lbrace G_{k}(\\alpha _{k,\\mathrm {min}},\\hat{\\beta }_{k},w_{k}),G_{k}(\\alpha _{k,\\mathrm {max}},\\hat{\\beta }_{k},w_{k})\\rbrace $ .", "Similarly, given the estimated azimuth angle $\\hat{\\alpha }_{k}$ , the minimum beam gain on the elevation plane with the elevation angle error range $[\\beta _{k,\\mathrm {min}},\\beta _{k,\\mathrm {max}}]$ is defined as $G_{k}(w_{k})_{\\text{e,min}}=\\min \\lbrace G_{k}(\\hat{\\alpha }_{k},\\beta _{k,\\mathrm {min}},w_{k}),G_{k}(\\hat{\\alpha }_{k},\\beta _{k,\\mathrm {max}},w_{k})\\rbrace $ .", "Since the beamwidth is determined by the size of the activated subarray $\\mathcal {S}_{k}^{\\mathrm {r}}\\left(m_{s,k},n_{s,k},p_{k}\\left(\\hat{\\alpha }_{k}\\right)\\right)$ , the beamwidth control is translated to the codeword selection with a proper subarray size $(M_{\\text{act},k}=m_{s,k},\\,N_{\\text{act,}k}=n_{s,k})$ .", "Note that the beamwidth on the azimuth plane and elevation plane is mainly determined by $n_{s,k}$ and $m_{s,k}$ jointly.", "For feasibility, we propose a suboptimal two-step scheme to find the codeword layer indexes $n^*_{s,k}$ and $m^*_{s,k}$ , sequentially.", "More specifically, let us define $\\mathcal {M}(\\mathcal {V}_{r})$ and $\\mathcal {N}(\\mathcal {V}_{r})$ as the sets of all layers' indexes $\\left\\lbrace m_{s}\\right\\rbrace $ and $\\left\\lbrace n_{s}\\right\\rbrace $ in the CCA codebook used by the r-UAV.", "Given $M_{\\text{max},k}:=\\text{max}\\,\\left\\lbrace \\mathcal {M}(\\mathcal {V}_{r})\\right\\rbrace $ and $\\forall n_{s,k}\\in \\mathcal {N}(\\mathcal {V}_{r})$ , the optimal codeword $v(i_{k}^{*},j_{k}^{*},\\mathcal {S}\\left(M_{\\text{max},k},n_{s,k}\\right)):=v^{*}\\left(n_{s,k}|M_{\\text{max},k}\\right)$ can be selected based on the predicted beam angles $(\\hat{\\alpha _{k}},\\hat{\\beta _{k}})$ according to (REF ).", "Then $G_{k}\\left(v^{*}\\left(n_{s,k}|M_{\\text{max},k}\\right)\\right)_{\\text{a,min}}$ can be calculated for all possible $n_{s,k}\\in \\mathcal {N}(\\mathcal {V}_{r})$ , and the one with the largest beam gain can return the optimal $n_{s,k}^{*}$ .", "Now, with the fixed $n_{s,k}^{*}$ and $\\forall m_{s,k}\\in \\mathcal {M}(\\mathcal {V}_{r})$ , we can get $v(i_{k}^{*},j_{k}^{*},\\mathcal {S}(m_{s,k},n_{s,k}^{*})):=v^{*}(m_{s,k}|n_{s,k}^{*})$ , and by finding the largest $G_{k}(v^{*}(m_{s,k}|n_{s,k}^{*}))_{\\mathrm {e,min}}$ in $\\mathcal {M}(\\mathcal {V}_{r})$ , we can get the optimal $m_{s,k}^{*}$ given $n_{s,k}^{*}$ .", "In summary, the TE-aware CCA codeword selection algorithm is given by Algorithm REF , and the corresponding complexity analysis is given as follows.", "Remark 2 The maximum-resolution layer of the multi-resolution CCA codebook with the fixed minimum beamwidth is used in Algorithm REF for the optimal solution.", "To solve the TE-aware beam tracking problem, the search space of Algorithm REF is the whole multi-resolution CCA codebook since the beamwidth is selected according to the estimated TE.", "Complexity Analysis: The complexity of searching for optimal $n_s$ and $m_s$ in our codebook is $\\mathcal {O}((|\\mathcal {M}(\\mathcal {V})|+|\\mathcal {N}(\\mathcal {V})|)M_{\\text{act}}N_{\\text{act}}K)$ , where $|\\cdot |$ is the cardinality of a set; the complexity of the updating procedure is similar with Algorithm 1.", "In total, the complexity of Algorithm REF at each slot is $\\mathcal {O}((|\\mathcal {M}(\\mathcal {V})|+|\\mathcal {N}(\\mathcal {V})|)M_{\\text{act}}N_{\\text{act}}K+K^3)$ in the worst case.", "[h] Tracking-Error-Aware codeword selection with 3D Beamwidth Control [1] Input: $(\\hat{\\alpha _{k}},\\hat{\\beta }_{k})$ , $[\\hat{\\alpha }_{\\text{min},k},\\hat{\\alpha }_{\\text{max},k}]$ , $[\\hat{\\beta }_{\\text{min},k},\\hat{\\beta }_{\\text{max},k}]$ , $\\mathcal {V}_{r}$ $M_{\\text{max},k}=\\text{argmax}_{m\\in \\mathcal {M}(\\mathcal {V}_{r})}\\,m$ .", "$n_{s,k}\\in \\mathcal {N}(\\mathcal {V}_{r})$ Select the optimal codeword $v^{*}(n_{s,k}|M_{\\text{max},k})$ in codebook $\\mathcal {V}_{k}$ according to (REF ).", "Calculate $G_{k}(v^{*}(n_{s,k}|M_{\\text{max},k}))_{\\text{a,min}}$ .", "$n_{s,k}^{*}=\\text{arg max}_{n_{s,k}\\in \\mathcal {N}(\\mathcal {V}_{r})}G(v^{*}(n_{s,k}|M_{\\text{max},k}))_{\\text{a,min}}$ .", "$m_{s,k}\\in \\mathcal {M}(\\mathcal {V}_{r})$ Given $n_{s,k}^{*}$ , select the optimal codeword $v^{*}(m_{s,k}|n_{s,k}^{*})$ in $\\mathcal {V}_{k}$ according to (REF ).", "Calculate $G_{k}(v^{*}(m_{s,k}|n_{s,k}^{*}))_{\\mathrm {e,min}}$ .", "$m_{s,k}^{*}=\\text{arg max}_{m_{s,k}\\in \\mathcal {M}(\\mathcal {V}_{r})}G_{k}(v^{*}(m_{s,k}|n_{s,k}^{*}))_{\\mathrm {e,min}}$ .", "Codeword $w_{k}(\\mathcal {S}_{k}^{r})=v(i_{k}^{*},j_{k}^{*},\\mathcal {S}(m_{s,k}^{*},n_{s,k}^{*})$ is selected.", "Detect confliction and update $w_{k}(\\mathcal {S}_{k}^{r})$ by using Algorithm REF .", "Output: $w_{k}(\\mathcal {S}_{k}^{r})$ .", "Simulation In this section, numerical results are provided to evaluate the effectiveness of the proposed codebook based SPAS algorithm and TE-aware beamwidth control for beam tracking.", "The simulation setups are given as follows.", "The UAV mmWave network with a carrier frequency 60 GHz is considered and the carrier wavelength is ${\\lambda }_{c}=0.005$  m. The inter-UAV mmWave channel follows the model in (REF ).", "The Smooth-Turn mobility model [33] is used to generate the UAV's trajectory on the $xy$ -plane, where the mean of the duration is set as $1/{\\lambda }=1$  s and the variance is set as $\\sigma _r^2=0.05$ .", "The distance between UAVs are limited to no less than $D_{\\text{r,min}}=10$  m. Referring to the composite wing UAV of CHC P316 technical parameters [36], the horizontal velocity is no more than $v_{xy}=20$  m/s, the minimum and maximum vertical velocities are set as $v_{t(r),z,\\text{min}}=2$  m/s and $v_{t(r),z,\\text{max}}=3$  m/s, respectively.", "The time slot duration is set as $\\delta {t}=10$  ms.", "Thus, the horizontal distance of the UAV navigation is approximately no more than $0.2$  m which can be almost neglected during the prediction process.", "The UAV MSI exchanging period is set as $T=50$ , i.e., 500 ms.", "In the simulation, the size of the t-UAV's DRE-covered CCA is set as $N_t=64$ and $M_t=16$ for the consideration of the computational complexity.", "Meanwhile, the radius $R_{\\text{cyl}}$ is set as $R_{\\text{cyl}}=0.0509$  m to achieve the desired beam pattern and the array response.", "Hence, the maximum number of the activated elements on the $xy$ -plane is $N_{\\text{act,max}}=21$ .", "The radiation range of the directive elements on the azimuth plane and the elevation plane is set as $\\Delta \\alpha =\\frac{2\\pi }{3}$ and $\\Delta \\beta =\\pi $ .", "The specific radiation range of the DRE on the azimuth plane and the elevation plane $(m,n)$ for CCA is written as $[\\alpha _{n,\\text{min}},\\alpha _{n,\\text{max}}]\\!=\\!\\left[\\text{mod}(\\phi _{n}\\!-\\!\\frac{\\pi }{3},2\\pi ),\\text{mod}(\\phi _{n}\\!+\\!\\frac{\\pi }{3},2\\pi )\\right],$ where $\\phi _{n}=\\frac{2n-1-N}{2}\\Delta \\phi $ and $[\\beta _{m,\\text{min}},\\beta _{m,\\text{max}}]=[0,\\pi ]$ .", "For comparison, the radiation range of the DRE for the UPA is $[\\alpha _{n,\\text{min}},\\alpha _{n,\\text{max}}]=[-\\frac{\\pi }{3},\\frac{\\pi }{3}]$ and $[\\beta _{m,\\text{min}},\\beta _{m,\\text{max}}]=[0,\\pi ]$ .", "Figure: Spectral efficiency vs. transmit power with different numbers of users.Codebook Based Beam Tracking: CCA Versus UPA Figure: The SNR outage vs. SNR threshold with K=2K=2 t-UAVs, the transmit power p k =0.06p_k=0.06 W and p k =0.08p_k=0.08 W, N act,max =10N_{\\text{act,max}}=10.In this subsection, two beam tracking schemes with different types of antenna array are illustrated by simulation results.", "One is the proposed DRE-covered CCA scheme where all the t-UAVs are equipped with the CCA of the size $N_t=64$ , $M_t=16$ , and the r-UAV is equipped with the CCA of the size $N_r=64$ , $M_r=112$ .", "The other is the DRE-covered UPA scheme where all the t-UAVs are equipped with the UPA of the size $N_t=64$ , $M_t=16$ and the r-UAV is equipped with UPA of the size $N_r=64$ , $M_r=112$ .", "The AOAs and AODs are predicted by using the GP-based position-attitude prediction algorithm in the prediction schemes.", "The real AOAs and AODs are used in the real schemes.", "Without the TE-aware beamwidth control, the initial size of subarray is selected as $M=M_{\\text{act,max}}$ and $N=N_{\\text{act,max}}$ .", "In the CCA scheme, the codeword selection algorithm is performed by t-UAVs and r-UAV.", "In the UPA scheme, the UPA is equally partitioned into subarrays with the size of $M_r/K \\times N_r$ and the beamforming/combining vector is calculated by the UAV position-attitude prediction based beam tracking algorithm in [31].", "The SE is calculated according to (REF ).", "Fig.", "REF shows the sum SE against transmitting power with different number of t-UAVs.", "It is observed that the codeword selection algorithm is effective and the CCA scheme achieves higher SE than the UPA scheme obviously with different t-UAV number $K$ .", "The main reason is that the UPA with DREs can only receive/transmit the signal within a limited angular range at a certain time slot while the CCA does not have such limitation.", "It is also shown that the gap between the real schemes and the prediction schemes is small.", "Thus, the codebook-based beam tracking algorithm is effective in the considered CCA-enabled UAV mmWave network.", "Next, we compare the outage probabilities achieved by the DER-covered CCA and UPA schemes in the UAV mmWave network.", "Here, the network outage probability is defined as [19] $\\text{Pr}(\\underset{1\\le k \\le K}{\\min }{\\text{SNR}_k(t)}<\\text{SNR}_{\\text{th}}),$ where $\\text{SNR}_{\\text{th}}$ is a certain SNR threshold.", "If an arbitrary link between the t-UAV and the r-UAV is in the outage, there is an outage in the network.", "Hence, the outage probability is determined by the minimum SNR among $K$ t-UAVs.", "Fig.", "REF shows the outage probability against the SNR threshold to further demonstrate the coverage of the CCA scheme and the UPA scheme with $K=2$ , and the fixed power $p_k=0.06$  W and $p_k=0.08$  W. It is shown that the CCA scheme is always superior to the UPA scheme in the coverage.", "In some cases, the outage remains unchanged with the threshold increasing.", "The reason is that the outage in these cases is mainly determined by the coverage ability of the arrays.", "Figure: The spectral efficiency vs. the number of t-UAVs with different transmit power.Figure: The spectral efficiency vs. transmit power with different codebooks.In this paper, we mainly focus on the analog beam tracking without considering the inter-UAV interference.", "The sum SE calculated by (REF ) and (REF ) with different numbers of t-UAVs and the given transmit power is shown in Fig.", "REF , respectively, to verify the influence of the inter-UAV interference.", "It is shown that the sum SE of the scheme without interference calculated by (REF ) is similar with that of the scheme with interference calculated by (REF ) with the appropriate number of t-UAVs and the limited transmit power.", "The gap between the schemes increases as the power and the number of t-UAVs increase.", "Therefore, the inter-UAV interference can be neglected in the considered scenario.", "As shown in Fig.", "REF , the SE of the CCA codebook scheme and the traditional codebook scheme is compared.", "The proposed DRE-covered CCA codebook is used in the CCA codebook scheme.", "In the traditional codebook scheme, the codebook without subarray partition is used.", "The CCA on the r-UAV is equally partitioned into $K$ fixed subarrays with the size of $M_r/K\\times N_r$ and the CCA on the t-UAVs is not partitioned.", "The CCA scheme performs better than the traditional scheme with different numbers of t-UAVs and different transmit power.", "The reason is that only a part of activated DREs' radiation range covers the determined radiation direction.", "TE-aware CCA Codeword Selection with 3D Beamwidth Control Figure: The position prediction and the corresponding error range.Although the GP-based UAV's position and attitude prediction results fit well with the position and attitude data, the prediction performance is effected by UAV's mobility.", "When the UAV has higher mobility such as the more random trajectory and high velocity, the prediction error may influence the beam tracking.", "The covariance of the turning radius is set as $\\sigma _r^2=0.06$ which determines the randomness of the trajectory, and the velocity is set as no more than $v_{xy}=20$  m/s.", "The UAV's position and attitude prediction results and the corresponding error range is shown in Fig.", "REF , where the error range with $99\\%$ confidence covers the error between the real data and the predicted data in most cases.", "The SEs of two array schemes against the transmit power with $K=2$ t-UAVs are illustrated in Fig.", "REF .", "The TE-aware codeword selection uses the proposed Algorithm REF and Algorithm REF .", "Serving as a reference, the minimum-beamwidth scheme always select the minimum beamwidth, i.e., the maximum number of antenna elements for an activated subarray.", "To evaluate the performance of the proposed two-step scheme, the exhaustive searching scheme for the optimal layer index is also simulated as a comparison, where the traversal of all codebook layers is executed.", "As shown in Fig.", "REF , the sum SE of the TE-aware codeword selection scheme is better than the minimum-beamwidth codeword selection scheme.", "In addition, the curve of the two-step scheme almost overlaps that of the optimal scheme, as the tracking error of elevation angle is relatively small and the optimal $m$ varies in a relatively small range.", "Figure: Spectral efficiency vs. transmit power under TE-aware beam selection scheme.In order to verify the feasibility of the proposed scheme, the average latency in an e-slot/t-slot/frame duration with different UAV numbers are evaluated as follows: The total latency in e-slot is given by $t_{\\text{total,e}}=t_{\\text{MSI}}+t_{\\text{tra}}+t_{\\text{pro}}+t_{\\text{local,e}}$ and the total latency in t-slot is $t_{\\text{total,t}}=t_{\\text{tra}}+t_{\\text{pro}}+t_{\\text{local,t}}$ , where the transmission time of MSI $t_{\\text{MSI}}=\\frac{B_{\\text{MSI}}}{C_{\\text{LB}}}$ , the transmission time with mmWave band $t_{\\text{tra}}=\\frac{B_{\\text{data}}}{C_{\\text{ave}}}$ , the propagation time $t_{\\text{pro}}=\\frac{D_{k,\\max }}{c}$, and the local processing times $t_{\\text{local,t}}$ and $t_{\\text{local,e}}$ are calculated by the time cost of the proposed algorithms.", "As a frame duration is composed of an e-slot and $T$ t-slots, the average latency over each slot in a frame duration is defined as $t_{\\text{ave}}=\\frac{T\\times t_{\\text{total,t}}+t_{\\text{total,e}}}{T+1}$ .", "Moreover, the data block of MSI is set as $B_{\\text{MSI}}=n_{\\text{MSI}}\\times T\\times B_{\\text{MSI}}$  bits, where $n_{\\text{MSI}}=6$ is the dimension of MSI at each slot, $T=50$ is the number of slots between the adjacent MSI exchanging, and each dimension of MSI at each slot is represented by $B_{\\text{MSI}}=4$  bits.", "The transmission rate of lower band is set as $C_{\\text{LB}}=500$  kbps [37], the data block is set as $B_{\\text{data}}=1$  Mbit, $C_{\\text{ave}}$ is the average rate of mmWave band, $D_{k,\\max }$ is the maximum distance between the t-UAV and the r-UAV, and $c$ is the velocity of light.", "As the computational complexity of the algorithms for the r-UAV is higher than that of t-UAVs, the local processing time mainly depends on the time for the r-UAV to perform the beam tracking algorithms, which is estimated based on the times of multiplication and addition, and the CPU of UAVs.", "The CPU Intel i7-8550u [38] with processor base frequency 1.8 GHz is considered in the simulation, which is adopted by a commonly-used onboard computer “Mainfold 2” supporting many types of UAVs such as DJI Matrice 600 pro, DJI Matrice 600 210 series, and so on [39].", "Note that the accurate MSI is known in e-slot by MSI exchanging and only the codeword selection without beamwidth control is performed in e-slot.", "The local processing times in e-slot and t-slot without TE are given by $t_{\\text{local,e}}=t_{AL2,neb}+t_{AL1}$ and $t_{\\text{local,t}}=t_{AL1}$ , respectively, where $t_{AL2,neb}$ is the computational time of Algorithm 2 without error bounding process and $t_{AL1}$ is the computational time of Algorithm 1.", "When the TE is considered, $t_{\\text{local,e}}=t_{AL2}+t_{AL1}$ and $t_{\\text{local,t}}=t_{AL3}$ , where $t_{AL2}$ is the computational time of Algorithm 2 and $t_{AL3}$ is the computational time of Algorithm 3.", "The schemes with TE and without TE are both evaluated in the simulation.", "In particular, the TE-aware scheme is adopted for the UAV movements with higher randomness ($\\sigma ^{2}=0.06$ ), while the scheme without TE is adopted for the UAV movements with lower randomness ($\\sigma ^{2}=0.05$ ).", "As shown in Fig.", "REF , the maximum average latency is less than 4 ms, during which the movement distance of the UAV navigation is less than 0.08 m according to the UAV's velocity of 20 m/s [36], and hence the impact of the latency on beam angles' calculation for beam tracking can be approximately neglected.", "Therefore, the average latency is tolerable in the actual UAV mmWave networks.", "Moreover, as shown in Fig.", "14, due to the relatively long interval between adjacent MSI exchanging, i.e., 500 ms ($T=50$ ), the average latency in a frame duration is close to that in t-slot, which is considerably lower than e-slot.", "Thus, the proposed algorithms can be applied to high-mobility scenarios.", "Figure: The average latency vs. the number of t-UAVs in e-slot, t-slot, and a frame duration consisting of an e-slot and TT t-slots.", "Conclusion In this paper, we propose a new mmWave beam tracking framework for the CA-enabled UAV mmWave network.", "A specialized hierarchical codebook has been constructed which fully exploits the properties of the DRE-covered CCA, wherein each codeword has a supporting subarray and the corresponding angular domain beam pattern.", "Then, the basic codeword selection principles have been developed for the t-UAV and the r-UAV, respectively; given the estimation of the AOA/AODs, the codeword can be quickly selected that achieves the optimized joint subarray activation and array weighting vector selection for the DRE-covered CCA.", "Moreover, the GP-based UAV position/attitude prediction has been proposed to track the angular information of UAV for the fast codebook-based beam tracking.", "The tracking error (TE) has been carefully bounded and a TE-aware codeword selection scheme has been proposed to adapt the beamwidth for better immunity against the high mobility of UAVs.", "Simulation results validate the performance advantages of the CA-enabled UAV mmWave network over the counterpart employing conventional UPA.", "Driven by our proposed codebook and codeword selection strategies, DRE-covered CCA can significantly improve the SE and coverage of the UAV mmWave network over the conventional UPA, and thus enabling agile and robust beam tracking in the highly dynamic scenarios.", "Proof of Theorem 1 The minimum number of the activated element is denoted as $n_1$ given by $\\begin{aligned}&n_{1}=\\mathop {\\text{arg min}}\\limits _n |\\alpha _0+2l\\pi -\\alpha _{n,\\max }|\\\\&\\text{s.t.", "}\\ {\\alpha _0+2l\\pi } \\ge \\alpha _{n,\\max }.\\end{aligned}$ and the maximum number of the activated element given by $\\begin{aligned}&n_{2}=\\mathop {\\text{arg min}}\\limits _n |\\alpha _0+2l\\pi -\\alpha _{n,\\min }|\\\\&\\text{s.t.", "}\\ {\\alpha _0+2l\\pi } \\le \\alpha _{n,\\min }.\\end{aligned}$ Let $\\alpha _0+2l\\pi -\\alpha _{n,\\max }=0$ and substitute (REF ) and $\\phi _c(n)$ into this equation, $n=\\frac{\\alpha _0+2l\\pi -\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}$ .", "Considering the constraint in (REF ), the optimal solution $n_1$ should satisfy $\\alpha _{n,\\max } \\le \\alpha _{n_1,\\max }$ .", "Therefore, the optimal solution of problem in (REF ) is given by $n_1=\\lceil {n}\\rceil $ .", "The problem in (REF ) can be solved similarly.", "Proof of Theorem 2 The element coverage for the $M \\times N$ -element subarray can be rewritten as $\\begin{aligned}\\mathcal {CV}_{\\text{a(e),element}}&=\\lbrace \\alpha (\\beta )|\\lambda _s(\\alpha ,\\beta )>0\\rbrace \\\\&=\\underset{n(m)}{\\cup }\\lbrace \\alpha (\\beta )|\\left[{{\\Lambda }}\\left( {{\\alpha }},{\\beta } \\right) \\right]_{(m,n)}>0\\rbrace \\\\&=\\underset{n(m)}{\\cup }[\\alpha _{n,\\min }(\\beta _{m,\\min }),\\alpha _{n,\\max }(\\beta _{m,\\max })].\\nonumber \\end{aligned}$ When $\\Delta \\phi _{\\text{c}}\\le \\Delta \\alpha $ , $\\begin{aligned}&\\alpha _{n,\\min }-\\alpha _{n-1,\\max }\\\\&=\\phi _n-\\frac{\\Delta \\alpha }{2}+2l_1\\pi -(\\phi _{n-1}+\\frac{\\Delta \\alpha }{2}+2l_2\\pi )\\\\&=\\Delta \\phi _c-\\Delta \\alpha +2l\\pi \\le 2l\\pi .", "\\nonumber \\end{aligned}$ Therefore, $[\\alpha _{n-1,\\min },\\alpha _{n-1,\\max }]\\cap [\\alpha _{n,\\min },\\alpha _{n,\\max }]=[\\alpha _{n-1,\\min },\\alpha _{n,\\max }],\\nonumber $ and the element coverage is $\\mathcal {CV}_{\\text{a,element}}=[\\alpha _{1,\\min },\\alpha _{N,\\max }]$ .", "The DREs coverage of the $M \\times N$ -element subarray of the CCA on the azimuth plane is $\\begin{aligned}BW_{\\text{a,element}}&=\\alpha _{N,\\max }-\\alpha _{1,\\min }\\\\&=\\phi _c(N)+\\frac{\\Delta \\alpha }{2}+2l_1\\pi -(\\phi _c(1)-\\frac{\\Delta \\alpha }{2}+2l_2\\pi )\\\\&=\\Delta \\alpha +\\phi _c(N)-\\phi _c(1)+2l\\pi \\\\&=\\Delta \\alpha +(N-1)\\Delta \\phi +2l\\pi ,\\nonumber \\end{aligned}$ As all elements on $z$ -axis has the same elevation angle coverage, the DREs coverage of the subbarray is equal to the coverage of each element given by $BW_{\\text{e,element}}=\\Delta \\beta .\\nonumber $ Proof of Theorem 3 According to (REF ), when $N_{\\text{act}}=N_{\\text{act,max}}$ , the position of the center element of all activated elements on the $xy$ -plane is $\\begin{aligned}n_{c}^{\\text{max}}&=\\biggl \\lceil {\\frac{n_1+n_2}{2}}\\biggr \\rceil \\\\&=\\biggl \\lceil \\frac{1}{2}\\left(\\biggl \\lceil {\\frac{\\alpha _0+2l\\pi -\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}}\\biggr \\rceil +\\right.\\\\&\\left.\\biggl \\lceil {\\frac{\\alpha _0+2l\\pi +\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}}\\biggr \\rceil \\right)\\biggr \\rceil \\\\&=\\biggl \\lceil {\\frac{\\alpha +2l\\pi }{\\Delta \\phi }+(N+1)/2}\\biggr \\rceil , l\\in \\mathbb {Z}.\\nonumber \\end{aligned}$ When $M_{\\text{act}}=M_{\\text{act,max}}$ , all elements on the $z$ -axis need to be activated.", "Therefore, the position of the center element of all activated elements on the $z$ -axis is $m_{c}^{\\text{max}}=\\biggl \\lceil {\\frac{1+M}{2}}\\biggr \\rceil .\\nonumber $ When $N_{\\text{act}}<N_{\\text{act,max}}$ and $M_{\\text{act}}<M_{\\text{act,max}}$ , $n_c=n_c^{\\max }$ and $m_c=m_c^{\\max }$ can still allow all elements of the subarray activated as less elements need to be activated.", "Proof of Theorem 4 Note that as the number of the activated elements increases, the array beamwidth increases and the element beamwidth decreases.", "When $BW_{\\text{a(e),array}}\\le BW_{\\text{a(e),element}}$ , $\\mathcal {CV}_{\\text{a(e)}}(i,j,\\mathcal {S}):=\\mathcal {CV}_{\\text{a(e),array}}(i,j,\\mathcal {S})$ and the corresponding beamwidth $BW_{\\text{a(e)}}:=BW_{\\text{a(e),array}}$ ; Otherwise, $\\mathcal {CV}_{\\text{a(e)}}(i,j,\\mathcal {S}):=\\mathcal {CV}_{\\text{a(e),element}}(i,j,\\mathcal {S})$ and the corresponding beamwidth $BW_{\\text{a(e)}}:=BW_{\\text{a(e),element}}$ .", "The element coverage of the $(i,j)$ -the codeword in the $(m_s,n_s)$ -th layer $\\mathcal {CV}_{\\text{a(e),element}}(i,j,\\mathcal {S})$ is given by $\\begin{aligned}&\\mathcal {CV}_{\\text{a,element}}(i,j,\\mathcal {S})\\\\&=[\\alpha _{1,\\min }+(i-1)BW_{\\text{a,element}},\\alpha _{1,\\min }+iBW_{\\text{a,element}}],\\\\&\\mathcal {CV}_{\\text{e,element}}(i,j,\\mathcal {S})=[\\beta _{\\min },\\beta _{\\max }],\\end{aligned}$ where $BW_{\\text{a,element}}$ is given by (REF ) in Theorem REF .", "$\\mathcal {CV}_{\\text{a(e),array}}(i,j,\\mathcal {S})$ is the array coverage of the $(i,j)$ -th codeword in the $(m_s,n_s)$ -th layer, which is given by $\\mathcal {CV}_{\\text{a,array}}(i,j,\\mathcal {S})=[(i-1)BW_{\\text{a,array}},iBW_{\\text{a,array}}],\\\\\\nonumber \\mathcal {CV}_{\\text{e,array}}(i,j,\\mathcal {S})=[(j-1)BW_{\\text{e,array}},jBW_{\\text{e,array}}].$ The array beamwidth is set as $BW_{\\text{a,array}}=\\frac{2\\pi }{n_s}$ and $BW_{\\text{e,array}}=\\frac{2\\pi }{m_s}$ in the $(m_s,n_s)$ -th layer.", "When $BW_{\\text{a,array}}\\le BW_{\\text{a,element}}$ , i.e., $\\frac{2\\pi }{n_s}\\le \\Delta \\alpha +(n_s-1)\\Delta \\phi $ , the union of the beam coverage in the azimuth plane of all codewords in each layer is $\\begin{aligned}&\\underset{i\\in \\mathcal {I}}{\\cup }\\mathcal {CV}_{\\text{a}}(i,j,\\mathcal {S})=[0,I*BW_{\\text{a,array}}]=[0,2\\pi ].\\nonumber \\end{aligned}$ Otherwise, the union of the beam coverage is given by $\\begin{aligned}&\\underset{i\\in \\mathcal {I}}{\\cup }\\mathcal {CV}_{\\text{a}}(i,j,\\mathcal {S})\\\\&=[\\alpha _{1,\\min },\\alpha _{1,\\min }+I*BW_{\\text{a,element}}]\\\\&=[\\alpha _{1,\\min },\\alpha _{1,\\min }+2\\pi ].\\end{aligned}$ Therefore, the union of the beam coverage of all codewords in each layer covers the whole azimuth angular domain.", "Revisiting Theorem REF , $BW_{\\text{e,element}}=\\Delta \\beta =\\pi $ and $BW_{\\text{e,array}}\\le BW_{\\text{e,element}}$ hold for all layers.", "The union of the beam coverage in the elevation plane of all codewords in each layer is $\\begin{aligned}&\\underset{j\\in \\mathcal {J}}{\\cup }\\mathcal {CV}_{\\text{e}}(i,j,\\mathcal {S})=[0,J*BW_{\\text{e,array}}]=[0,2\\pi ].\\nonumber \\end{aligned}$ Hence, the beam coverage of all codewords in each layer of the designed codebook covers the whole angular domain.", "The CCA codebook based SPAS algorithm is proposed in the previous section to solve the joint CCA subarray partition and AWV selection problem.", "In this section, the TE-aware beam tracking problem is addressed based on the CCA codebook based SPAS algorithm.", "Tracking the AOAs and AODs is essential for beam tracking, which is closely connected with the position and attitude of the t-UAVs and r-UAV.", "The position and attitude compose the UAV's motion state information (MSI).", "In this section, the MSI prediction based AOAs and AODs estimation scheme and the protocol for beam tracking are introduced in Section REF .", "Then the TE estimation algorithm which exploits the MSI prediction error is proposed in Section REF .", "The TE-aware CCA codebook based 3D beamwidth selection algorithm is developed based on the TE estimation to achieve effective beam tracking in Section REF ." ], [ "UAV Motion State Information", "The AOAs and AODs of the LOS channel in (REF ) are mainly determined by the position and attitude of the t-UAVs and r-UAV.", "The t-UAV and r-UAV motion state information (MSI) mainly consists of the t-UAV and r-UAV's position and attitude, denoted as ${{X}_{t(r)}}=\\left( {{x}_{t(r)}},{{y}_{t(r)}},{{z}_{t(r)}} \\right)$ and ${{\\Theta }_{t(r)}}=\\left( {{\\psi }_{t(r)}},{{\\theta }_{t(r)}},{{\\phi }_{t(r)}} \\right)$ , respectively.", "The velocity and acceleration vectors are given by $v_{t(r)}(t)=\\frac{X_{t(r)}(t)-X_{t(r)}(t-1)}{\\delta {t}}$ and $a_{t(r)}(t)=\\frac{v_{t(r)}(t)-v_{t(r)}(t-1)}{\\delta {t}}$ , where $\\delta {t}$ is the time slot duration.", "The position and attitude of CCA related to the UAV is denoted as $X_{\\text{CCA}}$ and $\\Theta _{\\text{CCA}}$ .", "The UAVs' trajectory on the $xy$ -plane is assumed to follow the Smooth-Turn mobility model [33] that can capture the mobility of UAVs in the scenarios like patrolling.", "In this model, the UAV circles around a certain point on the horizontal plane (xy-plane) for an exponentially distributed duration until the UAV selects a new center point with the turning radius whose reciprocal obeys the normal distribution $\\mathcal {N}(0,\\sigma ^2_r)$ .", "According to [33], $\\sigma ^2_r$ plays an important role in the degree of randomness.", "The UAVs are in the state of uniform linear motion in the vertical direction with different velocity $v_{t(r),z}$ , where $v_{t(r),z}$ obeys the uniform distribution $v_{t(r),z}\\sim \\mathcal {U}(v_{t(r),z,\\text{min}},v_{t(r)z,\\text{max}})$ .", "Moreover, aiming to maintain the communication link with the r-UAV, the t-UAVs keep their positions in a limited region at arbitrary time where the distance between the t-UAV and the r-UAV is less than $D_{\\text{r,max}}$ .", "The distance between UAVs is also limited no less than $D_{\\text{r,min}}$ to ensure the flight safety.", "The relationship between the position and attitude (equations (8)-(10) in [34]) is used to determine the UAVs' attitude.", "Thanks to the integrated sensors, such as inertial measurement unit (IMU) and global position system (GPS), the UAV is able to derive its own MSI.", "However, the r-UAV also needs the MSI of all t-UAVs and each t-UAV needs the r-UAV's MSI for beam tracking, which is challenging for the r-UAV/t-UAVs.", "The GP-based MSI prediction is proposed to solve the problem in [31].", "Specifically, the r-UAV/t-UAV's historical MSI is first exchanged with the t-UAV/r-UAV over a lower-frequency band and then the t-UAV will predict the future MSI of the r-UAV based on the historical MSI by using the GP-based MSI prediction model.", "However, the MSI prediction error causes the beam tracking error, which has a negative effect on the sum SE of UAV mmWave network and is not addressed by [31].", "In this paper, a new TE-aware transmission protocol is proposed to solve the problem as shown in Fig.", "REF .", "Figure: Our frame structure design for high-mobility UAV mmWave networks.", "Therein, exchanging slot contains MSI exchanging, codeword selection, prediction, error bounding, and data transmission, while tracking slot contains beamwidth selection, TE-aware codeword selection, and data transmission.A conceptual frame structure is designed which contains two types of time slots.", "One is the exchanging slot (e-slot) and the other is the tracking slot (t-slot).", "Let us first focus on the e-slot.", "It is assumed that UAVs exchange MSI every $T$ t-slots, i.e., in an e-slot, to save resource for payload transmission.", "In the MSI exchanging period of the e-slot $t$ , the r-UAV exchanges its historical MSI with each t-UAV and the t-UAV only exchanges its historical MSI with r-UAV over the low-rate control links that work in the lower-frequency band [35].", "Then t-UAVs and r-UAV perform codeword selection.", "Employing the GP-based MSI prediction algorithm proposed in [31], each t-UAV predicts the MSI of r-UAV and r-UAV predicts the MSI of all t-UAVs in the next $T$ t-slots.", "In the tracking error bounding period, the UAVs estimate the TE of AOAs and AODs based on the GP prediction error.", "Compared to e-slot, t-slot does not have the MSI exchanging, prediction and error bounding, but has the TE-aware codeword selection.", "Specifically, in t-slot the t-UAVs and r-UAV achieve the adaptive beamwidth control against AODs/AOAs prediction errors by employing the TE-aware codeword selection.", "Compared to the motion-aware protocol in [31], the new TE-aware protocol can be applied to the UAV mmWave network with higher mobility including random trajectories and high velocity.", "Since the new TE-aware protocol contains the error bounding and TE-aware codeword selection periods, it is able to deal with the beam tracking error caused by high mobility of UAVs.", "Next, we will detail how to bound the TE and how to select the proper codeword with suitable beamwidth against the TE in the following subsections." ], [ "Tracking Error Bounding", "The tracking error of beam angles has a negative influence on the beam gain obtained by CCA.", "The proposed tracking error bounding algorithm uses the position/attitiude prediction error of the GP-based MSI prediction to obtain the beam angle tracking error, wherein the geometry relationship between UAVs and the Monte-Carlo method is utilized.", "First, the algorithm used to predict MSI in the next $T$ t-slots after MSI exchanging is introduced.", "Due to the movement inertia, the MSI between adjacent slots is correlated with each other.", "Hence, the historical MSI can be used to predict the future MSI.", "According to the GP-based MSI prediction algorithm, the predicted position and attitude are estimated by the mean of the predictive distribution of the outputs (the future MSI) on the specific test dataset.", "The predictive distribution of the output (the future MSI) is given by ${\\left\\lbrace \\begin{array}{ll}& ({{Y}^{*}}|{{X}^{*}},\\mathcal {D})\\sim \\mathcal {N}(m(Y^{*}),K(Y^{*}))\\\\m(Y^{*}) & =K({{X}^{*}},X)K{{(X,X)}^{-1}}Y,\\\\K(Y^{*}) & =-K({{X}^{*}},X)K{{(X,X)}^{-1}}K(X,{{X}^{*}})\\\\& \\,\\,\\,\\,\\,+K({{X}^{*}},{{X}^{*}})\\end{array}\\right.", "}$ where $Y^{*}$ is the output of the test dataset, $X^{*}$ is the input of the test dataset, $\\mathcal {D}=\\lbrace X,Y\\rbrace $ is the training dataset, $m(Y^{*})$ is a mean matrix of the test data whose $i$ -th element is the mean function $m(y^{*}_i)$ , and $K(X,X)$ is a covariance matrix whose element is the kernel function $K(i,j)=k(x_i,x_j)$ .", "The input matrix of the training data is $X=[{x}_{1},\\ldots ,{x}_{K}]^T$ , where ${x}_{k}$ is the input of the $k$ -th training data.", "The output matrix of the training data is $Y=[{y}_{1},\\ldots ,{y}_{K}]^T$ , where ${y}_{k}$ is the output of the $k$ -th training data.", "The input and output of the test dataset $X^{*}$ and $Y^{*}$ can be defined similarly.", "Next, the specific training dataset $(X,Y)$ and the test dataset $(X^{*},Y^{*})$ for MSI prediction and tracking error bounding are introduced.", "We focus on the error bounding for the x-coordinate of UAV position and the yaw angle of UAV attitude to explain the general idea, and other positions and attitudes can be developed similarly.", "Let us first define the inputs and outputs of the $k$ -th training data for the x-coordinate of r-UAV prediction as column vectors $x_k=\\lbrace x_r(t-T),\\ldots ,x_r(t-1)\\rbrace $ and $y_k=\\lbrace x_{r}(t),\\ldots ,x_{r}(t+T_f)\\rbrace $ , respectively, where $T_f$ represents the number of slots predicted ahead of the reference time $t$ .", "Define the inputs and outputs of the $k$ -th training data for UAV yaw angle prediction as $x_k=\\lbrace \\phi _r(t-T),\\ldots ,\\phi _r(t-1),v_{r}(t),a_{r}(t)\\rbrace $ and $y_k=\\lbrace \\phi _r(t),\\ldots ,\\phi _r(t+T_f)\\rbrace $ , and the inputs and outputs of the $q$ -th test data can be defined in a similar way.", "Then we continue to bound the tracking error range.", "Note that the covariance represents the statistical uncertainty of the prediction.", "According to the properties of Gaussian distribution, the $99\\%$ confidence region can be bounded by adding and subtracting three times of the standard deviation to/from the mean value.", "That is to say, the GP-based MSI prediction algorithm can bound the prediction error within a certain range of a confidence value.", "More specifically, the covariance of the output of the $q$ -th test data is written by $k(y^{*}_{p})=\\lbrace k({y^{*}(t)}),\\ldots ,k({y^{*}(t+T_f)})\\rbrace $ .", "Then the predicted x-coordinate of r-UAV in the slot $t$ is $\\hat{x}_{r}(t)=m({x}_{r}(t))$ .", "The real position on the x-coordinate obeys the Gaussian distribution with the mean $m({x}_{r}(t))$ and the covariance $k(x_r(t))$ .", "It belongs to the range $[m({x}_{r}(t))-3\\sqrt{k(x_{r}(t))},m({x}_{r}(t))+3\\sqrt{k(x_{r}(t))}]$ which is called the error range of x-coordinate with $99\\%$ confidence.", "The distribution of the yaw angle can be given in a similar way.", "The predictive MSI $(X_r(t),\\Theta _r(t))$ and the corresponding error range is obtained after GP-based prediction.", "Meanwhile, the UAV's real position is assumed to follow the Gaussian distribution with the mean $m(X_r(t))=[m(x_r(t)),m(y_r(t)),m(z_r(t))]$ and covariance $K(X_r(t))=[k(x_r(t)),k(y_r(t)),k(z_r(t))]$ for the prediction problem, the real UAV's attitude is also assumed to obey the Gaussian distribution with the mean $m(\\Theta _r(t))=[m(\\psi _r(t)),m(\\theta _r(t)),m(\\phi _r(t))]$ and covariance $K(\\Theta _r(t))=[k(\\psi _r(t)),k(\\theta _r(t)),k(\\phi _r(t))]$ .", "The Monte-Carlo method is used to estimate the beam angle error.", "To achieve this goal, each UAV generates two random variables $\\overline{X}_{r}(t)\\sim \\mathcal {N}(m(X_r(t)),K(X_r(t)))$ and $\\overline{\\Theta }_{r}(t)\\sim \\mathcal {N}(m(\\Theta _r(t)),K(\\Theta _r(t)))$ with reference to the GP-predicted distributions.", "The corresponding beam angles $\\overline{\\alpha }(t),\\overline{\\beta }(t)$ are calculated by the geometry based beam tracking algorithm proposed in [31].", "Let us focus on the transmitting beam angle of a t-UAV to explain the basic idea, and other beam angles can be calculated similarly.", "First, we establish different coordinate frames, including the global coordinate frame (g-frame), the t-UAV coordinate frame (a-frame), the r-UAV coordinate frame (b-frame), and the antenna array coordinate frame (c-frame).", "Then, the coordinate frame transformation can be obtained by the predicted MSI of the r-UAV and the MSI of t-UAVs given by sensors, such as IMU, GPS, and so on.", "Finally, the transmitting beam angle of the t-UAV can be calculated in c-frame by using the geometric relationship between the CCA of the r-UAV and the CCA of t-UAVs.", "Repeat the above process and a lot of samples of beam angles will be generated.", "Then the empirical probability distribution functions (EDFs) of beam angles and the mean value $\\hat{\\alpha }_{\\text{mean}}(t)$ and $\\hat{\\beta }_{\\text{mean}}(t)$ are obtained based on the statistics over the set of samples $\\overline{\\alpha }(t)$ and $\\overline{\\beta }(t)$ .", "Subsequently the error range of the azimuth angle $[\\hat{\\alpha }_{\\text{min}},\\hat{\\alpha }_{\\text{max}}]$ can be derived with reference to the following probability $\\text{Pr}(\\overline{\\alpha }(t)\\in [\\hat{\\alpha }_{\\text{min}}(t),\\hat{\\alpha }_{\\text{max}}(t)])=\\text{P}_{\\alpha },$ and $[\\hat{\\alpha }_{\\text{min}},\\hat{\\alpha }_{\\text{max}}]$ is uniquely determined by assuming the range is centered by the mean value of $\\overline{\\alpha }(t)$ , i.e., $(\\hat{\\alpha }_{\\text{min}}+\\hat{\\alpha }_{\\text{max}})/2=\\hat{\\alpha }_{\\text{mean}}$ .", "In a similar way, the error range of the elevation angle $[\\hat{\\beta }_{\\text{min}},\\hat{\\beta }_{\\text{max}}]$ can be derived with reference to the following probability $\\text{Pr}(\\overline{\\beta }(t)\\in [\\hat{\\beta }_{\\text{min}}(t),\\hat{\\beta }_{\\text{max}}(t)])=\\text{P}_{\\beta }.$ In summary, the error bounding algorithm is shown in Algorithm REF , and the corresponding complexity analysis is given below.", "Complexity Analysis: For each slot, the complexity of the prediction with trained GP model is $\\mathcal {O}(N_K)$ , where $N_K$ is the dimension of kernel matrix $N_K$ .", "Generating $\\overline{X}_{r}(t)$ , $\\overline{\\Theta }_{r}$ and computing the corresponding beam angles takes $\\mathcal {O}(I_{\\text{max}})$ operations.", "Computing the EDF of the beam angles and obtaining their error ranges takes $\\mathcal {O}(I_{\\text{max}})$ operations.", "Consequently, the complexity of Algorithm REF at each slot is $\\mathcal {O}(N_K+I_{\\text{max}})$ .", "[t] UAV Position-Attitude Prediction Error Bounding Algorithm [1] Optimize the hyperparameters in the GP model using the training data.", "$t_0=1:T:T_{\\text{max}}$ Given MSI before $t$ , predict $X_r(t)$ , $X_t(t)$ , ${\\Theta }_{r}(t)$ and ${\\Theta }_{t}(t)$ , derive the covariation $K(X_r(t))$ and $K(\\Theta _r(t))$ , $t\\in [t_0,t_0+T]$ .", "each $t\\in [t_0,t_0+T]$ $i=1:I_{\\text{max}}$ Generate $\\overline{X}_{r}(t)\\sim \\mathcal {N}(m(X_r(t)),K(X_r(t)))$ , $\\overline{\\Theta }_{r}(t)\\sim \\mathcal {N}(m(\\Theta _r(t)),K(\\Theta _r(t)))$ .", "Compute $\\overline{\\alpha }_i(t),\\overline{\\beta }_i(t)$ using the geometry based beam tracking algorithm in [31].", "Compute the PDF of the beam angles based on $\\overline{\\alpha }_i(t),\\overline{\\beta }_i(t), 1\\le i\\le I_{\\max }$ .", "$[\\hat{\\alpha }_{\\text{min}},\\hat{\\alpha }_{\\text{max}}]$ and $[\\hat{\\beta }_{\\text{min}},\\hat{\\beta }_{\\text{max}}]$ according to (REF ) and (REF )." ], [ "Tracking-Error-Aware Codeword Selection with 3D Beamwidth Control", "At the r-UAV side, in the presence of beam tracking error for a t-UAV, the optimal beamwidth is not always the narrowest one that makes full use of all the possible DREs.", "A wider beamwidth (a smaller subarray size) should be selected to improve the beam gain with the increase of TE.", "However, too wide beamwith (too small subarray size) can cause the beam gain reduction due to the decrease in beam directivity.", "Therefore, the appropriate beamwidth (subarray size/layer in the codebook) should be selected.", "To this end, the TE-aware codeword selection using the multi-resolution CCA codebook is proposed to achieve adaptive 3D beamwidth control for more robust beam tracking at the r-UAV side.", "Without loss of generality, let us focus on the TE-aware codeword selection for the $k$ -th t-UAV at the r-UAV side.", "The beam gain is selected as the optimization objective, and the problem of beamwidth control is translated to choose the appropriate subarray size, which corresponds to the appropriate layer in the codebook, to optimize the beam gain.", "A two-step scheme is proposed to find the suboptimal layer index in the codebook and the codewords can be selected at the same time.", "Specifically, to start with, r-UAV can get the estimation of $(\\hat{\\alpha }_{k},\\hat{\\beta }_{k})$ according to (REF ) and obtain the subarray-dependent $MN\\times 1$ combining vector $w_{k}(\\hat{\\alpha }_{k},\\hat{\\beta }_{k},\\mathcal {S}_{k}^{\\mathrm {r}})$ .", "Then the corresponding beam gain along an arbitrary angle $(\\alpha ,\\beta )$ is expressed as $G_{k}(\\alpha ,\\beta ,w_{k})=\\sqrt{N_{\\text{act}}M_{\\text{act}}}a_{k}(\\alpha ,\\beta )^{H}w_{k}\\left(\\hat{\\alpha }_{k},\\hat{\\beta }_{k},\\mathcal {S}_{k}^{\\mathrm {r}}\\right),$ where $N_{\\text{act,}k}M_{\\text{act},k}$ is the size of the activated subarray $\\mathcal {S}_{k}^{\\mathrm {r}}$ to support $v$ , and $a_{k}(\\alpha ,\\beta )$ is the array steering vector along the angle $(\\alpha ,\\beta )$ (cf.(1)).", "When the estimated elevation angle $\\hat{\\beta }_{k}$ is fixed, the minimum beam gain achieved by CCA on the azimuth plane with the azimuth angle error range $[\\alpha _{k,\\mathrm {min}},\\alpha _{k,\\mathrm {max}}]$ is defined as $G_{k}(w_{k})_{\\text{a,min}}=\\min \\lbrace G_{k}(\\alpha _{k,\\mathrm {min}},\\hat{\\beta }_{k},w_{k}),G_{k}(\\alpha _{k,\\mathrm {max}},\\hat{\\beta }_{k},w_{k})\\rbrace $ .", "Similarly, given the estimated azimuth angle $\\hat{\\alpha }_{k}$ , the minimum beam gain on the elevation plane with the elevation angle error range $[\\beta _{k,\\mathrm {min}},\\beta _{k,\\mathrm {max}}]$ is defined as $G_{k}(w_{k})_{\\text{e,min}}=\\min \\lbrace G_{k}(\\hat{\\alpha }_{k},\\beta _{k,\\mathrm {min}},w_{k}),G_{k}(\\hat{\\alpha }_{k},\\beta _{k,\\mathrm {max}},w_{k})\\rbrace $ .", "Since the beamwidth is determined by the size of the activated subarray $\\mathcal {S}_{k}^{\\mathrm {r}}\\left(m_{s,k},n_{s,k},p_{k}\\left(\\hat{\\alpha }_{k}\\right)\\right)$ , the beamwidth control is translated to the codeword selection with a proper subarray size $(M_{\\text{act},k}=m_{s,k},\\,N_{\\text{act,}k}=n_{s,k})$ .", "Note that the beamwidth on the azimuth plane and elevation plane is mainly determined by $n_{s,k}$ and $m_{s,k}$ jointly.", "For feasibility, we propose a suboptimal two-step scheme to find the codeword layer indexes $n^*_{s,k}$ and $m^*_{s,k}$ , sequentially.", "More specifically, let us define $\\mathcal {M}(\\mathcal {V}_{r})$ and $\\mathcal {N}(\\mathcal {V}_{r})$ as the sets of all layers' indexes $\\left\\lbrace m_{s}\\right\\rbrace $ and $\\left\\lbrace n_{s}\\right\\rbrace $ in the CCA codebook used by the r-UAV.", "Given $M_{\\text{max},k}:=\\text{max}\\,\\left\\lbrace \\mathcal {M}(\\mathcal {V}_{r})\\right\\rbrace $ and $\\forall n_{s,k}\\in \\mathcal {N}(\\mathcal {V}_{r})$ , the optimal codeword $v(i_{k}^{*},j_{k}^{*},\\mathcal {S}\\left(M_{\\text{max},k},n_{s,k}\\right)):=v^{*}\\left(n_{s,k}|M_{\\text{max},k}\\right)$ can be selected based on the predicted beam angles $(\\hat{\\alpha _{k}},\\hat{\\beta _{k}})$ according to (REF ).", "Then $G_{k}\\left(v^{*}\\left(n_{s,k}|M_{\\text{max},k}\\right)\\right)_{\\text{a,min}}$ can be calculated for all possible $n_{s,k}\\in \\mathcal {N}(\\mathcal {V}_{r})$ , and the one with the largest beam gain can return the optimal $n_{s,k}^{*}$ .", "Now, with the fixed $n_{s,k}^{*}$ and $\\forall m_{s,k}\\in \\mathcal {M}(\\mathcal {V}_{r})$ , we can get $v(i_{k}^{*},j_{k}^{*},\\mathcal {S}(m_{s,k},n_{s,k}^{*})):=v^{*}(m_{s,k}|n_{s,k}^{*})$ , and by finding the largest $G_{k}(v^{*}(m_{s,k}|n_{s,k}^{*}))_{\\mathrm {e,min}}$ in $\\mathcal {M}(\\mathcal {V}_{r})$ , we can get the optimal $m_{s,k}^{*}$ given $n_{s,k}^{*}$ .", "In summary, the TE-aware CCA codeword selection algorithm is given by Algorithm REF , and the corresponding complexity analysis is given as follows.", "Remark 2 The maximum-resolution layer of the multi-resolution CCA codebook with the fixed minimum beamwidth is used in Algorithm REF for the optimal solution.", "To solve the TE-aware beam tracking problem, the search space of Algorithm REF is the whole multi-resolution CCA codebook since the beamwidth is selected according to the estimated TE.", "Complexity Analysis: The complexity of searching for optimal $n_s$ and $m_s$ in our codebook is $\\mathcal {O}((|\\mathcal {M}(\\mathcal {V})|+|\\mathcal {N}(\\mathcal {V})|)M_{\\text{act}}N_{\\text{act}}K)$ , where $|\\cdot |$ is the cardinality of a set; the complexity of the updating procedure is similar with Algorithm 1.", "In total, the complexity of Algorithm REF at each slot is $\\mathcal {O}((|\\mathcal {M}(\\mathcal {V})|+|\\mathcal {N}(\\mathcal {V})|)M_{\\text{act}}N_{\\text{act}}K+K^3)$ in the worst case.", "[h] Tracking-Error-Aware codeword selection with 3D Beamwidth Control [1] Input: $(\\hat{\\alpha _{k}},\\hat{\\beta }_{k})$ , $[\\hat{\\alpha }_{\\text{min},k},\\hat{\\alpha }_{\\text{max},k}]$ , $[\\hat{\\beta }_{\\text{min},k},\\hat{\\beta }_{\\text{max},k}]$ , $\\mathcal {V}_{r}$ $M_{\\text{max},k}=\\text{argmax}_{m\\in \\mathcal {M}(\\mathcal {V}_{r})}\\,m$ .", "$n_{s,k}\\in \\mathcal {N}(\\mathcal {V}_{r})$ Select the optimal codeword $v^{*}(n_{s,k}|M_{\\text{max},k})$ in codebook $\\mathcal {V}_{k}$ according to (REF ).", "Calculate $G_{k}(v^{*}(n_{s,k}|M_{\\text{max},k}))_{\\text{a,min}}$ .", "$n_{s,k}^{*}=\\text{arg max}_{n_{s,k}\\in \\mathcal {N}(\\mathcal {V}_{r})}G(v^{*}(n_{s,k}|M_{\\text{max},k}))_{\\text{a,min}}$ .", "$m_{s,k}\\in \\mathcal {M}(\\mathcal {V}_{r})$ Given $n_{s,k}^{*}$ , select the optimal codeword $v^{*}(m_{s,k}|n_{s,k}^{*})$ in $\\mathcal {V}_{k}$ according to (REF ).", "Calculate $G_{k}(v^{*}(m_{s,k}|n_{s,k}^{*}))_{\\mathrm {e,min}}$ .", "$m_{s,k}^{*}=\\text{arg max}_{m_{s,k}\\in \\mathcal {M}(\\mathcal {V}_{r})}G_{k}(v^{*}(m_{s,k}|n_{s,k}^{*}))_{\\mathrm {e,min}}$ .", "Codeword $w_{k}(\\mathcal {S}_{k}^{r})=v(i_{k}^{*},j_{k}^{*},\\mathcal {S}(m_{s,k}^{*},n_{s,k}^{*})$ is selected.", "Detect confliction and update $w_{k}(\\mathcal {S}_{k}^{r})$ by using Algorithm REF .", "Output: $w_{k}(\\mathcal {S}_{k}^{r})$ .", "Simulation In this section, numerical results are provided to evaluate the effectiveness of the proposed codebook based SPAS algorithm and TE-aware beamwidth control for beam tracking.", "The simulation setups are given as follows.", "The UAV mmWave network with a carrier frequency 60 GHz is considered and the carrier wavelength is ${\\lambda }_{c}=0.005$  m. The inter-UAV mmWave channel follows the model in (REF ).", "The Smooth-Turn mobility model [33] is used to generate the UAV's trajectory on the $xy$ -plane, where the mean of the duration is set as $1/{\\lambda }=1$  s and the variance is set as $\\sigma _r^2=0.05$ .", "The distance between UAVs are limited to no less than $D_{\\text{r,min}}=10$  m. Referring to the composite wing UAV of CHC P316 technical parameters [36], the horizontal velocity is no more than $v_{xy}=20$  m/s, the minimum and maximum vertical velocities are set as $v_{t(r),z,\\text{min}}=2$  m/s and $v_{t(r),z,\\text{max}}=3$  m/s, respectively.", "The time slot duration is set as $\\delta {t}=10$  ms.", "Thus, the horizontal distance of the UAV navigation is approximately no more than $0.2$  m which can be almost neglected during the prediction process.", "The UAV MSI exchanging period is set as $T=50$ , i.e., 500 ms.", "In the simulation, the size of the t-UAV's DRE-covered CCA is set as $N_t=64$ and $M_t=16$ for the consideration of the computational complexity.", "Meanwhile, the radius $R_{\\text{cyl}}$ is set as $R_{\\text{cyl}}=0.0509$  m to achieve the desired beam pattern and the array response.", "Hence, the maximum number of the activated elements on the $xy$ -plane is $N_{\\text{act,max}}=21$ .", "The radiation range of the directive elements on the azimuth plane and the elevation plane is set as $\\Delta \\alpha =\\frac{2\\pi }{3}$ and $\\Delta \\beta =\\pi $ .", "The specific radiation range of the DRE on the azimuth plane and the elevation plane $(m,n)$ for CCA is written as $[\\alpha _{n,\\text{min}},\\alpha _{n,\\text{max}}]\\!=\\!\\left[\\text{mod}(\\phi _{n}\\!-\\!\\frac{\\pi }{3},2\\pi ),\\text{mod}(\\phi _{n}\\!+\\!\\frac{\\pi }{3},2\\pi )\\right],$ where $\\phi _{n}=\\frac{2n-1-N}{2}\\Delta \\phi $ and $[\\beta _{m,\\text{min}},\\beta _{m,\\text{max}}]=[0,\\pi ]$ .", "For comparison, the radiation range of the DRE for the UPA is $[\\alpha _{n,\\text{min}},\\alpha _{n,\\text{max}}]=[-\\frac{\\pi }{3},\\frac{\\pi }{3}]$ and $[\\beta _{m,\\text{min}},\\beta _{m,\\text{max}}]=[0,\\pi ]$ .", "Figure: Spectral efficiency vs. transmit power with different numbers of users.Codebook Based Beam Tracking: CCA Versus UPA Figure: The SNR outage vs. SNR threshold with K=2K=2 t-UAVs, the transmit power p k =0.06p_k=0.06 W and p k =0.08p_k=0.08 W, N act,max =10N_{\\text{act,max}}=10.In this subsection, two beam tracking schemes with different types of antenna array are illustrated by simulation results.", "One is the proposed DRE-covered CCA scheme where all the t-UAVs are equipped with the CCA of the size $N_t=64$ , $M_t=16$ , and the r-UAV is equipped with the CCA of the size $N_r=64$ , $M_r=112$ .", "The other is the DRE-covered UPA scheme where all the t-UAVs are equipped with the UPA of the size $N_t=64$ , $M_t=16$ and the r-UAV is equipped with UPA of the size $N_r=64$ , $M_r=112$ .", "The AOAs and AODs are predicted by using the GP-based position-attitude prediction algorithm in the prediction schemes.", "The real AOAs and AODs are used in the real schemes.", "Without the TE-aware beamwidth control, the initial size of subarray is selected as $M=M_{\\text{act,max}}$ and $N=N_{\\text{act,max}}$ .", "In the CCA scheme, the codeword selection algorithm is performed by t-UAVs and r-UAV.", "In the UPA scheme, the UPA is equally partitioned into subarrays with the size of $M_r/K \\times N_r$ and the beamforming/combining vector is calculated by the UAV position-attitude prediction based beam tracking algorithm in [31].", "The SE is calculated according to (REF ).", "Fig.", "REF shows the sum SE against transmitting power with different number of t-UAVs.", "It is observed that the codeword selection algorithm is effective and the CCA scheme achieves higher SE than the UPA scheme obviously with different t-UAV number $K$ .", "The main reason is that the UPA with DREs can only receive/transmit the signal within a limited angular range at a certain time slot while the CCA does not have such limitation.", "It is also shown that the gap between the real schemes and the prediction schemes is small.", "Thus, the codebook-based beam tracking algorithm is effective in the considered CCA-enabled UAV mmWave network.", "Next, we compare the outage probabilities achieved by the DER-covered CCA and UPA schemes in the UAV mmWave network.", "Here, the network outage probability is defined as [19] $\\text{Pr}(\\underset{1\\le k \\le K}{\\min }{\\text{SNR}_k(t)}<\\text{SNR}_{\\text{th}}),$ where $\\text{SNR}_{\\text{th}}$ is a certain SNR threshold.", "If an arbitrary link between the t-UAV and the r-UAV is in the outage, there is an outage in the network.", "Hence, the outage probability is determined by the minimum SNR among $K$ t-UAVs.", "Fig.", "REF shows the outage probability against the SNR threshold to further demonstrate the coverage of the CCA scheme and the UPA scheme with $K=2$ , and the fixed power $p_k=0.06$  W and $p_k=0.08$  W. It is shown that the CCA scheme is always superior to the UPA scheme in the coverage.", "In some cases, the outage remains unchanged with the threshold increasing.", "The reason is that the outage in these cases is mainly determined by the coverage ability of the arrays.", "Figure: The spectral efficiency vs. the number of t-UAVs with different transmit power.Figure: The spectral efficiency vs. transmit power with different codebooks.In this paper, we mainly focus on the analog beam tracking without considering the inter-UAV interference.", "The sum SE calculated by (REF ) and (REF ) with different numbers of t-UAVs and the given transmit power is shown in Fig.", "REF , respectively, to verify the influence of the inter-UAV interference.", "It is shown that the sum SE of the scheme without interference calculated by (REF ) is similar with that of the scheme with interference calculated by (REF ) with the appropriate number of t-UAVs and the limited transmit power.", "The gap between the schemes increases as the power and the number of t-UAVs increase.", "Therefore, the inter-UAV interference can be neglected in the considered scenario.", "As shown in Fig.", "REF , the SE of the CCA codebook scheme and the traditional codebook scheme is compared.", "The proposed DRE-covered CCA codebook is used in the CCA codebook scheme.", "In the traditional codebook scheme, the codebook without subarray partition is used.", "The CCA on the r-UAV is equally partitioned into $K$ fixed subarrays with the size of $M_r/K\\times N_r$ and the CCA on the t-UAVs is not partitioned.", "The CCA scheme performs better than the traditional scheme with different numbers of t-UAVs and different transmit power.", "The reason is that only a part of activated DREs' radiation range covers the determined radiation direction.", "TE-aware CCA Codeword Selection with 3D Beamwidth Control Figure: The position prediction and the corresponding error range.Although the GP-based UAV's position and attitude prediction results fit well with the position and attitude data, the prediction performance is effected by UAV's mobility.", "When the UAV has higher mobility such as the more random trajectory and high velocity, the prediction error may influence the beam tracking.", "The covariance of the turning radius is set as $\\sigma _r^2=0.06$ which determines the randomness of the trajectory, and the velocity is set as no more than $v_{xy}=20$  m/s.", "The UAV's position and attitude prediction results and the corresponding error range is shown in Fig.", "REF , where the error range with $99\\%$ confidence covers the error between the real data and the predicted data in most cases.", "The SEs of two array schemes against the transmit power with $K=2$ t-UAVs are illustrated in Fig.", "REF .", "The TE-aware codeword selection uses the proposed Algorithm REF and Algorithm REF .", "Serving as a reference, the minimum-beamwidth scheme always select the minimum beamwidth, i.e., the maximum number of antenna elements for an activated subarray.", "To evaluate the performance of the proposed two-step scheme, the exhaustive searching scheme for the optimal layer index is also simulated as a comparison, where the traversal of all codebook layers is executed.", "As shown in Fig.", "REF , the sum SE of the TE-aware codeword selection scheme is better than the minimum-beamwidth codeword selection scheme.", "In addition, the curve of the two-step scheme almost overlaps that of the optimal scheme, as the tracking error of elevation angle is relatively small and the optimal $m$ varies in a relatively small range.", "Figure: Spectral efficiency vs. transmit power under TE-aware beam selection scheme.In order to verify the feasibility of the proposed scheme, the average latency in an e-slot/t-slot/frame duration with different UAV numbers are evaluated as follows: The total latency in e-slot is given by $t_{\\text{total,e}}=t_{\\text{MSI}}+t_{\\text{tra}}+t_{\\text{pro}}+t_{\\text{local,e}}$ and the total latency in t-slot is $t_{\\text{total,t}}=t_{\\text{tra}}+t_{\\text{pro}}+t_{\\text{local,t}}$ , where the transmission time of MSI $t_{\\text{MSI}}=\\frac{B_{\\text{MSI}}}{C_{\\text{LB}}}$ , the transmission time with mmWave band $t_{\\text{tra}}=\\frac{B_{\\text{data}}}{C_{\\text{ave}}}$ , the propagation time $t_{\\text{pro}}=\\frac{D_{k,\\max }}{c}$, and the local processing times $t_{\\text{local,t}}$ and $t_{\\text{local,e}}$ are calculated by the time cost of the proposed algorithms.", "As a frame duration is composed of an e-slot and $T$ t-slots, the average latency over each slot in a frame duration is defined as $t_{\\text{ave}}=\\frac{T\\times t_{\\text{total,t}}+t_{\\text{total,e}}}{T+1}$ .", "Moreover, the data block of MSI is set as $B_{\\text{MSI}}=n_{\\text{MSI}}\\times T\\times B_{\\text{MSI}}$  bits, where $n_{\\text{MSI}}=6$ is the dimension of MSI at each slot, $T=50$ is the number of slots between the adjacent MSI exchanging, and each dimension of MSI at each slot is represented by $B_{\\text{MSI}}=4$  bits.", "The transmission rate of lower band is set as $C_{\\text{LB}}=500$  kbps [37], the data block is set as $B_{\\text{data}}=1$  Mbit, $C_{\\text{ave}}$ is the average rate of mmWave band, $D_{k,\\max }$ is the maximum distance between the t-UAV and the r-UAV, and $c$ is the velocity of light.", "As the computational complexity of the algorithms for the r-UAV is higher than that of t-UAVs, the local processing time mainly depends on the time for the r-UAV to perform the beam tracking algorithms, which is estimated based on the times of multiplication and addition, and the CPU of UAVs.", "The CPU Intel i7-8550u [38] with processor base frequency 1.8 GHz is considered in the simulation, which is adopted by a commonly-used onboard computer “Mainfold 2” supporting many types of UAVs such as DJI Matrice 600 pro, DJI Matrice 600 210 series, and so on [39].", "Note that the accurate MSI is known in e-slot by MSI exchanging and only the codeword selection without beamwidth control is performed in e-slot.", "The local processing times in e-slot and t-slot without TE are given by $t_{\\text{local,e}}=t_{AL2,neb}+t_{AL1}$ and $t_{\\text{local,t}}=t_{AL1}$ , respectively, where $t_{AL2,neb}$ is the computational time of Algorithm 2 without error bounding process and $t_{AL1}$ is the computational time of Algorithm 1.", "When the TE is considered, $t_{\\text{local,e}}=t_{AL2}+t_{AL1}$ and $t_{\\text{local,t}}=t_{AL3}$ , where $t_{AL2}$ is the computational time of Algorithm 2 and $t_{AL3}$ is the computational time of Algorithm 3.", "The schemes with TE and without TE are both evaluated in the simulation.", "In particular, the TE-aware scheme is adopted for the UAV movements with higher randomness ($\\sigma ^{2}=0.06$ ), while the scheme without TE is adopted for the UAV movements with lower randomness ($\\sigma ^{2}=0.05$ ).", "As shown in Fig.", "REF , the maximum average latency is less than 4 ms, during which the movement distance of the UAV navigation is less than 0.08 m according to the UAV's velocity of 20 m/s [36], and hence the impact of the latency on beam angles' calculation for beam tracking can be approximately neglected.", "Therefore, the average latency is tolerable in the actual UAV mmWave networks.", "Moreover, as shown in Fig.", "14, due to the relatively long interval between adjacent MSI exchanging, i.e., 500 ms ($T=50$ ), the average latency in a frame duration is close to that in t-slot, which is considerably lower than e-slot.", "Thus, the proposed algorithms can be applied to high-mobility scenarios.", "Figure: The average latency vs. the number of t-UAVs in e-slot, t-slot, and a frame duration consisting of an e-slot and TT t-slots.", "Conclusion In this paper, we propose a new mmWave beam tracking framework for the CA-enabled UAV mmWave network.", "A specialized hierarchical codebook has been constructed which fully exploits the properties of the DRE-covered CCA, wherein each codeword has a supporting subarray and the corresponding angular domain beam pattern.", "Then, the basic codeword selection principles have been developed for the t-UAV and the r-UAV, respectively; given the estimation of the AOA/AODs, the codeword can be quickly selected that achieves the optimized joint subarray activation and array weighting vector selection for the DRE-covered CCA.", "Moreover, the GP-based UAV position/attitude prediction has been proposed to track the angular information of UAV for the fast codebook-based beam tracking.", "The tracking error (TE) has been carefully bounded and a TE-aware codeword selection scheme has been proposed to adapt the beamwidth for better immunity against the high mobility of UAVs.", "Simulation results validate the performance advantages of the CA-enabled UAV mmWave network over the counterpart employing conventional UPA.", "Driven by our proposed codebook and codeword selection strategies, DRE-covered CCA can significantly improve the SE and coverage of the UAV mmWave network over the conventional UPA, and thus enabling agile and robust beam tracking in the highly dynamic scenarios.", "Proof of Theorem 1 The minimum number of the activated element is denoted as $n_1$ given by $\\begin{aligned}&n_{1}=\\mathop {\\text{arg min}}\\limits _n |\\alpha _0+2l\\pi -\\alpha _{n,\\max }|\\\\&\\text{s.t.", "}\\ {\\alpha _0+2l\\pi } \\ge \\alpha _{n,\\max }.\\end{aligned}$ and the maximum number of the activated element given by $\\begin{aligned}&n_{2}=\\mathop {\\text{arg min}}\\limits _n |\\alpha _0+2l\\pi -\\alpha _{n,\\min }|\\\\&\\text{s.t.", "}\\ {\\alpha _0+2l\\pi } \\le \\alpha _{n,\\min }.\\end{aligned}$ Let $\\alpha _0+2l\\pi -\\alpha _{n,\\max }=0$ and substitute (REF ) and $\\phi _c(n)$ into this equation, $n=\\frac{\\alpha _0+2l\\pi -\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}$ .", "Considering the constraint in (REF ), the optimal solution $n_1$ should satisfy $\\alpha _{n,\\max } \\le \\alpha _{n_1,\\max }$ .", "Therefore, the optimal solution of problem in (REF ) is given by $n_1=\\lceil {n}\\rceil $ .", "The problem in (REF ) can be solved similarly.", "Proof of Theorem 2 The element coverage for the $M \\times N$ -element subarray can be rewritten as $\\begin{aligned}\\mathcal {CV}_{\\text{a(e),element}}&=\\lbrace \\alpha (\\beta )|\\lambda _s(\\alpha ,\\beta )>0\\rbrace \\\\&=\\underset{n(m)}{\\cup }\\lbrace \\alpha (\\beta )|\\left[{{\\Lambda }}\\left( {{\\alpha }},{\\beta } \\right) \\right]_{(m,n)}>0\\rbrace \\\\&=\\underset{n(m)}{\\cup }[\\alpha _{n,\\min }(\\beta _{m,\\min }),\\alpha _{n,\\max }(\\beta _{m,\\max })].\\nonumber \\end{aligned}$ When $\\Delta \\phi _{\\text{c}}\\le \\Delta \\alpha $ , $\\begin{aligned}&\\alpha _{n,\\min }-\\alpha _{n-1,\\max }\\\\&=\\phi _n-\\frac{\\Delta \\alpha }{2}+2l_1\\pi -(\\phi _{n-1}+\\frac{\\Delta \\alpha }{2}+2l_2\\pi )\\\\&=\\Delta \\phi _c-\\Delta \\alpha +2l\\pi \\le 2l\\pi .", "\\nonumber \\end{aligned}$ Therefore, $[\\alpha _{n-1,\\min },\\alpha _{n-1,\\max }]\\cap [\\alpha _{n,\\min },\\alpha _{n,\\max }]=[\\alpha _{n-1,\\min },\\alpha _{n,\\max }],\\nonumber $ and the element coverage is $\\mathcal {CV}_{\\text{a,element}}=[\\alpha _{1,\\min },\\alpha _{N,\\max }]$ .", "The DREs coverage of the $M \\times N$ -element subarray of the CCA on the azimuth plane is $\\begin{aligned}BW_{\\text{a,element}}&=\\alpha _{N,\\max }-\\alpha _{1,\\min }\\\\&=\\phi _c(N)+\\frac{\\Delta \\alpha }{2}+2l_1\\pi -(\\phi _c(1)-\\frac{\\Delta \\alpha }{2}+2l_2\\pi )\\\\&=\\Delta \\alpha +\\phi _c(N)-\\phi _c(1)+2l\\pi \\\\&=\\Delta \\alpha +(N-1)\\Delta \\phi +2l\\pi ,\\nonumber \\end{aligned}$ As all elements on $z$ -axis has the same elevation angle coverage, the DREs coverage of the subbarray is equal to the coverage of each element given by $BW_{\\text{e,element}}=\\Delta \\beta .\\nonumber $ Proof of Theorem 3 According to (REF ), when $N_{\\text{act}}=N_{\\text{act,max}}$ , the position of the center element of all activated elements on the $xy$ -plane is $\\begin{aligned}n_{c}^{\\text{max}}&=\\biggl \\lceil {\\frac{n_1+n_2}{2}}\\biggr \\rceil \\\\&=\\biggl \\lceil \\frac{1}{2}\\left(\\biggl \\lceil {\\frac{\\alpha _0+2l\\pi -\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}}\\biggr \\rceil +\\right.\\\\&\\left.\\biggl \\lceil {\\frac{\\alpha _0+2l\\pi +\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}}\\biggr \\rceil \\right)\\biggr \\rceil \\\\&=\\biggl \\lceil {\\frac{\\alpha +2l\\pi }{\\Delta \\phi }+(N+1)/2}\\biggr \\rceil , l\\in \\mathbb {Z}.\\nonumber \\end{aligned}$ When $M_{\\text{act}}=M_{\\text{act,max}}$ , all elements on the $z$ -axis need to be activated.", "Therefore, the position of the center element of all activated elements on the $z$ -axis is $m_{c}^{\\text{max}}=\\biggl \\lceil {\\frac{1+M}{2}}\\biggr \\rceil .\\nonumber $ When $N_{\\text{act}}<N_{\\text{act,max}}$ and $M_{\\text{act}}<M_{\\text{act,max}}$ , $n_c=n_c^{\\max }$ and $m_c=m_c^{\\max }$ can still allow all elements of the subarray activated as less elements need to be activated.", "Proof of Theorem 4 Note that as the number of the activated elements increases, the array beamwidth increases and the element beamwidth decreases.", "When $BW_{\\text{a(e),array}}\\le BW_{\\text{a(e),element}}$ , $\\mathcal {CV}_{\\text{a(e)}}(i,j,\\mathcal {S}):=\\mathcal {CV}_{\\text{a(e),array}}(i,j,\\mathcal {S})$ and the corresponding beamwidth $BW_{\\text{a(e)}}:=BW_{\\text{a(e),array}}$ ; Otherwise, $\\mathcal {CV}_{\\text{a(e)}}(i,j,\\mathcal {S}):=\\mathcal {CV}_{\\text{a(e),element}}(i,j,\\mathcal {S})$ and the corresponding beamwidth $BW_{\\text{a(e)}}:=BW_{\\text{a(e),element}}$ .", "The element coverage of the $(i,j)$ -the codeword in the $(m_s,n_s)$ -th layer $\\mathcal {CV}_{\\text{a(e),element}}(i,j,\\mathcal {S})$ is given by $\\begin{aligned}&\\mathcal {CV}_{\\text{a,element}}(i,j,\\mathcal {S})\\\\&=[\\alpha _{1,\\min }+(i-1)BW_{\\text{a,element}},\\alpha _{1,\\min }+iBW_{\\text{a,element}}],\\\\&\\mathcal {CV}_{\\text{e,element}}(i,j,\\mathcal {S})=[\\beta _{\\min },\\beta _{\\max }],\\end{aligned}$ where $BW_{\\text{a,element}}$ is given by (REF ) in Theorem REF .", "$\\mathcal {CV}_{\\text{a(e),array}}(i,j,\\mathcal {S})$ is the array coverage of the $(i,j)$ -th codeword in the $(m_s,n_s)$ -th layer, which is given by $\\mathcal {CV}_{\\text{a,array}}(i,j,\\mathcal {S})=[(i-1)BW_{\\text{a,array}},iBW_{\\text{a,array}}],\\\\\\nonumber \\mathcal {CV}_{\\text{e,array}}(i,j,\\mathcal {S})=[(j-1)BW_{\\text{e,array}},jBW_{\\text{e,array}}].$ The array beamwidth is set as $BW_{\\text{a,array}}=\\frac{2\\pi }{n_s}$ and $BW_{\\text{e,array}}=\\frac{2\\pi }{m_s}$ in the $(m_s,n_s)$ -th layer.", "When $BW_{\\text{a,array}}\\le BW_{\\text{a,element}}$ , i.e., $\\frac{2\\pi }{n_s}\\le \\Delta \\alpha +(n_s-1)\\Delta \\phi $ , the union of the beam coverage in the azimuth plane of all codewords in each layer is $\\begin{aligned}&\\underset{i\\in \\mathcal {I}}{\\cup }\\mathcal {CV}_{\\text{a}}(i,j,\\mathcal {S})=[0,I*BW_{\\text{a,array}}]=[0,2\\pi ].\\nonumber \\end{aligned}$ Otherwise, the union of the beam coverage is given by $\\begin{aligned}&\\underset{i\\in \\mathcal {I}}{\\cup }\\mathcal {CV}_{\\text{a}}(i,j,\\mathcal {S})\\\\&=[\\alpha _{1,\\min },\\alpha _{1,\\min }+I*BW_{\\text{a,element}}]\\\\&=[\\alpha _{1,\\min },\\alpha _{1,\\min }+2\\pi ].\\end{aligned}$ Therefore, the union of the beam coverage of all codewords in each layer covers the whole azimuth angular domain.", "Revisiting Theorem REF , $BW_{\\text{e,element}}=\\Delta \\beta =\\pi $ and $BW_{\\text{e,array}}\\le BW_{\\text{e,element}}$ hold for all layers.", "The union of the beam coverage in the elevation plane of all codewords in each layer is $\\begin{aligned}&\\underset{j\\in \\mathcal {J}}{\\cup }\\mathcal {CV}_{\\text{e}}(i,j,\\mathcal {S})=[0,J*BW_{\\text{e,array}}]=[0,2\\pi ].\\nonumber \\end{aligned}$ Hence, the beam coverage of all codewords in each layer of the designed codebook covers the whole angular domain." ], [ "Simulation", "In this section, numerical results are provided to evaluate the effectiveness of the proposed codebook based SPAS algorithm and TE-aware beamwidth control for beam tracking.", "The simulation setups are given as follows.", "The UAV mmWave network with a carrier frequency 60 GHz is considered and the carrier wavelength is ${\\lambda }_{c}=0.005$  m. The inter-UAV mmWave channel follows the model in (REF ).", "The Smooth-Turn mobility model [33] is used to generate the UAV's trajectory on the $xy$ -plane, where the mean of the duration is set as $1/{\\lambda }=1$  s and the variance is set as $\\sigma _r^2=0.05$ .", "The distance between UAVs are limited to no less than $D_{\\text{r,min}}=10$  m. Referring to the composite wing UAV of CHC P316 technical parameters [36], the horizontal velocity is no more than $v_{xy}=20$  m/s, the minimum and maximum vertical velocities are set as $v_{t(r),z,\\text{min}}=2$  m/s and $v_{t(r),z,\\text{max}}=3$  m/s, respectively.", "The time slot duration is set as $\\delta {t}=10$  ms.", "Thus, the horizontal distance of the UAV navigation is approximately no more than $0.2$  m which can be almost neglected during the prediction process.", "The UAV MSI exchanging period is set as $T=50$ , i.e., 500 ms.", "In the simulation, the size of the t-UAV's DRE-covered CCA is set as $N_t=64$ and $M_t=16$ for the consideration of the computational complexity.", "Meanwhile, the radius $R_{\\text{cyl}}$ is set as $R_{\\text{cyl}}=0.0509$  m to achieve the desired beam pattern and the array response.", "Hence, the maximum number of the activated elements on the $xy$ -plane is $N_{\\text{act,max}}=21$ .", "The radiation range of the directive elements on the azimuth plane and the elevation plane is set as $\\Delta \\alpha =\\frac{2\\pi }{3}$ and $\\Delta \\beta =\\pi $ .", "The specific radiation range of the DRE on the azimuth plane and the elevation plane $(m,n)$ for CCA is written as $[\\alpha _{n,\\text{min}},\\alpha _{n,\\text{max}}]\\!=\\!\\left[\\text{mod}(\\phi _{n}\\!-\\!\\frac{\\pi }{3},2\\pi ),\\text{mod}(\\phi _{n}\\!+\\!\\frac{\\pi }{3},2\\pi )\\right],$ where $\\phi _{n}=\\frac{2n-1-N}{2}\\Delta \\phi $ and $[\\beta _{m,\\text{min}},\\beta _{m,\\text{max}}]=[0,\\pi ]$ .", "For comparison, the radiation range of the DRE for the UPA is $[\\alpha _{n,\\text{min}},\\alpha _{n,\\text{max}}]=[-\\frac{\\pi }{3},\\frac{\\pi }{3}]$ and $[\\beta _{m,\\text{min}},\\beta _{m,\\text{max}}]=[0,\\pi ]$ .", "Figure: Spectral efficiency vs. transmit power with different numbers of users." ], [ "Codebook Based Beam Tracking: CCA Versus UPA", "In this subsection, two beam tracking schemes with different types of antenna array are illustrated by simulation results.", "One is the proposed DRE-covered CCA scheme where all the t-UAVs are equipped with the CCA of the size $N_t=64$ , $M_t=16$ , and the r-UAV is equipped with the CCA of the size $N_r=64$ , $M_r=112$ .", "The other is the DRE-covered UPA scheme where all the t-UAVs are equipped with the UPA of the size $N_t=64$ , $M_t=16$ and the r-UAV is equipped with UPA of the size $N_r=64$ , $M_r=112$ .", "The AOAs and AODs are predicted by using the GP-based position-attitude prediction algorithm in the prediction schemes.", "The real AOAs and AODs are used in the real schemes.", "Without the TE-aware beamwidth control, the initial size of subarray is selected as $M=M_{\\text{act,max}}$ and $N=N_{\\text{act,max}}$ .", "In the CCA scheme, the codeword selection algorithm is performed by t-UAVs and r-UAV.", "In the UPA scheme, the UPA is equally partitioned into subarrays with the size of $M_r/K \\times N_r$ and the beamforming/combining vector is calculated by the UAV position-attitude prediction based beam tracking algorithm in [31].", "The SE is calculated according to (REF ).", "Fig.", "REF shows the sum SE against transmitting power with different number of t-UAVs.", "It is observed that the codeword selection algorithm is effective and the CCA scheme achieves higher SE than the UPA scheme obviously with different t-UAV number $K$ .", "The main reason is that the UPA with DREs can only receive/transmit the signal within a limited angular range at a certain time slot while the CCA does not have such limitation.", "It is also shown that the gap between the real schemes and the prediction schemes is small.", "Thus, the codebook-based beam tracking algorithm is effective in the considered CCA-enabled UAV mmWave network.", "Next, we compare the outage probabilities achieved by the DER-covered CCA and UPA schemes in the UAV mmWave network.", "Here, the network outage probability is defined as [19] $\\text{Pr}(\\underset{1\\le k \\le K}{\\min }{\\text{SNR}_k(t)}<\\text{SNR}_{\\text{th}}),$ where $\\text{SNR}_{\\text{th}}$ is a certain SNR threshold.", "If an arbitrary link between the t-UAV and the r-UAV is in the outage, there is an outage in the network.", "Hence, the outage probability is determined by the minimum SNR among $K$ t-UAVs.", "Fig.", "REF shows the outage probability against the SNR threshold to further demonstrate the coverage of the CCA scheme and the UPA scheme with $K=2$ , and the fixed power $p_k=0.06$  W and $p_k=0.08$  W. It is shown that the CCA scheme is always superior to the UPA scheme in the coverage.", "In some cases, the outage remains unchanged with the threshold increasing.", "The reason is that the outage in these cases is mainly determined by the coverage ability of the arrays.", "Figure: The spectral efficiency vs. the number of t-UAVs with different transmit power.Figure: The spectral efficiency vs. transmit power with different codebooks.In this paper, we mainly focus on the analog beam tracking without considering the inter-UAV interference.", "The sum SE calculated by (REF ) and (REF ) with different numbers of t-UAVs and the given transmit power is shown in Fig.", "REF , respectively, to verify the influence of the inter-UAV interference.", "It is shown that the sum SE of the scheme without interference calculated by (REF ) is similar with that of the scheme with interference calculated by (REF ) with the appropriate number of t-UAVs and the limited transmit power.", "The gap between the schemes increases as the power and the number of t-UAVs increase.", "Therefore, the inter-UAV interference can be neglected in the considered scenario.", "As shown in Fig.", "REF , the SE of the CCA codebook scheme and the traditional codebook scheme is compared.", "The proposed DRE-covered CCA codebook is used in the CCA codebook scheme.", "In the traditional codebook scheme, the codebook without subarray partition is used.", "The CCA on the r-UAV is equally partitioned into $K$ fixed subarrays with the size of $M_r/K\\times N_r$ and the CCA on the t-UAVs is not partitioned.", "The CCA scheme performs better than the traditional scheme with different numbers of t-UAVs and different transmit power.", "The reason is that only a part of activated DREs' radiation range covers the determined radiation direction." ], [ "TE-aware CCA Codeword Selection with 3D Beamwidth Control", "Although the GP-based UAV's position and attitude prediction results fit well with the position and attitude data, the prediction performance is effected by UAV's mobility.", "When the UAV has higher mobility such as the more random trajectory and high velocity, the prediction error may influence the beam tracking.", "The covariance of the turning radius is set as $\\sigma _r^2=0.06$ which determines the randomness of the trajectory, and the velocity is set as no more than $v_{xy}=20$  m/s.", "The UAV's position and attitude prediction results and the corresponding error range is shown in Fig.", "REF , where the error range with $99\\%$ confidence covers the error between the real data and the predicted data in most cases.", "The SEs of two array schemes against the transmit power with $K=2$ t-UAVs are illustrated in Fig.", "REF .", "The TE-aware codeword selection uses the proposed Algorithm REF and Algorithm REF .", "Serving as a reference, the minimum-beamwidth scheme always select the minimum beamwidth, i.e., the maximum number of antenna elements for an activated subarray.", "To evaluate the performance of the proposed two-step scheme, the exhaustive searching scheme for the optimal layer index is also simulated as a comparison, where the traversal of all codebook layers is executed.", "As shown in Fig.", "REF , the sum SE of the TE-aware codeword selection scheme is better than the minimum-beamwidth codeword selection scheme.", "In addition, the curve of the two-step scheme almost overlaps that of the optimal scheme, as the tracking error of elevation angle is relatively small and the optimal $m$ varies in a relatively small range.", "Figure: Spectral efficiency vs. transmit power under TE-aware beam selection scheme.In order to verify the feasibility of the proposed scheme, the average latency in an e-slot/t-slot/frame duration with different UAV numbers are evaluated as follows: The total latency in e-slot is given by $t_{\\text{total,e}}=t_{\\text{MSI}}+t_{\\text{tra}}+t_{\\text{pro}}+t_{\\text{local,e}}$ and the total latency in t-slot is $t_{\\text{total,t}}=t_{\\text{tra}}+t_{\\text{pro}}+t_{\\text{local,t}}$ , where the transmission time of MSI $t_{\\text{MSI}}=\\frac{B_{\\text{MSI}}}{C_{\\text{LB}}}$ , the transmission time with mmWave band $t_{\\text{tra}}=\\frac{B_{\\text{data}}}{C_{\\text{ave}}}$ , the propagation time $t_{\\text{pro}}=\\frac{D_{k,\\max }}{c}$, and the local processing times $t_{\\text{local,t}}$ and $t_{\\text{local,e}}$ are calculated by the time cost of the proposed algorithms.", "As a frame duration is composed of an e-slot and $T$ t-slots, the average latency over each slot in a frame duration is defined as $t_{\\text{ave}}=\\frac{T\\times t_{\\text{total,t}}+t_{\\text{total,e}}}{T+1}$ .", "Moreover, the data block of MSI is set as $B_{\\text{MSI}}=n_{\\text{MSI}}\\times T\\times B_{\\text{MSI}}$  bits, where $n_{\\text{MSI}}=6$ is the dimension of MSI at each slot, $T=50$ is the number of slots between the adjacent MSI exchanging, and each dimension of MSI at each slot is represented by $B_{\\text{MSI}}=4$  bits.", "The transmission rate of lower band is set as $C_{\\text{LB}}=500$  kbps [37], the data block is set as $B_{\\text{data}}=1$  Mbit, $C_{\\text{ave}}$ is the average rate of mmWave band, $D_{k,\\max }$ is the maximum distance between the t-UAV and the r-UAV, and $c$ is the velocity of light.", "As the computational complexity of the algorithms for the r-UAV is higher than that of t-UAVs, the local processing time mainly depends on the time for the r-UAV to perform the beam tracking algorithms, which is estimated based on the times of multiplication and addition, and the CPU of UAVs.", "The CPU Intel i7-8550u [38] with processor base frequency 1.8 GHz is considered in the simulation, which is adopted by a commonly-used onboard computer “Mainfold 2” supporting many types of UAVs such as DJI Matrice 600 pro, DJI Matrice 600 210 series, and so on [39].", "Note that the accurate MSI is known in e-slot by MSI exchanging and only the codeword selection without beamwidth control is performed in e-slot.", "The local processing times in e-slot and t-slot without TE are given by $t_{\\text{local,e}}=t_{AL2,neb}+t_{AL1}$ and $t_{\\text{local,t}}=t_{AL1}$ , respectively, where $t_{AL2,neb}$ is the computational time of Algorithm 2 without error bounding process and $t_{AL1}$ is the computational time of Algorithm 1.", "When the TE is considered, $t_{\\text{local,e}}=t_{AL2}+t_{AL1}$ and $t_{\\text{local,t}}=t_{AL3}$ , where $t_{AL2}$ is the computational time of Algorithm 2 and $t_{AL3}$ is the computational time of Algorithm 3.", "The schemes with TE and without TE are both evaluated in the simulation.", "In particular, the TE-aware scheme is adopted for the UAV movements with higher randomness ($\\sigma ^{2}=0.06$ ), while the scheme without TE is adopted for the UAV movements with lower randomness ($\\sigma ^{2}=0.05$ ).", "As shown in Fig.", "REF , the maximum average latency is less than 4 ms, during which the movement distance of the UAV navigation is less than 0.08 m according to the UAV's velocity of 20 m/s [36], and hence the impact of the latency on beam angles' calculation for beam tracking can be approximately neglected.", "Therefore, the average latency is tolerable in the actual UAV mmWave networks.", "Moreover, as shown in Fig.", "14, due to the relatively long interval between adjacent MSI exchanging, i.e., 500 ms ($T=50$ ), the average latency in a frame duration is close to that in t-slot, which is considerably lower than e-slot.", "Thus, the proposed algorithms can be applied to high-mobility scenarios.", "Figure: The average latency vs. the number of t-UAVs in e-slot, t-slot, and a frame duration consisting of an e-slot and TT t-slots." ], [ "Conclusion", "In this paper, we propose a new mmWave beam tracking framework for the CA-enabled UAV mmWave network.", "A specialized hierarchical codebook has been constructed which fully exploits the properties of the DRE-covered CCA, wherein each codeword has a supporting subarray and the corresponding angular domain beam pattern.", "Then, the basic codeword selection principles have been developed for the t-UAV and the r-UAV, respectively; given the estimation of the AOA/AODs, the codeword can be quickly selected that achieves the optimized joint subarray activation and array weighting vector selection for the DRE-covered CCA.", "Moreover, the GP-based UAV position/attitude prediction has been proposed to track the angular information of UAV for the fast codebook-based beam tracking.", "The tracking error (TE) has been carefully bounded and a TE-aware codeword selection scheme has been proposed to adapt the beamwidth for better immunity against the high mobility of UAVs.", "Simulation results validate the performance advantages of the CA-enabled UAV mmWave network over the counterpart employing conventional UPA.", "Driven by our proposed codebook and codeword selection strategies, DRE-covered CCA can significantly improve the SE and coverage of the UAV mmWave network over the conventional UPA, and thus enabling agile and robust beam tracking in the highly dynamic scenarios." ], [ "Proof of Theorem 1", "The minimum number of the activated element is denoted as $n_1$ given by $\\begin{aligned}&n_{1}=\\mathop {\\text{arg min}}\\limits _n |\\alpha _0+2l\\pi -\\alpha _{n,\\max }|\\\\&\\text{s.t.", "}\\ {\\alpha _0+2l\\pi } \\ge \\alpha _{n,\\max }.\\end{aligned}$ and the maximum number of the activated element given by $\\begin{aligned}&n_{2}=\\mathop {\\text{arg min}}\\limits _n |\\alpha _0+2l\\pi -\\alpha _{n,\\min }|\\\\&\\text{s.t.", "}\\ {\\alpha _0+2l\\pi } \\le \\alpha _{n,\\min }.\\end{aligned}$ Let $\\alpha _0+2l\\pi -\\alpha _{n,\\max }=0$ and substitute (REF ) and $\\phi _c(n)$ into this equation, $n=\\frac{\\alpha _0+2l\\pi -\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}$ .", "Considering the constraint in (REF ), the optimal solution $n_1$ should satisfy $\\alpha _{n,\\max } \\le \\alpha _{n_1,\\max }$ .", "Therefore, the optimal solution of problem in (REF ) is given by $n_1=\\lceil {n}\\rceil $ .", "The problem in (REF ) can be solved similarly." ], [ "Proof of Theorem 2", "The element coverage for the $M \\times N$ -element subarray can be rewritten as $\\begin{aligned}\\mathcal {CV}_{\\text{a(e),element}}&=\\lbrace \\alpha (\\beta )|\\lambda _s(\\alpha ,\\beta )>0\\rbrace \\\\&=\\underset{n(m)}{\\cup }\\lbrace \\alpha (\\beta )|\\left[{{\\Lambda }}\\left( {{\\alpha }},{\\beta } \\right) \\right]_{(m,n)}>0\\rbrace \\\\&=\\underset{n(m)}{\\cup }[\\alpha _{n,\\min }(\\beta _{m,\\min }),\\alpha _{n,\\max }(\\beta _{m,\\max })].\\nonumber \\end{aligned}$ When $\\Delta \\phi _{\\text{c}}\\le \\Delta \\alpha $ , $\\begin{aligned}&\\alpha _{n,\\min }-\\alpha _{n-1,\\max }\\\\&=\\phi _n-\\frac{\\Delta \\alpha }{2}+2l_1\\pi -(\\phi _{n-1}+\\frac{\\Delta \\alpha }{2}+2l_2\\pi )\\\\&=\\Delta \\phi _c-\\Delta \\alpha +2l\\pi \\le 2l\\pi .", "\\nonumber \\end{aligned}$ Therefore, $[\\alpha _{n-1,\\min },\\alpha _{n-1,\\max }]\\cap [\\alpha _{n,\\min },\\alpha _{n,\\max }]=[\\alpha _{n-1,\\min },\\alpha _{n,\\max }],\\nonumber $ and the element coverage is $\\mathcal {CV}_{\\text{a,element}}=[\\alpha _{1,\\min },\\alpha _{N,\\max }]$ .", "The DREs coverage of the $M \\times N$ -element subarray of the CCA on the azimuth plane is $\\begin{aligned}BW_{\\text{a,element}}&=\\alpha _{N,\\max }-\\alpha _{1,\\min }\\\\&=\\phi _c(N)+\\frac{\\Delta \\alpha }{2}+2l_1\\pi -(\\phi _c(1)-\\frac{\\Delta \\alpha }{2}+2l_2\\pi )\\\\&=\\Delta \\alpha +\\phi _c(N)-\\phi _c(1)+2l\\pi \\\\&=\\Delta \\alpha +(N-1)\\Delta \\phi +2l\\pi ,\\nonumber \\end{aligned}$ As all elements on $z$ -axis has the same elevation angle coverage, the DREs coverage of the subbarray is equal to the coverage of each element given by $BW_{\\text{e,element}}=\\Delta \\beta .\\nonumber $" ], [ "Proof of Theorem 3", "According to (REF ), when $N_{\\text{act}}=N_{\\text{act,max}}$ , the position of the center element of all activated elements on the $xy$ -plane is $\\begin{aligned}n_{c}^{\\text{max}}&=\\biggl \\lceil {\\frac{n_1+n_2}{2}}\\biggr \\rceil \\\\&=\\biggl \\lceil \\frac{1}{2}\\left(\\biggl \\lceil {\\frac{\\alpha _0+2l\\pi -\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}}\\biggr \\rceil +\\right.\\\\&\\left.\\biggl \\lceil {\\frac{\\alpha _0+2l\\pi +\\Delta \\alpha /2}{\\Delta \\phi }+\\frac{N+1}{2}}\\biggr \\rceil \\right)\\biggr \\rceil \\\\&=\\biggl \\lceil {\\frac{\\alpha +2l\\pi }{\\Delta \\phi }+(N+1)/2}\\biggr \\rceil , l\\in \\mathbb {Z}.\\nonumber \\end{aligned}$ When $M_{\\text{act}}=M_{\\text{act,max}}$ , all elements on the $z$ -axis need to be activated.", "Therefore, the position of the center element of all activated elements on the $z$ -axis is $m_{c}^{\\text{max}}=\\biggl \\lceil {\\frac{1+M}{2}}\\biggr \\rceil .\\nonumber $ When $N_{\\text{act}}<N_{\\text{act,max}}$ and $M_{\\text{act}}<M_{\\text{act,max}}$ , $n_c=n_c^{\\max }$ and $m_c=m_c^{\\max }$ can still allow all elements of the subarray activated as less elements need to be activated." ], [ "Proof of Theorem 4", "Note that as the number of the activated elements increases, the array beamwidth increases and the element beamwidth decreases.", "When $BW_{\\text{a(e),array}}\\le BW_{\\text{a(e),element}}$ , $\\mathcal {CV}_{\\text{a(e)}}(i,j,\\mathcal {S}):=\\mathcal {CV}_{\\text{a(e),array}}(i,j,\\mathcal {S})$ and the corresponding beamwidth $BW_{\\text{a(e)}}:=BW_{\\text{a(e),array}}$ ; Otherwise, $\\mathcal {CV}_{\\text{a(e)}}(i,j,\\mathcal {S}):=\\mathcal {CV}_{\\text{a(e),element}}(i,j,\\mathcal {S})$ and the corresponding beamwidth $BW_{\\text{a(e)}}:=BW_{\\text{a(e),element}}$ .", "The element coverage of the $(i,j)$ -the codeword in the $(m_s,n_s)$ -th layer $\\mathcal {CV}_{\\text{a(e),element}}(i,j,\\mathcal {S})$ is given by $\\begin{aligned}&\\mathcal {CV}_{\\text{a,element}}(i,j,\\mathcal {S})\\\\&=[\\alpha _{1,\\min }+(i-1)BW_{\\text{a,element}},\\alpha _{1,\\min }+iBW_{\\text{a,element}}],\\\\&\\mathcal {CV}_{\\text{e,element}}(i,j,\\mathcal {S})=[\\beta _{\\min },\\beta _{\\max }],\\end{aligned}$ where $BW_{\\text{a,element}}$ is given by (REF ) in Theorem REF .", "$\\mathcal {CV}_{\\text{a(e),array}}(i,j,\\mathcal {S})$ is the array coverage of the $(i,j)$ -th codeword in the $(m_s,n_s)$ -th layer, which is given by $\\mathcal {CV}_{\\text{a,array}}(i,j,\\mathcal {S})=[(i-1)BW_{\\text{a,array}},iBW_{\\text{a,array}}],\\\\\\nonumber \\mathcal {CV}_{\\text{e,array}}(i,j,\\mathcal {S})=[(j-1)BW_{\\text{e,array}},jBW_{\\text{e,array}}].$ The array beamwidth is set as $BW_{\\text{a,array}}=\\frac{2\\pi }{n_s}$ and $BW_{\\text{e,array}}=\\frac{2\\pi }{m_s}$ in the $(m_s,n_s)$ -th layer.", "When $BW_{\\text{a,array}}\\le BW_{\\text{a,element}}$ , i.e., $\\frac{2\\pi }{n_s}\\le \\Delta \\alpha +(n_s-1)\\Delta \\phi $ , the union of the beam coverage in the azimuth plane of all codewords in each layer is $\\begin{aligned}&\\underset{i\\in \\mathcal {I}}{\\cup }\\mathcal {CV}_{\\text{a}}(i,j,\\mathcal {S})=[0,I*BW_{\\text{a,array}}]=[0,2\\pi ].\\nonumber \\end{aligned}$ Otherwise, the union of the beam coverage is given by $\\begin{aligned}&\\underset{i\\in \\mathcal {I}}{\\cup }\\mathcal {CV}_{\\text{a}}(i,j,\\mathcal {S})\\\\&=[\\alpha _{1,\\min },\\alpha _{1,\\min }+I*BW_{\\text{a,element}}]\\\\&=[\\alpha _{1,\\min },\\alpha _{1,\\min }+2\\pi ].\\end{aligned}$ Therefore, the union of the beam coverage of all codewords in each layer covers the whole azimuth angular domain.", "Revisiting Theorem REF , $BW_{\\text{e,element}}=\\Delta \\beta =\\pi $ and $BW_{\\text{e,array}}\\le BW_{\\text{e,element}}$ hold for all layers.", "The union of the beam coverage in the elevation plane of all codewords in each layer is $\\begin{aligned}&\\underset{j\\in \\mathcal {J}}{\\cup }\\mathcal {CV}_{\\text{e}}(i,j,\\mathcal {S})=[0,J*BW_{\\text{e,array}}]=[0,2\\pi ].\\nonumber \\end{aligned}$ Hence, the beam coverage of all codewords in each layer of the designed codebook covers the whole angular domain." ] ]

[ [ "General Probabilistic Theories with a Gleason-type Theorem" ], [ "Abstract Gleason-type theorems for quantum theory allow one to recover the quantum state space by assuming that (i) states consistently assign probabilities to measurement outcomes and that (ii) there is a unique state for every such assignment.", "We identify the class of general probabilistic theories which also admit Gleason-type theorems.", "It contains theories satisfying the no-restriction hypothesis as well as others which can simulate such an unrestricted theory arbitrarily well when allowing for post-selection on measurement outcomes.", "Our result also implies that the standard no-restriction hypothesis applied to effects is not equivalent to the dual no-restriction hypothesis applied to states which is found to be less restrictive." ], [ "Introduction", "More than sixty years ago, Mackey [1] asked whether density operators represent the most general notion of a quantum state that is consistent with the standard description of observables as self-adjoint operators.", "Gleason [2] responded with a proof that—in separable Hilbert spaces of dimension greater than two—every state must admit an expression in terms of a density operator if it is to consistently assign probabilities to the measurement outcomes of such observables.", "In 2003, Busch [3] (and then Caves et al.", "[4]) generalized the idea of Gleason's theorem to observables represented by positive-operator measures.", "The resulting Gleason-type theorem (GTT) is much simpler to prove and it also applies to two-dimensional Hilbert spaces, since the assumptions being made are stronger than in Gleason's case.", "In this paper, we investigate whether the Gleason-type theorem is special to quantum theory.", "Imagine that a theory different from quantum theory had been found to successfully describe Nature.", "Would a GTT still exist?", "Our question is made explicit by posing it within the family of general probabilistic theories (GPTs) which have emerged as natural generalizations of quantum theory [5], [6], [7].", "The framework of GPTs derives from operational principles and it encompasses both quantum and classical models.", "One of the motivations to explore these alternative theories has been to identify features which single out quantum theory among others of comparable structure.", "Our study contributes to that fundamental quest.", "The results of Gleason and Busch establish a bijection between frame functions and density operators in quantum theory.", "Frame functions associate probabilities to the mathematical objects representing the possible outcomes of measurements in such a way that the probabilities assigned to all disjoint outcomes of a given measurement sum to unity.", "The rationale behind a frame function is that the probabilities of all measurement outcomes for all observables should define a unique state.", "If this were not the case, then two “different” states would be indistinguishable, both practically and theoretically.", "Our strategy will be to generalise the concept of frame functions to GPTs in order to investigate whether they are in exact correspondence with the objects that represent states in these theories.", "The main result of this paper is a proof that the correspondence continues to hold if and only if the state and effect spaces of a GPT satisfy the no-restriction hypothesis [8], [9], or a “noisy” version thereof (see Theorem REF , Section REF ).", "In other words, we identify exactly all general probabilistic theories which admit a Gleason-type theorem.", "The existence of a GTT for GPTs such as quantum theory or real-vector-space quantum theory [10], [11], [12], [13] has a number of consequences.", "It becomes possible, for example, to modify the axiomatic structure of the theories as it is no longer necessary to, separately and independently, stipulate both the state space and the observables of the theory.", "Our result can also be used to derive the standard GPT framework from operational assumptions different to those found in the literature.", "More specifically, the standard GPT framework is recovered if—after motivating the standard description of observables in GPTs—states are assumed to correspond to frame functions of these observables.", "To make the paper self-contained and to introduce the notation, we will first review concepts of the GPT framework relevant here.", "In Section , we define frame functions for GPTs and prove Theorem REF .", "In Section , we provide three examples to demonstrate the simplification of the postulates required to specify an individual GPT.", "Section strengthens Theorem REF by defining frame functions only on a proper subset of all observables, the analog of projective-simulable observables.", "The stronger result makes possible an alternative operational motivation of the GPT framework.", "In Section we summarize and discuss our results." ], [ "General probabilistic theories", "The GPT framework allows one to define a broad family of theories of which quantum theory (in finite dimensional Hilbert spaces) is a member.", "Any (real or fictitious) system described by a GPT has the following fundamental property: there exists a finite set of fiducial measurement outcomes, the probabilities of which uniquely determine its stateThe restriction to a finite set of fiducial measurement outcomes has been relaxed by e.g.", "Nuida et al.", "[14], allowing the framework to encompass quantum theory in toto.. For example, the state of a spin-$1/2$ particle is determined by the probabilities of the $+1$ outcome of measuring spin observables in three orthogonal directions, as demonstrated by the Bloch vector description.", "There are many different yet equivalent ways to formulate the GPT framework.", "To make this paper self-contained, let us briefly outline an intuitive approach to GPTs which is based on an operational derivation [7]." ], [ "States", "If a system has a minimal fiducial set consisting of $d$ outcomesMinimal meaning that there is no fiducial set for the system with fewer than $d$ outcomes., its state space $\\mathcal {S}$ is given by a convex, compact set of vectors of the form $\\omega =\\begin{pmatrix}p_{1}\\\\\\vdots \\\\p_{d}\\\\1\\end{pmatrix}\\in \\mathbb {R}^{d+1},$ where $p_{k}\\in \\left[0,1\\right],k=1\\ldots d$ , are the probabilities of the fiducial outcomes.", "The extra dimension of the “ambient” vector space simplifies the description of measurement outcomes, as explained below.", "The convexity of the state space is derived from the assumption that if one were to prepare the system in the states $\\omega $ and $\\omega ^{\\prime }=\\left(p_{1}^{\\prime },\\ldots ,p_{d}^{\\prime },1\\right)^{T}$ with probabilities $\\lambda $ and $(1-\\lambda )$ , respectively, then the probability of observing the $k$ -th fiducial measurement outcome should equal $p_{k}^{\\prime \\prime }(\\lambda )=\\lambda p_{k}+\\left(1-\\lambda \\right)p_{k}^{\\prime }\\,,\\qquad \\lambda \\in [0,1],\\quad k=1\\ldots d\\,,$ and, therefore, this mixed state should be represented by the vector $\\omega ^{\\prime \\prime }(\\lambda )=\\lambda \\omega +\\left(1-\\lambda \\right)\\omega ^{\\prime }\\,.$ A state $\\omega $ is extremal if it cannot be written as a (non-trivial) convex combination of other states.", "The state space is assumed to be compact since, firstly, it must be bounded if the entries of the vector are to be between zero and one.", "Secondly, as an arbitrarily good approximation of a state would be operationally indistinguishable from the state itself, we also assume the state space is closed in the topological sense.", "As an example, consider a classical bit which may reside in one of two states called “0” and “1”, or in a mixture of the two.", "If we know that the bit is in state 0 with probability $p$ then it is in state 1 with probability $(1-p)$ ; in other words, the number $p\\in [0,1]$ determines the state of the system.", "When performing the measurement which asks “Is the bit in state 0 or 1?”, the outcome “The bit is in state 0.” forms a complete set of fiducial measurement outcomes.", "Thus, the state space $\\mathcal {S}_{b}$ of the bit can be represented by the line segment between $(0,1)^{T}$ and $(1,1)^{T}$ , as displayed in Figure REF (see Section REF ).", "The end points of the segment correspond to the states 0 and 1, respectively, and their convex hull defines the state space ${\\cal S}_{b}$ ." ], [ "Effects and observables", "The possible outcomes of measuring an observable in a GPT correspond to effects which are linear maps $e:\\mathbb {R}^{d+1}\\rightarrow \\mathbb {R}$ such that $0\\le e\\left(\\omega \\right)\\le 1$ for all states $\\omega \\in \\mathcal {S}$ ; here $e\\left(\\omega \\right)$ denotes the probability of observing the outcome $e$ when a measurement $\\mathbb {M}$ (with $e$ as a possible outcome) is performed on a system in state $\\omega $ .", "Due to the linearity of the map $e$ , any effect can be uniquely expressed in the form $e\\left(\\omega \\right)=e\\cdot \\omega ,$ for some vector $e\\in \\mathbb {R}^{d+1}$ .", "We will also use the term “effect” to refer to the vector $e$ characterizing a map $e$ .", "The linearity of effects is motivated by the assumption that they should respect the mixing of states with some parameter $\\lambda \\in [0,1]$ .", "More specifically, the following two events should occur with the same probability: observing the outcome $e$ of a measurement $\\mathbb {M}$ performed on a system in a mixed state $\\omega ^{\\prime \\prime }(\\lambda )=\\lambda \\omega +\\left(1-\\lambda \\right)\\omega ^{\\prime }$ ; observing the outcome $e$ when the measurement $\\mathbb {M}$ is performed with probability $\\lambda $ on a system in state $\\omega $ and with probability $(1-\\lambda )$ on a system prepared in state $\\omega ^{\\prime }$ .", "This assumption implies that the map $e$ should satisfy $e\\left(\\lambda \\omega +\\left(1-\\lambda \\right)\\omega ^{\\prime }\\right)=\\lambda e\\left(\\omega \\right)+\\left(1-\\lambda \\right)e\\left(\\omega ^{\\prime }\\right)\\,,\\qquad \\omega ,\\omega ^{\\prime }\\in \\mathcal {S}\\,,$ and thus be an affine function on the state space $\\mathcal {S}$ which can be extended to a linear function on the vector space $\\mathbb {R}^{d+1}$ containing $\\mathcal {S}$ .", "The set of all effects associated with measurement outcomes in a specific GPT is known as its effect space which will be denoted by $\\mathcal {E}$ .", "The effect space $\\mathcal {E}$ corresponds to a convex subset of $\\mathbb {R}^{d+1}$ , as does the state space $\\mathcal {S}$ .", "It necessarily contains the zero and unit vectors, $0=\\begin{pmatrix}0\\\\\\vdots \\\\0\\\\0\\end{pmatrix}\\qquad \\text{ and }\\qquad u=\\begin{pmatrix}0\\\\\\vdots \\\\0\\\\1\\end{pmatrix},$ as well as the vector $(u-e)$ for every $e\\in \\mathcal {E}$ [9], which arises automatically as a valid effect.", "We also assume that the effect space spans the full $(d+1)$ dimensions of the vector space; otherwise the model would contain states which result in identical probabilities for all effects in the effect space, making them indistinguishable and hence operationally equivalent.", "Note that a $d$ -dimensional state space comes with a $\\left(d+1\\right)$ -dimensional effect space.", "Extremal effects are defined by the property that they cannot be written as a (non-trivial) convex combination of other effects.", "Observables are given by tuples $\\left.e_{1},e_{2},\\cdots \\right.$ of elements of the effect space that sum to the unit effect $u$ , with each effect in the tuple corresponding to a different possible outcome when measuring the observable.", "The position of an effect in the tuple encodes the label of the corresponding outcome.", "Given the observable $\\mathbb {D}_{e}=\\left.e,u-e\\right.$ , for example, we will say effect $e$ represents the first possible outcome of measuring $\\mathbb {D}_{e}$ since $e$ occupies the first position in the tuple.", "A GPT should also specify which tuples of effects correspond to observables (or in the language of [15], the GPT should specify the set of meters).", "We will assume throughout (except in Section ) that any finite tuple of effects $\\left.e_1,\\ldots ,e_n\\right.$ satisfying $\\sum _{j=1}^ne_j=u$ and, $\\sum _{j\\in J}e_j\\in \\mathcal {E}$ for any subset $J\\subset \\left\\lbrace 1,\\ldots ,n\\right\\rbrace $ , in a GPT corresponds to an observableThis assumption is equivalent to considering GPTs with a restriction of type (R1) in [15], whereas the GPTs considered in Section can be of type (R2) or (R3).. Eq.", "REF ensures that the set of observables is closed under coarse-graining of outcomes.", "The effect space $\\mathcal {E}_{b}$ of the classical bit with state space $\\mathcal {S}_{b}$ is given by the parallelogram depicted in Figure REF .", "The two-outcome measurement $\\mathbb {B}$ determining “Is the bit in state 0 or 1?” is represented by $\\mathbb {B}=\\left.\\begin{pmatrix}-1\\\\1\\end{pmatrix},\\begin{pmatrix}1\\\\0\\end{pmatrix}\\right..$ Figure: State and effect spaces of the classical-bit GPT:(a) formulated as described in Sections and and using Eq.", "(); (b) after applying thetransformation given in Eq.", "()." ], [ "Equivalent GPTs", "When considering a specific GPT it is sometimes useful to linearly transform its state and effect spaces.", "The description of a qubit in terms of a Bloch vector is a simple example since its components do not necessarily take values in the range $\\left[0,1\\right]$ .", "Explicitly, the Bloch vector representation of a qubit density operator $\\rho =\\frac{1}{2}\\left(I+x_{x}+y_{y}+z_{z}\\right),$ is given by the vector $\\left(x,y,z,1\\right)^{T}$ , with the fourth component being the coefficient of the identity matrix.", "Each of the other coefficients $r\\in \\left\\lbrace x,y,z\\right\\rbrace $ is linearly related to the probability of the outcome $+1$ of the observable $_{r}$ , i.e.", "$p_{r}=\\left(1+r\\right)/2$ .", "Any linear transformation which preserves the inner product between states and effects of a given GPT gives rise to an alternative representation.", "Suppose that we transform the state space $\\mathcal {S}$ by an invertible $\\left(d+1\\right)\\times \\left(d+1\\right)$ matrix $\\mathbf {M}$ to the space $\\mathcal {S}_{\\mathbf {M}}\\equiv \\mathbf {M}\\mathcal {S}$ .", "Then we must apply the inverse transpose transformation $\\mathbf {M}^{-T}\\equiv \\left(\\mathbf {M}^{-1}\\right)^{T}$ to the effect space, $\\mathcal {E}_{\\mathbf {M}}\\equiv \\mathbf {M}^{-T}\\mathcal {E}$ , in order that the probabilities remain invariant, $e_{\\mathbf {M}}\\cdot \\omega _{\\mathbf {M}}=\\left(\\mathbf {M}^{-T}e\\right)\\cdot \\left(\\mathbf {M}\\omega \\right)=e\\cdot \\omega \\,.$ The transformed state and effect spaces continue to be convex subsets of $\\mathbb {R}^{d+1}$ , and they can even be thought of as a convex subset of a vector space isomorphic to $\\mathbb {R}^{d+1}$ .", "GPTs are often presented in this way (cf.", "[16], [9] and references therein).", "The standard formulation of quantum theory in finite dimensions is an example of representing the state and effect spaces of a theory as subsets of a vector space isomorphic to $\\mathbb {R}^{d+1}$ .", "Quantum states are represented by density operators on ${d}$ which form a convex subset of the real vector space of Hermitian operators on ${d}$ , which is isomorphic to $\\mathbb {R}^{d^{2}}$ .", "Quantum effects can also be embedded in this space with $e\\left(\\omega \\right)=\\operatorname{Tr}\\left(e\\omega \\right)$ for an operator $e$ satisfying $0\\le \\langle \\psi |e|\\psi \\rangle \\rangle \\le \\langle \\psi |\\psi \\rangle $ for all rays $|\\psi \\rangle \\in {d}$ .", "Using this representation of the state and effect spaces is essential for $d>2$ since it is highly non-trivial to explicitly describe the set of density operators by vectors of the form given in Eq.", "(REF )) (see e.g.", "[17], [18]).", "As an explicit example, let us transform the GPT description of a classical bit with state space $\\mathcal {S}_{b}$ by the matrix $\\mathbf {M}=\\begin{pmatrix}2 & -1\\\\0 & 1\\end{pmatrix}\\,.$ The new state space, $\\mathcal {S}_{B}\\equiv \\mathbf {M}\\mathcal {S}_{b}$ , is now the convex hull of the images of the extremal states 0 and 1 (previously located at $(0,1)^{T}$ and $(1,1)^{T}$ , respectively), i.e.", "$\\mathcal {S}_{B}=\\mbox{Conv}\\left\\lbrace \\begin{pmatrix}-1\\\\1\\end{pmatrix},\\begin{pmatrix}1\\\\1\\end{pmatrix}\\right\\rbrace \\,.$ Similarly, the effect space, $\\mathcal {E}_{B}\\equiv \\mathbf {M}^{-T}{\\cal E}_{b}$ , is given by the convex hull of the zero effect $\\mathbf {0}$ , the unit effect $u$ and two other extremal effects, ${\\cal E}_{B}=\\mbox{Conv}\\left\\lbrace \\begin{pmatrix}0\\\\0\\end{pmatrix},\\begin{pmatrix}1\\\\0\\end{pmatrix},\\frac{1}{2}\\begin{pmatrix}-1\\\\1\\end{pmatrix},\\frac{1}{2}\\begin{pmatrix}1\\\\1\\end{pmatrix}\\right\\rbrace \\,,$ as pictured in Figure REF ." ], [ "Cones in GPTs ", "The notion of a positive cone is useful when studying the properties of state and effect spaces of a GPT.", "A positive cone is a subset of $\\mathbb {R}^{d+1}$ that contains all non-negative linear combinations of its elements.", "Positive cones may, for example, be generated from convex subsets of real vector spaces.", "Definition 1 The positive cone $A^{+}$ of a convex subset $A$ of a real vector space is the set of vectors $A^{+}=\\left\\lbrace xa|x\\ge 0,a\\in A\\right\\rbrace \\,.$ Positive cones also arise from considering the space dual to a subset of vectors in an inner product space.", "Definition 2 The dual cone $A^{*}$ of a subset $A$ of a real inner product space $V$ is the positive cone $A^{*}=\\left\\lbrace b\\in V|\\left\\langle a,b\\right\\rangle \\ge 0\\text{ for all }a\\in A\\right\\rbrace \\,.$ Figure (REF ) illustrates, for a classical bit, the dual cone $\\mathcal {S}_{B}^{*}$ of the state space $\\mathcal {S}_{B}$ .", "It is easy to see that, in general, the effect space ${\\cal E}$ of a GPT must be contained within the dual cone $\\mathcal {S}^{*}$ of the state space in order that the effects assign non-negative probabilities to every state in the state space.", "The following lemma describes a simple but important property of effect spaces related to the fact that the elements of its dual cone effectively span the ambient space.", "Lemma 1 For any effect space $\\mathcal {E}$ and any vector $c\\in \\mathbb {R}^{d+1}$ , we have $c=a-b$ for some $a,b\\in \\mathcal {E}^{+}$ .", "Firstly, the interior of $\\mathcal {E}^{+}$ is non-empty since $\\mathcal {E}$ is convex and spans $\\mathbb {R}^{d+1}$ .", "Let $e$ be an interior point of $\\mathcal {E}^{+}$ .", "As $e$ is an interior point of $\\mathcal {E}^{+}$ , we have that $e+\\varepsilon c\\in \\mathcal {E}^{+}$ for some $\\varepsilon >0$ and we may take $a=\\left(e+\\varepsilon c\\right)/\\varepsilon $ and $b=e/\\varepsilon $ .", "Figure: State and effect spaces 𝒮 B \\mathcal {S}_{B} and ℰ B \\mathcal {E}_{B}(the horizontal line and the dark square, respectively) of the classical-bitGPT showing (a) the dual cone 𝒮 B * \\mathcal {S}_{B}^{*} (shaded) of thestate space; (b) the effect space ℰ B \\mathcal {E}_{B} is given by theintersection of the cones 𝒮 B * \\mathcal {S}_{B}^{*} and u-𝒮 B * u-\\mathcal {S}_{B}^{*}since the bit is an unrestricted GPT (see Section ).Two further lemmata, which we will need later on, establish relations between positive cones and their dual cones.", "Lemma 2 Let $A$ be a compact, convex subset of $\\mathbb {R}^{d+1}$ , then $A^{**}=A^{+}$ .", "This result is a consequence of the hyperplane separation theorem and has been shown as Theorem 14.1 in [19], for example.", "Lemma 3 For a compact and convex subset $A\\subset \\mathbb {R}^{d+1}$ , we have $A^{*}=\\left(A^{+}\\right)^{*}$ .", "By Definition REF , a vector $b$ is in the dual cone $A^{*}$ of $A$ if and only if $b\\cdot a\\ge 0$ for all $a\\in A$ .", "Equivalently, we may require $x\\left(b\\cdot a\\right)=b\\cdot \\left(xa\\right)\\ge 0$ for all vectors $a$ in the set $A$ and $x\\ge 0,$ which holds if and only if $b\\in A^{+}$ ." ], [ "The no-restriction hypothesis ", "A particularly close relationship between state and effect spaces exists in GPTs that satisfy the no-restriction hypothesis [9], i.e.", "GPTs with effect spaces consisting of all linear maps $e:\\mathbb {R}^{d+1}\\rightarrow \\mathbb {R}$ such that $0\\le e\\left(\\omega \\right)\\le 1$ for all $\\omega \\in \\mathcal {S}$ .", "In such an unrestricted theory the state space defines a unique effect space, and vice versa.", "The effect space of an unrestricted GPT with state space ${\\cal S}$ is given by $E\\left(\\mathcal {S}\\right) & =\\left\\lbrace e\\in \\mathbb {R}^{d+1}|0\\le e\\cdot \\omega \\le 1,\\text{ for all }\\omega \\in \\mathcal {S}\\right\\rbrace \\nonumber \\\\& =\\mathcal {S}^{*}\\cap \\left(u-\\mathcal {S}^{*}\\right)\\,,$ where $u-\\mathcal {S}^{*}=\\left\\lbrace u-e|e\\in \\mathcal {S}^{*}\\right\\rbrace $ .", "The classical bit is an example of an unrestricted GPT.", "The cones $\\mathcal {S}^{*}$ and $\\left(u-\\mathcal {S}^{*}\\right)$ as well as their intersection are illustrated in Figure REF .", "Conversely, if an unrestricted GPT has an effect space $\\mathcal {E}$ then a unique state space is associated with it, namely: $W\\left(\\mathcal {E}\\right) & =\\left\\lbrace \\omega \\in \\mathbb {R}^{d+1}|e\\cdot \\omega \\ge 0\\text{ for all }e\\in \\mathcal {E}\\text{ and }\\omega \\cdot u=1\\right\\rbrace \\nonumber \\\\& =\\mathcal {E}^{*}\\cap 1_{d+1}\\,,$ where $1_{d+1}=\\left\\lbrace \\omega \\in \\mathbb {R}^{d+1}|u\\cdot \\omega =1\\right\\rbrace $ ; we will omit the subscript $d+1$ whenever the dimension is clear from the context.", "We have introduced the maps $E$ and $W$ in the context of unrestricted GPTs but they are well-defined for the state and effect space of any GPT.", "The maps will play an important role in the derivation of our main result (see Section )." ], [ "Noisy unrestricted GPTs", "The main result of this paper establishes a Gleason-type theorem for a class of GPTs which we will now introduce, namely noisy unrestricted (NU) GPTs.", "Formally, the class of NU GPTs consists of all unrestricted GPTs along with a special subset of restricted GPTs.", "The included restricted GPTs are those that can be thought of as unrestricted GPTs in which some (or all) of the observables can only be measured with a limited efficiency, or with some inherent noise.", "Definition 3 A GPT with state space ${\\cal S}$ and effect space $\\mathcal {E}$ is a noisy unrestricted (NU) GPT if for every vector $e\\in E\\left(\\mathcal {S}\\right)$ there exists a number $p_{e}\\in \\left(0,1\\right]$ such that the rescaled vector $p_{e}e$ is contained in the effect space $\\mathcal {E}$ .", "This definition implies that each NU GPT is closely related to an unrestricted GPT in the following way: for each observable $\\mathbb {O}=\\left.e_{1},e_{2},\\ldots ,e_{n}\\right.$ in the unrestricted GPT, there exists an observable $\\mathbb {O}_{p}=\\left.pe_{1},pe_{2},\\ldots ,pe_{n},\\left(1-p\\right)u\\right.,$ of the NU GPT, for some $p\\in \\left(0,1\\right]$ , while the state spaces of the two GPTs are given by the same set.", "Thus, measuring the observable $\\mathbb {O}_{p}$ of the NU GPT can be thought of as successfully measuring the observable $\\mathbb {O}$ (of the associated unrestricted GPT) with probability $p$ , and observing no outcome with probability $(1-p)$ , regardless of the state of the system.", "For later convenience, the case in which $p_{e}=1$ for all vectors in $e\\in E\\left(\\mathcal {S}\\right)$ is included in Definition REF ; in other words, “noiseless” unrestricted GPTs—i.e.", "those in which $\\mathcal {E}=E\\left(S\\right)$ holds—are also considered to be NU GPTs.", "All other NU GPTs, however, are restricted, i.e.", "they violate the no-restriction hypothesis.", "Figure REF shows two modified versions of the bit GPT that violate the no-restriction hypothesis, one of which is a NU GPT while the other is not.", "Further examples of the three different varieties of GPTs—restricted, unrestricted and noisy unrestricted—can be found in Section .", "Figure: State and effect spaces for two restrictionsof the classical-bit GPT: (a) resulting in a NU GPT and (b) notresulting in a NU GPT (see Definition ).We conclude this section by pointing out an alternative characterisation of NU GPTs.", "Definition 4 A NU GPT has a state space ${\\cal S}$ and an effect space ${\\cal E}$ which are related by $E\\left(\\mathcal {S}\\right)=\\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)$ .", "The equivalence of Definition REF with Definition REF can be seen as follows.", "Consider a GPT with state space $\\mathcal {S}$ and effect space $\\mathcal {E}$ .", "Assume that the GPT satisfies Definition REF so that $E\\left(\\mathcal {S}\\right)=\\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)$ holds.", "Then if $e\\in E\\left(\\mathcal {S}\\right)$ , we have $e\\in \\mathcal {E}^{+}$ hence there exists $p\\in \\left(0,1\\right]$ such that $pe\\in \\mathcal {E}$ which means that the GPT satisfies Definition REF .", "Conversely, assume that the GPT satisfies Definition REF .", "Hence, for every vector $e\\in E\\left(\\mathcal {S}\\right)$ there exists $p\\in \\left(0,1\\right]$ such that $pe\\in \\mathcal {E}$ .", "This implies $E\\left(\\mathcal {S}\\right)\\subset \\mathcal {E}^{+}$ .", "Firstly, if $e\\in E\\left(\\mathcal {S}\\right)$ then $u-e\\in E\\left(\\mathcal {S}\\right)$ thus $e,\\left(u-e\\right)\\in \\mathcal {E}^{+}$ and we have $E\\left(\\mathcal {S}\\right)\\subseteq \\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)$ .", "Secondly, if $e\\in \\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)$ then $0\\le e\\cdot \\omega \\le 1$ for all $\\omega \\in \\mathcal {S}$ , so that $e\\in E\\left(\\mathcal {S}\\right)$ , by the definition of $E\\left(\\mathcal {S}\\right)$ .", "Combining these two arguments leads to $E\\left(\\mathcal {S}\\right)=\\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)$ , and thus the GPT satisfies Definition REF ." ], [ "GPTs and Gleason-type theorems ", "Gleason's theorem was motivated by the idea that a state of a quantum system should be uniquely identified by the probabilities of the outcomes of any measurement performed on the system.", "By this reasoning every state should have a corresponding frame function, that is a probability assignment on the space of projections (and later, quantum effects [3]), such that the probabilities of the disjoint outcomes of any measurement sum to unity.", "In order to formulate a GTT for GPTs, we need to generalize the concept of a frame function.", "Definition 5 A frame function on an effect space $\\mathcal {E}$ of a GPT is a map $v:\\mathbb {R}^{d+1}\\rightarrow \\mathbb {R}$ satisfying $0\\le v\\left(e\\right)\\le 1$ for all effects $e\\in \\mathcal {E}$ ; $v\\left(e_{1}\\right)+v\\left(e_{2}\\right)+\\ldots +v\\left(e_{n}\\right)=1$ for all sequences of effects $e_{1},e_{2},\\ldots ,e_{n}\\in \\mathcal {E}$ such that $\\left.e_{1},e_{2},\\ldots ,e_{n}\\right.$ is an observble in the GPT.", "Note that considering measurements with only a finite number of possible outcomes is sufficient for our purposes; thus, assumption (V2) is only required to hold for finite sequences of effects.", "Countable sequences of effects may be required if one considers infinite-dimensional systems.", "In quantum theory the results of Gleason and Busch show any frame function must correspond to a density operator.", "In other words, there are no states beyond those we already believe to exist under the assumption that states must correspond to frame functions.", "We will take the analog of this idea as the definition of a GTT for a GPT.", "Definition 6 A GPT with state space $\\mathcal {S}$ and effect space $\\mathcal {E}$ admits a Gleason-type theorem if and only if every frame function on $\\mathcal {E}$ can be represented by a state in $\\mathcal {S}$ .", "Such a GTT would allow the set of all possible states of a GPT to follow from the effect space via the natural assumption that a state can be uniquely defined by its propensity to take each possible value of every observable.", "The requirement that all mathematically possible states are realised in a theory could be thought of as analogous to the no-restriction hypothesis, i.e.", "requiring that all effects have a corresponding measurement outcome.", "We will show, however, that the classes of GPTs that satisfy these requirements are not the same." ], [ "A Gleason-type theorem for NU GPTs", "After these preliminaries, let us state the main result of this paper which identifies the condition under which Gleason-type theorems exist for general probabilistic theories.", "Theorem 1 Let $\\mathcal {S}$ and $\\mathcal {E}$ be the state and effect spaces, respectively, of a GPT.", "Any frame function $v:\\mathcal {E}\\rightarrow \\left[0,1\\right]$ admits an expression $v\\left(e\\right)=e\\cdot \\omega $ for some $\\omega \\in \\mathcal {S}$ if and only if $\\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)=E\\left(\\mathcal {S}\\right),$ i.e.", "a GPT admits a Gleason-type theorem if and only if it is a noisy unrestricted GPT.", "With quantum theory in finite dimensions being a GPT which obeys the non-restriction hypothesis, Busch's result [3] is an immediate consequence of Theorem REF ; the infinite-dimensional case is, however, not treated here.", "Before presenting the proof of Theorem REF , we briefly explain how a GTT allows one to simplify the postulates used to describe a specific GPT in an axiomatic approach.", "A simple way to state the postulates, often used for quantum theory, is to describe the mathematical objects that represent observables and states along with the rule for calculating the probabilities of measurement outcomes (supplemented by postulates describing the composition of systems and, possibly, the evolution of the system in time).", "In general, for some GPT with effect space $\\mathcal {E}$ and state space $\\mathcal {S}$ , such postulates would take the following form: The observables of the system correspond exactly to the tuples of vectors $\\left.e_{1},e_{2},\\ldots \\right.$ in $\\mathcal {E}$ that sum to the vector $u$ , with each vector corresponding to a possible disjoint outcome of measuring the observable.", "The states of the system correspond exactly to vectors $\\omega \\in \\mathcal {S}$ .", "When measuring the observable $\\left.e_{1},e_{2},\\ldots \\right.$ on a system in state $\\omega \\in \\mathcal {S}$ , the probability to obtain outcome $e_{j}$ is given by $p_{j}(\\omega )=e_{j}\\cdot \\omega $ .", "If there exists a GTT for the GPT in hand, then it could be recovered by replacing the postulates (S) and (P) by the operationally motivated assumption that every state must have a corresponding frame function defining its outcome probabilities, along with the converse assumption that every frame function must have a corresponding state in the theory.", "Consequently, one only needs to supplement postulate (O) with a single new postulate.", "There exists a state of the system for every frame function on the effect space $\\mathcal {E}$ .", "Combined with the GTT and operational reasoning, postulates (O) and (F) lead to the same theory as the postulates (O), (S) and (P).", "Thus, the Gleason-type theorem would simplify the axiomatic formulation of the GPT, just as the theorems by Gleason or Busch do in the case of quantum theory.", "It could be argued that the operational assumptions present in the derivation of the GPT framework are better suited to simplifying the postulates (O), (S) and (P).", "In Appendix we compare this strategy with the simplification achieved using Theorem REF .", "Our result also opens up an alternative approach to establish the GPT framework as a whole.", "In Section we reviewed the derivation of the framework from operational principles which motivates the structure of state spaces in GPTs.", "One may, however, arrive at the same framework by operationally motivating the structure of effect spaces in GPTs (as convex, compact subsets of a real vector space containing the zero vector, and a vector $u$ such that $u-e$ is in the set for every effect $e$ ), combined with a minimal set of observables formed from the effects in the space (namely, two-outcome observables and their convex combinations, see Section and [20], [21]).", "At this point a corollary to our Gleason-type theorem may be used to indeed recover the structure of the state space (as a subset of $W\\left(\\mathcal {E}\\right)$ where $\\mathcal {E}$ is the effect space of the model) by simply assuming that a state must have a corresponding frame function on this minimal set of observables.", "In other words, assuming that states are frame functions offers an alternative to the mathematically stronger assumption that states are linear functionals on the real vector space containing the effect space.", "The details of this approach are described in Appendix ." ], [ "Proof of Theorem ", "Using Definition REF of a GTT, Theorem REF shows that NU GPTs are exactly the class of GPTs that admit GTTs.", "We will prove this result in two steps: (i) in Proposition REF , a frame function on a GPT effect space $\\mathcal {E}$ is found to correspond to a vector in the set $W\\left(\\mathcal {E}\\right)$ defined in Section REF ; (ii) the set $W\\left(\\mathcal {E}\\right)$ is found to correspond to the state space of the GPT if and only if the GPT is in the class of NU GPTs, in Lemmata REF and REF .", "Step (i): The proof of the following propositionThis proposition was independently shown in [22], where it is considered as a GTT for GPTs.", "is inspired by the method used for the quantum case given in [3].", "Proposition 1 Let $\\mathcal {E}$ be an effect space of a GPT.", "Any frame function $v$ on $\\mathcal {E}$ admits an expression $v\\left(e\\right)=e\\cdot \\omega ,$ for some vector $\\omega \\in W\\left(\\mathcal {E}\\right)$ and all effects $e\\in \\mathcal {E}$ .", "Consider a finite set of effects $e_{1},e_{2},\\ldots ,e_{n}\\in \\mathcal {E}$ such that $\\sum _{j\\in J}e_{j}\\in \\mathcal {E}$ for any subset $J\\subseteq \\left\\lbrace 1\\ldots ,n\\right\\rbrace $ .", "First we show that a frame function $v$ must be additive on any such set, i.e.", "$v\\left(e\\right)+v\\left(e_{2}\\right)+\\ldots +v\\left(e_{n}\\right)=v\\left(e_{1}+e_{2}+\\ldots e_{n}\\right).$ We have that the tuples $\\left.e_1,\\ldots ,e_n,u-\\sum _{j=1}^ne_j\\right.\\text{, and }\\left.\\sum _{j=1}^ne_j,u-\\sum _{j=1}^ne_j\\right.,$ are both observables in the GPT since they satisfy Eqs.", "(REF ) and (REF ).", "Hence, by property (V2) of a frame function we find $\\sum _{j=1}^{n}v\\left(e_{j}\\right)+v\\left(u-\\sum _{j=1}^{n}e_{j}\\right)=v\\left(\\sum _{j=1}^{n}e_{j}\\right)+v\\left(u-\\sum _{j=1}^{n}e_{j}\\right)=1,$ and Eq.", "(REF ) follows.", "The next step is to show the homogeneity of $v$ on $\\mathcal {E}$ , i.e.", "$\\alpha v\\left(e\\right)=v\\left(\\alpha e\\right)\\quad \\text{for all }e\\in \\mathcal {\\mathcal {E}}\\quad \\text{and }\\alpha \\in \\left[0,1\\right]\\,.$ Note that the convexity of $\\mathcal {E}$ ensures that rescaling an effect $e$ by a factor $\\alpha \\le 1$ produces another effect: $\\alpha e=\\alpha e+\\left(1-\\alpha \\right)0\\in \\mathcal {E}$ .", "For any integer number $n\\in \\mathbb {N}$ , Eq.", "(REF ) implies $v\\left(e\\right)=v\\left(\\frac{n}{n}e\\right)=v\\left(\\frac{1}{n}e+\\ldots +\\frac{1}{n}e\\right)\\stackrel{\\text{}}{=}nv\\left(\\frac{1}{n}e\\right)\\:;$ then, letting $m\\in \\mathbb {N}$ with $m\\le n$ , Eqs.", "(REF ) and (REF ) lead to the homogeneity of $v$ over the rationals, $v\\left(\\frac{m}{n}e\\right)=v\\left(\\frac{1}{n}e+\\ldots +\\frac{1}{n}e\\right)\\stackrel{\\text{}}{=}mv\\left(\\frac{1}{n}e\\right)\\stackrel{}{=}\\frac{m}{n}v\\left(e\\right).$ Now consider two rational numbers $p,q\\in \\left[0,1\\right]$ with $p\\le q$ .", "Using property (V1) of a frame function with argument $\\left(q-p\\right)e\\in \\mathcal {E}$ guarantees that $v\\left(\\left(q-p\\right)e\\right)\\ge 0$ .", "Also we find by property (V2) of a frame function that $v\\left(qe\\right)=v\\left(qe-pe+pe\\right)\\stackrel{\\text{}}{=}v\\left(\\left(q-p\\right)e\\right)+v\\left(pe\\right)\\,.$ Thus, the values of frame functions on multiples of a given effect respect the ordering induced by the scale factors, $v\\left(pe\\right)\\le v\\left(qe\\right)\\,.$ Next, let $p_{\\mu }$ and $q_{\\nu }$ be sequences of rational numbers in the interval $\\left[0,1\\right]$ that tend to $\\alpha $ from below and above, respectively.", "Then we have $p_{\\mu }v\\left(e\\right)\\stackrel{}{=}v\\left(p_{\\mu }e\\right)\\stackrel{}{\\le }v\\left(\\alpha e\\right)\\stackrel{}{\\le }v\\left(q_{\\nu }e\\right)\\stackrel{\\text{ }}{=}q_{\\nu }v\\left(e\\right)\\,,$ so that the homogeneity of $v$ claimed in Eq.", "(REF ) follows from taking the limit of both sequences.", "Thirdly, we construct a well-defined extension of the frame function $v$ to $\\mathcal {E}^{+}$ , the positive cone associated with $\\mathcal {E}$ (see Definition REF ) such that $v\\left(a+b\\right)=v\\left(a\\right)+v\\left(b\\right)$ holds for all $a,b\\in \\mathcal {E}^{+}$ .", "To do so, consider two effects $e_{1},e_{2}\\in \\mathcal {E}$ which give rise to the same vector in the positive cone via $a=a_{1}e_{1}=a_{2}e_{2}\\in \\mathcal {E}^{+}$ , with $1<a_{1}<a_{2}$ .", "Then we have $v\\left(e_{2}\\right)=v\\left(\\frac{a_{1}}{a_{2}}e_{1}\\right)\\stackrel{\\text{}}{=}\\frac{a_{1}}{a_{2}}v\\left(e_{1}\\right),$ hence $a_{2}v\\left(e_{2}\\right)=a_{1}v\\left(e_{1}\\right)$ , and we may uniquely define the frame function on arbitrary vectors in the positive cone by $v\\left(a\\right):=a_{1}v\\left(e_{1}\\right).$ Additivity of the extended frame function is easily seen to hold for vectors in the positive cone: consider vectors $a=ae_{a}$ and $b=be_{b}$ for $e_{a},e_{b}\\in \\mathcal {E}$ and $a,b>1$ and let $c=a+b$ .", "Noting that $\\left(a+b\\right)/c\\in \\mathcal {E}$ is an effect, we obtain $v\\left(a+b\\right)\\stackrel{}{=}cv\\left(\\frac{1}{c}\\left(a+b\\right)\\right)\\stackrel{}{=}cv\\left(\\frac{1}{c}a\\right)+cv\\left(\\frac{1}{c}b\\right)=v\\left(a\\right)+v\\left(b\\right).$ A linear extension of a frame function $v$ to the whole of $\\mathbb {R}^{d+1}$ follows from the fact that any $c\\in \\mathbb {R}^{d+1}$ outside the positive cone $\\mathcal {E}^{+}$ may be decomposed into $c=a-b$ with $a,b\\in \\mathcal {E}^{+}$ by Lemma REF .", "If the decomposition is not unique, $c=a-b=a^{\\prime }-b^{\\prime }$ , we have $a+b^{\\prime }=a^{\\prime }+b$ leading to $v\\left(a+b^{\\prime }\\right)=v\\left(a^{\\prime }+b\\right).$ It then follows from Eq.", "(REF ), that $v\\left(a\\right)+v\\left(b^{\\prime }\\right)=v\\left(a^{\\prime }\\right)+v\\left(b\\right)$ and hence $v\\left(a\\right)-v\\left(b\\right)=v\\left(a^{\\prime }\\right)-v\\left(b^{\\prime }\\right).$ Therefore we may uniquely define the value of the frame function on the vector $c$ via $v\\left(c\\right):=v\\left(a\\right)-v\\left(b\\right)\\,,$ i.e.", "independently of the decomposition of the vector $c$ .", "Since this extension of any frame function $v$ on $\\mathcal {E}$ to $\\mathbb {R}^{d+1}$ is linear (see Appendix ), the extended map admits an expression as $v\\left(a\\right)=a\\cdot \\omega ,$ for some vector $\\omega =\\sum _{j=1}^{d+1}v\\left(x_{j}\\right)x_{j}\\in \\mathbb {R}^{d+1}$ , where $\\left\\lbrace x_{1},\\ldots x_{d+1}\\right\\rbrace $ is a basis of $\\mathbb {R}^{d+1}$ .", "Finally, requirements (V1) and (V2) on the behaviour of the frame function $v$ on the effect space $\\mathcal {E}$ imply that $\\omega \\in W\\mathcal {\\left(E\\right)}$ which concludes the proof.", "Proposition REF shows that if one defines states as frame functions on an effect space $\\mathcal {E}$ , then the associated state space must be $W\\left(\\mathcal {E}\\right)$ .", "Step (ii): We will now prove that the set $W\\left(\\mathcal {E}\\right)$ corresponds to the state space of a GPT with effect space $\\mathcal {E}$ if and only if the GPT is a NU GPT.", "Two lemmata will be needed to show that $W\\left(E\\left(\\mathcal {S}\\right)\\right)=\\mathcal {S}$ holds for all GPTs while the relation $E\\left(W\\left(\\mathcal {E}\\right)\\right)=E\\left(\\mathcal {S}\\right)$ only holds for NU GPTs.", "Lemma 4 For any GPT with state space $\\mathcal {S}$ , we have $W\\left(E\\left(\\mathcal {S}\\right)\\right)=\\mathcal {S}$ .", "Firstly, by the definitions of the maps $W$ and $E$ in Section REF , we have $W\\left(E\\left(\\mathcal {S}\\right)\\right)=\\left(\\mathcal {S}^{*}\\cap \\left(u-\\mathcal {S}^{*}\\right)\\right)^{*}\\cap 1,$ which, using Lemma REF , implies $W\\left(E\\left(\\mathcal {S}\\right)\\right)=\\left(\\left(\\mathcal {S}^{*}\\cap \\left(u-\\mathcal {S}^{*}\\right)\\right)^{+}\\right)^{*}\\cap 1.$ Secondly, we will show that $\\left(\\mathcal {S}^{*}\\cap \\left(u-\\mathcal {S}^{*}\\right)\\right)^{+}=\\mathcal {S}^{*}.$ The set on the left of this equation is clearly contained in that on the right since we have $\\left(\\mathcal {S}^{*}\\cap \\left(u-\\mathcal {S}^{*}\\right)\\right)^{+}\\subseteq \\left(\\mathcal {S}^{*}\\right)^{+}=\\mathcal {S}^{*}$ .", "Conversely, we can show that $\\mathcal {S}^{*}\\subseteq \\left(\\mathcal {S}^{*}\\cap \\left(u-\\mathcal {S}^{*}\\right)\\right)^{+}$ as follows.", "If $e\\in \\mathcal {S}^{*}$ , then non-negative rescalings of $e$ are also contained in $\\mathcal {S}^{*}$ : $xe\\in \\mathcal {S}^{*}$ for all $x\\ge 0$ .", "Since $u\\cdot \\omega =1$ for all $\\omega \\in \\mathcal {S}$ , $u$ is an internal point of $\\mathcal {S}^{*}$ .", "Thus, there exists an open ball $\\mathfrak {B}\\left(u,\\varepsilon \\right)$ around $u$ of radius $\\varepsilon $ in $\\mathcal {S}^{*}$ for some $\\varepsilon >0$ .", "Therefore, for $x<\\varepsilon /\\left\\Vert e\\right\\Vert $ we have $\\left\\Vert u-\\left(u-xe\\right)\\right\\Vert <\\varepsilon $ and hence $u-xe\\in \\mathcal {S}^{*}$ .", "By definition, $xe\\in \\left(u-\\mathcal {S}^{*}\\right)$ , hence we have $xe\\in \\mathcal {S}^{*}\\cap \\left(u-\\mathcal {S}^{*}\\right)$ and $e\\in \\left(\\mathcal {S}^{*}\\cap \\left(u-\\mathcal {S}^{*}\\right)\\right)^{+}$ .", "Finally, Eqs.", "(REF ) and (REF ) give $\\begin{aligned}W\\left(E\\left(\\mathcal {S}\\right)\\right) & =\\mathcal {S}^{**}\\cap 1=\\mathcal {S}^{+}\\cap 1\\\\& =\\left\\lbrace \\omega \\in \\mathbb {R}^{d+1}|\\omega =x\\omega ^{\\prime }\\text{ for some }\\omega ^{\\prime }\\in \\mathcal {S}\\text{ and }\\omega \\cdot u=1\\right\\rbrace ,\\end{aligned}$ and since $\\omega \\cdot u=x$ we find $\\mathcal {S}^{+}\\cap 1=\\mathcal {S}\\,,$ as required for Lemma REF to hold.", "Lemma 5 Given a GPT with state and effect spaces $\\mathcal {S}$ and $\\mathcal {E}$ , respectively, the relation $E\\left(W\\left(\\mathcal {E}\\right)\\right)=E\\left(\\mathcal {S}\\right)$ holds if and only if $E\\left(\\mathcal {S}\\right)=\\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)$ .", "Firstly, by the definitions of the maps $W$ and $E$ given in Eqs.", "(REF ) and (REF ), respectively, we have $E\\left(W\\left(\\mathcal {E}\\right)\\right)=\\left(\\mathcal {E}^{*}\\cap 1\\right)^{*}\\cap \\left(u-\\left(\\mathcal {E}^{*}\\cap 1\\right)^{*}\\right),$ as well as $\\left(\\mathcal {E}^{*}\\cap 1\\right)^{*}=\\left(\\left(\\mathcal {E}^{*}\\cap 1\\right)^{+}\\right)^{*}$ , by Lemma REF .", "Secondly, we will show that $\\left(\\mathcal {E}^{*}\\cap 1\\right)^{+}=\\mathcal {E}^{*}.$ If $\\omega \\in \\mathcal {E}^{*}$ then $\\omega \\cdot u\\ge 0$ , which gives $\\frac{1}{\\omega \\cdot u}\\omega \\in \\mathcal {E}^{*}\\cap 1\\,;$ therefore, we conclude that $\\omega \\in \\left(\\mathcal {E}^{*}\\cap 1\\right)^{+}$ .", "Conversely, if $\\omega \\in \\left(\\mathcal {E}^{*}\\cap 1\\right)^{+}$ , then $x\\omega \\in \\mathcal {E}^{*}$ for some $x\\ge 0$ , hence $\\omega \\in \\mathcal {E}^{*}$ .", "Finally, combining Eqs.", "(REF ) and (REF ), we have $E\\left(W\\left(\\mathcal {E}\\right)\\right)=\\mathcal {E}^{**}\\cap \\left(u-\\mathcal {E}^{**}\\right)=\\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)\\,,$ completing the proof.", "We are now in a position to prove our main result, Theorem REF , announced in the previous section.", "It states that a general probabilistic theory admits a Gleason-type theorem if and only if it is noisy unrestricted.", "The result is an immediate consequence of the lemmata just shown.", "By Lemma REF we know that frame functions must be of the form $v\\left(e\\right)=e\\cdot \\omega $ for some $\\omega \\in W\\left(\\mathcal {E}\\right)$ .", "Thus, to conclude the proof we need to show that the state space of a GPT coincides with the set $W\\left(\\mathcal {E}\\right)$ exactly in NU GPTs, i.e.", "$W\\left(\\mathcal {E}\\right)=\\mathcal {S}$ , if and only if Eq.", "(REF ) holds.", "Let $W\\left(\\mathcal {E}\\right)=\\mathcal {S}^{\\prime }$ .", "Firstly, assume that Eq.", "(REF ) holds.", "By Lemma REF we have $E\\left(\\mathcal {S}^{\\prime }\\right)=E\\left(W\\text{$\\left(\\mathcal {E}\\right)$}\\right)=E\\left(\\mathcal {S}\\right).$ Now, by applying the $W$ map to both sides of this equation and using Lemma REF , we find $W\\left(E\\left(\\mathcal {S}^{\\prime }\\right)\\right)=\\mathcal {S}^{\\prime }=W\\left(E\\left(\\mathcal {S}\\right)\\right)=\\mathcal {S}.$ Secondly, assume that Eq.", "(REF ) does not hold, i.e.", "$\\mathcal {E}^{+}\\cap \\left(u-\\mathcal {E}^{+}\\right)\\ne E\\left(\\mathcal {S}\\right)$ , then by Lemma REF $E\\left(\\mathcal {S}^{\\prime }\\right)\\ne E\\left(\\mathcal {S}\\right),$ and $\\mathcal {S}^{\\prime }\\ne \\mathcal {S}$ , which is the content of Theorem REF ." ], [ "Examples and applications", "In this section, we will consider examples of NU GPTs to show how their axiomatic formulation simplifies due to the Gleason-type theorem they allow.", "We also explain that GPTs with a GTT can be defined in a “measurement-first” approach, in contrast with the standard “states-first” approach.", "Finally, we point out that a simple well-known non-quantum model does not belong to the class of NU GPTs and, therefore, does not come with a GTT." ], [ "Simplified axioms for a rebit and other unrestricted GPTs", "Unrestricted GPTs are a well-studied class of GPT.", "The rebit [10], for example, is a GPT with a disc-shaped state space.", "The state space can be equivalently modeled (see Section REF ) by the subset of real density matrices of a qubit.", "Rebits are convenient low-dimensional building blocks for a toy model of quantum theory, giving rise to many characteristic features such as superposition, entanglement and non-locality [12], [10], [23].", "Using our notation, the state space of a rebit is given by $\\mathcal {S}_{R}=\\mbox{Conv}\\left\\lbrace \\omega _{\\theta }\\right\\rbrace _{\\theta \\in [0,2\\pi )},$ where $\\omega _{\\theta }=\\begin{pmatrix}\\cos \\theta \\\\\\sin \\theta \\\\1\\end{pmatrix}.$ The convex hull of the zero effect 0, the unit effect $u$ and a continuous ring of effects $\\mathbf {e}_{\\theta }=\\frac{1}{2}\\begin{pmatrix}\\cos \\theta \\\\\\sin \\theta \\\\1\\end{pmatrix},\\quad \\theta \\in [0,2\\pi )\\,,$ form the rebit effect space $\\mathcal {E}_{R}=\\mbox{Conv}\\left\\lbrace 0,u,\\mathbf {e}_{\\theta }\\right\\rbrace _{\\theta \\in [0,2\\pi )},$ illustrated in Figure REF .", "The rebit satisfies the no-restriction hypothesis since the effect space $\\mathcal {E}_{R}$ is as large as is permitted in the GPT framework (see Eq.", "REF ).", "Figure: State and effect spaces: (a) of the rebit GPT and (b) of the squitGPT.Let us now apply the general argument given at the end of Section REF to a rebit, as a first example of an unrestricted GPT.", "Instead of using the GPT framework, we could have described the rebit of this hypothetical world in an axiomatic fashion, i.e.", "by assuming the axioms (O), (S) and (P) from Section REF , using the effect and state spaces $\\mathcal {E}_{R}$ and $\\mathcal {S}_{R}$ .", "Then, Theorem REF states that, alternatively, we could postulate the rebit observables and, by considering the frame functions associated with them, recover both the state space $\\mathcal {S}_{R}$ and the probability rule.", "More explicitly, we replace postulates (S) and (P) by a single postulate with operational motivation.", "The states of a rebit correspond exactly to the frame functions on the effect space $\\mathcal {E}_{R}$ .", "In other words, we effectively introduce the states of the rebit as probability assignments on the outcomes of measurements.", "The model created by the postulates (O) and (F) is equivalent to the the original one in the sense that it makes exactly the same predictions.", "Classical bits, qubits and qudits as well as square bits (or squits, for short) are other unrestricted GPTs for which identical arguments also result in a smaller set of axioms by means of our Gleason-type theorem.", "The state and effect spaces of bits and qudits were described in Section REF while a squit or gbit [6] is a GPT with a square state space and an octahedral effect space, as illustrated in Figure REF .", "Pairs of squits are often considered in the study of non-local correlations since they are capable of producing the super-quantum correlations of a PR-box [24]." ], [ "Simplified axioms for a noisy rebit and other NU GPTs", "Next, let us consider a noisy rebit characterized by the property that the extremal rebit observables $\\mathbb {D}_{\\mathbf {e}_{\\theta }}=\\left.e_{\\theta },u-e_{\\theta }\\right.,\\theta \\in [0,2\\pi ),$ can be measured only imperfectly, i.e.", "with some efficiency $p\\in \\left(0,1\\right)$ ; see Eq.", "(REF ) for the definition of the effects $\\mathbf {e}_{\\theta }$The noisy rebit GPT also results from applying the general (shifted) depolarizing channel for a generic state to the rebit effect space; see [15] for details of the analogous qubit case..", "The state space of this NU GPT coincides with that of the rebit, $\\mathcal {S}_{R}^{n}\\equiv \\mathcal {S}_{R}$ .", "In order to define its effect space , let us introduce two continuous rings of effects, $\\mathbf {e}_{\\theta }^{+}=\\frac{p}{2}\\begin{pmatrix}\\cos \\theta \\\\\\sin \\theta \\\\1\\end{pmatrix}\\quad \\text{and}\\quad \\mathbf {e}_{\\theta }^{-}=\\frac{p}{2}\\begin{pmatrix}\\cos \\theta \\\\\\sin \\theta \\\\2/p-1\\end{pmatrix},\\quad \\theta \\in [0,2\\pi )\\,.$ These rings, along with the zero effect 0 and the unit effect $u$ , form the extremal points of the noisy rebit effect space, $\\mathcal {E}_{R}^{n}=\\mbox{Conv}\\left\\lbrace 0,u,\\mathbf {e}_{\\theta }^{+},\\mathbf {e}_{\\theta }^{-}\\right\\rbrace _{\\theta \\in [0,2\\pi )},$ depicted in Figure REF .", "While still being a GPT, the model does not satisfy the no-restriction hypothesis: the effect space $\\mathcal {E}_{R}^{n}$ is restricted to a proper subset of $\\mathcal {E}_{R}$ shown in Figure REF .", "Nevertheless, Theorem REF continues to apply: the noisy rebit admits a GTT which is effectively due to the fact that there exist finite neighbourhoods of the zero effect and the unit effect in which $\\mathcal {E}_{R}^{n}$ and $\\mathcal {E}_{R}$ coincide.", "Repeating the argument presented in Section REF , we are able to simplify the definition of the noisy rebit in terms of postulates (O), (S), and (P) which introduce its effect space $\\mathcal {E}_{R}^{n}$ , its state space $\\mathcal {S}_{R}^{n}$ , and the Born rule, respectively.", "The alternative axiomatic formulation in terms of only two postulates only rests on the effect space of the system, The observables of a noisy rebit correspond exactly to the tuples of vectors $\\left.e_{1},e_{2},\\ldots \\right.$ in $\\mathcal {E}_{R}^{n}$ that sum to the vector $u$ , with each vector corresponding to a possible disjoint outcome of measuring the observable.", "The states of a noisy rebit correspond exactly to frame functions on the effect space $\\mathcal {E}_{R}^{n}$ .", "Mutatis mutandis, this procedure applies to any other NU GPT.", "Figure: The effect space ℰ R n {\\cal E}_{R}^{n}and the state space 𝒮 R \\mathcal {S}_{R} of the noisy rebit, an exampleof a low-dimensional NU GPT; for comparison, the ring of extremalrebit effects e θ e_{\\theta } is shown as a thick line." ], [ "A GPT without a GTT: the Spekkens toy model", "In 2007, a toy theory was introduced [25] capable of reproducing a number of important quantum features such as the existence of non-commuting observables, the impossibility of cloning arbitrary states and the presence of entanglement, while simultaneously admitting a description in terms of local hidden variables.", "Originally, Spekkens' model had been introduced without reference to the GPT framework.", "Here, we will consider its reformulation as a GPT, as described in [9].", "Considered as a GPT, Spekkens' model comes with a restricted effect space, and it is not a NU GPT which can be seen as follows.", "Its state space is given by a regular octahedron $\\mathcal {S}_{S}=\\mbox{Conv}\\left\\lbrace x_{\\pm },y_{\\pm },z_{\\pm }\\right\\rbrace ,$ with vertices $x_{\\pm }=\\begin{pmatrix}\\pm 1\\\\0\\\\0\\\\1\\end{pmatrix},\\quad y_{\\pm }=\\begin{pmatrix}0\\\\\\pm 1\\\\0\\\\1\\end{pmatrix},\\quad z_{\\pm }=\\begin{pmatrix}0\\\\0\\\\\\pm 1\\\\1\\end{pmatrix}.$ Under the no-restriction hypothesis the extremal effects associated with the space $\\mathcal {S}_{S}$ would be the vertices of a cube (plus the zero and unit effects).", "In Spekkens' model, however, they are taken to be the vertices of another octahedron inscribed into this cube, as depicted in Figure REF .", "More explicitly, the effect space is the convex hull of the zero and unit effects and the six extremal effects given by the (rescaled) vectors in Eq.", "(REF ), $\\mathcal {E}_{S}=\\mbox{Conv}\\left\\lbrace 0,u,\\frac{x_{\\pm }}{2},\\frac{y_{\\pm }}{2},\\frac{z_{\\pm }}{2}\\right\\rbrace .$ Not being a NU GPT, Theorem REF tells us that the toy theory does not admit a GTT.This fact was, of course, clear without Theorem REF since every vector in $W\\left(\\mathcal {E}_{S}\\right)\\supset \\mathcal {S}_{S}$ yields frame function by definition.", "It is impossible to reproduce this GPT by assuming that the states of the system are in one-to-one correspondence with the frame functions on the effect space.", "There are, in fact, more frame functions than states in $\\mathcal {S}_{S}$ .", "The frame functions correspond to all vectors in the set $W\\left(\\mathcal {E}_{S}\\right)$ which is a strict superset of $\\mathcal {S}_{S}$ forming a cube around $\\mathcal {S_{S}}$ , in the same way that $E\\left(\\mathcal {S}_{S}\\right)$ encloses $\\mathcal {E}_{S}$ in Figure REF .", "In order to recover the original model, one would have to place a restriction on which frame functions correspond to allowed states.", "This restriction can be considered analogous to relaxing the no-restriction hypothesis on the effect space.", "Figure: The octahedral effect space ℰ S \\mathcal {E}_{S} of Spekkens' toy theoryprojected into the hyperplane of ℝ 4 \\mathbb {R}^{4} obtained by fixing thefourth entry of the vectors to 1/21/2; ℰ S \\mathcal {E}_{S} is a propersubset of the cubic effect space E𝒮 S E\\left(\\mathcal {S}_{S}\\right)(boundary depicted by black lines) required by the no-restrictionhypothesis given the state space 𝒮 S \\mathcal {S}_{S} defined in Eq.", "()." ], [ "A Gleason-type theorem for NU GPTs based on two-outcome observables\n", "The definition of a frame function used in Section is based on the idea that every sequence of effects $e_1,e_2,\\ldots \\in \\mathcal {E}$ satisfying Eqs.", "(REF ) and (REF ) corresponds to an observable.", "In quantum theory, however, a Gleason-type theorem can already be derived by involving only a specific subset of all POMs [26] known as projective-simulable observables [21].", "We will show now that a similar weakening of the assumptions continues to imply the result of Theorem REF in the context of GPTs.", "Let us begin by introducing the idea of simulating the measurement of an observable by measuring other observables.", "This is achieved in a GPT by classically mixing observables and post-processing measurement outcomes [20].", "For example, to simulate the observable $\\mathbb {G}=\\left.\\frac{1}{3}\\left(e_{1}+e_{2}+2f\\right),u-\\frac{1}{3}\\left(e_{1}+e_{2}+2f\\right)\\right..$ we may measure the observables $\\mathbb {E}=\\left.e_{1},e_{2},u-e_{1}-e_{2}\\right.\\quad \\text{and}\\quad \\mathbb {F}=\\left.f,0,u-f\\right.$ with probabilities $1/3$ and and $2/3$ , respectively, to simulate $\\mathbb {G}^{\\prime }=\\frac{1}{3}\\mathbb {E}+\\frac{2}{3}\\mathbb {F}=\\left.\\frac{1}{3}\\left(e_{1}+2f\\right),\\frac{1}{3}e_{2},u-\\frac{1}{3}\\left(e_{1}+e_{2}+2f\\right)\\right.\\,$ followed by coarse-graining the first two outcomes to produce the dichotomic observable $\\mathbb {G}$ .", "The only post-processing necessary in the proof to follow is to add outcomes to an observable that occur with probability zero.", "For example, the two-outcome observable $\\left.e,u-e\\right.$ simulates the three-outcome observable $\\left.e,u-e,0\\right.$ if one considers there to be a third outcome of measuring $\\left.e,u-e\\right.$ which never occurs.", "Now consider the set of observables which may be simulated by dichotomic extremal observables, i.e.", "those described by an extremal effect $e$ and its complement $u-e$ .", "For brevity we will refer to such observables as simulable.", "Next, let us call a frame function simulable if the property (V2) in Definition REF is required to hold for simulable observables only.", "Definition 7 A simulable frame function on an effect space $\\mathcal {E}$ is a map $v:\\mathbb {R}^{d+1}\\rightarrow \\mathbb {R}$ satisfying $0\\le v\\left(e\\right)\\le 1$ for all effects $e\\in \\mathcal {E}$ ; $v\\left(e_{1}\\right)+v\\left(e_{2}\\right)+\\ldots +v\\left(e_{n}\\right)=1$ for all sequences of effects $e_{1},e_{2},\\ldots ,e_{n}\\in \\mathcal {E}$ which give rise to simulable observables $\\mathbb {O}=\\left.e_{1},e_{2},\\ldots ,e_{n}\\right.$ .", "Theorem REF can now be strengthened because the properties of simulable frame functions are sufficient for a proof.", "Theorem 2 Let $\\mathcal {S}$ and $\\mathcal {E}$ be the state and effect spaces, respectively, of a NU GPT.", "Any simulable frame function $v$ on $\\mathcal {E}$ admits an expression $v\\left(e\\right)=e\\cdot \\omega ,$ for some $\\omega \\in \\mathcal {S}$ and all $e\\in \\mathcal {E}$ .", "See Appendix .", "This theorem can be used to provide an alternative operational derivation of the GPT framework as described in Section REF , with full details given in Appendix ." ], [ "Summary and Discussion ", "From a conceptual point of view, the results of this paper imply that each general probabilistic theory belongs to one of two distinct classes: either it admits, like quantum theory, a Gleason-type theorem which allows us to infer a description of the possible states of the theory, or it does not admit a GTT.", "In Proposition REF (see Section ) frame functions were found to be linear functionals on the effect space.", "If one considers this fact to be the main content of the Gleason-type theorems in quantum theory then the proposition proves that Gleason-type theorems exist for all GPTs.", "In this paper we have, however, taken the view that a Gleason-type theorem establishes a bijection between frame functions and states in the theory under consideration.", "Interpreting GTTs in this way, Theorem REF shows that a GPT admits such a theorem if and only if it is a noisy unrestricted GPT, of which classical and quantum models are examples.", "Requiring that there is a state in a theory for every frame function could be considered as an analog of the no restriction hypothesis which demands that to every effect there should correspond a measurement outcome.", "However, we have shown that the no-restriction hypothesis is more restrictive than requiring the existence of a GTT, since there are NU GPTs that admit a GTT but violate the no-restriction hypothesis.", "In Section REF we describe how a Gleason-type theorem can be used to derive the state space in a given GPT from the set of observables.", "The postulates (O), (S) and (P), which specify a given GPT, can be replaced by two postulates, namely (O) and (F) when the description of states as frame functions is assumed.", "This reduction is only possible in NU GPTs.", "The current work relies heavily on the convex structure of GPTs.", "In future work we would like to establish which GPTs admit an analog of Gleason’s original theorem, in the sense that the frame functions would only be defined on extremal effects where convexity arguments would no longer be available.", "In recent work [27], [28], [29], alternatives to simplifying the postulates of quantum theory have been put forward by assuming, for example, the postulates of pure states and their dynamics in combination with operational reasoning.", "It would be interesting to study whether similar approaches also hold for other GPTs, or whether they are unique to quantum theory.", "Finally, it might be possible to establish an unexpected link between Gleason-type theorems and the set of almost-quantum correlations [30].", "It is known that GPTs satisfying the no-restriction hypothesis cannot produce the set of almost quantum correlations in Bell scenarios [31].", "If this result could be extended to NU GPTs then the existence of a GTT for a GPT would also preclude the possibility of that GPT producing the set of almost quantum correlations.", "Acknowledgement We acknowledge partial support by the Foundation for Polish Science (IRAP project, ICTQT, contract no.", "2018/MAB/5, co-financed by EU within Smart Growth Operational Programme)." ], [ "Proof of Proposition ", "Here we show that the extension of a frame function $v$ described in the proof of Proposition REF is linear.", "First we show additivity; let $\\mathbb {R}^{d+1}\\ni c_{j}=a_{j}-b_{j}$ for $a_{j},b_{j}\\in \\mathcal {E}^{+}$ , then $\\begin{aligned}v\\left(c_{1}+c_{2}\\right) & =v\\left(a_{1}-b_{1}+a_{2}-b_{2}\\right)\\\\& =v\\left(a_{1}+a_{2}-\\left(b_{1}+b_{2}\\right)\\right)\\\\& \\stackrel{}{=}v\\left(a_{1}+a_{2}\\right)-v\\left(b_{1}+b_{2}\\right)\\\\& \\stackrel{}{=}v\\left(a_{1}\\right)+v\\left(a_{2}\\right)-v\\left(b_{1}\\right)-v\\left(b_{2}\\right)\\\\& \\stackrel{}{=}v\\left(c_{1}\\right)+v\\left(c_{2}\\right).\\end{aligned}$ Then to show homogeneity let $\\mathbb {R}^{d+1}\\ni c=a-b$ for $a,b\\in \\mathcal {E}^{+}$ , firstly consider $\\gamma \\ge 0$ , in which case we have $\\begin{aligned}v\\left(\\gamma c\\right) & =v\\left(\\gamma a-\\gamma b\\right)\\\\& \\stackrel{}{=}v\\left(\\gamma a\\right)-v\\left(\\gamma b\\right)\\\\& \\stackrel{\\text{}}{=}\\gamma \\left(v\\left(a\\right)-v\\left(b\\right)\\right)\\\\& =\\gamma v\\left(c\\right).\\end{aligned}$ Secondly, consider $\\gamma <0$ , $\\begin{aligned}v\\left(\\gamma c\\right) & =v\\left(\\left(-\\gamma \\right)\\left(-c\\right)\\right)\\\\& =v\\left(\\left(-\\gamma \\right)\\left(b-a\\right)\\right)\\\\& \\stackrel{}{=}\\gamma \\left(v\\left(a\\right)-v\\left(b\\right)\\right)\\\\& =\\gamma v\\left(c\\right).\\end{aligned}$" ], [ "An alternative simplification of the axioms defining a GPT", "Let us briefly mention an alternative approach to simplifying the postulates (S), (O) and (P) providing an equivalent definition a GPT which closely follows the operational assumptions of the GPT framework.", "It should be compared with the simplification using Theorem REF outlined at the end of Section REF .", "The starting point is a single postulate about the states of the model at hand.", "There exist $d$ fiducial measurement outcomes of observables whose probabilities determine the state of the system.", "These states are restricted to being represented by vectors in $\\mathcal {S}$ .", "The first part of the postulate, the existence of $d$ fiducial measurement outcomes, determines that the state space can be embedded in $\\mathbb {R}^{d}$ and is convex, with convex combinations of vectors representing classical mixtures of the corresponding states.", "However, this assumption does not determine the “shape” of the state space, hence the inclusion of the second part of the postulate restricting the state space to $\\mathcal {S}$ .", "For a specific GPT, the second part of the postulate may take a more natural-sounding form such as state vectors having modulus less than or equal to one.", "From (S'), using the standard operational assumption that effects must respect classical mixtures and the no-restriction hypothesis (see Section ), the postulates (O) and (P) are recovered easily.", "Let us conclude by comparing this approach to our approach of using Theorem REF in order to reduce the postulates (O), (S) and (P).", "First, postulate (O) does not assume that there exists $d$ fiducial outcomes.", "This property is a consequence in our approach once the states are identified as linear functionals on the effect space.", "Therefore, postulate (O) is not simply a stronger version of (S').", "Second, in order to postulate the existence of $d$ fiducial measurement outcomes, as is done in (S'), one assumes some knowledge of all the observables of the system; otherwise one would not know that the two outcomes in question form a complete fiducial set.", "Therefore, axiom (S') makes assumptions about both the states and the observables of the system whereas (O) only concerns observables.", "Finally, in the approach based on (S'), additional assumptions would be necessary to reconstruct a NU GPT which does not satisfy the no-restriction hypothesis because one could not use the no-restriction hypothesis to recover the postulates (O) and (P).", "However, such a GPT does admit a GTT, as Theorem REF shows, and hence the first method would still be valid." ], [ "Proof of Theorem ", "Due to the convexity of the effect space $\\mathcal {E}$ , we can express any effect $e\\in \\mathcal {E}$ as a convex combination $e=\\sum _{j}p_{j}e_{j}$ , for some extremal effects $e_{j}$ and real numbers $p_{j}\\in [0,1]$ which sum to one.", "Thus we may simulate the observable $\\mathbb {D}_{e}=\\left.e,u-e\\right.$ by measuring the observables $\\mathbb {D}_{e_{j}}=\\left.e_{j},u-e_{j}\\right.,j\\in J,$ with probability $p_{j}$ .", "Furthermore, for any effects $e,e^{\\prime }\\in \\mathcal {E}$ , we may simulate the observable $\\mathbb {T}_{e,e^{\\prime }}=\\left.\\frac{1}{2}e,\\frac{1}{2}e^{\\prime },u-\\frac{1}{2}\\left(e+e^{\\prime }\\right)\\right.$ by performing either $\\mathbb {T}_{2e,\\mathbf {0}}=\\left.e,0,u-e\\right.$ or $\\mathbb {T}_{\\mathbf {0},2e^{\\prime }}=\\left.0,e^{\\prime },u-e^{\\prime }\\right.$ , with equal probability.", "Firstly, applying Definition REF to Eqs.", "(REF ) and (REF ) with $e=e^{\\prime }$ gives $v\\left(e\\right)+v\\left(u-e\\right)=1=v\\left(\\frac{1}{2}e\\right)+v\\left(\\frac{1}{2}e\\right)+v\\left(u-e\\right),$ and hence $v\\left(e/2\\right)=v\\left(e\\right)/2.$ Secondly, for any effects $e,e^{\\prime }\\in \\mathcal {E}$ such that $e+e^{\\prime }\\in \\mathcal {E}$ , the observable $\\mathbb {D}_{\\frac{1}{2}(e+e^{\\prime })}=\\left.\\frac{1}{2}\\left(e+e^{\\prime }\\right),u-\\frac{1}{2}\\left(e+e^{\\prime }\\right)\\right.,$ is simulable by Eq.", "(REF ).", "Comparing with Eq.", "(REF ) gives $v\\left(\\frac{1}{2}e\\right)+v\\left(\\frac{1}{2}e^{\\prime }\\right)=v\\left(\\frac{1}{2}\\left(e+e^{\\prime }\\right)\\right),$ so that $v\\left(e\\right)+v\\left(e^{\\prime }\\right)=v\\left(e+e^{\\prime }\\right)$ follows, using Eq.", "(REF ).", "By induction, any simulable frame function $v$ is a frame function as defined in Definition REF .", "Thus, by Theorem REF , any simulable frame function $v$ admits the expression given in Eq.", "(REF )." ], [ "Deriving the GPT framework starting with measurements ", "The GPT framework is typically derived, as in Section , by considering the states of a system first, followed by a treatment of observables and their measurement.", "However, this order may be reversed, i.e.", "the framework may be derived, using equivalent operational assumptions, by first considering all possible measurements and their outcomes then finding the compatible mathematical description of states.", "Proceeding in this second manner the structure of effect spaces is established first then Theorem REF presents an alternative method for deriving the structure of state spaces, compared with the standard argument involving mixtures of measurement outcomes.", "We begin by summarising the “measurement first” derivation of the GPT framework in parallel with Section .", "Consider all the possible outcomes of the measurements of all the observables of a given system.", "We will assume that there exists a finite set of fiducial states such that any one of these outcomes, $\\zeta $ , is uniquely determined by the probabilities of $\\zeta $ being observed after a measurement (of which $\\zeta $ is a possible outcome) is performed on the system in each of the fiducial states.", "In other words, for a system with $d$ states in its fiducial set, an outcome may be identified by the vector $e\\in \\mathbb {R}^{d}$ such that $e=\\begin{pmatrix}p_{1}\\\\\\vdots \\\\p_{d}\\end{pmatrix},$ where $p_{j}$ is the probability of observing the outcome for a system in the $j$ th fiducial state.", "This representation of measurement outcomes is derived from the operational assumption that one should be able to distinguish two distinct measurement outcomes by their statistics on a finite number of states, in analogy to assuming the possibility of distinguishing two distinct states from the probabilities of a finite number of measurement outcomes in the “states first” approach.", "In line with GPT terminology we will call the set of vectors corresponding to outcomes in a model the effect space and the vectors within this set effects.", "Note that the effects are now simply vectors and not linear functionals.", "For brevity, we will often refer to a measurement outcome as the effect by which it is represented.", "In the bit example from Section , the fiducial set of states could be the “0” and “1” states.", "Thus the effect space would be a subset of $\\mathbb {R}^{2}$ .", "We will assume the existence of an outcome that occurs with probability one for any state of the system.", "This outcome must be represented by the effect $u=\\begin{pmatrix}1\\\\\\vdots \\\\1\\end{pmatrix}.$ Similarly, we assume the existence of an outcome that never occurs, represented by the effect $0=\\begin{pmatrix}0\\\\\\vdots \\\\0\\end{pmatrix}.$ Any outcome $e$ must have a complement, namely the outcome “not $e$ ” necessarily occurring with probability $(1-p_{j})$ when the measurement of “$e$ or not $e$ ” is performed on the $j$ th fiducial state.", "Therefore, for any effect $e=\\left(p_{1},\\ldots ,p_{d}\\right)^{T}$ the vector $u-e=\\begin{pmatrix}1-p_{1}\\\\\\vdots \\\\1-p_{d}\\end{pmatrix}$ must also be in the effect space.", "Consider two measurements on the system each with a discrete set of possible outcomes and label the outcomes of each measurement with positive integers such that the first measurement has outcomes $\\left\\lbrace e_{1},e_{2},\\ldots \\right\\rbrace $ and the second $\\left\\lbrace e^{\\prime }_{1},e^{\\prime }_{2},\\ldots \\right\\rbrace $ (if the measurement has a finite number, $n$ , of possible outcomes the labels $j$ for $j>n$ are assigned the zero effect).", "If a classical mixture of these measurements is performed then possible outcomes of this procedure can be represented by convex combinations of effects.", "Specifically, if the first measurement is performed with probability $p$ and the second with probability $1-p$ , then observing an outcome labeled $j$ from this procedure must be represented by the vector $pe_{j}+(1-p)e^{\\prime }_{j}$ in order to be consistent with the fiducial state set.", "Therefore we assume the effect space is convex.", "Finally, since an arbitrarily good approximation of an effect would operationally be indistinguishable from the effect itself we assume the effect space is a closed subset of $\\mathbb {R}^{d}$ .", "Returning to the bit example, we can build our effect space from the requirement of having a measurement that perfectly distinguishes “0” and “1”, and must therefore have outcomes, $(1,0)^{T}$ and $(0,1)^{T}$ .", "Combined with the other requirements for an effect space we find the bit effect space to be the square in Figure REF , a transformation of the bit effect space described in Section REF .", "Figure: State and effect spaces 𝒮 B ' \\mathcal {S}_{B^{\\prime }} (diagonalblack line) and ℰ B ' \\mathcal {E}_{B^{\\prime }} (grey square), respectively, ofthe classical-bit GPT when formulated in the “measurement-first”method.We have arrived at the same requirements for the structure of an effect space as were described in Section (a convex, compact subset of a real vector space containing the zero vector, and a vector $u$ such that $u-e$ is in the set for every $e$ in the set).", "We may now consider how states should be represented in the framework.", "We assume a state will be represented by a map $\\omega $ from an outcome $e$ to the probability of observing $e$ when a measurement (of which $e$ is a possible outcome) is performed on a system in state $\\omega $ .", "From here we may derive the state space structure of the GPT framework using the standard operational assumptions or the alternative presented by Theorem REF .", "One the one hand, the standard method for deriving the structure of the state space is to exploit the fact that we wish for outcome probabilities to respect mixtures, in analogy with the reasoning behind (REF ), to find $\\omega \\left(pe+(1-p)e^{\\prime }\\right)=p\\omega \\left(e\\right)+(1-p)\\omega \\left(e^{\\prime }\\right),$ for $p\\in \\left[0,1\\right]$ and all effects $e,e^{\\prime }$ .", "Thus each map $\\omega $ admits an expression $\\omega (e)=e\\cdot \\omega ,$ for all effects $e$ and some $\\omega \\in W\\left(\\mathcal {E}\\right)\\in \\mathbb {R}^{d}$ .", "One the other hand, we have already assumed that a pair $\\left\\lbrace e,u-e\\right\\rbrace $ form a measurement and have introduced the formalism for describing mixtures of measurements, therefore the simulable measurements from Section are already included in the framework.", "Theorem REF then tells us that if a state $\\omega $ is to assign probabilities to the possible outcomes of these measurements such that the probabilities of all the outcomes sum to one then $\\omega (e)=e\\cdot \\omega ,$ for all effects $e$ and some $\\omega \\in W\\left(\\mathcal {E}\\right)\\in \\mathbb {R}^{d}$ .", "Both of these approaches lead to the conclusion that the state space of a GPT with effect space $\\mathcal {E}$ must be a subset of $W\\left(\\mathcal {E}\\right)$ .", "Although the conditions are mathematically different there is no clear conceptual advantage to either argument.", "The “measurement first” derivation of the framework highlights the existence of a relative of the no-restriction hypothesis, which we will call the no-state-restriction hypothesis: the inclusion of all $\\omega \\in \\mathbb {R}^{d}$ satisfying $e\\cdot \\omega $ and $u\\cdot \\omega =1$ in the state space.", "Note that this is not equivalent to the no-restriction hypothesis in all cases, for example the noisy bit model in Figure REF satisfies the no-state-restriction hypothesis but not the no-restriction hypothesis.", "Continuing the bit example, employing either the no-restriction or no-state-restriction hypothesis leads to the state space $\\mathcal {S}_{B^{\\prime }}$ , the convex hull of the points $(0,1)^{T}$ and $(1,0)^{T}$ pictured in Figure REF .", "The pair of state and effect spaces $\\mathcal {S}_{B^{\\prime }}$ and $\\mathcal {E}_{B^{\\prime }}$ are a transformation of the state and effect spaces $\\mathcal {S}_{B}$ and $\\mathcal {E}_{B}$ in Figure REF ." ] ]

[ [ "Instant preheating in a scale invariant two measures theory" ], [ "Abstract The instant preheating mechanism in the framework of a scale invariant two measures theory is studied.", "We introduce this mechanism into a non oscillating inflationary model as another possible solution to the reheating of the universe in this theory.", "In this framework, we consider that the model includes two scalar matter fields, the first a dilaton field, that transforms under scale transformations and it will be considered also as the field that drives inflation and the second, a scalar field which will interact with the inflaton through an effective potential.", "By assuming this interaction term, we obtain a scenario of instant radiation or decay of particles according to the domain the effective mass of the field that interacts with the inflaton.", "Also, we consider a scale invariant Yukawa interaction and then after performing the transition to the physical Einstein frame we obtain an expression for the decay rate from our scalar field going into two fermions.", "Besides, from specific decay rates, different constraints and bounds for the coupling parameters associated with our model are found." ], [ "Introduction", "It is well known that the inflationary universe models have solved some problems present in the hot big bang model, such as the horizon, flatness, monopole problem etc.", "[1], [2], [3].", "However, the biggest feature of the inflationary stage is that it provides a causal interpretation to explicate the observed anisotropy of the cosmic microwave background (CMB) radiation[4] and moreover this framework gives account of the distribution of large scale structures [5].", "In order to study the inflationary stage, the scalar field or inflaton plays a fundamental role in the evolution of the early universe.", "In this context, the inflationary epoch can be described as a regime with a rapid accelerated expansion occurred during the early universe produced by the inflaton[1], [2].", "To describe the inflationary epoch, we have different gravitation theories and models that give account of the evolution of the early universe.", "In particular we can distinguish the scale invariant two measures theory[6], [7], [8] that produces an accelerated expansion of the universe by means of the evolution of a single scalar field or inflaton field with an effective potential [9], [10], [11].", "In relation to the two measures theories models, these utilize a non Riemannian measure of integration in the frame of the action.", "In particular in the situation of a scale invariant theory, the scale invariance was spontaneously broken from the equations of motion related with the degrees of freedom on the non Riemannian measure of integration in the framework of the action.", "In this sense, we can mention that the degrees of freedom that determine a non Riemannian measure of integration in four dimensions can be represented by scalar fields[9], [10].", "In this sense, utilizing the measure of integration and also in the frame of the action different models with several scalar fields in four dimensions have been studied in the literature[12], [13].", "The application of this scale invariant two measures theory to an emergent universe scenario was developed in ref.[13].", "It corresponds to a non singular cosmological type of stage previous to inflation (emergent scenario) in which the universe begins as a static universe to later connect with the inflationary epoch[14].", "In order to use the two measures theory to describe the dark energy in the present universe, in ref.", "[14] was considered that the two measures of integration leading to two independent integration constants and these constants break scale invariance, and characterize the strength of the dark energy density.", "Additionally, in ref.", "[15] the curvaton reheating mechanism in a scale invariant two measures theory defined in terms of two independent non-Riemannian volume forms was studied.", "In this context, the model has two scalar matter fields, a dilaton and it transforms under scale transformations and it corresponds to the inflaton field of the inflationary model while the other scalar field does not transform under scale transformations played the role of a curvaton field[15].", "The introduction of the curvaton field in this scenario occurs due to the problematic of connecting the inflationary epoch in the framework of the scale invariant two measures theory, with the reheating of the universe and its subsequent connection with the radiation era [16], [17].", "In relation to the reheating of the universe, we have that at the end of inflationary epoch the energy density of the universe can be interpreted as a combination of kinetic and potential energies of the inflaton to late dominate the kinetic energy [18].", "In the process of reheating of the universe, the matter and radiation of the universe are produced generally through the decay of the scalar field or another field (decay parameter), while the temperature increases in many orders of magnitude and then the universe connects with the radiation regime of the standard big-bang model[19], [20].", "In order to study the reheating of the universe the scenario of oscillations of the inflaton field (at the minimum of the potential) is an important part for the standard mechanism of reheating.", "Nevertheless, it is possible to find some inflationary models where the effective potential associated to the inflaton does not have a minimum and then the scalar field does not oscillate and then the standard mechanism of reheating does not work[21].", "Thus, these kinds of models with these effective potentials are known in the literature as non-oscillating models, or simply NO models[22].", "Interesting examples of these are the Quintessential inflation models which connect an early inflation with a late slowly accelerated phase, as the models considered by Peebles and Vilenkin[23] as well as [8], [9], [10], [13], [24], [25].", "Originally, in order to solve this problematic for these NO models was the introduction of a mechanism that incorporates the gravitational particle production[26].", "Nevertheless, this mechanism of reheating of the universe becomes inefficient and it presents several problems associated with the observational data, see ref.[27].", "The introduction of the curvaton field as other mechanism of reheating after inflation in these NO models was considered in refs.", "[16], [17], [22], [20].", "In this context, the decay rate of the scalar curvaton field into conventional matter gives account a mechanism of reheating of the universe.", "In this sense, introducing an effective potential associated to the curvaton field is possible to reheat the universe[20], [28].", "This model of reheating does not need to introduce an interaction between the scalar field that drives inflation and other scalar field[20], [17].", "Another mechanism of reheating known in the literature is called instant preheating[29].", "In this scenario, after of the inflationary regime the inflaton field moves quickly producing particles which can be bosons and/or fermions.", "This mechanism corresponds to a non-perturbative process and it happens almost instantly[29] and also this scenario does not need oscillations or parametric resonance of the inflaton field.", "In this sense, because the production of particles can happen immediately after the end of inflationary regime, within less than one oscillation of the field that drives inflation, the reheating of the universe can occurs efficiently.", "In order to study the instant reheating is indispensable to consider the interaction between the scalar field that drives inflation and another scalar field $\\sigma $ .", "Depending of the interaction between the inflaton field and the field $\\sigma $ (via an effective potential) the effective masses of the particles $\\sigma $ can be small or large at the moment when the particles are produced for later increase or decrease when the inflaton field moves to large values.", "In this mechanism the production of particles $\\sigma $ begins nearly instantaneously assuming the nonadiabatically condition given by the ratio between the evolution of the effective masses of the particles-$\\sigma $ and the square of these[29].", "For a review of reheating see refs.", "[29], [30], [31] and for instant preheating, see [32], [24].", "The goal of this investigation is to analyze the instant preheating in a scale invariant two independent non Riemannian volume-forms.", "In this sense, we investigate how the interaction term between the inflaton field that drives inflation and other scalar field (from the effective potential) in this theory modifies the results on the produced particles in this scenario and preheating of the universe.", "In this form, we will analyze the instant preheating in our model and in particular the energy density of produced particles and the decay rates in order to in account of the temperature and constraints on the parameters given by the observations.", "For the application of the developed formalism, we will analyze some examples assuming two decay parameters.", "From these decay rates, we will study the different conditions of time, in order to obtain the bounds on the coupling parameters associated to these decay rates.", "The outline of the paper goes as follow: in Sect.", "II we give a brief description of two independent non-Riemannian volume-forms.", "In Sect.", "III the instant preheating scenario is analyzed.", "The Sect.", "IV describes the instant radiation in which the energy density of the field-$\\sigma $ decays as radiation.", "The Sect.", "V explains the radiation from the decay.", "In Sect.", "VI we obtain the decay rate for the particles $\\sigma $ going into two fermions.", "The Sect.", "VII analyzes the decay rate and constraints on the parameters of our model, and in Sect.", "VIII includes our conclusions." ], [ "Two independent non-Riemannian volume-forms", "In this section, we discuss a brief description of the two independent non-Riemannian volume-forms.", "We follow the general structure of the references [9], [10], but now we will enrich the field content of the theory with a new field $\\sigma $ which will not transform under scale transformations, so we write, $S = \\int d^4 x\\,\\Phi _1 (A) [ R + L^{(1)}] +\\int d^4 x\\,\\Phi _2 (B)\\left[ L^{(2)} + \\epsilon R^2 +\\frac{\\Phi (H)}{\\sqrt{-g}}\\right] \\; ,$ where $\\Phi _1(A)$ and $\\Phi _2(B)$ are two independent non-Riemanniam volume-forms and defined as $\\Phi _1 (A) = \\frac{1}{3!", "}\\varepsilon ^{\\mu \\nu \\kappa \\lambda } \\partial _\\mu A_{\\nu \\kappa \\lambda }, \\quad \\mbox{and}\\quad \\Phi _2 (B) = \\frac{1}{3!", "}\\varepsilon ^{\\mu \\nu \\kappa \\lambda }\\partial _\\mu B_{\\nu \\kappa \\lambda } \\; , $ respectively.", "The quantities $L^{(1,2)}$ correspond to two different Lagrangians of two scalar fields, the dilaton $\\varphi $ , which will play the role of an inflaton and an additional scalar field $\\sigma $ .", "In this form, the Lagrangians can be written as $L^{(1)} = -g̉^{\\mu \\nu } \\partial _\\mu \\varphi \\partial _\\nu \\varphi -g̉^{\\mu \\nu } \\partial _\\mu \\sigma \\partial _\\nu \\sigma -\\frac{\\mu ^2\\sigma ^2}{2} \\exp \\lbrace -\\alpha \\varphi \\rbrace - V(\\varphi ),$ where $V(\\varphi ) = f_1 \\exp \\lbrace -\\alpha \\varphi \\rbrace $ and the Lagrangian $L^{(2)} = -\\frac{b}{2} e^{-\\alpha \\varphi } g^{\\mu \\nu } \\partial _\\mu \\varphi \\partial _\\nu \\varphi + U(\\varphi ),$ in which $U(\\varphi ) = f_2 \\exp \\lbrace -2\\alpha \\varphi \\rbrace $ .", "Here the quantities $\\alpha , f_1, f_2$ are dimension full positive parameters and the parameter $b$ is a dimensionless one.", "Also, the quantity $\\Phi (H)$ denotes the dual field strength of a third auxiliary 3-index antisymmetric tensor gauge field defined as $\\Phi (H) =\\frac{1}{3!", "}\\varepsilon ^{\\mu \\nu \\kappa \\lambda } \\partial _\\mu H_{\\nu \\kappa \\lambda } \\; .", "$ We mention that the scalar potentials have been chosen such that the action given by eq.", "(REF ) is invariant under global Weyl-scale transformations with which $g_{\\mu \\nu } \\rightarrow \\lambda g_{\\mu \\nu } \\;\\; ,\\;\\; \\Gamma ^\\mu _{\\nu \\lambda } \\rightarrow \\Gamma ^\\mu _{\\nu \\lambda } \\;\\; ,\\;\\;\\varphi \\rightarrow \\varphi + \\frac{1}{\\alpha }\\ln \\lambda \\;\\;, \\sigma \\rightarrow \\sigma ,$ $A_{\\mu \\nu \\kappa } \\rightarrow \\lambda A_{\\mu \\nu \\kappa } \\;\\; ,\\;\\; B_{\\mu \\nu \\kappa } \\rightarrow \\lambda ^2 B_{\\mu \\nu \\kappa }\\;\\; ,\\;\\; H_{\\mu \\nu \\kappa } \\rightarrow H_{\\mu \\nu \\kappa } \\; .$ Analogously, from the invariant under we have multiplied by an exponential factor the scalar kinetic term in $L^{(2)}$ and also by the scalar curvature $R$ and $R^2$ couple to the two different modified measures.", "The equations of motions of the measure fields lead to several simple relations.", "First, the variation of the tensor field $H$ implies that the ratio between the measure $\\Phi _2$ and $\\sqrt{-g}$ is a constant: $\\frac{\\Phi _2}{\\sqrt{-g}} = \\chi _2 = \\mbox{constant}.$ Likewise, the variation with respect to $\\Phi _1$ and $\\Phi _2$ leads to the the Lagrangians coupling to $\\Phi _1$ and $\\Phi _2$ being constants that we may call $M_1$ and $M_2$ : $R + L^{(1)} =- M_1 = \\mbox{constant},$ and $L^{(2)} + \\epsilon R^2 +\\frac{\\Phi (H)}{\\sqrt{-g}} =- M_2 = \\mbox{constant},$ while equation () does not break scale invariance, since the two measures $\\Phi _2$ and $\\sqrt{-g}$ transform identically under scale transformations.", "The same cannot be said however concerning (REF ) and (REF ) while the left hand side in these equations transforms, the right hand side ($M_1$ and $M_2$ )are constants and does not transform.", "We get then spontaneous breaking of scale invariance.", "We proceed in the so called first order formalism, where the connection is at the action level independent of the metric, in this case we can vary with respect to the metric and the consistency with the equations (REF ) and (REF ) allows us to solve for $\\frac{\\Phi _1}{\\sqrt{-g}} = \\chi _1$ , which is given by, $\\frac{1}{\\chi _1} =\\frac{(V+\\frac{\\mu ^2\\sigma ^2}{2} e^{-\\alpha \\varphi }-M_1)}{2\\chi _2(U+M_2)}.$ Here we have considered the case in which $\\epsilon =0$ and $b=0$ , respectively.", "Defining Einstein frame by a conformal transformation, we obtain an effective action, which in the case of $\\epsilon =0$ and $b=0$ is governed by a canonical minimally coupled scalar field with the following effective Lagrangian given by $L_{eff}=-\\frac{1}{2}\\bar{g}^{\\mu \\nu }\\partial \\varphi _\\mu \\partial \\varphi _\\nu -\\frac{1}{2}\\bar{g}^{\\mu \\nu }\\partial \\sigma _\\mu \\partial \\sigma _\\nu -U(\\varphi ,\\sigma ),$ where the Weyl-rescaled metric $\\bar{g}_{\\mu \\nu }$ is defined as $\\bar{g}_{\\mu \\nu }=\\chi _1\\;g_{\\mu \\nu },$ and the effective potential is given by $U_{\\rm eff} (\\varphi , \\sigma )=\\frac{(V+ \\frac{\\mu ^2 \\sigma ^2}{2}\\; e^{-\\alpha \\varphi } - M_1)^2}{4\\chi _2 [ U + M_2]}=\\frac{(f_1 e^{-\\alpha \\varphi }+ \\frac{\\mu ^2 \\sigma ^2}{2}\\; e^ {-\\alpha \\varphi } -M_1)^2}{4\\chi _2\\,[f_2\\; e^{-2\\alpha \\varphi } + M_2 ]}.", "$" ], [ "Instant preheating", "In order to explain the instant preheating scenario in our model we will consider that the effective potential given by Eq.", "(REF ) presents the interaction term given by $U_{\\rm eff} (\\varphi , \\sigma )\\approx \\frac{\\mu ^2\\beta ^2\\,\\sigma ^2}{2}\\:e^{-2\\alpha \\varphi }=\\frac{m_1^2\\;\\sigma ^2}{2}\\:e^{-2\\alpha \\varphi },$ where the constant $\\beta $ is defined as $\\beta ^2=\\frac{f_1}{2\\chi _2M_2}$ and $m_1=\\mu \\beta $ .", "Here we have assumed that $M_2\\gg f_2 e^{-2\\alpha \\varphi }$ and $f_1e^{-\\alpha \\varphi }\\gg M_1$ , since during the inflationary scenario we have used the values $M_1\\sim 10^{-60}$ , $f_1\\simeq f_2\\sim 10^{-8}$ and $M_2\\sim 1$ , from observational data, see ref.[9].", "Thus, the effective mass of the scalar field $\\sigma $ becomes $m_{\\sigma }=\\mu \\beta \\;e^{-\\alpha \\varphi }=m_1\\,e^{-\\alpha \\varphi },$ since the effective mass of $\\sigma $ is defined as $m_\\sigma ^2=\\partial ^2U_{\\rm eff}(\\varphi ,\\sigma )/\\partial \\sigma ^2$ .", "Following refs.", "[36], [37] we will consider that the production of particles $\\sigma $ starts to change nonadiabatically under the condition $|\\dot{m_\\sigma }|\\ge m_\\sigma ^2$ with which the scalar field $\\varphi $ can be written as $\\varphi \\sim -\\frac{1}{\\alpha }\\;\\ln \\left(\\frac{\\alpha |\\dot{\\varphi }_{0}|}{\\mu \\beta }\\right),$ where $\\dot{\\varphi }_{0}$ denotes the value of the velocity of the scalar field when this field rolls on the asymptotically flat potential after of the inflationary epoch.", "During this stage the mechanism of particle production starts nearly instantaneously in the time interval given by $\\triangle \\,t\\sim \\,\\frac{|\\varphi |}{|\\dot{\\varphi _0}|}\\sim \\frac{1}{\\alpha |\\dot{\\varphi _0}|}\\;\\big |\\ln \\left(\\frac{\\alpha |\\dot{\\varphi }_{0}|}{\\mu \\beta }\\right)\\big |>0.$ Also, we mention that during this time all effects associated to the expansion of the universe can be ignored in the process of particle production.", "Now, in order to determine the velocity of the scalar field $\\dot{\\varphi }_0$ , we can consider the break down approximation in which $\\ddot{\\varphi }\\simeq -\\frac{\\partial V(\\varphi )}{\\partial \\varphi }=\\frac{\\alpha \\,f_1^2}{2\\chi _2M_2}e^ {-2\\alpha \\varphi },$ where we have considered that the potential $V(\\varphi )=\\frac{f_1^2}{4\\chi _2M_2}e^ {-2\\alpha \\varphi }$ from eq.", "(REF ), see ref.[9].", "Under this approximation, we find that the solution of the eq.", "(REF ) for the scalar field $\\varphi (t)$ results $\\varphi (t)=\\frac{1}{\\alpha }\\,\\ln \\left[ \\frac{e^{-\\alpha \\sqrt{C_1}(t+C_2)}}{2}\\left(k_1^2+\\frac{e^{2\\alpha \\sqrt{C_1}(t+C_2)}}{\\alpha C_1}\\right) \\right],$ where $C_1$ and $C_2$ are two integration constants and $k_1$ is defined as $k_1^2=\\frac{\\alpha f_1^2}{2\\chi _2M_2}$ .", "From this solution we can find that the velocity of the scalar field $\\dot{\\varphi }$ is given by $\\dot{\\varphi }(t)=\\frac{\\sqrt{C_1}\\,(e^{2\\alpha \\sqrt{C_1}(t+C_2)}-\\alpha C_1k_1^2)}{e^{2\\alpha \\sqrt{C_1}(t+C_2)}+\\alpha C_1k_1^2},$ which for $\\alpha $ big the above expression big quickly approaches the asymptotic value $\\sqrt{C_1}$ i.e., $\\dot{\\varphi _0}\\sim \\sqrt{C_1} $ .", "Thus, in order to obtain the value of the $\\dot{\\varphi _0}$ , we can consider that the initial conditions for the scalar field and its velocity can be fixed at the end of inflationary epoch.", "In this way, we assume the slow roll approximation in which at the end of inflation we have $\\varphi _{end}=-\\alpha ^{-1}\\ln (2\\alpha M_1/f_1)$ and $\\dot{\\varphi }_{end}=\\frac{2M_1\\alpha ^2}{\\sqrt{3\\chi _2M_2}}$ , see ref.[9].", "From these initial conditions and considering the eqs.", "(REF ) and (REF ), we find that the asymptotic velocity of the scalar field becomes $\\dot{\\varphi }_0\\simeq \\,\\sqrt{C_1}\\,\\simeq \\dot{\\varphi }_{end}\\;\\,\\sqrt{1+\\frac{3}{2\\alpha ^2}}.$ Here we note that for large-$\\alpha $ the velocity $\\dot{\\varphi _0}\\simeq \\dot{\\varphi }_{end}$ .", "Thus, the time interval given by eq.", "(REF ) can be approximated to $\\triangle \\,t\\sim \\, (\\alpha |\\dot{\\varphi }_{end}|)^{-1}\\;\\ln \\left(\\frac{\\alpha |\\dot{\\varphi }_{end}|}{\\mu \\beta }\\right)$ .", "On the other hand, the occupation number $n_k$ of the particles $\\sigma $ with momentum $k$ in the time interval $\\triangle \\,t$ is defined as [36], [37], [38] $n_k=\\exp [-\\pi \\,(k\\,\\triangle \\,t)^2],$ and then considering eq.", "(REF ) we obtain that the occupation number can be written as $n_k=\\exp \\left[-\\frac{\\pi \\,k^2}{\\alpha ^2\\,\\dot{\\varphi _0}^2}\\,\\left(\\ln \\left[\\frac{\\alpha |\\dot{\\varphi _0}|}{\\mu \\beta }\\right]\\right)^2\\right].$ In fact, we can assume that the definition of the occupation number given by eq.", "(REF ) still is valid for massive particles of the scalar field $\\sigma $ of effective mass $m_\\sigma $ under replacement of the momentum $k^2$ by $k^2+m_\\sigma ^2$ [37].", "Thus, eq.", "(REF ) can be modified as $n_k=\\exp [-\\pi \\,(k^2+m_\\sigma ^2)\\,\\triangle \\,t^2]$ with which the occupation number becomes $n_k=\\exp \\left[-\\frac{\\pi \\,(k^2+m_\\sigma ^2)}{\\alpha ^2\\,\\dot{\\varphi _0}^2}\\,\\left(\\ln \\left[\\frac{\\alpha |\\dot{\\varphi _0}|}{\\mu \\beta }\\right]\\right)^2\\right].$ Now, this quantity can be integrated to establish the density of $\\sigma $ particles denotes by $n_\\sigma $ and defined as $n_\\sigma =\\frac{1}{2\\pi ^2}\\int _0^\\infty \\,d k\\,k^2\\,n_k$ .", "In this way, we find that the density of $\\sigma $ particles $n_\\sigma $ results $n_\\sigma =\\frac{1}{8\\pi ^3}\\,\\left[\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\big |\\ln \\left(\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\mu \\beta }\\right)\\big |}\\right]^{3}\\,\\exp \\left[-\\frac{\\pi \\,m_\\sigma ^2}{\\alpha ^2\\,\\dot{\\varphi _0}^2}\\,\\left(\\ln \\left[\\frac{\\alpha |\\dot{\\varphi _0}|}{\\mu \\beta }\\right]\\right)^2\\right].$ We note that naturally in our model the number of produced particles is not exponentially suppressed, since the mass of the scalar field $\\sigma $ decreases for large-$\\varphi $ ($m_\\sigma \\propto e^{-\\alpha \\varphi }$ ).", "Thus, the number density of particles during their creation results $\\frac{1}{8\\pi ^3}\\,\\left[\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\big |\\ln \\left(\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\mu \\beta }\\right)\\big |}\\right]^{3}$ , however it decreases as $a^{-3}(t)$ with which the number of produced particles in terms of the time can be written as $n_\\sigma =\\frac{1}{8\\pi ^3}\\,\\left[\\frac{\\alpha \\,|\\dot{\\varphi _0}|a_0}{a(t)\\,\\big |\\ln \\left(\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\mu \\beta }\\right)\\big |}\\right]^{3}.$ Here we have used that at the moment of particle production the scale factor is given by $a_0$ .", "Additionally, we have that the energy density of produced particles $\\rho _\\sigma $ is defined as [33], [34] $\\rho _\\sigma =\\frac{1}{(2\\pi \\,a)^3}\\int _0^\\infty \\,\\,n_k\\,\\,\\sqrt{\\frac{k^2}{a^2}+m_\\sigma ^2}\\,\\,\\;(4\\pi k^2) d k.$ Here, we can note that interestingly there are two limit cases given by $m_\\sigma \\gg k/a$ and $m_\\sigma \\ll k/a$ , because the effective mass of the $\\sigma $ -field $m_\\sigma $ depends of the time.", "Thus, initially after of the inflationary stage, we can consider that the dominant term becomes the mass $m_\\sigma $ over the physical momentum $k/a$ .", "Later, product of the decrease in the time of the mass $m_\\sigma $ , the dominant term corresponds to the momentum i.e., $k/a\\gg m_\\sigma $ .", "In the following, we will analyze these two limits separately." ], [ "Instant radiation", "In this section we will study the process in which the energy density of produced particles of the field $\\sigma $ decays as radiation.", "We call this process as instant radiation and it occurs for large-time when the mass of the $\\sigma -$ field decreases and then the effective mass tends to zero with which $m_\\sigma \\ll k/a$ .", "In this situation we find that energy density of the $\\sigma -$ field from Eq.", "(REF ) becomes $\\rho _\\sigma =B a^{-4},$ where the constant $B$ is defined as $B=\\left[\\frac{\\alpha \\,\\dot{\\varphi _0}}{\\left(2^{1/2}\\pi \\ln \\left[\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\mu \\beta }\\right]\\right)}\\right]^4.$ In this context, the equation of motion for the inflation field $\\varphi $ including backreaction of produced $\\sigma $ particles on the field $\\varphi $ can be written as $\\ddot{\\varphi }+3H\\dot{\\varphi }=\\alpha \\mu ^2\\beta ^2\\,e^{-2\\alpha \\varphi }\\langle \\sigma ^2\\rangle ,$ where the expectation value $\\langle \\sigma ^2\\rangle $ is defined as [35] $\\langle \\sigma ^2\\rangle \\approx \\frac{1}{2\\pi ^2}\\int \\frac{n_k\\, k^2\\, d k}{\\sqrt{(k/a)^2+m_\\sigma ^2}}.$ Thus, for the case of the instant radiation ($m_\\sigma \\ll k/a$ ) we find that $\\langle \\sigma ^2\\rangle $ becomes $\\langle \\sigma ^2\\rangle \\approx \\frac{1}{2\\pi ^2}\\int n_k\\, k\\, d k=\\,\\frac{B_1}{a^{2}(t)},$ where the constant $B_1$ is given by $B_1=\\sqrt{B}/(2\\pi )$ .", "In this way, the equation of motion for the inflaton field in the situation in which the effective mass $ m_\\sigma \\ll k/a$ including the backreaction term becomes $\\ddot{\\varphi }+3H\\dot{\\varphi }=\\alpha \\,\\mu ^2\\,\\beta ^2\\,B_1\\,\\frac{e^{-2\\alpha \\varphi (t)}}{a^2(t)}.$ We observe that the backreaction effect decreases very quickly due to exponential decay product of evolution of $\\varphi (t)$ that appears in the right hand side of this equation.", "Thus, the backreaction of produced $\\sigma $ particles on $\\varphi $ disappears naturally from the effective potential given by eq.", "(REF ).", "Also, from the condition $m_\\sigma \\ll k/a$ and considering that the scale factor $a(t)\\propto t^{1/3}$ together with neglecting the backreation of eq.", "(REF ), we find that the constraint for the $\\alpha $ parameter becomes $\\alpha >(2\\sqrt{3})^{-1}$ , if we want the condition $m_\\sigma \\ll k/a$ to be maintained during the time.", "Additionally, we note that in the scenario of instant radiation, if nothing else happens, meaning non-decay of the $\\sigma $ particles, and since the mass of these particles approaches to zero, then we obtain that these particles will asymptotically behave as radiation, as can be seen from eq.", "(REF ).", "However, we mention that this spectrum is not thermal becomes the distribution in the occupation number is not Boltzmann distribution (see eq.", "(REF )), since the spectrum is not thermal in order to obtain a real thermal spectrum a thermalization process is required.", "The thermalization should bring all particle species[39], [40]." ], [ "Radiation from decay", "In this section, we can analyze the case where the mass $m_\\sigma \\gg k/a$ , for the dominant range of integration of the momentum.", "In this limit we have $\\rho _\\sigma =m_\\sigma \\,n_\\sigma =\\frac{\\mu \\beta }{8\\pi ^3}\\,\\left[\\frac{\\alpha \\,|\\dot{\\varphi _0}|a_0}{\\big |\\ln \\left(\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\mu \\beta }\\right)\\big |}\\right]^{3}\\,\\frac{e^{-\\alpha \\,\\varphi (t)}}{a^{3}(t)}\\propto \\frac{ e^{-\\alpha \\,\\varphi (t)}}{a^{3}(t)},$ here, we have called to this stage as radiation from decay.", "On the other hand, the equation of motion for the inflation field $\\varphi $ after of the particles production can be written as $\\ddot{\\varphi }+3H\\dot{\\varphi }=\\alpha \\mu ^2\\beta ^2\\,e^{-2\\alpha \\varphi }\\langle \\sigma ^2\\rangle ,$ where the expectation value $\\langle \\sigma ^2\\rangle $ from eq.", "(REF ) and assuming $m_\\sigma \\gg k/a$ can be written as $\\langle \\sigma ^2\\rangle \\approx \\frac{1}{2\\pi ^2}\\int \\frac{n_k\\, k^2\\, d k}{\\sqrt{(k/a)^2+m_\\sigma ^2}}\\approx \\frac{n_\\sigma }{m_\\sigma }\\approx \\,A\\,\\,\\frac{e^{\\alpha \\varphi (t)}}{a^{3}(t)},$ in which the constant $A$ is defined as $A=\\frac{1}{8\\mu \\beta \\pi ^3}\\,\\left[\\frac{\\alpha \\,|\\dot{\\varphi _0}|a_0}{\\big |\\ln \\left(\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\mu \\beta }\\right)\\big |}\\right]^{3}.$ In this form, using eq.", "(REF ) we find that eq.", "(REF ) can be rewritten as $\\ddot{\\varphi }+3H\\dot{\\varphi }=\\alpha \\, m_\\sigma \\,\\,n_\\sigma =\\frac{\\alpha \\mu \\beta }{8\\pi ^3}\\,\\left[\\frac{\\alpha \\,|\\dot{\\varphi _0}|a_0}{\\,\\big |\\ln \\left(\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\mu \\beta }\\right)\\big |}\\right]^{3}\\frac{e^{-\\alpha \\varphi (t)}}{a^3(t)}.$ In order to analyze the behavior of the field $\\varphi (t)$ from eq.", "(REF ), we will consider as a first approximation neglects backreaction of produced particles.", "In this context, we can consider that the energy density of the $\\sigma -$ particles ($\\rho _\\sigma =m_\\sigma \\,n_\\sigma $ ) becomes subdominant and then the right hand side of eq.", "(REF ) can be negligible.", "In fact, from eq.", "(REF ) (or analogously of eq.", "(REF )), we note that naturally our effective potential gives rise to a force which produces that the inflaton field continues its movement to infinity.", "Under this approximation the scale factor is given by $a(t)\\sim t^{1/3}$ and then the Hubble parameter becomes $H=(3t)^{-1}$ .", "Thus, if one neglects backreaction, the solution for the scalar field as a function of the time can be written as $\\varphi (t)=\\frac{2}{\\sqrt{3}}\\,\\ln \\left(\\frac{t}{t_0}\\right),$ where the constant $t_0$ is defined as $t_0=(3H_0)^{-1}$ and it corresponds to the initial time during the phase transition time between inflation and kination regime.", "In this way, replacing eq.", "(REF ) in the equation for the energy density of produced particles given by eq.", "(REF ) we have $\\rho _\\sigma =\\alpha _1\\left(\\frac{t}{t_0}\\right)^{-2\\alpha /\\sqrt{3}}\\,\\left(\\frac{a_0}{a}\\right)^3,$ where the constant $\\alpha _1$ is defined as $\\alpha _1=\\frac{\\mu \\beta }{8\\pi ^3}\\,\\left[\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\big |\\ln \\left(\\frac{\\alpha \\,|\\dot{\\varphi _0}|}{\\mu \\beta }\\right)\\big |}\\right]^{3}.$ Additionally, during the kination regime the energy density of the background decreases as $\\rho (t)\\sim \\dot{\\varphi }^2\\sim a^{-6}$ or $\\rho (t)=6H_0^2\\,(a_0/a)^6$ and we can consider that both densities achieve equilibrium i.e., $\\rho \\sim \\rho _\\sigma $ .", "In this sense, if the densities $\\rho $ and $\\rho _\\sigma $ are of the same order, we can assume that this situation occurs at the equilibrium time $t_{eq}$ given by $t_{eq}=\\left[\\frac{2}{3\\alpha _1}\\,t_0^{\\delta _1}\\right]^{1/\\delta _2},$ where the constants $\\delta _1$ and $\\delta _2$ are defined as $\\delta _1=-(1+2\\alpha /\\sqrt{3}),\\,\\;\\;\\;\\;\\mbox{and}\\,\\,\\,\\,\\delta _2=(1-2\\alpha /\\sqrt{3}),$ respectively.", "In this way, the value of the scalar field at the time $t_{eq}$ becomes $\\varphi (t=t_{eq})=\\varphi _{eq}=\\frac{2}{\\sqrt{3}\\;\\delta _2}\\left[\\ln \\left(\\frac{2}{3\\alpha _1}\\right)+(\\delta _1-\\delta _2)\\ln \\,t_0)\\right].$ We note that in particular for values of $\\alpha \\gg 1$ , we have $\\varphi _{eq}\\simeq -\\frac{1}{\\alpha }\\,\\ln \\left[\\frac{2}{3\\alpha _1\\,t_0^2}\\right]=-\\frac{1}{\\alpha }\\,\\ln \\left[\\frac{\\dot{\\varphi _0}^2}{2\\alpha _1}\\right].$ Here, we have used that $t_0=2/(\\sqrt{3}\\,\\dot{\\varphi _0})$ , in which $\\dot{\\varphi }_0\\simeq \\sqrt{C_1}$ , see eq.", "(REF )." ], [ "Scale invariant coupling of $\\sigma $ and {{formula:e1544cbc-b0d4-44f5-a880-de89f112a2f3}} fields:\ndecay rate of the {{formula:b2761db3-c22e-43f7-8e3d-b4a96749c922}} particles to fermions", "In this section, we want to analyze now a coupling of the field $\\sigma $ to a fermionic spin $1/2$ field $\\Psi $ .", "We will consider possible couplings while respecting scale invariance.", "Let us consider first the $\\Psi $ kinetic term coupled to the measure $\\Phi _{1}$ $S= \\int \\Phi _{1} L_{kin},$ where $ L_{kin}$ is given by, $L_{kin}=\\frac{i}{2}\\bar{\\Psi }(\\gamma ^a e_{a}^{\\mu }{\\overrightarrow{\\bigtriangledown }}_{\\mu }\\Psi - \\bar{\\Psi }{\\overleftarrow{\\bigtriangledown }}_{\\mu }\\gamma ^a e_{a}^{\\mu } \\Psi ),$ where ${\\overrightarrow{\\bigtriangledown }}_{\\mu }\\Psi = \\partial _{\\mu }\\Psi + \\frac{1}{2}\\omega _{\\mu }^{ab}\\sigma _{ab}\\Psi ,$ and $\\bar{\\Psi }{\\overleftarrow{\\bigtriangledown }}_{\\mu } = \\partial _{\\mu }\\bar{\\Psi } - \\bar{\\Psi }\\frac{1}{2}\\omega _{\\mu }^{ab}\\sigma _{ab}.$ The $\\gamma ^a$ matrices are metric independent (m.i.)", "while $\\bar{\\Psi }= \\Psi ^{\\dagger } \\gamma ^0$ is as well m.i.", "Since under a scale transformation we have $\\Phi _{1} \\rightarrow e^\\theta \\Phi _{1}$ , then $S_{kin}$ is invariant under $\\omega _{\\mu }^{ab} \\rightarrow \\omega _{\\mu }^{ab},$ $\\Psi \\rightarrow e^{-\\frac{\\theta }{4}}\\Psi ,$ $\\bar{\\Psi } \\rightarrow e^{-\\frac{\\theta }{4}}\\bar{\\Psi },$ and $g_{\\mu \\nu } \\rightarrow e^{\\theta }g_{\\mu \\nu },$ which is equivalent also to $e_{a}^{\\mu } \\rightarrow e^{-\\frac{\\theta }{2}}e_{a}^{\\mu } $ and $e_{\\mu }^{a} \\rightarrow e^{\\frac{\\theta }{2}}e_{\\mu }^{a} $ .", "Thus, the bilinear quantity $\\bar{\\Psi }\\Psi $ transforms as $\\bar{\\Psi }\\Psi \\rightarrow e^{-\\frac{\\theta }{2}}\\bar{\\Psi }\\Psi $ .", "So, since $\\sigma $ is invariant under scale transformations, we see that a coupling to the measure $\\Phi _{1}$ must also require a factor $e^{\\alpha \\varphi }$ $\\sigma \\Phi _{1}e^{\\frac{\\alpha \\varphi }{2}}\\bar{\\Psi }\\Psi .$ Like wise, the coupling to the measure $\\Phi _{2}$ (or $\\sqrt{-g}$ which transforms the same way and which is proportional to $\\Phi _{2}$ ), must contain a factor of $e^{\\frac{3\\alpha \\varphi }{2}}$ leading to an invariant term $\\sigma \\Phi _{2}e^{3\\frac{\\alpha \\varphi }{2}}\\bar{\\Psi }\\Psi .$ Thus, the \"scale invariant Yukawa type interaction\" between the field $\\sigma $ and the fermions must include $\\varphi $ in the following way, $\\int \\sigma ( g_1 \\Phi _{1}e^{\\frac{\\alpha \\varphi }{2}}\\bar{\\Psi }\\Psi +g_2 \\Phi _{2}e^{3\\frac{\\alpha \\varphi }{2}}\\bar{\\Psi }\\Psi )d^4x.$ To properly use this interaction, we must transform to the Einstein Frame, use the Einstein Frame metric $\\bar{g}_{\\mu \\nu }$ and the Einstein Frame fermion field $\\Psi _{e.f}$ .", "For the case $\\epsilon = b = 0$ , we have that $\\bar{g}_{\\mu \\nu } = \\chi _{1}g_{\\mu \\nu }$ , or equivalently, $\\bar{e}_{\\mu }^a = \\chi _{1}^{\\frac{1}{2}}e_{\\mu }^a$ , and $\\bar{e}^a_{\\mu } = \\chi _{1}^{-\\frac{1}{2}}{e}^a_{\\mu }$ .", "Additionally, the Einstein Frame fermion field satisfies the normal Dirac equation in the curved space $\\bar{g}_{\\mu \\nu }$ and must be defined as $\\Psi _{e.f.} = \\chi _{1}^{-\\frac{1}{4}} \\Psi ,$ and one can also check that $\\Psi _{e.f.}$ is scale invariant.", "We now will look at the interaction terms after transformation to Einstein Frame, in different phases of the theory.", "For this we look $\\Psi $ field as a test field which is produced in a $\\sigma $ and $\\varphi $ background.", "That means that we will not consider the effects of the $\\Psi $ field in the equation for $\\chi _{1}$ .", "In this way, we can mention that there are two interesting cases: 1.", "Let us consider the limit $\\varphi \\rightarrow -\\infty $ , which corresponds to the inflationary period and in this case, the constants $M_1$ and $M_2$ can be ignored.", "Therefore, we can obtain that the quantity $\\chi _{1}= \\frac{2\\chi _{2}f_2}{f_1}e^{\\alpha \\varphi }$ , under such conditions, we can look at the $g_2$ coupling: $g_2 \\sigma \\Phi _2 e^{\\frac{3}{2}\\alpha \\varphi }\\bar{\\Psi }\\Psi $ becomes in E.F. $g_2 \\frac{f_1}{2} \\sigma \\sqrt{\\bar{g}} \\bar{\\Psi }_{e.f.}\\Psi _{e.f.}$ , which is therefore $\\varphi $ independent.", "Similar effect takes place for the $g_1$ coupling in the inflationary limit $\\varphi \\rightarrow -\\infty $ in which the quantity $g_1 \\sigma \\Phi _1 e^{\\frac{\\alpha \\phi }{2}}\\bar{\\Psi }\\Psi $ transforms to E.F. as $g_1 (\\frac{f_1}{2 \\chi _2 f_2})^{\\frac{1}{2}} \\sigma \\sqrt{\\bar{g}} \\bar{\\Psi }_{e.f.}\\Psi _{e.f.}$ , which is therefore again $\\varphi $ independent.", "2.", "Now let us do the same calculation in the inflationary regime in which we study particle creation.", "In this case, the quantity $1/\\chi _1$ becomes $\\frac{1}{\\chi _1}=\\frac{1}{2\\chi _2}\\frac{V-M_1}{U+M_2},$ and we can neglect $M_1$ in the numerator and $U$ in the denominator, obtaining therefore $\\frac{1}{\\chi _1}=\\frac{1}{2\\chi _2}\\frac{V}{M_2},$ which implies that $\\chi _1$ results $\\chi _1 =\\frac{2M_2 \\chi _2}{f_1} e^{-\\alpha \\varphi }.$ Here we see that this dependence is inverse to that of the one in the inflationary phase (where one can ignore the constants of integration $M_1$ and $M_2$ ) and as a result we will get a strong $\\varphi $ dependence of the $g_1$ and $g_2$ couplings.", "Let us start with the $g_2$ coupling: $g_2 \\sigma \\Phi _2 e^{\\frac{3}{2}\\alpha \\varphi }\\bar{\\Psi }\\Psi $ becomes in E.F. $g_2\\,\\chi _2 \\left(\\frac{f_1}{2M_2\\chi _2}\\right)^{\\frac{3}{2}} e^{3 \\alpha \\varphi }\\sigma \\sqrt{\\bar{g}} \\bar{\\Psi }_{e.f.}\\Psi _{e.f.}$ We see a very strong growth of the coupling as $\\varphi $ increases in this regime.", "Now for the $g_1$ coupling: $g_1 \\sigma \\Phi _1 e^{\\frac{\\alpha \\varphi }{2}}\\bar{\\Psi }\\Psi $ becomes in E.F. $g_1 (\\frac{f_1}{2 \\chi _2 M_2})^{\\frac{1}{2}}e^{\\alpha \\varphi } \\sigma \\sqrt{\\bar{g}} \\bar{\\Psi }_{e.f.}\\Psi _{e.f.}$ , which again grows as $\\varphi $ grows.", "In this form, we can define that the decay rate for $\\sigma $ going into two fermions becomes $\\Gamma (\\sigma \\rightarrow \\Psi \\Psi ) = \\frac{g^2 m_{\\sigma }}{8\\pi },$ where the $g$ -coupling is given by $g= g_1 \\left(\\frac{f_1}{2 \\chi _2 M_2}\\right)^{\\frac{1}{2}}e^{\\alpha \\varphi } +g_2 \\chi _2 \\left(\\frac{f_1}{2M_2\\chi _2}\\right)^{\\frac{3}{2}} e^{3 \\alpha \\varphi }.$ Here we note that the decay parameter $\\Gamma $ given by eq.", "(REF ) increases with the growth of the scalar field $\\varphi $ , (see eq.", "(REF )) and then the $\\sigma $ -particles tend to decay at large values of $\\varphi $ ." ], [ "Decay rates and constraints", "In this section we can study two decay rates in order to obtain different constraints on the parameters of our model.", "In the following, we will analyze the decay rate for the specific cases in which the coupling parameters $g_1=0$ and the another $g_2\\ne 0$ and vice versa.", "In this context, we consider the special cases in which the coupling parameter $g_1=0$ and the other coupling parameter $g_2\\ne 0$ with which the $\\Gamma $ -coefficient is reduced to $\\Gamma (\\sigma \\rightarrow \\Psi \\Psi ) =\\Gamma _2(\\sigma \\rightarrow \\Psi \\Psi )= c_2 g_2^2e^{5\\alpha \\varphi },$ where the constant $c_2$ is defined as $ c_2 = \\frac{\\mu \\beta \\chi _2^2}{8\\pi }\\left(\\frac{f_1}{2M_2\\chi _2}\\right)^{3}$ .", "Here, we have tagged the decay rate in this situation as $\\Gamma _2$ .", "For the other instance in which the coupling parameter $g_2=0$ , we have that the decay rate results $\\Gamma (\\sigma \\rightarrow \\Psi \\Psi ) =\\Gamma _1 (\\sigma \\rightarrow \\Psi \\Psi )= c_1 g_1^2 e^{\\alpha \\varphi },$ in which $ c_1 =\\frac{\\mu \\beta }{8\\pi }\\left(\\frac{f_1}{2 \\chi _2 M_2}\\right)$ .", "Now, as we discussed earlier we have considered that both densities become equivalent and this occurs at the equilibrium time given by eq.", "(REF ).", "At least during this time, we can consider that the inflaton field $\\varphi $ spends most of the time previous to the equilibrium time $t_{eq}$ in which the inflaton takes the value $\\varphi _{eq}$ .", "In fact, the backreaction is unimportant for times shorter than $t_{eq}$ and then we can assume that the decay rate at that time limit $t_{eq}$ denotes by $\\Gamma (\\varphi =\\varphi _{eq})=\\Gamma _{eq}$ satisfies the condition in which the particles $\\sigma $ will decays to fermions $\\Psi $ .", "At the equilibrium time, we find that the decay rate for the special case in which $g_1=0$ from eqs.", "(REF ) and (REF ) can be written as $\\Gamma _2(\\sigma \\rightarrow \\Psi \\Psi )\\Big |_{\\varphi =\\varphi _{eq}}\\simeq c_2g_2^2\\,\\left(\\frac{3\\alpha _1\\,t_0^2}{2}\\right)^5.$ Analogously, we obtain that the decay rate $\\Gamma _1$ at the equilibrium time for the special case $g_2=0$ results $\\Gamma _1(\\sigma \\rightarrow \\Psi \\Psi )\\Big |_{\\varphi =\\varphi _{eq}}\\simeq c_1g_1^2\\,\\left(\\frac{3\\alpha _1\\,t_0^2}{2}\\right).$ Additionally, we will assume that during the kinetic stage the Hubble factor decreases so that its value is similar to the decay rate $\\Gamma $ .", "Thus, we can consider that the scalar field $\\sigma $ decayed under the condition $H(t_{dec})=\\frac{1}{3t_{dec}}\\simeq \\Gamma $ , where $t_{dec}$ corresponds to the time when the scalar field $\\sigma $ decayed and as the field $\\varphi $ spends most of the time previous to the equilibrium time $t_{eq}$ we note that this time satisfies the condition $t_{dec}<t_{eq}$ .", "In particular for the case in which $g_1=0$ and considering the decay rate $\\Gamma _2$ given by eq.", "(REF ), we find that the time when the scalar field $\\sigma $ decayed $t_{dec}$ results $t_{dec}\\simeq \\frac{1}{3\\,c_2g_2^2}\\left(\\frac{2}{3\\alpha _1\\,t_0^2}\\right)^5$ .", "Similarly, for the situation in which $g_2=0$ , we obtain that the time $t_{dec}\\simeq \\frac{1}{c_1g_1^2}\\left(\\frac{2}{9\\alpha _1\\,t_0^2}\\right)$ , when we considered the decay rate $\\Gamma _1$ .", "Thus, under the condition $t_{dec}<t_{eq}$ we find that for the case $g_1=0$ we have $g_2^2>\\frac{\\sqrt{3}}{6\\,c_2}\\,\\left(\\frac{2}{3\\alpha _1\\,t_0^2}\\right)^5 |\\dot{\\varphi _0}|,$ and for the case in which $g_2=0$ , we obtain that the lower limit for the coupling $g_1$ becomes $g_1^2>\\frac{\\sqrt{3}|\\dot{\\varphi _0}|}{9\\,c_1\\,\\alpha _1\\,t_0^2},$ here we have considered eq.", "(REF ) for the equilibrium time.", "Note that in both cases the lower bounds for the coupling parameters are proportional to the velocity at the end of inflation, since for large $\\alpha $ we have $\\dot{\\varphi _0}\\simeq \\dot{\\varphi }_{end} $ .", "On the other hand, in order to obtain the temperature at the equilibrium time $T(t=t_{eq}^*)=T_{eq}(t_{eq}^*)$ , we can consider that previous to the equilibrium time, the scalar field $\\sigma $ has totally decayed.", "This situation occurs when the densities satisfy the condition $\\rho (t_{eq}^*)\\sim \\rho _\\sigma (t_{eq}^*)$ .", "Here we have used the notation $t_{eq}^*$ for the time when the scalar field $\\sigma $ has completely decayed and thus differentiate it from the equilibrium time $t_{eq}$ since this time is different according on whether $\\sigma $ field decays or not.", "As the energy density of the background $\\rho (a)$ decays during the kinetic epoch as $\\rho \\propto a^{-6}$ and the energy density $\\rho _\\sigma (a)$ as radiation i.e., $\\rho _\\sigma \\propto a^{-4}$ , we have $\\rho (t_{eq}^*)=\\rho (t_{dec})\\left(\\frac{a(t_{dec})}{a(t_{eq}^*)}\\right)^6,\\,\\,\\,\\,\\mbox{and}\\,\\,\\,\\,\\rho _\\sigma (t_{eq}^*)=\\rho _\\sigma (t_{dec})\\left(\\frac{a(t_{dec})}{a(t_{eq}^*)}\\right)^4,$ with which from condition $\\rho (t_{eq}^*)\\sim \\rho _\\sigma (t_{eq}^*)$ , we find that the temperature at the equilibrium $T_{eq}\\sim \\rho _\\sigma ^{1/4}(t_{eq}^*)$ can be written as $T_{eq}\\sim \\rho _\\sigma ^{1/4}(t_{eq}^*)=\\rho _\\sigma ^{1/4}(t_{dec})\\sqrt{\\frac{\\rho _\\sigma (t_{dec})}{\\rho (t_{dec})}}.$ On the other hand, as we have that the scalar field $\\sigma $ decayed under the condition in which $H(t_{dec})\\simeq \\Gamma $ , then we assume that the energy density of the background $\\rho (t_{dec})=6H^2=6\\Gamma ^2$ and for the energy density $\\rho _\\sigma (t_{dec})$ we have $\\rho _\\sigma (t_{dec})\\simeq \\frac{\\sqrt{3}}{3}\\,\\,\\Gamma \\,|\\dot{\\varphi _0}|.$ In this way, by using Eq.", "(REF ) we find that the temperature at the equilibrium can be written as $T_{eq}\\sim 10^{-1}\\,|\\dot{\\varphi _0}|^{3/4}\\,\\Gamma ^{-1/4}\\simeq 10^{-1}\\,|\\dot{\\varphi }_{end}|^{3/4}\\,\\Gamma ^{-1/4} ,$ where we have used that the velocity $\\dot{\\varphi _0}\\simeq \\dot{\\varphi }_{end}$ for values of $\\alpha \\gg 1$ .", "In this form, from eq.", "(REF ) we can analyze the temperature at the equilibrium for the specific cases of the coupling parameters $g_1$ and $g_2$ .", "Thus, in particular for the case in which coupling parameter $g_1=0$ , we find that the temperature $T_{eq}$ becomes $T_{eq}\\sim 10^{-2}\\,\\frac{|\\dot{\\varphi }_{end}|^{13/4}}{c_2^{1/4}\\,g_2^{1/2}\\,\\alpha _1^{5/4}},$ and combining with eq.", "(REF ), we obtain a lower bound for the velocity at the end of inflation $\\dot{\\varphi }_{end}$ in terms of the temperature at the equilibrium $T_{eq}$ given by $|\\dot{\\varphi }_{end}|>10^3\\;T_{eq}^2.$ However, this limit gives us a lower bound on the rate $\\alpha ^2/\\chi _2^{1/2}$ given by $\\frac{\\alpha ^2}{\\chi _2^{1/2}}>10^3\\;\\frac{M_2^{1/2}\\,}{M_1}\\,T_{eq}^2,$ or $\\alpha ^2>10^3\\;\\frac{M_2^{1/2}\\chi _2^{1/2}\\,}{M_1}\\,T_{eq}^2=10^3\\;\\frac{1}{2\\,U_{(+)}^{1/2}}\\,T_{eq}^2\\simeq 10^{63}\\,T_{eq}^2,$ where $U_{(+)}$ is defined as $U_{(+)}=M_1^2/(4\\chi _2\\,M_2)$ and it corresponds to the present vacuum energy density and its value is approximately $U_{(+)}\\sim 10^{-120}$ (in units of $M_{Pl}^4$ ), see ref.[9].", "From eq.", "(REF ) we can obtain different constraints on the parameter $\\alpha $ depending on the temperature $T_{eq}$ considered, since the lower bound for $\\alpha $ is given by $\\alpha >10^{31}\\, T_{eq}$ .", "As example, by assuming that temperature at the equilibrium corresponds to the big bang nucleosynthesis (BBN) temperature $T_{eq}\\sim T_{BBN}$ in which $ T_{BBN}\\sim 10^{-22}$ (in units of $M_{Pl}$ ), we obtain that the lower bound for the parameter $\\alpha $ results $\\alpha >10^{9}$ .", "Now if we assume that the temperature $T_{eq}$ corresponds to the electroweak temperature $T_{ew}\\sim 10^{-17}$ , we obtain that the lower limit for the parameter $\\alpha $ results $\\alpha >10^{14}$ .", "Note that these constraints for the parameter $\\alpha $ are consistent with considering large values of $\\alpha $ i.e., $\\alpha \\gg 1$ .", "Now for the special case in which the coupling term $g_2=0$ , we get that the temperature at the equilibrium becomes $T_{eq}\\sim 10^{-1}\\,\\frac{|\\dot{\\varphi }_{end}|^{5/4}}{c_1^{1/4}\\,g_1^{1/2}\\,\\alpha _1^{1/4}},$ and combining with eq.", "(REF ), we obtain a lower limit for $\\dot{\\varphi }_{end}$ as a function of the temperature $T_{eq}$ given by $|\\dot{\\varphi }_{end}|>10^{3/2}\\;T_{eq}^2.$ As before, this expression gives a lower bound on the ratio $\\alpha ^2/\\chi _2^{1/2}$ results $\\frac{\\alpha ^2}{\\chi _2^{1/2}}>10^{3/2}\\;\\frac{M_2^{1/2}\\,}{M_1}\\,T_{eq}^2,$ and then we have $\\alpha >10^{3/4}\\;\\frac{1}{\\sqrt{2}\\;U_{(+)}^{1/4}}\\,T_{eq}.$ As in the previous case, by considering that the temperature $T_{eq}\\sim T_{BBN}$ , we obtain that the parameter $\\alpha >10^{8}$ and for the case in which the temperature $T_{eq}$ corresponds to the $T_{ew}$ , we find that the constraint for $\\alpha >10^{13}$ .", "Again, we note that these results are consistent with assuming values of $\\alpha \\gg 1$ .", "On the other hand, we will obtain other constraints on the parameters of our model, by considering at least another conditions during the decay of the $\\sigma $ particles.", "In fact, we can consider the condition in which the time when the field-$\\sigma $ decayed $t_{dec}$ is such that $t_{dec}>t_{0}$ , where the time $t_{0}\\simeq H(t_0)^{-1}\\sim H(t_{end})^{-1}=H_{end}^{-1}$ .", "As at the end of inflationary epoch the Hubble parameter $H_{end}=(V(\\varphi _{end})/6)^{1/2}=(V_{end}/6)^{1/2}$ , in which the effective potential at the end of inflation is $V_{end}=f_1\\,e^{-2\\alpha \\varphi _{end}}/(2\\sqrt{\\chi _2M_2})$ , with which we find that the time $t_0$ is given by $t_0\\simeq (6/V_{end})^{1/2}=2e^{\\alpha \\varphi _{end}}\\,(3\\sqrt{\\chi _2M_2}/f_1)^{1/2}$ .", "In this way, considering the condition in which $t_{dec}>t_0$ , we find an upper bound for the coupling parameter $g_2$ associated to the decay rate $\\Gamma _2$ for the case in which the coupling parameter $g_1=0$ given by $g_2<8\\times 10^{3}\\left[\\frac{M_1}{\\chi _2^{1/3}f_1\\,U_{(+)}^{27/12}\\,\\alpha ^{17/3}\\, \\mu }\\right]^{3}.$ Here, we have used that the time when the field-$\\sigma $ decayed for the case in which the parameter $g_1=0$ becomes $t_{dec}\\simeq \\frac{1}{3\\,c_2g_2^2}\\left(\\frac{2}{3\\alpha _1\\,t_0^2}\\right)^5$ .", "In order to evaluate an upper limit for the parameter $g_2$ , we consider the special case in which $\\alpha =10^{10}$ results $g_2<10^{132}/\\mu ^{3}$ .", "For the particular case in which $\\alpha =10^{15}$ the upper bound for $g_2$ corresponds to $g_2<10^{45}/\\mu ^{3}$ .", "Note that by increasing the value of the parameter $\\alpha $ decreases the upper bound for the coupling parameter $g_2$ .", "Here, as before we have considered the values $M_1=4\\times 10^{-60}$ (in units of $M_{Pl}^4$ ), $U_{(+)}=10^{-120}$ (in units of $M_{Pl}^4$ ), $\\chi _2=10^{-3}$ and $f_1=2\\times 10^{-8}$ , respectively[9].", "Now, for the special case in which $g_2=0$ and considering that the time $t_{dec}$ is given by $t_{dec}\\simeq \\frac{1}{c_1g_1^2}\\left(\\frac{2}{9\\alpha _1\\,t_0^2}\\right)$ , we find that the upper bound on the coupling parameter $g_1$ associated to $\\Gamma _1$ becomes $g_1<15\\left[\\frac{M_1^2}{2^{1/2}\\,f_1\\,U_{(+)}\\,\\alpha ^2\\,\\mu }\\right].$ In particular, for the specific value $\\alpha =10^{10}$ we obtain that the upper bound for $g_1$ and it to $g_1<2\\times 10^{-10}/\\mu $ and for the value $\\alpha =10^{15}$ we get the bound $g_1<10^{-20}/\\mu $ .", "Again we have used the values of ref.", "[9], for $M_1$ , $f_1$ and $U_{(+)}$ .", "In this context, we will obtain a range for the coupling parameters $g_1$ and $g_2$ associated to the decay rates $\\Gamma _1$ and $\\Gamma _2$ , by using the condition in which the $\\sigma $ field decays (at the time $t_{dec}$ ) before reaching equilibrium (at the time $t_{eq}$ ) wherewith $t_{dec}<t_{eq}$ and from the time condition when the field-$\\sigma $ decayed at the time $t_{dec}$ is greater than the time $t_0\\sim H_{end}^{-1}\\sim V_{end}^{-1/2}$ i.e., $t_{dec}>t_0$ .", "In this form, unifying both time conditions, we find that the range for the parameter $g_2$ associated to decay parameter $\\Gamma _2$ for the specific case in which the parameter $g_1=0$ is given by $8\\times 10^{3}\\left[\\frac{M_1}{\\chi _2^{1/3}f_1\\,U_{(+)}^{27/12}\\,\\alpha ^{17/3}\\, \\mu }\\right]^{3}>g_2>\\frac{3^{1/4}}{\\sqrt{6\\,c_2}}\\,\\left(\\frac{2}{3\\alpha _1\\,t_0^2}\\right)^{5/2} |\\dot{\\varphi }_{end}|^{1/2}.$ Here we have used eqs.", "(REF ) and (REF ) together with the fact that $\\dot{\\varphi }_{0}\\simeq \\dot{\\varphi }_{end}$ for large $\\alpha $ .", "In order to find a numerical range for the parameter $g_2$ , we consider the special case in which $\\alpha =10^{10}$ results $(10^7/\\mu ^3)(\\ln [10^{-26}/\\mu ])^{15/2}<g_2<10^{132}/\\mu ^{3}$ .", "For the particular case in which $\\alpha =10^{15}$ the range for the coupling parameter $g_2$ corresponds to $(10^{-75}/\\mu ^3)(\\ln [10^{-11}/\\mu ])^{15/2}<g_2<10^{45}/\\mu ^{3}$ .", "We note that the range for the coupling parameter $g_2$ is very large.", "Here, as before we have considered that the time $t_0=(6/V_{end})^{1/2}$ together with the values $M_1=4\\times 10^{-60}$ (in units of $M_{Pl}^4$ ), $U_{(+)}=10^{-120}$ (in units of $M_{Pl}^4$ ), $\\chi _2=10^{-3}$ and $f_1=2\\times 10^{-8}$ , respectively[9].", "Analogously, for the specific case in which the parameter $g_2=0$ , we find that the range for the coupling parameter $g_1$ can be written as $15\\left[\\frac{M_1^2}{2^{1/2}\\,f_1\\,U_{(+)}\\,\\alpha ^2\\,\\mu }\\right]>g_1>\\frac{3^{1/4}\\,|\\dot{\\varphi }_{end}|^{1/2}}{3\\,t_0\\,\\sqrt{c_1\\,\\alpha _1}}.$ Here we have considered that $\\dot{\\varphi }_{0}\\simeq \\dot{\\varphi }_{end}$ together with limits given by eqs.", "(REF ) and (REF ), respectively.", "As before, in order to obtain a range for the parameter $g_1$ , we assume the special case in which the parameter $\\alpha =10^{10}$ results $(10^{-12}/\\mu )(\\ln [10^{-26}/\\mu ])^{3/2}<g_1<2\\times 10^{-10}/\\mu $ , where the quantity $(\\ln [10^{-26}/\\mu ])^{3/2}<10^2$ or $\\mu >4\\times 10^{-36}$ in order to satisfy the range for the parameter $g_1$ .", "For the particular case in which $\\alpha =10^{15}$ the range for the coupling parameter $g_1$ corresponds to $(10^{-24}/\\mu )(\\ln [10^{-11}/\\mu ])^{3/2}<g_1<10^{-20}/\\mu $ , with $\\mu >10^{-213}\\simeq 0$ .", "We note that the range for the parameter $g_1$ is very narrow in relation to $g_2$ .", "Here, as before we have used that the time $t_0=(6/V_{end})^{1/2}$ together with the values $M_1=4\\times 10^{-60}$ (in units of $M_{Pl}^4$ ), $U_{(+)}=10^{-120}$ (in units of $M_{Pl}^4$ ), $\\chi _2=10^{-3}$ and $f_1=2\\times 10^{-8}$ , respectively[9]." ], [ "Conclusion", "In this paper we have analyzed in detail the instant preheating mechanism in a scale invariant two measures theory.", "In this frame we have studied the instant preheating for a NO model where the potential associated to inflaton field does not have a minimum.", "Moreover, we have assumed that this preheating mechanism is applied to an effective potential that presents an interaction between the inflaton field $\\varphi $ and other scalar field $\\sigma $ given by eq.", "(REF ).", "In our analysis, we have noted that the instant preheating and in particular the particles production $\\sigma $ strongly depends on the interaction between the the fields $\\varphi $ and $\\sigma $ .", "From the energy density of the produced particles of the field-$\\sigma $ , we have obtained two limit decays that depend on the effective mass $m_\\sigma $ in relation to the physical momentum.", "In the first situation, we have found that the energy density of produced particles-$\\sigma $ decays as radiation, in a process that we called instant radiation and it occurs when the effective mass satisfied the condition $m_\\sigma \\ll k/a$ .", "In this stage we have observed that the backreaction of produced particles-$\\sigma $ on the equation of motion associated to the inflaton field $\\varphi $ disappears naturally product of the evolution of the inflaton and exponential decay of the backreaction term.", "For the situation in which the effective mass of the field $\\sigma $ satisfies the reverse situation in which $m_\\sigma \\gg k/a$ , we have analyzed the possibility that the energy density of produced particles $\\sigma $ is of the same order as the energy density of the background defining an equilibrium time.", "Further, we have studied the decay rate in the framework of the scale invariant coupling of the scalar fields $\\varphi $ and $\\sigma $ and as this last field decays to fermions.", "Here, after performing transition to the physical Einstein frame we have considered a Yukawa interaction and then we have found an expression for the decay rate from our scalar field going into two fermions, see eq.", "(REF ).", "From these results we have analyzed two decay rates separately assuming the values of the coupling parameters associated to the decay parameters.", "In this analysis, we have found different constraints on the coupling parameters of the decay $\\Gamma $ , considering the imposed conditions from the time when the scalar field decayed, the equilibrium time and the initial time of the kinetic epoch.", "Additionally, we have determined the temperature at the equilibrium for the different cases of the coupling parameters of $\\Gamma $ and as example we have compared our results with the nucleosynthesis and electroweak temperatures, respectively.", "Finally in this article, we have not addressed the process of the particles production considering other reheating mechanisms such as gravitational particle production from massless or heavy particles [26], [41].", "In this sense, we hope to return to this point in the near future.", "R.H. was supported by Proyecto VRIEA-PUCV N$_{0}$ 039.449/2020." ] ]

[ [ "The GlueX Beamline and Detector" ], [ "Abstract The GlueX experiment at Jefferson Lab has been designed to study photoproduction reactions with a 9-GeV linearly polarized photon beam.", "The energy and arrival time of beam photons are tagged using a scintillator hodoscope and a scintillating fiber array.", "The photon flux is determined using a pair spectrometer, while the linear polarization of the photon beam is determined using a polarimeter based on triplet photoproduction.", "Charged-particle tracks from interactions in the central target are analyzed in a solenoidal field using a central straw-tube drift chamber and six packages of planar chambers with cathode strips and drift wires.", "Electromagnetic showers are reconstructed in a cylindrical scintillating fiber calorimeter inside the magnet and a lead-glass array downstream.", "Charged particle identification is achieved by measuring energy loss in the wire chambers and using the flight time of particles between the target and detectors outside the magnet.", "The signals from all detectors are recorded with flash ADCs and/or pipeline TDCs into memories allowing trigger decisions with a latency of 3.3 $\\mu$s.", "The detector operates routinely at trigger rates of 40 kHz and data rates of 600 megabytes per second.", "We describe the photon beam, the GlueX detector components, electronics, data-acquisition and monitoring systems, and the performance of the experiment during the first three years of operation." ], [ " The ", "The GlueX experiment The search for Quantum ChromoDynamics (QCD) exotics uses data from a wide range of experiments and production mechanisms.", "Historically, the searches have looked for the gluonic excitations of mesons, searching for states of pure glue, glueballs, and hybrid mesons where the gluonic field binding the quark-anti-quark pair has been excited.", "Most experiments searching for glueballs looked for scalar mesons [1], where the searches relied on over-population of nonets, as well as unusual meson decay patterns.", "In the search for hybrid mesons [2], [3], efforts have focused on particles with exotic quantum numbers, i.e.", "systems beyond simple quark-anti-quark configurations.", "Good evidence exists for an isospin 1 state, the $\\pi _{1}(1600)$ .", "Looking collectively at past studies, data from high-statistics photoproduction experiments in the energy range above 6 GeV are lacking.", "Figure: (Color online)A cut-away drawing of the GlueX detector in Hall D, not to scale.The Gluonic Excitation (GlueX) experiment at the US Department of Energy's Thomas Jefferson National Accelerator Facility (JLab)Thomas Jefferson National Accelerator Facility, 12000 Jefferson Ave., Newport News, VA 23606, https://www.jlab.org.", "has been built to search for and map out the spectrum of exotic hybrid mesons using a 9-GeV linearly-polarized photon beam incident on a proton target[4].", "The GlueX detector and beamline are shown schematically in Figure REF .", "The detector is nearly hermetic for both charged particles and photons arising from reactions in the cryogenic target at the center of the detector, allowing for reconstruction of exclusive final states.", "A 2-T solenoidal magnet surrounds the drift chambers used for charged-particle tracking.", "Two electromagnetic calorimeters cover the central and forward regions, and a scintillation detector downstream provides particle-identification capability through time-of-flight measurements.", "The Hall-D complex The GlueX experiment is housed in the Hall-D complex at JLab (see Fig.REF ).", "This new facility starts with an extracted electron beam at the north end of the Continuous Electron Beam Accelerator Facility (CEBAF) [5], [6].", "The electron beam is delivered to the Tagger Hall, where the maximum energy is 12 GeV, due to one more pass through the north linac than the other experimental halls (A, B and C).", "Here, linearly-polarized photons are produced through coherent bremsstrahlung off a 50 $\\mu $ m thick diamond crystal radiator.", "The scattered electrons pass through a tagger magnet and are bent into tagging detectors.", "A high-resolution scintillating-fiber tagging array covers the 8 to 9 GeV energy range, and a tagger hodoscope covers photon energies both from 9 GeV to the endpoint, and from 8 GeV to 3 GeV.", "Electrons not interacting in the diamond are directed into a 60 kW electron beam dump.", "The tagged photons travel to the Hall-D experimental hall.", "The distance from the radiator to the primary collimator is 75 m. The collimator of 5 mm diameter removes off-axis incoherent photons.", "The front face of the collimator is instrumented with an active collimator to aid in beam tuning.", "The beamline and tagging system are described below in Section .", "Downstream of the primary collimator is a thin beryllium radiator used by both the Triplet Polarimeter, which measures the linear polarization of the photons, and a Pair Spectrometer, which is used to measure the flux of the photons.", "More information on the production, tagging and monitoring of the photon beam can be found in Section .", "The photon beam continues through to a liquid hydrogen target at the heart of the GlueX detector, and then to the end of the experimental hall where it enters the photon beam dump.", "Figure: (Color online) Schematic of the CEBAF accelerator showing the additions made during the 12-GeV project.", "The Hall-D complex is located at the north-east end.The layout of the GlueX detector is shown in Fig.", "REF .", "The spectrometer is based on a 4-m-long solenoidal magnet that is operated at a maximum field of 2 T, see Section .", "The liquid-hydrogen target is located inside the upstream bore of the magnet.", "The target consists of a 2-cm-diameter, 30-cm-long volume of hydrogen, as described in Section .", "Surrounding the target is the Start Counter, which consists of 30 thin scintillator paddles that bend to a nose on the down-stream end of the hydrogen target.", "The Start Counter is the primary detector that registers the time coincidence of the radio-frequency (RF) bunch containing the incident electron and the tagged photon producing the interaction.", "More information on this scintillator detector can be found in Section .", "Figure: GlueX spectrometer layout.", "Dimensions are given in mm.", "Thenumbers show the Z-coordinates of the detectors' centers, or ofthe front face of the FCAL modules.Glossary:SC - Start Counter (Section ),CDC - Central Drift Chamber (Section ),FDC - Forward Drift Chamber (Section ),BCAL - Barrel Calorimeter (Section ),TOF - Time-of-Flight hodoscope (Section ),FCAL - Forward Calorimeter (Section ).The Central Drift Chamber, a cylindrical straw-tube detector, starts at a radius of 10 cm from the beam line.", "The active volume of the chamber extends from 48 cm upstream to 102 cm downstream of the target center, and from 10 cm to 56 cm in radius.", "The Central Drift Chamber consists of 28 layers of straw tubes in axial and two stereo orientations.", "Downstream of the central tracker is the Forward Drift Chamber, which consists of four packages, each containing 6 planar layers in alternating $u$ -$y$ -$v$ orientations.", "Both cathodes and anodes in the Forward Drift Chamber are read out, providing three-dimensional space point measurements.", "More details on the tracking system are provided in Sections  and .", "Downstream of the magnet is the Time-of-Flight wall.", "This system consists of two layers of scintillator paddles in a crossed pattern, and, in conjunction with the Start Counter, is used to measure the flight time of charged particles.", "More information on the time-of-flight system is provided in Section .", "Photons arising from interactions within the GlueX target are detected by two calorimeter systems.", "The Barrel Calorimeter, located inside the solenoid, consists of layers of scintillating fibers alternating with lead sheets.", "The Forward Calorimeter is downstream of the Time-of-Flight wall, and consists of 2800 lead-glass blocks.", "More information on the the calorimeters can be found in Section ." ], [ "Experimental requirements ", "Experimental requirements The physics goals of the GlueX experiment require the reconstruction of exclusive final states.", "Thus, the GlueX detector must be able to reconstruct both charged particles ($\\pi ^{\\pm }$ , $K^{\\pm }$ and $p/\\bar{p}$ ) and particles decaying into photons ($\\pi ^{\\circ }$ , $\\eta $ , $\\omega $ and $\\eta ^{\\prime }$ ).", "For this capability, the charged particles and photons must be reconstructed with good momentum and energy resolution.", "The experiment must also be able to reconstruct the energy of the incident photon (8 to 9 GeV) with high accuracy ($0.1$ %) and have knowledge of the linear polarization (maximum $\\sim $ 40%) of the photon beam to an absolute precision of 1%.", "Finally, many interesting final states involve more than five particles.", "Thus, the GlueX detector must also be nearly hermetic for both charged particles and photons, with an acceptance that is reasonably uniform, well understood, and accurately modeled in simulation.", "In practice, the typical momentum resolution for charged particles is 1–$3\\%$ , while the resolution is 8-9% for very-forward high-momentum particles.", "For most charged particles, the tracking system has nearly hermetic acceptance for polar angles from $1^\\circ -2^{\\circ }$ to $150^{\\circ }$ .", "However, protons with momenta below about 250 MeV/c are absorbed in the hydrogen target and not detected.", "A further challenge is the reconstruction of tracks from charged pions with momenta under 200 MeV/c due to spiraling trajectories in the magnetic field.", "The measurement of energy loss ($dE/dx$ ) in the Central Drift Chamber enables the separation of pions and protons up to about 800 MeV/c, while time-of-flight determination allows separation of forward-going pions and kaons up to about 2 GeV/c.", "For photons produced from the decays of reaction products, the typical energy resolution is 5 to 6%$/\\sqrt{E_{\\gamma }}$ .", "Photons above 60 MeV can be detected in the Barrel Calorimeter, with some variation depending on the incident angle.", "The interaction point along the beam direction is determined by comparing the information from the readouts on the upstream and downstream ends of the detector.", "In the Forward Calorimeter, photons with energies larger than 100 MeV can be detected with uniform resolution across the face of the detector.", "There is a gap between the calorimeters at around $11^{\\circ }$ , where energy can be lost due to shower leakage.", "Both photon detection efficiency and energy resolution are degraded in this region." ], [ "Data requirements ", "The physics analyses need to be carried out in small bins of energy and momentum transfer, necessitating not only the ability to reconstruct exclusive final states but also to collect sufficient statistics.", "While exact cross sections are not known, the cross sections of interest will be in the 10 nb to 1 $\\mu $ b range.", "This paper describes the operation of GlueX Phase I.", "During this initial phase, the GlueX experiment has run with a data acquisition system capable of collecting data using photon beams of a few $10^{7}~\\gamma /$ s in the coherent peak (8.4-9 GeV), with an expectation to run with 2.5 times higher rates in the future.", "The data acquisition system ran routinely at 40 kHz with raw event sizes of 15-20 kilobytes, collecting about 600 megabytes of data per second.", "With firmware improvements, future running is expected at 90 kHz and 1 gigabyte per second.", "Details of the trigger and data acquisition are presented in Sections  and ." ], [ "Coordinate system ", "For reference, we introduce here the overall experiment coordinate system, which is used in this document and throughout the analysis.", "The z-axis is defined along the nominal beamline increasing downstream.", "The coordinate system is right-handed with the y-axis pointing vertically up and the x-axis pointing approximately north.", "The origin is located 50.8 cm (20 inches) downstream of the upstream side of the upstream endplate of the solenoid, placing the nominal center of the target at (0,0,65 cm)." ], [ "The coherent photon source and beamline ", "The coherent photon source and beamline Table: Electron beam parameters.The emittance, energy spread andrelated parameters are estimatesbased on a model of the transport line fromthe accelerator to the Hall D radiator.The dimensions of the beam spot at the position ofthe radiator are directly measured, and vary around thestated values by ±30%\\pm 30\\%depending on beam conditions.Values for image size at collimator,obtained by projection of the electron beamspot convergence forward to the position ofthe primary photon collimator, have relativeuncertainties of 50%." ], [ "CEBAF electron beam ", "CEBAF has a race track configuration with two parallel linear accelerators based on superconducting radio frequency (RF) technology [5].", "The machine operates at 1.497 GHz and delivers beam to Hall D at 249.5 MHz.Hall D beam at 499 MHz is possible, but not the norm.", "Precise timing signals for the accelerator beam bunches are available to the experiment and are used to determine the time that individual photon bunches pass through the target.", "The nominal properties for the CEBAF electron beam to the Tagger Hall are listed in Table REF .", "Figure: Schematic layout of the Hall-D complex, showing the Tagger Hall, Hall D, andseveral of the key beamline devices.Also indicated are the locations of the 5C11B and AD00C beam monitors." ], [ "Hall-D photon beam ", "The Hall-D complex, described in Section REF and shown schematically in Fig.", "REF , includes a dedicated Tagger Hall, an associated collimator cave, and Experimental Hall D itself.", "A linearly-polarized photon beam is created using the process of coherent bremsstrahlung [7], [8] when the electron beam passes through an oriented diamond radiator at the upstream end of the Tagger Hall.", "The electron beam position at the radiator is monitored and controlled using beam position monitors (5C11 and 5C11B) which are located at the end of the accelerator tunnel just upstream of the Tagger Hall (see Fig.", "REF .)", "The CEBAF electron beam is tuned to converge as it passes through the radiator, ideally so that the electron beam forms a virtual focus at the collimator located 75 m downstream of the radiator.", "At the collimator, the virtual spot size of 0.5 mm is small compared to the cm-scale size of the photon beam on the front face of the collimator, such that a cut on photon position at the collimator is effectively a cut on photon emission angle at the radiator.", "The convergence properties of the electron beam are measured by scanning the beam profile with vertical and horizontal wires.", "The wire scanners are referred to as \"harps.\"", "Examples of the horizontal and vertical convergence of the electron beam envelope (undeflected by the tagger magnet) measured using harp scans and projected downstream along the beamline are shown in Fig.", "REF .", "Figure: (Color online) Measurements of the root-mean-square width of the electron beamin horizontal (left)and vertical (right) projections as a function of position along the beamline, based onharp scans (data points) of the electron beam.", "The radiator position is just upstreamof the third data point.", "The primary collimator position is marked by the vertical lineindicated by the arrow.", "The curve downstream of the radiator is an extrapolation fromthe measured data points, with extrapolation uncertainty indicated by the shaded regions.The photon beam position on the collimator is monitored using an active collimator positioned just upstream of the primary photon beam collimator (described below in section REF ).", "The position stability of the photon beam is maintained during normal operation by a feedback system that locks the position of the electron beam at the 5C11B beam position monitor and, consequently, the photon beam at the active collimator.", "The stability of the electron beam current and position is monitored using an independent beam monitor (AD00C in Fig.", "REF ) located immediately upstream of the electron dump.", "The linearly-polarized photon beam is produced via a radiator placed in the electron beam just upstream of the Tagger (section REF ).", "A properly aligned 20–60 $\\mu $ m thick diamond crystal radiator produces linearly polarized photons via coherent bremsstrahlung in enhancements [7], [8], that appear as peaks at certain energies in the collimated bremsstrahlung intensity spectrum (Fig.", "REF ), superimposed upon the ordinary continuum bremsstrahlung spectrum from an aluminum radiator.", "The energies of the coherent photon peaks and the degree of polarization in each of those peaks depend on the crystal orientation with respect to the incident electron beam.", "Adjustment of the orientation of the diamond crystal with respect to the incoming electron beam permits production of essentially any coherent photon peak energy up to that of the energy of the incident electron beam, as well as the degree or direction of linear polarization.", "A choice of 9 GeV for the primary peak energy, corresponding to 40% peak linear polarization, was found to be optimum for the GlueX experiment with a 12-GeV incident electron beam.", "Figure: (color online) (a) Collimated photon beam intensity versus energy as measured by the Pair Spectrometer.", "(b) Collimated photon beam polarization as a function of beam energy,as measured by the Triplet Polarimeter, with data points offset horizontally by ±0.015\\pm 0.015 GeV for clarity.The labels PARA and PERP refer to orientations of the diamond radiator that result in polarizationplanes that are parallel and perpendicular to the horizontal, respectively.The degree of polarization for a coherent bremsstrahlung beam is greatest for photons emitted at small angles with respect to the incident electron direction.", "Collimation of the photon beam to a fraction of the characteristic bremsstrahlung angle exploits this correlation to significantly enhance the average polarization of the beam.", "In the nominal GlueX beamline configuration, a 5.0-mm-diameter collimator A 3.4 mm collimator is also available, and has been used for some physics production runs with the thinnest (20 $\\mu m$ ) diamond.", "positioned 75 m downstream of the radiator is used, corresponding to a cut at approximately 1/2 $m/E$ in characteristic angle, where $m$ is the electron rest mass and $E$ is the energy of the incident electron.", "The photon beam energy spectrum and photon flux after collimation are measured by the Pair Spectrometer (section REF ), located downstream of the collimator in Hall D. An example of the measured photon spectrum and degree of polarization with a 12-GeV electron beam is shown in Fig.", "REF .", "The spectrum labeled “Aluminum\" in Fig.", "REF (a) shows the spectrum of ordinary (incoherent) bremsstrahlung, normalized to the approximate thickness of the diamond radiator in terms of radiation lengths.", "The expected degree of linear polarization in the energy range of 8.4–9.0 GeV is $\\sim $ 40% after collimation.", "The photon beam polarization is directly measured by the triplet polarimeter (section REF ) located just upstream of the pair spectrometer.", "The stability of the beam polarization is independently monitored via the observed azimuthal asymmetry in various photoproduction reactions, particularly that for $\\rho $ photoproduction [9].", "Typical values for parameters and properties of the photon beam are given in Table REF .", "In the sections that follow, we describe in more detail how the linearly-polarized photon beam is produced, how the photon energy is determined using the tagging spectrometer, how the photon beam polarization spectrum and flux are measured with the Pair Spectrometer and Triplet Polarimeter, and how the photon flux is calibrated using the Total Absorption Counter.", "Table: Typical parameters for the GlueX photon beam,consistent with the electron beam properties listed in Table ,a diamond radiator of thickness 50 μ\\mu m, and the standard primary collimator of diameter 5.0 mmlocated at the nominal position.", "The electron beam current incident on the radiator is taken to be 150 nA.The hadronic rates are calculated for the GlueX 30 cm liquid hydrogen target." ], [ "Goniometer and radiators ", "For the linearly-polarized photon beam normally used in GlueX production running, diamond radiators are used to produce a coherent bremsstrahlung beam.", "This requires precise alignment of the diamond radiator, in order to produce a single dominant coherent peakDefined as 0.6 GeV below the coherent edge (nominally 9 GeV).", "The position of the edge scales approximately with the primary incident electron beam energy.", "with the desired energy and polarization by scattering the beam electrons from the crystal planes associated with a particular reciprocal lattice vector.", "A multi-axis goniometer, manufactured by Newport Corporation, precisely adjusts the relative orientation of the diamond radiator with respect to the incident electron beam horizontally, vertically and rotationally about the $X$ , $Y$ and $Z$ axes, respectively.", "The Hall-D goniometer holds several radiators, any of which may be moved into the beam for use at any time according to the requirements of the experiment.", "In addition to the diamond radiators, several aluminum radiators of thicknesses ranging from 1.5 to 40 $\\mu $ m are used to normalize the rate spectra measured in the Pair Spectrometer, correcting for its acceptance.", "A separate rail for these amorphous radiators is positioned 615 mm downstream of the goniometer." ], [ "Diamond selection and quality control ", "The properties of diamond are uniquely suited for coherent bremsstrahlung radiators.", "The small lattice constant and high Debye temperature of diamond result in an exceptionally high probability for coherent scattering in the bremsstrahlung process [10].", "Also, the high coherent scattering probability is a consequence of the small atomic number of carbon (Z = 6).", "At the dominant crystal momentum (9.8 keV) corresponding to the leading (2,2,0) reciprocal lattice vector, the small atomic number results in minimal screening of the nuclear charge by inner shell electrons.", "Diamond is the best known material in terms of its coherent radiation fraction, and its unparalleled thermal conductivity and radiation hardness make it well-suited for use in a high-intensity electron beam environment.", "The position of the coherent edge in the photon beam intensity spectrum is a simple monotonic function of the angle between the incident electron beam direction and the normal to the (2,2,0) crystal plane.", "The 12-GeV-electron beam entering the radiator has a divergence less than 10 $\\mu $ rad, corresponding to a broadening of the coherent edge in Fig.", "REF by just 7 MeV.", "However, if the incident electron beam had to travel through 100 $\\mu $ m of diamond material prior to radiating, the resulting electron beam emittance would increase by a factor of 10 due to multiple Coulomb scattering, resulting in a proportional increase in the width of the coherent edge.", "Such broadening of the coherent peak diminishes both the degree of polarization in the coherent peak as well as the collimation efficiency in the forward direction.", "Hence, diamond radiators for GlueX must be significantly thinner than 100 microns.", "The cross-sectional area of a diamond target must also be large enough to completely contain the electron beam so that the beam does not overlap with the material of the target holder.", "Translated to the beam spot dimensions from Table REF , GlueX requires a target with transverse size 5 mm or greater.", "Uniform single-crystal diamonds of this size are now available as slices cut from natural gems, HPHT (high-pressure, high-temperature) synthetics, and CVD (chemical vapor deposition) single crystals.", "Natural gems are ruled out due to cost.", "HPHT crystals had been thought to be far superior to CVD single crystals in terms of their diffraction widths, but our experience did not bear this out.", "GlueX measurements of the x-ray rocking curves of CVD crystals obtained from the commercial vendor Element SixElement Six, https://www.e6.com/en.", "routinely showed widths that were within a factor 2 of the theoretical Darwin width, similar to the results we found for the best HPHT diamonds that were available to us [11], [12].", "Figure: (color online) Rocking curve RMS width topograph taken of the (2,2,0) reflectionfrom a CVD diamond crystal using 15 keV X-rays at the C-line at CHESS.The bright diagonal lines in the cornersindicate regions of increased local strain, coinciding with growth boundaries radiatingoutward from the seed crystal used in the CVD growth process.Fig.", "REF shows a rocking curve topograph of a diamond radiator taken with 15 keV x-rays at the Cornell High Energy Synchrotron Source (CHESS).", "The instrumental resolution of this measurement is of the same order as the Darwin width for this diffraction peak, approximately 5 $\\mu $ rad.", "During operation, the electron beam spot would be confined to the relatively uniform central region.", "Any region in this figure with a rocking curve root-mean-square width of 20 $\\mu $ rad or less is indistinguishable from a perfect crystal for the purposes of GlueX.", "Regardless of whether or not better HPHT diamonds exist, these Element Six CVD diamonds have sufficiently narrow diffraction widths for our application.", "This, coupled with their lower cost relative to HPHT material, made them the obvious choice for the Hall-D photon source.", "The diamond radiator fabrication procedure began with procurement of the raw material in the form of $7\\times 7\\times 1.2$  mm$^3$ CVD single-crystal plates from the vendor.", "After x-ray rocking curve scans of the raw material were taken to verify crystal quality, the acceptable diamonds were shipped to a second vendor, Delaware Diamond Knives (DDK).", "At DDK, the 1.2-mm-thick samples were sliced into three samples of 250 $\\mu $ m thickness each, then each one was polished on both sides down to a final thickness close to 50 $\\mu $ m. The samples, now of dimensions $7\\times 7\\times 0.05$  mm$^3$ were fixed to a small aluminum mounting tab using a tiny dot of conductive epoxy placed in one corner.", "These crystals were then returned to the synchrotron light source for final x-ray rocking curve measurements prior to final approval for use in the GlueX photon source.", "The useful lifetime of a diamond radiator in the GlueX beamline is limited by the degradation in the sharpness of the coherent edge due to accumulation of radiation damage.", "Experience during the early phase of GlueX running showed that after exposure to about 0.5 C of integrated electron beam charge, the width of the coherent edge increased enough that the entire coherent peak was no longer contained within the energy window of the tagger microscope.", "When a crystal reached this degree of degradation, the radiator was regarded as no longer usable, and a new crystal was installed.", "During Phase 1 of GlueX, radiator crystals were replaced three times due to degradation, twice with 50 $\\mu $ m radiators and once with a 20 $\\mu $ m radiator.", "The 20-$\\mu $ m diamond was introduced to test whether the reduced multiple Coulomb scattering might result in an observable increase in peak polarization.", "This turned out not to be the case, for two reasons.", "The first is that to take full advantage of the reduced multiple scattering in the radiator for increased peak polarization, the collimator size must be reduced proportionally.", "A 3.4-mm-diameter collimator was available for this purpose, but variability observed in the convergence properties of the electron beam at the radiator overruled running with any collimator smaller than 5 mm, even when a thinner radiator was in use.", "The second reason is that any improvements from reduced multiple scattering that came with the smaller radiator thickness were more than offset by strong indications of radiation damage that appeared not long after the 20 $\\mu $ m crystal was put into production.", "The rapid appearance of radiation damage was partly due to the larger beam current (factor 2.5) that was needed to produce the same photon flux as with a 50 $\\mu $ m crystal, but that factor alone did not fully explain what was seen.", "Subsequent x-ray measurements showed that a large buckling of the 20 $\\mu $ m crystal had occurred in the region of the incident electron beam spot, evidently due to local differential expansion of the diamond lattice arising from radiation damage.", "Once the crystal buckled, the energy of the coherent peak varied significantly across the electron beam spot, effectively broadening the peak.", "Fortunately, the greater stiffness of a 50 $\\mu $ m crystal appears to suppress this local buckling under similar conditions of radiation damage.", "Based on these observations, 50 $\\mu $ m was selected as the optimum thickness for GlueX diamond radiators: thin enough to limit the effects of multiple scattering and thick enough to suppress buckling from internal stress induced by radiation damage.", "The effective useful lifetime of a 50 $\\mu $ m radiator in the photon source is about 0.5 C integrated incident electron charge.", "This lifetime might be extended somewhat by the use of thermal annealing to partially remove the effects of radiation damage.", "This possibility will be explored when the pace of diamond replacement increases with the start of full-intensity running (GlueX Phase 2) and the number of spent radiators starts to accumulate.", "Figure: Schematic diagram of the tagging spectrometer, showing the paths of the electronand photon beams.", "Dotted lines indicate post-radiation electron trajectories identified bythe energy the electron gave up to an associated radiated photon, as a fraction of the beam energy E 0 _0.The Tagger focal plane detector arrays TAGH and TAGM are described in the text." ], [ "Photon tagging system ", "After passing through the radiator, the combined photon and electron beams enter the photon tagging spectrometer (Tagger).", "The full-energy electrons are swept out of the beamline by a dipole magnet and redirected into a shielded beam dump.", "The subset of beam electrons that radiated a significant fraction of their energy in the radiator are deflected to larger angles by the dipole field.", "These post-bremsstrahlung electrons exit through a thin window along the side of the magnet, and are detected in a highly segmented array of scintillators called the Tagger Hodoscope, as shown in Fig.", "REF .", "The TAGH counters span the full range in energy from 25% to 97% of the full electron beam energy.", "A high-energy-resolution device known as the Tagger Microscope (TAGM) covers the energy range corresponding to the primary coherent peak, indicated by the denser portion of the focal plane in Fig.", "REF .", "The quadrupole magnet upstream of the Tagger dipole provides a weak vertical focus, optimizing the efficiency of the Tagger Microscope for tagging collimated photons.", "A 0.8 Tm permanent dipole magnet is installed downstream of the Tagger magnet on the photon beam line, in order to prevent the electron beam from reaching Hall D should the Tagger magnet trip.", "Both the TAGM and TAGH devices are used to determine the energy of individual photons in the photon beam via coincidence, using the relation $E_{\\gamma } = E_{0} - E_{e}$ , where $E_{0}$ is the primary electron beam energy before interaction with the radiator, and $E_{e}$ is the energy of the post-bremsstrahlung electron determined by its detected position at the focal plane.", "Multiple radiative interactions in a 50 $\\mu $ m diamond radiator ($3\\times 10^{-4}$ radiation lengths) produce uncertainties in $E_{\\gamma }$ of the same order as the intrinsic energy spread of the incident electron beam." ], [ "Tagger magnet ", "The Hall-D Tagger magnet deflects electrons in the horizontal plane, allowing the bremsstrahlung-produced photons to continue to the experimental hall while bending the electrons that produced them into the focal plane detectors.", "Electrons that lose little or no energy in the radiator are deflected by 13.4$^\\circ $ into the electron beam dump.", "The Hall-D Tagger magnet is an Elbek-type room temperature dipole magnet, similar to the JLab Hall-B tagger magnet [13], [14].", "The magnet is 1.13 m wide, 1.41 m high and 6.3 m long, weighing 80 metric tons, with a normal operating field of 1.5 T for a 12-GeV incident electron beam, a maximum field of 1.75 T, and a pole gap of 30 mm.", "The magnet design was optimized using the detailed magnetic field calculation provided by the TOSCA simulation package and ray tracing of electron beam trajectories [15], [16].", "The GlueX experiment requirements mandate that the scattered electron beam be measured with an accuracy of 12 MeV (0.1% of the incident electron energy).", "This requires that the magnetic field integrals along all useful electron trajectories be known to 0.1%.", "The magnetic field was mapped at Jefferson Lab and the detailed field maps were augmented by detailed TOSCA calculations, which have allowed us to meet these goals.", "Details of the magnet mapping and uniformity are found in Ref.", "[17]." ], [ "Tagger Microscope ", "The Tagger Microscope (TAGM) is a high-resolution hodoscope that counts post- bremsstrahlung electrons corresponding to the primary coherent peak.", "Normally the TAGM is positioned to cover between 8.2 and 9.2 GeV in photon energy, but the TAGM is designed to be movable should a different peak energy be desired.", "The microscope is segmented along the horizontal axis into 102 energy bins (columns) of approximately equal width.", "Each column is segmented in five sections (rows) along the vertical axis.", "The vertical segmentation allows the possibility of scattered electron collimation, which gives a significant increase in photon polarization when used in combination with photon collimation.", "The purpose of the quadrupole magnet upstream of the dipole is to provide the vertical focus needed to make the double-collimation scheme work efficiently.", "Summed signals are also available for each column for use in normal operation when electron collimation is not desired.", "The Tagger Microscope consists of a two-dimensional array of square scintillating fibers packed in a dense array of dimensions $102\\times 5$ .", "The fibers are multi-clad BCF-20 with a $2\\times 2$ mm$^2$ square transverse profile, manufactured by Saint-GobainSaint-Gobain, https://www.saint-gobain.com/en.", "The cladding varies in thickness from 100 microns near the corners to 70 microns in the middle of the sides, with an active area of $1.8\\times 1.8$  mm$^2$ per fiber.", "Variations at the level of 5% in the transverse size of the fibers impose a practical lower bound of 2.05 mm on the pitch of the array.", "The detection efficiency of the TAGM averages 75% across its full energy range, in good agreement with the geometric factor of 77%.", "Each scintillating fiber is 10 mm long, fused at its downstream end to a clear light guide of matching dimensions (Saint-Gobain BCF-98) that transmits the scintillation light from the focal plane to a shielded box where a silicon photomultiplier (SiPM) converts light pulses into electronic signals.", "The scintillators are oriented so that the electron trajectories are parallel to the fiber axis, providing large signals for electrons from the radiator, in contrast to the omni-directional electromagnetic background in the tagger hall.", "Figure: Conceptual overview of the tagger microscope design, showing the fiber bundles andlight guides (left), and the orientation of these bundles aligned with the incomingelectron beam direction in the tagger focal plane (right).", "The variation of thecrossing angle β\\beta is exaggerated for the sake of illustration.Because the electron trajectories do not cross the focal plane at right angles, the fiber array must be staggered along the dispersion direction.", "A staggering step occcurs every 6 columns, as illustrated in Fig.", "REF .", "The slight variation of the crossing angle $\\beta $ is taken into account by a carefully adjusted fan-out that is implemented by small evenly-distributed gaps at the rear ends of adjacent 6-column groups (bundles).", "A total of 17 such bundles comprise the full Tagger Microscope.", "The far ends of the scintillation light guides are coupled to Hamamatsu S10931-050P SiPMs.", "The SiPMs are mounted on a custom-built two-stage preamplifier board, with 15 SiPMs per board.", "In addition to the 15 individual signals generated by each preamplifier, the boards also produce three analog sum outputs, each the sum of five adjacent SiPMs corresponding to the five fibers in a single column.", "All 510 SiPMs are individually biased by custom bias control boards, one for every two preamplifier boards.", "The control boards connect to the preamplifiers over a custom backplane, and communicate with the experimental slow controls system over ethernet.", "Each control board has the capability to electronically select between two gain modes for the preamplifiers on that board: a low gain mode used during regular tagging operation, and a high gain mode used for triggering on single-pixel pulses during bias calibration.", "Each bias control board manages the control and biasing for two preamplifiers.", "The control board also measures live values for environmental parameters (voltage levels and temperatures) in the TAGM electronics, so that alarms can be generated by the experimental control system whenever any of these parameters stray outside predefined limits.", "Pulse height and timing information for 122 channels from the TAGM is provided by analog-to-digital converters (ADCs) and time-to-digital converters (TDCs).", "These 122 signals include the 102 column sums plus the individual fiber signals from columns 7, 27, 81, and 97.", "Here, each channel goes through a 1:1 passive splitter, with one output going to an ADC and the other through discriminators to a TDC.", "The ADCs are 250-MHz flash ADCs with 12-bit resolution and a full-scale pulse amplitude of 1 V. The TDCs are based on the F1 TDC chip [18], with a least-count of 62 ps.", "Pulse thresholds in both the ADC and discriminator modules are programmable over the range 1-1000 mV on an individual channel basis, covering the full dynamic range of the TAGM front end.", "The TAGM preamplifier outputs (before splitting) saturate at around 2 V pulse amplitude.", "The mean pulse charge in units of SiPM pixels corresponding to a single high-energy electron varies from 150 to 300 pC, depending on the fiber, with an average of 220 pC and standard deviation of 25 pC.", "During calibration, this yield is measured individually for each fiber by selectively biasing the SiPMs on each row of fibers, one row at a time, and reading out the column sums.", "Once all 510 individual fiber yields have been measured, the bias voltages within each column are adjusted to compensate for yield variations, so that the mean pulse height in a given column is the same regardless of which fiber in the column detected the electron.", "The ADC readout and discriminator thresholds are set individually for each column, for optimum efficiency and noise rejection.", "The ADC firmware provides an approximate time for each pulse, in addition to the pulse amplitude.", "During offline reconstruction, this time information is used to associate ADC and TDC pulse information from the same channel, so that a time-walk correction can be applied to the TDC time.", "Once this correction has been applied, a time resolution of 230 ps is achieved for the TAGM.", "This resolution is based on data collected at rates on the order of 1 MHz per column, while the typical rate in the tagger microscope is about 0.5 MHz.", "The readout was designed to operate at rates up to 4 MHz per column.", "A brief test above 2 MHz per column allowed visual inspection of the pulse waveforms from the TAGM, without change in the pulse shape or amplitude." ], [ "Broadband tagging hodoscope", "The Tagger Hodoscope (TAGH) consists of 222 scintillator counters distributed over a length of 9.25 m and mounted just behind the focal plane of the tagger magnet.", "The function of this hodoscope is to tag the full range of photon energy from 25% to 97% of the incident electron energy.", "A gap in the middle of that range is left open for the registration of the primary coherent peak by the Tagger Microscope.", "The geometry of the counters in the vicinity of the microscope is shown in Fig.", "REF .", "This broad coverage aids in alignment of the diamond radiator and expands the GlueX physics program reach to photon energies outside the range of the coherent peak.", "The coverage of the hodoscope counters in the region below 60% drops to half, with substantial gaps in energy between the counters.", "This was done because the events of primary interest to GlueX come from interactions of photons within and above the coherent peak; within and above the coherent peak the coverage is 100% up to the 97% $E_0$ cutoff.", "Each counter in the hodoscope is a sheet of EJ-228 scintillator, 6 mm thick and 40 mm high.", "The counter widths vary along the focal plane, from 21 mm near the end-point region down to 3 mm at the downstream end.", "The scintillators are coupled to a Hamamatsu R9800 photomultiplier tube (PMT) via a cylindrical acrylic (UVT-PMMA) light guide 22.2 mm in diameter and 120 mm long.", "Each PMT is wrapped in $\\mu $ -metal to shield the tube from the fringe field of the tagger magnet.", "Each PMT is instrumented with a custom designed active base [19], consisting of a high-voltage divider and an amplifier powered by current flowing through the divider.", "The base provides two signal outputs, one going to a flash ADC and the other through a discriminator to a TDC.", "Operating the amplifier with a gain factor of 8.5 allows the PMT to operate at a lower voltage of 900 V and reduce the PMT anode current, therefore improving the rate capability.", "The energy bite of each counter ranges between 8.5 and 30 MeV for a 12 GeV incident electron beam.", "Typical rates during production running are 1 MHz above the coherent peak and 2 MHz per counter below the coherent peak.", "The maximum sustainable rate per counter is about 4 MHz.", "The counters are mounted with their faces normal to the path of the scattered electrons in two or three rows slightly downstream of the focal plane, as shown in Fig.", "REF .", "This allows the counters to be positioned without horizontal gaps in the dispersion direction, enabling complete coverage of the entire tagged photon energy range.", "Figure: Schematic of electron trajectories in the region of the microscope.", "Shown are the three layers of hodoscope counters on either side of the microscope and theregion covered by the microscope.The mounting frame of the hodoscope is suspended from the ceiling of the Tagger Hall to provide full flexibility for positioning TAGH.", "The frame is constructed to also support the addition of counters to fill in the energy range currently occupied by the microscope when the TAGM location is changed.", "A similar procedure to that described in Section REF for the TAGM is used to apply a time-walk correction to the TDC times from the TAGH counters.", "Once this time-walk correction is applied, the time resolution of the TAGH is 200 ps.", "No significant degradation of this resolution is expected at the operating rates planned for Phase 2 running, which are on the order of 2 MHz per counter above the coherent peak.", "Under these conditions, the rates in the TAGH counters below the coherent peak would average around 4 Mhz, which is at the top of their allowed range.", "These counters will be turned off when running at full intensity." ], [ "Tungsten keV filter", "To reduce the photon flux in the $10-100$  keV range, a 100 $\\mu $ m tungsten foil ($3\\%$ of a radiation length) was installed in the beam line at the entrance of the collimator cave.", "We have studied the effect of different foil materials on the anode currents and random hits in the drift chambers (see Section ), as these factors limit the high-intensity operation of the experiment.", "By comparing the effect of different materials (Al, Cu, W) with fixed radiation lengths (see Fig.REF ) we learned that the drift chambers are mostly affected by photons in the 70-90 keV range.", "The analysis of the pulse shape of the random hits in the CDC confirmed that these photons directly produce hits in the inner layers of the chamber.", "The insertion of the tungsten foil reduced the number of random hits in the inner CDC layers by a factor of up to 8 and the anode current by $55\\%$ .", "The reduction of the current in the FDC was more moderate, about $25\\%$ .", "Note that the FDC sense wires are as close as 3 cm to the beam, while in the CDC the closest wires are at 10 cm.", "Figure: Attenuation of low-energy photons in foils with a thickness of 3%3\\% of a radiation length for different materials as a function of photon energy.", "The W foil was selected to reduce the random background hits in the detector drift chambers.", "The attenuation coefficients are taken from Ref.", "." ], [ "Beam profiler", "The beam profiler is located immediately upstream of the collimator (see Fig.", "REF ) and is used to measure the photon beam intensity in a plane normal to the incident photon beam.", "The profiler consists of two planes of scintillating fibers, giving information on the photon beam profile in the X and Y projections.", "Each plane consists of 64 square fibers, 2 mm in width, read out by four 16-channel multi-anode PMTs.", "The beam profiler is only used during beam setup until the photon beam is centered on the active collimator." ], [ "Active collimator ", "The active collimator monitors the photon beam position and provides feedback to micro-steering magnets in the electron beamline, for the purpose of suppressing drifts in photon beam position.", "The design of the active collimator for GlueX is based on a device developed at SLAC for monitoring the coherent bremsstrahlung beam there [21].", "The GlueX active collimator is located on the upstream face of the primary collimator, and consists of a dense array of tungsten pins attached to tungsten base plates.", "The tungsten plate intercepts off-axis beam photons before they enter the collimator, creating an electromagnetic shower that cascades through the array of pins.", "High-energy delta rays created by the shower in the pins (known as “knock-ons\") are emitted forward into the primary collimator.", "The resulting net current between the tungsten plates and the collimator is proportional to the intensity of the photon beam on the plate.", "The tungsten plates are mounted on an insulating support, and the plate currents are monitored by a preamplifier with pA sensitivity.", "The tungsten plate is segmented radially into two rings, and each ring is segmented azimuthally into four quadrants.", "The asymmetry of the induced currents on the plates in opposite quadrants indicates the degree of displacement of the photon beam from the intended center position.", "Typical currents on the tungsten sectors are at the level of 1.4 nA (inner ring) and 0.85 nA (outer ring) when running with a 50 $\\mu $ m diamond crystal and a 200-nA incident electron beam current.", "The current-sensitive preamplifiers used with the active collimator are PMT-5R devices manufactured by ARI CorporationAdvanced Research Instruments Corporation, http://aricorp.com..", "The PMT-5R has six remotely selectable gain settings ranging from $10^{12}$  V/A to $10^6$  V/A, selectable by powers of 10.", "This provides an excellent dynamic range for operation of the beam over a wide range of intensities, from 1 nA up to several $\\mu $ A.", "The preamplifier input stage exhibits a fixed gain-bandwidth product of about 2 Hz-V/pA which limits its bandwidth at the higher gain settings, for example 2 Hz at $10^{12}$  V/A, 20 Hz at $10^{11}$  V/A.", "In-situ electronic noise on the individual wedge currents is measured to be 1.5 pA/$\\sqrt{\\mbox{Hz}}$ on the inner ring, and 15 pA/$\\sqrt{\\mbox{Hz}}$ on the outer ring.", "The sensitivity of the current asymmetry to position is 0.160/mm for the inner ring and 0.089/mm for the outer.", "With a 50 micron diamond and 200 nA beam current, operating the active collimator at a bandwidth of 1 kHz yields a measurement error in the position of the beam centroid of 150 $\\mu $ m for the inner ring and 450 $\\mu $ m for the outer ring.", "The purpose of the outer ring is to help locate the beam when the beam location has shifted more than 2 mm from the collimator axis, where the response of the inner ring sectors becomes nonlinear.", "The maximum deviation allowed for the Hall D photon beam position relative to the collimator axis is 200 $\\mu $ m. The active collimator readout was designed with kHz bandwidth so that use in a fast feedback loop would suppress motion of the beam at 60 Hz and harmonics that might exceed this limit.", "Experience with the Hall-D beam has shown that the electron beam feedback system already suppresses this motion to less than 100 $\\mu $ m amplitude, so that fast feedback using the active collimator is not required during normal operation.", "Instead, the active collimator is used in a slow feedback loop which locks the photon beam position at the collimator with a correction time constant of a few seconds.", "This slow feedback system is essential for preventing long-term drifts in the photon beam position that would otherwise occur on the time scale of hours or days.", "The active collimator can achieve 200 $\\mu $ m position resolution down to beam currents as low as 2 nA when operated in this mode with noise averaging over a 5 s interval." ], [ "Collimator", "The photon beam produced at the diamond radiator contains both incoherent and coherent bremsstrahlung components.", "In the region of the coherent peak, where photon polarization is at its maximum, the angular spread of coherent bremsstrahlung photons is less than that of incoherent bremsstrahlung.", "The characteristic emission angle for incoherent bremsstrahlung is $m/E = 43$  $\\mu $ rad at 12 GeV, whereas the coherent flux within the primary peak is concentrated below 15 $\\mu $ rad with respect to the beam direction.", "Collimation increases the degree of linear polarization in the photon beam by suppressing the incoherent component relative to the coherent part.", "The Hall-D primary collimator provides apertures of 3.4 mm and 5.0 mm in a tungsten block mounted on an X-Y table.", "The 5.0 mm collimator is used under normal GlueX running conditions.", "The tungsten collimator is surrounded by lead shielding.", "The collimator may also be positioned to block the beam to prevent high-intensity beam from entering the experimental hall during tuning of the electron beam.", "Downstream of the primary collimator, a sweeping magnet and shield wall, followed by a secondary collimator with its sweeping magnet and shield wall, suppress charged particles and photon background around the photon beam that are generated in the primary collimator.", "The photon beam exiting the collimation system then passes through a thin pair conversion target.", "The resulting $e^+e^-$ pairs are used to continuously monitor the photon beam flux and polarization." ], [ "Triplet Polarimeter ", "The Triplet Polarimeter (TPOL) is used to measure the degree of polarization of the linearly-polarized photon beam [22].", "The polarimeter uses the process of $e^+e^-$ pair production on atomic electrons in a beryllium target foil, with the scattered atomic electrons measured using a silicon strip detector.", "Information on the degree of polarization of the photon beam is obtained by analyzing the azimuthal distribution of the scattered atomic electrons." ], [ "Determination of photon polarization ", "Triplet photoproduction occurs when the polarized photon beam interacts with the electric field of an atomic electron within a target material and produces a high energy $e^+e^-$ pair.", "When coupled with trajectory and energy information of the $e^+e^-$ pair, the azimuthal angular distribution of the recoil electron provides a measure of the photon beam polarization.", "The cross section for triplet photoproduction can be written as $\\sigma _t = \\sigma _0 [ 1 - P \\Sigma \\cos (2\\varphi )]$ for a polarized photon beam, where $\\sigma _0$ is the unpolarized triplet cross section, $P$ the photon beam polarization, $\\Sigma $ the beam asymmetry for the process, and $\\varphi $ the azimuthal angle of the recoil electron trajectory with respect to the plane of polarization for the incident photon beam.", "To determine the photon beam polarization, the azimuthal distribution of the recoil electrons is recorded and fit to the function $A [ 1- B \\cos (2\\varphi )]$ where the variables $A$ and $B$ are parameters of the fit, with $B = P \\Sigma $ .", "The value of $\\Sigma $ depends on the beam photon energy, the thickness of the converter target, and the geometry of the setup.", "The value of $\\Sigma $ was determined to be $0.1990 \\pm 0.0008$ at 9 GeV for the GlueX beamline and a 75 micron Be converter [22].", "The TPOL detects the recoil electron arising from triplet photoproduction.", "This system consists of a converter tray and positioning assembly, which holds and positions a beryllium foil converter where the triplet photoproduction takes place.", "A silicon strip detector (SSD) detects the recoil electron from triplet photoproduction, providing energy and azimuthal angle information for that particle.", "A vacuum housing, containing the pair production target and SSD, supplies a vacuum environment minimizing multiple Coulomb scattering between target and SSD.", "Preamplier and signal filtering electronics are placed within a Faraday-cage housing.", "The preamplifier enclosure is lined with a layer of copper foil to reduce exterior electromagnetic signal interference.", "Signals from the downstream (azimuthal sector) side of the SSD are fed to a charge-sensitive preamplifier located outside the vacuum.", "In operation, the TPOL vacuum box is coupled directly to the evacuated beamline through which the polarized photon beam passes.", "Upon entering TPOL, the photon beam passes into the beryllium converter, triplet photoproduction takes place, an $e^+e^-$ pair is emitted from the target in the forward direction, and a recoil electron ejected from the target at large angles with respect to the beam is detected by the SSD within the TPOL vacuum chamber.", "The recoil electron is ejected at large angles and detected by the SSD.", "The $e^+e^-$ pair, together with any beam photons that did not interact with the converter material, pass through the downstream port of the TPOL vacuum box into the evacuated beamline, which in turn passes through a shielding wall into the Hall-D experimental area.", "The $e^+e^-$ pair then enters the vacuum box and magnetic field of the GlueX Pair Spectrometer, while photons continue through an evacuated beamline to the target region of the GlueX detector.", "Accounting for all sources of uncertainty from this setup, the total estimated systematic error in the TPOL asymmetry $\\Sigma $ is 1.5% [22]." ], [ "Pair Spectrometer ", "The main purpose of the Pair Spectrometer (PS) [23] is to measure the spectrum of the collimated photon beam and determine the fraction of linearly polarized photons in the coherent peak energy region.", "The TPOL relies on the PS to trigger on pairs in coincidence with hits in the recoil detector.", "The PS is also used to monitor the photon beam flux, and for energy calibration of the tagging hodoscope and microscope detectors.", "The PS, located at the entrance to Hall D, reconstructs the energy of a beam photon by detecting the $e^+e^-$ pair produced by the photon in a thin converter.", "The converter used is typically the beryllium target housed within TPOL; otherwise the PS has additional converters that may be inserted into the beam with thicknesses ranging between 0.03% and 0.5% of a radiation length.", "The produced $e^+e^-$ leptons are deflected in a modified 18D36 dipole magnet with an effective field length of about 0.94 m and detected in two layers of scintillator detectors: a high-granularity hodoscope and a set of coarse counters, referred to as PS and PSC counters, respectively.", "The detectors are partitioned into two identical arms positioned symmetrically on opposite sides of the photon beam line.", "The PSC consists of sixteen scintillator counters, eight in each detector arm.", "Each PSC counter is 4.4 cm wide and 2 cm thick in the direction along the lepton trajectory and 6 cm high.", "Light from the PSC counters is detected using Hamamatsu R6427-01 PMTs.", "The PS hodoscope consists of 145 rectangular tiles (1 mm and 2 mm wide) stacked together.", "Hamamatsu SiPMs were chosen for readout of the PS counters  [24], [25], [26].", "Each detector arm covers an $e^\\pm $ momentum range between 3.0 GeV/c and 6.2 GeV/c, corresponding to reconstructed photon energies between 6 GeV and 12.4 GeV.", "The relatively large acceptance of the hodoscope enables energy determination for photons with energies from below the coherent peak to the beam endpoint energy near 12 GeV.", "The pair energy resolution of the PS hodoscope is about 25 MeV.", "The time resolution of the PSC counters is 120 ps, which allows coincidence measurements between the tagging detectors and the PS within an electron beam bunch.", "Signals from the PS detector are delivered to the trigger system, as described in Section .", "The typical rate of PS double-arm coincidences is a few kHz.", "Details about the performance of the spectrometer are given in [27], [28]." ], [ "Determination of photon flux ", "The intensity of beam photons incident on the GlueX target is important for the extraction of cross sections.", "The photon flux is determined by converting a known fraction of the photon beam to $e^\\pm $ pairs and counting them in the PS as a function of energy.", "Data from the PS are collected using a PS trigger, which runs in parallel to the main GlueX physics trigger, as described in Section .", "The number of beam photons integrated over the run period is obtained individually for each tagger counter (TAGH and TAGM), i.e., for each photon beam energy bin.", "The PS calibration parameter used in the flux determination, a product of the converter thickness, acceptance, and the detection efficiency for leptons, is determined using calibration runs with the Total Absorption Counter (TAC) [29].", "The TAC is a small calorimeter (see Section REF ) inserted directly into the photon beam immediately upstream of the photon beam dump to count the number of beam photons as a function of energy.", "These absolute-flux calibration runs are performed at reduced beam intensities in order to limit the rate of accidental tagging coincidences.", "Data are acquired simultaneously from the PS and TAC.", "These data enable an absolute flux calibration for the PS by measuring the number of reconstructed $e^+e^-$ pairs for a given number of photons of the same energy seen by the TAC.", "Uncertainties on the photon flux determinations are currently being investigated.", "The expected precision of the flux determination is on the level of $1\\%$ ." ], [ "Total Absorption Counter ", "The TAC is a high-efficiency lead-glass calorimeter, used at low beam currents ($<$ 5nA) to determine the overall normalization of the flux from the GlueX coherent bremsstrahlung facility.", "This device is intended to count all beam photons above a certain energy threshold, which have a matching hit in the tagger system.", "There would be a very large number of overlapping pulses in the TAC if it is used with the production photon flux, resulting in low detection efficiency and therefore large systematic uncertainties.", "Therefore, the TAC is only inserted into the beam during dedicated runs at very low intensities when the detector can run with near 100% efficiency.", "The TAC was originally developed for and deployed in Hall B, for photon beam operations with CLAS [30], [31], [32].", "Only a certain fraction of the photons produced at the radiator reach the target and causes an interaction that is seen in the GlueX detector.", "The count of tagged photons reaching the GlueX target is determined as a function of energy from individual TAC coincidence measurements with each tagging counter.", "Simultaneous with these counts, the coincidences between each of the tagging counters and converted pairs detected in the pair spectrometer are also recorded.", "The ratio between the count of tagged pairs and tagged TAC events thus determined for each tagging counter are used to convert the tagged rate in the pair spectrometer that is observed during normal operation into a total count of tagged photons for each tagging counter that were incident on the GlueX target." ], [ "Solenoid magnet\n\n", "Solenoid Magnet" ], [ "Overview \n", "Overview The core of the GlueX spectrometer is a superconducting solenoid with a bore diameter and overall yoke length of approximately 2 m and 4.8 m, respectively.", "The photon beam passes along the axis of the solenoid.", "At the nominal current of 1350 A, the magnet provides a magnetic field along the axis of about 2 T. The magnet was designed and built at SLAC in the early 1970's [33] for the LASS spectrometer [34].", "The solenoid employs a cryostatically stable design with cryostats designed to be opened and serviced with hand tools.", "The magnet was refurbished and modifiedThe front plate of the flux return yoke was modified, leading to a swap of the two front coils and modifications of the return flux yoke in order to keep the magnetic forces on the front coil under the design limit.", "The original gaps between the yoke's rings were filled with iron.", "The Cryogenic Distribution Box was designed and built for GlueX.", "for the GlueX experiment [35], [36].", "The magnet is constructed of four separate superconducting coils and cryostats.", "The flux return yoke is made of several iron rings.", "The coils are connected in series.", "A common liquid helium tank is located on top of the magnet, providing a gravity feed of the liquid to the coils.", "The layout of the coil cryostats and the flux return iron yoke is shown in Fig.", "REF .", "Table REF summarizes the salient parameters of the magnet.", "Table: Key parameters of the GlueX solenoid.", "Thecoils are listed in order along the beam direction." ], [ "Conductor and Coils\n\n", "Conductor and Coils The superconductor composite is made of niobium–titanium filaments in a copper substrate, twisted and shaped into a $\\sim $ 7.62$\\times $ 1 mm$^2$ rectangular band.", "The laminated conductor is made by soldering the superconductor composite band between two copper strips to form a rectangular cross section of 7.62$\\times $ 5.33 mm$^2$ .", "The measured residual resistivity ratio of the conductor at $\\sim {}300$ K and $\\sim {}15$ K is $\\approx {}$ 100.", "As the coil was wound, a 0.64 mm-thick stainless steel support band and two 0.2 mm-thick Mylar insulating strips were wound together with it for pre-tensioning and insulation.", "The liquid helium is in contact with the shorter (5.33 mm) sides of the cable.", "Each of the coils consists of a number of subcoils.", "Each subcoil contains a number of “double pancakes” with the same number of turns.", "Each double pancake is made from a single piece of conductor.", "The voltage across the subcoils is monitored using special wires.", "These pass through vertical cryostats, called chimneys, along with the helium supply pipes and the main conductor.", "The cold helium vessel containing the coil is supported within the warm cryostat vacuum vessel by a set of columns designed to provide sufficient thermal insulation.", "The columns are equipped with strain gauges for monitoring the stresses on the columns.", "The helium vessel is surrounded by a nitrogen-cooled thermal shield made of copper and stainless-steel panels.", "Super-insulation is placed between the vacuum vessel and the nitrogen shield.", "The vacuum vessels are attached to the matching iron rings of the yoke.", "The power supplyDanfysik System 8000 Type 854. provides up to 10 V DC for establishing the operating current while ramping.", "The supply also includes a protection circuit, which can be engaged by a quench detector as well as by other signals.", "During trips, a small dump resistor of 0.061 $\\Omega $ limits the maximum voltage on the magnet to 100 V. The dumping time constant of $L/R \\approx 7$  min is relatively long, but safe according to the original design of the magnet.", "A large copper mass and the helium bath are able to absorb a large amount of energy during a quench without overheating the solder joints.", "This permits the use of an “intelligent” quench detector with low noise sensitivity and a relatively slow decision time of 0.5 s. The quench detector compares the measured voltages on different subcoils in order to detect a resistive component.", "While ramping the current, such a voltage is proportional to the subcoil inductance.", "Relative values of inductance of various subcoils depend on the value of the current because of saturation effects in the iron yoke.", "Transient effects are also present at changes of the slew rate caused by Foucault currents in the yoke.", "The system includes two redundant detectors: one uses analog signals and a simplified logic, another is part of the PLC control system (see Section REF ) which uses digitized signals.", "The PLC digital programmable device is more sensitive since this monitoring system takes into account the dependence of the coils' inductance on the current and provides better noise filtering.", "The ramping slew rate is limited by the transient imbalance of the voltages on subcoils that may trigger the quench detector.", "Additionally, unexplained voltage spikes of 1 ms duration have been observed in coil 2 at high slew rates, which can trigger the quench detector.", "Powering up the magnet to 1350 A takes about 8 h. For diagnostic purposes two 40-turn pickup coils are installed on the bore surface of the vacuum vessel of each of the coils." ], [ "\nCooling System\n\n", "Cooling System The cooling system is described in detail in Ref. [37].", "A stand-alone helium refrigerator located in a building adjacent to Hall D provides liquid helium and nitrogen via a transfer line to the Cryogenic Distribution Box above the magnet.", "The transfer line delivers helium at 2.6 atm, and 6 K to a Joule-Thomson (JT) valve providing liquid to a cylindrical common helium tank in the Distribution Box.", "The level of liquid helium in the tank is measured with a superconducting wire probe;American Magnetics Model 1700 with HS-1/4-RGD-19\"/46\"-4LDCP-LL6-S sensor the liquid level is kept at about half of the tank diameter.", "The cold helium gas from the tank is returned to the refrigerator, which keeps the pressure at the top of the tank at 1.2 atm corresponding to about 4.35 K at the surface of the liquid.The original implementation at SLAC did not recycle the helium and operated at atmospheric pressure.", "Each coil is connected to the common helium tank by two vertical 2-inch pipes.", "One pipe is open at the bottom of the tank while the other one is taller than the typical level of helium inside the tank.", "The main conductor and the wires for voltage monitoring pass through the former pipe.", "Additionally, two $\\sim $ 6 m long, 3/8 inch ID pipes go outside the coil's helium vessel, from the Distribution Box to the bottom of the coil.", "One of those pipes, connected to a JT valve in the box, is used to fill the coil initially, but is not used during operation.", "The other pipe reaches the bottom of the common helium tank in order to provide a thermo-syphon effect essential for the proper circulation of helium in the coil.", "The main current is delivered into the helium tank via vapor-cooled leads, and is distributed to the coils by a superconducting cable.", "After cooling the leads, the helium gas is warmed and returned to the refrigeration system.", "The gas flow through the leads is regulated based on the current in the magnet; at 1350 A, the flow is about 0.25 g/s.", "The coils and the Distribution Box are equipped with various sensors for temperature, pressure, voltage, and flow rates." ], [ "\nMeasurements and Controls\n\n", "Measurements and Controls The control system for the superconducting solenoid, power supply, and cryogenic system, is based on Programmable Logic Controllers (PLC)Allen-Bradley Programmable Logic Controllers http://ab.rockwellautomation.com/Programmable-Controllers..", "The PLC system digitizes the signals from various sensors, communicates with other devices, reads out the data into a programmable unit for analysis, and sends commands to various devices.", "Additionally, the PLC is connected to EPICSExperimental Physics and Industrial Control System, https://epics.anl.gov.", "in order to display and archive the data (see Section ).", "The practical sampling limit for the readout of the sensor is a few Hz, which is too low for detection of fast voltage spikes on the coils due to motion, shorts, or other effects.", "Therefore, the voltage taps from the coils and the pickup coils are read out by a PXI systemNational Instruments, PXI Platform, http://www.ni.com/pxi/., which provides a sampling rate of about 100 kHz.", "The PXI system also reads out several accelerometers attached to the coils' chimneys, which can detect motion inside the coils.", "The PXI CPU performs initial integration and arranges the data in time-wise rows with a sampling rate of 10 kHz.", "The PLC system reads out the data from the PXI system.", "Additionally, the PXI data are read out by an EPICS server at the full 10 kHz sampling rate and are recorded for further analysis." ], [ "\nField calculation and measurement\n\n", "Field calculation and measurement The momentum resolution of the GlueX spectrometer is larger than 1% and is dominated by multiple scattering and the spatial resolution of the coordinate detectors.", "Thus, a fraction of a percent is sufficient accuracy for the field determination.", "The coils are axially symmetric, while the flux return yoke is nearly axially symmetric, apart from the holes for the chimneys.", "The field was calculated using a 2-dimensional field calculator Poisson/SuperfishPoisson/Superfish developed at LANL, https://laacg.lanl.gov/laacg/services/serv_codes.phtml#ps.", ", assuming axial symmetry.", "The model of the magnet included the fine structure of the subcoils and the geometry of the yoke iron.", "Different assumptions about the magnetic properties of the yoke iron have been used: the Poisson default AISI 1010 steel, the measurements of the original yoke iron made at SLAC, and the 1018 steel used for the filler plates.", "Since the results of the field calculations differ by less than 0.1%, the default Poisson AISI 1010 steel properties were used for the whole yoke iron in the final field map calculations.", "The three projections of the magnetic field have been measured along lines parallel to the axis, at four values of the radius and at up to six values of the azimuthal angle.", "The calculated field and the measured deviations are shown in Fig.", "REF .", "The tracking detectors occupy the volume of $R<56$  cm and $45<Z<340$  cm.", "In this volume the field deviation at $R=0$ does not exceed 0.2%.", "The largest deviation of 1.5% is observed at the downstream edge of the fiducial volume and at the largest radius.", "Such a field uncertainty in that region does not noticeably affect the momentum resolution.", "In most of the fiducial volume the measured field is axially symmetric to $\\approx $ 0.1% and deviates from this symmetry by $\\approx $ 2% at the downstream edge and the largest radius.", "The calculated field map is used for track reconstruction and physics analyses.", "Figure: The full field at 1350 A calculated with Poisson (leftscale) on the axis and at the edge of the tracking fiducial volume (R=56 cm).", "The deviations of the measurements from thecalculations are shown (right scale) on the axis, and at R=56 cm.", "The measurements were made at 6 azimuthalangles.", "We show the angles (0 ∘ ^\\circ and 90 ∘ ^\\circ ) with the largest deviations from thecalculations." ], [ "Target ", "Target A schematic diagram of the GlueX liquid hydrogen cryotarget is shown in Fig.", "REF .", "The major components of the system are a pulse tube cryocooler,Cryomech model PT415.", "a condenser, and a target cell.", "These items are contained within an aluminum and stainless steel `L'-shaped vacuum chamber with an extension of closed-cell foamRohacell 110XT, Evonik Industries AG.", "surrounding the target cell.", "In turn, the GlueX Start Counter (Sec.", "REF ) surrounds the foam chamber and is supported by the horizontal portion of the vacuum chamber.", "Polyimide foils, 100 $\\mu $ m thick, are used at the upstream and downstream ends of the chamber as beam entrance and exit windows.", "The entire system, including the control electronics, vacuum pumps, gas-handling system, and tanks for hydrogen storage, is mounted on a small cart that is attached to a set of rails for insertion into the GlueX solenoid.", "To satisfy flammable gas safety requirements, the system is connected at multiple points to a nitrogen-purged ventilation pipe that extends outside Hall D. Figure: Simplified process and instrumentation diagram for the GlueX liquid hydrogen target (not to scale).In the real system, the P-trap is above the level of the target cell and is used topromote convective cooling of the target cell from room temperature.Hydrogen gas is stored inside two 200 l tanks and is cooled and condensed into a small copper and stainless steel container, the condenser, that is thermally anchored to the second cooling stage of the cryocooler.", "The first stage of the cryocooler is used to cool the H$_2$ gas to about 50 K before it enters the condenser.", "The first stage also cools a copper thermal shield that surrounds all lower-temperature components of the system except for the target cell itself, which is wrapped in a few layers of aluminized-mylar/cerex insulation.", "The condenser is comprised of a copper C101 base sealed to a stainless steel can with an indium O-ring.", "Numerous vertical fins are cut into the copper base, giving a large surface area for condensing hydrogen gas.", "A heater and a pair of calibrated Cernox thermometersCernox, Lake Shore Cryotronics.", "are attached outside the condenser, and are used to regulate the heater temperature when the system is filled with liquid hydrogen.", "The target cell, shown in Fig.", "REF , is similar to designs used in Hall B at JLab [38].", "The cell walls are made from 100-$\\mu $ m-thick aluminized polyimide sheet wrapped in a conical shape and glued along the edge, overlapping into a 2 mm wide scarf joint.", "The conical shape prevents bubbles from collecting inside the cell, while the scarf joint reduces the stress riser at the glue joint.", "This conical tube is glued to an aluminum base, along with stainless steel fill and return tubes leading to the condenser, a feed-through for two calibrated Cernox thermometers inside the cell, and a polyamide-imide support for the reentrant upstream beam window.", "Both the upstream and downstream beam windows are made of non-aluminized, 100 $\\mu $ m thick polyimide films that have been extruded into the shapes indicated in Fig.", "REF .", "These windows are clearly visible in Fig.", "REF where reconstructed vertex positions are shown.", "All items are glued together using a two-part epoxy3M Scotch-Weld epoxy adhesive DP190 Gray.", "that has been in reliable use at cryogenic temperatures for long periods.", "A second heater, attached to the aluminum base, is used to empty the cell for background measurements.", "The base is attached to a kinematic mount, which is in turn supported inside the vacuum chamber using a system of carbon fiber rods.", "The mount is used to correct the pitch and yaw of the cell, while $X$ , $Y$ , and $Z$ adjustments are accomplished using positioning screws on the target cart.", "During normal operation, a sufficient amount of hydrogen gas is condensed from the storage tanks until the target cell, condenser, and interconnecting piping are filled with liquid hydrogen and an equilibrium pressure of about 19 psia is achieved.", "The condenser temperature is regulated at 18 K, while the liquid in the cell cools to about 20.1 K. The latter temperature is 1 K below the saturation temperature of H$_2$ , which eliminates boiling within the cell and permits a more accurate determination of the fluid density, $71.2 \\pm 0.3$  mg/cm$^3$ .", "The system can be cooled from room temperature and filled with liquid hydrogen in approximately six hours.", "Prior to measurements using an empty target cell, the liquid hydrogen is boiled back into the storage tanks in about five minutes.", "H$_2$ gas continues to condense and drain towards the target cell, but the condensed hydrogen is immediately evaporated by the cell heater.", "In this way, the cell does not warm above 40 K and can be re-filled with liquid hydrogen in about twenty minutes.", "Figure: Target cell for the liquid hydrogen target.", "Dimensions are in mm.Operation of the cryotarget is highly automated, requires minimal user intervention, and has operated in a very reliable and predictable manner throughout the experiment.", "The target controlsThe control logic uses National Instruments CompactRIO 9030. are handled by a LabVIEW program, while a standard EPICS softIOC running in Linux provides a bridge between the controller and JLab's EPICS enviroment (see Section ).", "Temperature readback and control of the condenser and target cell thermometers are managed by a four-input temperature controllerLake Shore Model 336. with PID control loops of 50 and 100 W. Strain gauge pressure sensors measure the fill and return pressures with 0.25% accuracy.", "When filled with subcooled liquid, the long-term temperature ($\\pm 0.2$  K) and pressure ($\\pm 0.1$  psi) stability of the liquid hydrogen enable a determination of the density to better than 0.5%." ], [ "Central drift chamber ", "Central drift chamber The Central Drift Chamber (CDC) is a cylindrical straw-tube drift chamber which is used to track charged particles by providing position, timing and energy loss measurements [39], [40].", "The CDC is situated inside the Barrel Calorimeter, surrounding the target and Start Counter.", "The active volume of the CDC is traversed by particles coming from the hydrogen target with polar angles between $6^{\\circ }$ and $168^{\\circ }$ , with optimum coverage for polar angles between $29^{\\circ }$ and $132^{\\circ }$ .", "The CDC contains 3522 anode wires of 20 $\\mu $ m diameter gold-plated tungsten inside Mylarwww.mylar.com straw tubes of diameter 1.6 cm in 28 layers, located in a cylindrical volume which is 1.5 m long, with an inner radius of 10 cm and outer radius of 56 cm, as measured from the beamline.", "Readout is from the upstream end.", "Fig.", "REF shows a schematic diagram of the detector.", "Figure: Cross-section through the cylindrically symmetric Central Drift Chamber, along the beamline.The straw tubes are arranged in 28 layers; 12 layers are axial, and 16 layers are at stereo angles of $\\pm 6^{\\circ }$ to provide position information along the beam direction.", "The stereo angle was chosen to balance the extra tracking information provided by the unique combination of stereo and axial straws along a trajectory against the size of the unused volume inside the chamber at each transition between stereo and axial layers.", "Fig.", "REF shows the CDC during construction.", "Figure: The Central Drift Chamber during construction.", "A partially completed layer of stereo straw tubes is shown, surrounding a layer of straw tubes at the opposite stereo angle.", "Part of the carbon fiber endplate, two temporary tension rods and some of the 12 permanent support rods linking the two endplates can also be seen.The volume surrounding the straws is enclosed by an inner cylindrical wall of 0.5 mm G10 fiberglass, an outer cylindrical wall of 1.6 mm aluminum, and two circular endplates.", "The upstream endplate is made of aluminum, while the downstream endplate is made of carbon fiber.", "The endplates are connected by 12 aluminum support rods.", "Holes milled through the endplates support the ends of the straw tubes, which were glued into place using several small components per tube, described more fully in [40].", "These components also support the anode wires, which were installed with 30 g tension.", "At the upstream end, these components are made of aluminum and were glued in place using conductive epoxyTIGA 920-H, www.loctite.com.", "This attachment method provides a good electrical connection to the inside walls of the straw tubes, which are coated in aluminum.", "The components at the downstream end are made of Noryl plasticwww.sabic.com and were glued in place using conventional non-conductive epoxy3M Scotch-Weld DP460NS, www.3m.com.", "The materials used for the downstream end were chosen to be as lightweight as feasible so as to minimize the energy loss of charged particles passing through them.", "At each end of the chamber, a cylindrical gas plenum is located outside the endplate.", "The gas supply runs in 12 tubes through the volume surrounding the straws into the downstream plenum.", "There the gas enters the straws and flows through them into the upstream plenum.", "From the upstream plenum the gas flows into the volume surrounding the straws, and from there the gas exhausts to the outside, bubbling through small jars of mineral oil.", "The gas mixture used is 50$\\%$ argon and 50$\\%$ carbon dioxide at atmospheric pressure.", "This gas mixture was chosen since its drift time characteristics provide good position resolution [39].", "A small admixture (approximately 1$\\%$ ) of isopropanol is used to prevent loss of performance due to aging[41], [42].", "Five thermocouples are located in each plenum and used to monitor the temperature of the gas.", "The downstream plenum is 2.54 cm deep, with a sidewall of ROHACELLwww.rohacell.com and a final outer wall of aluminized Mylar film, and the upstream plenum is 3.18 cm deep, with a polycarbonate sidewall and a polycarbonate disc outer wall.", "The readout cables pass through the polycarbonate disc and the upstream plenum to reach the anode wires.", "The cables are connected in groups of 20 to 24 to transition boards mounted onto the polycarbonate disc; the disc also supports the connectors for the high-voltage boards.", "Preamplifiers [43] are mounted on the high-voltage boards.", "The aluminum endplate, outer cylindrical wall of the chamber, aluminum components connecting the straws to the aluminum endplate and the inside walls of the straws are all connected to a common electrical ground.", "The anode wires are held at +2.1 kV during normal operation." ], [ "Forward Drift Chamber\n ", "Forward Drift Chamber The Forward Drift Chamber (FDC) consists of 24 disc-shaped planar drift chambers of 1 m diameter [44].", "They are grouped into four packages inside the bore of the spectrometer magnet.", "Forward tracking requires good multi-track separation due to the high particle density in the forward region.", "This is achieved via additional cathode strips on both sides of the wire plane allowing for a reconstruction of a space point on the track from each chamber.", "The FDC registers particles emitted into polar angles as low as $1^\\circ $ and up to $10^\\circ $ with all the chambers, while having partial coverage up to $20^\\circ $ .", "One FDC chamber consists of a wire plane with cathode planes on either sides at a distance of 5 mm from the wires (Fig.", "REF ).", "Figure: Artist rendering of one FDC chamber showing components.", "From top to bottom: upstream cathode, wire frame, downstream cathode, ground plane that separates the chambers.", "The diameter of the active area is 1 m.The frame that holds the wires is made out of ROHACELL with a thin G10 fiberglass skin in order to minimize the material and allow low energy photons to be detected in the outer electromagnetic calorimeters.", "The wire plane has sense ($20~\\mu $ m diameter) and field (80 $\\mu $ m) wires 5 mm apart, forming a field cell of $10\\times 10$  mm$^2$ .", "To reduce the effects of the magnetic field, a “slow\" gas mixture of $40\\%$  Ar and $60\\%$  CO$_2$ is used.", "A positive high voltage of about $2.2$  kV is applied to the sense wires and a negative high voltage of $0.5$  kV to the field wires.", "The cathodes are made out of 2-$\\mu $ m-thin copper strips on Kapton foil with a pitch of 5 mm, and are held at ground potential.", "The strips on the two cathodes are arranged at $30^\\circ $ relative to each other and at angles of $75^\\circ $ and $105^\\circ $ angle with respect to the wires.", "The six chambers of a package are separated by thin aluminized Mylar.", "Each chamber is rotated relative to the previous one by $60^\\circ $ .", "The total material of a package in the sensitive area corresponds to $0.43\\%$ radiation lengths, with about half of that in the area along the beam line that has no copper on the cathodes.", "The sense wires in the inner area of $6-7.8$  cm diameter (depending on the distance of the package to the target) are increased in thickness from 20 $\\mu $ m to $\\sim 80$  $\\mu $ m, which makes them insensitive to the high rates along the beam.", "The distance between the first and last package is $1.69$  m. All chambers are supplied with gas in parallel.", "In total, $2,304$ wires and $10,368$ strips are read using charge preamplifiers with 10 ns peaking time, with a gain of $0.77$  mV/fC for the wires and $2.6$  mV/fC for the strips." ], [ "Electronics ", "The high voltage (HV) supply units used are CAEN A1550Pwww.caen.it, with noise-reducing filter modules added to each crate chassis.", "The low voltage (LV) supplies are Wiener MPOD MPV8008www.wiener-d.com.", "The preamplifiers are a custom JLab design based on an ASIC [43] with 24 channels per board; the preamplifiers are charge-sensitive, capacitively coupled to the wires in the CDC and FDC, and directly coupled to strips in the FDC.", "Pulse information from the CDC anode wires and FDC cathode strips are obtained and read out using 72-channel 125 MHz flash ADCs (FADCs) [45], [46].", "These use Xilinxwww.xilinx.com Spartan-6 FPGAs (XC6SLX25) for signal digitization and data processing with 12 bit resolution.", "Each FADC receives signals from three preamplifiers.", "The signal cables from different regions of the drift chambers are distributed between the FADCs in order to share out the processing load as evenly as possible.", "The FADC firmware is activated by a signal from the GlueX trigger.", "The firmware then computes the following quantities for pulses observed above a given threshold within a given time window: pulse number, arrival time, pulse height, pulse integral, pedestal level preceding the pulse, and a quality factor indicating the accuracy of the computed arrival time.", "Signal filtering and interpolation are used to obtain the arrival time to the nearest 0.8 ns.", "The firmware performs these calculations both for the CDC and FDC alike, and uses different readout modes to provide the data with the precision required by the separate detectors.", "For example, the CDC electronics read out only one pulse but require both pulse height and integral, while the FDC electronics read out up to four pulses and do not require a pulse integral.", "The FDC anode wires are read out using the JLab pipeline F1 TDC[47] with a nominal least count of 120 ps." ], [ "Gas system ", "Gas system Both the CDC and FDC operate with the same gases, argon and CO$_{2}$ .", "Since the relative mixture of the two gases is slightly different for the two tracking chambers, the gas system has two separate but identical mixing stations.", "There is one gas supply of argon and CO$_{2}$ for both mixing stations.", "A limiting opening in the supply lines provides over-pressure protection to the gas system, and filters in the gas lines provide protection against potential pollution of the gas from the supply.", "Both gases are mixed using mass flow controllers (MFCs) that can be configured to provide the desired mixing ratio of argon and CO$_{2}$ .", "MFCs and control electronics from BROOKS InstrumentsBROOKS Instruments, https://www.brooksinstrument.com/en/products/mass-flow-controllers.", "are used throughout.", "The mixed gas is filled into storage tanks, with one tank for the CDC and another for the FDC.", "The pressures are regulated by controlling the operation of the MFCs with a logic circuit based on an Allen-Bradley ControlLogix systemAllen-Bradley, https://ab.rockwellautomation.com/ that keeps the pressure in the tank between 10 and 12 psi.", "The tank serves both as a reservoir and a buffer.", "A safety relief valve on each tank provides additional protection against over-pressure.", "While the input pressure to the MFC is at 40 psi, the pressure after the MFC is designed to always be less than 14 psi above atmospheric pressure.", "After the mixing tank, a provision is built into the system to allow the gas to pass through an alcohol bath to add a small amount of alcohol gas to the gas mixture.", "This small admixture of alcohol protects the wire chambers from aging effects caused by radiation exposure from the beam.", "This part of the gas system is located above ground in a separate gas shed, before the gas mixture is transported to the experimental hall via polyethylene pipes.", "Additional MFCs in the hall allow the exact amount of gas provided to the chambers to be specified: one MFC for the CDC and another four MFCs for the individual FDC packages.", "The CDC is operated with a flow of 1.0 l/m, while each FDC package is operated with a flow of 0.1 l/m.", "To protect the chambers from over-pressure, there is a bypass line at the input to the detectors that is open to the atmosphere following a bubbler containing mineral oil.", "The height of the oil level determines the maximum possible gas pressure at the input to the chambers.", "There is a second bubbler at the output to protect against possible air back-flow into the chamber.", "The height of the oil above the exhaust line determines the operating pressure inside the chambers.", "Valves are mounted at many locations in the gas system to monitor various pressures with a single pressure sensor.", "The pressures of all six FDC chambers are monitored, as well as the CDC gas at the input, downstream gas plenum and the exhaust.", "A valve in the exhaust line can be used to divert some gas from the chamber to an oxygen sensor.", "Trace quantities of oxygen will reduce the gas gain and reduce tracking efficiency.", "The oxygen levels in the chamber are below 100 ppm." ], [ "Calibration, performance and monitoring ", "Time calibrations for the drift chambers are used to remove the time offset due to the electronics, so that after calibration the earliest possible arrival time of the pulse signals is at 0 ns.", "These offsets and the function parameters used to describe the relationship between the pulse arrival time and the closest distance between the track and the anode wire are obtained for each session of data taking.", "The CDC measures the energy loss, $dE/dx$ , of tracks over a wide range of polar angles, including recoiling target protons as well as more forward-going tracks.", "Gain calibrations are made to ensure that $dE/dx$ is consistent between tracking paths through different straws and stable over time.", "The procedure entails matching the position of the minimum ionizing peak for each of the 3522 straws, and then matching the $dE/dx$ at 1.5 GeV/c to the calculated value of 2.0  keV/cm.", "This takes place during the early stages of data analysis.", "Gain calibration for the individual wires is performed each time the HV is switched on and whenever any electronics modules are replaced.", "Gain calibration for the chamber as a whole is performed for each session of data taking; these sessions are limited to two hours as the gain is very sensitive to the atmospheric pressure.", "Position calibrations were necessary to describe the small deflection of the straw tubes midway along their length; these were performed in 2016 and repeated in 2017, with no significant difference found between the two sets of results.", "Position resolution from the CDC is of the order of 130 $\\mu $ m and its detection efficiency per straw is over 98% for tracks up to 4 mm from the CDC wire.", "The efficiency decreases as the distance between the track and the wire increases, but the close-packing arrangement of the straw tubes and the large number of straws traversed by each track compensate for this.", "For the FDC system, an internal per-chamber calibration process is first performed to optimize the track position accuracy.", "In the FDC the avalanche created around the wire is seen in three projections: on the two cathodes and on the wires.", "The drift time information from the wires is used to reconstruct the hit position perpendicular to the wire.", "The strip charges from the two cathodes are used to reconstruct the avalanche position along the wire.", "The same strip information can be used to reconstruct the avalanche position perpendicular to the wire, which, due to the proximity of the avalanche to the wire, is practically the wire position, as illustrated in Fig REF .", "Figure: Wire (avalanche) positions reconstructed from the strip information on the two cathodes in one FDC chamber.", "Only one quarter of the chamber is shown in this figure.This strip information is used to align the strips on the two cathodes with respect to the wires.", "At the same time, the residuals of the reconstructed wire positions are an estimate of the strip resolution.", "The resolutions of the detector were reported earlier [44].", "The strip resolution along the wires, estimated from the wire position reconstruction, varies between 180 and 80 $\\mu $ m, depending on the total charge induced on the strips.", "The drift distance is reconstructed from the drift time with a resolution between 240 and 140 $\\mu $ m depending on the distance of the hit to the wire in the $0.5-4.5$  mm range.", "Position offsets and package rotations were determined for both drift chamber systems, first independently, and then together, using the alignment software MILLEPEDE[48] in a process described in [40] and in [49].", "Online monitoring software enables shift-takers to check that the number of channels recording data, the distribution of signal arrival times, and the $dE/dx$ distribution are as expected." ], [ "Performance of the charged-particle-tracking system ", "Performance of the charged-particle-tracking system" ], [ "Track reconstruction", "The first stage in track reconstruction is pattern recognition.", "Hits in adjacent layers in the FDC in each package are formed into track segments that are linked together with other segments in other packages to form FDC track candidates using a helical model for the track parameters.", "Hits in adjacent rings in the axial layers of the CDC are also associated into segments that are linked together with other segments in other axial layers and fitted with circles in the projection perpendicular to the beam line.", "Intersections between these circles and the stereo wires are found and a linear fit is performed to find a $z-$ position near the beamline and the tangent to the dip angle $\\lambda =\\pi /2-\\theta $ .", "These parameters, in addition to the circle fit parameters, form a CDC track candidate for each set of linked axial and stereo layers.", "Candidates that emerge from the target, and pass through both FDC and CDC in the $5^\\circ -20^\\circ $ range, are linked together.", "The second stage uses a Kalman filter [50], [51] to find the fitted track parameters {z,D,$\\phi $ ,$\\tan \\lambda $ ,$q/p_T$ } at the position of closest approach of the track to the beam line.", "The track candidate parameters are used as an initial guess, where D is the signed distance of closest approach to the beam line.", "The Kalman filter proceeds in steps from the hits farthest from the beam line toward the beam line.", "Energy loss and multiple scattering are taken into account at each step along the way, according to a map of the magnetic field within the bore of the solenoid magnet.", "For the initial pass of the filter, the drift time information from the wires is not used.", "Each particle is assumed to be a pion, except for low momentum track candidates ($p<0.8$  GeV/$c$ ), for which the fits are performed with a proton hypothesis.", "The third stage matches each fitted track from the second stage to either the Start Counter, the Time-of-Flight scintillators, the Barrel Calorimeter, or the Forward Calorimeter to determine a start time t0 so that the drift time to each wire associated with the track could be used in the fit.", "Each track is refitted with the drift information, separately for each value of mass for particles in the set {$e^\\pm ,\\pi ^\\pm ,K^\\pm ,p^\\pm $ }." ], [ "Momentum and vertex resolution", "The momentum resolution as a function of angle and magnitude for pions and protons is shown in Fig.", "REF .", "The angular resolution is shown in Fig.", "REF .", "Figure: (Left) Momentum resolution for π - \\pi ^- tracks.", "(Right) Momentum resolution for proton tracks.Figure: (Left) Polar angle resolution for π - \\pi ^- tracks.", "(Right) Azimuthal angle resolution for π - \\pi ^- tracks.The resolutions are plotted as a function of the polar angle, θ\\theta .The thin windows of the cryogenic target and the exit window of the target vacuum chamber provide a means to estimate the vertex resolution of the tracking system.", "Pairs of tracks from empty target measurements are used to reconstruct these windows as illustrated in Fig.", "REF .", "The distance of closest approach between two tracks, $d$ , was required to be less than 1 cm.", "The vertex position is at the mid-point of the line segment (of length $d$ ) defined by the points of closest approach for each track.", "The estimated $z$ -position resolution is 3 mm.", "Figure: Reconstructed vertex positions within 1 cm radialdistance with respect to the beam line for an empty target measurement.", "The curve shows the result of a fit to the vertex distribution used to determine the vertexresolution." ], [ "Electromagnetic calorimeters ", "Electromagnetic calorimeters" ], [ "Barrel Calorimeter ", "Barrel Calorimeter The Barrel Calorimeter (BCAL) is an electromagnetic sampling calorimeter in the shape of an open cylinder.", "Photon showers with energies between 0.05 GeV and several GeV, $11^{\\circ }$ –$126^{\\circ }$ in polar angle, and $0^{\\circ }$ –$360^{\\circ }$ in azimuthal angle are detected.", "The geometry is fairly unique with the production target located in the backward part of the cylinder, as shown in Fig.", "REF .", "The containment of showers depends on the angle of photon incidence, with a thickness of $15.3$ radiation lengths for particles entering normal to the calorimeter face and reaching up to 67 radiation lengths at $14^{\\circ }$ .", "Details of the design, construction and performance of the BCAL can be found in Ref.[52].", "The BCAL is constructed as a lead and scintillating-fiber matrix, consisting of 0.5 mm-thick corrugated lead sheets and 1.0 mm-diameter Kuraray SCSF-78MJ multi-clad scintillating fibers.", "The fibers run parallel to the cylindrical axis of the detector.", "Each module has approximately 185 layers and 15,000 fibers.", "The BCAL consists of 48 optically isolated modules, each with a trapezoidal cross section, forming a 3.9-m-long cylindrical shell having inner and outer radii of 65 cm and 90 cm, respectively.", "The light generated in the fibers is collected via small light guides at each end of the module, which transport the light to silicon photomultipliers (SiPMs), which were chosen due to their insensitivity to magnetic fields.", "The end of the calorimeter with light guides, light sensors and electronics is shown in Fig.", "REF .", "Figure: Three-dimensional rendition of the light guides mounted at the end of theBCAL, as well as the readout assemblies mounted over them.", "Thereadout assemblies contain theSiPMs and their electronics.", "(Color online)The SiPM light sensors are Hamamatsu S12045(X) Multi-Pixel-Photon Counter (MPPC) arrays Hamamatsu Corporation, Bridgewater, NJ 08807, USA (http://sales.hamamatsu.com/en/home.php).", ", which are $4\\times 4$ arrays of $3\\times 3$ mm$^2$ tiles [53].", "The SiPMs were accepted following extensive testing [54], [55], [56], [57], [58], [59].", "Four thousand units were purchased and 3840 are installed in the detector.", "The gain of the SiPM depends on the voltage above the breakdown voltage, about 70 V. These are operated at 1.4 V over the breakdown voltage, selected to reduce the effect of readout thresholds.", "Even at this relatively high overbias, the noise level is dominated by fluctuations in the electronics baseline and not by single-pixel noise.", "In order to keep a constant gain, the temperature is maintained within practical limits ($\\pm $ 2$^\\circ $ C) using a chilled-water system.", "The gain is stabilized using a custom circuit that adjusts the bias voltage based on the measured temperature.", "Two stages of preamplifiers and summing electronics are attached to the sensors.", "In order to reduce the number of signals that are digitized, circuits sum the outputs of the preamplifiers in groups of radial columns, with coarser granularity away from the target.", "The layer closest to the target employs a single SiPM, and the next three layers have two, three, and four SiPMs, respectively.", "On the end of each module, forty SiPMs generate sixteen signals that are delivered to FADCs and twelve signals that are discriminated and then recorded with pipeline TDCs.", "The FADCs and TDCs are housed in VXS crates located on the floor close to the detector (see Section ).", "Figure: Expanded view of a single FCAL module." ], [ "Forward Calorimeter ", "The Forward Calorimeter (FCAL) detects photon showers with energies ranging from 0.1 GeV to several GeV, and between $1^{\\circ }$ –$11^{\\circ }$ in polar angle.", "The front face of the FCAL is located 5.6 m downstream from the center of the GlueX target and consists of 2800 lead glass blocks stacked in a circular array that has a diameter of 2.4 m. Each lead glass block has transverse dimensions of $4\\times 4$ cm$^2$ and length of 45 cm.", "The material of the lead-glass blocks is equivalent to type F8 manufactured by the Lytkarino Optical Glass Factory.http://lzos.ru .", "The blocks and most of the PMTs were taken from the decommissioned experiments E852 at Brookhaven National Laboratory [60] and the RadPhi Experiment at JLab [61].", "To remove accumulated radiation damage, the glass was annealed by heat treatment prior to installation in GlueX.", "The detector is enclosed in a dark room.", "The light collection is accomplished via an Eljen EJ-560 optical interface “cookie” and a UVT acrylic cylindrical light guide glued to the PMT.", "The light guide recesses the magnetically sensitive photocathode of the PMT inside a dual layer of soft iron and mu-metal that attenuates the stray field of the GlueX solenoid ($\\lesssim $ 200 G).", "The sensors are FEU 84-3 PMTs with Cockcroft-Walton bases, each consuming 0.2 W. The design of the PMT base is similar to that noted in Ref.", "[62], and eliminates the need for a 2800-channel high-voltage power system.", "The bases communicate with a controller using the CAN protocol [63], with 100 bases on each of 28 CAN buses.", "The communication allows continuous monitoring of the PMT voltages, temperatures, and current draw.", "A schematic of a single FCAL module is shown in Fig.", "REF and more details may be found in Ref. [64].", "FCAL signals are routed to FADC electronics, situated on a platform, directly behind the FCAL dark room." ], [ "Electronics ", "Custom readout electronics for the two calorimeters are mounted in standard VXS crates and include JLab 12-bit 250 MHz FADCs [65], discriminators [66] and F1 TDCs [47].", "The maximum input scale of the FADCs (4095 counts) is set to 2 V. The FADCs sample each calorimeter channel every 4 ns and generate raw waveforms consisting of 100 samples (400 ns).", "The samples are available for further processing by the firmware upon a trigger signal, if the waveform exceeds a threshold voltage.", "The firmware computes several derived quantities of the pulse: pedestal, peak value, integral over a selected window, and time of the halfway point on the leading edge.", "At most one pulse is extracted from each readout window.", "These pulse features constitute the raw data that is nominally read out from the FADC.", "Optionally, the full waveforms can be read out for diagnostic purposes and to check the firmware output against the offline emulation of the parameter extraction; this is done for less than about 1% of the production runs.", "Pulses are identified by the first sample that exceeds a threshold, currently set to 5 (8) counts above the average pedestal for the BCAL (FCAL).", "These thresholds correspond to approximately 2.5 (12) MeV.", "The integral is determined using a fixed number of samples relative to the threshold crossing, which was determined by maximizing the ratio of signal to pedestal noise.", "The integration window begins one sample before the threshold time and extends to 26 (15) samples after the threshold time for the BCAL (FCAL).", "Typical pedestal widths are $\\sigma \\sim $ 1.2-1.3 (0.8) counts.", "For the BCAL, the pedestals are determined for each channel event-by-event, appropriately scaled, and then subtracted from the peak and integral to obtain signals proportional to the energy deposited in the calorimeter.", "For the FCAL, the average pedestal over a run period is determined offline for each channel and the pedestal contribution to the pulse integral is subtracted when the data are reconstructed.", "The algorithm that determines the time of the pulse is pulse-height independent and, therefore, time-walk correction is not required for the FADC times [67].", "The outputs of the three inner layers of the BCAL are also fanned out to leading-edge discriminators, which feed the JLab F1 TDCs.", "The discriminator thresholds are initially set to 35 mV and then adjusted channel by channel.", "The pulse times are recorded relative to the trigger in a 12-bit word.", "Multiple hits may be recorded per channel per event (up to eight), but are culled at a later time by comparison to FADC times.", "The nominal least count is configured to be 58 ps." ], [ "Calibration and monitoring ", "Calibration and monitoring The relative gains of the calorimeters are monitored using a modular LED-driver system [68].", "The control system is the same for both calorimeters, but the arrangement of LEDs is tailored to the respective detector geometries.", "In the BCAL, one LED is inserted into each light guide to monitor each individual SiPM and its partner at the far end of the module.", "Due to geometry, the illumination varies considerably from channel to channel.", "The average gain stability of the detector over a period of ten days is better than 1% and the fractional root-mean-square deviation of the mean for each SiPM during a single day from the average over the run period is typically less than 2%.", "For the FCAL, four acrylic panes were installed, each covering the upstream end of one quadrant of the FCAL.", "Each pane is illuminated by forty LEDs, ten violet, ten blue, and twenty green.", "In addition to monitoring the stability of the readout, the different colors are used to study the wavelength dependence of the transmission of light though the lead glass blocks.", "In particular, radiation damage to lead glass inhibits transmission at the blue end of the spectrum and tends to turn glass a brownish color [69].", "Throughout a several-month experiment, the response to the green LEDs was unchanged.", "However, the PMT response to violet LEDs degraded by about 10% in the blocks closest to the beam line, characteristic of radiation damage.", "Such damage is only evident in the first two layers of blocks surrounding the 12 cm$\\times $ 12 cm beam hole.", "This damage is likely confined to the upstream end of the block and does not significantly affect the response to particle showers in the body of the glass.", "The energy of a photon or lepton is obtained from the reconstructed electromagnetic shower.", "Here, a shower is reconstructed using an algorithm that finds a cluster by grouping signals close in time and space, called hits, that have been registered by individual detector elements.", "Details of the algorithms to obtain shower energies in the BCAL can be found in Ref.", "[52] and in Ref.", "[70] for the FCAL.", "The clustering in the FCAL requires that hits register within 15 ns of the primary hit, where the seed threshold is taken to be 35 MeV.", "Clusters with a single hit are discarded.", "In the event of overlapping showers, the hit energies are divided among the clusters in proportion to the partition predicted by a typical shower profile.", "Both detectors have sources of energy-dependent nonlinearities and empirical corrections are developed and applied to minimize the measured energy dependence of the measured $\\pi ^0$ mass." ], [ "Performance ", "The performance of the calorimeter is summarized by its ability to measure the energy, position and timing of electromagnetic showers.", "The energy resolution of each calorimeter was extracted from the measured $\\pi ^0$ and $\\eta $ mass distributions, yielding consistent results.", "To study the $\\eta $ mass resolution, events were selected using kinematic fits to $\\gamma p \\rightarrow p \\pi ^+ \\pi ^- \\gamma \\gamma $ , with $\\eta \\rightarrow \\gamma \\gamma $ and the photons having the same energies within 10%.", "The proton and pion tracks were used to determine the event vertex, needed to accurately reconstruct the two-photon invariant mass.", "This reaction provides a fairly clean sample of $\\eta $ 's with energy-symmetric photons recorded either both in the BCAL or both in the FCAL.", "The single-photon energy resolution was determined from Gaussian fits to the $\\eta $ invariant mass width, neglecting contributions from uncertainty in the opening angle.", "Monte Carlo simulation of $\\gamma p \\rightarrow p \\pi ^+ \\pi ^- \\eta $ events, with kinematics chosen to approximate the experimental distributions, were used to tune the MC resolution to match the data.", "The single-photon resolutions are shown in Fig.", "REF (a) for the BCAL and Fig.", "REF (b) for the FCAL as a function of the mean photon energy, both for data and simulation.", "A fit has been performed to the data for each calorimeter to estimate contributions to the width from stochastic and constant processes.", "The parameters in the fit are strongly correlated due to the limited range of energy available.For the BCAL these data constitute an average over many angles, resulting in a relatively large effective constant term that cannot be extrapolated to higher energy.", "See Ref.", "[52] Section 11 for details.", "The resolution of the position (Z) along the length of the BCAL ($\\sim $  2.5 cm) is computed from the timing resolution of the system, which was measured to be $\\sigma =150$  ps at 1 GeV.", "The transverse position resolution ($\\sigma $ ) obtained from simulation for 1 GeV showers in the FCAL is less than 1.1 cm.", "The performance of the calorimeters has been demonstrated in the reconstruction of neutral states including $\\pi ^0$ , $\\eta $ and $\\eta ^{\\prime }$ mesons for the first GlueX physics publications [71], [72].", "In addition, although the response of the calorimeters at high energy is still under evaluation, it has provided important electron-pion separation to identify the decays of $J/\\psi \\rightarrow e^+e^-$ [73] where electrons were recorded up to 8 GeV.", "Figure: The energy resolution, σ γ /E γ \\sigma _\\gamma /E_\\gamma , for single photons in the a) BCAL and b) FCAL calculated from the η\\eta mass distribution under the assumption that only the energy resolution contributes to its width.", "Solid black circles are data and open red squares are simulation.", "Fitted curves including the stochastic and constant terms are indicated.", "(Color online)" ], [ "Scintillation detectors ", "Scintillation detectors There are two scintillator-based detectors deployed in the GlueX spectrometer: a small barrel-shaped detector surrounding the target, referred to as the Start Counter (ST), and a two-plane hodoscope detector system in the forward direction, referred to as the Time-of-Flight (TOF) detector.", "Both detectors provide timing information.", "Charged-particle identification is derived from energy loss ($dE/dx$ ) in the ST and flight time from the TOF." ], [ "Start Counter ", "The ST, shown in Fig.", "REF , surrounds the target region and covers about 90% of the solid angle for particles originating from the center of the target.", "The ST is designed to operate at tagged photon beam intensities of up to $10^8$ photons per second in the coherent peak, and has a high degree of segmentation to limit the per-paddle rates.", "The time resolution must be sufficient to resolve the RF beam structure and identify the electron beam bunch from which the event originated (see Section REF ).", "The ST provides a timing signal that is relatively independent of particle type and trajectory (because of its proximity to the target) and can be used in the Level 1 trigger if necessary.", "The specific energy deposits $dE/dx$ in ST are used for charged-particle identification in combination with the flight-time from the TOF.", "Details of the design, construction and performance of the ST system can be found in Ref. [74].", "Figure: The GlueX Start Counter surrounding the liquid-hydrogentarget assembly.", "The incident beam travels from left to right down the centralaxis.The ST consists of 30 scintillator paddles arranged in a cylinder of radius 78 mm with a “nose” section that bends towards the beam line to a radius of 20 mm at the downstream end.", "EJ-200 scintillator from Eljen TechnologyEljen Technology, https://eljentechnology.com/products/plastic-scintillators.", "was selected for the ST paddles.", "EJ-200 has a decay time of 2.1 ns with a bulk attenuation length of 380 cm.", "Each scintillator paddle originated from stock 3 mm thick and 600 mm in length.", "The paddles were bent at Eljen to create the nose section, and then machined at McNeal Enterprises Inc.McNeal Enterprises Inc., http://www.mcnealplasticmachining.com to their final shape, including edges beveled at $6^\\circ $ to minimize loss of acceptance.", "The scintillator paddles are supported by a Rohacell closed-cell foam structure.", "The Rohacell is 11 mm thick and is rigidly attached to an aluminum support hub at the upstream end.", "The downstream support extends partially into the nose section.", "The cylindrical length of the Rohacell is further reinforced with three layers of carbon fiber, each layer being 650 $\\mu $ m thick.", "The assembly is made light-tight with a Tedlar wrapping, attached to a plastic collar at the upstream end.", "Silicon photomultiplier detectors are used as light sensors, as these are not affected by the magnetic field produced by the solenoid.", "The SiPMs were placed at the upstream end of each scintillator element with a 250 $\\mu $ m air gap.", "Each paddle is read out with an array of four SiPMs (Hamamatsu S109031-050P multi-pixel photon counters) whose signals are summed.", "The on-board electronics provides two signals per paddle, one delivered to an FADC, and the other to a 5$\\times $  amplifier that is sent to a discriminator and then to a TDC." ], [ "Time-of-flight counters ", "Time-of-Flight counters The TOF system delivers fast timing signals from charged particles passing through the detector, thereby providing information for particle identification.", "The TOF detector is a wall of scintillators located about 5.5 m downstream from the target, covering a polar angular region from 0.6$^{\\circ }$ to 13$^{\\circ }$ .", "The detector has two planes of scintillator paddles stacked in the horizontal and vertical direction.", "Most paddles are 252 cm long and 2.54 cm thick with a width of 6 cm.", "The scintillator material is EJ-200 from Eljen Technology.", "To allow the photon beam to pass through the central region, an aperture of 12$\\times $ 12 cm$^2$ is kept free of any detector material by using four shorter, single-PMT paddle detectors with a length of 120 cm around the beam hole in each detector plane.", "These paddles also have a width of 6 cm and a thickness of 2.54 cm.", "In order to keep the count rate of the paddles well below 2 MHz the two innermost full-length paddles closest to the beam hole on either side have a reduced width of 3 cm.", "Light guides built out of UV transmitting plastic provide the coupling between the scintillator and the PMT and allow the magnetic shielding to protect the photocathode by extending about 5 cm past the PMT entrance window.", "All paddles are wrapped with a layer of a highly reflective material (DF2000MA from 3M) followed by a layer of strong black Tedlar film for light tightness.", "The scintillator paddles are read out using PMTs from Hamamatsu.Hamamatsu Photonics, https://www.hamamatsu.com/us/en/index.html.", "Full-length paddles have a PMT at both ends, while the short paddles have a single PMT at the outer end of the detector.", "These 2\" H10534 tubes have ten stages and are complete assemblies with high voltage base, casing and $\\mu $ -metal shielding.", "Additional soft-iron external shielding protects each PMT from significant stray fields from the solenoid magnet." ], [ "Electronics ", "High voltage for the TOF PMTs is provided by CAEN HV modules of type A1535SN, initially controlled by a CAEN SY1527 main frame and later upgraded to a SY4527.", "The PMT outputs are connected to a passive splitter by a 55'-long RG-58 coaxial cables.", "The signal is split into two equal-amplitude signals.", "One signal is directly connected to a FADC [75], while the second signal passes first through a leading-edge discriminator and is then used as an input to a high resolution TDC.", "The digitizing modules are mounted in VXS crates as described in Section .", "The threshold of the leading-edge discriminator is controlled separately for each channel and has an intrinsic deadtime of about 25 ns.", "The sparsification threshold for the FADC is set to 120 (160) counts for the ST (TOF), with the nominal pedestal set at 100 counts.", "The high voltage of each TOF PMT is adjusted to generate a signal amplitude of at least 400 ADC counts above baseline from a minimum-ionizing particle.", "The data from the FADC are provided by the FPGA algorithm and consist of two words per channel with information about pedestal, signal amplitude, signal integral, and timing.", "The timing signals from the ST system are registered using the JLab F1 TDCs, which have a nominal least count of 58 ps.", "In order to take advantage of the higher intrinsic resolution of the TOF counters, this system uses the VX1290A TDCs from CAENCAEN, https://www.caen.it/, which are multi-hit high-resolution TDCs with a buffer of up to 8 words per channel and a nominal least count of 25 ps.", "Since these TDCs provide the best time measurements in the GlueX detector, the timing of the accelerator RF signal is also digitized using these TDCs." ], [ "Calibration and monitoring ", "The combined ST and TOF systems are used to determine the flight times of particles, the ST providing a precise start time in combination with the accelerator RF, and the TOF providing the stop time.", "Both systems may also be used to provide information on particle energy loss.", "Therefore, the signals in ST and TOF must be calibrated to determine corrections for the effects of time-walk, light propagation time offsets, and light attenuation.", "The procedures are slightly different for the two detectors because of the different geometries, intrinsic resolutions, and the advantages of the TOF system having two adjacent perpendicular planes.", "For the time-walk correction for each paddle of the ST, the detector signal is sent to both an FADC and a TDC.", "The time from the FADC, being independent of pulse amplitude, is the reference.", "The amplitude dependence of the difference between TDC and FDC times is used to measure the time walk; the resulting curve is fit to an empirical function for use in the correction.", "The propagation time is measured as a function of the hit position in a paddle as determined by well-reconstructed charged particle tracks.", "The propagation velocity is measured in three regions of the counter (“straight,” “bend,” and “nose”) and is not assumed to be a single value for all hits.", "The light attenuation is also measured at several positions along the counter using charged particle tracks.", "The energy-per-unit pathlength in the paddle as a function of distance from the SiPM is fit to a modified exponential, with different parameters allowed for the straight section and the nose section, with continuity enforced at the section boundary.", "The calibration procedures for the TOF system take advantage of the two planes of narrow paddles oriented orthogonal to each other, which permits calibration of the full TOF detector independently of any other external detector information.", "The overlap region of two full-length paddles from the two planes defines a 6$\\times $ 6 cm$^2$ area for most paddles, with a few 3$\\times $ 3 cm$^2$ areas close to the beam hole.", "The separation between the two detector planes is minimal as they are mounted adjacent to each other, separated only by wrapping material.", "While the time-difference (TD) between the two ends of a paddle is related to the hit position along the paddle, the mean-time (MT) is related to the flight time of a particle from the vertex to the paddle.", "Therefore, the MT for two overlapping paddles must be the same when hit by the same particle passing through both paddles, while the hit positions in the horizontal and vertical dimensions are defined by the TD of the two paddles.", "This relationship results in an internally consistent calibration of all paddles with respect to every other paddle.", "Prior to finding timing offsets for calibration, all times are corrected for the amplitude-dependent walk.", "The relation between time at threshold and signal amplitude is parameterized and used to correct for time slewing.", "After all full-length paddles have been calibrated, they can be used themselves as references to calibrate the remaining eight short paddles that only have single-ended readout.", "Again we use the fact that any overlap region of two paddles from different planes has the same particle flight time from the vertex.", "This coincidence produces peaks in the time difference distributions that can be used to determine the timing offsets of these single-ended readout paddles.", "To test the calibration, we take tracks that are incident on a paddle in one plane and compute the time difference between the MT of that paddle and the MT of every other full-length paddle in the other plane.", "The resulting distribution of these differences is shown in Fig.", "REF .", "Assuming that all paddles have the same timing resolution, we can compute the average time resolution to be $\\sigma $ = 105 ps$=\\frac{148}{\\sqrt{2}}$  ps, assuming a Gaussian distribution.", "Figure: Mean time difference between one TOF long paddle of one plane with all other long paddlesof the other plane.", "(Color online)" ], [ "Performance ", "The purpose of the ST is to select the electron beam bunch that generated the tagged photon which induced a reaction in the target.", "The corresponding time derived from a signal from the CEBAF accelerator, which is synchronized with the RF time structure of the machine, is used to determine the event start time.", "Therefore, the ST resolution does not contribute to the resolution of the flight time as long as the resolution is sufficient to pick out the correct beam bunch with high probability.", "The ST timing performance can be determined by comparing the event time at the target measured by the start counter and the accelerator RF time.", "The start counter time must be corrected for the flight path of the charged particle emerging from the event, and all instrumental corrections mentioned in the previous section must be applied.", "Fig.", "REF shows the distribution of this time difference.", "The average time resolution is about $\\sigma $ =234 ps, where the resolution varies depending on the position of the hit along the counter.", "Figure: Time difference distribution between the vertex time computed from the start counter and the accelerator RF.", "The time from the RF does not contribute significantly to the width of the distribution.", "The fit function is a double Gaussian plus a third-degree polynomial.The ST is also used to identify particles using $dE/dx$ .", "Fig.", "REF shows $dE/dx$ versus momentum, $p$ , for charged particles tracked to the Start Counter.", "Protons can be separated from pions up to $p=0.9$  GeV/$c$ .", "Figure: dE/dxdE/dx vs. pp for the Start Counter.", "The curved bandcorresponds to protons while the horizontal band corresponds toelectrons, pions, and kaons.", "Pion/proton separation is achievablefor tracks with p<0.9p < 0.9 GeV/cc.The performance of the TOF detector for particle identification (PID) was investigated by considering the relative number of particle types within the event sample.", "Events with at least three fully-reconstructed positively-charged tracks were selected, with at least one of these tracks intersecting the TOF detector.", "More pions are expected than protons, and more protons than kaons.", "Looking at the distribution of velocity, $\\beta $ , of these tracks as a function of momentum, the bands from protons, kaons and pions are identified (see Fig.", "REF ).", "The distributions of $\\beta $ at two specific track momenta, 2 GeV/c and 4 GeV/c (see Fig.", "REF ), are illustrative of the PID capability of the TOF detector.", "At $p=2$  GeV/c, the TOF detector provides about a 4$\\sigma $ separation between the pion/positron peak and the kaon peak, sufficient to identify tracks as kaons with $\\beta =0.97$ , or lower, with very high certainty.", "However, at $\\beta =0.98$ , the probability of the track being a kaon is less than 50%, due to the abundance of pions that is an order of magnitude larger than kaons.", "The protons, on the other hand, are very well separated from the other particle types and can be identified with high confidence over the full range in $\\beta $ .", "At a track momentum of 4 GeV/c, PID becomes much more difficult and represents the limit at which the time-of-flight measurement can identify protons with high confidence.", "The separation between the large peak containing pions, kaons and positrons from the proton peak is about 4$\\sigma $ , while the relative abundance in this case is about a factor of 4.", "As a consequence, a 4 GeV/c momentum track with $\\beta =0.975$ is most likely a proton, with a small probability of being a pion.", "At $\\beta =0.98$ , such a track has a similar probability for being a proton or a pion.", "Figure: β\\beta of positive tracks versus track momentum, showing bands for e + e^+, π + \\pi ^+, K + K^+ and pp for the TOF detector.", "The color coding of the third dimensionis in logarithmic scale.", "(Color online)Figure: β\\beta of positive tracks with 2 GeV/c momentum (left) and with 4 GeV/c (right)." ], [ "Trigger ", "Trigger The goal of the GlueX trigger is to accept most high-energy hadronic interactions while reducing the background rate induced by electromagnetic and low-energy hadronic interactions to the level acceptable by the data acquisition system (DAQ).", "The main trigger algorithm is based on measurements of energy depositions in the FCAL and BCAL as described in Ref.", "[76], [77].", "Supplementary triggers can also use hits from scintillator detectors, such as the PS, tagging detectors, ST, TOF, and TAC." ], [ "Architecture ", "The GlueX trigger system [78] is implemented on dedicated programmable pipelined electronics modules, designed at JLab using Field-Programmable Gate Arrays (FPGAs).", "The GlueX trigger and readout electronics are hosted in VXS (ANSI/VITA 41.0) crates.", "VXS is an extension of the VME/VME64x architecture, which uses high-speed backplane lines to transmit trigger information.", "A layout of the trigger system is presented in Fig.", "REF .", "Data from the FCAL and BCAL are sent to FADC modules [75], situated in 12 and 8 VXS crates, respectively, and are digitized at the sampling rate of 250 MHz.", "The digitized amplitudes are used for the trigger and are also stored in the FPGA-based pipeline for subsequent readout via VME.", "Digitized amplitudes are summed for all 16 FADC250 channels in each 4 ns sampling interval and are transmitted to the crate trigger processor (CTP) module, which sums up amplitudes from all FADC boards in the crate.", "The sub-system processor (SSP) modules located in the global trigger crate receive amplitudes from all crates and compute the total energy deposited in the FCAL and BCAL.", "The global trigger processor (GTP) module collects data from the SSPs and makes a trigger decision based on the encoded trigger equations.", "The core of the trigger system is the trigger supervisor (TS) module, which receives the trigger information from the GTP and distributes triggers to the electronics modules in all readout crates in order to initiate the data readout.", "The GlueX system has 55 VXS crates in total (26 with FADC250s, 14 with FADC125s, 14 with F1 TDCs, and 1 CAEN TDC).", "The TS also provides a synchronization of all crates and provides a 250 MHz clock signal.", "The triggers and clock are distributed through the trigger distribution (TD) module in the trigger distribution crate.", "The signals are received by the trigger interface (TI) module and signal distribution (SD) module in each crate.", "The GlueX trigger system provides a fixed latency.", "The longest trigger distribution time of about 3.3 $\\mu $ s is due to the distance of the tagger hall from Hall D. The smallest rewritable readout buffer, where hits from the detector are stored, corresponds to about 3.7 $\\mu $ s for the F1 TDC module.", "The trigger jitter does not exceed 4 ns.", "Figure: Schematic view of the Level-1 trigger system of the GlueX experiment.", "The electronics boards are described in the text." ], [ "Trigger types ", "The GlueX experiment uses two main trigger types: the pair spectrometer trigger, and the physics trigger based on energy depositions in the BCAL and FCAL.", "The pair spectrometer trigger is used to measure the flux of beam photons.", "This trigger requires a time coincidence of hits in the two arms of the PS detector, described in Section REF .", "The physics triggers are generated when the FCAL and BCAL energies satisfy the following conditions: $2\\times E_{\\rm FCAL} + E_{\\rm BCAL} > 1\\;{\\rm GeV}, E_{\\rm FCAL} > 0\\; {\\rm GeV}$ , and $E_{\\rm BCAL} > 1.2\\;{\\rm GeV}$ .", "The first condition defines the main trigger that uses the fact that most events produce forward-going energy.", "The second trigger type is used to accept events with large transverse energy released in the BCAL, such as decays of $J/\\psi $ mesons.", "Several other trigger types were implemented for efficiency studies and detector calibration.", "Efficiency of the main production trigger was studied using a trigger based on the coincidence of hits from the ST and TAGH, detectors not used in the main production trigger.", "A combination of the PS and TAC triggers was used for the acceptance calibration of the PS, described in Section REF .", "Ancillary minimum-bias random trigger and calorimeter LED triggers were collected concurrently with data taking." ], [ "Performance ", "The rate of the main physics triggers as a function of the PS trigger rate is shown in Fig.", "REF .", "The typical rate of the PS trigger in spring 2018 was about 3 kHz, which corresponds to a photon beam flux of $2.5\\cdot 10^7\\; \\gamma /{\\rm sec}$ in the coherent peak range.", "The total trigger rate was about 40 kHz.", "The rates of the random trigger and each of the LED calorimeter triggers were set to 100 Hz and 10 Hz, respectively.", "The electronics and DAQ were running with a livetime close to $100 \\%$ , collecting data at a rate of 600 MB per second.", "The trigger system can operate at significantly higher rates, considered for the next phase of the GlueX experiment.", "The combined dead time of the trigger and DAQ systems at the trigger rate of 80 kHz was measured to be about $10 \\%$ .", "The largest contribution to the dead time comes from the hit processing time of readout electronics modules.", "Figure: Rates of the main production triggers as a function of the PS rate: FCAL and BCAL trigger (boxes), BCAL trigger (triangles), the total trigger rate (circles).", "The vertical arrow indicates the run conditions during the spring of 2018 with a diamondradiator, 5 mm collimator and 75 μ\\mu m Be converter." ], [ "Data acquisition ", "Data Acquisition The GlueX data acquisition software uses the CEBAF Online Data Acquisition (CODA) framework.", "CODA is a software toolkit of applications and libraries that allows customized data acquisition systems based on distributed commercial networks.", "A detailed description of CODA software and hardware can be found in Ref. [79].", "The maximum readout capability of the electronics in the VME/VXS crate is 200 MB/s per crate and the number of crates producing data is about 55.", "The data from the electronic modules are read via the VME back-plane (2eSST, parallel bus) by the crate readout controller (ROC), which is a single-board computer running Linux.", "The GlueX  network layout and data flow are shown in Fig.", "REF .", "Typical data rates from a single ROC are in the range of 20–70 MB/s, depending on the detector type and trigger rate.", "The ROC transfers data over 1 Gbit Ethernet links to Data Concentrators (DC) using buffers containing event fragments from 40 triggers at a time.", "Data Concentrators are programs that build partial events received from 10-12 crates and run on a dedicated computer node.", "The DC output traffic of 200-600 MB/s is routed to the Event Builder (EB) to build complete events.", "The Event Recorder (ER), which is typically running on the same node as an Event Builder, writes data to local data storage.", "GlueX has been collecting data at a rate of 500–900 MB/s, which allows the ER to write out to a single output stream.", "The system is expandable to handle higher luminosity where rates rise to 1.5–2.5 GB/s.", "In this case, the ER must write multi-stream data to several files in parallel.", "All DAQ computer nodes are connected to both a 40 Gb Ethernet switch and a 56 Gb Infiniband switch.", "The Ethernet network is used exclusively for DAQ purposes: receiving data from detectors, building events, and writing data to disk, while the Infiniband network is used to transfer events for online data quality monitoring.", "This allows decoupling DAQ and monitoring network traffic.", "The livetime of the DAQ is in the range of 92–100%.", "The deadtime arises from readout electronics and depends on the trigger rate.", "The DAQ software does not cause dead time during an experimental run, but software-related dead time appears while stopping and starting the run, which takes between 2-8 minutes.", "Figure: Schematic DAQ configuration for GlueX.", "The high-speed DAQ connections between the ROCs and the ER are contained within an isolated network.", "The logical data paths are indicated by arrows,although physically they are routed through the 40 Gbit ethernet switch.", "The online monitoring system uses its own separate 56 Infiniband switch.Figure: Top-level graphical interface for the beamline.", "This screen provides information on beam currents and rates, radiators, magnet status, target condition, background levels, etc." ], [ "Slow controls ", "Slow controls GlueX must monitor and control tens of thousands of different variables that define the state of the experimental hardware.", "The values need to be acquired, displayed, archived, and used as inputs to control loops continually with a high degree of reliability.", "For GlueX, approximately 90,000 variables are archived, and many more are monitored." ], [ "Architecture ", "The GlueX slow control system consists of three layers.", "The first layer consists of the remote units such as high voltage or low voltage power chassis, magnet power supplies, temperature controller, LabView applications, and PLC-based applications, which directly interact with the hardware and contain almost the all the control loops.", "The second layer is the Supervisory Control and Data Acquisition (SCADA) layer, which is implemented via approximately 140 EPICS Input/Output Controllers (IOCs).", "This layer provides the interface between low level applications and higher level applications via the EPICS ChannelAccess protocol.", "The highest level, referred as the Experiment Control System (ECS), contains applications such as Human-Machine Interfaces, the alarm system, and data archiving system.", "This structure allows for relatively simple and seamless addition and integration of new components into the overall controls system." ], [ "Remote Units ", "GlueX uses a variety of commercial units to provide control over the hardware used in the experiment.", "For instance, most detector high voltages are provided by the CAEN SYx527 voltage mainframe,https://www.caen.it/subfamilies/mainframes/ while the low and bias voltages are provided by boards residing in a Wiener MPOD chassishttp://www.wiener-d.com/sc/power-supplies/mpod–lvhv/mpod-crate.html.", "These two power supply types provide most voltages for detector elements with the exception of the Tagger Microscope and the Forward Calorimeter.", "Here custom systems were developed that provide voltage regulation and interact with the EPICS-based layer through higher level interfaces using custom protocols.", "See Sections REF and REF for more details.", "Various beam line devices need to be moved during beam operations.", "Stepper motors are used to move motorized stages via Newport XPS universal multi-axis motion controllershttps://www.newport.com/c/xps-universal-multi-axis-motion-controller.", "that allow for execution of complex trajectories involving multiple axes.", "All stage referencing, motion profile computations, and encoder-based closed-loop control occurs within the controller chassis after the basic parameters, such as positions and velocities, are provided by the user via a TCP/IP-based interface to EPICS.", "Custom controls were often developed for each complex installation, such as a superconducting magnet that requires large numbers of input and output channels and sophisticated logic.", "For these cases, we used Allen-Bradley CompactLogix and ControlLogix PLC systemshttps://ab.rockwellautomation.com..", "These systems are designed for industrial operations, allow modular design, provide high reliability, and require minimal maintenance.", "All controls loops are programmed within the PLC application, and are interfaced with EPICS through a TCP/IP-EtherNet/IP-proprietary protocol to allow access by higher level applications to process variables delivered by the PLCs.", "The cryogenic target and the superconducting solenoid employ National Instruments LabView applications.", "The target controls use both custom-made and vendor-supplied hardware that include built-in remotely-accessible control systems and an NI CompactRIOhttps://www.ni.com/en-us/shop/compactrio.html chassis.", "This chassis communicates with the hardware and serves variables using an internal ChannelAccess server and an EPICS IOC running on the CompactRIO controller, as described in Sec. .", "A National Instruments PXI high-performance systemhttps://www.ni.com/en-us/shop/pxi.html is used to collect data from different sensors of the solenoid as described in Sec.", "." ], [ "Supervisory Control and Data Acquisition layer ", "The SCADA layer is the middle layer that distributes the process variables allowing the higher level –and sometimes lower level– applications to use various process variables of the Hall-D control system.", "This layer is based on EPICS and uses the ChannelAccess protocol to publish the values of the variables over Ethernet.", "Efficient exchange of the information between the experiment and accelerator operations is achieved because the accelerator controls also use EPICS.", "Several dozen software IOC processes, running on host computers of the experiment control process, collect data from different components of the lowest layer.", "Each IOC is configured to communicate using the protocol appropriate for the remote units with which data exchange is needed.", "For instance, the IOC controlling the voltage for the FDC detector needs to be able to communicate with the Wiener MPOD and CAEN SYx527 voltage chassis.", "The middle layer is primarily used to distribute data between different applications.", "This layer also contains some EPICS-based applications running on IOCs that provide different control loops and software interlocks.", "For instance, the low-voltage power supplies for the FDC detector (see Sec.", "REF ) are shut off if the temperature or the flow of the coolant in the chiller falls outside of required limits." ], [ "Experiment Control System ", "The highest level of controls contains applications that archive data, display data in interactive GUIs and as stripcharts, alarm and notify shift personnel and experts when problems occur, and interface with the CODA-based data acquisition system (Sec. ).", "An example of such a GUI is the beamline overview screen, shown in Fig.", "REF .", "Many of the buttons of the GUI are active and allow access to other GUIs.", "Display management and the alarm system for GlueX controls are based on Controls System Studio (CSS),http://controlsystemstudio.org/ which is an Eclipse-based toolkit for operating large systems.", "CSS is well suited for systems that use EPICS as an integral component.", "Although CSS provides an archiving engine and stripcharting tools, the MYA archiver,[80] provided by the JLab accelerator software group, was employed with its tools for displaying the archived data as a time-series.", "Display management for GlueX controls is within the CSS BOY [81] environment, which allows system experts to build sophisticated control screens using standard widgets.", "The alarm system is based on the CSS BEAST[82] alarm handler software, which alerts shift personnel of problems with the detector, and notifies a system expert if the problems are not resolved by shift personnel." ], [ "Online computing system ", "Online computing system This section describes the GlueX software and computing systems used for data monitoring and for transport to the tape system for permanent storage." ], [ "Monitoring ", "The Online Monitoring system consists of multiple stages that provide immediate monitoring of the data, as well as near-term monitoring (a few hours after acquisition).", "Immediate monitoring is based on the RootSpy system[83] written for use in GlueX, though its design is not experiment specific.", "Figure REF shows a diagram of the processes involved in the RootSpy system and how those processes are coupled to the DAQ system.", "The Event Transfer (ET) process is part of the CODA DAQ system [84] and is used to extract a copy of a portion of the datastream without interfering with data acquisition.", "The monitoring system uses a secondary ET to minimize connections to the RAID server running the Event Recorder process.", "Figure: Processes distributed across several computers in the online monitoring system.", "DC, EB, and ER are the Data Concentrator, Event Builder, and Event Recorder processes, respectively, in the CODA DAQ system.The monitoring system is run on a small computer farmThe online monitoring farm consists of eight 2012 era Intel x86_64 computers with 16 cores+16 hyper-threads (ht) plus six 2016 era Intel x86_64 computers with 36 cores + 36ht.", "The monitoring farm uses 40 Gbps (QDR) and 56 Gbps(FDR) IB for the primary interconnect.", "Note that the DAQ system uses a separate 40 Gbps ethernet network that is independent of the farm., with each computer processing a small part of the data stream.", "In total, about 10% of the data is processed for the low level occupancy plots while roughly 2% is fully reconstructed for higher level analysis.", "The CODA ET software system is used to distribute the data among the farm computers.", "Each farm node generates histograms, which RootSpy gathers and combines before display to shift workers in a GUI.", "Plots are displayed via a set of ROOT [85] macros, each responsible for drawing a single page.", "Most macros divide the page into multiple sections so that multiple plots can be displayed on a single page.", "Figure REF shows an example of a high-level monitoring plot, where four invariant-mass distributions are shown with fits.", "Values extracted from the fits are printed on the plots for easy quantitative comparison to a reference plot.", "Figure: Invariant mass distributions showing π ∘ \\pi ^\\circ , ω\\omega , ρ\\rho , and φ\\phi particles.", "These plots were generated online in about 1hr 40min by looking at roughly 2% of the data stream.There are several client programs that summarize the information available in the histograms produced by RootSpy and generate output that make it easy to assess the uniformity and quality of the data.", "One of these is the RSTimeSeries program, which periodically inserts data into an InfluxDB time series database.", "The database provides a web-accessible strip chart of detector hit rates and reconstructed quantities (e.g.", "number of $\\rho $ 's per 1k triggers).", "Another is the RSArchiver program that gathers summed histograms to be displayed in the Plot Browserhttps://halldweb.jlab.org/data_monitoring/Plot_Browser.html.", "website.", "Plot Browser provides easy comparison of plots between different runs and between different analysis passes.", "Jobs are automatically submitted to the JLab farm for full reconstruction of the first five files (100GB) of each run.", "The results are displayed in Plot Browser and may be compared directly with the online analysis of the same run." ], [ "Data transport and storage ", "GlueX Phase I generated production data at rates up to 650MB/s.", "The data were temporarily stored on large RAID-6 disk arrays, and then copied to an LT0 tape system in the JLab Computer Center for long term storage.", "Two RAID servers, each with four partitions, were used for staging the data.", "The partition being written was rotated between runs to minimize head thrashing on disks by only reading partitions not currently being written.", "Partitions were kept at approximately 80% capacity and older files were deleted to maintain this level, allowing the monitoring farm easy access to files when the beam was down.", "A copy of the first three files ($\\sim 1.5\\%$ ) of each run was also kept on the online computers for direct access to samples from each run.", "The data volumes stored to tape are shown in Table REF in units of petabytes (PB).", "Entries marked “actual” are values taken from the tape storage system.", "The line marked “model” comes from the GlueX computing model[86].", "Table: GlueX data volumes by year.", "All values are in petabytes (PB).", "Most years include two run periods.", "The line marked “model” gives calculated rates from the GlueX Computing Model based on the detector luminosity.", "“Raw data only” represents data generated by the DAQ system (not including the backup copy).", "“Production” represents all derived data including reconstructed values and ROOT trees." ], [ "Event reconstruction ", "Event reconstruction GlueX uses the computer center batch farm at JLab to perform data monitoring, event reconstruction, and physics analyses.", "For data monitoring, detector hit occupancies, calibration and reconstruction quality, and experimental yields and resolutions, are analyzed for several physics channels.", "A subset of the data is monitored automatically as it is saved to tape.", "Every few weeks, monitoring processes are launched on a subset of the data to study improvements from ongoing calibrations and reconstruction software improvements.", "The histograms produced by these monitoring jobs are displayed on a website and ROOT files are available for download, enabling the collaborators to easily study the quality of the data.", "Every few months, a major reconstruction launch over all of the data is performed, linking hits in the various detector systems to reconstruct particles in physics events.", "Monitoring plots from these launches are also published to the web.", "Finally, regular analysis launches over the reconstructed data are performed for the reactions requested by users on a web form.", "The results of these launches are saved in reaction-specific ROOT TTrees for further analysis.", "For all launches, the reconstruction is run in a multi-threaded mode to make efficient use of the available computing resources.", "Fig.", "REF shows the multithreaded scaling from our monitoring launches.", "The program performs near the theoretical limit for jobs that use a number of threads that is less than or equal to the number of physical cores on the processor.", "By using hyperthreads, a smaller but still significant gain is achieved.", "All file outputs are written to a write-through cache system, which is ultimately backed up to tape.", "Figure: The scaling of program performance as a function of the number of processing threads.", "The computer used for this test consisted of 24 full cores (Intel x86_64) plus 24 hyperthreads.", "The orange squares are from running multiple processes, each with 12 threads.GlueX  Phase I has recorded 1400 separate physics-quality runs, with a total data footprint of about 3 petabytes.", "Data were saved in 19-GB files, with all runs consisting of multiple files (typically 100 or more per run).", "Fig.", "REF shows an overview of the different production steps for GlueX data, which are described in more detail in the following subsections.", "Figure: Production flowchart for GlueX data, illustrating analysis steps." ], [ "Calibration ", "During the acquisition of data, a unique run number is assigned to a period of data corresponding to less than about 2 hours of clock time, which may result in writing a couple hundred files.", "It is assumed that the detector changes very little during this period and therefore there will be no changes in the calibration constants.", "Two types of calibration procedures are used, depending on the complexity of the calibration procedures.", "Simple, well-understood calibrations such as timing alignment between individual channels and subdetectors or drift chamber gain and time-to-distance calibrations, can be performed with one file of data per run.", "These procedures are executed either in the online environment or on the batch farm, and can be repeated as needed following any improvements in reconstruction algorithms or other calibrations.", "More complicated calibration procedures, such as calorimeter gain calibration, require more data and are often iterative procedures, requiring several passes through the data.", "The raw data are processed upon arrival on the batch farm, resulting in histograms or in selected event data files in EVIO [87] or ROOT-tree format.", "Many of these outputs require that charged particle tracks are reconstructed.", "However, the computationally intensive nature of track reconstruction makes it a challenge to fully reconstruct all raw data immediately.", "Therefore, the full suite of calibration procedures is only applied to 10 - 20% of the data.", "Processing of the remaining data is mostly focused on separating out, or “skimming,” events collected by calibration triggers." ], [ "Monitoring ", "In Fig.", "REF the “FULL RAW DATA” box represents experimental data that have been backed up to tape.", "The box labeled “subset\" represents the first five files of each run, which are run through offline monitoring processes.", "These monitoring jobs are first processed during the run to check the quality of the data, but are also processed after major changes to calibrations or software to validate those changes.", "The resulting Reconstructed Events Storage (REST) files and ROOT histogram files are used for checking the detector and reconstruction performance." ], [ "Reconstruction ", "When the data have been sufficiently well calibrated, a full (production) pass of the reconstructed software on the physics quality data is performed.", "In the current total GlueX data set, about 1400 runs were deemed “physics quality.\"", "The remaining runs were short runs related to engineering and commissioning tests of the experiment.", "The 1400 physics quality runs include the majority of the data recorded during the running period, representing about 3 petabytes.", "All these files were reconstructed using computing resources at several sites, equivalent to more than 20 million core-hours combined.", "This produced more than 500 terabytes of REST data files.", "The large reduction in size from collected event data to physics data files (about a factor of six) permits faster and more efficient physics analyses of the data.", "During the REST production, a series of detector studies were performed that required access to raw data and that would not be possible on the reconstructed data alone.", "Many improvements to software and detector calibration resulted from these studies.", "Similar studies can be made with simulated data to match and assess the detector acceptance." ], [ "Offsite reconstruction", "Production processing of GlueX data uses offsite high-performance computing resources in addition to the onsite computing farm at JLab, specifically, the National Energy Research Supercomputing Center (NERSC) and the Pittsburgh Supercomputing Center (PSC).", "For NERSC, the total allocation used for the academic year 2018-2019 was 53M NERSC units, which was used to process 70.5k jobs.", "This is equivalent to approximately 9M core-hours on a Intel x86_64 processor.", "The jobs were run on NERSC's Cori II system, which is comprised of KNL (Knight's Landing) processors.", "The PSC allocation was awarded through the XSEDEhttps://www.xsede.org.", "allocation system in the last quarter of calendar year 2019 for 5.9 MSU.", "Only 0.85M SU were used in 2019 to run 7k jobs on the PSC Bridges system or about 10% of the number processed at NERSC.", "Figure REF shows how the event processing rates scaled with the number of processing threads for both NERSC and PSC.", "Jobs run at both of those sites were assigned entire nodes so the number of processing threads used was equal to the total number of hardware threads.", "Figure: Event processing rate versus number of threads for reconstruction jobs on NERSC Cori II (left) and PSC Bridges (right).", "The slope changes in the NERSC plot are due to the KNL architecture, which had four hardware threads per core.", "For PSC Bridges, hyper-threading is disabled and the plot shows a single slope.Container and distributed file system technologies were used for offsite processing.", "The software binaries as well as calibration constants, field maps, etc.", "were distributed using the CERN-VM-file system (CVMFS).", "The binaries were all built at JLab using a CentOS7 system.", "A very lightweight Docker container was made based on CentOS7 that had only a minimal number of system RPMsRedHat Package Management, https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/5/html/deployment_guide/ch-rpm installed.", "All other software, including third-party packages such as ROOT, were distributed via CVMFS.", "This meant changes to the container itself were very rare (about once per year).", "The Docker container was pulled into NERSC's Shifter system without modification.", "The same container was used to create a Singularity container used at both PSC and on the Open Science Grid (OSG) for simulation jobs.", "Raw data ware transferred from JLab to the remote sites using Globushttps://opensciencegrid.org/technology/policy/globus-toolkit., which uses GridFTP.", "The Globus tasks were submitted and managed by the SWIF2 workflow tool written by the JLab Scientific Computing group.", "SWIF2 was needed to manage the data retrieval from tape, for transfer to the remote site, for submission of remote jobs, and for transfer of processed data back to JLab.", "Disk space limitations at both JLab and the remote sites meant only a portion of the data set could be on disk at any one time.", "Thus, SWIF2 had to manage the jobs through all stages of data transfer and job submission." ], [ "Analysis ", "The full set of reconstructed (REST) data is too large to be easily handled by individual analyzers.", "For that reason, a system was developed to analyze data at JLab and extract reaction-specific ROOT trees.", "This step is represented by the right-hand green box at the bottom of Fig.", "REF .", "Users can specify individual reactions via a web interface.", "Periodically, the submitted reactions are downloaded into a configuration file, which steers the analysis launch.", "For each reaction, the GlueX analysis library inside the JANA framework creates possible particle combinations from the reconstructed particle tracks and showers saved in the REST format.", "Common selection criteria are applied for exclusivity and particle identification before performing a kinematic fit, using vertex and four-momentum constraints.", "Displaced vertices and inclusive reactions are also supported.", "Objects representing successful particle combinations (e.g.", "$\\pi ^0 \\rightarrow \\gamma \\gamma $ ) and other objects are managed in memory pools, and can be reused by different channels to reduce the overall memory footprint of the process.", "With this scheme, up to one hundred different reactions can be combined into one analysis launch processing the reconstructed data.", "If the kinematic fit converged for one combination of tracks and showers, the event is stored into a reaction-specific but generic ROOT tree, made accessible to the whole collaboration.", "The size of the resulting ROOT trees for the full data set strongly depends on the selected reaction, but is usually small enough to be copied to the user's home institution for a more detailed analysis." ], [ "Monte Carlo simulation ", "Monte Carlo The detailed simulation of events in the Hall-D beamline and GlueX detector is performed with a GEANT-based software package.", "The package was originally developed within the GEANT3 framework [88] and then migrated to the GEANT4 framework [89], [90].", "The simulation framework uses the same geometry definitions and magnetic field maps as used in reconstruction.", "The geometry includes the full photon beamline, starting at the radiator and ending at the photon beam dump.", "Both internal and external event generators are supported by the framework.", "Internal sources include the coherent bremsstrahlung source and the single particle gun.", "Events read from any number of external generators are also supported.", "These input events specify one or more primary vertices to be simulated, which are randomized within the hydrogen target with timing that matches the RF structure of the beam.", "The Monte Carlo data flow is presented in Fig.", "REF .", "Events of interest are generated using either an internal or user-supplied event generator.", "The input event specification is fed to the Hall D GEANT simulation code, either hdgeant or hdgeant4, which tracks the particles through the experimental setup and records the signals they produce in the active elements of the detector.", "Behavior of the simulation is conditioned by a run number, which corresponds to a particular set of experimental conditions: beam polarization and intensity, beamline and detector geometry, magnetic field maps, etc.", "All this information is read by the simulation at run-time from the calibrations database, which functions as the single source for all time-dependent geometry, magnetic field, and calibration data relevant to the simulation.", "Events written by the simulation are processed by the detector response package mcsmear.", "It applies corrections to the simulated hits to account for detector system inefficiencies and resolution, and overlays additional hits from uncorrelated background events.", "Loss of hits from detector channels, multi-hit truncation, and electronic deadtime are also applied at this step.", "Information needed for this processing comes from the databases for calibrations and run-conditions, and from files containing real backgrounds sampled using random triggers.", "Events emerging from the smearing step are deemed to be faithful representations of what the detector would have produced for the given run in response to the specified input.", "These Monte Carlo events are then processed with the same reconstruction software as used for the real events, and the output is saved to a REST file.", "These REST files are then made available for physics analysis.", "Figure: The Monte Carlo data flow from event generators through physics analysis REST files.", "The ovals represent databases containing tables indexed by run number, providing a common configuration for simulation, smearing, and reconstruction.", "Background events represented by the circle marked bg are real events collected using a random trigger, which are overlaid on the simulated events to account for pile-up in the Monte Carlo." ], [ "Geometry specification", "Geometry specification The geometry and material descriptions for the experiment are common across simulation and reconstruction, residing in a family of xml files that follow a common schema called the Hall D Detector Specification, or HDDS [91], [92].", "Run-specific variations of the geometry xml records are maintained in the calibration database.", "The geometry and magnetic field map are also maintained in the calibration database.", "The output events from the simulation are written as a data stream, which may either be piped directly into the next step of the Monte Carlo pipeline or saved to a file.", "Events are passed between all stages of the Monte Carlo processing pipeline, shown in Fig.", "REF , using the common data format of the Hall-D Data Model, HDDM [93].", "HDDM is used for all intermediate input and output event streams." ], [ "Event generators ", "Simulation starts with the generation of events, which can be specific particles or reactions, or simply unbiased background events.", "A common toolset has been developed to minimize redundancy.", "These tools include standard methods to generate the distributions of primary photon beam energies and polarization.", "An output interface is used to produce files suitable as input to the GEANT simulation.", "The photon beam energy distribution can be produced using a coherent bremsstrahlung generator that accounts for the physical properties of the radiator and the photon beamline.", "This generator allows the user to select the orientation of the diamond radiator, and then calculates the linear polarization for each photon.", "Photons can also be generated according to the spectrum measured in the pair spectrometer during any actual data run by interfacing to the calibration data base.", "Here the user inputs the degree of linear polarization and the orientation.", "Finally, the user can provide a histogram of the photon energy spectrum and a second one of the degree of polarization to be used to generate the photon beam.", "One of the first generators was used to simulate the total photoproduction cross section.", "It is currently used to study backgrounds to physics reactions as well as develop analysis tools for extracting signals.", "This event generator, called bggen, is based on Pythia [94], and includes additions that describe the low-energy photoproduction cross sections.", "Other generators are tied to specific reactions, where the generator needs to describe the underlying physics." ], [ "HDGEANT ", "Both GEANT3 and GEANT4 versions are available for simulation of the experiment.", "Both versions have been tuned to reproduce the behavior of the experiment, but there are some differences arising from how the two versions decide when to stop tracking particles.", "In general, the simulation mimics the running conditions found across a range of runs, typically a large part of a single run period.", "The output from GEANT contains both hit times and energies deposited in detector volumes." ], [ "Detector response", "Detector response Converting time and energy deposits coming from GEANT into electronic detector responses that match the readout from the experiment is carried out by the detector response package mcsmear.", "The output of this digitization is identical to the real data with the exception that the so-called truth information about the data is retained to allow detailed performance studies.", "In addition to the digitization, at this stage the run-dependent efficiency effects are applied to the data, including both missing electronic channels and reduced efficiency of other channels.", "Additional smearing of some signals is also applied here to better match the performance of the Monte Carlo to data.", "The mcsmear package also folds measured backgrounds into the data stream.", "During regular data collection, random triggers are collected concurrently with data taking (see Section ).", "These are separated from the actual data and used to provide experimental background signals in the Monte Carlo, with rates based on the actual beam fluxes in the experiment." ], [ "Job submission ", "A large number of experimental conditions need to be matched in simulated data.", "The MCWrapper tool was developed to streamline the input specifications, implement consistency with corresponding data reconstruction, seamlessly access computer offsite resources, and produce Monte Carlo samples in proportion to the actual data taken.", "The goal is to model the differences between runs and provide a simulated data set, comparable to the real data.", "The primary system used for this phase is the Open Science Grid (OSG) in order to leverage resources in addition to the local JLab computing farm.", "Many automated checks are made to avoid flawed submission, and all aspects of the requests and jobs are monitored during running.", "Once completed, MCWrapper checks for expected output files to be returned as if the jobs were run on the JLab farm.", "If expected files are not found the system will automatically submit a replacement job.", "Once the jobs are verified completed and all data from the request have been properly moved, the user receives an automated email alerting them that their request has been fulfilled and providing the location where the user can access the event sample.", "Users are able to monitor and control their simulations via an online dashboard.", "The MCWrapper dashboard gives information about active projects and allows users (or administrators) to interact with their requests.", "Users may cancel, suspend, or declare projects complete.", "Detailed information is presented about the individual jobs, such as where the jobs are being run, basic usage statistics, and current status.", "This information gives individuals a near real-time look into the production of their Monte Carlo samples." ], [ "Detector performance ", "Detector performance The capability of the GlueX detector in reconstructing charged and neutral particles and assembling them into fully reconstructed events has been studied in data and simulation using several photoproduction reactions.", "The results of these studies are summarized in this section.", "Figure: Reconstructed mass distributions for the reaction γp→pπ 0 π ± (π ∓ )\\gamma p \\rightarrow p\\pi ^0\\pi ^{\\pm }(\\pi ^\\mp ) for a bin in φ\\phi .", "(Left) Distribution of the missing mass off the proton.", "(Right) Invariant mass distribution for the π + π - π 0 \\pi ^+\\pi ^-\\pi ^0 system.", "The blue curves show the resonant contributions, the blackcurve show the polynomial backgrounds, and the red curve shows the sum.", "(Color online)" ], [ "Charged-particle reconstruction efficiency ", "The track reconstruction efficiency was estimated by analyzing $\\gamma p \\rightarrow p \\omega $ , $\\omega \\rightarrow \\pi ^+\\pi ^-\\pi ^0$ events, where the proton, the $\\pi ^0$ , and one of the charged pions were used to predict the three-momentum of the other charged pion.", "Two methods were used to calculate this efficiency, $\\varepsilon =N_{found}/(N_{found}+N_{missing})$ .", "Events for which no track was reconstructed in the predicted region of phase space contributed to $N_{missing}$ , while events where the expected track was reconstructed contributed to $N_{found}$ .", "For the first method, the $\\omega $ yields for $N_{found}$ and $N_{missing}$ were estimated from the missing mass off the proton; for the second method, the invariant mass of the $\\pi ^+\\pi ^-\\pi ^0$ system was used to find $N_{found}$ .", "This analysis was performed for individual bins of track momentum, $\\theta $ , and $\\phi $ .", "Examples of mass histograms for a typical bin in $\\phi $ are shown in Fig.", "REF .", "The exercise was repeated for a sample of $\\omega $ Monte Carlo events.", "A comparison of the efficiency for pion reconstruction derived from the two methods for both Monte Carlo and experimental data is shown in Fig.", "REF .", "The efficiencies for Monte Carlo and experimental data agree to within 5%.", "While this reaction only allows the determination of track reconstruction efficiencies for $\\theta < 30^\\circ $ , this covers the majority of charged particles produced in GlueX due to its fixed-target geometry.", "Other reactions are being studied to determine the efficiency at larger angles.", "Figure: Tracking efficiency for π + \\pi ^+ tracks, determined by data and simulation using two methods.", "(Color online)" ], [ "Photon efficiency", "Photon-reconstruction efficiency has been studied using different methods for the FCAL and BCAL.", "In the FCAL, absolute photon reconstruction efficiencies have been determined using the “tag-and-probe” method with a sample of photons from the reaction $\\gamma p \\rightarrow \\omega p$ , $\\omega \\rightarrow \\pi ^+\\pi ^-\\pi ^0$ , $\\pi ^0 \\rightarrow \\gamma (\\gamma )$ , where one final photon is allowed but not required to be reconstructed.", "The yields with and without the reconstructed photon are determined using two methods.", "In the first method, the $\\omega $ yield is determined from the missing-mass spectrum, $M_X(\\gamma p \\rightarrow pX)$ , selecting on whether only one or both reconstructed photons are consistent with a final-state $\\pi ^0$ .", "In the second method, the count when both photons are found is determined from the $\\omega $ yield from the fully reconstructed invariant mass $M(\\pi ^+\\pi ^-\\gamma \\gamma )$ .", "If the photon is not reconstructed, the $\\omega $ yield is determined by a fit to the distribution of the missing mass off the proton.", "Both methods yield consistent results, with a reconstruction efficiency generally above 90%, and within 5% or less agree with the efficiencies determined from simulation.", "Figure: Photon reconstruction efficiency in FCAL determined from γp→ωp\\gamma p \\rightarrow \\omega p, ω→π + π - π 0 \\omega \\rightarrow \\pi ^+\\pi ^-\\pi ^0, π 0 →γ(γ)\\pi ^0 \\rightarrow \\gamma (\\gamma ) as a function of (left) photon energy and (right) photon polar angle.", "Good agreement between data and simulation is observed in the fiducial region θ=2 ∘ -10.6 ∘ \\theta = 2^\\circ - 10.6^\\circ .", "(Color online)A relative photon efficiency determination has been performed using $\\pi ^0\\rightarrow \\gamma \\gamma $ decays, which spans the full angular range detected in GlueX.", "A sample of fully reconstructed $\\gamma p \\rightarrow \\pi ^+\\pi ^-\\pi ^0 p$ events were inspected, taking advantage of the $\\pi ^0\\rightarrow \\gamma \\gamma $ decay isotropy in the center-of-mass frame.", "Thus, any anisotropy indicates an inefficiency in the detector.", "Results from this analysis are illustrated in Fig.", "REF .", "Generally, this relative efficiency is above 90%, and agrees within 5% of that determined from simulation.", "The models for the simulated response of both calorimeters are being updated, and the final agreement between photon efficiency determined in data and simulation is expected to improve.", "Figure: Ratios of relative photon reconstruction efficiency between data and simulation determined from π 0 →γγ\\pi ^0\\rightarrow \\gamma \\gamma decays in γp→π + π - π 0 p\\gamma p \\rightarrow \\pi ^+\\pi ^-\\pi ^0 p events.", "The efficiency ratios are shown for the cases where (left) both photons were measured in the BCAL, (middle) both photons were measured in the FCAL, and (right) one photon was measured in the BCAL and the other in the FCAL.Detailed studies of detector performance determined the standard fiducial region for most analyses to be $\\theta = 2^\\circ - 10.6^\\circ $ and $\\theta > 11.3^\\circ $ .", "These requirements avoid the region dominated by beam-related backgrounds at small $\\theta $ and the transition region between the BCAL and FCAL, where shower reconstruction is difficult." ], [ "Kinematic fitting ", "Kinematic fitting is a powerful tool to improve the resolution of measured data and to distinguish between different reactions.", "In GlueX, this method takes advantage of the fact that the initial state is very well known, with the target proton at rest, and the incident photon energy measured with very high precision ($<0.1\\%$ ).", "This knowledge of the initial state gives substantial improvements in the kinematic quantities determined for exclusive reactions.", "The most common kinematic fits that are performed are those that impose energy-momentum conservation between the initial and final-state particles.", "Additional optional constraints in these fits are for the four-momenta of the daughters of an intermediate particle to add up to a fixed invariant mass, and for all the particles to come from a common vertex (or multiple vertices, in the case of reactions containing long-lived, decaying particles).", "To illustrate the performance of the kinematic fit, we use a sample of $\\gamma p \\rightarrow \\eta p$ , $\\eta \\rightarrow \\pi ^+\\pi ^-\\pi ^0$ events selected using a combination of standard particle identification and simple kinematic selections.", "The use of the kinematic fit improves the $\\eta $ -mass resolution from 2.6 MeV to 1.7 MeV, which is typical of low-multiplicity meson production reactions.", "The quality of the kinematic fit is determined using either the probability calculated from the $\\chi ^2$ of the fit and the number of degrees-of-freedom or the $\\chi ^2$ of the fit itself.", "The distributions of the kinematic fit $\\chi ^2$ and probability are illustrated in Fig.", "REF for both reconstructed and simulated data.", "The agreement between the two distributions is good for small $\\chi ^2$ (large probability), and flat over most of the probability range, indicating good overall performance for most signal events.", "The disagreement between the two distributions at larger $\\chi ^2$ (probability $<0.2$ ) is due to a combination of background events and deficiencies in the modelling of poorly measured events with large resolution.", "The performance of the reconstruction algorithms and kinematic fit can be studied through investigating the “pull” distributions, where the pull of a variable $x$ is defined by comparing its measured values and uncertainties and those resulting from the kinematic fit as $\\text{pull}_x = \\frac{x_\\text{fitted} - x_\\text{measured}}{\\sqrt{\\sigma _{x,\\text{measured}}^2 - \\sigma _{x,\\text{fitted}}^2}}.$ If the parameters and covariances of reconstructed particles are Gaussian, are measured accurately, and the fit is performing correctly, then these pull values are expected to have a Gaussian distribution centered at zero with a width $\\sigma $ of 1.", "If the pull distributions are not centered at zero, this is an indication that there is a bias in the measurements or the fit.", "If $\\sigma $ varies from unity, this is an indication that the covariance matrix elements are not correctly estimated.", "As an example, the pull distributions for the momentum components of the $\\pi ^-$ in reconstructed $\\gamma p \\rightarrow \\eta p$ , $\\eta \\rightarrow \\pi ^+\\pi ^-\\pi ^0$ events are shown in Fig.", "REF .", "Both real and simulated data have roughly Gaussian shapes with similar widths.", "More insight into the stability of the results of the kinematic fit can be found by studying the variation of the means and widths of the fit distributions as a function of the fit probability.", "The results of such a study are summarized in Fig.", "REF , where broad agreement between the results from real and simulated data is seen.", "The means of the pull distributions are generally around zero, except for $p_x$ with a mean of roughly $-0.1$ , and the widths within about 20% of unity.", "This level of performance and agreement between data and simulation is acceptable for the initial analysis of data, where very loose cuts on the kinematic fit $\\chi ^2$ are performed, and steady improvement in the modeling of the covariance matrices of reconstructed particles is expected to continue.", "Figure: Distribution of kinematic fit (left) probability and (right) χ 2 \\chi ^2 for reconstructed γp→ηp\\gamma p \\rightarrow \\eta p, η→π + π - π 0 \\eta \\rightarrow \\pi ^+\\pi ^-\\pi ^0 events in data and simulation.", "Both distributions agree reasonably for well-measured events, and diverge due to additional background in data and differences in modeling poorly-measured events.", "(Color online)Figure: Pull distributions for momentum components of the π - \\pi ^- from reconstructed γp→ηp\\gamma p \\rightarrow \\eta p, η→π + π - π 0 \\eta \\rightarrow \\pi ^+\\pi ^-\\pi ^0 events in data and simulation for events with fit probability >0.01>0.01: (left) p x p_x, (center) p y p_y, (right) p z p_z.", "(Color online)Figure: Pull means (top) and sigmas (bottom) for the momentum components of each particle as a function of the minimum probability required of the fit from reconstructed γp→ηp\\gamma p \\rightarrow \\eta p, η→π + π - π 0 \\eta \\rightarrow \\pi ^+\\pi ^-\\pi ^0 events.", "(Color online)" ], [ "Invariant-mass resolution ", "The invariant-mass resolution for resonances depends on the momenta and angles of their decay products.", "This resolution has been studied using several different channels, which are illustrated in Figs.", "REF and REF .", "A typical meson production channel including both charged particles and photons, $\\omega \\rightarrow \\pi ^+\\pi ^-\\pi ^0$ from $\\gamma p \\rightarrow \\omega p$ , is shown in the left panel of Fig.", "REF .", "The distribution shows the strong peak due to $\\omega $ meson production.", "Other structures are also seen, such as peaks corresponding to the production of $\\eta $ and $\\phi $ mesons.", "The $\\omega $ peak resolution obtained is 26.1 MeV when using only the reconstructed particle 4-vectors, and improves to 16.4 MeV after a kinematic fit.", "The invariant-mass distribution of $\\pi ^+\\pi ^-$ from $\\gamma p \\rightarrow K_S K^+ \\pi ^- p$ , $K_S\\rightarrow \\pi ^+\\pi ^-$ exhibits the peak due to $K_S\\rightarrow \\pi ^+\\pi ^-$ decays (right panel of Fig.", "REF ).", "The $K_S$ peak resolution is 17.0 MeV using only the reconstructed charged particle 4-vectors, and improves to 8.6 MeV after a kinematic fit imposing energy and momentum conservation.", "The dependence of the $K_S\\rightarrow \\pi ^+\\pi ^-$ invariant-mass resolution as a function of $K_S$ momentum is shown in Fig.", "REF , both before and after an energy/momentum-constraint kinematic fit.", "The invariant mass of $\\Lambda ^0\\pi ^-$ from $\\gamma p \\rightarrow K^+ K^+ \\pi ^- \\pi ^- p$ is shown in the left panel of Fig.", "REF , illustrating the peak due to $\\Xi ^- \\rightarrow \\pi ^- \\Lambda ^0$ , $\\Lambda ^0 \\rightarrow p \\pi ^-$ .", "The $\\Xi ^-$ peak resolution obtained is 7.3 MeV when using only the reconstructed charged particle 4-vectors, and improves to 4.6 MeV after a kinematic fit imposing energy and momentum conservation and the additional constraint that the mass of the $p \\pi ^-$ pairs must be that of the $\\Lambda ^0$ mass.", "The $e^+e^-$ invariant mass distribution from kinematically fit $\\gamma p \\rightarrow e^+e^- p$ events is shown in the right panel of Fig.", "REF , illustrating the peak due to $J/\\psi \\rightarrow e^+e^-$ .", "The resolution of the peak is 13.7 MeV.", "Figure: (Left) π + π - π 0 \\pi ^+\\pi ^-\\pi ^0 invariant-mass distribution from γp→π + π - π 0 p\\gamma p \\rightarrow \\pi ^+\\pi ^-\\pi ^0 p (Right) π + π - \\pi ^+\\pi ^- invariant mass distribution from γp→K S K + π - p\\gamma p \\rightarrow K_S K^+ \\pi ^- p, K S →π + π - K_S\\rightarrow \\pi ^+\\pi ^-.", "(Color online)Figure: K S →π + π - K_S\\rightarrow \\pi ^+\\pi ^- invariant mass resolution for the events shown in Fig.", ", as a function of K S K_S momentum, both before and after a kinetic fit, which constrains energy and momentum conservation.", "(Color online)Figure: (Left) Λ 0 π - \\Lambda ^0\\pi ^- invariant mass distribution from γp→K + K + π - π - p\\gamma p \\rightarrow K^+ K^+ \\pi ^- \\pi ^- p. (Right) e + e - e^+e^- invariant mass distribution from kinematically fit γp→e + e - p\\gamma p \\rightarrow e^+e^- p events.", "(Color online)" ], [ "Particle identification ", "Particle identification in GlueX uses information from both energy loss in different detector systems and time-of-flight measurements.", "This information can be used for identification in several ways.", "The simplest method is to apply selections directly on the relevant PID variables.", "To include detector resolution information, one can create a $\\chi ^2$ variable comparing a measured value to the expected value for a particular hypothesis, that is $\\chi ^2(p) = \\left( \\frac{ X(\\mathrm {measured}) - X(\\mathrm {expected})_p}{\\sigma _X} \\right)^2$ where $X$ is the given PID variable, $p$ is the particle hypothesis, and $\\sigma _X$ is the resolution of this variable.", "Multiple PID variables can be combined into one probability, or a figure-of-merit.", "Standard, loose selections on time-of-flight and energy loss are sufficient for initial physics analyses, while the performance of more complicated selections is being actively studied.", "At sufficiently large $\\theta $ , the energy loss for charged particles in the central drift chamber $dE/dx$ can be used.", "Fig.", "REF illustrates these distributions for positively charged particles, showing a clear separation of pions and protons in the momentum range $\\lesssim 1$  GeV.", "The $dE/dx$ resolution is approximately 27%, with the separation between the pion and proton bands dropping from about $8\\sigma $ at $p=0.5$  GeV/$c$ to about $2\\sigma $ at $p=1.0$  GeV/$c$ , with both bands fully merged by $p=1.5$  GeV/$c$ .", "Figure: CDC energy loss (dE/dxdE/dx) for positively charged particles that have at least 20 hits in the detector, as a function of measured particle momentum.", "The band corresponding to protons curves upwards, showing a larger energy loss than pions and other lighter particles at low momentum.", "The two bands show a clear separation for momenta ≲1\\lesssim 1 GeV.", "A faint kaon band can be seen between them.The primary means of particle identification is through time-of-flight measurements, and information from several sources is combined to make the most accurate determination.", "The RF reference signal from the accelerator is used to define the time when each photon bunch enters the target.", "The reconstructed final-state particles are used to determine which photon bunch most likely generated the detected reaction, with the primary determination coming from the signals from the Start Counter associated with the charged particle tracks.", "The photon bunch determination has a resolution of $<10$  ps.", "Each charged particle is associated with additional timing information based on the hit in the highest resolution detector (for example the BCAL or TOF).", "The flight time to this measured hit $t_\\mathrm {meas}$ relative to the time of the photon bunch that generated the event $t_\\mathrm {RF}$ can be used to distinguish between particles of different mass.", "Two common variables that are used are the velocity ($\\beta $ ) determined using the measured time-of-flight and the momentum of the particle, and $\\Delta t_\\mathrm {RF}$ , the difference between the measured and RF times after they both have been extrapolated back to the center of the target, assuming some particle-mass hypothesis.", "An example of the separation between different particle types can be seen in Fig.", "REF .", "The loose selections used for initial analyses of this data placed on the $\\Delta t_\\mathrm {RF}$ distributions and the momentum dependence of the resolution of this variable in different detectors are shown in Fig.", "REF .", "Requiring reconstructed particles to have $\\Delta t_\\mathrm {RF} \\lesssim 1-2$  ns has been found to be sufficient for analyses of high-yield channels which are the focus of initial analysis.", "The study of the selections required for more demanding channels is ongoing.", "Figure: Resolution as a function of particle momentum for Δt RF \\Delta t_\\mathrm {RF} in various subdetectors: (left) BCAL, (center) FCAL, (right) TOF(Color online)Electrons are identified using the ratio of their energy loss in the electromagnetic calorimeters $E$ to the momentum reconstructed in the drift chambers $p$ .", "This $E/p$ ratio should be approximately unity for electrons and less for hadrons.", "The overall distributions of this variable are illustrated in Fig.", "REF .", "Other variables, such as the shape of the showers generated by the charged particles in the calorimeter, promise to provide additional information to separate electron and hadron showers.", "Figure: Electron identification in the calorimeters is performed using the E/pE/p variable, the ratio of the energy loss in the electromagnetic calorimeters (EE) to the momentum reconstructed in the drift chambers (pp).", "Left) This distribution was obtained using e ± e^{\\pm } showers reconstructed in the FCAL from the reaction γp→π 0 p\\gamma p \\rightarrow \\pi ^0 p, π 0 →e + e - γ\\pi ^0\\rightarrow e^+e^-\\gamma .", "Right) e ± e^{\\pm } showers reconstructed in the BCAL from photon conversions." ], [ "Summary and outlook ", "We have presented the design, construction, and performance, of the beamline and detector of the GlueX experiment in Hall D at Jefferson Lab during its first phase of operation.", "The experiment operated routinely at an incident photon flux of $2\\times 10^{7}$ photons/s in the coherent peak with an open trigger, taking data at 40 kHz, and recording 600 MB/s to tape with live time $>$ 95%.", "During this period the experiment accumulated 121.4 pb$^{-1}$ in the coherent peak and 319.4 pb$^{-1}$ total for $E_\\gamma >$ 8.1 GeV.", "Data were collected in two sets of orthogonal linear polarizations of the incident photons, with $\\sim $ 23% of the data in each of the four orientations.", "The remaining $\\sim $ 11% was collected with unpolarized photons.", "Approximately 270 billion triggers ($\\sim $ 3PB) were accumulated during this period, as shown in Fig.", "REF .", "Figure: Plot of integrated number of triggers versus the number of live days in 2017 and 2018.", "The legend provides the number of triggers for the four diamond orientations relative to the horizontal (0, 45, 90, 135 ∘ ^\\circ ) and the amorphous radiator.", "The trigger curves of the four diamond configurations fall on top of one another, as we attempted to match the amount of data taken for each configuration.", "(Color online)The operational characteristics of the charged and neutral particle detectors, trigger, DAQ, online and offline systems have been verified, and individual components performed as designed.", "The detector is able to reconstruct exclusive final states, reconstruction efficiencies have been determined, and Monte Carlo simulations compare well with experimental data.", "The infrastructure is in place to process our high volume of data both on the JLab computing farm as well on other offsite facilities, providing the ability to process the data in a timely fashion.", "Future running will include taking data at higher luminosity and with improved particle identification capability.", "The GlueX experiment has already implemented the necessary infrastructure to allow the experiment to operate at a flux of $5\\times 10^{7}$ photons/s in the coherent peak for the upcoming run periods and has added a new DIRC detectorFour “bar boxes\" from the BaBar DIRC[95] detector have been installed and tested.", "to extend particle identification of kaons to higher momenta." ], [ "Acknowledgments", "We gratefully acknowledge the outstanding efforts of technical support at all the collaborating institutions and the support groups at Jefferson Lab that completed the assembly, installation, and maintenance of the detector.", "We acknowledge the contributions of D. Bennett, M. Lara, A. Subedi and P. Smith to the construction and commissioning of the Forward Calorimeter.", "We thank E.C.", "Aschenauer, G. Young and all members of the JLab 12 GeV Project for guidance and direction during the design and construction phases of the project.", "This work was supported in part by the U.S. Department of Energy, the U.S. National Science Foundation, the Natural Sciences and Engineering Research Council of Canada (NSERC),the German Research Foundation, Forschungszentrum Jülich GmbH, GSI Helmholtzzentrum für Schwerionenforschung GmbH, the Russian Foundation for Basic Research, the UK Science and Technology Facilities Council, the Chilean Comisión Nacional de Investigación Científica y Tecnológica, the National Natural Science Foundation of China, and the China Scholarship Council.", "This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177." ] ]

[ [ "Measuring the eccentricity of GW170817 and GW190425" ], [ "Abstract Two binary neutron star mergers, GW170817 and GW190425, have been detected by Advanced LIGO and Virgo.", "These signals were detected by matched-filter searches that assume the star's orbit has circularized by the time their gravitational-wave emission is observable.", "This suggests that their eccentricity is low, but full parameter estimation of their eccentricity has not yet been performed.", "We use gravitational-wave observations to measure the eccentricity of GW170817 and GW190425.", "We find that the eccentricity at a gravitational-wave frequency of 10 Hz is $e \\leq 0.024$ and $e \\leq 0.048$ for GW170817 and GW190425, respectively (90% confidence).", "This is consistent with the binaries being formed in the field, as such systems are expected to have circularized to $e \\leq 10^{-4}$ by the time they reach the LIGO-Virgo band.", "Our constraint is a factor of two smaller that an estimate based on GW170817 being detected by searches that neglect eccentricity.", "However, we caution that we find significant prior dependence in our limits, suggesting that there is limited information in the signals.", "We note that other techniques used to constrain binary neutron star eccentricity without full parameter estimation may miss degeneracies in the waveform, and that for future signals it will be important to perform full parameter estimation with accurate waveform templates." ], [ "Introduction", "The Advanced LIGO and Virgo observatories have detected two binary neutron star mergers, GW170817 [5] and GW190425 [11].", "To date, 17 double neutron star systems have been observed through radio surveys of the Milky Way field [54], [78], [25], [75], [50].", "Observations of binary neutron stars allow us to determine their formation channels [73], [26], [63], [64], [45], [46], [19], [78], [61], [18], [82], [40], [52], [13], constrain the neutron-matter equation of state [17], [14], [36], [34], [7], [29], [79], [58], [65], [33], [38], and test the strong-field regime of general relativity [8].", "Although the eccentricity of double neutron stars in the Milky Way field ranges from $0.06$ to $0.828$ [86], [13], field binaries will circularize to eccentricity $e \\le 10^{-4}$ [62], [46], making them detectable by matched-filter searches that neglect eccentricity [53], [32], [23], [42].", "GW170817 and GW190425 were detected by searches that neglect eccentricity [5], [11], suggesting that their eccentricity is $e \\lesssim 0.05$ [42], however no full parameter estimation of their eccentricity of their eccentricity has been performed.", "The eccentricity of binary black hole observations [1], [2], [3], [4], [6], [10] has been explored in [69] and [84].", "[70] place a limit on the eccentricity of GW190425 by estimating the effect of eccentricity on the measured parameters of the signal.", "Here, we directly measure the eccentricity of GW170817 and GW190425 using Bayesian parameter estimation [20].", "We use the observations from the Gravitational-Wave Open Science Center [5], [11], waveform templates that include eccentricity [57], and Markov Chain Monte Carlo parameter estimation [39], [20] to measure the eccentricity of the GW170817 and GW190425 when they have a gravitational-wave frequency of 10 Hz.", "We find that the eccentricity of GW170817 is $e \\le 0.024$ and GW190425 is $e \\le 0.048$ at 90% confidence for a uniform prior on $e$ .", "Our limit on eccentricity of GW170817 is a factor of two smaller than the limit estimated by its detection with circular waveform templates.", "We note that when using a common prior on eccentricity, our limit on the eccentricity of GW190425 is a factor of three greater than the limit of [70].", "This is due to a degeneracy between the chirp mass and eccentricity that is not included in the analysis of [70].", "However, this difference does not invalidate their conclusions about the formation of GW190425.", "When considering the two priors we use in our analysis, the limit on the eccentricity of GW190425 changes by a factor of two, suggesting that the eccentricity-constraining information in the signal is limited.", "Dynamical interations may form binary neutron stars with residual eccentricity, although the rate of such mergers is expected to be small in current detectors [48], [85] and a search for eccentric binary neutron stars in the O1 and O2 observing runs did not yield any candidates [59].", "However, since eccentricity is an interesting probe of binary formation channels and eccentric binaries may produce different electromagnetic emission than circular binary neutron stars [66], [30], it is important to accurately constrain the eccentricity of binary neutron stars as the number of observed mergers increases in the coming years." ], [ "Methods", "We measure the parameters of GW170817 and GW190425 using Bayseian inference [37], [71].", "We use gravitational-wave data from Advanced LIGO and Virgo [21], [9], $d(t)$ , and a model of the gravitational waves, $H$ , to calculate the posterior probability density function, $p(\\theta |d(t),H)$ , given by $p(\\theta |d(t),H) = \\frac{p(\\theta |H) p(d(t)|\\theta ,H)}{p(d(t)|H)},$ where $\\theta $ denotes the parameters of the gravitational waveform, $p(\\theta |H)$ , is the prior distribution on the signal parameters, and $p(d(t)|\\theta ,H)$ , is the probability of observing the data, known as the likelihood.", "The likelihood models the noise in the detector as a Gaussian and depends upon a noise-weighted inner product between the gravitational waveform and gravitational-wave data, $d(t)$ .", "Markov Chain Monte Carlo (MCMC) techniques can be used to marginalize over the parameters to obtain the posterior probabilities [31].", "Our implementation of Bayesian inferences uses the PyCBC Inference software package [20], [60] and the parallel-tempered emcee sampler, emcee_pt [39], [83].", "For GW170817 and GW190425, the MCMC is performed over the component masses of the binary, $m_{1,2}$ , the component spins aligned with the orbital angular momentum, $\\chi _{1,2}$ , the time of coalescence, $t_c$ , the polarization of the GW, $\\psi $ , the inclination angle, $\\iota $ , and the eccentricity, $e$ .", "Table: Prior distributions and GPS time intervals for GW170817 and GW190425.We assume a uniform prior distribution on the component masses, component spins, and coalescence time around the trigger shown in Table REF .", "We assume an isotropic sky location for GW190425 and a prior uniform in $\\sin \\iota $ for the inclination angle of both detections.", "We fix the sky location of GW170817 to the observed EM counterpart using a Gaussian prior distribution on the distance [27].", "We explore the prior distribution on the eccentricity by running the MCMC with two prior distributions: a prior that is uniform in $e$ and a prior uniform in $\\log e$ to compare with the GW190425 results found by [70].", "We use the GW strain data from the Advanced LIGO and Virgo detectors for GW170817 and GW190425, available through the LIGO Open Science Center (LOSC) [81].", "The LOSC_CLN_4_V1 data that we use for GW170817 includes post-processing noise-subtraction performed by the LIGO/Virgo Collaboration [21], [35].", "The T1700406_v3 data that we use for GW190425 includes pre-processing glitch removal performed by the LIGO/Virgo Collaboration specifically for use in parameter estimation [11], [9].", "We high-pass the data using an eighth-order Butterworth filter with an attenuation of 0.1 at 15 Hz.", "To conserve the phase of the delay, the filter is applied forward and backwards.", "A low-pass finite impulse response filter is applied to the data prior to resampling.", "The data is decimated to 2048 Hz for the analysis.", "For computing the likelihood, we use Welch's method to estimate the detector's noise power spectral density (PSD).", "Welch's method is used with 16 second Hanning windowed segments that are overlapped by 8 seconds.", "The PSD is shortened to 8 seconds in the time domain [12].", "The gravitational-wave data, $d(t)$ , used in the likelihood is taken from the intervals shown in Table REF .", "The gravitational-wave likelihood is evaluated from a low-frequency cutoff of 20 Hz to the Nyquist Frequency of 1024 Hz.", "A variety of waveforms are available that model eccentricity [43], [77], [57], [44], [28], [41], [80], [56].", "From what we know of binary neutron star mergers, we expect them to have low mass, spin, and eccentricity making TaylorF2Ecc a suitable waveform.", "The waveform model, H, is TaylorF2Ecc, a TaylorF2 post-Newtonian (pN) model with eccentric corrections.", "We use the LIGO Algorithm Library implementation [47] accurate to 3.5 pN order in orbital phase [24], 3.5 pN order in the spin-orbit interactions [22], 2.0 pN order in spin-spin, quadrupole-monopole, and self-interactions of individual spins [55], [15], and 3.0 pN order in eccentricity [57].", "Since TaylorF2Ecc follows TaylorF2 in its construction, the waveform will terminate at twice the orbital frequency of a particle at the innermost stable circular orbit of a Schwarzschild black hole.", "As a check on our analysis, we estimate the parameters of GW170817 and GW190425 using two available waveforms: the TaylorF2Ecc waveform at $e=0$ and the TaylorF2 waveform.", "Our analyses are consistent with each other and with the parameters estimated by Advanced LIGO and Virgo [5], [11]." ], [ "Results", "We first constrain the level of the eccentricity by using the TaylorF2Ecc waveform and a prior uniform in $e$ .", "We find that the 90% credible intervals at 10 Hz for GW170817 and GW190425 are $e = 0.012^{+0.013}_{-0.012}$ and $e = 0.025^{+0.022}_{-0.025}$ respectively.", "A degeneracy between the chirp mass, $\\mathcal {M}$ , and eccentricity, $e$ and a small correlation between the effective spin, $\\chi _{\\mathrm {eff}}$ , and $e$ are shown in our posterior distributions in Figure REF and Figure REF .", "Since $\\mathcal {M}$ and $\\chi _{\\mathrm {eff}}$ are correlated [16], [72], this will create a small correlation between $e$ and $\\chi _{\\mathrm {eff}}$ .", "Figure: Posterior probability distribution of GW170817 at 10 Hz.", "The analysis used a prior uniform in ee.", "Each parameter is quoted with a median value (solid red line) and a 90% credible interval (dashed red lines).", "The chirp mass ℳ\\mathcal {M} is given in the detector frame.", "Note the degeneracy between ℳ\\mathcal {M} and ee.Figure: Posterior probability distribution of GW190425 at 10 Hz.", "The analysis used a prior uniform in ee.", "Each parameter is quoted with a median value (solid red line) and a 90% credible interval (dashed red lines).", "The chirp mass ℳ\\mathcal {M} is given in the detector frame.", "Note the degeneracy between ℳ\\mathcal {M} and ee.", "[70] estimated the eccentricity of GW190425 to determine if the formation channel was due to unstable BB mass transfer.", "They estimate the eccentricity induced by the supernova kick in this formation scenario to be between $10^{-6}$ and $10^{-3}$ at 10 Hz.", "To measure the eccentricity of GW190425, [70] reweight the posterior samples from the parameter estimation performed using circular binaries to estimate the limit of the eccentricity using the same method used to estimate the eccentricity of binary black holes [69].", "They estimate the eccentricity of GW190425 at 10 Hz to be $e \\le 0.007$ (90% confidence) using a prior uniform in $\\log e$ .", "They find no evidence for or against unstable BB mass transfer as their analysis is not able to distinguish the small residual eccentricity expected from the investigated formation channel.", "To more directly compare our limit on GW190425's eccentricity, we repeat our analysis using a $\\log e$ prior.", "In Figure REF we can see the differences in the posterior distributions of each prior.", "With the $\\log e$ prior we estimate the eccentricity at 10 Hz to be $e \\le 0.023$ .", "This is a factor of three larger than interval estimated by [70].", "By re-weighting the posterior samples rather than a full MCMC, the degeneracy between $\\mathcal {M}$ and $e$ is missed.", "We find that by excluding posterior samples with lower values of $\\mathcal {M}$ , we can recover the upper limit reported by [70].", "Although our limit on the eccentricity is larger than that of [70], our result does not change their conclusion: indeed the strong dependence of the eccentricity posterior on the prior seen in Figure REF agrees with their conclusion that the signal-to-noise ratio of GW190425 is not large enough to explore the eccentricities expected in BB mass transfer.", "We would need to be able to determine the eccentricity at lower frequencies to distinguish the formation channel.", "Figure: Eccentricity posteriors of GW190425 (solid black line) plotted against their priors (dotted line) for two choices of prior: uniform in ee (left) and uniform in log 10 (e)\\log _{10}(e) (right).", "We quote the median (solid red line) and 90% credible interval (dashed red lines) for e in each posterior." ], [ "Conclusion", "Our analysis used the gravitational-wave observations as well as a prior on the eccentricity to constrain the eccentricity of GW170817 and GW190425.", "Our 90% confidence limit using a uniform prior on $e$ for GW170817, $e \\le 0.024$ , and GW190425, $e \\le 0.048$ , are consistent with expectations since they were found by a circular search [62].", "We have constrained the eccentricity to a factor of two smaller than estimates obtained from circular searches [23], [42].", "Our 90% credible intervals on the eccentricity of GW190425 are a approximately a factor of six larger than the interval estimated by [70], which used a prior uniform in $\\log e$ .", "This demonstrates the impact of prior choice, and the importance of measuring the eccentricity of signals using full parameter estimation to account for the correlation between parameters.", "Unfortunately, based on current merger rate estimates the detection of an eccentric binary neutron star merger will be difficult with current observatories [48], [85], [59], but gravitational-wave capture binaries that have $e \\ge 0.8$ and could form in the LIGO-Virgo band [68], [76].", "However, since the eccentricity of the detections is expected to be low and negligible, $e \\le 0.02$ , a circular search is effective in detecting them [23], [42].", "The detection of a binary neutron star mergers with high eccentricity or spin in future observing runs or with third-generation detectors [51], [67] will reveal more about the formation channel of eccentric binaries and the existence of a dynamical formation channel." ], [ "Data availability", "The data underlying this article are available in the associated data release on GitHub, at https://github.com/gwastro/bns-eccentric-pe  [49]." ], [ "Acknowledgements", "We acknowledge the Max Planck Gesellschaft for support and the Atlas cluster computing team at AEI Hannover.", "This research was supported in part by the National Science Foundation under Grant No. PHY-1748958.", "DAB thanks National Science Foundation Grant No.", "PHY-1707954 for support.", "AL thanks National Science Foundation Grant No.", "AST-1559694 for support.", "This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration.", "LIGO is funded by the U.S. National Science Foundation.", "Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes." ] ]

[ [ "Backwards semi-martingales into Burgers' turtulence" ], [ "Abstract In fluid dynamics governed by the one dimensional inviscid Burgers equation $\\partial_t u+u\\partial_x(u)=0$, the stirring is explained by the sticky particles model.", "A Markov process $([Z^1_t,Z^2_t],\\,t\\geq0)$ describes the motion of random turbulent intervals which evolve inside an other Markov process $([Z^3_t,Z^4_t],\\,t\\geq0)$, describing the motion of random clusters concerned with the turbulence.", "Then, the four velocity processes $(u(Z^i_t,t),\\,t\\geq0)$ are backward semi-martingales." ], [ "Introduction", "Burgers equation is a simplified version of Navier-Stocks equations in fluid mechanics.", "It is well known that the entropy solution of the one dimensional inviscid Burgers equation $\\partial _t u+u\\partial _x(u)=0$ can be interpreted as the velocity field of fluid particles which evolve following a sticky dynamics ([6], [7]).", "The aim of this paper is the study of fluid turbulence via particular trajectories $s\\mapsto y(s)$ of the fluid particles.", "In the literature, the sticky particles dynamics was introduced, at a discrete level, by Zeldovich [10] in order to explain the formation of large structures in the universe.", "That is a finite number of particles which move with constant velocities while they are not collided.", "All the shocks are inelastic following the conservation laws of mass and momentum.", "At a continuous level, the initial state of particles is given by the support of a non negative measure $\\mu _0$ .", "A particle starts from position $x$ with velocity $u_0(x)$ and mass $\\mu _0(\\lbrace x\\rbrace )$ .", "The particles move with constant velocities and masses while not collided.", "All the shocks are inelastic, following the conservation laws of mass and momentum.", "Now and in the rest of the paper, our purpose concerns only a motion on the real line.", "In their pioneering work, Sinai et al [11] made this construction when the particles are every where in $\\mathbb {R}$ , $u_0$ is continuous and the mass of any interval $[a,b]$ is computed with a positive density $f$ , i.e.", "$\\mu _0([a,b])=\\int _a^bf(x)\\mathrm {d}x$ .", "At time $t$ , a particle of position $x(t)$ has the mass $\\mu _t(\\lbrace x(t)\\rbrace )$ and the velocity $u_t(x(t))$ , the momentum of any interval $[a(t),b(t)]$ is $\\int _{a(t)}^{b(t)}u_t(x)\\mathrm {d}\\mu _t(x)$ .", "The authors then solved the so called pressure-less gas system $\\partial _t \\mu +\\partial _x (u\\mu )=0$ , $\\partial _t (u\\mu )+\\partial _x (u^2\\mu )=0$ .", "At the same time and independently, Brenier and Grenier [9] considered the case of particles confined in a in interval $[\\alpha ,\\beta ]$ , i.e.", "$\\mu _0([\\alpha ,\\beta ]^c)=0$ .", "First, they remarked that at a discrete level of $n$ particles, the cumulative distribution function $M_n(a,t):=\\mu _{n,t}(]-\\infty ,a])$ and the momentum $A_n(M_n(a,t)):=\\int _{-\\infty }^au_{n,t}(x)\\mathrm {d}\\mu _{n,t}(x)$ solve the so called scalar conservation law $\\partial _t M_n+$ $\\partial _x (A_n(M_{n}))=0$ .", "Then, letting $n\\rightarrow +\\infty $ , they got, at the continuous level, a limit $(M,A)$ solution of $\\partial _t M+\\partial _x (A(M))=0$ .", "As a consequence, the Lebesgue-Stieltjes measure $\\partial _x (A(M))$ is absolutely continuous w.r.t.", "$\\partial _x M=:\\mu _t$ , of Randon-Nicodym derivative a function $u_t$ , then $(\\mu , u)$ solves again the above pressureless gas system.", "In [1], [5], Dermoune and Moutsinga constructed the sticky particles dynamics with initial mass distribution $\\mu _0$ , any probability measure, and initial velocity function $u_0$ , any continuous and locally integrable function such that $u_0(x)=o(x)$ as $x\\rightarrow \\infty $ .", "The authors united and generalized previous works of [11], [9].", "Moreover the particles paths define a Markov process $t\\mapsto X_t$ solution of the ODE $\\mathrm {d}X_t=u(X_t,t)\\mathrm {d}t,$ and the velocity process $t\\mapsto u(X_t,t)$ is a backward martingale.", "In [5], [6], Moutsinga extended the construction when $\\mu _0$ is any non negative measure and $u_0$ has no positive jump.", "He gave the description of different kinds of clusters $[\\alpha (x,t),\\beta (x,t)]$ , i.e the set of all the initial particles $y(0)$ which have the same position $y(t)=x$ at time $t$ .", "The author showed in [6] that if $\\mu _0$ is the Lebesgue measure, then the velocity field $u$ is the entropy solution of the inviscid Burgers equation $\\partial _t u+u\\partial _x(u)=0$ .", "In [7], Moutsinga showed the same connection with Burgers equation when $u_0$ is non increasing and $\\mu _0$ is the Stieltjes measure $-\\mathrm {d}u_0$ .", "In this case, the mass of any interval $[a(t),b(t)]$ , at time $t$ , is $u_t(a(t))-u_t(b(t))$ and its momentum is $(u_t^2(a(t))-u_t^2(b(t)))/2$ .", "The same year and independantly of Moutsinga, Eyink and Drivas [3] considered $\\mu _0=\\lambda $ , the Lebesgue measure (on a compact subset and renormalized to be a probability) and a random variable $\\tau $ denoting the first shock time.", "They connected Burgers turbulence to a Markov process $s\\mapsto Y_s$ solution of (REF ).", "The authors also showed the anomalous dissipativity of $u$ along the turbulences' paths.", "In fact, we show in section $\\ref {section dpc}$ that this process is the path of some sticky particle.", "Hence, the process of [3] coincides with particular paths of the sticky particles process $X$ defined in [6].", "A very interesting result of [3] is that the process $s\\mapsto u(Y(s),s)$ is a backward martingale, under the assumption of uniform distribution of $\\tau $ .", "Unfortunately, as it is shown in section $\\ref {section turbulent traj}$ , this assumption leads to the entire coincidence (undistinguishability) of both the processes $Y$ and $X$ , so the martingale property of $s\\mapsto u(Y(s),s)$ is obvious, since it was already stated in [6].", "The construction of [6] also allows us in subsection REF , under more general assumptions than in [3], to show the anomalous dissipativity of the system governed by Burgers equation.", "Without the assumption of uniform distribution of $\\tau $ , the processes $Y$ and $X$ are in general distinguishable.", "We study this general case in section where we give the main results of this paper.", "The velocity function $u_0$ is not necessarily derivable nor even continuous as considered in [3], but it is allowed to have negative jumps.", "We show that the process $t\\mapsto u(Y_t,t)$ is no longer a backward martingale but a semi-martingale.", "Furthermore, we concentrate on the birth and evolution of turbulence.", "We define a turbulent interval as a set $[\\alpha ,\\beta ]$ of initial positions of sticky particles from which rise a turbulence.", "The motion $s\\mapsto [Z^1(s),Z^2(s)]$ of random turbulent interval is given by two backward Markov processes $Z^1$ and $Z^2$ solutions of (REF ).", "Moreover, the velocity processes $s\\mapsto u(Z^1(s),s), u(Z^2(s),s)$ are semi-martingales.", "First, we recall the definition and the main properties of the sticky particles model ([5], [6], [7])." ], [ "The sticky particle dynamics", "The definition of one dimensional sticky particle dynamics requires a mass distribution $\\mu $ , any Radon measure (a measure finite on compact subsets) and a velocity function $u$ , any real function such that the couple $(\\mu ,\\,u)$ satisfies the Negative Jump Condition (NJC) defined in $\\cite {Convex-hullsMoutsinga}$ .", "Precisely, consider the support $\\mathcal {S}=\\left\\lbrace \\,x\\in \\mathbb {R}\\,:\\, \\mu (x-\\varepsilon ,\\,x+\\varepsilon )>0,\\,\\forall \\varepsilon >0\\right\\rbrace $ of $\\mu $ and the subsets $\\mathcal {S}_{-}=\\left\\lbrace \\,x\\in \\mathbb {R}\\,:\\, \\mu (x-\\varepsilon ,\\,x)>0\\right\\rbrace $ , $\\mathcal {S}_{+}=\\left\\lbrace \\,x\\in \\mathbb {R}\\,:\\ \\mu (x,\\,x+\\varepsilon )>0,\\,\\forall \\varepsilon >0\\right\\rbrace $ .", "Suppose that $u$ is $\\mu $ locally integrable and consider the generalized limits $u^{-}$ , $u^{+}$ : $u^{-}(x)=\\limsup \\limits _{\\varepsilon \\rightarrow 0}\\dfrac{\\int _{[x-\\varepsilon ,\\,x)}u(\\eta )\\mu (d\\eta )}{\\mu [x-\\varepsilon ,\\,x)},\\quad \\forall \\,x\\in \\mathcal {S}_{-},\\\\u^{+}(x)=\\liminf \\limits _{\\varepsilon \\rightarrow 0}\\dfrac{\\int _{(x,\\,x+\\varepsilon ]}u(\\eta )\\mu (d\\eta )}{\\mu (x,\\,x+\\varepsilon ]},\\quad \\forall \\,x\\in \\mathcal {S}_{+}.$ The Negative Jump Condition requires that $u^{-}(x)\\ge u(x)\\,\\,\\forall \\,x\\in \\mathcal {S}_{-},\\quad u(x)\\ge u^{+}(x)\\,\\,\\forall \\,x\\in \\mathcal {S}_{+}.$ In the whole paper, we mainly use $\\mu _0=\\lambda $ , the Lebesgue measure.", "That's why we always suppose that the support $\\mathcal {S}=\\mathbb {R}$ .", "Considering particles of initial mass distribution $\\mu _{0}$ and of initial velocity function $u_{0}$ , their sticky dynamics is defined in $\\cite {Moutsinga-burgers-sticky1}$ , when the couple $(\\mu _0,\\,u_0)$ satisfies (REF ) and $x^{-1}u(x)\\rightarrow 0$ as $|x|\\rightarrow +\\infty $ .", "The dynamics is characterized by a forward flow $(x,s,t)\\mapsto \\phi _{s,t}(x)$ defined on $\\mathbb {R}\\times \\mathbb {R}_{+}\\times \\mathbb {R}_{+}$ .", "Proposition 2.1 (Forward flow) Suppose that $\\mathcal {S}=\\mathbb {R}$ .", "For all $x,s,t$ : $\\phi _{s,s}(x)=x$ and $\\phi _{s,t}(\\cdot )$ is non-decreasing and continuous.", "The value $\\phi _{s,t}(x)$ is the position after supplementary time $t$ of the particle which occupied the position $x$ at time $s$ .", "More precisely : $\\phi _{s,t}(\\phi _{0,s}(y))=\\phi _{0,s+t}(y)\\;,\\quad \\forall \\,y.$ If $\\phi ^{-1}_{0,t}(\\left\\lbrace x \\right\\rbrace )=:\\left[ \\alpha (x,0,t),\\,\\beta (x,0,t)\\right]$ with $\\alpha (x,0,t)<\\beta (x,0,t)$ , then $x=\\frac{\\int _{[ \\alpha (x,0,t),\\,\\beta (x,0,t)]}(a+tu_0(a))\\mathrm {d}\\mu _0(a)}{\\mu _0([ \\alpha (x,0,t),\\,\\beta (x,0,t)])}\\;.$ In any case : $x=\\alpha (x,0,t)+tu_0(\\alpha (x,0,t))=\\beta (x,0,t)+tu_0(\\beta (x,0,t))\\;.$ If $\\mu _0([\\alpha (x,0,t),\\,y])>0$ and $\\mu _0(]y,\\,\\beta (x,0,t)])>0$ , then $\\frac{\\int _{]y, \\beta (x,0,t)]}(a+tu_0(a))\\mathrm {d}\\mu _0(a)}{\\mu _0(]y, \\beta (x,0,t)])}\\le x\\le \\frac{\\int _{[ \\alpha (x,0,t),y]}(a+tu_0(a))\\mathrm {d}\\mu _0(a)}{\\mu _0([\\alpha (x,0,t),y])}\\;.$ If $s\\le t$ , then $\\phi _{0,s}(\\alpha (x,0,t))&=\\alpha (x,0,t)+su_0(\\alpha (x,0,t))\\;,\\\\\\phi _{0,s}(\\beta (x,0,t))&=\\beta (x,0,t)+su_0(\\beta (x,0,t))\\;.$ For any compact subset $K=[a,b]\\times [0,T]$ , consider $A_T=\\alpha (\\phi _{s,T}(a),s,T)$ , $B_T=\\beta (\\phi _{s,T}(b),s,T)$ and the probability $\\mu _s^K=\\frac{1\\hspace{-3.69885pt}\\mathrm {I}_{[A_T,B_T]}}{\\mu _s([A_T,B_T])}\\mu _s$ .", "The sticky particle dynamics induced by $(\\mu _s^K,u_s)$ , during time interval $[0,T]$ , is characterized by the restriction of the function $(y,t)\\mapsto \\phi _{s,t}(y)$ on $[A_T,B_T]\\times [0,T]$ .", "The latter means that the restriction of flow on a compact subset of space-time does not depend of the whole matter, but only on the restriction of the matter (distribution) on a compact subset of space states.", "Assertion 5 shows that the graphs $[0,t]\\ni s\\mapsto \\phi _{0,s}(\\alpha (x,0,t)),\\,\\phi _{0,s}(\\beta (x,0,t))$ draw a delta-shock, well known in the literature (Figure REF ).", "Figure: The blue line on the left (resp right) of the middle shok wave represent the trajectory of the particle which start at the position α(x,0,t)\\alpha (x,0,t) (resp β(x,0,t)\\beta (x,0,t)) which is the trajectory of [0,t]∋s↦φ 0,s (α(x,0,t))[0,t]\\ni s\\mapsto \\phi _{0,s}(\\alpha (x,0,t)) (resp [0,t]∋s↦φ 0,s (β(x,0,t))[0,t]\\ni s\\mapsto \\phi _{0,s}(\\beta (x,0,t)))What about the velocity?", "Proposition 2.2 (Flow derivative) Suppose that $\\mathcal {S}=\\mathbb {R}$ .", "For all $y,s$ , the function $t\\mapsto \\phi _{s,t}(y)$ has everywhere left hand derivatives and right hand derivatives.", "Now and after, the notation $\\dfrac{\\partial }{\\partial t}\\phi _{s,t}(y)$ stands for the right hand derivative.", "There exists a function $(x,t)\\mapsto u_t(x)$ such that for all $(y,t)$ : $\\dfrac{\\partial }{\\partial t}\\phi _{0,t}(y)=u_t(\\phi _{0,t}(y))\\;.$ For any compact subset $K=[a,b]\\times [0,T]$ , consider $A_T=\\alpha (\\phi _{s,T}(a),s,T)$ , $B_T=\\beta (\\phi _{s,T}(a),s,T)$ and the probability $\\mu _s^K=\\frac{1\\hspace{-3.69885pt}\\mathrm {I}_{[A_T,B_T]}}{\\mu _s([A_T,B_T])}\\mu _s$ .", "Using conditional expectation under $\\mu _s^K$ , we have $ \\forall \\,(y,t)\\in K,\\quad \\dfrac{\\partial }{\\partial t}\\phi _{s,t}(y)=\\mathbb {E}_{\\mu _s^K}\\left[u_{s}|\\phi _{s,t}(\\cdot )= \\phi _{s,t}(y)\\right].$ We call a cluster at time $t$ all interval of the type $\\left[ \\alpha (x,0,t),\\,\\beta (x,0,t)\\right]$ .", "The last assertion of proposition REF implies an important property on the velocity of a cluster.", "Corollary 2.3 Suppose that $\\mathcal {S}=\\mathbb {R}$ .", "Let $(x,t)\\in \\mathbb {R}\\times \\mathbb {R}_+$ .", "If $\\alpha (x,0,t)<\\beta (x,0,t)$ , then $u_t(x)=\\frac{\\int _{[ \\alpha (x,0,t),\\,\\beta (x,0,t)]}u_0(a)\\mathrm {d}\\mu _0(a)}{\\mu _0([ \\alpha (x,0,t),\\,\\beta (x,0,t)])}\\;.$ If $\\alpha (x,0,t)=\\beta (x,0,t)$ , then $u_t(x)=u_0(\\alpha (x,0,t))$ .", "$u_0(\\beta (x,0,t))\\le u_t(x)\\le u_0(\\alpha (x,0,t))$ .", "If $\\mu _0([\\alpha (x,0,t),\\,y])>0$ and $\\mu _0(]y,\\,\\beta (x,0,t)])>0$ , then $\\frac{\\int _{]y, \\beta (x,0,t)]}u_0(a)\\mathrm {d}\\mu _0(a)}{\\mu _0(]y, \\beta (x,0,t)])}\\le u_t(x)\\le \\frac{\\int _{[ \\alpha (x,0,t),y]}u_0(a)\\mathrm {d}\\mu _0(a)}{\\mu _0([\\alpha (x,0,t),y])}\\,.$ $u_t^-(x)=u_0(\\alpha (x,0,t))$ and $u_t^+(x)=u_0(\\beta (x,0,t))$ ).", "If $u_0(\\alpha (x,0,t))=u_t(x)$ or $u_0(\\beta (x,0,t))=u_t(x)$ , then $\\alpha (x,0,t)=\\beta (x,0,t)$ .", "For all $t\\ge 0$ , we have $u_t(x)=o(x)$ as $|x|\\rightarrow +\\infty $ .", "For all $t>0$ : $\\underset{\\underset{y<x}{y\\rightarrow x}}{\\lim }\\,u_t(y)&=u_t^-(x)=u_0(\\alpha (x,0,t))\\;,\\\\\\underset{\\underset{y>x}{y\\rightarrow x}}{\\lim }\\,u_t(y)&=u_t^+(x)=u_0(\\beta (x,0,t))\\;.$" ], [ "Markov and martingale properties", "Let $(\\mu _{0},\\,u_{0})$ be as in theorem $\\ref {forward flow}$ .", "On abstract measure space $(\\Omega ,\\,\\mathcal {F},\\,P)$ we define a measurable function $X_{0}\\,:\\;\\Omega \\longrightarrow \\mathbb {R}$ with image-measure $P\\circ X^{-1}_{0}=\\mu _{0}$ .", "In practice, $(\\Omega ,\\,\\mathcal {F},\\,P)=(\\mathbb {R},\\mathcal {B}(\\mathbb {R}),\\mu _{0})$ and $X_0$ is the identity function.", "For all $t\\ge 0$ , we set $X_{t}=\\phi _{0,t}(X_{0})$ .", "As a consequence of theorem $\\ref {forward flow}$ , we have the following : Proposition 2.4 (Markov and martingale property) $\\forall s,t$ , we have $X_{s+t}=\\phi _{s,t}(X_{s})$ If $u_{0}$ is $\\mu _0$ integrable, then under the measure $\\mu _0$ (or $P$ ) : $\\dfrac{\\mathrm {d}}{\\mathrm {d}t}X_{t}=\\mathbb {E}[u_{0}(X_0)|X_{t}]=u_{t}(X_{t}).$ For any compact $K=[a,b]\\times [0,t]$ , consider $A_t=\\alpha (\\phi _{0,t}(a),0,t)$ , $B_t=\\beta (\\phi _{0,t}(a),0,t)$ and the probability $\\mu _0^K=\\frac{1\\hspace{-3.69885pt}\\mathrm {I}_{[A_t,B_t]}}{\\mu _0([A_t,B_t])}\\mu _0$ .", "If $\\phi _{0,t}(a)\\le X_{t}\\le \\phi _{0,t}(b)$ , then then under the conditional probability $\\mu _0^K$ (or knowing $A_t\\le X_0\\le B_t$ ) : $\\dfrac{\\mathrm {d}}{\\mathrm {d}t}X_{t}=\\mathbb {E}_{\\mu _0^K}[u_{0}(X_0)|X_{t}]=u_{t}(X_{t}).$ If $u_{0}$ is $\\mu _0$ integrable, then under the measure $\\mu _0$ (or $P$ ) : $u_{t+s}(X_{t+s})=\\mathbb {E}[u_{t}(X_{t})|\\mathcal {F}_{t+s}]\\;,\\quad \\mbox{with}\\quad \\mathcal {F}_{t}=\\sigma (X_{u},u\\ge t).$ For any compact $K=[a,b]\\times [0,t+s]$ , consider $A_{t+s}=\\alpha (\\phi _{0,t+s}(a),0,t+s)$ , $B_{t+s}=\\beta (\\phi _{0,t+s}(a),0,t+s)$ and the probability $\\mu _0^K=\\frac{1\\hspace{-3.69885pt}\\mathrm {I}_{[A_{t+s},B_{t+s}]}}{\\mu _0([A_{t+s},B_{t+s}])}\\mu _0$ .", "If $\\phi _{0,t+s}(a)\\le X_{t+s}\\le \\phi _{0,t+s}(b)$ , then then under the conditional probability $\\mu _0^K$ (or knowing $A_{t+s}\\le X_0\\le B_{t+s}$ ) : $\\dfrac{\\mathrm {d}}{\\mathrm {d}t}X_{t+s}=\\mathbb {E}_{\\mu _0^K}[u_{s}(X_s)|\\mathcal {F}_{t+s}]=u_{t+s}(X_{t+s}).$ Remark 2.5 Contrary to the conjecture of [3] (page 411) the properties of proposition REF do not ensure $u$ to be the inviscid Burgers solution.", "We can indeed give two examples (Figure REF ) where the matter is initially confined in an interval $[-A,A]$ and $u_0(x)=\\left\\lbrace \\begin{array}{rll}1 & \\mbox{if} & -A\\le x\\le 0\\\\0 & \\mbox{if} & 0<x\\le A.\\end{array}\\right.$ Example 1 (dealing with Burgers equation) : $\\mu _{0}=\\lambda _{[-A,A]}$ (the Lebesgue measure on $[-A,A]$ ).", "We have a single discontinuity line (shock wave) $t\\mapsto t/2$ starting at position 0, with velocity $1/2$ .", "At time $t$ , the position $x=t/2$ is the one of the cluster $[-t/2,t/2]$ .", "Moreover, $u(x,t)=\\left\\lbrace \\begin{array}{rll}1 & \\mbox{if} & A+t\\le x<t/2\\\\1/2 & \\mbox{if} & x=t/2\\\\0 & \\mbox{if} & t/2<x\\le A.\\end{array}\\right.$ Example 2 (not dealing with Burgers equation) : $\\mu _{0}=2\\lambda _{[-A,0]}+\\lambda _{[0,A]}$ .", "We have a single discontinuity line (shock wave) $t\\mapsto (2-\\sqrt{2})t$ starting at position 0, with velocity $2-\\sqrt{2}$ .", "At time $t$ , the position $x=(2-\\sqrt{2})t$ is the one of the cluster $[(1-\\sqrt{2})t,\\,(2-\\sqrt{2})t]$ .", "Moreover, $u(x,t)=\\left\\lbrace \\begin{array}{rll}1 & \\mbox{if} & A+t\\le x<(2-\\sqrt{2})t\\\\(2-\\sqrt{2}) & \\mbox{if} & x=(2-\\sqrt{2})t\\\\0 & \\mbox{if} & (2-\\sqrt{2})t<x\\le A.\\end{array}\\right.$ Figure: The green line is the discontinuous line in a lagrangian interval [A,B][A,\\,B] of the function uu described at example 1.", "The red line is the discontinuous line in a Lagrangian interval [A,B][A,\\,B] of the function uu described at example 2.", "[α 1 ,β 1 ][\\alpha _{1},\\,\\beta _{1}] is the cluster which contain the point zero for the discontinuous line in the first example and [α 2 ,β 2 ][\\alpha _{2},\\,\\beta _{2}] is the cluster which contain the point zero for the discontinuous line in the second example.", "The two functions u(x,t)u(x,t) described in the example 1 and example 2 don't coincide in the region of the plan which is red, blue and green.", "This region of the plan is not negligible for the Lebesgue measure." ], [ "Link with Burgers equation", "The inviscid Burgers solution of initial data $u_{0}$ is connected to sticky particles in two well known cases ($\\cite {Moutsinga-burgers-sticky1,Moutsinga-burgers-sticky2}$ ) as follows : Proposition 2.6 When $\\mu _{0}=\\lambda $ (the Lebesgue measure) and $(\\mu _{0},\\,u_{0})$ satisfies the NJC, the function $u(\\cdot ,t)=u_{t}$ defined at the relation $(\\ref {velocity-process})$ is the the entropy solution of the inviscid Burgers equation with initial data $u_{0}$ .", "Furthermore the distribution of matter at time $t\\ge 0$ is given by the relation $\\mu _{t}=\\lambda \\circ X^{-1}_{t}=\\lambda -t\\partial _{x}u(\\cdot ,t)$ where $\\partial _{x}u(\\cdot ,t)$ is the Stieltjes measure over $u(\\cdot ,t)$ .", "When $\\mu _{0}=-\\mathrm {d}u_{0}$ (the Stieltjes measure over $u_{0}$ ) and $u_{0}$ is non-increasing, the function $u(\\cdot ,t)=u_{t}$ defined at the relation $(\\ref {velocity-process})$ is the the entropy solution of the inviscid Burgers equation with initial data $u_{0}$ .", "Furthermore the distribution of matter at time $t\\ge 0$ is given by the relation $\\mu _{t}=(-\\mathrm {d}u_{0})\\circ X^{-1}_{t}=-\\partial _{x}u(\\cdot ,t)$" ], [ "Dissipativity", "Proposition 2.7 Let $u$ be the velocity field of sticky particles and $\\mu $ be the distribution field.", "Let $\\psi $ be a convex function and $0\\le s<t$ .", "If $\\int |\\psi (u(x,s))|\\mathrm {d}\\mu _s(x)+\\int |\\psi (u(x,t))|\\mathrm {d}\\mu _t(x)<+\\infty $ , then $\\int \\psi (u(x,t))\\mathrm {d}\\mu _t(x)\\le \\int \\psi (u(x,s))\\mathrm {d}\\mu _s(x)\\;,$ which is equivalent to $\\int \\psi (u(\\psi _{0,t}(a),t)\\mathrm {d}\\mu _0(a)\\le \\int \\psi (u(\\psi _{0,s}(a),s)\\mathrm {d}\\mu _0(a)\\;.$ Let $u$ be the entropy solution of the inviscid Burgers equation.", "Consider $\\mu _0=\\lambda $ , the Lebesgue measure and suppose that $\\lambda $ -essentially, $u_0(x)\\rightarrow 0$ as $x\\rightarrow \\pm \\infty $ .", "If $\\int [|\\psi (u_0(x))|+|\\psi (u(x,t))|]\\mathrm {d}x<+\\infty $ , then $\\int \\psi (u(x,t))\\mathrm {d}x=\\int \\psi (u(x,t))\\mathrm {d}\\mu _t(x)\\le \\int \\psi (u_0(x))\\mathrm {d}x\\;.$ Proof.", "Let us use the probabilistic notations of subsection REF : $(X_t,P)=(\\psi _{0,t},\\mathrm {d}\\mu _0)$ .", "For assertion 1), we first suppose that $u_0$ is $\\mu _0$ integrable.", "We use Jensen inequality : $\\psi (u(X_t,t))=\\psi (\\mathrm {E}[u(X_s,s)|X_t])\\le \\mathrm {E}[\\psi (u(X_s,s))|X_t]\\;.$ So $\\int \\psi (u(x,t))\\mathrm {d}\\mu _t(x)=\\mathrm {E}[\\psi (u(X_t,t))]\\le \\mathrm {E}[\\psi (u(X_s,s))]=\\int \\psi (u(x,s))\\mathrm {d}\\mu _s(x)\\;.$ If $u_0$ is not $\\mu _0$ integrable, we slightly modify the proof.", "Consider $y<z$ and $a_s=\\alpha (y,s,t-s)$ , $b_s=\\beta (z,s,t-s)$ and $K=[a_0,b_0]\\times [0,t]$ .", "Using Jensen inequality, we have $\\psi (u(X_t,t))1\\hspace{-3.69885pt}\\mathrm {I}_{[y,z]}(X_t)&=\\psi (\\mathrm {E}_{\\mu _0^K}[u(X_s,s)|X_t])1\\hspace{-3.69885pt}\\mathrm {I}_{[y,z]}(X_t)\\\\&\\le \\mathrm {E}_{\\mu _0^K}[\\psi (u(X_s,s))|X_t]1\\hspace{-3.69885pt}\\mathrm {I}_{[y,z]}(X_t)\\;.$ So $\\mathrm {E}_{\\mu _0^K}[\\psi (u(X_t,t))1\\hspace{-3.69885pt}\\mathrm {I}_{[y,z]}(X_t)]\\le \\mathrm {E}_{\\mu _0^K}[\\psi (u(X_s,s))1\\hspace{-3.69885pt}\\mathrm {I}_{[y,z]}(X_t)]$ .", "Since $1\\hspace{-3.69885pt}\\mathrm {I}_{[y,z]}(X_t)=1\\hspace{-3.69885pt}\\mathrm {I}_{[a_s,b_s]}(X_s)$ , this leads to $\\int _y^z\\psi (u(x,t))\\mathrm {d}\\mu _t(x)\\le \\int _{a_s}^{b_s}\\psi (u(x,s))\\mathrm {d}\\mu _s(x)\\;.$ When $y\\rightarrow -\\infty $ and $z\\rightarrow +\\infty $ , we have $a_s\\rightarrow -\\infty $ and $b_s\\rightarrow +\\infty $ .", "So we get $\\int \\psi (u(x,t))\\mathrm {d}\\mu _t(x)\\le \\int \\psi (u(x,s))\\mathrm {d}\\mu _s(x)\\;.$ Now, if $u$ is moreover the entropy solution of the inviscid Burgers equation and $\\mu _0=\\lambda $ , then (REF ) holds.", "Hence $\\int \\psi (u(x,t))\\mathrm {d}x&=\\int \\psi (u(x,t))\\mathrm {d}\\mu _t(x)+t\\int \\psi (u(x,t))\\partial _xu(x,t)\\\\&=\\int \\psi (u(x,t))\\mathrm {d}\\mu _t(x)+t[\\Psi (u(+\\infty ,t))-\\Psi (u(-\\infty ,t))]\\;,$ where $\\Psi $ is any primitive of $\\psi $ .", "As $x\\rightarrow \\pm \\infty $ , the $\\lambda $ -essential limit of $u_0$ is 0.", "Thus $u(x,t)\\rightarrow 0$ as $x\\rightarrow \\pm \\infty $ .", "The proof ends using assertion 1 with $s=0$ and $\\mu _0=\\lambda $ ." ], [ "Dynamics of Burgers turbulence", "In this section, we study the sticky particles dynamics from the point of view of turbulence.", "There emerge four more Markov processes solution of (REF ).", "A remarkable property is that their velocities are backward semi-martingales.", "We always suppose that the support $\\mathcal {S}=\\mathbb {R}$ ." ], [ "Turbulence time and semi-martingales", "Following the preoccupation of [3], we consider the first shock time and related delta-shocks.", "Consider the left hand limit function $u^{-}(\\cdot ,t)$ and the right hand limit function $u^{+}(\\cdot ,t)$ .", "Remark that if a particle of initial position $a$ is in a shock at time $t$ and position $x$ , then $x=\\phi _{0,t}(a)$ and $u^{-}(x,t)\\ne u^{+}(x,t)$ .", "That's why we define its first shock time as $\\tau (a)=\\inf \\left\\lbrace t:\\,u^{-}(\\phi _{0,t}(a),t)\\ne u^{+}(\\phi _{0,t}(a),t)\\right\\rbrace .$ In fact, as described in the sequel, this is more a turbulence time than a shock time.", "We define a turbulent interval containing $a$ as the greatest interval $[A,B]$ of initial positions of particles containing $a$ and which have first turbulence at same time and same position : $\\tau (a^{\\prime })=\\tau (a)=:t$ and $\\phi _{0,t}(a^{\\prime })=\\phi _{0,t}(a)$ $\\forall \\,a^{\\prime }\\in [A,B]$ .", "Because of the regularity of $u_0$ and $\\phi $ , a turbulent interval is effectively closed.", "At time $\\tau (a)$ , there is a turbulence located at $x=\\phi _{0,\\tau (a)}(a)=a+\\tau (a)u_0(a)$ .", "The triangle, in the space-time representation, delimited by $A,B,x$ in known in the literature as a delta-shock (see Figure REF ).", "As it is related to first shock, we call it a prime-delta-shock.", "Thus, in the space-time representation, the turbulences are entirely conditioned by prime-delta-shocks.", "Figure: The blue curve and red curve in (a)(a) and (b)(b) (resp.", "the green curve and red curve in (a)(a) and (b)(b)) represent the trajectory Z t 1 Z^{1}_{t} (resp.", "Z t 2 Z^{2}_{t}) for X 0 ∈[A,B]X_{0}\\in [A,\\,B].", "xx represent the position of the shock of particles from [A,B][A,\\,B]: the first shock position.", "In case (c), the red curve represents one trajectory of Z t 1 =Z t 2 Z^ {1}_{t}=Z^{2}_{t} in the discontinuous line.", "In this case, the point xx is a turbulent point.", "Immediately after the point xx, there is shocks, but there is not a real cluster which get the position xx .Turbulent intervals bring suddenly positive masses to shocks.", "If the turbulent interval $[A,B]$ is a cluster at time $\\tau (A)$ , a turbulence of length $B-A$ rises from $[A,B]$ ; it is born at time $\\tau (A)$ (see Figure REF a).", "If moreover $A=B$ , then the turbulence is not detectable when it appears, an infinitesimal colliding (agglomeration) process starts at position $x=A+\\tau (A)u_0(A)$ (see Figure REF c).", "If the turbulent interval $[A,B]$ is not a cluster at time $\\tau (A)$ , it simply aggregates (in the same way as above) a turbulence which was born earlier from another turbulent interval (see Figure REF b) : $\\exists a_0\\notin [A,B]$ , $\\exists t<\\tau (A)$ such that $\\phi _{0,\\tau (A)}(a_0)=\\phi _{0,\\tau (A)}(A)\\; \\mbox{ and }\\lbrace a_0\\rbrace \\subsetneq \\lbrace a^{\\prime }: \\phi _{0,t}(a^{\\prime })=\\phi _{0,t}(a_0)\\rbrace .$ Now, for any turbulent interval $[A,B]$ , consider $x=A+\\tau (A)u_0(A)$ .", "We define four processes : if $X_0\\in [A,B]$ , then $\\forall \\,t\\ge 0$ , $Z^1_t=\\phi _{0,t}(A)\\;,\\quad & Z^2_t=\\phi _{0,t}(B)\\;,\\\\Z^3_t=\\phi _{0,t}(\\alpha (x,0,\\tau (A))),\\quad & Z^4_t=\\phi _{0,t}(\\beta (x,0,\\tau (A)))\\;.$ See Figure REF for illustration.", "Figure: Merger of Two Shocks wavesIt is clear that the set of all the turbulent intervals is a partition of the whole initial state of particles.", "Hence, the stochastic processes $Z^1,Z^2,Z^3$ and $Z^4$ are well defined.", "They all give new point of views of the sticky particle dynamics.", "The process $[Z^1,Z^2]$ describes the motion turbulent interval while $[Z^3,Z^4]$ describes the motion of all the particles which are concerned in the turbulence.", "Moreover, $[Z^1,Z^2]\\subset [Z^3,Z^4]$ .", "The measurability of $Z^1$ and $Z^2$ comes from the fact that $\\sigma (Z^1_0,Z^2_0)\\subset \\sigma (X_0)$ .", "Indeed, for all $x\\in \\mathbb {R}$ , the event $\\lbrace Z^1_0\\le x\\rbrace =\\lbrace X_0\\le B\\rbrace $ and $\\lbrace Z^2_0\\le x\\rbrace =\\lbrace X_0\\le A\\rbrace $ , where $[A,B]$ is the turbulent interval such that $A\\le x\\le B$ .", "For $Z^3$ and $Z^4$ , we recall that for all $t$ , the functions $\\alpha (\\cdot ,0,t)$ and $\\beta (\\cdot ,0,t)$ are non decreasing, so they are Borel functions; then $\\alpha (X_t,0,t)$ , $\\beta (X_t,0,t)$ are $\\sigma (X_t)$ measurable.", "Furthermore, all the paths of $t\\mapsto \\alpha (X_t,0,t)$ , $\\beta (X_t,0,t)$ are càdlàg.", "The proof of the measurabilty of $Z^3_0$ and $Z^4_0$ is achieved by lemmas REF and REF , since $Z^3_0=\\alpha (X_{\\gamma },0,\\gamma ),\\quad Z^4_0=\\beta (X_{\\gamma },0,\\gamma ),$ where $\\gamma =\\tau (X_0)=\\inf \\left\\lbrace t:\\,u^{-}(X_t,t)\\ne u^{+}(X_t,t)\\right\\rbrace $ is the first shock time of $X_0$ .", "Remark that if $\\Omega =\\mathbb {R}$ and $X_0$ is the identity function, then $\\tau =\\gamma $ .", "By construction, we have a.e : Proposition 3.1 (Random delta-shock) Suppose that the support $\\mathcal {S}=\\mathbb {R}$ .", "Let $Z$ stand independently for $Z^1,Z^2,Z^3$ or $Z^4$ .", "$\\forall \\,t,s\\ge 0,\\quad Z_{s+t}=\\phi _{s,t}(Z_{s})\\;,\\quad \\frac{\\mathrm {d}}{\\mathrm {d}t}Z_{t}=u(Z_{t},t)\\;.$ $\\tau (Z_{0})=\\tau (X_{0})=\\gamma $ and $\\forall \\,t\\ge \\gamma $ , $Z_{t}=X_t$ ; $\\forall \\,t\\le \\gamma ,\\quad &Z_{t}=Z_{0}+t u_0(Z_{0})\\;,\\\\&Z^3_{t}\\le Z^1_{t}\\le X_t\\le Z^2_{t}\\le Z^4_{t}\\;.$ The graphs $[0,\\gamma ]\\ni t\\mapsto \\,Z^1,Z^2,Z^3,Z^4$ then draw nested delta-shocks (see figure REF ).", "Now we recall some properties well known in the the theory of stochastic processes ([2], [8]) for forward martingales and non-decreasing filtrations.", "By inversion of time, the following holds.", "Lemma 3.2 Let a process $Z$ be adapted to a non increasing filtration $\\mathcal {G}=(\\mathcal {G}_t,t\\ge 0)$ .", "Let $\\Gamma $ be an optional time with respect to $\\mathcal {G}$ , i.e.", "for all $t\\ge 0$ , the event $\\lbrace \\Gamma > t\\rbrace \\in \\mathcal {G}_t$ .", "The following holds.", "The set $\\mathcal {G}_{\\Gamma }:=\\lbrace A\\in \\mathcal {G}_0\\:: A\\cap \\lbrace \\Gamma > t\\rbrace \\in \\mathcal {G}_t\\rbrace $ is a sigma-algebra.", "If all the paths of $Z$ are either continuous on the right or on the left, then the r.v.", "$Z_{\\Gamma }1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma <\\infty }$ is $\\mathcal {G}_{\\Gamma }$ measurable.", "Suppose that $\\mathcal {G}$ is continuous on the right; that is, for all $t$ , $\\mathcal {G}_t=\\sigma \\left(\\underset{s>t}{\\cup }\\mathcal {G}_s\\right)$ .", "If $Z$ is a backward martingale with respect to $\\mathcal {G}$ , then for all $t$ , the right hand and left hand limits $Z_{t^+}$ , $Z_{t^-}$ exist a.s.", "Moreover, the process $t\\mapsto Z_{(\\Gamma \\vee t)^+}-\\Delta _{\\Gamma }1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma > t}$ is a backward martingale with respect to the completed filtration $\\overline{\\mathcal {G}}$ , with $\\Delta _{\\Gamma }=Z_{\\Gamma ^+}-Z_{\\Gamma ^-}$ .", "In the latter, the completed filtration is defined by $\\overline{\\mathcal {G}}_t=\\sigma (\\mathcal {N}\\cup \\mathcal {G}_t)$ and $\\mathcal {N}=\\lbrace A\\in \\mathcal {G}_0\\:: P(A)=0\\rbrace $ .", "The following is also a useful tool.", "Lemma 3.3 If a process $Z$ is such that $Z_{s+t}=\\phi _{s,t}(Z_{s})$ for all $t,s\\ge 0$ , then $\\tau (Z_0)=:\\Gamma $ is an optional time with respect to the natural non increasing filtration $\\mathcal {F}^{Z}$ of $Z$ .", "Moreover, $\\mathcal {F}^{Z}_0=\\mathcal {F}^{Z}_{\\Gamma }$ .", "Suppose that $\\lbrace \\Gamma \\le t\\rbrace \\in \\mathcal {F}^Z\\cap \\mathcal {F}^{Z^{\\prime }}$ for some $t\\ge 0$ .", "If $Z^{\\prime }_t1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}=Z_t1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}$ , then $\\mathrm {E}[F|Z^{\\prime }_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}=\\mathrm {E}[F|Z_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}$ for all integrable r.v.", "$F$ .", "From proposition REF , this lemma is satisfied by $Z^1,Z^2,Z^3,Z^4$ and $X$ , taking the role of $Z,Z^{\\prime }$ .", "Proof.", "We begin with the first assertion.", "$u^-(\\cdot ,t),u^+(\\cdot ,t)$ are Borel functions and it is well known that if $u$ is discontinuous in $(Z_t,t)$ , it is also discontinuous in $(Z_{t+s},t+s)$ .", "Then, $\\lbrace \\Gamma \\le t\\rbrace =\\lbrace u^-(Z_t,t)\\ne u^+(Z_t,t)\\rbrace \\cup [\\lbrace u^-(Z_t,t)= u^+(Z_t,t)\\rbrace \\cap \\lbrace \\Gamma = t\\rbrace ]\\;.$ Since $\\lbrace u^-(Z_t,t)= u^+(Z_t,t)\\rbrace &\\cap \\lbrace \\Gamma = t\\rbrace =\\lbrace u^-(Z_t,t)= u^+(Z_t,t)\\rbrace \\cap \\\\&\\left[\\underset{n\\ge 1}{\\cap }\\lbrace u^-(Z_{t+1/n},t+1/n)\\ne u^+(Z_{t+1/n},t+1/n)\\rbrace \\right],$ the proof of the first assertion is done.", "Remark that $Z_{t+1/n}=\\phi _{t,1/n}(Z_t)$ .", "So $\\lbrace \\Gamma \\le t\\rbrace =Z_t^{-1}(A_t)$ , with $A_t=\\lbrace u^-(\\cdot ,t)\\ne u^+(\\cdot ,t)\\rbrace \\cup &\\bigg (\\lbrace u^-(\\cdot ,t)= u^+(\\cdot ,t)\\rbrace \\cap \\\\&\\left[\\underset{n\\ge 1}{\\cap }\\lbrace u^-(\\phi _{t,1/n},t+1/n)\\ne u^+(\\phi _{t,1/n},t+1/n)\\rbrace \\right]\\bigg )$ Now we show that $\\mathcal {F}^{Z}_0=\\mathcal {F}^{Z}_{\\Gamma }$ .", "First remark that if $\\lbrace b\\rbrace \\ne \\phi _{0,t}^{-1}(\\phi _{0,t}(b))$ , then $\\tau (b)\\le t$ .", "Thus for all Borel subset $B$ and $t\\ge 0$ , we have $B\\cap \\lbrace \\tau >t\\rbrace =\\phi _{0,t}^{-1}(\\phi _{0,t}(B))\\cap \\lbrace \\tau >t\\rbrace $ and $Z_0^{-1}(B)\\cap \\lbrace \\tau (Z_0)>t\\rbrace &=Z_t^{-1}(\\phi _{0,t}(B))\\cap \\lbrace \\tau (Z_0)>t\\rbrace \\;,\\\\Z_0^{-1}(B)\\cap \\lbrace \\Gamma >t\\rbrace &=Z_t^{-1}(\\phi _{0,t}(B))\\cap \\lbrace \\Gamma >t\\rbrace \\in \\mathcal {F}^{Z}_t\\;.$ This means that $Z_0^{-1}(B)\\in \\mathcal {F}^{Z}_{\\Gamma }$ .", "For the second assertion, since $Z_t1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}=Z^{\\prime }_t1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}$ , it is easy to see that $\\mathrm {E}[F|Z^{\\prime }_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}$ is $\\sigma (Z^{\\prime }_t)\\cap \\sigma (Z_t)$ measurable; for all bounded Borel function $h$ , $\\mathrm {E}\\big (h(Z_t)\\mathrm {E}[F|Z^{\\prime }_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}\\big )=\\mathrm {E}\\big (h(Z^{\\prime }_t)\\mathrm {E}[F|Z^{\\prime }_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}\\big )=\\mathrm {E}\\big (h(Z^{\\prime }_t)F1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}\\big )\\\\=\\mathrm {E}\\big (h(Z_t)F1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}\\big )=\\mathrm {E}\\big (h(Z_t)\\mathrm {E}[F|Z_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}\\big ).$ Hence, $\\mathrm {E}[F|Z^{\\prime }_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}=\\mathrm {E}[F|Z_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\Gamma \\le t}$ a.s. Corollary 3.4 (Turbulence semi-martingales) Let $Z$ stand independently for $Z^1,Z^2,Z^3$ or $Z^4$ .", "The process $Z$ is Markovian.", "Suppose that the support of $\\mu _0$ is $\\mathcal {S}=\\mathbb {R}$ .", "$t\\mapsto u(Z_{t},t)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma >t}$ is a bounded variational process adapted to the natural non increasing filtration $\\mathcal {F}^Z$ of $Z$ .", "$t\\mapsto u(Z_{t},t)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}$ is a backward càdlàg semi-martingale with respect to $\\mathcal {F}^Z$ .", "Moreover, $t\\mapsto u(Z_{t},t)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}+\\mathrm {E}[u_0(X_0)|Z_0]1\\hspace{-3.69885pt}\\mathrm {I}_{t<\\gamma }$ is a backward martingale.", "$t\\mapsto u(Z_{t},t)$ is a backward càdlàg semi-martingale with respect to $\\mathcal {F}^Z$ .", "Moreover, $t\\mapsto u(Z_{t},t)-\\left(u_0(Z_0)-\\mathrm {E}[u_0(X_0)|Z_0]\\right)1\\hspace{-3.69885pt}\\mathrm {I}_{t<\\gamma }$ is a backward martingale.", "Proof.", "1) Obviously, $u(Z_{t},t)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma > t}=u_0(Z_0)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma > t}$ .", "2) For all $t$ , $u(Z_{t},t)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}&=u(X_{t},t)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}=\\mathrm {E}[u_0(X_0)|X_t]1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}=\\underbrace{\\mathrm {E}[u_0(X_0)|Z_t]}_{M_t}1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}=M_{t^+}1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}\\\\&=M_{(\\gamma \\vee t)^+}-M_{\\gamma ^+}1\\hspace{-3.69885pt}\\mathrm {I}_{t<\\gamma }\\;.$ But the process $t\\mapsto M_{\\gamma ^+}1\\hspace{-3.69885pt}\\mathrm {I}_{t<\\gamma }$ is adapted to $\\mathcal {F}^Z$ .", "Then, according to lemma REF , the process $t\\mapsto u(Z_{t},t)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma \\le t}+ M_{\\gamma ^-}1\\hspace{-3.69885pt}\\mathrm {I}_{t<\\gamma }$ is a backward martingale with respect to $\\mathcal {F}^Z$ .", "Moreover $M_{\\gamma ^-}&=\\underset{\\underset{s<\\gamma }{s\\rightarrow \\gamma }}{\\lim }M_s1\\hspace{-3.69885pt}\\mathrm {I}_{s<\\gamma }=\\underset{\\underset{s<\\gamma }{s\\rightarrow \\gamma }}{\\lim }\\mathrm {E}\\left[M_01\\hspace{-3.69885pt}\\mathrm {I}_{s<\\gamma }|Z_s\\right]\\\\&=\\underset{\\underset{s<\\gamma }{s\\rightarrow \\gamma }}{\\lim }M_01\\hspace{-3.69885pt}\\mathrm {I}_{s<\\gamma }=M_0=\\mathrm {E}[u_0(X_0)|Z_0]\\;.$ 3) $u(Z_{t},t)-(u_0(Z_0)-\\mathrm {E}[u_0(X_0)|Z_0])1\\hspace{-3.69885pt}\\mathrm {I}_{t<\\gamma }=M_{(\\gamma \\vee t)^+}-[M_{\\gamma ^+}-M_{\\gamma ^-}]1\\hspace{-3.69885pt}\\mathrm {I}_{t<\\gamma }$ .", "Now, in order to analyse the process of [3], let us define, for all $(a,t,s,r)\\in \\mathbb {R}\\times \\mathbb {R}^{3}_{+}$ $f(a,t,r) & = \\left\\lbrace \\begin{array}{ll}\\phi _{0,t} (a) & \\textrm {if t\\ge r}\\\\\\phi _{0,r} (a)-(r-t) u^{-}\\big (\\phi _{0,r} (a),\\,r\\big ) & \\textrm {if\\,t<r}\\\\\\end{array} \\right.\\\\g(a,t,r) & = \\left\\lbrace \\begin{array}{ll}\\phi _{0,t} (a) & \\textrm {if t\\ge r}\\\\\\phi _{0,r} (a)-(r-t) u^{+}\\big (\\phi _{0,r} (a),\\,r\\big ) & \\textrm {if\\,t<r}\\\\\\end{array} \\right.$ The definition of [3] is the following : $Y_{t}(a) = \\left\\lbrace \\begin{array}{ll}f(a,t,\\tau (a)) & \\textrm {if a enters in the shock from the left} \\\\g(a,t,\\tau (a)) & \\textrm {if a enters in the shock from the right}.\\end{array} \\right.$ In this definition, if a turbulent interval $[\\alpha ,\\beta ]$ is also a cluster at time $\\tau (\\alpha )$ , then the whole interval $[\\alpha ,\\beta [$ is considered as entering in the shock from the left, and $\\beta $ is considered as entering in the shock from the right.", "This definition is however ambiguous since it supposes that a particle can hurt only one discontinuity line at a time.", "In the sequel, we call this the assumption of simple shocks.", "In order to remove the ambiguity, we suggest to slightly modify the definition : $Y_{t}(a) = \\left\\lbrace \\begin{array}{ll}f(a,t,\\tau (a)) & \\textrm {if a enters in the shock from the left} \\\\& \\textrm {of all concerned discontinuity lines}\\\\g(a,t,\\tau (a)) & \\textrm {otherwise.", "}\\end{array} \\right.$ Of course, this definition matches the underlying assumption of simple shocks of [3].", "In that paper, the authors considered Lebesgue measure as initial distribution $\\mu _0$ of the matter.", "Proposition 3.5 (Link with delta-shocks) Suppose that the support of $\\mu _0$ is $\\mathcal {S}=\\mathbb {R}$ .", "For all $0\\le t\\le r$ and $a\\in \\mathbb {R}$ , let $x=\\phi _{0,r}(a)$ .", "We have $f(a,t,r) &=\\phi _{0,t}(\\alpha (x,0,r))\\\\&=\\alpha (x,0,r)+tu_0(\\alpha (x,0,r))\\;.\\nonumber \\\\g(a,t,r) & =\\phi _{0,t}(\\beta (x,0,r))\\\\&=\\beta (x,0,r)+tu_0(\\beta (x,0,r))\\;.\\nonumber $ If moreover $\\alpha (x,0,r)<\\beta (x,0,r)$ , then $r=\\tau (\\alpha (x,0,r))=\\tau (\\beta (x,0,r))=\\tau (a)\\;.$ Thus, $[0,\\tau (a)]\\ni t\\mapsto f(a,t,\\tau (a)),\\,g(a,t,\\tau (a))$ draw a delta-shock (figure REF ).", "We can (more generally) consider $Y$ on abstract set $\\Omega $ (instead of $\\mathbb {R}$ ) : $Y_{t} = \\left\\lbrace \\begin{array}{ll}f(X_0,t,\\gamma ) & \\textrm {if X_0 enters in the shock from the left} \\\\& \\textrm {of all concerned discontinuity lines}\\\\g(X_0,t,\\gamma ) & \\textrm {otherwise.", "}\\end{array} \\right.$ The process $Y$ then chooses one segment of delta-shock, at random.", "Note that the event \"$X_0$ enters the shock from the left\" coincides with $\\lbrace Z^1_0=Z^3_0\\rbrace \\cap [\\lbrace Z^2_0=Z^4_0,\\,X_0\\ne Z^2_0\\rbrace \\cup \\lbrace Z^2_0\\ne Z^4_0\\rbrace ]$ .", "Figure: Delta-shock.Proposition 3.6 (Motion on delta-shock) Suppose that the support of $\\mu _0$ is $\\mathcal {S}=\\mathbb {R}$ .", "$\\tau (Y_0)=\\tau (X_0)=\\gamma $ and $Y_t=\\left\\lbrace \\begin{array}{ll}Y_0+tu_0(Y_0)& \\textrm {if } t\\le \\gamma \\\\Z_t=X_t& \\textrm {if } t\\ge \\gamma \\;.\\end{array}\\right.$ $\\forall \\,s\\ge 0,\\,\\forall \\,t\\ge 0\\;,\\quad Y_{s+t}=\\phi _{s,t}(Y_s)=\\left\\lbrace \\begin{array}{ll}Z^3_{s+t}& \\textrm {if } Z^1_0=Z^3_0\\\\Z^4_{s+t}& \\textrm {if } Z^1_0\\ne Z^3_0\\;.\\end{array}\\right.$ If the assumption of simple shocks holds, then two events occur : $Y_0=Z^3_0=Z^1_0$ and in this case, for all $t\\ge 0$ , $Y_t=Z^3_t=Z^1_t$ .", "$Y_0=Z^4_0=Z^2_0$ and in this case, for all $t\\ge 0$ , $Y_t=Z^4_t=Z^2_t$ .", "If $\\gamma $ has no atom, i.e.", "$P(\\gamma =t)=0$ for all $t$ , then $X\\equiv Z^1\\equiv Z^2$ .", "And in this case, all the turbulent intervals are reduced to single points.", "If moreover the assumption of simple shocks holds, then $Y\\equiv X\\equiv Z^1\\equiv Z^2$ .", "Remark 3.7 (Delta-shock velocity as semi-martingale) From the first assertion of proposition REF , the process $t\\mapsto u(Y_{t},t)$ satisfies corollary REF (with $Z=Y$ ) and is a backward semi-martingale of $\\mathcal {F}^Y$ .", "The last assertion of proposition REF shows that the martingale $t\\mapsto u(Y_{t},t)$ of [3] (which required the assumption of uniformity of the law of $\\gamma $ ) was in fact already obtained in [4].", "The case of [3] is a particularity where all the turbulent intervals are reduced to single points.", "Now we precise, under more general assumptions, when the velocity of turbulence is a martingale." ], [ "Martingales and undetectability of turbulence", "In this part, we show that the martingality of the velocity of turbulence implies that any turbulent interval is a single point (single turbulent point).", "Corollary 3.8 (Turbulence martingales and prime-delta-shocks) Let $Z$ stand independently for $Z^1,Z^2,Z^3,Z^4$ , or $Y$ .", "Suppose that the support of $\\mu _0$ is $\\mathcal {S}=\\mathbb {R}$ .", "The process $t\\mapsto u(Z_{t},t)$ is a martingale if and only if a.e.", "$X\\equiv Z\\equiv Z^1\\equiv Z^2.$ Proof.", "The semi-martingale is a martingale iff its bounded variational part $t\\mapsto \\left(u_0(Z_0)-\\mathrm {E}[u_0(X_0)|Z_0]\\right)1\\hspace{-3.69885pt}\\mathrm {I}_{t<\\gamma }$ is constant.", "Letting $t$ tend to 0 and $+\\infty $ respectively gives $\\left(u_0(Z_0)-\\mathrm {E}[u_0(X_0)|Z_0]\\right)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma >0}&=\\left(u_0(Z_0)-\\mathrm {E}[u_0(X_0)|Z_0]\\right)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma =\\infty }=0$ since $\\gamma $ is $\\sigma (Z_0)$ -measurable and $X_01\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma =\\infty }=Z_01\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma =\\infty }$ .", "(In the same way, $X_01\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma =0}=Z_01\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma =0}$ .)", "Using the fact that $X_0+\\gamma u_0(X_0)=Z_0+\\gamma u_0(Z_0)$ , the NSC to have a martingale becomes $\\mathrm {E}\\left[\\gamma ^{-1}(X_0-Z_0)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma >0}|Z_0\\right]=\\mathrm {E}\\left[(u_0(Z_0)-u_0(X_0))1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma >0}|Z_0\\right]=0.$ Let us now study each case of $Z$ .", "For $Z=Z^3$ : we have $\\mathrm {E}[\\gamma ^{-1}(X_0-Z^3_0)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma >0}].$ But $X_0\\ge Z^3_0$ .", "So $X_0=Z^3_0=Z^1_0$ a.e.", "Thus $X\\equiv Z^3\\equiv Z^1$ a.e.", "Now, we show that $Z^1_0=Z^2_0$ .", "If $[Z^1_0,Z^2_0]=[\\alpha ,\\beta ]$ , then $\\mu _0(]\\alpha ,\\beta ])=P(\\alpha <X_0 \\le \\beta )\\le P(X_0\\ne Z^1_0)=0$ .", "This implies that $\\alpha =\\beta $ since there is no vacuum in the support.", "Hence $Z^1_0=Z^2_0$ .", "For $Z=Z^4$ , resp.", "$Z=Z^1$ , resp.", "$Z=Z^2$ : Analogous to previous case.", "For $Z=Y$ : let $L$ be the set of particles which enter the shock on the left.", "We have $\\mathrm {E}[\\gamma ^{-1}(X_0-Y_0)1\\hspace{-3.69885pt}\\mathrm {I}_{\\gamma >0}1\\hspace{-3.69885pt}\\mathrm {I}_L(Y_0)]=0$ , with $(X_0-Y_0)1\\hspace{-3.69885pt}\\mathrm {I}_L(Y_0)\\ge 0$ , so $(X_0-Y_0)1\\hspace{-3.69885pt}\\mathrm {I}_L(Y_0)=0$ .", "In the same way, $(X_0-Y_0)1\\hspace{-3.69885pt}\\mathrm {I}_{L^c}(Y_0)=0$ and we get $X_0=Y_0$ .", "Now, we show that $Z^1_0=Z^2_0$ .", "If $[Z^1_0,Z^2_0]=[\\alpha ,\\beta ]$ , then $\\mu _0(]\\alpha ,\\beta [)=P(\\alpha <X_0 <\\beta )\\le P(X_0\\ne Y_0)=0$ .", "This implies that $\\alpha =\\beta $ .", "Hence $Z^1_0=Z^2_0$ ." ] ]

[ [ "Injective Rota-Baxter operators of weight zero on $F[x]$" ], [ "Abstract Rota-Baxter operators present a natural generalisation of integration by parts formula for the integral operator.", "In 2015, Zheng, Guo, and Rosenkranz conjectured that every injective Rota-Baxter operator of weight zero on the polynomial algebra $\\mathbb{R}[x]$ is a composition of the multiplication by a nonzero polynomial and a formal integration at some point.", "We confirm this conjecture over any field of characteristic zero.", "Moreover, we establish a structure of an ind-variety on the moduli space of these operators and describe an additive structure of generic modality two on it.", "Finally, we provide an infinitely transitive action on codimension one subsets." ], [ "Introduction", "G. Baxter introduced the notion of a Rota–Baxter operator in 1960 [3] as a natural generalization of integration by parts formula for the integral operator.", "Further, such operators were studied from algebraic, combinatorial, topological, physical and many other points of view by G.-C. Rota, P. Cartier, L. Guo and others, see details in [5], [7].", "The integral operator is known to be injective on the algebra of continuous functions on $\\mathbb {R}$ .", "So, given an algebra, it is natural to classify injective Rota–Baxter operators of weight zero on it.", "In [5], it was proved that there are no injective Rota–Baxter operators of weight zero on a unital finite-dimensional algebra.", "The situation is completely different if we study Rota–Baxter operators of weight zero on the polynomial algebra $F[x]$ , where $F$  denotes the ground field of characteristic zero.", "In [13], S. H. Zheng, L. Guo, and M. Rosenkranz described all injective monomial Rota–Baxter operators of weight zero on $F[x]$ ; a monomial operator is an operator which maps each monomial to a monomial with some coefficient.", "More about monomial Rota–Baxter operators on polynomial algebras see in [6], [12].", "Representations of Rota–Baxter operators on $F[x]$ have been studied in [9], [11].", "A Rota–Baxter operator of weight zero on $F[x]$ is called analytically modeled if it equals a composition of the multiplication $l_r$ by a fixed nonzero polynomial $r$ and a formal integration $J_a$ at some fixed point $a$ .", "In [13], it was proved that up to a constant term every injective Rota–Baxter operator on $F[x]$ acts on each monomial as an analytically modeled Rota–Baxter operator.", "Based on this result, S.H.", "Zheng, L. Guo, and M. Rosenkranz conjectured that every injective Rota–Baxter operator of weight zero on $\\mathbb {R}[x]$ is analytically modeled.", "Throughout the work, we operate over the ground field $F$ of characteristic zero.", "We confirm the conjecture of S.H.", "Zheng, L. Guo, and M. Rosenkranz (Theorem 3.8).", "This allows us to describe the moduli space $\\mathcal {R}$ of the injective Rota–Baxter operators of weight zero on $F[x]$ (Proposition REF ), which we do in terms of ind-varieties and ind-groups introduced by I. Shafarevich in [10].", "Furthermore, we introduce natural regular actions of the additive group ${\\mathbb {G}}_a$ of the ground field.", "Given an affine variety $X$ and the group $\\operatorname{SAut}(X)$ generated by all the additive actions on $X$ , the $\\operatorname{SAut}(X)$ -action on $X$ is transitive if and only if it is infinitely transitive, see [2].", "Though $\\mathcal {R}$ is the quasi-affine infinite-dimensional ind-variety, we prove several results inspired by this equivalence.", "Namely, the introduced actions provide an additive structure on $\\mathcal {R}$ of complexity two (Corollary REF ), induce a transitive action on $\\mathcal {R}$ (Proposition REF ) and an infinitely transitive action on subsets of codimension one (Theorem REF ).", "In Section 2 we state all required preliminaries.", "In Section 3 we confirm the conjecture of S.H.", "Zheng, L. Guo, and M. Rosenkranz over any field of characteristic zero.", "Thus, we finish the description of all injective Rota–Baxter operators on $F[x]$ over a field of characteristic zero.", "In Section 4 we describe the structure of their moduli space $\\mathcal {R}$ in terms of ind-varieties, introduce families of ${\\mathbb {G}}_a$ -actions on $\\mathcal {R}$ and describe their properties.", "The authors are grateful to Ivan Arzhantsev for valuable remarks and suggestions." ], [ "Preliminaries", "There are two principally different classes of Rota–Baxter operators: those of nonzero weight and those of weight zero.", "In this work we consider only the latter ones.", "Definition 2.1 A linear operator $R$ defined on an algebra $A$ is called a Rota–Baxter operator of weight $\\lambda $, where $\\lambda \\in F$ , if the identity $R(f)R(g) = R( R(f)g + fR(g) )+\\lambda R(fg)$ holds for every $f,g\\in A$ .", "We are interested only in injective Rota–Baxter operators of weight zero.", "In particular, the identity (REF ) transforms into $R(f)R(g) = R( R(f)g + fR(g) ).$ Example 2.2 A linear operator $R$ on $F[x]$ sending $x^{2n} = 0$ into zero and $x^{2n+1}$ into $\\frac{x^{2n+2}}{2n+2}$ for any $n\\in {\\mathbb {Z}}_{\\ge 0}$ , is a nontrivial non-injective Rota–Baxter operator of weight zero.", "Let us introduce the following linear operators on $F[x]$ : (multiplication) $l_r\\colon f\\mapsto rf$ , where $r\\in F[x]$ ; (standard derivation) $\\delta \\colon x^n\\mapsto nx^{n-1};$ (formal integration at $a$ ) $J_a\\colon x^n\\mapsto \\dfrac{x^{n+1}-a^{n+1}}{n+1}.$ Note that $\\delta \\circ J_a={\\mathrm {id}}$ and $J_a$ is a Rota–Baxter operator on $F[x]$ for any $a\\in F$ .", "We also denote by $I$ the operator $J_0$ for brevity.", "By $\\circ $ we denote a composition of operators.", "Lemma 2.3 ([13]) Let $r\\in F[x]$ and let $R$ be a Rota–Baxter operator on $F[x]$ .", "Then a linear operator $R\\circ l_r$ is again a Rota–Baxter operator on $F[x]$ .", "In [13], S.H.", "Zheng, L. Guo, and M. Rosenkranz initiated a study of injective Rota–Baxter operators on $F[x]$ .", "They described completely all injective monomial Rota–Baxter operators of weight zero on $F[x]$ ; an operator on $F[x]$ is called monomial if it sends each monomial to a monomial with some coefficient.", "For the general case, the authors made a significant advance proving Theorem 2.4 (Zheng, Guo, Rosenkranz, 2015 [13]) Let $R$ be an injective Rota–Baxter operator on $F[x]$ , where $F$ is a field of characteristic 0.", "Then there exists a nonzero polynomial $r\\in F[x]$ such that $\\delta \\circ R = l_r$ .", "After Theorem REF , the following conjecture arises naturally (originally it was stated over $\\mathbb {R}$ ).", "Conjecture 2.5 (Zheng, Guo, Rosenkranz, 2015 [13]) Every injective Rota–Baxter operator on $F[x]$ over a field $F$ of characteristic 0 equals $J_a \\circ l_r$ for some nonzero polynomial $r\\in F[x]$ and $a\\in F$ ." ], [ "Proof of Conjecture", "Below we express Conjecture REF in terms of linear functionals on $F[x]$ .", "Proposition 3.1 Let $R$ be an operator on $F[x]$ such that $\\delta \\circ R=l_r$ for some nonzero polynomial $r(x)\\in F[x]$ .", "Let us introduce a linear functional $c\\colon F[x]\\rightarrow F$ by the formula $c\\colon f\\mapsto R(f)(0).$ Then $R$ is a Rota–Baxter operator if and only if $c(f)c(g) + c(I(rf)g+fI(rg)) = 0$ for all $f,g\\in F[x]$ .", "Moreover, this is a one-to-one correspondence between Rota–Baxter operators $R$ on $F[x]$ satisfying $\\delta \\circ R=l_r$ and linear functionals $c$ on $F[x]$ satisfying (REF ).", "Its inverse is defined by the formula $R=I\\circ l_r+c$ .", "Since $\\delta \\circ R=\\delta \\circ I\\circ l_r$ , there holds $R(f)-I(rf)\\in F$ for any $f\\in F[x]$ , and so $R(f) = I(rf)+R(f)(0)$ .", "Hence, $R = I\\circ l_r + c$ .", "Then the condition $(\\ref {RB})$ transforms into $0 = R( R(f)g + f R(g) ) - R(f)R(g)\\\\= R\\big (I(rf)g + I(rg)f + c(f)g + c(g)f\\big ) - (I(rf) + c(f))(I(rg) + c(g)) \\\\= I(I(rf)rg) + I(I(rg)rf) - I(rf) I(rg)+ c(f)c(g) + c\\big (I(rf)g+fI(rg)\\big )\\\\= c(f)c(g) + c\\big (I(rf)g+fI(rg)\\big ).$ The last equality holds since $I\\circ l_r$ is a Rota–Baxter operator by Lemma REF .", "Thus, $R$ is a Rota–Baxter operator if and only if $c$ satisfies (REF ).", "Since $R$ equals $I\\circ l_r+c$ , it is uniquely determined by $r$ and $c$ , hence the one-to-one equivalence.", "3.2 We denote by $M_r$ the set of linear functionals satisfying (REF ), where $r\\in F[x]$ ; by $c_{r,a}$ the linear functional $c_{r,a}\\colon f \\mapsto -I(rf)(a)$ ; by $N_r$ the family of linear functionals $\\lbrace c_{r,a}\\mid a\\in F\\rbrace $ parameterized by $a\\in F$ .", "Remark 3.3 The operator $R = I\\circ l_r + c_{r,a}$ , corresponding to $c_{r,a}\\in N_r$ , equals $J_{a}\\circ l_r$ .", "Indeed, $R(f)= I(rf)+c_{r,a}(f)=I(rf)-I(rf)(a)= J_{a}(rf).$ By Lemma REF , $J_{a}\\circ l_r$ is a Rota–Baxter operator.", "Thus, $N_r\\subseteq M_r$ .", "Remark 3.4 Let us introduce the coordinates $c_i=c(x^i)$ on the space of linear functionals and let $r=r_0+r_1x+\\cdots +r_kx^k$ .", "Then, after substituting $f=x^n,g=x^m$ , the equation (REF ) transforms into the system $c_n c_m + \\sum \\limits _{i=0}^k \\left(\\frac{1}{i+n+1}+\\frac{1}{i+m+1}\\right)r_ic_{i+n+m+1} = 0, \\quad n,m\\in \\mathbb {Z}_{\\ge 0}.$ Since $c$ is linear, it is defined by its values at $1,x,x^2\\ldots $ , so this system is equivalent to (REF ).", "In these coordinates $M_r\\subset F^{\\times \\infty }=\\lbrace (c_0,c_1,\\ldots )\\mid c_i\\in F\\rbrace $ .", "Lemma 3.5 Let $P(c_0,\\ldots ,c_n)=0$ be a polynomial equation on the space of linear functionals, $P\\in F[x_0,\\ldots , x_n],\\, n\\in {\\mathbb {N}}$ , such that $N_r\\subset \\lbrace P(c_0,\\ldots ,c_n)=0\\rbrace $ .", "Then $M_r\\subset \\lbrace P(c_0,\\ldots ,c_n)=0\\rbrace $ as well.", "The equation $P(c_0,\\ldots ,c_n)=0$ is equivalent on $M_r$ to some linear one $\\sum \\limits _{i=0}^s a_ic_i=0$ for some $s\\in {\\mathbb {Z}}_{\\ge 0}$ , which we denote by $L$ .", "In order to see this equivalence, it is enough to substitute products $c_nc_m$ repeatedly by linear combinations from corresponding equations in (REF ).", "Assume that $L$ is non-trivial and the highest coefficient $a_s$ is nonzero.", "Note that $c_{r,a}(x^n)=-I(rx^n)(a)$ is a polynomial of degree $n+k+1$ in $a$ .", "Thus, substituting $c_i=c_{r,a}(x^i)$ , $i=0,\\ldots ,s$ , into $L$ , we obtain a nonzero polynomial $\\sum \\limits _{i=0}^s a_i c_{r,a}(x^i)$ on $a$ of degree $s+k+1$ .", "Since $F$ is an infinite field, we get a contradiction.", "Define projections $\\pi _l\\colon F^{\\times \\infty }\\rightarrow F^{l+1}$ as follows, $\\pi _l((c_0,c_1,\\ldots )) = (c_0,c_1,\\ldots ,c_l).$ Lemma 3.6 Let $r\\in F[x]$ be a nonzero polynomial of degree $k=\\deg (r)$ .", "Then $\\pi _k$ is injective on $M_r$ and $N_r$ , $\\pi _k(M_r)$ and $\\pi _k(N_r)$ are Zariski closed subsets in $F^{k+1}$ .", "We will prove both assertions at first for $M_r$ and then do the same for $N_r$ .", "Let us transform the system (REF ) as follows.", "For any $c_t$ , where $t>k$ , we may transform the equation with $m=0$ and $n=t-1-k$ to the form $c_t=P_t(c_0,\\ldots ,c_{t-1})$ .", "So, $c_t$ can be expressed as a polynomial in coordinates with lower indices.", "Any other equation corresponding to some $m,n\\in \\mathbb {Z}_{\\ge 0}$ can be reduced to the form $g_{m,n}(c_0,\\ldots ,c_k)$ by consecutive substitutions $c_t=P_t(c_0,\\ldots ,c_{t-1})$ for all $t>k$ .", "Thus, the system (REF ) is equivalent to a system of polynomial equations $\\lbrace g_{m,n}(c_0,\\ldots ,c_k)=0\\mid m>0, n\\ge 0\\rbrace \\cup \\lbrace c_t=P_t(c_0,\\ldots ,c_{t-1})\\mid t>k\\rbrace .$ So, the subset $\\pi _k(M_r)\\subset F^{k+1}$ is defined by the polynomial system $\\lbrace g_{m,n}(c_0,\\ldots ,c_k)\\rbrace $ , hence is Zariski closed.", "The map $\\pi _k\\colon M_r\\rightarrow \\pi _k(M_r)$ is a one-to-one correspondence with the inverse map $(c_0,\\ldots $ , $c_k)\\mapsto (c_0,\\ldots ,c_k, P_{k+1},P_{k+2},\\ldots )$ .", "Since $N_r\\subset M_r$ , $\\pi _k$ is injective on $N_r$ as well.", "Consider the map $\\phi \\colon F\\rightarrow N_r$ acting as $\\phi (a)=c_{r,a}$ and $\\phi _k=\\pi _k\\circ \\phi \\colon a\\mapsto (I(r)(a),I(xr)(a),\\ldots ,I(x^kr)(a))$ .", "Then $\\phi _k$ is a morphism that maps an affine line into an affine space, hence its image $\\phi _k(\\mathbb {A}^1)=\\pi _k(N_r)$ is Zariski closed.", "Indeed, we may extend $\\phi _k$ to the morphism $\\bar{\\phi }_k\\colon \\mathbb {P}^1\\rightarrow \\mathbb {P}^{k+1}$ of a projective line into a projective space.", "Since $\\mathbb {P}^1$ is a complete algebraic variety, its image is closed.", "On the other hand, $\\mathbb {P}^1$ does not have non-constant regular functions and thus its image cannot lie in the affine space $\\mathbb {A}^{k+1}$ .", "So, $\\bar{\\phi }_k(\\mathbb {P}^1\\setminus \\mathbb {A}^1)\\in \\mathbb {P}^{k+1}\\setminus \\mathbb {A}^{k+1}$ , and the closure $\\phi _k(\\mathbb {A}^1)$ lies in $\\bar{\\phi }_k(\\mathbb {P}^1)\\cap \\mathbb {A}^{k+1}=\\phi _k(\\mathbb {A}^1)$ .", "Proposition 3.7 For any nonzero $r\\in F[x]$ there holds $M_r=N_r.$ By Remark REF , we have $N_r\\subset M_r$ .", "Assume that $c\\in M_r\\setminus N_r$ .", "Then $\\pi _k(c)\\in \\pi _k(M_r)\\setminus \\pi _k(N_r)$ , where $k=\\deg (r)$ .", "By Lemma REF , $\\pi _k(N_r)$ is Zariski closed, hence there exists a polynomial $P\\in F[c_0,\\ldots , c_k]$ that vanishes on $\\pi _k(N_r)$ but does not equal 0 at $\\pi _k(c)$ .", "Note that $P(b)=P(\\pi _k(b))$ for any $b\\in M_r$ .", "Thus, $P$ vanishes on $N_r$ but not on $M_r$ .", "This contradicts Lemma REF .", "In the following theorem we confirm Conjecture REF .", "Theorem 3.8 Every injective Rota–Baxter operator of weight zero on $F[x]$ over a field $F$ of characteristic 0 equals $J_a \\circ l_r$ for some nonzero polynomial $r\\in F[x]$ and $a\\in F$ .", "Let $R$ be an injective Rota–Baxter operator on $F[x]$ .", "By Theorem REF , there exists a nonzero $r\\in F[x]$ such that $\\delta \\circ R=l_r$ .", "Let $c\\colon f\\mapsto R(f)(0)$ .", "Proposition REF implies that $c\\in M_r$ , hence $c\\in N_r$ by Proposition REF .", "Due to Remark REF , the corresponding operator $R=I\\circ l_r+c$ equals $J_{a}\\circ l_r$ for some $a\\in F$ ." ], [ "The moduli space", "Given an ascending sequence of closed embeddings of algebraic varieties $X_1\\hookrightarrow X_2\\hookrightarrow \\ldots ,$ their inductive limit $X=\\varinjlim X_n$ is called an ind-variety.", "An morphism $\\phi \\colon X\\rightarrow Y$ of ind-varieties $X=\\varinjlim X_n$ and $Y=\\varinjlim Y_n$ is a collection of morphisms $\\phi _n\\colon X_n\\rightarrow Y_{n^{\\prime }}$ for each $n\\in {\\mathbb {N}}$ , where $n^{\\prime }\\in {\\mathbb {N}}$ depends on $n$ , such that $\\phi _{n+1}|_{X_n}=\\phi _n$ .", "Two sequences $\\varinjlim X_n$ and $\\varinjlim X_n^\\prime $ on $X$ are called equivalent, if the identity map $\\varinjlim X_n\\cong \\varinjlim X_n^\\prime $ is an isomorphism of ind-varieties.", "If all the $X_i$ , $i\\in {\\mathbb {N}}$ , are affine (resp.", "quasi-affine, projective, irreducible) up to an equivalence, then $X$ is called affine (resp.", "quasi-affine, projective, irreducible).", "Let $X=\\varinjlim X_n$ and $U=\\varinjlim U_n$ be ind-varieties such that $U_n\\subset X_n$ is an open (resp.", "closed) subset for each $n\\in {\\mathbb {N}}$ .", "Then $U$ is called an open (resp.", "closed) ind-subvariety of $X$ .", "The direct product of ind-varieties $X,Y$ is defined by $X\\times Y=\\varinjlim X_n\\times Y_n$ .", "An ind-variety $G=\\varinjlim X_n$ is called an ind-group if it is endowed with a group structure such that multiplication and inverse maps are ind-morphisms.", "An action of an ind-group $G$ on an ind-variety $X$ is called regular, if the action map $G\\times X\\rightarrow X$ is an ind-morphism.", "Ind-varieties and ind-groups were introduced by I. Shafarevich in 1965 [10], see e.g.", "[8] for precise definitions and properties and [4] for further details.", "By ${\\mathbb {G}}_a={\\mathbb {G}}_a(F)$ we denote the additive group of the field $F$ .", "In this section we describe the moduli space of injective Rota–Baxter operators of weight zero on $F[x]$ in terms of ind-varieties and additive actions on them." ], [ "Structure of an ind-variety", " 4.1 For each $a\\in F$ and $n\\in {\\mathbb {Z}}_{\\ge 0}$ we denote $\\mathcal {R}^a_n =\\lbrace J_a\\circ l_r\\mid r\\in F[x]\\setminus \\lbrace 0\\rbrace , \\deg (r)\\le n\\rbrace ,\\\\\\mathcal {R}_n=\\bigcup _{a\\in F}\\mathcal {R}^a_n,\\quad \\mathcal {R}^a =\\bigcup _{n\\in {\\mathbb {N}}}\\mathcal {R}^a_n,\\quad \\mathcal {R}=\\bigcup _{a\\in F}\\mathcal {R}^a.$ By Theorem REF , $\\mathcal {R}$ is the set of injective Rota–Baxter operators of weight zero on $F[x]$ .", "Remark 4.2 The set $\\mathcal {R}^a$ , where $a\\in F$ , together with the zero operator forms a subspace in the space of operators.", "Lemma 4.3 Let $R=J_a\\circ l_r\\in \\mathcal {R}$ , and let $b\\in F$ be such that $R(f)(b)=0$ for all $f\\in F[x]$ .", "Then $b=a$ .", "Up to the change of coordinates $x\\mapsto x-a$ we may assume that $a=0$ .", "Let $r=r_0+\\ldots +r_kx^k$ for some $k\\in {\\mathbb {Z}}_{\\ge 0}$ and substitute $f=x^j$ , where $j\\in {\\mathbb {Z}}_{\\ge 0}$ : $0=R(f)(b)=I(rx^j)(b)=\\sum _{i=0}^k \\frac{r_ib^{i+j+1}}{i+j+1}=b^{j+1}\\sum _{i=0}^k \\frac{r_ib^{i}}{i+j+1}.$ Assume that $b\\ne 0.$ Then the vector $v=(r_0,r_1b,\\ldots ,r_kb^k)\\in F^{k+1}$ is orthogonal to each of vectors $u_j=\\left(\\frac{1}{j+1},\\frac{1}{j+2},\\ldots ,\\frac{1}{j+k+1}\\right)\\in F^{k+1}$ , where $j\\in {\\mathbb {Z}}_{\\ge 0}$ .", "Since the vectors $u_0,\\ldots ,u_{k}$ form a basis of $F^{k+1}$ , we have $v=0$ and $r=0$ , a contradiction.", "Example 4.4 For any $d\\in {\\mathbb {N}}$ the set $F[x]_{\\le d}\\setminus \\lbrace 0\\rbrace $ of nonzero polynomials of degree at most $d$ is an irreducible quasi-affine variety isomorphic to ${\\mathbb {A}}^{d+1}\\setminus \\lbrace 0\\rbrace $ .", "Thus, $F[x]\\setminus \\lbrace 0\\rbrace =\\varinjlim F[x]_{\\le d}\\setminus \\lbrace 0\\rbrace $ is a quasi-affine ind-variety.", "Proposition 4.5 The correspondence $F\\times (F[x]\\setminus \\lbrace 0\\rbrace )\\rightarrow \\mathcal {R}$ , $(a,r)\\mapsto J_a\\circ l_r$ , is bijective.", "In particular, $\\mathcal {R}$ admits a natural structure of a quasi-affine ind-variety.", "The operator action map $\\mathcal {R}\\times F[x]\\rightarrow F[x]$ is an ind-morphism.", "Let $(a,r),(a^{\\prime },r^{\\prime })\\in \\mathcal {R}$ be such that $J_a\\circ l_r=J_{a^{\\prime }}\\circ l_{r^{\\prime }}$ .", "Applying this operator to 1, we obtain $J_a(r)=J_{a^{\\prime }}(r^{\\prime })$ .", "Discarding the constant term, we have $I(r)=I(r^{\\prime })$ , hence $r=r^{\\prime }$ .", "By Lemma REF , $a=a^{\\prime }$ .", "Therefore, we may identify $\\mathcal {R}$ with $F\\times (F[x]\\setminus \\lbrace 0\\rbrace )$ .", "For any $d\\in {\\mathbb {N}}$ we have an irreducible quasi-affine variety $F\\times (F[x]_{\\le d}\\setminus \\lbrace 0\\rbrace )$ isomorphic to ${\\mathbb {A}}^1\\times ({\\mathbb {A}}^{d+1}\\setminus \\lbrace 0\\rbrace )$ .", "Thus, $\\mathcal {R}=\\bigcup _{d\\in {\\mathbb {N}}}F\\times (F[x]_{\\le d}\\setminus \\lbrace 0\\rbrace )$ is an ind-variety.", "The first assertion follows.", "The operator action map $\\mathcal {R}_n\\times F[x]_{\\le n}\\rightarrow F[x]_{\\le 2n+1}$ is given by the morphism $((a,r_0+\\ldots +r_{n}x^n), f_0+\\ldots +f_{n}x^n)\\mapsto \\sum _{i=0}^n\\sum _{j=0}^n \\frac{r_if_j(x^{i+j+1}-a^{i+j+1})}{i+j+1}.$ The second assertion follows.", "Definition 4.6 Let $X$ be an ind-variety.", "A subset $Z\\subset \\operatorname{Aut}(X)$ of automorphisms of $X$ together with a structure of an algebraic variety on $Z$ is called an algebraic family of automorphisms of $X$ if the action map $Z\\times X\\rightarrow X$ is a morphism.", "We say that the action of an ind-group $G$ on $X$ is universal if for any algebraic family $Z$ of automorphisms of $X$ , which is a subset of $G$ , the trivial embedding $Z\\rightarrow G$ is a morphism.", "Remark 4.7 A universal structure of an ind-variety on $X$ is unique up to equivalence, if it exists.", "In particular, for any affine algebraic variety $X$ there exists a structure of an ind-group on $\\operatorname{Aut}(X)$ such that its action on $X$ is universal, see [4].", "Question 4.8 Does the introduced structure of an ind-group on $\\mathcal {R}$ provide the universal action on $F[x]$ ?", "4.9 Given $b\\in F$ and $s\\in F[x]$ such that $s(b)=0$ , we denote $h^b_s\\colon \\mathcal {R}\\rightarrow \\mathcal {R},\\ J_a\\circ l_r\\mapsto J_a\\circ l_{r+r(b)s}, \\\\\\mathcal {H}^b = \\lbrace h^b_s\\mid s\\in F[x], s(b)=0\\rbrace .$ Then $\\mathcal {H}^b$ is a group, because $h^b_s\\circ h^b_{s^{\\prime }}=h^b_{s+s^{\\prime }}$ .", "Further, for any $k\\in {\\mathbb {Z}}_{>0}$ we introduce the following one-parameter subgroup in $\\mathcal {H}^b$ : $H^b_k =\\lbrace h^b_{\\gamma (x^k-b^k)}\\mid \\gamma \\in F\\rbrace .$ The $H^b_k$ -action on $\\mathcal {R}$ is a ${\\mathbb {G}}_a$ -action.", "Definition 4.10 An action of an ind-group $G$ on an irreducible ind-variety $X$ is said to be of generic modality $k$ , if there exists an open ind-subvariety $U\\subset X$ and a morphism $U\\rightarrow Z$ , where $Z$ is an algebraic variety of dimension $k$ , such that fibers of $\\pi $ are $G$ -orbits.", "If $X$ is an algebraic variety, then $k=\\mathrm {tr.deg.", "}F(X)^G$ .", "So, this definition conforms with [1].", "The following lemma shows that this definition is correct.", "Lemma 4.11 Consider the regular action of an ind-group $G$ on an irreducible ind-variety $X$ .", "Assume that there exist open ind-subvarieties $U,U^{\\prime }\\subset X$ and morphisms to algebraic varieties $\\pi \\colon U\\rightarrow Z$ and $\\pi ^{\\prime }\\colon U^{\\prime }\\rightarrow Z^{\\prime }$ such that fibers of both $\\pi $ and $\\pi ^{\\prime }$ are $G$ -orbits.", "Then $\\dim Z=\\dim Z^{\\prime }$ .", "Since $X$ is irreducible, the intersection $W=U\\cap U^{\\prime }$ is a non-empty open ind-subvariety in $X$ , as well as in both $U$ and $U^{\\prime }$ .", "Consider the restrictions of $\\pi $ and $\\pi ^{\\prime }$ on $W$ .", "Their images are dense subsets $\\pi (W)$ in $Z$ and $\\pi ^{\\prime }(W)$ in $Z^{\\prime }$ respectively.", "Let $X=\\varinjlim X_n$ , then $W=\\varinjlim W_n$ , where $W_n=X_n\\cap W$ for each $n\\in {\\mathbb {N}}$ .", "So, for some $n\\in {\\mathbb {N}}$ the images $\\pi (W_n)$ and $\\pi ^{\\prime }(W_n)$ are dense in $Z$ and $Z^{\\prime }$ respectively.", "Since the generic fibers of $\\pi |_{W_n}$ and $\\pi ^{\\prime }|_{W_n}$ have the same dimension, the assertion follows.", "Definition 4.12 By an additive structure of generic modality $k$ on an ind-variety $X$ we mean an action of an abelian unipotent ind-group $G$ on $X$ of generic modality $k$ .", "Proposition 4.13 Let $a,b\\in F$ .", "The group $\\mathcal {H}^b$ is an abelian unipotent ind-group.", "Namely, $\\mathcal {H}^b= \\varinjlim _k (H^b_1\\times \\ldots \\times H^b_k) \\cong \\varinjlim _k ({\\mathbb {G}}_a)^k.$ Consider the evaluation map $\\pi _b\\colon \\mathcal {R}^a\\rightarrow F[x]/(b)\\cong F$ , $J_a\\circ l_r \\mapsto r(b)$ .", "Then $\\pi _b^{-1}(c)$ is an $\\mathcal {H}^b$ -orbit for any $c\\ne 0$ and $\\pi _b^{-1}(c)\\cap \\mathcal {R}_k$ is an $H_1^b\\times \\ldots \\times H_k^b$ -orbit.", "Thus, $\\mathcal {H}^b$ (resp.", "$H^b_k$ ) acts on $\\mathcal {R}^a$ (resp.", "$\\mathcal {R}^a_k$ ) with generic modality one.", "For any $k\\in {\\mathbb {N}}, c_1,\\ldots ,c_k\\in F$ , $r\\in F[x]$ , there holds $\\big (H_1^b(c_1)\\circ \\ldots \\circ H_k^b(c_k)\\big )(J_a\\circ l_r) = J_a\\circ l_{r^{\\prime }},$ where $r^{\\prime } = r+r(b)\\sum \\limits _{i=1}^k c_i(x^i-b^i)$ .", "The commutativity follows.", "Thus, $ H_1^b\\times \\ldots \\times H_k^b\\cong ({\\mathbb {G}}_a)^k$ , which is known to be unipotent.", "So, we have (i).", "Consider the map $\\phi ^k\\colon \\mathcal {R}^a_k\\rightarrow F$ , $J_a\\circ l_r\\mapsto r(b)$ .", "Each fiber of this map, except the preimage of 0, is an $\\big (H_1^b\\times \\ldots \\times H_k^b\\big )$ -orbit.", "The maps $\\lbrace \\phi ^k\\mid k\\in {\\mathbb {Z}}_{>0}\\rbrace $ provide the ind-morphism $\\mathcal {R}^a\\rightarrow F$ , a generic fiber of which is an $\\mathcal {H}^b$ -orbit, hence (ii).", "Corollary 4.14 The ind-variety $\\mathcal {R}$ admits an additive structure of generic modality two.", "Let us fix $b\\in F$ and consider an open ind-subvariety $U=\\lbrace J_a\\circ l_r\\mid a\\in F,\\, r\\in F[x],\\, r(b)\\ne 0\\rbrace $ in $\\mathcal {R}$ .", "The fibers of the map $\\pi \\colon U\\rightarrow F\\times F^\\times ,\\, J_a\\circ l_r\\mapsto (a,r(b))$ are $\\mathcal {H}^b$ -orbits.", "The assertion follows.", "Here we establish a transitive action on ordered tuples of operators in $\\mathcal {R}^a$ .", "Proposition 4.15 The group $\\mathcal {H}=\\langle \\mathcal {H}^b\\mid b\\in F\\rangle $ acts transitively on ordered $m$ -tuples of linearly independent operators in $\\mathcal {R}^a$ for any $m\\in {\\mathbb {N}}$ .", "4.16 For any pairwise distinct $b_1,\\ldots ,b_m\\in F$ and arbitrary $c_1,\\ldots $ , $c_m\\in F^\\times $ we define a set $A(b_1,\\ldots ,b_m\\mid c_1,\\ldots ,c_m)= \\left\\lbrace (J_a\\circ r_1,\\ldots ,J_a\\circ r_m)\\mid r_i\\in F[x]\\setminus \\lbrace 0\\rbrace ,\\ r_i(b_j)= \\delta _{ij}c_i\\right\\rbrace ,$ where $\\delta _{ij} = {\\left\\lbrace \\begin{array}{ll} 1, & i=j, \\\\0, & i\\ne j.", "\\end{array}\\right.", "}$ Lemma 4.17 Any $m$ -tuple of linearly independent operators in $\\mathcal {R}^a$ can be sent to some $A(b_1,\\ldots ,b_m\\mid c_1,\\ldots ,c_m)$ by an element of $\\mathcal {H}$ .", "Consider such an $m$ -tuple $(R_1,\\ldots ,R_m)$ and let $R_i=J_a\\circ l_{r_i}$ for some $r_i\\in F[x]$ , where $i=1,\\ldots ,m$ .", "Then $r_1,\\ldots ,r_m$ are linearly independent.", "There exist $b_1,\\ldots ,b_m$ such that $\\det ((r_i(b_j)))\\ne 0$ .", "Given $k,l\\in \\lbrace 1,\\ldots ,m\\rbrace $ such that $k\\ne l$ , let us denote $s_l=\\prod _{j\\ne l}\\frac{x-b_j}{b_l-b_j}$ and consider the element $h^{b_k}_{\\lambda s_l}\\in \\mathcal {H}^{b_k}$ .", "Applying it to $R_1,\\ldots ,R_m$ , we obtain an elementary transformation of the matrix $(r_i(b_j))$ : the $k$ th column is added to the $l$ th one with the coefficient $\\lambda $ .", "Using such transformations for suitable $k,l$ , and $\\lambda $ , we may leave just one nonzero value at each row of $(r_i(b_j))$ .", "The assertion follows.", "Lemma 4.18 The group $\\mathcal {H}$ acts on $A(b_1,\\ldots ,b_m\\mid c_1,\\ldots ,c_m)$ transitively.", "Let $(R_1,\\ldots ,R_m),(R_1^\\prime ,\\ldots ,R_m^\\prime )\\in A(b_1,\\ldots ,b_m\\mid c_1,\\ldots ,c_m).$ We have $R_1=J_a\\circ l_r$ and $R_1^\\prime =J_a\\circ l_{r^{\\prime }}$ for some $r,r^{\\prime }\\in F[x]$ such that $r(b_1)=r^{\\prime }(b_1)=c_1$ .", "Then $r=q\\prod _{i=2}^m(x-b_i),\\quad r^{\\prime }=q^{\\prime }\\prod _{i=2}^m(x-b_i)$ for some $q,q^{\\prime }\\in F[x]$ .", "Thus, $h^{b_1}_s$ for $s=\\frac{r^{\\prime }-r}{c_1}$ sends $(R_1,R_2,\\ldots ,R_m)$ to $(R_1^\\prime ,R_2,\\ldots ,R_m)$ .", "Acting in a similar way successively for $R_2,\\ldots , R_m$ , we may send $(R_1,\\ldots ,R_m)$ to $(R_1^\\prime ,\\ldots ,R_m^\\prime )$ .", "Lemma 4.19 The sets $A(b_1,\\ldots ,b_m\\mid c_1,\\ldots ,c_m)$ and $A(b_1^\\prime ,\\ldots ,b_m^\\prime \\mid c_1^\\prime ,\\ldots ,c_m^\\prime )$ have nonempty intersection if the sets $\\lbrace b_1,\\ldots ,b_m\\rbrace $ and $\\lbrace b_1^\\prime ,\\ldots ,b_m^\\prime \\rbrace $ are disjoint.", "It is enough to take $r_k$ such that $r_k(b_i)=r_k(b_i^\\prime )=0$ for $i\\ne k$ and $r_k(b_k)=c_k$ , $r_k(b_k^\\prime )=c_k^\\prime $ for each $k=1,\\ldots ,m$ .", "Then the tuple $(J_a\\circ l_{r_1},\\ldots ,J_a\\circ l_{r_m})$ belongs to both $A(b_1,\\ldots ,b_m\\mid c_1,\\ldots ,c_m)$ and $A(b_1^\\prime ,\\ldots ,b_m^\\prime \\mid c_1^\\prime ,\\ldots ,c_m^\\prime )$ .", "Let $R=(R_1,\\ldots ,R_m)$ be an $m$ -tuple of linearly independent operators in $\\mathcal {R}^b$ .", "By Lemma REF , we may send it to some tuple $R^{\\prime }$ in $A_1=A(b_1,\\ldots ,b_m\\mid c_1,\\ldots ,c_m)$ .", "Let us take a set $\\lbrace b_1^\\prime ,\\ldots ,b_m^\\prime \\rbrace $ disjoint with both $\\lbrace b_1,\\ldots ,b_m\\rbrace $ and $\\lbrace 1,\\ldots ,m\\rbrace $ .", "By Lemmas REF and REF , we may send $R^{\\prime }$ to some tuple $R^{\\prime \\prime }\\in A(b_1^\\prime ,\\ldots ,b_m^\\prime \\mid 1,\\ldots ,1)$ and then to a tuple $R^{(3)}\\in A(1,\\ldots ,m\\mid 1,\\ldots ,1)$ .", "Thus, any tuple can be sent to the set $A(1,\\ldots ,m\\mid 1,\\ldots ,1)$ , which enjoys the transitive $\\mathcal {H}$ -action.", "Proposition is proved.", "Remark 4.20 The action of the group $\\mathcal {H}$ is linear.", "In particular, if a nontrivial linear combination of operators $R_1,\\ldots ,R_m$ in $\\mathcal {R}^a$ is zero, then the same linear combination of $h.R_1,\\ldots ,h.R_m$ is zero for any element $h\\in \\mathcal {H}$ as well.", "Definition 4.21 A group $G$ is said to act on a set $S$ infinitely transitively if it acts transitively on the set of ordered $m$ -tuples of pairwise distinct points in $S$ for any $m\\in {\\mathbb {N}}$ .", "4.22 For each $a\\in F$ and $s\\in F[x]$ such that $s(a)=0$ we introduce the linear operator $h^{b,2}_s\\colon \\mathcal {R}\\rightarrow \\mathcal {R},\\; J_a\\circ l_r\\mapsto J_a\\circ l_{r+r(b)^2s}.$ Theorem 4.23 The group generated by ${\\mathbb {G}}_a$ -actions $\\big \\lbrace h^{b,2}_s,h^b_s\\mid b\\in F,\\,s\\in F[x],\\,s(b)=0\\big \\rbrace $ acts on $\\mathcal {R}^a$ infinitely transitively for any $a\\in F$ .", "Let us prove by induction by $m$ that for any $m\\in {\\mathbb {N}}$ the introduced group acts transitively on $m$ -tuples of pairwise distinct operators in $\\mathcal {R}$ .", "If $m=1$ , then the assertion follows from Proposition REF .", "Assume that the assertion holds for $(m-1)$ -tuples of operators.", "Using Proposition REF , it is enough to prove that any $m$ -tuple $(R_1,\\ldots ,R_m)$ of pairwise distinct operators can be sent to an $m$ -tuple of linearly independent ones.", "By induction, we may send $R_1,\\ldots $ , $R_{m-1}$ to $J_a\\circ l_1,J_a\\circ l_x,\\ldots ,J_a\\circ l_{x^{m-2}}$ respectively.", "So, we assume that $R_i=J_a\\circ l_{x^{i-1}}$ for $i=1,\\ldots ,m-1$ .", "If $R_m$ is linearly independent with $R_1,\\ldots ,R_{m-1}$ , we are done by Proposition REF .", "Otherwise, we have $R_m=J_a\\circ l_r$ for some $r=\\sum _{i=0}^{m-2} r_ix^i\\in F[x]_{\\le m-2}$ .", "The group $\\mathcal {H}^0$ fixes $R_2,\\ldots ,R_{m-1}$ .", "The element $h^{0,2}_{x^m}$ sends $R_1$ and $R_m$ to $J_a\\circ l_{1+x^m}$ and $J_a\\circ l_{r+r_0^2x^m}$ respectively.", "If $r_0\\notin \\lbrace 0,1\\rbrace $ , then the images of $R_1,\\ldots ,R_m$ are linearly independent.", "If $r_0=0$ , then by the inductive hypothesis we may send $R_2,\\ldots ,R_m$ into any $(m-1)$ -tuple of linearly independent operators.", "This is an open condition, hence we may keep $R_1$ linearly independent with $R_2,\\ldots ,R_m$ as well.", "Then we are done.", "Thus, we assume that $r_0=1$ .", "By Proposition REF , we can permute operators $R_1,\\ldots ,R_{m-1}$ .", "By Remark REF , coefficients $r_0,\\ldots ,r_{m-2}$ are permuted accordingly.", "Hence we may repeat the argument above for each coefficient and assume that $r_0=r_1=\\ldots =r_{m-2}=1$ , so $r=1+x+\\ldots +x^{m-2}$ .", "Then the element $h^{1,2}_{x^m}$ sends $R_1,\\ldots ,R_m$ into a tuple of linear independent operators.", "Indeed, for the map $\\varphi \\colon F[x]\\rightarrow F[x],\\quad f\\mapsto f+f(1)^2(x^m-1),$ we have $\\varphi (x^i) & = x^i + (x^m-1),\\ i\\in \\lbrace 0,\\ldots ,m-2\\rbrace ,\\\\\\varphi (r) & = 1+x+\\ldots +x^{m-2} + (m-1)^2(x^m-1).$ Thus, the images $\\varphi (1),\\varphi (x),\\ldots ,\\varphi (x^{m-2}),\\varphi (r)$ are linearly independent.", "Here we describe the induced action of $\\operatorname{Aut}(F[x])$ on $\\mathcal {R}$ .", "This allows us to present a collection of ${\\mathbb {G}}_a$ -actions on $\\mathcal {R}$ such that the group generated by them acts transitively on $\\mathcal {R}$ .", "4.24 Consider the action of $\\operatorname{Aut}(F[x])$ on $\\mathcal {R}$ by conjugation.", "Since $\\operatorname{Aut}(F[x])=\\lbrace x\\mapsto \\mu x+\\nu \\mid \\mu \\in F^\\times , \\nu \\in F\\rbrace \\cong {\\mathbb {G}}_a\\rtimes {\\mathbb {G}}_m,$ this provides a ${\\mathbb {G}}_a$ - and ${\\mathbb {G}}_m$ -action on $\\mathcal {R}$ , which we denote by $G_a$ and $G_m$ respectively.", "More formally, we have $G_a\\colon & {\\mathbb {G}}_a\\times \\mathcal {R}\\rightarrow \\mathcal {R},\\quad (\\nu ,J_{a}\\circ l_{r(x)})\\mapsto J_{a-\\nu }\\circ l_{r(x+\\nu )},\\\\G_m\\colon & {\\mathbb {G}}_m\\times \\mathcal {R}\\rightarrow \\mathcal {R},\\quad (\\mu ,J_{a}\\circ l_{r(x)})\\mapsto J_{\\frac{a}{\\mu }}\\circ l_{r(\\mu x)}.$ Remark 4.25 The $\\operatorname{Aut}(F[x])$ -orbits on $\\mathcal {R}$ are parameterized by polynomials in $F[x]$ , which leading coefficient equals one.", "Given such a polynomial $r=x^n+\\ldots \\in F[x]$ , the corresponding orbit is $\\lbrace J_a\\circ l_{r(\\mu (x-a))}\\mid a\\in F, \\mu \\in F^\\times \\rbrace .$ Proposition 4.26 The action map $\\operatorname{Aut}(F[x])\\times \\mathcal {R}\\rightarrow \\mathcal {R}$ is an ind-morphism.", "Moreover, an element $g\\in \\operatorname{Aut}(F[x])$ sends each subset $\\mathcal {R}^a$ , where $a\\in F$ , to some subset $\\mathcal {R}^{a^{\\prime }}$ , where $a^{\\prime }\\in F$ .", "It is easy to check that $g$ restricted to $\\mathcal {R}^k$ , where $k\\in {\\mathbb {Z}}_{\\ge 0}$ , is an automorphism of a quasi-affine variety.", "The first statement follows.", "Assume that $g$ sends an operator $J_a\\circ l_r$ to $J_{a^{\\prime }}\\circ l_{r^{\\prime }}$ for some nonzero $r,r^{\\prime }\\in F[x]$ .", "Then $a^{\\prime }$ does not depend on $r$ , and the second statement follows.", "Proposition 4.27 The group $\\langle G_a,\\mathcal {H}\\rangle $ acts transitively on $\\mathcal {R}$ .", "We have to prove that an element $R_1=J_{a_1}\\circ l_{r_1}$ can be sent to $R_2=J_{a_2}\\circ l_{r_2}$ for any $a_1,a_2\\in F$ and $r_1,r_2\\in F[x]\\setminus \\lbrace 0\\rbrace $ .", "Since $G_a(a_1-a_2)(R_1)\\in \\mathcal {R}^{a_2}$ , we may assume that $a_1=a_2$ .", "Since $r_1$ and $r_2$ are nonzero and the field $F$ is infinite, there exist $a,a^{\\prime }\\in F$ which are not roots of the polynomials $r_1,r_2$ .", "Then for some $\\gamma \\in F$ there holds $r_1(a^{\\prime })+\\gamma r_1(a)(a^{\\prime }-a)=r_2(a^{\\prime }).$ Thus, by Proposition REF , $h^a_\\gamma (R_1)$ and $R_2$ belong to the same orbit of $\\mathcal {H}^{a^{\\prime }}$ .", "The statement follows." ] ]

[ [ "Learning LWF Chain Graphs: an Order Independent Algorithm" ], [ "Abstract LWF chain graphs combine directed acyclic graphs and undirected graphs.", "We present a PC-like algorithm that finds the structure of chain graphs under the faithfulness assumption to resolve the problem of scalability of the proposed algorithm by Studeny (1997).", "We prove that our PC-like algorithm is order dependent, in the sense that the output can depend on the order in which the variables are given.", "This order dependence can be very pronounced in high-dimensional settings.", "We propose two modifications of the PC-like algorithm that remove part or all of this order dependence.", "Simulation results under a variety of settings demonstrate the competitive performance of the PC-like algorithms in comparison with the decomposition-based method, called LCD algorithm, proposed by Ma et al.", "(2008) in low-dimensional settings and improved performance in high-dimensional settings." ], [ "Introduction", "Probabilistic graphical models (PGMs) are now widely accepted as a powerful and mature tool for reasoning and decision making under uncertainty.", "A PGM is a compact representation of a joint probability distribution, from which we can obtain marginal and conditional probabilities .", "In fact, any PGM consists of two main components: (1) a graph that defines the structure of that model; and (2) a joint distribution over random variables of the model.", "Two types of graphical representations of distributions are commonly used, namely, Bayesian networks and Markov networks.", "Both families encompass the properties of factorization and independence, but they differ in the set of independencies they can encode and the factorization of the distribution that they induce.", "Currently systems containing both causal and non-causal relationships are mostly modeled with directed acyclic graphs (DAGs).", "Chain graphs (CGs) are a type of mixed graphs, admitting both directed and undirected edges, which contain no partially directed cycles.", "Figure: Learning LWF CGs with PC-like algorithm.So, CGs may contain two types of edges, the directed type that corresponds to the causal relationship in DAGs and a second type of edge representing a symmetric relationship .", "LWF Chain graphs were introduced by Lauritzen, Wermuth and Frydenberg , as a generalization of graphical models based on undirected graphs and DAGs and widely studied e.g., in , , , , , , , , , , , among others.. From the causality point of view, in an LWF CG directed edges represent direct causal effects, and undirected edges represent causal effects due to interference , , and .", "One important aspect of PGMs is the possibility of learning the structure of models directly from sampled data.", "Six constraint-based learning algorithms, that use a statistical analysis to test the presence of a conditional independency, exist for learning LWF CGs: (1) the inductive causation like (IC-like) algorithm , (2) the decomposition-based algorithm called LCD (Learn Chain graphs via Decomposition) , (3) the answer set programming (ASP) algorithm , (4) the inclusion optimal (CKES) algorithm , and (5) the local structure learning of chain graphs with the false discovery rate control , (5) the local structure learning of chain graphs with the false discovery rate control , and (6) the Markov blanket discovery (MbLWF) algorithm .", "Similar to the inductive causation (IC) algorithm , the IC-like algorithm cannot be applied to large numbers of variables because for testing whether there is a set separating $X$ and $Y$ in the skeleton recovery, the IC-like algorithm might search all $2^{n-2}$ subsets of all $n$ random variables not including $X$ and $Y$ .", "In order to overcome the scalability of the IC-like algorithm, we propose a constraint-based method for learning the structural of chain graphs based on the idea of the PC algorithm proposed by Peter Spirtes and Clark Glymour , which is used for learning the structure of Bayesian networks (BNs).", "Our method modifies the IC-like algorithm to make it computationally feasible in the phase of skeleton recovery and to avoid the time consuming procedure of complex recovery.", "We prove that the proposed PC-like algorithm in this paper is order dependent, in the sense that the output can depend on the order in which the variables are given.", "We propose several modifications of the PC-like algorithm that remove part or all of this order dependence, but do not change the result when perfect conditional independence information is used.", "When applied to data, the modified algorithms are partly or fully order independent.", "Our proposed algorithm, called the SPC4LWF (Stable PC-like for LWF CGs), similarly to the LCD algorithm, is able to exploit parallel computations for scaling up the task of learning LWF CGs.", "This will enable effective LWF chain graph discovery on large/high-dimensional datasets.", "In fact, lower complexity, higher power of computational independence test, better learned structure quality, along with the ability of exploiting parallel computing make our proposed algorithm in this paper more desirable and suitable for big data analysis when LWF chain graphs are being used.", "Our main contributions are the following: (1) We propose a PC-like algorithm for learning the structure of LWF CGs under the faithfulness assumption that includes two main procedures: (i) a feasible method for learning CG skeletons following the same idea of the PC algorithm, (ii) a polynomial time procedure for complex recovery similar to the proposed approach in .", "The whole procedure is shown in Figure REF .", "The algorithm and its soundness are discussed in section .", "(2) In section , we show that our proposed PC-like algorithm in section is order dependent.", "Then, we propose two modifications of this algorithm that remove part or all of this order dependence.", "The soundness of modified algorithms are discussed in section .", "(3) We experimentally compare the performance of our proposed PC-like algorithms with the LCD algorithm in section , and we show that the PC-like algorithms are comparable to the LCD algorithm in low-dimensional settings and superior in high-dimensional settings in terms of error measures and runtime.", "(4) We release supplementary material (https://github.com/majavid/PC4LWF2020) including data and an R package that implements the proposed algorithms." ], [ "Definitions and Concepts", "Below, we briefly list some of the central concepts used in this paper (see for more details).", "In this paper, we consider graphs containing both directed ($\\rightarrow $ ) and undirected ($-$ ) edges and largely use the terminology of , where the reader can also find further details.", "Below we briefly list some of the central concepts used in this paper.", "A path in $G$ is a sequence of its distinct nodes $v_1,v_2,\\dots ,v_k, k\\ge 1,$ such that $\\lbrace v_i,v_{i+1}\\rbrace $ is an edge in $G$ for every $i = 1,\\dots , k - 1$ .", "It is called a cycle if $v_{k+1}\\equiv v_1$ , and $k\\ge 3$ .", "A chord of a cycle $C$ is an edge not in $C$ whose endpoints lie in $C$ .", "A chordless cycle in $G$ is a cycle of length at least 4 in $G$ that has no chord (that is, the cycle is an induced subgraph).", "A cycle of length 3 is both chordal and chordless.", "A partially directed cycle (or semi-directed cycle) in a graph $G$ is a sequence of $n$ distinct vertices $v_1,v_2,\\dots ,v_n (n\\ge 3)$ , and $v_{n+1}\\equiv v_1$ , such that (a) $\\forall i (1\\le i\\le n)$ either $v_i-v_{i+1}$ or $v_i\\rightarrow v_{i+1}$ , and (b) $\\exists j (1\\le j\\le n)$ such that $v_j\\rightarrow v_{j+1}$ .", "If there is a path from $a$ to $b$ we say that $a$ leads to $b$ and write $a\\mapsto b$ .", "The vertices $a$ such that $a\\mapsto b$ and $b\\lnot \\mapsto a$ are the ancestors $an(b)$ of $b$ , and the descendants $de(a)$ of $a$ are the vertices $b$ such that $a\\mapsto b$ and $b\\lnot \\mapsto a$ .", "The non-descendants are $nd(a) = V\\setminus (de(a) \\cup \\lbrace a\\rbrace )$ .", "If there is an arrow from $a$ pointing towards $b$ , $a$ is said to be a parent of $b$ .", "If there is an undirected edge between $a$ and $b$ , $a$ and $b$ are said to be adjacent or neighbors.", "The boundary $bd(A)$ of a subset $A$ of vertices is the set of vertices in $V\\setminus A$ that are parents or neighbors to vertices in $A$ .", "The closure of $A$ is $cl(A)=bd(A)\\cup A$ .", "If $bd(a)\\subseteq A$ , for all $a\\in A$ we say that $A$ is an ancestral set.", "The smallest ancestral set containing $A$ is denoted by $An(A)$ .", "An LWF chain graph is a graph in which there are no partially directed cycles.", "The chain components $\\mathcal {T}$ of a chain graph are the connected components of the undirected graph obtained by removing all directed edges from the chain graph.", "A minimal complex (or simply a complex or a U-structure) in a chain graph is an induced subgraph of the form $a\\rightarrow v_1-\\cdots \\cdots -v_r\\leftarrow b$ .", "The skeleton (underlying graph) of an LWF CG $G$ is obtained from $G$ by changing all directed edges of $G$ into undirected edges.", "For a chain graph $G$ we define its moral graph $G^m$ as the undirected graph with the same vertex set but with $\\alpha $ and $\\beta $ adjacent in $G^m$ if and only if either $\\alpha \\rightarrow \\beta $ , or $\\alpha - \\beta $ , or $\\beta \\rightarrow \\alpha $ or if there are $\\gamma _1,\\gamma _2$ in the same chain component such that $\\alpha \\rightarrow \\gamma _1$ and $\\beta \\rightarrow \\gamma _2$ .", "Global Markov property for LWF chain graphs: For any triple $(A, B,S)$ of disjoint subsets of $V$ such that $S$ separates $A$ from $B$ in $(G_{An(A\\cup B\\cup S)})^m$ , in the moral graph of the smallest ancestral set containing $A\\cup B\\cup S$ , we have $A \\!\\perp \\!\\!\\!\\perp B | S$ i.e., $A$ is independent of $B$ given $S$ .", "We say $S$ $c$ -separates $A$ from $B$ in the chain graph $G$ .", "We say that two LWF CGs $G$ and $H$ are Markov equivalent or that they are in the same Markov equivalence class if they induce the same conditional independence restrictions.", "Two CGs $G$ and $H$ are Markov equivalent if and only if they have the same skeletons and the same minimal complexes .", "Every class of Markov equivalent CGs has a unique CG with the greatest number of undirected edges.", "This graph is called the largest CG (LCG) of the corresponding class of Markov equivalent CGs ." ], [ "PC4LWF: a PC-Like Algorithm for Learning LWF CGs", "In this section, we discuss how the IC-like algorithm can be modified to obtain a computationally feasible algorithm for LWF CGs recovery.", "A brief review of the IC-like algorithm is presented first, then we present a PC-like algorithm, called PC4LWF, which is a constraint-based algorithm that learns a CG from a probability distribution faithful to some CG.", "The IC-like algorithm is a constraint-based algorithm proposed for LWF CGs and is based on three sequential phases.", "The first phase finds the adjacencies (skeleton recovery), the second phase orients the edges that must be oriented the same in every CG in the Markov equivalence class (complex recovery), and the third phase transforms this graph into the largest CG (LCG recovery).", "The skeleton recovery of the IC-like algorithm works as follows: construct an undirected graph $H$ such that vertices $u$ and $v$ are connected with an undirected edge if and only if no set $S_{uv}$ can be found such that $u\\!\\perp \\!\\!\\!\\perp v|S_{uv}$ .", "This procedure is very inefficient because this requires a number of independence tests that increases exponentially with the number of vertices.", "In other words, to determine whether there is a set separating $u$ and $v$ , we might search all $2^{n-2}$ subsets of all $n$ random variables excluding $u$ and $v$ .", "So, the complexity for investigating each possible edge in the skeleton is $O(2^n)$ and hence the complexity for constructing the skeleton is $O(n^22^n)$ , where $n$ is the number of vertices in the LWF CG.", "Since it is enough to find one $S$ making $u$ and $v$ independent to remove the undirected edge $u\\mathrel {[baseline=-.5ex][-] (0,0)--(.5,0);}v$ , one obvious short-cut is to do the tests in some order, and skip unnecessary tests.", "In the PC algorithm for BNs the revised edge removal step is done as shown in Algorithm .", "[t] Edge-removal step of the PC algorithm for BNs $i\\leftarrow 0$ $|V_H|-2$ possible Select any ordered pair of nodes $u$ and $v$ in $H$ such that $u\\in ad_H(v)$ , $|ad_H(u)\\setminus v|\\ge i$ ($ad_H(x):=\\lbrace y\\in V| x\\rightarrow y, y\\rightarrow x, \\textrm { or }x-y\\rbrace $ ) there exists $S\\subseteq (ad_H(u)\\setminus v)$ s.t.", "$|S|=i$ and $u\\perp \\!\\!\\!\\perp _p v|S$ (i.e., $u$ is independent of $v$ given $S$ in the probability distribution $p$ ) Set $S_{uv} = S_{vu} = S$ Remove the edge $u - v$ from $H$ Since the PC algorithm only looks at adjacencies of $u$ and $v$ in the current stage of the algorithm, rather than all possible subsets, the PC algorithm performs fewer independence tests compared to the IC algorithm.", "The complexity of the PC algorithm for DAGs is difficult to evaluate exactly, but with the sparseness assumption the worst case is with high probability bounded by $O(n^q)$ , where $n$ is the number of vertices and $q$ is the maximum number of the adjacent vertices of the true underlying DAG .", "Our main intuition is that replacing the skeleton recovery phase in the IC-like algorithm with a PC-like approach will speed up this phase and make it computationally scalable when the true underlying LWF CG is sparse (see the skeleton recovery phase of Algorithm ).", "[t] PC-like algorithm for LWF CGs a set $V$ of nodes and a probability distribution $p$ faithful to an unknown LWF CG $G$ .", "The pattern of $G$ .", "Let $H$ denote the complete undirected graph over $V$ Skeleton Recovery $i\\leftarrow 0$ $|V_H|-2$ possible Select any ordered pair of nodes $u$ and $v$ in $H$ such that $u\\in ad_H(v)$ and $|ad_H(u)\\setminus v|\\ge i$ there exists $S\\subseteq (ad_H(u)\\setminus v)$ s.t.", "$|S|=i$ and $u\\perp \\!\\!\\!\\perp _p v|S$ (i.e., $u$ is independent of $v$ given $S$ in the probability distribution $p$ ) Set $S_{uv} = S_{vu} = S$ Remove the edge $u - v$ from $H$ Complex Recovery from Initialize $H^*=H$ each vertex pair $\\lbrace u,v\\rbrace $ s.t.", "$u$ and $v$ are not adjacent in $H$ each $u-w$ in $H^*$ $u\\lnot \\perp \\!\\!\\!\\perp _p v|(S_{uv}\\cup \\lbrace w\\rbrace )$ Orient $u - w$ as $u\\rightarrow w$ in $H^*$ Take the pattern of $H^*$ To get the pattern of $H^*$ in line 19, at each step, we consider a pair of candidate complex arrows $u_1 \\rightarrow w_1$ and $u_2\\rightarrow w_2$ with $u_1 \\ne u_2$ , then we check whether there is an undirected path from $w_1$ to $w_2$ such that none of its intermediate vertices is adjacent to either $u_1$ or $u_2$ .", "If there exists such a path, then $u_1 \\rightarrow w_1$ and $u_2\\rightarrow w_2$ are labeled (as complex arrows).", "We repeat this procedure until all possible candidate pairs are examined.", "The pattern is then obtained by removing directions of all unlabeled as complex arrows in $H^*$ .", "The looping procedure of the IC-like algorithm for complex recovery is computationally expensive.", "We use a polynomial time approach similar to the proposed algorithm by to reduce the computational cost of the complex recovery (see the complex recovery phase of Algorithm ).", "Finally, the IC-like algorithm uses three basic rules, namely the transitivity rule, the necessity rule, and the double-cycle rule, for changing the obtained pattern in the previous phase into the corresponding largest CG (see for details).", "When we have perfect conditional independence, both IC-like and LCD algorithms recover the structure of the model correctly if the probability distribution of the data is faithful to some LWF CGs i.e., all conditional independencies among variables can be represented by an LWF CG.", "The entire process is formally described in Algorithm .", "The correctness of Algorithm is proved in Appendix .", "Computational Complexity Analysis of Algorithm .", "The complexity of the algorithm for a graph $G$ is bounded by the largest degree in $G$ .", "Let $k$ be the maximal degree of any vertex and let $n$ be the number of vertices.", "Then in the worst case the number of conditional independence tests required by the algorithm is bounded by $2\\binom{n}{2}\\sum _{i=0}^k\\binom{n-2}{i}\\le \\frac{n^2(n-2)^k}{(k-1)!", "}$ To derive the inequality, use induction on $k$ .", "So, Algorithm has a worst-case running time of $O(n^{k+2})$ .", "This is a loose upper bound even in the worst case; it assumes that in the worst case for $n$ and $k$ , no two variables are c-separated by a set of less than cardinality $k$ , and for many values of $n$ and $k$ we have been unable to find graphs with that property.", "The worse case is rare, and the average number of conditional independence tests required for graphs of maximal degree $k$ is much smaller.", "In practice it is possible to recover sparse graphs with as many as a hundred variables as shown in section ." ], [ "STABLE PC-LIKE ALGORITHM", "In this section, we show that the PC-like algorithm proposed in the previous section is order dependent, in the sense that the output can depend on the order in which the variables are given.", "Proof of theorems in this section can be found in Appendix .", "In applications, we do not have perfect conditional independence information.", "Instead, we assume that we have an i.i.d.", "sample of size $n$ of variables $V = (X_1,\\dots ,X_p)$ .", "In the PC-like algorithm all conditional independence queries are estimated by statistical conditional independence tests at some pre-specified significance level (p value) $\\alpha $ .", "For example, if the distribution of $V$ is multivariate Gaussian, one can test for zero partial correlation, see, e.g., .", "Hence, we use the $\\mathsf {gaussCItest()}$ function from the R package $\\mathsf {pcalg}$ throughout this paper.", "Let order($V$ ) denote an ordering on the variables in $V$ .", "We now consider the role of order($V$ ) in every step of the Algorithm .", "In the skeleton recovery phase of the PC-like algorithm, the order of variables affects the estimation of the skeleton and the separating sets.", "In particular, as noted for the special case of BNs in , for each level of $i$ , the order of variables determines the order in which pairs of adjacent vertices and subsets $S$ of their adjacency sets are considered (see lines 4 and 5 in Algorithm ).", "The skeleton $H$ is updated after each edge removal.", "Hence, the adjacency sets typically change within one level of $i$ , and this affects which other conditional independencies are checked, since the algorithm only conditions on subsets of the adjacency sets.", "When we have perfect conditional independence information, all orderings on the variables lead to the same output.", "In the sample version, however, we typically make mistakes in keeping or removing edges, because conditional independence relationships have to be estimated from data.", "In such cases, the resulting changes in the adjacency sets can lead to different skeletons, as illustrated in Example .", "Moreover, different variable orderings can lead to different separating sets in the skeleton recovery phase.", "When we have perfect conditional independence information, this is not important, because any valid separating set leads to the correct U-structure decision in the complex recovery phase.", "In the sample version, however, different separating sets in the skeleton recovery phase may yield different decisions about U-structures in the complex recovery phase.", "This is illustrated in Example REF .", "(Order dependent skeleton of the PC4LWF algorithm.)", "Suppose that the distribution of $V = \\lbrace a,b,c,d,e\\rbrace $ is faithful to the DAG in Figure REF (a).", "This DAG encodes the following conditional independencies with minimal separating sets: $a\\perp \\!\\!\\!\\perp d|\\lbrace b,c\\rbrace $ and $a\\perp \\!\\!\\!\\perp e|\\lbrace b,c\\rbrace $ .", "Suppose that we have an i.i.d.", "sample of $(a,b,c,d,e)$ , and that the following conditional independencies with minimal separating sets are judged to hold at some significance level $\\alpha $ : $a\\perp \\!\\!\\!\\perp d|\\lbrace b,c\\rbrace $ , $a\\perp \\!\\!\\!\\perp e|\\lbrace b,c,d\\rbrace $ , and $c\\perp \\!\\!\\!\\perp e|\\lbrace a,b,d\\rbrace $ .", "Thus, the first two are correct, while the third is false.", "We now apply the skeleton recovery phase of the PC-like algorithm with two different orderings: $\\textrm {order}_1(V)=(d,e,a,c,b)$ and $\\textrm {order}_2(V)=(d,c,e,a,b)$ .", "The resulting skeletons are shown in Figures REF (b) and REF (c), respectively.", "Figure: Performance of the LCD and PC-like algorithms (original (OPC) and stable (SPC)) for randomly generated Gaussian chain graph models:over 30 repetitions with 50 (the first two columns) and 300 (the last two columns) variables, expected degree N = 2, and significance levels α=0.05,0.005\\alpha =0.05,0.005." ] ]

[ [ "Barbero-Immirzi Value from Experiment" ], [ "Abstract We consider General Relativity as a limit case of the Scalar-Tensor theory with Barbero-Immirzi field when the field tends to a constant.", "We use Shapiro time delay experimental limit of $1/w = (2.1 \\pm 2.3)10^{-5}$ provided by the Cassini spacecraft to find the Barbero-Immirzi parameter value." ], [ "Barbero-Immirzi Value from Experimental Data", "This is a short note on obtaining Barbero-Immirzi value from the experimental data by using the PPN (Parametric Post Newtonian) framework [6].", "Currently there are only two types of theories that still agree with experiment: General Relativity and the Scalar-Tensor theory [16].", "The Barbero-Immirzi parameter plays the major role in the Loop Quantum Gravity and in Ashtekar's GR formulation [12],[13], [14].", "This paper determines Barbero-Immirzi parameter's value directly from the experimental data.", "Barbero-Immirzi parameter appears in the Holst action [15] that is used in LQG and in Ashtekar's GR formulation as a coefficient of the topological term that vanishes due to the first Bianchi identity.", "Formally Barbero-Immirzi parameter can be real or complex valued.", "$S = \\frac{1}{16\\pi G} \\int d^4x \\; e e^{\\mu }_I e^{\\nu }_J (R^{IJ}_{\\;\\;\\mu \\nu } - \\frac{\\beta }{2}\\epsilon ^{IJ}_{\\;\\;\\;KL}R^{KL}_{\\;\\;\\mu \\nu })$ Thus, it is impossible to obtain its value directly from the classical GR theory.", "In this paper we consider GR as a limit of the Scalar-Tensor theory with Barbero-Immirzi field (BI) and non-zero torsion, when torsion tends to zero causing BI field tend to a constant.", "The Barbero-Immirzi field theory was introduced and studied in $\\cite {Mercuri}$ , $\\cite {Krasnov}$ , $\\cite {Taveras}$ , $\\cite {Montani}$ , and $\\cite {Oscar}$ .", "Its action obtained from ($\\ref {1}$ ) when $\\beta (x)$ is a field is as follows $\\cite {Mercuri}$ : $S = \\frac{1}{16\\pi G} \\int \\sqrt{-g}\\; d^4x \\; \\left[R - \\frac{3}{2} \\left(\\frac{1}{1+ \\beta ^2(x)}\\right) \\; \\partial _{\\mu } \\beta (x) \\partial ^{\\mu } \\beta (x) \\right]$ ,where $R$ is a torsion-free Riemann scalar.", "The Scalar-Tensor theories are rare types of theories besides GR that are still in agreement with the experimental data $\\cite {Will}, \\cite {Will2}$ .", "The Scalar-Tensor theory action written in the PPN ”Einstein frame” contains a scalar field $\\phi (x)$ and a coupling function $w(\\phi (x))$ $\\cite {Will}, \\cite {Brans}, \\cite {Weinberg}$ : $S = \\frac{1}{16\\pi G} \\int \\sqrt{-\\tilde{g}} \\; d^4x \\; \\left[ \\tilde{R} - \\frac{3+ 2w(\\phi (x))}{2{\\phi ^2(x)}} \\partial _{\\mu } \\phi (x)\\partial ^{\\mu } \\phi (x) \\right]$ , where $\\tilde{R}$ is a Riemann curvature after the conformal transformation $g_{\\mu \\nu } = \\tilde{g}_{\\mu \\nu }/\\phi (x)$ In the limit, when $w(\\phi (x))$ is constant and tends to infinity, the theory becomes pure GR [16], [18]: $\\phi (x) = const + O(\\frac{1}{w})$ , $R_{\\mu \\nu } - \\frac{1}{2}g_{\\mu \\nu }R = -8\\pi G T_{\\mu \\nu } + O(\\frac{1}{w})$ The recent experimental data $\\cite {Freire}$ , $\\cite {Bertotti}$ provied the following value for $w(\\phi (x)) = 10^5/(2.1\\pm 2.3)$ .", "It's worth mentioning that back in 1972 the experimental value was much lower, equal to 6 [16], $\\cite {Will2}$ , and it seems that the experimental data will keep pushing it further up to become GR.", "Today's experimental value comes from Shapiro time delay data provided in 2003 by the Cassini spacecraft on its way to Saturn $\\cite {Bertotti}$ , $\\cite {Will}$ with the result $\\gamma - 1 = (2.1 \\pm 2.3) \\times 10^{-5}$ , where $w(\\phi (x)) = 1 + \\frac{\\gamma }{\\gamma -1} = 10^5/(2.1\\pm 2.3)$ .", "For the graph showing different available experimental data we refer to $\\cite {Freire}$ .", "The values provided by the most recent binary pulsars are not competitive with the solar-system Shapiro time delay measurement provided by Cassini due to the near equality of the star masses suppressing dipole radiation $\\cite {Will}$ .", "By comparing ($\\ref {12}$ ) with ($\\ref {22}$ ) we see that ($\\ref {12}$ ) is a particular case of ($\\ref {22}$ ) when the scalar field $\\phi (x)$ is a Barbero-Immirzi field $\\beta (x)$ : $ \\phi (x) = \\beta (x)$ with one difference: ($\\ref {12}$ ) contains the original Riemann curvature $R$ , while in ($\\ref {22}$ ) it is conformally transformed $\\tilde{R}$ .", "By performing in ($\\ref {12}$ ) the same conformal transformation $g_{\\mu \\nu } = \\tilde{g}_{\\mu \\nu }/\\phi (x)$ , we obtain: $S = \\frac{1}{16\\pi G} \\int \\sqrt{-\\tilde{g}}\\; d^4x \\; \\left[\\frac{\\tilde{R}}{\\beta (x)} - \\frac{3}{2} \\left(\\frac{1}{1+ \\beta ^2(x)}+ \\frac{1}{\\beta ^3(x)}\\right) \\; \\partial _{\\mu } \\beta (x) \\partial ^{\\mu } \\beta (x) \\right]$ Now ($\\ref {22}$ ) and ($\\ref {32}$ ) are almost in the same form, the difference is only in $\\tilde{R}$ being divided by $\\beta (x)$ .", "We rewrite ($\\ref {22}$ ) by dividing the action integrand by $\\phi (x)$ .", "It does not change the dynamics equation as $\\phi (x)$ does not depend on metric: $S = \\frac{1}{16\\pi G} \\int \\sqrt{-\\tilde{g}} \\; d^4x \\; \\left[ \\frac{\\tilde{R}}{\\phi (x)} - \\frac{3+ 2w(\\phi (x))}{2{\\phi ^3(x)}} \\partial _{\\mu } \\phi (x)\\partial ^{\\mu } \\phi (x) \\right]$ Then we rewrite it once again substituting our case: $\\phi (x) = \\beta (x)$ : $S = \\frac{1}{16\\pi G} \\int \\sqrt{-\\tilde{g}} \\; d^4x \\; \\left[ \\frac{\\tilde{R}}{\\beta (x)} - \\frac{3+ 2w(\\beta (x))}{2{\\beta ^3(x)}} \\partial _{\\mu } \\beta (x)\\partial ^{\\mu } \\beta (x) \\right]$ By comparing ($\\ref {32}$ ) and ($\\ref {58}$ ) we can equate the second terms: $\\frac{3}{2}\\left(\\frac{1}{(1+{\\beta ^2(x)})} + \\frac{1}{\\beta ^3(x)}\\right) = \\frac{3 + 2w(\\beta (x))}{2{\\beta ^3(x)}}$ solving for $\\beta (x)$ we obtain the following equation: $3\\beta ^3(x) - 2w(\\beta (x)) \\beta ^2(x) -2w(\\beta (x)) = 0$ When $w(x)$ tends to infinity, we receive in the limit equation by dividing by $w(x)$ $\\frac{3\\beta ^3(x)}{w(\\beta (x))} - 2\\beta ^2(x) - 2 = 0$ $\\beta ^2(x) = -1 \\;\\;\\; \\mbox{or} \\;\\;\\; \\beta (x) = \\pm i$ If we use Cassini spacecraft experimental value $1/w(\\beta (x)) = (2.1 \\pm 2.3) 10^{-5}$ then, by using the numerical methods to solve ($\\ref {7}$ ), we obtain three solutions: $\\beta (x) = \\pm i + (0.59 \\pm 0.92) 10^{-6}$ corresponding to the above limit case $\\beta (x) = \\pm i$ , while the third root of the equation $ \\beta (x) =( -1.575 \\pm 1.725) 10^5 $ does not correspond to any limit cases.", "It is expected that $w(\\beta (x))$ experimental value will keep growing as it has been doing since 1961, starting from the value 6 [16], reaching thousands at the end of 1970th, and becoming Cassinin's $10^5/(2.1\\pm 2.3)$ today.", "The ongoing BepiColombo mission to Mercury, launched in 2018, will improve factor of 4 the Cassini's result in measuring PPN $\\gamma $ in July 2022 when it will experience the first solar conjunction $\\cite {Imperi}$ .", "The bigger $w(\\beta (x))$ the closer the Scalar-Tensor theory is to GR.", "When $w(\\beta (x))$ is constant, which is the original Brans-Dicke theory [16], the limit $w \\rightarrow \\infty $ implies $\\beta (x) = const + O(\\frac{1}{w})$ [18].", "As we see from ($\\ref {8}$ ) the limit of $\\beta = \\beta (x) \\rightarrow const $ in this case is $\\pm i$ , which corresponds to the Ashtekar sefl-dual GR formalism.", "At the end we would like to address the solution uniqueness question.", "Indeed, by looking at ($\\ref {12}$ ) we see that when $\\beta (x)$ is any constant, not necessarily $\\pm i$ , the action becomes GR.", "However, if we want the theory to be in agreement with the experiment all the time, while going to a limit, then the limit value of $\\beta (x)$ cannot be a random constant, but necessary $\\pm i$ .", "To conclude, today's value of the Barbero-Immirzi parameter detected from experiment is $ \\beta (x) = \\pm i + (0.59 \\pm 0.92) 10^{-6}$ , and with further experiments it will be going closer to Ashtekar's limit value $\\pm i$ .", "Acknowledgment I am very grateful to Michael Bukatin for reviewing this note and for his supportive enthusiastic spirit." ] ]

[ [ "A Complete Model of Cosmological Evolution of Scalar Field with Higgs\n Potential and Euclidian Cycles" ], [ "Abstract The revision of the Author's results with respect to possibility of existence of the so-called Euclidian cycles in cosmological evolution of a system of Higgs scalar fields has been performed.", "The assumption of non-negativity of the Universe's extension velocity, which contradicts in certain cases to complete system of Einstein equations, has been removed.", "It has been shown that in cases when effective energy of the system tends to zero, a smooth transition of the model to the range of negative values of the extension velocity occurs, i.e.", "it occurs the transition to collapse stage rather than winding of the phase trajectories on the boundary of prohibited area.", "This process has been researched with a help of numerical simulation methods for the model based on classical scalar Higgs field." ], [ " [ A Complete Model of Cosmological Evolution of Scalar Field with Higgs Potential and Euclidian Cycles[1] Yu.G.", "Ignat'ev, D.Yu.", "Ignatyev Institute of Physics, Kazan Federal University, Kremlyovskaya str., 18, Kazan, 420008, Russia" ] ]

[ [ "Heatmap-Based Method for Estimating Drivers' Cognitive Distraction" ], [ "Abstract In order to increase road safety, among the visual and manual distractions, modern intelligent vehicles need also to detect cognitive distracted driving (i.e., the drivers mind wandering).", "In this study, the influence of cognitive processes on the drivers gaze behavior is explored.", "A novel image-based representation of the driver's eye-gaze dispersion is proposed to estimate cognitive distraction.", "Data are collected on open highway roads, with a tailored protocol to create cognitive distraction.", "The visual difference of created shapes shows that a driver explores a wider area in neutral driving compared to distracted driving.", "Thus, support vector machine (SVM)-based classifiers are trained, and 85.2% of accuracy is achieved for a two-class problem, even with a small dataset.", "Thus, the proposed method has the discriminative power to recognize cognitive distraction using gaze information.", "Finally, this work details how this image-based representation could be useful for other cases of distracted driving detection." ], [ "Introduction", "Recent technological achievements have contributed to making vehicles greener, safer and smarter.", "However, despite all the efforts made regarding safety, the number of people who lose their lives due to road accidents is still rising.", "According to the who road safety report from 2018 [1], an average of 3700 people die on the road every day, which amounts to 1.35 million victims of car crashes per year (i.e., the eighth leading cause of death of people of all ages, and the primary cause of death for children and young adults between 5 and 29 years old).", "The growth in the number of available vehicles on open roads is naturally a contributing factor to the rise of accident occurrences; however, the main reason is distracted driving [1].", "Distracted driving is described as being occupied by any activity which is unnecessary for the task of driving, such as talking or texting on the phone, eating and drinking, talking to people in the vehicle, interacting with the stereo and entertainment or navigation system—i.e., anything that takes attention away from the task of safe driving [2].", "Based on the who's source, a driver's probable distractions are clustered as follows [3]: Visual distraction: taking the eyes off the road; Manual distraction: taking the hands of the wheel; Cognitive distraction: taking the mind off the driving task.", "Passive safety systems to combat visual and manual distraction are already widely used in commercial vehicles.", "These systems track the driver's eye-gaze.", "Once the driver looks anywhere other than the road, they are judged to be distracted [4].", "The downside of this is that if the driver is looking at the road but daydreaming (a phenomenon known as the mind wandering [5]), they are misjudged as attentive.", "Cognitive distracted driving is a dangerous situation which vehicles should be able to detect to increase road safety.", "It has been highlighted as one of the issues to resolve in the euroncap 2022 requirements (driver inattentiveness) [6].", "Figure: The overlay of the generated heatmap with the front camera's view, presented in a 3D illustration.", "The blue sphere represents the driver’s head position.", "The driver’s eye-gaze vector is represented as a red line protruding from the blue sphere.On the imaginary surface (virtual wall), the dark red spot is where the driver’s gaze activity is concentrated, and the dark blue represents the absence of any gaze activity.This work proposes to detect the cognitive load of the driver with a novel image-based representation of the driver's eye-gaze dispersion (see Figure REF ), called a heatmap.", "Features are extracted from this representation and a svm classifier is trained to estimate cognitive distracted driving.", "Additionally, the designed data collection protocol is presented.", "Section details the scientific foundation for the eye movements and the cognitive load, as well as the state-of-the-art method; the following section, , explains our experimental protocol and the data acquisition process.", "Then, section presents the obtained results, and finally section presents the conclusion and further discussions." ], [ "The State of The Art", "Both biological and physiological approaches naturally influence human behavior by nature (aspects of behavior that are inherited) and nurture (aspects of behavior that are acquired).", "The cognitive approach deals with how people process information and how data is centered on the concept of memory by encoding, storing and retrieving information [7].", "Scheme, perception and working memory concepts have been proposed to reveal cognitive processes using physiological behavior.", "The Multi-Store Model [8] proposes that memory consists of a process including a sensory register, stm and ltm.", "stm is developed as working memory, which is a system for temporarily storing and managing required information to carry out complex cognitive tasks such as learning, reasoning, and comprehension [9], [10].", "Cognitive load refers to the used amount of working memory resources.", "It is a variable which is used to assess and measure the demands on working memory and can be of the following types: intrinsic (relative complexity), extraneous (ineffective or unnecessary) and germane (effective) [11].", "With the increased demand on working memory placed by an abundance of novel information or by interactions of present elements, the cognitive load rises.", "Existing cognitive load measurement techniques are divided into three categories; self-reports, performance measures, and physiological measures [12].", "The self-report method cannot be used as a feature by a real time vehicle application.", "For performance and physiological measures, numerous clues from different sources contain information about the cognitive load of the driver.", "For instance, a combination of vehicle data, environment data and the knowledge of the current task is used to estimate the workload placed on the vehicle driver [13]; the merging of the driver's eye movement, eye-gaze direction, eye-closure blinking movement, head movement, head position, head orientation, movable facial features and facial temperature image into this method has been proposed [14].", "Bio-physiological signals such as driver-facing sensors and relay features such as the hands, fingers, head, eye gaze, feet, facial expression, voice tone, brain activity, heart rate, skin conductance, steering-wheel grip force, muscle activity and skin/body temperature are other signals which could be used for cognitive load estimation [15].", "Other methods based on an ecg assume that heart rhythms, controlled by the autonomic nervous system, can fluctuate with cognitive load [16] or on the eda [17].", "However, observing brain activity is extremely efficient to detect cognitive distraction by identifying frequency bands which are likely to capture the cognitive load and brain locations related to it [18].", "It has been reported up to 98% accuracy of cognitive distraction recognition, while driving is a simulator, by analyzing eeg dynamics [19].", "Nevertheless, this type of measurement needs numerous electrodes to be in direct contact with the head of the driver, which is not very ergonomic in a commercial vehicle (sixteen electrodes in [19]).", "In addition, the size of the pupils increases in cases of high cognitive load, and the latter also has an impact on blinking speed [20].", "In a simulator-based experiment, the cognitive load was detected by the pupil size while the drivers were involved in spoken dialogues [21].", "However, the blinking speed and pupil sizes are also influenced by light conditions.", "In a vehicle application, the cognitive distraction is also been detected by combining steering angle, vehicle speed, gaze location and head heading angle [22].", "Among all these available information sources, our work concentrates on a method which relies on only eye-gaze data, which is obtainable with contactlessly sensors.", "When the driver is distracted and experiences an increasing cognitive load, the rapid, ballistic eye movements—called saccades—of his eyes are altered, and their speed might reveal cognitive distraction.", "Saccades become quicker and more random with high cognitive load [23].", "Specific eye-related measurements such as blinks, saccades, pupils, and fixations provide a relevant and reliable assessment of cognitive load [24].", "An observer’s visual scanning behavior tends to narrow during periods of increased cognitive demand [25], which is in parallel to the fact that mental tasks produce impairments of spatial gaze concentration and visual-detection [26].", "In this work, based on this knowledge, instead of detecting and analyzing all eye-related movements individually, a method which sums all the gaze activity is proposed.", "Thus, the driver's eye-gaze vector is projected on an imaginary distant surface.", "By following the temporal variation of this projection, an image-based representation is created.", "These shapes are expected to reveal the cognitive distraction of the driver.", "Similar to our study, Friedman et al.", "[27] explored another image-based representation of the movements of eye pupils (without the gaze projection on an imaginary distant surface) and achieved 86.1% accuracy with 3D cnn.", "To the best of our knowledge, our method of gaze projection on a distant surface remains original.", "This method spatially represents all the summed gaze activity, i.e., where the driver looks, and can be extended with additional information, such as through the projection of the positions of other vehicles, pedestrians and road signs on the same imaginary surface (see Section REF )." ], [ "Cognitive Distraction and Eye Movements", "In this work, the link between short-term memory and distraction while driving is explored.", "Cognitive load, inattention and distraction are three different concepts.", "Cognitive load refers to the percentage of used resources in working memory, inattention is the state in which the driver is losing attention from the driving task to other secondary tasks, and distraction refers to the involvement of the driver in other tasks.", "Distraction leads to inattention from a particular task, and this causes a high cognitive load (in a driving task, this is of the germane type).", "Therefore, we obtained the following assumption: during neutral driving, the driver has sufficient cognitive resources to explore the environment and performs normal tasks related to driving, such as regularly checking the mirrors, other vehicles, road signs, etc.", "Among the vestibulo-ocular eye movements (fixations), saccades (rapid, ballistic movements) and smooth pursuits (slower tracking movements) should be observed [28].", "However, during distracted driving, the driver has fewer cognitive resources for the driving task; thus, the gaze traces cover a smaller area.", "As a result, a variation of the eye movements is expected.", "The experimental session was composed of driving two consecutive laps on the same route (see Section REF ).", "The first round (Neutral Driving) constituted the baseline, in which the driver performed the driving task naturally.", "The driver was told to relax and drive carefully.", "This lap was important as it allowed us to determine the baseline eye-gaze variation of the participants.", "The second lap (Distracted Driving) was performed immediately after the first one: in the second lap, the driver had to perform secondary tasks (see Section REF ) designed to cognitively overload them." ], [ "Path and Driving Conditions", "An important aspect of the experimental protocol was to recreate driving conditions (road, weather, traffic jams) which were as similar as possible between sessions and for both laps completed by a single participant.", "Therefore, a highway road near to Bobigny in France was defined as the experimental path for each participant.", "The speed limit on this highway was constant (90 km/h), and it took 22 minutes to complete a single lap.", "Driving was performed during the day-time between 10am and 5pm in order to minimize the variation in weather and traffic conditions." ], [ "The Expert", "The expert was in charge of the experiment protocol, launching the secondary tasks, annotating events and guiding the driver on the driving path.", "He was also in charge of momentarily pausing the secondary tasks whenever the road situation became dangerous (i.e., when another vehicle overtook the test vehicle).", "This expert is called the accompanist in the following sections." ], [ "User Group", "Five drivers participated in the data collection protocol.", "All of them were volunteers working in the automotive industry; however, they were not aware of the purpose of the driving session.", "All the participants were male, with an average age of 29.4 years." ], [ "Secondary Tasks", "The aim of the secondary tasks was to increase the mental workload of the driver.", "In the literature, distinct secondary tasks have been cited such as foot tapping (secondary task) while learning (primary task) and measuring the rhythmic precision [29] or measuring the drt while driving [12].", "In a simulator-based experiment, drivers had to accomplish visual, manual, auditory, verbal and haptic secondary tasks.", "Results of the eye-glance analysis showed that the visual drt were more efficient than the other ones [30].", "A vehicle oriented study used visuospatial secondary tasks (the participants should visualize the location of this time’s hour and minute hands on the face of an imaginary analog clock) [22].", "However, in our study, in order to keep the eye-gaze patterns as neutral as possible, visual and visuospatial secondary tasks were discarded.", "Immersive and fun secondary tasks have been designed in order to attempt to reach a more natural experimental procedure.", "The following four games were designed, all for the n-back task strategy.", "The n-back tasks are cognitively distracting tasks in which the participants have to recall successive instructions.", "Recalling these successive instructions increases their mental workload [31].", "Each game was designed to last four minutes with one minute of pause between them.", "Neither Yes nor No: This game was based on avoiding the words \"yes\", \"no\" and their alternatives such as \"yeah\", or \"oui\".", "The accompanist asked successive questions to force the participants to pronounce these words.", "In My Trunk There Is: The game consisted of citing \"In my trunk there is\" followed by an item’s name.", "The participant and the accompanist, turn by turn, had to recall all the past objects and add a new one to the list.", "Guess Who?", ": The participant thought about a real or imaginary character and the accompanist tried to determine the identity of the character by asking questions from a mobile application.", "The participant had to answer the questions correctly.", "The 21: The accompanist started to count and stated 1, 2 or 3 digits in numerical order (e.g., 2 digits: 1, 2).", "The driver followed the numerical order and stated it, and added a different number of digits than the accompanist (e.g., 3 digits: 3, 4, 5).", "The game continued in this manner; however, it was forbidden to say the number \"21\".", "When the counter arrived to \"21\", instead of saying \"21\", a new rule had to be added to the game (e.g., do not say multiples of 4) and the counter was reset to zero." ], [ "Data Acquisition", "The position of the vehicle's interior parts, such as the mirrors and the instrument cluster, were measured and illustrated in a 3D world representation (see objects 2, 3, 4 and 5 in Figure REF ).", "While driving, the driver was monitored with a nir camera, placed in front of the instrument cluster.", "This sensor, part of the Valeo dms [32], extracted the head position and eye position and their direction.", "These data were also imported to the 3D world representation (see Figure REF ).", "Thus, it was possible to detect if the driver was looking towards one of the objects present in the scene.", "In addition, an imaginary plane surface was placed in front of the vehicle as if it were one of the vehicle's interior parts (object 1 in Figure REF ).", "The eye-gaze vector was projected on this surface, and their intersection point was tracked for a given time window.", "By following the variations of the intersection point over this surface, image-based representations were generated (see Section REF ).", "This representation, called a heatmap, was used to detect the cognitive load of the driver.", "The vehicle was also equipped with a frontal RGB camera providing an image with a 1280 x 800 pixel resolution.", "The position and the dimensions of the imaginary surface were set to maximize the junction of this surface with the RGB camera's field of view and the area in which gaze detection was available.", "In our vehicle's configuration, these conditions were met when the virtual wall was placed 4 meters in front of the vehicle (point zero was selected the navigation screen of the car).", "Then, the vehicle was physically placed in front of a real wall, at the computed distance, and the camera's field of view was measured in meters (4.15 m x 2.59 m).", "In conclusion, the first step of the data acquisition process was to detect the location of the projection of the eye-gaze on the 3D imaginary surface (which was 4.15 m x 2.59 m) and convert it to pixels (i.e., 1280 x 800).", "The generated heatmaps were down-sampled to 640 x 400 to increase computational speed.", "The RGB camera was located at the center of the vehicle, whereas the driver was sitting on the front left seat.", "Thus, the driver's gaze activity seemed to be concentrated on the left side of the image on the overlays and heatmaps." ], [ "Heatmap Generation", "The heatmap is a data visualization technique used in different studies and solutions.", "Heatmaps are often used to highlight areas of interest; therefore, we can explore several situations which arise from it.", "The heatmap (visible in Figure REF was used for both visualization and feature extraction after performing the following steps: Figure: Successive steps for creating a heatmap mask.", "Point acquisition: The timestamped raw intersection points for x and y between the eye-gaze vector and the imaginary surface were the heatmap generator's input.", "These data were acquired every 50 ms, if the driver was looking through this imaginary plane (if the driver was not looking through the plane—i.e., checking his phone—see Section REF ).", "Buffering—window size: A single intersection point was not sufficiently meaningful for this specific problem.", "Therefore, the points were buffered as sliding windows (see Figure REF ).", "Section REF compares 12 window sizes from 5 to 60 seconds.", "Field of view: With the aim of covering the field of view of the driver, a circle of 15 pixels was placed, centered on the intersection points (Figure REF ).", "The choice of the circle diameter that represents the gaze fixation was mainly influenced by the pixel dimensions of our heatmaps (640 x 400).", "Opacity: After the normalization of the field of view circles, the obtained mask was used to vary the opacity of intersections (see Figure REF ).", "Blurring: Finally, a Gaussian filter was applied to reduce the noise due to the gaze activity and to concentrate on the most explored area (see Figure REF )." ], [ "Feature Extraction", "Feature engineering was applied on the generated heatmaps in order to reduce the data dimension.", "From each heatmap, the following feature sets, based on their pixel intensities and shape, were extracted:" ], [ "Appearance Features", "The pixel intensity variation of a heatmap contains information on the area checked by the driver.", "The histogram is an efficient tool to visualize the data distributions.", "Figure: Computed histogram of the heatmap presented in Figure .", "(a) Pixel intensities are represented on the abscissa, and the number of pixels on the ordinate.", "(b) The values from aa are distributed into 6 bins.During distracted driving, it is expected that we see a higher concentration on higher intensities than during natural driving, as the driver should cover a wider area, it is expected that the histogram should exhibit a shift towards low-intensity bins.", "Hence, a six-bin-histogram of the pixel number in terms of pixel intensity is generated per heatmap (see Figure REF )." ], [ "Geometric Features", "Beyond the raw pixel intensities, during distracted driving, the dispersion of the gaze activity is expected to vary differently on the abscissa and ordinate axes.", "Thus, their geometric form also has to be considered.", "The generated heatmap is divided into contours according to the differences in pixel intensities: blobs (see Figure REF ).", "Figure: Gaze activity blobs.", "(a) Driver’s gaze activity heatmap.", "The red area represents the most fixated area and the blue region is the less fixated area.", "(b) Thresholded heatmap, converted to a grayscale image with distinguished contours of focus.", "The following figures are extracted contours from the thresholded heatmap.", "Each contour is defined by the pixel intensities.", "A binary threshold is performed for each zone.In order to understand the information about the driver's gaze dispersion across the imaginary plane, the following features are extracted as statistical measures from all blobs: Standard deviation on $x$ and $y$ ; Coordinates of the centroid; Boundaries of each zone (min.", "and max.", "of $x$ and $y$ ); First quartile, median and third quartile on $x$ and $y$ ; Area of the contour; Perimeter of the contour." ], [ "Looking Ahead Confidence", "If the driver does not always look through the imaginary plane during the heatmap generation time window (i.e., they are engaging in activities such as checking their phone) or if the camera is not able to detect the driver's gaze (i.e., the driver might cover the camera with his arm while manipulating the steering wheel), the observation will contain less relevant data.", "Therefore, the information regarding how much time the driver spent looking ahead is another feature which determines the quality of that heatmap, called lac.", "Finally, all the extracted features are standardized by removing the mean and scaling to unit variance per heatmap." ], [ "Classifier Training", "The svm supervised binary classification algorithm is trained with the extracted features (scikit-learn implementation on Python computer language [33] with radial basis function kernel).", "Data collected during neutral driving have been annotated as neutral and data collected during the secondary tasks have been annotated as distracted.", "The classification is validated through a stratified k-fold cross-validation technique, with 10 iterations ($k=10$ ).", "The leave-one-driver out technique is used to ensure the test data are always different from the training data.", "Stratification seeks to ensure that each fold is representative of all strata of the data, which aims to ensure that each class is equally represented across each test fold and consists of splitting the data set into samples." ], [ "Shape Visualization", "In accordance with the initial expectations, the variation of the obtained shapes is visually different between neutral and distracted driving (see Figure REF , columns a and b).", "These shapes occupy a wide area in neutral driving, as the driver checks his environment often.", "However, in the presence of cognitive distraction, the covered area narrows as the driver fixates more on a single zone.", "For a heatmap gained by longer observation times, a better separable visual pattern is obtained.", "This is due to the fact that with a longer observation time, the driver has more time to explore his environment in neutral driving, whereas in distracted driving, as he often fixates on a narrowed zone, observing for a longer period does not greatly change the heatmap.", "As a result, the difference between neutral and distracted driving patterns becomes more obvious with a longer observation time (see window size in Figure REF ).", "Nevertheless, safety-oriented solutions should warn of dangerous situations as quickly as possible.", "Figure: Heatmaps during 5, 15 and 30 seconds in both distracted (left) and neutral (right) driving scenarios." ], [ "Scores", "The relationships between the observed window size and the classification result are presented in Table REF .", "A window of 5 seconds achieved 63% accuracy, whereas a window of 60 seconds achieved 85% of accuracy.", "These results are in accordance with the expectations based on the previous heatmap observations (see Figure REF ).", "Table: Performances in terms of window size in seconds.The presented results were obtained by averages of scores from 10 random training–testing splits (stratified k-fold cross validation) in which the subjects in the training sets were always distinct from the subjects in the testing set to prevent over-fitting.", "The confusion matrix obtained by averaging these folds, based on heatmaps of 30 seconds, is presented in Table REF .", "Table: Confusion matrix for the 30 second heatmap-based classifier (accuracy: 81.408%, F1 Score: 0.808.)" ], [ "Conclusion", "The field of human-centered artificial intelligence is tackling its current issues and aims to increasingly assist humans in their daily life.", "Specifically, intelligent systems are now part of vehicles and assist the driver to increase road safety.", "In this work, we have investigated the problem of the detection of the high cognitive load of drivers through an image-based representation created by tracking the driver's eye-gaze projection on an imaginary plane surface (heatmaps).", "The variation of the obtained shapes revealed the driver's cognitive distraction.", "These shapes occupy a wide area in neutral driving, as the driver checks his environment often.", "However, in the presence of cognitive distraction, the covered area narrows, as the driver fixates more on a single zone (see Figure REF ).", "The trained svm-based classifiers achieved 85.2% accuracy; thus, the proposed method has good discriminative power between neutral and distracted driving scenarios.", "For a heatmap obtained by longer observation times, a better separable visual pattern was obtained.", "Nevertheless, safety-oriented solutions should warn of dangerous situations as quickly as possible.", "Thus, a window size compromise should be selected between the algorithmic performance and alerting time.", "For the real participants, we selected two classifiers working in parallel, with different window sizes.", "The first one classified with a short window size to warn of problems as fast as possible ($t=10 sec$ ), and the second one used a long window size in order not to miss any dangerous situations ($t=30 sec$ )." ], [ "Further Discussions", "Future work in this context should involve increasing the participant numbers and collecting more data; however, the scientific background, the obtained heatmap shapes for neutral and distracted driving and the used validation technique shows that this result could be generalized to a wider population.", "Further studies should also include other road types and conditions, as in this work, the driver's cognitive load estimation was studied only under similar conditions (on highway roads with speed limited to 90 km/h, during the day-time, with low traffic and good weather conditions).", "Once more data are collected, further studies should investigate cnn-based classifiers, and ablation tests per feature set should be presented.", "Due to the end-user's needs, modern vehicles are equipped with only a single central nir camera.", "In parallel with this demand, our method is based on a single central nir camera.", "However, multiple cameras would open the possibility of implementing a wider and curved imaginary surface, which would increase the data availability.", "The 3D View (see Figure REF ) extracts other gaze-related features such as the mirror checking frequency.", "These data should also be added to the feature set.", "Figure: Overlay of the generated heatmap and the front camera's view.", "We can observe the low activity for the front vehicle and the high activity for the worker next to the traffic light, the presence of which is an unexpected event.Finally, the real positions of other vehicles, pedestrians and road signs could be taken into account in the heatmap creation process; additionally, we could change the weights for specific zones in the heatmap.", "Figure REF shows an uncommon case, in which the expected heatmap would be different from the default ones." ], [ "ACKNOWLEDGMENT", "The authors would like to thank Kevin Nguyen for his help over the entire course of the project, to Omar Islas-Ramirez for reviewing this article, to all collaborators—Julien, Pantelis, Gaëlle, Vincenzo, Joao, Philippe, Emmanuel and Gabriel—and to the students from Sorbonne University—Antoine, Rodolphe, William and Anes.", "=0mu plus 1mu" ] ]

[ [ "Wavefront prediction using artificial neural networks for open-loop\n Adaptive Optics" ], [ "Abstract Latency in the control loop of adaptive optics (AO) systems can severely limit performance.", "Under the frozen flow hypothesis linear predictive control techniques can overcome this, however identification and tracking of relevant turbulent parameters (such as wind speeds) is required for such parametric techniques.", "This can complicate practical implementations and introduce stability issues when encountering variable conditions.", "Here we present a nonlinear wavefront predictor using a Long Short-Term Memory (LSTM) artificial neural network (ANN) that assumes no prior knowledge of the atmosphere and thus requires no user input.", "The ANN is designed to predict the open-loop wavefront slope measurements of a Shack-Hartmann wavefront sensor (SH-WFS) one frame in advance to compensate for a single-frame delay in a simulated $7\\times7$ single-conjugate adaptive optics (SCAO) system operating at 150 Hz.", "We describe how the training regime of the LSTM ANN affects prediction performance and show how the performance of the predictor varies under various guide star magnitudes.", "We show that the prediction remains stable when both wind speed and direction are varying.", "We then extend our approach to a more realistic two-frame latency system.", "AO system performance when using the LSTM predictor is enhanced for all simulated conditions with prediction errors within 19.9 to 40.0 nm RMS of a latency-free system operating under the same conditions compared to a bandwidth error of $78.3\\pm4.4$ nm RMS." ], [ "Introduction", "In adaptive optics (AO) systems, time lag between wavefront detection and correction induces the bandwidth error.", "For Extreme AO (XAO) systems for high contrast imaging (HCI) of exoplanets, the bandwidth error results in broadening of the point spread function (PSF) along dominant wind directions, which severely degrades contrast, especially at small star separations [20], [27].", "For wide-field AO systems dominated by tomographic errors, to keep bandwidth error tolerable, the integration time of wavefront sensing and thus guidable star magnitude (either natural or laser) is limited, which then limits the sky coverage [4], [16].", "One way to overcome this problem is to predict the future wavefront from recent past wavefront measurements.", "Under the frozen flow hypothesis [42], [37], the turbulence volume is modeled as a linear composition of static, independent layers, each translating across the telescope aperture with certain velocity as a result of dominant wind at that layer.", "Because of this spatial and temporal correlation, it is possible that the future wavefronts can be partially predicted using past measurements.", "This hypothesis is a reasonable simplification of the turbulence for wavefront prediction purposes.", "Predictive control in AO is an active research area that incorporates wavefront prediction based on the frozen flow hypothesis into controller design.", "One of the most popular schemes is the Kalman filter based Linear Quadratic Gaussian (LQG) control [31], [24].", "Under this framework, the whole system (both turbulence and AO system) is represented by a small set of state variables.", "Linear models are used to describe temporal evolution of those variables as well as their links with system measurements.", "Priors from system telemetry and noise statistics are then combined to obtain the control law.", "Because of its flexibility in structure, LQG predictors allow for additional consideration of other system error sources such as static error and vibration.", "Numerical and laboratory implementations focusing on a single or a few Zernike modes show great improvement in terms of overall residual phase error or Strehl ratio [24], [22] and especially vibration filtering [32], [33].", "[36] developed a computationally efficient Fourier based LQG predictive controller, which can be extended to non-integer loop delays [35], facilitating graceful formulation of wind-blown turbulence evolution under Fourier basis.", "Laboratory tests demonstrate a reduction of around 67% in bandwidth error using a full Fourier LQG controller [39].", "[4] incorporates open-loop wavefront prediction into a minimum mean square error (MMSE) tomographic reconstructor design for multi-object AO systems.", "This tomographic predictor allows for use of one-magnitude fainter guide stars (corresponding to an increase in the density of available stars by a factor of 1.8) in end-to-end simulations of RAVEN [1], which is expected to be further improved if deployed within a LQG framework.", "LQG based predictive control has been deployed for AO systems on HCI instrument SPHERE [34] for both turbulence correction and vibration filtering in tip-tilt modes.", "Stability and robustness of LQG controller in full-mode single-conjugate AO (SCAO) control has also been verified on sky [40], showing overall performance improvement over a standard integrator controller in conditions where bandwidth error is not dominant.", "The recently proposed Empirical Orthogonal Functions framework [13] for predictive control aims at fully exploiting linear spatio-temporal correlations within input telemetry and improving controller robustness by assuming no physical model of turbulence evolution.", "Numerical HCI simulations demonstrate significantly improved contrast and robustness against sensor noise.", "Although this feature can significantly simplify practical implementation, frequent re-learning and update is unavoidable, for such data-driven predictor and above-mentioned LQG approach, to adapt to varying turbulence conditions.", "In this paper, we exploit the potential of artificial neural networks (ANNs) as a nonlinear framework for wavefront prediction.", "Early numerical simulations adopting a feed-forward multi-layer perceptron (MLP) network demonstrate promise for using this nonlinear tool for slope prediction based on a time series of past noisy measurements by a Shack-Hartmann wavefront sensor (SH-WFS) [17], [18], with further improvement over a linear predictor when signal-to-noise ratio (SNR) of wavefront sensing gets lower [26].", "The last few decades have seen significant advances in both the theory and applications of ANNs [25], among which the Long Short-Term Memory (LSTM) network is well-suited to time series modeling and prediction by design [14], [8]." ], [ "Artificial neural networks", "ANNs are computational models inspired by biological neural networks.", "They are composed of a series of computing elements called neurons that are interconnected in a layered structure.", "Each neuron receives inputs, either as an input to the entire network or as outputs of connected neurons in the former layer, then transmits mathematically processed input information to connected neurons in the next layer.", "This forward transmission continues until the final output neurons are reached.", "Information flow in each type of ANNs is thus specified.", "A thorough tutorial on ANNs can be found in [10].", "A detailed description of a MLP network and its successful application for tomographic wavefront reconstruction can be found in [30].", "Long Short-Term Memory (LSTM) is an advanced architecture of recurrent neural networks (RNNs) that are specially designed for processing sequential data [11].", "Compared with MLPs with forward transmissions only, RNNs have dynamic feedback connections and shared parameters across all time steps.", "An internal state vector is transferred through time to maintain memories.", "LSTMs can especially cope well with long-term dependencies [10], which otherwise renders training in normal RNNs much more difficult.", "It has been successfully applied in fields such as speech recognition [12], machine translation [41] and image captioning [19].", "LSTMs have two desirable features for wavefront prediction: No user input.", "No prior knowledge of the atmosphere is assumed for the training process.", "No user input is required either during application.", "No user tuning.", "The fluid nature of the memory elements within allows the network to learn temporal behaviours of turbulence of varied time constants and to adapt to changes in these without user tuning.", "The nonlinearity of LSTMs enables the agility and robustness when dealing with non-frozen flow turbulence evolution (such as fluctuations in wind velocities), WFS noise or change of turbulence strength." ], [ "Methodology", "We exploit the potential of ANNs for wavefront prediction in numerical simulations based on a SCAO system.", "More specifically, the ANN predictor is trained in simulation to predict uncorrected wavefront slopes at the next time step based on a time series of past noisy slope measurements by a SH-WFS operating in open loop.", "The simulated SCAO system serves two purposes.", "The wavefront sensing subsystem is used to generate a series of time sequences of wavefront slopes as training data, with the last frame of slopes in each sequence being the training target.", "After training, the predictor is incorporated into the AO correction loop for evaluation.", "To quantify the efficacy of the ANN predictor, we compare AO corrections in terms of root-mean-square wavefront errors (RMS WFE) under three operating conditions, depending on which WFS measurement is applied to DM at time step $t$ : Zero-delay or delay-compensated loop, where the current measurement $\\textbf {s}_{t}$ is used immediately.", "One-frame delay loop, where the prior measurement $\\textbf {s}_{t-1}$ is used.", "ANN predictive loop, where the predicted current measurement $\\tilde{\\textbf {s}}_{t}$ from $(\\textbf {s}_{1}, \\textbf {s}_{2},...\\ ,\\textbf {s}_{t-1})$ is used." ], [ "SCAO simulation", "The AO simulation tool used is Soapy (Simulation `Optique Adaptative' with Python) [38].", "Soapy is highly modular, enabling both end-to-end simulations and fast experimentation using a subset of its modules.", "New modules can also be easily integrated.", "The architecture of the simulated SCAO system is shown in Fig.", "REF .", "Throughout the simulation we use a point source at infinity to act as a natural guide star (GS).", "To generate the training data, one single turbulence layer is assumed.", "Here, the use of a single layer is for the ease of training.", "We will show that a ANN predictor trained using one turbulence layer is capable of predicting in multi-layer conditions.", "A large random phase screen with Von Karman statistics is generated within the atmosphere module Atmos at the start of each loop run.", "Pure frozen flow is assumed, under which the large phase screen is translated over the telescope aperture with a given velocity due to the wind.", "At each time step, a smaller portion of the large phase screen, the part of which is seen by the telescope aperture, is output to SH-WFS.", "SH-WFS then outputs measured noisy wavefront slopes from the image plane using thresholding centre of gravity (TCoG).", "The thresholding value is a flux cutoff described by a factor of the maximum intensity within a subaperture to suppress photon noise and readout noise.", "A single frame delay can be used in Soapy simulations (the center loop in Fig.", "REF ) to account for the inevitable WFS integration time.", "This time lag between wavefront measurement and correction can be compensated either by applying slope measurements immediately (the lower loop) or by sending the prior slopes to a ANN predictor to extrapolate the current measurements (the upper loop).", "A reconstructor module (Recon) combines noisy slopes (either delayed, predictive, or delay-compensated) and control matrix generated during calibration to output DM commands, which are used by DM to generate the corrected phase.", "RMS error between the phase distortion and DM shape is then output as RMS WFE.", "Principal simulation parameters are listed in Table REF .", "The configuration is adopted from CANARY low-order SCAO mode [28].", "We train the predictor under similar atmospheric and system conditions where it will be applied.", "The impact of the WFS SNR on ANN training and the predictor's robustness against changes in input statistics will be explored in Section .", "Figure: Composition of the simulated SCAO system and its data flow.", "RMS wavefront error of the predictive correction (upper) is expected to be between the delayed (center) and delay-compensated (lower) corrections.The wavefront sensing subsystem consisting of Atmos and SH-WFS modules is used to generate the first 100,000 training samples.", "Each sample is a time sequence of thirty 72-element vectors $(\\textbf {s}_{1}, \\textbf {s}_{2},...\\ ,\\textbf {s}_{30})$ , with each vector, $\\textbf {s}_{i}$ , being the X and Y slope for each of the 36 subapertures.", "$(\\textbf {s}_{1}, \\textbf {s}_{2},...\\ ,\\textbf {s}_{29})$ will be ANN inputs sequentially during training, and $\\textbf {s}_{30}$ will be the targeted output.", "Wind velocity corresponding to each sample is a random vector, with its magnitude uniformly sampled from the range 10 to 15 m/s and its direction uniformly sampled from the range 0 to 360.", "Wind velocity is constant within each sequence.", "We then reverse each sequence to form the other half of the training set, with the last frame being the first and first being last.", "This corresponds to reversing the wind direction.", "We use this data augmentation approach to introduce variability in training data to improve model robustness.", "Resulting training input set and target set are tensors of shape ($2\\times 10^5$ , 29, 72) and ($2\\times 10^5$ , 72) respectively.", "The amount of training data is decided by trial and error to match both ANN architecture complexity and problem complexity to balance between training data fitting and model generalisation.", "No further training data pre-processing is implemented.", "Table: Principal parameters used with the Soapy SCAO simulation for ANN training and optimisation." ], [ "ANN training and optimisation", "We use Keras [3] library written in Python for ANN training.", "The ANN architecture consists of stacked LSTM cells and a final fully-connected (FC) output layer.", "The depth of neural networks is associated with the depth of representations that can be learnt [10], thus the stacking of LSTM cells in our case.", "The ANN topology comprising two LSTM cells and a FC layer is shown in Fig.", "REF .", "The display is unrolled in time, which means all components in the same colour (or row) are duplicates in time and essentially identical to inputs at any time step.", "At each time step $t$ ($t\\ge 2$ ), the network can output a slope prediction ${\\tilde{\\textbf {s}}}_{t}$ based on the current input $\\textbf {s}_{t-1}$ and two state vectors, the cell state (also called the internal state) and the cell output (also called the hidden state).", "Both states are either initialised as all-zero vectors ($t=2$ ) or updated at each time step ($t>2$ ) using information in the input sequence so far.", "Figure: The ANN predictor structure unrolled in time.", "The predictor can start predicting from the 2 nd 2^{\\textrm {nd}} time step, although initial predictions can be unstable and inaccurate due to limited temporal information.", "The two LSTM cells have the same inner structure, but different sets of parameters after training.Parameters (also called trainable weights) of the network determine how inputs are processed mathematically layer by layer.", "ANN training is the process to optimise these parameters iteratively to minimise a training error.", "These parameters are initialised using a Gaussian distribution.", "10% dropout is deployed for each LSTM cell [7].", "10% training samples form a validation set.", "The remaining 90% samples are randomly split into batches of size 128 before each epoch.", "The training error is mean squared error (MSE) between the targeted output ${\\textbf {s}}_{30}$ and the actual output ${\\tilde{\\textbf {s}}}_{30}$ evaluated and averaged on the current batch.", "The Adam optimisation algorithm is used to optimise the network parameters in a direction that minimises the training error [21].", "It is a first-order gradient descent algorithm and features adaptive learning rate.", "During one epoch, every batch is evaluated once and the network parameters are updated accordingly multiple times.", "At the end of each epoch, the updated network is evaluated on the validation set.", "The initial learning rate is 1e-3.", "If MSE of the validation set shows no improvement for consecutive 10 epochs, the learning rate is reduced to its 1/5 unless reaching 1e-5.", "The reduced learning rate allows only small updates of the network parameters to prevent this optimisation process from early stagnation.", "Training is terminated after 40 epochs, at which point both training and validation errors have stagnated.", "The ANN optimisation process, also called hyperparameter tuning, is coupled with ANN training.", "Hyperparameters determine either the structure of the network or the training process.", "These are fixed before training starts.", "We tune two hyperparameters that determine the physical capacity of the network: number of stacked LSTM cells (1 or 2) and length of output vectors of each LSTM cell (a random integer between 100 and 250, different for each cell).", "Every time a set of these two hyperparameters are chosen, the model is recompiled, re-initialised and re-trained as is described above.", "The model that achieves the lowest validation MSE at the end of the $40^{\\textrm {th}}$ epoch is composed of two LSTM cells and a final FC layer (as is shown in Fig.", "REF ).", "The output vector of the first LSTM cell has 247 elements and the second cell has 226 element.", "The resulting model has 761,000 trainable parameters in total.", "Breakdown of the number of floating point operations (FLOP) of the optimised ANN structure is shown in Table REF .", "The resulting computational load is $2.3\\times 10^8$ FLOPS (FLOP per second) for the CANARY-scale $7\\times 7$ subaperture system operating at 150 Hz.", "Table: Breakdown of computational load within the optimised ANN architecture." ], [ "Results", "After training, the optimised predictor is inserted between SH-WFS and Recon to form part of a predictive correction loop.", "From this stage, the parameters within the network are fixed and inputs are now processed in a deterministic way.", "We test the predictor's generalisation and extrapolation capabilities in five different scenarios: The predictor is tested on unseen data generated within the parameter boundaries used for the training regime.", "GS magnitude is increased from 10 (on which the predictor is trained) to 6, which increases the SNR of input slopes.", "In this scenario, we also investigate the SNR of training data on the predictor's performance.", "A time-variant turbulence is considered by changing either the wind speed or the direction every 10 frames (15 Hz) after the predictor stabilises.", "A multi-layer turbulence is considered to test the predictor's ability to track multiple wind vectors.", "We extend our approach to account for a more realistic two-frame latency, where we trained a separate ANN predictor to predict two frames in advance directly, and compare that with applying the single-latency predictor twice.", "In most scenarios, statistics of the input slopes to the predictor are different to what was used during training.", "In each scenario, we use 1,000 test slope sequences each of 100 frames (0.67 s).", "We have found that our predictor will remain stable during a 2-minute period if both system and turbulence parameters remain unchanged, which is the case in most of the scenarios.", "Thus, in this paper we only include results obtained from 100-frame sequences to strengthen different aspects of the performance.", "The predictor's memory (both internal and hidden states) is zeroed before a new slope sequence.", "The predictor is expected to build up its memory and output stable predictions in 30 frames as the training is designed.", "Other simulation parameters are mostly the same as listed in Table REF , unless stated otherwise." ], [ "Performance within the ANN training regime", "Fig.", "REF shows RMS WFEs averaged over 1,000 test atmospheric turbulence sequences.", "Shaded areas indicate the standard error of the mean RMS WFE [15].", "The atmospheric statistics of this test set lie within the bounds of the training regime, though the test set did not form part of the training data set and had not been observed by the network before.", "Wind speed is 15 m/s in a single direction.", "All other simulation parameters are the same as during training, thus in this case the predictor is expected to reach its optimal performance.", "The predictor output stabilises after approximately 12 frames.", "The prediction is stable after this time span as the input statistics remain unchanged afterwards.", "This implies using shorter sequences for training and thus an alternative network that converges faster is possible.", "Mean RMS WFEs of the delayed, predictive and delay-compensated correction loop (averaged after the $12^{\\textrm {th}}$ frame and across all sequences) are 253.9 nm, 244.3 nm and 243.4 nm respectively, showing an overall performance improvement brought by the predictor." ], [ "Performance with varying WFS SNR", "In Fig.", "REF we show the results from three ANNs when observing a bright guide star of magnitude 6.", "In addition to the ANN used in section REF that was trained on a guide star of magnitude 10, we include results from two networks have been trained at different signal to noise levels.", "These three predictors are denoted as Mag-10, Mag-8 (trained with a GS of magnitude 8) and Noise-free (trained without WFS noise) respectively.", "The training procedure and other simulation parameters were the same as detailed in Section REF , except the thresholding value that was reduced to 0.02 for the Mag-8 predictor, and 0 for the Noise-free predictor.", "The resulting ANN architectures and computational loads are listed in Table REF .", "For each network, we see that prediction error decreases until the $20^{\\textrm {th}}$ frame, after which the performance of each ANN stabilises.", "However, we note that the ANN trained with the lowest SNR performs far better than the ANNs trained in higher SNR regime and this behaviour was observed irrespective of guide star magnitude.", "Figure: RMS slope error (mas) per subaperture compared with zero-delay measurements by SH-WFS as the WFS SNR varies.", "This quantity is the root of the ANN training metric.", "All predictors have lower errors around the corresponding training regimes.Figure: RMS slope error (mas) per subaperture with reference to measurements by the idealised WFS, which removes noise, aliasing and centroiding errors in the measurement of the first 36 Zernike orders compared with zero-delay measurements by SH-WFS.", "This along with Figs.", "and demonstrates the filtering of aliasing and centroiding errors apart from the reduction in bandwidth error brought by Mag-10 predictor.Table: Training conditions and structures of the three ANN predictors.In Fig.", "REF we show the RMS slope error (mas) per subaperture compared with zero-delay measurements as the WFS SNR changes.", "This quantity is the root of the ANN training metric.", "For each guide star magnitude, we generated 1,000 slope sequences each of 100 frames.", "Values shown in Fig.", "REF give the mean slope error across all subapertures after the $30^{\\textrm {th}}$ frame (by when all predictors have stabilised under all SNR conditions) in all sequences.", "All predictors have lower errors around the corresponding training regimes compared with slopes with one-frame delay, which shows the prediction power of ANN predictors of such type.", "In lower SNR regime (GS magnitude > 6), Mag-10 predictor achieves lowest slope errors.", "However, at smaller GS magnitudes (GS magnitude $\\le $ 6) or in the noise-free condition, the performance of the Mag-8 predictor is closer to that of the Mag-10 predictor, rather than the Noise-free predictor.", "This is inconsistent with Fig.", "REF .", "To assist understanding of this discrepancy in the brighter regime, we compare slope errors with reference to the slope measurements using an idealised WFS where centroiding and aliasing errors have been minimised.", "This WFS is defined as follows.", "The phase screen seen by the telescope aperture at each time step is firstly decomposed into its low- and high-order components, $\\phi = \\phi _l + \\phi _h = \\sum _{i=2}^{36}a_i\\textbf {z}_i + \\phi _h,$ where $a_i=\\textbf {z}_i^{\\textrm {T}}\\phi $ , $\\textbf {z}_i$ is the $i^{\\textrm {th}}$ Zernike term in Noll's notation [29].", "The number of Zernikes is chosen to match the structure of the subapertures.", "Measurements of $\\phi $ by the ideal WFS is computed as $\\textbf {s}^i \\equiv \\textbf {Da},$ where D is a perfectly calibrated interaction matrix and a is the Zernike vector ($a_2, a_3, ..., a_{36}$ ) given in Eq.", "1.", "Within D, the slope measurements of each Zernike term are calculated directly from the corresponding high-resolution phase grid instead of from the WFS image plane.", "Fig.", "REF shows the RMS slope error with reference to the ideal WFS measurements in each correction loop.", "Errors in zero-delay measurements are non-zero due to aliasing, centroiding and noise errors compared to the ideal WFS.", "Among the three predictors, Mag-10 predictor achieves significantly lower slope errors under all SNR conditions, which is now consistent with Fig.", "REF .", "In addition we see that in low-SNR regimes, Mag-10 predictor has even lower errors than zero-delay measurements.", "This implies that the reduction in WFE brought by Mag-10 predictor also accounts for some aliasing and/or centroiding errors in addition to the reduction in bandwidth error.", "We think that this is due to being exposed to much lower SNR training data where the temporal correlations within the data are less obvious and noise terms must be learnt to be ignored.", "Prediction error $\\sigma _{\\textrm {pred}}$ is defined as the RMSE between WFEs in the predictive loop and in the zero-delay loop, $\\sigma _{\\textrm {pred}} = \\sqrt{{{\\overline{WFE}}_{\\textrm {pred}}}^2 - {{\\overline{WFE}}_{\\textrm {zero-delay}}}^2},$ where ${\\overline{WFE}}_{*}$ is the average after the $30^{\\textrm {th}}$ frame and across all sequences.", "Bandwidth error is defined in a similar fashion, $\\sigma _{\\textrm {BW}} = \\sqrt{{{\\overline{WFE}}_{\\textrm {delay}}}^2 - {{\\overline{WFE}}_{\\textrm {zero-delay}}}^2}.$ $\\sigma _{\\textrm {pred}}$ of Mag-10 predictor ranges from 40.0 nm to 19.9 nm, decreasing as the WFS SNR is lowered due to the increasing filtering of aliasing and/or centroiding errors.", "The mean value of bandwidth error across all SNR conditions is 78.3 nm, with a standard deviation of 4.4 nm.", "This quantity also decreases slightly as GS gets fainter, due to the increasing correlation between bandwidth error and noise error.", "In the following three scenarios, we show the results obtained with our optimal Mag-10 predictor only.", "We also use a brighter guide star of magnitude 6 to reduce the performance variations brought by wavefront sensor noise." ], [ "Performance with time-variant wind velocity", "In the above scenarios, we have assumed stationary turbulence.", "In this section, we demonstrate the agility and robustness of our ANN predictor against fluctuations in wind velocity.", "Here we use a synthetic wind speed sequence (upper panel in Fig.", "REF ) in a relatively short time scale of 100 consecutive WFS frames (0.67 s).", "Wind speed changes every 10 frames (15 Hz) within 10 and 15 m/s after the first 20 frames during which time the predictor stabilises.", "This fluctuation is reflected in the dynamics of the delayed correction, as a faster translation of the phase screen induces increased phase variations between adjacent frames under frozen flow.", "Fig.", "REF demonstrates robustness of the predictor against wind direction fluctuations between 0 and 45 degrees every 10 frames (upper panel).", "This corresponds to a maximum instantaneous change of 8.4 m/s in wind speed along a single direction.", "Figure: Robustness of the predictor against wind direction fluctuations between 0 and 45 degrees every 10 frames.", "Wind speed is 15 m/s.", "Guide star magnitude is 6.Recently [43] have used typical wind profiles from the Thirty Metre Telescope (TMT) site to demonstrate effects of wind velocity variations in a data-driven linear minimum mean square error (LMMSE) predictor over a period of 5 seconds in numerical simulations.", "Wind data are linearly interpolated to system frequency to allow for per-frame fluctuation.", "Two adaptive variations, resetting-batch LMMSE and forgetting LMMSE, along with LMMSE were tested.", "Compared with these linear predictors, variances in WFEs of the ANN predictive loop before and after the disturbance are on the same order as that of the delay-compensated loop.", "This robustness can be explained as the ANN predictor is allowed to use more spatial and temporal information when making inferences.", "Furthermore, the updating and forgetting mechanisms of our predictor are not fixed, but can constantly self-adjust according to the inputs, which by design allows for more flexible control on data flow." ], [ "Performance with multi-layer turbulence", "Though we train the predictor with a single turbulence layer, there usually exists several layers at high altitudes in addition to a strong ground layer [5], [6].", "It is thus meaningful to test the predictor's sensitivity to multiple layers moving with different velocities.", "Here we show the results obtained with ESO (European Southern Observatory) median 35-layer profile.", "$r_0$ is 0.157 m, slightly worse than during ANN training.", "We generated 1,000 slope sequences each of 100 frames with this profile.", "For comparison, we also generated the same amount of test data of a single ground layer and of a four-layer profile (detailed in Table REF ), both moving at 9.21 m/s (slightly slower than the training range), which is equivalent to the dynamics of the 35-layer profile.", "Fig.", "REF shows residual WFEs when wind vectors of multi-layer profiles (either the 4-layer or the 35-layer) move in different directions.", "For the 35-layer profile, the moving direction of each layer is a random integer between 0 and 360 degrees.", "For the 4-layer profile, wind directions are listed in Table REF .", "The delayed and the delay-compensated correction loops behave similarly regardless of the number of layers, thus only values obtained from the single-layer profile are shown here.", "Mean RMS WFEs of the delayed, 35-layer predictive, 4-layer predictive, 1-layer predictive and delay-compensated correction loop after the $20^{\\textrm {th}}$ frame are 167.9 nm, 166.4 nm, 164.6 nm, 161.9 nm and 159.2 nm respectively.", "Fig.", "REF shows improved ANN performance when all layers in either multi-layer profile move in the same direction (wind speeds are the same as used in Fig.", "REF ).", "Mean RMS WFE of the 35-layer predictive loop decreases to 164.0 nm, slightly better than the 4-layer predictive loop when wind vectors are largely distinct from each other.", "Mean RMS WFE of the 4-layer predictive loop decreases to 162.4 nm, approaching that of the 1-layer predictive loop.", "We think that the wind directions adopted represent two extreme conditions, and that performance with real turbulence profiles would fall within these two cases.", "These results show that the predictor trained on a single layer frozen-flow conditions is capable of providing performance improvement even when complex profiles with random wind directions are encountered.", "Figure: ANN performance with multiple turbulence layers moving along the same direction.", "Compared with Fig.", ", the ANN performance suffers from the increased number of wind vectors, but mainly from the variety among those vectors.Table: Four-layer turbulence profile used within test dataset.", "r 0 r_0 is 0.157 m. L 0 L_0 is 25 m. Two sets of wind directions corresponding to Figs.", "and respectively are examined." ], [ "Performance with two-frame latency", "So far we have considered only single-frame delay in an AO loop, where we have accounted for WFS integration time only but ignored the time taken for real-time processing and the update of the surface shape of the DM.", "In Fig.", "REF we show the ANN performance when a more realistic loop delay of two frames is considered.", "We trained a separate ANN that was designed to predict two frames in advance in a single step.", "The training dataset described in Section REF was re-utilised in the way that $(\\textbf {s}_{1}, \\textbf {s}_{2},...\\ ,\\textbf {s}_{28})$ in each sequence is the ANN input and $\\textbf {s}_{30}$ is the training target.", "The training and hyperparameter searching setup follows that described in Section REF .", "The resulting network comprises two stacked LSTM cells and a final FC layer.", "The output vector sizes of the two LSTMs are 122 and 171 respectively, with a computational load of $0.9\\times 10^8$ FLOPS.", "The resulting mean RMS WFE of this single-step predictive loop after the $30{^\\textrm {th}}$ frame is significantly reduced to 166.9 nm, compared with 225.6 and 157.1 nm of the two-frame delayed and zero-delayed loop respectively.", "As a comparison, the single-frame predictor was also used twice to provide a two-frame prediction: first, the measured $(\\textbf {s}_{1}, \\textbf {s}_{2},...\\ ,\\textbf {s}_{t})$ ($t\\ge 2$ ) is fed into the predictor to generate the predicted $\\tilde{\\textbf {s}}_{t+1}$ as it was designed; second, $\\tilde{\\textbf {s}}_{t+1}$ is treated as its truth value $\\textbf {s}_{t+1}$ and forms part of the ANN input vector $(\\textbf {s}_{1}, \\textbf {s}_{2},...\\ ,\\textbf {s}_{t}, \\tilde{\\textbf {s}}_{t+1})$ , which is then used to generate $\\tilde{\\textbf {s}}_{t+2}$ .", "This resulted in a WFE of 174.4 nm, worse than the two frame prediction, however still significantly better than the two-frame delay.", "Figure: In a simulated system with a two-frame latency, the methodology adopted for the single-latency prediction is extended to training a separate ANN predictor (single-step prediction).", "In this case, the single-latency predictor can also be used twice (two-step prediction), albeit with worse performance.", "Both predictors improve the system performance significantly.", "The blue line representing the single-frame delay performance is the same as that shown in Fig.", ", and is depicted here for comparison with the two-frame delay performance.", "Guide star magnitude for test is 6.", "Wind speed is 15 m/s along a single direction.The results presented within this paper are based on simulations, which do not consider many of the practical issues relating to the implementation of the LSTM architecture within a real AO system.", "In this section we discuss issues relating to calibration and control within a real system.", "The training method presented here uses simulated Shack-Hartmann WFS slope data but could be applied to real data from any WFS data.", "However, the sensitivity of the ANN performance to the training regime and the requirement for a large number of wind velocities required for a robust ANN training means that the collection of a real WFS datasets may take a significant amount of time.", "It may therefore be best to initially train in simulation and convert real WFS slopes to ensure that the subaperture geometry and pixel scale matches that encoded within the ANN.", "Unlike other ANN approaches proposed for multiple guide star AO [30], the single WFS LSTM ANN does not require retraining for different targets, greatly reducing the calibration overhead of implementation within a real system.", "The ANN predictor proposed here may not be applicable to all closed-loop AO systems where imperfect POLC (pseudo open loop control) can introduce additional noise terms within the system, affecting performance and loop stability [9].", "To adapt the training regime here to closed-loop operation the training dataset would have to be expanded to include the range of potential closed-loop gain values.", "This will increase both training time and the size of the resulting ANN, with no guarantee that the resulting ANN would be more resistant to the errors that can affect POLC stability such as misalignments and open-loop DM errors.", "An on-sky implementation of the ANN presented here requires an additional processing step for each WFS before reconstruction within the system.", "The system simulated here uses a $7\\times 7$ subaperture Shack-Hartmann system that we selected such that can be rapidly trained and tested.", "Furthermore, wind profiles can be recovered from recorded off-axis WFS data of the CANARY demonstrator [23].", "By matching the configuration of CANARY, future comparison of multi-layer predictions using real data and in simulation is feasible.", "Extending beyond this low-order system is possible but implies additional training time and an increase in real-time computational load.", "Due to the hyperparameter tuning approach adopted here, the precise computational load of a higher-order system cannot be easily predicted, but implementation of this approach within any existing astronomical non-XAO system is feasible using existing hardware.", "There do however exist possibilities to reduce the computational load, including operating in actuator space where computational load is lower [2], or taking advantage of the sparsity of the ANN.", "The technique proposed here inherently scales to multiple guide star systems through parallelism." ], [ "Conclusions", "We have shown in extensive numerical simulations the potential of artificial neural networks as a nonlinear framework for wavefront prediction.", "The memory elements within the LSTM network give it the ability to learn information such as wind velocity vectors from the data and to use that information in its prediction.", "The fluid nature of the memory allows the network to adapt to changes in such information without user tuning.", "The residual wavefront error of the simulated $7\\times 7$ subaperture SCAO system with one frame delay improves significantly after the predictor is incorporated irrespective of guide star magnitude and wind velocity.", "In addition to accurately predicting the wavefront we have also provided evidence that the ANN predictor also compensates for some centroiding and/or aliasing errors that can be temporally filtered from the wavefront.", "This behaviour however is dependent on the ANN training regime and was only observed when the system was trained on a low SNR $10{^\\textrm {th}}$ magnitude guide star.", "The selection of the training regime has the greatest impact on the performance of the ANN prediction.", "We have shown that the ANN predictor is robust to changes in wind velocity on sub-second timescales, and that the ANN approach taken within this paper is transferable to systems with a two-frame delay.", "The ANN predictor trained on a single atmospheric turbulence layer is also capable of predicting under more complex conditions with multiple layers with independent wind vectors, albeit with reduced performance.", "Whilst we believe it is likely that a more realistic multi-layer training environment and/or the use of multiple wavefront sensors to allow identification of layer altitudes will improve ANN performance on multi-layer turbulence, this is subject to further study.", "Our next steps will be to investigate ANN performance on recorded CANARY data to investigate ANN stability and training in a real-world system." ], [ "Acknowledgements", "The authors gratefully acknowledge James Osborn from CfAI, Durham University for his valuable suggestions regarding the manuscript.", "We thank sincerely Ollie Farley from CfAI for the thought-provoking discussions with him.", "We sincerely appreciate the comments from the reviewer, which helped us improve the paper.", "X. Liu is funded by Durham University through their doctoral scholarship program and China Scholarship Council.", "CGG and JDCJ acknowledge financial support from the I+D 2017 project AYA2017-89121-P and JDCJ acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the H2020-INFRAIA-2018-2020 grant agreement No 210489629." ] ]

[ [ "Joint Reconstruction and Low-Rank Decomposition for Dynamic Inverse\n Problems" ], [ "Abstract A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements.", "In this work we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition.", "We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data.", "The proposed framework allows for flexible incorporation of separate regularisers for spatial and temporal features.", "For the special case of a stationary operator, we can effectively use the decomposition to reduce the computational complexity and obtain a substantial speed-up.", "The proposed methods are evaluated for two simulated phantoms and we compare the obtained results to a separate low-rank reconstruction and subsequent decomposition approach based on the widely used principal component analysis." ], [ "Introduction", "Several inverse problems are concerned with reconstruction of solutions in multiple physical dimensions such as space, time and frequency.", "Generally, such problems require very large datasets in order to satisfy conditions for accurate reconstruction, whereas in practice only subsets of such complete data can be measured.", "Furthermore, the information content of the solutions from such reduced data may be much less than suggested by the complete set.", "In these cases, regularisation in the reconstruction process is required to compensate for the reduced information content, for instance by correlating features between auxiliary physical dimensions.", "For instance, dynamic inverse problems have gained considerable interest in recent years.", "This development is partly driven by the increase in computational resources and the possibility to handle large data size more efficiently, but also novel and more efficient imaging devices enabling wide areas of applications in medicine and industrial imaging.", "For instance in medical imaging, dynamic information is essential for accurate diagnosis of heart diseases or for applications in angiography to determine blood flow by injecting a contrast agent to the patient's blood stream.", "But also in nondestructive testing and chemical engineering, tomographic imaging has become increasingly popular to monitor dynamic processes.", "The underlying problem in these imaging scenarios is often, that a fine temporal sampling, i.e.", "in the discrete setting a large number of channels, is only possible under considerable restrictions to sampling density at each time instance.", "This limitation often renders time-discrete (static) reconstructions insufficient.", "Additionally, an underlying problem in many dynamic applications is given by the specific temporal dynamics of the process, which are often non-periodic and hence prevents temporal binning approaches.", "Thus, it is essential to include the dynamic nature of the imaging task in the reconstruction process.", "With increasing computational resources, it has become more feasible to address the reconstruction task as a fully dynamic problem in a spatio-temporal setting.", "In these approaches it is essential to include the dynamic information in some form into the reconstruction task.", "This could be done for instance by including a regularisation on the temporal behaviour as penalty in a variational setting [36], [37].", "Such approaches have been used in a wide variety of applications, such as magnetic resonance imaging [15], [31], [38], X-ray tomography [4], [32] and applications to process monitoring with electrical resistance tomography [8].", "More advanced approaches aim to include a physical motion model and estimate the motion of the target from the measurements itself.", "This can be done for instance by incorporating an image registration step into the reconstruction algorithm and reformulate the reconstruction problem as a joint motion-estimation and reconstruction task [5], [6], [14], [30].", "Another possibility is the incorporation of an explicit motion model by methamorphsis as considered in [9], [20].", "In this work we consider another possibility to incorporate regularisation, and in particular temporal regularity, to the reconstruction task by assuming a low-dimensional representation of the unknown.", "This leads naturally to a low-rank description of the underlying inverse problem and is especially suitable to reduce data size in cases where we have much fewer basis functions to represent the unknown than the temporal sampling.", "In a continuous setting, this yields the analysis of low-rank approximations in tensor product of Hilbert spaces, for which we refer the reader to [22], [42].", "We rather focus on low-rank approximation methods in a discretised framework, which leads to the use of specific matrix factorisation approaches and their optimisation techniques.", "In particular, in this work we propose a joint reconstruction and decomposition in a variational framework using non-negative matrix factorisation, which naturally represents the physical assumption of nonnegativity of the dynamic target and allows for a variety of regularising terms on spatial and temporal basis functions.", "Following this framework, we propose two algorithms, that either jointly recover the reconstruction and the low-rank decomposition, or alternatively recovers only the low-rank representation of the unknown without the need to construct the full spatio-temporal target in the reconstruction process.", "Here, the second approach effectively incorporates the dimension reduction and can lead under certain assumptions on the forward operator to a significant reduction in computational complexity.", "This can be particularly useful, if one is only interested in the dynamics of the system and not the full reconstruction.", "This paper is organised as follows.", "In Section we discuss our setting for dynamic inverse problems and continue to discuss low-rank decomposition approaches.", "Specifically, principal component analysis (PCA) and non-negative matrix factorisation (NMF), which is the focus in this study.", "As a baseline we first present a low-rank reconstruction method followed by either of the decomposition methods.", "We then continue to present the proposed framework of joint reconstruction and decomposition with the NMF.", "In particular, we prove that the proposed framework leads to a monotonic decrease of the cost functions.", "We then proceed in Section to evaluate the algorithms under considerations with the use case of dynamic X-ray tomography and two simulated phantoms with different characteristics.", "We conclude the study in Section with some thoughts on the extension of the proposed framework." ], [ "A Setting for Dynamic Inverse Problems", "In this work, we consider a general multi-dimensional inverse problem, where the unknown $x(s,\\tau )$ is defined on a spatial domain $\\Omega _1 \\subset \\mathbb {R}^{d_1}$ with dependence on a secondary variable $t\\in \\mathbb {R}_{\\ge 0}$ defined in a bounded interval $\\mathcal {T}:=[0,T]$ .", "This setting admits some quite general applications where the secondary variable could have other physical interpretations, notably wavelength for hyper-spectral problems; however, to fix our ideas, we henceforth consider $t$ to explicitly represent time, and our application to be that of dynamic inverse problems.", "Consequently, the underlying equation of the resulting inverse problem can be described in the following form $\\mathcal {A}(x(s,t);t) = y(\\sigma ,t) \\quad \\text{for } t \\in \\mathcal {T},$ where $\\mathcal {A}:\\Omega _1\\times \\mathcal {T}\\rightarrow \\Omega _2\\times \\mathcal {T}$ is a time-dependent linear bounded operator between suitable Hilbert spaces and $\\Omega _2\\subset \\mathbb {R}^{d_2}$ is the domain for the measurement data.", "We will primarily consider the non-stationary case here, where the forward operator $\\mathcal {A}$ is dependent on $t$ .", "In the special case of a stationary operator $\\mathcal {A}(\\cdot ;t) \\equiv \\mathcal {A}$ for all $t\\in \\mathcal {T}$ , where for each $t$ the operator follows the same sampling process, we can achieve possible computational improvements.", "The resulting implications will be discussed later in Section REF .", "Furthermore, the underlying assumption in this work is that the unknown $x:\\Omega _1\\times \\mathcal {T}\\rightarrow \\mathbb {R}_{\\ge 0 }$ can be decomposed into a set of spatial $b^k:\\Omega _1\\rightarrow \\mathbb {R}_{\\ge 0}$ and channel basis functions $c^k(t):\\mathcal {T}\\rightarrow \\mathbb {R}_{\\ge 0}$ for $1 \\le k\\le K$ .", "Then the unknown can be represented by the decomposition $x(s,t) = \\sum _{k=1}^K b^k(s) c^k(t).$ This formulation naturally gives rise to the reconstruction and low-rank decompostion framework to extract the relevant features given by $b^k$ and $c^k$ .", "An illustration for a possible phantom represented by (REF ) is shown in Figure REF .", "We intentionally keep the formulation general here to allow for applications different to dynamic inverse problems, such as multi-spectral imaging.", "Nevertheless, the derived reconstruction and feature extraction framework in this paper will be used in Section for the specific application to dynamic computed tomography.", "Figure: Illustration of a phantom that can be represented by the decomposition in ().", "The phantom consists of K=3K=3 components: the background and two dynamic components with periodically changing intensity (left and right plot).", "As such, this phantom can be efficiently represented by a low-rank decomposition considered in this study.Furthermore, a suitable discretisation of the continuous formulation (REF ) is needed to introduce the feature extraction methods in the forthcoming sections.", "Let us first discretise the secondary variable, such that $t\\in \\mathbb {N}$ with $1\\le t \\le T$ .", "For the spatial domain, we assume a vectorised representation such that the resulting unknown can be represented as a matrix $X\\in \\mathbb {R}^{N\\times T}$ , which leads to the matrix equation $A_t X_{\\bullet , t} = Y_{\\bullet , t} \\quad \\text{for} \\quad 1\\le t\\le T,$ where $A_t\\in \\mathbb {R}^{M\\times N} $ is the discretised forward operator, $X_{\\bullet , t}$ the $t$ -th column of $X$ and $Y_{\\bullet , t}$ the $t$ -th column of the data matrix $Y\\in \\mathbb {R}^{M\\times T}.$ Analogously, we will write $M_{n, \\bullet }$ for the $n$ -th row of an arbitrary matrix $M.$ Suitable restrictions to the matrices in Equation (REF ) will be made in the following sections to ensure the applicability of the considered frameworks and, if possible, to properly represent the decomposition (REF )." ], [ "Feature Extraction Methods", "In this section, we introduce two feature extraction methods, namely the PCA (PCA) and the NMF (NMF).", "These approaches are used to compute the latent components of the reconstruction $X.$ The NMF will be used in Section REF to introduce a joint reconstruction and low-rank decomposition framework to tackle the problem stated in (REF )." ], [ "PCA", "Large and high dimensional datasets demand modern data analysis approaches to reduce the dimensionality and increase the interpretability of the data while keeping the loss of information as low as possible.", "Many different techniques have been developed for this purpose, but PCA is one of the most widely used and goes back to [35].", "For a given matrix $X\\in \\mathbb {R}^{N\\times T}$ with $N$ different observations of an experiment and $T$ features, the PCA is a linear orthogonal transformation given by the weights $C_{\\tilde{k}, \\bullet } = (C_{\\tilde{k}1}, \\dots , C_{\\tilde{k}T})$ with $C \\in \\mathbb {R}^{\\tilde{K}\\times T},$ which transforms each observation $X_{n, \\bullet }$ to principal component scores given by $B_{n\\tilde{k}} \\sum _t X_{nt} C_{\\tilde{k}t} $ with $B=[B_{\\bullet , 1}, \\dots , B_{\\bullet , \\tilde{K}}]\\in \\mathbb {R}^{N\\times \\tilde{K}}$ and $\\tilde{K}=\\min (N-1, T),$ such that the sample variance $\\operatorname{Var}(B_{\\bullet , \\tilde{k}})$ is maximised for all $\\tilde{k},$ each row $C_{\\tilde{k}, \\bullet }$ is constrained to be a unit vector and the sample covariance $\\operatorname{cov}(B_{\\bullet , k}, B_{\\bullet , \\tilde{k}}) = 0 $ for $k\\ne \\tilde{k}.$ Together with the usual assumption that the number of observations is higher than the underlying dimension, this leads to $\\tilde{K}=T$ and the full transformation $B = XC^\\intercal ,$ where $C$ is an orthogonal matrix.", "The $t$ -th column vector $(C_{t, \\bullet })^\\intercal $ defines the $t$ -th principal direction and is an eigenvector of the covariance matrix $S=X^\\intercal X/(N-1).$ The corresponding $t$ -th largest eigenvalue of $S$ denotes the variance of the $t$ -th principal component.", "The above transformation is equivalent to the factorisation of the matrix $X$ given by $X = BC,$ which allows to decompose each observation into the principal components, such that $X_{n, \\bullet } = \\sum _{t=1}^T B_{nt} C_{t, \\bullet }.$ Therefore, we also have $X = \\sum _{t=1}^{T} B_{\\bullet , t} C_{t, \\bullet }$ similarly to the continuous case in (REF ).", "Furthermore, it is possible to obtain an approximation of the matrix $X$ by truncating the sum at the first $K<T$ principle components for all $n,$ which yields a rank $K$ matrix $X^{(K)}$ given by $X^{(K)} = \\sum _{k=1}^K B_{\\bullet , k} C_{k, \\bullet }.$ Based on the Eckart–Young–Mirsky theorem [19], $X^{(K)}$ is the best rank $K$ approximation of $X$ in the sense that it minimises the discrepancy $\\Vert X - X^{(K)}\\Vert $ for both the Frobenius and spectral norm.", "One typical approach to compute the PCA is based on the SVD of the data matrix $X=U\\Sigma V^\\intercal $ and will be used in this work.", "Setting $BU\\Sigma $ and $C=V^\\intercal $ gives already the desired factorisation in (REF ) based on the PCA." ], [ "NMF", "NMF, originally introduced as positive matrix factorisation by Paatero and Tapper in 1994 [34], is an established tool to obtain low-rank approximations of nonnegative data matrices.", "It has been widely used in the machine learning and data mining community for compression, basis learning, clustering and feature extraction for high-dimensional classification problems with applications in music analysis [17], document clustering [13] and medical imaging problems such as tumor typing in matrix-assisted laser desorption/ionisation (MALDI) imaging in the field of bioinformatics [28].", "Different from the PCA approach above, the NMF enforces nonnegativity constraints on the factor matrices without any orthogonality restrictions.", "This makes the NMF the method of choice for application fields, where the underlying physical model enforces the solution to be nonnegative assuming that each datapoint can be described as a superposition of some unknown characteristic features of the dataset.", "The NMF makes it possible to extract these features while constraining the matrix factors to have nonnegative entries, which simplifies their interpretation.", "These data assumptions are true for many application fields including the ones mentioned above but also especially our considered problem of dynamic computed tomography, where the measurements consist naturally of the nonnegative absorption of photons.", "Mathematically, the basic NMF problem can be formulated as follows: For a given nonnegative matrix $X\\in \\mathbb {R}_{\\ge 0}^{N\\times T},$ find nonnegative matrices $B\\in \\mathbb {R}_{\\ge 0}^{N\\times K}$ and $C\\in \\mathbb {R}_{\\ge 0}^{K\\times T}$ with $K\\ll \\min \\lbrace N, T\\rbrace ,$ such that $X\\approx BC.$ The factorisation allows to approximate the rows $X_{n, \\bullet }$ and columns $X_{\\bullet , t}$ of the data matrix as a superposition of the $K$ columns $B_{\\bullet , k}$ of $B$ and rows $C_{k, \\bullet }$ of $C$ respectively, such that $X_{n,\\bullet } \\approx \\sum _{k=1}^{K} B_{nk}C_{k,\\bullet } $ and $ X_{\\bullet , t} \\approx \\sum _{k=1}^{K} C_{kt} B_{\\bullet , k}.$ Similarly, it holds that $X \\approx BC = \\sum _{k=1}^{K} B_{\\bullet , k} C_{k, \\bullet },$ where the $K$ terms of the sum are rank-one matrices.", "Hence, the sets $\\lbrace B_{\\bullet , k} \\rbrace _k$ and $ \\lbrace C_{k, \\bullet } \\rbrace _k $ can be interpreted as a low-dimensional basis to approximate the data matrix, i.e.", "the NMF performs the task of basis learning with additional nonnegativity constraints.", "The usual approach to compute the factorisation is to define a suitable discrepancy term $\\mathcal {D}_{\\text{NMF}},$ which has to be chosen according to the noise assumption of the underlying problem, and to reformulate the NMF as a minimisation problem.", "Typical discrepancies include the default case of the Frobenius Norm on which we will focus on, the Kullback-Leibler divergence, the Itakura-Saito distance or other generalized divergences [10].", "Furthermore, NMF problems are usually ill-posed due to the non-uniqueness of the solution [21] and require the application of suitable regularisation techniques.", "One common method is to include penalty terms in the minimisation problem to tackle the ill-posedness of the problem but also to enforce desirable properties of the factorisation matrices.", "Typical examples range from $\\ell _1, \\ell _2$ and total variation regularisation terms [25] to more problem specific terms, which enforce additional orhogonality of the matrices or even allow supervised classification workflows if the NMF is used as a prior feature exctraction method [16], [28].", "Hence, the general regularised NMF problem can be written as $\\min _{B, C\\ge 0} \\mathcal {D}_{\\text{NMF}}(X, BC) + \\sum _{\\ell = 1}^{L} \\gamma _\\ell \\mathcal {P}_\\ell (B, C) \\min _{B, C\\ge 0} \\mathcal {F}(B, C),$ where $\\mathcal {P}_\\ell $ denote the penalty terms, $\\gamma _\\ell \\ge 0$ the corresponding regularisation parameters and $\\mathcal {F}$ the cost function of the NMF.", "The considered optimisation approach in this work is based on the so-called Majorise-Minimisation principle and gives rise to multiplicative update rules of the matrices in (REF ), which automatically preserve the nonnegativity of the iterates provided that they are initialised nonnegative.", "For more details on this optimisation technique, we refer the reader to Appendix .", "The idea of the feature extraction procedure based on the NMF can be well illustrated by considering the example from Figure REF that satisfies the decomposition assumption from (REF ).", "Here, the highlighted spatial regions change their intensities according to the given dynamics.", "The NMF allows a natural interpretation of the factorisation matrices $B$ and $C$ as the spatial and temporal basis functions for this case, as illustrated in Figure REF .", "The column $X_{\\bullet , t}$ of $X$ denotes the reconstruction of the $t$ -th time step of the inverse problem in (REF ).", "The NMF allows to decompose the spatial and temporal features of the data: The matrix $B$ contains the spatial features in its columns with the corresponding temporal features in the rows of $C.$ Figure: Structure of the NMF in the context of the dynamic Shepp-Logan phantom as shown in Figure .", "Here, the nonnegative spatial and temporal basis functions can be naturally represented by the matrices BB and CC." ], [ "Separated Reconstruction and Low-rank Decomposition", "Let us first discuss a separated reconstruction and feature extraction approach to solve the inverse problem in (REF ), that means we compute first a reconstruction and perform then subsequently the feature extraction with one of the previously discussed methods.", "We consider this method as baseline for our comparison.", "The considered reconstruction method for this separated framework involves a basic gradient descent approach together with a regularisation step and a subsequent total variation denoising, which will be henceforth referred to as gradTV.", "The details on the algorithm are provided in Algorithm REF .", "In particular, we aim to compute solutions to the least squares problem and incorporate the low-rank assumptions as additional penalty of the nuclear norm of $X_{\\bullet , t}$ , that is $\\min _{X_{\\bullet , t}\\ge 0} \\Vert Y_{\\bullet , t} - A_t X_{\\bullet , t} \\Vert ^2_2 + \\alpha \\Vert X_{\\bullet , t}\\Vert _*$ for all $t$ ; see e.g.", "[29], [41].", "This can then be efficiently solved by a proximal gradient descent scheme with a soft-thresholding on the singular values and hence enforcing the low-rank structure.", "Ideally, one would like to include the total variation regularisation as penalty term, but as this tends to be computationally expensive for the fine temporal sampling, we included this as a subsequent denoiser.", "In practice, after a suitable initialisation of the reconstruction matrix, the gradient descent step is computed with an, a priori defined, fixed stepsize $\\rho _{\\text{grad}}.$ For the proximal step, the truncated SVD of $X$ is computed and a soft thresholding of the singular values is performed with a fixed threshold $\\rho _{\\text{thr}}.$ Afterwards, we enforce the nonnegativity with a projection step on the reconstruction $X.$ When the stopping criterion is satisfied, a TV denoising algorithmhttps://www.mathworks.com/matlabcentral/fileexchange/36278-split-bregman-method-for-total-variation-denoising based on [18], [40] with the corresponding parameter $\\rho _{\\text{TV}}$ is applied.", "gradTV [1] Initialise: $X$ Input: $\\rho _{\\text{grad}}, \\rho _{\\text{thr}}, \\rho _{\\text{TV}} >0$ $X_{\\bullet , t} \\leftarrow X_{\\bullet , t} - \\rho _{\\text{grad}} (A_t^\\intercal A_tX_{\\bullet , t} - A_t^\\intercal Y_{\\bullet , t}) \\ \\ \\ \\text{for all}\\ t$ $(U,\\Sigma , V) \\leftarrow \\textsc {SVD}(X)$ $\\Sigma \\leftarrow \\textsc {SoftThresh}_{\\rho _{\\text{thr}}}(\\Sigma )$ $X \\leftarrow U\\Sigma V^\\intercal $ $X\\leftarrow \\max (X, 0) $ StoppingCriterion satisfied $X \\leftarrow \\textsc {TVDenoiser}_{\\rho _{\\text{TV}}}(X)$ $X$ After the reconstruction procedure given by Algorithm REF , we perform the feature extraction of the reconstruction $X$ via both the PCA and the NMF and call the approach gradTV_PCA and gradTV_NMF respectively.", "For gradTV_PCA, we simply compute the PCA of $X$ based on its SVD.", "Concerning the method gradTV_NMF, we consider the standard NMF model $\\min _{B, C\\ge 0} \\Vert X - BC\\Vert _F^2 + \\dfrac{\\tilde{\\mu }_C}{2} \\Vert C \\Vert _F^2$ with the parameter $\\tilde{\\mu }_C.$ The $\\ell _2$ regularisaton penalty term on $C$ is motivated by our application in Section .", "The corresponding multiplicative algorithms to solve (REF ) are well-known [11], [16] and a special case of the derived update rules in the next Section." ], [ "Joint Reconstruction and Low-rank Decomposition", "Instead of the previously discussed separated reconstruction, we now aim to include the feature extraction into the reconstruction procedure.", "This gives rise to consider a joint reconstruction and low-rank decomposition approach based on the NMF, rather than one based on a low-rank plus sparsity approach based on PCA [7], [39], [44].", "The basic idea of the method is to incorporate the reconstruction procedure of the inverse problem in (REF ) into the NMF workflow.", "To do this, we have to additionally assume that $A_t\\in \\mathbb {R}_{\\ge 0}^{M\\times N}, Y\\in \\mathbb {R}_{\\ge 0}^{M\\times T}$ and $X\\in \\mathbb {R}_{\\ge 0}^{N\\times T}$ to ensure the desired nonnegativity of the factorisation matrices $B$ and $C,$ which corresponds to the assumptions of the decomposition in (REF ).", "The main motivation is that this joint approach allows the reconstruction process to exploit the underlying latent NMF features of the dataset, which can therefore enhance the quality of the reconstructions by enabling regularisation of temporal and spatial features separately.", "This can be achieved by including a discrepancy term $\\mathcal {D}_{\\text{IP}}(Y_{\\bullet , t}, A_t X_{\\bullet , t})$ of the inverse problem into the NMF cost function in (REF ).", "This leads together with some possible penalty terms for the reconstruction $X$ to the model $ \\min _{B, C, X\\ge 0} \\mathcal {D}_{\\text{IP}}(Y_{\\bullet , t}, A_t X_{\\bullet , t}) + \\alpha \\mathcal {D}_{\\text{NMF}}(X, BC) + \\sum _{\\ell = 1}^{L} \\gamma _\\ell \\mathcal {P}_\\ell (B, C, X),$ with $\\alpha \\ge 0$ for the joint reconstruction and low-rank decomposition problem, which we will call BC-X.", "Furthermore, we can enforce $XBC$ as a hard constraint, such that the reconstruction matrix will have at most rank $K.$ In this case, the discrepancy $\\mathcal {D}_{\\text{NMF}}$ vanishes and we end up with the model BC: $ \\min _{B, C\\ge 0} \\mathcal {D}_{\\text{IP}}(Y_{\\bullet , t}, A_t (BC)_{\\bullet , t}) + \\sum _{\\ell = 1}^{L} \\gamma _\\ell \\mathcal {P}_\\ell (B, C).$" ], [ "Considered NMF Models", "For both models (REF ) and (REF ), we use the standard Frobenius norm for both the discrepancy terms $\\mathcal {D}_{\\text{NMF}}$ and $\\mathcal {D}_{\\text{IP}}.$ Furthermore, the optimisation method discussed in Section REF allows to include a variety of penalty terms into the cost function.", "This makes it possible to construct suitable regularised NMF models and to enforce additional properties to the matrices depending on the specific application.", "In this work, we will consider standard $\\ell _1$ and $\\ell _2$ regularisation terms on each matrix and an isotropic total variation penalty on the matrix $B.$ The latter is motivated by our considered application, which denoises the spatial features and thus also the reconstruction matrix.", "Hence, we will focus on the following NMF models in the remainder of this work: $ \\begin{aligned}&\\scalebox {0.9}{\\text{\\qquad \\mathrm {\\texttt {BC-X}}$\\displaystyle \\min _{B, C, X\\ge 0} \\Bigg \\lbrace \\sum _{t=1}^{T} \\frac{1}{2} \\Vert A_t X_{\\bullet , t} - Y_{\\bullet , t} \\Vert _2^2 + \\frac{\\alpha }{2} \\Vert BC - X \\Vert _F^2 + \\lambda _{B} \\Vert B \\Vert _1 + \\frac{\\mu _B}{2} \\Vert B\\Vert _F^2 $ }} \\\\&\\hspace{42.5pt}\\scalebox {0.9}{\\text{$\\displaystyle + \\lambda _{C} \\Vert C \\Vert _1 + \\frac{\\mu _C}{2} \\Vert C\\Vert _F^2 + \\lambda _X \\Vert X \\Vert _1 + \\frac{\\mu _X}{2} \\Vert X\\Vert _F^2 + \\dfrac{\\tau }{2} \\operatorname{TV}(B)\\Bigg \\rbrace , $ }}\\end{aligned}$ $ \\begin{aligned}&\\min _{B, C \\ge 0} \\Bigg \\lbrace \\sum _{t=1}^{T} \\frac{1}{2} \\Vert A_t (BC)_{\\bullet , t} - Y_{\\bullet , t} \\Vert _2^2 \\!+\\!", "\\lambda _{C} \\Vert C \\Vert _1 + \\frac{\\mu _{C}}{2} \\Vert C\\Vert _F^2 + \\lambda _{B} \\Vert B \\Vert _1\\\\&\\hspace*{38.25pt} + \\frac{\\mu _{B}}{2} \\Vert B\\Vert _F^2 + \\dfrac{\\tau }{2} \\operatorname{TV}(B) \\Bigg \\rbrace .\\end{aligned}\\qquad \\mathrm {\\texttt {BC}}$ The regularisation parameters $ \\alpha , \\lambda _{C}, \\mu _{C}, \\lambda _{B}, \\mu _{B}, \\lambda _{X}, \\mu _{X}, \\tau \\ge 0, $ are chosen a priori.", "Furthermore, $ \\Vert \\cdot \\Vert _F $ denotes the Frobenius norm, $ \\Vert M \\Vert _1\\sum _{ij} \\vert M_{ij} \\vert $ the 1-norm for matrices $ M $ and $ \\operatorname{TV}(\\cdot ) $ is the following smoothed isotropic total variation [12], [16], [25].", "Definition 2.1 The total variation of a matrix $ B\\in \\mathbb {R}^{N\\times K} $ is defined as $\\operatorname{TV}(B) \\sum _{k=1}^K \\sum _{n=1}^N \\vert \\nabla _{nk}B\\vert \\sum _{k=1}^K \\sum _{n=1}^N \\sqrt{\\varepsilon _{\\operatorname{TV}}^2 + \\sum _{\\ell \\in \\mathcal {N}_n} (B_{nk}-B_{\\ell k})^2},$ where $ \\varepsilon _{\\operatorname{TV}}>0 $ is a small positive constant and $ \\mathcal {N}_n $ are index sets referring to spatially neighboring pixels.", "A typical example for the neighbourhood of the pixel $ (0,0) $ in two dimensions is $ \\mathcal {N}_{(0,0)} = \\lbrace (1,0), (0,1) \\rbrace $ to get an estimate of the gradient components in both directions of the axes.", "The parameter $ \\varepsilon _{\\operatorname{TV}} $ ensures the differentiability of the TV penalty term.", "In the following section, we will present the multiplicative update rules for the NMF models REF and REF and derive the algorithms in Appendix based on the Majorise-Minimisation principle." ], [ "Algorithms", "In this section, we present in Theorem REF and REF the multiplicative algorithms for the NMF problems in REF and REF .", "As mentioned in Section REF , the multiplicative structure of the iteration scheme ensures automatically the nonnegativity of the matrices $B$ and $C$ as long as they are initialised nonnegative.", "The derivation of such algorithms in this work are based on the so-called Majorise-Minimisation principle.", "The main idea of this approach is to replace the considered NMF cost function $\\mathcal {F}$ with a suitable auxiliary function $\\mathcal {Q}_{\\mathcal {F}}$ , whose minimisation is much easier to handle and leads to a monotone decrease of $\\mathcal {F}.$ Furthermore, specific construction techniques of these surrogate functions lead to the desired multiplicative update rules which fulfill the nonnegativity constraint.", "We provide a short description of the main principles in Appendix .", "A more detailed discussion of different construction methods for various kinds of discrepancy and penalty terms of $\\mathcal {F}$ can be found in the survey paper [16].", "For better readability, we present only the main results here and a detailed construction of the surrogate functions as well as derivation of the algorithms for both cost functions REF and REF can be found in Appendix .", "Consequently, we will only state the main results in Theorem REF and REF here.", "Nevertheless, due to the construction of a suitable surrogate function for the TV penalty term (see Appendix and [16] for more details), we first introduce the following matrices $ P(B), Z(B)\\in \\mathbb {R}_{\\ge 0}^{N\\times K} $ as P(B)n k 1nk B Nn 1 + Nn 1k B , Z(B)n k 1P(B)n k ( 1nk B Nn Bn k + Bk2 + Nn Bn k + Bk2 k B ), where $ \\bar{\\mathcal {N}}_n $ is the set of the so-called adjoint neighbourhood pixels, which is given by the relation $\\ell \\in \\bar{\\mathcal {N}}_n \\Leftrightarrow n\\in \\mathcal {N}_\\ell .$ Furthermore, we write ${\\mathit {1}}_{M\\times N}$ for an $M\\times N$ matrix with ones in every entry.", "We then obtain the two algorithms for both models under consideration.", "First for the REF model that jointly obtains the reconstruction $X$ and the decomposition: Theorem 2.1 (Algorithm for REF ) For $A_t\\in \\mathbb {R}_{\\ge 0}^{M\\times N}, Y\\in \\mathbb {R}_{\\ge 0}^{M\\times T}$ and initialisations $ X^{[0]}\\in \\mathbb {R}_{> 0}^{N\\times T}, B^{[0]}\\in \\mathbb {R}_{> 0}^{N\\times K}, C^{[0]}\\in \\mathbb {R}_{> 0}^{K\\times T}, $ the alternating update rules X, t[d+1] = X, t[d] AtY, t + B[d]C, t[d]AtAtX, t[d] + (X + )X, t[d] + X 1N1 B[d+1] = B[d] X[d+1]C[d]+ P(B[d]) Z(B[d])B[d]C[d]C[d]+ B B[d] + B 1NK + B[d] P(B[d]) C[d+1] = C[d] B[d+1]X[d+1]B[d+1]B[d+1] C[d] + C C[d] + C 1KT lead to a monotonic decrease of the cost function in REF .", "Similarly, for the REF model we obtain the updates rules without constructing the matrix $X$ during the reconstruction process: Theorem 2.2 (Algorithm for REF ) For $A_t\\in \\mathbb {R}_{\\ge 0}^{M\\times N}, Y\\in \\mathbb {R}_{\\ge 0}^{M\\times T}$ and initialisations $ B^{[0]}\\in \\mathbb {R}_{> 0}^{N\\times K}, C^{[0]}\\in \\mathbb {R}_{> 0}^{K\\times T}, $ the alternating update rules $B^{[d+1]}$ =$ B^{[d]} \\circ \\dfrac{\\sum _{t=1}^T A_t^\\intercal Y_{\\bullet , t} \\cdot ({C^{[d]}}^\\intercal )_{t,\\bullet } + \\tau P(B^{[d]}) \\circ Z(B^{[d]})}{\\sum _{t=1}^T A_t^\\intercal A_t (B^{[d]} C^{[d]})_{\\bullet , t} \\cdot ({C^{[d]}}^\\intercal )_{t,\\bullet } + \\mu _{B} B^{[d]} + \\lambda _{B}{\\mathit {1}}_{N\\times K} + \\tau B^{[d]} \\circ P(B^{[d]})}$ C[d+1],t = C[d],t B[d+1]AtY,tB[d+1]AtAt (B[d+1]C[d]),t + C C[d],t + C 1K1 lead to a monotonic decrease of the cost function in REF .", "We remind that the derivation is described in Appendix , which leads to the update rules in the Theorems above.", "Due to the multiplicative structure of the algorithms, zero entries in the matrices stay zero during the iteration scheme and can cause divisions by zero.", "This issue is handled via the strict positive initialisation in both Theorems.", "Furthermore, very small or high numbers can cause numerical instabilities and lead to undesirable results.", "As a standard procedure, this problem is handled by suitable projection steps after every iteration step [10]." ], [ "Complexity Reduction for Stationary Operator", "Let us now consider the case of a stationary operator, i.e.", "$\\mathcal {A}(\\cdot ;t)$ in equation (REF ) does not change with $t$ .", "Then we simply write $\\mathcal {A}$ or $A$ for the matrix representation in (REF ).", "If further the number of channels $T$ is large, the application of the forward operator represented a major computational burden per channel.", "In particular, we make use here of the assumption $T\\gg K$ , i.e.", "the number of channels is much larger than the basis functions for the decomposition.", "In this case, we can effectively reduce the computational cost by shifting the application of the forward operator to the spatial basis functions contained in $B$ .", "That means, we make essential use of the decomposition $X\\approx BC$ in the reconstruction task and as such avoid to construct the approximation to $X$ .", "Consequently, we will only consider the case of REF here.", "Since $A$ is independent from $t,$ the NMF model REF becomes $ \\begin{aligned}&\\min _{B, C \\ge 0} \\Big \\lbrace \\frac{1}{2} \\Vert ABC - Y \\Vert _F^2 + \\lambda _{C} \\Vert C \\Vert _1 + \\frac{\\mu _{C}}{2} \\Vert C\\Vert _F^2 + \\lambda _{B} \\Vert B \\Vert _1\\\\&\\hspace*{38.25pt} + \\frac{\\mu _{B}}{2} \\Vert B\\Vert _F^2 + \\dfrac{\\tau }{2} \\operatorname{TV}(B) \\Big \\rbrace .\\end{aligned}\\qquad \\mathrm {\\texttt {sBC}}$ To illustrate this, let us consider the update equation in Theorem REF for $B,$ where we can simplify the first term in the denominator as follows: $\\scalebox {0.86}{\\text{$\\sum _{t=1}^T A^\\intercal A (B^{[d]} C^{[d]})_{\\bullet , t} \\cdot ({C^{[d]}}^\\intercal )_{t,\\bullet } = A^\\intercal A \\sum _{t=1}^T (B^{[d]} C^{[d]})_{\\bullet , t} \\cdot ({C^{[d]}}^\\intercal )_{t,\\bullet } = A^\\intercal A B^{[d]} C^{[d]} {C^{[d]}}^\\intercal .$ }}$ The other terms in the update rules can be simplified similarly, such that we obtain the following reduced update equations: Corollary 2.1 (Algorithm for REF ) For $A\\in \\mathbb {R}_{\\ge 0}^{M\\times N}, Y\\in \\mathbb {R}_{\\ge 0}^{M\\times T}$ and initialisations $ B^{[0]}\\in \\mathbb {R}_{> 0}^{N\\times K},$ $C^{[0]}\\in \\mathbb {R}_{> 0}^{K\\times T}, $ the alternating update rules B[d+1] = B[d] AY C[d]+ P(B[d]) Z(B[d])AA B[d] C[d] C[d]+ B B[d] + B1NK + B[d] P(B[d]) C[d+1] = C[d] B[d+1]AYB[d+1]AA B[d+1]C[d] + C C[d] + C 1KT .", "lead to a monotonic decrease of the cost function in REF .", "Finally, the order of application is essential here to obtain the complexity reduction.", "In particular, we implemented the algorithm such that $A$ acts on the basis functions in $B$ .", "That means, we compute first the product $A^\\intercal A B$ followed by multiplication with $C$ .", "That means, we can expect a reduction of computational complexity by a factor $T/K$ with the REF model and hence is especially useful for dimension reduction under fine temporal sampling." ], [ "Application to Dynamic CT", "In the following we will apply the presented methods to the use case of dynamic computerised tomography (CT).", "Here, the quantity of interest is given as the attenuation coefficient $x(s,t)$ at time $t\\in [0,T]$ on a bounded domain in two dimensions $s\\in \\Omega _1\\subset \\mathbb {R}^2$ .", "Following the formulation in (REF ), the time-dependent forward operator is given by the Radon transform $y(\\theta , \\sigma , t) (\\mathcal {R}_{\\mathcal {I}(t)} x(s, t))(\\theta , \\sigma ) = \\int _{s\\cdot \\theta = \\sigma } x(s, t) \\mathop {} \\!", "\\mathrm {d}s $ Here, the measurement $y(\\theta ,\\sigma ,t)$ consist of line integrals over the domain $\\Omega _1$ for each time point $t\\in \\mathcal {T}$ , and is referred to as the sinogram.", "This measurement depends on two parameters, the angle $\\theta \\in S^1$ on the unit circle and a signed distance to the origin $\\sigma \\in \\mathbb {R}$ .", "Consequently, the measurements depend on a set of angles at each time step $\\mathcal {I}(t)$ , such that $(\\theta ,\\sigma )\\in \\mathcal {I}(t)$ at time $t$ , we will refer to this as the sampling patterns.", "In a slight abuse of notation, we will use $| \\mathcal {I}(t) |$ for the number of angles, i.e.", "directions for the line integrals, at each time point.", "In the following we consider two scenarios for the choice of angles in $\\mathcal {I}(t)$ and by that defining the nature of the forward operator, as discussed in Section REF .", "In the general case of a nonstationary forward operator, that means the sampling patterns are time-dependent, we assume that the angles change but the amount of angles is constant over time $| \\mathcal {I}(t) | \\equiv c$ .", "Additionally, we will consider the case for stationary operators, which in our setting means that the set of angles does not change over time, we can write for instance $\\mathcal {I}(t) \\equiv \\mathcal {I}(t=0)$ , and hence this leads to a stationary measurement operator of the dynamic process in (REF ).", "We note that even though the measurement process is stationary, the obtained measurement $y(\\theta ,\\sigma ,t)$ itself is still time dependent.", "For the computations, we discretise (REF ) to obtain a matrix vector representation as in (REF ).", "In the following we will write $R_t$ for the discrete Radon transform with respect to the sampling pattern $\\mathcal {I}(t)$ at time point $t$ , which gives rise to the discrete reconstruction problem for dynamic CT $R_t X_{\\bullet , t} = Y_{\\bullet , t} \\quad \\text{for} \\quad 1\\le t\\le T.$ We note, that due to the definition of the Radon transform by line integrals, the matrix $R_t\\in \\mathbb {R}_{\\ge 0}^{M\\times N}$ has only nonnegative entries and hence satisfies the assumption for Theorem REF and REF .", "Furthermore, $N$ denotes here the number of pixels in the original image and $M$ is given by the product $M\\vert \\mathcal {I}(t)\\vert n_S,$ where $n_S$ is the number of detection points." ], [ "Results and Discussion", "For a qualitative evaluation of the proposed NMF approaches, we consider in the following sections two simulated datasets.", "Due to the known ground truth in both cases, we are able to measure the performance of each method via computing the mean of the PSNR and the mean of the SSIM index [3] over all time steps for every experiment.", "For each dataset, the parameters of all methods are chosen empirically to provide good reconstructions.", "For the NMF models of the joint reconstruction and low-rank decomposition approach, we restrict ourselves to the total variation penalty term on $B$ to provide some denoising effect on the spatial features and the $\\ell ^2$ penalty on $C$ for the time features, since we expect and enforce smooth changes in time.", "We consider the standard case for the TV term with the default pixel neighbourhood and choose the smoothing parameter $\\varepsilon _{\\text{TV}} = 10^{-5}$ relatively small.", "Furthermore, for both datasets we measure different angles at each time step based on a tiny golden angle sampling [43] using consecutive projections with increasing angle of $\\varphi = 32.039\\dots $ , such that projection angles are not repeated.", "Nevertheless, we remind that we keep the total number of observed angles constant for each time step.", "For all considered approaches we use the unfiltered backprojection, given by the adjoint of the Radon transform, applied to the noisy data matrix $Y$ as the initialisation for the reconstruction matrix $X.$ In case of the NMF approaches, the matrices $B$ and $C$ are initialised via SVD of $X$ based on [2].", "After the initialisation and at every iteration of the NMF algorithm, a suitable projection step for small values is performed to prevent numerical instabilities and zero entries during the multiplicative algorithm [10].", "The algorithms were implemented with MATLAB® R2019b and run on an Intel® Core™ i7-7700K quad core CPU @4.20 GHz with 32 GB of RAM.", "To this end we present a summary and short explanation of all considered algorithms in this experimental section in Table REF .", "Table: Summary and short explanation of considered algorithms in the experimental section." ], [ "Shepp-Logan Phantom", "This synthetic dataset consists of a dynamic two-dimensional Shepp-Logan phantom with $T=100$ and spatial size $ 128 \\times 128$ , see Figure REF for the ground-truth.", "During the whole time, two of the inner ellipsoids change their intensities sinusoidally with different frequencies while the rest of the phantom remains constant.", "In the following, we perform a variety of experiments for $\\vert \\mathcal {I}_t\\vert \\in \\lbrace 2,\\dots , 12\\rbrace $ with 1% and 3% Gaussian noise respectively.", "For all cases, we choose $K=5$ for the number of the NMF features.", "In such a way, the NMF is also capable to approximate minor characteristics such as noise or other artefacts of the reconstruction matrix besides the three main features.", "The parameters of all methods were determined empirically and are displayed in Table REF in Appendix for both noise levels.", "The stopping criterion for all methods is met, if 1200 iteration steps are reached or if the relative change of all matrices $B, C$ and $X$ goes below $5\\cdot 10^{-5}.$ Figure: Results for the dynamic Shepp-Logan phantom with |ℐ t |=6\\vert \\mathcal {I}_t \\vert =6 angles per time step and 1% Gaussian noise.", "Shown are the leading extracted features for the model (left) and for (right).Figure: Results for the dynamic Shepp-Logan phantom with |ℐ t |=6\\vert \\mathcal {I}_t \\vert =6 angles per time step and 1% Gaussian noise.", "Shown are the leading extracted features for the gradTV_PCA model (left) and for gradTV_NMF (right).We show first some results for the case with $\\vert \\mathcal {I}_t \\vert =6$ and 1% Gaussian noise in Figure REF for the joint NMF methods and Figure REF for the separate reconstruction and extraction.", "The order of shown features is based on the singular values of $B$ for gradTV_PCA and on the $\\ell _2$ -norm of the spatial features for NMF approaches.", "In this case, all considered approaches are able to successfully identify the constant and dynamic parts of the dataset and extract meaningful spatial and temporal features.", "The extracted spatial features of REF , REF and gradTV_NMF show very clearly the dynamic and non-dynamic parts of the Shepp-Logan phantom.", "However, the spatial features of gradTV_NMF are slightly more blurred and affected by minor artefacts especially in both dynamic features.", "This underlines the positive effect of the separate TV regularisation on the spatial feature matrix $B$ in the joint methods.", "In contrast, gradTV_PCA is able to identify the main components of the dataset correctly, but there is a clear corruption of the dynamic features with other parts from the phantom.", "Furthermore, all spatial features contain negative parts due to the non-existent nonnegativity constraint of the gradTV_PCA approach which makes their interpretation more challenging.", "Hence, the additional nonnegativity constraint of the NMF methods improve significantly the quality and interpretability of the extracted components in comparison with the PCA based extraction method.", "The temporal features of all methods are clearly extracted and are consistent with the underlying ground truth of the dataset.", "However, we note that REF and REF have a slight difficulty to resolve the lower intensity part close to 0, which is probably caused by the multiplicative structure of the algorithms.", "Figure: Results for the dynamic Shepp-Logan phantom with |ℐ t |=6\\vert \\mathcal {I}_t \\vert =6 angles per time step and 3% Gaussian noise.", "Shown are the leading extracted features for the model (left) and for gradTV_PCA (right).Figure: Results for the dynamic Shepp-Logan phantom with |ℐ t |=3\\vert \\mathcal {I}_t \\vert =3 angles per time step and 1% Gaussian noise.", "Shown are the leading extracted features for the model (left) and for (right).Similar observations can be made for the case $\\vert \\mathcal {I}_t \\vert =6$ and 3% Gaussian noise.", "We present the reconstructed features in Figure REF for REF and gradTV_PCA only.", "The higher amount of noise can be observed especially in the spatial features of gradTV_PCA, whereas it only has a slight effect in the REF model.", "Finally, we present the reconstructed features with REF and REF in Figure REF for $\\vert \\mathcal {I}_t \\vert =3$ , i.e.", "only three three angles per time step with a noise level of 1%.", "The major difference to the previous cases can be seen in the results of the REF model.", "Here, the method splits up the dynamics of the right ellipse into two different temporal features, such that the true dynamics are not retained.", "However, the REF approach perform remarkably well with respect to the feature extraction despite the rather low number of projection angles.", "This might indicate, that enforcing the reconstruction $X$ to have small data error helps in the REF model to stabilise the reconstruction in highly sparse data settings.", "Let us shortly discuss other considered values of $\\vert \\mathcal {I}_t \\vert ,$ that are not shown here.", "First of all, the performance of gradTV_PCA and gradTV_NMF with respect to the feature extraction behaves very similar for both noise cases.", "Besides the above mentioned drawbacks, both approaches give remarkably consistent results especially for low number of angles and do not tend as much to split up features like in REF and REF .", "The latter occurs in different degrees for several numbers of angles.", "For 1% noise, it occurs for $\\vert \\mathcal {I}_t \\vert \\in \\lbrace 3, 7, 8, 10\\rbrace $ in REF and for $\\vert \\mathcal {I}_t \\vert = 10 $ in REF .", "In the case of a noise level of 3%, the split up effect only occurs for $\\vert \\mathcal {I}_t \\vert = 10 $ in REF .", "However, for $\\vert \\mathcal {I}_t \\vert = 10, $ it is possible to partially recover the correct temporal feature by simply adding up both features.", "Nevertheless, both approaches provide better reconstruction quality of $X$ than gradTV as we will discuss in the following." ], [ "Quantitative Evaluation", "Let us now discuss the quantitative reconstruction quality for all methods.", "In Figure REF and REF , we show the mean PSNR and SSIM of the reconstructions for 1% and 3% noise over all time steps for all considered numbers of projection angles.", "Note that for the NMF model REF , we compute the quality measures for $X.$ The same goes for gradTV, where we only compute the quality measures of $X$ after the reconstruction procedure independently of the subsequent feature extraction method.", "In the case of REF , the reconstruction is computed as $X=BC$ .", "As expected, the reconstruction quality tends to get better if more angles per time step are considered.", "More importantly, we see that it is possible to obtain reasonable reconstructions with just a few projections per time step especially in the case of the joint reconstruction and feature extraction method via the NMF approach.", "In particular, we reach a stable reconstruction quality already with 5 or more angles for both joint methods and 1% noise.", "Figure: Mean PSNR and SSIM values of the reconstructions of the dynamic Shepp-Logan phantom with 1% Gaussian noise for different numbers of projection angles.Figure: Mean PSNR and SSIM values of the reconstructions of the dynamic Shepp-Logan phantom with 3% Gaussian noise for different numbers of projection angles.Figure: Needed time in seconds for the reconstruction and feature extraction of the dynamic Shepp-Logan phantom with 1% Gaussian noise.The REF model clearly performs best with respect to the reconstruction quality.", "For almost every number of angles, the mean PSNR and SSIM values outperform the ones of the REF and gradTV method for both noise levels.", "In the case of 3% noise (see Figure REF ) we can see that gradTV performs slightly better than REF in most of the cases in terms of their SSIM values.", "Still, the mean PSNR values of gradTV are significantly lower than the ones in REF for all numbers of angles.", "A selection of reconstructions for the experiments in Figure REF and REF are provided as videos in the Supplementary files.", "Note that for REF , it is also possible to compute the reconstruction based on the decomposition $B\\cdot C$ instead of the joint reconstruction $X$ in the algorithm.", "Interestingly, our experiments showed that the reconstruction quality of $B\\cdot C$ is in almost all cases better than the one of the matrix $X$ itself and also mostly outperforms the gradTV approach.", "We believe, that this is due to the stronger regularising effect on the components $B$ and $C$ , which especially influences SSIM.", "The computation times for the reconstruction and feature extraction with 1% noise for all algorithms until the stopping criterion is fulfilled are shown in Figure REF .", "As expected, the computation time tends to increase with the number of projection angles and, considering all methods, ranges approximately from 1 to 5 minutes.", "For $ \\vert \\mathcal {I}_t \\vert \\le 8, $ the REF method is the fastest while it is outperformed by gradTV_PCA for $ \\vert \\mathcal {I}_t \\vert \\ge 9.$ gradTV_NMF and REF needs more time in all experiments compared to gradTV_PCA.", "The significant temporal difference between REF and REF is due to its higher computational complexity: Owing to the model formulation of REF with the discrepancy term $\\Vert R_t (BC)_{\\bullet , t} - Y_{\\bullet , t} \\Vert _2^2,$ the update rules in Theorem REF for both matrices $B$ and $C$ contain the discretised Radon transform $R_t.$ This is in contrast to the REF algorithm, where $R_t$ only appears in the update rule of $X.$ Based on the presented results for the dynamic Shepp-Logan phantom, we can conclude that the joint approaches REF and REF outperform both other methods with respect to the reconstruction quality and for most cases of the extracted features.", "Nevertheless, the models gradTV_PCA and gradTV_NMF give remarkably consistent and stable results of the extracted components throughout all numbers of angles.", "Furthermore, the nonnegativity constraint of the NMF improves significantly the interpretability and quality of the extracted spatial features." ], [ "Stationary Operator", "As we have seen, the computational complexity of the REF model with the non-stationary operator is clearly higher than for all other cases.", "Thus, let us now consider the possibility to speed up the reconstructions with a stationary operator, which leads us to the complexity reduced formulation presented in Corollary REF as the REF model.", "Here, we present similarly to the case above experiments with the dynamic Shepp-Logan phantom for $\\vert \\mathcal {I}_t \\vert \\in \\lbrace 2,\\dots ,30 \\rbrace $ and 1% Gaussian noise, as we primarily aim to illustrate the reduction of the computational cost.", "Furthermore, the same hyperparameters and stopping criteria are used as before.", "The reconstructed features for the cases $\\vert \\mathcal {I}_t \\vert = 6 $ and $\\vert \\mathcal {I}_t \\vert = 30$ are shown in Figure REF .", "In particular, comparing the results in Figure REF to the corresponding results of REF in Figure REF , one can immediately see a significant difference between the extracted spatial features.", "This is clearly due to the fact that the same projection angles are used at every time step and the individual projection directions are clearly visible for the stationary model REF .", "Consequently, the details in the Shepp-Logan phantom are not well recovered, such that the extracted constant feature is significantly inferior to the one of REF .", "As one would expect, more projection angles per time step are needed to reconstruct finer details.", "This effect can be clearly seen for 30 angles in Figure REF .", "However, all temporal basis functions with REF for $\\vert \\mathcal {I}_t \\vert = 6 $ are remarkably well reconstructed despite the low number of projection angles.", "This is also true for the other considered values of $\\vert \\mathcal {I}_t \\vert .$ Moreover, we observe that REF is able to extract the correct three main features for every $\\vert \\mathcal {I}_t \\vert \\in \\lbrace 2,\\dots ,30 \\rbrace .$ Even for $\\vert \\mathcal {I}_t \\vert = 2,$ the quality of the dynamic temporal features are similar to the ones in Figure REF .", "This behaviour is different from the dynamic case discussed above.", "The reason for this is probably based on the different projection directions at every time step in the dynamic case, which results in directional dependencies of the occurring reconstruction artefacts in contrast to the stationary case.", "This can make it difficult for the NMF to distinguish the main features in the non-stationary case and thus leads to a more stable feature extraction in the here presented stationary case.", "Figure: Results for the dynamic Shepp-Logan phantom with a stationary operator and 1% Gaussian noise.", "Shown are the leading extracted features with the model for |ℐ t |=6\\vert \\mathcal {I}_t \\vert = 6 angles per time step (left) and |ℐ t |=30\\vert \\mathcal {I}_t \\vert = 30 (right).The quantitative measures are shown in Figure REF for all experiments.", "Comparing the computation time of REF with the one of REF , we obtain a clear speed-up by a factor of 10–20 with the stationary model.", "However, as expected, comparing Figure REF and REF with the quality measures of REF in Figure REF , one can observe that significantly more projection angles per time step are needed in the stationary case to provide a sufficient reconstruction quality.", "In conclusion, we can say that the REF model is especially recommended if one is primarily interested in the dynamics of the system under consideration, as we could extract the temporal basis functions stably for all considered angles with $\\vert \\mathcal {I}_t| \\ge 2$ .", "Figure: Needed time in seconds, mean PSNR and mean SSIM values of the reconstructions of the dynamic Shepp-Logan phantom with 1% Gaussian noise for the stationary case and different numbers of projection angles." ], [ "Vessel Phantom", "The second test case is based on a CT scan of a human lungThe phantom is based on the CT scans in the ELCAP Public Lung Image database: http://www.via.cornell.edu/lungdb.html, see Figure REF .", "Here, the decomposition is given by the constant background and a segmented vessel that exhibits a sudden increase in attenuation followed by an exponential decay.", "This could for instance represent the injection of a tracer to the blood stream.", "Figure: Illustration of the vessel phantom dataset consisting of T=100 T=100 phantoms of dimension 264×264,264\\times 264, where the intensity of the blue highlighted area changes according to blue curve on the left.In contrast to the previous dataset, we perform only selected experiments for specific choices of noise levels and numbers of projection angles.", "More precisely, we present results for 1% Gaussian noise together with $\\vert \\mathcal {I}_t \\vert \\in \\lbrace 7, 12\\rbrace $ and 3% Gaussian noise with $\\vert \\mathcal {I}_t \\vert =12.$ In all cases, we choose $K=4$ NMF features.", "Furthermore, the stopping criterion from the experiments with the dynamic Shepp-Logan phantom is changed for this dataset in such a way, that the maximum number of iterations is raised to 1400 to ensure sufficient convergence.", "The regularisation parameters of all methods are chosen empirically and are displayed in Table REF in Appendix .", "Figure: Results for the vessel phantom with |ℐ t |=12\\vert \\mathcal {I}_t \\vert =12 angles per time step and 1% Gaussian noise.", "Shown are the leading extracted features for the model (left) and for (right).Figure: Results for the vessel phantom with |ℐ t |=12\\vert \\mathcal {I}_t \\vert =12 angles per time step and 1% Gaussian noise.", "Shown are the leading extracted features for the gradTV_PCA model (left) and for gradTV_NMF (right).Figure REF and REF show the feature extraction results for the noise level of 1% and $\\vert \\mathcal {I}_t \\vert =12,$ where all approaches are able to extract both the main constant and dynamic component of the underlying ground truth.", "The order of the features here is based on a manual sorting.", "Similar to the results of the Shepp-Logan phantom in Section REF , the joint methods REF and REF have difficulties to recover the lower intensities in the temporal features, whereas gradTV_PCA produce slight artefacts in the dynamic spatial feature due to the missing nonnegativity constraint.", "In addition, gradTV_NMF is able to recover more details in the vessel compared to the joint approaches.", "This is due to the relatively high choice of the total variation regularisation parameter $\\tau $ in REF and REF to ensure a sufficient denoising effect on the matrix $B.$ The low peak in the second temporal feature of gradTV_NMF is likely caused by the choice of the $\\ell _2$ regularisation parameter $\\tilde{\\mu }_C.$ Figure: Results for the vessel phantom with |ℐ t |=12\\vert \\mathcal {I}_t \\vert =12 angles per time step and 3% Gaussian noise.", "Shown are the leading extracted features for the model (left) and for gradTV_PCA (right).Further experiments show that the quality of the extracted components of REF decreases steadily for lower angles until the main features cannot be identified anymore for $\\vert \\mathcal {I}_t \\vert \\le 8.$ REF produces inferior results and cannot extract reasonable components anymore for $\\vert \\mathcal {I}_t \\vert \\le 10.$ In comparison, both separated approaches gradTV_PCA and gradTV_NMF are still able to extract decent features for $\\vert \\mathcal {I}_t \\vert = 7.$ For $\\vert \\mathcal {I}_t \\vert \\le 6,$ the performance of both methods decreases significantly.", "Similar results for gradTV_PCA could be obtained for 3% noise and $\\vert \\mathcal {I}_t \\vert = 12,$ which are shown in Figure REF .", "Its constant feature is inferior to the one of REF in Figure REF due to the additional nonnegativity constraint of the NMF model.", "However, the details of the vessel in the dynamic spatial feature of REF are lost due to the choice of large regularisation parameter $\\tau $ and the temporal features are affected by several disturbances.", "Further tests with the noise level of 3% showed that both joint methods are not able to recover the underlying features for $\\vert \\mathcal {I}_t \\vert \\le 10,$ while the separated approaches gives still acceptable results for $\\vert \\mathcal {I}_t \\vert = 6.$ Table: Mean PSNR and SSIM values of the reconstruction results of the vessel phantom for different noise levels and numbers of projection angles.", "Values in brackets indicate that the dynamic part of the dataset in the corresponding experiment could not be reconstructed sufficiently well.The reconstruction quality of the experiments are shown in Table REF .", "Similar to the Shepp-Logan phantom, the joint approach REF produces the best results compared to all other methods in terms of the mean PSNR and SSIM values.", "Further experiments confirm this observation for $4 \\le \\vert \\mathcal {I}_t \\vert \\le 11.$ However, these observations have to be treated with caution.", "REF is not able to recover the dynamics for $\\vert \\mathcal {I}_t \\vert \\le 10$ and 1% noise.", "In the case of REF , the dynamics can be reconstructed to some degree within the angle range $9 \\le \\vert \\mathcal {I}_t \\vert \\le 11,$ but are not recognizable anymore for $\\vert \\mathcal {I}_t \\vert \\le 8.$ In the case of 3% Gaussian noise, gradTV is still able to give acceptable reconstruction results for $\\vert \\mathcal {I}_t \\vert = 10.$ For less angles, the reconstructed dynamics of gradTV get constantly worse until they are not apparent anymore for $\\vert \\mathcal {I}_t \\vert \\le 6 .$ The computation times of the experiments in Table REF range approximately from 7 to 15 minutes.", "The corresponding reconstructions can be found as video files in the Supplementary information." ], [ "Conclusion", "In this work we consider dynamic inverse problems with the assumption that the target of interest has a low-rank structure and can be efficiently represented by spatial and temporal basis functions.", "This assumption leads naturally to a reconstruction and low-rank decomposition framework.", "In particular, we concentrate here on the NMF as decomposition because it exhibits three main advantages: It naturally incorporates the physical assumption of nonnegativity Basis functions are not restricted to being strictly orthogonal and therefore correspond more naturally to actual components It allows the flexibility to incorporate separate regularisation on each of the factorisation matrices In particular, the last point is of importance, as it allows to consider different regularisers for spatial and temporal basis functions, and as such can be tailored to different applications.", "We then proposed two approaches to obtain a joint reconstruction and low-rank decomposition based on the NMF, termed REF and REF .", "Both methods performed better than a baseline method, that computes a reconstruction with low-rank constraint followed by a subsequent decomposition.", "In particular, the second REF model has shown to have a stronger regularising effect on the reconstructed features as well as the reconstruction, which can be simply obtained as $X=BC.$ We believe this is due to the fact, that only the decomposition is recovered during the reconstruction without the need to build the reconstruction $X$ explicitly and hence the resulting features at the end exhibit a higher regularity.", "More importantly, if one considers a stationary operator in the complexity reduced REF model we can obtain a considerable computational speed-up.", "Even though, due to constant projection angles the spatial basis functions are not as well recovered as in the non-stationary case, but the temporal features can be nicely extracted even for as low as 2 angles.", "This might be especially of interest in applications, where one is primarily interested in the underlying dynamics of the imaged target.", "The primary limitation of the presented approach is the assumption on the decomposition of the target into spatial and temporal basis functions, as this does not allow for spatial movements in the target.", "However, it opens up the possibility of combination with other methods, that do in fact allow for movements but assume a brightness consistency in the target, such as the optical flow constraint in CT [5].", "Furthermore, the presented low-rank decomposition may be combined with a morphological motion model [20] to allow for a flexible and general model for dynamic inverse problems." ], [ "Acknowledgments", "This project was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the framework of RTG “$\\pi ^3$ : Parameter Identification – Analysis, Algorithms, Applications” – Projektnummer 281474342/ GRK2224/1.", "This work was partially supported by the Academy of Finland Project 312123 (Finnish Centre of Excellence in Inverse Modelling and Imaging, 2018–2025), EPSRC grant EDCLIRS (EP/N022750/1) as well as CMIC-EPSRC platform grant (EP/M020533/1)." ], [ "Optimisation Techniques for NMF Problems", "The majority of optimisation techniques for NMF problems are based on alternating minimisation schemes.", "This is due to the fact that the corresponding cost function in (REF ) is usually convex in $B$ for fixed $C$ and $C$ for fixed $B$ and non-convex in $(B, C)$ together, which yields algorithms of the form B[d+1] B0 F(B, C[d]), C[d+1] C0 F(B[d+1], C).", "Typical minimisation approaches are based on alternating least squares methods, multiplicative algorithms as well as projected gradient descent and quasi-newton methods [10].", "In this work, we focus on the derivation of multiplicative update rules based on the so-called Majorise-Minimisation (MM) principle [23].", "This approach allows the derivation of multiplicative update rules for non-standard NMF cost functions and gives therefore the flexibility to adjust the discrepancy and penalty terms according to the NMF model motivated by the corresponding application [16].", "What is more, the update rules consist only of multiplications and summations of matrices, which allow very simple implementations of the algorithms and ensure automatically the nonnegativity of the iterates $ B $ and $ C $ without the need of any inversion process, provided they are initialised nonnegative." ], [ "Multiplicative Algorithms", "The works of Lee and Seung [26], [27] brought much attention to NMF methods in general and, in particular, the multiplicative algorithms, which they derived based on the MM principle for the standard case with the Frobenius norm and the Kullback-Leibler divergence as discrepancy terms.", "The main idea of the MM approach is to replace the original cost function $ \\mathcal {F} $ by a majorizing so-called surrogate function $ \\mathcal {Q}_\\mathcal {F}, $ which is easier to minimise and leads to the desired multiplicative algorithms due to its tailored construction.", "Definition A.1 (Surrogate Function) Let $ \\Omega \\subset \\mathbb {R}^n $ be an open subset and $ \\mathcal {F}:\\Omega \\rightarrow \\mathbb {R} $ a function.", "Then $ \\mathcal {Q}_\\mathcal {F}:\\Omega \\times \\Omega \\rightarrow \\mathbb {R} $ is called a surrogate function or surrogate of $ \\mathcal {F}, $ if it fulfills the following properties: $ \\mathcal {Q}_\\mathcal {F}(x, \\tilde{x}) \\ge \\mathcal {F}(x)$ for all $ x, \\tilde{x} \\in \\Omega $ $ \\mathcal {Q}_\\mathcal {F}(x, x) = \\mathcal {F}(x) $ for all $ x\\in \\Omega $ The minimisation step of the MM approach is then defined by the update rule $x^{[d+1]} \\arg \\min _{x\\in \\Omega } \\mathcal {Q}_\\mathcal {F}(x, x^{[d]}),$ assuming that the $\\arg \\min _{x\\in \\Omega } \\mathcal {Q}_\\mathcal {F}(x, \\tilde{x})$ exists for all $\\tilde{x} \\in \\Omega .", "$ Due to the defining properties of a surrogate function in Definition REF , the monotonic decrease of the cost function $ \\mathcal {F} $ is easily shown: $ \\mathcal {F}(x^{[d+1]}) \\le \\mathcal {Q}_\\mathcal {F}(x^{[d+1]}, x^{[d]}) \\le \\mathcal {Q}_\\mathcal {F}(x^{[d]}, x^{[d]}) = \\mathcal {F}(x^{[d]}).$ This principle is also illustrated in Figure REF .", "Figure: Illustration of two iteration steps of the MM principle for a cost function ℱ\\mathcal {F} with bounded curvature and a surrogate function 𝒬 ℱ ,\\mathcal {Q}_\\mathcal {F}, which is strictly convex in the first argument.Typical construction techniques lead to surrogate functions, which are strictly convex in the first component to ensure the unique existence of the corresponding minimiser.", "Furthermore, the surrogates must be constructed in such a way, that the minimisation in Equation (REF ) yields multiplicative updates to ensure the nonnegativity of the matrix iterates.", "Finally, another useful property is the separability of $\\mathcal {Q}_\\mathcal {F}$ with respect to the first variable.", "This ensures, that $\\mathcal {Q}_\\mathcal {F}(x,\\tilde{x})$ can be written as a sum, where each component just depends on one entry of $ x $ and allows the derivation of the multiplicative algorithm via the zero gradient condition $\\nabla _x \\mathcal {Q}_\\mathcal {F} = 0.$ One typical construction method is the so-called Quadratic Upper Bound Principle (QUBP) [1], [23], which forms one of the main approaches to construct suitable surrogate functions for NMF problems.", "Nice overviews of other construction principles, which will not be used in this work, can be found in [23], [24].", "The QUBP is described in the following Lemma.", "Lemma A.1 Let $ \\Omega \\subset \\mathbb {R}^n $ be an open and convex subset, $ \\mathcal {F}:\\Omega \\rightarrow \\mathbb {R} $ twice continuously differentiable with bounded curvature, i.e.", "there exists a matrix $ \\Lambda \\in \\mathbb {R}^{n\\times n}, $ such that $ \\Lambda - \\nabla ^2\\mathcal {F}(x) $ is positive semi-definite for all $ x\\in \\Omega .", "$ We then have F(x) F(x) + F(x)(x-x) + 12 (x-x)(x-x)    x, x QF(x, x), where $ \\mathcal {Q}_\\mathcal {F} $ is a surrogate function of $ \\mathcal {F}.", "$ This is a classical result based on the second-order Taylor polynomial and will not be proven here.", "If the matrix $ \\Lambda $ is additionally symmetric and positive definite, it can be shown [16] that the update rule for $ x $ according to (REF ) via the zero gradient condition $ \\nabla _{x}\\mathcal {Q}_\\mathcal {F}(x, \\tilde{x})=0 $ gives the unique minimiser $ x_{\\tilde{x}}^* = \\tilde{x} - \\Lambda ^{-1} \\nabla \\mathcal {F}(\\tilde{x}).$ In this work, we will only apply the QUBP for quadratic cost functions $F,$ whose Hessian is automatically a constant matrix.", "For these functions, typical choices of $ \\Lambda $ are diagonal matrices of the form $ \\Lambda (\\tilde{x})_{i i} \\dfrac{(\\nabla ^2 f (\\tilde{x})\\ \\tilde{x})_i + \\kappa _i}{\\tilde{x}_i},$ which are dependent on the second argument of the corresponding surrogate $\\mathcal {Q}_\\mathcal {F}(x, \\tilde{x}).$ The parameters $ \\kappa _i\\ge 0, $ are constants and have to be chosen depending on the considered penalty terms of the NMF cost function.", "The diagonal structure of $ \\Lambda (\\tilde{x}) $ ensures its simple invertibility, the separability of the corresponding surrogate and the desired multiplicative algorithms based on (REF ).", "Hence, the update rule in (REF ) can be viewed as a gradient descent approach with a suitable stepsize defined by the diagnoal matrix $ \\Lambda .", "$" ], [ "Derivation of the Algorithms", "In this section, we derive the multiplicative update rules for the NMF minimisation problems in REF and REF ." ], [ "Algorithm for X", "We start first of all with the NMF model REF and the minimisation with respect to $X.$ The cost function of the NMF problem in REF for the minimisation with respect to $ X $ reduces to $ \\mathcal {F}(X) \\underbrace{\\sum _{t=1}^{T} \\frac{1}{2} \\Vert A_t X_{\\bullet , t} - Y_{\\bullet , t} \\Vert _2^2 + \\frac{\\mu _{X}}{2} \\Vert X\\Vert _F^2 + \\lambda _{X} \\Vert X \\Vert _1}_{\\mathcal {F}_1(X)} + \\underbrace{\\frac{\\alpha }{2} \\Vert X - BC \\Vert _F^2}_{\\mathcal {F}_2(X)}$ by neglecting the constant terms.", "To apply the QUBP and to avoid fourth-order tensors during the computation of the Hessians, we use the separability of $ \\mathcal {F}_1 $ with respect to the columns of $ X, $ i.e.", "it can be written as sum, where each term depends only on the respective column $X_{\\bullet , t}.$ Hence, we write $\\mathcal {F}_1(X) = \\sum _{t=1}^{T} \\left[ \\frac{1}{2} \\Vert A_t X_{\\bullet , t} - Y_{\\bullet , t} \\Vert _2^2 + \\frac{\\mu _{X}}{2} \\Vert X_{\\bullet , t}\\Vert _2^2 + \\lambda _{X} \\Vert X_{\\bullet , t} \\Vert _1 \\right] \\sum _{t=1}^{T} f_t(X_{\\bullet , t}).$ We can assume that $ X $ contains only strictly positive entries due to the strict positive initialisations of the multiplicative algorithms.", "Hence, the functions $ f_t $ are twice continuously differentiable despite the occuring $ \\ell ^1 $ regularisation term.", "The computations of the gradient and the Hessian of $ f_t $ are straightforward and we obtain ft(X, t) = AtAt X, t - AtY, t + XX, t + X 1N1, 2 ft(X, t)= AtAt + X INN, where $ I_{N\\times N} $ is the $ N\\times N $ identity matrix.", "Choosing $ \\kappa _n=\\lambda _{X} $ for all $ n $ in (REF ), we define the surrogate $ \\mathcal {Q}_{f_t} $ according to Lemma REF .", "It is then easy to see, that $\\mathcal {Q}_{\\mathcal {F}_1}(X, \\tilde{X}) \\sum _{t=1}^T \\mathcal {Q}_{f_t} (X_{\\bullet , t}, \\tilde{X}_{\\bullet , t})$ defines a separable and convex surrogate function for $ \\mathcal {F}_1.", "$ For $ \\mathcal {F}_2, $ we set simply $ \\mathcal {Q}_{\\mathcal {F}_2}(X,\\tilde{X}) {\\alpha }{2} \\Vert X-BC\\Vert _F^2, $ such that we end up with $\\mathcal {Q}_\\mathcal {F}(X, A) \\mathcal {Q}_{\\mathcal {F}_1}(X,A) + \\mathcal {Q}_{\\mathcal {F}_2}(X,A)$ as a suitable surrogate for $ F. $ Based on the update rule in (REF ), we consider the zero gradient condition $ \\nabla _{X}\\mathcal {Q}_\\mathcal {F}(X, \\tilde{X})=0 $ and compute QFXnt(X,X) =ftXnt (X,t) + ( (X,t) (X, t - X, t) )n + 2 Xnt X-BC F2 =( AtAt X, t )n - ( AtY, t )n + X Xnt + X +$\\displaystyle \\dfrac{\\left( (A_t^\\intercal A_t + \\mu _{X} I_{N\\times N})\\tilde{X}_{\\bullet ,t} \\right)_n + \\lambda _{X}}{\\tilde{X}_{nt}}(X_{nt} - \\tilde{X}_{nt} ) + \\alpha (X_{nt} - (BC)_{nt})$ =$\\displaystyle - \\left( A_t^\\intercal Y_{\\bullet , t} \\right)_n + X_{nt} \\dfrac{\\left( A_t^\\intercal A_t \\tilde{X}_{\\bullet , t} \\right)_n + \\mu _{X} \\tilde{X}_{nt} + \\lambda _{X}}{\\tilde{X}_{nt}} + \\alpha (X_{nt} - (BC)_{nt})$ =0.", "Rearranging the equation leads to $X_{nt} = \\dfrac{\\left( A_t^\\intercal Y_{\\bullet , t} \\right)_n + \\alpha (BC)_{nt}}{ \\dfrac{\\left( A_t^\\intercal A_t \\tilde{X}_{\\bullet , t} \\right)_n + \\mu _{X}\\tilde{X}_{nt} + \\lambda _{X}}{\\tilde{X}_{nt}} + \\alpha }.$ We therefore have $X_{\\bullet , t} = \\tilde{X}_{\\bullet , t} \\circ \\dfrac{A_t^\\intercal Y_{\\bullet , t} + \\alpha BC_{\\bullet , t}}{A_t^\\intercal A_t \\tilde{X}_{\\bullet , t} + (\\mu _{X} + \\alpha )\\tilde{X}_{\\bullet , t} + \\lambda _{X} {\\mathit {1}}_{N\\times 1}},$ which yields the multiplicative update rule $X_{\\bullet , t} \\leftarrow X_{\\bullet , t} \\circ \\dfrac{A_t^\\intercal Y_{\\bullet , t} + \\alpha BC_{\\bullet , t}}{A_t^\\intercal A_t X_{\\bullet , t} + (\\mu _{X} + \\alpha )X_{\\bullet , t} + \\lambda _{X} {\\mathit {1}}_{N\\times 1}}$ based on (REF ).", "Note that the correct choice of the matrix $ \\Lambda $ together with the $ \\kappa _i $ is crucial to ensure the multiplicative structure of the algorithm." ], [ "Algorithm for B", "The minimisation with respect to $ B $ reduces the cost function in REF to $ \\mathcal {F}(B) \\underbrace{\\frac{\\alpha }{2} \\Vert BC - X \\Vert _F^2 + \\frac{\\mu _{B}}{2} \\Vert B\\Vert _F^2 + \\lambda _{B} \\Vert B \\Vert _1}_{\\mathcal {F}_1(B)} + \\underbrace{\\dfrac{\\tau }{2} \\operatorname{TV}(B)}_{\\mathcal {F}_2(B)}$ and involves the TV regularisation on $ B $ of the NMF model.", "Analogously to the previous section, we use the separability of $ \\mathcal {F}_1 $ and write $\\displaystyle \\mathcal {F}_1(B) = \\sum _{n=1}^{N} \\Big [ \\dfrac{\\alpha }{2} \\Vert X_{n,\\bullet } - B_{n,\\bullet } C \\Vert _F^2 + \\dfrac{\\mu _{B}}{2} \\Vert B_{n,\\bullet } \\Vert _2^2 + \\lambda _{B} \\Vert B_{n,\\bullet }\\Vert _1 \\Big ] \\sum _{n=1}^{N} f_n(B_{n,\\bullet }).$ By computing the gradients fn(Bn,) = (Bn, C - Xn,) C+ B Bn, + B 11K 2 fn(Bn,) = CC+ B IKK and choosing $ \\kappa _k = \\lambda _{B} $ in (REF ), we define analogously the surrogates $ \\mathcal {Q}_{f_n}, $ which leads to the convex surrogate $\\mathcal {Q}_{\\mathcal {F}_1} (B, \\tilde{B}) \\sum _{n=1}^{N} \\mathcal {Q}_{f_n}(B_{n,\\bullet }, \\tilde{B}_{n,\\bullet })$ for $ \\mathcal {F}_1.", "$ The derivation of a suitable surrogate for the TV regularisation term $\\mathcal {F}_2$ is based on an approach different from the QUBP and shall not be discussed in detail.", "We just state the result and refer the reader for details to [33], [12], [16].", "A convex and separable surrogate function for $ \\mathcal {F}_2 $ is given by $ \\mathcal {Q}_{\\mathcal {F}_2}(B, \\tilde{B}) = \\dfrac{\\tau }{2} \\sum _{k=1}^K \\sum _{n=1}^N \\left[ P(\\tilde{B})_{nk} (B_{nk} - Z(\\tilde{B})_{nk})^2 \\right] + \\mathcal {G}(\\tilde{B}),$ with the matrices $ P(\\tilde{B}), Z(\\tilde{B})\\in \\mathbb {R}_{\\ge 0}^{N\\times K} $ defined in (REF ) and (REF ) and a function $ \\mathcal {G} $ depending only on the matrix $ \\tilde{B}.$ Hence, we finally end up with $ \\mathcal {Q}_\\mathcal {F}(B, \\tilde{B}) \\mathcal {Q}_{\\mathcal {F}_1}(B, \\tilde{B}) + \\mathcal {Q}_{\\mathcal {F}_2}(B, \\tilde{B}) $ as a suitable surrogate for $\\mathcal {F}$ .", "Similar to the computations in the previous paragraph, the zero gradient condition yields then $\\scalebox {0.83}{\\text{$\\displaystyle \\dfrac{\\partial \\mathcal {Q}_\\mathcal {F}}{\\partial B_{nk}}(B, \\tilde{B}) = - \\alpha (XC^\\intercal )_{nk} + B_{nk} \\dfrac{\\alpha (\\tilde{B}CC^\\intercal )_{nk} \\!+\\!", "\\mu _{{\\mathit {B}}}\\tilde{B}_{nk} \\!+\\!", "\\lambda _{{\\mathit {B}}}}{\\tilde{B}_{nk}} + \\tau P(\\tilde{B})_{nk} (B_{nk} \\!-\\!", "Z(\\tilde{B})_{nk}) = 0$}}$ and therefore $B_{nk} = \\tilde{B}_{nk} \\cdot \\dfrac{\\alpha (XC^\\intercal )_{nk} + \\tau P(\\tilde{B})_{nk}Z(\\tilde{B})_{nk} }{\\alpha (\\tilde{B}CC^\\intercal )_{nk} + \\mu _{B} \\tilde{B}_{nk} + \\lambda _{B} + \\tau P(\\tilde{B})_{nk}\\tilde{B}_{nk} }.$ Hence, we have the update rule $B \\leftarrow B \\circ \\dfrac{\\alpha XC^\\intercal + \\tau P(B) \\circ Z(B) }{\\alpha BCC^\\intercal + \\mu _{B} B + \\lambda _{B} {\\mathit {1}}_{N\\times K} + \\tau P(B) \\circ B }.$" ], [ "Algorithm for C", "The optimisation with respect to the matrix $ C $ can be tackled analogously with the QUBP and will not be described in detail.", "In this case, the cost function can be reduced to well-known regularised NMF problems [11], which leads to the update rule C C BXBBC + C C + C 1KT." ], [ "Model ", "In this section, we discuss the computation of the optimisation algorithms for the NMF model REF ." ], [ "Algorithm for B", "In this case, the cost function reduces to F(B) t=1T 12 At (BC),t - Y, t 22 + B2 B F2 + B B1F1(B) + 2 TV(B)F2(B).", "Analogously to the previous cases, we analyze the functions $ \\mathcal {F}_1 $ and $ \\mathcal {F}_2 $ separately.", "The difference is here, that $ \\mathcal {F}_1 $ is not separable with respect to the rows of $ B $ due to the discrepancy term and therefore, it is necessary to compute the gradient and the Hessian of the whole function $ \\mathcal {F}_1.", "$ Hence, the gradient $ \\nabla \\mathcal {F}_1(B)$ is an $ N\\times K $ matrix and the Hessian $ \\nabla ^2 \\mathcal {F}_1(B) $ a fourth-order tensor, which are given by their entries F1(B)n k = $\\displaystyle \\sum _{t=1}^T C_{k t} \\left( A_t^\\intercal A_t (B C)_{\\bullet , t} \\right)_{n} - \\sum _{t=1}^T C_{k t} \\left(A_t^\\intercal Y_{\\bullet , t}\\right)_{n} + \\mu _{B}B_{n k} + \\lambda _{B}$, 2 F1(B)(n, k), (n, k) = t=1T Ck t Ck t (AtAt)n n + B (n, k), (n, k), where $ \\delta _{(n, k), (\\tilde{n}, \\tilde{k})}=1 $ if and only if $ (n, k) = (\\tilde{n}, \\tilde{k}).", "$ The natural expansion of the quadratic upper bound principle given in Lemma REF is the ansatz function QF1(B, B) F1(B) + B-B, F1(B)F + 12 (n, k) (n, k) (B-B)n k (B)(n, k), (n, k) (B-B)n k with the fourth order tensor $\\Lambda (\\tilde{B})_{(n, k), (\\tilde{n}, \\tilde{k})} {\\left\\lbrace \\begin{array}{ll}\\dfrac{\\sum _{(i, j)} \\nabla ^2 \\mathcal {F}_1(\\tilde{B})_{(n, k), (i, j)} \\tilde{B}_{ij} + \\lambda _{B}}{\\tilde{B}_{n k}} &\\text{for} \\quad (n, k) = (\\tilde{n}, \\tilde{k}), \\\\0 &\\text{for} \\quad (n, k) \\ne (\\tilde{n}, \\tilde{k}),\\end{array}\\right.", "}$ where $ \\langle \\cdot , \\cdot \\rangle _F $ denotes the Frobenius inner product.", "Taking the same surrogate $ \\mathcal {Q}_{\\mathcal {F}_2} $ for the TV penalty term as in (REF ), we end up with the surrogate function $\\mathcal {Q}_\\mathcal {F}(B, \\tilde{B}) \\mathcal {Q}_{\\mathcal {F}_1}(B, \\tilde{B}) + \\mathcal {Q}_{\\mathcal {F}_2}(B, \\tilde{B})$ for $ \\mathcal {F}.", "$ Its partial derivative with respect to $ B_{nk} $ is given by QFBnk(B) = $\\displaystyle - \\sum _{t=1}^T C_{k t} \\left(A_t^\\intercal Y_{\\bullet , t}\\right)_{n} + B_{nk} \\frac{\\sum _{t=1}^T C_{k t} \\left( A_t^\\intercal A_t (\\tilde{B} C)_{\\bullet , t} \\right)_{n} + \\mu _B \\tilde{B}_{nk} + \\lambda _B}{\\tilde{B}_{nk}}$ + P(B)nk (Bnk - Z(B)nk).", "The zero-gradient condition gives then the equation Bnk = Bnk ( t=1T Ck t (AtY, t)n + P(B)nk Z(B)nkt=1T Ck t ( AtAt (B C), t )n + B Bnk + B + Bnk P(B)nk ), which can be extended to the whole matrix $ B.", "$ Therefore, based on (REF ), we have the update rule $B \\leftarrow B \\circ \\Bigg ( \\dfrac{\\sum _{t=1}^T A_t^\\intercal Y_{\\bullet , t} (C^\\intercal )_{t,\\bullet } + \\tau P(B) \\circ Z(B)}{\\sum _{t=1}^T A_t^\\intercal A_t (B C)_{\\bullet , t} \\cdot (C^\\intercal )_{t,\\bullet } + \\mu _{B} B + \\lambda _{B}{\\mathit {1}}_{N\\times K} + \\tau B \\circ P(B)} \\Bigg ).$" ], [ "Algorithm for C", "In this case, the cost function is separable with respect to the columns of $C,$ such that F(C) t=1T 12 At BC,t - Y, t 22 + C2 C,t 22 + C C,t 1 t=1T ft(C,t).", "Hence, we can split the minimisation into the columns of $ C $ to use the standard QUBP without considering higher order tensors.", "We compute ft (C,t) = BAtAt (B C),t - BAtY,t + C C,t + C 1K1, 2 ft (C,t) = BAtAt B + C IKK.", "By choosing $ \\kappa _k=\\lambda _{C} $ for all $ k $ in (REF ), we define $\\mathcal {Q}_{f_t} (C_{\\bullet ,t}, \\tilde{C}_{\\bullet ,t})$ as a surrogate function for $f_t$ according to Lemma REF .", "The update rule in (REF ) gives then $C_{\\bullet ,t} = \\tilde{C}_{\\bullet ,t} - \\Lambda ^{-1}(\\tilde{C}_{\\bullet ,t}) \\nabla f_t (\\tilde{C}_{\\bullet ,t}),$ which leads to $C_{\\bullet ,t} \\leftarrow C_{\\bullet ,t} \\circ \\dfrac{B^\\intercal A_t^\\intercal Y_{\\bullet ,t}}{B^\\intercal A_t^\\intercal A_t (BC)_{\\bullet ,t} + \\mu _{C} C_{\\bullet ,t} + \\lambda _{C} {\\mathit {1}}_{K\\times 1} }.$" ] ]

[ [ "Comparison of stellar populations in simulated and real post-starburst\n galaxies in MaNGA" ], [ "Abstract Recent integral field spectroscopic (IFS) surveys have revealed radial gradients in the optical spectral indices of post-starburst galaxies, which can be used to constrain their formation histories.", "We study the spectral indices of post-processed mock IFS datacubes of binary merger simulations, carefully matched to the properties of the MaNGA IFS survey, with a variety of black hole feedback models, progenitor galaxies, orbits and mass ratios.", "Based on our simulation sample, we find that only major mergers on prograde-prograde or retrograde-prograde orbits in combination with a mechanical black hole feedback model can form galaxies with weak enough ongoing star formation, and therefore absent H$\\alpha$ emission, to be selected by traditional PSB selection methods.", "We find strong fluctuations in nebular emission line strengths, even within the PSB phase, suggesting that H$\\alpha$ selected PSBs are only a subsample of the underlying population.", "The global PSB population can be more robustly identified using stellar continuum-based approaches.", "The difficulty in reproducing the very young PSBs in simulations potentially indicates that new sub-resolution star formation recipes are required to properly model the process of star formation quenching.", "In our simulations, we find that the starburst peaks at the same time at all radii, but is stronger and more prolonged in the inner regions.", "This results in a strong time evolution in the radial gradients of the spectral indices which can be used to estimate the age of the starburst without reliance on detailed star formation histories from spectral synthesis models." ], [ "Introduction", "The galaxy population in the local Universe shows a clear bimodality in the colour-magnitude diagram , .", "The majority of massive galaxies fall into two distinct groups: the “blue cloud” (star-forming, spiral galaxies) and the “red sequence” (quiescent, elliptical galaxies).", "The total mass of the stars living in quiescent galaxies, as well as the total number of quiescent galaxies, has grown steadily since a redshift of at least $z\\sim 4$ , , which implies a steady conversion of star-forming to quiescent galaxies through the quenching of their star formation.", "There may be multiple evolutionary processes by which galaxies migrate from the blue cloud to the red sequence.", "Analysis of the star formation histories (SFHs) of quiescent galaxies has revealed at least two different quenching mechanisms that operate on different timescales, “fast” and “slow” quenching.", "At high redshift, galaxies of all stellar masses consistently show a fast rise and decline of their star formation rate (SFR).", "At low redshift, low-mass galaxies grow slowly and quench quickly, whereas high-mass galaxies grow quickly and quench slowly .", "Post-starburst (PSB) galaxies, also sometimes referred to as E+A or K+A galaxies, may occupy an important transitional state between the star-forming and quiescent populations, and could be an important candidate for the “fast” quenching route.", "These galaxies have experienced a rapid decline in their star formation following a previous period of very rapid star formation.", "Catching these galaxies as they transition to quiescence may provide important constraints on the causes of the rapid quenching which is not possible from archaeological studies of already-quiescent galaxies.", "They are identified by their excess population of intermediate-age (A- or F-type stars), but deficit or absence of hotter, younger stars (O- and B- types) leading to strong Balmer absorption lines, a strong Balmer break, and weak or absent nebular emission lines from star formation.", "PSBs account for only about 1 per cent of local galaxies , , however they may account for a significantly larger fraction of the galaxy population at higher redshifts , , , , , , .", "Despite being rare, their number density is sufficient to account for a significant fraction of the growth of the red-sequence at $z<2$ , although the precise fraction is still debated , , , .", "At high redshift, the morphologies of PSB galaxies are strongly suggestive of an extremely intense and rapid quenching process, such as a major gas-rich merger, multiple mergers or protogalactic collapse.", "[3] find that massive PSBs at redshift ($z>1$ ) are significantly smaller than comparable quiescent galaxies at the same stellar mass and epoch which is strongly suggestive of a recent dissipative collapse event associated with the shut off in star formation .", "This appears to depend on stellar mass and epoch, with lower mass and lower redshift PSBs more similar in structure to equivalent quiescent galaxies , .", "At lower redshift, galaxy mergers have long been proposed as a possible formation mechanism for PSBs shortly after the discovery of E+A galaxies .", "A substantial fraction of PSBs are found with faint tidal features or companion galaxies , , .", "In addition, PSBs at low-redshift can have surprisingly high fractions of mass formed in the starburst of typically 20%, but occasionally as much as 70% , , which again supports major mergers as the only mechanism known in the local Universe to be able to feed sufficient gas into the central regions of a galaxy in a short enough time .", "Such a scenario is consistent with the results from the cosmological hydrodynamic simulation EAGLE, where find that local ($z\\sim 0$ ) simulated post-starburst galaxies are predominantly caused by major mergers.", "used the same EAGLE simulations, post-processed to produce mock optical spectra, to find that in addition to classical major mergers, a range of different processes can create weak PSB features detectable in modern spectroscopic surveys.", "These include harassment by multiple smaller galaxies as well as rejuvenation from gas brought in by infalling small satellites.", "During a galaxy merger, the discs of the galaxies are typically destroyed and the gas components are funnelled into the galaxy centre, leading to a powerful centralised starburst followed by rapid quenching of the star formation , , .", "However, this induced starburst may not cause sufficient quenching to produce remnants with no ongoing star formation, as without a mechanism to expel the gas directly, the remnant typically continues to form stars , , .", "The more or less complete prevention of star formation in quiescent galaxies is the primary motivation for black hole feedback in the current generation of galaxy evolution simulations.", "In this model, an active galactic nucleus (AGN) can heat up the cold gas in the galaxies by thermal feedback and/or expel the cold gas by mechanical feedback, resulting in a lack of material for further star formation.", "Several studies suggest that PSBs may harbour an excess fraction of AGN compared to normal galaxies , , , , however the short duty cycles of AGN, compared to the observed post-starburst features, hamper any direct comparison.", "It is also possible that external conditions play a role in the ultimate quenching of post-merger remnants, which may explain why PSB galaxies are more common in (some) clusters and intermediate density environments , , , , .", "Ram-pressure stripping, strangulation or harassment may all play a role in suppressing further star formation in merger remnants that find themselves in dense environments.", "The development of spectral synthesis models allowed the first theoretical explorations of the unusual spectral properties of PSB galaxies , , leading to a basic understanding of their star formation histories.", "However, significant progress on understanding the formation mechanisms was only made once the dynamics of the gas in galaxies could be tracked in detail.", "Early hydrodynamic gas-rich merger simulations combined with simple star formation laws showed that gas rich mergers could lead to star formation histories characterized by strong, short bursts , .", "Combining hydrodynamical galaxy merger simulations with spectral synthesis models, led to reproduce the positive colour gradient and negative radial H$\\delta $ gradient observed in local PSBs by .", "A similar analysis was carried out by , at higher resolution and using more sophisticated star formation and feedback models, including a comparison between models with and without BH feedback.", "They also found that the stellar continuum properties of PSB galaxies at $0.5<z<1$ could be reproduced by merger simulations with starburst mass fractions larger than $\\sim 5-10$ % and decay times shorter than $\\sim 10^8$  yr. BH feedback was not required to reproduce the stellar continuum features of PSB galaxies, although they did not investigate the emission line properties.", "Similar simulations were carried out by , who again concluded that the role of AGN feedback in ceasing star formation and producing PSB features was negligible.", "Since these studies were undertaken the sub-resolution recipes used in the hydrodynamic simulations have become much more sophisticated and the spatial resolution has increased, motivating a revisit of the question of how the unusual spectral properties of post-starburst galaxies can be reproduced.", "The recent advent of large integral field spectroscopic (IFS) surveys of local galaxies, including CALIFA , SAMI , and MaNGA , has led to greater interest in the spatial properties of galaxies.", "In this paper, we revisit the question of radial gradients in the spectral properties of MaNGA post-starburst galaxies , using up to date simulations to investigate the quenching mechanisms at play in these galaxies.", "The outline of the paper is as follows: in Section we motivate our analysis by presenting the radial gradients in spectral indices observed in three example MaNGA galaxies; in Section we introduce the merger simulations; in Section we describe how we create mock MaNGA datacubes for the simulated galaxies; we present our analysis of the mock PSBs in Section , including their global and radial gradient properties; finally in Sections and we discuss and summarise our results." ], [ "Observational Motivation", "Traditionally, post-starburst galaxies are selected based on strong Balmer absorption lines, typically H$\\delta $ , with weak or absent nebular emission lines, commonly either the H$\\alpha $ emission line , , or the [OII] line if the H$\\alpha $ line is unavailable, as is common in higher redshift observations , , , .", "However, selection based on nebular emission will result in incomplete samples, as narrow line nebular emission can also be caused by shocks and AGN, both of which appear to be prevalent in PSBs , , , [2], .", "Additionally, in order to give us a complete picture of the galaxy quenching mechanisms we may be interested in objects that have had a recent starburst, but that have not (yet) completely quenched their star formation.", "More recently, alternative methods to select PSBs have been developed, that relies on the stellar continuum shape alone.", "developed a principal component analysis (PCA) method to select PSBs from high quality spectral observations of the stellar continuum around 4000Å.", "Similarly, photometric identification is possible in the case of the strongest spectral features , , .", "identified $>300$ galaxies with PSB regions based on traditional cuts on Balmer absorption line and H$\\alpha $ emission line strengths, out of over 4000 galaxies in the MaNGA Product Launch 6 An internal team data release including the first 3 years of MaNGA survey data.", "Of these, they identified 31 galaxies with centralised PSB signatures (CPSB).", "For these 31 galaxies, we downloaded the publicly available spectral data cubes and spectral measurement maps from the SDSS Data Release 15 [1], which were created by the MaNGA Data Reduction and Analysis pipelines.", "In both cases we used the “HYB10” spaxel binning where the stellar-continuum analysis is performed on Voronoi binned cells with a minimum signal-to-noise ratio (SNR) of 10 , while the emission-line and spectral-index measurements are performed on the individual spaxels.", "From the DAP maps, we extracted the H$\\alpha $ emission line equivalent width (W(H$\\alpha $ )), H$\\delta _{\\rm A}$ and H$\\gamma _{\\rm A}$ stellar continuum absorption line indices , the mean g-band weighted signal-to-noise ratio per pixel map and stellar velocity map.", "In all that follows we masked spaxels with spaxel $g$ -band SNR smaller than 10.", "During the MaNGA data analysis the emission and absorption lines are fit simultaneously, and the emission component is subtracted from the spectrum before the absorption indices are calculated .", "In this way the stellar and gas components of the spectrum are separated and are therefore independent of one another.", "The W(H$\\alpha $ ) measurement is then calculated from a Gaussian fit to the residual emission line spectrum.", "The H$\\delta _{\\rm A}$ and H$\\gamma _{\\rm A}$ absorption line indices are defined as the continuum normalised total flux difference between the absorption feature and a “pseudo-continuum” defined as a straight line between two flanking bandpasses, with the bandpasses as defined in .", "They can therefore be negative in old stellar populations whose spectra exhibit only weak Balmer absorption.", "From the summary “DRPall” file we obtained the effective radius, measured as the radius that contains half of the Petrosian flux in $r$ -band elliptical apertures from the NASA-Sloan Atlas cataloguehttp://nsatlas.org, and the Galactic extinction.", "From the DAP spectral data cubes we obtained the flux and variance, which we corrected for Galactic extinction and shifted to the rest frame using the redshift and stellar velocity maps.", "We then linearly interpolated the flux and error arrays onto the eigenbasis wavelength array of , in order to measure principal component based spectral indices that describe the shape of the 4000Å break region of the optical stellar continuum spectrum, using a normalised-gappy PCA algorithm to take account of the error arrays and any masked spaxels.", "Further details on the PCA spectral indices are given in Section REF .", "Following the method of , we mask all spaxels with SNR$<10$ and those with “DONOTUSE” flags, and calculate the distance of every spaxel from the centre of the galaxy, as a function of the effective radius.", "In fig:observations we show the spectral index distribution as a function of distance from the centre for three good quality CPSB galaxies, in both W(H$\\alpha $ ) vs. Balmer absorption line strength (top), and PC1 vs. PC2 (bottom).", "These three galaxies were selected to illustrate the range in radial gradients observed in the full sample.", "By definition, the central regions of the three galaxies lie within the traditional W(H$\\alpha $ ) vs. Balmer absorption line strength selection box, which we define as (H$\\delta _{\\rm A}$ +H$\\gamma _{\\rm A}$ )$/2 > 3$ and W(H$\\alpha $ ) $< 2.2\\times $ (H$\\delta _{\\rm A}$$+$ H$\\gamma _{\\rm A}$ )$/2 - 0.3$ .", "We clearly see strong radial gradients in the Balmer absorption line strength, with the central regions much stronger than the outer regions.", "In the PCA spectral indices we see a range of radial gradients, from marginally positive to strongly negative.", "We note that neither 8933-3704 nor 8623-9102 would be identified as a PSB by the PCA spectral indices, within the selection box defined by , which suggests that a very recent and rapid shut down in their star formation has occurred, before the Balmer absorption lines have had a chance to strengthen sufficiently to reach the PCA selection box.", "Simple toy models indicate that these observations are consistent with two simple scenarios: a single co-eval burst which was stronger in the central regions, or a starburst that has progressed from outside-in (Weaver et al.", "in prep).", "However, the toy models are unable to distinguish between the two options.", "These results form the motivation for this paper: why do we observe radial gradients in the spectral indices of PSB galaxies, and can we use them to tell us something about the physical processes that led to the formation of these unusual galaxies?" ], [ "Simulations", "Our binary merger simulations are run with the N-body smoothed particle hydrodynamics (SPH) code, SPHGal , , which is an updated version of the Gadget-3 code .", "Compared with the original Gadget-3 code, SPHGal replaces the spline kernel with a Wendland $C^4$ kernel and increases the number of neighbours in the SPH kernel to 100.", "The code employs a set of new features including the pressure-entropy formulation of SPH, an updated estimate of velocity gradients, a modified artificial viscosity switch with a modified strong limiter, artificial conduction of thermal energy and a time step limiter .", "Together these changes reduce numerical artefacts in the fluid mixing and improve the convergence rate of the SPH calculation." ], [ "Stellar sub-resolution physics", "The subresolution astrophysics models in SPHGal are based on those by , , updated by , and include gas cooling, star formation, chemical evolution and stellar feedback.", "In this model, the gas component cools with a rate dependent on its temperature, density and metal abundance.", "Assuming that the gas is optically thin and in ionisation equilibrium, the cooling rates are calculated following , on an element-by-element basis.", "The effects of a uniform redshift-dependent ionising UV/X-ray background are also included assuming $z = 0$ .", "The cooling rates are calculated over a temperature range of $10^2 \\le T \\le 10^9\\ K$ .", "Dense and cold gas particles are able to form stars, once the gas density $\\rho _g$ is greater than $\\rho _{crit}=1.6\\times 10^{-23} \\ \\rm g/cm^3$ , i.e $n_H=10 \\ \\rm cm^{-3}$ , and the gas temperature is less than $T=12000 \\ \\rm K$ .", "The probability that a gas particle converts into a stellar particle is $1-e^{-p}$ , where p is given by: $p=\\epsilon _{SFR}\\frac{\\Delta t}{t_{dyn}} = \\epsilon _{SFR} \\Delta t \\sqrt{4 \\pi G \\rho _g}.$ $\\Delta t$ is the length of the current time step, $t_{dyn}$ is the local dynamical time, and $\\epsilon _{SFR}$ is the star formation efficiency.", "We adopt a fixed efficiency of $\\epsilon _{SFR} =0.02$ .", "To track the chemical evolution in the simulations, every baryonic particle contains 11 elements (H, He, C, Mg, O, Fe, Si, N, Ne, S, Ca) that evolve based on models of chemical release rates from for supernovae type Ia (SNIa), for supernovae type II (SNII) and for asymptotic giant branch (AGB) stars.", "The star particles distribute the metals to the surrounding gas particles by the stellar feedback of SNIa, SNII and AGB stars.", "The metallicity diffusion implementation by is included here to smooth the variations in the metallicity between neighbouring gas particles.", "The stellar particles provide feedback the surrounding gas via SNII 3 Myr after their formation, followed by the SNIa for which feedback is released repeatedly every 50 Myr from a stellar age of 50 Myr until 10 Gyr, with the assumption that the ejecta mass decays proportionally to $t^{-1}$ and the total release is 2 SNIa per $1000 M\\odot $ of stellar mass.", "The total feedback energy by SN to the interstellar matter (ISM) is given by $E_{SN} = \\frac{1}{2}m_{eject}v_{SN}^2$ where $m_{eject}$ is the mass of SN ejecta, $v_{SN}$ is the velocity of SN ejecta (we adopt $v_{SN} = 4000\\ {\\rm km/s}$ here) and the SN ejecta in our model is metallicity dependent.", "The thermal and kinetic feedback of SN are achieved by a distance dependent multiphase method.", "The phases here are the free expansion (FE) phase, the adiabatic SedovTaylor (ST) phase , and the snowplow (SP) phase , .", "The gas particles that are closest to the SN receive the feedback in the FE phase with momentum conservation and the feedback energy is in kinetic form only.", "At greater distances, the shocked ISM mass exceeds the SN ejecta mass, gas particles in this outer region then receive feedback in the ST phase with heating.", "In the ST phase, the feedback energy are in both thermal (70%) and kinetic (30%) forms.", "At even larger distances, the velocity of the SN ejecta decreases further, finally dispersing the shock.", "Gas particles in this outermost region receive feedback in the SP phase with efficient radiative cooling.", "Again both thermal (70%) and kinetic (30%) energy is assumed, however the total amount of energy decreases with distance from the SN.", "Both the energy and enrichment feedback from AGB stars are dealt with in the same fashion as that from the type Ia SNe.", "However, as the wind velocity of AGB ejecta is just 25 km/s, much smaller than $v_{SN}$ , only the FE phase is included for AGB feedback." ], [ "Thermal black hole feedback model", "We run a set of merger simulations without black hole (BH) feedback and find that extra centrally injected energy is required in order to suppress the star formation in the merger remnants sufficiently to obtain the small H$\\alpha $ equivalent widths observed in many post-starburst galaxies.", "We investigate two different black hole models: a “classical” model with thermal energy injection only , , and an “updated” model with both thermal and mechanical energy injection .", "In the classic model the unresolved gas accretion onto the BH is parameterised by the Bondi–Hoyle–Lyttleton model , , , in which the BH accretion rate is given by: $\\dot{M}_B=\\frac{4\\pi \\alpha M_{BH}^2\\rho }{ (c_s^2+v^2)^{3/2}}$ where $\\rho $ is the gas density, $c_s$ is the sound speed of the surrounding gas, $v$ is the velocity of the BH relative to the surrounding gas, and $\\alpha $ is a dimensionless efficiency parameter set to enable the self-regulation of the BH mass growth and ensure that the BH accretion reaches the Eddington regime in a gas-rich environment .", "Physically, we should have $\\alpha =1$ , however and use a value of $\\alpha = 100$ to account for the underestimated gas density or the overestimated gas temperature near the Bondi radius.", "As the spatial resolution of the simulations is increased, the local gas density and temperature is more accurately modelled, and $\\alpha $ must be decreased.", "Here we set $\\alpha = 25$ and confirmed that this value allows the BHs in the merger remnants to grow onto, and stay on, the $M_{BH}-\\sigma $ relation .", "The accretion rate is limited to the Eddington accretion rate: $\\dot{M}_{edd}=\\frac{4\\pi G M_{BH} m_p}{\\epsilon _r \\sigma _T c}$ where $m_p$ is the proton mass, $\\sigma _T$ is the Thomson cross-section and $\\epsilon _r$ is the radiative efficiency.", "Here we adopt $\\epsilon _r =0.1$ , the mean value of radiatively efficient accretion onto a Schwarzschild BH .", "Thus, the final (inflowing) accretion rate is given by: $\\dot{M}_{in}=min (\\dot{M}_B, \\ \\dot{M}_{edd}).$ The radiated luminosity of the BH is given by: $L_r=\\epsilon _r \\dot{M}_{in} c^2.$ We assume that a fraction, $\\epsilon _f$ , of radiated energy couples with the surrounding gas leading to the feedback energy: $E_{\\rm feed}=\\epsilon _f L_r=\\epsilon _f \\epsilon _r \\dot{M}_{in} c^2.$ We adopt a fixed value of $\\epsilon _f = 0.05 $ , a value that is widely chosen in the literature , , , , .", "The choice of this value for $\\epsilon _f$ is motivated by the fact that in combination with the thermal feedback model the simulations are able to reproduce the observed $M_{BH}-\\sigma $ relation.", "In this classical thermal black hole feedback model, all the feedback energy is distributed as thermal energy into the $\\sim 100$ gas particles closest to the BH, weighted by the SPH kernel.", "The gas around the BH is therefore heated by the BH feedback and expands, reducing its density, which leads to a cessation in star formation.", "In our merger simulations, we assume that the binary BHs merge as soon as their separation drops below the smoothing length and their relative velocity drops below the local sound speed of the surrounding gas.", "Due to the limited spatial resolution the BHs can wander away from the centre of the galaxies, especially in unequal-mass mergers.", "To ensure the successful merging of the BHs during the final coalescence of their host galaxies, at every time step we re-locate the BH to the minimum potential in the central region of the galaxy ." ], [ "Mechanical black hole feedback model", "It has been suggested that a strong wind from an accreting BH might convey energy, mass and momentum from the centre to the surrounding gas , , , causing the gas outflows and the regulation of star formation in the host galaxies , .", "To simulate this effect we create a third set of simulations in which we adopt the mechanical BH feedback model developed by Choi et al.", "(, ).", "In this model the total accretion rate onto the BH is given by: $\\dot{M}_{acc}= \\dot{M}_{in} - \\dot{M}_{out}$ where $\\dot{M}_{out}$ is the outflowing mass loss rate, and $\\dot{M}_{in}$ is calculated as above (Eqn.", "REF ) with $\\dot{M}_B$ computed with an `alternative averaging (AA)' method .", "We further define the kinetic energy rate of the outflowing wind: $\\dot{E}_{w}= \\epsilon _f \\epsilon _r \\dot{M}_{acc} c^2 = \\frac{1}{2}\\dot{M}_{out}v_w^2$ where $\\epsilon _r$ and $\\epsilon _f$ are, respectively, the radiative efficiency and fraction of the radiated energy that couples to the surrounding gas (as above), and $v_w$ is the velocity of the wind assuming energy conservation and a single wind velocity.", "Following we define the ratio of the mass outflow rate to the mass accretion rate as: $\\psi \\equiv \\dot{M}_{out}/\\dot{M}_{acc} = 2\\epsilon _f \\epsilon _r c^2/v_w^2.$ Since the momentum is conserved, the total accretion rate $\\dot{M}_{acc}$ and the kinetic energy rate $\\dot{E}_w$ can be solved out as: $\\dot{M}_{acc}= \\dot{M}_{in}\\frac{1}{1+\\psi }$ $\\dot{E}_w=\\epsilon _f \\epsilon _r c^2 \\dot{M}_{in}\\frac{1}{1+\\psi } $ Inspired by observations of broad absorption line winds , , we assume a wind velocity of $v_w = 10,000\\,{\\rm km/s}$ .", "With the widely adopted feedback efficiencies of $\\epsilon _f \\epsilon _r = 0.1 \\times 0.05 =0.005$ , Eqn.", "REF gives $\\psi =0.9$ , i.e., 90% of the inflowing mass is ejected in an AGN wind, with both energy and mass carried from the BH to the outskirts of the galaxy.", "This outflowing gas collides with the ambient ISM on its way out, resulting in a momentum-driven flow.", "We simulate this phenomenon by allowing the emitted wind particle to share its momentum with its 2 nearest neighbouring gas particles.", "As the gas particles all have the same mass in our simulation, all three gas particles gain the same velocity increment of $\\Delta v \\sim (v_w/3)\\ {\\rm km/s}$ .", "This treatment conserves the momentum but decreases the total kinetic energy.", "To conserve the total energy, the residual energy is deposited into these three particles in form of thermal energy.", "include an additional thermal feedback component in their simulations to simulate the effect of X-ray radiation from the AGN on the surrounding gas.", "However, we found that this was too effective at suppressing star formation, eliminating the starburst during the merger phase and preventing the appearance of the strong Balmer absorption lines seen in PSB galaxies.", "We therefore only ran one simulation with this effect included, which is presented in Section below.", "We note that the updated stellar feedback model used in this paper, compared to the stellar feedback model used in , may in part be responsible for the dramatic suppression of star formation seen in our simulations when including this additional thermal X-ray BH feedback component." ], [ "Galaxy models", "The progenitor galaxies are set up following the method given in assuming a $\\Lambda $ CDM cosmology with $\\Omega _m=0.30$ , $\\Omega _\\Lambda =0.70$ , and $H_0=71\\,{\\rm km/s/Mpc}$ , with the aim to mimic disc galaxies observed in the local Universe.", "A summary of the progenitor galaxy parameters used in this paper can be found in Table REF .", "The primary galaxies have a virial velocity of $v_{vir}=160\\,{\\rm km/s}$ , resulting in a total virial mass of $M_{vir}=v_{vir}^3/10GH_0=1.34\\times 10^{12}\\ M_\\odot $ .", "The majority of this mass ($M_{DM} = 1.286 \\times 10^{12} M_\\odot $ ) is in the dark matter (DM) halo, which has a Hernquist density profile with a concentration parameter of $ c = 9 $ .", "The baryonic mass fraction is 0.041, distributed between a gaseous disc ($M_{gas}$ ), a stellar disc ($M_{disc,*}$ ) and a stellar bulge ($M_{bulge}$ ), with $f_{gas} = M_{gas}/(M_{gas}+M_{disc,*})$ determining the gas fraction in the disc.", "Each progenitor galaxy has a total particle number of $8\\times 10^5$ .", "Half are DM particles and half baryonic, yielding a mass resolution of $1.4\\times 10^5$ M$_\\odot $ and $3.2\\times 10^6$ M$_\\odot $ for baryonic and DM particles respectively.", "The gravitational softening lengths are $\\epsilon _{\\rm bar}=28\\,{\\rm pc}$ and $\\epsilon _{\\rm DM}=137\\,{\\rm pc}$ for the baryonic and dark matter components respectively.", "The disc has an exponential mass profile with scale length $r_{disc}$ , determined by assuming that the disc material conserves specific angular momentum during the disc formation with a constant halo spin of $\\lambda =0.033$ .", "The scale height of the stellar disc is set to $z_{disc}=0.2\\,r_{disc}$ .", "The gaseous disc has the same scale length as that of the stellar disc and the vertical structure of the gaseous disc is determined such that it is in hydrostatic equilibrium .", "The stellar bulges follow a profile with a scale length of $r_{bulge}=0.2r_{disc}$ .", "The mass of the bulges are determined by the parameter $B/T$ , the stellar bulge-to-total stellar mass ratio.", "By choosing different bulge-to-total stellar mass ratios and corresponding gas fractions, we create 3 progenitor galaxies of different morphology types, aiming to roughly mimic the average properties of Sa, Sc, and Sd galaxies.", "The specific values are chosen loosely based on values for local SDSS galaxies, by combining the range of $i$ -band luminosity B/T values from , with the range of atomic and molecular gas mass fractions from .", "For the simulations with BH feedback the progenitor galaxies host BHs at their potential minima.", "The mass of the BH is given by the $M_{BH}-\\sigma $ relation, with $\\sigma $ calculated from the distribution of the bulge star velocities .", "In order to build accurate spectral models, as well as include the feedback from old stars, we must initialise the stellar particles with their ages and metallicities.", "For the stellar particles in the bulge, we adopt an exponentially decaying star formation rate (SFR) over the time: $SFR_{b} (t)=Ce^{- (t-t_0)/\\tau },$ where $t_0=10.2$  Gyr is the start of the simulation, $C$ is a normalisation factor, and $\\tau = 1$ Gyr is the timescale of the exponential decline.", "We assume that the star formation in the bulge started at a cosmic time of $t=0.5$  Gyr, leading to a negligible SFR compared to the disc by $t_0$ .", "For the stellar particles in the disc, we follow and adopt a linearly decaying SFR with an initial SFR at $t_0=10.2$  Gyr similar to that found in local star-forming disc galaxies.", "We then run the galaxies in isolation and iteratively adjust the initial SFR estimate until the SFR transitions continuously from the initial value to the actual SFR at the start of the simulation (see Table REF ).", "By requiring that the integration of the star formation rate history ove time is equal to the total stellar mass at the start of the simulation, the age distribution of the stellar particles can be obtained.", "To be fully consistent with the employed sub-resolution models, we must also initialise the metallicity distribution.", "Here we adopt a uniform, log-linear radially decaying metallicity profile with the Milky Way as a reference .", "Using the observed Oxygen gradient of 0.0585 dex/kpc, we set up our galaxies with roughly solar total metallicities.", "When assigning metal elements to individual stellar particles, we assume a scatter of 0.2 dex motivated by the the maximum measurement error in and fix an upper limit on the metal mass fraction of 5% to prevent the stellar particles from becoming unphysically metal heavy.", "Due to element recycling, stars with later cosmic formation times tend to have higher metallicities.", "To reproduce this phenomenon, we firstly randomly sample the age of individual stellar particles from the age distributions with an age scatter of $\\Delta t = 100$  Myr.", "We then sort the stellar particles by their assigned metal mass fraction, and re-assign their ages keeping the same overall age distribution.", "We additionally create a lower mass galaxy (Scp3), a smaller version of the Sc galaxy with a mass of one third that of the Sc galaxy.", "This is done by reducing $v_{vir}$ to $160/\\@root 3 \\of {3}\\,{\\rm km/s}\\sim 111\\,{\\rm km/s}$ .", "The particle numbers of each component are reduced by one third to keep the mass resolution per particle constant." ], [ "Merger Simulations", "After initialising the progenitor galaxies, we run the galaxy models in isolation for 0.5 Gyr to even out any numerical artefacts from the initial idealised setup.", "We then set up the merger simulation with these relaxed galaxies at $t_{m0} = 10.7$  Gyr, on parabolic trajectories with three different initial orbital configurations as introduced by : G00: $i_1=0^{\\circ }$ , $i_2=0^{\\circ }$ , $\\omega _1=0^{\\circ }$ , $\\omega _2=0^{\\circ }$ ; G07: $i_1=-109^{\\circ }$ , $i_2=71^{\\circ }$ , $\\omega _1=-60^{\\circ }$ , $\\omega _2=-30^{\\circ }$ ; G13: $i_1=-109^{\\circ }$ , $i_2=180^{\\circ }$ , $\\omega _1=60^{\\circ }$ , $\\omega _2=0^{\\circ }$ .", "where $i_1$ and $i_2$ denote the inclinations of the two discs relative to the orbital plane, and $\\omega _1$ and $\\omega _2$ denote the arguments of the orbits' pericenter.", "G00 is a symmetric prograde-prograde orbit.", "Both galaxies are orientated in the orbit plane and their angular momenta are parallel to the orbital angular momentum.", "G07 is a retrograde-prograde orbit with both galaxies inclined with respect to the orbital plane.", "G13 is retrograde-retrograde orbit, with one galaxy inclined while the other is not.", "The angular momentum of the second galaxy is anti-parallel to the orbital angular momentum.", "The initial separation $r_{\\rm sep}$ of the two progenitor galaxies is given by their average virial radii while the pericentre distance $r_p$ is given by the sum of their disc scale lengths.", "Thus, for the equal-mass mergers (1:1) the initial separation is $r_{\\rm sep} = (160+160)/2\\,{\\rm kpc/h} \\sim 225\\,{\\rm kpc}$ , the pericentre distance is $r_p=2\\times 2.7\\,{\\rm kpc/h} \\sim 7.6\\,{\\rm kpc}$ .", "For the unequal-mass merger (3:1), the initial separation is $r_{\\rm sep} = (160+111)/2\\,{\\rm kpc/h} \\sim 135.5\\,{\\rm kpc}$ , and the pericentre distance is $r_p= (2.7+1.90)\\,{\\rm kpc/h}\\sim 6.5\\,{\\rm kpc}$ .", "The progenitor galaxies approach each other following nearly parabolic orbits and interact under their own gravity.", "We run a set of simulations with different progenitor galaxies and orbits.", "For the equal-mass mergers we select 6 illustrative combinations of progenitor galaxies (2xSa, Sa_Sc, Sa_Sd, 2xSc, Sc_Sd, 2xSd), and for the 3:1 mass ratio mergers we choose one high mass progenitor of each type and merge it with the lower mass Sc galaxy giving three further combinations (Sa_Scp3, Sc_Scp3, Sd_Scp3).", "Each pair is set on 3 different orbits G00, G07, and G13 (Section REF ), giving 18 equal-mass and 9 unequal-mass merger simulations in total.", "The exhaustive table of the parameters for the merger simulations can be found in the Appendix (Table REF ).", "Each merger simulation is run for 3 Gyr until the current cosmic time $t=13.7$  Gyr.", "Snapshots are output every $2\\times 10^7$ years, giving 150 snapshots per simulation." ], [ "Building mock MaNGA datacubes", "We develop the SEDmorph codeThe project web page: https://github.com/SEDMORPH.", "The code for mock datacube creation https://github.com/SEDMORPH/YZCube.", "standing for Spectral Energy Distribution and morphology, to turn the particle data into mock images and integral field datacubes that closely match the properties of the MaNGA survey.", "We place the mock galaxies at a redshift of $z\\sim 0.04$ , close to the median redshift of the MaNGA survey.", "The 1 radius MaNGA fibres are equivalent to 0.79 kpc at this redshift.", "Here we provide details on the components of the code that are used in this paper." ], [ "Stellar continuum creation", "To build the stellar continuum spectra we use the spectral synthesis models, updated to 2016These are available at http://www.bruzual.org/~gbruzual/bc03/Updated_version_2016/.", "These models provide integrated light spectra for simple stellar populations (SSPs) which represent coeval, single metallicity stellar populations, assuming an initial mass function (IMF), evolutionary tracks and stellar input spectra.", "This version of the BC03 models are built from both observed and theoretical stellar spectra, assuming a IMF and “Padova-1994” evolutionary tracks [4], , , , .", "The observed spectra most relevant for the wavelength range of the MaNGA spectra are taken from the MILES stellar library , in the wavelength range 3540Å- 7350Å, extended with the STELIB stellar library out to 8750Å.", "We assume that each stellar particle in our simulation is composed of a single coeval, fixed metallicity stellar population, with an age and metallicity corresponding to when the progenitor gas particle becomes a stellar particle.", "The SSPs have already been interpolated onto an optimum age grid, we therefore refrain from any further interpolation in age and each stellar particle is assigned the closest SSP in age.", "The BC03 models provide SSPs for seven different metallicity values (with metal mass fractions of $Z=$ 0.0001, 0.0004, 0.004, 0.008, 0.02, 0.05, 0.1, where $Z_\\odot =0.02$ ).", "The metallicity distribution of particles in the simulations is such that most particles have metallicities in the 2 bins around solar metallicity ($0.08<Z<0.05$ ), which is not sufficient to accurately reproduce the changes in spectral indices caused by the metallicity gradient of the galaxies.", "We therefore interpolate the spectra linearly in log flux between the central metallicity bins, aiming to distribute the stellar particles between metallicity bins as equally as possible.", "This leads to additional SSPs with $Z=$ 0.012, 0.016, 0.023, 0.026, 0.035 and 0.04.", "We verify the success of the interpolation by computing the H$\\delta _{\\rm A}$ and H$\\gamma _{\\rm A}$ spectral indices of the 1 Gyr old interpolated spectrum, finding that the indices vary smoothly between the new SSPs as expected.", "The stellar particles contain 11 different elements as described in Section REF .", "We calculate the metallicity of each stellar particle as $Z_* = (1- M_H - M_{He})/M_*$ , where $M_H$ and $M_{He}$ are the total masses of Hydrogen and Helium respectively and $M_*$ is the mass of the star particle, and assign the closest interpolated SSP in metallicity.", "Finally, the integrated spectrum (luminosity density) of the simulated galaxy or part thereof, can be written as: $l_{\\lambda } = \\sum _{i=1}^{N_{*}} M_i \\times l_{\\lambda ,SSP} (t_{SSP, i}, Z_{SSP,i})$ where $M_{i}$ is the mass of the stellar particle, $l_{\\lambda ,SSP}$ is the luminosity density of the assigned SSP and the sum is over all relevant star particles in the region of interest.", "We employ a two-component dust attenuation model , in which the optical depth is contributed by the interstellar medium for all stars, as well as stellar birth clouds for stellar populations younger than $10^7$ years.", "The final effective optical depths as a function of wavelength for young and old stellar populations are thus given by: $\\begin{aligned}\\hat{\\tau }_{young} &=& \\mu _{d} \\tau _{v} \\left(\\frac{\\lambda }{5500}\\right)^{-0.7} &+ \\left(1-\\mu _{d} \\right) \\tau _{v} \\left(\\frac{\\lambda }{5500}\\right)^{-1.3} \\\\\\hat{\\tau }_{old} &=& \\mu _{d} \\tau _{v} \\left(\\frac{\\lambda }{5500}\\right)^{-0.7} &\\end{aligned}$ where $\\lambda $ is the wavelength in Å, $\\mu _{d}$ is the fraction of optical depth contributed by the ISM, which is set as 0.3 and $\\tau _v$ is the effective optical depth at 5500Å, which we set to a fiducial value of 1.0, typical for local star-forming galaxies.", "We note that by not using radiative transfer to calculate the attenuation we are not correctly accounting for the full 3D geometry of the stars and gas in the mock galaxies.", "Additionally, we do not attempt to link dust content to the metallicity of the gas particles.", "However, we choose this approach to avoid further sub-resolution recipes given the relatively high mass of the individual gas and star particles compared to giant molecular clouds in which stars are formed in reality.", "It is important to keep in mind the limitations of any dust modelling in mock galaxies with this level of spatial resolution." ], [ "Data Cube Creation", "The first step in creating mock datacubes is to locate the centre of the simulated galaxies in each snapshot.", "For simulations with black holes we simply take the position of the supermassive black hole as the centre of the galaxy as the black hole is repositioned to the minimum of the potential at every time step.", "For the simulations without black holes we use the distribution of dark matter particles to identify the centre of the potential well in which the galaxy resides.", "We create a single datacube for each merger simulation, focussing on the centre of the first galaxy, which is the highest mass galaxy in the case of unequal-mass mergers.", "This is sufficient for our purposes, as we are predominantly interested in the post-merger remnants.", "The basic steps to create a realistic MaNGA datacube are to: (1) apply a seeing (atmospheric) point spread function (PSF); (2) sum the light that falls down each circular fibre in a single pointing; (3) apply a dither pattern to build the Row Stacked Spectrum (RSS) file; (4) combine the RSS into a regularly gridded datacube.", "We also explored other algorithms for creating datacubes and found that they are inferior to the one described above.", "One of alternative algorithms is to apply the PSF after the cube is built.", "This method is computationally less expensive by $\\sim $ 20% but does not lead to sufficiently accurate results.", "Another algorithm is to bin the simulation directly onto a square grid and apply the well characterised combined atmospheric and instrumental PSF after the binning.", "This process is simpler to be realised but turns out to be more computationally expensive.", "More details on the comparison of different algorithms can be found in Appendix .", "In order to measure the spectrum received by a single MaNGA fibre, we must first apply a PSF to account for the blurring of the light as it travels through the atmosphere.", "We assume a two-component Gaussian profile with parameters taken from the median of SDSS DR7 imaging fields.", "For the purposes of this paper we are particularly interested in the spectra of local galaxies around 4000Å, therefore we apply the typical $g$ -band PSF taken from the median values for 500 000 SDSS DR7 fields (specifically, the $\\sigma $ widths of the two Gaussians are $0.54$ and $1.21$ respectively, and the peak amplitude ratio of the two Gaussians is $0.081$ ).", "We calculate the integrated spectrum that falls within the $j$ th fibre by summing over stellar particles in the simulation accounting for their distance from the centre of the fibre, $d$ , and the shape of the PSF: $l_{\\lambda ,j} = \\sum _{i=1}^{N_{star}}W_{\\rm PSF} (d)_i l_{\\lambda ,i}$ where $l_{\\lambda ,i}$ is the attenuated stellar continuum luminosity density assigned to the $i$ th stellar particle, as described above, and $W_{\\rm PSF} (d)$ is a weight function that accounts for the shape of the PSF integrated over the fibre area (assuming a MaNGA fibre radius of 1).", "In order to reduce the computing time we set $W_{\\rm PSF} (d)=0$ once it drops below a value of $10^{-6}$ .", "The MaNGA fibres are arranged in hexagonal bundles, and sets of three dithered exposures are taken to fill the gaps between the fibres using a triangular pattern with a side length of 1.44 .", "In each MaNGA observation, the number of dither sets depends on the on the time it takes to reach the target depth, leading to the an integer multiple of 3 exposures (3,6,9,12, etc).", "For our mock observation, one dither set of three exposures will be sufficient.", "To know where to place the circular apertures on the simulated galaxy, we take a representative metrology file for the largest MaNGA IFU bundle of 127 fibres (ma134-56995-2.parAn slightly older version that's publically accessible can be found here: https://svn.sdss.org/public/repo/manga/mangacore/tags/v1_6_2/metrology/ma134/ma134-56995-1.par.", "The older version is off by a slight scaling factor from the one used in this paper.)", "which records the positions of all the fibres in one dither.", "We then shift the fibre positions following the triangular pattern for the fibre positions in the other two dithers.", "Once we have calculated the spectra in each individual exposure, we combine them into a 3D datacube using the same flux-conserving algorithm as used in the MaNGA survey .", "We do not include noise due to e.g.", "sky background, therefore we do not calculate error arrays or masks.", "Neither do we model the line spread function, and the resulting spectra remain at the spectral resolution of the SSPs." ], [ "Spectral analysis", "In order to identify post-starburst galaxies we are looking for galaxies with an excess of A/F stars left over from a recent starburst, however ordinary star-forming galaxies have plenty of A/F stars and therefore some knowledge about the current star formation is required in all but the most extreme cases.", "The traditional method uses a Balmer absorption line (typically H$\\delta $ ) which is strong in A and F star spectra, alongside an absence of nebular emission to ensure no ongoing star formation.", "The disadvantage of this method is that it will not select post-starburst galaxies with low-level residual star formation, or those that contain a narrow line AGN , , and if a blue emission line is used then samples can be contaminated by dust obscured star-forming galaxies , .", "Alternatively we can use the stellar continuum to provide an estimate of the recent star formation, e.g.", "using a combination of the 4000Å break strength and H$\\delta $ absorption line strength .", "In more extreme cases, even photometric data can identify the strong Balmer break exhibited by post-starburst galaxies , , .", "Here we compare two methods for identifying post-starburst regions in our simulated galaxies." ], [ "The traditional method", "We sum the H$\\delta _{\\rm A}$ and H$\\gamma _{\\rm A}$ absorption line indices and combine this with the equivalent width of the H$\\alpha $ emission line to investigate when the simulated galaxies obey the “traditional” selection criteria for post-starburst galaxies.", "We measure the H$\\delta _{\\rm A}$ and H$\\gamma _{\\rm A}$ absorption indices using the same method as in the MaNGA survey (see Section ).", "We note that these are calculated at the resolution of the MILES-SSPs, $\\sim $ 58 km/s, which is similar to the MaNGA instrumental dispersion around the 4000 Å break.", "As we are dealing with small regions of galaxies, the velocity dispersion is not as large as for whole galaxies.", "Hence, the slight difference in resolution between the simulated spectra and the MaNGA data has little effect on the spectral indices used in this paper.", "In order to calculate the equivalent width of the H$\\alpha $ emission line consistently from the spectral synthesis models, we firstly measure the ionising photon luminosity $Q (H^0)$ from the dust-free continuum luminosity density $l(\\lambda )$ (Eqn.", "REF ) at wavelengths shorter than the Lyman limit ($\\lambda _{ly}$ ): $Q ({\\rm H}^0) = \\int _{}^{\\lambda _{ly}}l(\\lambda ) \\frac{\\lambda }{hc} \\mathrm {d}\\lambda $ where $h$ is the Planck constant and $c$ is the speed of light.", "We convert this into an integrated H$\\alpha $ emission line luminosity ($L (H\\alpha )$ ) assuming Case B recombination at $T_e = 10,000K$ : $L ({\\rm H}\\alpha ) [\\textrm {ergs s}^{-1}]=1.37x10^{-12}\\times Q ({\\rm H}^0) [\\textrm {s}^{-1}].$ We then attenuate the luminosity due to dust by the same amount as for young stars in our continuum model (Eqn.", "REF ).", "To calculate the equivalent width (EW) of the H$\\alpha $ emission line as used in observations, we extract the mean stellar continuum luminosity density from the attenuated stellar continuum between $\\pm 5$ Å from the wavelength of H$\\alpha $ ($l_{6563}$ ).", "The EW is then given by: $W({\\rm H}\\alpha ) = \\frac{L({\\rm H}\\alpha )}{l_{6563}}.$" ], [ "Principal Component Analysis", "We compare the traditional method with a second spectroscopic diagnostic which uses the stellar continuum alone to identify galaxies with stronger Balmer absorption than expected.", "This method applies a Principal Component Analysis (PCA) to the stellar continuum between 3750 and 4150Å, finding the first component (PC1) to correlate strongly with the 4000Å break strength, while the second component (PC2) provided the excess Balmer absorption line strength over that expected for the 4000Å break strength .", "The method is analogous to using $D_n4000$ vs. H$\\delta _{\\rm A}$ , however has the added benefit of being able to combine information from all the Balmer absorption lines as well as the shape of the continuum in order to increase the SNR of the measurement, as well as rotating the parameter space so that the PSBs are easily identified lying above the star-forming main sequence.", "This means that older and weaker PSBs can be detected with the PCA compared to H$\\delta _{\\rm A}$ alone .", "As the emission lines are not used, the method is sensitive to galaxies that contain narrow line AGN, and those galaxies that do not completely shut off their star formation.", "In order to calculate the principal component amplitudes for the spectra in the mock data cubes, we first convert from air to vacuum wavelengths, convolve to the common velocity dispersion of 150 km/s used by the eigenbasis (assuming an intrinsic dispersion of 58 km/s for the MILES library) and then project the spectra onto the same eigenvectors as calculated in The eigenvectors can be downloaded here: http://www-star.st-and.ac.uk/~vw8/downloads/DR7PCA.html.", "Code to perform the projection is available here: https://github.com/SEDMORPH/VWPCA (IDL) or https://github.com/astroweaver/pygappy (python)." ], [ "Results", "In this section we compare the impact of different black hole feedback models, progenitor galaxy properties and orbits on the star formation histories and spectral measurements of the simulated galaxies as a function of time from pre- to post-merger.", "We then focus in on one particular simulation to study the origin of the radial gradients in the spectral indices." ], [ "Comparison between different BH feedback models", "In this subsection we compare the quenching progress and observational signatures of quenching in simulations with very different BH models in order to constrain the type of feedback that might be needed to create the post-starburst galaxies observed in the real Universe.", "Here we consider four models: No BH: the BH feedback is turned off; Thermal BH: the “classical” thermal BH feedback model (Section REF ); With RFB: the mechanical BH feedback model with additional X-ray radiative feedback turned on (Section REF ); No RFB: the mechanical BH feedback model with additional X-ray radiative feedback turned off.", "We compare simulations with the same progenitor galaxies (2xSc) and orbits (G00); this orbital arrangement produces the strongest tidal forces and therefore cleanest star formation histories making them ideal to compare the impact of the different feedback mechanisms.", "The star formation histories (SFH) are derived from the change in the total stellar mass of the galaxies between different snapshots and the spectral indices are measured as described in Section REF .", "The results are shown in fig:comparemodels.", "All simulations are run for 3 Gyr, however some simulations experience a very sudden quenching phase caused by a complete exhaustion of cold star-forming gas.", "This is likely caused by a combination of the employed supernova$+$ BH feedback model and the relatively low particle resolution of our simulations.", "We argue that such \"numerical induced quenching\" is not astrophysical.", "To avoid confusion we remove these results from the figures.", "In all cases apart from the model with both mechanical and radiative feedback (With RFB) the SFH is dominated by a strong burst of star formation at coalescence ($t\\sim $ 12.3 Gyr), which leads to post-starburst features in one or both of the observational index spaces.", "However, the models differ in the amount that they quench and therefore the final SFR as we will discuss below.", "It is clear that the mechanical BH feedback model with radiative feedback turned on is too effective at suppressing the star formation.", "During the first encounter of the galaxies at $t\\sim 11.5$  Gyr some gas is funnelled into the galaxy centre and feeds the BH.", "The radiative feedback strongly suppresses the star formation below 0.1 $M_\\odot /yr$ , which prevents the small increase in SFR seen in each of the other simulations.", "Moreover, the radiative feedback completely eliminates the starburst during the final coalescence, preventing the remnant galaxy from showing observational post-starburst features.", "After the final merger, the galaxy structure becomes relative stable and the gas inflow becomes weaker, which limits the strength of the BH feedback.", "In observational index space, the galaxy sits predominantly in the “green-valley” throughout the simulation, temporarily reaching the red sequence after the first encounter.", "The majority of the gas in the galaxy remains unconsumed by the end of the simulation allowing a relative high final SFR of about 1 $M_\\odot /yr$ , and it quickly returns to the “green valley” following coalescence.", "In the other three simulations, the SFHs show a similar trend before the final merger.", "All the galaxies experience enhanced star formation during the first encounter of the two progenitor galaxies and a strong starburst during the final merger.", "Note that although the initial SFRs are set to be the same when creating the model galaxies, the SFRs end up with slightly different values when the galaxies reach equilibrium states after the 0.5 Gyr isolated run (see Section REF ).", "Shortly after coalescence the SFRs decline quickly from the peak, even in the case of no BH feedback.", "As noted previously in the literature, this initial decline is caused by gas consumption and stellar feedback from O and B-type stars, rather than BH feedback processes.", "All three galaxies enter the fiducial PSB region in PCA space, due to the spectral signatures of the excess of A/F stars created in the starburst.", "Following coalescence we see the divergence in the SFH caused by the different BH feedback models.", "Without BH feedback (NoBH) the remnant's SFR remains at a level comparable to that before the merger.", "Introducing thermal feedback suppresses star formation slightly in comparison, but only to the level of 1 $M_\\odot /yr$ (i.e.", "similar to the Milky Way).", "Only the mecahnical BH feedback model No RFB is able to completely quench star formation in the galaxy.", "Due to the residual star formation in the No BH and Thermal BH models these galaxies retain a level of star formation that leads to significant H$\\alpha $ emission and prevents the galaxy from passing into the traditional PSB selection region.", "Only the mechanical feedback model is able to suppress the H$\\alpha $ emission sufficiently that the galaxy would be selected as a PSB using the traditional H$\\alpha $ emission vs. Balmer absorption indices.", "The merger remnant evolves onto the red sequence in both index spaces as the simulation ends.", "Based on the discussion above, for the remainder of this paper we focus on simulations with mechanical BH feedback implemented, but without radiative feedback, as these are the only runs that would be selected using a traditional PSB selection method.", "We note that the fact that the PCA selection includes objects where the quenching is incomplete may well explain the much larger number of PCA-selected PSBs compared to those selected with the traditional cut on emission line luminosities ." ], [ "Comparison between different progenitor galaxies and orbits", "We now run the complete set of simulations presented in Table REF with different progenitor galaxies and orbits, using the mechanical BH feedback model.", "There are 6 combinations of progenitor galaxies for equal-mass merger (2xSa, Sa_Sc, Sa_Sd, 2xSc, Sc_Sd, 2xSd) and 3 combinations for 1:3 mass ratio mergers (Sa_Scp3, Sc_Scp3, Sd_Scp3).", "Each pair is set on 3 different orbits G00, G07, and G13 (Section REF giving 27 different mergers in total.", "In fig:allSFH we show the SFH of all 27 simulations.", "We refrain from showing all evolutionary plots of the spectral indices, as the results can largely be inferred from the SFHs given the knowledge gained in the previous subsection.", "In this subsection we summarise the results for all 27 simulations, and in the following subsection we focus on one key example showing the full evolution of the model in the observational parameter space.", "Almost all equal-mass mergers have strong starbursts, and the higher the gas fraction the stronger the burst.", "However, despite the strong mechanical BH feedback the starburst can have a long decay time following the merger, especially for the retrograde-retrograde (G13) merger or the retrograde-prograde (G07) merger with lower gas fractions.", "In these cases, the SFR displays only a gradual decline, and the H$\\alpha $ EW never declines sufficiently to place the galaxy into the PSB selection box.", "For unequal-mass mergers, the starburst is significantly weaker compared to the equal-mass merger with the same orbits and progenitors with the same Hubble type.", "A higher gas fraction again leads to a stronger starburst, however only the Sd_Scp3_00 and Sd_Scp3_07 mergers enter the PCA PSB region, although with a lower PC2 than the major mergers.", "Moreover, in all of the nine unequal-mass merger simulations, the post-merger SFR is comparable to the SFR before the merger, leading to strong residual H$\\alpha $ emission.", "The prograde-prograde orbit (G00) is the easiest orbit to reproduce a strong starburst and subsequent quenching.", "In this orbit, the stellar disc and bulge of the two progenitors collide with each other violently, causing the strongest tidal forces which drive a rapid flow of gas into the galaxy centre forming a large number of new stars in a short time.", "The gas is quickly consumed, leaving less material for further star formation.", "The gas flow drives rapid AGN accretion and the resulting BH feedback also reduces the efficiency for the remaining gas to transform into stars.", "A sharp decline in SFR can be seen in every pair of the equal-mass prograde-prograde progenitors.", "However, it seems unlikely that this orbital configuration occurs regularly in the real Universe.", "Comparison with very large cosmological simulations would be required to clarify this.", "We conclude that, within the scope of these simulations, sharp and sustained quenching of star formation caused by mergers is only achieved in particular circumstances: relatively gas rich progenitor galaxies with similar mass, approaching each other in either prograde-prograde or retrograde-prograde orbits.", "In these simulations, neither unequal-mass nor retrograde-retrograde mergers can reproduce features completely consistent with the full range of observed PSB galaxies, with the presence of nebular emission due to ongoing residual star formation being a key constraint." ], [ "Evolution of global spectral properties for a representative merger", "The previous two subsections have shown that post-starburst features can be reproduced with specific progenitor galaxies and certain orbits, in the presence of significant BH feedback.", "We now choose one representative merger simulation and focus on its evolving spectral properties.", "Though the prograde-prograde orbit (G00) produces the strongest starburst and sharpest quenching, such an orbital configuration with both progenitor galaxies having angular momenta parallel to the orbital angular momentum is likely to be rare in the local Universe.", "We therefore select the next best G07 orbit.", "Similarly, mergers between two massive disc-dominated galaxies with high gas fractions are unlikely to be common enough to be the most representative cases.", "We therefore select the Sc progenitors.", "In this and the following subsections we thus focus on the 2xSc_07 simulation.", "The SFH and BH accretion rate of the combined simulation cube are plotted in the top panel of fig:galpro, with the three coloured vertical lines indicating the different quenching stages.", "At the blue snapshot ($t=12.56$  Gyr), the SFR starts to decrease (the quenching just begins); at the cyan snapshot ($t=12.76$  Gyr), the SFR drops to a value that is comparable to that before the merger; at the red snapshot ($t=13$  Gyr), the SFR drops below 0.1$M_\\odot /yr$ .", "The dotted grey line indicates the merger of two BHs at $t=12.53$  Gyr, which is considered as the formation point of the merger remnant.", "The central and lower panels of fig:galpro show the evolution of the simulated galaxies in spectral index space, integrating the entire spectrum of both galaxies.", "The simulated galaxy starts from within the blue cloud in both index spaces at the beginning of the simulation, with the first encounter causing a very small change in the spectral indices.", "During the starburst we observe very strong H$\\alpha $ emission and weak 4000Å break strength (PC1) due to significant contribution to the spectrum from O and B-type stars.", "The Balmer absorption lines remain strong in starburst galaxies, due to the large number of A and F-type stars formed which are only partially outshone by the O and B stars.", "However, this translates into a low PC2 as the Balmer absorption line strength is slightly weaker than expected when compared to normal star-forming galaxies.", "As the star formation declines below its initial level, the remnant quickly moves into the PSB region in both spectral index spaces, ending up on the red sequence.", "The remnant galaxy can be found in the PCA-defined PSB region consistently from from 12.74 to 13.22 Gyr, while it is found less consistently in the H$\\alpha $ -defined PSB region between 12.92 and 13.16 Gyr.", "While the stellar continuum based PCA indices show a stable evolution with time, the H$\\alpha $ emission line strength fluctuates significantly during the PSB phase.", "The purple line in the top panel of fig:galpro shows the smoothed BHAR averaged over 20 Myr, which fluctuates substantially with peaks in BHAR followed by dips in the SFR.", "We therefore suggest that the fluctuating BH feedback strength causes the galaxy to shift in and out of the H$\\alpha $ -defined PSB region even after entering the PSB phase at $t\\sim 12.92$  Gyr.", "In general for our simulations, we find that the stellar continuum based PCA selection is a more consistent identifier of PSB galaxies.", "The very strong fluctuations in the H$\\alpha $ EW will cause an emission line-selected sample to be incomplete, even before the loss of objects from the sample due to narrow line emission from the AGN.", "However, we note that some observed PSB galaxies selected using the traditional nebular emission based method are not selected with the PCA, in particular the two examples presented in fig:observations.", "We also note that this is not an intrinsic property of the PCA analysis, but caused by not using emission lines in the selection.", "It would be equally true of any PSB selection method that does not use emission lines.", "We will return to these points in the discussion section below." ], [ "Evolution of radial gradients in spectral properties", "To analyse the spatial properties of the PSB remnant in the 2xSc_07 simulation, we create datacubes for the snapshots between $t= 12.56$  Gyr, and $t= 13.08$  Gyr, from the early quenching stage to the late quenching (i.e.", "the time range between the blue and red lines in the top panel of fig:galpro).", "To simplify the analysis we bin the spectra in the mock datacubes according to their distance to the galaxy centre, in annuli of 0–1 kpc, 1–2 kpc, 2–3 kpc, and 3–4 kpc respectively, and calculate the spectral indices from the binned spectra.", "The evolution in the radial gradients of the spectral indices are plotted in fig:gradient.", "As the H$\\alpha $ emission line is very sensitive to the fluctuating residual SFR in the post-merger galaxies, the radial gradient in H$\\alpha $ -H$\\delta _{\\rm A}$ fluctuates substantially with time during the PSB phase.", "There is a very strong radial gradient in the Balmer absorption lines at all snapshots, with the central region showing the strongest Balmer absorption lines.", "In all but a few snapshots, the central region also shows the strongest nebular emission lines.", "In PCA space the evolution is more stable with time, evolving rapidly from a positive to negative gradient, with the galaxy entering the fiducial selection box at the same time for all radii.", "In both spectral index figures, only the inner regions of the galaxy are clearly identified as post-starburst, while the outer regions move from star-forming towards quiescent stellar populations.", "The ability of the simulation to reproduce the very different radial gradients observed in different PSB galaxies shown in fig:observations suggests that the difference in observed gradients may simply be due to the time at which we catch the galaxy following coalescence.", "In Section we noted that the radial gradients could be produced by a single co-eval burst which was stronger in the central regions, or a starburst that has progressed from outside-in.", "However, simple toy models are unable to distinguish between these two hypotheses (Weaver et al.", "in prep.).", "In fig:SFHradii we show the star formation history of the different radial annuli, taken from the final snapshot shown in fig:gradient with $t=13.08$  Gyr.", "The star formation rates peak at roughly the same time at all the radii but the burst is both stronger and more prolonged at the galaxy centre i.e.", "a combination of the co-eval starburst and the outside-in starburst hypothesese.", "While the peaks of the star formation are co-eval, the star formation is quenched in the outer regions more quickly than in the inner regions, presumably due to the rapid gas flows continuing to feed the central regions." ], [ "Spectral index maps", "For completeness, we present the 2D maps of one snapshot of merger 2xSc_07 in fig:mockmaps.", "The snapshot was chosen for the maximal strength of its post-starburst spectral features.", "We see that the spatial distributions of the spectral indices are entirely smooth, with very little residual structure remaining following the catastrophic event that caused them." ], [ "Discussion", "A better model of the physical mechanisms responsible for creating post-starburst galaxies may help to improve our understanding of how and why star formation completely quenches in galaxies throughout cosmic time.", "Through a careful comparison of real observations from state-of-the-art IFU surveys and mock observations from advanced simulations, we hope to be able to constrain the external and internal processes responsible for quenching star formation.", "The galaxy merger simulations presented in this paper were designed to reproduce low redshift central post-starburst galaxies, where there is significant evidence that major mergers are the only plausible mechanism for the creation of the strongest spectral features and the high fraction of morphological disturbance (see Section ).", "However we have shown that with the galaxy merger simulations presented here it is difficult to create a sharp enough quenching event that leads to both the strong Balmer absorption lines and the complete absence of nebular emission lines due to ongoing star formation.", "While the strong gas inflow is required to drive the starburst, it is hard to completely shut off the star formation in the centre of the merger remnant without very significant BH feedback, for which there is limited evidence for galaxies in the very local Universe .", "Whether this is due to the limited size of our simulation set, or limitations in the sub-resolution physics of the models is unclear.", "Comparison with a full cosmological hydrodynamical simulation would address the former concern, however the latter is much more difficult to tackle: the simulations presented here use very similar recipes to the EAGLE simulation, and most of the currently available simulations do not differ significantly in any of the key ISM or feedback aspects.", "One obvious concern is the observed presence of cold gas in a large fraction of PSBs which appears to have very low star formation efficiencies , , , .", "For simulations that assume a Kennicutt-Schmidt style star formation law, cold gas forms stars, without any exceptions.", "In addition, our simulations assume a relatively smooth dust distribution, while it is possible that the observed EW of emission lines is reduced by a highly clumpy dust distribution.", "Improved recipes for treating the detailed sub-resolution physics of the ISM in these unusual galaxies may thus be required.", "Our results clearly disagree with the conclusions of and : with the improved simulation resolution and sub-resolution physics recipes we find that stellar feedback is only able to cause the initial decline of SFR needed to halt the starburst.", "Without BH feedback the galaxies return to the blue-cloud or green valley, but cannot further quench the star formation to become entirely quiescent.", "This leads us to the question of whether rapid quenching is related to mergers at all.", "While it is hard to envisage an alternative scenario in the local Universe, the higher gas mass fractions of galaxies at higher redshift may provide the fuel needed for galaxy wide starbursts, without the need for strong centralised inflows which are difficult to halt.", "used the SIMBA cosmological simulations to show that in most cases the quenching of the galaxy was not related to a recent major merger, over a wide range of cosmic time.", "A key additional constraint not investigated here is the morphologies of the remnants: highly compact at high redshift , [3] and tidally disturbed at low redshift , , .", "carried out a first look at the morphological asymmetries of mock images created from very similar simulations to those used in this paper, finding them to decay very rapidly at SDSS-like image depths.", "Further progress may be possible with deeper imaging datasets, such as will become available in the near future with the Vera C. Rubin Observatory.", "With the advent of IFU datasets, the differing radial gradients in the spectral indices of PSB galaxies have become apparent , .", "A negative gradient in the H$\\delta $ absorption line strength (i.e.", "stronger absorption in the centre) as seen in the vast majority of central PSBs , has commonly been suggested to be an indication of a merger origin due to the gas inflows leading to an excess of A and F stars in the centre.", "This has been borne out by merger simulations, including those presented here , .", "Galaxies with PSB spectral features only in their outskirts (so-called `ring' PSBs) are not produced at all by our suite of simulations, and are more likely caused by external processes which have disrupted the outer disc star formation .", "Interestingly, by decoupling the excess Balmer absorption from the past averaged star formation rate, the radial gradients in PCA spectral index space show an intriguing feature, starting out positive, flattening and then becoming increasingly negative with time.", "This should provide an excellent way to measure the age of PSB galaxies, independent of obtaining precise star formation histories, and suggests that the three different gradients observed in MaNGA galaxies in Section could well be caused by the same physical process caught at different observed times.", "Throughout this paper, we have combined the analysis of two very different spectral index spaces: the equivalent widths of the H$\\alpha $ emission line and Balmer absorption lines, and the PCA indices of which focus on the stellar continuum alone.", "The sensitivity of the H$\\alpha $ emission line to residual ongoing star formation, which in our simulations fluctuates wildly dependent on the fluctuating BH feedback strength, provides a natural explanation for the much larger number of PCA-selected PSBs than traditional selection .", "As stated previously, the PCA selection is insensitive to a small amount of residual star formation, which provides an important complementary and more inclusive approach to PSB selection.", "However, it is evident from fig:observations that the PCA-selection does not identify all observed PSBs, with 2/3 of the example MaNGA PSBs selected using the traditional method falling outside the PCA PSB region.", "According to their gradients in PC space, these are likely younger objects, where the Balmer absorption has not yet had time to strengthen and 4000Å break strength to increase sufficiently for PCA selection (i.e.", "the PCA selection box as used in this paper selects only older PSBs, as noted already previously in ).", "The two objects with slightly positive or flat gradients in PC1/2 highlight yet another discrepancy with the simulated mergers: the star formation must have been quenched incredibly rapidly for these to be identified as PSBs through their lack of H$\\alpha $ , and such rapid quenching of H$\\alpha $ is never observed in our simulations.", "This would require an immediate impact of the BH feedback, which as we see from the top panel of fig:galpro takes a little while to get going following the coalescence of the BHs.", "This again suggests an important missing ingredient to the sub-resolution star formation recipes in these simulations.", "Finally, we note that the radial gradients in the spectral indices are caused by a stronger, longer duration starburst occurring in the central regions of the galaxy.", "The starburst in the outer regions happens co-evally with that in the centre, i.e.", "not supporting either inside-out nor outside-in growth scenarios.", "The quenching occurs first in the outer regions, however, supporting outside-in quenching for these extreme galaxies." ], [ "Summary", "We use Gadget-3 to run a set of binary merger simulations with different black hole feedback models, progenitor galaxies, and orbits.", "We develop the SEDMorph code to build mock SDSS-like spectra for the simulated galaxy, by combining the star formation history and metallicity of each particle with stellar population synthesis models.", "We create mock datacubes following the MaNGA observational strategy including a PSF and dithering pattern.", "The spatial distribution of H$\\alpha $ emission, Balmer absorption lines, and stellar continuum shape indices are investigated in the post-merger galaxies, to diagnose recent and ongoing star formation.", "A summary of our findings is as follow: To create mock MaNGA datacubes we found it was necessary to strictly follow the MaNGA observation strategy.", "Short cuts such as gridding data directly, or convolving with the PSF after data gridding produce significant residuals or larger computational resources compared to a full treatment.", "To completely shut down the star formation in our model PSB galaxies, mechanical AGN feedback is required to expel the gas from the galaxy, while the more traditional thermal feedback BH model is not sufficiently efficient.", "The star formation histories of the merger simulations demonstrate that sharp quenching leading to PSB-like remnants is only achieved in particular circumstances: progenitor galaxies with similar mass, approaching each other in either prograde-prograde or retrograde-prograde orbits.", "Neither unequal-mass mergers nor retrograde-retrograde mergers lead to quenching that is significant or rapid enough to lead to PSB spectral features.", "The traditional PSB selection method, which identifies PSB galaxies via their absence of nebular emission lines as well as strong Balmer absorption lines, is highly sensitive to the complete shut down in star formation, which in turn appears to be sensitive to the accretion rate of the BH shortly before the time of observation.", "A stellar continuum based method, such as the PCA method used here, is much less sensitive to rapid fluctuations in the SFR of the galaxy and therefore is likely to lead to more complete samples.", "However, two of the three example MaNGA galaxies presented in this paper are not selected by the PCA method as their H$\\alpha $ emission has shut off before the Balmer absorption lines are strong enough to be identified cleanly by the PCA.", "Such a rapid shut off in star formation following the starburst is not found in any of our merger simulations.", "Combined with the fact that many PSBs are known to have cold gas that is not forming stars efficiently, this points to a missing ingredient in the sub-resolution star formation recipes or the ISM structure employed in these simulations.", "In agreement with previous work, the simulated post-starburst galaxies show a strong radial gradient in the Balmer absorption line strength, with stronger absorption in the inner region, as seen in the majority of local PSB galaxies where the integrated light is dominated by the central PSB region.", "This appears to be a defining feature of merger-origin PSBs, caused by the inflow of gas to the central regions.", "In PCA space an evolution in the radial gradient becomes apparent, that is masked by traditional methods using nebular emission lines due to their sensitivity to fluctuations in ongoing star formation.", "This indicates that the range of gradients observed in MaNGA PSB galaxies is simply due to different times of observation rather than different underlying processes.", "Our simulations show that the galaxies undergo a single co-eval burst which was stronger and longer lived in the central regions.", "This does not support either inside-out nor outside-in growth during the star-formation episode, but rather outside-in quenching.", "Clearly much more remains to be understood about the formation of PSB galaxies, with this paper raising questions about the effectiveness of the implementations of BH feedback in the current generation of hydrodynamic simulations.", "Comparison with cosmological simulations may help us to understand the range of different orbits and gas properties as a function of stellar mass and epoch; however they are still fundamentally limited by resolution as well as the employed sub-resolution star formation and stellar feedback recipes.", "Observationally, further progress on understanding the causes of both the starburst and final quenching in PSB galaxies may come from combining the analysis of radial gradients in spectral indices with the morphology in deep imaging data, or stellar kinematics from high quality IFU data." ], [ "Acknowledgments", "The authors would like to thank Ena Choi for helpful discussions and guidance on the mechanical BH models, thank Ariel Werle for helping performing PCA with Python, and Anne-Marie Weijmans for suggestions on Appendix .", "YZ acknowledges support of a China Scholarship Council - University of St Andrews Scholarship.", "VW and NJ acknowledge support from the European Research Council Starting Grant SEDMorph (P.I.", "V. Wild).", "NL and PHJ acknowledge support by the European Research Council via ERC Consolidator Grant KETJU (no.", "818930) Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions.", "SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah.", "The SDSS web site is www.sdss.org.", "SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.", "The related simulation data, including snapshots, black hole activity, mock spectra and mock MaNGA datacube are available at: https://doi.org/10.17630/ff244265-6540-494e-af3e-0969fdc5ff24.", "The related MaNGA data are available at the SDSS data base(https://www.sdss.org/dr16/).", "Other data underlying this article are publicly available from the web as listed in the footnote." ] ]

[ [ "Fantastic flat bands and where to find them: The CoSn-type compounds" ], [ "Abstract Quantum interference on the kagome lattice generates electronic bands with narrow bandwidth, called flat bands.", "Crystal structures incorporating this lattice can host strong electron correlations with non-standard ingredients, but only if these bands lie at the Fermi level.", "In the six compounds with the CoSn structure type (FeGe, FeSn, CoSn, NiIn, RhPb, and PtTl) the transition metals form a kagome lattice.", "The two iron variants are robust antiferromagnets so we focus on the latter four and investigate their thermodynamic and transport properties.", "We consider these results and calculated band structures to locate and characterize the flat bands in these materials.", "We propose that CoSn and RhPb deserve the community's attention for exploring flat band physics." ], [ "Introduction", "The behavior of electrons in solids is strongly determined by the constituent atoms and the lattices on which they are arranged [1], [2].", "In some lattices (e.g.", "dice, Lieb, kagome), destructive quantum interference of different hopping paths generates localized electronic states in otherwise itinerant systems [3], [4].", "The resulting energy bands have narrow bandwidths and appear as nearly horizontal dispersion plots, so-called flat bands (Fig.", "REF c).", "The interaction of relatively localized states drives some of the most curious phenomena in condensed matter systems; magnetism, unconventional superconductivity, and heavy-fermion physics [2].", "Flat bands can be built with elements not usually associated with magnetism or localized states because the localization arises from the connectivity of the lattice.", "This provides platforms to explore strongly correlated physics with unorthodox ingredients if these flat bands are at the Fermi level.", "An illustrative example is superconductivity and correlated ground states in twisted bilayer graphene which arise from flat bands [5], [6].", "In real materials, the localized character of flat bands is modified by other couplings.", "Notably, the spin orbit coupling can result in non-trivial band topology [7].", "Figure: a, kagome lattice with shaded CoSn unit cell.", "b, the CoSn structure with labeled sites drawn in Vesta .", "c, tight binding dispersion of kagome lattice from Ref. .", "d, CoSn Brillouin zones with labeled high symmetry points.The kagome lattice (Fig.", "REF a) can generate flat bands (Fig.", "REF c) due to hopping interference [10], [11], [12].", "It is composed of a pretty arrangement of hexagons and triangles depicted in Fig.", "REF  a.", "This lattice can be found in Fe$_3$ Sn$_2$ , Co$_3$ Sn$_2$ S$_2$ , and compounds with the HfFe$_6$ Ge$_6$ or CoSn structures.", "CoSn adopts a hexagonal B35 structure type depicted in Fig.", "REF b with space group $P6$ /$mmm$ (No. 191).", "It is composed of a kagome lattice of Co atoms (Wyckoff site $3f$ ) with a Sn1 (site $1a$ ) centered in the hexagons.", "These layers are stacked along the $c$ -axis and separated by honeycomb layers of Sn2 on site $2d$ .", "In CoSn, cobalt atoms generate multiple copies of the flat band states, each associated with different $d$ -orbitals [12].", "In total, six intermetallic compounds have been reported with the CoSn structure type [13], [14]: hexagonal-FeGe [15], FeSn [16], CoSn [16], NiIn [17], RhPb [18], and PtTl[19].", "In all but PtTl, the transition-metal is reported to take the Co site forming the kagome lattice with $p$ -block element occupying the $1a$ and $2d$ sites.", "The site assignment in PtTl has not been determined [19], but in section REF we argue this compound is ordered and Pt occupies the $3f$ Wycoff site.", "We divide these metals into two groups based on valence electron count.", "The antiferromagnets FeGe and FeSn are composed of Fe (group 8) and a group 14 element suggesting 22 valence electrons per formula unit (F.U.).", "The other four compounds have one additional electron, totaling 23.", "CoSn and RhPb have elements from groups 9 and 14 while NiIn and PtTl are pairs from groups 10 and 13.", "For simplicity, we will refer to these two sub-families as the 22 and 23 electron compounds, respectively.", "The antiferromagnetism of FeGe and FeSn has been thoroughly examined [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [28], [34].", "A series of recent angular resolved photo-emission spectroscopy (ARPES) studies have investigated the electronic bands arising from the kagome lattice in FeSn [10], [35] and CoSn [11], [12].", "The remaining three compounds are poorly explored and the character of their flat bands is unknown.", "In this family of compounds, we have $3d$ , $4d$ , and $5d$ transition-metals represented.", "We have an opportunity to examine trends across the periodic table with increasing period and spin orbit coupling strength.", "In FeGe and FeSn, the Fermi level lies within the flat bands and they reduce degeneracy by becoming antiferromagnets.", "Electron doping FeSn by swapping Co for Fe systematically suppresses magnetic order [30], [36].", "Rh and Pt are rarely magnetic.", "If the Fermi level can be tuned to the flat bands in RhPb and PtTl by hole doping, how will the systems reduce degeneracy?", "Will we find magnetism, charge density waves, or superconductivity?", "In this paper we want to lay the groundwork for exploring flat band physics in the CoSn-type kagome metals.", "To this end, we have two main goals.", "First, to present and compare the thermodynamic and electrical transport properties of CoSn-type compounds.", "We focus on the four 23 electron compounds; CoSn, NiIn, RhPb and PtTl.", "Second, to discuss the energy and character of the flat bands in these kagome metals in light of our data and density functional theory (DFT) results.", "In section REF , we recount our procedures for synthesizing the six compounds and we describe the products in section REF .", "Next, we present the low-temperature lattice parameters, thermal expansion, heat capacity, magnetic susceptibility, and resistivity of these metals in sections REF through REF .", "We do not observe any phase transitions in CoSn, NiIn, RhPb or PtTl down to 1.9 K. In the calculated band structures of these four (section REF ), we note that CoSn has flat bands closest to the Fermi energy ($\\varepsilon _\\mathrm {Fermi}$ ).", "This is supported by CoSn’s unusual magnetic susceptibility which we model as temperature dependence of Pauli paramagnetism resulting from flat bands about 20 meV below $\\varepsilon _\\mathrm {Fermi}$ (section REF ).", "Finally, we compare the relative attractiveness of these compounds as platforms for exploring flat band physics and provide broader guidelines for other systems in section REF .", "Single crystals of CoSn, FeSn, RhPb and PtTl were grown from molten metal fluxes.", "CoSn and FeSn were both grown from liquid Sn, RhPb from Pb and PtTl from Tl.", "Polycrystalline FeGe and NiIn was synthesized by arc melting the constituent elements and annealing.", "FeGe There are three polymorphs of FeGe with increasing temperature; hexagonal with the B35 CoSn structure, cubic with a B20 FeSi structure, and monoclinic [15].", "Crystals of each phase can be obtained by chemical vapor transport.", "We were unable to obtain the hexagonal phase and chose to create the hexagonal phase by solid state reaction instead.", "Equimolar iron granules (Alfa Aesar 99.98%) and germanium pieces (Alfa Aesar Puratronic 99.9999+%) totaling 2 g were arc melted together in argon.", "The iron pieces were placed on top of the Ge to prevent the latter shattering due to thermal shock in the arc.", "The button was flipped and remelted 3 times.", "Powder x-ray diffraction on this material showed a mix of FeGe$_2$ and Fe$_{6.5}$ Ge$_4$ .", "Pieces of the arc melted button were annealed on alumina in an argon filled, fused-silica ampoule.", "This assembly was annealed at 700°C for 112 h then quenched into water.", "X-ray diffraction revealed that it was composed of mostly hexagonal FeGe with about 5 wt.% Fe$_6$ Ge$_5$ .", "FeSn Crystals of FeSn were grown from Sn flux as in Refs. Sales20192DMagnetismFeSn,Meier2019AFMReorientationCoDopedFeSn.", "Iron granules (Alfa Aesar 99.98%) and tin shot (Alfa Aesar, Puratronic 99.9999%) in an atomic ratio of Fe : Sn = 2 : 98 were loaded into a 10 mL alumina crucible and capped with a second upside-down crucible loosely filled with silica wool to catch the crystals.", "These crucibles were expertly sealed in a fused-silica ampoule under vacuum by the ORNL glass shop.", "The sealed silica ampoule was heated in a box furnace to 1100°C at 120°C/h and held for 12 h, then quickly cooled at 6°C/h to 1000°C and held for 48 h. During this hold, the furnace was opened and the ampoule was shaken with tongs to mix the melt [29].", "Then the furnace is cooled quickly at 6°C/h to 800°C and held for 1 h. Finally we slowly cool to about 600°C at 1°C/h to grow the crystals.", "The ampoule was removed from the hot furnace, placed upside down in a centrifuge and spun.", "This “spinning\" process is intended to fling the liquid flux off the crystals and essentially quenches the ampoule assembly in air.", "CoSn Crystals of CoSn were prepared in a 31 g batch with cobalt pieces (Alfa Aesar 99.95%) and tin shot (Alfa Aesar Puratronic 99.9999%).", "An atomic ratio of Co : Sn = 8 : 92 was assembled in the same 10 mL alumina crucible setup as FeSn.", "The ampoule assembly was heated in a box furnace to 1130°C at 120°C/h, held for 24 h, then cooled at 6°C/h to 900°C and held for 1 h. Then the furnace was slowly cooled to 618°C at 1°C/h.", "The ampoule was then removed and centrifuged to separate the liquid from the crystals.", "NiIn The reported Ni-In phase diagrams suggest NiIn cannot be precipitated from a Ni-In liquid [37], [38].", "We obtained polycrystalline material by arc melting equiatomic nickel and indium metal (totaling 1.9 g) together then annealing.", "Indium and its alloys can stick to the copper hearth plate of the arc melter.", "To mitigate this, Ni slugs (Alfa Aesar 99.995%) and In shot (Alfa Aesar 99.9995%) were wrapped in a piece of nickel foil (Alfa Aesar 99.95%) to prevent direct contact between the indium and Cu hearth-plate.", "This procedure worked well but arc power needed to be kept low to reduce the vaporization of indium evident by its dark-blue arc (which gave indium its name[39]).", "The pellet was melted and flipped 5 times.", "Powder X-ray diffraction revealed the product was mostly NiIn with minor Ni$_{2}$ In$_{5}$ and Ni$_{13}$ In$_{9}$ .", "Annealing in an argon-filled fused-silica ampoule at 725°C for 134 h removed any trace of the diffraction peaks from these impurities.", "RhPb Single crystals of RhPb were grown from a lead flux in a 5.4 g batch.", "Rhodium sponge (Alfa Aesar 99.95%) and lead slugs (Alfa Aesar Puratronic 99.999%) were placed in a 2 mL alumina Canfield Crucible Set (CCS) [40] with an atomic ratio Rh : Pb = 1 : 2 based on the binary phase diagram [18], [41].", "The crucibles were sealed in a fused silica ampoule under vacuum using a oxygen-hydrogen torch.", "The assembly was heated in a box furnace over 6 h to 1000°C or 1100°C and held for 2 h, then quickly cooled to 900°C over 4 h. To grow the crystals, the furnace was cooled slowly to 750°C over 320 h. The remaining Pb-rich liquid was separated from the crystals by spinning in a centrifuge.", "PtTl Single crystals of PtTl were obtained from a melt with an atomic ratio of Pt : Tl = 1 : 3 based on existing phase diagrams [42], [43].", "Inside a He glove box, pieces of Pt sheet (Alfa Aesar 99.9%) and chunks of Tl (Alfa Aesar 99.999%) totaling 6.6 g were loaded into a 2 mL alumina CCS.", "The crucibles were assembled in a fused silica tube and a valve for our gas manifold was attached and closed in the helium atmosphere.", "This procedure reduced the exposure of thallium metal to air while sealing the tube with a torch.", "Our first reaction of Pt and Tl had a disturbing result.", "We heated the ampoule to 1050°C for a 24 h hold to dissolve the platinum.", "When the assembly was removed from the furnace to spin in the centrifuge the inner surface of silica ampoule had darkened dramatically.", "We assume this is due to a reaction with the thallium vapor and could cause the ampoule to break and release the poisonous vapor.", "To reduce the risk associated with thallium weakening the silica ampoule, we urge the reader not to exceed 1000°C and to use the following furnace schedule that worked just as well.", "In addition, the furnace should be placed in a fume hood.", "The furnace was heated at 120°C/hr to 1000°C for a 24 h hold, quickly cooled at 6°C/h to 900°C followed by a 1 h hold.", "To grow crystals, the furnace was slowly cooled toward 500 °C at 2 °C/h.", "At 527 °the ampoule was removed and spun in a centrifuge to fling the thallium-rich melt from the crystals." ], [ "Measurements", "Powder x-ray diffraction measurements were preformed on a PANalytical X'pert Pro diffractometer with a Cu K$\\alpha $ tube and an incident beam monochrometer.", "Low-temperature diffraction measurements were done in an Oxford PheniX closed-cycle helium cryostat.", "Lattice parameters were determined from fits of powder x-ray diffraction data with HighScore Plus.", "The high temperature diffraction data were collected using an Anton Paar XRK900 reactor stage on an PANalytical Empyrian diffractometer employing Cu K$\\alpha $ radiation under flowing UHP He gas.", "The stage height was initially aligned using computer controlled height adjustment, and the height was subsequently auto-corrected for thermal expansion of the Macor sample holder during the measurement.", "Heat capacity of each sample was measured with Apiezon N-grease using the heat capacity option of a Quantum Design Physical Property Measurement System (PPMS) or DynaCool PPMS.", "Magnetization measurements were performed at 10 kOe in a Quantum Design Magnetic Property Measurement System (MPMS) in plastic drinking straws.", "Resistivity vs temperature measurements were also performed in the PPMS using the ac-transport option.", "Both single crystals and polycrystalline samples were polished into bars and 4 platinum wires were attached with silver epoxy (EPO-TEK H20E)." ], [ "Density functional theory", "Electronic band structures and density of states were calculated for CoSn, NiIn, RhPb, and PtTl based on density functional theory (DFT) with Perdew-Burke-Ernzerhof exchange-correlation functional [44], as implemented in the VASP code [45].", "Spin orbit coupling was included in these calculations self-consistently.", "The interaction between ions and electrons was described by the projector augmented wave method [46].", "The kinetic energy cutoffs for the plane-wave basis are 350 eV for CoSn, NiIn, and RhPb and 308 eV for PtTl.", "12$\\times $ 12$\\times $ 12 and 20$\\times $ 20$\\times $ 20 $k$ -point meshes were used for structural optimization and density of states calculations, respectively.", "The optimized lattice parameters reported in table REF compare well with the measured lattice parameters at 15 K." ], [ "Products", "The crystals of FeSn, CoSn, RhPb and PtTl obtained by flux growth (Fig.", "REF ) were generally blocky, hexagonal prisms often with mirror-like faces and a bright metallic luster.", "Crystals of CoSn tended to be more elongated and had a subtle blue-gray hue.", "The annealed polycrystalline sample of FeGe had a metallic luster and a pale gray color.", "Notably, the annealed polycrystalline NiIn was metallic with an unusual pale pink color.", "This doesn't appear to be a surface film as it is still present on freshly broken surfaces and the ground powder had a distinct violet tint.", "Both NiIn and PtTl are in a group of intermetallics expected to be colored like red PdIn and purple AuAl$_2$[47], [48].", "All compounds are stable in air for months.", "Crystals of FeSn, CoSn, RhPb, and PtTl exhibit good (001) cleavage." ], [ "Thermal Expansion", "The lattice parameters of the kagome metals are reported in table REF and the thermal expansion of the 23 electron compounds are presented in Fig.", "REF .", "NiIn shows nearly isotropic relative expansion, but the other three show notably faster expansion along the $c$ -axis.", "The room temperature thermal expansion coefficients, $\\alpha $ , in table REF were calculated using the slope of a linear fit of $a(T)$ and $c(T)$ between 200 and 300 K. Figure REF emphasizes how antiferromagnetism leads to strikingly different lattice evolution in the 22 electron metals FeGe and FeSn.", "In contrast to the 23 electron metals, these compounds show smaller expansion along the $c$ -axis than $a$ .", "Strong magnetic coupling between the kagome sheets of Fe [29] seems to resist thermal expansion along the $c$ -axis.", "The inset shows that the thermal $c$ -axis expansion of FeSn is suppressed until magnetic order melts at the Néel temperature evident as change in slope near 365 K. A moment reorientation near 100 K appears as an anomaly in the FeGe data." ], [ "Heat Capacity", "The heat capacity vs temperature of the CoSn-type compounds is depicted in Fig.", "REF .", "The curves shift to lower temperatures with increasing molar mass from FeGe $\\rightarrow $ (FeSn, CoSn, and NiIn) $\\rightarrow $ RhPb $\\rightarrow $ PtTl, reflecting a decrease in the Debye temperature, $\\theta _\\textrm {D}$ (Table REF ).", "The inset in this figure shows low-temperature fits used to estimate the Sommerfeld coefficient, $\\gamma $ .", "These values are less certain for FeGe due to a subtle feature around 2.5 K. We attribute this to a minor second phase observed in the polycrystalline sample.", "Despite this, the antiferromagnets FeGe and FeSn have markedly larger $\\gamma $ 's than compounds without order.", "The values of $\\gamma $ we obtain for FeGe, FeSn, and CoSn are comparable to those estimated from low-temperature heat capacity in Ref.", "Larsson1974SpecificHeatFeGeFeSnCoSn (8.46(2), 10.95(1), and 4.57(3) mJ/mol f.u.", "K$^2$ respectively).", "They obtained different estimates of $\\theta _\\textrm {D}$ (341, 303, and 299 K respectively) from their more detailed analysis.", "FeGe and FeSn are both antiferromagnetic across the entire measured temperature range[21], [20], [26], [23], [29], [50], [28].", "The moment reorientation of FeGe is evident as a change in slope of $C_p(T)$ around 100 K [21], [20], [23]." ], [ "Magnetic susceptibility", "The magnetic susceptibility data for the CoSn-type compounds held some surprises for us.", "We anticipated relatively isotropic and temperature-independent susceptibility for the 23 electron materials.", "Figure REF reveals the situation is more complicated.", "Polycrystalline NiIn shows the smallest response with weak temperature dependence.", "RhPb and PtTl are diamagnetic and show only a subtly rising $\\chi (T)$ apart from a minor Curie contribution at low temperatures.", "CoSn shows a stronger Curie tail and an unusual rising susceptibility contribution above 50 K which is nearly identical parallel and perpendicular to $c$ -axis.", "Finally, we observe an essentially temperature-independent difference between the $c$ -axis and in-plane susceptibility of CoSn, RhPb, and PtTl (see Table REF ).", "In all three cases the susceptibility is smaller with ${H}\\parallel {c}$ and the difference is the same magnitude as the powder averaged susceptibility.", "Figure REF illustrates the dramatically weaker magnetic response of CoSn relative to the antiferromagnets FeGe and FeSn.", "This relatively small susceptibility suggests that magnetic order has no role in the magnetic response of CoSn, NiIn, RhPb or PtTl.", "The anisotropic susceptibility of FeGe [21], [20], [23] and FeSn[29], [50], [28] has been reported and explored elsewhere.", "The Néel transition of FeSn is evident at 365 K, as is the moment reorientation of FeGe near 100 K [21], [20], [23].", "We attribute the clear Curie contribution to the FeGe data below 60 K to the second phase in the sample." ], [ "Resistivity", "The anisotropic resistivity vs temperature, $\\rho (T)$ , of FeSn, CoSn, RhPb and PtTl are presented in Fig.", "REF along with the resistivity of polycrystalline FeGe and NiIn (dashed).", "The most obvious common characteristic of these anisotropic data is that the in-plane resistivity rises more quickly with temperature than it does for the $c$ -axis.", "In addition, the residual resistivity values in table REF are markedly small (order 1 µ$\\Omega $  cm).", "There are two outliers in this respect; CoSn with ${J}\\perp {c}$ and polycrystalline FeGe.", "In the latter case we believe this large residual resistivity value is also a result of our impure sample and grain boundaries.", "Finally, we note that there are no discontinuities or changes in slope in any of the $\\rho (T)$ curves suggestive of phase transitions." ], [ "Band structure", "The DFT band structures calculated for the four compounds with 23 valence electrons are presented in Fig.", "REF .", "The orbital character of each state is represented by the width of the colored lines on each band.", "The density of states, $D$ , of each compound is plotted along the right side of each panel.", "In these band structures we can identify replicas of the kagome dispersion (see Fig.", "REF c) for each $d$ -orbital character (indicated by color) at the $\\Gamma -M-K$ and $A-L-H$ planes (see Fig.", "REF  d).", "These are most distinct for CoSn, as noted in Refs.", "[11] and [12].", "The flat bands nearest the Fermi energy are of greatest interest.", "We estimate the position and extent of these bands across the Brillouin zone with red and green bars in the density of states plots (for $d_{xy}, d_{x^2-y^2}$ and $d_{xz}, d_{yz}$ flat bands respectively).", "These generally line up with the peaks in the density of states and are accompanied by the estimated bandwidth (in eV).", "None of the compounds have flat bands at the Fermi energy.", "We wish to compare three characteristics of the flat bands: bandwidth, band energy, and distinctness.", "Bandwidth is important as it determines the strength of electron correlations.", "First we note that the green $d_{xz}, d_{yz}$ flat bands generally are narrower than the red ($d_{xy}, d_{x^2-y^2}$ ).", "This is more pronounced for NiIn and PtTl where the $d_{xy}, d_{x^2-y^2}$ band nearest the Fermi energy disperses strongly in the $A-L-H$ plane.", "In addition, the heavier compounds also show relatively larger bandwidths.", "These broader bands are also reflected in the small magnitude of the corresponding peaks in the density of states.", "To explore flat band physics the bands of interest must be tuned to the Fermi level.", "Therefore, compounds with flat bands at smaller binding energies will require less doping.", "CoSn has the nearest flat bands starting at -0.13 and -0.38 eV.", "These shift to progressively lower energies in RhPb, NiIn, and PtTl.", "The band locations are also clearly visible in the position of peaks in the density of states.", "Finally, the “distinctness\" of the flat bands is a qualitative measure of how independent they are from other bands.", "For this purpose, a distinct band is one that maintains its orbital character (color) and small dispersion over most of the Brillouin zone.", "By this criteria, we would say that the green $d_{xz}, d_{yz}$ band at -0.5 eV in CoSn is the most distinct.", "Broadly, the CoSn and NiIn have more distinct flat bands than their heavier counterparts.", "In PtTl the bands are constantly swapping orbital character and it becomes difficult to identify any features of the kagome lattice dispersion." ], [ "Discussion", "Our data indicates that CoSn, NiIn, RhPb, and PtTl undergo no phase transitions below room temperature.", "Bulk superconductivity was not observed in any of the six compounds down to 1.9 K. In FeGe and FeSn the large density of states from the $3d$ -bands lies at the Fermi energy, facilitating robust itinerant antiferromagnetism [51].", "The other four compounds have an additional valence electron per formula unit, which fills these bands.", "The reduced density of states at $\\varepsilon _\\mathrm {Fermi}$ (Fig.", "REF ) is likely why magnetic order is not observed in these compounds.", "In fact, antiferromagnetism is systematically suppressed as electron count increases across the (Fe$_{1-x}$ Co$_x$ )Sn series [30], [36].", "The anisotropic resistivity of FeSn, CoSn, RhPb, and PtTl clearly show electronically 3-dimensional metals.", "We suspect smaller resistivities along the $c$ -axes are connected to the strongly dispersing bands passing the Fermi level along the $\\Gamma -A$ line in Fig.", "REF ." ], [ "Site order in PtTl", "PtTl presents a unique challenge for structural refinement.", "Its two constituents have relatively little difference in atomic number ($_{78}$ Pt $_{81}$ Tl).", "This creates significant uncertainty in the assignment of elements on the 3 sites in the CoSn structure and Zintl did not suggest a site assignment in the original report [19].", "We suspect that the transition metal, Pt, occupies only the Co-site (3f) forming the kagome lattice as observed in all the other members of this family.", "We have two pieces of evidence for this configuration.", "The small residual resistivity of PtTl suggests it is an ordered compound.", "If it was a random alloy (Pt and Tl equally likely on all sites), we estimate a residual resistivity of order 30-100 µ$\\Omega $  cm (using method in Refs.", "[53], [52], [54]).", "This is significantly larger than the measured 0.23 and 0.815 µ$\\Omega $  cm along the $c$ -axis and in-plane, respectively.", "In addition to this resistivity argument, first principals calculations suggest that PtTl with an inverted CoSn structure (with Tl forming the kagome lattice) has a formation energy of 5.25 eV/unit cell higher.", "Anti-site defect pairs (one pair of Pt and Tl atoms swapped) in the standard CoSn structure are also unlikely with formation energies of 2.20 eV.", "Therefore, we believe that PtTl is likely well ordered with the CoSn structure and Pt forming the kagome lattice." ], [ "Susceptibility model", "Lets explore our magnetic susceptibility data for these kagome metals in greater depth as it provides additional insights into the electronic bands structure of these materials.", "Consider some common contributions to a metal's magnetic response: Core diamagnetism Landau diamagnetism Pauli paramagnetism Curie-Weiss Ordered moments First, the antiferromagnets FeGe and FeSn are dominated by the last two contributions giving them relatively large, positive susceptibilities (Fig.", "REF ).", "Magnetic order is not evident in the other four compounds so we suspect that the small Curie contributions to the susceptibility at low-temperature (Fig.", "REF ) arise from impurity phases or defects in the crystals.", "Each of these compounds should also have a core diamagnetic contribution we estimate based on the noble gas cores in each compound in Table REF from a table in Ref. [1].", "The most striking feature in Fig.", "REF is the pronounced anisotropy between the ${H}\\parallel {c}$ and $\\perp {c}$ data.", "The difference is essentially temperature-independent and the susceptibility along the $c$ -axis is always smaller.", "In the absence of ordered moments, we ascribe this anisotropy to Landau diamagnetism.", "For a spherical Fermi surface, this is expected to be isotropic and proportional to the density of states at the Fermi energy[1], [2].", "Anisotropy can arise when the totaled magnetization contributions of Landau-orbits on non-spherical Fermi surfaces are strongly dependent on field direction[55].", "This had little bearing on our investigation of flat bands so we did not investigate further.", "The temperature dependence of CoSn's susceptibility data is markedly different from the others in Fig.", "REF .", "It shows a smoothly increasing, S-shaped $\\chi (T)$ on heating above 50 K. Critically, this contribution is basically identical in both directions.", "We attribute this isotropic susceptibility enhancement in CoSn to additional Pauli paramagnetic contributions of kagome flat bands near the Fermi energy.", "In brief, as temperature increases, electron states farther from the $\\varepsilon _\\mathrm {Fermi}$ are thermally depopulated and begin contributing to the Pauli susceptibility.", "In this way, the jump in the density of states from the top of the flat bands near the Fermi energy (see density of states plots in Fig.", "REF  a) can cause larger susceptibility at high temperatures.", "This makes temperature dependent magnetization measurements a useful tool for screening materials for the peaks and discontinuities in the density of states, $D(\\varepsilon )$ , near the Fermi energy[56], [57], [58], [59].", "Now we will present a model for the density of states to estimate the energy difference between the top of the kagome flat bands and the Fermi energy in CoSn.", "The expression for Pauli spin susceptibility can be written [60], [57], [58], $\\chi _\\mathrm {Pauli} = - \\mu _\\mathrm {B}^2 \\int _{-\\infty }^{\\infty } \\frac{df}{d\\varepsilon }(\\varepsilon )\\,D(\\varepsilon )\\,d\\varepsilon .$ $\\mu _\\mathrm {B}$ is the Bohr magneton, $\\frac{df}{d\\varepsilon }$ is the energy derivative of the Fermi-Dirac distribution, $f(\\varepsilon ) = 1 / (e^{\\frac{\\varepsilon -\\mu }{k_\\mathrm {B}\\,T}} + 1)$ and $D(\\varepsilon )$ is the density of states including spin degeneracy.", "$\\mu $ is the chemical potential, $k_\\mathrm {B}$ is Boltzmann constant and $T$ is temperature.", "Looking at the density of states plot for CoSn in Fig.", "REF , we approximate the dramatic increase in $D(\\varepsilon )$ just below the Fermi energy by a step function depicted in Fig.", "REF  a; $D_\\mathrm {S}(\\varepsilon ) =\\left\\lbrace \\begin{array}{ll}D_0 & \\varepsilon > \\varepsilon _S \\\\R\\,D_0 & \\varepsilon < \\varepsilon _S \\\\\\end{array}\\right.", ",$ where $D_0$ is the density of states at the Fermi energy and $R$ is the ratio of the larger density of states to $D_0$ .", "Fortunately, plugging this into Eq.", "REF yields an analytical result: $\\chi _\\mathrm {Pauli}(T) = \\mu _\\mathrm {B}^2\\,D_0\\,(R-1)\\,\\dfrac{e^{\\frac{\\varepsilon _\\mathrm {S}-\\mu }{k_\\mathrm {B}\\,T}}}{e^{\\frac{\\varepsilon _\\mathrm {S}-\\mu }{k_\\mathrm {B}\\,T}} + 1} + \\mu _\\mathrm {B}^2\\,D_0 .$ Figure REF sketches the origin of the rising $\\chi (T)$ .", "At low temperatures (panel a) only the states at the Fermi energy can contribute to Pauli susceptibility.", "At higher temperatures (Fig.", "REF  b) states below $\\varepsilon _S$ begin to thermally depopulate and the Pauli susceptibility increases as more electrons can participate in the spin polarization in field.", "These extra excited electrons push the chemical potential, $\\mu $ , higher (see appendix).", "Figure REF  c presents magnetic susceptibility is enhanced with temperature in this model.", "This S-shaped curve closely matches that seen in the susceptibility data for CoSn.", "We can fit susceptibility data with $\\chi (T) = \\chi _\\mathrm {Pauli}(T) + \\chi _\\mathrm {CW}(T) + \\chi _0,$ which incorporates a Curie term, $\\chi _\\mathrm {CW}(T) = C/(T-\\theta )$ , and a temperature-independent contribution, $\\chi _0$ (including Landau and core diamagnetism).", "Unfortunately, multiple temperature-independent contributions mean we can only extract $\\varepsilon _S$ from fits to the CoSn $\\chi (T)$ data as $\\chi _0$ , $D_0$ , and $R$ are not independent.", "Fitting Eq.", "REF (with $\\mu =0$  meV) to the CoSn data for ${H}\\parallel {c}$ and $\\perp {c}$ yielded values of $\\varepsilon _\\mathrm {S}$ of -19 and -15 meV, respectively.", "These values are of the same small magnitude as flat bands centered at -70 meV in the ARPES results in Ref. [12].", "The susceptibility of CoSn further increases above 400 K (see appendix ).", "This hints at additional discontinuities in the $D(\\varepsilon )$ in the -100 to -200 meV range.", "The $\\chi (T)$ data for NiIn, RhPb, and PtTl show far weaker temperature dependence and no inflection point like that expected from Equ.", "REF .", "In the context of our Pauli susceptibility model, this suggests that either the flat bands are farther from the Fermi energy or the change in the density of states is less dramatic.", "With $\\mu =0$  meV, the inflection point of Equ.", "REF is expected at $k_\\mathrm {B}\\,T = 0.42\\,|\\varepsilon _\\mathrm {S}|$ .", "No inflection is observed in these 3 compounds below 300 K suggesting there are no sharp discontinuities in the density of states within 60 meV of the Fermi level.", "This is consistant with the locations of the flat bands in Fig.", "REF ." ], [ "Where the flat bands are", "Based on our results, CoSn, NiIn, RhPb, and PtTl do not have flat bands at the Fermi energy.", "The DFT results in Fig.", "REF suggest this situation.", "A smaller density of states at $\\varepsilon _\\mathrm {Fermi}$ is supported by the smaller $\\gamma $ values measured for these four compounds compared to the two antiferromagnets.", "Of the four 23-electron compounds, CoSn appears to have flat bands closest to $\\varepsilon _\\mathrm {Fermi}$ .", "DFT results in Fig.", "REF show the top of the low-dispersion bands about 100 meV below.", "CoSn shows an increasing magnetic susceptibility with temperature which we model to arise from a discontinuity in the $D(\\varepsilon )$ about 20 meV below $\\varepsilon _\\mathrm {Fermi}$ .", "ARPES studies support this interpretation [12], [11].", "Our DFT calculations and magnetic susceptibility data suggest the flat bands are more distant in NiIn, RhPb, an PtTl.", "Their nearly-flat $\\chi (T)$ data suggests no abrupt variations in $D(\\varepsilon )$ within about 60 meV of $\\varepsilon _\\mathrm {Fermi}$ .", "Next, lets examine how the 23 electron CoSn-type compounds compare as platforms for future studies of flat band physics.", "Of these systems, CoSn is the best starting point for exploring flat band physics with its narrow flat band near the $\\varepsilon _\\mathrm {Fermi}$ .", "Hole doping should drop the Fermi energy into the flat bands.", "The large $D(\\varepsilon _\\mathrm {Fermi})$ this creates is likely the origin of antiferromagnetism in the (Fe,Co)Sn series [30], [36].", "Other doping series might favor different ground states.", "NiIn is a less attractive candidate for study.", "First, it isn't obvious how to obtain single crystals (see section REF ).", "On top of this, the exciting flat bands are deeper in energy and the nearest band is quite broad (see Fig.", "REF ).", "Magnetic order is one way to reduce the large degeneracy associated with flat bands at the Fermi level.", "This degeneracy likely drives the antiferromagnetic ground states of FeGe, FeSn and (Fe,Co)Sn.", "Notably, Rh and Pt do not frequently form magnetic compounds.", "If we succeed in tuning $\\varepsilon _\\mathrm {Fermi}$ into flat bands composed of Rh $4d$ or Pt $5d$ states, how will these systems reduce this degeneracy?", "Can we induce a more interesting ground state such as a structural distortion, charge density waves, or superconductivity?", "RhPb deserves some additional attention despite its expensive reactant and flat bands lying below -400 meV.", "In Fig.", "REF these bands still have relatively small bandwidths.", "We can estimate the doping required to tune these bands to the Fermi energy by integrating the density of states from $\\varepsilon _\\mathrm {Fermi}$ down to the top of the flat bands (-400 meV).", "This suggests we need to dope RhPb by 0.4 holes/formula unit to bring these bands to $\\varepsilon _\\mathrm {Fermi}$ .", "In addition to the potential for unusual ground states, RhPb is a heavier analog of CoSn and will allow us to explore the impact of stronger spin-orbit coupling on flat band physics.", "PtTl is the least attractive candidate.", "The kagome lattice flat bands are even broader and lower in energy than RhPb.", "It not only includes expensive Pt but also toxic and reactive Tl.", "On top of this, we estimate at least 0.8 holes/formula unit would be required to raise the flat bands about 1 eV to the Fermi energy.", "Now, lets generalize what we have learned here to other systems where flat bands are generated by geometric frustration.", "$3d$ transition metals tend to give narrower bandwidths for these bands than $4d$ and $5d$ versions, but the heavier atoms offer stronger spin orbit coupling.", "Next, CoSn and NiIn are entirely iso-electronic but NiIn has broader flat bands farther from $\\varepsilon _\\mathrm {Fermi}$ .", "This means that more favorable flat band energies might be obtained by exploring variations of existing flat band systems with different element pairs." ], [ "Conclusions", "Kagome lattice materials can host flat bands resulting from geometric frustration in their structure.", "The CoSn-type materials host stacked kagome lattices of transition-metals atoms.", "In this paper we explore the six compounds with the CoSn structure type.", "We make samples of each CoSn-type compound including single crystals of FeSn, CoSn, RhPb, and PtTl.", "Curiously, we find CoSn and NiIn are subtly colored metals.", "We report the low-temperature thermal expansion, heat capacity, magnetic susceptibility, and resistivity of these compounds.", "FeGe and FeSn are antiferromagnets, but we do not observe any phase transitions in CoSn, NiIn, RhPb and PtTl between 1.9 and 380 K. The single crystals of these metals exhibit both resistive and pronounced magnetic anisotropies.", "DFT band structure calculations reveal the location and character of flat bands in the CoSn-type materials without magnetic order.", "Critically, these calculations and the experimental data suggest that none of these have flat bands at the Fermi level.", "Of the four, CoSn has flat bands nearest the Fermi energy.", "We also note that the flat bands have increasing bandwidth for $3d\\rightarrow 4d\\rightarrow 5d$ transition-metals.", "CoSn shows an increasing magnetic susceptibility on heating above 50 K. We model this with additional Pauli paramagnetic contributions by thermal depopulation of flat bands about 20 meV below the Fermi energy.", "The magnetic data of NiIn, RhPb, and PtTl show far weaker variations with temperature.", "This suggests flat bands are more than 60 meV below $\\varepsilon _\\mathrm {Fermi}$ in these cases, in agreement with our DFT calculations.", "We favor CoSn for future investigations of flat band physics because its flat bands are narrow and near the Fermi energy.", "RhPb also deserves attention because of its heavier constituents and relatively-distinct flat bands.", "This makes it a good system for investigating flat band physics with $4d$ orbitals and stronger spin orbit coupling.", "NiIn and PtTl are less attractive starting points.", "The CoSn family of kagome metals provides a nice playground to explore physics in lattice-derived flat bands.", "The diverse family members include $3d$ , $4d$ , and $5d$ elements facilitating investigation of the role of $d$ -orbital character and spin orbit coupling strength.", "The authors thank Anna Böhmer, Dmitry Chichinadze, Anchal Padukone, and Jiaqiang Yan for their helpful discussions and insights.", "We would also like to acknowledge that NiIn is a palindrome.", "Research supported by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (under contract number DE-AC05-00OR22725).", "High-temperature X-ray diffraction experiments (C.A.B.)", "were sponsored by the Laboratory Directed Research and Development program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy.", "GDS was supported as part of the Energy Dissipation to Defect Evolution (EDDE), an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences under Contract Number DE-AC05-00OR22725." ], [ "High temperature CoSn susceptibility", "In section REF we model CoSn's rising magnetic susceptibility as a temperature dependent Pauli paramagnetic response.", "Figure REF shows how the magnetic susceptibility of CoSn continues to increase above 400 K. In our model, this suggests further discontinuities in the density of states 100-200 meV below $\\varepsilon _\\mathrm {Fermi}$ ." ], [ "Pauli Susceptibility model", "For our simple $D(\\varepsilon )$ model (Equ.", "REF ) we can derive an analytical expression for the change in chemical potential with temperature (setting $\\mu =0$ at $T=0$ ).", "When the density of states step is below the Fermi level ($\\varepsilon _\\mathrm {S}<0$ ) then the chemical potential $\\mu $ is the solution to; $\\mu = (R-1)\\,k_\\mathrm {B}\\,T\\, \\ln \\left(e^{\\frac{\\varepsilon _\\mathrm {S}-\\mu }{k_\\mathrm {B}\\,T}} + 1\\right)$ When the step is above, the chemical potential $\\mu $ is the solution to; $\\mu =(R-1)\\,\\left(-\\varepsilon _\\textrm {S}+\\,k_\\mathrm {B}\\,T\\, \\ln \\left(e^{\\frac{\\varepsilon _\\mathrm {S}-\\mu }{k_\\mathrm {B}\\,T}} + 1\\right)\\right)$ The magnetic susceptibility (Equ.", "REF ) and chemical potential (Equ.", "REF ) both begin to increase at the same temperature.", "As a result, the inflection point in Fig.", "REF  c is a good measure of the $\\varepsilon _S$ ." ] ]

[ [ "Comparability in the graph monoid" ], [ "Abstract Let $\\Gamma$ be the infinite cyclic group on a generator $x.$ To avoid confusion when working with $\\mathbb Z$-modules which also have an additional $\\mathbb Z$-action, we consider the $\\mathbb Z$-action to be a $\\Gamma$-action instead.", "Starting from a directed graph $E$, one can define a cancellative commutative monoid $M_E^\\Gamma$ with a $\\Gamma$-action which agrees with the monoid structure and a natural order.", "The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable.", "We comprehensively pair up these element features with the graph-theoretic properties of the generators of the element.", "We also characterize graphs such that every element of $M_E^\\Gamma$ is comparable, periodic, graphs such that every nonzero element of $M_E^\\Gamma$ is aperiodic, incomparable, graphs such that no nonzero element of $M_E^\\Gamma$ is periodic, and graphs such that no element of $M_E^\\Gamma$ is aperiodic.", "The Graded Classification Conjecture can be formulated to state that $M_E^\\Gamma$ is a complete invariant of the Leavitt path algebra $L_K(E)$ of $E$ over a field $K.$ Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of $E$ are well reflected by the structure of $M_E^\\Gamma.$ Our work also implies that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be row-finite." ], [ "Introduction", "There are several different ways to associate an algebra over a field $K$ to a directed graph $E$ .", "For example, one can form the path algebra $P_K(E)$ which is a vector space over $K$ based on paths multiplied using concatenation.", "If one wants to add a natural involutive structure to this algebra (as, for example, when completing the path algebra over complex numbers to obtain the graph $C^*$ -algebra $C^*(E)$ ), then every vertex naturally becomes a self-adjoint idempotent, a projection, and every edge $e$ becomes a partial isometry making the projections $ee^*$ and $e^*e$ equivalent.", "If $\\mathbf {s}$ and $\\mathbf {r}$ are the source and range maps of $E$ respectively, and $\\mathbf {s}(e)=v,$ then $ve=e$ so that $vee^*=ee^*$ and, hence, $v\\ge ee^*$ (recall that the projections are ordered by $p\\le q$ if $pq=p$ ).", "On the other hand, if $w=\\mathbf {r}(e),$ then $ew=e$ and so $w\\ge e^*e.$ The requirement that $w=e^*e$ is called the (CK1) axiom.", "One also aims to have that the projections $v$ and $w$ are equivalent if $e$ is the only edge from $v$ to $w$ and if $v$ does not emit any other edges.", "This is achieved by an additional requirement, the (CK2) axiom, stating that $v=\\sum _{e\\in \\mathbf {s}^{-1}(v)} ee^*$ if $v$ emits at least one and only finitely many edges.", "The axioms (CK1) and (CK2) imposed on the involutive closure of the path algebra produce the Leavitt path algebra $L_K(E).$ If $L_K(E))$ is the monoid of the isomorphism classes of finitely generated projective modules (or conjugation classes of idempotent matrices), the (CK1) and (CK2) axioms imply that $[v]=\\sum _{e\\in \\mathbf {s}^{-1}(v)} [\\mathbf {r}(e)]$ holds in $L_K(E))$ for every vertex $v$ which emits at least one and only finitely many edges.", "If $E$ is such that every vertex emits only finitely many edges, in which case we say that $E$ is row-finite, one of the first papers on Leavitt path algebras [5] shows that elements $[v]$ generate $L_K(E))$ and that the above relations are the only relations which hold on $L_K(E))$ .", "Thus, to capture $L_K(E))$ entirely, it is sufficient to consider a free commutative monoid $M_E,$ called the graph monoid, generated by $[v]$ where $v$ is a vertex of $E$ subject to the above relations.", "In [3], the authors generalized this construction to arbitrary graphs.", "To handle vertices which emit infinitely many edges (infinite emitters), one adds two natural relations to the one listed above (the details are reviewed in Section REF ) to obtain $M_E.$ The monoid $M_E$ is not necessarily cancellative which is easy to see: if $v$ is a vertex emitting two edges to itself, then the relation $[v]+[v]=[v]$ holds in the monoid but the generator $[v]$ is nonzero.", "So, when one forms the Grothendieck group $G_E$ of the monoid $M_E$ a lot of information can get lost.", "In particular, if $E$ is a graph consisting only of the vertex and edges from the previous example, then $G_E=0.$ In addition to the above mentioned downside, very different graphs give rise to isomorphic monoids and, consequently, isomorphic Grothendieck groups.", "For example, $\\bullet $ and ${ {\\bullet } @(ur,dr)}\\;\\;\\;\\;\\;.$ In addition, consider the graphs $E_1$ and $E_2$ below, for example.", "${\\bullet ^{v_1} [r] &\\bullet ^{w_1}}\\hspace{56.9055pt}{\\bullet ^{v_2} [r] & \\bullet [r] &\\bullet ^{w_2}}$ The relation $[v_1]=[w_1]$ holds in the first and the relation $[v_2]=[w_2]$ holds in the second graph monoid regardless of the fact that the length of the only path from $v_1$ to $w_1$ is 1 in $E_1$ while the length of the only path from $v_2$ to $w_2$ is 2 in $E_2.$ So, this type of information is also lost in the Grothendieck group.", "These downsides can be avoided by taking the natural grading of a Leavitt path algebra into consideration.", "Namely, the elements $pq^*$ where $p$ and $q$ are paths, generate the entire algebra as a $K$ -vector space and if $p$ and $q$ are such that the difference of the length of $p$ and the length of $q$ is an integer $n,$ the generator $pq^*$ is considered to be in the $n$ -th component of $L_K(E).$ This produces a $\\mathbb {Z}$ -graded structure of $L_K(E)$ where $\\mathbb {Z}$ is the set of integers.", "For a ring $R$ graded by a group $\\Gamma ,$ the monoid $\\Gamma (R)$ of the graded isomorphism classes of finitely generated graded projective modules (or conjugation classes of certain homogeneous idempotent matrices) is a natural analogue of $R).$ The monoid $\\Gamma (R)$ has a canonical $\\Gamma $ -action and we refer to a monoid with this type of structure as a $\\Gamma $ -monoid.", "To avoid confusion when working with structures which are $\\mathbb {Z}$ -modules but also have an additional $\\mathbb {Z}$ -action, we let $\\Gamma =\\lbrace x^n\\mid n\\in \\mathbb {Z}\\rbrace $ and consider the $\\mathbb {Z}$ -action to be a $\\Gamma $ -action instead.", "The $\\Gamma $ -action on $\\Gamma (L_K(E))$ is such that the relation $[v]=\\sum _{e\\in \\mathbf {s}^{-1}(v)} [\\mathbf {r}(e)]$ becomes $[v]=\\sum _{e\\in \\mathbf {s}^{-1}(v)} x[\\mathbf {r}(e)]$ if $\\mathbf {s}^{-1}(v)$ is nonempty and finite.", "The power 1 of $x$ in this relation indicates the length of the path $e$ from $v$ to $\\mathbf {r}(e).$ With analogous modifications of the other defining relations, we let the graph $\\Gamma $ -monoid $M_E^\\Gamma $ be the quotient of a free $\\Gamma $ -monoid $F_E^\\Gamma $ with basis elements labeled by the vertices and the elements related to the infinite emitters subject to the defining relations (Section REF contains more details).", "Alternatively, if $\\rightarrow _1$ is a binary relation of $F_E^\\Gamma $ given by these defining relations, $\\rightarrow $ is the reflexive and transitive closure of $\\rightarrow _1,$ and $\\sim $ is the congruence closure of $\\rightarrow ,$ then $M_E^\\Gamma $ is the quotient $\\Gamma $ -monoid $F_E^\\Gamma /\\sim .$ The $\\Gamma $ -monoid $M_E^\\Gamma $ is naturally isomorphic to $\\Gamma (L_K(E))$ .", "The monoid $M_E^\\Gamma $ has several important advantages over $M_E.$ First, it is always cancellative by [4] (we give an alternative proof in Proposition REF ) and so it is exactly the positive cone of its Grothendieck group $G_E^\\Gamma .$ This group inherits the $\\Gamma $ -action from $M_E^\\Gamma $ so we refer to it as the Grothendieck $\\Gamma $ -group.", "Second, the information on the lengths of paths from a vertex to vertex is not lost.", "For example, if $E_1$ and $E_2$ are the above two graphs, the relations $[v_1]=[w_1]$ and $[v_2]=[w_2]$ of $M_{E_1}$ and $M_{E_2}$ become $[v_1]=x[w_1]\\;\\;$ and $\\;\\;[v_2]=x^2[w_2]$ in $M_{E_1}^\\Gamma $ and $M_{E_2}^\\Gamma $ respectively.", "Here, the powers of $x$ indicate that the length of the (only) path from $v_1$ to $w_1$ is 1 in $E_1$ and that the length of the (only) path from $v_2$ to $w_2$ is 2 in $E_2.$ In addition, very different graphs $\\bullet $ and ${ {\\bullet } @(ur,dr)}\\;\\;\\;\\;\\;$ have different Grothendieck $\\Gamma $ -groups: $G_E^\\Gamma $ of the first graph is isomorphic to $\\mathbb {Z}[\\Gamma ]$ with the natural action of $\\Gamma $ while $G_E^\\Gamma $ of the second graph is isomorphic to $\\mathbb {Z}$ with the trivial action of $\\Gamma .$ Because of these favorable properties of $M_E^\\Gamma $ and $G_E^\\Gamma ,$ it was conjectured in [9] that $G_E^\\Gamma ,$ considered with a natural pre-order and an order-unit, is a complete invariant of a row-finite graph $E.$ Since the monoid $M_E^\\Gamma $ is always cancellative, this conjecture can also be phrased in terms of $M_E^\\Gamma $ instead of $G_E^\\Gamma .$ In addition, the restriction on row-finiteness can be deleted and we refer to the following statement as the Graded Classification Conjecture.", "For any two graphs $E$ and $F$ and any field $K$ , $L_K(E)$ and $L_K(F)$ are isomorphic as $\\Gamma $ -graded algebras if and only if $M_E^\\Gamma $ and $M_F^\\Gamma $ are isomorphic as pre-ordered $\\Gamma $ -monoids with order-units.", "Since $M_E^\\Gamma $ is cancellative, the natural pre-order is, in fact, an order.", "In [11], the authors show that the relation $a< x^na$ is impossible for any $a\\in M_E^\\Gamma $ and any positive integer $n$ if $E$ is row-finite.", "In Proposition REF , we show that this holds for all graphs $E.$ Hence, there are two remaining cases.", "$a\\ge x^na$ for some positive integer $n.$ In this case, we say that $a$ is comparable.", "$a$ and $x^na$ incomparable for any positive integer $n.$ In this case, we say that $a$ is incomparable.", "If $a$ is comparable, there are two possibilities.", "$a=x^na$ for some positive integer $n.$ In this case, we say that $a$ is periodic.", "$a>x^na$ for some positive integer $n.$ In this case, we say that $a$ is aperiodic.", "In this paper, we provide complete characterizations of all four types of elements (comparable, incomparable, periodic and aperiodic) in terms of the graph-theoretic properties of the generators of an element.", "We obtain this by three groups of results.", "First, in Section , we obtain a graph-theoretic characterization of the relation $\\rightarrow $ (Proposition REF ).", "Second, in Sections REF and REF , we introduce and study certain well-behaved building blocks of comparable elements, the stationary elements.", "Third, in Section REF , we produce a graph-theoretic characterization of a stationary element (Proposition REF ).", "This enables us to prove Theorem REF , the main result of Section , which characterizes a comparable element in terms of the graph-theoretic properties of its generators.", "In Section , we characterize periodic and aperiodic elements in Theorems REF and REF .", "We have already found a use of Theorem REF : it was used in [13] to characterize Leavitt path algebras which are crossed products in terms of the properties of the underlying graphs.", "We also characterize graphs such that every element of $M_E^\\Gamma $ is comparable (Theorem REF ), periodic (Theorem REF ), graphs such that every nonzero element of $M_E^\\Gamma $ is aperiodic (Theorem REF ), incomparable (Corollary REF ), graphs such that no nonzero element of $M_E^\\Gamma $ is periodic (Corollary REF ), and graphs such that no element of $M_E^\\Gamma $ is aperiodic (Corollary REF ).", "These characterizations comprehensively pair up the monoid and the graph properties and are summarized in the table below.", "In the table, c$(a)$ , p$(a)$ , ap$(a)$ , ic$(a)$ shorten the statements that $a\\in M_E^\\Gamma $ is comparable, periodic, aperiodic, and incomparable respectively.", "The formula “$(\\exists a\\ne 0)$ c$(a)$ ”, for example, shortens “There is a nonzero comparable element in $M_E^\\Gamma $ ”.", "Table: NO_CAPTIONIn Section REF , we relax the assumptions of statements in [11].", "In particular, we show that the main results of [11] hold without the requirement that the graph is row-finite (Corollaries REF , REF , REF and the first part of Corollary REF ).", "The second part of Corollary REF lists further properties of graphs which are preserved if the graph $\\Gamma $ -monoids are isomorphic.", "Our work focuses on graphs and their graph $\\Gamma $ -monoids.", "Leavitt path algebras, often mentioned in the introduction to illustrate wider context, do not appear often in the rest of the paper and no prior knowledge of Leavitt path algebras is needed for understanding our main results." ], [ "Prerequisites, notation and preliminaries", "In this section only, we use $\\Gamma $ to denote an arbitrary group with multiplicative notation.", "In the other sections of the paper, $\\Gamma $ stands for the infinite cyclic group generated by an element $x.$" ], [ "Pre-ordered $\\Gamma $ -monoids and {{formula:860e24c1-7c8a-4905-9e32-57a08c997f92}} -groups", "If $M$ is an additive monoid with a left action of $\\Gamma $ which agrees with the monoid operation, we say that $M$ is a $\\Gamma $ -monoid.", "If $G$ an abelian group with a left action of $\\Gamma $ which agrees with the group operation, we say that $G$ is a $\\Gamma $ -group.", "Such action of $\\Gamma $ uniquely determines a left $\\mathbb {Z}[\\Gamma ]$ -module structure on $G,$ so $G$ is also a left $\\mathbb {Z}[\\Gamma ]$ -module.", "Let $\\ge $ be a reflexive and transitive relation (a pre-order) on a $\\Gamma $ -monoid $M$ ($\\Gamma $ -group $G$ ) such that $g_1\\ge g_2$ implies $g_1 + h\\ge g_2 + h$ and $\\gamma g_1 \\ge \\gamma g_2$ for all $g_1, g_2, h$ in $M$ (in $G$ ) and $\\gamma \\in \\Gamma .$ We say that such monoid $M$ is a pre-ordered $\\Gamma $ -monoid and that such a group $G$ is a pre-ordered $\\Gamma $ -group.", "If $G$ is a pre-ordered $\\Gamma $ -group, the set $G^+=\\lbrace x\\in G\\mid x\\ge 0\\rbrace ,$ called the positive cone of $G,$ is a $\\Gamma $ -monoid.", "Any additively closed subset $M$ of $G$ which contains 0 and is closed under the action of $\\Gamma ,$ defines a pre-order $\\Gamma $ -group structure on $G$ such that $G^+=M$ .", "Such set $G^+$ is strict if $G^+\\cap (-G^+)=\\lbrace 0\\rbrace $ and this condition is equivalent with the pre-order being a partial order.", "In this case, we say that $G$ is an ordered $\\Gamma $ -group.", "For example, $\\mathbb {Z}[\\Gamma ]$ is an ordered $\\Gamma $ -group with the positive cone $\\mathbb {Z}^+[\\Gamma ]$ consisting of elements $a=\\sum _{i=1}^n k_i\\gamma _i\\in \\mathbb {Z}[\\Gamma ]$ such that $k_i\\ge 0$ for all $i=1,\\ldots , n.$ An element $u$ of a pre-ordered $\\Gamma $ -monoid $M$ is an order-unit if for any $x\\in M$ , there is a nonzero $a\\in \\mathbb {Z}^+[\\Gamma ]$ such that $x\\le au.$ An element $u$ of a pre-ordered $\\Gamma $ -group $G$ is an order-unit if $u\\in G^+$ and for any $x\\in G$ , there is a nonzero $a\\in \\mathbb {Z}^+[\\Gamma ]$ such that $x\\le au.$ If $G$ and $H$ are pre-ordered $\\Gamma $ -groups, a $\\mathbb {Z}[\\Gamma ]$ -module homomorphism $f\\colon G\\rightarrow H$ is order-preserving or positive if $f(G^+)\\subseteq H^+.$ If $G$ and $H$ are pre-ordered $\\Gamma $ -groups with order-units $u$ and $v$ respectively, an order-preserving $\\mathbb {Z}[\\Gamma ]$ -module homomorphism $f\\colon G\\rightarrow H$ is order-unit-preserving if $f(u)=v.$ A $\\Gamma $ -order-ideal of a pre-ordered $\\Gamma $ -monoid $M$ is a $\\Gamma $ -submonoid $I$ of $M$ such that $a\\le b$ and $b\\in I$ implies $a\\in I.$ If $G$ is a pre-ordered $\\Gamma $ -group, a $\\Gamma $ -subgroup $J$ of $G$ is a $\\Gamma $ -order-ideal of $G$ if $J\\cap G^+$ is a $\\Gamma $ -order-ideal of $G^+$ and $J=\\lbrace x-y\\mid x,y\\in J\\cap G^+\\rbrace $ (equivalently, $J$ is a directed and convex $\\Gamma $ -subgroup of $G$ using definitions of a directed set and a convex set from [8]).", "The lattices of $\\Gamma $ -order-ideals of $G^+$ and $\\Gamma $ -order-ideals of $G$ are isomorphic by the map $I\\mapsto \\lbrace x-y \\mid x,y\\in I\\rbrace $ with the inverse $J\\mapsto J\\cap G^+.$" ], [ "Graded rings", "We briefly review the concept of graded rings for context only.", "Other than a part of the statement of Corollary REF , no result of this paper refers to graded rings or requires any knowledge of their properties.", "A ring $R$ is $\\Gamma $ -graded if $R=\\bigoplus _{ \\gamma \\in \\Gamma } R_{\\gamma }$ where $R_{\\gamma }$ is an additive subgroup of $R$ and $R_{\\gamma } R_{\\delta } \\subseteq R_{\\gamma \\delta }$ for all $\\gamma , \\delta \\in \\Gamma $ .", "The standard definitions of graded right $R$ -modules, graded module homomorphisms and isomorphisms, and graded projective right modules can be found in [15] and [10].", "If $M$ is a graded right $R$ -module and $\\gamma \\in \\Gamma ,$ the $\\gamma $ -shifted graded right $R$ -module $(\\gamma )M$ is defined as the module $M$ with the $\\Gamma $ -grading given by $(\\gamma )M_\\delta = M_{\\gamma \\delta }$ for all $\\delta \\in \\Gamma .$ If $R$ is a $\\Gamma $ -graded ring, let ${\\Gamma }(R)$ denote the monoid of graded isomorphism classes $[P]$ of finitely generated graded projective right $R$ -modules $P$ with the direct sum as the addition operation and the left $\\Gamma $ -action given by $(\\gamma , [P])\\mapsto [(\\gamma ^{-1})P].$If $M$ is a graded left $R$ -module and $\\gamma \\in \\Gamma ,$ the $\\gamma $ -shifted graded left $R$ -module $M(\\gamma )$ is the module $M$ with the $\\Gamma $ -grading given by $M(\\gamma )_\\delta = M_{\\delta \\gamma }$ for all $\\delta \\in \\Gamma .$ The monoid $\\Gamma (R)$ can be represented using the classes of left modules in which case the corresponding formula is $(\\gamma , [P])\\mapsto [P(\\gamma )].$ Two representations are equivalent (see [15] or [10]).", "In particular, the definitions and results of [10] carry to the case when $\\Gamma $ is not necessarily abelian as it is explained in [18].", "The Grothendieck $\\Gamma $ -group $K_0^{\\Gamma }(R)$ is defined as the group completion of the $\\Gamma $ -monoid ${\\Gamma }(R)$ which naturally inherits the action of $\\Gamma $ from ${\\Gamma }(R)$ .", "The monoid ${\\Gamma }(R)$ is a pre-ordered $\\Gamma $ -monoid and the group $K_0^{\\Gamma }(R)$ is a pre-ordered $\\Gamma $ -group for any $\\Gamma $ -graded ring $R$ .", "If $\\Gamma $ is the trivial group, $K_0^{\\Gamma }(R)$ is the usual $K_0$ -group." ], [ "Graphs", "If $E$ is a directed graph, let $E^0$ denote the set of vertices, $E^1$ the set of edges and $\\mathbf {s}$ and $\\mathbf {r}$ the source and the range maps of $E.$ The graph $E$ is finite if both $E^0$ and $E^1$ are finite and $E$ is row-finite if $\\mathbf {s}^{-1}(v)$ is finite for every $v\\in E^0.$ A vertex $v\\in E^0$ is a sink if $\\mathbf {s}^{-1}(v)=\\emptyset $ and a source if $\\mathbf {r}^{-1}(v)=\\emptyset .$ A vertex of $E$ is regular if $\\mathbf {s}^{-1}(v)$ is finite and nonempty.", "We use the standard definitions of a path, a closed simple path and a cycle (see [1]).", "A path $q$ is a prefix of a path $p$ if $p=qr$ for some path $r.$ If $q=\\mathbf {s}(p),$ then $q$ is a trivial prefix.", "If $r\\ne \\mathbf {r}(p),$ then $q$ is a proper prefix.", "If $E$ has no cycles, $E$ is acyclic.", "A cycle $c$ has an exit if a vertex on $c$ emits an edge outside of $c.$ The graph $E$ satisfies Condition (NE) (and $E$ is a no-exit graph in this case) if $v$ emits just one edge for every vertex $v$ of every cycle.", "The graph $E$ satisfies Condition (L) if every cycle has an exit (equivalently if every closed simple path has an exit) and $E$ satisfies Condition (K) if for each vertex $v$ which lies on a closed simple path, there are at least two different closed simple paths based at $v$ .", "An infinite path is a sequence of edges $e_1e_2\\ldots $ such that $\\mathbf {r}(e_i)=\\mathbf {s}(e_{i+1})$ for $i=1,2,\\ldots $ .", "Such infinite path ends in a cycle if there is a positive integer $n$ and a cycle $c$ such that $e_ne_{n+1}\\hdots $ is equal to $cc\\hdots .$ If $E$ is a finite and acyclic graph, it is well-established that it has a source.", "Since we were not aware of a reference for this fact and we use it in the proof of Lemma REF , we provide a quick proof for it.", "Lemma 1.1 If $E$ is a finite and acyclic graph, it has a source.", "If the graph $E$ does not have any edges, then each of its vertices is both a source and a sink.", "If $E$ has edges, pick any of them, say $e_0.$ If $\\mathbf {r}^{-1}(\\mathbf {s}(e_0))$ is empty, then $\\mathbf {s}(e_0)$ is a source.", "If $\\mathbf {r}^{-1}(\\mathbf {s}(e_0))$ is nonempty, take $e_1\\in \\mathbf {r}^{-1}(\\mathbf {s}(e_0)).$ Then $e_0\\ne e_1$ since otherwise $\\mathbf {r}(e_0)=\\mathbf {s}(e_0)$ and $e_0$ would be a cycle.", "If $\\mathbf {r}^{-1}(\\mathbf {s}(e_1))$ is empty, then $\\mathbf {s}(e_1)$ is a source.", "If $\\mathbf {r}^{-1}(\\mathbf {s}(e_1))$ is nonempty, continue the process.", "At any step of the process, we obtain a different edge than any of the edges considered previously otherwise $E$ has a cycle.", "Since $E$ is finite, this process eventually ends.", "If it ends at the $n$ -th step, then $\\mathbf {s}(e_n)$ is a source." ], [ "Leavitt path algebras", "We review the concept of a Leavitt path algebra for context only.", "No result of this paper except one part of Theorem REF refers to Leavitt path algebras or requires any knowledge of these algebras.", "If $K$ is any field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is a free $K$ -algebra generated by the set $E^0\\cup E^1\\cup \\lbrace e^* \\mid e\\in E^1\\rbrace $ such that, for all vertices $v,w$ and edges $e,f,$ Table: NO_CAPTIONBy the first four axioms, $L_K(E)$ is a $K$ -linear span of the elements of the form $pq^*$ for paths $p$ and $q.$ If $L_K(E)_n$ is the $K$ -linear span of $pq^*$ for paths $p$ and $q$ with $|p|-|q|=n$ where $|p|$ denotes the length of a path $p,$ then it is the $n$ -component of $L_K(E)$ producing a natural grading of $L_K(E)$ by the group of integers $\\mathbb {Z}.$ One can also grade $L_K(E)$ by any group $\\Gamma $ as follows.", "Any function $w\\colon E^1\\rightarrow \\Gamma ,$ called the weight function, extends by $w(e^*)=w(e)^{-1}$ for $e \\in E^1$ and $w(v)=\\varepsilon $ for $v\\in E^0,$ and, ultimately, by $w(pq^*)=w(p)w(q)^{-1}$ for any generator $pq^*$ of $L_K(E)$ (see [10]).", "Thus, $L_K(E)$ becomes $\\Gamma $ -graded with $L_K(E)_\\gamma $ being the $K$ -linear span of the elements $pq^*$ with weight $\\gamma .$" ], [ "The graph monoid and the Grothendieck group of a graph", "If $E$ is a graph, the graph monoid $M_E$ was defined for row-finite graphs in [5] and for arbitrary graphs in [3].", "We briefly review this definition.", "Any edge $e\\in E^1$ is a partial isometry of $ee^*$ and $\\mathbf {r}(e)=e^*e$ so that $[ee^*]$ and $[\\mathbf {r}(e)]$ are the same element in $L_K(E)).$ Hence, the relation below holds in $L_K(E))$ by the (CK2)-axiom if $v$ is regular.", "$[v]=\\sum _{e\\in \\mathbf {s}^{-1}(v)}[\\mathbf {r}(e)]\\qquad \\mathrm {(1)}$ For any infinite emitter $v$ and any finite and nonempty $Z\\subseteq \\mathbf {s}^{-1}(v),$ one considers the element $q_Z$ representing $v-\\sum _{e\\in Z}ee^*.$ We refer to the elements of the form $q_Z$ as the improper vertices (and we note that this term was not used before).", "When we need to emphasize that $q_Z$ is related to the infinite emitter $v$ (in the sense that $Z\\subseteq \\mathbf {s}^{-1}(v)$ ) we write $q_Z^v$ for $q_Z.$ Also, whenever the notation $q_Z$ appears, it is to be understood that there is an infinite emitter $v$ and that $Z$ is a finite and nonempty subset of $\\mathbf {s}^{-1}(v).$ For any finite sets $Z$ and $W$ such that $\\emptyset \\subsetneq Z\\subsetneq W \\subsetneq \\mathbf {s}^{-1}(v),$ it is direct to check that the relations $[v]=[q_Z]+\\sum _{e\\in Z}[\\mathbf {r}(e)]\\;\\;\\mbox{ and }\\;\\;[q_{Z_1}]=[q_{Z_2}]+\\sum _{e\\in W-Z}[\\mathbf {r}(e)]\\qquad \\mathrm {(2 and 3)}$ also hold in $L_K(E)).$ So, one aims to define $M_E$ so that the relations (1), (2 and 3) are the only relations which hold in $M_E.$ This is achieved in the following way.", "Let $F_E$ be a free commutative monoid generated by the elements indexed by the proper and improper vertices of $E.$ To be consistent with [3], [4] and [11], we abuse the notation and refer to the generator indexed by a proper vertex $v\\in E^0$ as $v$ and, similarly, to the generator indexed by $q_Z$ by $q_Z$ .", "The monoid $M_E,$ called the graph monoid, is the quotient of $F_E$ with respect to the the congruence closure $\\sim $ of the relation $\\rightarrow _1$ defined on $F_E-\\lbrace 0\\rbrace $ by $a+v \\rightarrow _1 a+ \\sum _{e\\in s^{-1}(v)}r(e),$ whenever $v$ is a regular vertex and $a\\in F_E$ and by $a+ v\\rightarrow _1 a+ q_Z+\\sum _{e\\in Z}\\mathbf {r}(e)\\;\\;\\mbox{ and }\\;\\;a+q_Z\\rightarrow _1 a+ q_W+\\sum _{e\\in W-Z}\\mathbf {r}(e)$ whenever $v$ is an infinite emitter and $Z$ and $W$ are finite and such that $\\emptyset \\subsetneq Z \\subsetneq W\\subsetneq \\mathbf {s}^{-1}(v).$ One often considers an intermediate step of this construction and lets $\\rightarrow $ be the transitive and reflexive closure of $\\rightarrow _1$ on $F_E$ so that $\\rightarrow $ is a pre-order.", "In this case, $\\sim $ is the congruence on $F_E$ generated by the relation $\\rightarrow $ (i.e.", "the symmetric closure of the pre-order $\\rightarrow $ ).", "We use the notation $[v]$ for the congruence class of $v$ as an element of $M_E.$ As a side note, we add that the map $[v]\\mapsto [vL_K(E)]$ extends to a pre-ordered monoid isomorphism of $M_E$ and $L_K(E))$ (here $L_K(E))$ is given using the finitely generated projective right modules) by [1] (or [3]).", "So, the Grothendieck group completion $G_E$ of $M_E$ is isomorphic to $K_0(L_K(E)).$" ], [ "The graph $\\Gamma $ -monoid and the Grothendieck {{formula:c397f883-205c-48a1-84ff-38e1deb298f3}} -group of a graph", "Let $\\Gamma $ be a group and $w\\colon E^1\\rightarrow \\Gamma $ be a function which we refer to as a weight determining a $\\Gamma $ -grading of $L_K(E).$ The following relations hold in the $\\Gamma $ -monoid ${\\Gamma }(L_K(E)).$ For every regular vertex $v,$ $\\gamma [v]=\\sum _{e\\in \\mathbf {s}^{-1}(v)}\\gamma w(e)[\\mathbf {r}(e)],$ and for every infinite emitter $v$ and finite $Z$ and $W$ such that $\\emptyset \\subsetneq Z\\subsetneq W\\subsetneq \\mathbf {s}^{-1}(v),$ $\\gamma [v]=\\gamma [q_Z]+\\sum _{e\\in Z}\\gamma w(e)[\\mathbf {r}(e)]\\;\\;\\mbox{ and }\\;\\;\\gamma [q_Z]=\\gamma [q_W]+\\sum _{e\\in W-Z}\\gamma w(e)[\\mathbf {r}(e)].$ To adapt the original construction of $M_E$ to this setting, the authors of [4] replaced generators $v$ and $q_Z$ of $F_E$ by $v(\\gamma )$ and $q_Z(\\gamma )$ for any $\\gamma \\in \\Gamma $ and considered a free commutative monoid $F_E^\\Gamma $ with the action of $\\Gamma $ given by $\\delta v(\\gamma )=v(\\delta \\gamma )$ and $\\delta q_Z(\\gamma )=q_Z(\\delta \\gamma )$ for all $\\gamma ,\\delta \\in \\Gamma .$ Then $M^\\Gamma _E$ is the quotient of $F_E^\\Gamma $ subject to the congruence closure $\\sim $ of relation $\\rightarrow _1$ defined just as in the previous section but with the three relations modified accordingly so that $a+\\gamma v \\rightarrow _1 a+ \\sum _{e\\in s^{-1}(v)}\\gamma w(e)r(e),$ whenever $v$ is a regular vertex and $a\\in F_E$ and by $a+ \\gamma v\\rightarrow _1 a+ \\gamma q_Z+\\sum _{e\\in Z}\\gamma w(e)\\mathbf {r}(e)\\;\\;\\mbox{ and }\\;\\;a+\\gamma q_Z\\rightarrow _1 a+\\gamma q_W+\\sum _{e\\in W-Z}\\gamma w(e)\\mathbf {r}(e)$ whenever $v$ is an infinite emitter and $Z$ and $W$ are finite and such that $\\emptyset \\subsetneq Z\\subsetneq W\\subsetneq \\mathbf {s}^{-1}(v)$ .", "One downside of this approach is that $M^{\\Gamma }_E$ is still considered to be a commutative monoid, not a commutative $\\Gamma $ -monoid.", "For example, if $E$ is a single vertex, $M^\\Gamma _E$ is a direct sum of $|\\Gamma |$ -many copies of $\\mathbb {Z}^+$ (with a natural action of $\\Gamma $ ) instead of being a single copy of $\\mathbb {Z}^+[\\Gamma ].$ Also, the abundance of generators can make some proofs less direct.", "Because of this, we adopt a simpler and more intuitive approach here: we let $M^{\\Gamma }_E$ be defined by the same set of generators as when the weight function is trivial, but we let $F_E^\\Gamma $ be a free commutative $\\Gamma $ -monoid, not a free commutative monoid.", "In this case, if $E$ is a single vertex, then $M_E$ is a single copy of $\\mathbb {Z}^+$ and $M^\\Gamma _E$ is a single copy of $\\mathbb {Z}^+[\\Gamma ].$ The equivalence of ours and the construction from [4] can be seen considering the graph covering $\\overline{E}$ of $E$ .", "So, we let $F_E^\\Gamma $ be a free commutative $\\Gamma $ -monoid generated by proper and improper vertices.", "A nonzero element $a$ of $F_E^\\Gamma $ has a representation, unique up to a permutation, as $\\sum _{j=1}^n \\alpha _ig_i$ , where $g_i$ are different generators of $F_E^\\Gamma $ and $\\alpha _i\\in \\mathbb {Z}^+[\\Gamma ]$ .", "The support $\\operatorname{supp}(a)$ of $a$ is the set $\\lbrace g_i\\mid i=1,\\ldots , n\\rbrace .$ Let $k_\\gamma \\in \\mathbb {Z}^+$ be the coefficient of $\\gamma \\in \\Gamma $ in $\\alpha _i\\in \\mathbb {Z}^+[\\Gamma ]$ in the above representation.", "By writing each $k_\\gamma >0$ as the sum $1+1+\\ldots +1,$ one obtains the format $a=\\sum _{j=1}^m \\gamma _j g_j$ for some positive integer $m$ and $\\gamma _j\\in \\Gamma , j=1,\\ldots , m.$ We allow the generators $g_j$ and $g_k$ to be possibly equal for $j\\ne k$ in this form, also unique up to a permutation.", "We refer to it as a normal representation of $a$ and we say that each summand $\\gamma _j g_j$ of this representation is a monomial of $a$ .", "We can still write $\\operatorname{supp}(a)=\\lbrace g_j \\mid j=1,\\ldots , m\\rbrace $ because any possible repetition of an element does not impact $\\operatorname{supp}(a)$ as a set.", "For example, if $\\Gamma $ is the infinite cyclic group generated by $x,$ $v$ is a vertex of $E,$ and $a=xv+3v,$ then $(x+3)v$ is a representation of $a$ and $xv+v+v+v$ is a normal representation of $a.$ To shorten some statements, we say that a vertex $v,$ considered as a generator of $F_E^\\Gamma ,$ is regular if $v$ is regular as a vertex of $E.$ We also say that a generator $v\\in F_E^\\Gamma $ is a sink or an infinite emitter, if $v$ is a sink or an infinite emitter as a vertex of $E$ .", "An element $a\\in F_E^\\Gamma $ is regular if every element of $\\operatorname{supp}(a)$ is regular.", "We define the graph $\\Gamma $ -monoid $M^\\Gamma _E$ as a quotient of $F^\\Gamma _E$ subject to the congruence closure $\\sim $ of the relation $\\rightarrow _1$ on $F_E^\\Gamma -\\lbrace 0\\rbrace $ defined by (A1), (A2) and (A3) below for any $\\gamma \\in \\Gamma $ and $a\\in F_E^\\Gamma .$ If $v$ is a regular vertex, then $a+\\gamma v\\rightarrow _1 a+\\sum _{e\\in s^{-1}(v)}\\gamma w(e)\\mathbf {r}(e).$ If $v$ is an infinite emitter and $Z$ a finite and nonempty subset of $\\mathbf {s}^{-1}(v),$ then $a+\\gamma v\\rightarrow _1 a+\\gamma q_Z+\\sum _{e\\in Z}\\gamma w(e)\\mathbf {r}(e).$ If $v$ is an infinite emitter and $Z\\subsetneq W$ are finite and nonempty subsets of $\\mathbf {s}^{-1}(v),$ then $a+\\gamma q_Z\\rightarrow _1 a+\\gamma q_W+\\sum _{e\\in W-Z}\\gamma w(e)\\mathbf {r}(e).$ So, if $\\rightarrow $ is the reflexive and transitive closure of $\\rightarrow _1$ on $F_E^\\Gamma ,$ then $\\sim $ is the congruence on $F^\\Gamma _E$ generated by the relation $\\rightarrow $ .", "This means that the relation $a\\sim b$ holds for some $a,b\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ if and only if there is a nonnegative integer $n$ and $a=a_0,\\ldots ,a_n=b\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ such that $a_i\\rightarrow _1 a_{i+1}$ or $a_{i+1}\\rightarrow _1a_i$ for all $i=0,\\ldots ,n-1.$ We refer to such $n$ as the length of the sequence $a_0, \\ldots , a_n$ and we write $a\\sim ^n b$ to emphasize the length.", "In particular, if $a\\rightarrow b,$ the sequence can be chosen so that $a_i \\rightarrow _1 a_{i+1}$ for all $i=0,\\ldots , n-1.$ In this case, we write $a\\rightarrow ^n b.$ Note that $a\\rightarrow ^1 b$ is just $a\\rightarrow _1 b$ and that $a\\rightarrow ^0b$ is just $a=b.$ To shorten the notation in multiple proofs, if $g$ is a generator of $F_E^\\Gamma ,$ and one of the three axioms is applied to $g,$ we use $\\mathbf {r}(g)$ to denote the resulting term on the right side of relation $\\rightarrow _1:$ $\\sum _{e\\in \\mathbf {s}^{-1}(v)}w(e)\\mathbf {r}(e)$ if $g=v$ is a regular vertex, $q_Z+\\sum _{e\\in Z}w(e)\\mathbf {r}(e)$ for some finite and nonempty subset $Z$ of $\\mathbf {s}^{-1}(v)$ if $g=v$ is an infinite emitter, or $q_W+\\sum _{e\\in W-Z}w(e)\\mathbf {r}(e)$ for some finite $Z$ and $W$ such that $\\emptyset \\subsetneq Z\\subsetneq W\\subsetneq \\mathbf {s}^{-1}(v)$ if $g=q^v_Z$ for an infinite emitter $v.$ The element $\\mathbf {r}(g)$ is uniquely determined just for (A1).", "However, for a fixed use of (A2) or (A3) which is not changed within a proof, the notation $\\mathbf {r}(g)$ is a well-defined shortening.", "Such uniform treatment enables us to condense some proofs by avoiding considerations of three separate cases depending on which axiom is used.", "Another benefit of our approach is that the proofs of many known statements in the case when $\\Gamma $ is trivial directly transfer to the case when $\\Gamma $ is not trivial.", "For example, if $[g]$ denotes the congruence class of a generator $g$ of $F_E^\\Gamma ,$ the map $[g]\\mapsto [gL_K(E)]$ extends to a pre-ordered $\\Gamma $ -monoid isomorphism of $M^\\Gamma _E$ and $\\Gamma (L_K(E))$ and the proof of the case when $\\Gamma $ is trivial (see, for example, [3]) directly adapts to the case when $\\Gamma $ is arbitrary.", "In [4], this monoid isomorphism is shown to exist by considering the graph covering.", "Lemma REF greatly simplifies many proofs which involve handling relation $\\sim .$ Parts of this lemma can be shown by directly generalizing the proofs of [3].", "We add some new elements in part (1) of Lemma REF to control the length of sequences for certain relations.", "We also note that part (2), the Confluence Lemma, is shown for general $\\Gamma $ in [4] but using the graph covering.", "The Confluence Lemma is key for showing that the monoid $M_E^\\Gamma $ has the refinement property (see [3]).", "Lemma 1.2 Let $E$ be a graph, $\\Gamma $ a group, $w\\colon E^1\\rightarrow \\Gamma $ a weight function and $a,b\\in F^\\Gamma _E-\\lbrace 0\\rbrace $ .", "(The Refinement Lemma) If $a=a^{\\prime }+a^{\\prime \\prime }$ for some $a^{\\prime }, a^{\\prime \\prime }\\in F^\\Gamma _E$ and if $a\\rightarrow ^n b$ , then $b$ has summands $b^{\\prime }, b^{\\prime \\prime }\\in F^\\Gamma _E$ and $n$ has summands $i,j$ such that $b=b^{\\prime }+b^{\\prime \\prime },$ $i+j=n,$ $a^{\\prime }\\rightarrow ^i b^{\\prime },$ and $a^{\\prime \\prime }\\rightarrow ^j b^{\\prime \\prime }.$ (The Confluence Lemma) The relation $a\\sim b$ holds if and only if $a \\rightarrow c$ and $b\\rightarrow c$ for some $c \\in F_E-\\lbrace 0\\rbrace .$ We show (1) by induction on $n.$ If $n=0$ then $a=b$ and we can take $b^{\\prime }=a^{\\prime }=a=b,$ $b^{\\prime \\prime }=a^{\\prime \\prime }=0,$ and $i=j=0.$ Assuming the induction hypothesis, let $a=a_0\\rightarrow _1 a_1\\rightarrow _1\\ldots \\rightarrow _1 a_n=b$ and let $\\gamma g$ be a monomial of $a$ so that $a_1$ is obtained by replacing $\\gamma g$ by $\\gamma \\mathbf {r}(g).$ Since $a=a^{\\prime }+a^{\\prime \\prime },$ $\\gamma g$ is a summand of either $a^{\\prime }$ or $a^{\\prime \\prime }.$ Say it is $a^{\\prime }$ (the case when it is $a^{\\prime \\prime }$ is analogous) and let $a^{\\prime }=c+\\gamma g$ for some $c\\in F_E^\\Gamma .$ For $a_1^{\\prime }=c+\\gamma \\mathbf {r}(g)$ and $a_1^{\\prime \\prime }=a^{\\prime \\prime },$ $a^{\\prime }\\rightarrow ^1 a_1^{\\prime }$ and $a^{\\prime \\prime }\\rightarrow ^0 a_1^{\\prime \\prime }.$ The induction hypothesis implies the existence of $b^{\\prime }, b^{\\prime \\prime }\\in F^\\Gamma _E$ and $i,j$ such that such that $b=b^{\\prime }+b^{\\prime \\prime },$ $i+j=n-1,$ $a_1^{\\prime }\\rightarrow ^i b^{\\prime }$ and $a_1^{\\prime \\prime }\\rightarrow ^j b^{\\prime \\prime }.$ Thus, $i+1+j=n,$ $a^{\\prime }\\rightarrow ^1 a_1^{\\prime }\\rightarrow ^i b^{\\prime },$ and $a^{\\prime \\prime }\\rightarrow ^0 a_1^{\\prime \\prime }\\rightarrow ^j b^{\\prime \\prime }$ and so $a^{\\prime }\\rightarrow ^{i+1}b^{\\prime }$ and $a^{\\prime \\prime }\\rightarrow ^j b^{\\prime \\prime }.$ The direction $\\Leftarrow $ of (2) is direct since if $a\\rightarrow c$ and $b\\rightarrow c,$ then $a\\sim c$ and $b\\sim c$ so that $a\\sim b.$ First, we show the direction $\\Rightarrow $ of (2) for finite graphs using induction on $n$ for $a\\sim ^n b.$ If $n=0,$ $a=b$ and we can take $c=a=b.$ Assuming the induction hypothesis, let $a\\sim ^n b,$ $a_0=a, a_n=b$ and let $a_i\\rightarrow _1 a_{i+1}$ or $a_{i+1}\\rightarrow _1 a_i$ for some $a_i\\in F_E^\\Gamma $ for $i=0,\\ldots , n-1.$ Since $a_1\\sim ^{n-1} b,$ there is $d$ such that $a_1\\rightarrow d$ and $b\\rightarrow d.$ Then either $a\\rightarrow _1 a_1$ or $a_1\\rightarrow _1 a.$ In the first case, we can take $c=d.$ In the second case, there is a monomial $\\gamma g$ of $a_1$ so that $a_1=a^{\\prime }+\\gamma g$ for some $a^{\\prime }$ and $a=a^{\\prime }+\\gamma \\mathbf {r}(g).$ By part (1), $d=d^{\\prime }+d^{\\prime \\prime }$ for some $d^{\\prime }$ and $d^{\\prime \\prime }$ such that $a^{\\prime }\\rightarrow d^{\\prime }$ and $\\gamma g\\rightarrow ^l d^{\\prime \\prime }$ for some $l\\ge 0$ .", "If $l=0,$ then $d^{\\prime \\prime }=\\gamma g$ so $d=d^{\\prime }+\\gamma g.$ Let $c=d^{\\prime }+\\gamma \\mathbf {r}(g).$ Then we have that $a=a^{\\prime }+\\gamma \\mathbf {r}(g)\\rightarrow d^{\\prime }+\\gamma \\mathbf {r}(g)=c$ and $b\\rightarrow d=d^{\\prime }+\\gamma g\\rightarrow _1 d^{\\prime }+\\gamma \\mathbf {r}(g)=c.$ If $l$ is positive, we use the assumption that $E$ is finite to conclude that there are no infinite emitters so that $g$ is necessarily a regular vertex and $a_1\\rightarrow _1 a$ is an application of (A1.)", "Hence, the relation $\\gamma g\\rightarrow d^{\\prime \\prime }$ necessarily decomposes as $\\gamma g\\rightarrow _1 \\gamma \\mathbf {r}(g)\\rightarrow d^{\\prime \\prime }$ and we have that $a_1=a^{\\prime }+\\gamma g\\rightarrow _1 a=a^{\\prime }+\\gamma \\mathbf {r}(g)\\rightarrow d^{\\prime }+d^{\\prime \\prime }=d.$ So, in this case we can also take $c=d.$ To complete the proof in the case when $E$ is an arbitrary graph, we use the argument of the proof of [4] relying on [3].", "If $R(E)$ denotes the set of regular vertices of $E$ , the pair $(E, R(E)),$ considered as an element of an appropriate category from [3], can be represented as a direct limit of pairs $(E^{\\prime },S)$ where $E^{\\prime }$ is a finite subgraph of $E$ and $S$ is a subset of $R(E^{\\prime })$ (see [3] for details).", "The pair $(E^{\\prime },S)$ gives rise to the relative graph $E^{\\prime }_S$ of $E^{\\prime }$ with respect to $S$ such that the bijection on the generators of the corresponding free $\\Gamma $ -monoids produces a natural $\\Gamma $ -monoid isomorphism (see [14] and the graded version in [17]).", "Hence, if $a,b\\in F_E^\\Gamma $ correspond to elements $a^{\\prime }$ and $b^{\\prime }$ of $F^\\Gamma _{E^{\\prime }_S}$ for some finite subgraph $E^{\\prime }$ and some subset $S$ of $R(E^{\\prime }),$ then the relation $a\\sim b$ holds in $F_E^\\Gamma $ if and only if $a^{\\prime }\\sim b^{\\prime }$ holds in $F_{E^{\\prime }_S}^\\Gamma .$ Assuming that $a\\sim b$ holds, we have that $a^{\\prime }\\sim b^{\\prime }$ holds.", "By the proven claim for finite graphs, there is $c^{\\prime }\\in F_{E^{\\prime }_S}^\\Gamma $ such that the relations $a^{\\prime }\\rightarrow c^{\\prime }$ and $b^{\\prime }\\rightarrow c^{\\prime }$ hold in $F^\\Gamma _{E^{\\prime }_S}.$ If $c\\in F_E^\\Gamma $ corresponds to $c^{\\prime },$ these relations imply that $a\\rightarrow c$ and $b\\rightarrow c$ hold in $F_E^\\Gamma .$ One can also show the Confluence Lemma directly, by considering an arbitrary graph $E$ and discussing possibilities that the relation $a_1\\rightarrow _1 a$ in the above proof is obtained by (A2) or (A3).", "We conclude this section by a remark: the Graded Classification Conjecture is false if the pre-ordered $\\Gamma $ -monoids (equivalently $\\Gamma $ -groups) of the graphs are replaced by the free $\\Gamma $ -monoids.", "Indeed, let $E$ and $F$ be the graphs below and $\\Gamma $ be the group of integers.", "${ {\\bullet }@(ul,dl) @/^1pc/[r] & {\\bullet } @/^1pc/[l] @(ur,dr)}\\hspace{113.81102pt}{ {\\bullet }@(ul,dl) @(ur,dr)}$ The graph $E$ is an out-split of the graph $F$ , so the Leavitt path algebras of $E$ and $F$ are graded isomorphic (see [2]).", "Hence, $M_E^\\Gamma $ and $M_F^\\Gamma $ are isomorphic and so are $G_E^\\Gamma $ and $G_F^\\Gamma .$ Alternatively, one can show the existence of these isomorphisms by noting that $M_E^\\Gamma $ and $M_F^\\Gamma $ are both isomorphic to $\\mathbb {Z}^+[\\frac{1}{2}]$ and, consequently, $G_E^\\Gamma $ and $G_F^\\Gamma $ are both isomorphic to $\\mathbb {Z}[\\frac{1}{2}]$ .", "However, $F_E^\\Gamma $ and $F_F^\\Gamma $ are not isomorphic as $\\Gamma $ -monoids since one has two while the other has one generator.", "This example illustrates that the $\\Gamma $ -monoid $F_E^\\Gamma $ of a graph $E$ is informative only when considered together with the relation $\\sim .$" ], [ "Connectivity", "In this section and the rest of the paper, $\\Gamma =\\lbrace x^n \\mid n\\in \\mathbb {Z}\\rbrace $ is the infinite cyclic group with generator $x$ and $E$ is an arbitrary graph.", "To simplify the terminology in some of the proofs, we say that $n$ is the degree of the monomial $x^{n}g$ where $g$ is a generator of $F_E^\\Gamma .$ First, we characterize the relation $\\rightarrow $ in terms of the graph-theoretic properties (Proposition REF ).", "If $v$ and $w$ are vertices of $E$ and $p$ a path from $v$ to $w,$ one can apply (A1) or (A2) to the vertices on $p$ to obtain that $v\\rightarrow x^{|p|}w+a$ for some $a\\in F_E^\\Gamma .$ Indeed, if $p$ is trivial, then $v=w$ and one can take $a=0.$ If $p=e_1e_2\\ldots e_n,$ one can apply (A1) if $v$ is regular and (A2) if it is not, and then apply (A1) to $\\mathbf {r}(e_1)$ if it is regular and (A2) if it is not.", "Continuing this process, one obtains a sequence for $v\\rightarrow x^{|p|}w+a$ for some $a\\in F_E^\\Gamma ,$ where the “change” $a$ reflects the existence of bifurcations from $p.$ For example, in the graph below with $p=f,$ we have that $v\\rightarrow xw+xu$ so $a=xu.$ ${{\\bullet }^{u} & {\\bullet }^{v} [r]^f[l]_e & {\\bullet }^{w}}$ We generalize this process to improper vertices also.", "The terminology introduced below allows uniform treatment of generators of $F_E^\\Gamma $ of both types and enables us to express the comparability properties in terms of the properties of the graph $E$ .", "Definition 2.1 Let $g$ and $h$ be generators of $F_E^\\Gamma .$ We say that $g$ connects to $h$ by a path $p$ (written $g\\rightsquigarrow ^p h$ ) if one of the following conditions hold.", "$g=v$ and $h=w$ are proper vertices and $p$ is a path from $v$ to $w.$ In this case, $v\\rightarrow x^{|p|}w+a$ holds for some $a\\in F_E^\\Gamma $ as we pointed out above.", "$g=v$ is a proper vertex, $h=q^w_Z$ for an infinite emitter $w$ and some $Z,$ and $p$ is a path from $v$ to $w.$ In this case, $v\\rightarrow x^{|p|}w+a^{\\prime }\\rightarrow x^{|p|}q_Z+a$ for some $a^{\\prime }\\in F_E^\\Gamma $ and $a=a^{\\prime }+\\sum _{e\\in Z}x^{|p|+1}\\mathbf {r}(e).$ Note that if $v=w$ and $p$ is trivial then $\\rightarrow $ can be chosen to be a single application of (A2).", "If $v=w$ and $p$ has positive length, then $v$ is necessarily on a cycle.", "$g=q^v_Z$ for an infinite emitter $v$ and some $Z,$ $h=w$ is a proper vertex, and $p=eq$ is a path from $v$ to $w$ such that $e\\notin Z.$ In this case, $q_Z\\rightarrow q_{Z\\cup \\lbrace e\\rbrace }+x\\mathbf {r}(e)\\rightarrow q_{Z\\cup \\lbrace e\\rbrace }+x^{|p|}w+a^{\\prime }=x^{|p|}w+a$ for some $a^{\\prime }\\in F_E^\\Gamma $ and for $a=a^{\\prime }+ q_{Z\\cup \\lbrace e\\rbrace }.$ If $v=w,$ then $v$ is on a cycle.", "$g=q^v_Z$ for some $v$ and $Z,$ $h=q^w_W$ for some $w$ and $W,$ $p$ is a path from $v$ to $w,$ and one of the following two scenarios hold.", "If $p$ is trivial, then $v=w$ and $Z\\subseteq W.$ If $Z=W,$ then $q_Z\\rightarrow x^0q_Z$ and if $Z\\subsetneq W$ and $a=\\sum _{e\\in W-Z}x\\mathbf {r}(e),$ then $q_Z\\rightarrow x^0q_W+\\sum _{e\\in W-Z}x\\mathbf {r}(e)=x^0q_W+a.$ If $p$ has positive length, then $p=eq$ for some $e\\notin Z.$ In this case, $q_Z\\rightarrow q_{Z\\cup \\lbrace e\\rbrace }+x\\mathbf {r}(e)\\rightarrow q_{Z\\cup \\lbrace e\\rbrace }+x^{|p|}w+a^{\\prime }\\rightarrow q_{Z\\cup \\lbrace e\\rbrace }+x^{|p|}q_W+\\sum _{f\\in W}x^{|p|+1}\\mathbf {r}(f)+a^{\\prime }=x^{|p|}q_W+a$ for some $a^{\\prime }\\in F_E^\\Gamma $ and $a=a^{\\prime }+q_{Z\\cup \\lbrace e\\rbrace }+\\sum _{f\\in W}x^{|p|+1}\\mathbf {r}(f).$ If $v=w,$ then $v$ is on a cycle.", "Definition REF enables us to deal with every generator of $F_E^\\Gamma $ in a uniform way.", "In particular, in any of the above four cases, we have that $g\\rightarrow x^{|p|}h+a$ for some element $a\\in F_E^\\Gamma $ and a path $p.$ In this case, we say that $h$ is obtained from $g$ following the path $p$.", "The element $a$ reflects the existence of bifurcations from $p$ .", "In Corollary REF , we show the converse: $g\\rightarrow x^{n}h+a$ implies that $g\\rightsquigarrow ^p h$ for a path $p$ of length $n$ .", "We say that $g$ connects to $h$ , written $g\\rightsquigarrow h,$ if there is a path $p$ such that $g\\rightsquigarrow ^p h.$ If $v$ and $w$ are vertices, $v\\rightsquigarrow w$ is usually written $v\\ge w$ (see [1]).", "However, we reserve the relation $\\ge $ for the order on the monoid $M_E^\\Gamma .$ It is direct to check that $\\rightsquigarrow $ is reflexive and transitive.", "Note that a proper vertex $v$ is on a cycle if and only if $v$ connects to $v$ by a path of positive length.", "Definition REF enables us to talk about improper vertices being on cycles: we say that any generator $g$ of $F_E^\\Gamma $ is on a cycle if $g$ connects to $g$ by a path of positive length.", "We say that $g$ is on an exit from a cycle $c$ if $g$ is not on $c$ and there is a generator $h$ of $F_E^\\Gamma $ which is on $c$ such that $h$ connects to $g.$ By Definition REF , $q^v_Z$ is on a cycle if and only if there is $e\\in \\mathbf {s}^{-1}(v)-Z$ and a path $p$ with $\\mathbf {r}(p)=v, \\mathbf {s}(p)=\\mathbf {r}(e)$ such that $ep$ is a cycle.", "If $a\\rightarrow b$ and $a=\\sum _{i=1}^k x^{m_i}g_i$ and $b=\\sum _{j=1}^l x^{t_j}h_j$ are normal representations of $a$ and $b$ respectively, repeated use of the Refinement Lemma REF (1) ensures the existence of a partition $\\lbrace I_1, \\ldots , I_k\\rbrace $ of $\\lbrace 1, \\ldots , l\\rbrace $ and summands $b_i$ of $b$ such that $b=\\sum _{i=1}^k b_i,$ $b_i=\\sum _{j\\in I_i}x^{t_j}h_j,$ and $x^{m_i}g_i\\rightarrow b_i.$ This implies that $t_j\\ge m_i$ for all $j\\in I_i.$ Proposition REF implies the existence of paths $p_{ij}$ with $|p_{ij}|=t_j-m_i$ and $g_i\\rightsquigarrow ^{p_{ij}} h_j$ in this case.", "We introduce this idea of partitioning $b$ according to $a$ if $a\\rightarrow b$ in Proposition REF and use it again in Section REF .", "Proposition REF describes the relation $a\\rightarrow b$ in terms of the properties of the generators in the supports of $a$ and $b$ and the length of the paths connecting them.", "Proposition 2.2 Let $a,b\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ and $a=\\sum _{i=1}^k x^{m_i}g_i$ and $b=\\sum _{j=1}^l x^{t_j}h_{j}$ be normal representations of $a$ and $b$ respectively.", "The following conditions are equivalent.", "The relation $a\\rightarrow b$ holds.", "There is a partition $\\lbrace I_1, \\ldots , I_k\\rbrace $ of $\\lbrace 1, \\ldots , l\\rbrace $ and finitely many paths $p_{ij}, j\\in I_i, i=1,\\ldots , k,$ such that $g_i\\rightsquigarrow ^{p_{ij}}h_j,$ $|p_{ij}|=t_j-m_i$ for all $j\\in I_i, i=1,\\ldots , k,$ and $b=\\sum _{j=1}^l x^{t_j}h_{j}=\\sum _{i=1}^k \\sum _{j\\in I_i}x^{m_i+|p_{ij}|}h_{j}.$ If $p$ is a prefix of $p_{ij}$ and $v=\\mathbf {r}(p),$ let $P_p=\\lbrace e\\in \\mathbf {s}^{-1}(v)\\mid e \\mbox{ is on } p_{ij^{\\prime }}\\mbox{ for some }j^{\\prime }\\in I_i\\rbrace .$ Then the following hold.", "If $v$ is regular and $P_p$ nonempty, then $P_p=\\mathbf {s}^{-1}(v).$ If $v$ is an infinite emitter and $P_p$ nonempty, then there is $j^{\\prime }\\in I_i$ such that $h_{j^{\\prime }}=q_Z^v$ for some $Z$ such that $P_p\\subseteq Z.$ The relation $t_j=|p|+m_i$ holds if and only if $p=p_{ij}$ and $h_j=q_Z^v$ for some $Z$ implies $P_p\\subseteq Z.$ Before presenting the proof, let us motivate it by some examples.", "Example 2.3 In the graph below, $u\\rightarrow xv$ and $w\\rightarrow xv$ so $u+w\\rightarrow xv+xv.$ For this last relation, $k=2, l=2$ and one can take $I_1=\\lbrace 1\\rbrace , I_2=\\lbrace 2\\rbrace ,$ $p_{11}=e,$ and $p_{22}=f$ so condition (2) holds.", "${{\\bullet }^{u} [r]^e & {\\bullet }^{v} & {\\bullet }^{w}[l]_f}$ By condition (2) also, $u\\rightarrow x^2v+a$ fails for any $a$ since there is no path of length 2 from $u$ to $v.$ In the graph below, $v\\rightarrow xu+xw.$ For this relation, $k=1, l=2$ and one can take $I_1=\\lbrace 1,2\\rbrace ,$ $p_{11}=e,$ and $p_{12}=f$ so condition (2) holds.", "${{\\bullet }^{u} & {\\bullet }^{v} [r]^f[l]_e & {\\bullet }^{w}}$ Although $v$ connects to $w$ by a path of length one, $v\\rightarrow \\alpha w$ fails for any $\\alpha \\in \\mathbb {Z}^+[\\Gamma ]$ since the path from $v$ to $w$ has a bifurcation towards $u$ so $u$ must appear in any “result” obtained following a path from $v$ to $w$ by condition (2)(i).", "The relation $v_0\\rightarrow a$ fails for any $a$ with $\\operatorname{supp}(a)$ consisting of sinks only in the graph below.", "${\\bullet & \\bullet & \\bullet & & \\\\\\bullet _{v_0} [r][u] & \\bullet _{v_1} [r] [u] & \\bullet _{v_2} [r][u] & \\bullet _{v_3} @{.>}[r] @{.>}[u] &}$ Indeed, all paths from $v_0$ to finitely many sinks have a bifurcation on a path which does not end in any sink.", "Hence, if $v_0\\rightarrow a$ then $a$ necessarily has $v_i$ in its support for some $i\\ge 0.$ If $v\\rightarrow ^n a$ for $n>0$ in the graph below, ${{\\bullet }^{v} @{.}", "@/_1pc/ [r] _{\\mbox{ } } @/_/ [r] [r] @/^/ [r] @/^1pc/ [r] & {\\bullet }^{w}}$ condition (2) implies the existence of an improper vertex in $\\operatorname{supp}(a).$ Hence, $v\\rightarrow \\alpha w$ fails for any $\\alpha \\in \\mathbb {Z}^+[\\Gamma ]$ .", "Let us show direction $\\Rightarrow $ by induction on $n$ for $a\\rightarrow ^nb.$ If $n=0,$ then $a=b$ so $k=l$ and one can permute the monomials in the normal representation of $b$ if necessary to get that $t_i=m_i$ for all $i=1,\\ldots , k.$ In this case, one can take $I_i=\\lbrace i\\rbrace $ and $p_{ii}$ to be the trivial path which connects $g_i$ to $g_i$ for all $i=1,\\ldots , k.$ In this case any prefix $p$ of $p_{ij}$ is trivial and relation $t_i=|p_{ij}|+m_i=|p|+m_i$ holds.", "Since $P_p=\\emptyset ,$ conditions (i) to (iii) hold.", "Considering the case $n=1$ shortens the arguments in the inductive step.", "If $n=1,$ reorder the terms of the normal representation of $a$ if necessary to assume that $b$ is obtained by applying an axiom to $x^{m_k}g_k$ and let $x^{m_k}\\mathbf {r}(g_k)$ denote the result of this application.", "Thus, $g_k$ is not a sink.", "Reorder the terms of the normal representation of $b$ to have that $b=\\sum _{i=1}^{k-1} x^{m_i}g_i+x^{m_k}\\mathbf {r}(g_k)$ and let $x^{m_k}\\mathbf {r}(g_k)=\\sum _{j\\in J}x^{t_j}h_j$ for some finite subset $J$ of $\\lbrace 1,\\ldots ,l\\rbrace .$ Let $I_i=\\lbrace i\\rbrace $ for $i=1,\\ldots , k-1$ and $p_{ii}$ be the trivial path which connects $g_i$ and $g_i$ if $k>1.$ Let $I_k=J$ and $p_{kj}$ be the path (of length zero or one) which connects $g_k$ and $h_j$ .", "Since $g_k$ is not a sink, there are just three possible cases, listed below, for $g_k$ .", "$g_k$ is a regular vertex $v.$ In this case, $|J|=|\\mathbf {s}^{-1}(v)|$ and we can label the elements of $\\mathbf {s}^{-1}(v)$ such that $h_j=\\mathbf {r}(e_j)$ for $j\\in J.$ Then $x^{t_j}h_j=x^{m_k+1}\\mathbf {r}(e_j)$ and so $t_j=m_k+1.$ Let $p_{kj}=e_j.$ If $p$ is a prefix of $e_j,$ then either $p=v$ in which case $t_j>|p|+m_i$ and $P_p=\\mathbf {s}^{-1}(v),$ or $p=e_j$ in which case $t_j=|p|+m_i.$ In this case, if $\\mathbf {r}(e_j)$ is regular and $\\mathbf {r}(e_j)\\ne v,$ then $P_p=\\emptyset $ and if $\\mathbf {r}(e_j)=v,$ then $P_p=\\mathbf {s}^{-1}(v).$ $g_k$ is an infinite emitter $v.$ In this case, $\\mathbf {r}(g_k)=q_Z+\\sum _{e\\in Z}x\\mathbf {r}(e)$ for some $Z$ and $|J|=|Z|+1.$ We can label the elements of $Z$ such that $h_j=\\mathbf {r}(e_j)$ for $j\\in J-{j_0}$ and $h_{j_0}=q_Z.$ Thus, $t_{j_0}=m_k$ and $t_j=m_k+1$ for $j\\in J-\\lbrace j_0\\rbrace .$ Let $p_{kj_0}=v$ and $p_{kj}=e_j$ so that $|p_{kj}|=t_j-m_k$ for all $j\\in J.$ If $p$ is a prefix of $p_{kj_0}=v,$ then $p=v$ , $t_j=|p|+m_i,$ $h_{j_0}=q_Z^v$ and $P_p=Z.$ If $p$ is a prefix of $e_j,$ then either $p=v$ or $p=e_j.$ In the first case, $t_j>|p|+m_i,$ and condition (ii) holds with $j=j_0.$ In the second case, $t_j=|p|+m_i,$ if $\\mathbf {r}(e_j)$ is regular, then $\\mathbf {r}(e_j)\\ne v$ and so $P_p=\\emptyset .$ For $j\\ne j_0,$ $h_j$ is a proper vertex and $h_{j_0}=q_Z$ with $P_p=Z.$ Thus, condition (iii) holds.", "$g_k$ is an improper vertex $q^v_Z.$ In this case, $\\mathbf {r}(g_k)=q_W+\\sum _{e\\in W-Z}x\\mathbf {r}(e)$ for some $W\\supsetneq Z$ and $|J|=|W-Z|+1.$ We can label the elements of $W-Z$ such that $h_j=\\mathbf {r}(e_j)$ for $j\\in J-{j_0}$ and $h_{j_0}=q_W.$ Thus, $t_{j_0}=m_k$ and $t_j=m_k+1$ for $j\\in J-\\lbrace j_0\\rbrace .$ Let $p_{kj_0}=v$ and $p_{kj}=e_j$ so that $|p_{kj}|=t_j-m_k$ for all $j\\in J.$ If $p$ is a prefix of $p_{kj_0}=v,$ then $p=v$ , $t_j=|p|+m_i,$ $h_{j_0}=q_W,$ and $P_p=W-Z\\subseteq W.$ If $p$ is a prefix of $e_j,$ then either $p=v$ or $p=e_j.$ In the first case, $t_j>|p|+m_i,$ and condition (ii) holds with $j=j_0.$ In the second case, $t_j=|p|+m_i,$ if $\\mathbf {r}(e_j)$ is regular, then $\\mathbf {r}(e_j)\\ne v$ and so $P_p=\\emptyset .$ For $j\\ne j_0,$ $h_j$ is a proper vertex and $h_{j_0}=q_W$ with $P_p=W-Z\\subseteq W.$ Thus, condition (iii) holds.", "By construction, $b=\\sum _{i=1}^{k-1} x^{m_i}g_i+\\sum _{j\\in J}x^{t_j}h_j =\\sum _{i=1}^{k-1} x^{m_i+|p_{ii}|}g_i+\\sum _{j\\in I_k}x^{m_k+|p_{kj}|}h_j=\\sum _{i=1}^k \\sum _{j\\in I_i}x^{m_i+|p_{ij}|}h_{j}.$ Assuming the induction hypothesis, let us consider a sequence $a_0=a\\rightarrow _1 a_1\\rightarrow _1\\ldots \\rightarrow _1 a_n=b.$ Let $a_{n-1}=\\sum _{j^{\\prime }=1}^{l^{\\prime }} x^{t^{\\prime }_{j^{\\prime }}}h^{\\prime }_{j^{\\prime }}.$ By induction hypothesis, there is a partition $\\lbrace I^{\\prime }_1, \\ldots , I^{\\prime }_k\\rbrace $ of $\\lbrace 1, \\ldots , l^{\\prime }\\rbrace $ and finitely many paths $p_{ij^{\\prime }}, j^{\\prime }\\in I^{\\prime }_i, i=1,\\ldots , k,$ such that $g_i\\rightsquigarrow ^{p_{ij^{\\prime }}}h^{\\prime }_{j^{\\prime }},$ $|p_{ij^{\\prime }}|=t^{\\prime }_{j^{\\prime }}-m_i,$ and the required conditions hold for any prefix of $p_{ij^{\\prime }}$ for all $j^{\\prime }\\in I^{\\prime }_i$ and $i=1,\\ldots , k.$ The element $b$ is obtained from $a_{n-1}$ by application of one of the axioms to exactly one monomial $x^{t^{\\prime }_{j^{\\prime }}}h^{\\prime }_{j^{\\prime }}.$ Reordering the terms of $a_{n-1}$ if necessary, we can assume that it is the last one $x^{t^{\\prime }_{l^{\\prime }}}h^{\\prime }_{l^{\\prime }}.$ Reorder the terms of $b$ if necessary to have that $b=\\sum _{j^{\\prime }=1}^{l^{\\prime }} x^{t^{\\prime }_{j^{\\prime }}}h^{\\prime }_{j^{\\prime }}+x^{t^{\\prime }_{l^{\\prime }}}\\mathbf {r}(h^{\\prime }_{l^{\\prime }})$ and let $x^{t^{\\prime }_{l^{\\prime }}}\\mathbf {r}(h^{\\prime }_{l^{\\prime }})=\\sum _{j\\in J}x^{t_j}h_j$ for some finite subset $J$ of $\\lbrace 1,\\ldots ,l\\rbrace .$ By construction, we have that $l=l^{\\prime }+|J|$ and that $l^{\\prime }$ is in $I^{\\prime }_{i_0}$ for exactly one $i_0.$ So we let $I_i=I^{\\prime }_i, \\mbox{ if }i\\ne i_0,\\mbox{ and }I_{i_0}=J.$ If $i\\ne i_0,$ for each $j\\in I_i,$ $x^{t_j}h_j=x^{t^{\\prime }_{j^{\\prime }}}h^{\\prime }_{j^{\\prime }}$ for exactly one $j^{\\prime }\\in I^{\\prime }_i.$ So, for such $j$ and $j^{\\prime },$ we let $p_{ij}=p_{ij^{\\prime }}$ so that $|p_{ij}|=|p_{ij^{\\prime }}|=t^{\\prime }_{j^{\\prime }}-m_i=t_j-m_i.$ For $i_0,$ we let $p_{i_0j}$ be the concatenation of $p_{i_0l^{\\prime }}$ and the path $p_{l^{\\prime }j}$ constructed as in the case $n=1$ for $h^{\\prime }_{l^{\\prime }}$ and $h_j$ for $j\\in J=I_{i_0}.$ Since $g_{i_0}\\rightsquigarrow ^{p_{i_0l^{\\prime }}} h^{\\prime }_{l^{\\prime }}$ and $h^{\\prime }_{l^{\\prime }}\\rightsquigarrow ^{p_{l^{\\prime }j}}h_j$ for all $j\\in J=I_{i_0},$ we have that $g_{i_0}\\rightsquigarrow ^{p_{i_0j}}h_j$ for all $j\\in I_{i_0}.$ We have that $|p_{i_0l^{\\prime }}|=t^{\\prime }_{l^{\\prime }}-m_{i_0}$ and $|p_{l^{\\prime }j}|=t_j-t^{\\prime }_{l^{\\prime }}$ and so $|p_{i_0j}|=|p_{i_0l^{\\prime }}|+|p_{l^{\\prime }j}|=t^{\\prime }_{l^{\\prime }}-m_{i_0}+t_j-t^{\\prime }_{l^{\\prime }}=t_j-m_{i_0}\\;\\;\\mbox{ and }$ $b=\\sum _{j=1}^l x^{t_j}h_{j}=\\sum _{j^{\\prime }=1}^{l^{\\prime }} x^{t^{\\prime }_{j^{\\prime }}}h^{\\prime }_{j^{\\prime }}+\\sum _{j\\in J}x^{t_j}h_j=\\sum _{i=1, i\\ne i_0}^k \\sum _{j\\in I_i}x^{m_i+|p_{ij}|}h_{j}+ \\sum _{j\\in I_{i_0}}x^{m_{i_0}+|p_{i_0j}|}g_{i_0}=\\sum _{i=1}^k \\sum _{j\\in I_i}x^{m_i+|p_{ij}|}h_{j}.$ If $p$ is a prefix of $p_{i_0j},$ then it is either a prefix of $p_{i_0l^{\\prime }}$ or $p=p_{i_0l^{\\prime }}q$ for some prefix $q$ of $p_{l^{\\prime }j}$ and one of the following three cases holds: first, $p$ is a proper prefix of $p_{i_0l^{\\prime }},$ second, $q$ is a proper prefix of $p_{l^{\\prime }j}$ or, third, $q=p_{l^{\\prime }j}$ thus $p=p_{i_0j}.$ In the first case, $t^{\\prime }_{l^{\\prime }}>|p|+m_{i_0}$ and so $t_j\\ge t^{\\prime }_{l^{\\prime }}>|p|+m_{i_0}.$ In the second case, $t_j>|q|+t^{\\prime }_{l^{\\prime }}=|q|+|p_{i_0l^{\\prime }}|+m_{i_0}=|p|+m_{i_0}.$ In the last case, $t_j=|p|+m_{i_0}$ and if $h_j=q_Z^v$ for some $Z$ then $P_p \\subseteq Z$ since this condition holds for $a_{n-1}\\rightarrow _1 b$ by the first induction step.", "In all three cases, if $\\mathbf {r}(p)$ is regular and $P_p\\ne \\emptyset ,$ we can use induction hypothesis to conclude that $P_p=\\mathbf {s}^{-1}(\\mathbf {r}(p))$ and, if $\\mathbf {r}(p)$ is an infinite emitter $v$ and $P_p\\ne \\emptyset ,$ we can use induction hypothesis to conclude that there is $j^{\\prime }$ such that $h_{j^{\\prime }}=q_Z^v$ for some $Z$ such that $P_p\\subseteq Z.$ Thus, in any case, conditions (i) to (iii) hold.", "Let us use induction on $k$ to show direction $\\Leftarrow .$ If $k=1$ and $a=x^mg,$ let $p_j, j=1,\\ldots , l$ denote the paths which exist by condition (2).", "We show the claim using induction on $n=\\sum _{j=1}^l|p_j|.$ If this length is zero, then we claim that $b=a.$ Indeed, since $|p_j|=0,$ the relation $g\\rightsquigarrow ^{p_j}h_j$ implies that either $g=h_j,$ or $g=v$ for some infinite emitter $v$ and $h_j=q_Z^v,$ or that $g_i=q_W^v$ and $h_j=q_Z^v$ for some $v$ and $W\\subsetneq Z.$ However, in the second and third case we would have that $P_v\\subseteq Z$ by condition (2) so there would have to be some paths $p_{j^{\\prime }}$ of length at least one which cannot happen since $n=0$ .", "Hence, $a=b$ and, thus, $a\\rightarrow b.$ Assuming the induction hypothesis, let $n=\\sum _{j=1}^l|p_j|>0.$ Since $n>0,$ $a\\ne b$ and there is $j=1,\\ldots , l$ such that $|p_j|>0.$ If $p_j=e_0p$ for an edge $e_0$ and a path $p,$ let $v=\\mathbf {s}(e_0).$ Since $e_0\\in P_v,$ $P_v\\ne \\emptyset .$ We have exactly three possibilities for $g,$ listed below.", "$g=v$ is regular.", "Since $P_v$ is nonempty, $P_v=\\mathbf {s}^{-1}(v)$ by (i).", "Let $a_1=\\mathbf {r}(v)=\\sum _{e\\in P_v}x^{m+1}\\mathbf {r}(e).$ Note that $a\\rightarrow _1 a_1$ by (A1).", "We claim that condition (2) holds for $a_1$ and $b.$ For $e\\in P_v=\\mathbf {s}^{-1}(v),$ there is some $j=1,\\ldots , l$ such that $p_j=eq_j$ for some path $q_j$ and so the set $I_e=\\lbrace j\\in \\lbrace 1, \\ldots , l\\rbrace \\mid e\\mbox{ is the first edge of }p_j\\rbrace $ is nonempty.", "Since the first edge of $p_j$ is in $P_v=\\mathbf {s}^{-1}(v)$ for any $j,$ we have that $\\bigcup _{e\\in P_v} I_e=\\lbrace 1, \\ldots , l\\rbrace .$ If $j\\in I_e\\cap I_{e^{\\prime }},$ then $e=e^{\\prime }$ since the first edge of a path is unique.", "As $t_j=|p_j|+m,$ we have that $t_j=|q_j|+1+m$ and $b=\\sum _{j=1}^l x^{t_j}h_j=\\sum _{j=1}^l x^{|p_j|+m}h_j=\\sum _{j=1}^l x^{|q_j|+1+m}h_j.$ If $q$ is a prefix of $q_j,$ then $eq$ is a prefix of $p_j$ and conditions (i) to (iii) hold for $q$ because they hold for $eq.$ Thus, we have that $a_1\\rightarrow b$ by induction hypothesis.", "Since $a\\rightarrow _1 a_1,$ we have that $a\\rightarrow b.$ $g=v$ is an infinite emitter.", "In this case, let $a_1=x^m q_{P_v}+\\sum _{e\\in P_v}x^{m+1}\\mathbf {r}(e).$ So that $a\\rightarrow _1 a_1$ by (A2).", "Since $P_v\\ne \\emptyset ,$ there is $j$ such that $h_j=q_Z^v$ for some $Z$ with $P_v\\subseteq Z$ by (ii).", "By (iii), such $j$ can be found so that $t_j=|p_j|+m.$ Reorder the terms of $b$ if necessary so that we can assume that $j=1.$ We check that condition (2) holds for $a_1$ and $b.$ For $e\\in P_v$ , there is $j=2,\\ldots , l$ such that $p_j=eq_j$ for some path $q_j$ and so the sets $I_e, e\\in P_v,$ defined as in the previous case, are nonempty and mutually disjoint.", "Let $I_1=\\lbrace 1\\rbrace $ and $q_1=p_1.$ Since the first edge of $p_j$ is in $P_v$ for every $j=2,\\ldots , l,$ $\\bigcup _{e\\in P_v} I_e=\\lbrace 2, \\ldots , l\\rbrace ,$ so $I_1\\cup \\bigcup _{e\\in P_v} I_e=\\lbrace 1, 2, \\ldots , l\\rbrace .$ Hence, $\\lbrace I_1\\rbrace \\cup \\lbrace I_e| e\\in P_v\\rbrace $ is a partition of $\\lbrace 1,\\ldots ,l\\rbrace .$ For $j=2,\\ldots , l,$ $t_j=|p_j|+m=|q_j|+1+m,$ $t_1=|p_1|+m=|q_1|+m,$ and $b=\\sum _{j=1}^l x^{|p_j|+m}h_j=x^{|q_1|+m}h_1+\\sum _{j=2}^l x^{|q_j|+1+m}h_j.$ If $q$ is a prefix of $q_j$ for $j>1,$ then $e_iq$ is a prefix of $p_j$ and conditions (i) to (iii) hold for $q$ because they hold for $e_iq.$ If $q$ is a prefix of $q_1=p_1,$ then the requirements also hold.", "Thus, $a_1\\rightarrow b$ by induction hypothesis.", "Since $a\\rightarrow _1 a_1,$ we have that $a\\rightarrow b.$ $g=q_Z^v$ for some $Z.$ In this case, $P_v$ is a proper superset of $Z.$ Let $a_1=x^m q_{P_v}+\\sum _{e\\in P_v-Z}x^{m+1}\\mathbf {r}(e)$ so that $a\\rightarrow _1 a_1$ by (A3).", "By (ii), there is $j$ such that $h_j=q_W^v$ for some $W$ such that $P_v\\subseteq W$ and, by (iii), such $j$ can be found so that $t_j=|p_j|+m.$ Reorder the terms of $b$ if necessary so that we can assume that $j=1.$ We check that condition (2) holds for $a_1$ and $b.$ For $e\\in P_v-Z,$ there is some $j=2,\\ldots , l$ such that $p_j=eq_j$ for some path $q_j$ and so the sets $I_e, e\\in P_v-Z,$ defined as in the previous cases, are nonempty and mutually disjoint.", "If $I_1=\\lbrace 1\\rbrace $ and $q_1=p_1,$ one shows that $\\lbrace I_1\\rbrace \\cup \\lbrace I_e \\mid e\\in P_v-Z\\rbrace $ is a partition of $\\lbrace 1,\\ldots ,l\\rbrace $ as in the previous case.", "Since $t_j=|p_j|+m=|q_j|+1+m$ for $j=2,\\ldots , l$ and $t_1=|p_1|+m=|q_1|+m,$ $b=\\sum _{j=1}^l x^{|p_j|+m}h_j=x^{|q_1|+m}h_1+\\sum _{j=2}^l x^{|q_j|+1+m}h_j.$ The requirements on prefixes of $q_j$ can be checked just as in the previous case.", "Thus, we have that $a_1\\rightarrow b$ by induction hypothesis.", "Since $a\\rightarrow _1 a_1,$ we have that $a\\rightarrow b.$ This concludes the proof of the case $k=1$ .", "Assuming the induction hypothesis, let us show the claim for $a$ with $k$ terms in its normal decomposition.", "Note that if condition (2) holds, then it holds for $a^{\\prime }=\\sum _{i=1}^{k-1}x^{m_i}g_i$ and $b^{\\prime }=\\sum _{i=1}^{k-1}\\sum _{j\\in I_i}x^{m_i+|p_{ij}|}h_j$ and for $x^{m_k}g_k$ and $\\sum _{j\\in I_k}x^{m_k+|p_{kj}|}h_j.$ By the induction hypothesis, we have that $a^{\\prime }\\rightarrow b^{\\prime }$ and that $x^{m_k}g_k\\rightarrow \\sum _{j\\in I_k}x^{m_k+|p_{kj}|}h_h.$ Hence, $a=a^{\\prime }+x^{m_k}g_k\\rightarrow b=b^{\\prime }+\\sum _{j\\in I_k}x^{m_k+|p_{kj}|}h_j.$ We show two corollaries of Proposition REF which we use in Section REF .", "Recall that Definition REF implies that $g\\rightsquigarrow ^ph$ implies $g\\rightarrow x^{|p|}h+a$ for some $a.$ By the first corollary, the converse also holds.", "Corollary 2.4 Let $g,h$ be generators of $F_E^\\Gamma ,$ $a\\in F_E^\\Gamma ,$ and $m$ a nonnegative integer.", "Then $g\\rightarrow x^mh+a$ holds if and only if there is a path $p$ of length $m$ such that $g\\rightsquigarrow ^ph.$ If $g\\rightarrow x^mh+a$ , condition (2) of Proposition REF holds by Proposition REF , so there is a path $p$ of length $m$ from $g$ to $h.$ The converse holds by Definition REF (see the sentence following Definition REF ).", "By the next corollary, if $a\\rightarrow b,$ then each monomial of $b$ is obtained by a monomial of $a.$ This complements the Confluence Lemma.", "Corollary 2.5 If $g$ is a generator of $F_E^\\Gamma ,$ $a,b\\in F_E^\\Gamma ,$ and $m$ an integer, then $a\\rightarrow x^mg+b$ implies that there is $h\\in \\operatorname{supp}(a)$ and $k\\le m$ such that $x^kh$ is a monomial of $a$ and that $x^kh\\rightarrow x^mg+c$ for some $c\\in F_E^\\Gamma .$ If $a\\rightarrow x^mg+b$ holds, Proposition REF guarantees the existence of a monomial $x^kh$ of $a$ and a path $p$ such that $k+|p|=m$ and such that $h\\rightsquigarrow ^p g.$ Hence, $m-k=|p|\\ge 0$ and $x^kh\\rightarrow x^{k+|p|}g+c=x^mg+c$ for some $c\\in F_E^\\Gamma $ by Corollary REF ." ], [ "Connectivity of the supports", "Next, we associate the relation $a\\rightarrow b$ to the properties of the supports of $a$ and $b$ .", "The next definition is condition (2) of Proposition REF stripped down from any mention of degrees.", "Definition 2.6 Let $G$ and $H$ be finite and nonempty sets of generators of $F_E^\\Gamma .$ We write $G\\rightarrow H$ if there are $k\\ge |G|$ and $l\\ge |H|$ such that the elements of $G$ and $H$ can be indexed as $g_1,\\ldots , g_k$ and $h_1,\\ldots , h_l$ (with repetitions allowed) respectively and there is a partition $\\lbrace I_1, \\ldots , I_k\\rbrace $ of $\\lbrace 1,\\ldots , l\\rbrace $ and finitely many paths $p_{ij}, j\\in I_i, i=1,\\ldots , k,$ such that $g_i\\rightsquigarrow ^{p_{ij}}h_j$ for all $j\\in I_i, i=1,\\ldots , k$ and such that if $p$ is a prefix of $p_{ij}$ and $P_p$ is as in condition (2) of Proposition REF then (2)(i) and (2)(ii) of Proposition REF and condition (iii) below hold.", "If $v$ is an infinite emitter and $h_j=q_Z^v$ for some $Z,$ then $P_{p_{ij}}\\subseteq Z.$ Corollary 2.7 If $a,b\\in F_E^\\Gamma -\\lbrace 0\\rbrace ,$ then $a\\rightarrow b$ implies $\\operatorname{supp}(a)\\rightarrow \\operatorname{supp}(b).$ Let $a,b\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ be such that $\\operatorname{supp}(a)\\rightarrow \\operatorname{supp}(b).$ Then, there is $c\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ such that $\\operatorname{supp}(c)\\subseteq \\operatorname{supp}(b)$ and that $a\\rightarrow c.$ Part (1) directly follows from Proposition REF .", "To show part (2), assume that $a=\\sum _{i=1}^{k} x^{m_i}g_i$ and $b=\\sum _{j=1}^{l} x^{t_j}h_j$ be such that $\\operatorname{supp}(a)\\rightarrow \\operatorname{supp}(b).$ Let $m,n$ be the cardinalities of $\\operatorname{supp}(a)$ and $\\operatorname{supp}(b)$ respectively and $k^{\\prime }\\ge m, l^{\\prime }\\ge n,$ $\\lbrace I_1, \\ldots , I_{k^{\\prime }}\\rbrace $ and $p_{i^{\\prime }j^{\\prime }}, j^{\\prime }\\in I_{i^{\\prime }}, i^{\\prime }=1,\\ldots , k^{\\prime }$ be as in Definition REF for $\\operatorname{supp}(a)$ and $\\operatorname{supp}(b)$ .", "Then, we let $a^{\\prime }=\\sum _{i^{\\prime }=1}^{k^{\\prime }} g_{i^{\\prime }},\\;\\; \\;\\;b^{\\prime }_{i^{\\prime }}=\\sum _{j^{\\prime }\\in I_{i^{\\prime }}}x^{|p_{i^{\\prime }j^{\\prime }}|}h_{j^{\\prime }}\\mbox{ for }i^{\\prime }=1,\\ldots , k^{\\prime },\\;\\;\\mbox{ and }\\;\\;b^{\\prime }=\\sum _{i^{\\prime }=1}^{k^{\\prime }}b^{\\prime }_{i^{\\prime }}.$ By construction, $\\operatorname{supp}(a^{\\prime })=\\operatorname{supp}(a),$ $\\operatorname{supp}(b^{\\prime })=\\operatorname{supp}(b)$ and $g_{i^{\\prime }}\\rightarrow b_{i^{\\prime }}$ so that $a^{\\prime }\\rightarrow b^{\\prime }$ holds by Proposition REF .", "For any $i=1,\\ldots , k,$ there is $i^{\\prime }=1,\\ldots , k^{\\prime }$ such that $g_i=g_{i^{\\prime }}.$ For such $i,$ let $c_i=\\sum _{j^{\\prime }\\in I_{i^{\\prime }}}x^{m_i+|p_{i^{\\prime }j^{\\prime }}|}h_{j^{\\prime }}\\;\\;\\mbox{ and let }\\;\\;c=\\sum _{i=1}^{k}c_i.$ We have that $\\operatorname{supp}(c)\\subseteq \\operatorname{supp}(b^{\\prime })=\\operatorname{supp}(b)$ and $x^{m_i}g_i\\rightarrow c_i$ so that $a=\\sum _{i=1}^{k} x^{m_i}g_i\\rightarrow c=\\sum _{i=1}^{k}c_i.$ The relation $x^{m_i}g_i\\rightarrow c_i$ also implies that $c_i\\ne 0$ so $c\\ne 0.$ We note that the converse of part (1) of Corollary REF does not have to hold.", "Also, for an element $c$ as in part (2) of Corollary REF , the relation $b\\sim c$ does not have to hold even if $\\operatorname{supp}(c)=\\operatorname{supp}(b)$ .", "Indeed, in the graph below, $v\\rightarrow xw$ so $\\lbrace v\\rbrace \\rightarrow \\lbrace w\\rbrace .$ ${{\\bullet }^{v} [r] & {\\bullet }^{w}}$ However, for $a=v$ and $b=w,$ we have that $\\operatorname{supp}(a)\\rightarrow \\operatorname{supp}(b)$ but $a\\rightarrow b$ fails since there are no paths of length zero from $v$ to $w.$ If $c=xw,$ then $\\operatorname{supp}(c)=\\operatorname{supp}(b),$ but we do not have that $b=w\\sim c=xw$ since $w$ is a sink and the relation $w\\sim d$ for some $d$ implies $d=w$ or $d=x^{-1}v$ by the Confluence Lemma.", "Also, using Theorem REF , it is direct that $w\\sim xw$ cannot hold since $w$ is not a periodic element." ], [ "Connecting using (A1) only", "To emphasize that $a\\rightarrow b$ is such that only (A1) is used, we write $a\\rightarrow ^{\\hspace{-11.9501pt}A1}b.$ If $E$ is a row-finite graph, then $\\rightarrow ^{\\hspace{-11.9501pt}A1}$ is just the relation $\\rightarrow .$ If $V$ is a finite and nonempty set of regular vertices and $W$ a finite and nonempty set of proper vertices such that $V\\rightarrow W$ , we write $V\\rightarrow ^{\\hspace{-11.9501pt}A1}W.$ Corollary REF implies the corollary below.", "Corollary 2.8 Let $a,b\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ such that $a\\rightarrow ^{\\hspace{-11.9501pt}A1}b$ and that $\\operatorname{supp}(a)$ consists of regular vertices.", "Then $\\operatorname{supp}(a)\\rightarrow ^{\\hspace{-11.9501pt}A1}\\operatorname{supp}(b).$ Let $a,b\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ be such that $\\operatorname{supp}(a)\\rightarrow ^{\\hspace{-11.9501pt}A1}\\operatorname{supp}(b).$ Then, there is $c\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ such that $\\operatorname{supp}(c)\\subseteq \\operatorname{supp}(b)$ and that $a\\rightarrow ^{\\hspace{-11.9501pt}A1}c.$ To show (1), assume that $a\\rightarrow ^{\\hspace{-11.9501pt}A1}b$ and that $\\operatorname{supp}(a)$ consists of regular vertices.", "By Corollary REF , $\\operatorname{supp}(a)\\rightarrow \\operatorname{supp}(b).$ Since only (A1) is used in $a\\rightarrow ^{\\hspace{-11.9501pt}A1}b$ , $\\operatorname{supp}(b)$ does not contain any improper vertices, so $\\operatorname{supp}(a)\\rightarrow ^{\\hspace{-11.9501pt}A1}\\operatorname{supp}(b)$ by definition of $\\rightarrow ^{\\hspace{-11.9501pt}A1}$ for sets of vertices.", "To show (2), let $\\operatorname{supp}(a)\\rightarrow ^{\\hspace{-11.9501pt}A1}\\operatorname{supp}(b).$ By Corollary REF , there is $c\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ such that $\\operatorname{supp}(c)\\subseteq \\operatorname{supp}(b)$ and $a\\rightarrow c.$ Since the support of $a$ consists of regular vertices and the support of $b,$ thus of $c$ as well, of proper vertices, only (A1) can be applied in a sequence for $a\\rightarrow c$ .", "Hence, $a\\rightarrow ^{\\hspace{-11.9501pt}A1}c.$" ], [ "Cancellative property", "First, we show that the monoid $M_E^\\Gamma $ is cancellative by a direct proof.", "This was shown in [4] using the graph covering.", "Note that $M_E^\\Delta $ may not be cancellative for a group $\\Delta \\ne \\Gamma .$ In particular, if $E$ is a graph with a cycle with an exit and $\\Delta $ is trivial, then $M_E^\\Delta $ is not cancellative by [4].", "Proposition 3.1 The $\\Gamma $ -monoid $M_E^\\Gamma $ is cancellative.", "Assume that $a+c\\sim b+d$ holds in $F_E^\\Gamma $ for some $d\\in F^\\Gamma _E$ such that $c\\sim d.$ So, we have that $a+c\\sim b+d\\sim b+c.$ We show that $a\\sim b$ using induction on $n$ for $a+c\\sim ^n b+c.$ If $n=1,$ then either $a+c\\rightarrow _1 b+c$ or $b+c\\rightarrow _1 a+c.$ In the first case, there is a generator $g$ in the support of $a$ or $c$ such that $b+c$ is obtained by replacing a summand $x^mg$ of $a+c$ by $x^m\\mathbf {r}(g)$ and keeping the rest of the monomials intact.", "By the nature of the three axioms, the number of monomials of the form $x^m g$ in $a+c$ is larger than in $b+c$ and each of the monomials in $x^m\\mathbf {r}(g)$ appears one time less in $a+c$ than in $b+c.$ Since these terms appear equal number of times in $c,$ this means that $a$ contains a monomial $x^mg$ and that $x^m\\mathbf {r}(g)$ is a summand of $b.$ Hence, $a=a^{\\prime }+x^mg$ and $b=a^{\\prime }+x^m\\mathbf {r}(g)$ for some $a^{\\prime }\\in F_E^\\Gamma $ so that $a\\rightarrow _1 b.$ The case $b+c\\rightarrow _1 a+c$ is similar and the induction step is analogous.", "Remark 3.2 Proposition REF highlights an important difference between $M_E$ and $M_E^\\Gamma $ : while $M_E$ can be much larger than the positive cone of $G_E,$ the monoid $M_E^\\Gamma $ is equal to the positive cone of $G_E^\\Gamma .$ Thus, the monoid $M_E$ can carry some information which is lost under formation of its Grothendieck group but $M_E^\\Gamma $ carries no additional information than $G_E^\\Gamma .$ In other words, using the language of [11], the group $G_E^\\Gamma $ is equally “talented” as the monoid $M_E^\\Gamma .$" ], [ "The order", "The relation $\\sim $ on the monoid $F_E^\\Gamma $ enables one to define a relation $\\precsim $ as follows.", "$a\\precsim b\\mbox{ if there is $c\\in F^\\Gamma _E$ such that }a+c\\sim b$ for all $a,b\\in F^\\Gamma _E.$ If $a\\precsim b$ and $a\\nsim b$ , we write $a\\prec b.$ Using Proposition REF , it is direct to show that $a\\prec b$ is equivalent with $a+c\\sim b$ for some nonzero $c$ in $F_E^\\Gamma .$ The relation $\\precsim $ defines an order on $M_E^\\Gamma $ given by $[a]\\le [b]\\;\\;\\mbox{ if and only if }\\;\\;a\\precsim b.$ It is direct to show that $\\le $ is reflexive and transitive.", "The antisymmetry holds by Proposition REF .", "The relation $\\le $ induces an order on the Grothendieck group $G^\\Gamma _E.$ In [11], it is shown that $a\\prec x^n a$ is not possible for any $a$ and any positive $n$ if $E$ is row-finite.", "After the lemma below, we show that this statement holds for an arbitrary graph in Proposition REF .", "Lemma 3.3 If $a\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ is such that $a\\precsim x^na$ for some positive integer $n,$ then the following hold.", "No vertex in the support of $a$ is a sink.", "No vertex in the support of $a$ is an improper vertex.", "All vertices in the support of $a$ are regular (so $a$ is regular).", "Since $a\\precsim x^na,$ $a+b\\sim x^na$ for some $b\\in F_E^\\Gamma .$ Then $a+b+x^nb\\sim x^na+x^nb\\sim x^{2n}a$ so, by induction, $a+\\sum _{i=0}^kx^kb\\sim x^{(k+1)n}a.$ Hence, we can find $n$ large enough so that $n$ is larger than the degrees of all monomials in a normal representation of $a.$ Assume that $n$ is such and that $a+b\\sim x^na$ for some $b\\in F_E^\\Gamma .$ By the Confluence Lemma REF (2), there is $c\\in F^\\Gamma _E$ such that $a+b \\rightarrow c$ and $x^n a\\rightarrow c.$ (1) Assume that a sink $v$ is in $\\operatorname{supp}(a)$ and let $\\sum _{i=1}^l x^{m_i}v$ be the sum of all monomials in a normal representation of $a$ which contain $v.$ By construction, $m_i<n$ for every $i=1,\\ldots , l.$ Since the relation $\\rightarrow _1$ cannot be applied to $v,$ the relation $a+b\\rightarrow c$ implies that $x^{m_i}v$ is a summand of $c$ for every $i=1,\\ldots , l.$ On the other hand, the relation $x^n a\\rightarrow c$ implies that every monomial of $c$ has degree larger than or equal to $n$ so $x^{m_1}v$ cannot be a summand of $c.$ This is a contradiction.", "(2) Assume that an improper vertex $q^v_Z$ is in $\\operatorname{supp}(a)$ for some $v$ and some $Z.$ Let $\\sum _{i=1}^l x^{m_i}q_{Z_i}$ be the sum of all monomials in a normal representation of $a$ which contain $q^v_{Z_i}$ for some nonempty and finite $Z_i\\supseteq Z.$ Since an application of $\\rightarrow _1$ does not change the power of a monomial with $q^v_W$ for some $W\\supseteq Z,$ the relation $a+b\\rightarrow c$ implies that $c$ contains a summand of the form $\\sum _{i=1}^l x^{m_i}q_{W_i}$ for some $W_i\\supseteq Z_i, i=1,\\ldots , l.$ On the other hand, the relation $x^na\\rightarrow c$ implies that every monomial of $c$ has degree larger than or equal to $n$ so $x^{m_1}q_{W_1}$ cannot be a summand of $c.$ This is a contradiction.", "(3) By part (1), to show that a vertex $v$ in the support of $a$ is regular, it is sufficient to show that $v$ is not an infinite emitter.", "Assume that an infinite emitter $v$ is in the support of $a$ and let $\\sum _{i=1}^l x^{m_i}v$ be the sum of all monomials in a normal representation of $a$ which contain $v.$ Since axioms (A1) and (A3) are not applicable to any monomials with $v$ in them, the relation $a+b\\rightarrow c$ implies that $\\sum _{i=1}^l x^{m_i}g_i,$ where each $g_i$ is either $v$ or $q^v_Z$ for some $Z,$ is a summand in a normal representation of $c.$ On the other hand, the relation $x^n a\\rightarrow c$ implies that every monomial of $c$ has degree larger than or equal to $n$ so $x^{m_1}g_1$ cannot be a summand of $c$ which is a contradiction.", "Proposition 3.4 The relation $a\\prec x^na$ is not possible for any nonnegative $n$ and any $a\\in F_E^\\Gamma .$ Since $0\\prec 0$ is false, it is sufficient to consider $a\\ne 0.$ Also, since $a\\prec a$ is false, it is sufficient to consider positive $n.$ Assume that $a\\prec x^na$ for some positive $n$ and some nonzero $a\\in F_E^\\Gamma .$ By Lemma REF , all elements in the support of $a$ are regular and proper vertices.", "Let $m$ be the maximum of degrees of the monomials in a normal representation of $a.$ If a monomial $x^lv$ in a normal representation of $a$ is such that $l<m,$ apply (A1) to $x^lv$ to replace this monomial by $\\sum _{e\\in \\mathbf {s}^{-1}(v)}x^{l+1}\\mathbf {r}(e).$ We obtain an element $a_1$ such that $a_1\\sim a$ so the relation $a_1\\prec x^na_1$ also holds and, as a consequence, all vertices in the support of $a_1$ are regular also.", "Keep repeating this process until all monomials of some $a_k$ have the same degree $m$ so that we can write $a_k=x^m b$ where $b$ is a sum of regular vertices.", "Since $x^mb\\prec x^{n+m}b$ we have that $b\\prec x^nb$ and so $b+c\\sim x^nb$ for some nonzero $c\\in F_E^\\Gamma .$ By the Confluence Lemma REF (2), there is $d$ such that $b+c\\rightarrow d$ and $x^nb\\rightarrow d.$ The relation $x^nb\\rightarrow d$ implies that $x^{-n}d\\prec d$ so $d\\prec x^{n}d$ and all vertices in the support of $d$ are regular by Lemma REF .", "Using the same argument as when obtaining $x^mb$ from $a,$ we can show that there is an element $f$ such that $d\\rightarrow f$ and such that $f$ is a sum of monomials of the same degree $m^{\\prime }.$ Hence, $b+c\\rightarrow d\\rightarrow f$ and $x^nb\\rightarrow d\\rightarrow f.$ Since $\\rightarrow $ either increases the degree of a monomial or leaves it the same, the relation $x^nb\\rightarrow f$ implies that $m^{\\prime }\\ge n>0.$ Let $h=x^{-n}f$ so that $h$ is a sum of monomials of the same nonnegative degree $m^{\\prime }-n$ and that $b+c\\rightarrow x^nh$ and $b\\rightarrow h.$ We use induction on the length of a sequence for $b\\rightarrow h$ to show that $h+c\\rightarrow x^nh.$ If $b=h,$ the claim holds.", "Assume that it holds for length smaller than $k$ and let $b=b_0\\rightarrow _1 b_1\\rightarrow _1\\ldots \\rightarrow _1 b_k=h.$ Since $b$ is regular, $b\\rightarrow _1 b_1$ is an application of (A1).", "Hence, $b=b^{\\prime }+v$ and $b_1=b^{\\prime }+\\sum _{e\\in \\mathbf {s}^{-1}(v)}x\\mathbf {r}(e)$ for some regular vertex $v.$ Since the degree of every monomial in $x^nh=f$ is strictly larger than zero, $v$ has to be changed in the process of obtaining $x^nh$ from $b+c=b^{\\prime }+v+c.$ Reorder the terms of the sequence for $b+c\\rightarrow x^nh$ so that an application of (A1) to $v$ is the first step.", "Hence, $b+c=b^{\\prime }+v+c\\rightarrow b^{\\prime }+\\sum _{e\\in \\mathbf {s}^{-1}(v)}x\\mathbf {r}(e)+c=b_1+c\\rightarrow x^nh.$ We can now apply the induction hypothesis to $b_1$ to obtain that $h+c\\rightarrow x^nh.$ Lastly, we show that the relation $h+c\\rightarrow x^nh$ leads to a contradiction.", "Indeed, since $h$ is a sum of monomials of the same nonnegative degree and $n$ is strictly larger than zero, we have that $h+c\\ne x^nh$ so at least one of the three axioms is used.", "If normal representations of $h$ and $c$ have $n_h$ and $n_c$ monomials respectively, then the number of terms in the resulting $x^nh$ is larger than or equal to $n_h+n_c.$ But since $x^nh$ has the same number of monomials as $h,$ we necessarily have that $n_c=0$ which implies that $c=0.$ This is a contradiction since $c$ is chosen to be nonzero such that $b+c\\sim x^nb.$" ], [ "Comparable, periodic, aperiodic and incomparable elements", "Proposition REF implies that there are just two possibilities for $a\\in F_E^\\Gamma :$ either $a\\succsim x^na$ for some positive $n$ or $a$ and $x^na$ are not comparable for any positive $n.$ In the case when $a\\succsim x^na$ for some positive $n$ we have that either $a\\sim x^na$ or $a\\succ x^na.$ We introduce the following terminology.", "Definition 3.5 Let $a\\in F_E^\\Gamma .$ If $a\\succsim x^na$ for some positive integer $n,$ the element $a$ is comparable.", "If $a\\sim x^na$ for some positive integer $n,$ the element $a$ is periodic.", "If $a\\succ x^na$ for some positive integer $n,$ the element $a$ is aperiodic.", "If $a$ and $x^na$ are not comparable for any positive integer $n,$ the element $a$ is incomparable.", "For $[a]\\in M_E^\\Gamma ,$ we say that $[a]$ is comparable, periodic, aperiodic or incomparable if any $b$ such that $a\\sim b$ is such.", "Note that 0 is periodic by this definition.", "An element of $F_E^\\Gamma $ clearly cannot be both comparable and incomparable.", "We also note that a comparable element of $F_E^\\Gamma $ cannot be both periodic and aperiodic.", "Indeed, if $x^m a\\sim a\\succ x^n a$ for some positive integers $m$ and $n,$ let $n$ be the least positive integer such that $a\\succ x^na$ .", "Since $x^m a\\succ x^n a$ implies $x^{m-n}a\\succ a,$ $m-n$ is negative by Proposition REF so $n>m$ .", "On the other hand, the relation $x^m a\\sim a\\succ x^na$ also implies that $a\\sim x^{-m}a\\succ x^{n-m}a$ so $n-m\\ge n$ by the assumption that $n$ is the smallest possible such that $a\\succ x^na.$ The relation $n-m\\ge n$ implies that $m\\le 0$ which is in contradiction with the assumption that $m$ is positive." ], [ "Stationary elements", "Next, we prove a series of claims which bring us to Theorem REF .", "Lemma REF leads us to the notion of a stationary element introduced in Definition REF .", "Lemma 3.6 Let $a\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ be such that $a\\sim x^na+b$ for some positive integer $n$ and some $b\\in F_E^\\Gamma .$ There are $c\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ and $d\\in F_E^\\Gamma $ such that $c\\rightarrow x^nc+d,$ $a\\rightarrow c$ and $b\\rightarrow d.$ Note that the assumption of the lemma is exactly that $a$ is comparable, the case $b=0$ corresponds exactly to the case that $a$ is periodic, and the case $b\\ne 0$ to the case that $a$ is aperiodic.", "Since $a\\sim x^na+b\\sim x^{2n}a+x^nb+b\\sim \\ldots ,$ we can choose $n$ as large as needed.", "Let us choose $n$ larger than the degree of every monomial in a normal representation of $a.$ By the Confluence Lemma REF (2), $a\\rightarrow f$ and $x^na+b\\rightarrow f$ and by the Refinement Lemma REF (1), $f=f_1+f_2$ such that $x^na\\rightarrow f_1$ and $b\\rightarrow f_2.$ Let $c=x^{-n}f_1$ so that $a\\rightarrow x^{-n}f_1=c$ and that $a\\rightarrow f= x^nc+f_2.$ We use induction on $k$ for $a\\rightarrow ^k c.$ If $k=0,$ then $a=c.$ Let $d=f_2$ so that $b\\rightarrow d.$ Assuming the induction hypothesis, let us consider $a\\rightarrow ^k c$ with $a=a_0\\rightarrow _1 a_1\\rightarrow _1\\ldots \\rightarrow _1 a_k=c.$ Let $a=a^{\\prime }+x^mg$ for some generator $g$ such that $a_1=a^{\\prime }+x^m\\mathbf {r}(g).$ Consider the following two cases for the relation $a\\rightarrow x^nc+f_2.$ There is an application of the same axiom used for $a\\rightarrow _1 a_1$ to $x^mg$ at some point such that $x^mg$ is not changed prior to this point.", "Changing the order of applications of axioms in the sequence for $a\\rightarrow x^nc+f_2,$ we can assume that this application of the axiom happened first.", "In this case $a\\rightarrow a_1\\rightarrow x^nc+f_2.$ Thus, we can apply the induction hypothesis to $a_1$ instead of $a$ and obtain the relation $c\\rightarrow x^nc+d$ for some $d$ such that $f_2\\rightarrow d.$ Hence, $b\\rightarrow f_2\\rightarrow d.$ There is no application of the axiom used for $a\\rightarrow _1 a_1$ to $x^mg$ at any point.", "Since $n$ is larger than $m,$ then $x^mg$ has to be a summand of $f_2.$ Say $f_2=d^{\\prime }+x^mg.$ Then $a=a^{\\prime }+x^mg\\rightarrow x^nc+d^{\\prime }+x^mg.$ Replacing the terms $x^mg$ by $x^{m}\\mathbf {r}(g)$ on both sides of the relation $\\rightarrow ,$ we obtain that $a_1=a^{\\prime }+x^{m}\\mathbf {r}(g)\\rightarrow x^nc+d^{\\prime }+x^{m}\\mathbf {r}(g).$ Since we have $a_1\\rightarrow ^{k-1} c,$ we can apply the induction hypothesis to $a_1$ and obtain that $c\\rightarrow x^nc+d$ for some $d$ such that $d^{\\prime }+x^{m}\\mathbf {r}(g)\\rightarrow d.$ Hence, $b\\rightarrow f_2=d^{\\prime }+x^mg\\rightarrow d^{\\prime }+x^{m}\\mathbf {r}(g)\\rightarrow d.$ The properties of an element such as element $c$ of Lemma REF are significant in the characterization of a comparable element so we assign a name to such an element.", "Definition 3.7 An element $a\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ is a stationary element if $a\\rightarrow x^na+b$ for some positive integer $n$ and some $b\\in F_E^\\Gamma .$ Example 3.8 If $E$ is the graph ${{\\bullet }^v [r] & {\\bullet }^w@(ru,rd) }\\;\\;\\;\\;,$ then $w$ is stationary since $w\\rightarrow xw$ .", "One can directly check that if $v\\rightarrow a$ for some $a\\in F_E^\\Gamma ,$ then either $a=v$ or $a$ is of the form $x^nw$ for some positive integer $n.$ Hence, $v$ is not stationary.", "Let $E$ be the Toeplitz graph $\\;\\;\\;\\;{{\\bullet }^v@(lu,ld) [r] & {\\bullet }^w}$ and $a=v+w\\in F_E^\\Gamma .$ Since $a=v+w\\rightarrow xv+xw+w=x(v+w)+w=xa+w,$ $a$ is stationary.", "Note that $b=v+xw$ has the same support as $a$ but $b$ is not stationary.", "Indeed, if $b\\rightarrow c,$ then $c=b$ or $c=x^nv+x^nw+x^{n-1}w+\\ldots +xw+w+xw$ for some positive $n.$ So, assuming that $b\\rightarrow x^nb+d$ for some $d$ and positive $n$ leads to a contradiction.", "We note also that adding $xv$ to $b,$ we obtain a stationary element again since it is a sum of stationary elements $x(v+w)$ and $v.$ The next lemma describes the support of a stationary element.", "Recall that a generator $g$ is on a cycle if $g\\rightsquigarrow ^p g$ for some $p$ with $|p|>0.$ Lemma 3.9 Let $a\\in F_E^\\Gamma $ be stationary such that $a\\rightarrow x^na+b$ for some positive integer $n$ and some $b\\in F_E^\\Gamma $ .", "For any positive integer $k,$ $a\\rightarrow x^{kn}a+\\sum _{i=0}^{k-1}x^{in}b.$ The support of $a$ contains an element which is on a cycle.", "Each element of the support of $a$ which is not on a cycle is on a path exiting a cycle which contains another element of $\\operatorname{supp}(a).$ This condition can be described also in terms of the tree $T(g)=\\lbrace h\\mid g\\rightsquigarrow h\\rbrace $ of a generator $g$ as follows: $\\operatorname{supp}(a)\\subseteq \\bigcup \\lbrace T(g)\\mid g\\in \\operatorname{supp}(a)$ and $g$ on a cycle$\\rbrace .$ Each element of the support of $a$ is either on a cycle or on a path exiting a cycle which contains another element of $\\operatorname{supp}(a).$ To show (1), note that if $a\\rightarrow x^na+b,$ then $a\\rightarrow x^na+b \\rightarrow x^{2n}a+x^nb+b\\rightarrow x^{3n}a+x^{2n}b+x^nb+b\\rightarrow \\ldots \\rightarrow x^{kn}a+\\sum _{i=0}^{k-1} x^{in}b.$ To show (2), we use part (1) to choose $n$ larger than $k-m$ for any degrees $k$ and $m$ of any monomials in a normal representation of $a.$ Let $l$ be the number of monomials in a normal representation of $a.$ If all generators in $\\operatorname{supp}(a)$ are on cycles, there is nothing to prove.", "So, suppose that there is $g_1\\in \\operatorname{supp}(a)$ such that $x^{m_1}g_1$ is a monomial of $a$ and that $g_1$ is not on a cycle.", "Let $a=a_1+x^{m_1}g_1.$ By the Refinement Lemma REF (1), there are $c_{11}, c_{12}$ such that $a_1+x^{m_1}g_1\\rightarrow x^na_1+x^{n+m_1}g_1+b=c_{11}+c_{12},\\;\\;$ $a_1\\rightarrow c_{11}$ and $x^{m_1}g_1\\rightarrow c_{12}.$ The monomial $x^{n+m_1}g_1$ is a summand of either $c_{11}$ or $c_{12}.$ In the second case, $x^{m_1}g_1\\rightarrow x^{n+m_1}g_1+c$ for some $c$ and Corollary REF implies that there is a path of length $n>0$ from $g_1$ to $g_1$ which means that $g_1$ is on a cycle.", "This is a contradiction with the choice of $g_1.$ Hence, $x^{n+m_1}g_1$ is a summand of $c_{11}.$ This implies that $c_{11}\\ne 0$ and so $a_1\\ne 0$ also which means that $l>1$ and $a_1$ has $l-1$ terms.", "By Corollary REF , there is a monomial $x^{m_2}g_2$ of $a_1$ such that $a_1=a_2+x^{m_2}g_2$ (so $a_2$ has $l-2\\ge 0$ terms) and that $x^{m_2}g_2\\rightarrow x^{n+m_1}g_1+c$ for some $c.$ The choice of $n$ guarantees that $n+m_1-m_2>0$ so that there is a path of positive length from $g_2$ to $g_1$ by Corollary REF .", "If $g_2$ is on a cycle, we are done.", "If not, consider whether the term $x^{n+m_2}g_2$ is a summand of $c_{11}$ or $c_{12}.$ If it is a summand of $c_{12},$ then $x^{m_1}g_1\\rightarrow x^{n+m_2}g_2+d$ for some $d$ and so there is a path of positive length from $g_1$ to $g_2.$ As there is a path of positive length from $g_2$ to $g_1,$ $g_1$ is on a cycle.", "Since this is not the case, $x^{n+m_2}g_2$ is a summand of $c_{11}.$ Apply the Refinement Lemma REF (1) again to decompose $c_{11}$ as $c_{21}+c_{22}$ such that $a_2\\rightarrow c_{21}$ and $x^{m_2}g_2\\rightarrow c_{22}.$ Since $g_2$ is not on a cycle, $x^{n+m_2}g_2$ is a summand of $c_{21}$ which implies that $c_{21}\\ne 0$ and so $a_2\\ne 0$ which means that $l-2>0.$ By Corollary REF , there is a summand $x^{m_3}g_3$ of $a_2$ such that $a_2=a_3+x^{m_3}g_3$ (so that $a_3$ has $l-3\\ge 0$ terms) and that $x^{m_3}g_3\\rightarrow x^{n+m_2}g_2+d$ for some $d.$ The choice of $n$ guarantees that $n+m_2-m_3$ is positive so we can conclude that there is a path of positive length from $g_3$ to $g_2$ by Corollary REF .", "If $g_3$ is on a cycle, we are done.", "If not, the term $x^{n+m_3}g_3$ must be a summand of $c_{21}$ as otherwise $g_3$ is on a cycle which is not the case.", "So, $x^{n+m_3}g_3$ is a summand of $c_{21},$ $a_3+x^{m_3}g_3\\rightarrow c_{21},$ and we can continue the decomposition process $c_{21}=c_{31}+c_{32}$ as in the previous step.", "Since $l$ is finite, this process eventually stops.", "If it stops at the $k$ -th step, $g_k$ is on a cycle and (2) holds.", "Note that the proof of part (2) implies that if $g_1$ is not on a cycle, then $g_1$ is on a path leaving a cycle which contains $g_k.$ This is because the proof shows that there is a path from $g_{i+1}$ to $g_i$ for all $i=1,\\ldots , k-1.$ Hence, this automatically shows part (3).", "Part (4) is a direct corollary of part (3).", "The last part of Lemma REF describes the support of a stationary element.", "The properties of such set are relevant and we introduce some terminology for it.", "First, we say that a finite and nonempty set of generators of $F_E^\\Gamma $ is stationary if every $g\\in V$ is either on a cycle or on a path exiting a cycle which contains some generator $h\\in V.$ By Lemma REF , the support of every stationary element is a stationary set.", "For a stationary set $V,$ let $V_c$ denote the set of those $g\\in V$ which are on cycles (thus $V_c\\ne \\emptyset $ ).", "We say that $V_c$ is the core of $V$ and that $g\\in V_c$ is a core generator.", "We say that the cycles which contain core generators are the core cycles of $V$ .", "Let $V_e$ denote $V-V_c$ (so $V_e$ is possibly empty).", "We call this set the exit set of $V$ and we say that $g\\in V_e$ is an exit generator.", "For a core generator $g\\in V_c$ , let $n_g$ be the minimum of the set of lengths of cycles on which $g$ is.", "Let $n$ be the least common multiple of $n_g$ for $g\\in V_c.$ We show that $n$ has a special significance for a stationary set $V$ which consists of core generators only so we call it the core period of such $V.$ If $a$ is stationary, let $a=a_c+a_e$ such that the support of $a_c$ is $\\operatorname{supp}(a)_c$ and the support of $a_e$ is $\\operatorname{supp}(a)_e$ (thus $a_c\\ne 0$ and $a_e$ is possibly zero).", "We call $a_c$ and $a_e$ the core part and the exit part of $a$ respectively.", "The next example illustrates these newly introduced concepts.", "Example 3.10 If $E$ is the graph ${{\\bullet }^v [r] & {\\bullet }^w@(ru,rd) }\\;\\;\\;\\;,$ then the stationary element $w$ has the core part $w$ and the exit part 0.", "The set $\\lbrace w\\rbrace $ is a stationary set with the core equal to the entire set $\\lbrace w\\rbrace .$ The core period of the core $\\lbrace w\\rbrace $ is 1.", "If $E$ is the Toeplitz graph $\\;\\;\\;\\;{{\\bullet }^v@(lu,ld) [r] & {\\bullet }^w},$ then the stationary element $a=v+w$ has the core part $v$ and the exit part $w.$ The stationary set $\\lbrace v,w\\rbrace $ has the core $\\lbrace v\\rbrace $ and the exit set $\\lbrace w\\rbrace .$ The core period of the core $\\lbrace v\\rbrace $ is 1.", "If $a\\in F_E^\\Gamma $ is such that each $g\\in \\operatorname{supp}(a)$ is on a cycle, then $\\operatorname{supp}(a)$ is a stationary set by definition and $a=a_c.$ In the next lemma, we show that such element $a$ is necessarily stationary.", "Lemma 3.11 Let $a\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ be such that each $g\\in \\operatorname{supp}(a)$ is on a cycle, and let $n$ be the core period of $\\operatorname{supp}(a).$ The following hold.", "The element $a$ is stationary and $a\\rightarrow x^na+b$ for some $b\\in F_E^\\Gamma $ .", "The element $a$ is periodic if and only if the core cycles have no exits.", "If $g\\in \\operatorname{supp}(a),$ then $g\\rightsquigarrow ^{c_g} g$ where $c_g$ is a cycle of length $n_g,$ where $n_g$ is the minimum of the set of lengths of cycles which contain $g.$ Hence, $g\\rightarrow x^{n_g}g+b_g^{\\prime }$ for some $b_g^{\\prime }\\in F_E^\\Gamma $ such that $b_g^{\\prime }=0$ if and only if $c_g$ has no exits.", "Since $n$ is a multiple of $n_g,$ $g\\rightarrow x^ng+b_g$ for some $b_g$ such that $b_g=0$ if and only if $b_g^{\\prime }=0.$ If $a=\\sum _{j=1}^l x^{k_j}g_j$ is a normal representation of $a,$ then we have that $x^{k_j}g_j\\rightarrow x^{n+k_j}g_j+x^{k_j}b_{g_j}.$ Adding these relations together produces $a\\rightarrow \\sum _{j=1}^l x^{n+k_j}g_j+\\sum _{j=1}^l x^{k_j} b_{g_j}=x^n\\sum _{j=1}^l x^{k_j}g_j+\\sum _{j=1}^l x^{k_j}b_{g_j}=x^na+b$ for $b=\\sum _{j=1}^l x^{k_j}b_{g_j}$ so (1) holds.", "To show (2), note that $a$ is periodic if and only if $b=0$ and $b=0$ if and only if any core cycle has no exits.", "We note the following corollary of Lemmas REF , REF , and REF .", "Corollary 3.12 The following conditions are equivalent.", "There is a comparable generator of $F_E^\\Gamma .$ There is a nonzero comparable element of $F_E^\\Gamma .$ The graph $E$ has a cycle.", "The implication (1) $\\Rightarrow $ (2) is direct.", "If (2) holds, there is a stationary element $a$ by Lemma REF .", "Since $a_c\\ne 0$ by Lemma REF , there is at least one core cycle so (3) holds.", "If (3) holds, any vertex of a cycle is a comparable generator of $F_E^\\Gamma $ by Lemma REF so (1) holds." ], [ "The Core Lemma", "The following lemma highlights an important property of a stationary element and justifies our terminology “core” – if $a$ is stationary and $x^na$ can be produced from $a$ with some possible “change” $b$ , then $x^{kn}a$ , for some positive $k,$ can be produced by using the core part $a_c$ only with possibly some other “change” $c$ such that $c=0$ and $a_e=0$ if and only if $b=0.$ Lemma 3.13 (The Core Lemma) Let $a\\in F_E^\\Gamma $ be a stationary element with the core part $a_c$ and the exit part $a_e.$ If $a\\rightarrow x^na+b$ for some positive integer $n$ and some $b\\in F_E^\\Gamma ,$ then $a_c\\rightarrow x^{kn}a+c$ for some positive integer $k$ and some $c\\in F_E^\\Gamma $ such that $c+a_e\\sim \\sum _{i=0}^{k-1}x^{in}b.$ If $a_e=0,$ the claim trivially holds with $k=1$ and $c=b.$ If $a_e\\ne 0,$ let $V=\\operatorname{supp}(a)$ so that $V_e$ is nonempty.", "Let also $V_c=V_c^{\\prime }\\cup V_c^{\\prime \\prime }$ where $V_c^{\\prime }$ consists of the core generators in $V$ such that no exit generator connects to them and $V_c^{\\prime \\prime }$ consists of the core generators in $V$ such that some exit generators connect to them.", "By these definitions, no $g\\in V_c^{\\prime \\prime }$ connects to any $h\\in V_c^{\\prime }$ (otherwise $h$ would be in $V_c^{\\prime \\prime }$ ).", "Also, note that $V_c^{\\prime }$ is nonempty since otherwise some exit generator would be on a cycle which would make it a core, not an exit generator.", "Let also $a_c=a_c^{\\prime }+a_c^{\\prime \\prime }$ so that $\\operatorname{supp}(a_c^{\\prime })=V_c^{\\prime }$ and $\\operatorname{supp}(a_c^{\\prime \\prime })=V_c^{\\prime \\prime }.$ Choose $n$ to be larger than the difference of degrees of any two monomials in a normal representation of $a$ by using Lemma REF (1) if $n$ is not already such.", "We construct a sequence of finite acyclic graphs $F_0\\supsetneq F_1\\supsetneq \\ldots \\supsetneq F_l\\supsetneq \\emptyset $ such that the sequence terminates exactly when the claim is shown.", "Graph $\\mathbf {F_0.", "}$ Let us define a graph $F_0$ such that $V_e$ is the set of vertices of $F_0$ and that there is an edge from $g$ to $h$ for some $g,h\\in V_e$ if $g$ connects to $h$ in $E.$ Since no $g\\in V_e$ is on a cycle, the graph $F_0$ is acyclic.", "Since $F_0$ is a finite and acyclic graph, it has a source by Lemma REF .", "Let $V_{e0}$ be the set of sources of $F_0$ and $a_e=a_{e0}+a_{e0}^{\\prime }$ such that $\\operatorname{supp}(a_{e0})=V_{e0}$ and $\\operatorname{supp}(a_{e0}^{\\prime })=V_e-V_{e0}.$ By the Refinement Lemma REF (1), there are $a_1,a_2,a_3\\in F_E^\\Gamma $ such that $a=a_c^{\\prime }+(a_c^{\\prime \\prime }+a_{e0}^{\\prime })+a_{e0}\\rightarrow x^na+b=a_1+a_2+a_3\\;\\mbox{ and }\\;a_c^{\\prime }\\rightarrow a_1,\\; a_c^{\\prime \\prime }+a_{e0}^{\\prime }\\rightarrow a_2,\\; a_{e0}\\rightarrow a_3.$ If $x^mg$ is any monomial of $x^na_{e0}$ for $g\\in V_{e0},$ then $x^mg$ is a summand of either $a_1, a_2$ or $a_3.$ By Corollary REF and by the choice of $n,$ if $x^mg$ is a summand of $a_3$ then either $g$ is on a cycle or there is a path from another source of $F_0$ to $g$ and each of these options leads to a contradiction.", "If $x^mg$ is a summand of $a_2,$ then there is either a nontrivial path from some $g^{\\prime }\\in V_e$ to $g$ or a nontrivial path from some $h\\in V_c^{\\prime \\prime }$ to $g$ also by Corollary REF and by the choice of $n$ .", "In the second case, there is $g^{\\prime }\\in V_e$ and a path from $g^{\\prime }$ to $h$ and, hence, a nontrivial path from $g^{\\prime }$ to $g$ as well.", "Thus, both cases lead to a contradiction since $g$ is a source of $F_0.$ Hence, $x^mg$ has to be a summand of $a_1.$ Since the monomial $x^mg$ was arbitrary, $x^na_{e0}$ is a summand of $a_1.$ In addition, if $x^mh$ is any monomial of $x^na_c^{\\prime },$ $x^mh$ is a summand of $a_1$ also.", "Indeed, assuming that $x^mh$ is a summand of either $a_3$ or $a_2$ implies that $h$ is in $V_c^{\\prime \\prime }$ not $V_c^{\\prime }.$ Hence, for some $b_0\\in F_E^\\Gamma ,$ $a_c^{\\prime }\\rightarrow a_1=x^na_c^{\\prime }+ x^na_{e0}+b_0.$ If $a_{e0}^{\\prime }=0,$ we claim that the process is complete.", "In this case, $a_e=a_{e0}.$ The support of $a_c^{\\prime \\prime }$ consists of core generators so $a_c^{\\prime \\prime }$ is stationary by Lemma REF .", "Let $m$ be the least common multiple of $n$ and the core period of $a_c^{\\prime \\prime }$ and let $m=kn.$ Let $b^{\\prime \\prime }_0$ be such that $a_c^{\\prime \\prime }\\rightarrow x^{kn} a_c^{\\prime \\prime }+b^{\\prime \\prime }_0.$ After repeated use of the relation $a_c^{\\prime }\\rightarrow x^na_c^{\\prime }+x^na_e+b_0$ for $k$ times, we have that $a_c^{\\prime }\\rightarrow x^{kn}a_c^{\\prime }+x^{kn}a_e+\\sum _{i=1}^{k-1}x^{in}a_e+\\sum _{i=0}^{k-1} x^{in}b_0.$ Thus, $a_c=a_c^{\\prime }+a_c^{\\prime \\prime } \\rightarrow x^{kn}a_c^{\\prime }+x^{kn}a_e+\\sum _{i=1}^{k-1}x^{in}a_e+\\sum _{i=0}^{k-1} x^{in}b_0+ x^{kn} a_c^{\\prime \\prime }+b^{\\prime \\prime }_0=x^{kn}a+\\sum _{i=1}^{k-1}x^{in}a_e+\\sum _{i=0}^{k-1} x^{in}b_0+b^{\\prime \\prime }_0=x^{kn}a+c$ for $c=\\sum _{i=1}^{k-1}x^{in}a_e+\\sum _{i=0}^{k-1} x^{in}b_0+b^{\\prime \\prime }_0.$ Thus, $a=a_c+a_e\\rightarrow x^{kn}a+c+a_e.$ On the other hand, $a\\rightarrow x^{kn}a+\\sum _{i=0}^{k-1}x^{in}b$ holds by part (1) of Lemma REF .", "Thus, $x^{kn}a+c+a_e\\sim x^{kn}a+\\sum _{i=0}^{k-1}x^{in}b\\;\\;\\;\\mbox{ which implies }\\;\\;\\;c+a_e\\sim \\sum _{i=0}^{k-1}x^{in}b.$ If $a_{e0}^{\\prime }\\ne 0,$ we construct $F_1.$ Graph $\\mathbf {F_1.", "}$ Let $F_1$ be the graph obtained by eliminating the sources and all edges they emit from $F_0.$ Then $F_1$ is a finite acyclic graph which is a proper subgraph of $F_0.$ Let $V_{e1}$ be the set of the sources of $F_1$ and $a_e=a_{e0}+a_{e1}+a_{e1}^{\\prime }$ be such that $\\operatorname{supp}(a_{e1})=V_{e1}$ and $\\operatorname{supp}(a_{e1}^{\\prime })=V_e-V_{e0}-V_{e1}.$ Let also $a_c^{\\prime \\prime }=a_{c0}^{\\prime \\prime }+a_{c1}^{\\prime \\prime }$ such that $a_{c0}^{\\prime \\prime }$ consists of those monomials $x^mh$ of $a_c^{\\prime \\prime }$ such that $g\\rightsquigarrow h$ for some $g\\in V_{e0}$ and $a_{c1}^{\\prime \\prime }$ consists of all other monomials of $a_c^{\\prime \\prime }.$ Using the Refinement Lemma REF (1) again, there are $a_1^{\\prime }, a_2^{\\prime }, a_3^{\\prime }\\in F_E^\\Gamma $ such that $a=(a_c^{\\prime }+a_{c0}^{\\prime \\prime }+a_{e0})+(a_{c1}^{\\prime \\prime }+a_{e1}^{\\prime })+a_{e1}\\rightarrow x^na+b=a_1^{\\prime }+a_2^{\\prime }+a_3^{\\prime }$ and that $a_c^{\\prime }+a_{c0}^{\\prime \\prime }+a_{e0}\\rightarrow a_1^{\\prime }$ , $a_{c1}^{\\prime \\prime }+a_{e1}^{\\prime }\\rightarrow a_2^{\\prime }$ , $a_{e1}\\rightarrow a_3^{\\prime }.$ If $x^mg$ is any summand of $x^na_c^{\\prime }+x^na_{c0}^{\\prime \\prime }+x^na_{e0}+x^na_{e1},$ we can repeat the arguments from before to show that the assumption that $x^mg$ is a summand of $a_2^{\\prime }$ or $a_3^{\\prime }$ leads to a contradiction.", "Hence, $x^mg$ is a summand of $a_1^{\\prime }$ and so $a_c^{\\prime }+a_{c0}^{\\prime \\prime }+a_{e0}\\rightarrow a_1^{\\prime }=x^na_c^{\\prime }+x^na_{c0}^{\\prime \\prime }+x^na_{e0}+x^na_{e1}+b_1^{\\prime }$ for some $b_1^{\\prime }\\in F_E^\\Gamma .$ If $k_1n$ is the least common denominator of $n$ and the core period of $a_{c0}^{\\prime \\prime }$ , there is $b_1^{\\prime \\prime }\\in F_E^\\Gamma $ such that $a_{c0}^{\\prime \\prime }\\rightarrow x^{k_1n}a_{c0}^{\\prime \\prime }+b_1^{\\prime \\prime }.$ Using the last two relations and the relation $a_c^{\\prime }\\rightarrow x^na_c^{\\prime }+ x^na_{e0}+b_0$ from the first step for $k_1$ times, we have that $a_c^{\\prime }+a_{c0}^{\\prime \\prime }\\rightarrow x^{k_1n}(a_c^{\\prime }+ a_{e0})+\\sum _{i=1}^{k_1-1}x^{in}a_{e0}+\\sum _{i=0}^{k_1-1}x^{ni}b_0+x^{k_1n}a_{c0}^{\\prime \\prime }+b_1^{\\prime \\prime }\\rightarrow $ $x^{(k_1+1)n}(a_c^{\\prime }+a_{c0}^{\\prime \\prime }+a_{e0}+a_{e1})+x^{k_1n}b_1^{\\prime }+\\sum _{i=1}^{k_1-1}x^{in}a_{e0}+\\sum _{i=0}^{k_1-1}x^{ni}b_0+b_1^{\\prime \\prime }=x^{(k_1+1)n}(a_c^{\\prime }+a_{c0}^{\\prime \\prime }+a_{e0}+a_{e1})+b_1$ for $b_1=x^{k_1n}b_1^{\\prime }+\\sum _{i=1}^{k_1-1}x^{in}a_{e0}+\\sum _{i=0}^{k_1-1}x^{ni}b_0+b_1^{\\prime \\prime }.$ If $a_{e1}^{\\prime }=0,$ then $a_e=a_{e0}+a_{e1}.$ Let $kn$ be the least common multiple of $(k_1+1)n$ and the core period of $a_{c1}^{\\prime \\prime }.$ Arguing as in the case $a_{e0}^{\\prime }=0,$ we have that $a_c\\rightarrow x^{kn}a+c$ for some $c\\in F_E^\\Gamma $ such that $\\sum _{i=0}^{k-1}x^{in}b\\sim c+a_e$ and this finishes the proof.", "If $a_{e1}^{\\prime }\\ne 0,$ we construct $F_2$ and continue the process.", "This process eventually terminates since $V_e$ is a finite set.", "Hence, there is a positive integer $l$ such that $a_{el}^{\\prime }=0$ so that $a_c\\rightarrow x^{kn}a+c$ for some $k$ and some $c.$ The relations $a\\rightarrow x^{kn}a+\\sum _{i=0}^{k-1}x^{in}b$ and $a\\rightarrow x^{kn}a+c+a_e$ imply that $\\sum _{i=0}^{k-1}x^{in}b\\sim c+a_e$ which proves the lemma.", "The Core Lemma has the following corollary, characterizing a stationary and periodic element, which we use in the proof of Theorem REF .", "Corollary 3.14 A stationary element $a$ is periodic if and only if the support of $a$ consists of regular vertices on cycles without exits.", "Let $a$ be such that $a\\rightarrow x^na+b$ for some $b$ and positive $n$ .", "If $a$ is periodic, then $b=0.$ By the Core Lemma REF , $a_c\\rightarrow x^{kn}a+c$ for some $k$ and some $c$ such that $\\sum _{i=0}^{k-1}x^{in}b\\sim a_e+c.$ So $b=0$ implies that $a_e=0$ (and $c=0$ ).", "Hence, $a=a_c.$ This enables us to use Lemma REF which implies that the support of $a$ consists of generators on cycles without exits so that these generators are regular vertices.", "For the converse, assume that the support of $a$ consists of core vertices on cycles without exits.", "If $n$ is the core period, then $a\\rightarrow x^na$ so $a$ is both stationary and periodic." ], [ "The stationary-partition", "By Lemma REF , the support of a stationary element is a stationary set.", "By Lemma REF , the converse is true if a stationary set contains no exit generators.", "It would be convenient to have the converse of this fact in general.", "However, the exit generators can complicate the situation as the next example shows.", "Part (2) of Example REF shows that we need additional requirements for any element with a stationary support to be stationary.", "In particular, these requirements impose restrictions on powers of $x$ which appear in the normal form of such element.", "Let $a$ be stationary such that $a\\rightarrow x^na+b$ holds for some positive $n$ and some $b.$ If $a=\\sum _{i=1}^{k}x^{m_i}g_i$ is a normal representation of $a,$ by repeated use of the Refinement Lemma REF (1), there are mutually disjoint subsets $I_1, \\ldots , I_k$ of $\\lbrace 1, 2, \\ldots , k\\rbrace $ whose union is $\\lbrace 1, 2, \\ldots , k\\rbrace $ and there are $b_1,\\ldots , b_k$ such that $a= \\sum _{i=1}^{k}\\sum _{j\\in I_i} x^{m_j}g_j\\;\\mbox{ and }\\;\\; b=\\sum _{i=1}^{k}b_i$ and that for every $i=1, \\ldots , k$ $ x^{m_i}g_i\\rightarrow \\sum _{j\\in I_i} x^{m_j+n}g_j+b_i.\\qquad \\mathrm {(Rel.", "1)}$ The set $I_i$ can be empty if $i$ is in $I_{i^{\\prime }}$ for some $i^{\\prime }\\ne i$ (see also Example REF below).", "If $I_i\\ne \\emptyset ,$ Corollary REF applied to relation (REF ) ensures the existence of a path $p_{ij}$ connecting $g_i$ and $g_j$ such that $ m_i+|p_{ij}|=m_j+n.\\qquad \\mathrm {(Rel.", "2)}$ By Lemma REF (1), we can choose $n$ such that $n>m_i-m_j$ so that $|p_{ij}|=n+m_j-m_i>0$ for all $i,j=1,\\ldots , k.$ The requirement that $p_{ij}$ has positive length justifies the following definition and implies direction $\\Rightarrow $ of Proposition REF .", "Definition 3.15 Let $a\\in F_E^\\Gamma $ have a stationary support $V$ and a normal representation $a=\\sum _{i=1}^{k}x^{m_i}g_i.$ We say that $a$ has a stationary-partition if there is a positive integer $n,$ mutually disjoint subsets $I_1, \\ldots , I_k$ of $\\lbrace 1, 2, \\ldots , k\\rbrace $ with $\\bigcup _{i=1}^k I_i=\\lbrace 1, 2, \\ldots , k\\rbrace $ and paths $p_{ij}$ of positive length for $i=1,\\ldots , k$ and $j\\in I_i$ with $\\mathbf {s}(p_{ij})=g_i$ and $\\mathbf {r}(p_{ij})=g_j$ and such that relation (REF ) holds for each $i=1, \\ldots , k$ and $j\\in I_i.$ The following example shows that a stationary-partition does not have to be unique.", "Example 3.16 Let $E$ be the following graph $\\;\\;{{\\bullet }^{v_1}@(lu,ld) [r] & {\\bullet }^{v_2}@(ru,rd)}\\;\\;\\,.$ Then $v_1+v_2$ is stationary since $v_1\\rightarrow xv_1+xv_2$ and so $v_1+v_2\\rightarrow x(v_1+v_2)+v_2$ and $k=2$ in this case.", "We can take $I_1=\\lbrace 1,2\\rbrace $ and $I_2=\\emptyset $ since $v_1$ “produces” both terms of $x(v_1+v_2).$ In this case, relations (REF ) are $v_1\\rightarrow xv_1+xv_2\\;\\;\\mbox{ and }\\;\\;v_2\\rightarrow v_2.$ However, $v_2\\rightarrow xv_2$ also, so the summand $xv_2$ can be “produced” by $v_2$ also.", "Hence, $v_1+v_2$ is stationary also because $v_1+v_2\\rightarrow xv_1+xv_2+v_2\\rightarrow xv_1+xv_2+xv_2=x(v_1+v_2)+xv_2.$ So, we can also take $I_1=\\lbrace 1\\rbrace ,$ $I_2=\\lbrace 2\\rbrace .$ In this case, relations (REF ) are $v_1\\rightarrow xv_1+xv_2\\;\\;\\mbox{ and }\\;\\;v_2\\rightarrow xv_2.$ We characterize a stationary element in terms of the properties of the generators in its support which is the final and key step towards Theorem REF .", "Proposition 3.17 Let $a\\in F_E^\\Gamma $ be an element such that $\\operatorname{supp}(a)=V$ is stationary.", "Then $a$ is stationary if and only if $a$ has a stationary-partition.", "We showed that direction $\\Rightarrow $ holds before Definition REF .", "To summarize, if $a=\\sum _{i=1}^{k}x^{m_i}g_i\\rightarrow x^na+b$ holds for some $n$ and some $b,$ use Lemma REF (1) to choose $n>m_i-m_j$ for all $i,j=1,\\ldots , k.$ Repeated use of the Refinement Lemma REF (1) produces required sets $I_1, \\ldots , I_k$ such that relations (REF ) hold for $i=1, \\ldots , k.$ Using Corollary REF produces paths $p_{ij}$ and our choice of $n$ ensures that the paths $p_{ij}$ have positive length so that relations (REF ) hold.", "Thus, $a$ has a stationary-partition.", "Conversely, let $a$ have a stationary-partition and let $n$ , $I_1,\\ldots , I_k$ and $p_{ij}$ be as in Definition REF .", "Starting with $x^{m_i}g_i$ and applying the axioms following the paths $p_{ij}$ for all $i=1,\\ldots , k$ and all $j\\in I_i,$ we obtain $x^{m_i}g_i\\rightarrow \\sum _{j\\in I_i} x^{m_i+|p_{ij}|}g_j+b_i$ for some $b_i\\in F_E^\\Gamma $ for $i=1,\\ldots , k.$ By relations (REF ), $x^{m_i}g_i\\rightarrow \\sum _{j\\in I_i} x^{m_i+|p_{ij}|}g_j+b_i =\\sum _{j\\in I_i} x^{m_j+n}g_j+b_i$ which shows that relations (REF ) hold for all $i.$ Adding these relations together produces $a=\\sum _{i=1}^{k}x^{m_i}g_i\\rightarrow \\sum _{i=1}^{k}\\left(\\sum _{j\\in I_i} x^{m_j+n}g_j+b_i\\right)=x^n\\sum _{i=1}^{k}\\sum _{j\\in I_i} x^{m_j}g_j+\\sum _{i=1}^{k}b_i=x^na+\\sum _{i=1}^{k}b_i$ where the last equality holds since $I_1,\\ldots , I_k$ are disjoint and their union is $\\lbrace 1,\\ldots , k\\rbrace .$ Letting $b=\\sum _{i=1}^{k}b_i,$ we have that $a\\rightarrow x^na+b.$ Hence, $a$ is stationary.", "Remark 3.18 Since the relation $a\\rightarrow x^na+b$ holds for some $n$ and $b$ if and only if $a_c\\rightarrow x^ma+c$ holds for some $m$ and $c$ by the Core Lemma REF , we can also consider a partition of $x^ma+c$ based on a normal representation of $a_c$ instead of $a.$ If Definition REF is modified accordingly, Proposition REF can be formulated to state that $a$ is stationary if and only if $a$ has a partition based on its core part $a_c$ .", "Proposition REF also reaffirms Lemma REF since if an element $a$ has the stationary support consisting of core generators only, then $a$ has a stationary-partition.", "Indeed, if $a=a_c,$ one can take $n$ to be the core period and $I_i=\\lbrace i\\rbrace .$ If $n_{g_i}$ is the minimum of the set of lengths of cycles on which $g_i$ is and if $n=l_in_{g_i},$ one can take $p_{ii}$ to be the path obtaining by traversing a cycle of length $n_{g_i}$ $l_i$ times starting at $g_i$ so that $|p_{ii}|=n.$ Thus, relation (REF ) holds trivially for each $i$ since $m_i+n=m_i+n$ and so $a$ has a stationary-partition." ], [ "Characterization of comparability", "Using Propositions REF and REF , we prove Theorem REF characterizing a comparable element.", "Theorem 3.19 The following conditions are equivalent for an element $a\\in F^\\Gamma _E.$ The element $a$ is nonzero and comparable.", "There is a stationary element $b$ such that $a\\rightarrow b.$ There is an element $b$ with a stationary support and a stationary-partition such that condition (2) from Proposition REF holds for $a$ and $b.$ The implication (1) $\\Rightarrow $ (2) holds by Lemma REF .", "Conversely, if (2) holds, then $b$ is nonzero and comparable.", "The relation $a\\rightarrow b$ implies $a\\sim b$ so $a$ is nonzero and comparable as well.", "The equivalence (2) $\\Leftrightarrow $ (3) follows directly from Propositions REF and REF .", "In Theorem REF , we characterize when every element of $F_E^\\Gamma $ is comparable.", "First, we show the following corollary of Proposition REF and Lemmas REF and REF which we use in the proof of Theorem REF .", "Corollary 3.20 Let $v$ be an infinite emitter.", "If $v$ connects to $q^v_Z$ by a path of positive length, then $v$ is on a cycle.", "If $q_Z^v$ connects to $q_W^v$ by a path of positive length, then $q_Z^v$ is on a cycle.", "If $q_W^v$ is on a cycle, then $v$ is on a cycle and $q^v_Z$ is on a cycle for every $\\emptyset \\ne Z\\subseteq W.$ If $v$ is comparable, then $v$ is on a cycle.", "If $q^v_Z$ is comparable, then $q_Z^v$ is on a cycle.", "To show (1), assume that $v\\rightsquigarrow ^p q_Z=q_Z^v$ for some path $p$ of positive length $n.$ Then $v\\rightarrow x^n q_Z+a$ for some $a\\in F_E^\\Gamma .$ By the nature of axioms (A2) and (A3), there has to be a term $x^nv$ produced at some point.", "Hence, $v\\rightarrow x^nv+b$ for some $b$ which implies that $v$ is on a cycle.", "To show (2), assume that $q_Z=q_Z^v\\rightsquigarrow ^p q_W=q_W^v$ for some path $p$ of positive length $n.$ Then $q_Z\\rightarrow x^n q_W+a$ for some $a.$ By the nature of axioms (A2) and (A3), there has to be a term $x^nv$ produced at some point using a cycle $c$ based at $v$ such that the first edge of $c$ is not in $Z$ .", "Hence, $q_Z\\rightarrow x^nv+b\\rightarrow x^nq_Z+c$ for some $b$ and $c$ by (A2) and so $q_Z$ is on a cycle.", "By Definition REF , if $q_W^v$ is on a cycle, then there is a cycle based at $v$ such that the first edge $e$ of that cycle is not in $W.$ So, $v$ is on a cycle.", "If $Z\\subseteq W,$ then $e\\notin Z$ and so $q_Z$ is also on a cycle by Definition REF .", "This shows (3).", "To show (4), let $v$ be comparable.", "By Lemma REF , there is a stationary element $a$ such that $v\\rightarrow a$ .", "If $a=v,$ then $v$ is stationary and it is necessarily on a cycle by part (2) of Lemma REF .", "So, assume that $a\\ne v.$ By Proposition REF , if $a=\\sum _{j=1}^l x^{t_j}h_j$ is a normal representation of $a,$ there are paths $p_{j}, j=1,\\ldots , l$ such that $v\\rightsquigarrow ^{p_j} h_j,$ $t_j=|p_j|,$ and at least one of $h_j$ is $q_Z^v$ for some $Z.$ Reordering the terms we can assume that $j=1.$ If $p_1$ has positive length, then $v$ is on a cycle by part (1).", "If $q_Z$ is on a cycle, then $v$ is on a cycle by part (3).", "So, let us consider the remaining case when $p_1$ is trivial and $q_Z$ is not on a cycle.", "In this case, $q_Z$ has to be on an exit from a core cycle by part (4) of Lemma REF .", "So, there is $j>1$ such that $h_j$ is on a cycle and $h_j\\rightsquigarrow ^p q_Z$ for some path $p.$ Hence, $v\\rightsquigarrow ^{p_j}h_j \\rightsquigarrow ^p q_Z.$ If $|p|>0,$ then $v$ connects to $q_Z$ by a path $p_jp$ of positive length and so $v$ is on a cycle by part (1).", "If $|p|=0,$ then either $h_j=v,$ in which case $v$ is on a cycle, or $h_j=q_{Z^{\\prime }}$ for some $Z^{\\prime }\\subsetneq Z$ in which case $v$ is also on a cycle by part (3).", "To show (5), let $q_Z$ be comparable.", "By Lemma REF , there is a stationary element $a$ such that $q_Z\\rightarrow a.$ If $a=q_Z,$ then $q_Z$ is stationary and it is necessarily on a cycle by part (2) of Lemma REF .", "So, assume that $a\\ne q_Z.$ By Proposition REF , if $a=\\sum _{j=1}^l x^{t_j}h_j$ is a normal representation of $a,$ there are paths $p_{j}, j=1,\\ldots , l$ such that $q_Z\\rightsquigarrow ^{p_j} h_j,$ $t_j=|p_j|,$ and at least one of $h_j$ is $q_W^v$ for some $W\\supsetneq Z.$ Reordering the terms we can assume that $j=1.$ If $p_1$ has positive length, then $q_Z$ is on a cycle by part (2).", "If $q_W$ is on a cycle, then $q_Z$ is on a cycle by part (3).", "So, let us consider the remaining case when $p_1$ is trivial and $q_W$ is not on a cycle.", "By part (4) of Lemma REF , there is $j>1$ such that $h_j$ is on a cycle and $h_j\\rightsquigarrow ^p q_W$ for some path $p.$ So, $q_Z\\rightsquigarrow ^{p_j}h_j \\rightsquigarrow ^p q_W.$ If $|p|>0,$ $q_Z$ connects to $q_W$ by a path $p_jp$ of positive length and so $q_Z$ is on a cycle by part (2).", "If $|p|=0,$ then either $h_j=v$ or $h_j=q_{Z^{\\prime }}$ for some $Z^{\\prime }\\subsetneq W.$ In the first case, $q_Z\\rightsquigarrow ^{p_j}v$ so there is a cycle based at $v$ such that its first edge $e$ is not in $Z$ and so $q_Z$ is on that cycle by Definition REF .", "In the second case, $q_Z\\rightsquigarrow ^{p_j}q_{Z^{\\prime }}.$ If $|p_j|>0,$ then $q_Z$ is on a cycle by part (2).", "If $|p_j|=0,$ then $Z\\subseteq Z^{\\prime }.$ Since $q_{Z^{\\prime }}$ is on a cycle, there is a cycle based at $v$ such that the first edge $e$ of it is in $\\mathbf {s}^{-1}(v)-Z^{\\prime }.$ Hence, $e\\notin Z$ and so $q_Z$ is on a cycle by Definition REF .", "Theorem 3.21 The following conditions are equivalent.", "Every element $a\\in F_E^\\Gamma $ is comparable.", "Every generator of $F_E^\\Gamma $ is comparable.", "For every generator $g$ of $F_E^\\Gamma ,$ $g\\rightarrow a$ for some stationary element $a.$ The following hold for every generator $g$ of $F_E^\\Gamma .$ The generator $g$ is not a sink and it connects to a cycle.", "If $g$ is an infinite emitter or an improper vertex, then $g$ is on a cycle.", "If $g$ is regular, there is stationary $a\\in F_E^\\Gamma $ with the exit part zero such that $g\\rightarrow a.$ The following hold for every vertex $v$ of $E.$ The vertex $v$ is not a sink and it connects to a cycle.", "If $v$ is an infinite emitter, then it is on a cycle.", "If $v$ is regular, there are finitely many proper or improper vertices $h_1,\\ldots , h_l$ on cycles and paths $p_j, j=1,\\ldots , l$ such that $v\\rightsquigarrow ^{p_j}h_j$ and such that for every prefix $p$ of $p_j$ the conditions (i), (ii) and (iii) of Proposition REF hold with $t_j=|p_j|.$ The implication (1) $\\Rightarrow $ (2) is direct and the implication (2) $\\Rightarrow $ (1) holds since a finite sum of comparable elements is comparable.", "The equivalence (2) $\\Leftrightarrow $ (3) follows directly from Theorem REF .", "To complete the proof, we show (3) $\\Rightarrow $ (4) $\\Rightarrow $ (5) $\\Rightarrow $ (3).", "Assume that (3) holds and let $g$ be any generator.", "Let $a$ be stationary such that $g\\rightarrow a.$ Since $g$ connects to all generators in the support of $a_c\\ne 0,$ $g$ connects to a generator on a cycle so $g$ is not a sink and (a) holds.", "Part (b) holds by parts (4) and (5) of Corollary REF .", "To show part (c), let $g=v\\in E^0$ be regular.", "We claim that there is an element $b\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ with support containing only vertices on cycles such that $v\\rightarrow b$ .", "We prove this claim using induction on the minimum $n$ of lengths of paths from $v$ to cycles which exist by part (a).", "If this length $n$ is zero, $v$ is on a cycle and one can take $b=v.$ Assuming the induction hypothesis, consider $v$ with $n>0.$ For every $e\\in \\mathbf {s}^{-1}(v),$ either $\\mathbf {r}(e)$ is on a cycle, in which case we let $b_e=\\mathbf {r}(e)$ or $\\mathbf {r}(e)$ is not on a cycle in which case $\\mathbf {r}(e)$ is necessarily regular by parts (a) and (b).", "In this case, the minimum of lengths of paths from $\\mathbf {r}(e)$ to cycles is less than $n$ and we can use induction hypothesis to obtain $b_e$ with vertices in the support on cycles and $\\mathbf {r}(e)\\rightarrow b_e$ .", "Then $b=\\sum _{e\\in \\mathbf {s}^{-1}(v)}xb_e$ has vertices in the support on cycles and $v\\rightarrow _1 \\sum _{e\\in \\mathbf {s}^{-1}(v)}x\\mathbf {r}(e)\\rightarrow \\sum _{e\\in \\mathbf {s}^{-1}(v)}xb_e=b.$ Since $\\operatorname{supp}(b)$ consists of generators on cycles, $b$ is stationary by Lemma REF and its exit part is zero.", "Assume that (4) holds and let $v$ be any vertex of $E$ .", "Parts (5a) and (5b) directly hold by (4a) and (4b).", "If $v$ is regular, let $a$ be stationary with exit part zero such that $v\\rightarrow a$ which exists by (4c).", "If $a=\\sum _{j=1}^l x^{t_j}h_j,$ then $h_j$ are on cycles since the exit part of $a$ is zero.", "Part (5c) then follows from the relation $v\\rightarrow a$ by Proposition REF .", "Assume that (5) holds and let $g$ be any generator.", "By (5a), $g$ is not a sink.", "If $g$ is an infinite emitter, then $g$ is on a cycle by (5b) and so it is stationary.", "If $g$ is an improper vertex and $g=q^v_Z,$ then $v$ is on a cycle by (5b) so $g$ is on a cycle by Definition REF and, again $g$ is stationary.", "In both of these cases, (3) holds since $g\\rightarrow g.$ If $g$ is a regular vertex, (5c) and Proposition REF imply that $g\\rightarrow a$ for $a=\\sum _{j=1}^l x^{|p_j|}h_j.$ Since the elements $h_j$ are on cycles, $a$ is stationary by Lemma REF .", "Hence, (3) holds.", "Part (5) with any of the conditions (a), (b), or (c) deleted is not equivalent with the other conditions of Theorem REF as the next set of examples shows.", "Example 3.22 If $E$ is the Toeplitz graph (see part (2) of Example REF ), then (b) and (c) hold.", "There is a sink so (a) fails and the sink is not comparable.", "If $E$ is the graph below, then (a) and (c) hold.", "The infinite emitter $v$ is not on a cycle, so (b) fails and $v$ is not comparable by Corollary REF (4).", "${{\\bullet }^{v} @{.}", "@/_1pc/ [r] _{\\mbox{ } } @/_/ [r] [r] @/^/ [r] @/^1pc/ [r] & {\\bullet }^{w}@(ru,rd)}$ Let $E$ be the graph below.", "${\\bullet @(ul,ur) & \\bullet @(ul,ur) & \\bullet @(ul,ur) & & \\\\\\bullet _v [r][u] & \\bullet [r] [u] & \\bullet [r][u] & \\bullet @{.>}[r] @{.>}[u] &}$ If $a$ is any element whose support consists only of vertices on cycles, then $v\\rightarrow a$ fails since there is a path originating at $v$ which does not connect to $\\operatorname{supp}(a)$ (analogous argument is used in part (3) of Example REF ).", "The conditions (a) and (b) hold for $E$ , but (c) fails and $v$ is not comparable." ], [ "Characterizations of periodic, aperiodic and incomparable elements", "Next, we show characterizations of periodic, aperiodic and incomparable elements as well as other properties discussed in the introduction.", "We start by Theorem REF which characterizes a nonzero periodic element of $F_E^\\Gamma $ .", "Theorem REF has already been used in [13] to characterize Leavitt path algebras which are crossed products in terms of the properties of the underlying graphs.", "Theorem 4.1 The following conditions are equivalent for an element $a\\in F^\\Gamma _E-\\lbrace 0\\rbrace .$ The element $a$ is periodic.", "There is an element $b$ whose support consists of vertices on cycles without exits such that $a\\rightarrow ^{\\hspace{-11.9501pt}A1}b.$ Any path originating at a generator in the support of $a$ is a prefix of a path $p$ ending in one of finitely many cycles with no exits and such that all vertices of $p$ are regular.", "Every infinite path originating at a vertex in the support of $a$ ends in a cycle with no exits.", "If (1) holds, then $a$ is comparable so $a\\rightarrow b$ for some stationary element $b$ by Lemma REF .", "The relation $a\\rightarrow b$ implies $a\\sim b$ so $b$ is periodic as well.", "Hence, the supports of both $a$ and $b$ consists of regular vertices only by Lemma REF .", "Thus, $a\\rightarrow ^{\\hspace{-11.9501pt}A1}b.$ By Corollary REF , the support of $b$ consists of regular vertices on cycles without exits which shows (2).", "If (2) holds, the element $b$ as in (2) is stationary and periodic by Lemma REF .", "Since the core cycles of $b$ do not have exits, each generator in $\\operatorname{supp}(b)$ is proper, emits exactly one edge and, hence, it is regular.", "As $a\\rightarrow ^{\\hspace{-11.9501pt}A1}b,$ any element of $\\operatorname{supp}(a)$ is proper and regular also.", "Let $a=\\sum _{i=1}^k x^{m_i}v_i,$ $b=\\sum _{j=1}^l x^{t_j}w_j,$ and $I_i$ and $p_{ij}$ be as in Proposition REF for $a\\rightarrow b.$ Since only (A1) is used, each vertex of any path $p_{ij}$ is regular.", "If $p$ is a path with $\\mathbf {s}(p)=v_i,$ we use induction on $|p|$ to show that there is a path $q$ such that $p$ is a prefix of $q,$ $q$ ends in one of the core cycles and all vertices of $q$ are on some $p_{ij}$ for $j\\in I_i$ (thus regular) or on cycles without exits (thus also regular).", "If $p=v_i,$ $q$ can be taken to be $p_{ij}$ for any $j\\in I_i.$ Assuming that the claim holds for $p,$ let us consider $pe$ for some edge $e.$ By the induction hypothesis, all vertices of $p$ are regular, on $p_{ij}$ for some $j\\in I_i$ or on a core cycle.", "If $\\mathbf {r}(e)$ is on a core cycle, then it emits exactly one edge so it is regular and we can take $q$ to be $pe.$ So, let us consider the case that $\\mathbf {r}(e)$ is not on a core cycle in which case $\\mathbf {r}(p)$ is not on a core cycle also and so $\\mathbf {r}(p)$ is on $p_{ij}$ for some $j\\in I_i.$ Since $\\mathbf {r}(p)$ is not on a cycle, there is a proper prefix $r$ of $p_{ij}$ which ends in $\\mathbf {r}(p).$ Thus $P_r\\ne \\emptyset $ and so $P_r=\\mathbf {s}^{-1}(\\mathbf {r}(p))$ by part (2)(i) of Proposition REF .", "In particular, $e\\in P_r.$ Hence, there is $j^{\\prime }\\in I_i$ such that $e$ is in $p_{ij^{\\prime }}.$ Let $q$ be $pe$ up to $\\mathbf {r}(e)$ and the suffix of $p_{ij^{\\prime }}$ after $pe.$ Thus, $pe$ is a prefix of $q,$ $q$ ends in a core cycle and each vertex of $q$ is on $p_{ij}$ for some $j\\in I_i.$ It remains to show the condition on the infinite path.", "Let $e_1e_2\\ldots $ be an infinite path originating at $v_i.$ For any $n,$ each vertex of the path $e_1e_2\\ldots e_n$ is on $p_{ij}$ for some $j\\in I_i$ or in a core cycle.", "Let $n$ be strictly larger than the length of $p_{ij}$ for all $j\\in I_i.$ Then $\\mathbf {r}(e_n)$ must be in a core cycle and so $e_ne_{n+1}\\ldots $ is on that same cycle since the cycle has no exits.", "This shows that (3) holds.", "If condition (3) holds, then the support of $a$ consists of regular vertices such that every path they emit connects to finitely many cycles without exits by paths which contain regular vertices only.", "Let $\\operatorname{supp}(a)=\\lbrace v_1,\\ldots , v_k\\rbrace $ and let $n_i$ be the number of paths $p$ from $v_i$ to the finitely many cycles from condition (3) such that no vertex of any of the paths from (3) is on the cycle except the range of $p.$ Index the paths originating at $v_i$ as $p_{i1},\\ldots , p_{in_i}$ for some positive $n_i$ and let $w_{ij}=\\mathbf {r}(p_{ij}).$ Let $J$ be the set of $(i, j)$ with $i=1,\\ldots , k$ and $j=1,\\ldots , n_i$ and let $I_i$ be the set of those $(i^{\\prime },j)\\in J$ such that $i^{\\prime }=i.$ By construction, $\\lbrace I_1,\\ldots , I_k\\rbrace $ is a partition of $J$ and, by considering a bijection between $J$ and the set $\\lbrace 1,\\ldots , l\\rbrace $ for $l=|J|,$ this partition corresponds to a partition of $\\lbrace 1,\\ldots , l\\rbrace .$ If $p$ is a prefix of $p_{ij},$ let us use the notation $P_p$ in the same sense as in Proposition REF and Definition REF .", "If $p$ is a proper prefix of $p_{ij},$ then $\\mathbf {r}(p)$ is regular and $P_p$ is nonempty as it contains the first edge of $p_{ij}$ not on $p.$ If $e\\in \\mathbf {s}^{-1}(\\mathbf {r}(p)),$ then $pe$ is a prefix of some $p_{ij^{\\prime }}$ by condition (3) and so $e\\in P_p.$ Hence, $P_p=\\mathbf {s}^{-1}(\\mathbf {r}(p)).$ If $p=p_{ij},$ then $P_p$ is empty by construction.", "Thus condition (i) of Definition REF holds and conditions (ii) and (iii) are trivially satisfied.", "So, for $W=\\lbrace w_{ij}\\mid (i,j)\\in J\\rbrace ,$ $\\operatorname{supp}(a)\\rightarrow W.$ Moreover, $\\operatorname{supp}(a)\\rightarrow ^{\\hspace{-11.9501pt}A1}W$ since $\\operatorname{supp}(a)$ and $W$ contain regular vertices only.", "Hence, there is $c\\ne 0$ such that $a\\rightarrow ^{\\hspace{-11.9501pt}A1}c$ and $\\operatorname{supp}(c)\\subseteq W$ by Corollary REF .", "The set $W$ is stationary and, by part (1) of Lemma REF , every element with support contained in $W$ is stationary and, part (2) of Lemma REF , periodic.", "Thus, $c$ is periodic.", "Since $a\\sim c,$ $a$ is also periodic.", "Hence, (1) holds.", "We note that the sources of graphs in parts (1) and (3) of Example REF are such that condition (2) fails, so that these vertices are not periodic by Theorem REF (and incomparable by Theorem REF ).", "In Theorem REF , we characterize when every element of $F_E^\\Gamma $ is periodic in terms of the properties of $E$ , in terms of the form of the Leavitt path algebra, as well as in terms of the form of the Grothendieck $\\Gamma $ -group.", "Theorem 4.2 The following conditions are equivalent.", "Every element $a\\in F_E^\\Gamma $ is periodic.", "Every vertex is periodic.", "For every vertex $v,$ $\\lbrace v\\rbrace \\rightarrow ^{\\hspace{-11.9501pt}A1}V$ for some stationary set $V$ which contains core vertices only and every core cycle has no exits.", "Each path is a prefix of a path $p$ ending in a cycle with no exits and such that the vertices on $p$ are regular.", "Every infinite path ends in a cycle with no exits.", "$E$ is a row-finite, no-exit graph without sinks such that every infinite path ends in a cycle.", "For any field $K$ , the Leavitt path algebra $L_K(E)$ is graded isomorphic to an algebra of the form $\\bigoplus _{i\\in I}\\operatorname{\\mathbb {M}}_{\\mu _i}(K[x^{n_i}, x^{-n_i}])(\\overline{\\gamma }_i)$ where $I$ is a set, $\\mu _i$ are cardinals, $n_i$ positive integers, and $\\overline{\\gamma }_i$ maps $\\mu _i\\rightarrow \\mathbb {Z}$ for $i\\in I.$ The graph $\\Gamma $ -monoid is isomorphic to $\\bigoplus _{i\\in I} \\mathbb {Z}^+[x]/\\langle x^{n_i}=1\\rangle $ where $I$ is a set and $n_i$ are positive integers for $i\\in I$ .", "The Grothendieck $\\Gamma $ -group $G_E^\\Gamma $ is isomorphic to $\\bigoplus _{i\\in I} \\mathbb {Z}[x]/\\langle x^{n_i}=1\\rangle $ where $I$ is a set and $n_i$ are positive integers for $i\\in I$ .", "The implication (1) $\\Rightarrow $ (2) is direct.", "If (2) holds, then every vertex of $E$ is regular by Lemma REF .", "If a vertex $v$ is periodic, $v\\rightarrow a$ for some stationary element $a$ by Lemma REF .", "Since $v$ is periodic, $a$ is periodic also and the support of $a$ consists of regular vertices on cycles without exits by Corollary REF .", "Thus, $v\\rightarrow a$ implies that $v\\rightarrow ^{\\hspace{-11.9501pt}A1}a.$ If $V=\\operatorname{supp}(a),$ condition (3) follows by Corollary REF .", "If (3) holds, then all vertices of $E$ are regular.", "If $p$ is any finite or infinite path, $\\lbrace \\mathbf {s}(p)\\rbrace \\rightarrow ^{\\hspace{-11.9501pt}A1}V$ for some $V$ as in condition (3).", "By Corollary REF , there is $a\\in F_E^\\Gamma -\\lbrace 0\\rbrace $ such that $\\mathbf {s}(p)\\rightarrow ^{\\hspace{-11.9501pt}A1}a$ and $\\operatorname{supp}(a)\\subseteq V.$ Since $V$ consists of vertices on cycles without exits, $a$ is stationary and periodic and so $\\mathbf {s}(p)$ is periodic also.", "Then (4) holds by Theorem REF .", "If (4) holds, then all vertices of $E$ are regular so $E$ is a row-finite graph.", "Every vertex connects to cycles so there are no sinks.", "Every infinite path ends in a cycle and no cycle has an exit.", "So, (5) holds.", "Conditions (5) and (6) are equivalent by [17].", "The implications (6) $\\Rightarrow $ (7) and (7) $\\Rightarrow $ (8) are rather direct.", "Condition (8) directly implies that every element of $G_E^\\Gamma $ has a finite orbit.", "Hence, every element of $F_E^\\Gamma $ is periodic and (1) holds.", "Using Theorem REF , we characterize when no nonzero element of $F_E^\\Gamma $ is periodic.", "Corollary 4.3 The following conditions are equivalent.", "No nonzero element of $F_E^\\Gamma $ is periodic.", "The graph $E$ satisfies Condition (L).", "If $E$ has a cycle with no exits, any vertex on this cycle is periodic.", "Conversely, if Condition (L) holds, the core cycles of any stationary element have exits.", "By Theorem REF , no nonzero element is periodic.", "We characterize aperiodic elements next.", "Theorem 4.4 The following conditions are equivalent for an element $a\\in F^\\Gamma _E.$ The element $a$ is aperiodic.", "The element $a$ is comparable and not periodic.", "There is a stationary element $b$ such that $a\\rightarrow b$ and at least one of the core cycles of $b$ has an exit.", "It is direct that (1) $\\Leftrightarrow $ (2).", "The equivalence (2) $\\Leftrightarrow $ (3) holds by Theorems REF and REF .", "We also characterize when every element of $F_E^\\Gamma $ is aperiodic.", "Theorem 4.5 The following conditions are equivalent.", "Every nonzero element $a\\in F_E^\\Gamma $ is aperiodic.", "Every generator of $F_E^\\Gamma $ is aperiodic.", "Every generator of $F_E^\\Gamma $ is comparable and every cycle has an exit.", "For every generator $g$ of $F_E^\\Gamma ,$ $g\\rightarrow a$ for some stationary element $a$ such that all core cycles have exits.", "The implication (1) $\\Rightarrow $ (2) is direct.", "The converse holds since a sum of aperiodic elements is comparable and, if at least one of them is aperiodic, aperiodic.", "If (2) holds and $g$ is a generator on an arbitrary cycle (which exists by Corollary REF ), then $g$ is aperiodic if and only if the cycle has an exit by Lemma REF .", "Hence, (3) holds.", "If (3) holds and $g$ is an arbitrary generator, then $g\\rightarrow a$ for a stationary element $a.$ By assumption (3) all core cycle of $a$ have exits so (4) holds.", "Finally, let us assume that (4) holds and show (2).", "If $g$ is an arbitrary generator and $a$ a stationary element such that $g\\rightarrow a$ and all core cycles have exit, then $a\\rightarrow x^na+b$ for some nonzero $b$ .", "Hence, $a$ is aperiodic and, since $g\\rightarrow a,$ $g$ is aperiodic also.", "We also characterize when no element of $F_E^\\Gamma $ is aperiodic.", "Corollary 4.6 The following conditions are equivalent.", "No element of $F_E^\\Gamma $ is aperiodic.", "The graph $E$ is no-exit (i.e.", "satisfies Condition (NE)).", "If $E$ is not a no-exit graph, there is a cycle with an exit and any vertex on that cycle is an aperiodic element of $F_E^\\Gamma .$ Conversely, if $a$ is an aperiodic element of $F_E^\\Gamma ,$ then $a\\rightarrow b$ for some stationary element $b$ such that at least one of the core cycles of $b$ must have an exit by Theorem REF .", "Hence, $E$ is not no-exit.", "Since every element which is not comparable is incomparable, Theorem REF implies a characterization of an incomparable element in $F_E^\\Gamma $ also.", "The following characterization of graphs such that all elements of $F_E^\\Gamma $ are incomparable follows directly from Corollary REF .", "Corollary 4.7 The following conditions are equivalent.", "Every nonzero element $a\\in F_E^\\Gamma $ is incomparable.", "Every generator of $F_E^\\Gamma $ is incomparable.", "The graph $E$ is acyclic." ], [ "Strengthening results of {{cite:73a6fbb9b5692e632f690814a396ec9df25c02ee}}", "By Proposition REF , a result of [11] holds without the assumption that the graph under consideration is row-finite.", "In this section, we show that the same assumption can be deleted from some of the main results of [11].", "The second part of Corollary REF shows that our results provide some further progress towards a positive answer to the Graded Classification Conjecture.", "First, we show that Theorems REF and REF and Corollary REF imply [11] without assuming that the graph is row-finite.", "We formulate this in the following corollary.", "Corollary 4.8 The graph $E$ has a cycle with no exit if and only if some nonzero element of $F^\\Gamma _E$ is periodic.", "The graph $E$ has a cycle with an exit if and only if some element of $F^\\Gamma _E$ is aperiodic.", "The graph $E$ is acyclic if and only if every nonzero element of $F^{\\Gamma }_E$ is incomparable.", "One direction of parts (1) and (2) follows by Theorems REF and REF .", "The other follows by Lemma REF which implies that a vertex on a cycle is periodic if the cycle has no exits and it is aperiodic if the cycle has an exit.", "Part (3) directly follows from Corollary REF .", "By [16], a $\\Gamma $ -order-ideal of $M_E^\\Gamma $ uniquely determines certain subset of vertices.", "We briefly review this construction.", "A subset $H$ of $E^0$ is said to be hereditary if for any $v\\in H$ and a path $p$ with $\\mathbf {s}(p)=v,$ $\\mathbf {r}(p)$ is in $H$ and it is saturated if $\\mathbf {r}(\\mathbf {s}^{-1}(v))\\subseteq H$ for a regular vertex $v$ implies that $v\\in H.$ For a hereditary and saturated set $H$ , let $G(H)=\\lbrace v\\in E^0-H\\mid v\\mbox{ is not regular and }\\mathbf {s}^{-1}(v)\\cap \\mathbf {r}^{-1}(E^0-H)\\mbox{ is nonempty and finite} \\rbrace .$ For $G\\subseteq G(H),$ the pair $(H, G)$ is said to be an admissible pair.", "The set of all such pairs is a lattice by $(H_1, G_1)\\le (H_2, G_2)\\; \\mbox{ iff }\\; H_1\\subseteq H_2,\\;\\; G_1\\subseteq G_2\\cup H_2$ (see [16] or [3]).", "By [16], this lattice is isomorphic to the lattice of graded ideals of $L_K(E)$ and by [3], this lattice is isomorphic to the set of order-ideals of $M_E.$ If $(H,G)\\mapsto I(H,G)$ denotes this isomorphism, then $M_E/I(H,G)\\cong M_{E/(H,G)}$ and both [16] and [3] contain details.", "By [4], the lattices of order-ideals of $M_E$ and of $\\Gamma $ -order-ideals of $M_E^\\Gamma $ are isomorphic.", "Moreover, if the assumption that $E$ is row-finite is deleted and hereditary and saturated set replaced by an admissible pair, the proof of [11] establishes that $M^\\Gamma _E/I(H,G)\\cong M^\\Gamma _{E/(H,G)}$ for an admissible pair $(H,G).$ Next, we show that the assumption that $E$ is row-finite can be removed from [11].", "Corollary 4.9 The following conditions are equivalent.", "The graph $E$ satisfies Condition (L).", "No nonzero element of $F_E^\\Gamma $ is periodic.", "$\\Gamma $ acts freely on $M_E^\\Gamma .$ The following conditions are equivalent.", "The graph $E$ satisfies Condition (K).", "No nonzero element of $M_E^\\Gamma /I$ is periodic for any $\\Gamma $ -order-ideal $I$ of $M_E^\\Gamma .$ The group $\\Gamma $ acts freely on $M_E^\\Gamma /I$ for any $\\Gamma $ -order-ideal $I$ of $M_E^\\Gamma .$ Part (1) directly follows from Corollary REF .", "By [16], $E$ satisfies Condition (K) if and only if $E/(H, G)$ satisfies Condition (L) for any admissible pair $(H,G).$ Since every such pair uniquely determines a $\\Gamma $ -order-ideal of $M_E^\\Gamma ,$ part (1) and Corollary REF imply the equivalences of conditions in part (2).", "[11] focuses on the monoid properties of $M_E^\\Gamma $ which are equivalent with various forms of simplicity of $L_K(E).$ We show these properties without requiring that $E$ is row-finite.", "Corollary 4.10 Let $K$ be any field.", "The following conditions are equivalent.", "The algebra $L_K(E)$ is graded simple.", "The $\\Gamma $ -monoid $M_E^\\Gamma $ is simple.", "The $\\Gamma $ -group $G_E^\\Gamma $ is simple as an ordered $\\Gamma $ -group.", "The following conditions are equivalent.", "The algebra $L_K(E)$ is simple.", "The $\\Gamma $ -monoid $M_E^\\Gamma $ is simple and no nonzero element of $M_E^\\Gamma $ is periodic.", "The $\\Gamma $ -monoid $M_E^\\Gamma $ is simple and every nonzero comparable element of $M_E^\\Gamma $ is aperiodic.", "The following conditions are equivalent.", "The algebra $L_K(E)$ is purely infinite simple.", "The $\\Gamma $ -monoid $M_E^\\Gamma $ is simple, no nonzero element of $M_E^\\Gamma $ is periodic and some element of $M_E^\\Gamma $ is aperiodic.", "Part (1) directly follows from the fact that the lattices of graded ideals of $L_K(E),$ $\\Gamma $ -order-ideals of $M_E^\\Gamma $ and $\\Gamma $ -order-ideals of $G_E^\\Gamma $ are isomorphic.", "By [1], $L_K(E)$ is simple if and only if it is graded simple and $E$ satisfies Condition (L).", "By part (1) and Corollary REF , this is equivalent with $M_E^\\Gamma $ being simple and without a nonzero periodic element.", "This last condition is equivalent with the requirement that every nonzero comparable element is aperiodic.", "By [1], $L_K(E)$ is purely infinite simple if and only if it is simple and $E$ has a cycle with an exit.", "By Corollary REF , $E$ has a cycle with an exit if and only if $M_E^\\Gamma $ has an aperiodic element.", "Lastly, we show Corollary REF .", "Parts (1) and (3) show that the first part of [11] holds without the condition that $E$ is row-finite.", "Parts (4) to (8) are further corollaries of our results.", "Corollary 4.11 Let $E$ and $F$ be arbitrary graphs.", "If there is a $\\Gamma $ -monoid isomorphism $M_E^\\Gamma \\rightarrow M_F^\\Gamma ,$ then the following hold.", "The graph $E$ satisfies Condition (L) if and only if $F$ satisfies Condition (L).", "The graph $E$ satisfies Condition (K) if and only if $F$ satisfies Condition (K).", "The lattices of graded ideals of $L_K(E)$ and $L_K(F)$ are isomorphic.", "$E$ is acyclic if and only if $F$ is acyclic.", "There is a cycle without an exit in $E$ if and only if there is a cycle without an exit in $F.$ There is a cycle with an exit in $E$ if and only if there is a cycle with an exit in $F.$ None of the cycles of $E$ have exits if and only if none of the cycles of $F$ have an exit.", "$E$ satisfies the condition below if and only if $F$ satisfies the condition below.", "The graph is row-finite, no-exit, has no sinks and it is such that every infinite path ends in a cycle.", "Parts (1) and (2) directly follow from Corollary REF .", "To show part (3), note that a $\\Gamma $ -monoid isomorphism $M_E^\\Gamma \\rightarrow M_F^\\Gamma $ induces a lattice isomorphism on the lattices of $\\Gamma $ -order-ideals.", "Since these lattices are isomorphic to lattices of graded ideals of $L_K(E)$ and $L_K(F),$ part (3) holds.", "Part (4) holds since $E$ has a cycle if and only if there is a nonzero comparable element in $M_E^\\Gamma $ by Corollary REF .", "Part (5) holds since $E$ has a cycle with no exit if and only if there is a nonzero periodic element in $M_E^\\Gamma $ by Corollary REF (1).", "Part (6) holds since $E$ has a cycle with an exit if and only if there is an aperiodic element in $M_E^\\Gamma $ by Corollary REF (2).", "Part (7) holds by Corollary REF and part (8) by Theorem REF .", "Corollary REF asserts that many relevant properties of two graphs match if the graphs have isomorphic graph $\\Gamma $ -monoids.", "Together with our previous results, Corollary REF indicates that the Graded Classification Conjecture may have a positive answer since the properties of the graph are well reflected by the structure of its graph $\\Gamma $ -monoid.", "The Graded Classification Conjecture was shown for finite polycephaly graphs in [9] and for a certain class of countable, row-finite, no-exit graphs in [12].", "In [7], it was shown for countable graphs such that for any two vertices the set of edges from one to the other is either empty or infinite.", "We also note that a weaker version of the conjecture was shown for finite graphs with neither sources nor sinks in [6]." ] ]

[ [ "Coherent charge carrier dynamics in the presence of thermal lattice\n vibrations" ], [ "Abstract We develop the coherent state representation of lattice vibrations to describe their interactions with charge carriers.", "In direct analogy to quantum optics, the coherent state representation leads from quantized lattice vibrations (phonons) naturally to a quasiclassical field limit, i.e., the deformation potential.", "To an electron, the deformation field is a sea of hills and valleys, as ``real'' as any external field, morphing and propagating at the sound speed, and growing in magnitude with temperature.", "In this disordered potential landscape, the charge carrier dynamics is treated nonperturbatively, preserving their coherence beyond single collision events.", "We show the coherent state picture agrees exactly with the conventional Fock state picture in perturbation theory.", "Furthermore, it goes beyond by revealing aspects that the conventional theory could not explain: transient localization even at high temperatures by charge carrier coherence effects, and band tails in the density of states due to the self-generated disorder (deformation) potential in a pure crystal.", "The coherent state paradigm of lattice vibrations supplies tools for probing important questions in condensed matter physics as in quantum optics." ], [ "5 colorlinks=true, linkcolor=black, citecolor=black, filecolor=black, urlcolor=black, 5 colorlinks=true, linkcolor=black, citecolor=black, filecolor=black, urlcolor=black, red APS/123-QED Semiclassical Theory of Thermal Resistivity in Metals and Semimetals Eric J. Heller Department of Physics, Harvard University, Cambridge, MA 02138 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Alvar Daza Department of Physics, Harvard University, Cambridge, MA 02138 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Donghwan Kim Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 K. Nasiri Avanaki Department of Physics, Harvard University, Cambridge, MA 02138 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 We extend the traditional semiclassical approaches for electrons in metals and semimetals to include thermal deformation potentials exerting forces on electrons, just as do external fields.", "The slowly traveling deformations nonperturbatively participate in smooth quasielastic exchange of energy and momentum with electrons.", "Analytic estimates and numerical simulations reveal resistivity in 2D has an approximate $T^4$ rise at low temperature, followed by a rollover to linear in $T$ rise at high temperature.", "The analytic formula is derived by classical perturbation theory and gives an expression behaving very much like the Bloch-Grüneisen formula.", "The deformation potential takes the form of a blackbody classical electromagnetic field.", "The electron momentum distribution shows non-Gaussian diffusion arising from long range deformation potential correlations related to the Hanbury Brown-Twiss effect.", "Traditionally, electrons in crystals under external electromagnetic fields are treated semiclassically , .", "Band energy is promoted to the status of a Hamiltonian and external classical fields are added.", "This program is ultimately phenomenological, but very successful.", "Since the 1950 paper of Bardeen and Shockley  the notion of a deformation potential resulting from phonon induced lattice strain has been firmly established and used in inelastic single electron-phonon perturbation theory.", "Current practice if both deformation potentials and external fields are important is to use a mixed approach, treating the electron-phonon interaction by perturbation theory, and using semiclassical equations for the external fields .", "Might there be a unified treatment for both, i.e., can the deformation potential be treated semiclassically, as a random pseudo-electric field generating a classical force on the electron?", "The answer seems to be yes.", "Semiclassical deformation model.", "Consider a uniform crystal at 0 Kelvin.", "For given band energy $E(\\vec{k})$ , the electron group velocity is $ d\\vec{r} /dt = \\partial E(\\vec{k})/\\partial (\\hbar \\vec{k})$ ; $\\vec{r}$ and $\\hbar \\vec{k}$ are single electron position and momentum, respectively.", "If there are externally applied electric and magnetic fields $\\vec{\\mathcal {E}}$ and $\\vec{\\mathcal {B}}$ , the traditional phenomenological semiclassical model gives $ d(\\hbar \\vec{k}) /dt = -e \\mathcal {\\vec{E}}(\\vec{r},t) -\\frac{e}{c} \\dot{\\vec{r} }\\times \\mathcal {\\vec{B}}(\\vec{r},t); $ $c$ is the speed of light , .", "Responsibility for resistivity in a pure crystal lies mainly with acoustic phonons, with inhomogeneities generated by strain - i.e.", "deformation potentials.", "These can be considered to generate pseudo-electric fields and put on a par with external fields.", "In fact, the thermal equilibrium deformation potential (equation (REF ), below) considered as an electric potential generates forces on the electron that follow the blackbody Planck radiation law, except for magnitude.", "Moreover, electrons are in a semiclassical regime with respect to the deformation potential, with short Fermi level electron wavelengths compared to the length scale of the pseudo-electric deformation fields.", "This supports the idea that the deformation potential can be taken into the semiclassical domain.", "The deformation potential arises because band structure is affected locally by strain induced by acoustic phonons.", "We may take those phonon modes that are active at a given temperature (in the sense of Bose occupation) to be collected into coherent states, so an electron interacts smoothly and quasielastically with well defined, moving inhomogeneous pseudo-electric field gradients.", "There is an exact prescription for a thermal ensemble of coherent states (that have classical parameters for mean position and momentum) for a harmonic lattice that gives all equilibrium expectation values of operators .", "This follows the role of the coherent states in classical electromagnetic fields, including blackbody radiation.", "When interference effects are important, the full quantum version of the single electron dynamics is an option, as is the semiclassical Van Vleck-Morette-Gutzwiller Green function (the stationary phase version of the Feynman path integral).", "This is a program laid out for external fields long ago by Lifshitz and Koesvich .", "The deformation potential.", "The ingredients for constructing a deformation potential begin with phonon induced lattice displacements.", "The important parameter is changes in atom-atom distances, i.e., the strain field ; rigid movement of groups of atoms are of little consequence.", "The deformation potential at temperature $T$ becomes a time dependent sum over all longitudinal acoustic phonon modes of all propagation directions and with a weighting given by (the square root of) Bose occupations, $\\begin{aligned}V_{D}(\\vec{r}, t)&=\\dfrac{\\mathbb {E}_{d}}{v_{s}}\\sum _{\\begin{array}{c}\\vec{q}\\\\{\\vec{q}}<q_{max}\\end{array}}\\sqrt{\\dfrac{2\\hbar \\omega _{\\vec{q}}}{\\rho _m V}}\\dfrac{ \\cos {(\\vec{q}\\cdot \\vec{r}-\\omega _{\\vec{q} }t + \\phi _{\\vec{q}})}}{\\sqrt{\\exp ({\\hbar \\omega _{\\vec{q}}}/{k_{B}T})-1}}\\end{aligned}$ where $\\vec{q}$ is a phonon wavevector, $\\phi _{\\vec{q}}$ is an arbitrary phase shift of each mode, $v_{s}$ is the longitudinal acoustic sound speed and $\\omega _{\\vec{q}} = v_{s} \\vert \\vec{q}\\vert $ , $\\mathbb {E}_{d}$ is the deformation potential constant that varies from several eV to tens of eV, $\\rho _m$ and $V$ are the mass density and the volume (or area in case of 2D) of metals or semimetals, respectively, $\\hbar $ is the reduced Planck constant, and $k_{B}$ is the Boltzmann constant.", "One of two limits cuts off the phonon wavenumber, i.e., either the Debye wavenumber $q_{D}$ or twice the Fermi wavenumber $2k_{F}$ determines the cutoff $q_{max}=\\min \\lbrace q_{D},2k_{F}\\rbrace $ .", "Undulations in the deformation potential shorter than $\\lambda _F/2 = \\pi /k_{F}$ (or in momentum space, $q>2k_{F}$ ) have little refractive influence on the electrons at the Fermi level, as if they were not present.", "This is a well known ballistic transparency effect for small scale fluctuations of a medium that are uniform when averaged over a wavelength or so , familiar for visible light in clear glass for example.", "Fig.", "REF gives numerical examples demonstrating the transparency specific to short wave components of the deformation potential, uniform on larger scales, but random on small scales.", "Metals normally have large Fermi surfaces, so $q_{D}<2k_{F}$ and the cutoff occurs at the Debye wavenumber, $q_{max}=q_{D}$ .", "Semimetals have small Fermi surfaces, then $q_{D}>2k_{F}$ , and the cutoff occurs at the (twice the) Fermi wavenumber, $q_{max}=2k_{F}$  .", "Thus, the critical temperatures separating the low and high temperature behavior of resistivity are Debye temperature $T_D=\\hbar v_{s} q_{D}/k_{B}$ for normal metals, and Bloch-Grüneisen temperature $T_{BG}=\\hbar v_s 2k_{F}/k_{B}$ for semimetals.", "Figure: Quantum wavepacket propagation in refractive (upper left) and ballistically transparent (lower left) random potentials.", "The potentials on the left are components of deformation potentials of the same form, differing in horizontal scale but not in height.", "The same wavepacket is sent into the viewing area from the upper right, starting in a flattened region of the potentials.", "Wavefront shaping and deflection (leading to branched flow if continued) is clearly seen in the upper right, whereas the wave at the lower right ignores the shorter correlation length potential.Constructing a random electromagnetic potential implied by Planck's Law of thermal radiation yields the same form (in 2D and in 3D) as the deformation potential; the strength of the potential and the wave speed are of course different.", "The classical electromagnetic version of the deformation potential appeared already as a vector potential in Hanbury Brown-Twiss .", "Acoustic phonons also have a linear dispersion and thermally populated harmonic degrees of freedom, so the blackbody electromagnetic potential connection is not surprising, although it seems to have gone unnoticed.", "In fact the controversy surrounding the early days of the Hanbury Brown-Twiss effect involving the success of a classical wave explanation for what seemed manifestly to be a quantum phenomenon is precisely relevant to the present paper.", "As became clear in that discussion, leading no less than to the birth of quantum optics, quantum optical and classical field results are fully compatible, with now well understood connections.", "The subject of electrical resistivity of metals and semimetals has so far proceeded within a quantum, “phonon as particle,” single quantum perturbative paradigm.", "It can also be cast in terms of phonon coherent states, and from there it is a small step to “phonon as field”, as we have done here.", "We expect the (phenomenological) Hamiltonian equations $\\frac{d(\\hbar \\vec{k})}{dt} = -\\frac{\\partial E(\\vec{k}, \\vec{r},t)}{\\partial \\vec{r}}; \\ \\ \\ \\frac{d\\vec{r}}{dt} = \\ \\frac{\\partial E(\\vec{k}, \\vec{r},t)}{\\partial (\\hbar \\vec{k})},$ where $E(\\vec{k}, \\vec{r},t)\\equiv E(\\vec{k}, a(\\vec{r},t))$ is a band energy in terms of the local atomic spacing $a(\\vec{r},t)$ , or better, the local strain field derived from the atomic spacings.", "We take the spacial dependence of $E(\\vec{k}, \\vec{r},t)$ to be proportional to the deformation potential.", "Properties and influence of the deformation potential.", "In our semiclassical approach, the deformation potential generates a classical force deflecting electron ray paths.", "The picture is one of gently curving rays suffering the unruly (but not completely random - see discussion below) time dependent forces of the deformations.", "From the phase and coordinate space perspective, the motion is expected to be a branched flow , , .", "In Fig.", "REF we see contour maps of the two dimensional deformation field at the top at two temperatures and the branched flow evolution of the same manifold of trajectories riding over the potential.", "The trajectories were launched from a point in space over a small range of angles as seen at the left in each panel.", "There are no hard collisions or other defects, but the finite mobility is evident.", "Figure: (Top) Deformation potential contours calculated with equation ().", "The peaks and valleys get higher and deeper as temperature increases, and the length scale of potential decreases in proportion to increasing temperature.", "(Bottom) Semiclassical electron pathways reveal branched flow typical of weak random scattering , , .", "Notice the difference in scale between top and bottom: trajectories need to interact with many bumps to produce branched flow.", "We used the same random phases for the four panels, which allows us to visualize the effects of temperature.Classical ray path tests.", "Now we investigate the ray momentum diffusion numerically.", "Including screening, we take the electric potential to be proportional to the deformation potential: $E(\\vec{k},\\vec{r},t) = E(\\vec{k}) + \\lambda V_D(\\vec{r}, t)$ .", "For each temperature, we run thousands of trajectories with the same initial kinetic energy, with random initial directions and positions, using several realizations of the random deformation potential.", "We use $v_{s}\\sim {5e3}{m/s}$ as the speed of longitudinal acoustic phonons, $a\\sim {e-10}{m}$ and $k_{F}\\sim {e8}{m^{-1}}$ as typical values of lattice spacing (length scale) and the Fermi momentum, respectively, for metals.", "For semimetals, $k_{F}$ depends on carrier density.", "We use a fourth-order symplectic scheme for integration.", "The function $c(t)=\\vec{p} (0)\\cdot \\vec{p} (t)$ measures the correlation between the initial momentum and the momentum at a time $t$ .", "In a Gaussian random diffusive process we have $\\langle c(t) \\rangle ={\\vec{p}(0)}^2 e^{-\\frac{t}{\\tau }}$ ; we suppose for now that we can treat the deformation potential as causing random diffusion of momentum, although there may be refinements needed, caused by spatial correlations in the deformation potential.", "It possesses a nonrandom averaging property: straight sections of trajectories find that the perpendicular forces from the higher frequency modes average to 0.", "This is easy to show from equation (REF ).", "The results in Fig.", "REF show an approximate $T^4$ low temperature rise in resistivity in 2D, rolling over to near $T$ at high temperature.", "The inset of Fig.", "REF shows the exponential decay of the correlation for the given time window.", "Changing this interval of observation leads to somewhat different power laws.", "If very short times are used, somewhat higher powers of $T$ are obtained.", "For longer observation windows the power law goes closer to $T^3$ (presumably $T^4$ in 3D but we have not checked this).", "This is an analogue of the nonrandom Hanbury Brown-Twiss correlations in the deformation potential.", "In fact only for a Markovian Gaussian random process should the sampling time not matter, and our numerical results show a strongly nongaussian momentum distribution for short to moderate times.", "The concept of a single relaxation time or free path may need re-examining, but this is beyond the scope of this paper.", "Therefore, it would be surprising if integer power laws for temperature dependence of the resistivity emerged, except in limiting approximations, as derived below.", "Figure: In a log-log plot, the dependence of the resistivity with temperature is shown, from numerical experiments in 2D for the carrier densityn=5e12cm -2 n={5e12}{cm^{-2}}and the Bloch-Grüneisen temperature T BG =30KT_{BG}={30}{K}.", "The inset shows a typical computation of the correlation as a function of time, revealing an exponential decay for the given time window.", "The slope of this plot is taken as the relaxation time τ\\tau and the resistivity from electron-phonon interaction was calculated from Δρ=ρ-ρ 0 =m ne 2 τ\\Delta \\rho =\\rho -\\rho _0=\\frac{m}{ne^2\\tau }; mm is the electron mass, ρ 0 \\rho _0 is the resistivity at T=0KT={0}{K}, which is the resistivity from impurities.", "In our simulation, ρ 0 =0\\rho _0=0 since there is no impurity.", "A key experimental paper by Efetov and Kim  showed very similar plots in graphene with doping varying the carrier density almost a factor of 10.Thermal relaxation and fluctuation.", "(under construction) Energy is exchanged because electrons collide quasielastically with moving deformation hills and valleys (the phonon bath is at a fixed temperature).", "It is neither possible nor necessary to keep track of individual phonons.", "Without time dependence of the deformations, we have electron energy conserved $dE(\\vec{k}(t),\\vec{r}(t))/dt = 0$ .", "However, given the $\\omega _{\\vec{q}} t$ term in the argument of the cosines, $dE(\\vec{k}(t),\\vec{r}(t),t)/dt = \\partial E(\\vec{k}(t),\\vec{r}(t),t)/\\partial t\\ne 0$ .", "This brings down a factor of $\\omega _{\\vec{q}}$ , and on the other hand the spatial derivative for mean squared momentum dispersion brings down a factor of $q_y$ in 2D, if the trajectory is temporarily moving along $x$ -direction.", "Since $\\omega _{\\vec{q}} = v_{s} \\sqrt{q_x^2+q_y^2}$ , the two quantities are very closely related.", "It is reassuring that the semiclassical quasielastic approach thus comes with its own mechanism for thermalization, although as yet we have not yet checked this numerically.", "The relaxation time for momentum and energy is the same, so the Weidemann-Franz law should follow without the usual free electron model and unspecified relaxation mechanism.", "Semiclassical derivation of temperature dependence of resistivity.", "(now under construction) For the frozen deformation potential $V_D(\\vec{r},t)=V_D(\\vec{r})$ , the momentum correlation function can be calculated analytically using classical perturbation theory.", "One knows $\\vec{p}(t)=\\vec{p}(0)+\\delta \\vec{p}(t)$ where $\\delta \\vec{p}(t)=\\int _0^tt^{\\prime }\\left(-{V_D(\\vec{r}(t^{\\prime }))}{\\vec{r}(t^{\\prime })}\\right)$ .", "Then, $\\vec{p}(0)\\cdot \\vec{p}(t)={\\vec{p}(0)}{\\vec{p}(t)}\\cos \\chi (t)\\approx \\vec{p}(0)^2(1-\\frac{\\delta \\vec{p}(t)^2}{2\\vec{p}(0)^2})$ where $\\chi (t)$ is the angle between the two momenta, and the quasielasticity of scattering ${\\vec{p}(t)}\\approx {\\vec{p}(0)}$ and the law of cosine for angle $\\chi (t)$ were used.", "Take ensemble average (average over possible realizations of deformation potentials specified by $\\lbrace \\phi _{\\vec{q}}\\rbrace $ ) to obtain $c(t)={\\vec{p}(0)\\cdot \\vec{p}(t)}=\\vec{p}(0)^2\\left(1-\\frac{{\\delta \\vec{p}(t)^2}}{2\\vec{p}(0)^2}\\right)$ where ${\\delta \\vec{p}(t)^2}=\\mathbb {E}_d^2\\sum \\limits _{\\begin{array}{c}\\vec{q}\\\\{\\vec{q}}<q_{max}\\end{array}}\\frac{2\\hbar q}{\\rho _m V v_s}\\left(\\frac{\\vec{q}}{\\vec{q}\\cdot \\vec{p}(0)/m}\\right)^2\\frac{1-\\cos (\\vec{q}\\cdot \\vec{p}(0)t/m)}{e^{\\hbar v_s q/k_BT}-1}$ using unperturbed trajectory $\\vec{r}^{(0)}(t^{\\prime })=\\vec{p}(0)t^{\\prime }/m$ .", "Note sinusoidal oscillation with different phases vanishes due to the ensemble average, i.e., ${\\cos (A+\\phi _{\\vec{q}})\\cos (B+\\phi _{\\vec{q^{\\prime }}})}=0$ if $\\vec{q}\\ne \\vec{q^{\\prime }}$ .", "For 2D, use polar coordinate for $\\vec{q}$ where angle $\\theta $ is chosen such that $\\vec{q}\\cdot \\vec{p}(0)=q{\\vec{p}(0)}\\sin \\theta $ , and use ${\\vec{p}(0)}=mv_F$ , then ${\\delta \\vec{p}(t)^2}&=\\mathbb {E}_d^2\\int _0^{q_{max}}\\frac{q\\;q}{(2\\pi /L)^2}\\int _0^{2\\pi }\\theta \\frac{2\\hbar q}{\\rho _m V v_s}\\left(\\frac{1}{v_F\\sin \\theta }\\right)^2\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\frac{1-\\cos (qv_Ft\\sin \\theta )}{e^{\\hbar v_s q/k_BT}-1}\\\\&=\\mathbb {E}_d^2\\int _0^{q_{max}}\\frac{q\\;q}{(2\\pi )^2}\\frac{2\\hbar q}{\\rho _m v_s}\\frac{1}{v_F^2}\\frac{fn(qv_Ft)}{e^{\\hbar v_s q/k_BT}-1}$ where $fn(A)=A \\pi \\lbrace J_1(A)[-2+A\\pi H_0(A)]+A J_0(A)[2-\\pi H_1(A)]\\rbrace \\approx 2\\pi A\\text{ for }A\\gg 1$ , and $H$ 's and $J$ 's are Struve and Bessel functions, respectively.", "Look at sufficiently long time correlation $t\\gg 1$ such that $A=qv_Ft\\gg 1$ holds.", "One can obtain momentum relaxation time from $c(t)=\\vec{p}(0)^2(1-t/\\tau )$ to obtain semiclassical resistivity $\\Delta \\rho _{SC}(T)=\\frac{m}{ne^2\\tau }$ in 2D $\\Delta \\rho _{SC}(T)=\\frac{m}{ne^2}\\frac{1}{2\\vec{p}(0)^2}\\mathbb {E}_d^2\\int _0^{q_{max}}\\frac{q\\;q}{(2\\pi )^2}\\frac{2\\hbar q}{\\rho _m v_s v_F^2}\\frac{2\\pi qv_F}{e^{\\hbar v_s q/k_BT}-1}$ The factor of $q^3$ has three origins.", "First there is a $q$ for the density of states in the magnitude of the wavevector; a second factor of $q$ arises from the square of the $(\\sqrt{\\omega }_q)^2 = v_{s} q$ in the numerator of equation (REF ), and finally we just discussed another factor of coming from the ensemble and angular average with long time.", "For specificity, one can look at graphene where $k_F^2=\\pi n$ , ${\\vec{p}(0)}=mv_F=\\hbar k_F$ , $q_{max}=2k_F$ and substitute (renormalized wavenumber) $x=q/q_{max}$ , (renormalized inverse temperature) $z=T_{BG}/T$ to obtain $\\Delta \\rho _{SC}(T)=\\frac{8\\mathbb {E}_d^2k_F}{e^2\\rho _m v_s v_F^2}\\int _0^1x\\frac{x^3}{e^{zx}-1}$ This is comparable to the generalized Bloch-Grüneisen formula , $\\Delta \\rho _{BG}(T)=\\frac{8\\mathbb {E}_d^2k_F}{e^2\\rho _m v_s v_F^2}\\int _0^1x\\frac{zx^4\\sqrt{1-x^2} e^{zx}}{(e^{zx}-1)^2}$ At low temperature limit $T\\rightarrow 0$ , $\\Delta \\rho _{SC}(T)=\\frac{8\\mathbb {E}_d^2k_F}{e^2\\rho _m v_s v_F^2}\\frac{\\pi ^4}{15}\\left(\\frac{T}{T_{BG}}\\right)^4\\\\\\Delta \\rho _{BG}(T)=\\frac{8\\mathbb {E}_d^2k_F}{e^2\\rho _m v_s v_F^2}\\frac{4\\pi ^4}{15}\\left(\\frac{T}{T_{BG}}\\right)^4$ At high temperature limit $T\\rightarrow \\infty $ , $\\Delta \\rho _{SC}(T)=\\frac{8\\mathbb {E}_d^2k_F}{e^2\\rho _m v_s v_F^2}\\frac{1}{3}\\left(\\frac{T}{T_{BG}}\\right)\\\\\\Delta \\rho _{BG}(T)=\\frac{8\\mathbb {E}_d^2k_F}{e^2\\rho _m v_s v_F^2}\\frac{\\pi }{16}\\left(\\frac{T}{T_{BG}}\\right)$ From these results, one can see $\\Delta \\rho _{SC}(T)$ and $\\Delta \\rho _{BG}(T)$ differ only by a factor of unity.", "In our model, Fermi distribution was not considered and that's why there is difference.", "If one considers Fermi distribution, one obtains identical result.", "In 3D the straight line trajectory is still one dimensional, and there is an extra factor of $q$ in the density of states.", "Then, finally we have $\\int q^4 dq /(e^{\\hbar v_{s} q/k_{B} T}-1)\\sim T^5.$ There is a correlation between successive events: the forces on a trajectory from the deformation potential self-average, nearly vanishing over straight paths.", "Pushes one way are followed by opposing pushes; this is not a simple random walk in momentum.", "Actual classical ray paths are not perfectly straight and the averaging is not complete.", "However, the averaging encourages nearly linear trajectories, which feeds more averaging.", "This will perhaps lead to interesting nonlinear behavior as a function of potential strength.", "The consequences are already present in our numerical studies but ignored in the analytical estimates.", "Conclusion.", "We have given a non-perturbative semiclassical and quantitatively successful theory of the temperature dependence of electrical resistivity in metals and semimetals.", "The approach suggests a new model of electron-phonon interactions, involving quasielastic collisions of electrons with slowly moving (as seen by the electrons) deformation potential hills and valleys.", "When we make classical perturbation theory approximations to our expressions, we find $T^4$ in 2D and $T^5$ in 3D as limiting low temperature behaviors.", "Although an approximate $T^4$ low temperature rise in resistivity in 2D also appeared numerically, there are strong reasons to believe the temperature rise need not be analytic.", "Even the exactness of $T^1$ at high temperature is in question, since although the classical perturbation theory gives that power, there are known corrections with the exact trajectories.", "Indeed the experimental rise is typically more like $T^{1.2}$ , a significant deviation (see Table 2.1 of Kasap ).", "The model of electron-phonon interactions has been expanded from one phonon inelastic events to smooth quasielastic ones, accounting for both energy and momentum changes with the same forces.", "The notion of quasi-elastic electron interactions with thermal distributions of phonon coherent states achieves a certain parity with electrons in classical electromagnetic fields-a long overdue development we believe.", "There are single photon (phonon) processes at one limit, and interactions with coherent states of a thermal radiation (deformation) blackbody field at the other.", "In certain limits, one picture is far more useful than the other.", "The semiclassical approach to deformation potential scattering has many implications for future work.", "Simulations of temperature dependent single particle magnetoresistance may now be possible without other assumptions about relaxation mechanisms.", "Full thermalization theory will however need more work, at the very least to somehow incorporate Pauli blocking effects.", "Acknowledgements.", "We thank Profs.", "Efthimios Kaxiras, Philip Kim, Peter Milonni, and Bertrand Halperin for stimulating and informative discussions surrounding the issues raised in this paper.", "Chris Fechinsn was very helpful with our understanding of the thermal coherent state ensemble.", "We thank The National Science Foundation for supporting this research, through the NSF the Center for Integrated Quantum Materials (CIQM) Grant No.", "DMR-12313 19, and NSF CHE 1800101.", "We would like to acknowledge the support of Real Colegio Complutense (RCC) through the Faculty Research Fellowship that supported the research of Alvar Daza at the Harvard Faculty of Arts and Sciences." ] ]

[ [ "Phonon counting thermometry of an ultracoherent membrane resonator near\n its motional ground state" ], [ "Abstract Generation of non-Gaussian quantum states of macroscopic mechanical objects is key to a number of challenges in quantum information science, ranging from fundamental tests of decoherence to quantum communication and sensing.", "Heralded generation of single-phonon states of mechanical motion is an attractive way towards this goal, as it is, in principle, not limited by the object size.", "Here we demonstrate a technique which allows for generation and detection of a quantum state of motion by phonon counting measurements near the ground state of a 1.5 MHz micromechanical oscillator.", "We detect scattered photons from a membrane-in-the-middle optomechanical system using an ultra-narrowband optical filter, and perform Raman-ratio thermometry and second-order intensity interferometry near the motional ground state ($\\bar{n}=0.23\\pm0.02$ phonons).", "With an effective mass in the nanogram range, our system lends itself for studies of long-lived non-Gaussian motional states with some of the heaviest objects to date." ], [ "Introduction", "Single photon Raman scattering from a system initiated at or near its quantum ground state is a powerful method for generation of highly non-classical states.", "Addition or subtraction of a countable number of excitations is a common way to generate Fock states or Schrödinger cat states [1], [2], [3], [4], [5].", "Of particular interest are situations where such operations are carried out by optical photon scattering from material systems with long coherence times, combined with high photon counting efficiency [6], since the memory capability effectively converts the heralded system into a deterministic source of single quanta.", "Indeed, heralded schemes have been used to great effect in atomic systems [7], [8], where fidelities and generation rates have improved steadily over the past few decades [9], [10], [11], [12].", "In the meantime, quantum systems based on mechanical resonators have shown great promise in the context of quantum transduction and communication, with devices exhibiting millisecond coherence times emerging in recent years [13].", "As a natural step in expanding the toolbox available to cavity optomechanical systems, heralded schemes have recently been developed and implemented for mechanical devices [14], [15], serving as a fundamentally new source of single quanta, namely phonons.", "blackIn parallel, recent advances in quantum electromechanics have also allowed for generation of highly non-classical states of motion [16].", "Apart from applications in quantum information processing, where phonons can be mapped onto flying qubits, such non-Gaussian states of macroscopic mechanical resonators [17] have been suggested as a way to study gravitational decoherence processes [18], [19], [20], [21].", "Here we report on a system which combines a number of versatile capabilities relevant for those applications.", "We realize phonon counting measurements [22] of a single high-$Q$ mechanical mode of motion of a membrane resonator [23], [24], [25], [13] blackin an optical cavity (see Fig.", "REF ).", "Thanks to cavity-based engineering of the optomechanical coupling, the interaction of light and mechanics is effectively dominated by a beamsplitter-like interaction between phonons and anti-Stokes photons: $\\hat{H}_\\mathrm {BS} \\propto \\hat{b}\\hat{a}^{\\dagger }_\\text{AS} + \\hat{b}^{\\dagger }\\hat{a}_\\text{AS} \\:,$ which leads to an exchange of optical and mechanical quanta, equivalent to anti-Stokes scattering of pump photons.", "This strong conversion of phonons into photons is the mechanism that both cools the mechanical resonator and maps its state onto light.", "Due to finite sideband resolution of the optomechanical cavity, there exists a small amount of two-mode squeezing interaction between phonons and Stokes photons ($\\propto \\hat{b}^{\\dagger }\\hat{a}^{\\dagger }_\\text{S} + \\hat{b}\\hat{a}_\\text{S}$ ), as illustrated in Fig.", "[fig:expset]fig:expset(d).", "This process introduces a small amount of heating that limits the minimum possible occupation of the mechanical oscillator under optical cooling [26].", "In practice, however, frequency-resolved detection of Stokes and anti-Stokes photons, particularly for low-frequency mechanical oscillators, is challenging due to the presence of numerous nearby mechanical modes and a strong optical pump field.", "Here we address this challenge with an ultra-narrowband spectral filter based on four cascaded free-space Fabry-Pérot cavities.", "This filter provides extremely efficient suppression ($>{155}{}$ ) of the Raman pumping light combined with highly selective detection of Stokes or anti-Stokes photons (Fig.", "[fig:expset]fig:expset(a,b)) detuned from the pump laser by the mechanical frequency.", "Importantly, the filter system efficiently suppresses other sources of spurious signals, such as nearby mechanical modes.", "The excellent passive stability and high transmission of the system (30% through the cascaded filter system), as well as robust and easily reproducible implementation of the filter cavity design, allow us to select sideband photons scattered by a single high-$Q$ mechanical mode.", "In parallel, we are able to perform high-efficiency heterodyne detection of sideband photons, thus demonstrating capabilities for versatile state engineering and characterization in both continuous-variable and discrete-variable domains.", "Using the narrowband properties of the filter system, we perform Raman-ratio thermometry of a membrane resonator [27] by counting scattered Stokes- and anti-Stokes photons, and demonstrate the efficiency of this method for characterizing the effective mode temperature and coherence properties near the motional ground state.", "Finally, we analyze statistical properties of the Raman-scattered light and show single-mode thermal statistics with a coherence time matching the dynamical optical broadening of the mode, thus verifying its spectral purity.", "The mechanical system employed here is a soft-clamped membrane resonator [24] which has recently emerged as a viable platform for generation and storage of long-lived quantum states [13].", "Already at moderate cryogenic temperatures, these resonators show millisecond coherence times.", "When combined with high detection efficiencies [25], which is necessary for optical quantum state tomography [28], a membrane-in-the-middle system based on soft-clamped resonators lends itself ideal for studies of non-Gaussian states of motion [29].", "However, the inherent multimode nature of these resonators poses a major challenge in this pursuit, since mechanical modes in close proximity to the mode of interest (less than 50 away) require strong spectral filtering.", "As a consequence, all major experiments in the -frequency regime have thus far relied on homodyne or heterodyne detection, and photon counting techniques in optomechanics remained exclusive to the -frequency systems [22].", "In the following section we describe our implementation of a system enabling phonon counting in the MHz-frequency regime.", "Here, we use a soft-clamped silicon nitride membrane resonator as our mechanical system.", "The membrane resonator is 12-thick, square-shaped with 3.1-long edges, and is patterned with a phononic crystal structure, which includes a defect ($\\sim {200}{}$ in diameter) in the center of the membrane [24].", "As shown in Fig.", "[fig:expset]fig:expset(c), the defect hosts a localized radial vibrational mode at a mechanical frequency of $\\Omega _\\mathrm {m}/2\\pi = {1.48}{}$ , which lies within the bandgap of the phononic structure.", "Soft-clamping and dissipation dilution provide a mechanical quality factor $Q = \\Omega _\\mathrm {m}/\\Gamma _\\mathrm {m}={380(10)d6}$ for this defect mode, which has an effective mass of $\\sim {2}{ŋ}$ .", "The membrane is inserted into a high-finesse ($\\mathcal {F}\\approx 22000$ , linewidth $\\kappa /2\\pi ={2.75}{}$ ) optical cavity with single-photon coupling rate of $g_0/2\\pi \\approx {50}{}$ .", "blackThe cavity is placed blackinside blacka liquid helium flow cryostat.", "Under these conditions, the linewidth of the cavity is sufficiently small to allow for ground-state blacksideband cooling, where a minimum achievable occupation of $\\sim 0.18$ phonons is given by the back-action limit [30].", "The cavity is pumped at a wavelength of $\\lambda \\sim {852}{}$ using a continuous-wave Ti:Sapphire laser (M Squared SolsTiS) allowing low-phase-noise operation.", "The laser drive is detuned to the red side of the cavity response in order to enable cooling and readout [30].", "Our optomechanical membrane-in-the-middle system allows reaching quantum cooperativities of $C_\\mathrm {q}=4g^2/\\Gamma _\\text{m} \\kappa \\bar{n}_\\mathrm {th}\\sim 100$ [13], with $\\bar{n}_\\mathrm {th}$ being the phonon bath occupation and $g$ being the light-enhanced optomechanical coupling $g = g_0 \\sqrt{\\bar{n}_\\mathrm {cav}}$ , where $\\bar{n}_\\mathrm {cav}$ is the intracavity photon number of the drive light.", "In our case the bath phonon number is $\\bar{n}_\\mathrm {th}\\approx k_\\mathrm {B}T/\\hbar \\Omega _\\mathrm {m}\\approx {1.3d5}$ due to the membrane substrate thermalization temperature estimated to be $T\\approx {9}{}$ .", "blackCombined with blacka long energy decay time blackof $T_1=1/\\Gamma _\\mathrm {M}\\approx {40}{}$ , we blackestimate a decoherence blacktime of $T_2=1/(\\bar{n}_\\mathrm {th}\\Gamma _\\mathrm {M})\\approx {300}{}$ , which corresponds to the average time for one phonon to enter the high-$Q$ mode from the thermal bath.", "Simultaneously, we achieve high scattered photon out-coupling efficiency of $\\sim 70\\%$ via properly engineered over-coupling.", "Similar systems have been used in experiments in the continuous-variable domain [31], [13], [25].", "We characterize the optomechanical system by phase-modulating the drive light in a frequency range that covers the cavity resonance.", "This allows us to determine the cavity parameters, such as detuning $\\Delta $ and optical linewidth $\\kappa $ .", "Furthermore, we use the same sweep to characterize the mechanical response based on the optomechanically-induced transparency (OMIT, see Ref.", "[32]), which allows us to precisely determine the mechanical frequency $\\Omega _\\mathrm {m}$ , light-enhanced coupling rate $g$ and optomechanical broadening $\\Gamma _\\mathrm {opt}$ .", "These parameters, when combined with the knowledge of the outcoupling efficiency and the amount of transmitted optical power, allow us to estimate the mean intracavity photon number $\\bar{n}_\\mathrm {cav}$ and hence the single-photon coupling rate $g_0$ ." ], [ "Filtering system", "The light emerging from the optomechanical cavity consists of the unscattered pump light (carrier), as well as the Stokes and anti-Stokes sidebands generated by mechanical motion.", "The sidebands generated by the mechanical mode of interest are not only close to the carrier frequency (merely 1.48 away), but are also significantly weaker.", "The latter is due to the fact that the probability of Raman scattering for a mechanical resonator in the ground state is $4g_0^2/\\kappa ^2 < 10^{-8}$ [29].", "In addition, blackas will be further discussed in the Results section, excited out-of-bandgap vibrational modes of the membrane are less than 70 away from the mode of interest.", "The combinations of these factors places very stringent requirements on a filtering system that should be able to fully isolate the photons scattered by the mechanical mode of interest.", "Using a single Fabry-Pérot cavity for this task would not be practical: a cavity linewidth of 300 would be needed to achieve the barely sufficient 80 of rejection at a detuning of 1.5.", "Apart from technical difficulties in implementation, such a filter would become increasingly inefficient as the optical mode becomes spectrally broader than 300 due to dynamical backaction [30], which is bound to happen during efficient optomechanical readout in our system.", "Finally, the $\\sim {0.5}{}$ time delay introduced by a filter with 300 bandwidth would exceed the estimated $\\sim {0.3}{}$ decoherence time of our system, severely limiting our choices of experimental protocols.", "The difficulties above can be avoided by using a cascade of several Fabry-Pérot cavities [14], [22].", "The rejection of a series of optical filters grows exponentially with the number of filters, being the product of the individual rejections, while the time delay scales only linearly with the number of cavities, being the sum of the individual delays.", "And importantly, the passband of the composite system remains sufficiently wide to accommodate broader signals.", "This approach allows us to use optical cavities with much more manageable linewidths.", "Our filter system consists of four Fabry-Pérot cavities positioned in series, with each cavity having a linewidth of approximately 30.", "As described in more detail in Supplementary Material, the filter system is locked to a desired center frequency by sending an auxiliary locking beam to the system and sequentially locking the cavities.", "During measurement, the locking light is temporarily disabled to prevent saturation of the photon counter, and the photon counting of the filtered signal can take place.", "The intensity transmission of a single filter cavity, normalized to a peak value of 1, is given by the Lorentzian $L(\\Omega )={1/[1+(2\\Omega /\\kappa _\\mathrm {f})^2]}$ , where $\\kappa _\\mathrm {f}$ is filter cavity linewidth and $\\Omega $ is detuning from the filter's resonance.", "The complete filtering system consisting of four cavities is therefore expected to have a transmission of $L(\\Omega )^4$ .", "The output of the optomechanical cavity is fiber-coupled to the input of the first filter as shown in Fig.", "[fig:expset]fig:expset(b).", "At the end of the filter array we position a single-photon counting module (SPCM, avalanche photodiode COUNT-20C from Laser Components).", "The filter can then be tuned to observe either anti-Stokes ($\\Delta _\\mathrm {f}=\\Omega _\\mathrm {m}$ ) or Stokes ($\\Delta _\\mathrm {f}=-\\Omega _\\mathrm {m}$ ) emission.", "A mechanical shutter [33] is used to isolate the photon counter from the strong locking light present when the cavities are being actively locked.", "Figure: Rejection of the cascaded filtering system as a function of detuning, where the L(Ω) 4 L(\\Omega )^4 model (violet) closely follows the measured response (gray).", "For frequencies above 200 the heterodyne signal becomes too weak to measure, but is expected to continue following the L(Ω) 4 L(\\Omega )^4 response.The characterization of the filter system is done at lower frequencies by applying phase-modulation sidebands on the incoming resonant light.", "By measuring the output from the filtering system with heterodyne detection, we recover both the resonant drive and the highly attenuated sidebands, which allows us to calculate the rejection of the whole filtering system with no need for additional calibration.", "As shown in Fig.", "REF , the rejection of the system is enough to strongly suppress ($>{30}{}$ ) the spurious mechanical modes which are only tens of away from the high-$Q$ mode.", "In addition, the strong optical drive detuned by the mechanical resonance frequency of $\\approx {1.5}{}$ is estimated to experience a much greater attenuation of $>{155}{}$ , as given by the $L(\\Omega )^4$ model.", "This makes the drive completely negligible compared to the scattering by the mechanical resonator.", "For resonant light, the filter system has a transmission of $\\sim 30\\%$ , currently limited mostly by losses at cavity mirrors and polarization optics.", "We believe that an overall transmission of 50% should be possible by improving the cavity incoupling efficiencies and transmission between cavities, with further improvements requiring cavity mirrors with lower intrinsic losses." ], [ "Detection of filtered mechanical sidebands", "Unfiltered optomechanical spectra can be easily and efficiently measured using direct detection, i.e.", "using a single photodiode placed directly at the output of the optomechanical cavity.", "This is due to the fact that the cavity transduces membrane motion into light intensity fluctuations when the optical drive is correctly detuned [30].", "As seen in Fig.", "[fig:forestfilter]fig:forestfilter(a), the spectrum consists of the main high-$Q$ mode surrounded by a phononic bandgap and dense regions of modes outside of the bandgap.", "In order to confirm the performance of the filtering system, we apply it to the output of the OM cavity and detect the filtered light with a heterodyne measurement.", "The effect of filtering on the light spectrum is easily seen on Figs.", "[fig:forestfilter]fig:forestfilter(b,c), where we tune the filter to be resonant with distinct parts of the spectrum.", "When the filter is tuned to the main mechanical mode [Fig.", "[fig:forestfilter]fig:forestfilter(b)], it efficiently isolates it from closely neighboring out-of-bandgap modes, which is a necessary condition for single-photon-based measurements.", "We can also select a part of the spectrum containing many out-of-bandgap modes, as in Fig.", "[fig:forestfilter]fig:forestfilter(c), which clearly reveals the envelope of the filter system's response.", "Figure: Filtering verified by heterodyne detection of filtered light.", "(a) Power spectral density (PSD) of light emitted directly from the cavity as registered by the direct-detection photodetector.", "PSD is calibrated in shot noise (SN) units.", "The spectrum shows the bandgap (between roughly 1.42{1.42}{} and 1.59{1.59}{}) provided by the phononic crystal structure and the high-QQ mechanical defect mode at Ω m /2π=1.48\\Omega _\\mathrm {m}/2\\pi ={1.48}{}.", "The overlaid shaded curves are transmission functions L(Ω-Δ f ) 4 L(\\Omega -\\Delta _\\mathrm {f})^4 of the filtering system positioned at Δ f =Ω m =2π×1.48\\Delta _\\mathrm {f}=\\Omega _\\mathrm {m}=2\\pi \\times {1.48}{} and at Δ f =2π×1.69\\Delta _\\mathrm {f}=2\\pi \\times {1.69}{}.", "(b) and (c) show the PSD of filtered light, with the filter centered on the high-QQ mechanical defect mode frequency,(b) and on the dense out-of-bandgap part of the spectrum (c).", "Note the different scale between (b) and (c).Figure: Photon counting of anti-Stokes sidebands, as a function of filter detuning, with count rates registered by the SPCM (blue dots), predicted count rates (solid blue line) with uncertainty (shaded teal area) and scaled directly-measured PSD (gray) for visual reference.", "The bandgap is effectively observed via photon counting.", "The shaded gray vertical strip shows the region surrounding the main mechanical mode.In a different measurement, we direct the output of the filtering system to the SPCM and lock the center frequency at different detunings across the bandgap (Fig.", "REF ).", "We observe greatly reduced photon scattering rates inside the bandgap, and large scattering when approaching low-$Q$ out-of-bandgap modes that are strongly coupled to the thermal bath.", "As a consistency check, we estimate the expected count rates by convolving the $L(\\Omega )^4$ response of the filters with a directly measured spectrum, where shot noise has been subtracted.", "The predicted and measured rates are in good agreement, with visible uncertainty only inside of the bandgap, where the scattering is low and shot noise level estimation errors can lead to increased uncertainties in the predicted count rate, as shown in Fig.", "REF .", "blackThe residual discrepancy at 1.51 MHz is due to one of the higher-order defect modes that is very weakly damped by light and prone to mechanical excitation due to e.g.", "unstable liquid helium flow through the cryostat.", "This mode does not affect the measurements of the main mode, as shown on Fig.", "[fig:forestfilter]fig:forestfilter(b), as it is being suppressed by the filter systems by 30 (see Fig.", "REF ).", "Importantly, when the filters are tuned to the main mechanical frequency, the relative photon flux contribution of out-of-bandgap modes is expected to be less than 1% compared to the photon flux due to the main mechanical mode." ], [ "Raman-ratio phonon thermometry", "An asymmetry between Stokes and anti-Stokes sidebands is a direct signature of near-zero occupation of the mechanical mode responsible for these sidebands.", "In particular, the ratio of powers in the two sidebands can be directly used to infer the residual phonon occupation or, equivalently, the mode temperature.", "For mechanical resonators in the frequency range, the most commonly employed method for measuring sideband powers has been heterodyne detection [27], [26], [34], [35].", "In particular, it is fully sufficient to perform heterodyning of photons scattered from red-detuned cooling light.", "In this case, as cooling increases, the two sidebands move from being asymmetric due to the optomechanical cavity response, to being equally strong, indicating the balance of Stokes and anti-Stokes scattering in the quantum back-action dominated regime.", "In the mechanical frequency range, a more direct method of measuring sideband power based on photon counting (and thus effectively phonon counting) [22], [36] has been demonstrated, where Stokes and anti-Stokes photons are filtered and subsequently detected by a single-photon detector.", "This method is not affected by the local oscillator noise [37], although it may suffer from dark counts of the photon counting detectors [22].", "The calibration-free nature of Raman-ratio thermometry, in both resonant and red-detuned cases, is one of its advantages as compared to the more commonly employed technique based on spectral calibration using external phase modulation [38].", "In particular, one does not need to pre-calibrate the optomechanical single-photon coupling rate strength $g_0$ .", "In the context of our work, phonon counting thermometry demonstrates the feasibility of efficient counting of single-phonon excitations, a fundamentally non-Gaussian operation.", "Here we apply the phonon counting thermometry technique to our $\\Omega _\\mathrm {m}/2\\pi = {1.48}{}$ -frequency mechanical mode.", "A single beam detuned from the optomechanical cavity resonance by approximately optimal detuning $\\Delta /2\\pi = {-1.85}{}$ , is used to both cool the membrane motion by dynamical back-action and simultaneously probe the system as it reaches the quantum back-action dominated regime.", "Figure: Ground-state cooling measured by photon counting.", "(a) Measured Stokes (red) and anti-Stokes (blue) scattering rates as a function of optical broadening Γ opt \\Gamma _\\mathrm {opt} of the mechanical mode and theoretical prediction with calibrated efficiency, as given in the legend.", "At lower driving powers, corresponding to smaller broadenings, the rates are set by the Lorentzian cavity response (left inset).", "At higher driving powers we observe that asymmetry is reduced due to the mechanical oscillator approaching the ground state (right inset), with scattering dominated by the quantum back-action.", "(b) Inferred thermal occupation n ¯ est \\bar{n}_\\mathrm {est} of the mechanical mode, along with the theoretical prediction (for bulk thermalization temperature of T=8.8±0.5T = {8.8\\pm 0.5}{}, which is the only free parameter in the fit) and the back-action limit n ¯ ba \\bar{n}_\\mathrm {ba} (dashed horizontal line).", "Error bars are inferred from statistical uncertainties from photon counting and fitting of other parameters used in Eq.", "().The transition rates for the mechanical system can be calculated following Refs.", "[39], [40], [30] as: $A_\\pm = g_0^2 \\bar{n}_\\mathrm {cav}\\frac{ \\kappa }{(\\Delta \\mp \\Omega _\\mathrm {m})^2+\\frac{\\kappa ^2}{4}},$ where $+$ ($-$ ) denotes upward (downward) transitions in the quantum harmonic oscillator ladder.", "The expected Stokes and anti-Stokes rates are then given by: $\\Gamma _\\mathrm {AS}= \\bar{n} A_-,\\quad \\Gamma _\\mathrm {S}= (\\bar{n} +1) A_+,$ with the dynamical optical broadening given by ${\\Gamma _\\mathrm {opt}= A_- - A_+}$ .", "Remarkably, in the ground state ($\\bar{n}=0$ ) the rates become highly asymmetric, regardless of $A_\\pm $ .", "The ratio between Stokes and anti-Stokes count rates is unaffected by the overall system efficiency, thus we can estimate the residual phonon occupancy from these rates as: $\\begin{aligned}\\bar{n}_\\mathrm {est}&=\\frac{R A_+}{A_- - R A_+}\\\\& =\\frac{R ((\\Delta +\\Omega _\\mathrm {m})^2+\\kappa ^2/4)}{((\\Delta -\\Omega _\\mathrm {m})^2+\\kappa ^2/4) - R ((\\Delta +\\Omega _\\mathrm {m})^2+\\kappa ^2/4)},\\end{aligned}$ with $R=\\Gamma _\\mathrm {AS}/\\Gamma _\\mathrm {S}$ .", "Notably, both $g_0$ and the cavity photon number $\\bar{n}_\\mathrm {cav}$ cancel out in the estimator.", "Other parameters of the cavity are found via OMIT measurements, as described in the Sec.", "2.REF .", "The theoretical prediction for the expected final phonon occupancy can be calculated as: $\\bar{n} = \\frac{A_+ + \\bar{n}_\\mathrm {th}\\Gamma _\\mathrm {m}}{\\Gamma _\\mathrm {opt}+\\Gamma _\\mathrm {m}}.$ In our case of $\\Gamma _\\mathrm {opt}\\gg \\Gamma _\\mathrm {m}$ , two regimes can be distinguished.", "In the thermally-dominated regime, corresponding to $C_\\mathrm {q}\\ll 1$ , the ratio between the rates is determined only by the cavity response, leading to ${R\\rightarrow ((\\Delta - \\Omega _\\mathrm {m})^2+\\kappa ^2/4)/((\\Delta + \\Omega _\\mathrm {m})^2+\\kappa ^2/4)}$ and ${\\Gamma _\\mathrm {AS}=\\bar{n}_\\mathrm {th}\\Gamma _\\mathrm {m}(1-1/R)^{-1}}$ .", "In the sideband-resolved regime, $R\\rightarrow \\infty $ and the anti-Stokes rate becomes equal to the phonon flux into the mechanical resonator coming from the thermal bath.", "In the other extreme (i.e.", "$C_\\mathrm {q}\\gg 1$ ) the two rates equalize and we have $\\Gamma _\\mathrm {S}\\approx \\Gamma _\\mathrm {AS}\\rightarrow g_0^2 \\bar{n}_\\mathrm {cav}\\kappa /(4 |\\Delta | \\Omega _\\mathrm {m})\\propto C_\\mathrm {q}$ , which shows that the scattering rates of two processes become equal and are dominated by quantum back-action.", "We now proceed to demonstrate this behavior in our experimental setting.", "Figure [fig:asymmetry]fig:asymmetry(a) demonstrates the measured Stokes and anti-Stokes rates both growing with the intracavity photon number, quantified in terms of induced optical broadening $\\Gamma _\\mathrm {opt}$ measured using OMIT.", "For the lowest broadening of $\\Gamma _\\mathrm {opt}/2\\pi ={255}{}$ we observe sideband scattering rates of ${20}{}$ for Stokes and ${100}{}$ for anti-Stokes, corresponding the the ratio $R=5$ .", "As we increase the broadening to $\\Gamma _\\mathrm {opt}/2\\pi ={11}{}$ , the detected rates arrive at ${215}{}$ for Stokes and ${260}{}$ for anti-Stokes corresponding to a ratio of $R=1.2$ .", "Increased optical broadening also leads to reduced transmission through the filter setup, as compared with raw rates given by Eq.", "(REF ), which is due to the optically-broadened scattered light getting slightly \"clipped\" by the filtering system response.", "We model this loss by integrating a normalized Lorentzian spectrum with a width $\\Gamma _\\mathrm {opt}$ centered around $\\Omega _\\mathrm {m}$ with $L(\\Omega -\\Omega _\\mathrm {m})^4$ : $\\begin{aligned}t(\\Gamma _\\mathrm {opt}) &= \\int L(\\Omega -\\Omega _\\mathrm {m})^4 \\frac{2}{\\pi \\Gamma _\\mathrm {opt}}\\frac{\\Gamma _\\mathrm {opt}^2/4}{(\\Omega -\\Omega _m)^2+\\Gamma _\\mathrm {opt}^2/4} \\mathrm {d}\\Omega \\\\&=\\frac{\\kappa _\\mathrm {f}\\left(5 \\Gamma _\\mathrm {opt}^3+20 \\Gamma _\\mathrm {opt}^2 \\kappa _\\mathrm {f}+29 \\Gamma _\\mathrm {opt}\\kappa _\\mathrm {f}^2+16 \\kappa _\\mathrm {f}^3\\right)}{16 (\\Gamma _\\mathrm {opt}+\\kappa _\\mathrm {f})^4}.\\end{aligned}$ This reduction is the same for Stokes and anti-Stokes sidebands, and thus it does not affect the ratio $R$ .", "We find that the optical spring effect shifting the effective mechanical resonance frequency is below 3 has a negligible effect on count rates for a fixed detuning of the filter system.", "In all cases we subtract the independently measured dark count rate of 15.5(5).", "The overall detection efficiency of the entire system is estimated to be $\\eta \\approx 2.5\\%$ , consisting of optomechanical cavity outcoupling ($75\\%$ ), fiber transmission/coupling ($60\\%$ ), filtering system ($30\\%$ for cavities and $50\\%$ for incoupling/outcoupling) and SPCM efficiency ($35\\%$ ).", "We note that the room for improvement of these efficiencies lies mostly in optics of the filtering system and SPCM efficiency.", "Finally, we calculate the ratio $R$ and estimate the mean phonon occupancy as given by Eq.", "(REF ) and plotted in Fig.", "[fig:asymmetry]fig:asymmetry(b).", "The estimated phonon occupation $\\bar{n}_\\mathrm {est}$ is accurately described following a fit of Eq.", "(REF ).", "The mechanical occupation finally reaches a value of ${\\bar{n}_\\mathrm {est}=0.23\\pm 0.02}$ at ${\\Gamma _\\mathrm {opt}/2\\pi = {11.0}{}}$ corresponding to ${C_\\mathrm {q}\\approx 22}$ , as estimated from calibrated parameters.", "The only free parameter of the theory is the the temperature of the phononic bath, determined to be $T={8.8\\pm 0.5}{}$ , which is consistent with previous works involving similar mechanical systems [13].", "The minimum occupation achievable with sideband cooling, often referred to as the back-action limit, lies at ${\\bar{n}_\\mathrm {ba}=(A_-/A_+-1)^{-1}\\approx 0.185}$ for our case.", "We thus observe a strong suppression of the classical sideband asymmetry due to the mechanical oscillator motion being primarily driven by the radiation-pressure shot noise." ], [ "Phonon correlation interferometry", "Lastly, we concentrate on statistical properties of light emitted from the high-$Q$ mechanical mode.", "We set $\\Gamma _\\mathrm {opt}/2\\pi ={2.1}{}$ and park the filter at the anti-Stokes sideband.", "We collect a total of ${18d3}$ counts at a count rate of 90, and look at coincidences between counts as a function of the delay time $\\tau $ .", "Since we only use a single detector, we reject the events for which $|\\tau | < {500}{}$ , in order to avoid effects of dead time of the SPCM and afterpulsing.", "blackThis time is still much shorter than any dynamics present in the system, and thus we can extrapolate our results on the coincidence rate to the zero-delay value.", "We analyze the coincidences in terms of the second-order Glauber correlation function ${g^{(2)}(\\tau )=\\langle \\hat{a}^\\dagger (0)\\hat{a}^\\dagger (\\tau ) \\hat{a}(0)\\hat{a}(\\tau )\\rangle /\\langle \\hat{a}^\\dagger (0) \\hat{a}(0) \\rangle \\langle \\hat{a}^\\dagger (\\tau ) \\hat{a}(\\tau ) \\rangle }$ .", "Light scattered by a single mechanical mode in thermal equilibrium has thermal statistics, as described by the following second-order correlation function: $g^{(2)}(\\tau ) = 1+A \\exp (-2|\\tau |/\\tau _\\text{C})=1+A \\exp (-\\Gamma _\\mathrm {opt}|\\tau |),$ where $\\tau _\\text{C}=2/\\Gamma _\\mathrm {opt}$ is the coherence time of light, with $g^{(2)}(0)=2$ blackfor $A=1$ .", "For a multimode thermal state, one would expect a multi-exponential or oscillatory decay.", "Our measurement, shown in Fig.", "REF , shows a single-exponential decay with a decay time matching the optically-broadened linewidth and exhibits $g^{(2)}(0)=1.88\\pm 0.08$ (obtained from a fit of Eq.", "(REF ) blackwith $A$ and $\\tau _\\mathrm {C}$ as free parameters), which is close to the theoretical value of 2, indicating the high purity and single-mode behavior of the measured thermal state of light.", "The optical coherence time of $\\tau _\\text{C}={143(18)}{}$ (corresponding to an optical linewidth of 2.2(3)), closely matches the optical broadening of the mechanical oscillator, independently measured by OMIT to be $\\Gamma _\\mathrm {opt}/2\\pi = {2.1}{}$ .", "This feature confirms our system's potential for producing non-Gaussian quantum states of light and of motional degrees of freedom.", "We attribute the residual discrepancy to the dark counts that exhibit Poissonian counting statistics.", "Figure: Second-order autocorrelation measurements of the spectrally-filtered anti-Stokes photons from the high-Q mechanical mode.", "We estimate g (2) (0)=1.88±0.08{g^{(2)}(0)=1.88\\pm 0.08} and optical coherence time τ C =143(18)\\tau _\\text{C}={143(18)}{}.", "Error bars in the plot are inferred from statistical uncertainties of Poissonian counts, while shading for the fitted curve corresponds to three s.d.", "confidence bounds." ], [ "Conclusions and Outlook", "We have demonstrated a versatile optomechanical system where an ultracoherent high-$Q$ mechanical resonator is subject to both discrete and continuous variable measurements, paving the road towards generation of a wide range of non-classical states of motion.", "We have directly demonstrated selection of photons scattered from a single mechanical mode by heterodyne spectroscopy, as well as by second-order single-photon intensity interferometry.", "The mode has been optically cooled to a final phonon occupation of $0.23\\pm 0.02$ , which has been measured using Raman-ratio thermometry via photon counting.", "Our work marks the first application of phonon counting techniques to low-frequency mechanical resonators, paving the way towards generation of non-Gaussian mechanical states [29], [17] and studying related decoherence processes [18], [19], [20], blackfor which the relatively larger mass of our system, as compared to GHz-frequency resonators [41], [15], is a blackclear asset.", "blackThe path towards generating non-Gaussian macroscopic quantum states blackin our system blackpresents blackadditional technical challenges.", "The mean phonon occupation of $0.23$ demonstrated here is at best borderline for demonstrating non-classical features [29].", "The main step will be employing a narrower optomechanical cavity, to simultaneously allow better sideband cooling to at least $0.1$ phonons in the fully sideband-resolved regime ($\\kappa /2\\pi \\sim {300}{}$ for which $\\bar{n}_\\mathrm {ba}\\sim 0.003$ ) and high degree of selectivity of Stokes or anti-Stokes processes.", "In such a regime, the technical noise of the laser is expected to start limiting the occupation [42].", "Furthermore, performing Raman-ratio thermometry will require photon detectors with very low ($<1/\\mathrm {s}$ ) dark count rates.", "The ultra-narrowband filtering technique we developed can become useful in many different optomechanical systems, ranging from other low-frequency devices such as trampoline resonators [43], to macroscopic levitated particles [44], as well as atomic ensembles [45], [46] and ionic or defect emitters in solid state [47].", "In the case of atomic ensembles, optical cavities are routinely used to distill weak quantum light [48], but ultra-narrowband filters, such as demonstrated here, would be required to employ photon counting techniques for quantum memories operating in the spin-exchange relaxation free (SERF) regime [49], [50] or based on motional averaging [51], [12], for example.", "Narrowband filtering can be beneficial for solid-state emitters as well, allowing better understanding of their optical properties [52], [53], as well as enabling spectrally-based selection of single emitters from an ensemble [54].", "Narrowband photonic states demonstrated here can be directly interfaced with material systems of long coherence times, facilitating long-distance quantum communication.", "Ultimately, they can also be used in hybrid quantum networks [31], [55] to generate entangled states via heralded photon counting.", "Funding information.", "ERC Advanced Grant QUANTUM-N and Villum Investigator Grant QMAC.", "Acknowledgments.", "We acknowledge contributions to soft-clamped membrane design by A. Schliesser, discussions with J. Appel, J. H. Müller, M. Zugenmaier, K. B. Dideriksen, B. Albrecht, contributions to laser stabilization and electronic design by T. Zwettler, technical assistance by D. Wistisen, and early-stage development of the experiment by A. Barg.", "M. P. was partially supported by the Foundation for Polish Science (FNP).", "Disclosures.", "The authors declare no conflicts of interest.", "See Supplementary Material for supporting content." ], [ "Construction and material", "We have chosen Invar 36 (1.3912) as the material for the construction of the spacers for our filtering resonators.", "Invar provides a low coefficient of thermal expansion of $<{1.5}{ppm/}$ [56] at room temperature, while being significantly easier to machine, more robust, available and economic than higher-grade low-expansion materials such as ULE glass [57], [58].", "In order to achieve a length of 60, we have constructed each cavity out of three pieces that are joined by two threaded connections as shown in Fig.", "REF .", "Figure: Design of a filtering cavity.", "The resonator's main body (top) is composed of 3 parts joined by threads (central inset).", "One of the mirrors is pressed directly against the main body (left inset), while the other is positioned on a piezo-transducer to allow for cavity locking (right inset)." ], [ "Vibration isolation", "The cavities are placed inside four long cylindrical vacuum tubes (KF40) that are part of the same vacuum system.", "Given the substantial length of each filtering resonator, they are susceptible to vibrations at their first and second bending mode frequencies (260 and 720 respectively).", "To isolate the system from external perturbations, we employ damped spring supports.", "There, each cavity is supported from below at its Airy points by two mounts of two soft springs each (5/), as shown in Fig.", "REF .", "This method gives a cavity-spring oscillation frequency of approximately 10 and translates into a rejection of more than 52 of power of external vibrations at the frequency of the first flexural mode of the spacer.", "To dampen the remaining 10 oscillations, we employ a cellulose (cotton) layer on top of each cavity, has proven to have minimal outgassing in our vacuum conditions of 1d-4.", "The resulting stability and vibration immunity is excellent, allowing normal alignment work to be performed on any given cavity while locking other cavities and using light transmitted through them.", "Figure: Damped spring cavity support.", "The two soft springs on the bottom give the cavity-spring system a resonance frequency of 10 and block external high-frequency vibrations.", "The cellulose damper (top of the cavity) dampens the remaining 10 cavity-spring vibrations.", "This damper is tightly inserted between the top of the filter cavity and the inner surface of the vacuum tube." ], [ "Dither-locking", "In order to lock all four cavities on resonance, we use the well-known technique of dither-locking, functionally equivalent to low-frequency Pound-Drever-Hall locking [59].", "Here, a periodic $\\sim {1}{}$ modulation is applied to each filter's piezoelectric transducer, therefore modulating each cavity's resonance frequency.", "The reflected light intensity is detected and demodulated, as shown in Fig.", "REF .", "This produces an error signal that is directly related to the derivative of each cavity's reflection function.", "By using slightly different modulation frequencies for different cavities, interference of signals between the different dither drives is avoided.", "This technique, when compared to the conventional Pound-Drever-Hall scheme, has the advantage of not requiring an electro-optical modulator before each cavity, which would otherwise introduce significant losses and complexity into the optical setup.", "Low-level locking functionality, including modulation, demodulation, and feedback, is implemented digitally in ARM-based microcontroller boards (Arduino Due) attached to each filter cavity, while high-level coordination of the four microcontroller boards is performed by a Python application running on computer.", "The full locking procedure consists of the following steps.", "In the beginning, the first cavity is scanned using its piezo transducer in order to find several of its TEM00 fundamental optical resonances (3 to 4 free spectral ranges are scanned).", "Locking light is generated from a common laser with a pair of frequency-tuned acousto-optic modulators (AOMs).", "Once the optical resonances have been identified, the first cavity is locked on the side of one of those resonances, i.e.", "with a detuning of $\\Delta = \\kappa _\\mathrm {f}/2$ .", "Once lock is confirmed to be stable, dithering is enabled and the new dither-derived error signal is used to lock the cavity on its resonance ($\\Delta = 0$ ).", "With light being transmitted through the first cavity, the next cavity is now locked via the same procedure, and so on.", "While lock light is being used, one mechanical shutter [33] blocks this light from going to the fragile SPCM, and a second shutter prevents any spurious back-reflections of lock light from propagating into the sensitive optomechanical system.", "When performing photon-counting measurements, lock light is disabled by turning off the AOM drives that generate it, and the locks of the filter cavities are switched into the dead-reckoning (,,frozen lock”) mode.", "The optical paths to the SPCM and the optomechanical setup are subsequently opened to allow the signal to propagate through the filters.", "After several seconds of passive stability and photon collection, we close the mechanical shutter to block the input to the SPCM and re-enable the locking light.", "The dither-locking is then re-engaged and the four cavities, still being in the vicinity to their resonances, are promptly brought back to resonance within less than 500, as shown in Fig.", "REF .", "Figure: Dither-lock scheme for filter cavities.", "The light reflected from the cavity is redirected towards the photodetector.", "The length of the cavity is modulated by the piezoelectric transducer (PZT).", "The resulting reflection signal is digitally demodulated and used to lock the resonator on resonance." ], [ "Measurement of passive stability", "To be useful in pulsed protocols involving photon counting, the filtering setup must be stable enough to operate without any reference light input (\"frozen lock\" regime) while photons are being counted.", "In this regime, active piezo feedback is paused and the reference lock light is disabled.", "The cavities' length then evolves freely, and it is crucial that all cavities remain sufficiently close to their resonances during the relevant timescale.", "On average, the relative transmission of the entire filter system decays to 50% in more than 4 seconds, as shown in Fig.", "REF , and in 90% of the cases, the relative transmission of the full filter system stays above 80% for approximately 1 second.", "Importantly, 1 second is more than 3000 times longer than the next slowest experimental timescale, namely the 0.3 decoherence time of our mechanical resonator, given by $T_1 = \\hbar Q / (k_\\mathrm {B}T)$ [30], where we assume a bath temperature of 9 and a $Q$ -factor of 380d6.", "blackDuring data acquisition, a lock freezing time of 1.5 was used.", "With this setting, the relative efficiency of the filters is $92 \\pm 6 \\%$ (averaged over the freezing interval).", "Notably, a typical experimental run is comprised of many lock-freeze cycles, which permits us to average out technical fluctuations (e.g.", "filter system transmission) and Poissonian count rate fluctuations.", "Statistical properties of the ensemble of these cycles then allow us to accurately estimate the uncertainty on photon rates, as well as dark count rate.", "Figure: Transmission of the four-cavity filtering system over 4 after freezing the locking feedback loop.", "Feedback is active for times before 0 and after 4, showing the high passive stability of the system during dead-reckoning.", "The background color plot corresponds to the probability density of relative transmission." ] ]

[ [ "Experimental Demonstration of Dynamic Topological Pumping Across\n Incommensurate Bilayered Acoustic Metamaterials" ], [ "Abstract A Thouless pump can be regarded as a dynamical version of the integer quantum Hall effect.", "In a finite-size configuration, such topological pump displays edge modes that emerge dynamically from one bulk-band and dive into the opposite bulk-band, an effect that can be reproduced with both quantum and classical systems.", "Here, we report the first un-assisted dynamic energy transfer across a metamaterial, via pumping of such topological edge modes.", "The system is a topological aperiodic acoustic crystal, with a phason that can be fast and periodically driven in adiabatic cycles.", "When one edge of the metamaterial is excited in a topological forbidden range of frequencies, a microphone placed at the other edge starts to pick up a signal as soon as the pumping process is set in motion.", "In contrast, the microphone picks no signal when the forbidden range of frequencies is non-topological." ], [ "Experimental Demonstration of Dynamic Topological Pumping Across Incommensurate Bilayered Acoustic Metamaterials Wenting Cheng Department of Physics, New Jersey Institute of Technology, Newark, NJ, USA Emil Prodan prodan@yu.edu Department of Physics, Yeshiva University, New York, NY, USA Camelia Prodan cprodan@njit.edu Department of Physics, New Jersey Institute of Technology, Newark, NJ, USA A Thouless pump can be regarded as a dynamical version of the integer quantum Hall effect.", "In a finite-size configuration, such topological pump displays edge modes that emerge dynamically from one bulk-band and dive into the opposite bulk-band, an effect that can be reproduced with both quantum and classical systems.", "Here, we report the first un-assisted dynamic energy transfer across a metamaterial, via pumping of such topological edge modes.", "The system is a topological aperiodic acoustic crystal, with a phason that can be fast and periodically driven in adiabatic cycles.", "When one edge of the metamaterial is excited in a topological forbidden range of frequencies, a microphone placed at the other edge starts to pick up a signal as soon as the pumping process is set in motion.", "In contrast, the microphone picks no signal when the forbidden range of frequencies is non-topological.", "Figure: Dynamic topological pumping.", "(a) Photograph of a fully assembled acoustic bilayer consisting of top/bottom periodic arrays of cylindrical acoustic resonators with incommensurate lattice constants.", "The labels Sii and Mii indicate the positioning of the speaker and microphone during various experiments referenced in the text.", "The middle red bar indicates the presence of an inner chamber, which connects the top and bottom resonators and is referred to as the spacer.", "For dynamical pumping, additional resonators are mounted on the top left side.", "(b) Photograph of the inner structure, with the spacer now fully visible.", "(c) Cross-section showing only the domain of wave propagation, together with relevant parameters.", "Note that the very left resonator is completed decoupled from the main structure.", "(d) Bulk resonant spectrum as function of top-bottom relative alignment, when the top and bottom lattice constants are equal.", "In this case, all spectral gaps are trivial.", "(e) Bulk resonant spectrum as function of d 2 /d 1 d_2/d_1.", "In this case, additional gaps open in the spectrum, which are all topological.", "(f) Schematic of the pumping process as well as a simulation of the air pressure at the beginning of the pumping cycle.", "(g) Microphone reading when the source frequency is adjusted in a topological gap.", "(h) Microphone reading when the source frequency is adjusted in a non-topological gap.More than 35 years ago, Thouless asked himself what happens with a filled sea of fermions when the underlying potential is slowly and periodically modulated in time [1].", "He predicted that a precise non-fluctuating number of particles will be effectively transported from one side of the system to the other and that this number is determined by a topological invariant computed for a virtual system of one dimension higher than the original.", "The effect has been directly demonstrated recently, with both fermions and bosons [2], [3].", "It is now well established [4], [5] that augmentation of a parameter space to a $d$ -dimensional quantum or classical system can give access to topological effects that, in normal conditions, are observed in $d+1$ or higher dimensions.", "The prototypical example is the periodic 1-dimensional Rice-Mele model [6], where an adiabatic deformation of the parameters leads to a virtual 2-dimensional system whose energy bands support non-trivial Chern numbers [4].", "As a result, the system displays chiral edge bands when driven in an adiabatic cycle and edge-to-edge topological pumping becomes possible.", "The existing experimental works on edge-to-edge topological pumping can be classified in three groups.", "The ones in the first group [7], [8], [9], [10], [11], [12] report only renderings of the resonant spectra as functions of the adiabatic parameters.", "The parameters are not varied continuously but rather the measurements are interrupted and the systems are adjusted by hand or other means to achieve the next parameter values.", "Acquisition of the spectra for a single adiabatic cycle can take days.", "In the experimental works from the second group [13], [14], [15], [16], [17], [18], [19], the systems with different adiabatic parameters are rendered and coupled in space and the profiles of the resonant modes are mapped in space rather than time.", "The connection with a real dynamical Thouless pump is done through a mathematical argument that involves simplifications and assumptions [13].", "Lastly, the experimental works in the third group [20], [21] report assisted dynamical edge-to-edge pumping.", "We call it assisted primarily because energy was pumped into the mode to keep it alive as it traversed from one edge to another.", "Without such external intervention, the pumping would have succumbed to the dissipation and nothing would have been observed at the receiving end of the system.", "These experiments also contain a large number of active components controlled by an expensive layer of electronics, whose complexity grows with the size of the system.", "For this reason, the system in [20] had only eight unit cells.", "While valuable demonstrations, these approaches do not offer yet a path towards practical implementations.", "We demonstrate an un-assisted edge-to-edge topological pumping of sound, where the human intervention is completely absent once the mode is loaded at one end of the system.", "The key innovation is the use of an aperiodic meta-material structure that has a simple built-in mechanism that implements global structural changes resulting in rapid and repeated cyclings of its phason.", "This mechanism is simply the relative sliding of two coupled incommensurate periodic acoustic crystals.", "All the previous experimental works based on aperiodic structures employ the quasi-periodic pattern originally proposed in [13], where the dynamical matrices can be directly connected with the Aubry-André and Harper models [22], [23].", "This is not the case for an incommensurate bilayered acoustic crystal, yet we show in the Supplementary Materials that its dynamical matrix belongs to an algebra isomorphic to that of magnetic translations.", "As such, the spectral gaps carry Chern numbers and the bulk-boundary correspondence principle is the same as for the Chern insulators.", "However, our topological system is distinct from the standard Chern mechanical crystals [24], [25], [26] because one dimension is virtual.", "Using the phason of a quasi-periodic structure to generate topological edge modes is an appealing strategy because the bulk resonant spectrum is independent of the phason [27], hence all the bulk spectral gaps are preserved during the phason cycles.", "This remarkable characteristic is un-matched by any other design principle [28].", "Furthermore no fine-tuning of the systems is required [7], because the presence of topological edge modes steams from the aperiodic pattern alone.", "In [29], we supplied an algorithmic method to generate aperiodic system with phasons living on generic topological spaces.", "In particular, [29] idenfied the phason space for incommensurate bilayered patterns.", "The principles discovered in [7] and [29] made the present work possible.", "Figure: Principles and mechanism of our dynamical pumping.", "(a) The configuration of the system at the beginning of a pumping cycle.", "The top array is uniformly displaced to the right and, after a total displacement d 2 d_2, the system returns in its original configuration and completes a full pumping cycle.", "A speaker is inserted in resonator S1 and is kept on at all times, while a microphone is inserted in the resonator M0.", "(b) Simulation of the resonant spectrum as function of displacement.", "Chiral left and right edge bands are observed, which both connect two disjoint parts of the bulk spectrum.", "(c-h) Rendering of pumping mechanism: The left edge mode is loaded when the source frequency matches the mode frequency (c); The mode self-oscillates while its frequency is pushed up (d); The character of the mode changes from left-localized to delocalized (e); The character of the mode changes again from delocalized to right-localized (f); The mode self-oscillates as its frequency is pushed down (g); The cycle repeats itself as the top array is further pushed to the right (h).", "The microphone starts to pick signal after the event (e).", "The simulations in panels (c-h) show the spatial profile of the resonant mode highlighted in the sub-panel below it.", "The shown microphone outputs are not from real measurements.", "The pumping parameter is ϕ\\varphi in all panels.Our main results are summarized in Fig.", "REF , where we present direct evidence of energy transfer from one end of a bulk structure to the other end, even though the frequency of the source falls in a forbidden wave-propagation range.", "This energy transfer happens in pumping conditions and when the source frequency is in a topological spectral gap of the metamaterial.", "In contradistinction, when the frequency is adjusted in a non-topological spectral gap, there is no energy transfer even though same conditions of pumping are applied.", "Our experimental platform consists of the two incommensurate periodic arrays of acoustic resonators described in Fig.", "REF (a-c).", "The dimensions have been optimized to maximize the size of the topological gap.", "This type of patterned resonators was theoretically studied in [29], where it was found to support topological spectral gaps and topological edge modes.", "However, to our knowledge, this is the first time when coupled incommensurate chains are experimentally used to engineer chiral edge bands for topological pumping.", "Key to our experimental design was the replacement of any elaborate interconnections between acoustic resonators with a thin uniform spacer, extending from one end of the structure to the other.", "The resonators are attached to and coupled through this spacer.", "Note that this type of coupling does not allow fine-tuning but, as we mentioned, that is not necessary when using aperiodic principles, as long as the coupling is strong [7].", "Furthermore, edges can be created by simply filling the spacer with solid material.", "The adiabatic parameter of the system is $\\varphi = x/d_2$ , where $x$ and $d_2$ are specified in Fig.", "REF (c).", "Note that $\\varphi $ lives on the circle, which is only a part of the total phason space [30].", "The advantages of our design are: a) $\\varphi $ can be driven in an adiabatic cycle by simply sliding the top array while holding the bottom one fixed; b) Since the bottom array is fixed, we can continuously pump energy at one edge (and only on the that edge) by placing a source on the first bottom resonator; c) The left and right edges can be independently adjusted to achieve the optimal dispersion of the edge modes.", "The numerically simulated topological pumping process is reported in Fig.", "REF , where we also explain its mechanism.", "Sure enough, the left and right chiral edge bands are present in the topological gap.", "Note their particular and optimal dispersion, which made the dynamical pumping possible.", "Indeed, it is important that the right chiral edge band emerges from the top bulk-band shortly after the left chiral edge dived into the same band.", "This is because the non-adiabatic effects cannot be prevented when the pumping of energy is through the bulk states.", "As such, one has to optimize the pumping cycle such that there is a rapid change of the mode character from left-localized to extended and to right-localized, exactly as it can be seen in Fig.", "REF (c-h), where our pumping cycle was broken down into steps.", "In a standard topological edge-to-edge pumping, the mode self-oscillates after being loaded at the left edge, hence the pumping cycle must be performed fast enough to overcome dissipation.", "Given the particular engineering of our system, the pumping cycle can be performed extremely fast and repetitively, even without any external intervention.", "This enabled us to achieve the first un-assisted dynamical energy pumping via topological edge modes.", "Its dramatic manifestation is documented in Fig.", "REF (g), where a receiver placed opposite to an acoustic source is shown to pick up acoustic signal when the excitation frequency is in a topological resonant gap.", "In this experiment, 10 resonators were added beyond the edge to the left side of the top array, which resulted in the 10 pumping cycles visible in Fig.", "REF (g).", "The time period of the pumping cycle is approximately 0.12 seconds in Fig.", "REF (g).", "We have experimented with the time period of the cycle and found that the energy transfer is completely cut out when the period is about 1 second.", "This demonstrates that the pumping process is indeed essential for the energy transfer across the acoustic meta-crystal.", "Furthermore, when the source frequency is adjusted in a non-topological spectral gap, the receiver picks no signal whatsoever.", "We have experimented with different source frequencies inside the non-topological gap and we can confirm that the receiver does not pick any signal even when the frequency is very close to the bulk spectrum.", "This demonstrates that the chiral edge bands, formed inside the topological spectral gap, play an essential role for the energy transfer phenomena detected in our experiments.", "The sound of the pumping reported in Fig.", "REF (h) can be played here, $[poster=Button.jpg,repeat=10]{0.03}{0.03}{Fig1h.wav}$ , or from the audio files available online.", "As one can see, there is stark difference between the two pumpings reported in Fig.", "REF .", "Taking into account all the above facts, there can be no doubt that the energy transfer across the meta-crystal was through a classic topological pumping process.", "Figure: Experimental mapping of the bulk resonant spectrum.", "(a) Simulated resonant spectrum reproduced from Fig.", "(e), with arrows indicating the topological gaps.", "The vertical marking identifies d 1 =20d_1 = 20 mm and d 2 =16d_2 = 16 mm used in experiments.", "(b) Measurement of the spaced-resolved density of states (See Supplementary Material for Experimental Protocols).", "(c) Collapse of the data in panel (b) on the frequency axis.", "Two spectral gaps can be clearly identified in the experimental data, which are well aligned with the theoretical calculations.As we already mentioned, the resonator coupling through the spacer does not allow fine-tuning but that is not necessary when using aperiodic principles, as long as the coupling is strong [7].", "To understand the mechanism of topological gap generation in our system, we show first in Fig.", "REF (d) the evolution of the simulated resonant spectrum with respect to the relative alignment of two identical arrays of resonators.", "As expected in any 1-dimensional periodic system, gaps appear in the resonant spectrum and, as the system switches between period-one and period-two, some of these spectral gaps close while other remain open.", "Regardless of that behavior, all these gaps are topologically trivial because the resonant bands seen in Fig.", "REF (d) result from dispersion-induced thickening of the discrete resonances of the individual resonators.", "However, when the lattice constant of the bottom array is varied and the system becomes aperiodic, these trivial bands are seen in Fig.", "REF (e) to become fragmented in sub-bands, exactly as it happens when a magnetic field is turned on a two dimensional electronic system [31].", "In the Supplementary Material, we in fact show that the dynamical matrix behind the resonant spectrum belongs to an algebra of observables generated by two operators obeying the same commutation relations as the magnetic translations.", "The conclusion is that the spectrum seen in Fig.", "REF (e) is a representation of the Hofstadter butterfly [31].", "In particular, the sub-bands carry non-zero Chern numbers [7], [29] and the presence of the chiral edge bands can be explained by the standard bulk-boundary correspondence [32], [33].", "The simulated bulk spectrum is reproduced with high fidelity by the experimental measurements, as demonstrated in Fig.", "REF .", "In particular, well defined bulk-spectral gaps can be identified in the measured local density of states, which are well aligned with the theoretical predictions.", "The frequency 5.4 kHz used for topological pumping in Fig.", "REF (g) falls in the middle of one such gap.", "Furthermore, the signature of the non-zero Chern numbers, that is, the chiral edge bands, are also detected experimentally, as reported in Fig.", "REF .", "By comparing the panels (a) and (b), one can see that the experiment reproduces the simulations with very high fidelity.", "Figure: Experimental measurement of the topological edge bands.", "(a) Simulated resonant spectrum for the finite acoustic meta-crystal shown in Fig.", "(a), as function of the pumping parameter.", "The latter is the ratio between the displacement x x, shown in Fig.", "(c), and d 2 d_2.", "(b) Experimental measurement of the resonant spectrum as function of the pumping paramter.", "(c) Bulk-spectrum measurements reproduced from Fig.", "(c), used here to pin-point the position of the bulk bands shown by dashed lines.Having demonstrated an un-assisted energy transfer via a topological pumping process, we have laid down a set of specific principles which could facilitate the engineering of the effect in many other contexts.", "The most important one is that fine-tuning is not necessary which, together with the many different ways of engineering phason spaces [29], relaxes the design constraints, hence giving scientists better chances with finding optimal and practical meta-structures.", "While for meta-materials this process is now more or less straightforward, it will be extremely interesting if these aperiodic principles can be successfully applied to mesoscopic systems and achieve electron pumping in conventional insulators.", "All authors acknowledge support from the W. M. Keck Foundation.", "E. P. acknowledges additional support from the National Science Foundation through the grant DMR-1823800." ] ]

[ [ "On time-domain NRBC for Maxwell's equations and its application in\n accurate simulation of electromagnetic invisibility cloaks" ], [ "Abstract In this paper, we present analytic formulas of the temporal convolution kernel functions involved in the time-domain non-reflecting boundary condition (NRBC) for the electromagnetic scattering problems.", "Such exact formulas themselves lead to accurate and efficient algorithms for computing the NRBC for domain reduction of the time-domain Maxwell's system in $\\mathbb R^3$.", "A second purpose of this paper is to derive a new time-domain model for the electromagnetic invisibility cloak.", "Different from the existing models, it contains only one unknown field and the seemingly complicated convolutions can be computed as efficiently as the temporal convolutions in the NRBC.", "The governing equation in the cloaking layer is valid for general geometry, e.g., a spherical or polygonal layer.", "Here, we aim at simulating the spherical invisibility cloak.", "We take the advantage of radially stratified dispersive media and special geometry, and develop an efficient vector spherical harmonic (VSH)-spectral-element method for its accurate simulation.", "Compared with limited results on FDTD simulation, the proposed method is optimal in both accuracy and computational cost.", "Indeed, the saving in computational time is significant." ], [ "Introduction", "Numerical simulation of electromagnetic wave propagations in anisotropic and dispersive medium is of fundamental importance in many scientific applications and engineering designs.", "The model problem of interest is the time-dependent three-dimensional Maxwell's system: t D(r, t)-H(r,t)=J(r,t)     in R3,   t>0, t B(r,t)+E(r,t)=0           in R3,   t>0, with the constitutive relations $D=\\varepsilon _0\\varepsilon E,\\quad B=\\mu _0\\mu H,$ where $r=(x,y,z)\\in {\\mathbb {R}}^3$ , $E, H$ are respectively the electric and magnetic fields, $D, B$ are the corresponding electric displacement and magnetic induction fields, and ${J}$ is the electric current density.", "In (REF ), $\\varepsilon _0, \\mu _0$ are the electric permittivity and magnetic permeability in vacuum and $\\varepsilon , \\mu $ are the relative permittivity and permeability tensors of the material.", "Throughout the paper, we denote $c=1/{\\sqrt{\\varepsilon _0\\mu _0}}$ and $\\eta =\\sqrt{ \\mu _0/\\varepsilon _0}.", "$ Without loss of generality, we assume that the inhomogeneity or dispersity of the medium is confined in a bounded domain $\\Omega $ and ${J}$ is compactly supported.", "As illustrated in Figure REF , both $\\Omega $ and ${\\rm supp}(f)$ are contained in a ball $\\Omega _b$ of radius $b$ .", "The Maxwell's system () is supplemented with the initial conditions: $E(r,0) =E_0(r),\\quad H(r,0)=H_0(r) \\quad {\\rm in} \\;\\;\\; {\\mathbb {R}}^3,$ where $E_0$ and $H_0$ are also assumed to be compactly supported in the ball $\\Omega _b.$ As usual, we impose the far-field Silver-Müller radiation boundary condition on the scattering fields: ${E}^{\\rm sc}=E-E^{\\rm in}$ and ${H}^{\\rm sc}=H-H^{\\rm in}$ as follows $\\partial _t {E}^{\\rm sc}_T-\\eta \\,\\partial _t {H}^{\\rm sc}\\times \\hat{r}=o(|r|^{-1})\\quad {\\rm as}\\;\\; |r|\\rightarrow \\infty ,\\;\\; t>0, \\;\\;$ where $\\hat{r}=r/|r|,$ and $E_T^{\\rm sc}:= \\hat{r}\\times E^{\\rm sc}\\times \\hat{ r}$ is the tangential component of $E^{\\rm sc}.$ Here, $E^{\\rm in}, H^{\\rm in}$ are the incident fields.", "Despite its seemly simplicity, the system ()-(REF ) is notoriously difficult to solve numerically.", "Some of the major numerical issues are (i) unboundedness of the computational domain; (ii) the incompressibility implicitly implied by () (i.e., ${\\rm div}(D)={\\rm div}(B)=0$ ); and (iii) the coefficients $\\varepsilon $ and $\\mu $ might be singular or frequency-dependent (see (REF ) and (REF )).", "In this paper, we shall address all these three aspects.", "In regards to the first issue, the method of choice typically includes the perfectly matched layer (PML) technique [5] or the artificial boundary condition [8], [11], [12].", "In particular, the latter is known as the absorbing boundary condition (ABC), if it leads to a well-posed initial-boundary value problem (IBVP) and the reflection near the boundary is controllable.", "Ideally, if the solution of the reduced problem coincides with that of the original problem, then the underlying artificial boundary condition is called a transparent (or nonreflecting) boundary condition (TBC) (or NRBC).", "In this paper, we resort to the NRBC to reduce the problem ()-(REF ) to an IBVP inside a spherical bounded domain $\\Omega _b:=\\lbrace r:|r|<b\\rbrace $ : t D-H=J;    t B+E=0    in b,   t>0, E=E0,   H=H0   in b,   t=0, tET-tHr-Tb[E]=tETin-tHinr-Tb[Ein]:=h    at r=b, where the NRBC () involves the capacity operator ${T}_b$ to be specified in Theorem REF .", "It is important to point out that the NRBC is formulated upon the scattering fields ${E}^{\\rm sc}, {H}^{\\rm sc}$ , so $h$ inevitably contains ${T}_b[{E}^{\\rm in}].$ As such, it is rather complicated to implement and computationally time-consuming due to the involvement of the vector spherical harmonics (VSH) expressions of $E^{\\rm in}$ and history dependence in time induced by the temporal convolution (see (REF ) and Remark REF ).", "To avoid the serious problem of high costs of computing $h$ , we instead solve the total fields in the subdomain $\\Omega _{b_0}:=\\lbrace r: r<b_0\\rbrace $ $\\subseteq $ $\\Omega _b$ (see Figure REF ), and compute the outgoing scattering fields in a narrow spherical shell $\\Omega _b \\setminus \\Omega _{b_0}.$ More precisely, we reformulate () as t D-H=J,   t B+E=0,    r<b0,   t>0, t Dsc-Hsc=J,   t Bsc+Esc=0,    b0<r<b,   t>0, (E-Esc)r=Einr;   (H-Hsc)r=Hinr,   at r=b0,    t>0, tETsc-tHscr-Tb[Esc]=0,   at r=b,    t>0, E=E0,H=H0,   0<r<b0, Esc=E0sc,H=H0sc,    b0<r<b, t=0.", "As a consequence, the NRBC () depends solely on the scattering fields, which leads to more efficient algorithm.", "Note that () is obtained from the classic transmission conditions (see, e.g., [31] and [26]), that is, the continuity of the tangential components of the total fields $E$ and $H$ at the artificial interface $r=b_0$ .", "Figure: An illustration of the geometryOne of the main purposes of this paper is devoted to deriving new formulas of the NRBC by using the compact VSH expansion of the scattering field.", "For the convolution kernel in the NRBC, an explicit expression in time domain is obtained based on a direct inversion of the Laplace transform (see Theorem REF below).", "As shown in [3], [40], the explicit expressions of NRBKs allow for a rapid and accurate evaluation of the convolution in NRBC.", "The second main purpose of this paper is to propose an accurate and efficient numerical method for the simulation of the electromagnetic invisibility cloaks by using the new NRBC formula and the compact VSH expansion.", "Transformation optics originated from the seminal works [33], [18] offers an effective approach to design novel and unusual optical devices such as the invisibility cloaks (see, e.g., [33], [9]), superlens (see, e.g., [42], [39]) and beam splitters (see, e.g., [34]), etc.", "Numerical simulation plays a crucial role in modelling of the electromagnetic wave interaction with these devices since it serves as a reliable tool to the justification of expensive physical experiments and validation of theoretical predictions.", "Over the recent years, intensive simulations and analysis have been devoted to the frequency domain (see, e.g., [7], [35], [46], [47], [25], [17], [44], [45]).", "Due to the fact that metamaterials used for manufacturing such kind of devices are unavoidably dispersive (cf.", "[32]), i.e., $\\varepsilon $ and $\\mu $ are frequency-dependent, time-domain mathematical models and simulations of anisotropic and dispersive electromagnetic devices are of fundamental importance.", "However, only limited works are available for the time-domain simulations including the FDTD [48], [49], [29] and the FETD [19], [21], [22], [43], [24].", "Because of the computational complexity, so far the numerical simulation of spherical cloaking structures has only been examined by [49] with a parallel implementation of FDTD method.", "In this paper, we propose a new formulation of the spherical cloak model in the time domain using the Drude dispersion model (cf.", "[31]).", "This new formulation allows us to use the symmetry of the problem together with the compact VSH expansions to provide an efficient VSH-spectral-element method for the simulation.", "Compared with the classic FDTD based algorithm, the VSH-spectral-element method can produce accurate numerical results in much less computational cost.", "The rest of the paper is organised as follows.", "In section , we present some new formulas of the NRBC and derive an explicit expression for the underlying convolution kernel.", "In section , we first derive a new time-domain model for the spherical dispersive cloaks by using Drude model.", "Then, an VSH-spectral-element method with Newmark's time integration scheme is proposed for efficient simulation of spherical cloaks.", "Ample interesting simulations for the spherical dispersive cloaks are presented in section to show the accuracy and efficiency of the proposed numerical scheme." ], [ "Computation of time-domain NRBC ", "In this section, we present the formulations of the capacity operator ${T}_b$ involved in the time-domain NRBC, and then derive some analytically perspicuous formulas for the associated temporal convolution kernels (dubbed as NRBKs), which are crucial for efficient and accurate computation of the NRBC, and in return for its seamless integration with the interior solvers." ], [ "Formulation of time-domain NRBC", "Let $L^2(\\Omega )$ be the usual space of square integrable functions on $\\Omega ,$ and denote $ L^2(\\Omega )=(L^2(\\Omega ))^3.$ We introduce the spaces $\\begin{split}{\\mathbb {H}}({\\rm div};\\Omega )=\\big \\lbrace v \\in {L}^2(\\Omega ) : {\\rm div}v\\in L^2(\\Omega ) \\big \\rbrace ; \\;\\;{\\mathbb {H}}({\\bf curl};\\Omega )=\\big \\lbrace v\\in (L^2(\\Omega ))^3: \\nabla \\times {v}\\in {L}^2(\\Omega ) \\big \\rbrace ,\\end{split}$ which are equipped with the graph norms as defined in [26].", "We further define ${\\mathbb {H}}_0({\\rm div};\\Omega )=\\big \\lbrace v\\in {\\mathbb {H}}({\\rm div};\\Omega ) : {\\rm div} v=0 \\big \\rbrace .\\;$ For $0\\ne x \\in \\mathbb {R}^3,$ let ${e}_r=x/|x|.$ Recall that the VSH $\\big \\lbrace Y_l^m , \\Psi _{l}^m,\\Phi _l^m\\big \\rbrace :=\\big \\lbrace Y_l^m {e}_r, \\nabla _SY_{l}^m,\\nabla _S Y_l^m {e}_r\\big \\rbrace $ used in the Spherepack [37] forms a complete orthogonal basis of ${L}^2(S):=(L^2(S))^3,$ where $\\lbrace Y_l^m\\rbrace $ are the spherical harmonic basis defined on the unit sphere $S$ as in [28].", "Nevertheless, the following compact form of the VSH expansion of a solenoidal or divergence-free field in (REF ) can simplify the derivation of NRBC.", "Moreover, it will lead to more efficient spectral-element algorithm for the 3D spherical cloaking simulation in Section .", "Proposition 2.1 For $u\\in {\\mathbb {H}}_0({\\rm div};\\Omega )$ , we can write ${u}= u_{00}\\,Y_0^0 + \\sum _{l=1}^\\infty \\sum _{|m|=0}^l \\Big \\lbrace u_{lm}\\, {\\Phi }_l^m+ \\nabla \\times \\big (\\tilde{u}_{lm}\\, {\\Phi }_l^m\\big )\\Big \\rbrace ,$ where $u_{00}$ satisfies $\\Big (\\frac{d}{dr}+\\frac{2}{r} \\Big )u_{00} = 0\\;\\;\\; {\\rm or}\\;\\;\\; u_{00}=\\frac{C}{r^2},$ for a constant $C$ depends on the average value of the $e_r$ component of $u$ on $S$ .", "The expansion (REF ) can be reformulated in terms of the VSH (REF ) as follows ${u}= u_{00}\\,Y_0^0 + \\sum _{l=1}^\\infty \\sum _{|m|=0}^l \\Big \\lbrace \\frac{\\beta _l}{r} \\tilde{u}_{lm} \\, Y_l^m+\\hat{\\partial }_r \\tilde{u}_{lm} \\,\\Psi _l^m +u_{lm}\\, {\\Phi }_l^m\\Big \\rbrace ,$ where $\\beta _l:=l(l+1)$ and $\\begin{split}& u_{lm}(r)= {\\beta _l^{-1}}\\big \\langle u, {\\Phi }_l^m\\big \\rangle _S,\\quad r^{-1} \\tilde{u}_{lm}(r)= \\beta _l^{-1}\\big \\langle u, Y_l^m \\big \\rangle _S.\\end{split}$ We first show that if (REF ) holds, then the expansion (REF ) automatically satisfies ${\\rm div}\\, u =0.$ Note that ${\\rm div}(u_{lm}\\, {\\Phi }_l^m)=0$ (cf.", "(REF )).", "Performing the divergence operator on (REF ), and using (REF ), we have ${\\rm div}\\, u=0,$ if $u_{00}$ satisfies the equation in (REF ) which has explicit solution: $u_{00}=C/r^2$ .", "Thanks to (REF ), the expansion (REF ) follows immediately from (REF ).", "Then (REF ) is a direct consequence of the orthogonality of VSH.", "Remark 2.1 For any constant $C,$ the field $ C\\, {e}_r/r^2, r>0$ (note: $Y_0^0= 1/{(2\\sqrt{\\pi })}$ ) is solenoidal.", "Given a vector field, the VSH expansion coefficients in (REF ) can be evaluated accurately and efficiently by using discrete VSH-transforms in SpherePack [37].", "$\\Box $ The time-domain NRBC to be formulated below involves the modified spherical Bessel function (cf.", "[41]) defined by $k_{l}(z)=\\sqrt{\\frac{\\pi }{2 z}} K_{l+1/2}(z),$ with $K_{l+1/2}(\\cdot )$ being the modified Bessel function of the second kind of order $l+1/2$ , together with the temporal convolution and inverse Laplace transform: $ (f\\ast g)(t)=\\int _0^t f(\\tau )g(t- \\tau )d\\tau ,\\quad {L}^{-1}[F](t)= \\frac{1}{2\\pi {\\rm i}}\\int _{\\gamma -\\infty {\\rm i}}^{\\gamma +\\infty {\\rm i}}F(s) e^{s t} ds,$ where $F(s)$ is the Laplace transform of $f(t),$ and the integration is done along the vertical line $\\Re (s) = \\gamma $ in the complex plane such that $\\gamma $ is greater than the real part of all singularities of $F(s)$ .", "The formulation of the capacity operator can be found in e.g., [6], [28], [26], but the notation and normalisation are very different.", "Here, we feel compelled to sketch its derivation.", "Theorem 2.1 The time-domain capacity operator $T_b$ takes the form ${T}_b[E^{\\rm sc}]:=\\frac{c}{b} \\sum _{l=1}^\\infty \\sum _{m=-l}^l\\Big \\lbrace \\big (\\rho _l\\ast {\\psi }^{(1)}_{lm}\\big )\\,{\\Psi }_l^m+\\big (\\sigma _l\\ast {\\psi }^{(2)}_{lm}\\big )\\,{\\Phi }_l^m\\Big \\rbrace ,$ where the convolution kernels are given by the inverse Laplace transforms: $\\begin{split}\\rho _l(t)&={L}^{-1}\\bigg [z\\,\\bigg (\\frac{z\\, k_l(z)}{k_l(z)+z\\, k_l^{\\prime }(z)}+1\\bigg ) \\bigg ](t), \\quad \\sigma _l(t) ={L}^{-1}\\bigg [1+z+z\\frac{k_l^{\\prime }(z)}{k_l(z)}\\bigg ](t),\\end{split}$ with $z=\\frac{sb}{c}$ , and where $\\lbrace {\\psi }^{r}_{lm}, {\\psi }^{(1)}_{lm}, {\\psi }^{(2)}_{lm}\\rbrace $ are the VSH expansion coefficients of $E^{\\rm sc}$ at the spherical surface $r=b$ , i.e., $E^{\\rm sc}=\\psi _{00}\\,Y_0^0 + \\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\Big \\lbrace {\\psi }^{r}_{lm}\\, Y_l^m\\, +{\\psi }_{lm}^{(1)}\\Psi _{l}^m+{\\psi }_{lm}^{(2)}\\, {\\Phi }_l^m\\Big \\rbrace .$ Consider the Maxwell's equations exterior to the artificial ball $B:=\\lbrace r: |r|<b\\rbrace :$ 0 t Ee-He=0,   0 t He+Ee=0, in R3B, t>0, Eer =,    r=b, t>0 t ETe-   t Her =o(r-1),   |r|, t>0, Ee =He=0,    in R3B, t=0, where $\\lambda $ is a given field.", "It is known that this system can be solved analytically by using Laplace transform in time and separation of variables in space.", "For this purpose, we denote by $\\breve{E}^{\\rm e}, \\breve{H}^{\\rm e}$ and $\\breve{\\lambda }$ the Laplace transforms of $E^{\\rm e}, H^{\\rm e}$ and ${\\lambda }$ with respect to $t$ , respectively.", "As $\\breve{E}^{\\rm e}$ is a divergence-free vector field, we have from Proposition REF that $\\breve{E}^{\\rm e}=\\breve{u}_{00}Y_0^0+\\sum _{l=1}^\\infty \\sum _{|m|=1}^l\\big \\lbrace \\breve{u}_{lm} \\Phi _l^m+\\nabla \\times \\big (\\breve{v}_{lm} \\Phi _l^m\\big )\\big \\rbrace .$ According to [28], the Laplace transformed system of (REF ) in $s$ -domain $\\varepsilon _0 s \\breve{E}^{\\rm e} -\\nabla \\times \\breve{H}^{\\rm e}=0,\\quad \\mu _0 s \\breve{H}^{\\rm e} +\\nabla \\times \\breve{E}^{\\rm e}=0;\\quad \\breve{E}^{\\rm e}\\hat{r} =\\breve{\\lambda } \\;\\; {\\rm at}\\;\\; r=b,$ has the exact solution (REF ) with $\\breve{u}_{00}=0$ and $\\breve{u}_{lm}(r)=-\\frac{k_l(sr/c)}{k_l(sb/c)} \\breve{\\lambda }_{lm}^{(1)},\\quad \\breve{v}_{lm}=\\frac{k_l(sr/c)}{\\hat{\\partial }_rk_l(sb/c)}\\breve{\\lambda }_{lm}^{(2)},\\quad l\\ge 1,$ where $\\hat{\\partial }_r=\\frac{d}{dr}+\\frac{1}{r}$ , $\\lbrace \\breve{\\lambda }_{lm}^{(1)}, \\breve{\\lambda }_{lm}^{(2)}\\rbrace $ are the VSH expansion coefficients of $\\breve{\\lambda }$ (involving only the tangential components).", "Moreover, we can derive the electric-to-magnetic Calderon (EtMC) operator that maps the data $\\breve{\\lambda }$ to $\\breve{H}^{\\rm e}\\times \\hat{r}$ (cf.", "[6], [26]) as follows $\\breve{H}^{\\rm e} \\hat{r}=-\\frac{1}{s\\mu _0}\\sum _{l=1}^\\infty \\sum _{m=-l}^l\\bigg \\lbrace \\frac{ s^2k_l(sr/c)}{c^2\\hat{\\partial }_r k_l(sb/c)}\\breve{\\lambda }_{lm}^{(2)}\\Psi _l^m-\\frac{\\hat{\\partial }_rk_l(sr/c)}{k_l(sb/c)}\\breve{\\lambda }_{lm}^{(1)}\\Phi _l^m\\bigg \\rbrace ,$ which can be derived from the second equation in (REF ) and the properties of VSH in Appendix .", "By requiring the scattering fields $\\lbrace \\breve{E}^{\\rm sc}, \\breve{H}^{\\rm sc}\\rbrace $ to be identical to the exterior fields $\\lbrace \\breve{E}^{\\rm e}, \\breve{H}^{\\rm e}\\rbrace $ across the artificial boundary $r=b,$ and setting $\\breve{E}^{sc}\\times \\hat{r}|_{r=b}=\\breve{ \\lambda },$ we obtain $\\begin{split}\\mu _0 s\\breve{H} ^{\\rm sc}\\hat{r}\\big |_{r=b}&=-\\sum _{l=1}^\\infty \\sum _{m=-l}^l\\Big \\lbrace \\frac{s^2 k_l(sb/c)}{c^2\\hat{\\partial }_rk_l(sb/c)}\\breve{\\psi }^{(1)}_{lm}\\Psi _l^m+\\frac{\\hat{\\partial }_rk_l(sb/c)}{k_l(sb/c)}\\breve{\\psi }^{(2)}_{lm}\\Phi _l^m\\Big \\rbrace \\\\&=-\\sum _{l=1}^\\infty \\sum _{m=-l}^l\\Big \\lbrace \\Big (\\frac{1}{b}{L}[\\rho _l]-\\frac{s}{c}\\Big )\\breve{\\psi }^{(1)}_{lm}\\Psi _l^m+\\Big (\\frac{1}{b}{L}[\\sigma _l]-\\frac{s}{c}\\Big )\\breve{\\psi }^{(2)}_{lm}\\Phi _l^m\\Big \\rbrace \\\\&=-\\frac{1}{b}\\sum _{l=1}^\\infty \\sum _{m=-l}^l\\Big \\lbrace {L}[\\rho _l]\\breve{\\psi }^{(1)}_{lm}\\Psi _l^m+{L}[\\sigma _l]\\breve{\\psi }^{(2)}_{lm}\\Phi _l^m\\Big \\rbrace +\\frac{s}{c}\\breve{E}_{T}^{\\rm sc},\\end{split}$ where $\\sigma _l(t)$ and $\\rho _l(t)$ are defined in (REF ).", "Transforming the relation (REF ) back to $t$ -domain leads to the time-domain NRBC $\\partial _t{E}^{\\rm sc}_T-\\eta \\partial _tH^{\\rm sc}\\times \\hat{r}-{T}_b[{E}^{\\rm sc}]=0 \\quad {\\rm at}\\;\\;\\; r=b,$ and the capacity operator ${T}_b[E^{\\rm sc}]$ given by (REF ).", "Remark 2.2 Note that the exact NRBC in [13] expressed as a system of $E^{\\rm sc}$ and $H^{\\rm sc}$ , is actually equivalent to the formulation () by using the VSH $\\big \\lbrace Y_l^m , \\Psi _{l}^m, \\Phi _l^m\\big \\rbrace $ .", "From the NRBC () on scattering fields, the NRBC () on total field $\\lbrace E, H\\rbrace $ can be derived straightforwardly as follows $\\partial _t{E}_T-\\eta \\partial _tH\\times \\hat{r}-{T}_b[{E}]=\\partial _t{E}_T^{\\rm in}-\\eta \\partial _tH^{\\rm in}\\times \\hat{r}-{T}_b[{E}^{\\rm in}]:=h, \\quad {\\rm at}\\;\\;\\; r=b.$ We shall see from (REF ) and (REF ) that the source term $h$ needs to be precomputed from $\\lbrace E^{\\rm in}, H^{\\rm in} \\rbrace $ if the model () is used for solving the total field $\\lbrace E, H\\rbrace $ in the whole computational domain $B$ .", "However, the involved term ${T}_b[{E}^{\\rm in}]$ is computationally costly due to: (i) The VSH expansion coefficients of the incident wave are necessary for the computation; (ii) in the time direction, two convolutions need to be calculated numerically for each group of VSH expansion coefficients; (iii) the numerical scheme for the convolution needs to be more accurate than the time discretization scheme for the model problem due to sensitivity of the operator ${T}_b$ on the error.", "Thus, a great number of time consuming VSH expansion need to be performed.", "$\\Box $ It is seen from (REF ) that to compute ${T}_b[E^{\\rm sc}]$ at $t>0,$ it requires (i) accurate evaluation of the convolution kernel functions: $\\rho _l(t)$ and $\\sigma _l(t)$ defined in (REF ); (ii) fast computation of the temporal convolutions: $\\rho _l\\ast {\\psi }^{(1)}_{lm}$ and $\\sigma _l\\ast {\\psi }^{(2)}_{lm}.$ With these, we can compute ${T}_b[E^{\\rm sc}]$ on the sphere $r=b$ by using the VSH transform in the Spherepack [37].", "We now deal with the first issue.", "Interestingly, the kernel function $\\sigma _l(t)$ coincides with the NRBC kernel of the transient wave equation, which admits the following explicit formula (cf.", "[36], [13], [40]).", "Proposition 2.2 Let $\\sigma _{l}(t)$ be the kernel function defined in (REF ).", "Then we have $\\sigma _{l}(t)=\\frac{c}{b} \\sum _{j=1}^{l}z_j^l e^{\\frac{c}{b} z_j^lt},\\quad l\\ge 1,\\;\\; t\\ge 0,$ where $\\lbrace z_j^l\\rbrace _{j=1}^{l}$ are the zeros of $K_{l+1/2}(z)$ with $l\\ge 1$ .", "We remark that according to [41], $K_{l+1/2}(z)$ has exactly $l$ zeros in conjugate pairs which are simple and lie in the second and third quadrants.", "The interested readers might refer to [40] for more details.", "Remarkably, we can derive a very similar analytical formula for the kernel function $\\rho _l(t).$ Our starting point is to rewrite the ratio of the modified Bessel functions in (REF ) by using (REF ) as follows $\\begin{split}&z\\bigg (\\frac{zk_l(z)}{k_l(z)+zk^{\\prime }_l(z)}+1\\bigg )=z\\bigg (\\frac{zK_{l+1/2}(z)}{\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)}+1\\bigg ).\\end{split}$ We are interested in the poles of the ratio, i.e., zeros of $\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)$ .", "It is indeed very fortunate to find the following results in [38] (for more general combination of this form).", "Figure: Distributions of the zeros ofK l+1/2 (z)K_{l+1/2}(z) (left) and 1 2K l+1/2 (z)+zK l+1/2 ' (z)\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z) (right).Lemma 2.1 Let $l$ be a nonnegative integer.", "Then we have the following properties.", "(a) $\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)$ has exactly $l+1$ zeros.", "(b) If $z_*$ is a zero of $\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)$ , then its complex conjugate $\\bar{z}_*$ is also a zero.", "(c) All zeros of $\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)$ are simple and have negative real parts, so they lie in the left half of the complex plane.", "Figure: Contour LL for the inverse Laplace transform.We illustrate in Figure REF the distribution of zeros of $K_{l+1/2}(z)$ (left) and $\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)$ (right) for various $l.$ We observe that for a given $l$ , the zeros of $\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)$ have a distribution very similar to those of $K_{l+1/2}(z)$ , that is, sitting on the left half boundary of an eye-shaped domain that intersects the imaginary axis approximately at $\\pm l{\\rm i},$ and the negative real axis at $-la$ with $a\\approx 0.66274$ (see the vertical dashed coordinate grids).", "Such a behaviour is very similar to that of $K_{l+1/2}(z)$ (cf.", "[40]).", "With the above understanding, we now ready to present the analytical formula for the convolution kernel function $\\rho _l(t)$ .", "Theorem 2.2 Let $\\lbrace \\tilde{z}_j^l\\rbrace _{j=1}^{l+1}$ be the zeros of $\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)$ with integer $l\\ge 0$ .", "Then we can compute $\\rho _l(t)$ in (REF ) via $\\begin{split}\\rho _{l}(t)&=\\frac{c}{b} \\sum _{j=1}^{l+1}\\frac{(\\tilde{z}_j^l)^3}{l(l+1)+(\\tilde{z}_j^l)^2}\\, e^{ct \\tilde{z}_j^l /b}+\\delta (t)\\sum _{j=1}^{l+1}\\frac{(\\tilde{z}_j^l)^2}{l(l+1)+(\\tilde{z}_j^l)^2},\\end{split}$ where $\\delta (t)$ is the Dirac delta function.", "Using the property $L^{-1}[sf(s)](t)=f^{\\prime }(t)+f(0)\\delta (t),$ we obtain from (REF ) and (REF ) that $\\rho _l(t)=L^{-1}\\bigg [z\\,\\bigg (\\frac{z\\, k_l(z)}{k_l(z)+z\\, k_l^{\\prime }(z)}+1\\bigg )\\bigg ](t)=\\frac{b}{c}\\big (\\tilde{\\rho }^{\\prime }_{l}(t)+\\tilde{\\rho }_{l}(0)\\delta (t)\\big ),$ where $z={sb}/{c},$ and with a change of variable $s=cz/b,$ we have $\\begin{split}\\tilde{\\rho }_{l}(t)&=\\frac{1}{2\\pi {\\rm i}}\\frac{c}{b}\\int _{\\gamma -\\infty {\\rm i}}^{\\gamma +\\infty {\\rm i}}\\bigg [\\frac{zk_l(z)}{k_l(z)+zk^{\\prime }_l(z)}+1\\bigg ] e^{czt/b} dz:=\\frac{1}{2\\pi {\\rm i}}\\frac{c}{b}\\int _{\\gamma -\\infty {\\rm i}}^{\\gamma +\\infty {\\rm i}}F_{l}(z) e^{czt/b} dz.\\end{split}$ In view of the formula (see [30]): $k_{l}(z)=\\frac{\\pi }{2}\\sum \\limits _{k=0}^l\\frac{(l+k)!e^{-z}}{2^kk!", "(l-k)!z^{k+1}},\\quad l\\ge 0,$ and Lemma REF , we conclude that $F_l(z)$ is a meromorphic function.", "We introduce the closed contour $L$ as depicted in Figure REF .", "Using the residue theorem and Jordan's Lemma, we have $2\\pi {\\rm i}\\sum \\limits _{j=1}^{l+1}{\\rm Res}\\,\\big [F_{l}(z)e^{czt/b}; \\tilde{z}_{j}^{l}\\big ]=\\lim _{R\\rightarrow +\\infty }\\oint _L F_{l}(z)e^{czt/b}dz=\\int _{\\gamma -\\infty {\\rm i}}^{\\gamma +\\infty {\\rm i}}F_{l}(z) e^{czt/b} dz.$ From (REF ), we calculate that $\\begin{split}\\frac{b}{c}\\tilde{\\rho }_l(t)&= \\sum \\limits _{j=1}^{l+1}{\\rm Res}\\,\\big [F_{l}(z)e^{czt/b}; \\tilde{z}_{j}^{l}\\big ]\\\\&=\\sum \\limits _{j=1}^{l+1}\\lim _{z\\rightarrow \\tilde{z}_{j}^{l}}\\bigg \\lbrace \\big (z-\\tilde{z}_{j}^{l}\\big ) e^{ctz/b}\\bigg [\\frac{zK_{l+1/2}(z)}{\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)}+1\\bigg ]\\bigg \\rbrace \\\\&=\\sum \\limits _{j=1}^{l+1}\\frac{e^{ct\\tilde{z}_{j}^{l}/b}\\tilde{z}_{j}^{l}K_{l+1/2}(\\tilde{z}_{j}^{l})}{\\frac{3}{2}K^{\\prime }_{l+1/2}(\\tilde{z}_{j}^{l})+\\tilde{z}_{j}^{l}K^{\\prime \\prime }_{l+1/2}(\\tilde{z}_{j}^{l})}.\\end{split}$ Since $K_{l+1/2}(z)$ satisfies the equation (cf.", "[41]) $z^2\\frac{d^2w}{dz^2}+z\\frac{dw}{dz}-\\Big (z^2+(l+1/2)^2\\Big )w=0,$ we have $\\begin{split}z^2K^{\\prime \\prime }_{l+1/2}(z)+\\frac{3}{2}zK^{\\prime }_{l+1/2}(z)&=\\big (z^2+(l+1/2)^2\\big )K_{l+1/2}(z)-zK^{\\prime }_{l+1/2}(z)+\\frac{3}{2}zK^{\\prime }_{l+1/2}(z)\\\\&=\\big (z^2+l(l+1)\\big )K_{l+1/2}(z)+\\frac{1}{2}\\bigg (\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)\\bigg ).\\end{split}$ A combination of (REF ) and the fact that $\\lbrace \\tilde{z}_j^l\\rbrace _{j=1}^{l+1}$ are zeros of $\\frac{1}{2}K_{l+1/2}(z)+zK^{\\prime }_{l+1/2}(z)$ yields $\\begin{split}\\tilde{\\rho }_{l}(t)&=\\frac{c}{b}\\sum \\limits _{j=1}^{l+1}\\frac{e^{ct\\tilde{z}_{j}^{l}/b}\\big (\\tilde{z}_{j}^{l}\\big )^2K_{l+1/2}(\\tilde{z}_{j}^{l})}{\\big [\\big (\\tilde{z}_{j}^{l}\\big )^2+l(l+1)\\big ]K_{l+1/2}(\\tilde{z}_{j}^{l})}=\\frac{c}{b}\\sum \\limits _{j=1}^{l+1}\\frac{e^{ct\\tilde{z}_{j}^{l}/b}\\big (\\tilde{z}_{j}^{l}\\big )^2}{\\big (\\tilde{z}_{j}^{l}\\big )^2+l(l+1)}.\\end{split}$ Inserting (REF ) into (REF ) leads to the expression of $\\rho _l(t)$ in (REF ).", "Having addressed the first issue on how to compute the convolution kernel functions, we now introduce an efficient technique to alleviate the historical burden of temporal convolutions involved in the capacity operator (REF ).", "Observe from (REF ) and (REF ) that the time variable $t$ only presents in the complex exponentials.", "As a result, the temporal convolutions can be evaluated recursively as shown in e.g., [3], [40].", "More precisely, given a continuous function $g(t)$ , we define $f(t;z):=e^{ctz/b}\\ast g(t)=\\int _0^t e^{c(t-\\tau )z/b}g(\\tau )d\\tau .$ Then by (REF ) and (REF ), $(\\sigma _l\\ast g)(t)& =\\frac{c}{b} \\sum _{j=1}^{l}z_j^l f(t;z_j^l);\\\\(\\rho _l\\ast g)(t)&=\\frac{c}{b} \\sum _{j=1}^{l+1}\\frac{\\big (\\tilde{z}_j^l\\big )^3 f(t;\\tilde{z}_j^l)}{\\big (\\tilde{z}_j^l\\big )^2+l(l+1)}+g(t)\\sum _{j=1}^{l+1}\\frac{\\big (\\tilde{z}_j^l\\big )^2}{\\big (\\tilde{z}_j^l\\big )^2+l(l+1)}.$ One verifies readily that $f(t+\\Delta t;z)=e^{c\\Delta t\\,z/b}f(t; z)+\\int _t^{t+\\Delta t}e^{c(t+\\Delta t-\\tau )z/b}g(\\tau )\\,d\\tau ,$ so $f(t; z)$ can march in $t$ with step size $\\Delta t$ recursively.", "As a result, the temporal convolution in the NRBC can be computed efficiently with the explicit expressions (REF ) and (REF ), and with the above fast recursive algorithm.", "Next, we provide some numerical results to demonstrate the high accuracy in computing $\\rho _l(t)$ and the related convolution in (REF ).", "Let $\\phi (t), t\\ge 0$ be a given differentiable function such that $\\phi (0)=0.$ As shown in [40], $(\\sigma _l\\ast \\phi )(t)$ can be computed very accurately (i.e., using Proposition REF and (REF )).", "Let $\\psi _l(t)$ be a function associated with $\\phi (t)$ through ${L}[\\psi _l](s)=\\frac{k_l(z)+zk^{\\prime }_l(z)}{k_l(z)}{L}[\\phi ](s)=\\left(\\!1+z+z\\frac{k^{\\prime }_l(z)}{k_l(z)}\\right)\\!", "{L}[\\phi ](s)-z{L}[\\phi ](s),$ where $z=sb/c.$ Applying the inverse Laplace transform and using the definition of $\\sigma _l(t)$ in (REF ), we obtain from (REF ), $\\phi (0)=0$ and Proposition REF that $\\psi _l(t)=(\\sigma _l\\ast \\phi )(t)-\\frac{b}{c}\\phi ^{\\prime }(t)=\\frac{c}{b}\\sum _{j=1}^l (z_j^l)^2 e^{ct z_j^l /b}\\ast \\phi (t)+\\phi (t)\\sum _{j=1}^l z_j^l.$ We next present two ways to compute $(\\rho _l\\ast \\psi _l)(t),$ where the first one only requires the use of the formula for $\\sigma _l(t).$ Indeed, we have from (REF ), (REF ) and (REF ) that $\\begin{split}(\\rho _{l}\\ast \\psi _l)(t)&=L^{-1}\\bigg [z\\Big (\\frac{z\\, k_l(z)}{k_l(z)+z\\, k_l^{\\prime }(z)}+1\\Big ){L}[\\psi _l]\\bigg ]=L^{-1}\\big [z^2{L}[\\phi ] +z{L}[\\psi _l]\\big ]\\\\&=L^{-1}\\bigg [\\Big (1+z+z\\frac{k^{\\prime }_l(z)}{k_l(z)}\\Big )z{L}[\\phi ]\\bigg ]=\\sigma _l\\ast L^{-1}\\big [z{L}[\\phi ]\\big ]= (\\sigma _l\\ast \\phi ^{\\prime })(t).\\end{split}$ Then, by Proposition REF and integration by parts, we find $\\begin{split}(\\rho _{l}\\ast \\psi _l)(t)&= \\sum _{j=1}^l z_j^l e^{ct z_j^l /b}\\ast \\phi ^{\\prime }(t)=\\frac{c}{b}\\sum _{j=1}^l (z_j^l)^2 e^{ct z_j^l /b}\\ast \\phi (t)+\\phi (t)\\sum _{j=1}^l z_j^l.\\end{split}$ On the other hand, using Theorem REF and the relation (REF ) leads to $\\begin{split}(\\rho _{l}\\ast \\psi _l)(t)&=\\frac{c}{b} \\sum _{j=1}^{l+1}\\frac{\\big (\\tilde{z}_j^l\\big )^3}{\\big (\\tilde{z}_j^l\\big )^2+l(l+1)} e^{ct \\tilde{z}_j^l /b}\\ast \\psi _l(t)+\\psi _l(t)\\sum _{j=1}^{l+1}\\frac{\\big (\\tilde{z}_j^l\\big )^2}{\\big (\\tilde{z}_j^l\\big )^2+l(l+1)},\\end{split}$ where $\\psi _l$ is computed by (REF ).", "It is evident that the convolutions in (REF )-(REF ) can be evaluated by using (REF ).", "It is seen that (REF ) and (REF ) are equivalent, but the former solely involves $\\sigma _l(t),$ which can used as a reference to check the accuracy of $\\rho _l(t).$ We take $b=3, c=5$ and $\\phi (t)=\\sin ^6(8t)$ , so we can compute the two functions and convolutions with exponential functions in (REF )-(REF ) exactly.", "We tabulate in Table REF the relative errors $e_l(t)=\\frac{|f_l(t)-\\tilde{f}_l(t)|}{|\\tilde{f}_l(t)|},$ where $f_l(t)$ and $\\tilde{f}_l(t)$ denote the convolution computed by (REF ) and (REF ), respectively.", "We can see that the relative errors are of machine accuracy, which validate the formula (REF ).", "Table: The relative error e l (t)e_l(t) for different ll and tt.Remark 2.3 It is clear that the number of zeros to be used is determined by the truncation of the expansion (REF ).", "If $l$ is large, the pole compression algorithm (cf.", "[2], [16]) can be adopted to obtain approximations for the NRBKs: $\\sigma _l(t)$ and $\\rho _l(t)$ .", "The approximated kernels have the same form as in (REF ) and (REF ) while the number of poles in the summation has been reduced significantly." ], [ "An alternative formulation of the capacity operator", "It is seen from (REF ) that the capacity operator ${T}_b[E^{\\rm sc}]$ only involves the tangential component of the vector field $E^{\\rm sc}\\in {\\mathbb {H}}_0({\\rm div};\\Omega ).$ In fact, as shown in Proposition REF , the VSH expansion coefficients for a divergence-free field satisfy some relation that allows us to derive the following alternative representation of the capacity operator in Theorem REF .", "We find that it has certain advantage in the application in the forthcoming section.", "Theorem 2.3 The time-domain capacity operator ${T}_b$ in Theorem REF can also be formulated as ${T}_b[E^{\\rm sc}]=\\frac{c}{b} \\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\bigg \\lbrace \\frac{\\omega _l\\ast {\\psi }_{lm}^{r}}{l(l+1)}\\,{\\Psi }_l^m+\\big (\\sigma _l\\ast {\\psi }_{lm}^{(2)}\\big )\\,{\\Phi }_l^m\\bigg \\rbrace ,$ where $\\sigma _l$ is given in (REF ) and $\\begin{split}\\omega _l(t)=\\frac{b}{c}\\big (\\sigma _l^{\\prime }(t)+\\sigma _l(0)\\delta (t)\\big )=\\frac{c}{b} \\sum _{j=1}^{l}\\big (z_j^l\\big )^2 e^{\\frac{c}{b} z_j^lt}+\\delta (t)\\sum _{j=1}^{l}z_j^l.\\end{split}$ Here, the expression (REF ) involves two of the VSH expansion coefficients $\\lbrace {\\psi }^{r}_{lm}, {\\psi }^{(1)}_{lm}, {\\psi }^{(2)}_{lm}\\rbrace $ of $E^{\\rm sc}$ in (REF ).", "In view of Proposition REF , we can reformulate (REF ) as $\\breve{E}^{\\rm e}=\\breve{u}_{00}Y_0^0+\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\Big \\lbrace \\breve{u}_{lm} \\Phi _l^m+\\frac{\\beta _l}{r} \\breve{v}_{lm} \\, Y_l^m+\\hat{\\partial }_r \\breve{v}_{lm} \\,\\Psi _l^m \\Big \\rbrace ,$ which is a solution of the exterior problem (REF ).", "Let $\\big \\lbrace \\breve{\\psi }^r_{lm}, {\\breve{\\psi }}^{(1)}_{lm},{\\breve{\\psi }}^{(2)}_{lm}\\big \\rbrace $ be the Laplace transforms of $\\big \\lbrace {\\psi ^r_{lm}, \\psi }^{(1)}_{lm},{\\psi }^{(2)}_{lm}\\big \\rbrace $ , which is the VSH expansion coefficients of the scattering field $E^{\\rm sc}$ in (REF ).", "Note that $\\breve{E}^{\\rm e}=\\breve{E}^{\\rm sc}$ is the solution of the exterior problem (REF ) with boundary data $\\breve{\\lambda }=\\breve{E}^{\\rm sc}\\times \\hat{r}$ on the artificial boundary $r=b$ .", "Then by (REF ), (REF ) and (REF ), we arrive at the VSH expansion coefficients of Laplace transformed scattering field $\\breve{E}^{\\rm sc}:$ $\\breve{\\psi }^{r}_{lm}=\\frac{\\beta _l}{r}\\breve{v}_{lm}=\\frac{\\beta _l}{r} \\frac{ k_l(sr/c)}{\\hat{\\partial }_rk_l(sb/c)}\\breve{\\lambda }_{lm}^{(2)},\\quad \\breve{\\psi }^{(1)}_{lm}=\\hat{\\partial }_r\\breve{v}_{lm} =\\frac{\\hat{\\partial }_rk_l(sr/c)}{\\hat{\\partial }_rk_l(sb/c)}\\breve{\\lambda }_{lm}^{(2)},$ for $r\\ge b$ .", "This implies $\\breve{\\psi }^r_{lm}= \\frac{\\beta _l}{r} \\frac{ k_l(sr/c)}{\\hat{\\partial }_rk_l(sr/c)} \\breve{\\psi }^{(1)}_{lm}= \\frac{\\beta _l}{r} \\frac{ k_l(sr/c)}{ \\frac{s}{c} k_l^{\\prime }(sr/c)+\\frac{1}{r} k_l(sr/c)} \\breve{\\psi }^{(1)}_{lm},$ so at $r=b,$ we have $\\frac{\\breve{\\psi }^r_{lm}}{\\beta _l}=\\frac{ k_l(z)}{k_l(z)+z\\, k_l^{\\prime }(z)}\\breve{\\psi }_{lm}^{(1)},\\quad z=\\frac{sb}{c}.$ Multiplying both sides of (REF ) by $(1+z)k_l(z)+zk_l^{\\prime }(z)$ yields $z\\,\\bigg (\\frac{z\\, k_l(z)}{k_l(z)+z\\, k_l^{\\prime }(z)}+1\\bigg ) \\breve{\\psi }_{lm}^{(1)}=z\\,\\bigg (1+z+z\\frac{k_l^{\\prime }(z)}{k_l(z)}\\bigg )\\frac{\\breve{\\psi }^r_{lm}}{\\beta _l}.$ In view of the definitions of $\\rho _l$ and $\\sigma _l$ in (REF ) and using the property (REF ), we take the inverse Laplace transform on both sides of (REF ) and find $\\rho _l\\ast \\psi _{lm}^{(1)}=\\frac{\\omega _l\\ast {\\psi }_{lm}^{r}}{\\beta _l},$ where the kernel function $\\begin{split}\\omega _l(t)&={L}^{-1}\\bigg [z\\,\\bigg (1+z+z\\frac{k_l^{\\prime }(z)}{k_l(z)}\\bigg )\\bigg ](t)=\\frac{b}{c}\\big (\\sigma _l^{\\prime }(t)+\\sigma _l(0)\\delta (t)\\big ).", "\\end{split}$ Then, substituting (REF ) into (REF ) leads to the capacity operator ${T}_b[E^{sc}]$ in (REF ).", "Finally, the last formula in (REF ) can be obtained from (REF ) directly." ], [ "Simulation of three-dimensional dispersive invisibility cloak", "As already mentioned in the introductory section, the invisibility cloak is one of the most appealing examples in the field of transformation optics [33], [10].", "In this section, we focus on the time-domain modelling and efficient simulation of the electromagentic invisibility cloak first proposed in [33].", "Indeed, there exist very limited works in three-dimensional cloak simulations.", "Our contributions are twofold.", "(i) We shall derive a new mathematical formulation of the time-domain dispersive cloak.", "Different from the existing models based on some mixed forms of both $E$ and $H$ (see, e.g., [14], [49], [21], [23]), the new formulation only involves one unknown field $D$ (see Theorem REF ), where the seemingly complicated temporal convolutions in the form of (REF ) can be evaluated efficiently as shown in (REF ).", "Moreover, the proposed governing equation in the cloaking layer with special dispersive media is valid for other geometries (e.g., the polygonal layer) other than the spherical shell.", "(ii) To simulate the time-domain spherical cloaking designed in [33], we shall develop a very efficient VSH-spectral-element method for solving the reduced problem truncated by the NRBC (REF ) using the alternative formulation of the capacity operator in Theorem REF .", "The implementation of the new algorithm can run thousands of time steps in a few hours on a desktop with intel i7 CPU, while the parallel implementation of the classic FDTD method running on a cluster with 100 processors and 220 GB memory takes 45 hours for 13000 time steps (cf.", "[49])." ], [ "Dispersive modelling of 3D invisibility cloaks", "The key to the design of invisibility cloak is to fill the cloaking layer, denoted by $\\Omega _{\\rm cl}$ (see, e.g., $\\Omega _1$ in Figure REF ), with specially designed metamaterials, which can steer electromagnetic waves from penetrating into the enclosed region, and thereby render the interior “invisible” to the outside observer.", "According to the pioneering work by Pendry et al.", "[33], the cloaking parameters disperse with frequency and therefore can only be fully effective at a single frequency.", "To investigate this interesting phenomena, it is necessary to simulate the full wave and consider the non-monochromatic waves passing through such frequency-dependent materials in time domain.", "The central issue for time-domain modelling is to formulate the constitutive relations.", "The material parameters of an ideal spherical cloak are given by (cf.", "[33]): ${\\varepsilon }={\\mu }={\\rm diag}\\big (\\varepsilon (r), \\epsilon , \\epsilon \\big ),\\quad \\varepsilon (r)=\\frac{R_2}{R_2-R_1}\\Big (\\frac{r-R_1}{r}\\Big )^2,\\quad \\epsilon =\\frac{R_2}{R_2-R_1},$ in the cloaking layer $\\Omega _{\\rm cl}=\\lbrace R_1<r<R_2\\rbrace .$ Since $\\varepsilon (r) \\in [0,1)$ , same as in the left-handed materials (LHMs), the material parameters are often mapped by dispersive medium models, e.g., Drude model, Lorentz model [14].", "Here, we map $\\varepsilon (r)$ (to frequency $\\omega $ -dependent medium) via the Drude dispersion model: $\\varepsilon _k(r,\\omega )=1-\\frac{\\omega _{p,k}^2(r)}{\\omega (\\omega -{\\rm i}\\gamma _k)}, \\quad \\omega _{p,k}(r)=\\sqrt{\\omega _c(\\omega _c-{\\rm i}\\gamma _k)(1-\\varepsilon (r))},$ for $k=1,2,$ leading to the dispersive media in $\\Omega _{\\rm cl}:$ $\\hat{\\varepsilon }(r,\\omega )={\\rm diag} \\big (\\varepsilon _1(r, \\omega ), \\epsilon ,\\epsilon \\big ),\\quad \\hat{\\mu }(r,\\omega )={\\rm diag}\\big (\\varepsilon _2(r,\\omega ), \\epsilon ,\\epsilon \\big ).$ In the above expressions, $\\omega $ is the wave frequency, $\\lbrace \\omega _{p,k}\\rbrace $ are the plasma frequencies, $\\lbrace \\gamma _k\\rbrace $ are damping terms called collision frequencies and $\\omega _c>0$ is the operating frequency of the cloak.", "Indeed, if $\\omega =\\omega _c,$ then (REF ) reduces to (REF ).", "Although the ideal lossless case, i.e., $\\gamma _1=\\gamma _2=0$ has been adopted in [48], [49], [21], it is physically more reasonable to include the loss effect of the medium in the modeling.", "Hereafter, we assume that $\\gamma _1\\ne 0, \\gamma _2\\ne 0$ .", "Denote by $\\hat{f}$ the Fourier transform of a generic function $f(t)$ , i.e., $\\hat{f}(\\omega )={F}[f(t)](\\omega )=\\int _{-\\infty }^{+\\infty }f(t)e^{-{\\rm i}\\omega t}dt.$ Let ${v}=(v_r,v_{\\theta }, v_{\\phi })^t$ be a generic vector field, where $v_r$ , $v_{\\theta }$ and $v_{\\phi }$ are the components of ${v}$ in the coordinate units ${e}_r$ , ${e}_{\\theta }$ and ${e}_{\\phi }$ , respectively.", "Given (REF ), the constitutive relation in the cloaking layer $\\Omega _{\\rm cl}$ in the frequency domain reads $&\\widehat{D}=\\big (\\widehat{D}_r,\\widehat{D}_{\\theta }, \\widehat{D}_{\\phi }\\big )^t=\\varepsilon _0 \\hat{\\varepsilon }(r,\\omega )\\widehat{E}=\\varepsilon _0\\big (\\varepsilon _1(r,\\omega ) \\widehat{E}_r,\\epsilon \\widehat{E}_{\\theta }, \\epsilon \\widehat{E}_{\\phi }\\big )^t, \\\\&\\widehat{B}=\\big (\\widehat{B}_r,\\widehat{B}_{\\theta }, \\widehat{B}_{\\phi }\\big )^t=\\mu _0 \\hat{\\mu }(r,\\omega )\\widehat{H}=\\mu _0\\big (\\varepsilon _2(r,\\omega ) \\widehat{H}_r,\\epsilon \\widehat{H}_{\\theta }, \\epsilon \\widehat{H}_{\\phi }\\big )^t.", "$ Remark 3.1 The time-domain constitutive equations extensively used in [48], [49], [20], [23] reads $&\\frac{\\partial ^2 D_r}{\\partial t^2}+\\gamma _1\\frac{\\partial D_r}{\\partial t}=\\varepsilon _0\\Big (\\frac{\\partial ^2 E_r}{\\partial t^2}+\\gamma _1\\frac{\\partial E_r}{\\partial t}+\\omega _{p,1}^2E_r\\Big ), \\quad (D_{\\theta }, D_{\\phi })=\\varepsilon _0 \\epsilon (E_{\\theta }, E_{\\phi }), \\\\&\\frac{\\partial ^2 B_r}{\\partial t^2}+\\gamma _2\\frac{\\partial B_r}{\\partial t}=\\mu _0\\Big (\\frac{\\partial ^2 H_r}{\\partial t^2}+\\gamma _2\\frac{\\partial H_r}{\\partial t}+\\omega _{p,2}^2H_r\\Big ), \\quad (B_{\\theta }, B_{\\phi })=\\mu _0 \\epsilon (H_{\\theta }, H_{\\phi }), $ which can be obtained by simply applying the inverse Fourier transform to (REF )-().", "$\\Box $ Different from the existing models, we use (REF )-() to derive the following relations in time domain.", "Lemma 3.1 We have the constitutive relations in time domain of the form $&{E}=\\varepsilon _0^{-1}{D}_1[{D}],\\quad {H}=\\mu _0^{-1}{D}_2[{B}], $ where for $k=1,2,$ the operators ${D}_k[D]:=\\Big (D_r+\\int _{0}^{t} \\vartheta _k(r, t-\\tau )D_r(\\cdot ,\\tau )d\\tau ,\\, \\epsilon ^{-1}D_{\\theta },\\, \\epsilon ^{-1}D_{\\phi }\\Big )^t,$ with kernel functions given by $\\vartheta _k(r, t)=\\frac{{\\rm i}\\,\\omega _{p,k}^2(r)}{\\zeta ^0_k(r)-\\zeta ^1_k(r)}\\big (e^{{\\rm i}\\zeta ^0_k(r) t}-e^{{\\rm i}\\zeta ^1_k(r) t} \\big ).$ Here, $\\lbrace \\zeta ^0_k(r), \\zeta ^1_k(r)\\rbrace _{k=1}^2$ are the roots of the quadratic equation: $z^2-{\\rm i}\\gamma _kz-\\omega _{p,k}^2=0$ given by $\\begin{split}& \\zeta _k^0(r)=-\\frac{1}{\\sqrt{2}} \\sqrt{\\sqrt{\\xi _k^2+\\eta _k^2} + {\\xi _k} }+{\\rm i}\\bigg (\\frac{\\gamma _k}{2} +\\frac{1}{\\sqrt{2}} \\sqrt{ \\sqrt{\\xi _k^2+\\eta _k^2} -{\\xi _k} }\\bigg ),\\\\& \\zeta _k^1(r)=\\frac{1}{\\sqrt{2}} \\sqrt{\\sqrt{\\xi _k^2+\\eta _k^2} + {\\xi _k} }+{\\rm i}\\bigg (\\frac{\\gamma _k}{2} -\\frac{1}{\\sqrt{2}} \\sqrt{ \\sqrt{\\xi _k^2+\\eta _k^2} -{\\xi _k} }\\bigg ),\\end{split}$ where $\\xi _k= \\omega _c^2\\,(1-\\varepsilon (r))-\\frac{\\gamma ^2_k}{4},\\quad \\eta _k=-\\gamma _k\\omega _c\\,(1-\\varepsilon (r)),$ are the real and imaginary parts of $\\omega _{p,k}^2-\\frac{\\gamma ^2_k}{4}$ for $k=1,2.$ Using the definition of $\\varepsilon _k(r,\\omega )$ in (REF ), we derive from (REF )-() that $& \\widehat{E}_r=\\varepsilon _0^{-1} \\Big (1+\\frac{\\omega _{p,1}^2(r)}{\\omega ^2-{\\rm i}\\gamma _1\\omega -\\omega _{p,1}^2(r) }\\Big ) \\widehat{D}_r,\\quad (\\widehat{E}_{\\theta },\\widehat{E}_{\\phi })^t=(\\varepsilon _0 \\epsilon )^{-1}(\\widehat{D}_{\\theta },\\widehat{D}_{\\phi })^t\\;, \\\\&\\widehat{H}_r=\\mu _0^{-1}\\Big (1+\\frac{\\omega _{p,2}^2(r)}{ \\omega ^2-{\\rm i}\\gamma _2\\omega -\\omega _{p,2}^2(r) } \\Big )\\widehat{B}_r ,\\quad (\\widehat{H}_{\\theta },\\widehat{H}_{\\phi })^t=(\\mu _0 \\epsilon )^{-1}(\\widehat{B}_{\\theta },\\widehat{B}_{\\phi })^t.\\; $ Applying the inverse Fourier transform to (REF )-() leads to $\\begin{split}&E_r=\\varepsilon _0^{-1}D_r+\\varepsilon _0^{-1}\\omega _{p,1}^2(r){F}^{-1}\\Big [\\frac{1}{\\omega ^2-{\\rm i}\\gamma _1\\omega -\\omega _{p,1}^2(r)}\\Big ]\\ast D_r,\\;\\; \\\\&H_r=\\mu _0^{-1}B_r+\\mu _0^{-1}\\omega _{p,2}^2(r){F}^{-1}\\Big [\\frac{1}{\\omega ^2-{\\rm i}\\gamma _2\\omega -\\omega _{p,2}^2(r)}\\Big ]\\ast B_r, \\\\[3pt]&(E_{\\theta }, E_{\\phi })^t=(\\varepsilon _0 \\epsilon )^{-1}(D_{\\theta }, D_{\\phi })^t,\\quad (H_{\\theta }, H_{\\phi })^t=(\\mu _0 \\epsilon )^{-1}(B_{\\theta }, B_{\\phi })^t,\\end{split}$ where “ $\\ast $  \" is the usual convolution as before.", "The rest of the derivation is to explicitly evaluate two inverse Fourier transforms.", "Let $\\zeta ^0_k, \\zeta ^1_k$ be two roots of $z^2-{\\rm i}\\gamma _kz-\\omega _{p,k}^2=0.$ Then we immediately have $\\zeta _k^0+\\zeta _k^1={\\rm i}\\gamma _k$ and $\\zeta _k^0\\zeta _k^1=-\\omega _{p,k}^2,$ so we can write $\\frac{1}{\\omega ^2-{\\rm i}\\gamma _k\\omega -\\omega _{p,k}^2(r)}=\\frac{1}{\\zeta _k^0-\\zeta _k^1}\\Big (\\frac{1}{\\omega -\\zeta ^0_k}-\\frac{1}{\\omega -\\zeta ^1_k}\\Big ).$ Recall that (cf.", "[4]): ${F}^{-1}\\Big [\\frac{1}{{\\rm i}\\omega +a}\\Big ]=-{\\rm i}\\, {F}^{-1}\\Big [\\frac{1}{\\omega -a{\\rm i}}\\Big ]=e^{-at}H(t), \\quad \\hbox{if}\\;\\;\\mathfrak {R}\\lbrace a\\rbrace >0,$ where $H(t)$ is the Heaviside function.", "Suppose that we can show $ \\mathfrak {Im}\\lbrace \\zeta ^0_k\\rbrace >0,\\quad \\mathfrak {Im}\\lbrace \\zeta ^1_k\\rbrace >0.$ Then by (REF )-(REF ), ${F}^{-1}\\Big [\\frac{1}{\\omega ^2-{\\rm i}\\gamma _k\\omega -\\omega _{p,k}^2(r)}\\Big ]=\\frac{{\\rm i}}{\\zeta ^0_k-\\zeta ^1_k}\\big (e^{{\\rm i}\\zeta ^0_k t}-e^{{\\rm i}\\zeta ^1_k t} \\big )H(t).$ Consequently, we derive (REF )-(REF ) from (REF ) and (REF ).", "It remains to verify (REF ) and (REF ).", "It is evident that the quadratic equation has the roots: $z=\\frac{\\gamma _k}{2} {\\rm i}\\pm \\sqrt{\\omega _{p,k}^2-\\frac{\\gamma _k^2}{4}}=\\frac{\\gamma _k}{2} {\\rm i}\\pm \\sqrt{\\xi _k+ {\\rm i}\\eta _k}\\,.$ Setting $\\alpha _k+{\\rm i}\\beta _k=\\sqrt{\\xi _k+ {\\rm i}\\eta _k}$ , we find $\\alpha _k^2-\\beta _k^2=\\xi _k$ and $2\\alpha _k\\beta _k=\\eta _k.$ Solving this system yields $\\alpha ^2_k=\\frac{\\sqrt{\\xi _k^2+\\eta _k^2}+\\xi _k}{2},\\quad \\beta ^2_k=\\frac{\\sqrt{\\xi _k^2+\\eta _k^2}-\\xi _k}{2}.$ Noting that $\\alpha _k\\beta _k<0, $ we can determine $\\alpha _k,\\beta _k,$ and obtain (REF ) from (REF ).", "By (REF ), $\\mathfrak {Im}\\lbrace \\zeta ^0_k\\rbrace >\\mathfrak {Im}\\lbrace \\zeta ^1_k\\rbrace ,$ so we next show that $\\mathfrak {Im}\\lbrace \\zeta ^1_k\\rbrace >0,$ that is, $\\frac{\\gamma _k}{2} >\\frac{1}{\\sqrt{2}} \\sqrt{ \\sqrt{\\xi _k^2+\\eta _k^2} -{\\xi _k} }\\;\\;\\; {\\rm i.e.,}\\;\\;\\; \\gamma _k^4+4\\gamma _k^2 \\xi _k-4\\eta _k^2>0.$ Direct calculation from (REF ) leads to $\\gamma _k^4+4\\gamma _k^2 \\xi _k-4\\eta _k^2=4\\gamma _k^2 \\omega _c^2\\, \\varepsilon (r)\\, (1-\\varepsilon (r))>0,$ as $\\gamma _k\\ne 0$ , $\\omega _c>0$ and $0<\\varepsilon (r)<1$ (cf.", "(REF )).", "This verifies (REF ) and completes the proof.", "With the constitutive relations (REF )-(REF ) at our disposal, we represent $E,H$ in terms of $D, B$ and then eliminate $B$ , leading to the following equation in $\\Omega _{\\rm cl}.$ Theorem 3.1 Assume that the source term and initial fields vanish in the cloaking layer $\\Omega _{\\rm cl}$ .", "Then the governing equation in the cloaking layer takes the form $\\partial ^2_{t}{D}+c^2 \\nabla \\times \\big ( {D}_2[ \\nabla \\times ({D}_1[{D}])] \\big )=0\\quad {\\rm in}\\;\\; \\Omega _{\\rm cl}.$ First, we show that given the homogeneous initial condition $B(r, 0)=0$ , we have $\\partial _t {D}_2[B]={D}_2[\\partial _t B]$ , that is, the operators $\\partial _t$ and ${D}_2$ are commutable.", "Indeed, by $\\int _{0}^{t} \\frac{\\partial }{\\partial t}\\vartheta _2(r,t-\\tau ) B_r(r,\\tau )d\\tau =&-\\vartheta _2( r,t-\\tau )B_r(r,\\tau )\\big |_0^t+\\int _{0}^{t} \\vartheta _2(r,t-\\tau ) \\frac{\\partial B_r(r,\\tau )}{\\partial \\tau } d\\tau \\\\=&\\int _{0}^{t} \\vartheta _2(r,t-\\tau ) \\frac{\\partial B_r(r,\\tau )}{\\partial \\tau } d\\tau ,$ and (REF )-(REF ), we verify that $\\partial _t {D}_2[B]={D}_2[\\partial _t B]$ .", "Thus, taking time derivative on both sides of the second equation in (REF ), we obtain $\\partial _t {H}=\\mu _0^{-1} {\\partial _t({D}_2[B])}=\\mu _0^{-1} {{D}_2[\\partial _t B]}.$ By substituting the constitutive relation (REF ) in the second equation in (), we derive ${\\partial _t {B}}=-\\varepsilon _0^{-1}\\nabla \\times \\big ({D}_1[{D}]\\big ).$ Then, taking time derivative on the first equation in () and utilizing (REF )-(REF ) to eliminate ${H}$ leads to (REF ), which ends the proof.", "Remark 3.2 It is worthwhile to note that the mathematical model (REF ) is not limited to spherical dispersive cloaks.", "It is applicable to the modelling of many electromagnetic devices with symmetric non-diagonal $\\varepsilon $ and $\\mu $ made from metamaterials.", "Following the procedure in [29], we start with diagonalising the symmetric matrices $\\varepsilon $ and $\\mu ,$ i.e., $\\varepsilon ={P} {\\Lambda _1} {P}^t,\\quad \\mu ={Q} {\\Lambda _2} {Q}^t,\\quad {\\Lambda _i}={\\rm diag}(\\lambda _{i1}, \\lambda _{i2},\\lambda _{i3}),\\quad i=1, 2,$ and $\\lbrace P, Q \\rbrace =\\lbrace P_{ij}, Q_{ij} \\rbrace _{1\\le i,j \\le 3} $ are orthonormal matrices.", "Then, we use the Drude model to map $\\lbrace \\lambda _{ij}(r)\\rbrace $ less than 1 to $\\lbrace \\lambda _{ij}(r, \\omega ) \\rbrace $ similar with (REF ) and take inverse Fourier transform to (REF ) with replaced $\\lbrace \\lambda _{ij}(r, \\omega ) \\rbrace .$ As a result, we obtain the same constitutive relations as (REF ) ${E}=\\varepsilon _0^{-1}{D}_1[{D}],\\quad {H}=\\mu _0^{-1}{D}_2[{B}]$ with more complicated forms of $D_1$ and $D_2$ : $\\begin{split}{D}_1[D]:={P} \\widetilde{\\Lambda }^{-1}_1 {P}^t D+\\int _0^t {P} {\\Theta }_1(r, t-\\tau ){P}^t D(r, \\tau )d\\tau ,\\\\{D}_2[B]:={Q} \\widetilde{{\\Lambda }}^{-1}_2 {Q}^t B+\\int _0^t {Q} {\\Theta }_2(r, t-\\tau ) {Q}^t B(r, \\tau )d\\tau ,\\end{split}$ where $\\widetilde{\\Lambda }_i={\\rm diag}\\big (\\tilde{\\lambda }_{i1}(r),\\tilde{\\lambda }_{i2}(r),\\tilde{\\lambda }_{i3}(r) \\big )$ , $\\Theta _i={\\rm diag}(\\vartheta _{i1}, \\vartheta _{i2}, \\vartheta _{i3})$ are diagonal matrices with $\\begin{split}& \\tilde{\\lambda }_{ij}={\\left\\lbrace \\begin{array}{ll}\\displaystyle 1& {\\rm if}\\;\\; \\lambda _{ij}(r)\\in (0, 1),\\\\\\displaystyle \\lambda _{ij}(r) & {\\rm if}\\;\\; \\lambda _{ij}(r)\\in [1,\\infty ),\\end{array}\\right.", "}\\quad \\vartheta _{ij}={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{{\\rm i}(\\omega _{p,i}^j(r))^2\\big (e^{{\\rm i}\\zeta ^0_{ij} t}-e^{{\\rm i}\\zeta ^1_{ij} t} \\big )}{\\zeta ^0_{ij}-\\zeta ^1_{ij}}& {\\rm if}\\;\\; \\lambda _{ij}(r)\\in (0, 1),\\\\\\displaystyle 0 & {\\rm if}\\;\\; \\lambda _{ij}(r)\\in [1,\\infty ),\\end{array}\\right.", "}\\end{split}$ $\\omega _{p,i}^j(r)$ has a similar expression $\\omega _{p,i}^j(r) =\\sqrt{\\omega _c(\\omega _c-{\\rm i}\\gamma _i)(1-\\lambda _{ij}(r))},\\quad i=1, 2,$ and complex pairs $\\lbrace \\zeta ^0_{ij}, \\zeta ^1_{ij}\\rbrace $ are the roots of quadratic equations $\\omega ^2-{\\rm i}\\gamma _i\\omega -(\\omega _{p,i}^j(r))^2=0$ , $i=1, 2$ , respectively.", "$\\Box $ Figure: Sketch of the cross section." ], [ "Simulation of the spherical invisibility cloaks", "In what follows, we focus on the simulation of the spherical cloaks.", "We first present the full model with reduction of the unbounded domain by using the NRBC in Section .", "As sketched in Figure REF , we denote $\\begin{split}R_0=0;\\;\\; \\Omega _i=\\lbrace R_i<r<R_{i+1}\\rbrace ,\\;\\; i=0,1; \\\\\\Omega _2=\\Omega _{b_0}\\setminus (\\Omega _0\\cup \\Omega _1), \\;\\; \\Omega _3=\\Omega _b\\setminus \\Omega _{b_0}.\\end{split}$ Correspondingly, we further denote $F(r, t)=\\partial _tJ(r, t),$ and $\\begin{split}&\\Gamma _i=\\bar{\\Omega }_i \\cap \\bar{\\Omega }_{i+1},\\quad \\lbrace E^i, H^i, D^i, B^i,F^i \\rbrace =\\lbrace E, H, D,B,F \\rbrace |_{\\Omega _i}, \\;\\;\\;i=0,1,2;\\\\[2pt]&\\lbrace E^3, H^3, B^3, D^3, F^3\\rbrace =\\lbrace E^{\\rm sc} , H^{\\rm sc}, B^{\\rm sc}, D^{\\rm sc}, F \\rbrace |_{\\Omega _3}.\\end{split}$ We summarise below the assumptions (for usual scattering problems): (i) ${\\varepsilon }={\\mu }={I}_3 \\;\\;{\\rm in}\\;\\; \\Omega _b \\setminus \\Omega _1$ ; (ii) There is no wave in the truncated domain $\\Omega _b$ at time $t=0$ , that is, we shall have homogeneous initial condition; (iii) The source term $J$ is compactly supported in $\\Omega _2.$ Proposition 3.1 The full model for 3D cloak takes the form t2 Di+c2 Di=Fi    in i, i=0, 2, 3, t2 D1+c2 ( D2[ (D1[D1])] )=0   in 1, (D0 -D1[D1])r=0,(D0 -D2[D1[D1]])r=0 on 0, (D1[D1]-D2)r=0,(D2[D1[D1]]-D2)r=0 on 1, (D2-D3)r=Dinr,(D2-D3)r=Dinr on 2, t DT3+c( D3 )r-Tb[D3]=0 at r=b, D(r, 0)=0,   t D(r, 0)=0 in b, where $D_T^{3}:= \\hat{r}\\times D^{3}\\times \\hat{ r}$ is the tangential component of $D^{3}$ on the boundary $r=b$ .", "Note that (REF ) is a direct consequence of (), (REF ) is proved in Theorem REF , and (REF )-(REF ) are direct consequences of ()-() and the above assumption (i).", "For the jump conditions (REF )-(REF ), we recall the standard transmission conditions $E^i\\times \\hat{ r}=E^{i+1}\\times \\hat{ r},\\quad H^i\\times \\hat{ r}=H^{i+1}\\times \\hat{ r}\\quad {\\rm at}\\;\\;\\; \\Gamma _i, \\quad i=0, 1.$ The first jump conditions in (REF )-(REF ) can be obtained by directly applying the first constitutive relation between $E$ and $D$ to the above transmission condition on $E$ .", "Therefore, we focus on the first jump conditions in (REF )-(REF ).", "Inserting the constitutive relations (REF ) into (REF ) directly leads to (REF ) and $B^0\\times \\hat{ r}={D}_2[B^1]\\times \\hat{ r} \\quad {\\rm at}\\;\\;\\;\\Gamma _1;\\quad B^2\\times \\hat{ r}={D}_2[B^1]\\times \\hat{ r} \\quad {\\rm at}\\;\\;\\; \\Gamma _2.$ From (REF ), we derive ${\\partial _t B^0}\\times \\hat{ r}={D}_2\\big [{\\partial _t B^1}\\big ]\\times \\hat{ r} \\quad {\\rm at}\\;\\;\\;\\Gamma _1;\\quad {\\partial _t B^2}\\times \\hat{ r}={D}_2\\big [{\\partial _t B^1}\\big ]\\times \\hat{ r} \\quad {\\rm at}\\;\\;\\; \\Gamma _2,$ which, together with () and (REF ), yields the second jump conditions in (REF )-(REF ).", "This ends the derivation." ], [ "VSH-spectral-element discretization", "In view of the spherical geometry and radially stratified dispersive media, we can fully exploit these advantages to develop an efficient and accurate VSH-spectral-element solver for the Maxwell's system (REF ).", "Needless to say, it is optimal compared with the FDTD simulation in [49] for the time-domain Pendry's spherical cloak.", "The key is to employ the divergence-free VSH expansion of the fields and reduce the governing equations into two sequences of decoupled one-dimensional problems.", "By proposition REF , the solenoidal fields $D^i$ , $F^i$ and $D^{in}$ can have VSH expansions $\\lbrace D^i, F^i\\rbrace =\\lbrace u_{00}^i, f_{00}^i\\rbrace \\,Y_0^0 + \\sum _{l=1}^\\infty \\sum _{|m|=0}^l \\Big \\lbrace \\lbrace u_{lm}^i,f_{1,l}^{i,m}\\rbrace \\, \\Phi _l^m + \\nabla \\times (\\lbrace v_{lm}^i,f_{2,l}^{i,m}\\rbrace \\, \\Phi _l^m\\big )\\Big \\rbrace ,$ and $D^{\\rm in}=g_{00}\\,Y_0^0 + \\sum _{l=1}^\\infty \\sum _{|m|=0}^l \\Big \\lbrace g_{lm}\\, \\Phi _l^m+ \\nabla \\times (h_{lm}\\, \\Phi _l^m\\big )\\Big \\rbrace .$ It is worthy of pointing out that the capacity operator in (REF ) has two alternative expressions (REF ) and (REF ).", "Both use the usual VSH expansion coefficients.", "For example, we have $T_b[D^{3}]=\\frac{c}{b} \\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\bigg \\lbrace \\frac{\\omega _l\\ast D_{lm}^{r}}{l(l+1)}\\,{\\Psi }_l^m+\\big (\\sigma _l\\ast D_{lm}^{(2)}\\big )\\,{\\Phi }_l^m\\bigg \\rbrace ,$ according to (REF ), where $\\lbrace D_{lm}^r, D_{lm}^{(1)}, D_{lm}^{(2)}\\rbrace $ are the coefficients in the VSH expansion $D^{3}=D_{00}Y_0^0+\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\bigg \\lbrace D_{lm}^{r}\\,Y_{lm}+D_{lm}^{(1)}\\,{\\Psi }_l^m+D_{lm}^{(2)}\\,{\\Phi }_l^m\\bigg \\rbrace .$ In order to do dimension reduction using expansion (REF ), we re-express the formulation (REF ) using coefficients $\\lbrace u_{lm}^3, v_{lm}^3\\rbrace $ .", "From the Proposition REF , we have relations $D_{lm}^{r}=\\frac{l(l+1)}{r}v_{lm}^{3},\\quad D_{lm}^{(1)}=\\hat{\\partial }_rv_{lm}^{3}, \\quad D_{lm}^{(2)}=u_{lm}^{3}.$ A simple substitution in (REF ) gives ${T}_b[D^{3}]=\\frac{c}{b} \\sum _{l=1}^\\infty \\sum _{m=-l}^l\\bigg \\lbrace b^{-1}(\\omega _l\\ast v_{lm}^{3})\\,{\\Psi }_l^m+\\big (\\sigma _l\\ast u_{lm}^{3}\\big )\\,{\\Phi }_l^m\\bigg \\rbrace .$ Proposition 3.2 For $l\\ge 1$ , $|m|\\le l$ and $i=0, 1, 2, 3,$ denote $g=g_{lm},\\;\\; h=h_{lm},\\;\\; u^i=u_{lm}^i, \\;\\; v^i=v_{lm}^i, \\;\\; f_1^i=f_{1,l}^{i,m},\\;\\; f_2^i=f_{2,l}^{i,m}, \\;\\; I_i:=(R_i, R_{i+1}).$ With the simple variable substitution $\\tilde{u}^0=\\epsilon u^0,\\quad \\tilde{u}^1=u^1,\\quad \\tilde{u}^2=\\epsilon u^2, \\quad \\tilde{u}^3=\\epsilon u^3, \\quad \\tilde{g}=\\epsilon g,$ the Maxwell system (REF ) reduced to the following two sequences of one-dimensional problem for $v$ and $\\tilde{u}$ , respectively, for $l\\ge 1$ , $|m|\\le l$ : 2 vit2- c2r2r( r2 vir )+c2lr2vi=f2i,    rIi,   i=0, 2, 3, 2 v1t2-c22r2r( r2 v1r )+c2 lr2v1+c2 lr2 1v1(r,t)=0,   rI1, v0=v1,   r v1=r v0+(-1)r-1v0   at r=R1, v2=v1,   r v1=r v2+(-1)r-1v2   atr=R2, v2-v3=h,   r v2-rv3=rh   atr=R3, 1ct v3+v3 r+1bv3-1bl*v3=0   atr=b, v|t=0=t v|t=0=0.", "while $\\tilde{u}$ satisfies the same equations in (REF ) with $\\tilde{u},$ $f_1,$ $\\tilde{g}$ and $\\vartheta _2$ , in place of $v,$ $f_2,$ $h$ and $\\vartheta _1$ , respectively, and for $l=m=0,$ $\\partial ^2_t u_{00}^i=f_{00}^i,\\quad u^i_{00}|_{t=0}=\\partial _tu^i_{00}|_{t=0}=0,\\quad r\\in I_i.$ We postpone the detailed derivation in .", "The above proposition shows that $\\tilde{u}$ and $v$ can be obtained by solving (REF ) with different input data.", "Therefore, we only need to focus on the one dimensional problems (REF ).", "Note that the solution of (REF ) has a jump at $r=R_3$ .", "We introduce $\\tilde{v}(r, t)={\\left\\lbrace \\begin{array}{ll}\\displaystyle v(r, t), \\quad 0\\le r\\le R_3,\\quad t\\ge 0,\\\\[4pt]\\displaystyle v(r, t)+h(R_3, t)\\frac{b-r}{b-R_3},\\quad R_3<r\\le b,\\quad t\\ge 0,\\end{array}\\right.", "}$ and $\\tilde{f}_2(r, t)={\\left\\lbrace \\begin{array}{ll}\\displaystyle f_2(r, t), \\quad 0< r< R_3,\\;\\; t>0,\\\\[4pt]\\displaystyle f_2(r, t)+\\bigg \\lbrace \\frac{\\partial ^2h(R_3, t)}{\\partial t}+\\Big (\\frac{2bc^2}{r(b-r)}+\\frac{c^2\\beta _{l}}{r^2}\\bigg )h(R_3, t)\\bigg \\rbrace \\frac{b-r}{b-R_3},\\;\\; R_3<r< b,\\;\\; t>0,\\end{array}\\right.", "}$ to rewrite (REF ) into 2 vit2- c2r2r( r2 vir )+c2lr2vi=f2i,    rIi, i=0, 2, 3, 2 v1t2-c22r2r( r2 v1r )+c2 lr2v1+c2 lr2 1v1(r,t)=0,   rI1, v0=v1,   r v1=r v0+(-1)r-1v0   at r=R1, v2=v1,   r v1=r v2+(-1)r-1v2   atr=R2, v2=v3,   r v2-rv3=rh+1b-R3h(R3, t)   atr=R3, 1ctv3+v3 r+1bv3-1bl*v3=-1b-R3h(R3, t)   atr=b, v|t=0=h(R3, 0)b-rb-R3[R3, b],   t v|t=0=h(R3, 0)tb-rb-R3[R3, b].", "Here $\\chi _{[R_3, b]}$ is the indicator function which is equal to 1 inside the interval $[R_3,b]$ and vanish outside.", "Obviously, $\\tilde{v}(r, t)$ is continuous in $I$ .", "Multiplying (REF ) and (REF ) by test function $r^2\\phi $ and $\\epsilon r^2\\phi $ respectively for $\\phi \\in H^1(I)$ , using integration by parts and summing up the resulted equations, then applying the interface conditions and boundary condition we obtain the variational problem: Find $\\tilde{v}(\\cdot , t)\\in H^1(I)$ , s.t.", "$\\mathcal {B}(\\tilde{v}, \\phi )=(\\tilde{f}_2, \\phi )+\\Big (\\partial _rh(R_3, t)+\\frac{h(R_3,t)}{b-R_3}\\Big )c^2R_3^2\\phi (R_3)-\\frac{h(R_3,t)}{b-R_3}c^2b^2\\phi (b),$ for all $\\phi \\in H^1(I)$ , where $\\begin{split}\\mathcal {B}(\\tilde{v},\\phi ):=&\\int _{I\\backslash I_1}(r^2\\partial _{tt}\\tilde{v}\\phi +c^2r^2\\partial _r\\tilde{v}\\partial _r\\phi )dr+\\int _{I_1}\\Big (\\epsilon r^2\\partial _{tt}\\tilde{v}\\phi +\\frac{c^2r^2}{\\epsilon }\\partial _r\\tilde{v}\\partial _r\\phi \\Big )dr\\\\+&\\beta _lc^2\\Big (\\int _{I}\\tilde{v}\\phi dr+\\int _{I_1}\\vartheta _1\\ast \\tilde{v}(r,t)\\phi (r)dr\\Big )+c^2(\\epsilon -1)R_1\\tilde{v}(R_1,t)\\phi (R_1)\\\\-&c^2(\\epsilon -1)R_2\\tilde{v}(R_2,t)\\phi (R_2)+\\big \\lbrace cb^2\\partial _t\\tilde{v}(b,t)+c^2b(\\tilde{v}(b,t)-\\sigma \\ast \\tilde{v}(b, t))\\big \\rbrace \\phi (b).\\end{split}$ Based on the variational problem (REF ), we introduce the spectral-element discretization.", "Let $\\mathcal {I}_h:0=r_0<r_1<\\cdots <r_E=b$ be an interface conforming mesh of the interval $I$ and denote the element by $\\lbrace K_e=(r_{e-1}, r_e)\\rbrace _{e=1}^E$ .", "Here, the interface conforming mesh means that the points $r=R_1, R_2, R_3$ are mesh points, see Figure REF .", "Let $\\mathcal {P}_N(K_e)$ be the set of all complex valued polynomials of degree at most $N$ in each interval $K_e$ and define the spectral element approximation space as ${X}_N(\\mathcal {I}_h):=\\big \\lbrace u\\in H^1(I): u|_{K_e}\\in \\mathcal {P}_N(K_e)\\big \\rbrace .$ The spectral element discretization of (REF ) is to find $\\tilde{v}_N(r,t)\\in {X}_N(\\mathcal {I}_h)$ for all $t>0$ , such that $\\mathcal {B}(\\tilde{v}_N, \\phi )=(\\tilde{f}_2, \\phi )+\\Big (\\partial _rh(R_3, t)+\\frac{h(R_3,t)}{b-R_3}\\Big )c^2R_3^2\\phi (R_3)-\\frac{h(R_3,t)}{b-R_3}c^2b^2\\phi (b),$ for all $\\phi \\in {X}_N(\\mathcal {I}_h)$ .", "Figure: Interface conforming mesh used by the spectral element discretization.This spectral element discretization for $\\tilde{v}$ leads to the following integral differential system $\\begin{split}&\\mathbb {M}\\ddot{V}+\\mathbb {B}\\dot{V}+{\\mathbb {C}}{V}+G-\\frac{c}{b}\\mathbb {B}(\\sigma \\ast {V})={F}, \\quad V(0)=V_0,\\quad \\dot{V}(0)=V_1,\\end{split}$ where $\\mathbb {M}=(m_{ij})_{\\mathcal {N}\\times \\mathcal {N}},\\quad \\mathbb {A}=(a_{ij})_{\\mathcal {N}\\times \\mathcal {N}},\\quad G=(G_i)_{\\mathcal {N}},$ $\\mathbb {B}=cb^2\\mathbb {E}_{\\mathcal {N}\\mathcal {N}},\\quad \\mathbb {C}=\\mathbb {A}+c^2b\\mathbb {E}_{\\mathcal {N}\\mathcal {N}}+c^2(\\epsilon -1)(R_1\\mathbb {E}_{i_1i_1}-R_2\\mathbb {E}_{i_2i_2}),$ are matrices with entries given by $\\begin{split}& m_{ij}=\\int _{I\\backslash I_1}r^2\\phi _i\\phi _jdr+\\epsilon \\int _{I_1}r^2\\phi _i\\phi _jdr,\\quad G_i=\\beta _lc^2\\int _{I_1}{\\vartheta }_1\\ast \\tilde{v}_N(r, t)\\phi _i(r)dr,\\\\& a_{ij}=c^2\\int _{I\\backslash I_1}(r^2\\partial _r\\phi _j\\partial _r\\phi _i+\\beta _l\\phi _j\\phi _i)dr+c^2\\int _{I_1}\\Big (\\frac{r^2}{\\epsilon }\\partial _r\\phi _j\\partial _r\\phi _i+\\beta _l\\phi _j\\phi _i\\Big )dr,\\\\&F_i=\\int _{I}\\tilde{f}_2\\phi _i(r)dr+\\Big (\\partial _rh(R_3, t)+\\frac{h(R_3,t)}{b-R_3}\\Big )c^2R_3^2\\phi _i(R_3)-\\frac{h(R_3,t)}{b-R_3}c^2b^2\\phi _i(b).\\end{split}$ Here, $i_1$ , $i_2$ denote the global index of the freedom at $r=R_1, R_2$ (see Figure REF for illustration), respectively, $\\mathcal {N}$ is the degree of freedom and also the global index of the freedom attached to mesh point $r=b$ , and $\\mathbb {E}_{mn}=(E_{ij})_{\\mathcal {N}\\times \\mathcal {N}}$ is the matrix with only one non-zero entry $E_{mn}=1$ ." ], [ "Newmark's scheme for time discretization", "The spectral element discretization leads to the integral differential system (REF ) w.r.t $t$ .", "Noting that all the involved time integrations are actually convolutions of exponential functions with the unknown functions, fast algorithm based on formula (REF ) can be used.", "Let us first discuss the discretization of the convolution ${\\vartheta }_1\\ast \\tilde{v}_N(r, t)$ .", "Define $\\tilde{\\vartheta }_1^{\\alpha }(r, t)=e^{{\\rm i}\\zeta ^0_1(r) t},\\quad \\tilde{\\vartheta }_1^{\\beta }(r, t)=e^{{\\rm i}\\zeta ^1_1(r) t}.$ Then ${\\vartheta }_1\\ast \\tilde{v}_N(r, t)=\\frac{{\\rm i}(\\omega _{p,1}(r))^2}{\\zeta ^0_1(r)-\\zeta ^1_1(r)}\\big (\\tilde{\\vartheta }_1^{\\alpha }\\ast \\tilde{v}_N(r, t)-\\tilde{\\vartheta }_1^{\\beta }\\ast \\tilde{v}_N(r,t)\\big ).$ By using the trapezoidal rule and (REF ), we have the second-order approximations $\\begin{split}&\\tilde{\\vartheta }_1^{\\alpha }\\ast \\tilde{v}_N(r,t_{n+1})\\approx \\lambda _0(r)\\tilde{\\vartheta }_1^{\\alpha }\\ast \\tilde{v}_N(r, t_{n})+\\frac{\\Delta t}{2}(\\tilde{v}_N(r, t_{n+1})+\\lambda _0(r)\\tilde{v}_N(r, t_n)),\\\\&\\tilde{\\vartheta }_1^{\\beta }\\ast \\tilde{v}_N(r, t_{n+1})\\approx \\lambda _1(r)\\tilde{\\vartheta }_1^{\\beta }\\ast \\tilde{v}_N(r, t_{n})+\\frac{\\Delta t}{2}(\\tilde{v}_N(r, t_{n+1})+\\lambda _1(r)\\tilde{v}_N(r,t_n)),\\end{split}$ where $\\lambda _0(r)=e^{{\\rm i}\\zeta ^0_1(r)\\Delta t},\\quad \\lambda _1(r)=e^{{\\rm i}\\zeta ^1_1(r)\\Delta t}.$ Substituting (REF ) into (REF ), we obtain ${\\vartheta }_1\\ast \\tilde{v}_N(r, t_{n+1})\\approx \\frac{{\\rm i}(\\omega _{p,1}(r))^2}{\\zeta ^0_1(r)-\\zeta ^1_1(r)}\\big (\\tilde{v}^c_{N}(r,t_n)+\\frac{\\Delta t}{2}(\\lambda _0(r)-\\lambda _1(r))\\tilde{v}_N(r,t_n)\\big ),$ where $\\tilde{v}^c_N(r, t_n):=(\\lambda _0(r)\\tilde{\\vartheta }_1^{\\alpha }-\\lambda _1(r)\\tilde{\\vartheta }_1^{\\beta })\\ast \\tilde{v}_N(r, t_n).$ Thus, we get the discretization for $G(t_{n+1})$ given by $G^{n+1}:=(G_i^{n+1})$ with $G_i^{n+1}:=\\frac{{\\rm i}\\beta _lc^2(\\omega _{p,1}(r))^2}{\\zeta ^0_1(r)-\\zeta ^1_1(r)}\\int _{I_1}(v^c_{N}(r,t_n)+\\frac{\\Delta t}{2}(\\lambda _0(r)-\\lambda _1(r))v_N(r,t_n))\\phi _i(r)\\,dr.$ It is important to point out that $G^{n+1}$ is a vector obtained by using the solution before the current time step thus can be moved to the right hand side in the fully discretization scheme.", "We denote the new right hand side vector by $\\widetilde{F}^{n}=F^n-G^n$ .", "Next, we consider the discretization of the convolution term $(\\sigma _l\\ast V)(t)$ .", "For this purpose, we define ${V}_j(t):=\\int _0^te^{c(t-\\tau )z_j^l/b}V(\\tau )\\,d\\tau .$ By using the trapezoidal rule and (REF ) again, we obtain the second order approximations ${V}^{0}_j=0, \\quad {V}^{n+1}_j= e^{c\\Delta tz_j^l/b}{V}^{n}_j+\\frac{\\Delta t}{2}V^{n+1}+\\frac{\\Delta t}{2}e^{c\\Delta tz_j^l/b}V^n,\\;\\;$ of ${V}_j(t_{n+1})$ for $j=1, 2, \\cdots , l$ .", "Accordingly, we have $(\\sigma \\ast {V})^0=0,\\quad (\\sigma \\ast {V})^{n+1}=\\frac{\\Delta t}{2}\\alpha _1V^{n+1}+\\frac{\\Delta t}{2}\\alpha _2V^{n}+\\sum _{j=1}^{l}\\alpha _2^j{V}^n_j,$ with $\\alpha _1=\\frac{c}{b} \\sum _{j=1}^{l}z_j^l, \\quad \\alpha _2^j=\\frac{c}{b}z_j^le^{c\\Delta tz_j^l/b},\\quad \\alpha _2=\\sum _{j=1}^{l}\\alpha _2^j,$ is a second order discretization of the convolution term $(\\sigma \\ast {V})(t_{n+1})$ .", "For the dicretization of time derivatives, we adopt the new marks scheme (cf.", "[40]).", "The key idea is to use the approximations: ${V}^{n+1}&={V}^{n}+\\Delta t\\dot{V}^{n}+\\frac{\\Delta t^2}{2}(1-2\\beta )\\ddot{V}^{n}+\\beta \\Delta t^2\\ddot{V}^{n+1},\\\\\\dot{V}^{n+1}&=\\dot{V}^{n}+(1-\\gamma )\\Delta t\\ddot{V}^{n}+\\gamma \\Delta t\\ddot{V}^{n+1},$ where $\\beta $ and $\\gamma $ are given parameters.", "Using the approximations (REF ) and (REF ), we can formulate the time discretization of the system (REF ) at $t_{n+1}$ as $\\begin{split}\\mathbb {M}\\ddot{V}^{n+1}+\\mathbb {B}\\dot{V}^{n+1}+\\mathbb {C}{V}^{n+1}-\\frac{c}{b}\\mathbb {B}\\bigg \\lbrace \\frac{\\alpha _1\\Delta t}{2}V^{n+1}+\\frac{\\alpha _2\\Delta t}{2}V^{n}+\\sum \\limits _{j=1}^l\\alpha _2^j{V}^n_j\\bigg \\rbrace =\\widetilde{F}^{n+1}.\\end{split}$ Inserting () into (REF ) to eliminate $\\dot{V}^{n+1}$ leads to $\\big (\\mathbb {M}+\\gamma \\Delta t\\mathbb {B}\\big )\\ddot{V}^{n+1}+\\Big \\lbrace \\mathbb {C}-\\frac{\\alpha _1c\\Delta t}{2b}\\mathbb {B}\\Big \\rbrace {V}^{n+1}=\\widetilde{F}^{n+1}-W^n,$ where $W^{n}=(1-\\gamma )\\Delta t\\mathbb {B}\\ddot{V}^n+\\mathbb {B}\\dot{V}^{n}-\\frac{\\alpha _2c\\Delta t}{2b}\\mathbb {B}V^{n}-\\frac{c}{b}\\mathbb {B}\\sum \\limits _{j=1}^l\\alpha _2^j{V}^n_j.$ From (REF ), we have $\\beta \\Delta t^2\\ddot{V}^{n+1}=V^{n+1}-\\Big \\lbrace V^n+\\Delta t\\dot{V}^n+\\Delta t^2\\Big (\\frac{1}{2}-\\beta \\Big )\\ddot{V}^n\\Big \\rbrace :=V^{n+1}-\\widetilde{V}^n.$ Using (REF ) in (REF ) we arrive the fully discretization scheme $\\bigg \\lbrace \\mathbb {M}+\\Big (\\gamma \\Delta t-\\frac{\\alpha _1c\\beta \\Delta t^3}{2b}\\Big )\\mathbb {B}+\\beta \\Delta t^2\\mathbb {C}\\bigg \\rbrace {V}^{n+1}=\\beta \\Delta t^2\\big (\\widetilde{F}^{n+1}-W^n\\big )+(\\mathbb {M}+\\gamma \\Delta t\\mathbb {B})\\widetilde{V}^n.$ It is known that in general, the Newmark's scheme is of second-order and unconditionally stable, if the parameter satisfy $\\gamma \\ge \\frac{1}{2}$ and $\\beta \\ge \\frac{1}{4}(\\frac{1}{2}+\\gamma )^2$ ." ], [ "Numerical experiments", "In this section, we shall validate the feasibility and accuracy of the methodology for the simulation of 3D spherical cloaks via some numerical experiments.", "In all the experiments, we set $\\varepsilon _0=\\mu _0=1$ , $c=1/\\sqrt{\\varepsilon _0\\mu _0}=1$ , $R_3=0.95$ , $b=1$ , $E=20$ , $N=20$ , $\\triangle t=1.0e-3$ .", "All VSH expansions are truncated at $L=40$ and the parameters in Newmark's time discretization are set to $\\gamma =0.5, \\beta =0.25$ ." ], [ "Monochromatic incident wave", "Set $D^{\\rm in}(r, t)=(1-e^{-10t})\\cos (kx-\\omega t)A,\\quad A:=\\begin{bmatrix}0 & 0 & A\\end{bmatrix}^{\\rm T},$ where $k=\\omega =40$ and $A=1$ .", "Note that it gets close to a monochromatic wave very quickly as $t$ increases, e.g.", "$t>3, (1-e^{-10t})\\ge 0.999999999999906$ due to the exponential term.", "The parameters of the cloaking device are set $\\omega _c=40$ , $\\gamma _1=\\gamma _2=0.001$ , $R_1=0.15$ , $R_2=0.35$ .", "We plot the contours of $D_z$ at different time in Figure REF .", "It shows that the spherical domain $|r|<R_1$ is perfectly cloaked from monochromatic wave with angular frequency $\\omega =40$ .", "No waves are propagating inside the cloaked region.", "Figure: t=11In the analysis of [23], a constraint $R_2\\ge 2R_1$ is assumed.", "It was pointed out that it was unsure if this constraint is necessary and no numerical results regarding the case $R_2<2R_1$ were presented therein.", "Here, we shall do some numerical test for the case $R_2<2R_1$ .", "For this purpose, we set $R_1=0.15$ , $R_2=0.25<2R_1$ and plot the contours of $D_z$ at different time in Figure REF .", "It shows that the cloak works as well as in the case $R_2>2R_1$ .", "Figure: t=11Figure: t=11As discussed in [49], the cloak is relatively sensitive to the frequency of the incident wave.", "Here we use the monochromatic incident wave (REF ) with $k=\\omega =38$ to test the cloak device with parameters: $\\omega _c=40$ , $\\gamma _1=\\gamma _2=0.001$ , $R_1=0.15$ , $R_2=0.35$ .", "The contours of the numerical $D_z$ at different time are plotted in Figure REF .", "The numerical results show that there are waves propagating inside the cloaking region." ], [ "Polychromatic incident wave", "In this example, we use a pulse of plane wave given by $D^{\\rm in}=\\mathfrak {Re}\\lbrace e^{{\\rm i}k(x-t)}\\rbrace e^{-\\frac{(x-t+t_c)^2}{q}}A,\\quad A:=\\begin{bmatrix}0 & 0 & A\\end{bmatrix}^{\\rm T},$ as the incident wave with $A=1$ , $k=40$ , $t_c=4$ and $q=0.5$ .", "We first consider the cloaking device with parameters given by $\\omega _c=40$ , $\\gamma _1=\\gamma _2=0.001$ , $R_1=0.15$ , $R_2=0.35>2R_1$ .", "The contours of $D_z$ at different time are plotted in Figure REF .", "Then, the outer radius of the cloak is set to $R_2=0.25<2R_1$ and other parameters remain unchanged.", "The contours of $D_z$ at different time are plotted in Figure REF .", "In these tests, there are polychromatic EM waves interacting with the cloaking devices.", "We can see from the numerical results that there are waves propagating inside the cloaked region.", "Figure: t=6.5Figure: t=6.5" ], [ "Conclusion", "In this paper, we proposed accurate algorithms for computing the involved temporal convolutions of the NRBCs for the time-dependent Maxwell's equations on a spherical artificial surface.", "More precisely, we provided the explicit formulas of the convolution kernel functions defined by inverse Laplace transforms of special modified Bessel functions, and also derived a new formulation of the NRBC capacity operator.", "With these at our proposal, the temporal convolutions in the NRBCs can be computed in a fast manner which therefore could offer an accurate way to reduce Maxwell's system in $\\mathbb {R}^3$ to a bounded domain.", "As a direct application of the truncated model, we considered the modelling and accurate simulation of the time-domain invisibility cloaks.", "We derived a new model valid for general cloaking geometry for the design of time-domain full wave invisibility cloaks involving just one unknown field $D$ and seemingly complicated convolution operators that could be evaluated recursively in time again.", "In this work, we focused on the spherical invisibility cloaks designed in the first, original of Pendry et al (cf.", "[33]).", "We proposed an efficient VSH-spectral-element method for numerical simulation.", "The resulted algorithm could produce accurate numerical solution with far less computation cost compared with the simulations based on FDTD in literature." ], [ "Acknowledgments", "The research of the first author is supported by NSFC (grant 11771137), the Construct Program of the Key Discipline in Hunan Province and a Scientific Research Fund of Hunan Provincial Education Department (No.", "16B154).", "The research of the third author is supported by the Ministry of Education, Singapore, under its MOE AcRF Tier 2 Grants (MOE2018-T2-1-059 and MOE2017-T2-2-144).", "The authors would like to thank Dr. Xiaodan Zhao at the National Heart Centre in Singapore for the initial exploration of this topic when she was a research associate in NTU." ], [ "Vector spherical harmonics", "We adopt the notation and setting as in Nédélec [28].", "The spherical coordinates $(r,\\theta ,\\varphi )$ are related to the Cartesian coordinates ${r}=(x,y,z)$ via $x=r\\sin \\theta \\cos \\varphi ,\\quad y=r\\sin \\theta \\sin \\varphi ,\\quad z=r\\cos \\theta ,$ where $r\\ge 0,\\theta \\in [0,\\pi ]$ and $\\phi \\in [0,2\\pi ).$ The corresponding moving (right-handed) orthonormal coordinate basis $\\lbrace {e}_r,{e}_\\theta , {e}_\\varphi \\rbrace $ is given by $\\begin{split}&{e}_r=\\hat{r}={r}/{r},\\;\\; {e}_\\theta =(\\cos \\theta \\cos \\varphi , \\ \\cos \\theta \\sin \\varphi , \\ -\\sin \\theta ),\\;\\; {e}_\\varphi = (-\\sin \\varphi , \\ \\cos \\varphi , \\ 0).\\end{split}$ Let $\\lbrace Y_{l}^m\\rbrace $ be the spherical harmonics as normalized in [28], and let $S$ be the unit sphere.", "Recall that $\\nabla _S Y_l^m =\\frac{\\partial Y_l^m}{\\partial \\theta }{e}_\\theta +\\frac{1}{\\sin \\theta }\\frac{\\partial Y_l^m}{\\partial \\varphi }{e}_\\varphi .$ The VSH family $\\big \\lbrace Y_l^m , \\Psi _{l}^m,\\Phi _l^m\\big \\rbrace :=\\big \\lbrace Y_l^m {e}_r, \\nabla _SY_{l}^m,\\nabla _S Y_l^m {e}_r\\big \\rbrace ,$ which has been used in the Spherepack [37] (also see [27]) forms a complete orthogonal basis of ${L}^2(S):=(L^2(S))^3$ under the inner product: $\\langle u, v\\rangle _S=\\int _S u\\cdot \\bar{v}\\, dS=\\int _0^{2\\pi }\\hspace*{-3.0pt} \\int _0^{\\pi } u\\cdot \\bar{v}\\, \\sin \\theta \\, d\\theta d\\varphi .$ Define the subspace of ${L}^2(S),$ consisting of the tangent components of the vector fields on $S$ : ${L}_T^2(S)= \\big \\lbrace u\\in {L}^2(S) : u\\cdot \\hat{x}=0\\big \\rbrace .$ The VSH $\\lbrace \\Psi _{l}^m, {\\Phi }_l^m\\rbrace $ forms a complete orthogonal basis of ${L}_T^2(S).$ Consequently, the vector field expanded in terms of VSH has a distinct separation of tangential and normal components.", "For any vector fields $u\\in {L}^2(S)$ , we write ${u}=u_{00}Y_0^0+\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\big \\lbrace u_{lm}^{r}\\, Y_l^m\\, +u_{lm}^{(1)}\\Psi _{l}^m+u_{lm}^{(2)}\\, {\\Phi }_l^m\\big \\rbrace ,$ where we denote $\\beta _l=l(l+1),$ and have $u_{00}=\\langle u, Y_0^0 \\rangle _S,\\;\\;\\;u_{lm}^r= \\langle u, Y_l^m \\rangle _S, \\;\\;\\; u_{lm}^{(1)}= {\\beta _l^{-1}}\\langle u, \\Psi _{l}^m \\rangle _S, \\;\\;\\; u_{lm}^{(2)}= {\\beta _l^{-1}}\\langle u, {\\Phi }_l^m\\rangle _S.$ It is noteworthy that given $u$ , we can be computed $\\lbrace u_{lm}^r, u_{lm}^{(1)}, u_{lm}^{(2)}\\rbrace $ via the discrete VSH-transform using the Spherepack [37], and vice versa by the inverse transform.", "Moreover, the normal component solely involves the first term while the tangential component $E_T$ of $E$ involves the last two terms in (REF ).", "Now, we collect some frequently used vector calculus formulas.", "Define the differential operators: $d_l^{\\pm }=\\frac{d}{dr}\\pm \\frac{l}{r}, \\quad \\hat{\\partial }_r=\\frac{d}{dr}+\\frac{1}{r}, \\quad {\\mathcal {L}}_l=\\hat{\\partial }_r^2-\\frac{\\beta _l}{r^2}=\\frac{d^2}{dr^2}+\\frac{2}{r}\\frac{d}{dr}-\\frac{\\beta _l}{r^2},$ where $\\beta _l:=l(l+1)$ .", "For any given $f(r)$ , the following properties can be derived from [15]: For divergence operator ${\\rm div}\\big (f Y_{l}^m\\big )=\\Big (\\frac{d}{dr}+\\frac{2}{r}\\Big ) f\\,Y_l^m,\\quad {\\rm div}\\big (f \\Psi _{l}^m\\big )=-\\beta _l\\frac{f}{r} \\,Y_l^m,\\quad {\\rm div}\\big (f \\Phi _{l}^m\\big )=0;$ For curl operator $\\nabla \\times \\big (f Y_l^m \\big )=\\frac{f}{r} \\,\\Phi _l^m,\\quad \\nabla \\times \\big (f \\Psi _{l}^m\\big )=-\\hat{\\partial }_r f \\,\\Phi _l^m, \\quad \\nabla \\times \\big (f \\Phi _l^m\\big )=\\hat{\\partial }_r f \\,\\Psi _{l}^m+\\beta _l\\frac{f}{r} Y_l^m;$ For Laplace operator $\\Delta \\big (f{\\Phi }_l^m\\big )={\\mathcal {L}}_{l}(f){\\Phi }_l^m.$" ], [ "Proof of Proposition ", "Recall that if ${\\rm div} {u}=0,$ then $ \\nabla \\times \\nabla \\times {u}=-\\Delta {u}.", "$ Thus, from (REF )-(REF ), we derive $&\\nabla \\times \\nabla \\times \\big ( u{\\Phi }_l^m\\big )=-\\Delta \\big ( u {\\Phi }_l^m\\big )=-\\mathcal {L}_l(u){\\Phi }_l^m,\\\\&\\nabla \\times \\nabla \\times \\nabla \\times \\big ( v {\\Phi }_l^m\\big )=-\\nabla \\times \\big ( \\Delta \\big ( v {\\Phi }_l^m\\big ) \\big )=-\\nabla \\times \\big ( \\mathcal {L}_l\\big (v\\big ){\\Phi }_l^m\\big ).$ Therefore, (REF ) can be reduced to: $\\frac{\\partial ^2u^i_{lm}}{\\partial t^2}-c^2\\mathcal {L}_l( u^i_{lm} )=f_{1,l}^{i,m}\\quad \\frac{\\partial ^2v^i_{lm}}{\\partial t^2}-c^2\\mathcal {L}_l( v^i_{lm} )=f_{2,l}^{i,m}, \\quad r\\in I_i,\\quad i=0, 2, 3,$ for $|m|\\le l$ , $l=1, 2, \\cdots $ , by using the expansions (REF ).", "In spherical coordinates (cf.", "[1]): $\\nabla \\times {v}= \\frac{1}{r\\sin \\theta }\\Big ( \\frac{\\partial \\big (\\sin \\theta v_\\varphi \\big )}{\\partial \\theta } -\\frac{\\partial v_\\theta }{\\partial \\varphi }\\Big ){e}_r+ \\frac{1}{r} \\Big (\\frac{1}{\\sin \\theta } \\frac{\\partial v_r}{\\partial \\varphi }-\\frac{\\partial \\big (r v_\\varphi \\big )}{\\partial r} \\Big ){e}_\\theta +\\frac{1}{r} \\Big (\\frac{\\partial \\big (r v_\\theta \\big )}{\\partial r}-\\frac{\\partial v_r}{\\partial \\theta } \\Big ){e}_\\varphi ,$ for any vector field $v=v_r{e}_r+v_{\\theta }{e}_\\theta +v_{\\varphi }{e}_\\varphi $ .", "Apparently, we have $\\nabla \\times (u_{00}^i(r, t)Y_0^0)=0,$ as $Y_0^0={e}_r/\\sqrt{4\\pi }$ .", "For the coefficient $u_{00}^i$ , we then have $\\frac{\\partial ^2u_{00}^i}{\\partial t^2}=f_{00}^i,\\quad r\\in I_i,\\quad i=0, 2, 3.$ We now turn to the governing equation (REF ) in the cloaking layer $I_1=(R_1,R_2)$ .", "According to (REF ), the vector spherical harmonic expansion of $D^1$ can be rewritten as ${D}^1=u_{00}^1Y_0^0+\\sum _{l=1}^\\infty \\sum _{|m|=0}^l \\Big \\lbrace u_{lm}^1\\Phi _l^m+ \\hat{\\partial }_r v_{lm}^1 \\Psi _{l}^m+\\frac{\\beta _l}{r}v_{lm}^1 Y_{l}^m \\Big \\rbrace .$ Using (REF ) and the fact that ${D}_1$ defined in (REF ) is uniaxial, we have ${D}_1[{D}^1]=\\big (u_{00}^1+\\theta _1\\ast u_{00}^1\\big )Y_{0}^0+\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\Big \\lbrace \\epsilon ^{-1}u_{lm}^1\\Phi _l^m+\\epsilon ^{-1}\\hat{\\partial }_r v_{lm}^1 \\Psi _{l}^m +\\frac{\\beta _l}{r}\\big (v_{lm}^1+\\theta _1\\ast v_{lm}^1\\big )Y_{l}^m\\Big \\rbrace .$ Using formula (REF ), we have $\\nabla \\times \\Big (\\big (u_{00}^1+\\theta _1\\ast u_{00}^1\\big )Y_{0}^0\\Big )=0.$ Then, we calculate from (REF ) that $\\begin{split}\\nabla \\times \\big ({D}_1[{D}^1 ] \\big )=&\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\Big (\\frac{\\beta _l}{r^2}\\big (v_{lm}^1+\\theta _1\\ast v_{lm}^1\\big )-\\epsilon ^{-1} \\hat{\\partial }_r^2 v_{lm}^1\\Big )\\Phi _l^m\\\\ &+\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\Big (\\epsilon ^{-1} \\hat{\\partial }_ru_{lm}^1 \\Psi _{l}^m+\\epsilon ^{-1}\\frac{\\beta _l}{r}u_{lm}^1Y_{l}^m\\Big )\\end{split}$ by using formulas (REF ).", "Repeating the above calculation and using the definition of $D_2$ and (REF ), we obtain $\\begin{split}\\nabla \\times \\big ( {D}_2&\\big [\\nabla \\times \\big ({D}_1[{D}^1] \\big )\\big ] \\big )=\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\epsilon ^{-1}\\Big ( \\frac{\\beta _l}{r^2}(u_{lm}^1+\\theta _2\\ast u_{lm}^1)-\\epsilon ^{-1}\\hat{\\partial }_r^2 u_{lm}^1 \\Big )\\Phi _l^m\\\\&+\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\epsilon ^{-1}\\nabla \\times \\Big (\\Big ( \\frac{\\beta _l}{r^2}(v_{lm}^1+\\theta _1\\ast v_{lm}^1)- \\epsilon ^{-1}\\hat{\\partial }_r^2 v_{lm}^1 \\Big )\\Phi _l^m \\Big ),\\end{split}$ Inserting the above equation into (REF ), one immediately shows that the expansion coefficients $\\lbrace u_{lm}^1$ , $v_{lm}^1\\rbrace , |m|\\le l, l=1, 2, \\cdots $ satisfy the same governing equation (REF ) with different convolution kernels $\\theta _2$ and $\\theta _1$ , respectively.", "As in (REF ), $u_{00}^1$ satisfies the same differential equation.", "According to (REF ) and (REF ) and the facts $\\Psi _l^m\\times e_r=\\Phi _l^m,\\quad \\Phi _l^m\\times e_r=-\\Psi _l^m,$ we have $\\begin{split}&{D}^i\\times e_r= \\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\big (-u_{lm}^i\\Psi _l^m+ \\hat{\\partial }_r v_{lm}^i \\Phi _{l}^m\\big ),\\quad i=0, 2,3,\\\\&(\\nabla \\times {D}^i)\\times e_r= \\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\Big \\lbrace \\Big (\\hat{\\partial }_r^2 v_{lm}^i- \\frac{\\beta _l}{r^2}v_{lm}^i\\Big )\\Psi _l^m+\\hat{\\partial }_ru_{lm}^i \\Phi _{l}^m\\Big \\rbrace ,\\quad i=0, 2,3,\\end{split}$ and $\\begin{split}&{D}_1[{D}^1]\\times e_r=\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\big (-\\epsilon ^{-1}u_{lm}^1\\Psi _l^m+\\epsilon ^{-1}\\hat{\\partial }_r v_{lm}^1 \\Phi _{l}^m\\big ),\\\\&\\big (\\nabla \\times ({D}_1[{D}^1])\\big )\\times e_r= \\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\Big \\lbrace \\Big (\\epsilon ^{-1} \\hat{\\partial }_r^2 v_{lm}^1- \\frac{\\beta _l}{r^2}\\big (v_{lm}^1+\\theta _1\\ast v_{lm}^1\\big )\\Big )\\Psi _l^m +\\epsilon ^{-1} \\hat{\\partial }_ru_{lm}^1 \\Phi _{l}^m\\Big \\rbrace .\\end{split}$ Substituting the above equations into jump condition (REF ) and (REF ), we obtain jump conditions $\\begin{split}\\epsilon u_{lm}^0=u_{lm}^1,\\quad \\partial _r v_{lm}^1=\\epsilon \\partial _r v_{lm}^0+(\\epsilon -1)r^{-1}v_{lm}^0 \\quad {\\rm at}\\;\\;\\; r=R_1,\\\\\\epsilon u_{lm}^2=u_{lm}^1,\\quad \\partial _r v_{lm}^1=\\epsilon \\partial _r v_{lm}^2+(\\epsilon -1)r^{-1}v_{lm}^2 \\quad {\\rm at}\\;\\;\\; r=R_2,\\end{split}$ and $\\partial _r u_{lm}^1=\\epsilon ^2 \\partial _r u_{lm}^0+\\epsilon (\\epsilon -1)r^{-1}u_{lm}^0 \\quad {\\rm at}\\;\\;\\; r=R_1, \\\\\\partial _r u_{lm}^1=\\epsilon ^2 \\partial _r u_{lm}^2+\\epsilon (\\epsilon -1)r^{-1}u_{lm}^2 \\quad {\\rm at}\\;\\;\\; r=R_2,\\\\\\hat{\\partial }_r^2 v_{lm}^0- \\frac{\\beta _l}{r^2}v_{lm}^0=\\epsilon ^{-2} \\hat{\\partial }_r^2 v_{lm}^1- \\frac{\\beta _l}{\\epsilon r^2}\\big (v_{lm}^1+\\theta _1\\ast v_{lm}^1\\big ) \\quad {\\rm at}\\;\\;\\; r=R_1, \\\\\\hat{\\partial }_r^2 v_{lm}^2- \\frac{\\beta _l}{r^2}v_{lm}^2=\\epsilon ^{-2} \\hat{\\partial }_r^2 v_{lm}^1- \\frac{\\beta _l}{\\epsilon r^2}\\big (v_{lm}^1+\\theta _1\\ast v_{lm}^1\\big ) \\quad {\\rm at}\\;\\;\\; r=R_2.$ Noting that $\\hat{\\partial }_r^2u=\\frac{1}{r^2}\\frac{\\partial }{\\partial r}\\big (r^2\\frac{\\partial u}{\\partial r}\\big )$ , the governing equations (REF )-(REF ) then gives $\\begin{split}\\epsilon ^{-2} \\hat{\\partial }_r^2 v_{lm}^1- \\frac{\\beta _l}{\\epsilon r^2}\\big (v_{lm}^1+\\theta _1\\ast v_{lm}^1\\big )=\\frac{1}{c^2}\\frac{\\partial ^2v^1_{lm}}{\\partial t^2},\\quad \\hat{\\partial }_r^2 v_{lm}^i- \\frac{\\beta _l}{r^2}v_{lm}^i=\\frac{1}{c^2}\\frac{\\partial ^2v^i_{lm}}{\\partial t^2},\\end{split}$ for $i=0, 2, 3.$ Substituting (REF ) into the jump conditions ()-() and integrate w.r.t.", "$t$ and using homogeneous initial conditions (REF ), we derive $v^0_{lm}=v^1_{lm}\\quad {\\rm at}\\;\\;\\; r=R_1; \\quad v^2_{lm}=v^1_{lm}\\quad {\\rm at}\\;\\;\\; r=R_2.$ Note that the jump conditions at artificial interface $r=R_3$ are trivial.", "Thus, we consider the boundary condition at $ r=b$ .", "Applying expansion (REF ) in (REF ) and using identities (REF ), (REF ) and formulation (REF ), we obtain $\\begin{split}\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\!\\Big (\\partial _t\\hat{\\partial }_r v_{lm}^3+c\\Big (\\hat{\\partial }_r^2 v_{lm}^3- \\frac{\\beta _l}{b^2}v_{lm}^3\\Big )-\\frac{c}{b^2}\\omega _l\\ast v_{lm}^{3}\\Big )\\Psi _l^m&\\\\+\\sum _{l=1}^\\infty \\sum _{|m|=0}^l\\!\\Big (\\partial _tu_{lm}^3+c\\hat{\\partial }_ru_{lm}^3 -\\frac{c}{b}\\sigma _l\\ast u_{lm}^{3}\\Big )\\Phi _{l}^m&=0,\\end{split}$ which implies two boundary conditions $\\frac{1}{c}\\partial _tu_{lm}^3+\\frac{{\\partial }u_{lm}^3}{\\partial r}+\\frac{1}{b}u_{lm}^3 -\\frac{1}{b}\\sigma _l\\ast u_{lm}^{3}=0\\quad {\\rm at}\\;\\; r=b,\\\\\\frac{\\partial }{\\partial r}\\frac{\\partial v_{lm}^3}{\\partial t}+\\frac{1}{b}\\frac{\\partial v_{lm}^3}{\\partial t}+c\\Big (\\hat{\\partial }_r^2 v_{lm}^3- \\frac{\\beta _l}{b^2}v_{lm}^3\\Big )-\\frac{c}{b^2}\\omega _l\\ast v_{lm}^{3}=0\\quad {\\rm at}\\;\\; r=b.$ Here, the definition of differential operator $\\hat{\\partial }$ in (REF ) is applied.", "Obviously, the boundary condition for $u_{lm}^3$ is exactly the one we adopted in the model problem (REF ).", "Next, we will show that the equation () can be reformulated to the same form as (REF ).", "Indeed, we can directly calculate $\\frac{c}{b^2}\\omega _l(t)\\ast v_{lm}^3(b, t)=\\frac{1}{b}\\bigg (\\int _0^t\\sigma _l^{\\prime }(t-\\tau )v_{lm}^3(b, \\tau )\\,d\\tau +\\sigma _l(0)v_{lm}^3(b, t)\\bigg )=\\frac{1}{b}\\partial _t(\\sigma _l\\ast v_{lm}^3(b,t)),$ by using the expression of $\\omega _l(t)$ (REF ).", "Using the above equation and (REF ) in () gives $\\frac{\\partial }{\\partial t}\\Big \\lbrace \\frac{ \\partial v_{lm}^3}{\\partial r}+\\frac{1}{b}v_{lm}^3+\\frac{1}{c}\\frac{\\partial v_{lm}^3}{\\partial t}-\\frac{1}{b}\\sigma _l\\ast v_{lm}^{3}\\Big \\rbrace =0\\quad {\\rm at}\\;\\; r=b.$ Consequently, we obtain boundary condition (REF ) by the zero initial data assumption.", "Note that the initial boundary value problems for coefficients $u_{lm}^i$ and $v_{lm}^i$ have almost the same form except the interface conditions (REF )-() and (REF ).", "Apparently, by introducing the variable substitution (REF ), $\\lbrace \\widetilde{u}_{lm}^i\\rbrace $ satisfy the same governing equation as $\\lbrace u_{lm}^i$ and the same interface and boundary conditions as $v_{lm}^i$ ." ] ]

[ [ "Towards Question-based Recommender Systems" ], [ "Abstract Conversational and question-based recommender systems have gained increasing attention in recent years, with users enabled to converse with the system and better control recommendations.", "Nevertheless, research in the field is still limited, compared to traditional recommender systems.", "In this work, we propose a novel Question-based recommendation method, Qrec, to assist users to find items interactively, by answering automatically constructed and algorithmically chosen questions.", "Previous conversational recommender systems ask users to express their preferences over items or item facets.", "Our model, instead, asks users to express their preferences over descriptive item features.", "The model is first trained offline by a novel matrix factorization algorithm, and then iteratively updates the user and item latent factors online by a closed-form solution based on the user answers.", "Meanwhile, our model infers the underlying user belief and preferences over items to learn an optimal question-asking strategy by using Generalized Binary Search, so as to ask a sequence of questions to the user.", "Our experimental results demonstrate that our proposed matrix factorization model outperforms the traditional Probabilistic Matrix Factorization model.", "Further, our proposed Qrec model can greatly improve the performance of state-of-the-art baselines, and it is also effective in the case of cold-start user and item recommendations." ], [ "Introduction", "Online shopping on Internet platforms, such as Amazon, and eBay, is increasingly prevalent, and helps customers make better purchase decisions [44].", "The high demand for online shopping calls for task-oriented conversational agents which can interact with customers helping them find items or services more effectively [34].", "This greatly stimulates related research on conversational and question-based recommender systems  [34], [41].", "Traditional recommender systems infer user preferences based on their historical behaviors, with the assumption that users have static preferences.", "Unfortunately, user preferences might evolve over time due to internal or external factors [31].", "Besides, the quality of traditional recommendations suffers greatly due to the sparsity of users' historical behaviors [33].", "Even worse, traditional recommendation systems fail to generate recommendations for new users or new items, for which the historical data is entirely missing: the cold-start problem [33].", "Compared to the traditional approaches, question-based and conversational recommender systems overcome these issues by placing the user in the recommendation loop  [34], [41].", "By iteratively asking questions and collecting feedback, more accurate recommendations can be generated for the user.", "Work on conversational and question-based recommenders [7], [34], [21], [41] demonstrates the importance of interactivity.", "[7] presented a recommender system, which elicits user preferences over items.", "[34] and  [21] train their models on a large number of natural language conversations, either on the basis of predefined and well-structured facets  [34] or based on free-style dialogues but require dialogues to mention items  [21].", "[41] proposed a unified paradigm for product search and recommendation, which constructs questions on extracted item aspects, and utilizes user reviews to extract values as simulated user answers.", "While these works have developed a successful direction towards conversational recommendation, research in the field is still limited.", "[7] collects user preferences over items, which is inefficient when the item pool is large and continuously updated.", "[34],  [41] and  [21] make certain assumptions over their input data, most importantly the availability of historical conversational data, or the availability of hierarchical item facets and facet-value pairs.", "In our work, we drop these assumptions: we only hypothesize that items can be discriminated based on textual information associated with them, e.g.", "descriptions and reviews  [46], [44].", "Our model asks questions based on extracted descriptive terms in the related contents, and beliefs are updated based on collaborative filtering, which is one of the most successful technologies in recommender systems  [13], [33].", "In this work, we propose a novel Question-based recommendation method, Qrec, to assist users to find items interactively Source code: https://github.com/JieZouIR/Qrec.", "Our proposed model follows the works by  [46] and  [44], and generates questions over extracted informative terms; a question pool is constructed by entities (informative terms) extracted from the item descriptions and reviews.", "proposes a novel matrix factorization method to initialize the user and item latent factors offline by using user-item ratings; develops a belief-updating method to track the user's belief (preferences over items), and uses GBS  [28] to select a sequence of questions based on the tracked user belief, aiming at learning to ask discriminative questions to gain new information about the user; asks questions, receives answers, updates the user and item latent factors online accordingly by incorporating feedback from the user based on our proposed matrix factorization algorithm, and also renews the user belief to select the next question to ask.", "generates a recommendation list based on the final user and item latent factors.", "Our model combines the advantages of collaborative filtering based on matrix factorization and content analysis by querying users about extracted informative terms.", "The matrix factorization model is able to utilize the rating data and discover latent correlation between items, while incorporating question-answering over content information, provides explicit content discrimination to assist the recommender systems.", "By iteratively asking questions over informative terms and collecting the immediate feedback from the user, our question-based recommender can track the shifted user preferences, clarify the user needs, and improve capturing the true underlying user latent factors and item latent factors.", "Besides, the information gathered from the user constitutes the new observations to overcome the sparsity and cold-start problem.", "The main contribution of this paper is three-fold: We propose a novel question-based recommendation method, Qrec, that interacts with users by soliciting their preferences on descriptive item characteristics.", "We propose a novel framework, that incorporates the online matrix factorization and online users' belief tracking for sequential question asking.", "We propose a novel matrix factorization method which can incorporate the offline training and efficient online updating of the user and item latent factors.", "To the best of our knowledge, this is the first work that incorporates online matrix factorization and question asking for item related features.", "The evaluation results show that our Qrec model achieves the highest performance compared to state-of-the-art baselines and our model is effective in both user and item cold-start recommendation scenarios.", "Related Work Recommender systems can be classified into three categories: content-based  [29], collaborative filtering  [17], [13], and hybrid [42] systems.", "Conversational and question-based recommender systems can extend recommender systems in any of the three categories.", "Early related attempts include the work by  [2], [3], [24], [35], [25], [8].", "More recently, different ways of feedback are introduced  [43], [23], [10], [7], [40], [32], [15], [39].", "[43] studied the problem of interactive collaborative filtering, and proposed methods to extend PMF  [27] using linear bandits to select the item to ask feedback for and incorporate the rating back to the PMF output.", "[23] focused on set-based feedback, while  [10] focused on choice-based feedback to learn the latent factors and perform interactive preference elicitation online.", "Contrary to these works that update the individual user's latent representation,  [7] proposed a method to update all user and item latent factor parameters of a PMF variant at every feedback cycle, obtaining absolute and pairwise user feedback on items.", "We refer the reader to  [12] and  [16] for a literature review of interactive recommendation.", "Compared with  [7], our model also updates all user and item latent factor parameters but based on our own matrix factorization model.", "Further, while  [7] elicit user ratings on items, our Qrec model asks questions about extracted descriptive terms of the items, and learns a strategy of asking sequential questions.", "Furthermore, the selection of questions in Qrec is adaptive to the change of user preferences, instead of relying on the distribution of the items [7].", "Last,  [7] focus on rating prediction while our work focus on the top-N recommendation.", "They use semi-synthetic data for which they need to obtain the ground truth of the user's preference to every item (like/dislike) using bootstrapping, and thus simulate user's answers for each question, which is not available in our case.", "Figure: Framework of our proposed question-based recommendation model, Qrec.", "Cotton is an extracted entity (informative term), 𝐔,𝐕,𝐩,𝐪\\mathbf {U}, \\mathbf {V}, \\mathbf {p}, \\mathbf {q} are model variables, and α\\alpha is a hyper-parameter of user belief.", "[41] designed a unified framework for product search and recommendation, and proposed a Personalized Multi-Memory Network (PMMN) architecture for conversational search and recommendation by asking questions over “aspects” extracted from user reviews by the means of sentiment labeling.", "Their model obtains the opinion of the user (i.e.", "value of the aspect-value pair) for the “aspect” as feedback.", "They utilize the user query as an initial query and use the aspect-value pairs of the conversation to expand the representation of the user's query, and thus to match the search and recommendation results.", "Different from this work which uses only the content of user reviews, we incorporate user ratings by collaborative filtering based on our proposed matrix factorization model.", "Besides, their work trains a model using the data for each user while our online question answering can work without these training data for cold start users and items.", "Moreover, they query the aspect-value pairs extracted from user review and choose questions based on the log-likelihood of probability estimation over aspects, while we ask questions about descriptive terms of items and select questions based on the user belief tracking and GBS.", "Reinforcement learning and deep learning on dialogue agents have also been studied for recommendations  [11], [6], [22], [4], [20].", "[34] proposed a deep reinforcement learning framework to build a conversational recommendation agent, which queries users on item facets and focuses on the long-term utility of success or conversion rate.", "[21] presented a publicly available dataset called ReDial, and explored a neural method based on dialogue for composing conversational recommendations.", "They try to predict user opinion over the mentioned items based on the dialogue and sentiment classification to generate a recommendation.", "On the basis of the ReDial dataset,  [4] proposed a knowledge-based recommender dialog system framework, which incorporates a recommender into a dialog system by using knowledge graphs and transformers.", "All the aforementioned works are trained on usage data (i.e.", "existing natural language conversations or interactions with the recommender system).", "[34] require a large number of repeated interactions between the users and the information seeking system to train upon, while  [21] and  [4] require mentioning items during the natural language dialogue.", "Such kind of data is not always available.", "Different from these works, our method does not require such data with large numbers of repeated interactions and mentioned items.", "Learning to ask is another recent and related field of study  [14], [37].", "[14] presented a policy-based reinforcement learning method to identify the optimal strategy of question selection by continuously learning the probability distribution over all the objects on a 20 Questions game setup.", "They regard the learned probability distribution on confidence as a state and select the next question according to this state.", "Different from our work, their work introduces data-hungry techniques, which require having large numbers of labeled data and repeated interactions from multiple users for a target item to train upon.", "A recent line of work that also involves learning to ask is the work in dialogue and information seeking conversational systems  [5], [1].", "For example,  [37] studied how to ask good questions in large-scale, open-domain conversational systems with neural question generation mechanisms.", "These models need to be trained on existing natural language conversations, which is different from our setup that depends on user ratings.", "[44] proposed an interactive sequential Bayesian model for product search.", "They learn to ask a good question by a cross-user duet training, which learns a belief over product relevance and the rewards over question performances.", "Different from their work which focuses on a sequential Bayesian product search model based on a cross-user duet training, our model incorporates the user feedback into a matrix factorization model for the recommendation.", "Further, they require the question answering history and purchase behavior from the same input query for their duet training, while our model does not require having such data.", "Methodology In this section, we discuss how we build our question-based recommender system.", "Our framework shown in Figure REF comprises of five modules: (1) an offline initialization module (Section REF ); (2) a continuous updating module (Section REF ); (3) a question learning module (Section REF ); (4) a question asking module (Section REF ); and (5) a recommendation module (Section REF ).", "Latent Factor Recommendation In this section, we describe two of the subcomponents of our Qrec model (shown in Figure REF ): the offline initialization module and the continuous updating module.", "Let $\\mathbf {R} \\in \\mathbb {R}^{N \\times M}$ be a user-item matrix, and $\\mathbf {R}_{i.", "}$ represents the $i$ -th row of $\\mathbf {R}$ , $\\mathbf {R}_{.j}$ represents the $j$ -th column of $\\mathbf {R}$ .", "Here $N$ and $M$ are the number of users and the number of items, respectively.", "Similarly, we use $\\mathbf {Y}_{i.", "}$ to represent the $i$ -th row of our online affinity matrix $\\mathbf {Y} \\in \\mathbb {R}^{N \\times M}$ , which is for incorporating user feedback (will be discussed later), use $\\mathbf {Y}_{.j}$ to represent the $j$ -th column of $\\mathbf {Y}$ .", "$\\mathbf {U} = [\\mathbf {u_1}, \\mathbf {u_2}, \\ldots , \\mathbf {u_i}, \\ldots , \\mathbf {u_N}]$ , $\\mathbf {V} = [\\mathbf {v_1}, \\mathbf {v_2}, \\ldots , \\mathbf {v_j}, \\ldots , \\mathbf {v_M}]$ , where $\\mathbf {u_i}$ , $\\mathbf {v_j}$ are user and item latent factors respectively.", "$\\mathbf {u_i}$ and $\\mathbf {v_j}$ are column vectors.", "Unless mentioned otherwise, all the vectors in this paper are column vectors.", "$\\mathcal {D}$ is the item collection represented by item documents (descriptions and reviews).", "Matrix factorization recommendation techniques have proven to be powerful tools to perform collaborative filtering in recommender systems [19].", "Assume we have $N$ users and $M$ items, matrix factorization decomposes a partially-observed matrix $\\mathbf {R} \\in \\mathbb {R}^{N \\times M}$ into two low-rank matrices, the user latent factors $\\mathbf {U} \\in \\mathbb {R}^{N \\times K}$ and the item factors $\\mathbf {V} \\in \\mathbb {R}^{M \\times K}$ where $K$ is the dimension of user and item latent factors.", "The prediction of the unobserved entries in $\\mathbf {R}$ is performed as a matrix completion, i.e.", "$\\mathbf {R} \\approx \\mathbf {UV}^\\top $ .", "Matrix factorization-based methods have been proposed and successfully applied to various recommendation tasks [19], [17], [7], [27].", "In matrix factorization, users and items are mapped to the same latent space.", "Items that have been co-liked by users will lie close in a low dimensional embedding space (latent vector).", "In this paper, we propose a novel model to perform the matrix factorization recommendation, and we refer to it as QMF.", "The generative process for our model is: For each user $i=1,\\ldots ,M$ , draw a user latent factor $\\mathbf {u_i}\\sim \\mathcal {N}(0,\\lambda _u^{-1}\\mathbf {I})$ ; For each item $j=1,\\ldots ,N$ , draw an item latent factor $\\mathbf {v_j} \\sim \\mathcal {N} (0, \\lambda _v^{-1}\\mathbf {I})$ .", "For each user-item pair $(i, j)\\in \\mathbf {R}$ , draw $R_{ij} \\sim \\mathcal {N}(\\mathbf {p}^\\intercal (\\mathbf {u_i}\\circ \\mathbf {v_j}),1)$ .", "In each user session targeting at a certain item, for each user-item pair $(i, j^{\\prime })\\in \\mathbf {Y}$ , draw $Y_{ij^{\\prime }}\\sim \\mathcal {N}(\\mathbf {q}^\\intercal (\\mathbf {u_i}\\circ \\mathbf {v_{j^{\\prime }}}),\\gamma ^{-1}\\mathbf {I})$ for each question asked.", "In the above, $\\lambda _u, \\lambda _v$ are the hyper-parameters modeling the variances in latent vectors, and $\\gamma $ is a hyper-parameters modeling the variance in $Y_{ij^{\\prime }}$ .", "$\\mathbf {p}$ and $\\mathbf {q}$ are the free parameters of column vector with $K$ dimension for $R_{ij}$ and $Y_{ij}$ , respectively.", "The intuition behind is that $\\mathbf {p}$ and $\\mathbf {q}$ can capture some general information across users and items.", "Optimization When optimizing our model, the maximization of posterior distributions over $\\mathbf {U}$ and $\\mathbf {V}$ can be formulated as follows according to the generative process: $\\small \\max \\limits _{\\mathbf {U},\\mathbf {V},\\mathbf {p},\\mathbf {q}} p (\\mathbf {U},\\mathbf {V},\\mathbf {p},\\mathbf {q} | \\mathbf {R}, \\mathbf {Y}, \\lambda _u, \\lambda _v, \\lambda _p, \\lambda _q, \\gamma ) .$ Then the maximization of the posterior probability can be reformulated as the minimization of its negative logarithm, which is $\\small \\begin{aligned}&&& -\\log p(\\mathbf {U},\\mathbf {V}\\mid \\mathbf {R},\\mathbf {Y},\\Theta ) \\\\& \\propto && \\frac{1}{2}\\sum _{i,j\\in \\mathbf {R}} \\left(R_{ij}-\\mathbf {p}^\\intercal (\\mathbf {u_i}\\circ \\mathbf {v_j})\\right)^2+\\frac{\\gamma }{2}\\sum _{i,j\\in \\mathbf {Y}} \\left(Y_{ij}-\\mathbf {q}^\\intercal (\\mathbf {u_i}\\circ \\mathbf {v_j})\\right)^2+\\\\&&& \\sum _{i=1}^M \\frac{\\lambda _u}{2}\\Vert \\mathbf {u_i}\\Vert _2^2+\\sum _{j=1}^N \\frac{\\lambda _v}{2}\\Vert \\mathbf {v_j}\\Vert _2^2+\\frac{\\lambda _p}{2}\\Vert \\mathbf {p}\\Vert _2^2+\\frac{\\lambda _q}{2}\\Vert \\mathbf {q}\\Vert _2^2,\\end{aligned}$ where $\\Theta =\\left\\lbrace \\mathbf {p},\\mathbf {q}\\right\\rbrace $ are the parameters, and $\\gamma $ is a trade-off of the online affinity $\\mathbf {Y}$ for incorporating the user feedback.", "Offline Optimization When optimizing offline by using the historical ratings of all users, we use gradient descent with Adaptive Moment Estimation (Adam) optimizer  [18] for Eq.", "(REF ), with $\\gamma $ set to 0, since we do not have the historical question asking data and thus do not have $Y_{ij}$ for the question asking.", "Therefore, we do not train $\\mathbf {q}$ , instead set $\\mathbf {q}$ to all-ones vector in this paper, but one can also train $\\mathbf {q}$ using historical question asking data.", "That is, the model variables $\\mathbf {U}, \\mathbf {V}, \\mathbf {p}$ are learned by maximizing the log-posterior over the user and item latent vectors with fixed hyper-parameters, given the training observations $\\mathbf {R}$ .", "Online Optimization Since we aim to recommend items online, it is necessary to update the variables effectively and efficiently according to the user feedback.", "Thus, we optimize Eq.", "(REF ) by ALS technique to update the model variables $\\mathbf {u_i}$ , and $\\mathbf {v_j}$ in order to guarantee efficiency.", "Then we have our following derived closed-form solution: $& \\mathbf {u_i} = \\left(\\mathbf {V_p}^\\intercal \\mathbf {V_p}+\\gamma \\mathbf {V_q}^\\intercal \\mathbf {V_q}+\\lambda _u \\mathbf {I}\\right)^{-1}(\\mathbf {V_p}^\\intercal \\mathbf {R}_i+\\gamma \\mathbf {V_q}^\\intercal \\mathbf {Y}_i)\\\\& \\mathbf {v_j} = \\left(\\mathbf {U_p}^\\intercal \\mathbf {U_p}+\\gamma \\mathbf {U_q}^\\intercal \\mathbf {U_q}+\\lambda _v \\mathbf {I}\\right)^{-1}(\\mathbf {U_p}^\\intercal \\mathbf {R}_{.j}+\\gamma \\mathbf {U_q}^\\intercal \\mathbf {Y}_{.j})$ where $\\small \\begin{split}\\mathbf {V_p}&=\\mathbf {V} \\text{diag}(\\mathbf {p}) , \\\\\\mathbf {V_q}&=\\mathbf {V} \\text{diag}(\\mathbf {q}) , \\\\\\mathbf {U_p}&=\\mathbf {U} \\text{diag}(\\mathbf {p}) , \\\\\\mathbf {U_q}&=\\mathbf {U} \\text{diag}(\\mathbf {q}).\\end{split}$ ALS repeatedly optimizes one of $\\mathbf {U}$ and $\\mathbf {V}$ while temporarily fixing the other to be constant.", "After each question being asked and feedback received, we update $\\mathbf {U}$ and $\\mathbf {V}$ .", "We assume that there is a target item related document $d^* \\in \\mathcal {D}$ and define an indicator vector $y^l_j$ for the $l$ -th question, with each dimension $j$ corresponding to an item in the collection: $y^l_j &= \\mathbb {1}\\lbrace e^{d_j}_l = e^{d^*}_l\\rbrace , \\\\Y_{j} &=\\sum _{t = 0}^{l-1} y^t_j ,$ where $e^{d_j}_l$ is true if the item related document ${d_j}$ contains the $l$ -th requested entity $e_l$ (see details for the question construction in Section REF ), and $\\mathbb {1}\\lbrace \\cdot \\rbrace $ is an indicator function.", "$e^{d^*}_l$ expresses whether the target item contains the $l$ -th requested entity $e_l$ .", "This also represents the answer by the user, given that the user's answers are driven by a target item.", "Hence, for example if the question is “Are you seeking for a [cotton] item?” and the target item description includes “cotton” as an entity, then $y^l_j$ is 1 for all items that also have “cotton” as an important entity.", "If the question is “Are you seeking for a [beach towel] item?” and the target product does not contain a “beach towel” in its description or reviews (hence the answer of the user is “no”) then $y^l_j$ is 1 for all the items that are not beach towels.", "$Y_j$ is the accumulated $y_j$ with the dimension corresponding to $j$ -th item until the $l$ -th question.", "Based on whether or not the target item is relevant to the requested entity, the feedback from user becomes a new or an updated observation for our system, and hence it is used to update $\\mathbf {Y}$ related to the particular user, i.e.", "$\\mathbf {Y}_i$ , which is a vector of the online affinity for user $i$ , with each of the dimension $Y_{ij}$ corresponding to $j$ -th item.", "Then $\\mathbf {u_i}$ , and all item factors $\\mathbf {V}$ are updated by Eq.", "(REF ) and Eq. ().", "Note that this observation only affects the current user's interaction session, and not any follow-up user interactions.", "As we ask about an entity $e$ and observe the user's response, the user's preference over the items which are consistent with the answer increases.", "The variance of the inferred noisy preferences over these items which is consistent with the answer as well as the variance of the nearby items in the learned embedding are reduced.", "The model's confidence in its belief over the user's preference on these items increases.", "As the system keeps asking questions to user $i$ and incorporates his/her responses, the latent user feature vectors $\\mathbf {U}$ and latent item feature vectors $\\mathbf {V}$ change and move towards the true underlying user and item latent vectors.", "After updating our matrix factorization model, we use the final user latent factor $\\mathbf {U}$ and item latent factor $\\mathbf {V}$ to computing $\\mathbf {UV}^\\top $ to yield a ranking of items to generate the recommendation list, which constitutes the recommendation module in Figure REF .", "Question Learning In this section, we describe how we select the next question to ask from the question pool (see Section REF for the question pool construction).", "After the offline initialization by using all of the historical ratings, the user initiates an interaction with our recommender system, our system asks a few questions to learn about the user latent factor, the item latent factor, and the user's belief.", "During this interactive phase, it is important to select the most informative questions that lead to learning effectively the user's preference, so as to minimize the number of questions asked and locate the target item effectively.", "Similar to  [38] and  [46], we use the estimated user preferences to help the question learning module to learn the most discriminative question to ask next.", "We model the user preferences for the items by a (multinomial) probability distribution $\\pi ^*$ over items $\\mathcal {D}$ , and the target item is drawn i.i.d.", "from this distribution.", "We also assume that there is a prior belief $\\mathbb {P}$ over the user preferences $\\pi ^*$ , which is a probability density function over all the possible realizations of $\\pi ^*$ .", "$\\small \\mathbb {P}_{l} = Dir(\\alpha + \\mathbf {Y}_i),$ where $\\mathbb {P}$ is a Dirichlet distribution with parameter $\\alpha $ .", "Having applied the offline initialization of our matrix factorization model, items can be scored and ranked for each user, the rank of each item expresses our initial belief on the preference of items for each given user.", "This initial belief will be used to initialize the hyper-parameter $\\alpha $ of the Dirichlet distribution.", "In particular, we set $\\alpha _i$ for item $i$ to $1/(p_i+1)$ , where $p_i$ is the index of item $i$ in the ranked list.", "$\\mathbf {Y}_i$ is the vector for the user $i$ with each dimension corresponding to accumulated $y^l_j$ until the $l$ -th question.", "Let $\\mathbb {P}_l$ be the system’s belief over $\\pi ^*$ prior to the $l$ -th question.", "We compute the user preferences $\\pi ^*_l(d)$ prior to the $l$ -th question by: $\\small \\pi ^*_l(d) = \\mathbb {E}_{\\pi \\sim \\mathbb {P}_l} [\\pi (d)]\\forall d \\in \\mathcal {D} .$ The $\\pi ^*$ is a multinomial distribution over items $\\mathcal {D}$ , and $\\mathbb {P}$ is modeled by the conjugate prior of the multinomial distribution, i.e.", "the Dirichlet distribution.", "From the properties of the Dirichlet distribution, the user preferences $\\pi ^*_l$ can be updated by counting and re-normalization of $\\alpha $ and $\\mathbf {Y}_i$ .", "As the system keeps asking questions to the user and incorporates his/her response, the predicted belief and preferences about the user is updated accordingly.", "This belief tracker thus specifies the direction for moving towards the true underlying belief distribution and true user preferences.", "This predicted user preferences will be used for guiding the question selection.", "Same to  [38] and  [46], we apply GBS to find the entity that best splits the probability mass of predicted user preferences closest to two halves for the remaining of the items during the $l$ -th question, as the nearly-optimal entity to ask.", "$\\small e_l = \\arg \\min _{e}\\Big |\\sum _{d \\in {\\mathcal {C}_l}} (2 \\mathbb {1}\\lbrace e^d = 1\\rbrace - 1)\\pi _l^*(d) \\Big |$ where $e_l$ is the $l$ -th chosen entity, ${\\mathcal {C}_l}$ is the candidate version space containing the set of remaining items when asking the $l$ -th question; the initial ${\\mathcal {C}_l}$ is equal to $\\mathcal {D}$ , $e^d$ expresses whether the item $d$ contains the entity $e$ or not.", "Specifically, for the entity embedding in this paper, the entity is represented by one-hot encoding, i.e.", "if the entity appears in a certain item documents, the value of the dimension corresponding to this item is 1 ($e^d=1$ ), otherwise the value of the dimension corresponding to this item is 0 ($e^d=0$ ).", "After each question is asked and the answer is obtained, the user preferences $\\pi ^*$ are updated by the belief tracker module.", "GBS tend to select entities by minimizing the objective function of Eq.", "(REF ).", "This means, GBS selects the entity which is able to split the sum of calculated user preferences corresponding to the item with $e^d=1$ and the sum of user preferences corresponding to the item with $e^d=0$ closest to two halves.", "Question Asking The proposed method of learning informative questions to ask to users, depends on the availability of a pool of questions regarding informative terms.", "Given an item, the user should be able to answer questions about this item with a “yes” or a “no”, having a reference to the relevant item (or item in mind).", "In this work, we use the approach taken by  [46], and [44] to extract meaningful short-phrases – typically entities – from the surface text to construct the question pool using the entity linking algorithm TAGME  [9].", "These entities are recognized to comprise the most important characteristics of an item  [46], [44], and we generate questions about the presence or absence of these entities in the item related documents.", "One could also use other sources like labelled topics, extracted keywords, item categories and attributes, to construct questions.", "In TAGME, each annotated short-phrase in unstructured text is weighted using a probability, that measures the reliability of that substring being a significant mention.", "Only the short-phrases with high probability should be considered as entities.", "In this paper, similar to  [9], and after a set of preliminary experiments, we set the threshold to 0.1 and filter out the short-phrases whose probability is below 0.1.", "Prior to this, we also removed stop words such as “about”, “as well” etc.. Having extracted the most important entities from the corpus, the proposed algorithm asks a sequence of questions in the form of “Are you seeking for a [entity] related item?” to locate the target item.", "In this case, the users can respond with a “yes”, a “no” or a “not sure” according to their belief.", "[tb] Inputinput $l \\leftarrow 0$ $\\mathbf {Y}_i \\leftarrow \\mathbf {0}$ Offline intialization of our matrix factorization model: $\\mathbf {U}, \\mathbf {V} = QMF({\\mathbf {R}}) $ ${Ranking}_l= Sort(\\mathbf {U} {\\mathbf {V}}^\\top )$ $\\alpha \\leftarrow {Ranking}_l$ $l < N_q$ and $| {\\mathcal {C}_l}| > 1 $ Compute the user belief with $\\alpha $ : $\\mathbb {P}_l = Dir(\\alpha + \\mathbf {Y}_i)$ Compute the user preferences with $\\mathbb {P}_{l}(\\pi )$ : $\\pi ^*_l(d) = \\mathbb {E}_{\\pi \\sim \\mathbb {P}_l}[\\pi (d)] ~ \\forall d \\in \\mathcal {D}$ Find the optimal target entity by question learning: $e_l = \\arg \\min _{e}\\Big |\\sum _{d \\in {\\mathcal {C}_l}} (2 \\mathbb {1}\\lbrace e^d = 1\\rbrace - 1)\\pi _l^*(d) \\Big |$ Ask the question about $e_l$ , observe the reply $e^{d^*}_l$ Remove $e_l$ from question pool ${\\mathcal {C}_{l+1}} = {\\mathcal {C}_l} \\cap {d \\in \\mathcal {D} : e^d_l=e^{d^*}_l}$ Update $\\mathbf {Y}_i$ by the reply $e^{d^*}_l$ according to Eq.", "(REF ) and Eq.", "() Update $\\mathbf {U},\\mathbf {V}$ by ALS according to Eq.", "(REF ) and Eq.", "() $l \\leftarrow l + 1$ Generate recommendation list by updated $\\mathbf {U},\\mathbf {V}$ : $result = Sort(\\mathbf {U}_{N_q}{\\mathbf {V}_{N_q}}^\\top )$ font=small The proposed Qrec algorithm Question-based Recommender System The algorithm of our question based recommender system is provided in Algorithm  REF .", "Our Qrec model performs two rounds: the offline phase and the online phase.", "The offline phase includes line 3-5, and the online phase includes line 6-17 in Algorithm  REF .", "During the offline phase, we firstly initialize our model parameters offline by using the history rating data across all users.", "We make the assumption that we have access to historical user-item interaction data (e.g., rating or purchasing data), even though our system can work without it as well.", "When a new user session starts, we use the initialized user's latent factors and items' latent factors to yield the preliminary ranking of candidate items.", "We then utilize this ranking score to initialize the Dirichlet prior parameter $\\alpha $ .", "When there is a new user session starts in online phase, we calculate the user belief with this $\\alpha $ and $\\mathbf {Y}_i$ .", "After that, we compute the user preferences with prior belief equal to $\\mathbb {P}_l$ , and find the optimal entity $e_l$ by GBS.", "We ask whether the entity $e_l$ is present in the target item that the user wants to find, $d^*$ , observe the reply $e^{d^*}_l$ , remove $e_l$ from the question pool, and update the candidate version space ${\\mathcal {C}_l}$ .", "Then we update $\\mathbf {Y}_i$ by the user response, and update the user latent factors $\\mathbf {U}$ and the item latent factors $\\mathbf {V}$ using ALS based on the updated $\\mathbf {Y}_i$ .", "After the online question asking phase is over, the recommendation list is generated by sorting the inner product of the last updated user latent factors $\\mathbf {U}_{N_q}$ and item latent factors $\\mathbf {V}_{N_q}$ .", "Experiments and Analysis Experimental Setup Dataset.", "In our experiments we use a collection of Amazon items http://jmcauley.ucsd.edu/data/amazon/  [26].", "Each item contains rich metadata such as title, descriptions, categories, and reviews from users as well.", "Following  [36] and  [44], we use four different product domains from the Amazon product dataset, but due to the limited space, we only report two domains in this paper, which are \"Home and Kitchen\", and \"Pet Supplies\" respectively.", "The documents associated with every item consist of the item description and the reviews provided by Amazon customers.", "On the two item domains, we use the same item list Product list: https://github.com/cvangysel/SERT/blob/master/PRODUCT_SEARCH.md with  [36], and filtered those items and users that appeared in less than five transactions to construct the user-item recommendation matrix like most of Collaborative Filtering papers  [13].", "We randomly split the entire dataset of user-item interactions to a training, validation and testing set with 80%, 10% and 10% split similar to other recommendation papers, e.g.", "[34].", "Statistics on the dataset are shown in Table  REF .", "Table: Statistics of the dataset.", "#entity is the number of unique entities.", "Parameter Setting To learn the matrix factorization embedding, we set the hyper-parameters to the combination that achieved the highest pairwise accuracy in the offline observations: the maximum training iterations of PMF and our matrix factorization model is set to 100, and $\\lambda _u = \\lambda _v = \\lambda _p = \\lambda _q = 0.1$ .", "The parameters $\\gamma $ , the dimension of the latent factors $K$ , and the number of questions asked $N_q$ are decided in [section:setup:rq1]RQ1.", "Evaluation Metrics We use average Recall at cut-off 5 (recall$@$ 5), Average Precision at 5 (AP$@$ 5), and Mean Reciprocal Rank (MRR) and Normalized Discounted Cumulative Gain (NDCG) as our evaluation metrics, which are commonly used metrics for capturing accuracy in recommendation  [7], [43], [41].", "NDCG is calculated by top 100 items like other paper  [36].", "The ground truth used to compute the aforementioned metrics is constructed by looking at the historical buying behavior of the user; an item is considered relevant if the user wrote a review and gave a rating to it, similar to other works  [41], [36].", "Baselines We compare our method with five baselines; the first two are static baselines, while the other three are interactive baselines.", "In particular the baselines are: (1) PMF, which is a typical, static recommendation approach; (2) NeuMF  [13], which is one of the state of the art approaches of collaborative filtering and widely used as the baseline by other papers.", "(3) QMF+Random, which uses our proposed matrix factorization for offline initialization and then randomly chooses a question from the question pool to ask; (4) SBS, which is the sequential Bayesian search algorithm.", "We applied the SBS  [38] to our recommendation task and uses the same question asking strategy with our Qrec model, but with the uniform prior; and (5) PMMN  [41], the Personalized Multi-Memory Network model, which is a state-of-the-art conversational recommender system asking questions on aspect-value pairs.", "For the PMF, QMF+Random, and SBS baselines, we use the same parameter setting with our Qrec model.", "For the NeuMF and PMMN, we use the optimal parameters reported in the corresponding paper and tuned their hyper-parameters in the same way as they reported.", "Simulating Users Our experimentation depends on users responding to questions asked by our method.", "In this paper we follow recent work  [41], [34], [46], [45], [44] and simulate users.", "We also conduct a small user study described next.", "During the simulation, we follow the approach proposed by  [46] and  [44], i.e.", "we assume that the user will respond to the questions with full knowledge of whether an entity is present or not in the target item.", "Hence, we assume that the user will respond with “yes” if an entity is contained in the target item documents and “no” if an entity is absent.", "This simulation also follows the one used by  [41], which assumes that the user has perfect knowledge of the value of an aspect for the target product.", "Online User Study To confirm some of the assumptions made in this work and test how well our recommender system works “in-situ” we also conduct a small online user study.", "The ideal users would be ones who have actually bought a number of items on an online shopping platform and now converse with our system embedded in the platform to find their next target item.", "In the absence of such a user base and commercial recommender system we use a crowdsourcing platform.", "First, we let the crowd worker select a product category she feels familiar with.", "Then, we randomly sample a product from our test data as a target product.", "To let the user familiarize herself with the target product we provide her with a product image, title, description, and the entities extracted from the product reviews.", "After the user indicates that she is familiar with the product and the conversation with the system can start, the information of the target item disappears from the screen and a question is selected by our algorithm to be asked to the user.", "The user needs to provide an answer to the question according to the information she read in the previous step, and then our system updates according to the user answer.", "With each question being answered, the user is shown a grid (4-by-4) of the pictures of sixteen top ranked items.", "The user can stop answering questions any time during her interaction with the system.", "When stoping the interaction with the system, users are asked a number of exit questions about their experiences with the system.", "Figure: The impact of the trade-off parameter γ\\gamma (top), and the dimension of the latent factors KK (bottom) on \"Home and Kitchen\" (left) and \"Pet Supplies\" (right) categories.", "Research Questions.", "Through the experiments in this work we aim to answer the following research questions: [topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ1 What is the impact of the trade-off $\\gamma $ , the dimension of the latent factors $K$ , and the number of questions asked $N_q$ ?", "[topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ2 How effective is Qrec compared to prior works?", "[topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ3 How effective is Qrec for the cold-start user and the cold-start item problem?", "[topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ4 Does the offline initialization help?", "[topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ5 Are the assumptions made in this work along with the effectiveness of our algorithm confirmed by a user study?", "Table: The comparison with PMF, NeuMF, QMF+Random, SBS, and PMMN on the \"Home and Kitchen\" (top) and the \"Pet Supplies\" (bottom) categories.", "#.", "represents the number of asked questions.", "QMF+Rand.", "represents the QMF+Randam model.", "Our proposed model achieve highest results when compared with interactive baselines, and our model performs better than the state of the art collaborative filtering model NeuMF on all of four different metrics with less than 5 questions.", "Impact of Parameters ([section:setup:rq1]RQ1) In [section:setup:rq1]RQ1, we examine the impact of the trade-off parameter $\\gamma $ , the dimension of the latent factors $K$ , and the number of questions asked $N_q$ over the effectiveness of our model.", "We compare the performance for different parameters.", "When the comparison for the given parameter, we fix the other two parameters.", "The performance evolution of different $\\gamma $ and different dimension of the latent factors $K$ on the two categories is shown in Figure  REF , and the results of different number of questions on the two categories can be seen in \"Qrec\" column of Table  REF .", "The $\\gamma $ ranges from 0 to 5 with a step of 0.5, and the $K$ ranges from 1 to 10 with a step of 1.", "As one can observe, with the increase of $\\gamma $ , the performance first improves and then drops.", "The best $\\gamma $ is 0.5 on the two categories.", "$\\gamma $ can control how much online user feedback is incorporated into the user latent factor and item latent factor.", "In particular, when $\\gamma $ is 0, i.e.", "the online updating do not take the user feedback (i.e.", "$\\mathbf {Y}$ ) into account, as expected the performance is very bad.", "As for the dimension of the latent factors $K$ , the overall performance trend also rises and then goes down with the increase of $K$ .", "This suggests that the dimension of the latent factors $K$ should not be set too high or too low.", "In this paper, we set it to the optimal value, i.e.", "3.", "Unless mentioned otherwise, in the rest of research questions, we use the optimal parameter $\\gamma $ , which is 0.5, and $K$ we used is the optimal value 3.", "To figure out the impact of the number of asked questions, we vary $N_q$ and see the performance shown in \"Qrec\" column of Table  REF .", "As shown in Table  REF , the performance of our Qrec model increases on all metrics with the increase of the number of questions, as expected.", "The more questions asked, the better the user needs are captured, and the closer the modeled user latent factor and item latent factor are to the true real-time user and item latent factors.", "Furthermore, the performance of Qrec reaches very good performance already, within the first 10 questions, while asking more than 10 questions does not add much regarding the performance.", "Performance Comparison ([section:setup:rq2]RQ2) To answer how effective is our proposed method compared to prior works, we compare our results with five baselines, PMF, NeuMF, QMF+Random, SBS, and PMMN.", "The results on the two categories are shown in Table  REF .", "From Table  REF , we can see that our proposed model, Qrec, achieves the highest results on all four metrics compared with the interactive baselines QMF+Random, SBS, and PMMN, on these two categories, which suggests that our question-based recommender system Qrec is effective.", "Our Qrec model performs better than QMF+Random, this suggests that our question selection is effective.", "There are few fluctuations on some metrics for QMF+Random with different number of questions asked, this is because the uncertainty of random question selection in different number of questions asked.", "Our Qrec model is superior to the SBS model, this suggests that using the prior from the offline initialization is beneficial.", "We will further discuss this in [section:setup:rq4]RQ4.", "Further, our Qrec model performs better than PMMN  [41], especially after 5 questions asked.", "This might be explained by the fact that asking questions on extracted entities can gather more information from users and is able to better learn user true preferences than asking questions on aspect-value pairs.", "Further, what we indeed observed is that the results of all four metrics regarding PMMN do not increase much and the result differences between PMMN and our Qrec become big when the number of questions is larger than 10.", "The reason for this is the fact that it is rather difficult to extract more than 10 aspect-value pairs from each user review for a certain item.", "As a consequence, there are no more available questions to ask, and thus the metric results never increase.", "Overall, this suggests that asking question on extracted entities is more effective.", "It also can be observed that our proposed matrix factorization model achieves better performance than PMF on the four metrics, this suggests that our proposed matrix factorization model is rather helpful.", "The reason might be because that adding the parameter $P$ improves the model capability of fitting.", "The NeuMF model outperforms linear models PMF and QMF, this is because the nonlinear deep neural model can obtain more subtle and better latent representations.", "But note that the stacked neural network structures also make them difficult to train and incur a high computational cost.", "Specifically, our model is able to achieve better results than the NeuMF model on all of four different metrics with less than 5 questions.", "With more questions being asked, the result differences between NeuMF and our Qrec become bigger.", "This shows that interactive or question-based recommendation can improve the performance over static models as interactive or question-based recommendation can continuously learn from the user.", "Table: The results on cold-start tuples.", "The top table represents the cold-start user tuples on \"Home and Kitchen\" and bottom table represents the cold-start item tuples on \"Pet Supplies\" category.", "Our Qrec model can still achieve high performance for cold start users and cold start items.Table: Results for the effects of offline initialization on the \"Home and Kitchen\" (top) and the \"Pet Supplies\" (bottom) categories.", "Qrec_offl.", "represents the Qrec including offline initialization, Qrec_rand.", "represents the Qrec with random initialization (i.e, excluding offline initialization).", "The Qrec including offline initialization is superior to the Qrec excluding offline initialization.", "Cold Start Performance Analysis ([section:setup:rq3]RQ3) To explore if our proposed method is effective for the cold-start user and the cold-start item problem or not, we extract cold-start user tuples (i.e.", "user-item interactions in which the user never appear in the training set) and cold-start item tuples (i.e.", "user-item interactions in which the item never appear in the training set) from our testing dataset.", "Because there are very few cold-start item tuples in \"Home and Kitchen\" category, and very few cold-start user tuples in \"Pet Supplies\" category, to the extent that results would not be reliable, we only use cold-start user tuples from the \"Home and Kitchen\" category and cold-start item tuples from the \"Pet Supplies\" category to validate the cold-start addressing ability of our model.", "Statistics on two categories shows that there are about 84% cold-start user tuples on the \"Home and Kitchen\" category and about 7% cold-start item tuples on the \"Pet Supplies\" category.", "The results on the two categories are shown in Table  REF .", "As it is observed, our Qrec model can still achieve high recall$@$ 5, AP$@$ 5, NDCG, and MRR for cold start users and cold start items.", "As it is known, PMF does not really work for cold start users and cold start items, which is indeed what we observe.", "We conclude that our Qrec model is capable of tackling the cold-start recommendation problem.", "Contribution of Offline Initialization ([section:setup:rq4]RQ4) In this research question, we investigate the effect of our offline initialization.", "We compare the performance results including the offline initialization and the performance results excluding the offline initialization of our model (i.e.", "random initialization for the model parameters when the new user session starts).", "Our hypothesis is that the offline learned parameters from the historical ratings capture some general trend and provide a generic prior to guide the model.", "Indeed, the results shown in Table  REF demonstrates the model with offline initialization achieves higher performance than the one without offline initialization, especially when the early stage of question asking (here: the number of asked questions is less than 10).", "Based on the observation of performance improvements when initializing the model from the offline data, we conclude that using offline initialization is highly beneficial.", "Online User Study ([section:setup:rq5]RQ5) In this research question we first want to examine the assumptions made in this work.", "In particular, we first want to understand how many questions actual users are willing to answer, how well do they answer them, and how is their experience with the system.", "We collected 489 conversations made between our system and 21 crowd workers on 33 target items.", "From the collected data, we observe that users answered an average number of 15 questions per target item in the system (with the median being 12).", "Further, in the exit questionnaire, 71.4% of the users declare that they are willing to answer between 10 and 20 questions.", "Despite a median time of 5 seconds to answer a question, in the exit questionnaire, 95.2% of the users indicate that the system's questions were easy to answer.", "From the results we collected, most of the users think the conversational system is helpful and they will use it in the future.", "In particular, 81% of users found the experience positive, 14.3% neutral, and 4.7% negative.", "Last but not least, the users provided the correct answers to the system's question 95% of the time, they were not sure about their answers 3.5% of the time, and they gave the wrong answers (i.e.", "their answers disagreed with the description of the product) 1.5% of the time.", "The second important question is how well the system performed.", "We measured performance after 5, 10, 15, and 20 queries asked (for those conversations that had this number of questions), as well as the performance when the user indicated that she wanted to stop.", "The results are shown in Table REF , and are in agreement with the Qrec results of Table REF .", "Conclusions and Discussion In this paper, we propose a novel question-based recommendation method, Qrec, which directly queries users on the automatically extracted entities in relevant documents.", "Our model is initialized offline by our proposed matrix factorization model QMF and updates the user and item latent factors online by incorporating the modeling of the user answer for the selected question.", "Meanwhile, our model tracks the user belief and learns a policy to select the best question sequence to ask.", "Experiments on the Amazon product dataset demonstrate that the effectiveness of the Qrec model compared to existing baselines.", "In this work, the questions asked to users are based on the presence or absence of entities in the target items, following past work.", "Richer type of questions could be constructed by using other sources such as categories, keywords, labelled topics  [47], [48], structural item properties, and domain-specific informative terms.", "Also, we ignore the fact that entities may be semantically related to the target item even though they are not contained lexically in the item documents.", "Further, we leave the number of questions asked as a parameter to be predefined instead of algorithmically decided.", "Our work uses a stand-alone algorithm that learns the informativeness of questions to ask based on GBS.", "One can also use other techniques (e.g., reinforcement learning) to learn the optimal question asking strategy, or incorporate more factors, e.g., the relatedness and importance level of different informative terms, to extend the work.", "Still, the user may change their target item during the interaction with the system  [30].", "Theoretically our method is able to deal with this kind of situation, with new answers received gradually for the new target item.", "Last, we conduct a small user study, however a larger and in-situ user study by intervening at the interface of a commercial recommender system would be more informative.", "We leave all these as future work.", "Table: System effectiveness on user study.", "Results are in agreement with the Qrec results of Table ." ], [ "Related Work", "Recommender systems can be classified into three categories: content-based  [29], collaborative filtering  [17], [13], and hybrid [42] systems.", "Conversational and question-based recommender systems can extend recommender systems in any of the three categories.", "Early related attempts include the work by  [2], [3], [24], [35], [25], [8].", "More recently, different ways of feedback are introduced  [43], [23], [10], [7], [40], [32], [15], [39].", "[43] studied the problem of interactive collaborative filtering, and proposed methods to extend PMF  [27] using linear bandits to select the item to ask feedback for and incorporate the rating back to the PMF output.", "[23] focused on set-based feedback, while  [10] focused on choice-based feedback to learn the latent factors and perform interactive preference elicitation online.", "Contrary to these works that update the individual user's latent representation,  [7] proposed a method to update all user and item latent factor parameters of a PMF variant at every feedback cycle, obtaining absolute and pairwise user feedback on items.", "We refer the reader to  [12] and  [16] for a literature review of interactive recommendation.", "Compared with  [7], our model also updates all user and item latent factor parameters but based on our own matrix factorization model.", "Further, while  [7] elicit user ratings on items, our Qrec model asks questions about extracted descriptive terms of the items, and learns a strategy of asking sequential questions.", "Furthermore, the selection of questions in Qrec is adaptive to the change of user preferences, instead of relying on the distribution of the items [7].", "Last,  [7] focus on rating prediction while our work focus on the top-N recommendation.", "They use semi-synthetic data for which they need to obtain the ground truth of the user's preference to every item (like/dislike) using bootstrapping, and thus simulate user's answers for each question, which is not available in our case.", "Figure: Framework of our proposed question-based recommendation model, Qrec.", "Cotton is an extracted entity (informative term), 𝐔,𝐕,𝐩,𝐪\\mathbf {U}, \\mathbf {V}, \\mathbf {p}, \\mathbf {q} are model variables, and α\\alpha is a hyper-parameter of user belief.", "[41] designed a unified framework for product search and recommendation, and proposed a Personalized Multi-Memory Network (PMMN) architecture for conversational search and recommendation by asking questions over “aspects” extracted from user reviews by the means of sentiment labeling.", "Their model obtains the opinion of the user (i.e.", "value of the aspect-value pair) for the “aspect” as feedback.", "They utilize the user query as an initial query and use the aspect-value pairs of the conversation to expand the representation of the user's query, and thus to match the search and recommendation results.", "Different from this work which uses only the content of user reviews, we incorporate user ratings by collaborative filtering based on our proposed matrix factorization model.", "Besides, their work trains a model using the data for each user while our online question answering can work without these training data for cold start users and items.", "Moreover, they query the aspect-value pairs extracted from user review and choose questions based on the log-likelihood of probability estimation over aspects, while we ask questions about descriptive terms of items and select questions based on the user belief tracking and GBS.", "Reinforcement learning and deep learning on dialogue agents have also been studied for recommendations  [11], [6], [22], [4], [20].", "[34] proposed a deep reinforcement learning framework to build a conversational recommendation agent, which queries users on item facets and focuses on the long-term utility of success or conversion rate.", "[21] presented a publicly available dataset called ReDial, and explored a neural method based on dialogue for composing conversational recommendations.", "They try to predict user opinion over the mentioned items based on the dialogue and sentiment classification to generate a recommendation.", "On the basis of the ReDial dataset,  [4] proposed a knowledge-based recommender dialog system framework, which incorporates a recommender into a dialog system by using knowledge graphs and transformers.", "All the aforementioned works are trained on usage data (i.e.", "existing natural language conversations or interactions with the recommender system).", "[34] require a large number of repeated interactions between the users and the information seeking system to train upon, while  [21] and  [4] require mentioning items during the natural language dialogue.", "Such kind of data is not always available.", "Different from these works, our method does not require such data with large numbers of repeated interactions and mentioned items.", "Learning to ask is another recent and related field of study  [14], [37].", "[14] presented a policy-based reinforcement learning method to identify the optimal strategy of question selection by continuously learning the probability distribution over all the objects on a 20 Questions game setup.", "They regard the learned probability distribution on confidence as a state and select the next question according to this state.", "Different from our work, their work introduces data-hungry techniques, which require having large numbers of labeled data and repeated interactions from multiple users for a target item to train upon.", "A recent line of work that also involves learning to ask is the work in dialogue and information seeking conversational systems  [5], [1].", "For example,  [37] studied how to ask good questions in large-scale, open-domain conversational systems with neural question generation mechanisms.", "These models need to be trained on existing natural language conversations, which is different from our setup that depends on user ratings.", "[44] proposed an interactive sequential Bayesian model for product search.", "They learn to ask a good question by a cross-user duet training, which learns a belief over product relevance and the rewards over question performances.", "Different from their work which focuses on a sequential Bayesian product search model based on a cross-user duet training, our model incorporates the user feedback into a matrix factorization model for the recommendation.", "Further, they require the question answering history and purchase behavior from the same input query for their duet training, while our model does not require having such data." ], [ "Methodology", "In this section, we discuss how we build our question-based recommender system.", "Our framework shown in Figure REF comprises of five modules: (1) an offline initialization module (Section REF ); (2) a continuous updating module (Section REF ); (3) a question learning module (Section REF ); (4) a question asking module (Section REF ); and (5) a recommendation module (Section REF )." ], [ "Latent Factor Recommendation", "In this section, we describe two of the subcomponents of our Qrec model (shown in Figure REF ): the offline initialization module and the continuous updating module.", "Let $\\mathbf {R} \\in \\mathbb {R}^{N \\times M}$ be a user-item matrix, and $\\mathbf {R}_{i.", "}$ represents the $i$ -th row of $\\mathbf {R}$ , $\\mathbf {R}_{.j}$ represents the $j$ -th column of $\\mathbf {R}$ .", "Here $N$ and $M$ are the number of users and the number of items, respectively.", "Similarly, we use $\\mathbf {Y}_{i.", "}$ to represent the $i$ -th row of our online affinity matrix $\\mathbf {Y} \\in \\mathbb {R}^{N \\times M}$ , which is for incorporating user feedback (will be discussed later), use $\\mathbf {Y}_{.j}$ to represent the $j$ -th column of $\\mathbf {Y}$ .", "$\\mathbf {U} = [\\mathbf {u_1}, \\mathbf {u_2}, \\ldots , \\mathbf {u_i}, \\ldots , \\mathbf {u_N}]$ , $\\mathbf {V} = [\\mathbf {v_1}, \\mathbf {v_2}, \\ldots , \\mathbf {v_j}, \\ldots , \\mathbf {v_M}]$ , where $\\mathbf {u_i}$ , $\\mathbf {v_j}$ are user and item latent factors respectively.", "$\\mathbf {u_i}$ and $\\mathbf {v_j}$ are column vectors.", "Unless mentioned otherwise, all the vectors in this paper are column vectors.", "$\\mathcal {D}$ is the item collection represented by item documents (descriptions and reviews).", "Matrix factorization recommendation techniques have proven to be powerful tools to perform collaborative filtering in recommender systems [19].", "Assume we have $N$ users and $M$ items, matrix factorization decomposes a partially-observed matrix $\\mathbf {R} \\in \\mathbb {R}^{N \\times M}$ into two low-rank matrices, the user latent factors $\\mathbf {U} \\in \\mathbb {R}^{N \\times K}$ and the item factors $\\mathbf {V} \\in \\mathbb {R}^{M \\times K}$ where $K$ is the dimension of user and item latent factors.", "The prediction of the unobserved entries in $\\mathbf {R}$ is performed as a matrix completion, i.e.", "$\\mathbf {R} \\approx \\mathbf {UV}^\\top $ .", "Matrix factorization-based methods have been proposed and successfully applied to various recommendation tasks [19], [17], [7], [27].", "In matrix factorization, users and items are mapped to the same latent space.", "Items that have been co-liked by users will lie close in a low dimensional embedding space (latent vector).", "In this paper, we propose a novel model to perform the matrix factorization recommendation, and we refer to it as QMF.", "The generative process for our model is: For each user $i=1,\\ldots ,M$ , draw a user latent factor $\\mathbf {u_i}\\sim \\mathcal {N}(0,\\lambda _u^{-1}\\mathbf {I})$ ; For each item $j=1,\\ldots ,N$ , draw an item latent factor $\\mathbf {v_j} \\sim \\mathcal {N} (0, \\lambda _v^{-1}\\mathbf {I})$ .", "For each user-item pair $(i, j)\\in \\mathbf {R}$ , draw $R_{ij} \\sim \\mathcal {N}(\\mathbf {p}^\\intercal (\\mathbf {u_i}\\circ \\mathbf {v_j}),1)$ .", "In each user session targeting at a certain item, for each user-item pair $(i, j^{\\prime })\\in \\mathbf {Y}$ , draw $Y_{ij^{\\prime }}\\sim \\mathcal {N}(\\mathbf {q}^\\intercal (\\mathbf {u_i}\\circ \\mathbf {v_{j^{\\prime }}}),\\gamma ^{-1}\\mathbf {I})$ for each question asked.", "In the above, $\\lambda _u, \\lambda _v$ are the hyper-parameters modeling the variances in latent vectors, and $\\gamma $ is a hyper-parameters modeling the variance in $Y_{ij^{\\prime }}$ .", "$\\mathbf {p}$ and $\\mathbf {q}$ are the free parameters of column vector with $K$ dimension for $R_{ij}$ and $Y_{ij}$ , respectively.", "The intuition behind is that $\\mathbf {p}$ and $\\mathbf {q}$ can capture some general information across users and items.", "When optimizing our model, the maximization of posterior distributions over $\\mathbf {U}$ and $\\mathbf {V}$ can be formulated as follows according to the generative process: $\\small \\max \\limits _{\\mathbf {U},\\mathbf {V},\\mathbf {p},\\mathbf {q}} p (\\mathbf {U},\\mathbf {V},\\mathbf {p},\\mathbf {q} | \\mathbf {R}, \\mathbf {Y}, \\lambda _u, \\lambda _v, \\lambda _p, \\lambda _q, \\gamma ) .$ Then the maximization of the posterior probability can be reformulated as the minimization of its negative logarithm, which is $\\small \\begin{aligned}&&& -\\log p(\\mathbf {U},\\mathbf {V}\\mid \\mathbf {R},\\mathbf {Y},\\Theta ) \\\\& \\propto && \\frac{1}{2}\\sum _{i,j\\in \\mathbf {R}} \\left(R_{ij}-\\mathbf {p}^\\intercal (\\mathbf {u_i}\\circ \\mathbf {v_j})\\right)^2+\\frac{\\gamma }{2}\\sum _{i,j\\in \\mathbf {Y}} \\left(Y_{ij}-\\mathbf {q}^\\intercal (\\mathbf {u_i}\\circ \\mathbf {v_j})\\right)^2+\\\\&&& \\sum _{i=1}^M \\frac{\\lambda _u}{2}\\Vert \\mathbf {u_i}\\Vert _2^2+\\sum _{j=1}^N \\frac{\\lambda _v}{2}\\Vert \\mathbf {v_j}\\Vert _2^2+\\frac{\\lambda _p}{2}\\Vert \\mathbf {p}\\Vert _2^2+\\frac{\\lambda _q}{2}\\Vert \\mathbf {q}\\Vert _2^2,\\end{aligned}$ where $\\Theta =\\left\\lbrace \\mathbf {p},\\mathbf {q}\\right\\rbrace $ are the parameters, and $\\gamma $ is a trade-off of the online affinity $\\mathbf {Y}$ for incorporating the user feedback.", "When optimizing offline by using the historical ratings of all users, we use gradient descent with Adaptive Moment Estimation (Adam) optimizer  [18] for Eq.", "(REF ), with $\\gamma $ set to 0, since we do not have the historical question asking data and thus do not have $Y_{ij}$ for the question asking.", "Therefore, we do not train $\\mathbf {q}$ , instead set $\\mathbf {q}$ to all-ones vector in this paper, but one can also train $\\mathbf {q}$ using historical question asking data.", "That is, the model variables $\\mathbf {U}, \\mathbf {V}, \\mathbf {p}$ are learned by maximizing the log-posterior over the user and item latent vectors with fixed hyper-parameters, given the training observations $\\mathbf {R}$ ." ], [ "Online Optimization", "Since we aim to recommend items online, it is necessary to update the variables effectively and efficiently according to the user feedback.", "Thus, we optimize Eq.", "(REF ) by ALS technique to update the model variables $\\mathbf {u_i}$ , and $\\mathbf {v_j}$ in order to guarantee efficiency.", "Then we have our following derived closed-form solution: $& \\mathbf {u_i} = \\left(\\mathbf {V_p}^\\intercal \\mathbf {V_p}+\\gamma \\mathbf {V_q}^\\intercal \\mathbf {V_q}+\\lambda _u \\mathbf {I}\\right)^{-1}(\\mathbf {V_p}^\\intercal \\mathbf {R}_i+\\gamma \\mathbf {V_q}^\\intercal \\mathbf {Y}_i)\\\\& \\mathbf {v_j} = \\left(\\mathbf {U_p}^\\intercal \\mathbf {U_p}+\\gamma \\mathbf {U_q}^\\intercal \\mathbf {U_q}+\\lambda _v \\mathbf {I}\\right)^{-1}(\\mathbf {U_p}^\\intercal \\mathbf {R}_{.j}+\\gamma \\mathbf {U_q}^\\intercal \\mathbf {Y}_{.j})$ where $\\small \\begin{split}\\mathbf {V_p}&=\\mathbf {V} \\text{diag}(\\mathbf {p}) , \\\\\\mathbf {V_q}&=\\mathbf {V} \\text{diag}(\\mathbf {q}) , \\\\\\mathbf {U_p}&=\\mathbf {U} \\text{diag}(\\mathbf {p}) , \\\\\\mathbf {U_q}&=\\mathbf {U} \\text{diag}(\\mathbf {q}).\\end{split}$ ALS repeatedly optimizes one of $\\mathbf {U}$ and $\\mathbf {V}$ while temporarily fixing the other to be constant.", "After each question being asked and feedback received, we update $\\mathbf {U}$ and $\\mathbf {V}$ .", "We assume that there is a target item related document $d^* \\in \\mathcal {D}$ and define an indicator vector $y^l_j$ for the $l$ -th question, with each dimension $j$ corresponding to an item in the collection: $y^l_j &= \\mathbb {1}\\lbrace e^{d_j}_l = e^{d^*}_l\\rbrace , \\\\Y_{j} &=\\sum _{t = 0}^{l-1} y^t_j ,$ where $e^{d_j}_l$ is true if the item related document ${d_j}$ contains the $l$ -th requested entity $e_l$ (see details for the question construction in Section REF ), and $\\mathbb {1}\\lbrace \\cdot \\rbrace $ is an indicator function.", "$e^{d^*}_l$ expresses whether the target item contains the $l$ -th requested entity $e_l$ .", "This also represents the answer by the user, given that the user's answers are driven by a target item.", "Hence, for example if the question is “Are you seeking for a [cotton] item?” and the target item description includes “cotton” as an entity, then $y^l_j$ is 1 for all items that also have “cotton” as an important entity.", "If the question is “Are you seeking for a [beach towel] item?” and the target product does not contain a “beach towel” in its description or reviews (hence the answer of the user is “no”) then $y^l_j$ is 1 for all the items that are not beach towels.", "$Y_j$ is the accumulated $y_j$ with the dimension corresponding to $j$ -th item until the $l$ -th question.", "Based on whether or not the target item is relevant to the requested entity, the feedback from user becomes a new or an updated observation for our system, and hence it is used to update $\\mathbf {Y}$ related to the particular user, i.e.", "$\\mathbf {Y}_i$ , which is a vector of the online affinity for user $i$ , with each of the dimension $Y_{ij}$ corresponding to $j$ -th item.", "Then $\\mathbf {u_i}$ , and all item factors $\\mathbf {V}$ are updated by Eq.", "(REF ) and Eq. ().", "Note that this observation only affects the current user's interaction session, and not any follow-up user interactions.", "As we ask about an entity $e$ and observe the user's response, the user's preference over the items which are consistent with the answer increases.", "The variance of the inferred noisy preferences over these items which is consistent with the answer as well as the variance of the nearby items in the learned embedding are reduced.", "The model's confidence in its belief over the user's preference on these items increases.", "As the system keeps asking questions to user $i$ and incorporates his/her responses, the latent user feature vectors $\\mathbf {U}$ and latent item feature vectors $\\mathbf {V}$ change and move towards the true underlying user and item latent vectors.", "After updating our matrix factorization model, we use the final user latent factor $\\mathbf {U}$ and item latent factor $\\mathbf {V}$ to computing $\\mathbf {UV}^\\top $ to yield a ranking of items to generate the recommendation list, which constitutes the recommendation module in Figure REF .", "In this section, we describe how we select the next question to ask from the question pool (see Section REF for the question pool construction).", "After the offline initialization by using all of the historical ratings, the user initiates an interaction with our recommender system, our system asks a few questions to learn about the user latent factor, the item latent factor, and the user's belief.", "During this interactive phase, it is important to select the most informative questions that lead to learning effectively the user's preference, so as to minimize the number of questions asked and locate the target item effectively.", "Similar to  [38] and  [46], we use the estimated user preferences to help the question learning module to learn the most discriminative question to ask next.", "We model the user preferences for the items by a (multinomial) probability distribution $\\pi ^*$ over items $\\mathcal {D}$ , and the target item is drawn i.i.d.", "from this distribution.", "We also assume that there is a prior belief $\\mathbb {P}$ over the user preferences $\\pi ^*$ , which is a probability density function over all the possible realizations of $\\pi ^*$ .", "$\\small \\mathbb {P}_{l} = Dir(\\alpha + \\mathbf {Y}_i),$ where $\\mathbb {P}$ is a Dirichlet distribution with parameter $\\alpha $ .", "Having applied the offline initialization of our matrix factorization model, items can be scored and ranked for each user, the rank of each item expresses our initial belief on the preference of items for each given user.", "This initial belief will be used to initialize the hyper-parameter $\\alpha $ of the Dirichlet distribution.", "In particular, we set $\\alpha _i$ for item $i$ to $1/(p_i+1)$ , where $p_i$ is the index of item $i$ in the ranked list.", "$\\mathbf {Y}_i$ is the vector for the user $i$ with each dimension corresponding to accumulated $y^l_j$ until the $l$ -th question.", "Let $\\mathbb {P}_l$ be the system’s belief over $\\pi ^*$ prior to the $l$ -th question.", "We compute the user preferences $\\pi ^*_l(d)$ prior to the $l$ -th question by: $\\small \\pi ^*_l(d) = \\mathbb {E}_{\\pi \\sim \\mathbb {P}_l} [\\pi (d)]\\forall d \\in \\mathcal {D} .$ The $\\pi ^*$ is a multinomial distribution over items $\\mathcal {D}$ , and $\\mathbb {P}$ is modeled by the conjugate prior of the multinomial distribution, i.e.", "the Dirichlet distribution.", "From the properties of the Dirichlet distribution, the user preferences $\\pi ^*_l$ can be updated by counting and re-normalization of $\\alpha $ and $\\mathbf {Y}_i$ .", "As the system keeps asking questions to the user and incorporates his/her response, the predicted belief and preferences about the user is updated accordingly.", "This belief tracker thus specifies the direction for moving towards the true underlying belief distribution and true user preferences.", "This predicted user preferences will be used for guiding the question selection.", "Same to  [38] and  [46], we apply GBS to find the entity that best splits the probability mass of predicted user preferences closest to two halves for the remaining of the items during the $l$ -th question, as the nearly-optimal entity to ask.", "$\\small e_l = \\arg \\min _{e}\\Big |\\sum _{d \\in {\\mathcal {C}_l}} (2 \\mathbb {1}\\lbrace e^d = 1\\rbrace - 1)\\pi _l^*(d) \\Big |$ where $e_l$ is the $l$ -th chosen entity, ${\\mathcal {C}_l}$ is the candidate version space containing the set of remaining items when asking the $l$ -th question; the initial ${\\mathcal {C}_l}$ is equal to $\\mathcal {D}$ , $e^d$ expresses whether the item $d$ contains the entity $e$ or not.", "Specifically, for the entity embedding in this paper, the entity is represented by one-hot encoding, i.e.", "if the entity appears in a certain item documents, the value of the dimension corresponding to this item is 1 ($e^d=1$ ), otherwise the value of the dimension corresponding to this item is 0 ($e^d=0$ ).", "After each question is asked and the answer is obtained, the user preferences $\\pi ^*$ are updated by the belief tracker module.", "GBS tend to select entities by minimizing the objective function of Eq.", "(REF ).", "This means, GBS selects the entity which is able to split the sum of calculated user preferences corresponding to the item with $e^d=1$ and the sum of user preferences corresponding to the item with $e^d=0$ closest to two halves." ], [ "Question Asking", "The proposed method of learning informative questions to ask to users, depends on the availability of a pool of questions regarding informative terms.", "Given an item, the user should be able to answer questions about this item with a “yes” or a “no”, having a reference to the relevant item (or item in mind).", "In this work, we use the approach taken by  [46], and [44] to extract meaningful short-phrases – typically entities – from the surface text to construct the question pool using the entity linking algorithm TAGME  [9].", "These entities are recognized to comprise the most important characteristics of an item  [46], [44], and we generate questions about the presence or absence of these entities in the item related documents.", "One could also use other sources like labelled topics, extracted keywords, item categories and attributes, to construct questions.", "In TAGME, each annotated short-phrase in unstructured text is weighted using a probability, that measures the reliability of that substring being a significant mention.", "Only the short-phrases with high probability should be considered as entities.", "In this paper, similar to  [9], and after a set of preliminary experiments, we set the threshold to 0.1 and filter out the short-phrases whose probability is below 0.1.", "Prior to this, we also removed stop words such as “about”, “as well” etc.. Having extracted the most important entities from the corpus, the proposed algorithm asks a sequence of questions in the form of “Are you seeking for a [entity] related item?” to locate the target item.", "In this case, the users can respond with a “yes”, a “no” or a “not sure” according to their belief.", "[tb] Inputinput $l \\leftarrow 0$ $\\mathbf {Y}_i \\leftarrow \\mathbf {0}$ Offline intialization of our matrix factorization model: $\\mathbf {U}, \\mathbf {V} = QMF({\\mathbf {R}}) $ ${Ranking}_l= Sort(\\mathbf {U} {\\mathbf {V}}^\\top )$ $\\alpha \\leftarrow {Ranking}_l$ $l < N_q$ and $| {\\mathcal {C}_l}| > 1 $ Compute the user belief with $\\alpha $ : $\\mathbb {P}_l = Dir(\\alpha + \\mathbf {Y}_i)$ Compute the user preferences with $\\mathbb {P}_{l}(\\pi )$ : $\\pi ^*_l(d) = \\mathbb {E}_{\\pi \\sim \\mathbb {P}_l}[\\pi (d)] ~ \\forall d \\in \\mathcal {D}$ Find the optimal target entity by question learning: $e_l = \\arg \\min _{e}\\Big |\\sum _{d \\in {\\mathcal {C}_l}} (2 \\mathbb {1}\\lbrace e^d = 1\\rbrace - 1)\\pi _l^*(d) \\Big |$ Ask the question about $e_l$ , observe the reply $e^{d^*}_l$ Remove $e_l$ from question pool ${\\mathcal {C}_{l+1}} = {\\mathcal {C}_l} \\cap {d \\in \\mathcal {D} : e^d_l=e^{d^*}_l}$ Update $\\mathbf {Y}_i$ by the reply $e^{d^*}_l$ according to Eq.", "(REF ) and Eq.", "() Update $\\mathbf {U},\\mathbf {V}$ by ALS according to Eq.", "(REF ) and Eq.", "() $l \\leftarrow l + 1$ Generate recommendation list by updated $\\mathbf {U},\\mathbf {V}$ : $result = Sort(\\mathbf {U}_{N_q}{\\mathbf {V}_{N_q}}^\\top )$ font=small The proposed Qrec algorithm" ], [ "Question-based Recommender System", "The algorithm of our question based recommender system is provided in Algorithm  REF .", "Our Qrec model performs two rounds: the offline phase and the online phase.", "The offline phase includes line 3-5, and the online phase includes line 6-17 in Algorithm  REF .", "During the offline phase, we firstly initialize our model parameters offline by using the history rating data across all users.", "We make the assumption that we have access to historical user-item interaction data (e.g., rating or purchasing data), even though our system can work without it as well.", "When a new user session starts, we use the initialized user's latent factors and items' latent factors to yield the preliminary ranking of candidate items.", "We then utilize this ranking score to initialize the Dirichlet prior parameter $\\alpha $ .", "When there is a new user session starts in online phase, we calculate the user belief with this $\\alpha $ and $\\mathbf {Y}_i$ .", "After that, we compute the user preferences with prior belief equal to $\\mathbb {P}_l$ , and find the optimal entity $e_l$ by GBS.", "We ask whether the entity $e_l$ is present in the target item that the user wants to find, $d^*$ , observe the reply $e^{d^*}_l$ , remove $e_l$ from the question pool, and update the candidate version space ${\\mathcal {C}_l}$ .", "Then we update $\\mathbf {Y}_i$ by the user response, and update the user latent factors $\\mathbf {U}$ and the item latent factors $\\mathbf {V}$ using ALS based on the updated $\\mathbf {Y}_i$ .", "After the online question asking phase is over, the recommendation list is generated by sorting the inner product of the last updated user latent factors $\\mathbf {U}_{N_q}$ and item latent factors $\\mathbf {V}_{N_q}$ ." ], [ "Dataset.", "In our experiments we use a collection of Amazon items http://jmcauley.ucsd.edu/data/amazon/  [26].", "Each item contains rich metadata such as title, descriptions, categories, and reviews from users as well.", "Following  [36] and  [44], we use four different product domains from the Amazon product dataset, but due to the limited space, we only report two domains in this paper, which are \"Home and Kitchen\", and \"Pet Supplies\" respectively.", "The documents associated with every item consist of the item description and the reviews provided by Amazon customers.", "On the two item domains, we use the same item list Product list: https://github.com/cvangysel/SERT/blob/master/PRODUCT_SEARCH.md with  [36], and filtered those items and users that appeared in less than five transactions to construct the user-item recommendation matrix like most of Collaborative Filtering papers  [13].", "We randomly split the entire dataset of user-item interactions to a training, validation and testing set with 80%, 10% and 10% split similar to other recommendation papers, e.g.", "[34].", "Statistics on the dataset are shown in Table  REF .", "Table: Statistics of the dataset.", "#entity is the number of unique entities." ], [ "Parameter Setting", "To learn the matrix factorization embedding, we set the hyper-parameters to the combination that achieved the highest pairwise accuracy in the offline observations: the maximum training iterations of PMF and our matrix factorization model is set to 100, and $\\lambda _u = \\lambda _v = \\lambda _p = \\lambda _q = 0.1$ .", "The parameters $\\gamma $ , the dimension of the latent factors $K$ , and the number of questions asked $N_q$ are decided in [section:setup:rq1]RQ1." ], [ "Evaluation Metrics", "We use average Recall at cut-off 5 (recall$@$ 5), Average Precision at 5 (AP$@$ 5), and Mean Reciprocal Rank (MRR) and Normalized Discounted Cumulative Gain (NDCG) as our evaluation metrics, which are commonly used metrics for capturing accuracy in recommendation  [7], [43], [41].", "NDCG is calculated by top 100 items like other paper  [36].", "The ground truth used to compute the aforementioned metrics is constructed by looking at the historical buying behavior of the user; an item is considered relevant if the user wrote a review and gave a rating to it, similar to other works  [41], [36]." ], [ "Baselines", "We compare our method with five baselines; the first two are static baselines, while the other three are interactive baselines.", "In particular the baselines are: (1) PMF, which is a typical, static recommendation approach; (2) NeuMF  [13], which is one of the state of the art approaches of collaborative filtering and widely used as the baseline by other papers.", "(3) QMF+Random, which uses our proposed matrix factorization for offline initialization and then randomly chooses a question from the question pool to ask; (4) SBS, which is the sequential Bayesian search algorithm.", "We applied the SBS  [38] to our recommendation task and uses the same question asking strategy with our Qrec model, but with the uniform prior; and (5) PMMN  [41], the Personalized Multi-Memory Network model, which is a state-of-the-art conversational recommender system asking questions on aspect-value pairs.", "For the PMF, QMF+Random, and SBS baselines, we use the same parameter setting with our Qrec model.", "For the NeuMF and PMMN, we use the optimal parameters reported in the corresponding paper and tuned their hyper-parameters in the same way as they reported." ], [ "Simulating Users", "Our experimentation depends on users responding to questions asked by our method.", "In this paper we follow recent work  [41], [34], [46], [45], [44] and simulate users.", "We also conduct a small user study described next.", "During the simulation, we follow the approach proposed by  [46] and  [44], i.e.", "we assume that the user will respond to the questions with full knowledge of whether an entity is present or not in the target item.", "Hence, we assume that the user will respond with “yes” if an entity is contained in the target item documents and “no” if an entity is absent.", "This simulation also follows the one used by  [41], which assumes that the user has perfect knowledge of the value of an aspect for the target product." ], [ "Online User Study", "To confirm some of the assumptions made in this work and test how well our recommender system works “in-situ” we also conduct a small online user study.", "The ideal users would be ones who have actually bought a number of items on an online shopping platform and now converse with our system embedded in the platform to find their next target item.", "In the absence of such a user base and commercial recommender system we use a crowdsourcing platform.", "First, we let the crowd worker select a product category she feels familiar with.", "Then, we randomly sample a product from our test data as a target product.", "To let the user familiarize herself with the target product we provide her with a product image, title, description, and the entities extracted from the product reviews.", "After the user indicates that she is familiar with the product and the conversation with the system can start, the information of the target item disappears from the screen and a question is selected by our algorithm to be asked to the user.", "The user needs to provide an answer to the question according to the information she read in the previous step, and then our system updates according to the user answer.", "With each question being answered, the user is shown a grid (4-by-4) of the pictures of sixteen top ranked items.", "The user can stop answering questions any time during her interaction with the system.", "When stoping the interaction with the system, users are asked a number of exit questions about their experiences with the system.", "Figure: The impact of the trade-off parameter γ\\gamma (top), and the dimension of the latent factors KK (bottom) on \"Home and Kitchen\" (left) and \"Pet Supplies\" (right) categories." ], [ "Research Questions.", "Through the experiments in this work we aim to answer the following research questions: [topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ1 What is the impact of the trade-off $\\gamma $ , the dimension of the latent factors $K$ , and the number of questions asked $N_q$ ?", "[topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ2 How effective is Qrec compared to prior works?", "[topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ3 How effective is Qrec for the cold-start user and the cold-start item problem?", "[topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ4 Does the offline initialization help?", "[topsep=0pt,nosep=0pt, leftmargin=0.75cm, noitemsep,nolistsep] RQ5 Are the assumptions made in this work along with the effectiveness of our algorithm confirmed by a user study?", "Table: The comparison with PMF, NeuMF, QMF+Random, SBS, and PMMN on the \"Home and Kitchen\" (top) and the \"Pet Supplies\" (bottom) categories.", "#.", "represents the number of asked questions.", "QMF+Rand.", "represents the QMF+Randam model.", "Our proposed model achieve highest results when compared with interactive baselines, and our model performs better than the state of the art collaborative filtering model NeuMF on all of four different metrics with less than 5 questions." ], [ "Impact of Parameters (", "In [section:setup:rq1]RQ1, we examine the impact of the trade-off parameter $\\gamma $ , the dimension of the latent factors $K$ , and the number of questions asked $N_q$ over the effectiveness of our model.", "We compare the performance for different parameters.", "When the comparison for the given parameter, we fix the other two parameters.", "The performance evolution of different $\\gamma $ and different dimension of the latent factors $K$ on the two categories is shown in Figure  REF , and the results of different number of questions on the two categories can be seen in \"Qrec\" column of Table  REF .", "The $\\gamma $ ranges from 0 to 5 with a step of 0.5, and the $K$ ranges from 1 to 10 with a step of 1.", "As one can observe, with the increase of $\\gamma $ , the performance first improves and then drops.", "The best $\\gamma $ is 0.5 on the two categories.", "$\\gamma $ can control how much online user feedback is incorporated into the user latent factor and item latent factor.", "In particular, when $\\gamma $ is 0, i.e.", "the online updating do not take the user feedback (i.e.", "$\\mathbf {Y}$ ) into account, as expected the performance is very bad.", "As for the dimension of the latent factors $K$ , the overall performance trend also rises and then goes down with the increase of $K$ .", "This suggests that the dimension of the latent factors $K$ should not be set too high or too low.", "In this paper, we set it to the optimal value, i.e.", "3.", "Unless mentioned otherwise, in the rest of research questions, we use the optimal parameter $\\gamma $ , which is 0.5, and $K$ we used is the optimal value 3.", "To figure out the impact of the number of asked questions, we vary $N_q$ and see the performance shown in \"Qrec\" column of Table  REF .", "As shown in Table  REF , the performance of our Qrec model increases on all metrics with the increase of the number of questions, as expected.", "The more questions asked, the better the user needs are captured, and the closer the modeled user latent factor and item latent factor are to the true real-time user and item latent factors.", "Furthermore, the performance of Qrec reaches very good performance already, within the first 10 questions, while asking more than 10 questions does not add much regarding the performance." ], [ "Performance Comparison (", "To answer how effective is our proposed method compared to prior works, we compare our results with five baselines, PMF, NeuMF, QMF+Random, SBS, and PMMN.", "The results on the two categories are shown in Table  REF .", "From Table  REF , we can see that our proposed model, Qrec, achieves the highest results on all four metrics compared with the interactive baselines QMF+Random, SBS, and PMMN, on these two categories, which suggests that our question-based recommender system Qrec is effective.", "Our Qrec model performs better than QMF+Random, this suggests that our question selection is effective.", "There are few fluctuations on some metrics for QMF+Random with different number of questions asked, this is because the uncertainty of random question selection in different number of questions asked.", "Our Qrec model is superior to the SBS model, this suggests that using the prior from the offline initialization is beneficial.", "We will further discuss this in [section:setup:rq4]RQ4.", "Further, our Qrec model performs better than PMMN  [41], especially after 5 questions asked.", "This might be explained by the fact that asking questions on extracted entities can gather more information from users and is able to better learn user true preferences than asking questions on aspect-value pairs.", "Further, what we indeed observed is that the results of all four metrics regarding PMMN do not increase much and the result differences between PMMN and our Qrec become big when the number of questions is larger than 10.", "The reason for this is the fact that it is rather difficult to extract more than 10 aspect-value pairs from each user review for a certain item.", "As a consequence, there are no more available questions to ask, and thus the metric results never increase.", "Overall, this suggests that asking question on extracted entities is more effective.", "It also can be observed that our proposed matrix factorization model achieves better performance than PMF on the four metrics, this suggests that our proposed matrix factorization model is rather helpful.", "The reason might be because that adding the parameter $P$ improves the model capability of fitting.", "The NeuMF model outperforms linear models PMF and QMF, this is because the nonlinear deep neural model can obtain more subtle and better latent representations.", "But note that the stacked neural network structures also make them difficult to train and incur a high computational cost.", "Specifically, our model is able to achieve better results than the NeuMF model on all of four different metrics with less than 5 questions.", "With more questions being asked, the result differences between NeuMF and our Qrec become bigger.", "This shows that interactive or question-based recommendation can improve the performance over static models as interactive or question-based recommendation can continuously learn from the user.", "Table: The results on cold-start tuples.", "The top table represents the cold-start user tuples on \"Home and Kitchen\" and bottom table represents the cold-start item tuples on \"Pet Supplies\" category.", "Our Qrec model can still achieve high performance for cold start users and cold start items.Table: Results for the effects of offline initialization on the \"Home and Kitchen\" (top) and the \"Pet Supplies\" (bottom) categories.", "Qrec_offl.", "represents the Qrec including offline initialization, Qrec_rand.", "represents the Qrec with random initialization (i.e, excluding offline initialization).", "The Qrec including offline initialization is superior to the Qrec excluding offline initialization." ], [ "Cold Start Performance Analysis (", "To explore if our proposed method is effective for the cold-start user and the cold-start item problem or not, we extract cold-start user tuples (i.e.", "user-item interactions in which the user never appear in the training set) and cold-start item tuples (i.e.", "user-item interactions in which the item never appear in the training set) from our testing dataset.", "Because there are very few cold-start item tuples in \"Home and Kitchen\" category, and very few cold-start user tuples in \"Pet Supplies\" category, to the extent that results would not be reliable, we only use cold-start user tuples from the \"Home and Kitchen\" category and cold-start item tuples from the \"Pet Supplies\" category to validate the cold-start addressing ability of our model.", "Statistics on two categories shows that there are about 84% cold-start user tuples on the \"Home and Kitchen\" category and about 7% cold-start item tuples on the \"Pet Supplies\" category.", "The results on the two categories are shown in Table  REF .", "As it is observed, our Qrec model can still achieve high recall$@$ 5, AP$@$ 5, NDCG, and MRR for cold start users and cold start items.", "As it is known, PMF does not really work for cold start users and cold start items, which is indeed what we observe.", "We conclude that our Qrec model is capable of tackling the cold-start recommendation problem." ], [ "Contribution of Offline Initialization (", "In this research question, we investigate the effect of our offline initialization.", "We compare the performance results including the offline initialization and the performance results excluding the offline initialization of our model (i.e.", "random initialization for the model parameters when the new user session starts).", "Our hypothesis is that the offline learned parameters from the historical ratings capture some general trend and provide a generic prior to guide the model.", "Indeed, the results shown in Table  REF demonstrates the model with offline initialization achieves higher performance than the one without offline initialization, especially when the early stage of question asking (here: the number of asked questions is less than 10).", "Based on the observation of performance improvements when initializing the model from the offline data, we conclude that using offline initialization is highly beneficial." ], [ "Online User Study (", "In this research question we first want to examine the assumptions made in this work.", "In particular, we first want to understand how many questions actual users are willing to answer, how well do they answer them, and how is their experience with the system.", "We collected 489 conversations made between our system and 21 crowd workers on 33 target items.", "From the collected data, we observe that users answered an average number of 15 questions per target item in the system (with the median being 12).", "Further, in the exit questionnaire, 71.4% of the users declare that they are willing to answer between 10 and 20 questions.", "Despite a median time of 5 seconds to answer a question, in the exit questionnaire, 95.2% of the users indicate that the system's questions were easy to answer.", "From the results we collected, most of the users think the conversational system is helpful and they will use it in the future.", "In particular, 81% of users found the experience positive, 14.3% neutral, and 4.7% negative.", "Last but not least, the users provided the correct answers to the system's question 95% of the time, they were not sure about their answers 3.5% of the time, and they gave the wrong answers (i.e.", "their answers disagreed with the description of the product) 1.5% of the time.", "The second important question is how well the system performed.", "We measured performance after 5, 10, 15, and 20 queries asked (for those conversations that had this number of questions), as well as the performance when the user indicated that she wanted to stop.", "The results are shown in Table REF , and are in agreement with the Qrec results of Table REF ." ], [ "Conclusions and Discussion", "In this paper, we propose a novel question-based recommendation method, Qrec, which directly queries users on the automatically extracted entities in relevant documents.", "Our model is initialized offline by our proposed matrix factorization model QMF and updates the user and item latent factors online by incorporating the modeling of the user answer for the selected question.", "Meanwhile, our model tracks the user belief and learns a policy to select the best question sequence to ask.", "Experiments on the Amazon product dataset demonstrate that the effectiveness of the Qrec model compared to existing baselines.", "In this work, the questions asked to users are based on the presence or absence of entities in the target items, following past work.", "Richer type of questions could be constructed by using other sources such as categories, keywords, labelled topics  [47], [48], structural item properties, and domain-specific informative terms.", "Also, we ignore the fact that entities may be semantically related to the target item even though they are not contained lexically in the item documents.", "Further, we leave the number of questions asked as a parameter to be predefined instead of algorithmically decided.", "Our work uses a stand-alone algorithm that learns the informativeness of questions to ask based on GBS.", "One can also use other techniques (e.g., reinforcement learning) to learn the optimal question asking strategy, or incorporate more factors, e.g., the relatedness and importance level of different informative terms, to extend the work.", "Still, the user may change their target item during the interaction with the system  [30].", "Theoretically our method is able to deal with this kind of situation, with new answers received gradually for the new target item.", "Last, we conduct a small user study, however a larger and in-situ user study by intervening at the interface of a commercial recommender system would be more informative.", "We leave all these as future work.", "Table: System effectiveness on user study.", "Results are in agreement with the Qrec results of Table ." ] ]

[ [ "Joint Total Variation ESTATICS for Robust Multi-Parameter Mapping" ], [ "Abstract Quantitative magnetic resonance imaging (qMRI) derives tissue-specific parameters -- such as the apparent transverse relaxation rate R2*, the longitudinal relaxation rate R1 and the magnetisation transfer saturation -- that can be compared across sites and scanners and carry important information about the underlying microstructure.", "The multi-parameter mapping (MPM) protocol takes advantage of multi-echo acquisitions with variable flip angles to extract these parameters in a clinically acceptable scan time.", "In this context, ESTATICS performs a joint loglinear fit of multiple echo series to extract R2* and multiple extrapolated intercepts, thereby improving robustness to motion and decreasing the variance of the estimators.", "In this paper, we extend this model in two ways: (1) by introducing a joint total variation (JTV) prior on the intercepts and decay, and (2) by deriving a nonlinear maximum \\emph{a posteriori} estimate.", "We evaluated the proposed algorithm by predicting left-out echoes in a rich single-subject dataset.", "In this validation, we outperformed other state-of-the-art methods and additionally showed that the proposed approach greatly reduces the variance of the estimated maps, without introducing bias." ], [ "Introduction", "The magnetic resonance imaging (MRI) signal is governed by a number of tissue-specific parameters.", "While many common MR sequences only aim to maximise the contrast between tissues of interest, the field of quantitative MRI (qMRI) is concerned with the extraction of the original parameters [30].", "This interest stems from the fundamental relationship that exists between the magnetic parameters and the tissue microstructure: the longitudinal relaxation rate $R_1=1/T_1$ is sensitive to myelin content [28], [10], [27]; the apparent transverse relaxation rate $R_2^\\star =1/T_2^\\star $ can be used to probe iron content [22], [21], [12]; the magnetization-transfer saturation (MTsat) indicates the proportion of protons bound to macromolecules (in contrast to free water) and offers another metric to investigate myelin loss [31], [14].", "Furthermore, qMRI allows many of the scanner- and centre-specific effects to be factored out, making measures more comparable across sites [29], [9], [3], [33].", "In this context, the multi-parameter mapping (MPM) protocol was developed at 3 Tesla to allow the quantification of $R_1$ , $R_2^\\star $ , MTsat and the proton density (PD) at high resolutions (0.8 or 1 mm) and in a clinically acceptable scan time of 25 mins [14], [33].", "However, to reach these values, compromises must be made so that the signal-to-noise ratio (SNR) suffers, making the parameter maps noisy; Papp et al.", "[23] found a scan-rescan root mean squared error of about 7.5% for $R_1$ at 1mm, in the absence of inter-scan movement.", "Smoothing can be used to improve SNR, but at the cost of lower spatial specificity.", "Denoising methods aim to separate signal from noise.", "They take advantage of the fact that signal and noise have intrinsically different spatial profiles: the noise is spatially independent and often has a characteristic distribution while the signal is highly structured.", "Denoising methods originate from partial differential equations, adaptive filtering, variational optimisation or Markov random fields, and many connections exist between them.", "Two main families emerge: Optimisation of an energy: $\\textstyle \\hat{Y} = \\operatornamewithlimits{\\arg \\!\\min }_Y \\mathcal {E}_1\\left(X - \\mathcal {A}(Y)\\right) + \\mathcal {E}_2\\left(\\mathcal {G}(Y)\\right),$ where $X$ is the observed data, $Y$ is the unknown noise-free data, $\\mathcal {A}$ is an arbitrary forward transformation (e.g., spatial transformation, downsampling, smoothing) mapping from the reconstructed to the observed data and $\\mathcal {G}$ is a linear transformation (e.g., spatial gradients, Fourier transform, wavelet transform) that extracts features of interest from the reconstruction.", "Application of an adaptive nonlocal filter: $\\textstyle \\hat{Y}_i = \\sum _{j\\in \\mathcal {N}_i} w\\left(\\mathcal {P}_i(X),\\mathcal {P}_j(X)\\right) X_j,$ where the reconstruction of a given voxel $i$ is a weighted average all observed voxels $j$ in a given (possibly infinite) neighbourhood $\\mathcal {N}_i$ , with weights reflecting similarity between patches centred about these voxels.", "For the first family of methods, it was found that the denoising effect is stronger when $\\mathcal {E}_2$ is an absolute norm (or sum of), rather than a squared norm, because the solution is implicitly sparse in the feature domain [2].", "This family of methods include total variation (TV) regularisation [25] and wavelet soft-thresholding [11].", "The second family also leverages sparsity in the form of redundancy in the spatial domain; that is, the dictionary of patches necessary to reconstruct the noise-free images is smaller than the actual number of patches in the image.", "Several such methods have been developed specifically for MRI, with the aim of finding an optimal, voxel-wise weighting based on the noise distribution [7], [20], [6], [19].", "Optimisation methods can naturally be interpreted as a maximum a posteriori (MAP) solution in a generative model, which eases its interpretation and extension.", "This feature is especially important for MPMs, where we possess a well-defined (nonlinear) forward function and wish to regularise a small number of maps.", "In this paper, we use the ESTATICS forward model [32], which assumes a shared $R_2^\\star $ decay across contrasts, with a joint total variation (JTV) prior.", "JTV [26] is an extension of TV to multi-channel images, where the absolute norm is defined across channels, introducing an implicit correlation between them.", "TV and JTV have been used before in MR reconstruction (e.g., in compressed-sensing [15], quantitative susceptibility mapping [17], super-resolution[4]).", "JTV is perfectly suited for modelling the multiple contrasts in MPMs and increases the power of the implicit edge-detection problem.", "However, a challenge stems from the nonlinear forward model that makes the optimisation problem nonconvex.", "Our implementation uses a quadratic upper bound of the JTV functional and the surrogate problem is solved using second-order optimisation.", "Positive-definiteness of the Hessian is enforced by the use of Fisher's scoring, and the quadratic problem is efficiently solved using a mixture of multi-grid relaxation and conjugate gradient.", "We used a unique dataset – five repeats of the MPM protocol acquired, within a single session, on a healthy subject – to validate the proposed method.", "Our method was compared to two variants of ESTATICS: loglinear [32] and Tikhonov-regularised.", "We also compared it with the adaptive optimized nonlocal means (AONLM) method [20], which is recommended for accelerated MR images (as is the case in our validation data).", "In that case, individual echoes were denoised using AONLM, and maps were reconstructed with the loglinear variant of ESTATICS.", "In our validation, JTV performed consistently better than all other methods." ], [ "Methods", "Spoiled Gradient Echo.", "The MPM protocol uses a multi-echo spoiled gradient-echo (SGE) sequence with variable flip angles to generate weighted images.", "The signal follows the equation: S(,TR,TE) = S0(,TR)(-TE R2) , where $\\alpha $ is the nominal flip angle, $T_R$ is the repetition time and $T_E$ is the echo time.", "PD and T1 weighting are obtained by using two different flip angles, while MT weighting is obtained by playing a specific off-resonance pulse beforehand.", "If all three intercepts $S_0$ are known, rational approximations can be used to compute $R_1$ and MT$_{\\mathrm {sat}}$ maps [14], [13].", "ESTATICS.", "ESTATICS aims to recover the decay rate $R_2^\\star $ and the different intercepts from ().", "We therefore write each weighted signal (indexed by $c$ ) as: S(c,TE) = (c - TER2) ,   with   c = S0c .", "At the SNR levels obtained in practice ($> 3$ ), the noise of the log-transformed data is approximately Gaussian (although with a variance that scales with signal amplitude).", "Therefore, in each voxel, a least-squares fit can be used to estimate $R_2^\\star $ and the log-intercepts $S_c$ from the log-transformed acquired images.", "Regularised ESTATICS.", "Regularisation cannot be easily introduced in logarithmic space because, there, the noise variance depends on the signal amplitude, which is unknown.", "Instead, we derive a full generative model.", "Let us assume that all weighted volumes are aligned and acquired on the same grid.", "Let us define the image acquired at a given echo time $t$ with contrast $c$ as $\\mathbf {{s}}_{c,t} \\in \\mathbb {R}^{I}$ (where $I$ is the number of voxels).", "Let $\\mathbf {{\\theta }}_c \\in \\mathbb {R}^I$ be the log-intercept with contrast $c$ and let $\\mathbf {{r}} \\in \\mathbb {R}^{I}$ be the $R_2^\\star $ map.", "Assuming stationary Gaussian noise, we get the conditional probability: sc,tc, r = sc,tsc,t,  c2I  ,   sc,t = (c - tr)  .", "The regularisation takes the form of a joint prior probability distribution over $\\mathbf {{\\Theta }} = \\left[\\mathbf {{\\theta }}_1,~\\cdots , ~\\mathbf {{\\theta }}_C,~\\mathbf {{r}}\\right]$ .", "For JTV, we get: i (-c=1C+1 c cGiGic)  , where $\\mathbf {{G}}_i$ extracts all forward and backward finite-differences at the $i$ -th voxel and $\\lambda _c$ is a contrast-specific regularisation factor.", "The MAP solution can be found by maximising the joint loglikelihood with respect to the parameter maps.", "Quadratic Bound.", "The exponent in the prior term can be written as the minimum of a quadratic function [8], [2]: c c cGiGic = wi > 0 { wi2 + 12wi c c cGiGic } .", "When the weight map $\\mathbf {{w}}$ is fixed, the bound can be seen as a Tikhonov prior with nonstationary regularisation, which is a quadratic prior that factorises across channels.", "Therefore, the between-channel correlations induces by the JTV prior are entirely captured by the weights.", "Conversely, when the parameter maps are fixed, the weights can be updated in closed-form: wi = c c cGiGic  .", "The quadratic term in () can be written as $\\lambda _c\\mathbf {{\\theta }}_c{L}\\mathbf {{\\theta }}_c$ , with $\\mathbf {{L}} = \\sum _i \\frac{1}{w_i} \\mathbf {{G}}_i{G}_i$ .", "In the following sections, we will write the full (bounded) model negative loglikelihood as $\\mathcal {L}$ and keep only terms that depend on $\\mathbf {{\\Theta }}$ , so that: L = c,t Ldc,t + Lp,  Ldc,t c=12c2sc,t-sc,t2,  Lp c=12 c cLcc.", "Fisher's Scoring.", "The data term () does not always have a positive semi-definite Hessian (it is not convex).", "There is, however, a unique optimum.", "Here, to ensure that the conditioning matrix that is used in the Newton-Raphson iteration has the correct curvature, we take the expectation of the true Hessian, which is equivalent to setting the residuals to zero – a method known as Fisher's scoring.", "The Hessian of $\\mathcal {L}^{\\mathrm {d}}_{c,t}$ with respect to the $c$ -th intercept and $R_2^\\star $ map then becomes: Hdc,t = 12 diag(sc,t) [ccc1 -t -t t2]  .", "Misaligned Volumes.", "Motion can occur between the acquisitions of the different weighted volumes.", "Here, volumes are systematically co-registered using a skull-stripped and bias-corrected version of the first echo of each volume.", "However, rather than reslicing the volumes onto the same space, which modifies the original intensities, misalignment is handled within the model.", "To this end, equation () is modified to include the projection of each parameter map onto native space, such that $\\tilde{\\mathbf {{s}}}_{c,t} = \\exp (\\mathbf {{\\Psi }}_c\\mathbf {{\\theta }}_c - t\\mathbf {{\\Psi }}_c\\mathbf {{r}})$ , where $\\mathbf {{\\Psi }}_c$ encodes trilinear interpolation and sampling with respect to the pre-estimated rigid transformation.", "The Hessian of the data term becomes $\\mathbf {{\\Psi }}_c{H}^{\\mathrm {d}}_{c,t}\\mathbf {{\\Psi }}_c$ , which is nonsparse.", "However, an approximate Hessian can be derived [1], so that: Hdc,t 12 diag(csc,t) [ccc1 -t -t t2]  .", "Since all elements of $\\tilde{\\mathbf {{s}}}_{c,t}$ are strictly positive, this Hessian is ensured to be more positive-definite than the true Hessian in the Löwner ordering sense.", "Newton-Raphson.", "The Hessian of the joint negative log-likelihood becomes: H = Hd + L diag()  .", "Each Newton-Raphson iteration involves solving for $\\mathbf {{H}}^{-1}\\mathbf {{g}}$ , where $\\mathbf {{g}}$ is the gradient.", "Since the Hessian is positive-definite, the method of conjugate gradients (CG) can be used to solve the linear system.", "CG, however, converges quite slowly.", "Instead, we first approximate the regularisation Hessian $\\mathbf {{L}}$ as $\\tilde{\\mathbf {{L}}}=\\frac{1}{\\min \\left(\\mathbf {{w}}\\right)}\\sum _i \\mathbf {{G}}_i{G}_i$ , which is more positive-definite than $\\mathbf {{L}}$ .", "Solving this substitute system therefore ensures that the objective function improves.", "Since $\\mathbf {{H}}^{\\mathrm {d}}$ is an easily invertible block-diagonal matrix, the system can be solved efficiently using a multi-grid approach [24].", "This result is then used as a warm start for CG.", "Note that preconditioners have been shown to improve CG convergence rates [5], [34], at the cost of slowing down each iteration.", "Here, we have made the choice of performing numerous cheap CG iterations rather than using an expensive preconditioner." ], [ "Validation", "Dataset.", "A single participant was scanned five times in a single session with the 0.8 mm MPM protocol, whose parameters are provided in table REF .", "Furthermore, in order to correct for flip angles nonhomogeneity, a map of the $B_1^+$ field was reconstructed from stimulated and spin echo 3D EPI images [18].", "Table: Sequence parameters of the MPM protocol.", "The MTw sequence has an off-resonance prepulse (PP): 220 ∘ ^\\circ , 4ms duration, 2kHz off-resonance.Evaluated Methods.", "Three ESTATICS methods were evaluated: a simple loglinear fit (LOG) [32], a nonlinear fit with Tikhonov regularisation (TKH) and a nonlinear fit with joint total variation regularisation (JTV).", "Additionally, all echoes were denoised using the adaptive nonlocal means method (AONLM) [20] before performing a loglinear fit.", "The loglinear and nonlinear ESTATICS fit were all implemented in the same framework, allowing for misalignment between volumes.", "Regularised ESTATICS uses estimates of the noise variance within each volume, obtained by fitting a two-class Rice mixture to the first echo of each series.", "Regularised ESTATICS possesses two regularisation factors, one for each intercept and one for the $R_2^\\star $ decay, while AONLM has one regularisation factor.", "These hyper-parameters were optimised by cross-validation (CV) on the first repeat of the MPM protocol.", "Leave-One-Echo-Out.", "Validating denoising methods is challenging in the absence of a ground truth.", "Classically, one would compute similarity metrics, such as the root mean squared error, the peak signal-to-noise ratio, or the structural similarity index between the denoised images and noise-free references.", "However, in MR, such references are not artefact free: they are still relatively noisy and, as they require longer sequences, more prone to motion artefacts.", "A better solution is to use cross-validation, as the forward model can be exploited to predict echoes that were left out when inferring the unknown parameters.", "We fitted each method to each MPM repeat, while leaving one of the acquired echoes out.", "The fitted model was then used to predict the missing echo.", "The quality of these predictions was scored by computing the Rice loglikelihood of the true echo conditioned on the predicted echo within the grey matter (GM), white matter (WM) and cerebro-spinal fluid (CSF).", "An aggregate score was also computed in the parenchyma (GM+WM).", "As different echoes or contrasts are not similarly difficult to predict, Z-scores were computed by normalising across repeats, contrasts and left-out echoes.", "This CV was applied to the first repeat to determine optimal regularisation parameters.", "We found $\\beta =0.4$ without Rice-specific noise estimation to work better for AONLM, while for JTV we found $\\lambda _1=5 \\times 10^3$ for the intercepts and $\\lambda _2=10$ for the decay (in $s^{-1}$ ) to be optimal.", "Quantitative Maps.", "Rational approximations of the signal equations [14], [13] were used to compute $R_1$ and MT$_{\\mathrm {sat}}$ maps from the fitted intercepts.", "The distribution of these quantitative parameters was computed within the GM and WM.", "Furthermore, standard deviation (S.D.)", "maps across runs were computed for each method." ], [ "Results", "Leave-One-Echo-Out.", "The distribution of Rice loglikelihoods and Z-scores for each methods are depicted in Fig.", "REF in the form of Tukey's boxplots.", "In the parenchyma, JTV obtained the best score (mean log-likelihood: -$9.15\\times 10^6$ , mean Z-score: $1.19$ ) followed by TKH (-$9.26\\times 10^6$ and -$0.05$ ), AONLM (-$9.34\\times 10^6$ and -$0.41$ ) and LOG (-$9.35\\times 10^6$ and -$0.72$ ).", "As some echoes are harder to predict than others (typically, early echoes because their absence impacts the estimator of the intercept the most) the log-pdf has quite a high variance.", "However, Z-scores show that, for each echo, JTV does consistently better than all other methods.", "As can be seen in Fig.", "REF , JTV is particularly good at preserving vessels.", "Figure: Leave-one-echo out prediction.", "Left: the true PDw echo at T E =9.7T_E=9.7ms from the 5th repeat and three predicted images.", "Right: boxplots of the Rice log-pdf and corresponding Z-score computed for each method within GM, WM and CSF masks.Quantitative Maps.", "$R_1$ , MT$_{\\mathrm {sat}}$ and $R_2^\\star $ maps reconstructed with each method are shown in Fig.", "REF , along with mean intensity histograms within GM and WM.", "Note that these maps are displayed for qualitative purposes; low standard deviations are biased toward over-regularised methods and do not necessarily indicate a better predictive performance.", "It is evident from the histograms that all denoising methods sharpen the peaks without introducing apparent bias.", "It can be seen that JTV has lower variance than AONLM in the centre of the brain and higher in the periphery.", "This is because in our probabilistic setting, there is a natural balance between the prior and the quality of the data.", "In the centre of the brain, the SNR is lower than in the periphery, which gives more weight to the prior and induces a smoother estimate.", "The mean standard deviation of AONLM, LOG, JTV and TKH is respectively 9.5, 11.5, 11.5, 9.9 $\\times 10^{-3}$ in the GM and 8.6, 12, 9.6, 10 $\\times 10^{-3}$ in the WM for $R_1$ , 15, 2, 17, 20 in the GM and 11, 20, 10, 13 in the WM for $R_2^\\star $ , and 4.6, 5.8, 5.1, 4.5 $\\times 10^{-2}$ in the GM and 4.9, 8.2, 4.3, 4.7 $\\times 10^{-2}$ in the WM for MT$_{\\mathrm {sat}}$ .", "Once again, variance is reduced by all denoising methods compared to the nonregularised loglinear fit.", "Again, a lower variance does not necessarily indicate a better (predictive) fit, which can only be assessed by the CV approach proposed above.", "Figure: Quantitative maps.", "Left: example R 1 R_1, MT sat _{\\mathrm {sat}} and R 2 ☆ R_2^\\star maps obtained with each method, and standard deviation (S.D.)", "maps computed across runs.", "Right: mean intensity histograms computed within the GM (plain) and WM (dotted) masks." ], [ "Discussion & Conclusion", "In this paper, we introduce a robust, regularisation-based reconstruction method for quantitative MR mapping.", "The joint total variation prior takes advantage of the multiple MPM contrasts to increase its edge-detection power.", "Our approach was validated using an unbiased CV scheme, where it compared favourably over other methods, including a state-of-the-art MR denoising technique.", "It was shown to reduce the variance of the estimated parametric maps over non-regularised approaches, which should translate into increased power in subsequent cross-sectional or longitudinal voxel-wise studies.", "The use of a well-defined forward model opens the door to multiple extensions: the projection operator could be modified to include other components of the imaging process such as non-homogeneous receive fields or gridding, which would allow for joint reconstruction and super-resolution; parameters that are currently fixed a priori, such as the rigid matrices, could be given prior distribution and be optimised in an interleaved fashion; non-linear deformations could be included to account for changes in the neck position between scans; finally, the forward model could be unfolded further so that parameter maps are directly fitted, rather than weighted intercepts.", "An integrated approach like this one could furthermore include and optimise for other components of the imaging process, such as non-homogeneous transmit fields.", "In terms of optimisation, our approach should benefit from advances in conjugate gradient preconditioning or other solvers for large linear systems.", "Alternatively, JTV could be replaced with a patch-based prior.", "Nonlocal filters are extremely efficient at denoising tasks and could be cast in a generative probabilistic framework, where images are built using a dictionary of patches [16].", "Variational Bayes can then be used to alternatively estimate the dictionary (shared across a neighbourhood, a whole image, or even across subjects) and the reconstruction weights." ], [ "Acknowledgements:", "YB, MFC and JA were funded by the MRC and Spinal Research Charity through the ERA-NET Neuron joint call (MR/R000050/1).", "MB and JA were funded by the EU Human Brain Project’s Grant Agreement No 785907 (SGA2).", "MB was funded by the EPSRC-funded UCL Centre for Doctoral Training in Medical Imaging (EP/L016478/1) and the Department of Health NIHR-funded Biomedical Research Centre at University College London Hospitals.", "CL is supported by an MRC Clinician Scientist award (MR/R006504/1).", "The Wellcome Centre for Human Neuroimaging is supported by core funding from the Wellcome [203147/Z/16/Z]." ] ]

[ [ "Globular Multicategories with Homomorphism Types" ], [ "Abstract We introduce a notion of globular multicategory with homomorphism types.", "These structures arise when organizing collections of \"higher category-like\" objects such as type theories with identity types.", "We show how these globular multicategories can be used to construct various weak higher categorical structures of types and terms." ], [ "Introduction", "Suppose that we have a dependent type theory with identity types $\\mathcal {T}$ .", "Then both van den Berg and Garner [11] and Lumsdaine [9] have shown that the tower of identity types of each type in $\\mathcal {T}$ has the structure of a Batanin-Leinster weak $\\omega $ -category (see [1] and [7]).", "As an intermediate step van den Berg and Garner construct a monoidal globular category of contexts and terms from $\\mathcal {T}$ .", "Our goal in this paper is to understand and generalize this phenomenon.", "In section , we review Leinster's theory of generalized multicategories [7] and focus on the free strict $\\omega $ -category monad on globular sets.", "The associated notion of generalized multicategories are called globular multicategories.", "In the terminology of [3] these are the “virtual” analogues of monoidal globular categories.)", "We then describe a notion of homomorphism type within a globular multicategory in section .", "Every type theory with identity types gives rise to such a globular multicategory.", "However, we also capture directed examples such as the double category of categories, functors, profunctors and transformations.", "In general we believe that objects in a globular multicategory with homomorphism types can be seen as “higher category-like”.", "Future work will make comparisons with structures arising in the study of directed homotopy type theory.", "Section is a brief interlude in section to study the homotopy theory of globular multicategories.", "We develop the tools we require to study weak higher categorical structures in this setting, Finally in section we show that when a globular multicategory has homomorphism types, there is a precise sense in which: types behave like weak higher categories dependent types behave like profunctors terms behave like higher functors and transformations between profunctors the collection of types and terms has the structure of a weak $\\omega $ -category.", "These results generalize those of van den Berg and Garner [11] and Lumsdaine [9], [8].", "Weak higher categorical structures based on globular multicategories have previously been studied by Kachour.", "(See for instance [6].)" ], [ "Globular Pasting Diagrams", "Definition 2.1 The category of globes $\\mathbb {G}$ is freely generated by the morphisms $\\begin{tikzcd}0 [shift left]{r}{\\sigma _0} [swap, shift right]{r}{\\tau _0}&1 [shift left]{r}{\\sigma _1} [swap, shift right]{r}{\\tau _1}&\\cdots [shift left]{r}{\\sigma _{n-1}} [swap, shift right]{r}{\\tau _{n-1}}&n [shift left]{r}{\\sigma _n} [swap, shift right]{r}{\\tau _n}&\\cdots \\end{tikzcd}$ subject to the globularity conditions $\\begin{aligned}\\sigma _{n+1} \\circ \\sigma _n &= \\tau _{n+1} \\circ \\sigma _n\\\\\\sigma _{n+1} \\circ \\tau _{n} &= \\tau _{n+1} \\circ \\tau _{n}\\end{aligned}$ which ensure that for each $k < n$ , there are exactly two composite arrows in $\\mathbb {G}$ of the form $\\begin{aligned}\\sigma _k : k \\longrightarrow n\\\\\\tau _k : k \\longrightarrow n\\end{aligned}$ A globular object in a category $\\mathcal {C}$ is a functor $ A : \\mathbb {G}^{{\\operatorname{op}}} \\longrightarrow \\mathcal {C}$ .", "We denote the image of $\\sigma _n : n \\rightarrow n+1$ under such a functor by $\\begin{tikzcd}A(n)[shift left]{r}{s_{k}} [swap, shift right]{r}{t_{k}}&A(k)\\end{tikzcd}$ A presheaf on $\\mathbb {G}$ is called globular set.", "In this case, we refer to the elements of $A(n)$ as $n$ -cells and think of the maps $s_k, t_k$ as describing source and target $k$ -cells.", "We now define a notion of $n$ -pasting diagram for each $n$ .", "We will inductively define a globular set $\\mathbf {pd}$ whose $n$ -cells are $n$ -pasting diagrams.", "Definition 2.2 We define the globular set $\\mathbf {pd}$ together with, for each $k < n$ and each $\\pi \\in \\mathbf {pd}(n)$ , boundary inclusions $\\begin{aligned}\\sigma _k : s_k \\pi &\\longrightarrow \\pi \\\\\\tau _k : t_k \\pi &\\longrightarrow \\pi \\end{aligned}$ inductively: Every globe $n$ , identified with its image under the Yoneda embedding, is an $n$ -pasting diagram.", "In this case $\\sigma _k$ and $\\tau _k$ are induced by the corresponding maps in $\\mathbb {G}$ .", "Whenever $\\pi $ is an $n$ -pasting diagram, there is an $(n+1)$ -pasting diagram $\\pi ^+$ whose underlying globular set is just $\\pi $ .", "We define the boundary inclusions to be the identity map: $\\sigma _k = \\tau _k = \\operatorname{id}_\\pi .$ Given $n$ -pasting diagrams $\\pi _1, \\pi _2$ such that $t_k \\pi _1 = s_k \\pi _2 = \\rho $ the pushout $\\pi _1 +_k \\pi _2$ of the following diagram $\\begin{tikzcd}\\rho {r}{\\tau _k}[swap]{d}{\\sigma _k}&\\pi _1{d}{}\\\\\\pi _2[swap]{r}{}&\\pi _1 +_k \\pi _2\\end{tikzcd}$ is a globular pasting diagram.", "In other words we can “glue pasting diagrams along shared $k$ -cells”.", "When $j < k$ , we define $\\sigma _j$ , $\\tau _j$ to be the composites: $\\begin{aligned}s_j \\rho &\\overset{\\sigma _j}{\\longrightarrow } \\rho &\\longrightarrow \\pi _1 +_k \\pi _2\\\\t_j \\rho &\\overset{\\tau _j}{\\longrightarrow } \\rho &\\longrightarrow \\pi _1 +_k \\pi _2\\end{aligned}.$ When $j = k$ , we define $\\sigma _j$ , $\\tau _j$ to be the composites $\\begin{aligned}s_j \\pi _1 &\\overset{\\sigma _j}{\\longrightarrow } \\pi _1&\\longrightarrow \\pi _1 +_k \\pi _2\\\\t_j \\pi _2 &\\overset{\\tau _j}{\\longrightarrow } \\pi _2&\\longrightarrow \\pi _1 +_k \\pi _2\\end{aligned}$ When $j > k$ , we define $\\begin{aligned}s_j (\\pi _1 +_k \\pi _2) &= (s_j \\pi _1) +_k (s_j \\pi _2)\\\\t_j (\\pi _1 +_k \\pi _2) &= (t_j \\pi _1) +_k (t_j \\pi _2)\\end{aligned}$ and $\\sigma _j, \\tau _j$ are induced by the universal properties of these pushouts.", "This notion of pasting diagram allows us to describe the free strict $\\omega $ -category monad $T : \\mathbb {G}\\text{-Set}\\rightarrow \\mathbb {G}\\text{-Set}$ explicitly.", "For any globular set $A$ , an $n$ -cell of $TA$ consists of a pasting diagram $\\pi \\in \\mathbf {pd}(n)$ together with a map $\\begin{aligned}\\pi \\rightarrow A\\end{aligned}.$ Thus, we refer to these maps as pasting diagrams in $A$ .", "Suppose that $f, g \\in TA(n)$ are $n$ -cells such that such that $t_k f = s_k g$ .", "Then we have corresponding pasting diagrams $\\begin{aligned}f : \\pi _1 \\rightarrow A\\\\g : \\pi _2 \\rightarrow A\\end{aligned}$ such that $f \\tau _k = g \\sigma _k$ .", "We define $f \\odot _{k} g$ to be the induced map $\\begin{aligned}\\pi _1 +_{k} \\pi _2 \\rightarrow A\\end{aligned}.$ This defines another element of $TA(n)$ .", "More generally suppose that we have a pasting diagram $\\Gamma : \\rho \\rightarrow TA$ .", "Then the multiplication of $T$ gives us a new pasting diagram which we denote $\\begin{aligned}\\bigodot _{i \\in \\rho } \\Gamma _i\\end{aligned}.$ Suppose that $\\rho $ is the $n$ -globe, thought of as an $k$ -pasting diagram for some $k \\ge n$ .", "Then $\\Gamma $ is uniquely determined by the unique $n$ -cell $M \\in \\Gamma (n)$ .", "Hence we write $\\begin{aligned}\\Gamma = [M]\\end{aligned}$ and we have that $\\begin{aligned}\\bigodot _{i \\in \\rho } \\Gamma _i = M\\end{aligned}.$ A crucial property of the monad $T$ is that it is cartesian; i.e., its underlying functor preserves pullbacks and the naturality squares of its unit and multiplication are pullback squares (see [7]).", "Following Leinster [7], this allows us to define a notion of generalized multicategory." ], [ "Globular Multicategories", "We will assume for the rest of this paper that $T$ is the free strict $\\omega $ -category monad.", "In this section we review Leinster's [7] theory of $T$ -multicategories.", "Definition 2.3 A $T$ -span is a span of the following form.", "$\\begin{tikzcd}& X {dl} {dr}\\\\TA && B\\end{tikzcd}$ We can compose $T$ -spans $\\begin{tikzcd}& X [swap]{dl}{a}{dr}{b}& &&& Y[swap]{dl}{b^{\\prime }}{dr}{c}\\\\TA && B && TB && C\\\\\\end{tikzcd}$ by computing a pullback as in the following diagram $\\begin{tikzcd}&& TX \\times _{TB} Y {dl}{dr}\\\\& TX [swap]{dl}{Ta}{dr}{Tb}& & Y[swap]{dl}{b^{\\prime }}{dr}{c}\\\\T^{2}A [swap]{d}{\\mu }&& TB && C\\\\TA\\end{tikzcd}$ Let $\\eta _X$ be the unit of $T$ at $X$ .", "Then the identity $T$ -span at $X$ is the following diagram: $\\begin{tikzcd}& X [swap]{dl}{\\eta _X}{dr}{\\operatorname{id}_X}&\\\\TX && X\\end{tikzcd}$ Putting all this data together we obtain a bicategory $T$-Span.", "This allows us to present our main definition succinctly.", "Definition 2.4 A globular multicategory is a monad in the bicategory $T$-Span.", "Suppose that $X$ is a globular multicategory.", "Then there is a $T$ -span of the following form: $\\begin{tikzcd}& X_1 [swap]{dl}{d} {dr}{c}\\\\TX_0 && X_0\\end{tikzcd}$ We refer to these $T$ -spans as globular multigraphs .", "We will use type theoretic terminology to refer to the data contained in $X$ : An $n$ -type is an element of $X_0(n)$ .", "When $sM = A$ and $tM$ = B, we write $\\begin{aligned}M : A \\rightarrow B\\end{aligned}.$ A $\\pi $ -shaped $n$ -context $\\Gamma = (\\Gamma _i)_{i \\in \\operatorname{el}(\\pi )}$ is an element of $TX_0 (n)$ of the form $\\Gamma : \\pi \\longrightarrow X_0.$ A variable in $\\Gamma $ is an element $x \\in \\operatorname{el}(\\pi )$ .", "When $A = \\Gamma _x$ , we say that $A$ is the type of $x$ and write $\\begin{aligned}x : A\\end{aligned}.$ An $n$ -term $f$ is an element of $X_1(n)$ .", "Suppose that $\\Gamma = df$ is a $\\pi $ -shaped $n$ -context.", "Let $A = cf$ be an $n$ -type.", "Then $f$ can be thought of as a generalized arrow sending a $\\pi $ -shaped $n$ -context $\\Gamma $ (a pasting diagrams of typed input variables) to an $n$ -type $A$ .", "For this reason we say that $f$ is $\\pi $ -shaped and write $f : \\Gamma \\longrightarrow A$ When $n \\ge 0$ , $n$ -terms also have source and target $(n-1)$ -terms $sf$ and $tf$ .", "A substitution or context morphism is an element $f \\in TX_1(n).$ For some $\\pi \\in \\mathbf {pd}(n)$ , $f$ is a collection of terms $f_i : \\Gamma _i \\longrightarrow \\Delta _i$ for $i \\in \\operatorname{el}(\\pi )$ .", "(That is a $\\pi $ -shaped pasting diagram of terms.)", "By pasting together the domain contexts of these terms we obtain a context $\\begin{aligned}\\Gamma = \\bigodot _{i \\in \\pi } \\Gamma _i\\end{aligned}$ By pasting together the codomain types we obtain a $\\pi $ -shaped context $\\begin{aligned}\\Delta = \\bigodot _{i \\in \\pi } \\Delta _i\\end{aligned}$ In this case, we write $f : \\Gamma \\longrightarrow \\Delta $ .", "Suppose that we have a substitution $f : \\Gamma \\rightarrow \\Delta $ and a term $g : \\Delta \\rightarrow A$ .", "Then the multiplication of $X$ allows us to define a composite term $f ; g : \\Gamma \\longrightarrow \\Delta .$ We think of $f ; g$ as the result of substituting $f_i$ for the variable $i$ into (the domain context of) $g$ .", "Now suppose that $h : \\Delta \\rightarrow E$ is a substitution.", "Then we define $\\begin{aligned}f ; h : \\Gamma \\longrightarrow E\\end{aligned}$ to be the substitution such that $(f ; h)_j = (f_{i_j})_{i_j \\in d h_j} ; h_j$ .", "For each $n$ -type $A$ , the unit of $X$ gives us an identity $n$ -term $\\begin{aligned}\\operatorname{id}_A : [A] \\rightarrow A\\end{aligned}.$ For each context $\\Gamma $ , we define $\\begin{aligned}\\operatorname{id}_\\Gamma : \\Gamma \\rightarrow \\Gamma \\end{aligned}$ to be the substitution such that $\\begin{aligned}(\\operatorname{id}_\\Gamma )_i = \\operatorname{id}_{\\Gamma _i}\\end{aligned}$ We think of this as the trivial substitution.", "Associativity of $X$ tells us that, whenever it makes sense, $\\begin{aligned}(f ; g) ; h = f ; (g ; h)\\end{aligned}$ The unit laws of $X$ tell us that, whenever it makes sense, $\\begin{aligned}f ; \\operatorname{id}_A = f = \\operatorname{id}_\\Gamma ; f\\end{aligned}$ Example 2.5 Batanin's [1] globular operads are a particular important class of globular multicategories.", "A globular operad is a globular multicategory $X$ with $X_0 = \\top $ , the terminal globular set.", "In other words a globular operad has a unique $n$ -type for each $n \\in \\mathbb {G}$ .", "The terminal globular multicategory (operad) has a unique $\\pi $ -shaped $n$ -term for each $\\pi \\in \\mathbf {pd}(n)$ ,.", "Algebras of this operad are closely related to strict $\\omega $ -categories.", "Example 2.6 Let $\\mathcal {C}$ be a category with pullbacks.", "There is a globular multicategory $\\operatorname{Span}_{T}(\\mathcal {C})$ such that: An $n$ -type is a functor $A : \\operatorname{el}(n)^{{\\operatorname{op}}} \\rightarrow \\mathcal {C}$ from the category of elements of the representable globular set $n$ .", "Given a pasting diagram $\\pi \\in \\mathbf {pd}(n)$ , a context with shape $\\pi $ is a functor $\\Gamma : \\operatorname{el}(\\pi )^{{\\operatorname{op}}} \\longrightarrow \\mathcal {C}.$ Associated to such a context there is a canonical functor $\\Gamma _0 : \\operatorname{el}(n)^{{\\operatorname{op}}} \\rightarrow A$ , which sends $s : m \\rightarrow n \\in \\mathbb {G}&&\\longmapsto &&\\ \\operatorname{lim}\\left(\\operatorname{el}(s \\pi )^{{\\operatorname{op}}} \\overset{\\operatorname{el}(s)^{{\\operatorname{op}}}}{\\longrightarrow }\\operatorname{el}(\\pi )^{{\\operatorname{op}}} \\overset{\\Gamma }{\\longrightarrow }\\mathcal {C}\\right)$ and sends arrows to the canonical morphisms induced between these limits.", "A term $u : \\Gamma \\rightarrow A$ in $\\operatorname{Span}_{T}(\\mathcal {C})$ is a natural transformation $\\Gamma _0 \\rightarrow A$ .", "Remark 2.7 We refer to functors $\\operatorname{el}(n)^{{\\operatorname{op}}} \\rightarrow \\mathcal {C}$ as $n$ -spans in $\\mathcal {C}$ (see [1]).", "The globular multicategory $\\operatorname{Span}_{T}(\\mathcal {C})$ underlies the monoidal globular category $\\operatorname{Span}(\\mathcal {C})$ described ibid.", "A 1-span is a span in the usual sense.", "More generally, an $(n+1)$ -span is a “span between $n$ -spans”.", "For example a 3-span is a diagram of the following form: $\\begin{tikzcd}& \\bullet {dl} {dr} &\\\\\\bullet {d} {drr} &&\\bullet {d} {dll}\\\\\\bullet {d} {drr}&&\\bullet {d} {dll}\\\\\\bullet &&\\bullet \\end{tikzcd}$ Definition 2.8 A map of globular multigraphs is a pair of arrows $f_0, f_1$ making the following diagram commute: $\\begin{tikzcd}& X_1[swap]{dl}{d}{dr}{c}{ddd}{f_1} &\\\\TX_0 [swap]{ddd}{Tf_0} && X_0{ddd}{f_0}\\\\\\\\& Y_1{dl}{d}[swap]{dr}{c} &\\\\TY_0 && Y_0\\\\\\end{tikzcd}$ A homomorphism of globular multicategories is a map of globular multigraphs preserving the multiplication and unit of $X$ .", "We denote the category of globular multicategories and homomorphisms by ${\\operatorname{GlobMult}}$ .", "A globular multicategory can be also be seen as an algebraic theory whose operations have arities which are shapes of pasting diagrams.", "For this reason we also think of homomorphisms $X \\rightarrow Y$ as algebras of $X$ in $Y$ .", "Remark 2.9 Algebras of the terminal globular operad in $\\operatorname{Span}_{T}(\\mathcal {C})$ are strict $\\omega $ -categories internal to $\\mathcal {C}$.", "In particular, algebras of the terminal globular operad in $\\operatorname{Span}_{T}(\\operatorname{Set})$ are strict $\\omega $ -categories.", "Definition 2.10 Let $X : \\mathbb {G}^{\\operatorname{op}}\\rightarrow \\mathcal {C}$ be a globular object in $\\mathcal {C}$ .", "The endomorphism operad $\\operatorname{End}(X)$ is the collection of terms in $\\operatorname{Span}_{T}{\\mathcal {C}}$ whose types come from $X$ (see [1]).", "To be precise, let $\\hat{X} : \\mathbb {G}\\text{-Set}\\rightarrow \\mathcal {C}$ be the canonical cocontinuous extension of $X$ .", "Then a term $f : \\pi \\rightarrow n$ in $\\operatorname{End}(X)$ is a term $\\begin{aligned}\\hat{X}(\\pi ) \\longrightarrow X(n)\\end{aligned}$ in $\\operatorname{Span}_{T}(\\mathcal {C})$ respecting boundaries.", "This is the objects part of a functor $\\begin{aligned}\\operatorname{End}: \\mathcal {C}^{\\mathbb {G}^{\\operatorname{op}}} \\longrightarrow {\\operatorname{GlobMult}}\\end{aligned}$ Every globular multigraph can be viewed as a presheaf over a category whose objects are shapes of types and terms.", "Definition 2.11 The category $\\mathbb {G}^{+}$ of generic types and terms is defined as follows: Its set of objects is the the coproduct of sets $\\mathbb {G}+ \\operatorname{el}(\\mathbf {pd})$ .", "In this case, we say that $n \\in \\mathbb {G}$ is the generic n-type and that $\\pi \\in \\mathbf {pd}(n)$ is the generic $\\pi $ -shaped $n$ -term.", "There are four classes of arrows in $\\mathbb {G}^{+}$ : Every arrow in $\\mathbb {G}$ induces a corresponding arrow between generic types in $\\mathbb {G}^{+}$ .", "These arrows pick out the source and target types of generic types.", "Every arrow $\\operatorname{el}(\\mathbf {pd})$ induces a corresponding arrow between generic terms in $\\mathbb {G}^{+}$ .", "These arrows pick out the source and target terms of generic terms.", "Let $u$ be the generic $\\pi $ -shaped $n$ -pasting diagram and let $k \\in \\mathbb {G}$ .", "Then each map of globular sets $k \\rightarrow \\pi $ induces an arrow $\\begin{aligned}k \\rightarrow u\\end{aligned}$ in $\\mathbb {G}^{+}$ from the generic $k$ -type to the generic $\\pi $ -shaped term $u$ .", "These arrow pick out the types in the domain contexts of generic terms.", "For each generic $n$ -term $u$ and each arrow $k \\rightarrow n$ in $\\mathbb {G}$ there is an arrow $\\begin{aligned}k \\rightarrow u\\end{aligned}$ from the generic $k$ -type to the generic $n$ -term $u$ .", "These arrows pick out the codomain types of generic terms.", "Proposition 2.12 The category of globular multigraphs ${\\operatorname{GlobGraph}}$ is equivalent to the category of presheaves over $\\mathbb {G}^{+}$ .", "This follows from [7].", "Remark 2.13 The category $\\mathbb {G}^{+}$ is a direct category.", "There is an identity-reflecting functor $\\dim : \\mathbb {G}^{+}\\rightarrow \\mathbb {N}$ which sends the generic $n$ -types and all generic $n$ -terms to the natural number $n$ .", "Let $X$ be a globular multigraph.", "Let $u \\in \\mathbb {G}^{+}$ be a generic type or term, identified with its image under the Yoneda embedding.", "Then the boundary $\\partial u$ is the subpresheaf of $u$ such that $\\begin{aligned}w \\in \\partial u(v) \\iff \\dim v < \\dim u\\end{aligned}.$ We will denote the canonical inclusion by $\\begin{aligned}\\iota _u : \\partial u \\longrightarrow u\\end{aligned}.$ In Section we will use these boundary inclusions to describe a higher dimensional notion of weakness." ], [ "Homomorphism Types", "Definition 3.1 We say that a globular multigraph is reflexive when for each $n$ -type $A$ , we have an “identity” $(n+1)$ -type $\\begin{aligned}\\mathcal {H}_A : A \\rightarrow A\\end{aligned}$ and a reflexivity $(n+1)$ -term $\\begin{aligned}r_A : A \\rightarrow \\mathcal {H}_A\\end{aligned}$ Definition 3.2 Let $0 \\le k < n$ .", "Suppose that $\\Gamma $ is a $\\pi $ -shaped $n$ -context in a reflexive globular multigraph.", "Then for any $k$ -variable $x : A$ in $\\Gamma $ , since $\\mathcal {H}_A : A \\lnot \\rightarrow A$ , we can form a new context $\\Gamma \\oplus \\mathcal {H}_x$ by “adding an homomorphism type at $x$ ”.", "$\\begin{aligned}\\Gamma \\oplus \\mathcal {H}_x=\\bigodot _{i \\in \\operatorname{el}{\\pi }}{\\left\\lbrace \\begin{array}{ll}\\Gamma _i & \\text{if $i \\ne x$}\\\\\\mathcal {H}_{\\Gamma _x} & \\text{if $i = x$}\\end{array}\\right.", "}\\end{aligned}$ Similarly there is a canonical reflexivity substitution $r^\\Gamma _x : \\Gamma \\rightarrow \\Gamma \\oplus \\mathcal {H}_x$ defined by $\\begin{aligned}r^\\Gamma _x=\\bigodot _{i \\in \\operatorname{el}{\\pi }}{\\left\\lbrace \\begin{array}{ll}\\operatorname{id}_i & \\text{if $i \\ne x$}\\\\r_x & \\text{if $i = x$}\\end{array}\\right.", "}\\end{aligned}$ When $\\Gamma $ is clear from the context we will denote this substitution by $r_x$ .", "Definition 3.3 We say that a globular multicategory has homomorphism types when its underlying globular multigraph is reflexive and we have the following structure: For each $n$ -term $f : \\Gamma \\rightarrow A$ and each $(n-1)$ -variable $x$ , there is a J-term $\\begin{aligned}J_x(f) : \\Gamma \\oplus \\mathcal {H}_x \\rightarrow A\\end{aligned}$ such that $\\begin{aligned}r_x ; J_x(f) = f\\end{aligned}.$ Suppose that $0 < k < n$ .", "Let $f : \\Gamma \\rightarrow A$ be an $n$ -term.", "Then for each $k$ -variable $x$ and any $\\begin{aligned}j_s : s\\Gamma \\oplus \\mathcal {H}_x \\rightarrow sA\\\\j_t : t\\Gamma \\oplus \\mathcal {H}_x \\rightarrow tA\\end{aligned}$ such that $\\begin{aligned}{r_x ; j_s = sf}\\\\{r_x ; j_t = tf}\\end{aligned}$ we have a J-term $\\begin{aligned}J^{j_s, j_t}_x(f) : \\Gamma \\oplus \\mathcal {H}_x \\rightarrow A\\end{aligned}$ such that $\\begin{aligned}s J^{j_s, j_t}_x(f) &= j_s\\\\t J^{j_s, j_t}_x(f) &= j_t\\end{aligned}$ and $\\begin{aligned}r_x ; J^{j_s, j_t}_x(f) = f\\end{aligned}$ Example 3.4 Every type theory with identity types $\\mathcal {T}$ induces a globular multicategory $\\mathbb {G}_T(\\mathcal {T})$ with homomorphism types.", "The construction essentially follows van den Berg and Garner [11].", "We have that: A 0-type $A$ in $\\mathbb {G}_T(\\mathcal {T})$ is a type $\\begin{aligned}\\vdash {A}^{\\circ } : \\operatorname{Type}\\end{aligned}$ in $\\mathcal {T}$ .", "An $(n+1)$ -type $M : A \\lnot \\rightarrow B$ in $\\mathbb {G}T$ is a dependent type judgement $\\begin{aligned}x : {A}^{\\circ }, y : {B}^{\\circ } \\vdash {M}^{\\circ }(x, y) : \\operatorname{Type}\\end{aligned}$ in $\\mathcal {T}$ .", "Each globular context $\\Gamma : s\\Gamma \\lnot \\rightarrow t\\Gamma $ in $\\mathbb {G}_T(\\mathcal {T})$ corresponds to a list of dependent types in $\\mathcal {T}$ and thus induces a dependent context $\\begin{aligned}{s\\Gamma }^{\\circ }, {t \\Gamma }^{\\circ } \\vdash {\\Gamma }^{\\circ }\\end{aligned}$ in $\\mathcal {T}$ .", "A 0-term in $f : \\Gamma \\rightarrow A$ in $\\mathbb {G}_T(\\mathcal {T})$ is a term $\\vec{x} : {\\Gamma }^{\\circ }\\vdash {f}^{\\circ } \\vec{x} : {A}^{\\circ }$ in $\\mathcal {T}$ .", "Suppose that $\\Gamma $ is an $(n+1)$ -context such that $\\vec{x}_s : {s \\Gamma }^{\\circ }, \\vec{x}_t : {t \\Gamma }^{\\circ } \\vdash {\\Gamma }^{\\circ }(\\vec{x}_s, \\vec{x}_t)$ is a dependent context in $\\mathcal {T}$ .", "Then an $(n+1)$ -term in $f : \\Gamma \\rightarrow A$ in $\\mathbb {G}_T(\\mathcal {T})$ is a term $\\begin{aligned}\\vec{x} : {\\Gamma }^{\\circ }(\\vec{x}_s, \\vec{x}_t)\\vdash {f}^{\\circ } \\vec{x}: {A}^{\\circ }({sf}^{\\circ }(\\vec{x_s}), {tf}^{\\circ }(\\vec{x_t}))\\end{aligned}$ in $\\mathcal {T}$ .", "Composition of terms in $\\mathbb {G}_T(\\mathcal {T})$ is defined by substitution in $\\mathcal {T}$ .", "The homomorphism types, reflexivity terms and J-terms in $\\mathbb {G}_T(\\mathcal {T})$ are the corresponding terms in $\\mathcal {T}$ .", "Example 3.5 In forthcoming work we will construct a globular multicategory with homomorphism types whose: 0-types are strict $\\omega $ -categories $(n+1)$ -types are profunctors between $n$ -types 0-terms are strict $\\omega $ -functors $(n+1)$ -terms are transformations between profunctors Homomorphism type data comes from a version of the Yoneda Lemma for strict $\\omega $ -categories Definition 3.6 A globular category $A$ is a globular object in the category of categories.", "Following Batanin [1] a monoidal globular category is an $\\omega $ -category internal to the category of globular categories up to isomorphism.", "This amounts to a globular category $A$ together with: for each $k < n$ , composition functors $\\begin{aligned}\\otimes _k : A(n) \\times A(n) \\longrightarrow A_n\\end{aligned}$ for each $n$ , a unit functor $\\begin{aligned}Z : A(n) \\longrightarrow A(n-1)\\end{aligned}$ natural transformations and axioms, mimicking those of a strict $\\omega $ -category up to isomorphism.", "When these natural transformations are all identities we say that a globular multicategory is strict.", "We denote the category of strict monoidal globular categories by ${\\operatorname{MonGlobCat}}$ .", "Every monoidal globular category can be viewed as a globular multicategory (see [3]).", "We will show that we always have homomorphism types in this case.", "Proposition 3.7 The globular multicategory induced by a monoidal globular category has homomorphism types.", "[Proof sketch] We will work in the strict case.", "There is a coherence theorem for monoidal globular categories (see [1]) and so there should be no real loss of generality.", "an $n$ -type is an object of $A(n)$ $n$ -term $f : \\bigodot _{i} M_i \\rightarrow N$ is an arrow $\\begin{aligned}\\bigotimes _{i} M_i \\rightarrow N\\end{aligned}$ in $A(n)$ .", "Here $\\bigotimes $ is repeated application of $\\otimes $ and we follow the convention that a nullary product is a unit.", "This globular multicategory has homomorphism types.", "For any type $M$ , we set $\\begin{aligned}\\mathcal {H}_M = Z(M)\\end{aligned}$ .", "The $(n+1)$ -context $M$ corresponds to the object $Z(M)$ and so we define the reflexivity term $r_M : M \\rightarrow \\mathcal {H}_M$ to be the identity arrow $\\begin{aligned}\\operatorname{id}_{Z(M)} : Z(M) \\rightarrow Z(M)\\end{aligned}$ The J-terms can now be constructed using the coherence laws of $Z$ .", "For instance, whenever $f : A \\rightarrow B$ is an $n$ -term and $x : M$ is a $k$ -variable.", "We define $\\begin{aligned}J_x(f) : A \\odot _k \\mathcal {H}_x \\rightarrow B\\end{aligned}$ to be the composite $\\begin{aligned}A \\otimes _k Z(M) \\longrightarrow A \\overset{f}{\\longrightarrow }B\\end{aligned}.$ where the arrow on the left is a unit law.", "Definition 3.8 We say that a globular multicategory has strict homomorphism types when for any $n$ -terms $f, f^{\\prime }$ and any variable $x$ such that $\\begin{aligned}r_x ; f = r_x ; f^{\\prime }\\end{aligned},$ we have that $\\begin{aligned}f = f^{\\prime }\\end{aligned}$ Example 3.9 Every strict monoidal globular category induces a globular multicategory with strict homomorphism types.", "Many more familiar examples can be seen as low-dimensional globular multicategories.", "Definition 3.10 Let $X$ be a globular multicategory with identity types and let $n \\ge 0$ .", "We say that $X$ is $n$ -truncated, when for any $m$ -term $f : \\Gamma \\rightarrow A$ with $m > n$ , we have that: There exists an $n$ -context $\\Gamma ^{\\prime }$ and variables $x_1, \\ldots , x_l$ such that $\\begin{aligned}\\Gamma = \\Gamma ^{\\prime } \\oplus \\mathcal {H}_{x_1} \\oplus \\cdots \\oplus \\mathcal {H}_{x_l}\\end{aligned}.$ There exists an $n$ -type $A^{\\prime }$ such that $\\begin{aligned}A = \\mathcal {H}^{n-m} A^{\\prime }\\end{aligned}.$ There exists an $n$ -term $f^{\\prime } : \\Gamma ^{\\prime } \\rightarrow A^{\\prime }$ such that $\\begin{aligned}f = J_{x_1}(\\cdots (J_{x_l}(f^{\\prime })\\end{aligned}.$ Example 3.11 The category of 1-truncated strict monoidal globular categories is equivalent to the category of double categories.", "Example 3.12 The category of 1-truncated globular multicategories with strict homomorphism types is equivalent to the category of virtual double categories with horizontal units as described by Crutwell and Shulman [3].", "Ibid., the monoids and modules construction for virtual double categories is exhibited as the the right adjoint of the 2-functor which forgets horizontal units.", "Many familiar collections of “category-like” objects can be seen as the result of this construction.", "Hence any such collection gives rise to a globular multicategory with homomorphism types.", "In future work we will define a higher modules construction.", "This will allow us to study globular multicategories of “higher category-like” objects.", "Example 3.13 Riehl and Verity [10] describe a framework for reasoning about “$(\\infty , 1)$ -category'-like” objects using structures called $\\infty $ -cosmoi.", "Every $\\infty $ -cosmos gives rise to a virtual double category of modules whose horizontal units are arrow $\\infty $ -categories.", "This construction, throws away much higher dimensional data but still provides a convenient framework for a great deal of formal category theory.", "We expect to be able to construct more general globular multicategories with homomorphism types from $\\infty $ -cosmoi which retain this higher dimensional data.", "Remark 3.14 Starting with a category with a well behaved notion of path object, van den Berg and Garner [11] construct a globular multicategory with homomorphism types.", "In future work we will construct a globular multicategory with homomorphism types from a category with directed path objects." ], [ "The Homotopy Theory of Globular Multicategories", "By Remark REF .", "the category $\\mathbb {G}^{+}$ of generic types and terms is a direct category.", "This induces a weak factorization system on globular multicategories and related structures.", "Definition 4.1 Let us denote by $\\begin{aligned}I = \\lbrace \\iota _u : \\partial u \\longrightarrow u \\mid u \\in \\mathbb {G}^{+}\\rbrace \\end{aligned}.$ the set of boundary inclusions of $\\mathbb {G}^{+}$ .", "Then $I$ cofibrantly generates a weak factorization system $(\\mathcal {L}, \\mathcal {R})$ on ${\\operatorname{GlobGraph}}$ .", "We refer to maps in $\\mathcal {L}$ as cofibrations and maps in $\\mathcal {R}$ as acyclic fibrations.", "A map of globular multigraphs $f : X \\rightarrow Y$ is an acyclic fibration when, for any generic type or term $u$ , each commutative square $\\begin{tikzcd}\\partial u {r}[swap]{d}{\\iota _u} & X{d}{f}\\\\u {r}[dotted]{ur} & Y\\\\\\end{tikzcd}$ has a filler.", "A map of globular multigraphs $i : Z \\rightarrow W$ is a cofibration when for each acyclic fibration $f : X \\rightarrow Y$ , each commutative square $\\begin{tikzcd}\\partial Z {r}[swap]{d}{i} & X{d}{f}\\\\W {r}[dotted]{ur} & Y\\\\\\end{tikzcd}$ has a filler.", "Proposition 4.2 A map of globular multigraphs is a cofibration exactly when it is a monomorphism.", "Since $\\mathbb {G}^{+}$ is a direct category, it is skeletal and has no non-trivial automorphisms.", "The result now follows from [2].", "Our weak factorization system can be transferred to other categories of interest using the adjunctions induced by various forgetful functors We have the following commutative diagram of functors: $\\begin{tikzcd}[column sep = small]&&&{\\operatorname{GlobMult}}{rd}&\\\\{\\operatorname{MonGlobCat}}{r}&\\operatorname{GlobMult}_{\\mathcal {H}_{\\operatorname{Str}}}{r}&\\operatorname{GlobMult}_{\\mathcal {H}}{rd} {ru}&&{\\operatorname{GlobGraph}}\\\\&&&\\overline{\\operatorname{GlobGraph}}{ru}&\\end{tikzcd}$ Each of these forgets essentially algebraic data and so has a left adjoint.", "Let $U : \\mathcal {C} \\rightarrow \\mathcal {D}$ be one of these forgetful functors and let $F : \\mathcal {D} \\rightarrow \\mathcal {C}$ be its left adjoint.", "Then the weak factorization system of cofibrations and acyclic fibrations in $\\mathcal {C}$ is generated by $\\begin{aligned}F \\iota _u : F \\partial u \\longrightarrow F u\\end{aligned}$ for each generating cofibration in $\\iota _u$ in $\\mathcal {C}$ .", "A morphism $f : X \\rightarrow Y$ in $\\mathcal {C}$ is an acyclic fibration exactly when $Uf$ is an acyclic fibration in $\\mathcal {D}$ .", "Furthermore $\\mathcal {F}$ preserves cofibrations.", "Example 4.3 Suppose that $X$ and $Y$ are globular operads.", "Then every homomorphism of globular operads is bijective on types and so the lifting conditions for generic types are always satisfied.", "It follows that a homomorphism $f : X \\rightarrow Y$ is an acyclic fibration if and only if it satisfies the lifting conditions for generic terms.", "We say that a globular operad is normalized when it has a unique 0-term.", "(This term must be the identity term on the unique 0-type).", "Suppose that the globular operad $X$ is normalized and let $\\top $ be the terminal globular operad.", "Then the canonical map $\\begin{tikzcd}X [two heads]{r}{!}", "& \\top \\end{tikzcd}$ is an acyclic fibration exactly when $P$ is a normalized contractible globular operad.", "This follows from the observations of Garner [4].", "The algebras of $P$ are weak $\\omega $ -categories in the sense of Leinster [7].", "The terms defining the natural transformations in this example motivate the following definition: Definition 4.4 Suppose that $f,f^{\\prime } : \\Gamma \\rightarrow A$ are parallel $n$ -terms in a globular multicategory with identity types.", "Then a transformation $\\begin{aligned}\\phi : f \\longrightarrow f^{\\prime }\\end{aligned}$ is a term $\\phi : \\Gamma \\rightarrow \\mathcal {H}_A$ with $\\phi : f \\lnot \\rightarrow f^{\\prime }$ .", "Given a term $g : f^{\\prime } \\lnot \\rightarrow tg$ , the composite $\\begin{aligned}\\phi \\circ g : f \\lnot \\longrightarrow tg\\end{aligned}$ is defined to be $(\\phi \\odot _n g) ; m$ where $m = J_{tA}(\\operatorname{id}_A)$ .", "Given $h : sh \\lnot \\rightarrow f$ , we define $\\begin{aligned}h \\circ \\phi : sh \\lnot \\longrightarrow f^{\\prime }\\end{aligned}$ similarly.", "We have that $s (h \\circ \\phi ) = sh$ and $t (h \\circ \\phi ) = f^{\\prime }$ .", "The transformation $f ; r_A : f \\rightarrow f$ is the identity at $f$ with respect to $-\\circ -$ .", "A transformation between substitutions is a pasting diagram of transformations between terms.", "This definition immediately implies the following lemma.", "One interpretation of this lemma is that composition with reflexivity terms defines a sort of weak equivalence.", "Lemma 4.5 Suppose that $f, f^{\\prime } : \\Gamma \\oplus \\mathcal {H}_x \\rightarrow A$ are terms in a globular multicategory with identity types.", "Then whenever $\\begin{aligned}r_x ; f = r ; f^{\\prime }\\end{aligned}$ we have $\\begin{aligned}J_x^{f, f^{\\prime }}(r_x ; f ; r_A) : f \\longrightarrow f^{\\prime }\\end{aligned}$ When we have strict homomorphism types, we have that $f = f^{\\prime }$ and $J_x^{f, f^{\\prime }}(r_x ; f ; r_A) = f ; r_A$ .", "Transformations can be used to provide a useful alternative description of acyclic fibrations.", "Intuitively this description says that term-lifting properties of acyclic fibrations are satisfied exactly when, on terms, a homomorphism is“strictly surjective and weakly reflects identities”.", "Definition 4.6 Let $f : X \\rightarrow Y$ be a homomorphism of globular multicategories with identity types.", "We say that $f$ weakly reflects identities of terms if whenever $v, v^{\\prime } : \\Gamma \\rightarrow A$ are parallel terms in $X$ such that $f(v) = f(v^{\\prime })$ , we have a transformation $\\phi : v \\rightarrow v^{\\prime }$ such that $f(\\phi ) = f(v) ; r_A$ .", "In this case we say that $\\phi $ is an identification.", "We say that $f$ strictly reflects identities when all the corresponding identifications can be chosen to be identity transformations.", "Proposition 4.7 A homomorphism of globular multicategories with homomorphism types $f : X \\rightarrow Y$ is an acyclic fibration if and only if all the following conditions hold: The homomorphism $f$ has the right lifting property against the boundary-inclusions of types.", "The homomorphism $f$ is surjective on terms.", "The homomorphism $f$ weakly reflects identities of terms.", "First suppose that $f$ is an acyclic fibration.", "Then REF follows trivially.", "For each generic type or term $u$ , the unique map $\\emptyset \\rightarrow u$ is a cofibration.", "The lifting property of $f$ with respect to this map tells us that $f$ is surjective on types or terms with the same shape as $u$ .", "This proves REF .", "Now suppose that $v, v^{\\prime } : \\Gamma \\rightarrow A$ are parallel terms in $X$ and that $f(v) = f(v^{\\prime })$ .", "Let $u$ be the generic term with the same shape as $v$ and $v^{\\prime }$ .", "Let $I_u$ be the generic term with same shape as transformations between $v$ and $v^{\\prime }$ .", "Then$v$ and $v^{\\prime }$ together correspond to a map $[{v}, {v^{\\prime }}] : \\partial I_u \\rightarrow X$ of globular multigraphs.", "Furthermore, we have the following commutative square in ${\\operatorname{GlobGraph}}$ : $\\begin{tikzcd}\\partial I_u[r, \"[{v}, {v^{\\prime }}]\"][d, \"\\partial I_u\" left]&X[d, \"f\"]\\\\I_u [ur, dotted][r, \"f(v);r_A\" below]&Y\\end{tikzcd}$ Since $f$ is an acyclic fibration, this square has a filler.", "This filler defines the transformation $v \\rightarrow v^{\\prime }$ required by REF .", "Now suppose on the other hand that we have REF , REF and REF .", "Let $u$ be a generic term and fix a commutative square: $\\begin{tikzcd}\\partial u {r}{\\widetilde{\\partial v}}[swap]{d}{\\iota _u} & X{d}{f}\\\\u [swap]{r}{v} & Y\\\\\\end{tikzcd}$ Let the source and target of $\\widetilde{\\partial v}$ be $\\widetilde{sv}$ and $\\widetilde{tv}$ respectively.", "By REF , there is a term $w$ in $X$ with $f(w) = v$ .", "It follows that $f(sw) = sv = f(\\widetilde{sv})$ and $f(tw) = tv = f(\\widetilde{tv})$ .", "Hence, by REF there are transformations $\\phi : \\widetilde{sv} \\rightarrow sw$ and $\\psi : tw \\rightarrow \\widetilde{tv}$ such that $f(\\phi ) = sw ;r$ and $f(\\psi ) = tw ; r$ .", "We define $\\begin{aligned}\\tilde{v} = \\phi \\circ w \\circ \\psi \\end{aligned}$ By construction $\\partial \\tilde{v} = \\widetilde{\\partial v}$ .", "Furthermore, every homomorphism of globular multicategories with homomorphism types preserves $- \\circ -$ and so $f(\\tilde{v}) = f(w) = v$ .", "Hence $\\tilde{v}$ defines the required filler and so $f$ is an acyclic fibration.", "Our next result describes conditions under which, given a category $\\mathcal {C}$ and strictification functor $S$ which respects homotopy theoretic information in $\\mathcal {C}$ , there is an induced acyclic fibration of endomorphism operads.", "Theorem 4.8 Let $\\mathcal {C}$ be a category with a factorization system $(\\mathcal {L}, \\mathcal {R})$ .", "Let $X : \\mathbb {G}^{\\operatorname{op}}\\rightarrow \\mathcal {C^{\\operatorname{op}}}$ be a globular object in $\\mathcal {C^{\\operatorname{op}}}$ and let $S : C \\rightarrow D$ be a pullback-preserving functor.", "Suppose that there exists a functor $U : \\mathcal {D} \\rightarrow \\mathcal {C}$ together with a natural transformation $\\phi : \\operatorname{id}\\Rightarrow US$ .", "Suppose that all the following conditions hold: The functor $U$ is faithful.", "Each boundary inclusion $X(\\iota _n) : X(\\partial n) \\rightarrow X(n)$ is an $\\mathcal {L}$ -map.", "For each globular pasting diagram $\\pi $ , the arrow $\\phi _{X(\\pi )}$ is an $\\mathcal {R}$ -map and an epimorphism.", "Then the induced homomorphism of globular operads $\\begin{aligned}\\operatorname{End}(S^{\\operatorname{op}}) : \\operatorname{End}(X) \\longrightarrow \\operatorname{End}(S^{\\operatorname{op}}X)\\end{aligned}$ is an acyclic fibration of globular multicategories.", "Following Example REF it suffices to check the lifting conditions for generic terms.", "Let $u$ be the generic $\\pi $ -shaped $n$ -term and suppose that we have a commutative square of the following form: $\\begin{tikzcd}\\partial u {r}{\\widetilde{\\partial v}} [swap]{d}{\\iota _u}&\\operatorname{End}(X) {d}{\\operatorname{End}(S)}\\\\u [swap]{r}{v} & \\operatorname{End}(S^{\\operatorname{op}}X)\\end{tikzcd}$ By the Yoneda Lemma, the homomorphism $\\widetilde{\\partial }v$ corresponds to a term boundary $\\widetilde{\\partial f}$ in $\\operatorname{End}(X)$ and the homomorphism $v$ corresponds to a term $f$ in $\\operatorname{End}(S^{\\operatorname{op}}X)$ .", "Commutativity of the square tells us that $\\begin{aligned}S(\\widetilde{\\partial f}) = \\partial f\\end{aligned}.$ Expanding the definition of $\\operatorname{End}$ , we find that $\\widetilde{\\partial f}$ corresponds to a boundary-preserving arrow $\\widetilde{\\partial f} : X(\\partial n) \\rightarrow X(\\pi )$ in $\\mathcal {C}$ .", "(Here $X(\\pi )$ comes from the Yoneda extension of $X$ .)", "Similarly $f$ corresponds to a boundary-preserving arrow $f : SX(n) \\rightarrow SX(\\pi )$ in $\\mathcal {D}$ .", "Hence we have the following commutative diagram in $\\mathcal {C}$ : $\\begin{tikzcd}&X(\\partial n){r}{\\widetilde{\\partial f}}[swap]{d}{\\phi _{X(\\partial n)}}&X(\\pi ){d}{\\phi _{X(\\pi )}}\\\\X(\\partial n){r}{\\phi _{X(\\partial n)}}[swap]{d}{X(\\iota _n)}[ur, equal]&USX(\\partial n)[r, bend left = 20, \"{US(\\widetilde{\\partial (f)})}\"{name = top, above}][r, bend right = 20, \"{U \\partial f}\"{below, name=bottom}][swap]{d}{USX(\\iota _n)}\\ [equal, from = top, to = bottom, shorten <= 1ex, shorten >= 1ex]&USX(\\pi )\\\\X(n)[swap]{r}{\\phi _{X(n})}&USX(n)[ur, bend right = 30, \"U f\" swap]\\end{tikzcd}$ The squares commute by naturality and the bottom triangle commutes by definition of $\\partial f$ .", "Now, since $X(\\iota _n)$ is an $\\mathcal {L}$ -map and $\\phi _{\\pi }$ is an $\\mathcal {R}$ -map, the outer rectangle has a filler $\\begin{tikzcd}X(\\partial n){r}{\\widetilde{\\partial f}}[swap]{d}{X(\\iota _n)}&X(\\pi ){d}{\\phi _{X(\\pi )}}\\\\X(n)[swap]{r}{Uf \\circ \\phi _{X(n)}}[ur, dashed, \"{\\tilde{f}}\" description]&USX(\\pi )\\end{tikzcd}\\qquad \\mathrm {(\\dagger )}$ We know that $\\tilde{f}$ is boundary preserving because its boundary $\\widetilde{\\partial f}$ is boundary-preserving.", "Hence, using the definition of $\\operatorname{End}$ , we can view $\\tilde{f}$ as a $\\pi $ -shaped $n$ -term in $\\operatorname{End}(X)$ .", "By the Yoneda Lemma $\\tilde{f}$ corresponds to a map of globular multigraphs $\\begin{aligned}\\tilde{v} : u \\longrightarrow \\operatorname{End}(X)\\end{aligned}.$ Hence it remains to show that the following diagram commutes: $\\begin{tikzcd}\\partial u {r}{\\widetilde{\\partial v}} [swap]{d}{\\iota _u}&\\operatorname{End}(X) {d}{\\operatorname{End}(S^{\\operatorname{op}})}\\\\u [swap]{r}{v} [ur, dashed, \"\\tilde{v}\" description]& \\operatorname{End}(S^{\\operatorname{op}}X)\\end{tikzcd}\\qquad \\mathrm {(\\star )}$ Commutativity of the top triangle of (REF ) says that $\\begin{aligned}\\partial \\tilde{f} = \\widetilde{\\partial f}\\end{aligned}.$ This follows from commutativity of the top triangle of (REF ).", "Commutativity of the bottom triangle of (REF ) says that $\\begin{aligned}S(\\tilde{f}) = f\\end{aligned}.$ By naturality of $\\phi $ and commutativity of the bottom triangle of (REF ), we have that $\\begin{aligned}US(\\tilde{f}) \\circ \\phi _n= \\phi _{\\pi } \\circ \\tilde{f}= Uf \\circ \\phi _{n}\\end{aligned}.$ Since $\\phi _n$ is an epimorphism and $U$ is faithful, the result now follows." ], [ "Weak Higher Categorical Structure", "In this section we show that the types and terms of globular multicategories with homomorphism types have higher categorical structure.", "We will demonstrate two related results.", "Firstly each piece of data internal to a globular multicategory with homomorphism types has higher categorical structure.", "For example, each 0-type has the structure of a weak $\\omega $ -category and each 0-term has the structure of a weak $\\omega $ -functor.", "We will then combine this internal data and show that the external collection of types and terms has the structure of a weak $\\omega $ -category.", "Most of the work involved in moving from the internal to the external situation is done by Theorem REF .", "For the remainder of this section we will label the adjunctions between reflexive globular multigraphs, globular multicategories with homomorphism types and globular multicategories with strict homomorphism types as in the following diagram.", "$\\begin{tikzcd}{\\operatorname{MonGlobCat}}[r, bend right, \"V\" below][r, phantom, \"\\perp \" description]&[bend right, l, \"{G}\" above]\\operatorname{GlobMult}_{\\mathcal {H}_{\\operatorname{Str}}}[r, bend right, \"U\" below][r, phantom, \"\\perp \" description]&\\operatorname{GlobMult}_{\\mathcal {H}}[bend right, l, \"S\" above][bend right, r][r, phantom, \"\\perp \" description]&\\overline{\\operatorname{GlobGraph}}[bend right, l, \"F_{\\mathcal {H}}\" above]\\end{tikzcd}$ The leftmost adjunction adds composition of types.", "We denote the unit of this adjunction by $\\beta : \\operatorname{id}\\Rightarrow V {G}$ .", "The middle adjunction can be thought of as describing strictification of globular multicategories with homomorphism types.", "We denote the unit of this adjunction by $\\eta : \\operatorname{id}\\Rightarrow U S$ .", "We denote the unit of the composite adjunction ${G}S \\dashv UV$ by $\\theta : \\operatorname{id}\\Rightarrow UV {G}S$ .", "We write ${F_{\\operatorname{Cat}}}= {G}S F_{\\mathcal {H}}$ .", "Our “internal” result can now be stated precisely.", "Theorem 5.1 Let $X$ be a reflexive globular multigraph.", "Then the unit of the strictification adjunction at $F_{\\mathcal {H}}X$ $\\begin{tikzcd}F_{\\mathcal {H}}X [two heads, swap]{d}{\\eta _{F_{\\mathcal {H}}X}}\\\\U SF_{\\mathcal {H}}X\\end{tikzcd}$ is an acyclic fibration of globular multicategories with homomorphism types.", "Furthermore, $\\eta _X$ is bijective on types and 0-terms.", "We first describe the strictification adjunction $S \\dashv U$ explicitly.", "The globular multicategory $SY$ is the result of forcing $Y$ to satisfy the additional requirement that for any $n > 0$ , $n$ -terms $f, f^{\\prime }$ and any variable $x$ such that $\\begin{aligned}r_x ; f = r_x ; f^{\\prime }\\end{aligned},$ we have that $\\begin{aligned}f = f^{\\prime }\\end{aligned}.$ Hence, for every pair of contexts $\\Gamma , \\Delta $ , we define an equivalence relation $\\sim $ relating pairs of substitutions $f, f^{\\prime } : \\Gamma \\rightarrow \\Delta $ using the following inductively defined rules: This relation is reflexive, symmetric and transitive.", "For any variable $x$ and terms $f, f^{\\prime }$ , we have that $\\begin{aligned}r_x ; f \\sim r_x ; f^{\\prime } \\Rightarrow f \\sim f^{\\prime }\\end{aligned}.$ Whenever we have substitutions $f, f^{\\prime }, g, g^{\\prime }$ such that $t_k f = s_k g$ and $t_k f^{\\prime } = s_k g^{\\prime }$ , we have that $\\begin{aligned}f \\sim f^{\\prime } &&\\text{and}&& g \\sim g^{\\prime } \\Rightarrow f \\odot _k g \\sim f^{\\prime } \\odot _k g^{\\prime }\\end{aligned}.$ Whenever $f : \\Gamma \\rightarrow \\Delta $ and $g : \\Delta \\rightarrow A$ , we have that $\\begin{aligned}f \\sim f^{\\prime } &&\\text{and}&& g \\sim g^{\\prime } \\Rightarrow f ; g \\sim f^{\\prime } ; g^{\\prime }\\end{aligned}.$ Since terms are the same as substitutions whose target context is a type, this defines an equivalence relation on terms.", "The globular multicategory $SY$ has the same types as $Y$ and has equivalence classes of terms under $\\sim $ as terms.", "The rules defining $\\sim $ ensure that the composition and $J$ -rules of $SY$ are well defined and that the homomorphism types are strict.", "Furthermore, the unit $\\eta _Y : Y \\rightarrow U SY$ is the homomorphism that quotients by $\\sim $ .", "Using this description of $\\sim $ we see that pointwise $\\eta $ is bijective on types, bijective on 0-terms and surjective on terms.", "Hence by Proposition REF , it suffices to show that $\\eta _{F_{\\mathcal {H}}X}$ weakly reflects identities of terms.", "Lemma REF will do most of the work for us and allow us to construct a sort of “normalization procedure” for terms in $F_{\\mathcal {H}}X$ .", "The resulting “normal form” actually lives in the monoidal globular category ${F_{\\operatorname{Cat}}}X$ .", "In other words, by plugging in reflexivity terms, we can always obtain a formal composite of generators in $X$ .", "Consider the following commutative diagram: $\\begin{tikzcd}F_{\\mathcal {H}}X [two heads, swap]{d}{\\eta _{F_{\\mathcal {H}}X}}[r, equal]&F_{\\mathcal {H}}X[d, two heads, \"\\theta _{F_{\\mathcal {H}}X}\"]\\\\U SF_{\\mathcal {H}}X[r, hook, \"U \\beta _{SF_{\\mathcal {H}}X}\" below]&U V {G}S F_{\\mathcal {H}}X[r, equal]&UV {F_{\\operatorname{Cat}}}X\\end{tikzcd}$ Since $V$ is faithful and $U$ preserves limits, the bottom arrow is monic.", "It follows that $\\eta _{F_{\\mathcal {H}}X}$ weakly reflects identities of terms if $\\theta _{F_{\\mathcal {H}}X}$ weakly reflects identities of terms.", "This is Lemma REFREF below.", "Lemma 5.2 Let $X$ be a reflexive globular multigraph.", "Then we have the following results: The unit $\\begin{tikzcd}F_{\\mathcal {H}}X[d, two heads, \"\\theta _{F_{\\mathcal {H}}X}\"]\\\\UV {F_{\\operatorname{Cat}}}X\\end{tikzcd}$ weakly reflects identities of terms.", "The unit $\\begin{tikzcd}S F_{\\mathcal {H}}X[d, two heads, \"\\beta _{S F_{\\mathcal {H}}X}\"]\\\\V G X\\end{tikzcd}$ strictly reflects identities of terms.", "The furthermore part is easily verified.", "We will prove REF .", "The proof of REF is similar except all identifications are actual identities.", "Let $X$ be a reflexive globular set.", "Then explicitly: $F_{\\mathcal {H}}X$ is generated from $X$ by freely adding $-;-$ composition, reflexivity terms and J-terms.", "$UV {F_{\\operatorname{Cat}}}X$ is freely generated from $X$ by freely adding $- ; -$ composition and $-\\otimes _k$ - composition for each $k$ .", "Equivalently a term in $UV {F_{\\operatorname{Cat}}}X$ is a formal composite $\\gamma _1 ; \\cdots ; \\gamma _l$ of substitutions in $X$ .", "The homomorphism $\\theta _{F_{\\mathcal {H}}X}$ maps types in $F_{\\mathcal {H}}X$ to types in $UV {F_{\\operatorname{Cat}}}X$ up to “removing homomorphism types”.", "That is, for each $n$ -type $A$ , we set $\\begin{aligned}\\ [A] \\sim \\mathcal {H}_A\\end{aligned}.$ Substitutions in $F_{\\mathcal {H}}X$ are mapped to substitutions in $X$ up to “removing homomorphism types”.", "That is, for each term $f$ , we set $\\begin{aligned}\\ [f] \\sim [f ; r_A]\\end{aligned}$ and for each variable $x$ and any J-rule at $x$ , we set $\\begin{aligned}\\ [f] \\sim [J_x(f)]\\end{aligned}.$ We also require that $-;-$ and $-\\odot _k-$ are respected.", "Since terms are substitutions whose target is a type, this defines a homomorphism.", "Now let $f, f^{\\prime } : \\Gamma \\rightarrow \\Delta $ be substitutions in $F_{\\mathcal {H}}X$ .", "such that $\\theta _{F_{\\mathcal {H}}X}(f) = \\theta _{F_{\\mathcal {H}}X}(f^{\\prime })$ .", "Let $H$ be the set of homomorphism type variables in $\\Gamma $ .", "As stated above, the equivalence class $\\theta _{F_{\\mathcal {H}}X}(f)$ corresponds to a composable sequence $\\begin{aligned}\\gamma _1 ; \\gamma _2 ; \\cdots ; \\gamma _l\\end{aligned}$ of substitutions in $X$ .", "We will now prove the claim by induction on $l$ .", "First suppose that $l = 0$ .", "By induction on $f$ , we must have that $\\begin{aligned}r_{H} ; f = R\\end{aligned}$ where $R$ is some composite of identity and reflexivity terms.", "Similarly $\\begin{aligned}r_{H} ; f = R^{\\prime }\\end{aligned}$ where $R^{\\prime }$ is some composite of identity and reflexivity terms.", "Since $f$ and $f^{\\prime }$ have the same target, namely $\\Delta $ , it follows that $R = R^{\\prime }$ .", "Hence, since the source contexts of $f$ and $f^{\\prime }$ are the same, we have an identification $f \\rightarrow f^{\\prime }$ by Lemma REF .", "Now suppose that $l > 0$ .", "Let $H_{\\gamma _1}$ be the set of homomorphism type variables in the source context of $\\gamma _1$ .", "By induction on the rules defining $\\theta $ , it follows that $\\begin{aligned}r_H ; f = r_{H_{\\gamma _1}} ; \\gamma _1 ; f_{>1}\\end{aligned}$ for some term $f_{>1}$ such that $\\theta _{F_{\\mathcal {H}}X}(f_{>1}) = \\gamma _2 ; \\cdots ; \\gamma _l$ .", "Similarly we have that $\\begin{aligned}r_H ; f^{\\prime } = r_{H_{\\gamma _1}} ; \\gamma _1 ; f^{\\prime }_{>1}\\end{aligned}$ for some term $f^{\\prime }_{>1}$ such that $\\theta _{F_{\\mathcal {H}}X}(f^{\\prime }_{>1}) = \\gamma _2 ; \\cdots ; \\gamma _l$ .", "Now, by induction, we have an identification $\\begin{aligned}\\phi : f_{>1} \\longrightarrow f^{\\prime }_{>1}\\end{aligned}$ Hence we have an identification $\\begin{aligned}J_H^{f, f^{\\prime }}(r_{H_{\\gamma _1}} ; \\gamma _1 ; \\phi ) : f \\longrightarrow f^{\\prime }\\end{aligned}.$ The result follows by induction.", "Example 5.3 Let $X$ be a globular multicategory with strict homomorphism types and Let 0 denote the generic 0-type.", "Then it follows from our explicit descriptions that $U SF_{\\mathcal {H}}0$ is the terminal globular operad.", "Thus, Theorem REF tells us that $F_{\\mathcal {H}}0$ is a normalized contractible globular operad.", "By the Yoneda Lemma and adjointness, every 0-type $A$ in $X$ corresponds, to a homomorphism $A : F_{\\mathcal {H}}0 \\rightarrow X$ of globular multicategories with homomorphism types.", "It follows that every 0-type in $X$ with its tower of homomorphism types has the structure of a weak $\\omega $ -category.", "Example 5.4 Let $I_0$ be the generic 0-term.", "Then $SF_{\\mathcal {H}}I_0$ can be seen as the algebraic theory describing a strict functor between strict $\\omega $ -categories.", "The fact that there is an acyclic fibration $F_{\\mathcal {H}}I_0 \\rightarrow U SF_{\\mathcal {H}}I_0$ can be viewed as saying that $F_{\\mathcal {H}}I_0$ is a weak functor between weak $\\omega $ -categories.", "Consequently every 0-term in a globular multicategory with homomorphism types has the structure of a weak functor.", "More generally, Theorem REF can be seen as saying that, in a globular multicategory with homomorphism types, The 0-types are weak $\\omega $ -categories.", "The 1-types are weak profunctors.", "The 2-types are weak profunctors between weak profunctors.", "...", "The 0-terms are weak $\\omega $ -functors The 1-terms are weak transformations between profunctors The 2-terms are weak transformations between 2-types ... Recall that a transformation between parallel $n$ -terms $f, f^{\\prime } : \\Gamma \\rightarrow A$ is a term $\\phi : \\Gamma \\rightarrow \\mathcal {H}_A$ such that $s \\phi = f$ and $t\\phi = f^{\\prime }$ .", "In other words, a transformation is an abstract assignment taking an object in $\\Gamma $ and outputting an arrow in $A$ from the output of $f$ to the output of $f^{\\prime }$ .", "Hence transformations between 0-terms, can be seen as natural transformations.", "Similarly transformations between transformations between 0-terms can be seen as modifications.", "Continuing in this way, let $\\mathbb {T}_0 : \\mathbb {G}^{\\operatorname{op}}\\rightarrow \\overline{\\operatorname{GlobGraph}}^{\\operatorname{op}}$ be the globular object defined by: $\\begin{aligned}\\mathbb {T}_0(n) &={\\left\\lbrace \\begin{array}{ll}\\text{the generic $0$-type } \\bullet & \\text{if $n = 0$}\\\\\\text{the generic term } u_{n-1} : A \\rightarrow \\mathcal {H}^{n - 1} B & \\text{if $n > 0$}\\end{array}\\right.", "}\\\\\\mathbb {T}(\\sigma _n) & ={\\left\\lbrace \\begin{array}{ll}\\text{the inclusion of } A & \\text{if $n = 0$}\\\\\\text{the inclusion of }s u_n & \\text{if $n > 0$}\\end{array}\\right.", "}\\\\\\mathbb {T}(\\tau _n) &={\\left\\lbrace \\begin{array}{ll}\\text{the inclusion of } B & \\text{if $n = 0$}\\\\\\text{the inclusion of }t u_n & \\text{if $n > 0$}\\end{array}\\right.", "}\\end{aligned}$ For any globular multicategory with homomorphism types $X$ , we have a globular set $n \\mapsto \\hom (F_{\\mathcal {H}}^{\\operatorname{op}}\\mathbb {T}_0(n), X)$ of 0-types, 0-terms and transformations between them in $X$ .", "We will show that this globular set has the structure of a weak $\\omega $ -category.", "Theorem 5.5 The operad $\\operatorname{End}(F_{\\mathcal {H}}^{\\operatorname{op}}\\mathbb {T}_0)$ is normalized and contractible.", "The following result is an immediate application.", "Corollary 5.6 The 0-types, 0-terms and transformations between them in a globular multicategory with homomorphism types form a weak $\\omega $ -category.", "First note that Theorem REF gives us the following result: Proposition 5.7 The homomorphism $\\operatorname{End}(F_{\\mathcal {H}}^{\\operatorname{op}}\\mathbb {T}_0) \\rightarrow \\operatorname{End}(S^{\\operatorname{op}}F_{\\mathcal {H}}^{\\operatorname{op}}\\mathbb {T}_0)$ is an acyclic fibration of normalized globular multicategories.", "The globular object $ F_{\\mathcal {H}}^{\\operatorname{op}}\\mathbb {T}_0$ , together with the adjunction $\\begin{tikzcd}\\operatorname{GlobMult}_{\\mathcal {H}}^{\\operatorname{op}}[r, bend left, \"S\" above][r, phantom, \"\\bot \" description]&\\operatorname{GlobMult}_{\\mathcal {H}_{\\operatorname{Str}}}[l, bend left, \"U\" below]\\end{tikzcd}$ and the natural transformation $\\eta : \\operatorname{id}\\Rightarrow U S$ are easily seen to satisfy the conditions of Theorem REF .", "Thus, it suffices to show that $\\operatorname{End}(S^{\\operatorname{op}}F_{\\mathcal {H}}^{\\operatorname{op}}\\mathbb {T}_0)$ is the terminal globular operad.", "This is really an abstract expression of the well known fact that the collection of all strict $\\omega $ -categories is a strict $\\omega $ -category.", "Lemma 5.8 Suppose that $X$ is a reflexive globular multigraph and that $Y = S F_{\\mathcal {H}}X$ .", "Let $0 \\le k < n$ and suppose that $\\beta _Y(f), \\beta _Y(g)$ are parallel terms in $V G Y$ such that $t_k \\beta _Y(f) = s_k \\beta _Y(g)$ .", "Then we can always choose substitutions $f, g$ in $Y$ such that $t_k f = s_k g$ .", "In this case we have that $\\beta _Y(f \\odot _k g) = \\beta _Y(f) \\otimes \\beta _Y(g)$ .", "It follows from our analysis in the proof of REF that there exist substitutions $\\gamma _1, \\ldots , \\gamma _l$ , $\\delta _1, \\ldots , \\delta _l$ in $X$ such that $\\beta _Y(f) = \\gamma _1 ; \\cdots ; \\gamma _l$ and $\\beta _Y(g) = \\delta _1 ; \\cdots ; \\delta _{l}$ and $t_k \\gamma _i = s_k \\delta i$ for each $i$ .", "We can now use J-terms to find a composable list of substitutions $\\epsilon _1, \\ldots , \\epsilon _l$ in $Y$ such that $\\beta _Y(\\epsilon _i) = t_k \\gamma _i = s_k \\delta _i$ .", "We can then use J-terms again to find composable lists $\\tilde{\\gamma }_1, \\ldots \\tilde{\\gamma }_l$ and $\\tilde{\\delta }_1, \\ldots , \\tilde{\\delta }_l$ in $Y$ such that $\\beta _Y(\\tilde{\\gamma }_i) = \\gamma _i$ and $\\beta _Y(\\tilde{\\delta }_i) = \\delta _i$ and such that $t_k \\tilde{\\gamma }_i = \\epsilon _i = t_k \\tilde{\\delta }_i$ .", "Thus we can set $f = \\tilde{\\gamma }_1 ; \\cdots \\tilde{\\gamma }_l$ and $g = \\tilde{\\delta }_1 ; \\cdots \\tilde{\\delta }_l$ .", "Definition 5.9 Suppose that $X$ and $Y$ are as above.", "Suppose furthermore that every term of $X$ is of the form $f : A \\rightarrow \\mathcal {H}^n_B$ where $A$ and $B$ are 0-types.", "In this case we say that $Y$ is an arrow theory.", "We denote the category of arrow theories and homomorphism type preserving maps by $\\mathbb {A}$ .", "Proposition 5.10 Suppose that $f, g$ are terms in an arrow theory $Y$ such that $t_k f = s_k g$ .", "Then there exists a term $h$ in $Y$ such that $\\begin{aligned}\\beta _Y(h) = \\beta _Y(f) \\otimes \\beta _Y(g)\\end{aligned}.$ We must that $f : \\Gamma \\rightarrow \\mathcal {H}^n_A$ and $g : \\Delta \\rightarrow \\mathcal {H}^m_A$ are substitutions where $A$ is a 0-type in $Y$ and $0 \\le n \\le m$ .", "Suppose that $0 \\le k < m$ .", "Then using J-rules we can construct a term $m : \\mathcal {H}^n_A \\odot _k \\mathcal {H}^m_A \\rightarrow \\mathcal {H}^m_A$ such that $\\beta _Y(m) = \\beta _Y(r^m_A)$ .", "It follows that $\\begin{aligned}\\beta _Y((f \\odot _k g) ; m) = (\\beta _Y(f) \\odot _k \\beta _Y(g)) ; \\beta _Y(m)= \\beta _Y(f) \\odot _k \\beta _Y(g)\\end{aligned}.$ and this is the same as $\\beta _Y(f) \\otimes _k \\beta _Y(g)$ .", "Hence $\\beta _{Y}$ is surjective on types and terms.", "Corollary 5.11 We have a fully faithful functor $\\begin{aligned}\\mathbb {I}: \\mathbb {A}\\longrightarrow {\\operatorname{Str}\\omega \\operatorname{-Cat}}\\end{aligned}.$ It follows from Lemma REFREF that $\\beta _{Y}$ is injective on types and terms.", "Hence the homomorphism $\\beta _{Y}$ is bijective on types and terms.", "It follows that there is an induced strict monoidal globular category structure on $Y$ such that $\\beta : Y \\rightarrow VG Y$ is an isomorphism of strict monoidal globular categories.", "Moreover every map preserving homomorphism types preserves the $-\\otimes _k-$ operations defined using Lemma REF .", "Hence, we have a fully faithful functor $\\begin{aligned}B : \\mathbb {A}\\longrightarrow {\\operatorname{MonGlobCat}}\\end{aligned}.$ The monoidal globular categories in the image of $B$ are precisely those with no non-trivial $n$ -types for $n > 1$ and which are freely generated by a reflexive globular multigraph.", "However, such monoidal globular multicategories are the same as strict $\\omega $ -categories freely generated by reflexive globular sets.", "(We view the 0-types of an arrow theory as the 0-cells of a reflexive globular set.", "We view the $n$ -terms of an arrow theory as the $(n+1)$ -cells of this reflexive globular set.)", "Hence, we have a fully faithful functor $\\mathbb {I} : \\mathbb {A}\\rightarrow {\\operatorname{Str}\\omega \\operatorname{-Cat}}$ to the category of strict of $\\omega $ -categories.", "Proposition 5.12 The globular operad $\\operatorname{End}(S^{\\operatorname{op}}F_{\\mathcal {H}}^{\\operatorname{op}}\\mathbb {T}_0)$ is the terminal globular operad.", "By definition a $\\pi $ -shaped $n$ -term in this operad is a boundary preserving homomorphism of arrow theories $\\begin{aligned}v : SF_{\\mathcal {H}}\\mathbb {T}_0(n) \\longrightarrow SF_{\\mathcal {H}}\\mathbb {T}_0(\\pi )\\end{aligned}$ Using the fully faithful functor $\\mathbb {I} : \\mathbb {A}\\longrightarrow {\\operatorname{Str}\\omega \\operatorname{-Cat}}$ of Corollary REF , this homomorphism of arrow theories corresponds to a boundary preserving functor $\\begin{aligned}\\mathbb {I}(v) : T n \\longrightarrow T \\pi \\end{aligned}.$ from the strict $\\omega $ -category generated by a single $n$ -cell to the strict $\\omega $ -category generated by the pasting diagram $\\pi $ .", "In the terminology of [5], the functor $\\mathbb {I}(v)$ is active.", "It follows that there is a unique functor of this form and so $v$ is uniquely determined." ], [ "Acknowledgements", "I would like thank my supervisor Kobi Kremnitzer for many useful conversations.", "This work at the University of Oxford was supported by an EPSRC studentship." ] ]

[ [ "Suppression of superconducting parameters by correlated\n quasi-two-dimensional magnetic fluctuations" ], [ "Abstract We consider a clean layered magnetic superconductor in which a continuous magnetic transition takes place inside superconducting state and the exchange interaction between superconducting and magnetic subsystems is weak so that superconductivity is not destroyed at the magnetic transition.", "An example of such material is RbEuFe$_{4}$As$_{4}$.", "We investigate the suppression of the superconducting gap and superfluid density by correlated magnetic fluctuations in the vicinity of the magnetic transition.", "The influence of nonuniform exchange field on superconducting parameters is sensitive to the relation between the magnetic correlation length, $\\xi_{h}$, and superconducting coherence length $\\xi_{s}$ defining the 'scattering' ($\\xi_{h}<\\xi_{s}$) and 'smooth' ($\\xi_{h}>\\xi_{s}$) regimes.", "As a small uniform exchange field does not affect the superconducting gap and superfluid density at zero temperature, smoothening of the spatial variations of the exchange field reduces its effects on these parameters.", "We develop a quantitative description of this 'scattering-to-smooth' crossover for the case of quasi-two-dimensional magnetic fluctuations.", "Since the magnetic-scattering probability varies at the energy scale comparable with the gap, the quasiclassical approximation is not applicable in the crossover region and microscopic treatment is required.", "We find that the corrections to both the gap and superfluid density grow proportionally to $\\xi_{h}$ until it remains much smaller than $\\xi_{s}$.", "When $\\xi_{h}$ exceeds $\\xi_{s}$, both parameters have much weaker dependence on $\\xi_{h}$.", "Moreover, the gap correction may decrease with increasing of $\\xi_{h}$ in the vicinity of the magnetic transition.", "We also find that the crossover is unexpectedly broad: the standard scattering approximation becomes sufficient only when $\\xi_{h}$ is substantially smaller than $\\xi_{s}$." ], [ "Introduction", "Since the seminal work of Abrikosov and Gor’kov (AG)[1] and its extensions[2], [3], [4], the pair breaking by magnetic scattering has been established as a key concept in the physics of superconductivity.", "Its applications extend far beyond the original physical system for which the theory was developed, singlet superconductors with dilute magnetic impurities.", "In particular, the magnetic pair-breaking scattering strongly influences properties of superconducting materials containing an embedded periodic lattice of magnetic rare-earth ions.", "Several classes of such materials are known at present including magnetic Chevrel phases $\\mathit {RE}$ Mo$_{6}$X$_{8}$ ($\\mathit {RE}$ =rare-earth element and X=S, Se), ternary rhodium borides $\\mathit {RE}$ Rh$_{4}$ B$_{4}$[5], [6], [7], [8], the rare-earth nickel borocarbides $\\mathit {RE}$ Ni$_{2}$ B$_{2}$ C[9], [10], [11], and recently discovered Eu-based iron pnictides[12], [13], [14], [15], [16].", "Some of these compounds experience a magnetic-ordering transition inside the superconducting state.", "Depending on the strength of the exchange interaction between the rare-earth moments and conducting electrons, the magnetic transition may either destroy superconductivity or leave it intact.", "In any case, in the paramagnetic state, the fluctuating magnetic moments suppress superconductivity via magnetic scattering, similar to magnetic impurities.", "Near the ferromagnetic transition, the moments become strongly correlated which enhances the suppression.", "The AG theory has been generalized to describe this enhancement in several theoretical studies [17], [18].", "A straightforward generalization, however, is only possible when the magnetic correlation length $\\xi _{h}$ is shorter than the superconducting coherence length $\\xi _{s}$ and this condition was always assumed in all theoretical works.", "For a continuous magnetic transition, there is always temperature range where this condition is violated, see Fig.", "REF (a).", "A small uniform exchange field does not modify superconducting gap in clean materials at zero temperature [19], because, in absence of free quasiparticles, the exchange field does not generate spin polarization of the Cooper-pair condensate.", "This observation indicates that, once the exchange field becomes smooth at the scale of coherence length, its efficiency in suppressing superconducting parameters at low temperatures diminishes.", "We can conclude that the existing treatments of the impact of correlated magnetic fluctuations on superconductivity are incomplete.", "A full theoretical description of this phenomenon requires consideration of the crossover between the ‘scattering’ and ‘smooth’ regimes illustrated in Fig.", "REF (a).", "For most magnetic superconductors, however, such full theory would be a mostly academic exercise, because the coherence length is typically much larger than the separation between magnetic ions.", "Consider, for example, the magnetic nickel borocarbide ErNi$_{2}$ B$_{2}$ C, which has the superconducting transition at T$_{c}\\approx $ 11 K and magnetic transition at T$_{m}\\approx $ 6 K [9], [10].", "Its c-axis upper critical field has linear slope 0.3 T/K near T$_{c}$ [20], from which we can estimate the in-plane coherence length at T$_{m}$ as $\\xi _{s}(T_{m})\\approx $ 18 nm which is $\\sim 54$ times larger than the distance between the Er$^{3+}$ moments.", "Therefore, in this and similar materials the magnetic correlation length exceeds the coherence length only within an extremely narrow temperature range near the magnetic transition.", "The situation is very different, however, in Eu-based layered iron pnictides, such as RbEuFe$_{4}$ As$_{4}$[13], [15], [16], [21].", "The latter material has the superconducting transition at 36.5 K and the magnetic transition at 15K.", "The magnetism is quasi-two-dimensional: the Eu$^{2+}$ moments have strong ferromagnetic interactions inside the magnetic layers with easy-plane anisotropy [22] and weak interactions between the magnetic planes leading to helical interlayer order[23], [24].", "Due to the quasi-two-dimensional nature of magnetism, the in-plane magnetic correlation length smoothly grows within an extended temperature range.", "Another relevant material's property is a very short in-plane coherence length, $\\sim $ 1.5–2 nm, which is only 4–6 times larger than the distance between the magnetic ions.", "As a consequence, contrary to most magnetic superconductors, the magnetic correlation length exceeds the coherence length within a noticeable temperature range near the magnetic transition.", "Therefore, for the magnetic iron pnictides, the crossover between the ‘scattering’ and ‘smooth’ regimes is very relevant.", "Recent vortex imaging in RbEuFe$_{4}$ As$_{4}$ with scanning Hall-probe spectroscopy revealed a significant increase of the London penetration depth in the vicinity of the magnetic transition[25].", "This suggests that the exchange interaction between Eu$^{2+}$ moments and Cooper pairs leads to substantial suppression of superconducting parameters near $T_{m}$ .", "The goal of this paper is to develop a quantitative theoretical description of the influence of correlated magnetic fluctuations on the superconducting gap and supercurrent response with a proper treatment of the crossover at $\\xi _{h}\\sim \\xi _{s}$ .", "The problem occurs to be technically challenging because in the crossover region the probability of magnetic scattering varies at the energy scale comparable with the temperature or the gap.", "This forbids the standard energy integration necessary for the quasiclassical approximation and requires a full microscopic consideration.", "In this consideration, one has to include the self-energy correction to the electronic spectrum and maintain the energy dependence of the scattering probability.", "As this accurate analysis is rather complicated, we utilize several simplifying assumptions.", "We limit ourselves to the case of weak exchange interaction and consider only the lowest-order corrections.", "We also assume the static approximation for magnetic fluctuations.", "This assumption is justified when typical frequency scale for magnetic fluctuations is smaller than the superconducting gap.", "Due to the critical slowing down, this always becomes valid sufficiently close to the transition.", "In the scattering regime, the dynamic effects have been investigated in several theoretical papers, see, e.g., [17], [26], [27].", "The behavior is also sensitive to the dimensionality of magnetic fluctuations.", "Having in mind application to layered magnetic superconductors, such as RbEuFe$_{4}$ As$_{4}$ , we assume quasi-two-dimensional magnetic fluctuations.", "In this case the discussed effects are more pronounced than for three-dimensional magnetic fluctuations[18].", "The paper is organized as follows.", "In Sec.", ", we introduce the model for layered magnetic superconductors.", "In Sec.", ", we evaluate the self energy caused by scattering by correlated magnetic fluctuations for arbitrary relation between the magnetic correlation length and coherence length and develop a quantitative description of the crossover between the scattering and smooth regimes.", "In Sec.", ", we use these results to evaluate the exchange correction to the gap.", "In Sec.", ", we evaluate the leading correction to the electromagnetic kernel accounting for the vertex correction.", "Also, in Appendix this correction is evaluated in the scattering regime with quasiclassical approach.", "Finally, in Sec.", ", we discuss the results and illustrate them by plotting representative temperature dependences for the parameters roughly corresponding to RbEuFe$_{4}$ As$_{4}$ .", "Figure: (a)Schematic temperature dependences of the magnetic correlation lengthξ h \\xi _{h} and superconducting coherence length ξ s \\xi _{s}.", "The influenceof the fluctuating magnetic moments on superconductivity is very differentin the regions ξ h >ξ s \\xi _{h}>\\xi _{s} and ξ h <ξ s \\xi _{h}<\\xi _{s}.", "(b)Typicalscales in momentum space characterizing scattering on magnetic fluctuationsfor two relations between ξ h \\xi _{h}and ξ s \\xi _{s} in the case ξ h ≫k F -1 \\xi _{h}\\gg k_{F}^{-1}with ℏk F =p F \\hbar k_{F}=p_{F}.", "The small circle illustrates the small-anglescattering on magnetic fluctuations with the range p-p ' ∼ℏ/ξ h \\left|p-p^{\\prime }\\right|\\sim \\hbar /\\xi _{h}and the ring with width ℏ/ξ s \\hbar /\\xi _{s} illustrate the range relevantfor superconductivity." ], [ "Model", "We consider a layered material composed of superconducting and magnetic layers described by the Hamiltonian $\\mathcal {H}=\\hat{\\mathcal {H}}_{\\mathrm {S}}+\\hat{\\mathcal {H}}_{\\mathrm {M}}+\\hat{\\mathcal {H}}_{\\mathrm {MS}},$ where $\\hat{\\mathcal {H}}_{\\mathrm {S}} & =\\sum _{n,\\mathbf {p}_{\\parallel },\\sigma }\\xi _{\\mathrm {2D}}(\\mathbf {p}_{\\parallel })a_{n,\\sigma }^{\\dagger }(\\mathbf {p}_{\\parallel })a_{n,\\sigma }(\\mathbf {p}_{\\parallel })\\nonumber \\\\+ & \\!\\sum _{n,\\mathbf {p}_{\\parallel },\\sigma }\\!t_{\\bot }\\left[a_{n+1,\\sigma }^{\\dagger }(\\mathbf {p}_{\\parallel })a_{n,\\sigma }(\\mathbf {p}_{\\parallel })\\!+\\!a_{n-1,\\sigma }^{\\dagger }(\\mathbf {p}_{\\parallel })a_{n,\\sigma }(\\mathbf {p}_{\\parallel })\\right]\\nonumber \\\\- & \\sum _{n,\\mathbf {p}_{\\parallel }}\\left[\\Delta a_{n,\\uparrow }^{\\dagger }(\\mathbf {p}_{\\parallel })a_{n,\\downarrow }^{\\dagger }(-\\mathbf {p}_{\\parallel })\\!+\\!\\Delta ^{\\ast }a_{n,\\downarrow }(-\\mathbf {p}_{\\parallel })a_{n,\\uparrow }(\\mathbf {p}_{\\parallel })\\right]$ is the standard BCS Hamiltonian describing a layered superconductor.", "Here $\\sigma $ is spin index, $\\xi _{\\mathrm {2D}}(\\mathbf {p}_{\\parallel })=\\varepsilon _{\\mathrm {2D}}(\\mathbf {p}_{\\parallel })-\\mu $ is the single-layer spectrum, $t_{\\bot }$ is the interlayer hopping integral, and $\\Delta $ is the superconducting gap.", "The full 3D spectrum for this model is $\\xi _{\\mathbf {p}}\\!=\\!\\xi _{\\mathrm {2D}}(\\mathbf {p}_{\\parallel })\\!+\\!2t_{\\bot }\\cos \\left(p_{z}s\\right)$ .", "However, its exact shape has a very little effect on further consideration.", "The second term, $\\hat{\\mathcal {H}}_{\\mathrm {M}}$ , describes the quasi-two-dimensional magnetic subsystem leading to a continuous phase transition at $T_{m}$ .", "The last term $\\hat{\\mathcal {H}}_{\\mathrm {MS}} \\!=\\!\\!\\sum _{n,m,\\mathbf {R}}\\!\\int \\!\\!d^{2}r J_{nm}\\!\\left(\\mathbf {r}\\!-\\!\\mathbf {R}\\right)\\mathbf {S}_{m}(\\mathbf {R})\\hat{\\mathbf {\\sigma }}_{\\alpha \\beta }a_{n,\\alpha }^{\\dagger }(\\mathbf {r})a_{n,\\beta }(\\mathbf {r})$ describes the interaction between the magnetic and superconducting layers with the strength set by the nonlocal exchange constants $J_{nm}\\left(\\mathbf {r}\\!-\\!\\mathbf {R}\\right)$ .", "Here the index $m$ marks magnetic layers, $\\hat{\\mathbf {\\sigma }}$ is Pauli-matrix vector, and summation is assumed over the spin indices $\\alpha $ and $\\beta $ .", "We can rewrite the interaction term as $\\hat{\\mathcal {H}}_{\\mathrm {MS}}\\!=\\!-\\!\\sum _{n}\\int d^{2}ra_{n,\\alpha }^{\\dagger }(r)h_{n}(r)\\hat{\\sigma }_{\\alpha \\beta }a_{n,\\beta }(r),$ where $h_{n}(r)\\!=\\!-\\!\\sum _{m,R}J_{nm}(r\\!-\\!R)S_{m}(R)$ is the effective exchange field acting on spins of conducting electrons.", "It can be split into the average part $\\bar{h}$ due to either polarization of the moments by the magnetic field or spontaneous magnetization in the ordered state and the fluctuating part $\\tilde{h}_{n}(r)$ , $h_{n}(r)=\\bar{h}+\\tilde{h}_{n}(r)$ , $\\bar{h} & =-\\sum _{m,R}J_{nm}(r-R)\\bar{S},\\\\\\tilde{h}_{n}(r) & =-\\sum _{m,R}J_{nm}(r-R)\\tilde{S}_{m}(R).$ The fluctuating part of the exchange field also depends on time.", "We assume that the time scales of magnetic fluctuations exceeds time scales relevant for superconductivity and employ the quasistatic approximation.", "This assumption is justified near the transition due to the critical slowing down.", "The fluctuating part is characterized by the correlation function $& \\left\\langle \\tilde{h}_{n}(r)\\tilde{h}_{n^{\\prime }}(r^{\\prime })\\right\\rangle \\\\\\!", "& =\\!\\!\\sum _{m,R,R^{\\prime }}J_{nm}(r\\!-\\!R)J_{n^{\\prime }m}(r^{\\prime }\\!-\\!R^{\\prime })\\left\\langle \\tilde{S}_{m}(R)\\tilde{S}_{m}(R^{\\prime })\\right\\rangle .\\nonumber $ Here we neglected correlations between different magnetic layers.", "In the following, we limit ourselves to the case when the uniform field, $\\bar{h}$ , can be neglected.", "This corresponds to the paramagnetic state and ordered state near the transition in the absence of an external magnetic field.", "We will also neglect correlations between different conducting layers and drop the layer index, $\\left\\langle \\tilde{h}_{n}(r)\\tilde{h}_{n^{\\prime }}(r^{\\prime })\\right\\rangle \\rightarrow \\delta _{n,n^{\\prime }}\\left\\langle \\tilde{h}(r)\\tilde{h}(r^{\\prime })\\right\\rangle $ .", "This corresponds to the two-dimensional approximation for magnetic fluctuations.", "The spin correlation function is related to the nonlocal spin susceptibility $\\chi (r-r^{\\prime })$ .", "Sufficiently close to the magnetic transition, the spin correlation length exceeds the range of $J_{nm}(r-R)$ and we can approximate $\\left\\langle \\tilde{h}(r)\\tilde{h}(r^{\\prime })\\right\\rangle \\approx \\sum _{m}\\mathcal {J}_{nm}^{2}\\left\\langle \\tilde{S}_{m}(r)\\tilde{S}_{m}(r^{\\prime })\\right\\rangle .$ with $\\mathcal {J}_{nm}=\\sum _{R}J_{nm}(r-R)$ .", "Away from the transition, however, the nonlocality of the exchange interaction may have substantial influence on the amplitude and extent of the exchange-field correlations.", "We neglect these complications and assume the simplest shape of the correlation function of $\\tilde{h}(r)$ defined by a single length scale, the in-plane magnetic correlation length $\\xi _{h}$ , $\\left\\langle \\tilde{h}_{\\alpha }(r)\\tilde{h}_{\\beta }(r^{\\prime })\\right\\rangle & =\\frac{h_{0}^{2}}{2}\\delta _{\\alpha \\beta }f_{h}\\left(\\left|r\\!-\\!r^{\\prime }\\right|/\\xi _{h}\\right),$ where $f_{h}(0)\\!=\\!1$ , and the parameter $h_{0}^{2}\\!=\\!\\left\\langle \\tilde{h}^{2}\\right\\rangle \\!\\approx \\!\\sum _{m}\\mathcal {J}_{nm}^{2}\\left\\langle \\tilde{S}^{2}\\right\\rangle $ weakly depends on temperature.", "The Fourier transform of the correlation function is $\\left\\langle \\left|\\tilde{h}_{\\mathbf {\\mathbf {q}}}\\right|^{2}\\right\\rangle & \\!=\\!sh_{0}^{2}\\int \\!d^{2}rf_{h}\\left(\\frac{r}{\\xi _{h}}\\right)\\exp \\left(iqr\\right)\\!=\\!sh_{0}^{2}\\xi _{h}^{2}\\tilde{f}_{h}\\left(\\xi _{h}q\\right).$ Here we assume a conventional Lorentz shape for the $q$ dependence, $\\tilde{f}_{h}\\left(\\xi _{h}q\\right)\\!=\\!C_{h}/\\left(1\\!+\\!\\xi _{h}^{2}q^{2}\\right)$ with $C_{h}\\!=\\!2\\pi \\int _{0}^{\\infty }\\!f_{h}\\left(x\\right)xdx$ .", "In real space, this corresponds to $f_{h}\\left(\\frac{r}{\\xi _{h}}\\right)\\!=\\!\\xi _{h}^{2}\\!\\int \\!\\frac{d^{2}q}{(2\\pi )^{2}}\\frac{C_{h}}{1\\!+\\!\\xi _{h}^{2}q^{2}}\\exp (iqr)\\!=\\!\\frac{C_{h}}{2\\pi }K_{0}\\left(\\frac{r}{\\xi _{h}}\\right).$ The logarithmic divergency $K_{0}(r/\\xi _{h})\\!\\propto \\!\\ln \\left(\\xi _{h}/r\\right)$ has to be terminated at the distance between neighboring moments $r\\!\\sim \\!a$ .", "Since the function $f_{h}\\left(r/\\xi _{h}\\right)$ is normalized by the condition $f_{h}(0)=1$ , this means that $C_{h}\\!\\approx \\!2\\pi /\\ln \\left(\\xi _{h}/a\\right)$ .", "We will utilize the Green's functions formulation of the superconductivity theory [28], [29].", "For investigation of scattering by the magnetic fluctuations, we have to operate with the matrix $4\\times 4$ Green's function[3], $\\hat{G}(1,2)=-\\begin{pmatrix}\\left\\langle T_{\\tau }a_{\\alpha }^{\\dagger }(1)a_{\\beta }(2)\\right\\rangle & \\left\\langle T_{\\tau }a_{\\alpha }(1)a_{\\beta }(2)\\right\\rangle \\\\\\left\\langle T_{\\tau }a_{\\alpha }^{\\dagger }(1)a_{\\beta }^{\\dagger }(2)\\right\\rangle & \\left\\langle T_{\\tau }a_{\\alpha }(1)a_{\\beta }^{\\dagger }(2)\\right\\rangle \\end{pmatrix}.$ We will expand it with respect to the fluctuating exchange field.", "The unperturbed Green's function can be written as $\\hat{G}_{0} & =-\\frac{\\left(i\\omega _{n}\\hat{\\tau }_{0}+\\xi _{\\mathbf {p}}\\hat{\\tau }_{z}\\right)\\hat{\\sigma }_{0}-\\Delta \\hat{\\sigma }_{y}\\hat{\\tau }_{y}}{\\omega _{n}^{2}+\\xi _{\\mathbf {p}}^{2}+\\Delta ^{2}}$ where $\\hat{\\sigma }_{a}$ and $\\hat{\\tau }_{b}$ are the Pauli matrices in the spin and Nambu space, respectively.", "We see that the unperturbed Green's function can be expanded as $\\hat{G}=\\sum _{ab}\\hat{\\sigma }_{a}\\hat{\\tau }_{b}G_{ab}$ and, without the uniform exchange field, the only nonzero components are 00, $0z$ , and $yy$ .", "For the single-band BCS model, the gap equation is $\\Delta =UT\\sum _{\\omega _{n}}\\int \\frac{d^{3}\\mathbf {p}}{(2\\pi )^{3}}G_{yy}(\\mathbf {p}),$ where $U$ is the pairing interaction." ], [ "Scattering by fluctuating exchange field", "The Green's function renormalized by scattering is $\\hat{G}^{-1} & =\\hat{G}_{0}^{-1}-\\hat{\\Sigma }$ where $\\hat{G}_{0}^{-1}\\!=i\\omega _{n}\\hat{\\sigma }_{0}\\hat{\\tau }_{0}\\!-\\xi _{\\mathbf {p}}\\hat{\\sigma }_{0}\\hat{\\tau }_{z}\\!+\\Delta \\hat{\\sigma }_{y}\\hat{\\tau }_{y}$ and $\\hat{\\Sigma }(\\mathbf {p})=\\int \\frac{d^{3}\\mathbf {q}}{(2\\pi )^{3}}\\left\\langle \\left|\\tilde{h}_{q,i}\\right|^{2}\\right\\rangle \\hat{\\alpha }_{i}\\hat{G}(\\mathbf {p}+\\mathbf {q})\\hat{\\alpha }_{i}$ is the self-energy due to the scattering on the fluctuating exchange field with $\\hat{\\alpha }=\\left(\\hat{\\tau }_{z}\\hat{\\sigma }_{x},\\hat{\\tau }_{0}\\hat{\\sigma }_{y},\\hat{\\tau }_{z}\\hat{\\sigma }_{z}\\right)$[3].", "Using the expansion $\\hat{\\Sigma }(\\mathbf {p})=\\sum _{a,b}\\Sigma _{ab}\\hat{\\sigma }_{a}\\hat{\\tau }_{b}$ , we obtain that the relevant components with $ab=00,\\,0z,\\,yy$ are $\\Sigma _{ab}(\\mathbf {p}) & =\\int \\frac{d^{3}\\mathbf {\\mathbf {p}^{\\prime }}}{(2\\pi )^{3}}\\left\\langle \\left|\\tilde{h}_{p-\\mathbf {p}^{\\prime }}\\right|^{2}\\right\\rangle G_{ab}(\\mathbf {p}^{\\prime }).$ with $\\Sigma _{00}(\\mathbf {p}) & =-\\int \\frac{d^{3}\\mathbf {p}^{\\prime }}{(2\\pi )^{3}}\\left\\langle \\left|\\tilde{h}_{p-p^{\\prime }}\\right|^{2}\\right\\rangle \\frac{i\\omega _{n}}{\\omega _{n}^{2}+\\xi _{\\mathbf {p}^{\\prime }}^{2}+\\Delta ^{2}},\\\\\\Sigma _{0z}(\\mathbf {p}) & =-\\int \\frac{d^{3}\\mathbf {p}^{\\prime }}{(2\\pi )^{3}}\\left\\langle \\left|\\tilde{h}_{p-p^{\\prime }}\\right|^{2}\\right\\rangle \\frac{\\xi _{\\mathbf {p}^{\\prime }}}{\\omega _{n}^{2}+\\xi _{\\mathbf {p}^{\\prime }}^{2}+\\Delta ^{2}},$ and $\\Sigma _{yy}(\\mathbf {p})=-\\frac{\\Delta }{i\\omega _{n}}\\Sigma _{00}(\\mathbf {p})$ .", "The behavior of $\\hat{\\Sigma }_{\\mathbf {p}}$ depends on the relation between three length scales: the magnetic correlation length $\\xi _{h}$ , in-plane coherence length $\\xi _{s}$ , and inverse Fermi wave vector $k_{F}^{-1}$ .", "Consider first limiting cases qualitatively.", "For very long correlations $\\xi _{h}>\\xi _{s}$ , we have a slowly varying exchange field.", "In this case, we can neglect $\\mathbf {p}^{\\prime }$ dependence everywhere except $\\langle |\\tilde{h}_{p-p^{\\prime }}|^{2}\\rangle $ giving $\\hat{\\Sigma }(\\mathbf {p}) & \\approx h_{0}^{2}\\hat{G}_{0}(\\mathbf {p}).$ This corresponds to the correction due to the uniform exchange field equal to $h_{0}$ averaged over its directions.", "We make two observations from this simple result, which will be essential in the further consideration: (i) $\\hat{\\Sigma }(\\mathbf {p})$ has the same nonzero components as $\\hat{G}_{0}(\\mathbf {p})$ , i.e., 00, $yy$ , and $0z$ and (ii) the momentum dependence in $\\hat{\\Sigma }(\\mathbf {p})$ can not be neglected.", "In the case $\\xi _{h}\\!<\\!\\xi _{s}$ , we can integrate over $\\xi _{\\mathbf {p}^{\\prime }}$ and obtain the well-known Abrikosov-Gor'kov magnetic-scattering result [1], $\\hat{\\Sigma }(\\mathbf {p}) & \\approx \\frac{1}{2\\tau _{m}}\\frac{-i\\omega _{n}\\hat{\\sigma }_{0}\\hat{\\tau }_{0}+\\Delta \\hat{\\sigma }_{y}\\hat{\\tau }_{y}}{\\sqrt{\\omega _{n}^{2}+\\Delta ^{2}}}$ with the scattering rate $\\frac{1}{2\\tau _{m}} & =\\int \\frac{\\pi dS_{F}^{\\prime }}{(2\\pi )^{3}v_{F}^{\\prime }}\\left\\langle \\left|\\tilde{h}_{p-p^{\\prime }}\\right|^{2}\\right\\rangle ,$ which accounts for possibility that the range of $\\langle |\\tilde{h}_{p-p^{\\prime }}|^{2}\\rangle $ may be much smaller than the Fermi-surface size[18].", "Note that, in contrast to the case of long correlations, Eq.", "(REF ), (i) the $p$ dependence of $\\hat{\\Sigma }(\\mathbf {p})$ in Eq.", "(REF ) can be neglected and (ii)$\\Sigma _{0z}$ component can be omitted.", "These are standard approximations of the AG theory.", "In the regime $\\xi _{h}>k_{F}^{-1}$ the magnetic fluctuations give small-angle scattering, see illustration in Fig.", "REF (b).", "The dependence of the scattering rate on the correlation length following from Eq.", "(REF ) is sensitive to the dimensionality of scattering.", "For three-dimensional scattering, the scattering rate increases logarithmically with $\\xi _{h}$ [18].", "In our quasi-2D case, we assume that scattering occurs in the whole range of $p_{z}-p_{z}^{\\prime }$ but with small change of the in-plane momentum.", "In this case Eq.", "(REF ) gives $\\frac{1}{2\\tau _{m}}=\\frac{2\\pi }{s}\\int \\limits _{-\\infty }^{\\infty }\\frac{\\pi dq}{(2\\pi )^{3}v_{F}}\\frac{C_{h}sh_{0}^{2}\\xi _{h}^{2}}{1+\\xi _{h}^{2}q^{2}}=\\frac{C_{h}h_{0}^{2}\\xi _{h}}{4v_{F}}.$ In general case, the product $C_{h}h_{0}^{2}\\xi _{h}$ in this formula and in several results below, can be directly computed from the correlation function of the exchange field as $C_{h}h_{0}^{2}\\xi _{h}=\\int _{0}^{\\infty }dr\\left\\langle \\tilde{h}(r)\\tilde{h}(0)\\right\\rangle .$ This relation allows evaluation of the scattering rate from the spin-spin correlation function, see Eq.", "(REF ), which can be computed for a particular magnetic model.", "We can see that in the quasi-2D case the scattering rate increases linearly with $\\xi _{h}$ , much faster than in the 3D case [18].", "For completeness, we also present here the result for very short correlation $\\xi _{h}k_{F}<1$ when magnetic fluctuations scatter at all angles.", "In this case we can replace $|\\tilde{h}_{p-p^{\\prime }}|^{2}$ with $|\\tilde{h}_{0}|^{2}$ and obtain the Abrikosov-Gor'kov result for uncorrelated magnetic impurities $\\frac{1}{2\\tau _{m}}=C_{h}\\nu h_{0}^{2}s\\xi _{h}^{2},$ where $\\nu $ is the density of states.", "In particular, for quasi-2D electronic spectrum $\\nu =m/(2\\pi \\hbar ^{2}s)$ where $m$ is the effective mass.", "Away from the magnetic transition, the magnetic correlation length $\\xi _h$ is of the order of separation between the magnetic moments $a$ .", "For a continuous magnetic transition inside the superconducting state, the magnetic correlation length rapidly increases for $T\\rightarrow T_{m}$ and at some point exceeds the coherence length.", "At this crossover the impact of magnetic fluctuations on superconductivity modifies qualitatively.", "We now quantify the crossover between the regimes $\\xi _{h}>\\xi _{s}$ , Eq.", "(REF ), and $\\xi _{h}<\\xi _{s}$ , Eqs.", "(REF ) and (REF ).", "It is important to note that in the second (scattering) regime only two components of $\\hat{\\Sigma }$ are essential, 00 and $yy$ .", "In the first regime, however, also the $0z$ component describing spectrum renormalization has to be included.", "The latter component obviously also has to be taken into account in the description of the crossover.", "First, we consider the 00 component (the 00 and $yy$ components are related as $\\Sigma _{yy}=-\\frac{\\Delta }{i\\omega _{n}}\\Sigma _{00}$ ).", "As the scattering in the regime $k_{F}\\xi _{h}\\gg $ 1 is small angle, we need to consider only a small region at the Fermi surface near the initial momentum $p$ .", "Selecting the $x$ axis along this momentum and $y$ axis in the perpendicular direction [see Fig.", "REF (b)] and using $\\langle |\\tilde{h}_{\\mathbf {\\mathbf {q}}}|^{2}\\rangle $ in Eq.", "(REF ), we transform Eq.", "(REF ) as $\\Sigma _{00}(\\mathbf {p}) & =\\!-\\!\\int \\frac{dp_{x}^{\\prime }dp_{y}^{\\prime }}{(2\\pi )^{2}}\\frac{C_{h}h_{0}^{2}\\xi _{h}^{2}}{1\\!+\\!\\xi _{h}^{2}\\left(p_{x}^{\\prime }\\!-\\!p_{x}\\right)^{2}\\!+\\!\\xi _{h}^{2}p_{y}^{\\prime }{}^{2}}\\nonumber \\\\\\times & \\frac{i\\omega _{n}}{\\omega _{n}^{2}\\!+\\!v_{F}^{2}\\left(p_{x}^{\\prime }\\!-\\!p_{F}\\right)^{2}\\!+\\!\\Delta ^{2}}.$ Integrating with respect to $p_{y}^{\\prime }$ , we obtain $\\Sigma _{00}(\\mathbf {p}) & =-\\frac{C_{h}h_{0}^{2}\\xi _{h}}{4\\pi }\\int _{-\\infty }^{\\infty }dp_{x}^{\\prime }\\frac{1}{\\sqrt{1+\\xi _{h}^{2}{p_{x}^{\\prime }}^{2}}}\\nonumber \\\\\\times & \\frac{i\\omega _{n}}{\\omega _{n}^{2}\\!+\\!v_{F}^{2}\\left(p_{x}^{\\prime }\\!+\\!\\delta p_{x}\\right)^{2}\\!+\\!\\Delta ^{2}}\\nonumber \\\\& =-\\frac{C_{h}h_{0}^{2}}{4\\pi }\\frac{i\\omega _{n}}{\\omega _{n}^{2}+\\Delta ^{2}}U\\left(\\delta k_{x},g_{n}\\right),$ where $\\delta p_{x}=p_{x}-p_{F}$ , $g_{n}=\\frac{v_{F}/\\xi _{h}}{\\sqrt{\\omega _{n}^{2}\\!+\\!\\Delta ^{2}}},\\,\\delta k_{x}=\\!\\frac{v_{F}\\left(p_{x}\\!-\\!p_{F}\\right)}{\\sqrt{\\omega _{n}^{2}+\\Delta ^{2}}}=\\!\\frac{\\xi _{p}}{\\sqrt{\\omega _{n}^{2}\\!+\\!\\Delta ^{2}}},$ and the reduced function $U\\left(k,g\\right)$ is defined by the integral $U\\left(k,g\\right)=\\int \\limits _{-\\infty }^{\\infty }du\\frac{1}{\\sqrt{1+u^{2}}}\\frac{1}{1+\\left(gu+k\\right)^{2}},$ which can be taken analytically giving $U\\left(k,g\\right) & =\\mathrm {Re}\\left[W\\left(k,g\\right)\\right],\\\\W\\left(k,g\\right) & =\\!\\frac{2}{\\sqrt{\\left(ik\\!+\\!1\\right)^{2}\\!-\\!g^{2}}}\\ln \\!\\left(\\frac{ik\\!+\\!1\\!+\\!\\sqrt{\\left(ik\\!+\\!1\\right)^{2}\\!-\\!g^{2}}}{g}\\right).\\nonumber $ We note that the $k$ dependence of the function $U\\left(k,g\\right)$ corresponding to the $\\xi _{p}$ dependence of the self energy is essential only for $g\\lesssim 1$ .", "The value of the function $U\\left(k,g\\right)$ at $k=0$ has the simple analytical form $U\\left(0,g\\right) & ={\\left\\lbrace \\begin{array}{ll}\\frac{2}{\\sqrt{1-g^{2}}}\\ln \\frac{1+\\sqrt{1-g^{2}}}{g} & \\mathrm {for}\\:g<1\\\\\\frac{2}{\\sqrt{g^{2}-1}}\\arctan \\sqrt{g^{2}-1} & \\mathrm {for}\\:g>1\\end{array}\\right.", "}.$ The asymptotic $U\\left(0,g\\right)\\simeq \\pi /g$ for $g\\gg 1$ corresponds the scattering regime, Eqs.", "(REF ) and (REF ).", "In this limit the $k$ dependence of the function $U\\left(k,g\\right)$ can be neglected.", "On the other hand, the asymptotic for $g\\ll 1$ is $U\\left(k,g\\right)\\simeq 2\\left[\\frac{1}{1\\!+\\!k^{2}}\\ln \\left(2\\frac{\\sqrt{1\\!+\\!k^{2}}}{g}\\right)+\\frac{k}{1\\!+\\!k^{2}}\\arctan k\\right].$ It corresponds to the uniform-field result in Eq.", "(REF ) only for the main logarithmic term.", "Additional terms appear because the correlation function $\\langle \\tilde{h}_{\\alpha }(r)\\tilde{h}_{\\beta }(r^{\\prime })\\rangle $ in Eq.", "(REF ) is not a constant at $|r-r^{\\prime }|\\!<\\!\\xi _{h}$ but increases logarithmically as $\\ln \\left(\\xi _{h}/|r-r^{\\prime }|\\right)$ .", "As mentioned above, for the proper description of the crossover at $\\xi _{h}\\sim \\xi _{s}$ , we also need to take into account the $0z$ component of the self energy, $\\Sigma _{0z}(\\mathbf {p}) & =\\!-\\!\\int \\frac{dp_{x}^{\\prime }dp_{y}^{\\prime }}{(2\\pi )^{2}}\\frac{C_{h}h_{0}^{2}\\xi _{h}^{2}}{1\\!+\\!\\xi _{h}^{2}\\left(p_{x}^{\\prime }\\!-\\!p_{x}\\right)^{2}\\!+\\!\\xi _{h}^{2}p_{y}^{\\prime }{}^{2}}\\nonumber \\\\\\times & \\frac{v_{F}\\left(p_{x}^{\\prime }-p_{F}\\right)}{\\omega _{n}^{2}\\!+\\!v_{F}^{2}\\left(p_{x}^{\\prime }\\!-\\!p_{F}\\right)^{2}\\!+\\!\\Delta ^{2}}.$ Following the same route as in derivation of Eq.", "(REF ), we present it as $\\Sigma _{0z}(\\mathbf {p}) & =-\\frac{C_{h}h_{0}^{2}}{4\\pi }\\frac{1}{\\sqrt{\\omega _{n}^{2}+\\Delta ^{2}}}V\\left(\\delta k_{x},g_{n}\\right),$ where the parameters $g_{n}$ and $\\delta k_{x}$ are defined in Eq.", "(REF ), $V\\left(k,g\\right)\\!=\\!\\int \\limits _{-\\infty }^{\\infty }\\!du\\frac{1}{\\sqrt{1\\!+\\!u^{2}}}\\frac{gu+k}{1\\!+\\!\\left(gu\\!+\\!k\\right)^{2}}=\\!-\\mathrm {Im}\\left[W\\left(k,g\\right)\\right],$ and the function $W\\left(k,g\\right)$ is defined in Eq.", "(REF ).", "In particular, for $g\\rightarrow 0$ $V\\left(k,g\\right)\\simeq \\frac{2}{1+k^{2}}\\left[k\\ln \\left(\\frac{2}{g}\\right)-\\arctan k\\right].$ As follows from Eq.", "(REF ), the renormalized Green's function can be obtained by substitutions $i\\omega _{n}\\rightarrow i\\tilde{\\omega }_{n}=i\\omega _{n}-\\Sigma _{00}$ , $\\Delta \\rightarrow \\tilde{\\Delta }=\\Delta -\\Sigma _{yy}$ , and $\\xi _{\\mathbf {p}}\\rightarrow \\tilde{\\xi }_{\\mathbf {p}}=\\xi _{\\mathbf {p}}+\\Sigma _{0z}$ .", "The renormalized frequency, gap, and spectrum can be written as $\\tilde{\\omega }_{n} =\\omega _{n}(1+\\alpha _{n}),\\,\\tilde{\\Delta }=\\Delta _{0}(1-\\alpha _{n}),\\,\\tilde{\\xi }_{\\mathbf {p}}=\\xi _{\\mathbf {p}}(1-\\beta _{n})$ with $\\alpha _{n}=\\frac{C_{h}h_{0}^{2}}{4\\pi }\\frac{U\\left(\\delta k_{x},g_{n}\\right)}{\\omega _{n}^{2}+\\Delta _{0}^{2}},\\ \\beta _{n}=\\frac{C_{h}h_{0}^{2}}{4\\pi }\\frac{V\\left(\\delta k_{x},g_{n}\\right)/\\xi _{\\mathbf {p}}}{\\sqrt{\\omega _{n}^{2}+\\Delta _{0}^{2}}}.$ With derived results for the self-energy in Eqs.", "(REF ) and (REF ), we proceed with evaluation of correction to the gap parameter from Eq.", "(REF )." ], [ "Correction to the gap", "The superconducting gap is the most natural parameter characterizing the strength of superconductivity at a given temperature.", "In this section, we calculate the suppression of this parameter by correlated magnetic fluctuations.", "The key observation is that a small uniform exchange field has no influence on the gap at zero temperature [19].", "Therefore, one can expect that the suppression caused by correlated magnetic fluctuations at low temperatures diminishes when the magnetic correlation length exceeds the superconducting coherence length.", "The gap equation in Eq.", "(REF ) is determined by the integral $\\mathcal {I} =\\int \\frac{d^{3}\\mathbf {p}}{(2\\pi )^{3}}G_{yy}(\\mathbf {p}) =\\nu \\int _{-\\infty }^{\\infty }d\\xi \\frac{\\tilde{\\Delta }}{\\tilde{\\omega }_{n}^{2}\\!+\\!\\tilde{\\xi }^{2}\\!+\\!\\tilde{\\Delta }^{2}},$ where the parameters with “$\\sim $ ” are defined in Eqs.", "(REF ) and (REF ).", "We evaluate the linear correction to $\\mathcal {I}$ with respect to $\\alpha _{n},\\beta _{n}\\propto h_{0}^{2}$ as $\\mathcal {\\delta I} & =-\\nu \\int \\limits _{-\\infty }^{\\infty }\\!d\\xi \\frac{\\Delta _{0}}{\\omega _{n}^{2}\\!+\\!\\xi ^{2}\\!+\\!\\Delta _{0}^{2}}\\\\\\times & \\left(\\alpha _{n}\\!+\\!\\frac{2\\alpha _{n}\\left(\\omega _{n}^{2}\\!-\\!\\Delta _{0}^{2}\\right)\\!-\\!2\\beta _{n}\\xi ^{2}}{\\left(\\omega _{n}^{2}+\\xi ^{2}+\\Delta _{0}^{2}\\right)}\\right).$ Making the substitution $\\xi =z\\sqrt{\\omega _{n}^{2}+\\Delta _{0}^{2}}$ , we transform this correction to the following form $\\delta \\mathcal {I} & =-\\frac{C_{h}\\nu h_{0}^{2}}{4\\pi }\\frac{\\Delta _{0}}{\\left(\\omega _{n}^{2}+\\Delta _{0}^{2}\\right)^{3/2}}\\nonumber \\\\\\times & \\mathrm {Re}\\left[\\int \\limits _{-\\infty }^{\\infty }dz\\frac{W\\left(z,g_{n}\\right)\\left(\\left(z-i\\right)^{2}+\\frac{4\\omega _{n}^{2}}{\\omega _{n}^{2}+\\Delta _{0}^{2}}\\right)}{\\left(z^{2}+1\\right)^{2}}\\right].$ Calculation of the integral described in appendix yields the result $\\delta \\mathcal {I} & =-C_{h}\\nu h_{0}^{2}\\frac{\\Delta _{0}}{\\left(\\omega _{n}^{2}+\\Delta _{0}^{2}\\right)^{5/2}}\\left\\lbrace \\frac{\\Delta _{0}^{2}}{4-g_{n}^{2}}\\right.\\nonumber \\\\+ & \\left.\\frac{\\omega _{n}^{2}\\left(4\\!-\\!g_{n}^{2}\\right)\\!-\\!2\\Delta _{0}^{2}}{\\left(4-g_{n}^{2}\\right)^{3/2}}\\ln \\left(\\frac{2\\!+\\!\\sqrt{4\\!-\\!g_{n}^{2}}}{g_{n}}\\right)\\right\\rbrace .$ The corrected equation for the gap $\\Delta =UT\\sum _{\\omega _{n}}(\\mathcal {I}_{0}+\\delta \\mathcal {I})$ with $\\mathcal {I}_{0}=\\pi /\\sqrt{\\omega _{n}^{2}+\\Delta _{0}^{2}}$ gives the gap correction caused by the nonuniform exchange field $\\tilde{\\Delta } & =-C_{h}h_{0}^{2}T\\!\\sum _{\\omega _{n}}\\!\\frac{1}{\\left(\\omega _{n}^{2}\\!+\\!\\Delta _{0}^{2}\\right)^{5/2}}.\\\\\\times & \\mathrm {Re}\\!\\left[\\frac{\\Delta _{0}^{2}}{4\\!-\\!g_{n}^{2}}\\!+\\!\\frac{4\\omega _{n}^{2}\\!-\\!2\\Delta _{0}^{2}\\!-\\!\\omega _{n}^{2}g_{n}^{2}}{\\left(4-g_{n}^{2}\\right)^{3/2}}\\ln \\!\\left(\\!\\frac{2\\!+\\!\\sqrt{4\\!-\\!g_{n}^{2}}}{g_{n}}\\!\\right)\\!\\right]\\\\\\times & \\left[\\pi T\\!\\sum _{\\omega _{n}}\\frac{\\Delta _{0}}{\\left(\\omega _{n}^{2}\\!+\\!\\Delta _{0}^{2}\\right)^{3/2}}\\right]^{-1}.$ Substituting the definition of $g_{n}$ in Eq.", "(REF ), we finally obtain $\\tilde{\\Delta }& =\\!-\\left[\\pi T\\sum _{\\omega _{n}}\\frac{\\Delta _{0}}{_{n}^{3}}\\right]^{-1}\\!C_{h}h_{0}^{2}T\\sum _{\\omega _{n}}\\frac{1}{_{n}^{3}\\left(4_{n}^{2}-\\varepsilon _{h}^{2}\\right)}\\\\\\times & \\!\\left[\\!\\Delta _{0}^{2}\\!+\\!\\frac{2\\left(2\\omega _{n}^{2}\\!-\\!\\Delta _{0}^{2}\\right)_{n}^{2}\\!-\\!\\omega _{n}^{2}\\varepsilon _{h}^{2}}{_{n}\\sqrt{4_{n}^{2}-\\varepsilon _{h}^{2}}}\\ln \\!\\left(\\!\\frac{2_{n}\\!+\\!\\sqrt{4_{n}^{2}\\!-\\!\\varepsilon _{h}^{2}}}{\\varepsilon _{h}}\\right)\\!\\right]\\nonumber $ with $_{n}=\\sqrt{\\omega _{n}^{2}\\!+\\!\\Delta _{0}^{2}}$ and the magnetic-scattering energy scale $\\varepsilon _{h}\\!=\\!v_{F}/\\xi _{h}$ .", "Introducing the reduced variables $\\tilde{T}\\!=&2\\pi T/\\Delta _{0}(T),\\ \\tilde{\\omega }_n\\!=\\!\\tilde{T}(n\\!+\\!1/2),\\\\ \\alpha _{h}\\!=&\\varepsilon _{h}(T)/2\\Delta _{0}(T)\\!=\\!\\xi _{s}(T)/\\xi _{h}(T)$ with $\\xi _{s}(T)\\!=\\!v_{F}/2\\Delta _{0}(T)$ , and using the estimate $C_{h}\\!\\approx \\!", "2\\pi /\\ln \\left(\\xi _{h}/a\\right)$ , we rewrite this result in the form convenient for numerical evaluation $\\tilde{\\Delta }(T)= & -\\frac{h_{0}^{2}}{2\\Delta _{0}(T)\\ln \\left(\\xi _{h}(T)/a\\right)}\\mathcal {V}_{\\Delta }\\left(\\frac{2\\pi T}{\\Delta _{0}(T)},\\frac{\\xi _{s}(T)}{\\xi _{h}(T)}\\right),\\\\\\mathcal {V}_{\\Delta }\\left(\\tilde{T},\\alpha _{h}\\right) & =\\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\tilde{T}\\sum _{n=0}^{\\infty }R\\left[\\tilde{\\omega }_n,\\alpha _{h}\\right],\\\\\\mathcal {D}(\\tilde{T}) & =\\tilde{T}\\sum _{n=0}^{\\infty }\\left(\\tilde{\\omega }_n^{2}\\!+1\\right)^{-3/2},\\\\R\\left(z,\\alpha _{h}\\right) & =\\frac{1}{\\left(z^{2}\\!+\\!1\\right)^{3/2}\\left(z^{2}\\!+\\!1\\!-\\!\\alpha _{h}^{2}\\right)}\\left[1\\!+\\!\\left(2z^{2}\\!-\\!1\\!-\\!\\frac{2z^{2}\\alpha _{h}^{2}}{z^{2}\\!+\\!1}\\right)\\mathit {L}\\left(z,\\alpha _{h}\\right)\\right].\\\\\\mathit {L}\\left(z,\\alpha _{h}\\right) & ={\\left\\lbrace \\begin{array}{ll}\\frac{\\sqrt{z^{2}+1}}{\\sqrt{z^{2}+1-\\alpha _{h}^{2}}}\\ln \\left(\\frac{\\sqrt{z^{2}\\!+\\!1}\\!+\\!\\sqrt{z^{2}\\!+\\!1\\!-\\!\\alpha _{h}^{2}}}{\\alpha _{h}}\\right), & z^{2}\\!>\\!\\alpha _{h}^{2}\\!-\\!1\\\\\\frac{\\sqrt{z^{2}+1}}{\\sqrt{\\alpha _{h}^{2}-z^{2}-1}}\\arctan \\frac{\\sqrt{\\alpha _{h}^{2}\\!-\\!z^{2}\\!-\\!1}}{\\sqrt{z^{2}\\!+\\!1}}, & z^{2}\\!<\\!\\alpha _{h}^{2}\\!-\\!1\\end{array}\\right.", "}.$ Note that the function $L\\left(z,\\alpha _{h}\\right)$ does not have singularity at $z=\\sqrt{\\alpha _{h}^{2}-1}$ for $\\alpha _{h}>1$ , contrary to what its shape may suggest.", "We see that the gap correction has the amplitude $h_{0}^{2}/\\Delta _{0}$ and mostly depends on two dimensionless parameters: reduced temperature $T/\\Delta _{0}(T)$ and the ratio $\\alpha _h\\!=\\!\\xi _{s}(T)/\\xi _{h}(T)$ .", "It also weakly depends on the ratio $\\xi _{h}/a$ , which determines the logarithmic factor in the denominator of Eq.", "(REF ).", "Let us discuss asymptotic behavior of the reduced function $\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})$ and the gap correction it gives.", "In the range $\\alpha _{h}\\gg 1$ corresponding to the scattering regime, the function $R\\left(z,\\alpha _{h}\\right)$ in Eq.", "() behaves as $R\\left(z,\\alpha _{h}\\right)\\simeq \\frac{\\pi z^{2}}{\\alpha _{h}\\left(z^{2}+1\\right)^{2}}$ .", "This gives the following asymptotics of the function $\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})$ $\\mathcal {V}_{\\Delta }\\left(\\tilde{T},\\alpha _{h}\\right) & \\simeq \\frac{\\pi }{\\alpha _{h}}V_{\\Delta }(\\tilde{T}),\\\\V_{\\Delta }(\\tilde{T})= & \\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\!\\tilde{T}\\sum _{n=0}^{\\infty }\\frac{\\tilde{\\omega }_n^2}{\\left(\\tilde{\\omega }_n^2+1\\right)^{2}},$ where the limiting behaviors of the function $V_{\\Delta }(\\tilde{T})$ are $V_{\\Delta }(0)\\!=\\!\\pi /4$ and $V_{\\Delta }(\\tilde{T})\\!\\simeq \\!", "{\\pi ^{2}\\tilde{T}}/[{14\\zeta (3)}]$ for $\\tilde{T}\\!\\gg \\!1$ with $\\zeta (3)\\approx 1.202$ .", "Correspondingly, the correction to the gap in this regime simplifies to $\\tilde{\\Delta }(T)\\simeq -\\frac{\\pi \\xi _{h}h_{0}^{2}}{v_{F}\\ln \\left(\\xi _{h}/a\\right)}V_{\\Delta }(\\tilde{T})=-\\frac{1}{\\tau _{m}}V_{\\Delta }(\\tilde{T}).$ This is a well-known result for the gap correction caused by the magnetic scattering[1], [2].", "In the opposite regime $\\alpha _{h}\\rightarrow 0$ corresponding to the vicinity of the magnetic transition, the function $R\\left(z,\\alpha _{h}\\right)$ has a logarithmic dependence on $\\alpha _{h}$ $R\\left(z,\\alpha _{h}\\right) & =R_{0}(z)+R_{1}(z)\\ln \\left(\\frac{1}{\\alpha _{h}}\\right),\\\\R_{0}(z)= & \\frac{1\\!+\\!\\left(2z^{2}\\!-\\!1\\right)\\ln \\left(2\\sqrt{z^{2}\\!+\\!1}\\right)}{\\left(z^{2}+1\\right)^{5/2}},\\\\R_{1}(z)= & \\frac{2z^{2}-1}{\\left(z^{2}+1\\right)^{5/2}}$ meaning that $\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})$ also logarithmically diverges for $\\alpha _{h}\\rightarrow 0$ , $\\mathcal {V}_{\\Delta }\\left(\\tilde{T},\\alpha _{h}\\right) & =\\mathcal {A}(\\tilde{T})+\\mathcal {B}(\\tilde{T})\\ln \\left(\\frac{1}{\\alpha _{h}}\\right),\\\\\\mathcal {A}(\\tilde{T}) & =\\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\!\\tilde{T}\\sum _{n=0}^{\\infty }\\!R_{0}(\\tilde{\\omega }_n), \\\\\\mathcal {B}(\\tilde{T}) & =\\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\tilde{T}\\sum _{n=0}^{\\infty }R_{1}(\\tilde{\\omega }_n)$ with $\\mathcal {A}(0)=1$ , $\\mathcal {B}(0)=0$ .", "The plots of the coefficients $\\mathcal {A}(\\tilde{T})$ and $\\mathcal {B}(\\tilde{T})$ are shown in Fig.", "REF .", "The corresponding correction to the gap can be presented as $\\tilde{\\Delta }(\\tilde{T}) & =\\!-\\frac{h_{0}^{2}\\mathcal {B}(\\tilde{T})}{2\\Delta _{0}}\\left[1\\!-\\frac{\\ln \\left(\\frac{\\xi _{s}}{a}\\right)\\!-\\!\\mathcal {A}(\\tilde{T})\\!/\\!\\mathcal {B}(\\tilde{T})}{\\ln \\left(\\xi _{h}/a\\right)}\\right].$ Therefore, the absolute value of correction $|\\tilde{\\Delta }|$ decreases when $T$ approaches $T_{m}$ if the ratio $\\mathcal {A}(\\tilde{T}_{m})/\\mathcal {B}(\\tilde{T}_{m})$ exceeds $\\ln \\left(\\xi _{s}/a\\right)$ , which always occurs at sufficiently low temperatures, see inset in Fig.", "REF .", "In this case the overall dependence of the correction on $\\xi _{h}$ is nonmonotonic and maximum suppression of the gap occurs at $\\xi _{h}\\sim \\xi _{s}$ .", "The limiting value at $T=T_{m}$ , $\\tilde{\\Delta }(\\tilde{T}_{m})=-h_{0}^{2}\\mathcal {B}(\\tilde{T}_{m})/2\\Delta _{0}$ , corresponds to the correction from a uniform exchange field equal to $h_{0}$ .", "It vanishes for $T_{m}\\rightarrow 0$ as $\\exp \\left(-\\Delta _{0}/T_{m}\\right)$ .", "Figure: Temperature dependences of the coefficients 𝒜(T ˜)\\mathcal {A}(\\tilde{T})and ℬ(T ˜)\\mathcal {B}(\\tilde{T}) which determine the small-α h \\alpha _{h}asymptotics of the function 𝒱 Δ (T ˜,α h )\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})in Eq. ().", "The inset shows the temperature dependenceof the ratio 𝒜(T ˜)/ℬ(T ˜)\\mathcal {A}(\\tilde{T})/\\mathcal {B}(\\tilde{T}).At temperatures much smaller than $T_{c}$ , the summation over the Matsubara frequencies in Eq.", "() can be transformed into integration leading to $\\tilde{\\Delta }(0) & =-\\frac{h_{0}^{2}}{2\\Delta _{0}\\ln \\left(\\xi _{h}/a\\right)}\\mathcal {V}_{\\Delta }\\!\\left(\\frac{\\xi _{s}}{\\xi _{h}}\\right),$ where the reduced function $\\mathcal {V}_{\\Delta }\\left(\\alpha _{h}\\right)\\equiv \\mathcal {V}_{\\Delta }\\left(0,\\alpha _{h}\\right)$ is defined by the integral $\\mathcal {V}_{\\Delta }\\left(\\alpha _{h}\\right) & =\\int _{0}^{\\infty }R\\left(z,\\alpha _{h}\\right)dz.$ This is a monotonically-decreasing function with the asymptotics $\\mathcal {V}_{\\Delta }\\left(\\alpha _{h}\\right)\\!\\simeq \\!", "{\\left\\lbrace \\begin{array}{ll}1\\!+\\frac{1}{9}(6\\ln \\alpha _{h}\\!+\\!1)\\alpha _{h}^{2}, & \\mathrm {for}\\,\\alpha _{h}\\!\\ll \\!1\\\\\\frac{\\pi ^{2}}{4\\alpha _{h}}-\\frac{2\\ln \\alpha _{h}+1}{\\alpha _{h}^{2}}, & \\mathrm {for}\\,\\alpha _{h}\\!\\gg \\!1\\end{array}\\right.", "}.$ It also has the exact value $\\mathcal {V}_{\\Delta }(1)=\\frac{\\pi ^{2}}{8}-\\frac{1}{2}$ .", "The large-$\\alpha _{h}$ asymptotics corresponds to the magnetic-scattering regime[1], [2], [3].", "Substituting the first leading term into Eq.", "(REF ) yields the known result for the gap correction at zero temperature $\\tilde{\\Delta }(0)\\approx -\\pi /4\\tau _{m}$ , where $\\tau _{m}$ is given by Eq.", "(REF ).", "Plots of the numerically evaluated function $\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})$ are shown in Fig, REF for several values of the reduced temperature $\\tilde{T}$ .", "The function monotonically decreases with $\\alpha _{h}$ and increases with temperature.", "At zero temperature this function approaches a finite value for $\\alpha _{h}\\rightarrow 0$ while at finite temperatures it logarithmically diverges, as discussed above.", "For zero temperature, we also show the scattering-regime dependence by dashed line and more accurate asymptotic presented in Eq.", "(REF ) by dotted line.", "We can see that the scattering approximation noticeably overestimates the gap correction for rather large values of $\\alpha _{h}$ .", "The finite value of the function for $\\alpha _{h}\\rightarrow 0$ at zero temperature is in an apparent contradiction with the known result that a uniform exchange field does not change the gap at zero temperature [19].", "This finite value is the consequence of small-distance behavior of the exchange-field correlation function for the two-dimensional case: it does not approach a constant for $r\\ll \\xi _{h}$ but keeps growing logarithmically, see Eqs.", "(REF ) and (REF ).", "We note, however, that despite this small-$\\alpha _{h}$ saturation of the function $\\mathcal {V}_{\\Delta }\\left(0,\\alpha _{h}\\right)$ , the gap correction in Eq.", "(REF ) does have a nonmonotonic dependence on $\\xi _{h}$ and vanishes in the limit $\\xi _{h}\\rightarrow \\infty $ at low temperatures because of the logarithmic factor in the denominator.", "Figure: Plots of the function 𝒱 Δ (T ˜,α h )\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})in Eq.", "() determining the gap correction caused bynonuniform exchange field on the parameter α h =ξ s /ξ h \\alpha _{h}=\\xi _{s}/\\xi _{h}for several values of the reduced temperature T ˜=2πT/Δ 0 \\tilde{T}=2\\pi T/\\Delta _{0}.The corresponding relative temperatures for BCS superconductors areshown in parenthesis.", "For zero temperature, we also show the scattering-regimeasymptotics (dashed line) and more accurate asymptotics presentedin Eq.", "() (dotted line)." ], [ "Correction to the electromagnetic kernel and London penetration depth", "In this section, we investigate the correction to the superconducting current response caused by the exchange interaction with correlated magnetic fluctuations.", "As in the case of the gap parameter, there are two different regimes depending on the relation between the magnetic correlation length $\\xi _{h}$ and superconducting coherence length $\\xi _{s}$ .", "Our goal is to quantitatively describe the crossover between these two regimes.", "The case $\\xi _{h}<\\xi _{s}$ corresponds to the well-studied magnetic-scattering regime.", "Influence of magnetic scattering on the electromagnetic kernel, which determines the London penetration depth, was investigated by Skalski et al.", "[2], see also Ref. [3].", "Recently, a very detailed investigation of this problem has been performed within the quasiclassical approach [30].", "Most studies, however, have been done for isotropic magnetic scattering.", "The case of correlated magnetic fluctuation in the regime $k_{F}\\xi _{h}\\gg 1$ requires a proper accounting for the vertex correction to the kernel which is equivalent to accounting for the reverse scattering events in quasiclassical approach The vertex correction for arbitrary magnetic scattering has been considered in Ref. [2].", "The recipe to account for the vertex correction in the kernel given after Eq.", "(6.9), however, contains a mistake: the sign in front of $\\Gamma ^{t}$ is incorrect..", "The superconducting current response $j_{\\alpha }(q,\\omega )=-Q_{\\alpha \\beta }(q,\\omega )A_{\\beta }(q,\\omega )$ is determined by the electromagnetic kernel $Q_{\\alpha \\beta }(q,\\omega )$ .", "In strongly type-II superconductors, the screening of magnetic field is determined by the local static kernel $Q_{\\alpha \\beta }\\!\\equiv \\!Q_{\\alpha \\beta }(0,0)$ .", "The superfluid density $n_{s}$ introduced in the phenomenological London theory is related to $Q_{\\alpha \\beta }$ as $Q_{\\alpha \\beta }\\!=\\!e^{2}n_{s}/cm_{\\alpha \\beta }$ , where $m_{\\alpha \\beta }$ is the effective mass tensor.", "The London penetration depth components $\\lambda _{\\alpha }$ are related to the static uniform kernel as $Q_{\\alpha \\alpha }\\!=\\!c/(4\\pi \\lambda _{\\alpha }^{2})$ .", "In nonmagnetic superconductors the screening length $\\tilde{\\lambda }_{\\alpha }$ is identical to this 'bare' length $\\lambda _{\\alpha }$ defined via the electromagnetic kernel.", "In magnetic superconductors, however, the screening length $\\tilde{\\lambda }_{\\alpha }$ is reduced by the magnetic response of local moments as $\\tilde{\\lambda }_{\\alpha }\\!=\\!\\lambda _{\\alpha }/\\sqrt{\\mu }$ , where $\\mu $ is the magnetic permeability in the magnetic-field direction [31], [5].", "Note that the exchange and magnetic response have opposite influences on the screening length: the former enlarges and the latter reduces it.", "In the following, we concentrate on the calculation of the bare London penetration depth.", "In the Green's function formalism, the kernel can be evaluated as[3] $Q_{\\alpha \\beta }(q,\\omega _{\\nu }) & =\\frac{e^{2}n}{cm_{\\alpha \\beta }}+\\frac{e^{2}}{2c}T\\sum _{\\omega _{n}}\\!\\int \\!\\frac{d^{3}p}{(2\\pi )^{3}}v_{\\alpha }v_{\\beta }\\nonumber \\\\\\times & \\mathrm {Tr}\\left[\\hat{G}(p,\\omega _{n})\\hat{G}(p\\!-\\!q,\\omega _{n}\\!-\\!\\omega _{\\nu })\\right],$ where $n$ is total density and $v_\\alpha =\\partial \\xi _p/\\partial p_\\alpha $ are the velocity components Note that $n/m_{\\alpha \\beta }=2\\nu \\left\\langle v_{\\alpha }v_{\\beta }\\right\\rangle $ where $\\nu $ is the density of states per spin.", "In particular, for clean case $Q_{\\alpha \\beta }^{(0)} & =\\frac{2\\pi e^{2}}{c}\\nu \\left\\langle v_{\\alpha }v_{\\beta }\\right\\rangle T\\sum _{\\omega _{n}}\\frac{\\Delta _{0}^{2}}{\\left(\\omega _{n}^{2}+\\Delta _{0}^{2}\\right)^{3/2}}$ giving $Q_{\\alpha \\beta }^{(0)}=2\\frac{e^{2}}{c}\\nu \\left\\langle v_{\\alpha }v_{\\beta }\\right\\rangle $ at zero temperature.", "We first consider the scattering regime, $\\xi _{h}\\ll \\xi _{s}$ , within the quasiclassical approximation.", "The generalization of the isotropic-scattering calculations in Ref.", "[30] for arbitrary scattering described in Appendix gives the following result for the correction to $\\lambda _{\\alpha }^{-2}$ due to the magnetic scattering $\\lambda _{1\\alpha }^{-2}(T)= & \\!-\\!\\lambda _{0\\alpha }^{-2}(T)\\left[\\frac{1}{\\tau _{m}\\Delta _{0}(T)}V_{\\lambda ,m}\\left(\\frac{2\\pi T}{\\Delta _{0}(T)}\\right)\\right.\\nonumber \\\\+ & \\left.\\frac{1}{\\tau _{m}^{\\mathrm {tr}}\\Delta _{0}(T)}V_{\\lambda ,m}^{\\mathrm {tr}}\\left(\\frac{2\\pi T}{\\Delta _{0}(T)}\\right)\\right],$ with $V_{\\lambda ,m}\\!", "(\\tilde{T}) & \\!=\\!\\frac{1}{\\mathcal {D}(\\tilde{T})}\\tilde{T}\\!\\sum _{n=0}^{\\infty }\\left[\\!\\frac{2\\tilde{\\omega }_n^2\\!-\\!1}{\\left(1\\!+\\!\\tilde{\\omega }_n^2\\right)^{5/2}}V_{\\Delta }\\!", "(\\tilde{T})\\!+\\!\\frac{3\\tilde{\\omega }_n^2-1}{\\left(1\\!+\\!\\tilde{\\omega }_n^2\\right)^{3}}\\right],\\\\V_{\\lambda ,m}^{\\mathrm {tr}}\\!", "(\\tilde{T}) & \\!=\\frac{1}{2\\mathcal {D}(\\tilde{T})}\\tilde{T}\\sum _{n=0}^{\\infty }\\frac{1-\\tilde{\\omega }_n^2}{\\left(1+\\tilde{\\omega }_n^2\\right)^{3}},$ where the functions $\\mathcal {D}(\\tilde{T})$ and $V_{\\Delta }(\\tilde{T})$ are defined in Eqs.", "() and (), respectively.", "Here $\\tau _{m}$ is the magnetic-scattering lifetime, Eqs.", "(REF ) and (REF ), and $\\tau _{m}^{\\mathrm {tr}}$ is the corresponding transport time, $\\frac{1}{2\\tau _{m}^{\\mathrm {tr}}} & =\\int \\frac{\\pi dS_{F}^{\\prime }}{(2\\pi )^{3}v_{F}^{\\prime }}\\left(1-\\frac{v\\cdot v^{\\prime }}{\\left\\langle v^{2}\\right\\rangle }\\right)\\left\\langle \\left|\\tilde{h}_{p-p^{\\prime }}\\right|^{2}\\right\\rangle .$ The two terms in Eq.", "(REF ) can be referred to as the pair-breaking and transport contributions.", "In the case we consider, the transport scattering rate is much smaller than the total rate, $1/\\tau _{m}^{\\mathrm {tr}}\\!\\sim \\!1/\\left(\\xi _{h}k_{F}\\tau _{m}\\right)\\!\\ll \\!1/\\tau _{m}$ , and it does not increase when the temperature approaches the magnetic transition.", "We point, however, that the contribution from the total scattering rate vanishes at low temperatures, $V_{\\lambda ,m}(0)\\!=\\!0$ , while the transport contribution remains finite $V_{\\lambda ,m}^{\\mathrm {tr}}(0)\\!=\\!\\pi /16$ .", "Nevertheless, as our main goal is to understand suppression of the superconducting parameters near the magnetic transition, in the following consideration we mostly focus on the behavior of the pair-breaking term proportional to the total scattering rate.", "The above results are only valid until $\\xi _{h}<\\xi _{s}$ .", "We proceed with the consideration of the crossover to the opposite regime, which can not be treated within the quasiclassical approach.", "The total correction to the electromagnetic kernel is Figure: The diagrams for the lowest-order corrections to the electromagnetic kernel caused by the nonuniform exchange field in Eqs.", "() and ().", "The left-column diagrams represent the self-energy correction and the upper diagram in the right column gives the vertex correction.", "The lower diagram in the right columns illustrates the equation for the vertex, Eq.", "().$\\delta Q_{\\alpha \\beta } & =\\frac{e^{2}}{2c}T\\sum _{\\omega _{n}}\\mathcal {C}_{\\alpha \\beta }(\\omega _{n}),\\\\\\mathcal {C}_{\\alpha \\beta }(\\omega )=\\!", "& \\int \\!\\frac{d^{3}p}{(2\\pi )^{3}}v_{\\alpha }\\left\\lbrace 2\\mathrm {Tr}\\!\\left[\\hat{G}(p,\\omega )v_{\\beta }\\hat{G}_0(p,\\omega )\\hat{\\Sigma }(p,\\omega )\\hat{G}_0(p,\\omega )\\right]\\!+\\!\\mathrm {Tr}\\!\\left[\\hat{G}_0(p,\\omega )\\hat{\\upsilon }_{\\beta }\\hat{G}_0(p,\\omega )\\right]\\right\\rbrace ,$ where the first term in $\\mathcal {C}_{\\alpha \\beta }(\\omega )$ is the self-energy correction with $\\hat{\\Sigma }(p,\\omega )$ given by Eq.", "(REF ) and the second term is the vertex correction with $\\hat{\\upsilon }_{\\beta } & =\\sum _{i}\\int \\frac{d^{3}p^{\\prime }}{(2\\pi )^{3}}\\hat{\\alpha }_{i}\\hat{G}_{0}(p^{\\prime })v_{\\beta }^{\\prime }\\hat{G}_{0}(p^{\\prime })\\hat{\\alpha }_{i}\\left\\langle \\left|\\tilde{h}_{p-p^{\\prime },i}\\right|^{2}\\right\\rangle .$ Figure REF shows the diagrammatic presentation of these equations.", "We split the vertex correction into two contributions $\\hat{\\upsilon }_{\\beta } & =v_{\\beta }\\hat{\\Gamma }_{p}+\\delta \\hat{\\upsilon }_{\\beta },\\\\\\hat{\\Gamma }_{p} & =\\sum _{i}\\int \\frac{d^{3}p^{\\prime }}{(2\\pi )^{3}}\\hat{\\alpha }_{i}\\hat{G}_{0}(p^{\\prime })\\hat{G}_{0}(p^{\\prime })\\hat{\\alpha }_{i}\\left\\langle \\left|\\tilde{h}_{p-p^{\\prime },i}\\right|^{2}\\right\\rangle ,\\\\\\delta \\hat{\\upsilon }_{\\beta } & =\\sum _{i}\\!\\int \\!\\frac{d^{3}p^{\\prime }}{(2\\pi )^{3}}\\hat{\\alpha }_{i}\\hat{G}_{0}(p^{\\prime })\\left(v_{\\beta }^{\\prime }\\!-\\!v_{\\beta }\\right)\\hat{G}_{0}(p^{\\prime })\\hat{\\alpha }_{i}\\\\\\times &\\left\\langle \\left|\\tilde{h}_{p-p^{\\prime },i}\\right|^{2}\\right\\rangle .$ The second contribution $\\delta \\hat{\\upsilon }_{\\beta } $ is proportional to the transport scattering rate and in our situation is typically smaller than the first one.", "We therefore focus on the calculation of the first contribution.", "Using the relations $\\hat{G}_{0}\\hat{G}_{0} & =i\\frac{\\partial \\hat{G}_{0}}{\\partial \\omega },\\\\\\hat{\\Gamma }= & i\\frac{\\partial \\hat{\\Sigma }_{\\mathbf {p}}}{\\partial \\omega },$ where the second relation is usually called Ward identity, we can present $\\mathcal {C}_{\\alpha \\beta }(\\omega )$ as $\\mathcal {C}_{\\alpha \\beta }(\\omega ) & \\!=\\!\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {m}}(\\omega )+\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {tr}}(\\omega ),\\nonumber \\\\\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {m}}(\\omega ) & \\!=\\!i\\frac{\\partial }{\\partial \\omega }\\!\\int \\!\\frac{d^{3}p}{(2\\pi )^{3}}v_{\\alpha }v_{\\beta }\\mathrm {Tr}\\left[\\hat{G}_0(p,\\omega )\\hat{\\Sigma }(p,\\omega )\\hat{G}_0(p,\\omega )\\right],\\\\\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {tr}}(\\omega ) & \\!=\\!\\int \\frac{d^{3}p}{(2\\pi )^{3}}v_{\\alpha }\\mathrm {Tr}\\left[\\hat{G}_0(p,\\omega )\\delta \\hat{\\upsilon }_{\\beta }\\hat{G}_0(p,\\omega )\\right].\\nonumber $ The term $\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {tr}}(\\omega )$ corresponds to contribution in Eq.", "(REF ) proportional to the transport magnetic scattering rate $1/\\tau _{m}^{\\mathrm {tr}}$ .", "As discussed above, in our case this term is typically small and does not increase when the temperature approaches the magnetic transition.", "That is why we will neglect this term in the following consideration.", "As the term $\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {m}}(\\omega )$ is proportional to a full derivative with respect to $\\omega $ , it vanishes at zero temperature.", "To evaluate this term, we explicitly compute the trace inside the integral as $&\\mathrm {Tr}\\left[\\hat{G}_0(p,\\omega )\\hat{\\Sigma }(p,\\omega )\\hat{G}_0(p,\\omega )\\right] \\!=\\!4\\left[\\left(G_{00}^{2}\\!+\\!G_{0z}^{2}\\!+\\!G_{yy}^{2}\\right)\\Sigma _{00}\\right.\\\\&+\\!\\left.2G_{00}G_{0z}\\Sigma _{0z}\\!+\\!2G_{00}G_{yy}\\Sigma _{yy}\\right],$ where $\\Sigma _{00}$ and $\\Sigma _{0z}$ are given by Eqs.", "(REF ) and (REF ), respectively, and $\\Sigma _{yy}=-\\left(\\Delta /i\\omega \\right)\\Sigma _{00}$ .", "Substituting these results into Eq.", "(REF ), we transform $\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {m}}(\\omega )$ to $&\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {m}}(\\omega ) =\\frac{C_{h}h_{0}^{2}}{\\pi }\\nu \\left\\langle v_{\\alpha }v_{\\beta }\\right\\rangle \\\\&\\times \\frac{\\partial }{\\partial \\omega }\\frac{\\omega }{\\left(\\omega ^{2}\\!+\\!\\Delta _{0}^{2}\\right)^{3/2}}\\mathrm {Re}\\!\\!\\int \\limits _{-\\infty }^{\\infty }\\!dz\\frac{\\frac{-\\omega ^{2}+3\\Delta _{0}^{2}}{\\omega ^{2}+\\Delta _{0}^{2}}+z^{2}+2iz}{\\left(z^{2}+1\\right)^{2}}W\\left(z,g\\right),$ where $z\\!=\\!\\xi /\\sqrt{\\omega ^{2}\\!+\\!\\Delta _{0}^{2}}$ .", "The parameter $g\\equiv g_{n}$ and the function $W\\left(z,g\\right)$ are defined in Eqs.", "(REF ) and (REF ), respectively.", "Computation of the $z$ integral yields the result $\\mathcal {C}_{\\alpha \\beta }^{\\mathrm {m}}(\\omega ) & =-4C_{h}h_{0}^{2}\\nu \\left\\langle v_{\\alpha }v_{\\beta }\\right\\rangle \\frac{\\partial }{\\partial \\omega }\\left\\lbrace \\frac{\\omega \\Delta _{0}^{2}}{\\left(\\omega ^{2}+\\Delta _{0}^{2}\\right)^{5/2}\\left(4-g^{2}\\right)}\\left[1-\\frac{6-g^{2}}{\\sqrt{4-g^{2}}}\\ln \\left(\\frac{2+\\sqrt{4-g^{2}}}{g}\\right)\\right]\\right\\rbrace .$ Therefore, the corresponding correction to the kernel, Eq.", "(REF ), is $\\delta Q_{\\alpha \\beta }^{\\mathrm {m}}=-2\\frac{e^{2}}{c}C_{h}h_{0}^{2}\\nu \\left\\langle v_{\\alpha }v_{\\beta }\\right\\rangle T\\sum _{\\omega _{n}}\\frac{\\partial }{\\partial \\omega _{n}}\\left\\lbrace \\frac{\\omega _{n}\\Delta _{0}^{2}}{\\left(\\omega _{n}^{2}+\\Delta _{0}^{2}\\right)^{5/2}\\left(4-g_{n}^{2}\\right)}\\left[1-\\frac{6-g_{n}^{2}}{\\sqrt{4-g_{n}^{2}}}\\ln \\left(\\frac{2+\\sqrt{4-g_{n}^{2}}}{g_{n}}\\right)\\right]\\right\\rbrace .$ This result gives correction at the fixed gap parameter.", "The full correction also contains the contribution due to the shift of $\\Delta $ , $\\delta Q_{\\alpha \\beta }^{\\Delta }=\\tilde{\\Delta }dQ_{\\alpha \\beta }^{(0)}/d\\Delta $ , where $\\tilde{\\Delta }$ is given by Eq.", "(REF ).", "Using the same reduced variables as in Eq.", "(), we rewrite the corresponding correction to $\\lambda _{\\alpha }^{-2}\\propto Q_{\\alpha \\alpha }$ in the reduced form suitable for numerical evaluation $\\lambda _{1\\alpha }^{-2}(T) =\\!-\\lambda _{0\\alpha }^{-2}(T)\\frac{h_{0}^{2}}{2\\Delta _{0}^{2}\\ln \\left(\\xi _{h}/a\\right)}\\mathcal {V}_{Q}\\!\\left(\\frac{2\\pi T}{\\Delta _{0}},\\frac{\\xi _{s}}{\\xi _{h}}\\right)$ with $\\mathcal {V}_{Q}\\!\\left(\\tilde{T},\\alpha _{h}\\right)\\!", "& =\\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\!\\tilde{T}\\sum _{n=0}^{\\infty }\\left[ K_{Q}\\left(\\tilde{\\omega }_n\\right)\\mathcal {V}_{\\Delta }\\left(\\tilde{T},\\alpha _{h}\\right)\\right.\\nonumber \\\\&+\\left.R_{Q}\\left(\\tilde{\\omega }_n,\\alpha _{h}\\right)\\right] ,\\\\K_{Q}(z) & =-\\frac{\\partial }{\\partial z}\\frac{z}{\\left(z^{2}\\!+\\!1\\right)^{3/2}},\\\\R_{Q}\\left(z,\\alpha _{h}\\right) & =\\frac{\\partial }{\\partial z}\\frac{z\\left[1-\\left(3-\\frac{2\\alpha _{h}^{2}}{z^{2}\\!+\\!1}\\right)L(z,\\alpha _{h})\\right]}{\\left(z^{2}\\!+\\!1\\right)^{3/2}\\left(z^{2}\\!+\\!1\\!-\\!\\alpha _{h}^{2}\\right)},$ where the first term in the square brackets in Eq.", "(REF ) is due to the gap correction, the function $\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})$ is defined in Eq.", "(), and the function $L(z,\\alpha _{h})$ in the last definition is defined in Eq. ().", "For brevity, in Eq.", "(REF ) we omitted the $T$ dependences of $\\Delta _0(T)$ , $\\xi _s(T)$ , and $\\xi _h(T)$ .", "Plots of the function $\\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h})$ versus $\\alpha _{h}$ for different values of $\\tilde{T}$ are shown in Fig.", "REF .", "As the function $\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})$ shown in Fig.", "REF , this function also monotonically decreases with increasing of both $\\tilde{T}$ and $\\alpha _{h}$ .", "The essential difference is that the function $\\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h})$ vanishes for $\\tilde{T}\\!\\rightarrow \\!0$ while the function $\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})$ approaches the finite limit.", "The large-$\\alpha _{h}$ asymptotics of the function $\\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h})$ is $\\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h})\\approx \\frac{\\pi }{\\alpha _{h}}V_{\\lambda ,m}(\\tilde{T})$ , where the function $V_{\\lambda ,m}(\\tilde{T})$ is defined in Eq.", "(REF ).", "These asymptotics are also shown in Fig.", "REF by dashed lines.", "Noting also the relation $\\pi h_{0}^{2}/\\left(2\\alpha _{h}\\Delta _{0}\\ln \\left(\\xi _{h}/a\\right)\\right)=1/\\tau _{m}$ , we see that in the limit $\\alpha _{h}\\gg 1$ the above result reproduces the correction in Eq.", "(REF ) for the scattering regime.", "At small $\\alpha _{h}$ corresponding to the proximity of the magnetic transition, the function $R_{Q}\\left(z,\\alpha _{h}\\right)$ has logarithmic dependence on $\\alpha _{h}$ , $R_{Q}\\left(z,\\alpha _{h}\\right) & \\approx R_{Q,0}(z)+R_{Q,1}(z)\\ln \\left(\\frac{1}{\\alpha _{h}}\\right),\\\\R_{Q,0}(z) & =\\frac{\\partial }{\\partial z}\\frac{z\\left[1\\!-\\!3\\ln \\left(2\\sqrt{z^{2}\\!+\\!1}\\right)\\right]}{\\left(z^{2}\\!+\\!1\\right)^{5/2}},\\nonumber \\\\R_{Q,1}(z) & =-\\frac{\\partial }{\\partial z}\\frac{3z}{\\left(z^{2}\\!+\\!1\\right)^{5/2}}.\\nonumber $ The function $\\mathcal {V}_{\\Delta }(\\tilde{T},\\alpha _{h})$ describing the gap contribution also has logarithmic dependence on $\\alpha _{h}$ , Eq.", "(REF ).", "Correspondingly, the function $\\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h})$ also logarithmically diverges with $\\alpha _{h}\\rightarrow 0$ , $\\mathcal {V}_{Q}\\!\\left(\\tilde{T},\\alpha _{h}\\right) =\\mathcal {A}_{Q}(\\tilde{T})+\\mathcal {B}_{Q}(\\tilde{T})\\ln \\left(\\frac{1}{\\alpha _{h}}\\right),$ with $&\\mathcal {A}_{Q}(\\tilde{T})\\!= \\!\\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\!\\tilde{T}\\sum _{n=0}^{\\infty }\\!\\left[ K_{Q}(\\tilde{\\omega }_n)\\mathcal {A}(\\tilde{T})\\!+\\!R_{Q,0}(\\tilde{\\omega }_n)\\right], \\\\&\\mathcal {B}_{Q}(\\tilde{T})\\!= \\!\\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\!\\tilde{T}\\sum _{n=0}^{\\infty }\\!\\left[ K_{Q}(\\tilde{\\omega }_n)\\mathcal {B}(\\tilde{T})\\!+\\!R_{Q,1}(\\tilde{\\omega }_n)\\right],$ where the coefficients $\\mathcal {A}(\\tilde{T})$ and $\\mathcal {B}(\\tilde{T})$ are defined in Eqs.", "() and (), respectively.", "The small-$\\alpha _{h}$ asymptotics are plotted in Fig.", "REF with dotted lines and plots of the the coefficients $\\mathcal {A}_{Q}(\\tilde{T})$ and $\\mathcal {B}_{Q}(\\tilde{T})$ and their ratio are presented in Fig.", "REF .", "Note that the coefficient $\\mathcal {A}_{Q}(\\tilde{T})$ becomes negative for $\\tilde{T}\\!<\\!0.683$ .", "Even though the small-$\\alpha _{h}$ behavior in Eq.", "(REF ) looks similar to the behavior of the gap in Eq.", "(REF ), the essential difference is that both coefficients $\\mathcal {A}_{Q}(\\tilde{T})$ and $\\mathcal {B}_{Q}(\\tilde{T})$ vanish at $\\tilde{T}=0$ .", "Figure: Plots of the function 𝒱 Q (T ˜,α h )\\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h})in Eq.", "() which determines the correction to the electromagnetickernel and London penetration depth in Eq.", "().The dashed lines show large-α h \\alpha _{h} asymptotics, 𝒱 Q (T ˜,α h )∝1/α h \\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h})\\propto 1/\\alpha _{h},corresponding to the scattering regime.", "The dotted lines show small-α h \\alpha _{h}asymptotics, 𝒱 Q (T ˜,α h )∝ln1/α h \\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h})\\propto \\ln \\left(1/\\alpha _{h}\\right).Figure: Temperature dependence of the coefficients 𝒜 Q (T ˜)\\mathcal {A}_{Q}(\\tilde{T})and ℬ Q (T ˜)\\mathcal {B}_{Q}(\\tilde{T}) defined by Eqs.", "()and (), respectively, which determine the small-α h \\alpha _{h}asymptotics of the function 𝒱 Q (T ˜,α h )\\mathcal {V}_{Q}(\\tilde{T},\\alpha _{h}),Eq. ().", "The inset shows the temperature dependenceof their ratio.", "The coefficient 𝒜 Q (T ˜)\\mathcal {A}_{Q}(\\tilde{T}) changessign at T ˜=0.683\\tilde{T}=0.683.Similar to Eq.", "(REF ), we can present the correction in Eq.", "(REF ) in the limit $\\xi _{h}\\gg \\xi _{s}$ as $\\lambda _{1\\alpha }^{-2}\\!=\\!-\\!\\lambda _{0\\alpha }^{-2}\\frac{h_{0}^{2}\\mathcal {B}_{Q}\\!", "(\\tilde{T})}{2\\Delta _{0}^{2}}\\!\\left[1\\!-\\!\\frac{\\ln \\left(\\frac{\\xi _{s}}{a}\\right)\\!-\\!\\mathcal {A}_{Q}\\!", "(\\tilde{T})\\!/\\!\\mathcal {B}_{Q}\\!", "(\\tilde{T})}{\\ln \\left(\\xi _{h}/a\\right)}\\right].$ The ratio $\\mathcal {A}_{Q}\\!", "(\\tilde{T})\\!/\\!\\mathcal {B}_{Q}\\!", "(\\tilde{T})$ is of the order unity in the whole temperature range and becomes negative for $\\tilde{T}<0.683$ , see inset in Fig.", "REF , meaning that the nominator $\\ln \\left(\\xi _{s}/a\\right)\\!-\\!\\mathcal {A}_{Q}\\!", "(\\tilde{T})\\!/\\!\\mathcal {B}_{Q}\\!", "(\\tilde{T})$ is always positive.", "As a consequence, the correction to the superfluid density monotonically increases when temperature approaches $T_{m}$ .", "This is different from the behavior of the gap correction, Eq.", "(REF ), which becomes nonmonotonic at small temperatures.", "The maximum suppression of $\\lambda _{\\alpha }^{-2}$ for $\\xi _{h}\\rightarrow \\infty $ , $\\lambda _{1\\alpha ,\\mathrm {max}}^{-2}\\!=\\!-\\lambda _{0\\alpha }^{-2}\\left(h_{0}^{2}/2\\Delta _{0}^{2}\\right)\\mathcal {B}_{Q}\\!\\left(2\\pi T_{m}/\\Delta _{0}\\right)$ , corresponds to the correction from a uniform exchange field equal to $h_{0}$ ." ], [ "Discussion", "In summary, we evaluated the corrections to the gap, Eq.", "(REF ), and superfluid density, Eq.", "(REF ), caused by the exchange interaction with quasi-two-dimensional magnetic fluctuations in materials composed of superconducting and local-moment layers.", "Growth of the correlation length near the magnetic transition enhances spin-flip scattering leading to increasing suppression of superconducting parameters.", "This suppression significantly weakens when the magnetic correlation length exceeds the coherence length.", "In addition to dependence on the correlation length $\\xi _{h}(T)$ , the corrections have also direct regular dependence on the ratio $T/\\Delta _{0}(T)$ .", "Moreover, as one can see from Figs.", "REF and REF , in the paramagnetic state these dependences are opposite.", "While in the immediate vicinity of the magnetic transition the growth of $\\xi _{h}(T)$ dominates, in a wider range, the overall temperature dependence is determined by the interplay between both sources.", "To generate the parameter's temperature dependences for real materials from the derived general formulas, one need to specify the temperature dependent gap, coherence length, and magnetic correlation length, as well as the strength of the exchange field.", "Even though the consideration of this paper has been mostly motivated by physics of RbEuFe$_{4}$ As$_{4}$ , at present, there are too many uncertainties in the parameters of this material to make a reliable quantitative predictions.", "Therefore, we limit ourselves with showing expected qualitative behavior using representative parameters and illustrating general trends.", "Figure REF (left) shows the temperature dependences of the gap and $\\lambda ^{-2}$ for the parameters very roughly corresponding to RbEuFe$_{4}$ As$_{4}$ .", "Namely, we assume (i)the the Ginzburg-Landau coherence length $\\xi _{s0}^{\\mathrm {GL}}\\!=\\!1.46$ nm, following from the linear slope of the c-axis upper critical field [21], [22], (ii) the BCS value of the zero-temperature gap, $\\Delta _{0}(0)\\!=\\!1.76T_{c}\\!\\approx \\!5.6$ meV, (iii) the BCS temperature dependences for all unperturbed superconducting parameters, (iv) the amplitude of the exchange field $h_{0}\\!=\\!0.6T_{c}$ , (v) the magnetic transition at $t_{m}\\!\\equiv \\!T_{m}/T_{c}\\!=\\!0.4$ , and (vi) Berezinskii-Kosterlitz-Thouless (BKT) shape for the magnetic correlation length, $\\xi _{h}(T)\\!=\\!a\\exp [b\\sqrt{T_{m}/(T\\!-\\!T_{m})}]$ , where $a\\!=\\!0.39$ nm is the distance between the neighboring Eu$^{2+}$ moments and we take the value $b\\!=\\!0.5$ for nonuniversal numerical constant.", "For these parameters, $\\xi _{s}(T_{m})=6.6a$ and the 'scattering-to-smooth' crossover is nominally located at $t_{\\mathrm {cr}}\\!\\approx \\!0.43$ .", "We see, however, that above this temperature the behavior is not well described by the scattering-regime asymptotics shown by the dashed lines.", "This is related to the broad range of the crossover.", "Consequently, for selected parameters, the gap does not display a nonmonotonic behavior, expected from the analysis of asymptotics.", "In fact, due to the interplay between two competing temperature dependences, both corrections are almost temperature independent in the range $0.42<t<0.47$ .", "Nevertheless, we see that, according to the general predictions, $\\Delta (T)$ somewhat increases when $T$ approaches $T_{m}$ , while $[\\lambda (T)]^{-2}$ shows a noticeable drop.", "For illustrative purposes, we show in Fig.", "REF (right) the plots of $\\Delta (T)$ and $[\\lambda (T)]^{-2}$ for the same parameters as in the previous figure except for larger coherence length, $\\xi _{s0}^{\\mathrm {GL}}\\!=\\!10a\\!\\approx \\!3.9$ nm.", "In this case $\\xi _{s}(T_{m})\\!=\\!17.6a$ and the crossover nominally takes place much closer to $t_{m}$ , at $t_{\\mathrm {cr}}\\!\\approx \\!0.41$ .", "In this case the behavior at $t>0.43-0.44$ is already fairly well described by the scattering asymptotics.", "The gap in this case does have a nonmonotonic temperature dependence.", "Clearly, the plots in Fig.", "REF (left) do not literally describe the behavior of RbEuFe$_{4}$ As$_{4}$ and serve only as a qualitative illustration.", "This material has several additional features that influence the behavior of the parameters but substantially complicate an accurate analysis.", "Firstly, the assumed two-dimensional behavior always breaks down sufficiently close to the transition and the dimensional crossover to the three-dimensional regime takes place.", "In this 3D regime the correlations between the different magnetic layers emerge meaning that the assumption for two-dimensional scattering does not work any more.", "In addition, the magnetic correlation length does not follow the BKT temperature dependence assumed in Fig.", "REF .", "Secondly, due to spatial separation between the magnetic and conducting layers, we expect a significant nonlocality of the exchange interaction, see Eq.", "(REF ), ranging at least 2–3 lattice spacing.", "Consideration of this manuscript assumes that the magnetic correlation length exceeds this nonlocality range.", "This assumption is only justified close to the magnetic transition.", "The nonlocality significantly reduces the exchange corrections at higher temperatures, when $\\xi _{h}$ drops below the nonlocality range.", "Finally, our single-band consideration does not take into account a complicated multiple-band structure of RbEuFe$_{4}$ As$_{4}$ .", "In this paper, we developed a general theoretical framework for the analysis of the influence of correlated magnetic fluctuations on superconducting parameters.", "We focus on the behavior of the gap and superfluid density for the in-plane current direction, but the consideration can be directly extended to other thermodynamic and transport properties.", "For some properties, however, such as specific heat and magnetization, a reliable separation of the superconducting contribution from the magnetic background in experiment is challenging.", "This makes a theoretical analysis somewhat academic.", "Our result can be straightforwardly generalized to the case of a large exchange field leading to a strong suppression of superconductivity.", "Such generalization requires the development of a self-consistent scheme similar to the AG theory[1].", "For the problem considered here, this is a formidable theoretical task.", "I would like to thank U. Welp, S. Bending, D. Collomb, and V. Kogan for useful discussions.", "This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division." ], [ "Calculation of the integral for the gap correction ", "In this appendix we briefly describe calculation of the integral in Eq.", "(REF ) leading to result in Eq.", "(REF ).", "Substituting the function $W(z,g)$ defined in Eq.", "(REF ) into Eq.", "(REF ), we present $\\delta \\mathcal {I}$ as $\\delta \\mathcal {I}\\!= \\!-\\frac{C_{h}\\nu h_{0}^{2}}{4\\pi }\\frac{\\Delta _{0}}{\\left(\\omega _{n}^{2}\\!+\\!\\Delta _{0}^{2}\\right)^{3/2}}\\left[\\mathcal {T}_{1}(g_{n})\\!+\\!\\frac{4\\omega _{n}^{2}}{\\omega _{n}^{2}\\!+\\!\\Delta _{0}^{2}}\\mathcal {T}_{2}(g_{n})\\right],$ where $\\mathcal {T}_{1}(g) & =\\mathrm {Re}\\left[\\int \\limits _{-\\infty }^{\\infty }dz\\frac{2}{\\left(z+i\\right)^{2}\\sqrt{\\left(iz+1\\right)^{2}-g^{2}}}\\right.\\\\\\times &\\left.\\ln \\left(\\frac{iz+1+\\sqrt{\\left(iz+1\\right)^{2}-g^{2}}}{g}\\right)\\right],\\\\\\mathcal {T}_{2}(g) & =\\mathrm {Re}\\left[\\int \\limits _{-\\infty }^{\\infty }dz\\frac{2}{\\left(z^{2}+1\\right)^{2}\\sqrt{\\left(iz+1\\right)^{2}-g^{2}}}\\right.\\\\\\times &\\left.\\ln \\left(\\frac{iz+1+\\sqrt{\\left(iz+1\\right)^{2}-g^{2}}}{g}\\right)\\right].$ The integral for $\\mathcal {T}_{1}(g)$ has a pole at $z=-i$ and branches at the imaginary axis terminating at $z_{\\pm }=i\\left(1\\pm g\\right)$ .", "Deforming the integration contour into the complex plane, we reduce it to the integral along the square-root branch $z=ix$ , $1+g<x<\\infty $ , $\\mathcal {T}_{1}(g) & =-2\\pi \\int \\limits _{1+g}^{\\infty }dx\\frac{2}{\\left(x+1\\right)^{2}\\sqrt{\\left(x-1\\right)^{2}-g^{2}}}\\\\& =4\\pi \\left[\\frac{1}{4-g^{2}}-\\frac{2}{\\left(4-g^{2}\\right)^{3/2}}\\ln \\left(\\frac{2+\\sqrt{4-g^{2}}}{g}\\right)\\right].$ The integral for $\\mathcal {T}_{1}(g)$ has the same square-root branches and the poles at $z=\\pm i$ .", "Consequently, we split the integral into contribution from the pole at $z=i$ and square-root branch $z=ix$ , $1+g<x<\\infty $ which yields $\\mathcal {T}_{2}(g) & =\\frac{\\pi ^{2}}{2g}-\\frac{\\pi }{g^{2}}+4\\pi \\int \\limits _{1+g}^{\\infty }dx\\frac{1}{\\left(x^{2}-1\\right)^{2}\\sqrt{\\left(x-1\\right)^{2}-g^{2}}}\\\\& =-\\pi \\left[\\frac{1}{4-g^{2}}-\\frac{6-g^{2}}{\\left(4-g^{2}\\right)^{3/2}}\\ln \\left(\\frac{2+\\sqrt{4-g^{2}}}{g}\\right)\\right]$ Substituting the above results into Eq.", "(REF ), we arrive to Eq.", "(REF )." ], [ "Magnetic-scattering correction to the London penetration depth using\nquasiclassical approach ", "The London penetration depth $\\lambda $ in the presence of isotropic potential and magnetic scattering has been investigated within the quasiclassical approach in Ref. [30].", "Here we derive a general equation for $\\lambda $ for arbitrary magnetic scattering having in mind application to the case of correlated magnetic fluctuations.", "The Eilenberger equations for the quasiclassical Green's functions, $f(p,r)$ , $f^{\\dagger }(p,r)$ , and $g(p,r)$ for arbitrary scattering are [32] $& \\mathbf {v}\\mathbf {\\Pi }f=2\\Delta g-2\\omega _{n}f\\nonumber \\\\& +g\\left\\langle \\left[W(p,p^{\\prime })\\!-W_{m}(p,p^{\\prime })\\right]f{}^{\\prime }\\right\\rangle ^{\\prime }\\!\\nonumber \\\\& -\\!f\\left\\langle \\left[W(p,p^{\\prime })\\!+\\!W_{m}(p,p^{\\prime })\\right]g{}^{\\prime }\\right\\rangle ^{\\prime },\\\\-\\!", "& \\!\\mathbf {v}\\mathbf {\\Pi }^{*}\\!f^{\\dagger }\\!=\\!2\\Delta ^{*}g\\!-\\!2\\omega _{n}f^{\\dagger }\\!\\nonumber \\\\& +\\!g\\left\\langle \\left[W(p,p^{\\prime })\\!-\\!W_{m}(p,p^{\\prime })\\right]f^{\\dagger }{}^{\\prime }\\right\\rangle ^{\\prime }\\!\\nonumber \\\\& -\\!f^{\\dagger }\\left\\langle \\left[W(p,p^{\\prime })\\!+\\!W_{m}(p,p^{\\prime })\\right]g{}^{\\prime }\\right\\rangle ^{\\prime }\\!,$ where we used shortened notations $f\\!\\equiv \\!f(p,r)$ , $f^{\\prime }\\!\\equiv \\!f(p^{\\prime },r)$ , $\\mathbf {\\Pi }f\\!\\equiv \\!\\left(\\nabla \\!+\\!2\\pi i\\mathbf {\\mathbf {A}}/\\phi _{0}\\right)f$ , $\\left\\langle A(p^{\\prime })\\right\\rangle ^{\\prime }\\!\\equiv \\!\\int _{S_{F}}\\!d^{2}p^{\\prime }\\rho (p^{\\prime })A(p^{\\prime })$ , and $\\rho (p)\\!=\\!\\left[(2\\pi )^{3}\\nu v_{F}(p)\\right]^{-1}$ .", "Further, $W(p,p^{\\prime })$ and $W_{m}(p,p^{\\prime })$ are the probabilities of potential and magnetic scattering defining the corresponding scattering times as $\\frac{1}{\\tau }=\\left\\langle W(p,p^{\\prime })\\right\\rangle ^{\\prime },\\,\\frac{1}{\\tau _{m}}=\\left\\langle W_{m}(p,p^{\\prime })\\right\\rangle ^{\\prime }.$ For the model considered in this paper $W_{m}(p,p^{\\prime })=2\\pi \\nu \\left\\langle \\left|\\tilde{h}_{p-p^{\\prime }}\\right|^{2}\\right\\rangle $ .", "The above equations have to be supplemented with the normalization condition $g^{2}=1-ff^{\\dagger }$ , the gap equation $\\frac{\\Delta }{2\\pi T}\\ln \\frac{T_{c0}}{T}=\\sum _{\\omega _{n}>0}\\left(\\frac{\\Delta }{\\omega }-\\langle f\\rangle \\right),$ and formula for the current $\\mathbf {j}=4\\pi e\\nu T\\,{\\rm Im}\\sum _{\\omega _{n}>0}\\langle \\mathbf {v}g\\rangle .$ Our goal is to derive the response to weak supercurrents.", "In linear order, weak supercurrents do not modify the gap absolute value but only add the same phase $\\theta (\\mathbf {r})$ to $\\Delta $ and $f$ and the opposite phase to $f^{\\dagger }$ .", "Therefore, the linear-order solutions have the form, $\\Delta =\\Delta _{0}\\,e^{i\\theta },\\,f=(f_{0}+f_{1})\\,e^{i\\theta },\\\\f^{\\dagger }=(f_{0}+f_{1})e^{-i\\theta },\\,g=g_{0}+g_{1}\\,,$ where only the phase $\\theta $ has coordinate dependence, while $f_{1},f_{1}^{\\dagger },g_{1}$ are uniform meaning that $\\mathbf {\\Pi }f=i\\mathbf {P}f$ and $\\mathbf {\\Pi }^{*}f^{\\dagger }=-i\\mathbf {P}f^{\\dagger }$ with $\\mathbf {P}=\\nabla \\theta +2\\pi \\mathbf {\\mathbf {A}}/\\phi _{0}$ .", "Equations for the linear corrections are $2\\Delta _{0}&g_{1}\\!", "-\\!2\\omega _{n}f_{1}\\!+\\!\\frac{g_{1}(p)}{\\tau _{-}}f_{0}-\\frac{f_{1}(p)}{\\tau _{+}}g_{0}\\nonumber \\\\+ & g_{0}\\left\\langle \\left[W(p,p^{\\prime })\\!-\\!W_{m}(p,p^{\\prime })\\right]f_{1}(p^{\\prime })\\right\\rangle ^{\\prime }\\!\\nonumber \\\\-&f_{0}\\left\\langle \\left[W(p,p^{\\prime })\\!+\\!W_{m}(p,p^{\\prime })\\right]g_{1}(p^{\\prime })\\right\\rangle ^{\\prime }\\!=\\!iv_{\\alpha }P_{\\alpha }f_{0},\\\\& g_{0}g_{1}=-f_{0}f_{1}$ with $\\frac{1}{\\tau _{\\pm }}=\\frac{1}{\\tau }\\pm \\frac{1}{\\tau _{m}}.$ The remaining averages in the first equation account for the reverse scattering events.", "These averages vanish for the case of isotropic scattering.", "We assume that the solutions are proportional to $v_{\\alpha }P_{\\alpha }$ and define the corresponding averages as $\\left\\langle W(p,p^{\\prime })v_{\\alpha }^{\\prime }\\right\\rangle ^{\\prime }=\\frac{1}{\\tau ^{\\alpha }}v_{\\alpha },\\,\\left\\langle W_{m}(p,p^{\\prime })v_{\\alpha }^{\\prime }\\right\\rangle ^{\\prime }=\\frac{1}{\\tau _{m}^{\\alpha }}v_{\\alpha },$ giving $\\frac{1}{\\tau ^{\\alpha }}=\\frac{\\left\\langle \\left\\langle W(p,p^{\\prime })v_{\\alpha }v_{\\alpha }^{\\prime }\\right\\rangle \\right\\rangle ^{\\prime }}{\\left\\langle v_{\\alpha }^{2}\\right\\rangle },\\,\\frac{1}{\\tau _{m}^{\\alpha }}=\\frac{\\left\\langle \\left\\langle W_{m}(p,p^{\\prime })v_{\\alpha }v_{\\alpha }^{\\prime }\\right\\rangle \\right\\rangle ^{\\prime }}{\\left\\langle v_{\\alpha }^{2}\\right\\rangle }.$ These quantities determine the corresponding transport times in a standard way, $1/\\tau ^{\\mathrm {tr}}=1/\\tau -1/\\tau ^{\\alpha }$ and $1/\\tau _{m}^{\\mathrm {tr}}=1/\\tau _{m}-1/\\tau _{m}^{\\alpha }$ .", "In the case of correlated magnetic fluctuation which we consider in this paper, the transport rate is much smaller than the total scattering rate.", "The averagings in Eq.", "(REF ) can now be performed as, $\\left\\langle \\left[W(p,p^{\\prime })\\!\\pm W_{m}(p,p^{\\prime })\\right]f_{1}(p^{\\prime })\\right\\rangle ^{\\prime }\\!=\\frac{f_{1}(p)}{\\tau _{\\pm }^{\\alpha }}.$ with $1/\\tau _{\\pm }^{\\alpha }=1/\\tau ^{\\alpha }\\pm 1/\\tau _{m}^{\\alpha }$ .", "This allows us to rewrite Eq.", "(REF ) as $2\\Delta _{0}g_{1}\\!-\\!2\\omega _{n}f_{1}\\!+\\frac{g_{1}}{\\tau _{-}}f_{0}\\!+g_{0}\\frac{f_{1}}{\\tau _{-}^{\\alpha }}\\!-\\frac{f_{1}}{\\tau _{+}}g_{0}-f_{0}\\frac{g_{1}}{\\tau _{+}^{\\alpha }}=iv_{\\alpha }P_{\\alpha }f_{0},$ Substituting $g_{1}=-f_{0}f_{1}/g_{0}$ from Eq.", "(), we obtain the solutions $f_{1} & =-\\frac{\\!iv_{\\alpha }f_{0}P_{\\alpha }}{2\\omega _{n}\\!+\\!2\\Delta _{0}\\frac{f_{0}}{g_{0}}\\!+\\!\\frac{f_{0}^{2}}{g_{0}}\\left(\\frac{1}{\\tau _{-}}\\!-\\!\\frac{1}{\\tau _{+}^{\\alpha }}\\right)\\!+\\!g_{0}\\left(\\frac{1}{\\tau _{+}}\\!-\\!\\frac{1}{\\tau _{-}^{\\alpha }}\\right)},\\\\g_{1} & =\\frac{f_{0}^{2}}{g_{0}}\\frac{\\!iv_{\\alpha }P_{\\alpha }}{2\\omega _{n}\\!+\\!2\\Delta _{0}\\frac{f_{0}}{g_{0}}\\!+\\!\\frac{f_{0}^{2}}{g_{0}}\\left(\\frac{1}{\\tau _{-}}\\!-\\!\\frac{1}{\\tau _{+}^{\\alpha }}\\right)\\!+\\!g_{0}\\left(\\frac{1}{\\tau _{+}}\\!-\\!\\frac{1}{\\tau _{-}^{\\alpha }}\\right)}.$ We can rewrite the scattering-rate differences here in terms of scattering and transport times as $\\frac{1}{\\tau _{-}}\\!-\\!\\frac{1}{\\tau _{+}^{\\alpha }} & =\\frac{1}{\\tau ^{\\mathrm {tr}}}-\\frac{2}{\\tau _{m}}+\\frac{1}{\\tau _{m}^{\\mathrm {tr}}},\\\\\\frac{1}{\\tau _{+}}\\!-\\!\\frac{1}{\\tau _{-}^{\\alpha }} & =\\frac{1}{\\tau ^{\\mathrm {tr}}}+\\frac{2}{\\tau _{m}}-\\frac{1}{\\tau _{m}^{\\mathrm {tr}}}.$ We use a standard parametrization for the unperturbed Green's function components $f_{0}=\\frac{1}{\\sqrt{1+u^{2}}},\\:g_{0}=\\frac{u}{\\sqrt{1+u^{2}}}.$ The parameter $u$ obeys the Abrikosov-Gor'kov equation[1], [3] $u\\left(1-\\frac{1}{\\tau _{m}\\Delta \\sqrt{1+u^{2}}}\\right)=\\frac{\\omega _{n}}{\\Delta }.$ and determines the gap via equation $\\Delta \\ln \\frac{T_{c0}}{T}=2\\pi T\\sum _{\\omega _{n}>0}\\left(\\frac{\\Delta }{\\omega _{n}}-\\frac{1}{\\sqrt{1+u^{2}}}\\right).$ We proceed with the derivation of the current response using Eq.", "(REF ).", "Rewriting $g_{1}$ in Eq.", "() in terms of the parameter $u$ $g_{1} =\\frac{\\!iv_{\\alpha }P_{\\alpha }}{\\left(1+u^{2}\\right)\\left(2\\Delta \\sqrt{1+u^{2}}+\\!\\frac{1}{\\tau _{-}^{\\mathrm {tr}}}\\right)\\!-\\!\\frac{2}{\\tau _{m}^{\\alpha }}}.$ and substituting it into Eq.", "(REF ), we obtain the linear current response as $j_{\\alpha } \\!=\\!4\\pi e\\nu T\\sum _{\\omega _{n}>0}\\frac{\\langle v_{\\alpha }^{2}\\rangle }{\\left(1\\!+\\!u^{2}\\right)\\left(2\\Delta \\sqrt{1\\!+\\!u^{2}}+\\!\\frac{1}{\\tau _{-}^{\\mathrm {tr}}}\\right)\\!-\\!\\frac{2}{\\tau _{m}^{\\alpha }}}P_{\\alpha }.$ Using the definition $4\\pi j_{\\alpha }/c=-\\lambda _{\\alpha }^{-2}A_{\\alpha }$ and $P_{\\alpha }=2\\pi \\,A_{\\alpha }/\\phi _{0}$ , we finally obtain the result for $\\lambda _{\\alpha }^{-2}$ , $\\lambda _{\\alpha }^{-2}\\!=\\!\\frac{16\\pi ^{3}|e|\\nu \\langle v_{\\alpha }^{2}\\rangle }{c\\phi _{0}}T\\!\\!\\sum _{\\omega _{n}\\!>0}\\frac{1}{\\left(1\\!+\\!u^{2}\\right)\\left(\\Delta \\sqrt{1\\!+\\!u^{2}}+\\!\\frac{1}{2\\tau _{-}^{\\mathrm {tr}}}\\right)\\!-\\!\\frac{1}{\\tau _{m}^{\\alpha }}}.$ This result can be used for self-consistent evaluation of the London penetration depth of arbitrary scattering.", "Note that it is different from the similar result in Ref.", "[2] by the sign in front of $\\frac{1}{\\tau _{m}^{\\alpha }}$ in the denominator." ], [ "Small-scattering-rate expansion", "For comparison with the results in the main text, we derive small correction to $\\lambda _{\\alpha }^{-2}$ in the case of weak scattering.", "Expanding the parameter $u$ in Eq.", "(REF ), $u=\\frac{\\omega _{n}}{\\Delta }+u_{m},$ we obtain $u_{m}\\approx \\frac{\\omega _{n}}{\\tau _{m}\\Delta \\sqrt{\\omega _{n}^{2}+\\Delta ^{2}}}.$ Substituting this expansion into the gap equation, Eq.", "(REF ), we derive the correction to the gap, $\\Delta =\\Delta _{0}+\\tilde{\\Delta }$ , $\\tilde{\\Delta } & \\!=\\!-\\frac{2\\pi T}{\\tau _{m}\\Delta _{0}}\\!\\sum _{\\omega _{n}>0}\\frac{\\omega _{n}^{2}}{\\left(\\Delta _{0}^{2}\\!+\\!\\omega _{n}^{2}\\right)^{2}}\\left[2\\pi T\\!\\!\\sum _{\\omega _{n}>0}\\frac{1}{\\left(\\Delta _{0}^{2}\\!+\\!\\omega ^{2}\\right)^{3/2}}\\right]^{-1}\\!.$ In the reduced form, this correction is identical to Eq.", "(REF ).", "To derive the correction to $\\lambda _{\\alpha }^{-2}$ , we expand the fraction in Eq.", "(REF ) $& \\frac{1}{\\left(1\\!+\\!u^{2}\\right)\\left(\\Delta \\sqrt{1\\!+\\!u^{2}}\\!+\\!\\frac{1}{2\\tau _{-}^{\\mathrm {tr}}}\\right)\\!-\\!\\frac{1}{\\tau _{m}^{\\alpha }}}\\!\\approx \\!\\frac{\\Delta ^{2}}{\\left(\\Delta ^{2}\\!+\\!\\omega _{n}^{2}\\right)^{3/2}}\\\\& \\times \\left(\\!1\\!-\\frac{3\\omega _{n}^{2}-\\Delta ^{2}}{\\tau _{m}\\left(\\Delta ^{2}\\!+\\!\\omega _{n}^{2}\\right)^{3/2}}\\!+\\frac{\\omega _{n}^{2}-\\Delta ^{2}}{2\\tau _{m}^{\\mathrm {tr}}\\left(\\Delta ^{2}\\!+\\!\\omega _{n}^{2}\\right)^{3/2}}\\!\\right.\\\\& -\\left.\\frac{1}{2\\tau ^{\\mathrm {tr}}\\sqrt{\\Delta ^{2}\\!+\\!\\omega _{n}^{2}}}\\!\\right)$ and also separate the contribution from the gap correction $\\frac{\\Delta ^{2}}{\\left(\\Delta ^{2}\\!+\\!\\omega _{n}^{2}\\right)^{3/2}} & \\approx \\frac{\\Delta _{0}^{2}}{\\left(\\Delta _{0}^{2}+\\omega _{n}^{2}\\right)^{3/2}}+\\frac{\\Delta _{0}\\tilde{\\Delta }\\left(2\\omega _{n}^{2}\\!-\\!\\Delta _{0}^{2}\\right)}{\\left(\\Delta _{0}^{2}+\\omega _{n}^{2}\\right)^{5/2}}.$ This gives the correction to $\\lambda _{\\alpha }^{-2}$ , $\\lambda _{\\alpha }^{-2} & \\approx \\lambda _{0\\alpha }^{-2}+\\lambda _{1\\alpha }^{-2},\\nonumber \\\\\\lambda _{0\\alpha }^{-2} & =\\!\\frac{16\\pi ^{4}\\nu \\langle v_{\\alpha }^{2}\\rangle }{\\phi _{0}^{2}}T\\sum _{\\omega _{n}>0}\\frac{\\Delta _{0}^{2}}{\\left(\\Delta _{0}^{2}+\\omega _{n}^{2}\\right)^{3/2}},\\nonumber \\\\\\lambda _{1\\alpha }^{-2} & =\\!\\frac{16\\pi ^{4}\\nu \\langle v_{\\alpha }^{2}\\rangle }{\\phi _{0}^{2}}T\\!\\sum _{\\omega _{n}>0}\\!\\left[\\frac{\\Delta _{0}\\tilde{\\Delta }\\left(2\\omega _{n}^{2}\\!-\\!\\Delta _{0}^{2}\\right)}{\\left(\\Delta _{0}^{2}+\\omega _{n}^{2}\\right)^{5/2}}\\right.\\nonumber \\\\- & \\frac{\\Delta _{0}^{2}\\left(3\\omega _{n}^{2}\\!-\\!\\Delta _{0}^{2}\\right)}{\\tau _{m}\\left(\\Delta _{0}^{2}+\\omega _{n}^{2}\\right)^{3}}+\\frac{\\Delta _{0}^{2}\\left(\\omega _{n}^{2}\\!-\\!\\Delta _{0}^{2}\\right)}{2\\tau _{m}^{\\mathrm {tr}}\\left(\\Delta _{0}^{2}\\!+\\!\\omega _{n}^{2}\\right)^{3}}\\nonumber \\\\- & \\left.\\frac{\\Delta _{0}^{2}}{2\\tau ^{\\mathrm {tr}}\\left(\\Delta _{0}^{2}\\!+\\!\\omega _{n}^{2}\\right)^{2}}\\right].$ We see that, in contrast to the potential scattering, which only influences the London penetration depth via the transport time, the magnetic-scattering contribution to $\\lambda _{1\\alpha }^{-2}$ also has contributions proportional to the total scattering rate $1/\\tau _{m}$ , both direct and via the gap correction.", "This pair-breaking contribution, however, vanishes at zero temperature.", "For numerical convenience, we can rewrite the correction in the following reduced form $&\\lambda _{1\\alpha }^{-2} (T)=-\\lambda _{0\\alpha }^{-2}(T)\\left[\\frac{1}{\\tau _{m}\\Delta _{0}}V_{\\lambda ,m}\\!\\left(\\frac{2\\pi T}{\\Delta _{0}}\\right)\\nonumber \\right.\\\\&+\\left.\\frac{1}{\\tau _{m}^{\\mathrm {tr}}\\Delta _{0}}V_{\\lambda ,m}^{\\mathrm {tr}}\\!\\left(\\frac{2\\pi T}{\\Delta _{0}}\\right)\\!+\\!\\frac{1}{\\tau ^{\\mathrm {tr}}\\Delta _{0}}V_{\\lambda }^{\\mathrm {tr}}\\!\\left(\\frac{2\\pi T}{\\Delta _{0}}\\right)\\right],$ with $V_{\\lambda ,m}(\\tilde{T}) & \\!=\\!\\left[\\mathcal {D}\\!", "(\\tilde{T})\\right]^{-1}\\!\\tilde{T}\\sum _{n=0}^{\\infty }\\!\\left[\\frac{2\\tilde{\\omega }_n^2\\!-1}{\\left(1\\!+\\!\\tilde{\\omega }_n^2\\right)^{5/2}}V_{\\Delta }(\\tilde{T})\\!+\\!\\frac{3\\tilde{\\omega }_n^2\\!-1}{\\left(1\\!+\\!\\tilde{\\omega }_n^2\\right)^{3}}\\right]\\!,\\nonumber \\\\V_{\\lambda ,m}^{\\mathrm {tr}}(\\tilde{T}) & \\!=\\!\\frac{1}{2}\\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\!\\tilde{T}\\sum _{n=0}^{\\infty }\\frac{1-\\tilde{\\omega }_n^2}{\\left(1+\\tilde{\\omega }_n^2\\right)^{3}},\\nonumber \\\\V_{\\lambda }^{\\mathrm {tr}}(\\tilde{T}) & \\!=\\!\\frac{1}{2}\\left[\\mathcal {D}(\\tilde{T})\\right]^{-1}\\!\\tilde{T}\\sum _{n=0}^{\\infty }\\frac{1}{\\left(1+\\tilde{\\omega }_n^2\\right)^{2}},\\nonumber $ where $\\tilde{\\omega }_n\\!\\equiv \\!\\tilde{T}(n\\!+\\tfrac{1}{2})$ , $\\mathcal {D}(\\tilde{T})$ is defined in Eq.", "(), and $V_{\\Delta }(\\tilde{T})$ in the formula for $V_{\\lambda ,m}(\\tilde{T})$ is defined in Eq. ().", "For zero temperature, we derive the following result for the gap correction $\\lambda _{1\\alpha }^{-2}(0) & =-\\frac{\\pi }{8}\\lambda _{0\\alpha }^{-2}\\left(\\frac{1}{2\\tau _{m}^{\\mathrm {tr}}\\Delta _{0}}+\\frac{1}{\\tau ^{\\mathrm {tr}}\\Delta _{0}}\\right)$ with $\\lambda _{0\\alpha }^{-2}=8\\pi ^{3}\\nu \\langle v_{\\alpha }^{2}\\rangle /\\phi _{0}^{2}$ .", "Therefore, the correction to the London penetration depth at $T=0$ in the clean case is proportional to transport scattering rates for both scattering channels." ] ]

[ [ "Low-Cost Fiducial-based 6-Axis Force-Torque Sensor" ], [ "Abstract Commercial six-axis force-torque sensors suffer from being some combination of expensive, fragile, and hard-to-use.", "We propose a new fiducial-based design which addresses all three points.", "The sensor uses an inexpensive webcam and can be fabricated using a consumer-grade 3D printer.", "Open-source software is used to estimate the 3D pose of the fiducials on the sensor, which is then used to calculate the applied force-torque.", "A browser-based (installation free) interface demonstrates ease-of-use.", "The sensor is very light and can be dropped or thrown with little concern.", "We characterize our prototype in dynamic conditions under compound loading, finding a mean $R^2$ of over 0.99 for the $F_x, F_y, M_x$, and $M_y$ axes, and over 0.87 and 0.90 for the $F_z$ and $M_z$ axes respectively.", "The open source design files allow the sensor to be adapted for diverse applications ranging from robot fingers to human-computer interfaces, while the sdesign principle allows for quick changes with minimal technical expertise.", "This approach promises to bring six-axis force-torque sensing to new applications where the precision, cost, and fragility of traditional strain-gauge based sensors are not appropriate.", "The open-source sensor design can be viewed at http://sites.google.com/view/fiducialforcesensor." ], [ "Motivation", "Force-torque sensors are used extensively in both industry and research.", "We focus here on the use of these sensors in two examples: robotic grasping, where they are used to provide tactile feedback (e.g.", "detecting when contact is made), and in human computer interaction.", "However, commercial six-axis force-torque sensors can be both expensive and fragile.", "This combination makes them tricky to use for grasping, where controlled contact is desired, but a small coding error could easily smash and overload the sensor.", "One of the most common types of sensors, the ATI force/torque sensor, costs tens of thousands of dollars and relies on strain gauges that are fragile and have to be surrounded in a bulky package.", "For these reasons, we are motivated to consider new sensor designs that could promote the use of tactile data in the robotics community through being a combination of cheaper, easier to use, and more robust.", "Figure: Consumer webcams and a printed fiducial markers can be used to create asix-axis force-torque sensor.", "We used four springs to build aplatform free to move in all angular directions.", "We affixed twoprinted fiducials to the platform, and then aimed a consumercamera up at them.", "To the right, the camera viewreveals the tag location.", "The tags are glued to the lightshield, which is removable, allowing for easy design changes.Note that cardstock, which was removed for picture clarity, was usedto diffuse the LED and avoid overexposing the camera.", "Green bottle cap is for scale." ], [ "Related Work", "Multiple designs have emerged recently taking advantage of the rich information available from consumer webcams.", "Even low-end webcams will output 640x480 RGB images at 15 frames-per-second (fps).", "The webcam-based sensors are particularly easy to manufacture and wire.", "Notable examples include the Gelsight [1], GelForce [2], TacTip [3], the Fingervision [4], and others.", "These sensors rely on cameras facing markers embedded in transparent or semi-transparent elastomer (often with supplemental LED lighting).", "These can be used to estimate shear, slip, and force, but tend not to do well in cases where the object hits the side of the finger instead of dead on.", "They also require casting elastomers.", "Several MEMS multi-axis force-torque sensors have been developed, which use the same principle of creating a device free to deflect into multiple axes, but then measures them using capacitative [5] or piezoresistive [6] means.", "In [7] the deflection is measured using a camera as well, a CCD camera mounted to a microscope, however the device only measures two directions of force.", "Prior work used MEMS barometers to create six-axis force-torque sensors with very low parts cost and good durability [8].", "However, fabricating the sensor requires specialized lab equipment such as a degassing machine.", "Other work explored estimating fingertip force via video, but only for human fingers [9], [10].", "Commercial sensors like the Spacemouse and the OptoForce use similar ideas, but rely on custom circuitboards for a ranging sensor inside.", "In contrast, our work is straightforward to fabricate even for users unfamiliar with electronics." ], [ "Contributions", "In this paper, we investigate novel combinations of readily-accessible technologies to create six-axis force-torque sensors that are inexpensive, require minimal expertise to design and build, and are easily customized for diverse applications.", "The proposed novel type of sensor makes six-axis force-torque measurements by tracking position and orientation displacement using the 3D pose estimate from fiducial tags, and uses a linear fit between displacement and applied force-torque.", "Fiducials are markers used to help locate objects or serve as points of reference.", "They can be found in robotics and augmented reality applications, where they usually take the form of printed paper markers glued onto various objects of interest.", "Sensors employing these fiducials operate by detecting the sharp gradients that are created between black and white pixels, such as one might find on a checkerboard.", "An example of two fiducials can be found in the top right of the labelled diagram of our sensor at fig:introlabelled.", "Using the known geometry of the tag (e.g.", "perpendicular sides of checkeboard), as well as known tag size and pre-determined camera calibration matrix, the 3D object pose (location and orientation) of the object can be estimated.", "This calculation is known as the solving the Perspective-n-Point (PnP) problem.", "We created prototypes utilizing two open-source tag protocols, AprilTags [11] and ArUco markers [12]; pictured in fig:introlabelled are two ArUCo markers.", "In the following sections, we begin with the design and fabrication process for our sensor.", "We follow with a theoretical analysis of how the sensor design parameters affect resolution, sensitivity, measurement range, and bandwidth.", "We also present an analysis of data collected from a prototype sensor.", "We conclude with a discussion of the advantages and limitations of this sensor." ], [ "Sensor Design", "At a high level, the sensor consists of two main parts: a base and a platform above the base.", "The platform is connected to the base with 4 springs and can move in all directions with respect to the base.", "Two fiducial tags were glued to the underside of the platform.", "Then, a webcam pointed up at the tags was installed at the base.", "As force or torque is applied to the platform, the tags translate and rotate accordingly.", "The camera is used to track the 3D pose of the tags.", "Should there be a suitably linear relationship between the displacement and the force-torque applied, a short calibration procedure using known weights can be used to collect datapoints for regression.", "Given a known linear fit, the sensor can then output force and torque measurements.", "fig:diagram shows the principle behind this fiducial-based force sensor." ], [ "Design Goals", "When designing the sensor prototype, a few considerations were made.", "First and foremost, the sensor needs to be sensitive to all six degrees of freedom (displacement in $x$ , $y$ , $z$ and rotation in yaw, pitch, roll).", "For illustrative purposes, the following analysis is performed in terms of specific specification values that are appropriate for a sample robot gripper.", "Alternate values for other use cases such as human-computer interfaces can be easily substituted.", "For grasping, between $\\pm $ 40 N is realistic, and sensitivity of at least 1/10 N is desirable.", "Qualitatively, we want the sensor to be small (for grasping applications, the sensor should be roughly finger-sized), inexpensive, and robust.", "The sensor should allow for rapid prototyping and easy customization with minimal technical expertise.", "The sensor should be not only easy to fabricate, but also easy to use." ], [ "Physical Fabrication", "The four pieces in fig:pieces (figure includes dimensions) are 3D-printed in two to three hours on an inexpensive consumer-grade device (Select Mini V2, Monoprice).", "Epoxy is used to glue the springs into the camera cover and top plate.", "The tags are printed on paper and glued in.", "A small piece of white cardstock is used to diffuse the LED (in the future, this would be built into the 3D design).", "Conveniently, the pose estimate is relative to the camera frame, and the sensor relies only on relative measurements, so the tag placement can be imprecise.", "The LED is mounted in and connected to a 3.3 V power source.", "The heat-set thread inserts (for bolting the light shield to the platform) are melted in with a soldering iron.", "The camera is placed between the mounting plate and camera cover and then everything is bolted together.", "The springs are steel compression springs available online as part of an assortment pack from Swordfish Tools.", "The spring dimensions are 2.54 cm long, 0.475 cm wide, and wire width of 0.071 cm, with a stiffness of approximately ${0.7}{N/mm}$ .", "Fabrication can be completed in a day.", "The actual assembly, given a complete set of hardware and tools, can be completed in 30 minutes, depending on the epoxy setting time." ], [ "Usage and Software", "The only data cable used is the USB from the webcam to the computer.", "On the computer, the OpenCV Python library [13] (version 4.1.2) is used to detect the ArUco markers in the video feed.", "We used a commercial force-torque sensor to characterize our sensor, for which we used another freely available Python library (see [14]).", "The data from the commercial sensor (Model HEX-58-RE-400N, OptoForce, Budapest, Hungary) and the markers are read in parallel threads and timestamped, then recorded to CSV.", "Python is used for further analysis.", "By using a consumer webcam, sensor reading is also possible without installing Python.", "To demonstrate this, we developed a simple interface using a Javascript ArUco tag detector library (see [15]).", "fig:JSGUI shows a graphical user interface (GUI) that plots the $x$ , $y$ , and $z$ -axes of the 3D pose estimate for a single tag.", "Figure: Our prototype JavaScript-based interface (modified fromthe Js-aruco library example) .", "In this way,sensor data can be read just by loading a webpage.In theory, the sensor reading can be done on-the-go with a smartphone and a wireless or USB-C webcam (such as inexpensive endoscope inspection cameras found online)." ], [ "Calibration", "Although we calibrated using a commercial force-torque sensor, the same can be achieved with a set of weights and careful clamping.", "The sensor can be clamped sideways to a sturdy surface to calibrate the $x$ - and $y$ -axes.", "A set of known weights is then attached to the center bolts on the light shield piece via a string.", "The same procedure can be applied to calibrate the $z$ -axis, with the sensor clamping upside down to a tabletop.", "Finally, weights can be applied to the two side bolts to produce known torques while hanging upside down or sideways.", "Considering the above design goals, there are a few primary concerns amenable to theoretical analysis: the sensor resolution, sensitivity, force range, and bandwidth.", "Here, sensor resolution is defined in bits (relative terms) and sensitivity in millimeters and degrees." ], [ "Resolution", "Let us conservatively estimate the discernible resolution of the tag system to be $d_R = 1/4$ pixel, or $C= 4$ counts per pixel.", "This factor exists because we have more than just binary information (1 bit) for every pixel.", "For instance, if a black/white intersection is halfway between two pixels, the pixels will be gray.", "(Tag algorithms also use the known grid geometry to achieve subpixel resolution – see the cornerSubPix function in the OpenCV library).", "In that case, we can determine the resolution of the sensor itself geometrically, by looking at the number of pixels.", "The fact that the tags must stay on-screen limits the sensor resolution.", "We can characterize an approximate $y$ -axis resolution $r_y$ of the camera by taking the number of pixels available, multiplying by $C$ , and converting our counts into bits.", "$r_y &= \\lfloor \\log _2 \\left( C \\cdot (h_{frame} - h_{img}) \\right) \\rfloor + 1$ For instance, the calculations for our sensor prototype are as follows.", "In the $y$ -axis, $r_y &= \\lfloor \\log _2{\\left(4 \\cdot (480 - 240)\\right)} \\rfloor \\\\r_y &= 10 ~\\text{bits}$ In the $x$ -axis, repeating the same calculations we have $r_x &= \\lfloor \\log _2 \\left( C \\cdot (w_{frame} - w_{img}) \\right) \\rfloor + 1 \\\\r_x &= \\lfloor \\log _2{\\left(4 \\cdot (640- 150)\\right)} \\rfloor + 1 \\\\r_x &= 11 ~\\text{bits}$ In the $z$ -axis, our limitation is the same as the $y$ -axis, so we have $r_z = 11 ~\\text{bits}$ ." ], [ "Sensitivity", "Let us now calculate the sensitivity of the sensor.", "We will start by looking at the minimum detectable travel in each of the $x$ , $y$ , and $z$ -axes." ], [ "Translational Sensitivity", "In the $x$ and $y$ directions, we can measure the mm/px at rest (the sensor resolution varies a bit since the tag gets larger or smaller depending on the $z$ distance).", "Roughly, the tag measures ${4.5}{mm}$ and appears as $w_{tag} = 150$  pixels in the image.", "Assuming as above that we can discern 4 counts per pixel, the theoretical sensitivity is $s_y &= \\frac{h_{frame} ~{}{(mm)}}{h_{frame} ~{}{(px)}} d_R= \\frac{4.5}{150} \\cdot \\frac{1}{4}= {0.0075}{mm}$ For the $z$ -axis sensitivity, we consider that the tag will get smaller as it displaces in the $+z$ direction.", "Using a simple geometrical model (see fig:zres), given that the smallest detectable change in $xy$ plane is $1/4$  pixel, we can calculate what is the resulting change in $z$ .", "Using similar triangles, we see that $\\frac{d_1}{d_z} &= \\frac{d_2}{d_{tag}} \\\\d_1 + d_2 &= w_{img}/2 \\\\d_2 &= (w_{img}/2) - d_1$ We would like to work in ${}{mm}$ , therefore we use the fact that the tag is 4.5 mm and appears as 150 px.", "$d_1 = d_R = 1/4 \\; \\text{px} \\cdot \\frac{{4.5}{mm}}{{150}{px}}&= {0.0075}{mm} \\\\d_2 = \\frac{4.5}{2} - 0.0075 &= {2.2425}{mm} \\\\s_z = d_z = \\frac{d_1}{d_2} d_{tag} = \\frac{0.0075}{2.2425} \\cdot 21 &= {0.07}{mm}$" ], [ "Rotational Sensitivity", "For rotation about the $z$ axis, we can calculate the chord length in pixels traveled when a tag is rotated 45 degrees (about its center), and use the same assumption of four counts per pixel to estimate our rotational sensitivity.", "Geometrically, we know that $l_{chord} &= 2 ~r \\sin \\frac{\\theta }{2}$ In our case, with $w_{img} = {150}{px}$ , we see that $\\mathit {r} &= \\sqrt{2} \\cdot w_{img} /2 \\\\l_{chord} &= 2 \\sqrt{2} \\cdot 150/2 \\cdot \\sin \\frac{\\pi /4}{2} = {81.18}{px} \\\\s_{\\tau z} &= \\frac{\\theta }{l_{chord}} \\cdot d_R= \\frac{81.18}{150} \\cdot \\frac{1}{4}= {0.14}$ For rotation about the $x$ and $y$ -axes, the analysis becomes a matter of determining the $z$ -axis change in mm, and using that to determine the pixels changed in the $x$ -$y$ plane.", "Consider a 45 degree rotation around the $z$ -axis of a tag that starts out flat (facing the camera), as shown in fig:tauxyres.", "Using $w_{img} ={150}{px}$ as before, the $z$ sensitivity is as follows: $w_{img}/2 &= \\sqrt{2} \\cdot d_z \\\\d_z + w_x &= w_{img}/2 \\\\w_x &= 0.5 \\; w_{img} - \\frac{0.5 \\; w_{img}}{\\sqrt{2}} = {21.97}{px} \\\\s_{\\tau xy} &= \\frac{\\theta }{w_x} d_R= \\frac{{45}}{21.97} \\cdot \\frac{1}{4} = {0.51}$" ], [ "Notes on ", "Intuitively, we expect that the sensor is much less reliable in the $z$ displacement direction.", "For movement along the $x$ and $y$ -axes axes, the camera sees the entire set of black/white intersections moving left or right.", "For the same reason, in the single tag setup it would be easy to detect rotations about the $z$ -axis, and difficult to detect rotations around the $x$ and $y$ -axes.", "Data collected from this initial (single-tag) design exactly reflected the aforementioned issue.", "Consequently, the design was enhanced with two tags oriented at 45 degrees to the camera.", "This proved sufficient for recovering all six force/torque axes." ], [ "Force Range Versus Sensitivity", "There is a clear trade-off between sensitivity (minimum detectable change in force) and the maximum force range.", "As an example, for a desired force range $F_{range} = \\pm {1}{N} = {2}{N}$ (close to the observed force range for our prototype), and a maximum displacement of $y_{range} = h_{frame} - h_{img}$ , the $y$ sensitivity $s_y$ in Newtons is as follows.", "$s_{y} &= \\frac{F_{range}}{ C \\cdot y_{range}}= \\frac{2}{4 \\cdot (480 - 240)} = {0.0021}{N}$ Our $s_y$ is thus 2.1 mN (given our assumption of $d_R = 0.25$ ).", "Similarly, for the $x$ -axis we find a sensitivity $s_x$ = ${1.0}{mN}$ at this force range.", "Now consider instead the grasping use case, with a desired force range of $\\pm $ 40 N, and desired sensitivity of at least 0.1 N. If we scale the calculations in eqn:maxdisplacement by 40 to get a $\\pm $ 40 N force range while keeping the other parameters the same, the sensor has 0.04 N and 0.08 N sensitivities in the $x$ and $y$ directions respectively." ], [ "Linearity", "In order to evaluate the linearity (and therefore usefulness) of the sensor, we used a commercial force-torque sensor (Model HEX-58-RE-400N, OptoForce, Budapest, Hungary) to provide ground truth measurements.", "Although the OptoForce measures force and torque at a different origin than where the load is applied, the analysis of the linearity of the sensor holds.", "Data was collected with a Python script which used the OpenCV library to interface with the camera.", "The setup is shown in fig:datacollect.", "Figure: Left, the data collection setup is shown (with theLED off – note that out-of-frame, there is an Arduino supplying3.3V{3.3}{V} to the LED.", "Later designs used a 3.3 V coin cellbattery to make the sensor standalone).", "Right, a method tocalibrate the sensor without using the commercial sensor isdemonstrated.", "The sensor is mounted upside down and weights are hungby string from the sensor to apply force uniaxially to the +zz axis.Autocorrelation was used to determine the lag between our sensor and the OptoForce.", "The sensor lag between the prototype sensor and the OptoForce was roughly 40 milliseconds.", "Next, linear interpolation was used to match our sensor data with the OptoForce data, which were output at roughly 25 Hz and 125 Hz respectively.", "The sensor data was smoothed with an exponential filter with weight of 0.2 to improve the autocorrelation results.", "For calibration, we take a dataset of displacements $D$ and apply linear regression (with an affine term) against all six axes.", "$\\theta $ , $\\phi $ , and $\\gamma $ refer to rotation around the $x$ , $y$ , and $z$ axes respectively.", "$K$ then forms a 6-by-6 matrix as shown below.", "$\\begin{bmatrix}F_x \\\\ F_y \\\\ F_z \\\\ M_x \\\\ M_y \\\\ M_z\\end{bmatrix}=\\left[\\begin{array}{cccccc}\\cr \\\\& & K_{6\\times 6} & & & \\\\\\cr \\\\\\end{array}\\right]\\begin{bmatrix}D_x \\\\ D_y \\\\ D_z \\\\ D_{\\theta } \\\\ D_{\\phi } \\\\ D_{\\gamma }\\end{bmatrix}+\\begin{bmatrix}\\cr \\\\B\\\\\\cr \\\\\\end{bmatrix}$" ], [ "Bandwidth", "Sensor bandwidth is directly limited by the camera framerate.", "This must be physically measured since the Python script will output at unrealistically high framerate – the OpenCV library reads from a buffer of stale images and will return a result even if the camera has not physically delivered a new frame.", "The webcam is pointed at a display with high refresh rate.", "A script turns the screen black, and as soon as the camera detects the black color, the screen changes to white, and so forth, and the frames displayed is compared to system time to obtain the framerate of the webcam.", "Note that this calculates our maximum sensor bandwidth; our actual sensor bandwidth is determined by the tag detection rate.", "If dynamic instead of quasi-static loading is assumed, then motion blur can lead to tag detection failure." ], [ "Linearity", "In multiaxial loading, the sensor was manually moved around in all directions.", "As shown in fig:linfit, the fits had a $R^2$ of 0.991, 0.996, 0.875, 0.997, 0.997, and 0.902 for the $F_x, F_y, F_z, M_x, M_y$ , and $M_z$ axes respectively.", "The $F_z$ axis fit is notably worse than the $F_x$ and $F_y$ fits, which was expected as explained in sec:zres.", "Figure: The black line represents a perfectly linear response between oursensor and the commercial sensor.", "The red dots show the actualsensor measurements using the ArUco tags.For qualitative comparison, fig:linfit shows an example of a reconstructed dataset, where the linear fits are plotted against the original signal for qualitative comparison.", "This diagram shows the relatively large deviations in $F_z$ from the original signal, indicating noisiness in the tag measurements.", "Figure: For qualitative inspection, a compound-loading datasetis shown here.", "The commercial sensor measurements are in black, andthe interpolated and linearly fitted prototype sensor's measurements are shown in red." ], [ "Bandwidth", "Our maximum sensor bandwidth is experimentally determined to be 25 Hz.", "Additionally, the camera we used was one of three cameras bought by selecting for low cost, quick availability, and lack of external camera case.", "We also measured the other two cameras which, despite advertising similar framerates, exhibited noticeable differences in framerate.", "Operating at 640x480, we measured 25 fps, 33 fps, and 15 fps for the three cameras, as listed in tbl:camera.", "Table: Camera Specifications" ], [ "Discussion", "Our prototype sensor showed mostly linear responses under dynamic loading.", "While the linearity is not precise, these results still validate the underlying hypothesis that with fiducials it is possible to collect data on all three axes of force and three axes of torque.", "Further design iterations could improve on these results, although this approach is unlikely to achieve the  0.1 sensors." ], [ "Design Goals", "The sensor can now be evaluated against the goals specified previously in sec:designgoals.", "The sensor design is indeed responsive in all six axes (after our pivot from one tag to two tags, as well as using a much brighter LED).", "Additionally, for grasping applications, the calculations in eqn:maxdisplacement shows that if a much stiffer spring were chosen so that 40 N of load could be applied without exceeding the $y_{range}$ , the sensor would still have better than 0.1 N of sensitivity.", "The qualitative design goals were also met.", "The sensor is small, measuring only 3.6 cm by 3.1 cm by 5.1 cm in size.", "The sensor is inexpensive, with the majority of the cost being a $20 webcam.", "The sensor is robust and has survived multiple plane trips and the occasional throw or drop.", "The sensor is also easy to modify.", "The light shield can easily be unbolted to change the fiducials, or re-printed in an hour to accommodate different designs (e.g.", "a single-tag vs. dual-tag design).", "Fabrication is easy and non-toxic, requiring no degassing machine (as with elastomer-based sensors) nor electrical discharging machines (as with custom strain-gauge based designs).", "The sensor by design does not suffer from thermal considerations (as in [8]) or electrical noise (as with designs based on strain gauges)." ], [ "Error Sources", "An important consideration is the coordinate origin around which measurements are made.", "As load must be applied to the spring platform on which the tags are glued, the origin around which measurements are collected may be different than desired, although a linear offset matrix should suffice to correct for this.", "Our six-axis measurement reflects a combination of a camera pose estimation and mechanical coupling, each of which can introduce errors.", "In the following section on sensor improvement, we focus on camera sensor issues." ], [ "Fiducial Changes", "Unlike the standard use cases for ArUco markers, we do not care about distinguishing multiple objects and care more about the quality of the pose estimate for a tag guaranteed to be in-frame.", "A custom fiducial (perhaps solely a checkerboard) could improve the force-torque measurements." ], [ "Noise in z-axis", "The sensor is noisy in force and torque measurements along the $z$ -axis.", "To address this, one possibility is to use a mirror and two tags which are laid flat on the $xy$ plane and the $yz$ plane respectively.", "The “sideways\" tag (on the $yz$ plane) has good sensitivity to $z$ -axis displacements, and the flat $xy$ plane tag is addresses rotations around the $z$ -axis.", "A 45-degree mirror then allows the camera to also observe the \"sideways\" tag on the $yz$ plane.", "On the downside, the small mirror could make assembly difficult." ], [ "Sensor Size", "Closer placement of the tag, to minimize the size of the sensor, may also be desired – this would necessitate a custom lens for the camera to allow for closer focus (e.g.", "a macro lens).", "Miniaturization could also be accomplished with a smaller camera, as in [3]." ], [ "Replacing Springs", "The use of springs means that the sensor may behave poorly in high frequency domains.", "Replacing the springs with another mechanism, such as a Stewart platform, could allow custom tuning of the response.", "Another possibility would be to fill the gap between the camera and the tag with optically clear material that would be resistant to high frequency inputs.", "[16] used a similar idea with a magnet and hall effect sensor, for a three-axis force sensor.", "However, such a design would complicate fabrication and potentially make camera calibration difficult due to image warping.", "Table: List of components and approximate costs." ], [ "Conclusion", "We present a novel type of six-axis force-torque sensor using fiducial tags and a webcam.", "The design is fast to fabricate and simple to use, and is also strong enough to survive drops and crashes common in contact-rich tasks such as robotic grasping.", "With only 3D-printed custom components, the design needs minimal technical expertise to adapt to applications ranging from manipulation to human-computer interaction research.", "The open-source design also allows for direct integration in designs for tasks such as grasping where sensor size is important.", "This fiducial-based sensor is less accurate than commercial force-torque sensors, but is also orders-of-magnitude less expensive – commercial sensors can cost thousands of dollars, while the parts cost of our sensor is under $50 (see tbl:bom).", "These combined advantages of our prototype sensor validates the general design principle of using 3D pose estimates from printed fiducials to create a six-axis force-torque sensor.", "Future work on improving the $F_z$ and $M_z$ axes could allow for an inexpensive, user-friendly, and robust alternative to current commercial sensors, opening up a new range of use cases for six-axis force-torque sensors." ] ]

[ [ "Construction and Calibration of a Streaked Optical Spectrometer for\n Shock Temperature" ], [ "Abstract Here we describe the implementation and calibration of a streaked visible spectrometer (SVS) for optical pyrometry and emission/absorption spectroscopy on light gas gun platforms in the UC Davis Shock Compression Laboratory.", "The diagnostic consists of an optical streak camera coupled to a spectrometer to provide temporally and spectrally-resolved records of visible emission from dynamically-compressed materials.", "Fiber optic coupling to the sample enables a small diagnostic footprint on the target face and flexibility of operation on multiple launch systems without the need for open optics.", "We present the details of calibration (time, wavelength and spectral radiance) for absolute temperature determination and present benchmark measurements of system performance." ], [ "INTRODUCTION", "Thermodynamic properties of dynamically-compressed materials are important to understand across a wide range of pressure-temperature phase space.", "The UC Davis Shock Compression Laboratory studies fundamental material properties governing planetary structure and formation; areas in which accurate thermodynamic knowledge is crucial and yet largely undocumented for the major constituents of rocky planets.", "High-quality temperature measurements are notoriously difficult in dynamic high-pressure experiments due to the inherently short timescales, uncertainty in optical properties such as emissivity, and other complications introduced by the experimental environment that can bias or pollute observations of thermal emission from the sample.", "Carefully executed experiments with reliable calibrations are therefore critical for a number of disciplines.", "Pyrometry diagnostics and calibration techniques have varied widely in the shock-compression community [1], [2].", "While different approaches (e.g.", "discrete, multi-color vs spectrally integrated measurements) have unique strengths, a broadband, time-resolved diagnostic returns the most complete thermodynamic information about the sample and the evolution of its optical properties under compression.", "Diverse calibration sources and techniques are reported in the literature, from traditional, absolute methods which reference a known spectral radiance to the increasing popularity of quartz as an in-situ calibrant for relative temperature determination [e.g.", "[3], [4]].", "Regardless of the technique, it is imperative that the calibration be robust, insensitive to minor perturbations in the experiment, and ideally independent of secondary corrections to relate the experimental data to the calibrant.", "Here, we describe the construction and calibration of a fiber-coupled Streaked Visible Spectrometer (SVS) designed for optical pyrometry and emission/absorption spectroscopy with high sensitivity and wide spectral response.", "The system is versatile for a number of different types of measurements, has a small physical footprint on the sample, and is easily and robustly calibrated against a NIST-traceable standard.", "The system is sensitive to relatively low-intensity calibration sources, permitting direct comparison to experimental data without additional scaling, correction factors or other analytical transformations to relate observed thermal emission to the calibration source.", "We describe how the system response is determined and show examples of benchmark experiments to test the calibration and sensitivity over a range of timescales.", "While the system description is specific to the UC Davis SVS system, the calibration methods may apply to any similarly configured diagnostic.", "The streaked spectrometer consists of an Optronis SC20 streak camera [5], coupled to a Princeton Instruments HRS300 spectrometer [6] with protected Ag parabolic injection optics.", "A generalized schematic is given in Fig.", "REF .", "The system was conceived for maximum sensitivity to relatively low temperatures and components were therefore chosen for optimal sensitivity at the longest wavelengths allowed by the streak camera.", "Light is conveyed from the sample to the diagnostic via a single 300-$\\mu $ m, low-OH silica fiber.", "The target-facing end is a bare fiber so that the collection area on the sample is defined by the numerical aperture (0.39) and the standoff distance from the sample.", "Custom fiber rosettes are typically fabricated for simultaneous use of complimentary multi-color VIS/NIR pyrometry.", "A single, continuous fiber is used for the SVS, eliminating connection loss at the vacuum/air interface.", "The fiber remains connected to the spectrometer following calibration thereby ensuring that there is no alteration of the optical path between calibration and the science shot.", "The fiber output is imaged on the spectrometer entrance slit using two protected Ag parabolic reflectors which generate a short section of collimated beam for installation of neutral density or bandpass filters, as required.", "The Princeton Instruments HRS300 spectrometer includes a variable entrance slit and a rotating turret with 3 grating selections of 150, 300 or 600 grooves/mm, allowing spectral resolution of $\\sim $ 0.5 nm and variable central wavelength for detailed spectroscopy.", "A motorized mirror diverts the spectrometer output to either of two ports - one upon which a Thorlabs LC100 line CCD camera is mounted to check static throughput at the exit of the spectrometer without exposing the streak camera, and the second which diverts light directly to the slit of the streak camera.", "The system utilizes an Optronis SC20 optical streak camera equipped with a 35 mm $\\times $ 4 mm S25 photocathode which provides the best long-wavelength sensitivity of the available options [5].", "Two modular timing units allow sweep speeds from 112 ns to 700 $\\mu $ s total duration and a single-stage micro-channel plate permits variable gain.", "Readout is performed with a 2k $\\times $ 2k, 16-bit, chilled CCD from Spectral Instruments Inc. [7].", "The system response, shown in Fig.", "REF , spans $\\sim $ 375 to 875 nm with optimum uniformity between $\\sim $ 500 nm and 750 nm.", "Figure: Schematic of the streaked visible spectrometer including fiber probe from the experiment, injection optics, spectrometer, streak camera and readout." ], [ "Calibration Hardware", "A variety of spectral radiance, blackbody, and illumination standards are used for absolute temperature calibration of science frames, characterization of system response, and measurement of transmission spectra for neutral density and bandpass filters.", "Calibration equipment is permanently located adjacent to the spectrometer to permit easy data acquisition from any source by moving a single fiber connection.", "Two additional USB spectrometers (StellarNet BLK-CXR) allow real-time signal comparison between detectors or quick diagnostic spectra during calibrations.", "The SVS system response is absolutely calibrated for dynamic pyrometry measurements using a Gooch and Housego OL455 tungsten-halogen lamp with a NIST-traceable spectral radiance curve.", "An additional Mikron M360 tuneable blackbody source ($\\sim $ 873 K to 1373 K) or Optronics Labs OL550 tungsten ribbon lamp (also NIST traceable) are used for cross-comparison.", "Temporal calibration of streak images is performed using a Thorlabs NPL52B pulsed laser (450 nm, 1-10 MHz) which functions as an optical timing comb for in-situ characterization of streak linearity.", "Spectral calibration is carried out using an HgAr pencil lamp, which has emission lines spanning the spectral response of the overall system ($\\sim $ 375-875 nm).", "Finally, the SVS setup includes a laser-driven Xe plasma white light source (LDLS, Energetiq EQ99X).", "This high brightness source is used to characterize filters and may be used to correlate measured intensity at fast sweep speeds with absolute calibrations performed over longer durations (though this is not typically necessary for observations on $\\mu $ s timescales typical of light gas gun experiments).", "Tests are currently underway to apply it for sample illumination for dynamic broadband emissivity measurements." ], [ "Direct Calibration", "To determine absolute temperature in any pyrometry system, the measured signal must be related to a source for which the spectral radiance (emission, measured in W sr$^{-1}$  m$^{-2}$  nm$^{-1}$ ) is known as a function of wavelength.", "The relationship between the known spectral radiance and the measured signal defines the system response of the diagnostic.", "The system response is measured prior to every experiment to include variations in fiber-optic probe efficiency and user-selected choices such as camera settings and attenuation.", "The system response can be determined in a direct manner for the setup described above because it is sensitive enough to detect the calibration sources on experimental timescales, eliminating the need for additional corrections due to integration time.", "Furthermore, distant extrapolation between calibrated sources and experimental conditions are not required for the temperatures typically generated in light gas gun plate impact experiments.", "Calibration without signal level extrapolation is often not possible for other HED experiments at more extreme conditions.", "Consider an ideal greybody, defined by Planck's law, at some known temperature.", "The spectral radiance is defined as $S_\\lambda (\\lambda ,T) = \\epsilon \\frac{2h c^2}{\\lambda ^5} \\frac{1}{e^{\\frac{h c}{\\lambda k_B T}}-1},$ where $\\epsilon $ is the emissivity, $\\lambda $ is the wavelength, $T$ is temperature, $h$ is the Planck constant, $c$ is the speed of light, and $k_B$ is the Boltzmann constant.", "The number of counts recorded by the detector observing the greybody at temperature $T$ is simply given by $S_{Meas} = \\frac{S_{real}}{C},$ where $S_{Meas}$ is the measured number of counts, $S_{real}$ is the known emission spectrum, and $C$ is the optical response of the system.", "The optical response is then applied to the measured experiment image to generate a calibrated image.", "This simple correction is based on three assumptions: first, that the emitting surface is Lambertian (i.e., that the apparent radiance is independent of viewing angle); second, that the acceptance angle of the fiber does not include light from edge effects or other heterogeneities; and third, that emissivity, $\\epsilon $ , is wavelength independent.", "This latter assumption, though commonly applied, is rarely if ever true; however, the advantage of a spectrally resolved measurement is that the deviation of the raw data from an ideal greybody can be observed and additional forms of $\\epsilon $ that are not constant and are dependent on wavelength can be fit.", "Furthermore, by illuminating the sample and measuring dynamic reflectivity, the same diagnostic can be used to measure the wavelength dependence of $\\epsilon $ under experimental conditions.", "Figure: Optical Response of the SVS (CC, in Eq.", "), as determined with a tungsten-halogen calibration source via integrating sphere.", "The sharp rise above 875nm is due to the efficiency of the S25 photocathode in the streak camera and limits the lowest temperatures observable with the system.Corrections to the acquired image are minimal and include subtraction of dark current (determined by taking an acquisition with no incident light on the streak camera), and temporal and spectral image distortion corrections to correct for slight non-linearities induced by the streak tube.", "These are done using the in-situ timing comb and a separate pre-shot acquisition of the emission lines from the HgAr pencil lamp.", "For sources beyond the dynamic range of the diagnostic, neutral density (ND) filters may be used and must be accounted for multiplicatively.", "Each ND filter is calibrated individually to generate a wavelength-dependent correction.", "While not applied here, an `indirect' calibration technique can be used in cases where the integration time of a particular experiment is too short for the diagnostic to detect calibration sources on the same timescales.", "In this case, the LDLS white light source (described above) may be used to correlate measured intensities for different camera sweep rates.", "For the system described above, we have empirically determined that the relationship between measured counts and sweep rate is linear." ], [ "Sources of Uncertainty", "Several sources of uncertainty affect the accuracy of pyrometry experiments.", "Uncertainty in the spectral radiance of the Gooch and Housego OL455 tungsten-halogen source and the Mikron blackbody source is about 1%.", "The blackbody oven is assumed to have an emissivity of 1.", "ND filter corrections are also assumed to have an uncertainty of $\\sim $ 1% due to variations in the stability of the light source used to characterize them and the potential for degradation over time.", "For every acquired image, the uncertainty is assumed to scale as the $\\sqrt{S_{\\rm meas}}$ for each pixel.", "The random uncertainty in counts is propagated through every image operation (e.g., dark subtraction) and is on the order of 1% for most calibrated images.", "Variation in counts during the duration of interest in the experiment also contributes to uncertainty in the final temperature determination.", "Averages are taken over line-outs with a standard deviation that depends on the temporal width of the region of interest.", "Longer averages will therefore generate more precise measurements barring rapid changes in temperature over that interval.", "As described above, the system has been designed to eliminate the potential for changes in the optical path (hence system response) since these cannot be quantified if they occur after the calibration.", "All reported uncertainties are 1-sigma in this work." ], [ "PROOF OF CONCEPT: QUARTZ STANDARD AND BLACKBODY CROSS REFERENCE", "Two benchmark examples are shown here to demonstrate the performance of the SVS diagnostic.", "Both examples are fit to equation REF using the method of least-squares, with either $T$ or $T$ and $\\epsilon $ as fitting parameters, as discussed below.", "The first example was conducted statically as a benchtop experiment using a LAND blackbody oven (on loan from Lawrence Livermore National Laboratory) at 1453 K as a calibration source.", "The UC Davis Mikron blackbody oven, with an internal temperature of 1349 K was treated as an `unknown' sample.", "Both oven set points were verified separately using a Minolta Cyclops 52 handheld pyrometer.", "A sweep rate of 25 $\\mu $ s/mm was used on the camera to enable sensitivity below 1500 K. A fit averaged over 350 $\\mu $ s yields a calculated temperature of T=1348$\\pm $ 37 K. In this case, the calibration ovens are known to have a constant emissivity very near unity, so $T$ is the only fitted parameter and $\\epsilon $ is assumed constant.", "At timescales representative of most light gas gun experiments (several $\\mu $ s), the lowest observable temperatures are $\\sim $ 2000 K due to the shorter acquisition time.", "Figure: (Left) Observed spectral radiance for the MIKRON blackbody source (blue with magenta uncertainty bands) and the Plank fit (black with grey uncertainty bands) corresponding to a calculated temperature of 1348±\\pm 37 K.(Right) Spectral radiance spectrum of the quartz shock front.", "The inset shows the experiment region of interest with spectral radiance given by the hot map.", "The spectrum from which temperature is calculated is a line out of the bright region of the map.For the second example, a plate impact experiment was conducted on the UC Davis two-stage light gas gun.", "The flyer and driver were 304 Stainless steel with an impact velocity of 5400$\\pm \\ 65$ m/s.", "The sample was z-cut $\\alpha $ -quartz, with a measured shock velocity from sample transit time measurements of 8.46$\\pm .015$ km/s, corresponding to a pressure of 86.68$\\pm $ .68 GPa [8].", "A Gooch and Housego OL455 tungsten-halogen lamp was used as the calibration source.", "The sweep rate was 1 $\\mu $ s/mm and an ND3 filter was used to attenuate the signal.", "The spectrum from the shocked quartz was averaged over $\\sim $ 1 $\\mu $ s and is shown in Fig.", "REF (right).", "The shorter observation duration ($\\sim $ 1/28th of the entire streak length) results in lower signal-to-noise in the data.", "Faster sweep rates at higher time resolution are needed to average random uncertainty of the experiment.", "We present two fitting cases to illustrate the capabilities and representative uncertainties of this diagnostic.", "The first case lets $T$ and $\\epsilon $ be free parameters in fitting equation REF and the second only fits $T$ .", "In both cases, $\\epsilon $ is assumed to be independent of wavelength.", "For quartz shocked to 106.5 GPa, the emission spectrum appears as a graybody and the unshocked quartz is totally transparent [9].", "The experiment here is also of sufficiently low pressure that the emissivity of silica during the shock approaches 1 [10], [4].", "Over the wavelengths measured in this work, emissivity dependence on wavelength is not expected to be appreciable.", "In the first case, the measured temperature is 4716$\\pm $ 210 K, with $\\epsilon =0.864\\pm 0.091$ .", "The emissivity in this case is smaller than the measured $\\epsilon $ in [10]; however,uncertainty improves with faster sweep speeds, which improves signal-to-noise ratios (and temporal resolution) by spreading the data over a larger fraction of the CCD.", "For the second case, at the shock pressure of this experiment, the reflectivity is negligible [10], [4] and we assume reflecticivity $R=0$ and $\\epsilon =1$ .", "In this case, the measured temperature is 4661$\\pm $ 172 K. Both cases give temperatures that are consistent with previously measured quartz temperatures at 86$\\pm $ 1 GPa and 4860$\\pm $ 150 K [11]." ], [ "CONCLUSION", "We have built and commissioned a streaked visible spectrometer for optical pyrometry and emission/absorption spectroscopy in the UC Davis Shock Compression Laboratory.", "Calibration schemes have been cross-referenced using multiple sources, and we have shown that the system enables observation of relatively low dynamic temperatures with high fidelity.", "With a small fiber-optic footprint on the sample, it may be fielded in parallel with a number of other optical diagnostics, including complimentary, multi-channel visible and NIR pyrometry and velocimetry.", "Additional applications could extend to Raman spectroscopy, dynamic reflectivity and other studies of high-pressure optical and electronic properties." ], [ "ACKNOWLEDGMENTS", "This work was supported by NASA grant NNX15AH54G, NASA grant NNX16AP35H, LLNL contract No.", "B617085, and UC Office of the President grant LFR-17-449059." ] ]

[ [ "Fuzziness-based Spatial-Spectral Class Discriminant Information\n Preserving Active Learning for Hyperspectral Image Classification" ], [ "Abstract Traditional Active/Self/Interactive Learning for Hyperspectral Image Classification (HSIC) increases the size of the training set without considering the class scatters and randomness among the existing and new samples.", "Second, very limited research has been carried out on joint spectral-spatial information and finally, a minor but still worth mentioning is the stopping criteria which not being much considered by the community.", "Therefore, this work proposes a novel fuzziness-based spatial-spectral within and between for both local and global class discriminant information preserving (FLG) method.", "We first investigate a spatial prior fuzziness-based misclassified sample information.", "We then compute the total local and global for both within and between class information and formulate it in a fine-grained manner.", "Later this information is fed to a discriminative objective function to query the heterogeneous samples which eliminate the randomness among the training samples.", "Experimental results on benchmark HSI datasets demonstrate the effectiveness of the FLG method on Generative, Extreme Learning Machine and Sparse Multinomial Logistic Regression (SMLR)-LORSAL classifiers." ], [ "Introduction", "Hyperspectral Imaging (HSI) is concerned with the extraction of meaningful information form the objects of interest-based on the radiance acquired by the sensor from long, medium or short distance [1].", "HSI technology has been investigated in a wide variety of urban, mineral exploration, environmental, in-depth classification of forest areas, monitoring the pollution in city areas, investigating the coastal and domestic water zones, inspection for natural risks, i.e.", "flood, fires, earthquakes, eruptions [2], [3].", "Therefore, to enhance the applicability, robust, effective and automatic Hyperspectral Image Classification (HSIC) methods are required.", "The classical kernel-based methods [4] are effective and robust.", "Nevertheless, these methods are inadequate to handle ill-posed conditions [5].", "Furthermore, it is well established fact that the HSIC performance depends on the quality and size of training data [6].", "However, in the scenario of limited training samples, classical HSIC methods do not perform well [7], [8].", "Therefore, the main objective of this work is to develop a novel method to automatically extract the meaningful information captured by HSI sensors particularly in the case when the labeled training samples are not adequate or not fully reliable [9].", "In a nutshell, the following specific contributions are made in this work.", "FLG utilizes the fuzziness concept instead of uncertainty and couples it with the samples diversity while maximizing separability measure using total local and global for both within and between class discriminant information.", "Total global class information impairs the local topology and cannot satisfactorily characterize the local class discriminant information.", "This may lead to instability of within-class compact representation.", "A similar problem whereby local class discriminant information only considers local between and within-class information and ignore the global class information.", "To overcome the above two problems, we define a novel objective function that jointly considers the total global and local class discriminative information.", "The proposed discriminative objective function assesses the stationary behavior of samples in the spectral domain in a fine-grained manner.", "The objective function proposed in this work is adapted for deriving a set of spatially heterogeneous samples that jointly optimize the above terms.", "The objective function is based on the optimization of a multi-objective problem for the estimation of within and between class scatters while preserving the total local and global class discriminant information.", "Thus, FLG significantly increases the generalization capabilities and robustness of classical machine learning classifiers.", "The rest of the paper is structured as follows.", "Section examines the related works.", "Section presents the theoretical aspects of FLG.", "Section discusses the details of experimental settings.", "Section provides the details about HSI datasets, experimental results and comparison with the state-of-the-art methods.", "Finally, section summarizes the contributions and discuss future research directions." ], [ "Related Work", "In the last decade, kernel-based methods have been successfully applied for HSIC.", "However, kernel-based methods do not perform well when the ratio between the number of labeled training samples and spectral bands [10], [11], [12] is small [13].", "Several alternative methods have been proposed to address the issues related to the limited availability of labeled training samples.", "One of them is to iteratively enlarge the original training set in an interactive process, i.e, human-human interaction or human-machine interaction [14].", "Active Learning (AL) is an iterative process of selecting informative samples from a set of unlabeled samples.", "The selection choice is based on a ranking of scores that are computed from a model's outcome.", "Carefully selected samples are added to the training set and the classifier is retrained with the new training set.", "The training with selected samples is robust because it uses samples that are suitable for learning.", "Thus, the sample selection criteria are a key component of the AL framework [1], [15].", "The most common sample selection criteria are formalized into three different groups.", "Uncertainty of samples [16], [17] and query by committee [18], [19] Influence on the model such as length of gradients [20] and Fisher information ratio [21].", "Intrinsic structure and distribution of the unlabeled samples such as Gaussian similarity [22], Kullback-Leibler divergence similarity [23], manifold learning [24], clustering [25] and density-weighting [26] methods.", "AL combined with a special classifier such as AL combined with Logistic Regression [27], SVM [28] and Gaussian process regression [29].", "These sample selection methods have shown that the AL process significantly improves the performance of any classifier while querying the informative samples [30].", "However, in HSIC, the collection and labeling of queried samples are associated with a high cost in terms of time.", "Therefore, most of the previous studies focused on selecting the single sample in each iteration (stream-based), by assessing its uncertainty [31].", "This can be computationally expensive because the classifier has to be retrained for each new labeled sample.", "Pool-based (Batch-mode) methods have been proposed to address the above-mentioned issues by assessing the pool of samples.", "The major drawback of multiclass pool-based models is that the pool of selected samples brings redundancy i.e., no new knowledge or information is provided to the classifier in the retraining phase.", "This work addresses the above-mentioned issues by defining a multiclass fuzziness-based total local and global for both within and between-class scatter information preserving AL pipeline.", "This work explicitly considers spatial-spectral heterogeneity of the selected samples by defining a novel discriminative objective function and properly generalize it.", "The combination of the above criteria results in the choice of the potentially most informative and heterogeneous samples than ever possible.", "The proposed method is experimentally compared with state-of-the-art methods and based on the comparisons, some guidelines are derived to use AL techniques for HSIC." ], [ "Methodology", "A number of sample selection methods for AL have been proposed in the literature [7], [32], [33], [34] though uncertainty Most uncertain sample has similar posteriori probability for two possible classes.-based sample selection remains popular due to its simplicity.", "The probabilistic classification models can directly be used to compute the uncertainty whereas it's not that simple for non-probabilistic models [35].", "Let us assume a HSI cube can be represented as $X = [x_1, x_2, x_3, \\dots , x_L]^T \\in \\mathcal {R}^{L\\times (M \\times N)}$ composed of $(M\\times N)$ samples per band belonging to $C$ classes and $L$ bands [36], [37], [38].", "Further assume that $(x_i, y_j)$ be a sample of Hyperspectral cube in which $y_j$ is the class label of $x_i$ sample.", "We first randomly select $n$ number of labeled training samples to form a training set $X_T$ and rest has been selected for the test set $X_V$ .", "We further make sure that $n \\ll m$ and $X_T \\cap X_V = \\emptyset $ [1] for each iteration of our proposed AL method." ], [ "Fuzziness", "A probabilistic/non-probabilistic classifier produces the output $\\mu = \\mu _{ij}$ of $m \\times C$ matrix containing probabilistic/non-probabilistic outputs.", "There is no need to compute the marginal probabilities for probabilistic classifiers however it is mandatory for non-probabilistic classifiers and for that discriminative random field is used to compute the probabilities.", "These probabilistic outputs are used to construct a membership matrix which must satisfy the properties [1], [7] $\\sum _{j = 1}^{C} \\mu _{ij} = 1 ~~ and ~~ 0 < \\sum _{i = 1}^{N} \\mu _{ij} < 1$ where $\\mu _{ij} = \\mu _j(x_i) \\in [0,1]$ represent the membership of $x_i$ to the $y_j$ class.", "Later this membership matrix is used to compute the fuzziness of $m$ samples for $C$ class as; $E(\\mu ) = \\frac{-1}{C} \\sum _{i=1}^{N}\\sum _{j = 1}^{C} [\\mu _{ij}log(\\mu _{ij}) + \\\\(1 - \\mu _{ij})log(1-\\mu _{ij})]$" ], [ "Fuzziness Categorization", "We first make a matrix of fuzziness information associated with samples' spatial information and their predicted and actual class labels and a test set, i.e., $[E(\\mu ),~ X_V(s),~ y_j(a),~ y_j(p),~ X_V]$ where $X_V(s)$ stands for spatial information of test set, $y_j(a)$ and $y_j(p)$ represents the actual and predicted class labels.", "Later a median-based fuzziness categorization method is proposed to select the samples to compute the class scatter information.", "There are two ways to compute the median value to make $\\mathcal {F}_1$ and $\\mathcal {F}_2$ sets depending on the number of total samples.", "If the total number of samples is odd then the median can be computed as: $\\mathcal {M}(\\mathcal {F}_1) = \\frac{m_1+1}{2}$ $\\mathcal {M}(\\mathcal {F}_2) = \\frac{m_2+1}{2}$ If the total number of samples in the test set is even then the median can be computed as; $\\mathcal {M}(\\mathcal {F}_1) = \\frac{\\frac{m_1}{2} + \\frac{m_1+1}{2}}{2}$ $\\mathcal {M}(\\mathcal {F}_2) = \\frac{\\frac{m_2}{2} + \\frac{m_2+1}{2}}{2}$ where $m_1$ and $m_2$ refers to the total number of samples in $\\mathcal {F}_1$ and $\\mathcal {F}_2$ respectively.", "Now we have two sets of fuzziness then we will find the median values in both sets and keep these values in $\\mathcal {Q}_1$ and $\\mathcal {Q}_2$ .", "By this process, we create two fuzziness groups and place the samples in low ($[0-0.5]$ fuzziness magnitude) and high ($[0.5-1.0]$ fuzziness magnitude) fuzziness groups, respectively.", "From these two sets we select misclassified foreground samples to compute the local and global class discriminate information to select the heterogeneous spectral samples for the training set." ], [ "Global Class Discriminant Information", "The fuzziness categorization process returns $\\mathcal {X} = \\lbrace (\\textbf {\\textit {x}}_i~ | ~ i = 1, \\dots , \\mathcal {K} ~ and ~ y_j ~ | ~ j = 1, \\dots , C)\\rbrace $ , where $\\textbf {\\textit {x}}_i \\in \\mathcal {R}^{L \\times (M \\times N)}$ , $\\mathcal {K} < m$ and $n \\ll \\mathcal {K}$ .", "The global class discriminant information preserving process aims to attain a space $\\mathcal {R}^{L \\times (M \\times N)}$ in which each training sample $\\textbf {\\textit {x}}_i$ can be well represented by $\\textbf {\\textit {x}}_i\\rightarrow y_j \\in \\mathcal {R}^{L \\times (M \\times N)}$ .", "Therefore, linear discriminant analysis (LDA) is most efficient method to preserve the global information in computational efficient fashion.", "LDA seeks a linear projection matrix $\\textbf {U} \\in \\mathcal {R}^{L \\times (M \\times N)}$ that maximizes the Fisher discriminant ratio (FDR) as follows: $\\textbf {U} = \\underset{\\textbf {U}}{\\mathrm {argmax}} \\frac{|\\textbf {U}^T S_B \\ \\textbf {U}|}{|\\textbf {U}^T S_W \\ \\textbf {U}|}$ where $S_B$ , $S_W \\in \\mathcal {R}^{L*L}$ are the global between and within class scatter matrices respectively.", "The performance of FDR is highly dependent on the quality of the scatter matrices i.e.", "when the number of samples in each class is much smaller than the number of bands, $S_W$ would be singular.", "This makes the inverse of $S_W$ and the eigenvalue decomposition $S_W^{-1} S_B$ impracticable which is commonly known as small sample size problem [39].", "However, non-probabilistic LDA (NPLDA) [40] and probabilistic vector-based LDA (PLDA) [41] methods have been proposed to address the problems.", "In NPLDA, regularized LDA (RLDA) use PCA to reduce the dimensionality of HSI from $L$ to $L^* < L$ and performs the classical LDA.", "Therefore, the size of within-class scatters matrix $S_W$ is reduced from $L \\times L$ to $L^* \\times L^*$ , making $S_W$ well-conditioned in the PCA subspace as long as $L^*$ is small enough.", "Although PCA$+$ LDA solves the singularity problem with the expense to lose geospatial information due to the lossy compression of PCA.", "RLDA can hardly extract meaningful information when the size of the training samples becomes large.", "Unlike NPLDA counterparts, PLDA solves the singularity problem by capturing between and within-class variation under the probabilistic framework.", "Whereas, NPLDA extracts meaningful information by manipulating the scatter matrices.", "While the PLDA avoids the inverse of ill-conditioned $S_W$ through probabilistic modeling of within and between class variation, individually.", "Moreover, since it explicitly characterizes both the class and noise components, PLDA has a unique advantage in capturing discriminative information.", "It may be discarded or considered as less important by its NPLDA counterparts [41].", "Since NPLDA and PLDA successfully address the singularity problem of LDA, however, these are still far from solving the small sample size problem.", "In real-life, many HSI datasets are in the form of tensors and tensor structures are useful in alleviating the small sample size problem.", "Vectorization or reshaping consequently breaks the valuable tensor structures.", "To cope with these issues, multi-linear LDA (MLDA) approaches have been proposed to gain more robustness.", "According to different criteria used in projection learning, MLDA can be grouped into two categories, i.e.", "ratio-based MLDA (RMLDA) [42] and difference-based MLDA [43].", "RMLDA consider $\\mathcal {X} = \\lbrace \\lbrace \\textbf {\\textit {x}}_{ij} \\in \\mathcal {R}^{L_c * L_r}\\rbrace ^{n}_{i=1}\\rbrace ^{C}_{j=1}$ as a set of true class HSI samples obtained by fuzziness process.", "The aim of RMLDA is to find projections that maximize the ratio between and within class scatter.", "For instance, $2D$ -RMLDA learn two matrices $U_r \\in \\mathcal {R}^{L_r * q_r}$ and $U_s \\in \\mathcal {R}^{L_s * q_s}$ , which characterize the column and row spaces respectively.", "These matrices solved alternately based on the following scatter ratio criterion.", "By fixing $U_r$ , $U_s$ is solved by: $U_c = \\underset{U}{\\mathrm {argmax}} \\frac{tr(U^T {{S_B}_s} U)}{tr(U^T {{S_W}_s} U)}$ where ${S_W}_s \\in \\mathcal {R}^{L_s * L_s}$ and ${S_B}_s \\in \\mathcal {R}^{L_s * L_s}$ are the MLDAs within and between class scatter matrices respectively.", "These matrices are further decomposed as follows; ${S_B}_s = \\sum _{j=1}^{C} N_j (M_j - M) U_r U^T_r (M_j - M)^T$ ${S_W}_s = \\sum _{ij} (x_{ij} - M_j) U_r U^T_r (x_{ij} - M_j)^T$ where $N_j$ be the total number of samples in $j^{th}$ class, $M = \\frac{1}{N} \\sum _{ij} \\textbf {\\textit {x}}_{ij}$ is the overall mean matrix and $M_j = \\frac{1}{N_j} \\sum _{i=1}^{N_j} \\textbf {\\textit {x}}_{ij}$ is the class mean matrix.", "Analogous to LDA, the solution $U_s$ is given by the eigenvectors of ${S_W}_s^{-1}{S_B}_s$ associated with the $q_s$ largest eigenvalues.", "By fixing $U_s$ , the solution of the row projection $U_r$ can be obtain similarly.", "Similar to LDA, the above objective functions (Equations REF and REF ) are further be decomposed into the following two objective functions: $S_{GB} = \\sum _{j=1}^C N_j U^T(M_j - M) (M_j - M)^T U$ $S_{GW} = \\sum _{j=1}^C \\bigg ( \\sum _{i=1}^{K+1} U^T (x_{ij} - M_j) (x_{ij} - M_j)^T U \\bigg )$ Equation REF aims to ensure that the samples from the center are as close as possible if and only if they belong to the same class whereas equation REF makes sure that the center of each class from the total center is as distant as possible.", "Furthermore, MLDA aims to find projections that maximize the difference between within and between-class scatter.", "For instance, MLDA learns multi-linear projections alternately and maximizes the following objective function for the column projection: $U_s = \\underset{U}{\\mathrm {argmax}} \\bigg (U^T ({S_B}_s - \\xi {S_W}_s) U \\bigg )$ where $\\xi $ is the tuning parameter which is heuristically set to the largest eigenvalue of $({S_W}_s)^{-1}{S_B}_s$ .", "After a simple derivation, the solution $U_s$ is given by the eigenvectors of ${S_B}_c - \\xi {S_W}_c$ associated with the $q_s$ largest eigenvalues.", "By exploiting the tensor structures, MLDA can learn more reliable multi-linear scatter matrices that have smaller sizes and much better conditioning than LDA.", "MLDA solves the singularity problem of LDA and gains robustness in estimating global class discriminant information with a small sample size [44].", "Equation REF have a shortcoming, for instance, it may impair the local topology but cannot satisfactorily characterize the local class discriminant information of HSI data.", "This may lead to instability of within-class compact representation.", "To overcome these difficulties, this work explicitly computes the local class discriminant information." ], [ "Local Class Discriminant Information", "We incorporate the geometrical transformation method which preserves the local class discriminant information by building two adjacency matrices i.e.", "intrinsic and penalty matrix.", "The intrinsic matrix characterizes class compactness and connects each sample with its neighboring samples of the same class.", "While the penalty matrix connects marginal points characterize within class separability.", "Thus, the objective functions presented in [45] are redefined as follows.", "$U^* = \\underset{U}{\\mathrm {argmax}} \\frac{S_{LB}}{S_{LW}}$ In which, local within and between class compactness is characterized by the following criteria: $S_{LW} = \\sum _{i_{i \\in S_{k_1}^W}(j)} \\ \\sum _{j_{j \\in S_{k_1}^W}(i)} \\Vert U^T \\textbf {\\textit {x}}_i - U^T \\textbf {\\textit {x}}_j \\Vert ^2$ $S_{LB} = \\sum _{i_{i \\in S_{k_2}^B}(j)} \\ \\sum _{j_{j \\in S_{k_2}^B}(i)} \\Vert U^T \\textbf {\\textit {x}}_i - U^T \\textbf {\\textit {x}}_j \\Vert ^2$ $S_{LW} = 2 U^T \\mathcal {X} (D^{LW} - W^{LW}) \\mathcal {X}^T U$ $S_{LB} = 2 U^T \\mathcal {X} (D^{LB} - W^{LB}) \\mathcal {X}^T U$ where $S_{k_1}^W (i)$ and $S_{k_2}^W (i)$ indicates the index set of $k_1$ and $k_2$ nearest neighbors of $\\textbf {\\textit {x}}_i$ in the same class.", "$D^{LW}$ and $D^{LB}$ are diagonal matrix whose columns or rows are the sum of $W^{LW}$ and $W^{LB}$ because both are symmetric matrices, e.g.", "$D_{ii}^{LW} = \\sum _j W_{ji}^{LW}$ and $D_{ii}^{LB} = \\sum _j W_{ji}^{LB}$ .", "Thus, similar to equations REF and REF , the equation REF is decomposed into the following two objective functions.", "$S_{LW} = U^T \\mathcal {X} (D^{LW} - W^{LW}) \\mathcal {X}^T U$ $S_{LB} = U^T \\mathcal {X} (D^{LB} - W^{LB}) \\mathcal {X}^T U$ Equations REF and REF are local within and between-class scatters and can characterize the local topology of the samples.", "Equation REF aims to make samples compact if and only if they belong to the same class.", "Equation REF aims to make them separable if and only if they belong to the different classes.", "Equations REF and REF have a similar problem whereby they only consider local between and within-class scatters but ignore global class discriminant information.", "It is worth noting that Equation REF finds the optimal space by maximizing the ratio among within and between-class scatters and to address it, we rewrite equation REF as follows.", "$\\hat{U} = \\underset{U}{\\mathrm {argmax}} \\frac{U^T \\mathcal {X} (D^{LB} - W^{LB}) \\mathcal {X}^T U}{U^T \\mathcal {X} (D^{LW} - W^{LW}) \\mathcal {X}^T U}$ The optimal projection vector $U$ that maximizes equation REF is given by the solution of the maximum eigenvalue to the generalized eigenvalue as shown in the following.", "$\\mathcal {X} (D^{LB} - W^{LB}) \\mathcal {X}^T U = \\lambda \\mathcal {X} (D^{LW} - W^{LW}) \\mathcal {X}^T U$ The effectiveness is still limited such as $S_{LW}$ is singular in many cases and the optimal projection cannot be directly calculated.", "To overcome this problem, we define a novel objective function that jointly considers the total local and global class discriminative information in the following section." ], [ "Objective Function", "The parameters $\\Omega , \\lambda , \\Psi = 0.5$ is used to make the trade-off among total within class and total between class scatters and control their proportion as follows.", "$\\hat{U} = \\underset{U}{\\mathrm {argmax}} \\ S(U)$ $\\hat{U} = \\underset{U}{\\mathrm {argmax}} \\big (\\Omega \\ S_B(U) - (1 - \\Omega ) \\ S_W(U)\\big )$ $S_W(U) = \\lambda S_{GW} + (1 - \\lambda ) S_{LW}$ $S_B(U) = \\Psi S_{GB} + (1 - \\Psi ) S_{LB}$ In above equations $\\lambda , \\Psi , \\Omega = 0.5$ and if $\\lambda , \\Psi = 0$ the equations REF and REF reduces to keep only the local within and between class information.", "Whereas, if $\\lambda , \\Psi = 1$ then the equations REF and REF will reduce to keep only global within and between-class scatter information.", "If $\\Omega = 0$ , equation REF reduces to the only total between class scatter whereas, $\\Omega = 1$ reduces to the only total within-class scatter.", "Equation REF impairs the topological structure of samples and the local within class structure which ignores the global information.", "To remedy these problems we introduce trade-off coefficient $\\Omega = 0.5$ to balance both local and global within class information.", "The advantages of this integration are to preserve the global and local within class information in which we can achieve total local class discriminant information.", "Equation REF involves the global and local between class information, i.e.", "relationship among sample centers, but ignores local information e.g.", "relationship among samples if and only if they belong to a different class.", "However, a single characterization, either global or local between and within-class may be insufficient, which deteriorates classification performance.", "This work overcomes the above-said limitation by introducing a discriminative objective function, which preserves both global and local between and within-class scatter information as shown in equation REF .", "To gain more insight, we solve the equation REF as follows.", "$\\hat{U} = \\underset{U}{\\mathrm {argmax}} ({U}^T \\ \\mathcal {G} \\ U)$ where $\\mathcal {G} = \\Omega \\Psi S_B(U) - (1 - \\Omega ) \\lambda S_W(U) + \\mathcal {X}[\\Omega (1 - \\Psi )S_{LB} - (1 - \\Omega )(1 - \\lambda )S_{Lw}]\\mathcal {X}^T$ , in which $D^{LB}$ and $D^{LW}$ are the Laplacian matrices.", "Thus, the optimal solution of equation REF is given by $\\underset{U}{\\mathrm {Max}} \\ Tr \\lbrace U^T \\ \\mathcal {G} \\ U\\rbrace $ with the constraint $U^T U = I$ means the columns of $U$ are orthogonal projection matrix which is able to enhance the discrimination ability among samples and we also used it to reduce the redundancy among the selected samples.", "Furthermore, we assume that $\\mathcal {G}$ is a real symmetric matrix i.e.", "$\\mathcal {G} = \\mathcal {G}^T$ because $[\\mathcal {X}L^{LB}\\mathcal {X}^T]^T = \\mathcal {X}L^{LB}\\mathcal {X}^T$ and $[\\mathcal {X}^T L^{LB}\\mathcal {X}]^T = \\mathcal {X}^T L^{LB}\\mathcal {X}$ , moreover,$S_B = S_B^T$ and $S_W = S_W^T$ .", "The complete pipeline of our proposed method is presented in the Algorithm REF and in Figure REF .", "[!ht] Important steps of our proposed AL method.", "$X_T, X_V$ Initialization: $\\mathcal {X},~\\lambda = 0.5,~\\Psi = 0.5,~\\Omega = 0.5$ $|X_T| \\le Threshold$ $\\mu _{ij} \\leftarrow $ Compute the membership matrix $E(\\mu ) \\leftarrow $ Compute the fuzziness $D_V \\leftarrow $ Associate the fuzziness, actual and predicted class and spatial information with $X_V$ Categorize the $D_V$ based on Fuzziness into two groups and sort each group in descending order individually Pick $\\mathcal {K}$ misclassified classified samples from each group individually Compute local and global class information for within class scatter $S_W(U) = \\lambda S_{GW} + (1 - \\lambda ) S_{LW}$ Compute local and global class information for between class scatter $S_B(U) = \\Psi S_{GB} - (1 - \\Psi ) S_{LB}$ Formulate the objective function $\\mathcal {G} = \\Omega \\Psi S_B(U) - (1 - \\Omega ) \\lambda S_W(U) + \\mathcal {X}[\\Omega (1 - \\Psi )S_{LB} (1 - \\Omega )(1 - \\lambda )S_{Lw}]\\mathcal {X}^T$ Pick $h \\ll \\mathcal {K}$ most diverse and spectrally heterogeneous samples from $D_V$ , add them to $X_T$ and remove from $X_V$ Repeat until $|X_T| > Threshold$ Figure: FLG Flow-Graph: We first compute the fuzziness and then used a divide and conquer strategy to split the fuzziness values into two groups e.g.", "Low and High Fuzziness.", "We pick misclassified samples (𝒦)(\\mathcal {K}) to compute the global and local class discriminative information and pass this information to the objective function to select (h≪𝒦)(h \\ll \\mathcal {K}) heterogeneous samples." ], [ "Experimental Settings", "The Hyperspectral datasets used for experimental purposes belong to the wide variety of classification problems in which the number of classes, number of samples and types of sensors are different, i.e, AVIRIS, ROSIS and Hyperion EO-1 Satellite Sensors.", "The 5-fold-cross-validation process is adopted to evaluate the performance of our proposed pipeline.", "A necessary normalization between the range of $[0,1]$ is performed.", "All the stated experiments are conducted using Matlab $2016b$ installed on a Intel inside Core $i5$ with $8GB$ RAM.", "Experiments have been conducted on several benchmark Hyperspectral datasets using four different types of classifiers, i.e., Support Vector Machine (SVM) [1], K Nearest Neighbours (KNN) [1], Extreme Learning Machine (ELM) [1] and MLR-LORSAL [46] classifiers.", "All these classifiers are tuned according to the settings mentioned in their respective works.", "The first goal of this work is to compare the results of all these classifiers on “High” fuzziness samples while considering the total local and global class discriminant information.", "Later we compare the results of the best classifier on “Low” fuzziness samples again while considering the total local and global class discriminant information.", "Finally, the complete pipeline is compared with the state-of-the-art AL methods.", "For experimental evaluation, several tests have been conducted including but not limited to Kappa $(\\kappa )$ , overall accuracy, F1-Score, Precision, and Recall rate.", "All these evaluation metrics are calculated using the following mathematical formulations.", "$OA = \\frac{1}{C} \\sum _{i = 1}^C TP_i$ $\\kappa = \\frac{P_o - P_e}{1 - P_e}$ where $P_o = \\frac{TP + TN}{TP + FN + FP + TN}$ $P_e = P_{Y} + P_{N}$ $P_{Y} = \\frac{TP + FN}{TP + FN + FP + TN} \\times \\\\\\frac{TP + FN}{TP + FN + FP + TN}$ $P_{N} = \\frac{FP + TN}{TP + FN + FP + TN} \\times \\\\\\frac{FN + TN}{TP + FN + FP + TN}$ where TP and FP are true and false positive, TN and FN are true and false negative computed from the confusion matrix.", "$Precision = \\frac{1}{C} \\sum _{i = 1}^C \\frac{TP_i}{TP_i + FP_i}$ $Recall = \\frac{1}{C} \\sum _{i = 1}^C \\frac{TP_i}{TP_i + FN_i}$ $F1-Score = \\frac{2 \\times (Recall \\times Precision) }{(Recall + Precision)}$" ], [ "Experimental Results", "In all these experiments, the initial $n = 50$ training samples are selected randomly and the rest of the samples are used as a test example.", "We start preserving the total local and global class scatter information from these random samples and later used it for the queried samples.", "In each iteration of AL, $h = 100$ new samples are selected through the proposed pipeline and added back to the original training set." ], [ "Salinas", "The Salinas dataset (SD) was acquired over Salinas Valley California using AVIRIS sensor.", "SD is of size $512\\times 217\\times 224$ with a $3.7$ meter spatial resolution with $512\\times 217$ is spatial and 224 spectral dimensions.", "SD consists of vineyard fields, vegetables and bare soils.", "SD consist of 16 classes.", "A few water absorption bands $108-112, 154-167$ and 224 are removed before analysis.", "Further details about the dataset can be found at [15].", "SD class information is provided in Table REF and Figure REF with the respective class accuracies.", "Table: SD Class Description i.e., Class name, Test Samples out of which n=50n = 50 as initial and h=[50:100:2500]h = [50:100:2500] in each iteration, as training samples with their respective overall accuracies for SVM, KNN, ELM and MLR-LORSAL Classifiers over 5-cross validation.Figure: Salinas Dataset (SD): Initial Training and Test Ground Truths (Figure ) and Classification accuracy with different number of training samples for SVM, KNN, EML and MLR-LORSAL classifiers (Figure )." ], [ "Indian Pines Dataset", "Indian Pines Dataset (IPD) is obtained over northwestern Indiana’s test site by Airborne Visible / Infrared Imaging Spectrometer (AVIRIS) sensor.", "IPD is of size $145\\times 145\\times 224$ in the wavelength range $0.4-2.5\\times 10^{-6}$ meters where $145\\times 145$ is the spatial and 224 spectral dimensions.", "IPD consists of $1/3$ forest and $2/3$ agriculture area and other naturally evergreen vegetation.", "Some corps in the early stages of their growth is also present with approximately less than $5\\%$ of total coverage.", "Low-density housing, building and small roads, Two dual-lane highway and a railway line are also a part of IPD.", "The IPD ground truth comprised of 16 classes which are not mutually exclusive.", "The water absorption bands have been removed before the experiments thus the remaining 200 bands are used in this experiment.", "Further details about IPD can be found at [15].", "IDP class description is provided in Table REF and Figure REF with the respective class accuracies.", "Table: IPD Class Description i.e., Class name, Test Samples out of which n=50n = 50 as initial and h=[50:100:2500]h = [50:100:2500] in each iteration, as training samples with their respective κ\\kappa accuracies for SVM, KNN, ELM and MLR-LORSAL Classifiers over 5-cross validation.Figure: Indian Pines Dataset (IPD): Initial Training and Test Ground Truths (Figure ) and Classification accuracy with different number of training samples for SVM, KNN, EML and MLR-LORSAL classifiers (Figure )." ], [ "Kennedy Space Center", "Kennedy Space Center dataset (KSCD) acquired on March $23,~ 1996$ using NASA AVIRIS instrument over Kennedy Space Center, Florida.", "KSCD consists of 224 spectral bands of $10~nm$ in the wavelength range $400-2500~nm$ with 18 meter spatial resolution from an altitude of $20~km$ .", "176 out of 224 bands are used for the experimental purposes and remaining water absorption and low SNR bands have been removed.", "KSCD consist of 13 classes.", "KSCD class information is provided in Table REF and Figure REF with the respective class accuracies.", "Table: KSCD Class Description i.e., Class name, Test Samples out of which n=50n = 50 as initial and h=[50:100:2500]h = [50:100:2500] in each iteration, as training samples with their respective κ\\kappa accuracies for SVM, KNN, ELM and MLR-LORSAL Classifiers over 5-cross validation.Figure: Kennedy Space Center Dataset (KSCD): Initial Training and Test Ground Truths (Figure ) and Classification accuracy with different number of training samples for SVM, KNN, EML and MLR-LORSAL classifiers (Figure )." ], [ "Botswana", "Botswana Dataset (BSD) acquired over the Okavango Delta, Botswana on May $31,~ 2001$ through a Hyperion sensor mounted on NASA $EO-1$ Satellite.", "BSD consists of 242 bands of $10 nm$ in the wavelength range of $400-2500~nm$ with a 30 meter spatial resolution from an altitude of $7.7~km$ .", "The number of bands reduced to 145 from 242 by removing noisy and uncalibrated bands.", "The total ground truth classes are 14 that represent occasional swamps, drier woodlands and seasonal swamps.", "BSD class information is provided in Table REF and Figure REF with the respective class accuracies.", "Table: BSD Class Description i.e., Class name, Test Samples out of which n=50n = 50 as initial and h=[50:100:2500]h = [50:100:2500] in each iteration, as training samples with their respective κ\\kappa accuracies for SVM, KNN, ELM and MLR-LORSAL Classifiers over 5-cross validation.Figure: Botswana Dataset (BSD): Initial Training and Test Ground Truths (Figure ) and Classification accuracy with different number of training samples for SVM, KNN, EML and MLR-LORSAL classifiers (Figure )." ], [ "Pavia University", "Pavia University Dataset (PUD) gathered over Pavia in northern Italy using a Reflective Optics System Imaging Spectrometer (ROSIS) optical sensor.", "PUD consists of $610\\times 610$ spatial and 103 spectral bands with a spatial resolution of $1.3$ meters.", "PUD ground truth classes are 9.", "PUD class information is provided in Table REF and Figure REF with the respective class accuracies.", "Table: PUD Class Description i.e., Class name, Test Samples out of which n=50n = 50 as initial and h=[50:100:2500]h = [50:100:2500] in each iteration, as training samples with their respective κ\\kappa accuracies for SVM, KNN, ELM and MLR-LORSAL Classifiers over 5-cross validation.Figure: Pavia University Dataset (PUD): Initial Training and Test Ground Truths (Figure ) and Classification accuracy with different number of training samples for SVM, KNN, EML and MLR-LORSAL classifiers (Figure ).Table: Statistical Significance of our proposed Pipeline.Here we enlist the experimental results obtained in each iteration of the proposed AL framework for four different types of classifiers, i.e., SVM, KNN, ELM, and MLR-LORSAL.", "These classifiers have been rigorously used in the literature for comparative analysis.", "The comparative performance of our proposed AL pipeline using the aforementioned classifiers has been shown in Figures REF -REF .", "The tuning parameters of all the above-said classifiers have been explored very carefully in the first few experiments and chosen those which provide the best accuracy.", "To avoid bias, all the listed experiments are carried out in the same settings on the same machine.", "Before the experiments, we performed the necessary normalization between $[0, 1]$ and all the experiments are carried out using Matlab $2016b$ installed on an Intel inside Core $i5$ with $8GB$ RAM.", "Here we also enlisted the computational time for SVM, KNN, ELM and MLR-LORSAL classifiers in Figure REF .", "One can observe that the computational time is gradually increasing as the number of training samples increases for all classifiers except for the KNN classifier which exponentially increases.", "However, the trend is quite different for accuracy which increases exponentially for all classifiers as compared to the time.", "Computational complexity can significantly reduce for KNN classifiers while using any optimization methods.", "In our case, we retrain KNN classifier for $k = [2-20]$ in each iteration, which can be overcome using a grid-search type method.", "All other classifiers have less computational cost and better accuracies, however, SVM and MLR-LORSAL have higher generalization performance then ELM followed by KNN.", "Figure: Computational Time for all classifiers for each dataset used in this work.The results shown in the above Figures and Tables are based on 5 Monte Carlo runs with a different number of training samples with equal class representation and in each iteration, the training set size is increased with $h = 100$ of selected samples by our proposed pipeline.", "It is perceived form Figures and Tables that by including the samples back to the training set, the classification results are significantly improved for all the classifiers and datasets.", "Moreover, it can be seen that all the classifiers are robust except KNN and their generalization has been significantly increased as shown in Table REF .", "In this work, we start evaluating our hypotheses from $n = 50$ number of randomly selected labeled training samples and we demonstrate that adding more samples back into the training set significantly increases the accuracy.", "It is worth noting from experiments that the classifiers trained with selected samples produce better accuracy and improve the generalization performance on those samples which were initially misclassified.", "To experimentally observe a sufficient quantity of labeled training samples for each classifier, we evaluated the hypotheses with a different number of labeled training samples.", "Based on the experimental results we conclude that $h = 400-600$ samples obtained by FLG are good enough to produce the acceptable accuracy for Hyperspectral Image Classification." ], [ "Conclusion", "In this paper, a fuzziness-based total local and global class discriminant information preserving active learning method is proposed for HSIC.", "In this line of investigation, a classifier is trained with a very small set of labeled training samples and evaluated on a large number of unlabeled samples.", "From results, we observe that it is enough to create a classification model from a small sample instead of a complex model with too many labeled training samples and parameters.", "Moreover, a small portion of unlabeled samples selected from high fuzziness group to train the model can enhance the generalization performance of any classifier.", "The classification results obtained with a different number of labeled training samples prove that the selection of initial labeled training samples does not affect FLG labeling success and does not influence the final classification accuracies.", "The experimental results show that the FLG which exploits both labeled and unlabeled samples information performs better than standard methods that use only uncertainty information, especially with small sample sizes." ] ]

[ [ "Dissipative dynamics of an open quantum battery" ], [ "Abstract Coupling with an external environment inevitably affects the dynamics of a quantum system.", "Here, we consider how charging performances of a quantum battery, modelled as a two level system, are influenced by the presence of an Ohmic thermal reservoir.", "The latter is coupled to both longitudinal and transverse spin components of the quantum battery including decoherence and pure dephasing mechanisms.", "Charging and discharging dynamics of the quantum battery, subjected to a static driving, are obtained exploiting a proper mapping into the so-called spin-boson model.", "Analytic expressions for the time evolution of the energy stored in the weak coupling regime are presented relying on a systematic weak damping expansion.", "Here, decoherence and pure dephasing dissipative coupling are discussed in details.", "We argue that the former results in better charging performances, showing also interesting features reminiscent of the Lamb shift level splitting renormalization induced by the presence of the reservoir.", "Charging stability is also addressed, by monitoring the energy behaviour after the charging protocol has been switched off.", "This study presents a general framework to investigate relaxation effects, able to include also non Markovian effects, and it reveals the importance of controlling and, possibly, engineering system-bath coupling in the realization of quantum batteries." ], [ "Introduction", "Quantum properties can play a predominant role in determining the behaviour of micro- and nano-devices.", "Recently, both theoretical and experimental works considered thermodynamic aspects of small quantum systems, in the new research field called “quantum thermodynamics” [1], [2], [3], [4], [5], [6], [7], [8].", "In this context one of the major issues, also triggered by potential technological applications, is the possibility to efficiently store energy in small systems, exploiting quantum features, and using it on-demand providing power supply.", "This naturally leads to the idea of quantum batteries (QBs) [9], devices where the performances in terms of energy transfer and charging power, namely the energy stored or released in a given time interval, can be improved by exploiting quantum resources such as entanglement [10], [11], [12], [13].", "On the one hand the attention focused on the characterization of possible quantum advantage of single [12], [14], [15], [16], [17] and many-body QBs [18], [19], [20] over their classical counterparts [21].", "On the other hand, theoretical frameworks aiming at actual experimental implementations in several setups such as circuit-QED, already used for quantum computing purposes [22], are under investigation [23].", "Here, a paradigmatic playground is a quantum two-level systems (TLS) [24], [25], physically realized by means of superconducting qubits [22] or quantum dots [26] in semiconducting nanostructures.", "This, indeed, represents the elementary building block (cell) for realizing QBs.", "Charging of a TLS, namely the controlled transition between the ground and the excited state, can be induced by means of an external classical drive [15], [16], [17], by properly controlling the exchange interaction between different cells [9], [18], [19], or through cell-cell coupling mediated by interaction with an external cavity radiation [23], [27], [28], [29].", "Most of the literature on QBs focused on the dynamics of closed systems where the energy is coherently transferred from a charger to the battery [9], [12], [13], [29], leaving only marginal discussions on possible effects due to the presence of external environment.", "However, as it is well known, the unavoidable coupling with an environment is responsible for dissipation, leading to relaxation and dephasing of each two-level system [30], [31] and therefore should be properly addressed also in this context.", "First investigations on “open” quantum batteries have been based on the study of the time evolution of the reduced density matrix of the TLS following a conventional Lindblad approach of Markovian master equations [32], [33].", "Within this framework, and under Markov approximation, general constraint on the possible stored energy and averaged charging power associated to the QB have been recently introduced [34], [35].", "Moreover, some protocols, based on multiple projective or weak measurements, able to mitigate these detrimental effects on the efficiency of the QB, have been proposed [36], [37], [38], [39].", "However, a microscopic description of the physical processes involved in the energy dissipation and their impact on the performances of the QB is still lacking.", "The aim of this work is to investigate how environment-induced dissipation can affect charging (and discharging) dynamics of a quantum battery.", "To avoid complications due to collective behaviours and interactions between the single entities of a large quantum battery, that can lead to competing effects [9], [14], we will assume the cells to be independent quantities.", "We therefore inspect the dynamics of a single cell quantum battery (hereafter indicated as QB) when a static external classical drive is acting as a charger and in presence of a coupling to an external reservoir responsible for dissipation.", "We underline that several system-reservoir couplings have been investigated and engineered, depending on the actual implementation of the quantum device under study [22], [31], [40], [41].", "Among all, we will concentrate on the case of linear dissipative couplings described by an Ohmic spectrum, which well describes the low energy noise source of many solid-state devices [31], [40].", "Dissipative dynamics of open QB is systematically investigated by mapping the problem in the so-called spin-boson (SB) model [31], [40], [42], [43], [44].", "As one would expect, too strong dissipation inevitably lead to fast incoherent relaxation dynamics, undermining the potential use of TLS as quantum batteries.", "Therefore, in the following, specific results will be discussed in the case of weak dissipation strength (weak damping regime), where exact analytic results can be derived.", "Focussing on the average energy variation, we investigate both charging and discharging dynamics, underlining the different behaviour between decoherence and pure dephasing linear dissipative couplings.", "By identifing the energy flows, both in the charging and discharging process, we will show that the former possible coupling results in better QB performances at fixed dissipation strength.", "The paper is organized as follows.", "In Section  we present the model of a single cell QB coupled to a reservoir (thermal bath).", "Different QB-bath couplings, leading to decoherence or pure dephasing mechanisms, are taken into account, by means of proper mapping to the SB model.", "Section  focuses on the dissipative dynamics, recalling generalized master equations governing the time evolution of the TLS in the presence of Ohmic dissipation.", "Here, analytic closed expressions for the average energy associated to the QB in the weak damping regime are presented.", "Finally, Section  contains our main results on charging performances in presence of dissipation and Section  summarizes our main conclusions.", "Technical details, together with a discussion of the effect due to stronger dissipation, are reported in three Appendices." ], [ "Open quantum battery", "We consider a single cell QB described by the Hamiltonian (hereafter we set $\\hbar =1$ ) $H_{{\\rm QB}}=\\frac{\\Delta }{2}\\sigma _z~,$ where $\\Delta $ represents the level spacing between the ground and the excited state, which can be seen as the empty and the full cell configuration respectively.", "At time $t=0^+$ a coupling with the $\\sigma _x$ component of the QB with an external classical field $A$ (the charger) is switched on [12], [13], [23] $H_{{\\rm C}}=\\frac{A}{2}\\sigma _x~.$ In the above equations $\\sigma _k$ ($k=x,y,z$ ) indicates the usual $k$ -th Pauli matrix.", "The QB is also coupled to a reservoir (thermal bath), responsible for dissipation, modelled as an ensemble of harmonic oscillators of frequency $\\omega _j$  [31], [43], [44], [45].", "In terms of bosonic creation (annihilation) operators $a^{\\dagger }_j (a_j)$ it reads $H_{{\\rm R}}=\\sum _{j}\\omega _{j} a^{\\dagger }_ja_j ~.$ We consider a linear coupling with the reservoir along both the longitudinal ($z$ ) and transverse ($x$ ) directions $H_{{\\rm I}}=\\frac{1}{2}\\left[\\sigma _x\\cos (\\theta /2) + \\sigma _z\\sin (\\theta /2) \\right]\\cdot \\sum _{j}\\lambda _j\\left(a^{\\dagger }_j+a_j\\right),$ capturing both decoherence ($\\theta =0$ ) and pure dephasing ($\\theta =\\pi $ ) processes [40], [46], [47].", "Notice that different couplings can correspond to different and independent noise sources, for instance they can be linked to charge and flux noise in a superconducting Josephson realization of a TLS [40], [41], [48].", "The spectral properties of the reservoir are characterized by the density function [31] $J(\\omega ) = \\sum _j \\lambda _j^2 \\delta (\\omega -\\omega _j),$ which in the continuum limit and in the relevant case of Ohmic dissipation we are interested in, has the form [31], [44], [45] $J(\\omega ) =2 \\alpha \\omega \\, ^{-\\omega /\\omega _{ c}}.$ Here, $\\alpha $ is a dimensionless parameter which quantifies dissipation strength and $\\omega _c$ the high frequency cut-off of the bath [31], [49], [50].", "The total Hamiltonian governing the dynamics at $t\\ge 0$ is thus given by $H= H_{{\\rm QB}} + H_{{\\rm C}} + H_{{\\rm RI}}~,$ where we have defined the total contribution of the reservoir $H_{{\\rm RI}}=H_{{\\rm R}}+H_{{\\rm I}}$ .", "At time $t=0$ , we assume factorized initial conditions, with a total density matrix given by $\\rho _{{\\rm tot}}(0)=\\rho (0)\\times \\rho _{{\\rm R}}(0)$ .", "Here, the reservoir is at thermal equilibrium with density $\\rho _{{\\rm R}}(0)=e^{-\\beta H_{{\\rm R}}}/{\\rm Tr}[e^{-\\beta H_{{\\rm R}}}]$ , where $\\beta =1/(k_{{\\rm B}}T)$ is the bath inverse temperature.", "On the other hand, the QB is described by the reduced density matrix $\\rho (0)=\\left(\\begin{array}{cc}{p}_R & {a}-i{b} \\\\{a} +i{b} & {p}_L\\end{array}\\right),$ where we have included the possibility to have initial coherences (${a}$ and ${b}$ ).", "Normalization of $\\rho (0)$ imposes ${p}_R +{p}_L=1$ .", "In addition, with ${a}$ and ${b}$ real coefficients, the condition ${\\rm Tr}\\rho ^2\\le 1$ leads to the constraint ${a}^2+{b}^2\\le {p}_R {p}_L$ .", "Time evolution of the spin components $\\langle \\sigma _k(t)\\rangle $ ($k=x,y,z$ ) are written as averages over the time dependent total density matrix, $\\rho _{{\\rm tot}}(t)$ driven by the total Hamiltonian $H$ $\\langle \\sigma _k(t)\\rangle = {\\rm Tr}[\\rho _{{\\rm tot}}(t)\\sigma _k]~.$ These averages can be represented in terms of the time dependent reduced density matrix $\\rho (t)$ as $\\langle \\sigma _k(t)\\rangle ={\\rm Tr}[\\rho (t)\\sigma _k]~,$ where $\\rho (t)={\\rm Tr}_{{\\rm R}}[\\rho _{{\\rm tot}}(t)]$ with ${\\rm Tr}_{{\\rm R}}$ the trace over the bath degrees of freedom.", "Energy exchanges between the different subparts, and thus charging dynamics, at time $t$ can be then determined by studying the energy variations $\\langle E_{s} (t)\\rangle = \\langle H_s(t)\\rangle -\\langle H_s(0)\\rangle ~,$ where $s={\\rm QB}, {\\rm C}, {\\rm RI}$ .", "Recall that for $t>0$ , because of the static driving, we have $\\dot{H} =0$ , with an energy balance of the form $\\langle E_{{\\rm QB}}(t)\\rangle + \\langle E_{{\\rm C}}(t)\\rangle +\\langle E_{{\\rm RI}}(t)\\rangle =0$ that must hold for any driving amplitude and dissipation coupling strength.", "Before discussing the dissipative dynamics, it is instructive to recall that in absence of dissipation ($\\alpha =0$ ), all energy supplied by $H_{{\\rm C}}$ is transfered to the QB, whose maximum energy that can be stored is given by the energy level spacing $\\Delta $ .", "A static driving protocol then charges the QB according to [12], [17] $\\langle E_{{\\rm QB}}(t)\\rangle =\\frac{\\Delta }{2}\\frac{A^2}{\\Omega ^2}\\big [1-\\cos (\\Omega t)\\big ]~,$ where $\\Omega $ is the Rabi frequency $\\Omega =\\sqrt{\\Delta ^2+A^2}~.$ Here, the unitary evolution of the closed system implies a periodic behaviour, as evident from Equation (REF ), with the QB completely discharged (again empty and in the $|g\\rangle $ state) for even multiplies of $\\Omega t= 2\\pi n$ ($n>0$ integer).", "Notice that the maximum amount of the energy stored, is reached for very large amplitude $A\\gg \\Delta $ ." ], [ "Mapping to the spin boson model", "In order to study the time evolution of the different subparts we start by considering a unitary rotation in the spin space ${\\cal R}=e^{-i\\phi \\sigma _y}$ with the angle $\\phi $ chosen in such a way to project the interaction part $H_{\\rm I}$ only along the $z$ axis.", "It is easy to show (see  for details) that with $\\phi =(\\theta +\\pi )/4$ one has ${\\widetilde{H}}_{\\rm I}&\\equiv &\\mathcal {R}H_{\\rm I}\\mathcal {R}^{-1}=-\\frac{1}{2}\\sigma _z\\cdot \\sum _{j}\\lambda _j\\left(a^{\\dagger }_j+a_j\\right),\\\\{\\widetilde{H}}_{\\rm QB}&\\equiv &\\mathcal {R}H_{\\rm QB}\\mathcal {R}^{-1}=-\\frac{\\Delta }{2}\\left[\\sigma _z\\sin (\\theta /2)-\\sigma _x\\cos (\\theta /2) \\right]\\\\{\\widetilde{H}}_{\\rm C}&\\equiv &\\mathcal {R}H_{\\rm C}\\mathcal {R}^{-1}=-\\frac{A}{2}\\left[\\sigma _z\\cos (\\theta /2)+\\sigma _x\\sin (\\theta /2) \\right].$ In this way, the total Hamiltonian ${\\widetilde{H}}={\\widetilde{H}}_{\\rm QB}+{\\widetilde{H}}_{\\rm C}+H_R+{\\widetilde{H}}_{\\rm I}$ can be then recast in the standard form of a SB model [31], [51], which represents a prototypical quantum dissipative two state system that has been studied with several numerical [52], [53], [54], [55], [56], [57] and analytical approaches both in the weak and strong coupling regimes [31], [49], [50], [58], [59], [60], [61].", "We have indeed ${\\widetilde{H}}\\equiv H^{{(\\rm SB)}}=H^{(SB)}_0+ H_{{\\rm R}} + H^{(SB)}_{{\\rm I}}$ where $H^{(SB)}_0= -\\frac{\\Delta _0}{2}\\sigma _x-\\frac{\\epsilon _0}{2}\\sigma _z,$ is the bare TLS Hamiltonian, with $\\Delta _0$ the tunneling amplitude and $\\epsilon _0$ a constant bias, and $H^{(SB)}_{{\\rm I}}= {\\widetilde{H}}_{{\\rm I}}$ given in Equation (REF ).", "By comparing the above $H^{{(\\rm SB)}}$ with ${\\widetilde{H}}$ in Equation (REF ) we obtain the following identifications $\\Delta _0&=&\\left[A\\sin (\\theta /2)-\\Delta \\cos (\\theta /2)\\right]\\,\\\\\\epsilon _0&=&\\left[A\\cos (\\theta /2)+\\Delta \\sin (\\theta /2)\\right].$ To complete the mapping we still need to identify the reduced SB initial density matrix, as $\\rho _{\\rm SB}(0)=\\left(\\begin{array}{cc}\\bar{p}_R & \\bar{a}-i \\bar{b} \\\\\\bar{a}+i \\bar{b} & \\bar{p}_L\\end{array}\\right)\\,,$ with the rotated reduced initial matrix $\\widetilde{\\rho }(0)\\equiv \\mathcal {R}\\rho (0)\\mathcal {R}^{-1}$ of the QB.", "We therefore get (see ) $\\bar{p}_R&=&-p_R\\sin (\\theta /2)+\\frac{1+\\sin (\\theta /2)}{2}-a\\cos (\\theta /2),\\\\\\bar{p}_L &=&-p_L\\sin (\\theta /2)+\\frac{1+\\sin (\\theta /2)}{2}+a\\cos (\\theta /2),\\\\\\bar{a}&=&-a\\sin (\\theta /2)+\\frac{p_R-p_L}{2}\\cos (\\theta /2),\\\\\\bar{b}&=&b.$ From now on, the average energy of the QB, $\\langle H_{\\rm QB}(t)\\rangle $ and that of the charger $\\langle H_{\\rm C}(t)\\rangle $ will be directly obtained using the expressions () and () as $\\langle H_{\\rm QB}(t)\\rangle &=&-\\frac{\\Delta }{2}\\left[\\langle \\sigma _z(t)\\rangle \\sin (\\theta /2)-\\langle \\sigma _x(t)\\rangle \\cos (\\theta /2)\\right]\\\\\\langle H_{\\rm C}(t)\\rangle &=&-\\frac{A}{2}\\left[\\langle \\sigma _z(t)\\rangle \\cos (\\theta /2)+\\langle \\sigma _x(t)\\rangle \\sin (\\theta /2)\\right]$ where the time evolution of the spin components will be considered in the SB model as $\\langle \\sigma _k(t)\\rangle ={\\rm Tr}[\\rho _{{\\rm SB}}(t)\\sigma _k]$ with the proper identifications just discussed.", "As mentioned above, it is convenient to solve the dynamics of the QB by using the mapping to the SB model given in Equation (REF ) with initial density matrix in Equation (REF ).", "Indeed, for this model, the time dependent evolution of the spin components $\\sigma _{z}(t)$ and $\\sigma _{x}(t)$ , were studied with several numerical and analytical methods [31], [53], [56], [60], [61], [62].", "In particular, in the framework of the path integral approach [31], [42], [45], [58], [60] it was possible to represent $\\langle \\sigma _{z}(t)\\rangle $ in the form of an exact generalized master equations (GME), and to connect $\\langle \\sigma _{x}(t)\\rangle $ with $\\langle \\sigma _{z}(t)\\rangle $ by an exact integral relation [31], [50], [59].", "Usually in this approach the reduced initial density matrix is choosen to be diagonal, with $\\bar{a}=\\bar{b}=0$ .", "Here, following the procedure outlined in [58], we will relax this assumption by considering also the presence of non-diagonal terms, which are necessary for the mapping to the QB.", "Below we will briefly present the main steps, while the details are reported in .", "Let us start with the $z$ spin component which fulfills an exact integro-differential equation $\\hspace{-28.45274pt}\\frac{d\\langle \\sigma _{z}(t)\\rangle }{dt}= \\int _0^t dt^{\\prime } [K^{(-)}_{1,z} (t-t^{\\prime }) - K^{(+)}_{1,z}(t-t^{\\prime })\\langle \\sigma _{z}(t^{\\prime })\\rangle ] +2\\bar{a} K^{(-)}_{2,z}(t)-2\\bar{b} K^{(+)}_{2,z}(t),$ with initial condition $\\langle \\sigma _{z}(0)\\rangle = \\bar{p}_R-\\bar{p}_L$ .", "The kernels $K_{1,z}^{(\\pm )}(t-t^{\\prime })$ determine the spin evolution in the presence of an initial diagonal density matrix ($\\bar{a}=\\bar{b}=0$ ), on the other hand, $K_{2,z}^{(\\pm )}(t-t^{\\prime })$ are responsible for the additional contributions due to the initial coherence terms ($\\bar{a}\\ne 0$ , $\\bar{b}\\ne 0$ ).", "The upper label $(+)$ and $(-)$ indicate whether the kernel is an even or odd function of the bias $\\epsilon _0$ .", "All these kernels encorporate dissipative effects and they only depend on the time difference since the SB Hamiltonian is time independent.", "As shown in  they are expressed in terms of an infinite series over all possible tunneling processes, governed by $\\Delta _0$ .", "Notice that, due to the linearity of Equation (REF ), $\\langle \\sigma _z(t)\\rangle $ can be always decomposed as $\\langle \\sigma _z(t)\\rangle =\\langle \\sigma _{z,0}(t)\\rangle +\\langle \\sigma _{z,\\bar{a}}(t)\\rangle +\\langle \\sigma _{z,\\bar{b}}(t)\\rangle ,$ where $\\langle \\sigma _{z,0}(t)\\rangle $ is the contribution without the initial coherence terms ($\\bar{a}\\!=\\!\\bar{b}\\!=\\!0$ ), while $\\langle \\sigma _{z,\\bar{a}}(t)\\rangle $ and $\\langle \\sigma _{z,\\bar{b}}(t)\\rangle $ are the terms due to the presence of the coefficients $\\bar{a}$ and $\\bar{b}$ respectively.", "As demonstrated in  (see Equation (REF )) in the so called scaling limit, i.e.", "large cut-off frequency $\\omega _c$ , these two parts can be directly linked to $\\langle \\sigma _{z,0}(t)\\rangle $ in the following way $\\langle \\sigma _{z,\\bar{a}}(t)\\rangle &=& \\frac{2\\bar{a}}{\\Delta _0\\sin (\\pi \\alpha )}\\frac{d}{dt}\\langle {\\sigma }^{(-)}_{z,0}(t)\\rangle \\nonumber \\\\\\langle \\sigma _{z,\\bar{b}}(t)\\rangle &=& \\frac{2\\bar{b}}{\\Delta _0\\cos (\\pi \\alpha )(\\bar{p}_R-\\bar{p}_L)}\\frac{d}{dt}\\langle {\\sigma ^{(+)}_{z,0}(t)}\\rangle ,$ where again $\\pm $ indicates symmetric/antisymmetric term with respect to $\\epsilon _0$ .", "It is important to underline that this result is valid at any order in the dissipation coupling strength $\\alpha $ and in the tunneling amplitude $\\Delta _0$ .", "These relations are particularly helpfull since they allow to evaluate the full dynamics of $\\langle \\sigma _z(t)\\rangle $ starting from initial diagonal conditions ($\\bar{a}=\\bar{b}=0$ ), and deriving the full expressions also in the presence of coherent (off-diagonal) components of the initial density matrix.", "Let us comment now on the general structures of $\\langle \\sigma _{x}(t)\\rangle $ .", "As shown in [58], [59] this quantity is directly connected to $\\langle \\sigma _{z}(t)\\rangle $ via an exact integral relation $\\hspace{-28.45274pt}\\langle \\sigma _{x}(t)\\rangle = \\int _0^t dt^{\\prime } [K^{(+)}_{1,x}(t-t^{\\prime }) + K^{(-)}_{1,x}(t-t^{\\prime })\\langle \\sigma _{z}(t^{\\prime })\\rangle ] +2\\bar{a} K^{(+)}_{2,x}(t)+2\\bar{b} K^{(-)}_{2,x}(t).$ The Kernels $K_{1/2,x}^{(\\pm )}$ are again given in the form of a series and they are quoted in  (see Equation (REF )).", "Also this quantity can be decomposed as a sum of a term without $\\bar{a}$ and $\\bar{b}$ , called $\\langle \\sigma _{x,0}(t)\\rangle $ and the remaining parts $\\langle \\sigma _{x,\\bar{a}}(t)\\rangle $ and $\\langle \\sigma _{x,\\bar{b}}(t)\\rangle $ as $\\langle \\sigma _x(t)\\rangle =\\langle \\sigma _{x,0}(t)\\rangle +\\langle \\sigma _{x,\\bar{a}}(t)\\rangle +\\langle \\sigma _{x,\\bar{b}}(t)\\rangle .$ Similarly to the $z$ -component, for sufficiently large cut-off frequency, the two last terms are linked to $\\langle \\sigma _{x,0}(t)\\rangle $ as $\\langle \\sigma _{x,\\bar{a}}(t)\\rangle &=& \\frac{2\\bar{a}}{\\Delta _0\\sin (\\pi \\alpha )}\\frac{d}{dt}\\langle {\\sigma ^{(+)}_{x,0}(t)}\\rangle \\nonumber \\\\\\langle \\sigma _{x,\\bar{b}}(t)\\rangle &=& \\frac{2\\bar{b}}{\\Delta _0\\cos (\\pi \\alpha )(\\bar{p}_R-\\bar{p}_L)}\\frac{d}{dt}\\langle {\\sigma ^{(-)}_{x,0}(t)}\\rangle .$ Few remarks on the possible approaches to solve the above exact expressions are now in order.", "In general the complete resummation of the series expansions which determine the kernels $K_{1/2,z}^{(\\pm )}$ and $K_{1/2,x}^{(\\pm )}$ cannot be done in closed form.", "Therefore one needs to resort to suitable analytical approximations or numerical computations.", "A well-known example is the so-called non-interacting blip approximation (NIBA) [31], [42], [62], [63].", "As shown in the final part of  this scheme approximates the kernels $K_{1/2,z/x}^{(\\pm )}$ at their lowest order in $\\Delta _0$ truncating then their series expansions.", "On the other hands, it retains the dissipation strength $\\alpha $ at any order.", "It has been shown that NIBA is a good approximation for sufficiently high temperatures $\\beta \\sqrt{\\Delta _0^2 + \\epsilon _0^2}<1 $ , while strong deviations occur at low temperatures especially in the long time regime and in the presence of a finite bias [31].", "Another, complementary, approximation is the so called systematic weak damping expansion, valid for $\\alpha \\ll 1$ .", "This method evaluates the kernels at lowest order in $\\alpha $ with an exact resummation of the series in $\\Delta _0$ .", "This is a very powerful method to treat dissipative dynamics in the weak coupling regime at low temperature $\\beta \\sqrt{\\Delta _0^2+\\epsilon _0^2}>1$ , where quantum coherence and non markovianity effects may play an important role." ], [ "Weak damping dynamics", "Here we specify the spin dynamics in the weak damping regime in the SB model.", "In the limit $\\alpha \\ll 1$ and low temperature regime $\\beta \\sqrt{\\Delta _0^2+\\epsilon _0^2}>1$ it is possible to obtain closed analytical expressions for the time evolution of spin operators relying on a systematic weak damping espansion, valid at any order in the tunneling amplitude $\\Delta _0$ .", "Resummation of the infinite series expressions in the systematic weak damping expansion have been usually considered starting from diagonal initial conditions, neglecting coherence terms [31].", "However, by exploiting the general links derived in , from the knowledge of $\\langle \\sigma _{z,0}(t)\\rangle $ and $\\langle \\sigma _{x,0}(t)\\rangle $ it is easy to derive also the expressions related to finite off diagonal contributions $\\bar{a}$ and $\\bar{b}$ .", "We briefly remind the main steps behind this systematic expansion, for the $\\langle \\sigma _{z,0}(t)\\rangle $ contribution.", "The starting point is the formal series expression (reported in  ) of the kernels entering the generalized master equation (REF ) $K_{1,z}^{(\\pm )}(t-t^{\\prime })$ .", "The sum in Equation (REF ) can be resummed at all order in $\\Delta _0$ , considering the lowest order expansion in $\\alpha $ of the exponential factors present in (REF ) which enters in Equation (REF ).", "Notice that in this approach, the kernels $K_{1,z}^{(\\pm )}$ can be viewed as self-energy correction for the operator $\\langle \\sigma _{z,0}(t)\\rangle $ .", "Upon the expansion at lowest order in $\\alpha $ the series expression for the kernels can be summed and the resulting expression can be plugged into the generalized master equation which now admits a simple closed solution.", "Following a similar procedure it is possible to obtain a systematic weak damping expression also for the $x$ - component [31], [59].", "The averaged spin component $\\langle \\sigma _{k,0}(t)\\rangle $ with $k=z,x$ can be written in analytic form as $\\hspace{-28.45274pt}\\langle \\sigma _{k,0}(t)\\rangle = N^{(1)}_{k,0} e^{-\\Gamma ^{(r)} t} + \\left[N^{(2)}_{k,0} \\cos (\\Omega _{{\\rm SB}} t) + N^{(3)}_{k,0} \\sin (\\Omega _{{\\rm SB}} t)\\right] e^{-\\Gamma t}+ \\langle \\sigma _k(\\infty )\\rangle .$ Both quantities posses an oscillatory behaviour, with characteristic frequency $\\Omega _{{\\rm SB}}=\\sqrt{ \\Delta _{0,{\\rm eff}}^2+\\epsilon _0^2},$ with $\\Delta _{0,{\\rm eff}}=\\Delta _0 \\left[\\frac{\\Delta _0}{\\omega _c}\\right]^{\\alpha /(1-\\alpha )}\\cdot \\left[\\Gamma (1-2\\alpha )\\cos (\\pi \\alpha )\\right]^{1/[2(1-\\alpha )]}~.$ The level splitting gets renormalized with respect to the bare case ($(\\Delta _0^2+\\epsilon _0^2)^{1/2}$ ), with always $\\Omega _{{\\rm SB}}<\\sqrt{\\Delta _0^2+\\epsilon _0^2}$ .", "The oscillatory behaviour present in Equation (REF ) is modulated by exponential decay dictated by the incoherent relaxation $\\Gamma ^{(r)}$ and dephasing $\\Gamma $ rates, given by $\\Gamma ^{(r)} &=& \\frac{\\pi \\alpha \\Delta _{0,{\\rm eff}}^2}{\\Omega _{{\\rm SB}}}\\coth [\\beta \\Omega _{{\\rm SB}}/2]\\nonumber \\\\\\Gamma &=&\\frac{\\Gamma ^{(r)}}{2} + 2\\pi \\alpha \\frac{\\epsilon _0^2}{\\beta \\Omega _{{\\rm SB}}^2}.$ The amplitudes entering the above expressions are evaluated up to linear order in $\\alpha $ , (apart the renormalized frequency $\\Delta _{0,{\\rm eff}}$ ) and read $N^{(1)}_{z,0}&=& \\frac{(\\bar{p}_R-\\bar{p}_L)\\epsilon _0^2}{\\Omega _{{\\rm SB}}^2} - \\langle \\sigma _z(\\infty )\\rangle \\nonumber \\\\N^{(2)}_{z,0} &= &\\frac{ (\\bar{p}_R-\\bar{p}_L)\\Delta _{0,{\\rm eff}}^2}{\\Omega _{{\\rm SB}}^2}\\nonumber \\\\N^{(3)}_{z,0}&= & \\frac{\\Gamma ^{(r)} N^{(1)}_{z,0} +\\Gamma N^{(2)}_{z,0}}{\\Omega _{{\\rm SB}}}\\nonumber \\\\N^{(1)}_{x,0} &=& \\frac{\\epsilon _0\\Delta _{0,{\\rm eff}}}{\\Omega _{{\\rm SB}}^2} (\\bar{p}_R-\\bar{p}_L) - \\langle \\sigma _x(\\infty )\\rangle \\nonumber \\\\N^{(2)}_{x,0} &=&- \\frac{\\epsilon _0\\Delta _{0,{{\\rm eff}}}}{\\Omega _{{\\rm SB}}^2} (\\bar{p}_R-\\bar{p}_L)\\nonumber \\\\N^{(3)}_{x,0} &=&\\frac{1}{\\Omega _{{\\rm SB}}}\\left[\\Gamma ^{(r)} N^{(1)}_{x,0} +\\Gamma N^{(2)}_{x,0}+\\pi \\alpha \\Delta _{0,{\\rm eff}}\\right].$ In the above equations we have introduced the steady ($t\\rightarrow \\infty $ ) values $\\langle \\sigma _z(\\infty )\\rangle &=& \\frac{\\epsilon _0}{\\Omega _{{\\rm SB}}}\\tanh [\\beta \\Omega _{{\\rm SB}}/2]\\nonumber \\\\\\langle \\sigma _x(\\infty )\\rangle &=&\\frac{\\Delta _{0,{\\rm eff}}}{\\Omega _{{\\rm SB}}}\\tanh [\\beta \\Omega _{{\\rm SB}}/2].$ Exploiting Equation (REF )-(REF ), with the substitution $\\Delta _0\\rightarrow \\Delta _{{\\rm eff}}$ to take consistently into account the dissipation induced renormalization, we obtain $\\hspace{-56.9055pt}\\langle \\sigma _{z,\\bar{a}}(t)\\rangle &=& \\frac{2\\bar{a} \\Delta _{0,{\\rm eff}}\\epsilon _0}{\\Omega _{{\\rm SB}}^2}\\Big [e^{-\\Gamma ^{(r)} t} -e^{-\\Gamma t} \\cos (\\Omega _{{\\rm SB}} t)+ \\frac{\\Gamma }{\\Omega _{{\\rm SB}}}e^{-\\Gamma t}\\sin (\\Omega _{{\\rm SB}} t)\\Big ]\\\\\\hspace{-56.9055pt}\\langle \\sigma _{z,\\bar{b}}(t)\\rangle &=& \\frac{-2\\bar{b}}{\\Delta _{0,{\\rm eff}}\\Omega _{{\\rm SB}}^2}\\Big \\lbrace \\epsilon _0^2\\Gamma ^{(r)}[e^{-\\Gamma ^{(r)} t} - e^{-\\Gamma t}\\cos (\\Omega _{{\\rm SB}} t)] +\\Delta _{0,{\\rm eff}}^2\\Omega _{{\\rm SB}} e^{-\\Gamma t}\\sin (\\Omega _{{\\rm SB}} t)\\Big \\rbrace \\\\\\hspace{-56.9055pt}\\langle \\sigma _{x,\\bar{a}}(t)\\rangle &=&\\frac{2 \\bar{a}}{\\Omega _{{\\rm SB}}^2} \\Big [\\Delta _{0,{\\rm eff}}^2 e^{-\\Gamma ^{(r)} t}+ \\epsilon _0^2e^{-\\Gamma t}\\cos (\\Omega _{{\\rm SB}} t) -\\frac{\\epsilon _0^2\\Gamma }{\\Omega _{{\\rm SB}}} e^{-\\Gamma t}\\sin (\\Omega _{{\\rm SB}} t)\\Big ]\\\\\\hspace{-56.9055pt}\\langle \\sigma _{x,\\bar{b}}(t)\\rangle &=& -\\frac{2 \\bar{b}}{\\Omega _{{\\rm SB}}^2} \\Big \\lbrace \\epsilon _0\\Gamma ^{(r)} [e^{-\\Gamma ^{(r)} t}- e^{-\\Gamma t}\\cos (\\Omega _{{\\rm SB}} t)] - \\epsilon _0\\Omega _{{\\rm SB}}\\sin (\\Omega _{{\\rm SB}} t)e^{-\\Gamma t}\\Big \\rbrace .$ Finally the full time-evolution of $\\langle \\sigma _z(t)\\rangle $ and $\\langle \\sigma _x(t)\\rangle $ in the weak damping regime is obtained by plugging the above expressions (Equations (REF ),(REF )-()) into Equations (REF ) and (REF )." ], [ "Charging dynamics in the weak damping regime", "We now use the previous results in order to describe the dynamics of the QB.", "We focus on the weak damping limit in the low temperature regime, which seems to be the most promising regime in order to still achieve good performance of the QB, even in presence of a thermal bath.", "Moreover, we note that, in this regime, possible effects related to quantum coherences and non-Markovian contributions can play relevant role in determining the QB dynamics.", "We will determine the energy variation associated to the charging process considering the QB initially prepared in the ground state $|g\\rangle $ , i.e.", "$p_L=1$ and $p_R=a=b=0$ .", "Similar expressions can be obtained considering other initial conditions as well.", "Unless otherwise stated, we will discuss the two limiting case of decoherence coupling ($\\theta =0$ in Equation (REF )) and of pure dephasing coupling ($\\theta =\\pi $ in Equation (REF )).", "To distinguish between the two we introduce an index $_{\\theta }$ with $\\theta =0$ or $\\theta =\\pi $ whenever necessary.", "Using the expressions (REF ), () with the mapping in (REF ), () the average energies associated to the QB and to the charger C are directly expressed in terms of the time evolution of the spin components $\\langle \\sigma _{z}(t)\\rangle $ and $\\langle \\sigma _{x}(t)\\rangle $ of the SB model.", "As explained above their evolution can be exactly solved for weak damping (see Section REF ).", "Hereafter, for sake of clarity, we directly quote the closed expressions for the time evolution in terms of QB variables.", "We start considering the case of pure decoherence with $\\theta =0$ in Equation (REF ), where the thermal bath is coupled only to $\\sigma _x$ operator.", "We have $&&\\hspace{-45.52458pt}\\langle E_{{\\rm QB}}(t)\\rangle = \\frac{\\Delta }{2} \\left\\lbrace 1-\\frac{\\Delta _{\\rm eff}}{\\Omega _{0}}\\tanh (\\frac{\\beta \\Omega _{0}}{2})+\\frac{\\Delta _{\\rm eff}}{\\Omega _{0}}\\bigg (\\tanh (\\frac{\\beta \\Omega _{0}}{2})-\\frac{\\Delta _{\\rm eff}}{\\Omega _{0}}\\bigg )e^{-\\Gamma _{0}^{(r)} t}+\\right.", "\\nonumber \\\\&&\\hspace{-45.52458pt}+\\frac{e^{-\\Gamma _{0} t}}{\\Omega _0^2}\\left.\\bigg [-A^2\\cos (\\Omega _{0}t)+ \\bigg (\\Delta _{\\rm eff}\\Gamma _{0}^{(r)}\\tanh (\\frac{\\beta \\Omega _{0}}{2})+\\frac{A^2\\Gamma _{0}}{\\Omega _{0}}\\bigg )\\sin (\\Omega _{0}t)\\bigg ]\\right\\rbrace ,$ while the energy variation of the charger $\\langle E_{{\\rm C}}(t)\\rangle $ is $&&\\hspace{-45.52458pt}\\langle E_{{\\rm C}}(t)\\rangle =\\frac{A^2}{2\\Omega _{0}}\\left\\lbrace -\\tanh (\\frac{\\beta \\Omega _{0}}{2})+\\bigg [\\tanh (\\frac{\\beta \\Omega _{0}}{2})-\\frac{\\Delta _{\\rm eff}}{\\Omega _{0}}\\bigg ]e^{-\\Gamma _{0}^{(r)} t}+\\right.\\nonumber \\\\&&\\hspace{-45.52458pt}+\\left.\\frac{e^{-\\Gamma _{0} t}}{\\Omega _{0}}\\bigg [\\Delta _{\\rm eff}\\cos (\\Omega _{0}t)+\\bigg (\\Gamma _{0}^{(r)}\\tanh (\\frac{\\beta \\Omega _{0}}{2})-\\frac{\\Delta _{\\rm eff}\\Gamma _{0}}{\\Omega _{0}}\\bigg )\\sin (\\Omega _{0}t)\\bigg ]\\right\\rbrace .$ In the above equations we have introduced the renormalized characteristic energy $\\Omega _{0}=\\sqrt{\\Delta _{{\\rm eff}}^2+A^2}~,$ where $\\Delta _{{\\rm eff}}=\\Delta \\left(\\frac{\\Delta }{\\omega _c}\\right)^{\\alpha /(1-\\alpha )}\\left[\\Gamma (1-2\\alpha )\\cos (\\pi \\alpha )\\right]^{1/[2(1-\\alpha )]},$ is the QB level splitting renormalized by dissipation with respect to the bare one $\\Delta $ .", "This renormalization of the level splitting it is the analogue of the Lamb shift [31], [46], [47], and it is due to the presence of weak dissipation.", "Notice that the renormalized energy is always $\\Omega _{0}\\le \\Omega $ , with $\\Omega $ the bare characteristic energy of the QB in Equation (REF ).", "Dissipation effects are reflected in the incoherent relaxation rate $\\Gamma ^{(r)}_{0}$ and dephasing rate $\\Gamma _{0}$ , which are given by $\\Gamma _{0}^{(r)} &=& \\frac{\\pi \\alpha \\Delta _{\\rm eff}^2}{\\Omega _{0}}\\coth [\\beta \\Omega _{0}/2]\\nonumber \\\\\\Gamma _{0}&=&\\frac{\\Gamma _{0}^{(r)}}{2} + 2\\pi \\alpha \\frac{A^2}{\\beta \\Omega ^2_{0}} ~.$ The case of pure dephasing is described by setting $\\theta =\\pi $ in Equation (REF ), with the reservoir coupled to $\\sigma _z$ , which induces a different dissipative dynamics.", "Indeed, in this case the average energy associated to the QB can be written as $&&\\hspace{-45.52458pt}\\langle E_{{\\rm QB}}(t)\\rangle = \\frac{\\Delta }{2} \\left\\lbrace 1-\\frac{\\Delta }{\\Omega _{\\pi }}\\tanh (\\frac{\\beta \\Omega _{\\pi }}{2})+\\frac{\\Delta }{\\Omega _{\\pi }}\\bigg (\\tanh (\\frac{\\beta \\Omega _{\\pi }}{2})-\\frac{\\Delta }{\\Omega _{\\pi }}\\bigg )e^{-\\Gamma _{\\pi }^{(r)} t}+\\right.\\nonumber \\\\&&\\hspace{-45.52458pt}+\\left.\\frac{e^{-\\Gamma _{\\pi } t}}{\\Omega _{\\pi }^2}\\bigg [-A_{\\rm eff}^2\\cos (\\Omega _{\\pi }t)+\\bigg (\\Delta \\Gamma _{\\pi }^{(r)}\\tanh (\\frac{\\beta \\Omega _{\\pi }}{2}) -\\frac{\\Delta ^2\\Gamma ^{(r)}_{\\pi }+A_{\\rm eff}^2\\Gamma _{\\pi }}{\\Omega _{\\pi }}\\bigg )\\sin (\\Omega _{\\pi }t)\\bigg ]\\!\\right\\rbrace $ and the charger contribution is given by $&&\\hspace{-42.67912pt}\\langle E_{{\\rm C}}(t)\\rangle =\\frac{AA_{\\rm eff}}{2\\Omega _{\\pi }}\\bigg \\lbrace -\\tanh (\\frac{\\beta \\Omega _{\\pi }}{2})+\\bigg [\\tanh (\\frac{\\beta \\Omega _{\\pi }}{2})-\\frac{\\Delta }{\\Omega _{\\pi }}\\bigg ]e^{-\\Gamma _{\\pi }^{(r)} t}+\\nonumber \\\\&&\\hspace{-42.67912pt}+\\left.\\frac{e^{-\\Gamma _{\\pi } t}}{\\Omega _{\\pi }}\\bigg [\\Delta \\cos (\\Omega _{\\pi }t)+\\bigg (\\Gamma _{\\pi }^{(r)}\\tanh (\\frac{\\beta \\Omega _{\\pi }}{2})-\\frac{\\Delta (\\Gamma ^{(r)}_{\\pi }-\\Gamma _{\\pi })}{\\Omega _{\\pi }}\\bigg )\\sin (\\Omega _{\\pi }t)\\bigg ]\\right\\rbrace .$ Notice the presence of a different characteristic energy $\\Omega _{\\pi }=\\sqrt{\\Delta ^2+A^2_{{\\rm eff}}}~,$ where the bare amplitude $A$ , felt by the QB, gets renormalized by the presence of dissipation as $A_{{\\rm eff}}=A\\left(\\frac{A}{\\omega _c}\\right)^{\\alpha /(1-\\alpha )}\\left[\\Gamma (1-2\\alpha )\\cos (\\pi \\alpha )\\right]^{1/[2(1-\\alpha )]}~.$ Also here, one has $\\Omega _{\\pi }\\le \\Omega $ .", "The relaxation and dephasing rates responsible for the damped oscillations are given by $\\Gamma _{\\pi }^{(r)} &=& \\frac{\\pi \\alpha A^2_{\\rm eff}}{\\Omega _{\\pi }}\\coth [\\beta \\Omega _{\\pi }/2]\\nonumber \\\\\\Gamma _{\\pi }&=&\\frac{\\Gamma _{\\pi }^{(r)}}{2} + 2\\pi \\alpha \\frac{\\Delta ^2}{\\beta \\Omega ^2_{\\pi }}~.$ In passing, we comment on another particular value of the angle $\\theta $ .", "Indeed, looking at the mapping with the SB model, it is possible to fix a particular angle such that the tunneling amplitude $\\Delta _0$ in Equation (REF ) vanishes $\\Delta _0=0 \\rightarrow \\bar{\\theta }=2\\arctan \\left(\\frac{\\Delta }{A}\\right)~.$ Here, the competition between the two linear dissipative couplings $\\sigma _x$ and $\\sigma _z$ lead to a very peculiar behaviour.", "Interference between longitudinal and transverse noise have been discussed in related context of quantum dissipative systems [40], [54], [55], and are reflected in the following expressions for the incoherent relaxation and dephasing rates $\\Gamma ^{(r)}_{\\bar{\\theta }}&=&0\\nonumber \\\\\\Gamma _{\\bar{\\theta }}&=&\\frac{2\\pi \\alpha }{\\beta }$ Putting these values in the weak damping expressions of the SB (see Section REF ) we finally get $\\langle E_{\\rm QB}(t)\\rangle &=&\\frac{\\Delta }{2}\\left[1-\\frac{\\Delta ^2}{\\Omega ^2}-\\frac{A^2}{\\Omega ^2}e^{-\\Gamma _{\\bar{\\theta }} t}[\\cos (\\Omega t)-\\frac{\\Gamma _{\\bar{\\theta }}}{\\Omega }\\sin (\\Omega t)]\\right]\\\\\\langle E_{\\rm C}(t)\\rangle &=&-\\frac{A^2\\Delta }{2\\Omega ^2}\\left[1- e^{-\\Gamma _{\\bar{\\theta }} t}[\\cos (\\Omega t)-\\frac{\\Gamma _{\\bar{\\theta }}}{\\Omega }\\sin (\\Omega t)]\\right].$ Interestingly, at zero temperature also $\\Gamma _{\\bar{\\theta }}=0$ and the above expressions reduce to the same expression obtained for a closed QB under static driving (see Equation (REF )).", "This peculiar choice of $\\bar{\\theta }$ thus lead to a very strong suppression of the effect caused by dissipation.", "However, to achieve this optimal point a precise control on the various parameters is required and other sources of dissipation (such as due to non linear couplings or $1/f$ noise) can become relevant [40], [41], [48], [54], [64], [65]." ], [ "Effect of dissipation on average energy", "We now discuss the results obtained in the previous section, showing how the effects of dissipation can modify the charging dynamics of a QB, in the weak damping regime at low temperature (setting $\\beta \\Delta =10$ in all the plots).", "In Figure REF we show the time evolution of the average energy stored $\\langle E_{{\\rm QB}}(t)\\rangle $ considering the decoherence coupling in Equation (REF ) (see panels (a) and (b)) and pure dephasing one in Equation (REF ) (see panels (c) and (d)).", "Representative examples of driving amplitude $A=0.5\\Delta $ (Figure REF (a) and (c)) and $A=3\\Delta $ (Figure REF (b) and (d)) have been chosen.", "The closed QB system driven by a static bias (see Equation (REF )) is also reported as a reference limit (see black dashed curves).", "It is evident that, also in presence of dissipation, larger driving amplitude $A$ results in better charging of the QB (compare panels (b) and (d) versus panels (a) and (c)).", "Overall, all curves present a damped oscillatory behaviour, whose amplitude is modulated by the exponential decay dictated by the incoherent relaxation and dephasing rates given in Equations (REF )-(REF ).", "We recall that the rate expressions are different for the two dissipative couplings considered, and this gives rise to completely different relaxation dynamics as shown in the Figure.", "Figure: Time evolution of the average energy stored in the QB 〈E QB 〉\\langle E_{{\\rm QB}}\\rangle .", "Both decoherence coupling with θ=0\\theta =0 (upper panels) and pure dephasing coupling with θ=π\\theta =\\pi (lower panels) are shown as a function of Ωt\\Omega t. Panels (a) and (c) refer to charging amplitude A=0.5ΔA=0.5\\Delta , while panels (b) and (d) correspond to A=3ΔA=3\\Delta .", "Different curves represent different dissipation strength α\\alpha .", "Other parameters are βΔ=10\\beta \\Delta =10, ω c =500Δ\\omega _c=500 \\Delta .The positions of maxima and minima are slightly shifted to higher values of $\\Omega t$ with respect to the undamped case, effect particularly visible in panels (a) and panel (d).", "This can be understood by recalling that the characteristic energy gets renormalized with respect to the bare case $\\Omega $ by the presence of dissipation as reported in Equations (REF )-(REF ) (and is related to the Lamb shift).", "Remarkably, looking at the first maximum (around $\\Omega t\\sim \\pi $ ), an opposite behaviour between decoherence coupling $\\theta =0$ and pure dephasing $\\theta =\\pi $ is observed while increasing the coupling strength $\\alpha $ .", "Indeed, in the former case we observe an increase of the value of the maximum with respect to the closed system reference (black dashed curve).", "Again, this is a consequence of the renormalization of $\\Delta $ in Equation (REF ).", "Notice that this is true also for $A=3\\Delta $ , shown in panel (b), although it is less visible due to the larger value of the driving amplitude which partially mask the effect due to the renormalization of $\\Delta _{{\\rm eff}}$ .", "The opposite behaviour shown in panel (c)-(d) of Figure REF can be explained by recalling the different mapping of the QB parameters in the pure dephasing case, where now the driving amplitude is renormalized to $A_{{\\rm eff}}$ , see Equation (REF ).", "Direct comparison between the two dissipative coupling schemes is presented in Figure REF (a)-(b) for a fixed driving amplitude $A=3\\Delta $ and dissipation strength $\\alpha =0.03$ .", "The plot reports (see blue lines) the average energy stored in the QB as a function of time, showing that in the case $\\theta =0$ it is possible to achieve better charging of the QB (higher values of the maxima).", "Due to the different form of the rates in Equations (REF )-(REF ) decoherence coupling is less affected by dissipation.", "Indeed, for the chosen parameters of Figure REF (a) and (b) the corresponding rates are $\\Gamma ^{(r)}_{0}=0.021\\Delta $ , $\\Gamma _{0}=0.028\\Delta $ and $\\Gamma ^{(r)}_{\\pi }=0.23\\Delta $ , $\\Gamma _{\\pi }=0.12\\Delta $ , respectively.", "This is a general trend, namely pure dephasing coupling is less efficient for charging process.", "Indeed, as discussed in related context of quantum dissipative systems [40], [54], [55], [65], pure dephasing coupling induced dynamics can be explained effectively with a larger value of the dissipative coupling strength $\\alpha $ .", "Figure: Time evolution of the average energy variation (a) and the corresponding power (b) associated to the QB (blue curves) and to the charger (red curves).", "Solid and dashed lines correspond to the case of decoherence (θ=0\\theta =0) and pure dephasing (θ=π\\theta =\\pi ) interaction with the reservoir, respectively.", "Dissipation strength has been fixed to α=0.03\\alpha =0.03 and driving amplitude to A=3ΔA=3\\Delta .", "Other parameters as in Figure .", "In panel (c) pictorial representation of the energy flow during charging and discharging process for the two couplings with the reservoir.This relaxation process leads to a damping of the oscillations (see in Figure REF (a)) resulting in a lower amount of energy that can be stored in the QB (quantified as the difference between a given maximum of $\\langle E_{{\\rm QB}}(t)\\rangle $ and its preceding minimum).", "This is a general trend since dissipation induces relaxation dynamics toward a steady thermal state, whose values in the two discussed cases read $&&\\langle E_{{\\rm QB},\\theta =0}(t\\rightarrow \\infty )\\rangle = \\frac{\\Delta }{2} \\left[1-\\frac{\\Delta _{\\rm eff}}{\\Omega _{0}}\\tanh (\\frac{\\beta \\Omega _{0}}{2})\\right]\\nonumber \\\\&&\\langle E_{{\\rm QB},\\theta =\\pi }(t\\rightarrow \\infty )\\rangle = \\frac{\\Delta }{2} \\left[1-\\frac{\\Delta }{\\Omega _{\\pi }}\\tanh (\\frac{\\beta \\Omega _{\\pi }}{2})\\right]~.$ Being a thermal one, this steady state is equivalent to a passive state for the QB from which it is not possible to extract energy [12], [13].", "In Figure REF (a) it is also reported (see red curves) the variation of energy associated to the charger $\\langle E_{{\\rm C}}(t)\\rangle $ (see Equations (REF )-(REF )).", "Notice that curves associated to the QB and to the charger are not specular (with respect to the $x$ axis), reflecting the fact that a given amount of energy is also dissipated into the reservoir.", "This can be better visualized looking at the corresponding powers $\\langle P_s(t)\\rangle =\\langle \\dot{E}_s(t)\\rangle $ , shown in Figure REF (b).", "There, $\\langle P_s\\rangle >0$ indicates energy absorbed in the $s$ -th channel ($s={\\rm QB}, {\\rm C}, {\\rm RI}$ ), while $\\langle P_s\\rangle <0$ corresponds to energy released to the other subparts.", "The marked differences between the solid blue ($\\theta =0$ ) and dashed red ($\\theta =\\pi $ ) curves underlines the different charging (and discharging) process in the two cases, with a much better performance for the decoherence coupling case.", "Indeed for $\\theta =0$ , the amplitude of first positive maximum $\\langle P_{{\\rm QB}}\\rangle >0 $ corresponds almost to the first negative minimum $\\langle P_{{\\rm C}}\\rangle <0 $ (energy released from the charger to the QB).", "Conversely, for $\\theta =\\pi $ (dashed lines) one has lower values for $\\langle P_{{\\rm QB}}\\rangle $ and its value is also different from the corresponding $-\\langle P_{{\\rm C}}\\rangle $ , indicating that part of the energy is already transfered to the bath.", "Physically, this is due to the different energy flow (during charging and discharging dynamics) associated to the two different dissipative couplings.", "These can be visualized as sketched in Figure REF (c), where red arrows indicate charging process, while blue arrows refer to the reverse process.", "Notice that the difference between the two pictures in Figure REF (c) reflects the different spin operator whose reservoir is directly coupled to (see Equation (REF )).", "To better understand the arrows directions in the different coupling mechanisms, we consider the first charging and the first discharging process and how energy is transferred between the various channels (QB, C, or RI) during these time windows.", "Being $t_1$ and $t_2$ the value of the first maxima and the first minima, respectively, we can define the energy variation in the first charging process as ${\\cal E}_s(t,0)\\equiv \\langle E_s(t)\\rangle -\\langle E_s(0)\\rangle \\quad 0\\le t\\le t_1~,$ and in the first discharging process as ${\\cal E}_s(t,t_1)\\equiv \\langle E_s(t)\\rangle -\\langle E_s(t_1)\\rangle \\quad t_1\\le t\\le t_2~,$ with $s=\\rm {QB},\\rm {C},\\rm {RI}$ .", "These quantities are reported in Figure REF for a representative coupling strength $\\alpha =0.03$ and driving amplitude $A=3\\Delta $ , where upper panels show the case $\\theta =0$ and lower panels the case $\\theta =\\pi $ .", "Figure: Average energy flows in the first charging (panel (a) and (c)) and in the first discharging (panel (b) and (d)) windows.", "Upper panels refer to decoherence coupling (θ=0\\theta =0) and lower panels refer to pure dephasing (θ=π\\theta =\\pi ).", "Different curves indicate the average energy variation in the various channels during the charging/discharging process.", "Dissipation strength is α=0.03\\alpha =0.03 and driving amplitude A=3ΔA=3\\Delta .", "Other parameters as in Figure .Figure REF (a)-(c) consider the energy variations associated to the QB, the charger and the reservoir contributions in the first charging interval $0\\le t\\le t_1$ .", "As one can see looking at Figure REF (a) the QB absorbs energy (positive energy variation) while both the charger and the reservoir release energy (negative energy variation) in this time interval.", "In Figure REF (c), instead, the charger supplies energy (negative energy variation) both to the QB and the reservoir (positive energy variation), i.e.", "a certain amount of energy is already dissipated into the thermal bath, thus reducing the amount of energy stored in the QB.", "These two situations are represented by the red arrows in the sketch of Figure REF (c).", "Figure REF (b) and (d) report the first discharging process associated to decoherence (b) and pure dephasing (d) couplings, consistent with the blue arrows of Figure REF (c).", "Here, in both cases energy is transferred from the QB (negative energy variation) to both the charger and the reservoir (positive energy variation).", "However, the precise amount of energy transferred to different channels is different in the two cases, as one can see looking at the red and black curves in Figure REF (b) and (d).", "This stress once more that not only the value of dissipation coupling strength $\\alpha $ affects the charging dynamics, but also how the reservoir is coupled to the quantum system is crucial to determine both charging and discharging process of a QB.", "All the results confirm that the decoherence coupling has a more efficient charging dynamics with respect to the pure dephasing case.", "Analogous conclusions between the two linear dissipative couplings can be obtained for higher values of dissipation strength.", "Indeed, higher values of dissipation strengths have been also considered in the NIBA framework and reported in .", "Stronger dissipation strength would undermine charging performance of a QB, however the decoherence coupling channel remains the one that less influence QB dynamics.", "Notice also that high temperature regime $\\beta \\Delta <1$ would lead to faster relaxations, and weaker charging performances, due to larger values of the associated rates (see Equations (REF )-(REF ) and Figure REF (b) in  ).", "Before concluding this section, in Figure REF we show a comparison with the case of the optimal choice which lead to a strong suppression of linear dissipation (see Equation (REF )).", "Figure: Time evolution of the average energy stored in the QB.", "The plot shows comparison between decoherence coupling, pure dephasing and the optimal choice θ ¯\\bar{\\theta } for which linear coupling dissipation is strongly suppressed.", "Dissipative coupling strength is fixed to α=0.03\\alpha =0.03, A=3ΔA=3\\Delta and other parameters are the same as in Figure .As discussed in the previous section, by properly tuning the external parameters in presence of both longitudinal and transverse noise (with both decoherence and pure dephasing mechanisms) linear dissipation can be strongly suppressed.", "This results in a very slow relaxation, especially at long times, leading to very stable values of the maximum amount of average energy stored even after many cycles $\\omega t$ .", "However, it is worth to notice that considering the short time dynamics (confining to the first maxima shown in the Figure REF ), again decoherence coupling mechanism lead to better charging, due to the renormalization effect induced by weak dissipation, reminiscent of the Lamb shift phenomena [40], [46], [47].", "Moreover, the choice of the optimal regime requires a very accurate fine tuning of the external parameters." ], [ "Charging stability", "Another important question regards how long a QB can retain a given amount of energy that has been stored during a charging process, in presence of dissipation.", "To answer this question, the following protocol is inspected.", "At $t\\ge 0$ the QB is charged by interacting with the external charger with amplitude $A$ until a time $t_c>0$ .", "Then, the charger is switched off, $A=0$ , for $t>t_c$ .", "Of course, if one considers a closed system, the amount of energy stored in the QB until the time $t_c$ will remain constant at later times.", "However, in a more realistic situation the QB will dissipate part of its energy into the thermal bath.", "It is therefore important to quantify the amount of energy retained in the QB, after the charging field has been switched off.", "To model this protocol, the QB is initially prepared at $t=0$ in a given state described by Equation (REF ), e.g.", "starting from the ground state as considered before.", "The system then evolves under the action of the charger with amplitude $A$ until the time $t_c$ .", "At $t=t_c$ the reduced density matrix of the QB is $\\rho (t_c)=\\frac{1}{2}\\left(\\begin{array}{cc}1+\\langle \\sigma _z(t_c)\\rangle & \\langle \\sigma _x(t_c)\\rangle - i \\langle \\sigma _y(t_c)\\rangle \\\\\\langle \\sigma _x(t_c)\\rangle + i \\langle \\sigma _y(t_c)\\rangle & 1-\\langle \\sigma _z(t_c)\\rangle \\end{array}\\right)~.$ At times $t\\ge t_c$ the amplitude $A$ is switched off and therefore it produces a different dynamics.", "In the weak damping regime, this can be described in terms of an effective initial reduced density matrix $\\rho (t_c)$ .", "Due to the different coupling with the thermal bath for $t>t_c$ the dynamics of the QB will be still dissipative for a decoherence coupling, while the case of pure dephasing results in total decoupling between the QB and the reservoir, once the driving field $A$ is switched off.", "The charging storage will be then affected in a different way.", "Figure REF shows the time evolution of the average energy stored in the QB under this protocol (switching-off $A=0$ at time$t_c=t_1$ ).", "Different curves refer to different coupling strengths $\\alpha $ .", "As one can see looking at the solid curves (corresponding to $\\theta =0$ ), after the first maximum (at $t_c=t_1$ ) the average energy stored in the QB start to decrease.", "As mentioned above, the flat behaviour of the pure dephasing case is due to the effective decoupling of the QB from the bath (resembling the dynamics of a closed system).", "Interestingly, the decoherence coupling, for sufficiently weak dissipation strength, retains larger values of energy stored in the QB with respect to the other case in a wide time window.", "This confirm once more that decoherence coupling represents a good choice for the reservoir-quantum system engineering, resulting in both high values of average energy stored and both in terms of charging stability." ], [ "Conclusions", "In this work we have analized the dynamics of a single cell quantum battery, modelled as a quantum two level system, charged by a classical drive and coupled with an external thermal bath.", "We have focussed on the impact of dissipation on the charging performance of a QB, considering generic linear dissipative couplings with the environment, including both decoherence and pure dephasing mechanisms.", "Exploiting a mapping to the spin-boson model and relying on a systematic perturbative expansion, we have found analytical expressions for the charging dynamics in the weak damping regime.", "Here, charging and discharging dynamics at short times have been studied in details, showing that decoherence coupling between reservoir and QB results in better charging performance with respect to pure dephasing one.", "Indeed, in the former case, the QB achieves higher values of average energy stored, and its value is quite stable also after switching-off the charging protocol for sufficiently weak coupling.", "These findings, although based on a simple QB model, represent important hints for a realistic implementation of a QB in a solid state device, where the unavoidable presence of an external environment has to be properly considered and possibly engineered." ], [ "Acknowledgments", "M.C.", "acknowledges support from the Quant-EraNet project “Supertop\"." ], [ "Details on the unitary rotation", "To obtain Equation (REF ) we perform a unitary rotation $\\mathcal {R}$ on the Hamiltonian (REF ) with $\\mathcal {R}=e^{-i\\phi \\sigma _y}=(\\cos (\\phi ) )\\mathbf {1}-i (\\sin (\\phi )) \\sigma _y =\\left(\\begin{array}{cc}\\cos (\\phi ) & -\\sin (\\phi ) \\\\\\sin (\\phi ) & \\cos (\\phi )\\end{array}\\right).$ Here, $\\phi $ is a phase factor whose value has to be chosen in order to project the interaction part $H_{\\rm I}$ only along the $z$ axis.", "We thus consider the rotated interaction Hamiltonian $\\hspace{-28.45274pt}\\widetilde{H}_{{\\rm I}}= \\mathcal {R} H_{\\rm I} \\mathcal {R}^{-1} = \\frac{1}{2}\\left[\\sin \\bigg (\\frac{\\theta }{2}-2\\phi \\bigg )\\sigma _z+\\cos \\bigg (2\\phi -\\frac{\\theta }{2}\\bigg )\\sigma _x\\right]\\cdot \\sum _{j}\\lambda _j\\left(a^{\\dagger }_j+a_j\\right).$ In order to eliminate the contribution proportional to $\\sigma _x$ , we fix the value $\\phi =(\\theta +\\pi )/4$ , obtaining $\\widetilde{H}_{\\rm I}= -\\frac{1}{2} \\sigma _z\\sum _{j}\\lambda _j\\left(a^{\\dagger }_j+a_j\\right).$ Performing the same unitary rotation on the QB Hamiltonian one has $\\widetilde{H}_{{\\rm QB}}=\\mathcal {R} H_{{\\rm QB}} \\mathcal {R}^{-1} =\\frac{\\Delta }{2}\\left[\\sigma _z\\cos (2\\phi ) +\\sigma _x\\sin (2\\phi )\\right],$ which reduces to $\\widetilde{H}_{\\rm QB}= -\\frac{\\Delta }{2} \\left[\\sigma _z\\sin (\\theta /2) - \\sigma _x\\cos (\\theta /2)\\right]~,$ when $\\phi =(\\theta + \\pi )/4$ .", "Finally, the term $H_{\\rm C}$ related to the charger is transformed into $\\widetilde{H}_{{\\rm C}}=\\mathcal {R}H_{{\\rm C}}\\mathcal {R}^{-1}= \\frac{A}{2}\\left[-\\sigma _z\\sin (2\\phi )+\\sigma _x\\cos (2\\phi )\\right],$ and fixing $\\phi $ it becomes $\\widetilde{H}_{\\rm C}= -\\frac{A}{2}\\left[\\sigma _z\\cos (\\theta /2)+\\sigma _x\\sin (\\theta /2)\\right].", "$ Of course, the term $H_{\\rm R}$ is invariant under the rotation $\\mathcal {R}$ .", "We have therefore obtained Equation (REF ).", "By applying the same rotation $\\mathcal {R}$ on the density matrix of Equation (REF ) it is possible to obtain Equation (REF ).", "Taking advantage of the property $p_R+p_L=1$ we obtain $\\hspace{-65.44142pt}\\rho _{\\rm SB}(0)&\\equiv &\\widetilde{\\rho }(0)=\\mathcal {R}\\rho (0)\\mathcal {R}^{-1}=\\nonumber \\\\\\hspace{-65.44142pt}&=&\\left(\\!\\!\\begin{array}{cc}p_R\\cos (2\\phi ) +\\frac{1-\\cos (2\\phi )}{2}-a\\sin (2\\phi ) & \\frac{p_R-p_L}{2}\\sin (2\\phi )+a\\cos (2\\phi )-ib \\\\\\frac{p_R-p_L}{2}\\sin (2\\phi )+a\\cos (2\\phi )+ib & p_L\\cos (2\\phi ) +\\frac{1-\\cos (2\\phi )}{2}+a\\sin (2\\phi )\\end{array}\\!\\!\\right)$ where we have identified the reduced density matrix of the SB in Equation (REF ) with $\\begin{array}{rcl}\\bar{p}_R&=&-p_R\\sin \\frac{\\theta }{2}+\\frac{1+\\sin \\frac{\\theta }{2}}{2}-a\\cos \\frac{\\theta }{2}\\\\\\\\\\bar{p}_L&=&-p_L\\sin \\frac{\\theta }{2}+\\frac{1+\\sin \\frac{\\theta }{2}}{2}+a\\cos \\frac{\\theta }{2}\\\\\\\\\\bar{a}&=&-a\\sin \\frac{\\theta }{2}+\\frac{p_R-p_L}{2}\\cos \\frac{\\theta }{2}\\\\\\\\\\bar{b}&=&b,\\end{array}$ which coincide with Equations (REF )-() of the main text." ], [ "Dissipative Kernels", "This Appendix details the results introduced in Subsection REF .", "We will derive the general expressions for $\\langle \\sigma _{z/x}(t)\\rangle $ in the SB model, discussing the dissipative kernels $K_{1/2,z}^{(\\pm )}(\\tau )$ and $K_{1/2,x}^{(\\pm )}(\\tau )$ , which are the main building blocks to construct the generalized master equation.", "The useful links introduced in Equations (REF ) and (REF ) will be demonstrated.", "The dynamics of $\\langle \\sigma _{k}(t)\\rangle $ ($k=x,y,z$ ) can be studied evaluating the time evolution of the reduced density matrix, starting from the initial condition at $t=0$ .", "We have indeed $\\langle \\sigma _z(t)\\rangle &=& \\rho _{1,1}(t)-\\rho _{-1,-1}(t),\\nonumber \\\\\\langle \\sigma _x(t)\\rangle &=& \\rho _{1,-1}(t)+\\rho _{-1,1}(t),\\nonumber \\\\\\langle \\sigma _y(t)\\rangle &=& i[\\rho _{1,-1}(t)-\\rho _{-1,1}(t)]$ where $\\rho _{1,1}(t)$ and $\\rho _{-1,-1}(t)$ are the populations, while $\\rho _{1,-1}(t)$ and $\\rho _{-1,1}(t)$ are the so-called coherence terms.", "As shown in details in References [49], [31], [50], [58] the reduced density matrix can be expressed in terms of a real-time double path integral $\\rho _{\\sigma ,\\sigma ^{\\prime }}(t)=\\sum _{\\sigma _0,\\sigma _0^{\\prime }=\\pm 1}\\int {\\cal D}\\sigma \\int {\\cal D}\\sigma ^{\\prime } {\\cal A}[\\sigma ] {\\cal A}^*[\\sigma ^{\\prime }]{\\cal F}[\\sigma ,\\sigma ^{\\prime }]\\rho _{\\sigma _0,\\sigma _0^{\\prime }}(t=0).$ Here, the symbol $\\int {\\cal D}\\sigma \\int {\\cal D}\\sigma ^{\\prime }$ means the summation over all spin paths which for a two-state system are $\\sigma (t^{\\prime })=\\pm 1$ and $\\sigma ^{\\prime }(t^{\\prime })=\\pm 1$ with $0\\le t^{\\prime }\\le t$ .", "The boundary conditions are $\\sigma (t)=\\sigma $ , $\\sigma ^{\\prime }(t)=\\sigma ^{\\prime }$ at final time $t$ and $\\sigma (0)=\\sigma _0$ , $\\sigma ^{\\prime }(0)=\\sigma ^{\\prime }_0$ at inital time.", "The elements of the initial reduced density matrix are $\\rho _{\\sigma _0,\\sigma _0^{\\prime }}(0)$ and correspond to the matrix in Equation (REF ) $\\rho (0)=\\left(\\begin{array}{cc}\\bar{p}_R & \\bar{a}-i \\bar{b} \\\\\\bar{a}+i \\bar{b} & \\bar{p}_L\\end{array}\\right).$ The quantity ${\\cal A}[\\sigma ]$ is the probability amplitude for the free (undamped) two-level system to follow the path $\\sigma (t^{\\prime })$ .", "It consists of a contribution due to the tunneling processes which is $i\\Delta _0/2$ for each jump with spin changes $\\sigma =\\pm 1\\rightarrow \\mp 1$ .", "In addition it receives a contribution from the static bias given by $\\exp \\left\\lbrace {i\\epsilon _0\\int _0^{t}dt^{\\prime }\\sigma (t^{\\prime })}\\right\\rbrace $ .", "The effects of the bath are included in the Feynman-Vernon influence function ${\\cal F}[\\sigma ,\\sigma ^{\\prime }]$ which is obtained after tracing out the thermal reservoir [66] $\\hspace{-71.13188pt}{\\cal F}[\\sigma ,\\sigma ^{\\prime }]=\\exp {\\left\\lbrace -\\frac{1}{4}\\int _0^{t} dt^{\\prime }\\int _0^{t^{\\prime }} dt^{\\prime \\prime }[\\sigma (t^{\\prime })-\\sigma ^{\\prime }(t^{\\prime })][{\\cal L}(t^{\\prime }-t^{\\prime \\prime })\\sigma (t^{\\prime \\prime })- {\\cal L}(t^{\\prime \\prime }-t^{\\prime })\\sigma ^{\\prime }(t^{\\prime \\prime })]\\right\\rbrace }.$ Here, the Kernel ${\\cal L}(t)$ represents the autocorrelation function of the fluctuating force which the reservoir exerts on the two-level system and it has the following spectral representation [31] ${\\cal L}(t)=\\int _0^{\\infty }d\\omega J(\\omega )\\left[\\cos (\\omega t)\\coth (\\beta \\omega /2)-i\\sin (\\omega t)\\right],$ with $J(\\omega )$ the spectral density of the bath given in Equation (REF ).", "Notice that the presence of the bath induces interactions among the paths at different times, with a non-markovian time memory.", "Since the spin paths are piecewise constant with sudden jumps in between it is convenient to perform integration by parts in the influence functional (REF ) introducing the function $\\ddot{Q}(t)={\\cal L}(t)$ .", "Below we quote the result avoiding to explicitly writing the terms containing the boundaries of $\\sigma (t^{\\prime })$ and $\\sigma ^{\\prime }(t^{\\prime })$ in $t^{\\prime }=t,0$ $\\hspace{-28.45274pt}{\\cal F}[\\sigma ,\\sigma ^{\\prime }]&=&\\exp \\Big \\lbrace \\frac{1}{4}\\int _0^{t} dt^{\\prime }\\int _0^{t^{\\prime }} dt^{\\prime \\prime }[\\dot{\\sigma }(t^{\\prime })-\\dot{\\sigma }^{\\prime }(t^{\\prime })]Q^{\\prime }(t^{\\prime }-t^{\\prime \\prime })[\\dot{\\sigma }(t^{\\prime \\prime })-\\dot{\\sigma }^{\\prime }(t^{\\prime \\prime })]\\nonumber \\\\&+&i[\\dot{\\sigma }(t^{\\prime })-\\dot{\\sigma }^{\\prime }(t^{\\prime })]Q^{\\prime \\prime }(t^{\\prime }-t^{\\prime \\prime })[\\dot{\\sigma }(t^{\\prime \\prime })+\\dot{\\sigma }^{\\prime }(t^{\\prime \\prime })]\\Big \\rbrace ,$ where $\\hspace{-56.9055pt}Q(t)=Q^{\\prime }(t)+iQ^{\\prime \\prime }(t)=\\int _0^{\\infty }d\\omega \\frac{J(\\omega )}{\\omega ^2}\\Big [(1-\\cos (\\omega t))\\coth (\\beta \\omega /2)+i\\sin (\\omega t)\\Big ].$ This dissipative correlator can be evaluated for the Ohmic spectral density (REF ) and it reads ($\\beta \\omega _c\\gg 1$ ) $Q^{\\prime }(t)&=&2\\alpha \\ln \\left[\\frac{\\beta \\sqrt{1+\\omega _{{ c}}^2 t^2}}{\\pi t}\\sinh \\left(\\frac{\\pi t}{\\beta }\\right)\\right]\\\\Q^{\\prime \\prime }(t)&=&2\\alpha \\arctan (\\omega _{{ c}}t).$ Considering the scaling limit of large cut-off $\\omega _c$ , $Q(t)$ reduces to $Q^{\\prime }(t)=2\\alpha \\ln \\left[\\frac{\\beta \\omega _{{ c}}}{\\pi }\\sinh \\left(\\frac{\\pi |t|}{\\beta }\\right)\\right],\\qquad Q^{\\prime \\prime }(t)=\\pi \\alpha {\\rm sgn}(t).$ In the following we will consider this regime.", "The time evolution of the density matrix will be now written in terms of the summations over all possible paths which consists in an expansion in the number of tunneling transitions $\\Delta _0$ , weighted by the different factors discussed above.", "The details of this procedure are discussed in several references, see e.g.", "[31], [50], [58].", "For this reason, we decided to write and comment the series expansion for $\\langle \\sigma _z(t)\\rangle $ , which will allow us to demonstrate the important relation introduced in Equation (REF ).", "We choose as initial density matrix the one given in Equation (REF ).", "Inserting the different spin paths and considering the diagonal elements of $\\rho (t)$ one obtains the following decomposition for $\\langle \\sigma _z(t)\\rangle $ $\\langle \\sigma _z(t)\\rangle =\\langle \\sigma _{z,0}(t)\\rangle +\\langle \\sigma _{z,\\bar{a}}(t)\\rangle +\\langle \\sigma _{z,\\bar{b}}(t)\\rangle ,$ where [50] $\\langle \\sigma _{z,0}(t)\\rangle &=& (\\bar{p}_R-\\bar{p}_L)+\\sum _{n=1}^{\\infty }\\Delta _0^{2n}\\left(\\frac{-1}{2}\\right)^{n}\\int _0^t dt_{2n}\\cdots \\int _0^{t_2} dt_{1}\\cdot \\nonumber \\\\&&\\cdot \\sum _{\\xi _1\\cdots \\xi _n=\\pm 1}\\left[(\\bar{p}_R-\\bar{p}_L){F}_{n,1}^{(+)}C_{n,1}^{(+)}-{F}_{n,1}^{(-)}C_{n,1}^{(-)}\\right]$ and [58] $\\hspace{-56.9055pt}\\langle \\sigma _{z,\\bar{a}}(t)\\rangle &=&-2{\\bar{a}} \\sum _{n=1}^{\\infty }\\Delta _0^{2n-1}\\left(\\frac{-1}{2}\\right)^{n}\\int _0^t dt_{2n-1}\\cdots \\int _0^{t_2} dt_{1}\\sum _{\\xi _1\\cdots \\xi _n=\\pm 1}\\xi _1{F}_{n,2}C_{n,2}^{(-)}\\\\\\hspace{-56.9055pt}\\langle \\sigma _{z,\\bar{b}}(t)\\rangle &=& 2{\\bar{b}} \\sum _{n=1}^{\\infty }\\Delta _0^{2n-1}\\left(\\frac{-1}{2}\\right)^{n}\\int _0^t dt_{2n-1}\\cdots \\int _0^{t_2} dt_{1}\\sum _{\\xi _1\\cdots \\xi _n=\\pm 1}{F}_{n,2}C_{n,2}^{(+)}.$ Notice that the initial condition is $\\langle \\sigma _{z}(t=0)\\rangle =\\langle \\sigma _{z,0}(t=0)\\rangle =\\bar{p}_R-\\bar{p}_L$ .", "The integrations are over the flips times, related to the different transitions of a given spin path.", "These times are time-ordered with $0\\le t_1\\le \\cdots \\le t_{2n}\\le t$ .", "Notice that in $\\langle \\sigma _{z,\\bar{a}}(t)\\rangle $ and $\\langle \\sigma _{z,\\bar{b}}(t)\\rangle $ there is one integration left with $t_0=0$ .", "The sum $\\sum _{\\xi _j=\\pm 1}$ is over the integer variables $\\xi _1\\cdots \\xi _{n}$ which can assume, each, the values $\\pm 1$ .", "Let us now discuss the different terms inside the integrals.", "The factors $C_{n,1/2}^{(\\pm )}$ are the even and odd phase associated to the static bias term $\\epsilon _0$ $\\hspace{-56.9055pt}C_{n,1}^{(+)}&=&\\cos [\\epsilon _0\\sum _{j=1}^n\\xi _j(t_{2j}-t_{2j-1})]\\qquad C_{n,1}^{(-)}=\\sin [\\epsilon _0\\sum _{j=1}^n\\xi _j(t_{2j}-t_{2j-1})],\\\\\\hspace{-56.9055pt}C_{n,2}^{(+)}&=&\\cos [\\epsilon _0\\sum _{j=1}^n\\xi _j(t_{2j-1}-t_{2j-2})]\\qquad C_{n,2}^{(-)}=\\sin [\\epsilon _0\\sum _{j=1}^n\\xi _j(t_{2j-1}-t_{2j-2})].$ The influence of the bath is included in the functions ${F}_{n,1}^{(\\pm )}$ and ${F}_{n,2}$ , which correlate the spin's transitions at different times.", "In the scaling limit they are expressed in compact notations, using the expression (REF ) for the interaction $Q(t)$ .", "We have $\\hspace{-56.9055pt}{F}_{n,1}^{(+)}(t_{2n},\\cdots , t_1)&=&G_{n,1}\\cdot \\left[\\cos (\\pi \\alpha )\\right]^{n},\\qquad {F}_{n,1}^{(-)}(t_{2n},\\cdots , t_1)=\\xi _1\\tan (\\pi \\alpha ){F}_{n,1}^{(+)}\\\\\\hspace{-56.9055pt}{F}_{n,2}(t_{2n-1},\\cdots , t_0)&=&G_{n,2}\\cdot \\left[\\cos (\\pi \\alpha )\\right]^{n-1},$ where $G_{n,1}(t_{2n},\\cdots , t_1)&=&\\exp \\left( -\\sum _{j=1}^{n}Q^{\\prime }_{{2j},{2j-1}}-\\sum _{j=2}^{n}\\sum _{k=1}^{j-1}\\xi _j\\xi _k \\Lambda ^{(1)}_{j,k}\\right),\\\\G_{n,2}(t_{2n-1},\\cdots , t_0)&=&\\exp \\left(-\\sum _{j=0}^{n-1}Q^{\\prime }_{{2j+1},{2j}}-\\sum _{j=2}^{n}\\sum _{k=1}^{j-1}\\xi _j\\xi _k \\Lambda ^{(2)}_{j,k}\\right).$ Here, we defined $Q^{\\prime }_{{j},{k}}=Q^{\\prime }({t_{j}-t_{k}})$ and $\\Lambda ^{(1)}_{j,k}=Q^{\\prime }_{{2j},{2k-1}}+Q^{\\prime }_{{2j-1},{2k}}-Q^{\\prime }_{{2j},{2k}}-Q^{\\prime }_{{2j-1},{2k-1}}\\nonumber \\\\\\Lambda ^{(2)}_{j,k}=Q^{\\prime }_{{2j-1},{2k-2}}+Q^{\\prime }_{{2j-2},{2k-1}}-Q^{\\prime }_{{2j-1},{2k-1}}-Q^{\\prime }_{{2j-2},{2k-2}}.$ We are now in the position to demonstrate the important links quoted in Equations (REF ) between $\\langle \\sigma _{z,0}(t)\\rangle $ and $\\langle \\sigma _{z,{\\bar{a}}/{\\bar{b}}}(t)\\rangle $ .", "For this, we write the even an odd part of $\\langle \\sigma _{z,0}(t)\\rangle $ with respect to the bias $\\epsilon _0$ , $\\langle \\sigma _{z,0}(t)\\rangle =\\langle \\sigma _{z,0}^{(+)}(t)\\rangle +\\langle \\sigma _{z,0}^{(-)}(t)\\rangle $ , whose series is implicitly defined in Equation (REF ) $\\hspace{-65.44142pt}\\langle \\sigma _{z,0}^{(+)}(t)\\rangle &=&\\!", "(\\bar{p}_R-\\bar{p}_L)\\!\\left[1+\\sum _{n=1}^{\\infty }\\Delta _0^{2n}\\left(\\frac{-1}{2}\\right)^{n}\\int _0^t dt_{2n}\\cdots \\int _0^{t_2} dt_{1}\\sum _{\\xi _1\\cdots \\xi _n=\\pm 1}{F}_{n,1}^{(+)}C_{n,1}^{(+)}\\right]\\\\\\hspace{-65.44142pt}\\langle \\sigma _{z,0}^{(-)}(t)\\rangle &=&-\\sum _{n=1}^{\\infty }\\Delta _0^{2n}\\left(\\frac{-1}{2}\\right)^{n}\\int _0^t dt_{2n}\\cdots \\int _0^{t_2} dt_{1}\\sum _{\\xi _1\\cdots \\xi _n=\\pm 1}{F}_{n,1}^{(-)}C_{n,1}^{(-)}.$ The time derivative of these two parts is easily performed by fixing the last integration time $t_{2n}=t$ .", "By comparing these expressions to the ones written in Equations (REF ) and () for $\\langle \\sigma _{z,{\\bar{a}}/{\\bar{b}}}(t)\\rangle $ we identify, after a proper change of integration variables, the links quoted in the main text $\\langle \\sigma _{z,\\bar{a}}(t)\\rangle &=& \\frac{2\\bar{a}}{\\Delta _0\\sin (\\pi \\alpha )}\\frac{d}{dt}\\langle {\\sigma }^{(-)}_{z,0}(t)\\rangle \\nonumber \\\\\\langle \\sigma _{z,\\bar{b}}(t)\\rangle &=& \\frac{2\\bar{b}}{\\Delta _0\\cos (\\pi \\alpha )(\\bar{p}_R-\\bar{p}_L)}\\frac{d}{dt}\\langle {\\sigma ^{(+)}_{z,0}(t)}\\rangle .$ Notice that the first Equation is well defined for $\\alpha \\rightarrow 0$ since the quantity $\\frac{d}{dt}\\langle {\\sigma ^{(-)}_{z,0}(t)}\\rangle $ is proportional to $\\sin (\\pi \\alpha )$ .", "We now comment on the possibility to write an exact master equation for $\\langle \\sigma _{z}(t)\\rangle $ starting from its series expression obtained above.", "Although the series cannot be in general solved exactly, it is always possible to link the time derivative $d\\langle \\sigma _{z}(t)\\rangle /dt$ with $\\langle \\sigma _{z}(t)\\rangle $ itself.", "This is achieved via direct comparison, term by term, of their series expansions.", "Notice that $d\\langle \\sigma _{z}(t)\\rangle /dt$ has also a formal series expansions obtained directly by the time derivation of $\\langle \\sigma _{z}(t)\\rangle $ .", "This procedure allows us to write, on a general ground, the following generalized master equation [50], [58], [31] $\\hspace{-65.44142pt}\\frac{d\\langle \\sigma _{z}(t)\\rangle }{dt}= \\int _0^t dt^{\\prime } [K^{(-)}_{1,z} (t-t^{\\prime }) - K^{(+)}_{1,z}(t-t^{\\prime })\\langle \\sigma _{z}(t^{\\prime })\\rangle ] +2\\bar{a} K^{(-)}_{2,z}(t)-2\\bar{b} K^{(+)}_{2,z}(t).$ Here, the kernels $K^{(\\pm )}_{1/2,z}(t-t^{\\prime })$ are constructed by matching the iterative solution represented in Equation (REF ) with the exact formal series for $\\langle \\sigma _{z}(t)\\rangle $ in Equations (REF )-().", "They are expressed as series and they represent the irreducible components of the exact series for $\\langle \\sigma _{z}(t)\\rangle $ [31].", "Due to the links demonstrated in Equations (REF ) it directly follows that the kernels associated to the off diagonal terms of the initial density matrix are related to the kernel $K^{(\\pm )}_{1,z} (t)$ as $K_{2,z}^{(+)}(t)&=& \\frac{1}{\\Delta _0\\cos (\\pi \\alpha )}K_{1,z}^{(+)}(t)\\nonumber \\\\K_{2,z}^{(-)}(t)&=& \\frac{1}{\\Delta _0\\sin (\\pi \\alpha )}K_{1,z}^{(-)}(t).$ For this reason we quote below only the series expansions obtained for $K^{(\\pm )}_{1,z} (t)$ .", "We have ($t>t^{\\prime }$ ) [50], [31] $\\hspace{-65.44142pt}K^{(\\pm )}_{1,z}(t-t^{\\prime })= {\\cal K}^{(\\pm )}_{1,z}(t-t^{\\prime })-\\sum _{n=2}^{\\infty }\\Delta _0^{2n}\\left(\\frac{-1}{2}\\right)^{n}\\int _{t^{\\prime }}^t dt_{2n-1}\\cdots \\int _{t^{\\prime }}^{t_3} dt_{2}\\sum _{\\xi _1\\cdots \\xi _n=\\pm 1} {\\widetilde{\\cal F}}_{n,1}^{(\\pm )}.$ Here, the new functionals ${\\widetilde{\\cal F}}_{n,1}^{(\\pm )}(t_{2n},t_{2n-1},\\cdots t_2,t_1)$ , depend on $2n$ -times with the first and the last blocked at $t_1=t^{\\prime }$ $t_{2n}=t$ .", "They correspond to the irreducible influence functionals with $2n$ number of transitions, obtained from ${F}_{n,1}^{(\\pm )}C_{n,1}^{(\\pm )}$ by subtracting all possibilities of factorizing the influence functions into independent clusters (reducible components).", "Their expression is discussed in [31], for a path with $2n$ transitions they are $\\hspace{-65.44142pt}{\\widetilde{\\cal F}}_{n,1}^{(+)}&=&{F}_{n,1}^{(+)}C_{n,1}^{(+)}- \\sum _{j=2}^{n}(-1)^j\\!\\!\\sum _{m_1,\\cdots ,m_j}\\!\\!", "{F}_{m_1,1}^{(+)}C_{m_1,1}^{(+)}\\cdot {F}_{m_2,1}^{(+)}C_{m_2,1}^{(+)}\\cdots {F}_{m_j,1}^{(+)}C_{m_j,1}^{(+)}\\cdot \\delta _{m_1+\\cdots +m_j,n}\\nonumber \\\\\\hspace{-65.44142pt}{\\widetilde{\\cal F}}_{n,1}^{(-)}&=&{F}_{n,1}^{(-)}C_{n,1}^{(-)}- \\sum _{j=2}^{n}(-1)^j\\!\\!\\sum _{m_1,\\cdots ,m_j}\\!\\!", "{F}_{m_1,1}^{(+)}C_{m_1,1}^{(+)}\\cdot {F}_{m_2,1}^{(+)}C_{m_2,1}^{(+)}\\cdots {F}_{m_j,1}^{(-)}C_{m_j,1}^{(-)}\\cdot \\delta _{m_1+\\cdots +m_j,n}.\\nonumber \\\\\\hspace{-65.44142pt}&&$ The inner sum is over positive integers $m_k$ , and in the subtracting part the bath correlations are only inside each individual term ${F}_{m_k,1}^{(+)}C_{m_k,1}^{(+)}$ without correlations among them.", "Any terms has the time variables written with time growing from right to left.", "The first term ${\\cal K}^{(\\pm )}_{1,z}(t-t^{\\prime })$ in (REF ) are the contributions at order $\\Delta _0^2$ (the series starts with $\\Delta _0^4$ ) and are without internal integrations.", "For these terms the irreducible functional corresponds directly to the functional itself ${\\widetilde{\\cal F}}_{n=1,1}^{(\\pm )}={F}_{n=1,1}^{(\\pm )}C_{n=1,1}^{(\\pm )}$ .", "Their explicit expressions are ${\\cal K}^{(+)}_{1,z} (t-t^{\\prime })&=& \\Delta ^2_0 e^{-Q^{\\prime }(t-t^{\\prime })}\\cos (\\pi \\alpha )\\cos [\\epsilon _0(t-t^{\\prime })]\\\\{\\cal K}^{(-)}_{1,z} (t-t^{\\prime })&=& \\Delta ^2_0 e^{-Q^{\\prime }(t-t^{\\prime })}\\sin (\\pi \\alpha ) \\sin [\\epsilon _0(t-t^{\\prime })].$ We conclude by commenting on the properties of $\\langle \\sigma _{x}(t)\\rangle $ .", "The $x$ -component has also a series expansion in the tunneling amplitude $\\Delta _0$ , obtained similarly to what already discussed for the $z$ -component, for this reason we will omit the details quoting directly the most important results.", "First, $\\langle \\sigma _{x}(t)\\rangle $ can be decomposed as $\\langle \\sigma _x(t)\\rangle =\\langle \\sigma _{x,0}(t)\\rangle +\\langle \\sigma _{x,\\bar{a}}(t)\\rangle +\\langle \\sigma _{x,\\bar{b}}(t)\\rangle ,$ where $\\langle \\sigma _{x,0}(t)\\rangle $ corresponds to an intial density matrix without $\\bar{a}$ and $\\bar{b}$ while the remaining parts are due to the coefficients $\\bar{a}$ and $\\bar{b}$ .", "Similarly to the $z$ -component, for a large cut-off frequency, these two last terms are linked to the even and odd part, with respect to the bias, of $\\langle \\sigma _{x,0}(t)\\rangle $ $\\langle \\sigma _{x,\\bar{a}}(t)\\rangle &=& \\frac{2\\bar{a}}{\\Delta _0\\sin (\\pi \\alpha )}\\frac{d}{dt}\\langle {\\sigma ^{(+)}_{x,0}(t)}\\rangle \\nonumber \\\\\\langle \\sigma _{x,\\bar{b}}(t)\\rangle &=& \\frac{2\\bar{b}}{\\Delta _0\\cos (\\pi \\alpha )(\\bar{p}_R-\\bar{p}_L)}\\frac{d}{dt}\\langle {\\sigma ^{(-)}_{x,0}(t)}\\rangle .$ We then quote only the series expansion of the term $\\langle \\sigma _{x,0}(t)\\rangle $ .", "We have [31] $\\hspace{-28.45274pt}\\langle \\sigma _{x}(t)\\rangle &=&\\langle {\\sigma ^{(+)}_{x,0}(t)}\\rangle +\\langle {\\sigma ^{(-)}_{x,0}(t)}\\rangle =\\sum _{n=1}^{\\infty }\\Delta _0^{2n-1}(-1)^{n-1}2^{-n}\\int _0^t dt_{2n-1}\\cdots \\int _0^{t_2} dt_{1}\\cdot \\nonumber \\\\\\hspace{-28.45274pt}&&\\cdot \\sum _{\\xi _j=\\pm 1}\\xi _n\\left[{F}_{n,1}^{(-)}C_{n,1}^{(+)}+(\\bar{p}_R-\\bar{p}_L){F}_{n,1}^{(+)}C_{n,1}^{(-)}\\right].$ Comparing now term by term the two series expressions for $\\langle \\sigma _{x}(t)\\rangle $ and $\\langle \\sigma _{z}(t)\\rangle $ it is possible to connect them via an exact integral relation $\\hspace{-28.45274pt}\\langle \\sigma _{x}(t)\\rangle = \\int _0^t dt^{\\prime } [K^{(+)}_{1,x}(t-t^{\\prime }) + K^{(-)}_{1,x}(t-t^{\\prime })\\langle \\sigma _{z}(t^{\\prime })\\rangle ] +2\\bar{a} K^{(+)}_{2,x}(t)+2\\bar{b} K^{(-)}_{2,x}(t).$ Due to the relations (REF ), the kernels associated to the $\\bar{a}$ and $\\bar{b}$ are linked to $K^{(\\pm )}_{1,x} (t)$ as $K_{2,x}^{(+)}(t)&=& \\frac{1}{\\Delta _0\\sin (\\pi \\alpha )}K_{1,x}^{(+)}(t)\\nonumber \\\\K_{2,x}^{(-)}(t)&=& \\frac{1}{\\Delta _0\\cos (\\pi \\alpha )}K_{1,x}^{(-)}(t).$ The kernels $K^{(\\pm )}_{1,x} (t)$ are again obtained by matching the integral relation (REF ) with the series expansions, the explicit forms are then in terms of irreducible functionals $\\hspace{-65.44142pt}K^{(\\pm )}_{1,x}(t-t^{\\prime })= {\\cal K}^{(\\pm )}_{1,x}(t-t^{\\prime })-\\sum _{n=2}^{\\infty }\\Delta _0^{2n-1}\\left(\\frac{-1}{2}\\right)^{n}\\int _{t^{\\prime }}^t dt_{2n-1}\\cdots \\int _{t^{\\prime }}^{t_3} dt_{2}\\sum _{\\xi _1\\cdots \\xi _n=\\pm 1} {\\xi _n\\widetilde{\\cal L}}_{n,1}^{(\\pm )},$ with $\\hspace{-65.44142pt}{\\widetilde{\\cal L}}_{n,1}^{(+)}&=&{F}_{n,1}^{(-)}C_{n,1}^{(+)} - \\sum _{j=2}^{n}(-1)^j\\!\\!\\sum _{m_1,\\cdots ,m_j}\\!\\!", "{F}_{m_1,1}^{(+)}C_{m_1,1}^{(+)}\\cdot {F}_{m_2,1}^{(+)}C_{m_2,1}^{(+)}\\cdots {F}_{m_j,1}^{(-)}C_{m_j,1}^{(+)}\\cdot \\delta _{m_1+\\cdots +m_j,n}\\nonumber \\\\\\hspace{-65.44142pt}{\\widetilde{\\cal L}}_{n,1}^{(-)}&=&{F}_{n,1}^{(+)}C_{n,1}^{(-)}- \\sum _{j=2}^{n}(-1)^j\\!\\!\\sum _{m_1,\\cdots ,m_j}\\!\\!", "{F}_{m_1,1}^{(+)}C_{m_1,1}^{(+)}\\cdot {F}_{m_2,1}^{(+)}C_{m_2,1}^{(+)}\\cdots {F}_{m_j,1}^{(+)}C_{m_j,1}^{(-)}\\cdot \\delta _{m_1+\\cdots +m_j,n}.\\nonumber \\\\\\hspace{-65.44142pt}&&$ All the symbols were already explained discussing the $z-$ component case.", "The first terms ${\\cal K}^{(\\pm )}_{1,x}(t-t^{\\prime })$ , in () are the contributions at order $\\Delta _0$ .", "They reads ${\\cal K}^{(+)}_{1,x} (t-t^{\\prime })&=& \\Delta _0 e^{-Q^{\\prime }(t-t^{\\prime })}\\sin (\\pi \\alpha ) \\cos [\\epsilon _0(t-t^{\\prime })]\\\\{\\cal K}^{(-)}_{1,x} (t-t^{\\prime })&=& \\Delta _0 e^{-Q^{\\prime }(t-t^{\\prime })}\\cos (\\pi \\alpha ) \\sin [\\epsilon _0(t-t^{\\prime })].$ Before closing this part, we comment on the so-called non-interacting blip approximation (NIBA) [31], [49], [61].", "As already mentioned in the main text, this corresponds to the lowest order expansion of the kernels in the tunneling amplitude $\\Delta _0$ .", "Therefore the kernels are given by Equations (REF )-() and (REF )-() for the $z$ - and $x$ components, respectively.", "For sake of completeness, we report here the closed expressions at order $\\Delta _0^2$ for ${\\cal K}_{2,z/x}^{(\\pm )}$ that can be derived by exploiting the general links explained above.", "We have ${\\cal K}_{2,z}^{(+)}(t)&=&\\Delta _0 e^{-Q^{\\prime }(t)} \\cos (\\epsilon _0 t)\\\\{\\cal K}^{(-)}_{2,z}(t)&=& \\Delta _0 e^{-Q^{\\prime }(t)} \\sin (\\epsilon _0 t)$ and ${\\cal K}^{(+)}_{2,x} (t)&=& e^{-Q^{\\prime }(t)} \\cos (\\epsilon _0 t)\\\\{\\cal K}_{2,x}^{(-)} (t)&=&e^{-Q^{\\prime }(t)} \\sin (\\epsilon _0 t)~.$ The integro-differential equations posed by (REF )-(REF ) can be thus solved numerically using the above NIBA Kernels.", "We close by recalling that the NIBA approach is non perturbative in the dissipation strength $\\alpha $ and it is a reliable approximation scheme for sufficiently short times [31], [62].", "At long times, it is strictly consistent for temperature higher or of the order of the level splitting $\\Omega _{{\\rm SB}} =\\sqrt{\\Delta _0^2 +\\epsilon _0^2}$ ." ], [ "Charging dynamics in the NIBA framework", "In this Appendix we present results for the charging dynamics considering higher values of the dissipation strength $\\alpha $ .", "These have been obtained within the NIBA framework, by solving the integro-differential equations (REF )-(REF ), using the closed expressions for the dissipative kernels at order $\\Delta _0^2$ reported in Equations (REF )-(), (REF )-() and (REF )-().", "The time-evolution of the energy variation of the QB (for both linear dissipative couplings) is then obtained by resorting to the mapping discussed in the main text.", "In Figure REF we show results obtained in the NIBA framework for stronger dissipative coupling $\\alpha $ with respect to the ones discussed in the main text.", "As a representative example, we have fixed the driving amplitude at $A=3\\Delta $ and we have reported the case of decoherence coupling with the bath in the low temperature regime $\\beta \\Delta =10$ .", "As expected, stronger dissipation strengths induce faster relaxation dynamics, resulting in weaker charging performances (see panel (a)).", "As discussed in the main text, dissipation effects are even stronger in the case of pure dephasing (not shown), whereas in the case shown in Figure REF at short times still a sizeable amount of energy can be stored in the QB for $\\alpha =0.1$ .", "Figure REF (b) shows comparison between the NIBA predictions and exact expressions evaluated in the weak damping regime for $\\alpha =0.01$ , demonstrating that NIBA is still a very good approximation at very short times, even in the low temperature regime considered." ] ]

[ [ "Spintronics meets nonadiabatic molecular dynamics: Geometric spin torque\n and damping on noncollinear classical magnetism due to electronic open\n quantum system" ], [ "Abstract We analyze a quantum-classical hybrid system of steadily precessing slow classical localized magnetic moments, forming a head-to-head domain wall, embedded into an open quantum system of fast nonequilibrium electrons.", "The electrons reside within a metallic wire connected to macroscopic reservoirs.", "The model captures the essence of dynamical noncollinear and noncoplanar magnetic textures in spintronics, while making it possible to obtain the exact time-dependent nonequilibrium density matrix of electronic system and split it into four contributions.", "The Fermi surface contribution generates dissipative (or damping-like in spintronics terminology) spin torque on the moments, and one of the two Fermi sea contributions generates geometric torque dominating in the adiabatic regime.", "When the coupling to the reservoirs is reduced, the geometric torque is the only nonzero contribution.", "Locally it has both nondissipative (or field-like in spintronics terminology) and damping-like components, but with the sum of latter being zero, which act as the counterparts of geometric magnetism force and electronic friction in nonadiabatic molecular dynamics.", "Such current-independent geometric torque is absent from widely used micromagnetics or atomistic spin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation, where previous analysis of Fermi surface-type torque has severely underestimated its magnitude." ], [ "=1 Spintronics meets nonadiabatic molecular dynamics: Geometric spin torque and damping on noncollinear classical magnetism due to electronic open quantum system Utkarsh Bajpai Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA We analyze a quantum-classical hybrid system of steadily precessing slow classical localized magnetic moments, forming a head-to-head domain wall, embedded into an open quantum system of fast nonequilibrium electrons.", "The electrons reside within a metallic wire connected to macroscopic reservoirs.", "The model captures the essence of dynamical noncollinear and noncoplanar magnetic textures in spintronics, while making it possible to obtain the exact time-dependent nonequilibrium density matrix of electronic system and split it into four contributions.", "The Fermi surface contribution generates dissipative (or damping-like in spintronics terminology) spin torque on the moments, and one of the two Fermi sea contributions generates geometric torque dominating in the adiabatic regime.", "When the coupling to the reservoirs is reduced, the geometric torque is the only nonzero contribution.", "Locally it has both nondissipative (or field-like in spintronics terminology) and damping-like components, but with the sum of latter being zero, which act as the counterparts of geometric magnetism force and electronic friction in nonadiabatic molecular dynamics.", "Such current-independent geometric torque is absent from widely used micromagnetics or atomistic spin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation, where previous analysis of Fermi surface-type torque has severely underestimated its magnitude.", "One of the most fruitful applications of geometric (or Berry) phase [1] concepts is encountered in quantum-classical hybrid systems where separation of time scales makes it possible to consider fast quantum degrees of freedom interacting with the slow classical ones [2], [3].", "The amply studied example of this kind are fast electrons interacting [4], [5] with slow nuclei in molecular dynamics (MD) [6], [7], [8], [9] problems of physics, chemistry and biology.", "The parameters driving adiabatic evolution of quantum subsystem, with characteristic frequency smaller that its level spacing, are nuclear coordinates elevated to the status of dynamical variables.", "The electronic system then develops geometric phase in states evolving out of an instantaneous energy eigenstate, while also acquiring shifts in the energy levels.", "Conversely, nuclei experience forces due to back-action from electrons.", "The simplest force is the adiabatic Born-Oppenheimer (BO) force [4], [5] which depends only on the coordinates of the nuclei, and it is associated with electronic adiabatic potential surfaces [6], [7].", "Even small violation of BO approximation leads to additional forces—the first nonadiabatic correction generates forces linear in the velocity of the nuclei, and being Lorentz-like they are dubbed [2], [10] “geometric magnetism.” The “magnetism” is not a not a real magnetic field, but an emergent geometrical property of the Hilbert space [11], and akin to the true Lorentz force, the emergent geometric force is nondissipative.", "Figure: (a) Schematic view of a two-terminal system where a single classicalLMM, precessing steadily with frequency ω\\omega and cone angle θ\\theta , interacts with an open quantum system of conduction electron spins.", "The electrons hop along 1D infinite tight-binding chain which terminates into the left and right macroscopic reservoirs kept at the same chemical potential μ\\mu .", "Panel (c) depicts 7 LMMs, M 1 {M}_1–M 7 {M}_7 forming a head-to-head Bloch domain wall, which precess with the same frequency but are noncollinear and noncoplanar.", "Both (a) and (c) can be mapped in the rotating frame to a time-independent four-terminal system in (b) with an effective bias voltage ℏω/e\\hbar \\omega /e between the left or right pair of leads.Additional forces appear upon making the quantum system open by coupling it to a thermal bath [10], [12] (usually modeled as an infinite set of harmonic oscillators [13]) or to macroscopic reservoirs of particles [14].", "In the latter case, one can also introduce chemical potential difference between the reservoirs to drive particle flux (i.e., current) through the quantum system which is, thereby, pushed out of equilibrium [15], [14], [16], [18], [19].", "In both equilibrium and nonequilibrium cases, the energy spectrum of the quantum system is transformed into a continuous one, and frictional forces [10], [15], [14], [16], [8], [9], [17], [18], [19] linear in the velocity of the nuclei become possible.", "Also, due to continuous spectrum, adiabaticity criterion has to be replaced by a different one [14].", "Stochastic forces also appear, both in equilibrium and in nonequilibrium, where in the former case [10], [12] they are due to fluctuations at finite temperature while in the latter case they include additional contribution from nonequilibrium noise [15], [14], [16].", "Finally, specific to nonequilibrium is the emergence of nonconservative forces [15], [14], [16], [18], [19].", "The derivation of all of these forces is achieved by computing nonadiabatic corrections to the density matrix (DM) [10], [12], [15], [14], [16], [18], [19].", "This yields a non-Markovian stochastic Langevin equation, with nonlocal-in-time kernel describing memory effects [20], as the most general [16], [19] equation for nuclei in nonadiabatic MD.", "The analogous problem exists in spintronics, where the fast quantum system is comprised of conduction electron spins and slow classical system is comprised of localized-on-atoms spins and associated localized magnetic moments (LMMs) described by unit vectors $\\mathbf {M}_i(t)$ .", "The dynamics of LMMs is accounted by the Landau-Lifshitz-Gilbert (LLG) type of equation [21] $\\frac{\\partial \\mathbf {M}_i}{dt} & = & -g \\mathbf {M} \\times \\mathbf {B}^\\mathrm {eff}_i + \\lambda \\mathbf {M}_i \\times \\frac{\\partial \\mathbf {M}_i}{\\partial t} \\nonumber \\\\&& + \\frac{g}{\\mu _M}\\left( \\mathbf {T}_i \\left[I^{S_\\alpha }_\\mathrm {ext}\\right] + \\mathbf {T}_i\\left[\\partial \\mathbf {M}_i/\\partial t \\right] \\right).$ This includes phenomenological Gilbert damping, whose parameter $\\lambda $ can be measured or independently calculated [22] by using electronic Hamiltonian with spin-orbit coupling and impurities.", "It can also include Slonczewski spin-transfer torque (STT) term $\\mathbf {T}_i \\left[ I^{S_\\alpha }_\\mathrm {ext} \\right]$ due to externally supplied spin current $I^{S_\\alpha }_\\mathrm {ext}$ .", "The STT is a phenomenon [28] in which spin angular momentum of conduction electrons is transferred to local magnetization not aligned with electronic spin-polarization.", "Finally, some analyses  [23], [24], [25] also consider current-independent torque $\\mathbf {T}_i[\\partial \\mathbf {M}_i/\\partial t]$ as a back-action of electrons pushed out of equilibrium by time-dependent $\\mathbf {M}_i(t)$ .", "Nevertheless, such effects have been deemed negligible [23], [26] or easily absorbed into Eq.", "(REF ) by renormalizing $g$ and $\\lambda $  [23].", "Here $g$ is the gyromagnetic ratio; $\\mathbf {B}^{\\rm eff}_i = - \\frac{1}{\\mu _M} \\partial \\mathcal {H} /\\partial \\mathbf {M}_i$ is the effective magnetic field as the sum of external field, field due to interaction with other LMMs and magnetic anisotropy field in the classical Hamiltonian $\\mathcal {H}$ of LMMs; and $\\mu _M$ is the magnitude of LMM [21].", "The STT vector, $\\mathbf {T}=\\mathbf {T}^\\mathrm {FL} + \\mathbf {T}^\\mathrm {DL}$ , can be decomposed [Fig.", "REF (a)] into: (i) even under time-reversal or field-like (FL) torque, which affects precession of LMM around $\\mathbf {B}^{\\rm eff}_i$ ; and (ii) odd under time-reversal or damping-like (DL) torque, which either enhances the Gilbert damping by pushing LMM toward $\\mathbf {B}^{\\rm eff}_i$ or competes with Gilbert term as “antidamping.” For example, negative values of $T^\\mathrm {DL} = \\mathbf {T}^\\mathrm {DL} \\cdot \\mathbf {e}_\\mathrm {DL}$ in Figs.", "REF and  REF , where $\\mathbf {e}_\\mathrm {DL}=({M}_i \\times \\partial {M}_i/\\partial t)|{M}_i \\times \\partial {M}_i/\\partial t|^{-1}$ , means that $\\mathbf {T}^\\mathrm {DL}$ vector points away from the axis of precession which is antidamping action.", "Similarly, $T^\\mathrm {FL} = \\mathbf {T}^\\mathrm {FL} \\cdot \\mathbf {e}_\\mathrm {FL}$ , where $\\mathbf {e}_\\mathrm {FL}=(\\partial {M}_i/\\partial t)|\\partial {M}_i/\\partial t|^{-1}$ , is plotted in Figs.", "REF and  REF .", "Figure: The FL and DL components [Fig.", "()] of three spin torques contributions in Eq.", "() exerted by nonequilibrium spin density of electrons onto a single localized precessing magnetic moment in the setup of Fig.", "(a) as a function of coupling to the leads.", "Black dotted line is the sum of the three torques.", "In panels (a) and (c) J sd =0.1γJ_{sd} = 0.1 \\gamma , while in panels (b) and (d) J sd =20γJ_{sd} = 20 \\gamma ensures perfectly adiabatic regime , J sd /ℏω≫1J_{sd}/\\hbar \\omega \\gg 1, for the chosen precession frequency ℏω=0.001γ\\hbar \\omega =0.001 \\gamma .Figure: Spatial profile of FL and DL components of 𝐓 i geo \\mathbf {T}_i^\\mathrm {geo}, 𝐓 i sea \\mathbf {T}_i^\\mathrm {sea} and 𝐓 i surf \\mathbf {T}_i^\\mathrm {surf} spin torques on precessing LMMs depicted in Fig.", "(c) for closed or open electronic quantum system and for two different values of J sd J_{sd}.", "Insets on the top of each row mark positions and static configuration of LMMs within the Bloch DW, with their xx-component depicted by the colorbar next to panel (a).The current-driven STT $\\mathbf {T}_i \\left[ I^{S_\\alpha }_\\mathrm {ext} \\right]$ acts as the counterpart of nonconservative force in nonadiabatic MD.", "The Gilbert damping plus current-independent torque $\\mathbf {T}_i[\\partial \\mathbf {M}_i/\\partial t]$ appear as the counterpart of electronic friction [15], [14], [16], [8], [9], [17], [18], [19], but Gilbert damping requires agents [22] other than electrons alone considered in nonadiabatic MD.", "Thus, the geometric torque and damping, as counterparts of geometric magnetism force [2] and friction [10], are absent from standard modeling of classical magnetization dynamics.", "Geometric torque has been added ad hoc into the LLG equation applied to specific problems, such as spin waves within bulk magnetic materials [29], [30], [31].", "A recent study [32] of a single classical LMM embedded into a closed (i.e., finite length one-dimensional wire) electronic quantum system finds that nonequilibrium electronic spin density always generates geometric torque, even in perfectly adiabatic regime where electron-spin/LMM interaction is orders of magnitude larger than the characteristic frequency of LMM dynamics.", "It acts as a purely FL torque causing anomalous frequency of precession that is higher than the Larmor frequency.", "By retracing the same steps [14], [15] in the derivation of the stochastic Langevin equation for electron-nuclei system connected to macroscopic reservoirs, Ref.", "[33] derived the stochastic LLG equation [34], [35], [36], [37] for a single LMM embedded into an open electronic system out of equilibrium.", "The novelty in this derivation is damping, present even in the absence of traditional spin-flip relaxation mechanisms [23], [25], while the same conclusion about geometric torque changing only the precession frequency of LMM has been reached (in some regimes, geometric phase can also affect the stochastic torque [38]).", "However, single LMM is a rather special case, which is illustrated in Fig.", "REF (a) and revisited in Fig.", "REF , and the most intriguing situations in spintronics involve dynamics of noncollinear textures of LMMs.", "This is exemplified by current- or magnetic-field driven dynamics of domain walls (DWs) and skyrmions [25], [37], [39], [40], [41], [42], [43] where a much richer panoply of back-action effects from fast electronic system can be expected.", "In this Letter, we analyze an exactly solvable model of seven steadily precessing LMMs, $\\mathbf {M}_1(t)$ –$\\mathbf {M}_7(t)$ [Fig.", "REF (c)], which are noncollinear and noncoplanar and embedded into a one-dimensional (1D) infinite wire hosting conduction electrons.", "The model can be viewed as a segment of dynamical noncollinear magnetic texture, and it directly describes magnetic field-driven [43] head-to-head Bloch DW [44] but without allowing it to propagate [43], [41].", "Its simplicity makes it exactly solvable—we fins the exact time-dependent DM via the nonequilibrium Green function (NEGF) formalism [45] and analyze its contributions in different regimes of the ratio $J_{sd}/\\hbar \\omega $ of $sd$ exchange interaction $J_{sd}$  [23] between electron spin and LMM and frequency of precession $\\omega $ .", "In both Figs.", "REF (a) and REF (c), the electronic subsystem is an open quantum system and, although no bias voltage is applied between the macroscopic reservoirs, it is pushed into the nonequilibrium state by the dynamics of LMMs.", "For example, electronic quantum Hamiltonian becomes time-dependent due to $\\mathbf {M}_1(t)$ [Fig.", "REF (a)] or $\\mathbf {M}_1(t)$ –$\\mathbf {M}_7(t)$ [Fig.", "REF (c)], which leads to pumping [25], [27], [46] [Fig.", "REF (b),(c)] of spin current locally within the DW region, as well as into the leads [Fig.", "REF (a)].", "Pumping of charge current will also occur if the left-right symmetry of the device is broken statically [27] or dynamically [47].", "The electrons are modeled on an infinite tight-binding (TB) clean chain with Hamiltonian in the lab frame $\\hat{H}_\\mathrm {lab}(t) = -\\gamma \\sum _{\\langle ij\\rangle } \\hat{c}_i^\\dagger \\hat{c}_j-J_{sd}\\sum _{i}\\hat{c}_i^\\dagger \\hat{\\mathbf {\\sigma }} \\hat{c}_i \\cdot {M}_i(t).$ Here $\\hat{c}_i^\\dagger = (\\hat{c}_{i\\uparrow }^\\dagger , \\hat{c}_{i\\downarrow }^\\dagger )$ and $\\hat{c}_{i\\sigma }^\\dagger $ ($\\hat{c}_{i\\sigma }$ ) creates (annihilates) an electron of spin $\\sigma =\\uparrow , \\downarrow $ at site $i$ .", "The nearest-neighbor hopping $\\gamma = 1$ eV sets the unit of energy.", "The active region in Figs.", "REF (a) or  REF (c) consists of one or seven sites, respectively, while the rest of infinite TB chain is taken into account as the left (L) and the right (R) semi-infinite leads described by the same Hamiltonian in Eq.", "(REF ), but with $J_{sd}=0$ .", "The hopping between the leads and the active region is denoted by $\\gamma _c$ .", "The leads terminate at infinity into the macroscopic particle reservoirs with identical chemical potentials $\\mu _\\mathrm {L}=\\mu _\\mathrm {R}=E_F$ due to assumed absence of bias voltage, and $E_F=0$ is chosen as the Fermi energy.", "In contrast to traditional analysis in spintronics [23], [25], but akin to Refs.", "[32], [33], Hamiltonian in Eq.", "(REF ) does not contain any spin-orbit or impurity terms as generators of spin-flip relaxation.", "The spatial profile of Bloch DW is given by $M_i^x=-\\operatorname{sech}[(h_\\mathrm {DW}-z_i)/W]\\tanh [(Z_\\mathrm {DW}-z_i)]$ , $M_i^y=\\operatorname{sech}^2[(Z_\\mathrm {DW}-z_i)/W]$ and $M_i^z=\\tanh [(Z_\\mathrm {DW}-z_i)/W]$ , where $Z_\\mathrm {DW}=4$ and $W=0.9$ .", "Instead of solving LLG equations [Eq.", "(REF )] for ${M}_1(t)$ –${M}_7(t)$ , we impose a solution where LMMs precess steadily around the $z$ -axis: $M_i^x(t) = \\sin \\theta _i \\cos (\\omega t + \\phi _i)$ ; $M_i^y(t) = \\sin \\theta _i \\sin (\\omega t + \\phi _i)$ ; and $M_i^z(t) = \\cos \\theta _i$ .", "Using a unitary transformation into the rotating frame (RF), the Hamiltonian in Eq.", "(REF ) becomes time-independent [27], [25], $\\hat{H}_\\mathrm {RF} = \\hat{U}^\\dagger (t) \\hat{H}_\\mathrm {lab}(t) \\hat{U}(t) -i \\hbar \\hat{U}^\\dagger \\partial \\hat{U}/\\partial t = \\hat{H}_\\mathrm {lab}(t=0) - \\hbar \\omega \\hat{\\sigma }_\\alpha /2$ , with LMMs frozen at $t=0$ configuration from the lab.", "The unitary operator is $\\hat{U}(t) = \\exp (-i\\omega t \\hat{\\sigma }_\\alpha /2)$ for $\\alpha $ -axis of rotation.", "In the RF, the original two-terminal Landauer setup for quantum transport in Figs.", "REF (a) and  REF (c) is mapped, due to $\\hbar \\omega \\hat{\\sigma }_\\alpha /2$ term, onto an effective four-terminal setup [27] [illustrated for single LMM in Fig.", "REF (b)].", "Each of its four leads is an effective half-metal ferromagnet which accepts only one spin species, $\\uparrow $ or $\\downarrow $ along the $\\alpha $ -axis, and effective dc bias voltage $\\hbar \\omega /e$ acts between L or R pair of leads.", "In the RF, the presence of the leads and macroscopic reservoirs can be taken into account exactly using steady-state NEGFs [45] which depend on time difference $t-t^{\\prime }$ and energy $E$ upon Fourier transform.", "Using the retarded, $\\hat{G}(E)$ , and the lesser, $\\hat{G}^<(E)$ , Green functions (GFs), we find the exact nonequilibrium DM of electrons in the RF, $\\hat{\\rho }_\\mathrm {RF} = \\frac{1}{2\\pi i}\\int dE \\, \\hat{G}^<(E)$.", "Here the two GFs are related by the Keldysh equation, $\\hat{G}^<(E)=\\hat{G}(E)\\hat{\\Sigma }^<(E)\\hat{G}^\\dagger (E)$ , where $\\hat{\\Sigma }^<(E)$ is the lesser self-energy [45] due to semi-infinite leads and $\\hat{G}(E)=[E - \\hat{H}_\\mathrm {RF} -\\hat{\\Sigma }(E, \\hbar \\omega ) ]^{-1}$ with $\\hat{\\Sigma } (E, \\hbar \\omega ) = \\sum _{p=\\mathrm {L,R}, \\sigma =\\uparrow , \\downarrow } \\hat{\\Sigma }_p^\\sigma (E-Q_\\alpha ^\\sigma \\hbar \\omega )$ being the sum of retarded self-energies for each of the four leads $p$ , $\\sigma $ in RF.", "We use shorthand notation $Q_p^\\uparrow = -1/2$ and $Q_p^\\downarrow = +1/2$ .", "Since typical frequency of magnetization dynamics is $\\hbar \\omega \\ll E_F$, we can expand [48] $\\hat{\\rho }_\\mathrm {RF}$ in small $\\hbar \\omega /E_F$ and then transform it back to the lab frame, $\\hat{\\rho }_\\mathrm {lab}(t) = \\hat{U}(t)\\hat{\\rho }_\\mathrm {RF} \\hat{U}^\\dagger (t)$ to obtain $\\hat{\\rho }_\\mathrm {lab}(t) = \\hat{\\rho }^\\mathrm {ad}_t + \\hat{\\rho }_\\mathrm {geo}(t) + \\hat{\\rho }_\\mathrm {sea}(t) + \\hat{\\rho }_\\mathrm {surf}(t)$ where: $\\hat{\\rho }^\\mathrm {ad}_t & = & -\\frac{1}{\\pi } \\hat{U} \\int \\limits _{-\\infty }^{+\\infty } \\!", "dE \\mathrm {Im}\\hat{G}_0 f(E)\\hat{U}^\\dagger , \\\\\\hat{\\rho }_\\mathrm {geo}(t) & = & \\frac{1}{\\pi } \\hat{U} \\int \\limits _{-\\infty }^{+\\infty } \\!", "dE \\mathrm {Im}\\bigg [\\hat{G}_0\\bigg ( i\\hbar \\hat{U}^\\dagger \\frac{\\partial \\hat{U}}{\\partial t} \\bigg ) \\hat{G}_0 \\bigg ] f(E) \\hat{U}^\\dagger , \\\\\\hat{\\rho }_\\mathrm {sea}(t) & = & -\\frac{\\hbar \\omega }{2\\pi } \\hat{U} \\sum _p \\int \\limits _{-\\infty }^{+\\infty } \\!", "dE \\mathrm {Im}\\bigg [\\hat{G}_0\\bigg ( \\frac{\\partial \\hat{\\Sigma }^\\uparrow _p}{\\partial E}- \\frac{\\partial \\hat{\\Sigma }^\\downarrow _p}{\\partial E} \\bigg ) \\hat{G}_0 \\bigg ] \\nonumber \\\\&& \\times f(E)\\hat{U}^\\dagger , \\\\\\hat{\\rho }_\\mathrm {surf}(t) & = & \\frac{\\hbar \\omega }{4\\pi } \\hat{U} \\sum _{p} \\int \\limits _{-\\infty }^{+\\infty } \\!", "dE \\hat{G}_0 (\\hat{\\Gamma }^\\uparrow _p - \\hat{\\Gamma }_p^\\downarrow ) \\hat{G}_0^\\dagger \\frac{\\partial f}{\\partial E} \\hat{U}^\\dagger .", "$ We confirm by numerically exact calculations [39] that thus obtained $\\hat{\\rho }_\\mathrm {lab}(t)$ is identical to $\\hbar G^<(t,t)/i$ computed in the lab frame.", "Here $\\hat{G}_0(E) = [E - \\hat{H}_\\mathrm {RF} - \\hat{\\Sigma }(E, 0)]^{-1}$ is $\\hat{G}(E)$ with $\\hbar \\omega = 0$ ; $\\hat{\\Gamma }_p^{\\sigma }(E) = i[\\hat{\\Sigma }_p^{\\sigma }(E) - \\hat{\\Sigma }^{\\sigma }_p(E)^\\dagger ]$ is the level broadening matrix due the leads; and $f_p^\\sigma (E) = f(E - [E_F + Q_\\alpha ^\\sigma \\hbar \\omega ])$ is the the Fermi function of macroscopic reservoir $p$ , $\\sigma $ in the RF.", "The total nonequilibrium spin density, $\\langle \\hat{\\mathbf {s}}_i \\rangle (t)=\\mathrm {Tr}[\\hat{\\rho }_\\mathrm {lab}(t) |i\\rangle \\langle i| \\otimes \\hat{\\mathbf {\\sigma }}]=\\langle \\hat{\\mathbf {s}}_i \\rangle ^\\mathrm {ad}_t + \\langle \\hat{\\mathbf {s}}_i \\rangle _\\mathrm {geo}(t) + \\langle \\hat{\\mathbf {s}}_i \\rangle _\\mathrm {sea}(t) + \\langle \\hat{\\mathbf {s}}_i \\rangle _\\mathrm {surf}(t)$ , has the corresponding four contributions from DM contributions in Eq.", "(REF ).", "Here $\\langle \\hat{\\mathbf {s}}_i \\rangle ^\\mathrm {ad}_t$ is the equilibrium expectation value at an instantaneous time $t$ which defines `adiabatic spin density' [23], [25], [30], [31], [32].", "It is computed using $\\hat{\\rho }^\\mathrm {ad}_t$ as the grand canonical equilibrium DM expressed via the frozen (adiabatic) retarded GF [14], [15], [33], $\\hat{G}_t(E)=[E-\\hat{H}_t-\\hat{\\Sigma }]^{-1}$, for instantaneous configuration of $\\mathbf {M}_i(t)$ while assuming $\\partial \\mathbf {M}_i/\\partial t = 0$ [subscript $t$ signifies parametric dependence on time through slow variation of $\\mathbf {M}_i(t)$ ].", "The other three contributions—from $\\hat{\\rho }_\\mathrm {geo}(t)$ and $\\hat{\\rho }_\\mathrm {sea}(t)$ governed by the Fermi sea and $\\hat{\\rho }_\\mathrm {surf}(t)$ governed by the Fermi surface electronic states—contain first nonadiabatic correction [14], [15], [33] proportional to velocity $\\partial \\mathbf {M}_i/\\partial t$ , as well as higher order terms due to $\\hat{\\rho }_\\mathrm {lab}(t)$ being exact.", "These three contributions define STT out of equilibrium [23], [39], [48] $\\mathbf {T}_i = J_{sd} \\langle \\hat{\\mathbf {s}}_i \\rangle (t) \\times \\mathbf {M}_i(t) = \\mathbf {T}_i^\\mathrm {geo} + \\mathbf {T}_i^\\mathrm {sea} + \\mathbf {T}_i^\\mathrm {surf}.$ Each term $\\mathbf {T}_i ^\\mathrm {geo}$ , $\\mathbf {T}_i^\\mathrm {sea}$ , $\\mathbf {T}_i^\\mathrm {surf}$ can be additionally separated into its own DL and FL components [Fig.", "REF (a)], as plotted in Figs.", "REF and  REF .", "Note that $\\mathbf {T}_i ^\\mathrm {sea}$ is insignificant in both Figs.", "REF and  REF , so we focus on $\\mathbf {T}_i^\\mathrm {geo}$ and $\\mathbf {T}_i^\\mathrm {surf}$ .", "To gain transparent physical interpretation of $\\mathbf {T}_i^\\mathrm {geo}$ and $\\mathbf {T}_i^\\mathrm {surf}$ , we first consider the simplest case [33], [32]—a single $\\mathbf {M}_1(t)$ in setup of Fig.", "REF (a).", "The STT contributions as a function of the coupling $\\gamma _c$ to the leads (i.e., reservoirs) are shown in Fig.", "REF .", "We use two different values for $J_{sd}$ , where large ratio of $J_{sd}=20$ eV and $\\hbar \\omega =0.001$ eV is perfect adiabatic limit [30], [31], [32].", "Nevertheless, even in this limit and for $\\gamma _c \\rightarrow 0$ we find $\\mathbf {T}_1^\\mathrm {geo} \\ne 0$ in Fig.", "REF (b) as the only nonzero and purely FL torque.", "This is also found in closed system of Ref.", "[32] where $\\mathbf {T}_1^\\mathrm {geo}$ was expressed in terms of the spin Berry curvature.", "As the quantum system becomes open for $\\gamma _c>0$ , $\\mathbf {T}_1^\\mathrm {geo}$ is slightly reduced while $\\mathbf {T}_1^\\mathrm {surf}$ emerges with small FL [Fig.", "REF (b)] and large DL [Fig.", "REF (d)] components.", "The DL torque $\\mathbf {T}_1^\\mathrm {surf,DL}$ points toward the $z$ -axis and, therefore, enhances the Gilbert damping.", "In the wide-band approximation [49], the self-energy $\\hat{\\Sigma }(E) = -i\\Gamma \\hat{I}_2$ is energy-independent for $E$ within the bandwidth of the lead, which allows us to obtain analytical expression (at zero temperature) $\\mathbf {T}^\\mathrm {geo}_1(t) = \\frac{\\hbar \\omega }{2\\pi }\\bigg [ \\pi - 2\\tan ^{-1}\\bigg ( \\frac{\\Gamma }{J_{sd}}\\bigg )\\bigg ]\\sin \\theta ~ {e}_\\phi (t).$ Here ${e}_\\phi (t) = -\\sin \\omega t~{e}_x + \\cos \\omega t~ {e}_y$ .", "Thus, in perfect adiabatic limit, $J_{sd}/\\hbar \\omega \\rightarrow \\infty $ , or in closed system, $\\Gamma \\rightarrow 0$ , $\\mathbf {T}_1^\\mathrm {geo}$ is independent of microscopic parameters as expected from its geometric nature [29].", "The always present $\\mathbf {T}_i^\\mathrm {geo} \\ne 0$ means that electron spin is never along `adiabatic direction' $\\langle \\hat{\\mathbf {s}}_i \\rangle ^\\mathrm {ad}_t$ .", "Figure: (a) The zz-component of total DL torques which act on DW in Fig.", "(c) as a function of J sd J_{sd} for γ c =γ\\gamma _c=\\gamma .", "Circles show that sum of spin currents pumped into the leads matches ∑ i 𝐓 i surf , DL z ≡I L S z +I R S z \\left(\\sum _i \\mathbf {T}_i^\\mathrm {surf,DL} \\right)_z \\equiv I_\\mathrm {L}^{S_z} + I_\\mathrm {R}^{S_z}.", "Panel (b) and (c), which correspond to Fig.", "(g), show spatial profile of local spin currents I i→j S z I_{i \\rightarrow j}^{S_z} pumped between sites ii and jj for J sd =0.1γJ_{sd}=0.1 \\gamma , with their sum being identically zero in panel (c).", "Dashed black line in panels (a) and (b) is pumped local spin current by SMF , , I SMF S z (x)=gμ B ℏG 0 4e 2 [∂𝐌(x,t)/∂t×∂𝐌(x,t)/∂x] z I_\\mathrm {SMF}^{S_z}(x)=\\frac{g\\mu _B \\hbar G_0}{4e^2} [ \\partial {\\bf M}(x, t)/\\partial t \\times \\partial {\\bf M}(x, t)/\\partial x ]_z, where G 0 =G ↑ +G ↓ G_0 = G^\\uparrow + G^\\downarrow is the total conductivity.Switching to DW [Fig.", "REF (c)] embedded into a closed quantum system ($\\gamma _c=0)$ shows in Fig.", "REF (a)–(d) that only $\\mathbf {T}_i^\\mathrm {geo} \\ne 0$ , which also acquires DL component locally with damping or antidamping action depending on the position of LMM.", "Upon opening the quantum system ($\\gamma _c=\\gamma $ ), Fig.", "REF (e)–(h) shows emergence of additional $\\mathbf {T}^\\mathrm {surf}_i \\ne 0$ which, however, becomes negligible [Fig.", "REF (f),(h)] in the perfectly adiabatic limit $J_{sd}/\\hbar \\omega \\gg 1$ .", "At first sight, $\\mathbf {T}_i^\\mathrm {geo,DL} \\ne 0$ violates Berry and Robbins original analysis [2] according to which an isolated quantum system, with discrete energy spectrum, cannot exert friction onto the classical system.", "This apparent contradiction is resolved in Fig.", "REF (a) where we show that total $\\sum _i \\mathbf {T}_i^\\mathrm {geo,DL} \\equiv 0$ is always zero.", "Conversely, Fig.", "REF (a) confirms that total $\\left(\\sum \\mathbf {T}^\\mathrm {surf,DL}_i\\right)_z \\equiv I_\\mathrm {L}^{S_z} + I_\\mathrm {R}^{S_z}$ is identical to net spin current pumped into the leads via which the conduction electrons carry away excess angular momentum of precessing LMMs [46].", "Such identity underlies physical picture where spin current generated by time-dependent magnetization becomes DL torque [46], [24].", "Note that pumped spin current $I_{i \\rightarrow j}^{S_z}$ due to $\\hat{\\rho }_\\mathrm {geo}$ or $\\hat{\\rho }_\\mathrm {sea}$ in Fig.", "REF (c) can be nonzero locally, but they sum to zero.", "The nonuniform pumped spin current due to spatially and time varying magnetization has prompted proposals [24], [26] to amend the LLG equation by adding the corresponding DL torque $\\mathbf {M} \\times \\mathcal {D} \\cdot \\partial \\mathbf {M}/\\partial t$ with $3 \\times 3$ damping tensor $\\mathcal {D}$ whose spatial dependence is given by the so-called spin-motive force (SMF) formula.", "However, SMF correction was estimated to be small [26] in the absence of spin-orbit coupling in the band structure.", "We confirm its smallness in Fig.", "REF (a),(b) for our DW case, but this actually reveals that SMF formula produces incorrectly an order of magnitude smaller torque than obtained from our exact $\\hat{\\rho }_\\mathrm {surf}(t)$ .", "Due to possibly complex [40] time and spatial dependence of $\\mathbf {T}_i ^\\mathrm {surf}$ and $\\mathbf {T}_i^\\mathrm {geo}$ , the accurate path to incorporate them is offered by self-consistent coupling of electronic DM and LLG calculations, as proposed in Refs.", "[39], [42], [50] and in full analogy to how electronic friction is included in nonadiabatic MD [9], [14], [15], [16], [7], [8], [17], [18], [19].", "This research was supported in part by the U.S. National Science Foundation (NSF) under Grant No.", "CHE 1566074." ] ]

[ [ "Machine learning and excited-state molecular dynamics" ], [ "Abstract Machine learning is employed at an increasing rate in the research field of quantum chemistry.", "While the majority of approaches target the investigation of chemical systems in their electronic ground state, the inclusion of light into the processes leads to electronically excited states and gives rise to several new challenges.", "Here, we survey recent advances for excited-state dynamics based on machine learning.", "In doing so, we highlight successes, pitfalls, challenges and future avenues for machine learning approaches for light-induced molecular processes." ], [ "Introduction", "Photosynthesis, photovoltaics, the processes that enable our vision or photodamage of biologically relevant molecules, such as DNA or peptides – they all have one thing in common: The underlying processes are governed by a manifold of excited states after the absorption of light [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].", "They can be studied experimentally via several techniques, such as UV/visible spectroscopy, transient absorption spectroscopy, photoionization spectroscopy or ultrafast electron diffraction [19], [20], [21], [22], [23], [24], [25], [26], [27].", "However, experimental techniques are to some extent blind to the exact electronic mechanism of photo-induced reactions.", "In order to get a more comprehensive understanding, theoretical simulations can complement experimental findings and can provide explanations for observed reactions [9].", "For instance, simulated UV spectra can be used to unveil the states relevant for photodamage and -stability of molecules [28], [29], [30], [31], [32], [33], [34], [35], [36] and the temporal evolution of molecules can be studied via nonadiabatic molecular dynamics (NAMD) simulations [37], [38], [39], [40], [41], [42], [43], [44], [45], [46].", "The latter gives access to different reaction channels, branching ratios, and excited-state lifetimes and will be the main topic of discussion here.", "While experimental techniques require large and costly setups, theoretical simulations require high-performance computing facilities due to expensive electronic structure computations.", "Especially NAMD simulations are seriously limited by the underlying quantum chemical calculations, making long and experimentally relevant simulation times inaccessible with conventional ab initio methods.", "The larger the molecule becomes, the more electronically excited states are involved in reactions and the more complex their interactions become.", "This leads to non-linearly increasing costs of quantum chemical calculations and a compromise between accuracy and computational efficiency is indispensable.", "Relying on such expensive ab initio potentials, only a couple of picoseconds can be simulated and the exploration of rare reaction channels is restricted due to bad statistics [47], [43], [17].", "Technically, the nuclear part and the electronic part of the calculations can be separated to a large extent.", "First, the electronic problem is solved leading to potential energies for the nuclei.", "Afterwards, the nuclei move on these potentials classically or quantum chemically [48], [49], [50], [51], [52], [6], [8].", "These two subsequent steps can be carried out in every time step (on-the-fly), if classical trajectories are employed.", "Alternatively, the two steps are separated as much as possible by precomputing the potential energy surfaces (PESs) and then using these precomputed PESs in the subsequent nuclear dynamics.", "Experimental observables and macroscopic properties can be obtained in follow-up computations or analysis runs.", "Machine learning (ML) can accelerate the overall simulation process on different levels and at several points.", "A broad classification of how to use ML models to replace different parts of quantum chemistry to make simulations more efficient is given in Fig.", "REF  [53].", "Figure: A broad classification of how to use machine learning models to replace different parts of quantum chemistry .", "Simulations can be enhanced by providing (1) a wavefunction from a machine learning model, (2) a force field by fitting energies and forces or (3) other properties, such as energy gaps or reaction rates by learning the final output of a dynamics simulation directly.machine learning model and overview of the parts of a quantum chemical simulation.The probably most fundamental way is to use ML to solve the Schrödinger equation.", "This has been done for the ground state by representing the molecular wavefunction on a grid, in a molecular orbital basis or in a Monte-Carlo approach [54], [55], [56], [57], [58], [59], [60], [61], [62] and has recently also been applied for the excited states of a one-dimensional model [63].", "ML can also be used to reconstruct the wavefunction from near-field spectra [64] or to bypass the Kohn-Sham equation in density functional theory (DFT) [65].", "The external potential, functional, electronic density or local density of states can be learned [66], [65], [67], [68], [53], [69], [70], [71], [72].", "Very recently, Ceriotti and co-workers further introduced a smooth atomic density by defining an abstract chemical environment [73].", "By having access to the molecular wavefunction or the electron density, the secondary output, which are energies and forces for the ground state and additionally couplings for the excited states, can be derived efficiently with ML.", "The coefficients of the ML wavefunctions or the density can further be used as an input for quantum chemical simulations, reducing the number of SCF iterations substantially [59].", "Instead of learning the quantum chemistry of systems, the so-called \"secondary outputs\" [53] can also be mapped directly to a molecular structure, giving rise to so-called ML force fields.", "By training an ML model on ab initio data, the accuracy of quantum chemistry can be combined with the efficiency of conventional force fields for molecular dynamics (MD) simulations in the ground state [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96], [97], [98], [99], [100], [101], [102], [103], [104], [105], [106], [107], [108], [109].", "For the excited states, only a couple of studies are available [110], [111], [112], [113], [114], [115], [116], [117], [118], [119], [106], [120], [121], [122], [122], [123], [124], [125], [126], [120], [127], [127], [128], [129].", "Nevertheless, the first NAMD simulation with ML dates back to the year 2008, where the scattering of O$_2$ on Al(III) was studied in a mixed quantum-classical approach considering singlet-triplet transitions [130], [111].", "Having access to the excited state energies, \"tertiary properties\", such as UV spectra [131], band gaps [132], [133], [134], HOMO-LUMO gaps or vertical excitation energies [135], [136], [137], [138] of molecules can be derived.", "Again, this tertiary output can also be fit in a direct fashion, which has been done for instance for a light-harvesting system by learning the excitation energy transfer properties [139] or the output of NAMD simulations to find out about the relations of molecular structures and dynamic properties [140].", "Moreover, ML has been successfully applied for the inverse design of molecules and materials featuring specific properties, such as defined HOMO-LUMO gaps or catalytic activities.", "Examples range from the inverse design of photonic materials, to (photo-)catalysts, solar cells or (photo-active) drugs, to name only a few applications [141], [142], [143], [144], [107], [145], [146], [147], [148], [149], [150].", "Despite the opportunities of ML for the development of groundbreaking new methodologies, current techniques are often limited to certain molecules or specific problems.", "Methods exist, that extrapolate throughout chemical compound space, see e.g.", "Refs.", "[151], [152], [153], [154], [155], [156], [59], [157], [158], [159], [131], [160], but usually models fail to go beyond energies and related properties, such as forces, atomization or excitation energies.", "Further, it is challenging to predict compounds consisting of atom types strongly different from those inside of the training set.", "Especially the fitting of the excited-state PESs poses another obstacle, let alone the transferability of excited-state PESs: Not only are ML models restricted to certain molecules or materials [110], [111], [112], [113], [114], [115], [116], [117], [118], [119], [106], [120], [121], [122], [122], [123], [124], [125], [119], [120], [127], [127], [128], [129], more often the different energetic states are fit independently from each other with separately trained ML models.", "As it is clear that the PESs of molecules are not independent of each other, it might also be unsurprising that learning them simultaneously is advantageous for various applications, such as NAMD or spectra predictions.", "Only a few studies exist that include more energetic states in one ML model and even less treat related properties, such as the vectorial dipole moments or couplings between different PESs in one ML model [85], [131], [154], [118], [161], [162].", "However, in our view, the \"holy grail\" of ML for photochemistry is an ML model that provides all relevant energetic states, forces and properties at once, using derivatives where possible, rather than learning the properties independently.", "At the very best, this model should be transferable throughout chemical compound space [163] and could be used for molecules of any composition.", "Given the fact that ML models for the electron wavefunctions of different excited states (or related models within the DFT framework) do not exist for polyatomic systems, this dream has not yet come alive.", "Hence, we will focus this perspective on ML models that learn the secondary outputs, i.e., excited state PESs, corresponding forces, and nonadiabatic and spin-orbit couplings (NACs and SOCs, respectively) between them.", "We note that our discussion holds for different spin-multiplicities, although most studies focus on singlet states only.", "We try to address the recent achievements in the fields of photochemistry using ML and discuss the current challenges and future perspectives to get a step further to a transferable ML model for excited states that treats all properties on the same footing.", "We start by discussing the generation of a training set for the treatment of excited state PESs, corresponding forces and couplings and focus on their use in NAMD simulations.", "Especially, we aim to clarify the differences between excited-state and ground-state properties.", "We therefore describe the NACs and SOCs that couple different electronic states and highlight their importance for NAMD simulations.", "Subsequently, state-of-the-art ML models for excited-state PESs are considered along with the challenge of modelling a manifold of energetic states." ], [ "Generating a training set for excited states", "The basis of any successful ML model is a comprehensive and accurate training set that contains the molecular geometries in combination with the corresponding properties that need to be predicted.", "For the application of ML within NAMD simulations, the training set should contain a molecular geometry and the energies of all energetic states, corresponding forces, and couplings between the states.", "It is computed with the quantum chemistry method, whose accuracy one wants to obtain.", "The choice of the quantum chemistry method is a problem on its own and often requires expert knowledge [18], [164].", "Simply said, ML can be seen as efficient interpolation between data points with the accuracy of the reference method.", "Before we go into detail on how to efficiently create a training set for excited states, we will first discuss the differences to ground state potentials and properties that need to be considered.", "A major drawback is the fact that a manifold of excited states and thus also the properties between them have to be accounted for.", "These are NACs between states of same spin multiplicity and SOCs between states of different spin multiplicity as well as transition dipole moments.", "The fitting of such properties is problematic due to the arbitrary phase of the wavefunction [118], [161].", "Therefore, an additional pre-processing might be necessary.", "Either a diabatization [115], [116], [117], [119], [106], [120], [121], [122], [122], [123], [124], [125], [126], [120], [127], [127], a so-called phase correction [118], or a special learning algorithm [165] renders data learnable.", "The latter two are described in the following while further details on the former are given in section REF ." ], [ "Making excited-state data learnable", "Compared to energies and forces, NACs, SOCs as well as transition dipole moments result from the wavefunctions of two different electronic states.", "Due to the non-unique definition of the wavefunction itself, i.e., the fact that multiplication of the electronic wavefunction with a phase factor still gives a valid eigenfunction of the electronic Hamiltonian, leads to an arbitrary phase, which is initiated randomly in a quantum chemical calculation.", "Consequently, also the sign of the couplings, $C_{ij}$ , can be positive or negative.", "Here, $i$ and $j$ denote the indices of the involved states.", "The resulting inconsistencies in the coupling hypersurfaces make it challenging to find a good relation between an ML model, which is per definition a smooth function [166], and those discontinuous raw outputs.", "This problem can be illustrated with molecular orbitals of the methylenimmonium cation (Fig.", "REF reproduced from Ref. [118]).", "Panel A shows the molecular geometries, which are given as an input to a quantum chemical program.", "Two molecular orbitals are shown as placeholders for the wavefunctions of two electronic states, the $S_1$ and $S_2$ states.", "The color of the orbitals can be either blue or red and changes arbitrarily throughout the reaction coordinate.", "In the same way, also the overall wavefunction (which is difficult to plot) for the respective state changes its phase arbitrarily.", "As a consequence, also the sign of the couplings, where the product of the two wavefunctions' signs enters, may change randomly (see panel B).", "Figure: A set of quantum chemical calculations of the methylenimmonium cation, CH 2 _2NH 2 + _2^+, along the C-N bond.", "The molecular conformations, which are the input of a quantum chemical calculation, are given in panel A, the orbitals of the computed two electronic states, the S 1 S_1 and S 2 S_2 state, as well as the corresponding off-diagonal matrix elements between those two states, 〈S 1 ∣H ^∣S 2 〉\\langle S_1 \\mid \\hat{H}\\mid S_2\\rangle , are given in panel B.", "As can be seen, the sign of those values arbitrarily switches.", "Those sign jumps can be removed by applying a phase correction algorithm.", "Results are given for those elements in panel C. Reproduced from Ref.", "under CC-BY, https://creativecommons.org/licenses/by/3.0/.In order to allow for a meaningful ML description of elements resulting from two different electronic states, a data pre-processing is helpful.", "The former process is termed phase correction [167], [118] and is practicable to remove almost all such inconsistencies in the configurational space of the training set.", "This phase correction makes the use of conventional training algorithms possible.", "To carry out phase correction, a wavefunction overlap computation [168], ${S}=\\langle \\Psi _\\alpha \\mid \\Psi _\\beta \\rangle $ , has to be carried out between the wavefunction $\\Psi _\\beta $ at every geometry $\\beta $ inside the training set and the wavefunction $\\Psi _\\alpha $ at a reference geometry $\\alpha $ .", "The phase thus has to be tracked from a pre-defined reference geometry.", "It often happens that two geometries are dissimilar to each other, so that interpolation between them is necessary, making this process generally more expensive.", "So-called intruder states give rise to additional problems, since they are so high in energy at the reference geometry that they usually would not be included in the initial calculation.", "However, they enter the lower energy region at another geometry visited during the NAMD simulations and thus need to be considered in the current calculation step.", "Hence, they should have been included from the beginning for the phase correction algorithm to work.", "As a solution to this problem, many electronic states need to be computed from the start.", "In some cases, where many energetic states lie close to each other and where the photochemistry is complex, phase correction might even be infeasible.", "The problem of intruder states was also identified by Robertsson et al.", "and is well explained for a diabatization procedure in Ref. [169].", "For a more detailed discussion on phase correction, the reader is referred to Ref.s [118], [170], [168], [167].", "Nevertheless, as given in panel C of Fig.", "REF , smooth curves are obtained if phase correction is carried out correctly and these phase-corrected properties can be learned with conventional ML models.", "Similarly, a small set of data can be corrected manually and afterwards a cluster growing algorithm can be applied [171], [125].", "This algorithm uses Gaussian process regression to continuously add data points to the initially phase-corrected data set.", "This approach has been employed recently to obtain diabatic transition dipole moments [119].", "However, in systems containing many degrees of freedom and many electronic states, a manual correction of the sign of couplings is tedious and the approach has only been applied to small systems, yet [126], [119].", "In contrast to the phase-correction procedures described above, an ML-based internal phase-correction during training renders the learning of raw quantum chemical data possible and does not require any pre-processing.", "However, it requires a modification of the training process itself [165].", "In a recent study, we applied such a phase-free training using the deep continuous-filter convolutional-layer NN SchNet [156], [172] that we adapted for the treatment of excited states.", "In contrast to conventional algorithms, where the hyperparameters of the network are optimized to minimize the L$_1$ or L$_2$ loss function, here a phase-less loss function is applied.", "The latter allows the ML model to possess a different phase (or sign) for the learned property than the reference data.", "Since ML models intrinsically yield smooth curves, the algorithm will then automatically choose a phase for every data point such that smooth coupling curves are produced.", "This freedom of choice is achieved by calculating the errors between the ML value and all possible phase variations of the reference value and using only the smallest of these errors.", "The possibilities for phase conventions scales with $2^{N_S-1}$ , where $N_S$ is the number of considered states.", "Since the error is computed more often than in conventional ML training, the phase-less loss training becomes more expensive, when including more electronic states.", "Mathematically, instead of computing the mean squared error, $\\varepsilon _{L_2}$ , between reference couplings, $C_{ij}^{QC}$ , and predicted couplings, $C_{ij}^{ML}$ , $\\varepsilon _{L_2} = \\mid \\mid C_{ij}^{QC} - C_{ij}^{ML} \\mid \\mid ^2,$ the phase-free error, $\\varepsilon _{ph}$ , is computed as the minimum of $2^{N_S-1}$ computed errors: $\\varepsilon _{ph} = \\mid \\mid C^{QC}_{ij} \\cdot p_i^{k} \\cdot p_j^{k} - C^{ML}_{ij} \\mid \\mid ~\\text{with}~ 0 \\le k \\le 2^{N_S-1}.$ $p_i^k$ and $p_j^k$ are phase factors, giving rise to the sign of state $i$ and $j$ resulting in a phase vector with index $k$ .", "This adaption of the loss function can remove the influence of any phase during the training process, making the use of raw data possible.", "A variation of this approach with reduced cost is possible if only one property, i.e.", "NACs or SOCs, are trained for NAMD simulations.", "A detailed discussion can be found in Ref. [165].", "It is worth mentioning that, besides the arbitrary phase of the wavefunction, also the Berry phase (or geometric phase) [173] exists.", "Effects due to the Berry phase can not be accounted for with phase correction.", "Nevertheless, most often in mixed quantum classical NAMD simulations, the Berry phase can be neglected.", "As a drawback, some effects, such as interference of nuclear wavefunctions might not be described correctly with such methods, and thus prevents the application of the corresponding ML properties if those effects are important.", "In some other dynamics methods and reactions, the Berry phase plays a crucial role and can lead to path-dependent transition probabilities close to conical intersections.", "This effect is important in quantum dynamics simulations, where problems can be circumvented by using diabatic potentials, which will be described in section REF ." ], [ "Choosing the right reference method", "While many ML potentials for ground-state MD simulations are based on DFT training sets, see e.g.", "Refs.", "[174], [77], [79], [175], [86], the training sets for the excited states are mainly obtained with multi-reference methods.", "Examples are the complete active space self-consistent-field (CASSCF) method [127], [112], [114], [113], [120], [122], [140], [165] or multi-reference configuration interaction (MR-CI) schemes [176], [177], [178], [115], [179], [180], [181], [117], [118], [161], [182], [126], [119], [165].", "The advantage of multi-reference methods compared to single reference methods is that photo-dissociation, which is likely to occur in many molecules after their excitation by light, can be treated accurately.", "In contrast, single-reference methods fail to do so in many cases.", "However, multi-reference methods are seriously limited by their computational costs [7], [10], calling for an efficient and meaningful training set generation.", "Therefore, the training set should be as small as possible, but should cover the relevant conformational space of a molecule that is required for accurate NAMD simulations [183].", "Accordingly, many recent training sets for MD simulations are built by a so-called \"active-learning\" [184], [77] or iterative/adaptive sampling scheme [185], [85] that will be described in the following and can be adapted for excited states [118].", "From our point of view, it is most favorable to start by computing a small initial training set and to expand it via such an adaptive sampling scheme [77], [85], [118], [185], [184], [186]." ], [ "Initial training set", "If only static calculations are targeted, data bases can be generated efficiently by starting from already existing data sets.", "As an example, Schwilk et al.", "[182] constructed a large data set of 13k carbene structures by randomly choosing 4,000 geometries from the QM9 [187] data set (consisting of 130k organic molecular structures).", "Hydrogen-atoms were abstracted and singlet and triplet states were optimized.", "The MR-CI method was subsequently used to compute the energies of the singlet and triplet state and a data set of 13,000 different carbene structures, called QMspin, was obtained, opening avenues to investigate important intermediate geometries critical for organic reaction networks.", "As a starting point for all following training-set generation schemes aiming to investigate the temporal evolution of a system, the equilibrium geometry of a molecule can be computed and taken as a reference.", "The initial training set can then be built up by sampling conformations close to this molecular configuration.", "In general, every sampling method is possible.", "Since the normal modes of a molecule are generally important for dynamics, scans along these coordinates can be used to sample different conformations.", "In cases of small molecules with few degrees of freedom, this process might be a good starting guess for an initial training set [118].", "It also makes sense to optimize critical points like excited-state minima, conical intersections and state crossings and to include data along such optimization runs into the training set.", "The same is advisable for larger systems, but in addition, some other approaches like Wigner sampling [188] or sampling via MD simulations [189], [190] can be considered.", "To name a few approaches, umbrella sampling [191], trajectory-guided sampling [192], enhanced sampling [193] or metadynamics [194], using a cheap electronic structure method like the semi-empirical tight-binding based quantum chemistry method GFN2-xTB [195]), can be employed.", "Further, if literature or chemical intuition indicate that certain reactions, like dissociation, take place after photo-excitation, it is also favorable to include those reaction coordinates right from the beginning.", "The initial training set can easily comprise on the order of 1,000 data points, which might seem like a lot but is reasonable given the large number of data points in commonly used training sets [114], [112], [113], [161].", "The quality of the initial ML potentials can be assessed by carrying out short scans along different reaction coordinates, such as combinations of normal modes.", "As soon as the initial training set is large enough, the training set expansion via an adaptive sampling scheme can be started." ], [ "Adaptive sampling for excited-states", "ML models fail to predict regions with scarce training data, i.e., their extrapolation capabilities are faint [196].", "Since such regions are likely visited during a dynamics simulation, the initial training set then needs to be expanded.", "A quality control is needed to detect whether unknown conformational regions of the molecule are visited, such that the corresponding structures afterwards can be added to the training set.", "This concept was introduced already in 1992 as query by committee [197] and has been used in chemistry in the so-called GROW algorithm of Collins and coworkers [129], [128] as well as in the iterative sampling of Behler [185].", "The latter is nowadays well adapted for the ground state [184], [77], [85], [186] and was recently modified for the excited states [118].", "The scheme is described in more detail in the following.", "To apply the procedure of adaptive sampling, at least two ML models have to be trained independently, e.g., with slightly different hyperparameters or starting weights.", "An overview of this procedure with two neural networks (NNs) is given as an example in Fig.", "REF .", "Figure: Overview of the adaptive sampling scheme for excited states, reproduced from Ref.", "under CC-BY.", "Adaptive sampling is illustrated exemplarily with two NNs for the methylenimmonium cation, CH 2 _2NH 2 + _2^+.", "As a starting point, ML models are trained on the initial training set and ML-based dynamics are executed.", "At each time step, the predictions of the two deep NNs (NN1 and NN2) are compared to each other for energies (E) and gradients (G), nonadiabatic couplings (NACs), spin-orbit couplings (SOCs) and dipole moments (μ\\mu ).", "If the difference between the ML models overcomes a pre-defined, adaptive threshold, the geometry visited at this time step is re-computed with quantum chemistry, added to the training set after phase correction and the ML models are re-trained.", "Subsequently, a new dynamics cycle is started and this process is repeated until the ML models are deemed to be converged.At every time step during an MD simulation, the predictions ${Y}_M^p$ of at least two ML models, $M$ , for a property $p$ (e.g., a potential energy) are compared to each other.", "To this end, the standard deviation of these predictions with respect to the mean of each property, $\\overline{Y}^P$ , is computed according to $Y_\\sigma ^p = \\sqrt{\\frac{1}{M-1}\\sum _{m=1}^J ( Y^p_m-\\overline{Y}^p)^2}.$ This standard deviation is compared to a pre-defined threshold, $\\varepsilon ^p$ , for each trained property.", "If the standard deviation stays below the threshold, the mean of each property, $\\overline{Y}^P$ , is forwarded to the MD program to propagate the nuclei.", "If the threshold is exceeded, the ML prediction is assumed to stem from an undersampled or unknown region of the PESs and is deemed untrustworthy.", "This conformation has to be included into the training set to guarantee accurate ML PESs.", "Thus, a quantum chemical reference computation is carried out, the data point is added to the training set and the ML models are re-trained to execute ML-NAMD simulations on longer time scales.", "It is sensible to choose a large threshold, $\\varepsilon ^p$ , in the beginning and adaptively make it smaller as the robustness of the ML models increases, giving rise to the name adaptive sampling [85].", "Adaptive sampling for excited states differs from adaptive sampling in the ground state in the number of properties that are considered.", "As illustrated in Fig.", "REF , not only the energies must be accurately predicted, but also the couplings and, if necessary, dipole moments.", "Since more states are considered, an average standard deviation is taken as the mean of the standard deviations of each state in case of energies and gradients, $Y_\\sigma ^p = \\frac{1}{N_S}\\sum _i^{N_S}\\left(\\sqrt{\\frac{1}{M-1}\\sum _{m=1}^M ( Y^{p_i}_{m}-\\overline{Y}^{p_i})^2}\\right),$ and as the mean of the standard deviations of each pair of states for couplings or dipole moments, $Y_\\sigma ^p = \\frac{1}{2N_S^2}\\sum _{i}^{N_S}\\sum _j^{N_S}\\left(\\sqrt{\\frac{1}{M-1}\\sum _{m=1}^M ( Y^{p_{ij}}_{m}-\\overline{Y}^{p_{ij}})^2}\\right).$ A separate threshold is set for each of these averaged quantities.", "If any of the quantities is predicted inaccurately, the data point is recomputed with quantum chemistry, phase corrected and added to the training set.", "In order to make this process more efficient, not only one MD simulation, but an ensemble of trajectories can be computed.", "The ML models are only retrained after each of the independent trajectories has reached an untrustworthy region of the PES and after each of the reference calculations has been finished and included in the training set.", "This makes the parallelization of many trajectories possible [185], [85], [118], [198].", "The adaptive sampling scheme should be carried out until the relevant conformational space for photodynamics is sampled sufficiently.", "However, using more than one ML model for production runs is still favorable.", "One of us and coworkers observed that the error of predictions decreases with the number of ML models used [85], [198].", "We have seen the same trend in a recent study for NAMD simulations.", "With the adaptive sampling scheme for excited states, we generated a training set of 4,000 data points of the methylenimmonium cation, CH$_2$ NH$_2^+$ , to carry out long time-scale NAMD simulations with NNs [118]." ], [ "Additional sampling techniques for excited states", "Further training sets for NAMD simulations were generated for one-dimensional systems as well as polyatomic molecules.", "For example, Chen et al.", "have computed 90,000 data points via Born-Oppenheimer MD simulations and NAMD simulations, where emphasis was placed on the inclusion of geometries after a transition from one state to another took place [114].", "Deep NNs were trained on energies and gradients of this training set to accurately reproduce NAMD simulations.", "Hu et al.", "[112] used a very similar approach and obtained around 200,000 data points of 6-aminopyrimidine from Born-Oppenheimer MD simulations.", "They further carried out NAMD simulations with surface hopping [199], [200], where transitions from one state to another were allowed via so-called hops.", "The geometries that were visited shortly before a hop took place were used as a starting guess to optimize conical intersections, i.e.", "critical points of the PESs, where two states become degenerate.", "Those data points were included to comprehensively sample the regions around a conical intersection.", "However, the ML models were not accurate enough for NAMD simulations solely based on ML potentials and the authors had to resort to quantum chemistry calculations in critical regions of the PESs.", "Dral et al.", "[113] generated a training set for a one-dimensional two-state spin-boson model consisting of 10,000 data points with a grid-based method.", "The training data selection was then based on the structure, rather than on the energy of the molecules.", "For each data point, a molecular descriptor was computed and the distances of the descriptors were compared.", "Data points for the training set were chosen to sample the relevant space sufficiently [201], [113].", "Compared to random sampling, this method allowed a reduction of training set sizes up to 90 %, which was shown for static calculations of the methyl chloride molecule [201], [202].", "A similar structure-based sampling scheme was proposed by Ceriotti et al. [203].", "Additionally, a maximum and minimum value can be computed for each representation of a molecule inside the training set.", "Every new structure that is obtained throughout an MD run can be compared to those values to get a measure of reliability of ML predictions.", "If the configuration does not lie within the known region, it can be added to the training set [184], [204].", "Very recently, an active learning approach has been proposed to construct PESs without the need of running MD trajectories.", "The difference between two NN potentials was computed and points were iteratively added at the maxima of this difference surface (or, as phrased in the study, at the minima of the negative difference) [205].", "It becomes evident from the diversity of approaches and training set sizes that a general guide on how to compute the training set and how large it should be for NAMD simulations can not be given.", "It is rather a matter of the efficiency that should be achieved and the computational costs that can be justified.", "Further, the training set strongly depends on the molecule under investigation.", "Especially its size, flexibility and the complexity of the light-induced dynamics play an important role.", "A molecule, whose photodynamics can be described as a two-state problem, such as in a simplified case of ethylene [206], [207], or a molecule, which is rigid, where dynamics mostly lead towards one reaction channel, possibly requires less data points than molecules that exhibit several different reaction channels after photo-excitation." ], [ "Beyond Born-Oppenheimer dynamics", "With an accurate training set for excited states at hand, NAMD simulations can be enhanced with ML models in order to enable the dynamics on time scales otherwise unfeasible.", "The most accurate way to study the dynamics of a molecule would be the full quantum mechanical treatment, which is, however, expensive and limited to a few atoms, even if ML PESs are applied [49], [208], [209], [210], [121], [127], [211], [127], [120], [115], [116], [117], [122], [123], [124], [125], [119].", "A mixed quantum classical treatment is thus often preferred, where the motion of the nuclei are treated classically on one of the ML PESs.", "The mixed quantum classical MD simulation can then be interpreted as a mixed MLMD simulation [165].", "The Born-Oppenheimer approximation allows to separate the nuclear from the electronic degrees of freedom.", "However, this approximation is not valid in the vicinity of avoided state crossings of PESs (or conical intersections, as mentioned before), which play an important role in excited-state dynamics.", "In these critical regions of the PESs, ultrafast rearrangement of the motions of the electrons and the nuclei takes place due to strong couplings.", "As already mentioned, the relevant coupling elements are NACs and SOCs.", "The NACs (denoted as $C^{\\text{NAC}}$ ) are vectorial properties and can be computed as [212], [213], [48] $\\begin{array}{lr}C^{\\text{NAC}}_{ij} \\approx \\langle \\Psi _i \\mid \\frac{\\partial }{\\partial \\mathbf {R}}\\Psi _j \\rangle =\\\\\\frac{1}{E_i-E_j}{\\langle \\Psi _i \\mid \\frac{\\partial H_{el}}{\\partial \\mathbf {R}}\\mid \\Psi _j \\rangle } ~~~\\text{for}~ i \\ne j,\\end{array}$ neglecting the second order derivatives.", "Thus, in the vicinity of a conical intersection, the couplings become very large, whereas they are almost vanishing elsewhere.", "The singularities that arise when two states are degenerate do not only pose an obstacle to quantum chemistry, but consequently also to PESs fitted with ML [113], [112], [114], [118].", "NACs are nevertheless important properties to determine the direction and probability of internal conversion – a transition from one state to another, where the spin multiplicity does not change [48], [51], [52], [214], [8].", "In contrast, the SOCs (denoted as $C^{\\text{SOC}}$ ) are complex-valued properties that determine the rate of intersystem crossing, i.e., the transitions from one state to another, where spin multiplicity does change.", "In standard quantum chemistry programs, SOCs are given as the off-diagonal elements of the Hamiltonian matrix [52], [215], [8]: $C^{\\text{SOC}}_{ij}=\\langle \\Psi _i \\mid \\hat{H}^{SOC}\\mid \\Psi _j \\rangle .$" ], [ "Fitting diabatic potentials", "The numerical difficulties that arise due to discontinuous PESs and singularities of couplings at conical intersections can be circumvented by the use of diabatic potentials instead of adiabatic ones [216], [116], [217], [218], [219], [220].", "In the diabatic basis, the coupling elements are smooth properties and the arbitrary phase of the wavefunction does not have an impact.", "This favors the use of diabiatic PESs.", "Since the output of a quantum chemistry program is generally given in the adiabatic basis, a quasi-diabatization procedure is necessary.", "Strictly speaking, a diabatization procedure is not possible because e.g.", "an infinite number of states is needed for an accurate representation.", "If using a finite number of states, the term quasi-diabatic is employed.", "For simplicity, we still use the notation of diabatic potentials for quasi-diabatic potentials.", "Those have been generated with different methods [221] and for small molecules up to date.", "Examples are the propagation diabatization [222], diabatization by localization [223] or by ansatz [224], [115], diabatization based on couplings or other properties [225], [226], [227], [228], configuration uniformity [229], block-diagonalization [230], CI vectors [169] or (partly) on ML [231], [232], [233], [234], [224], [115], [116].", "Since several years, (modified) Shepard interpolation is used to fit diabatic potentials [235], [129], [236], [237], [238] and also least squares fitting was applied to study the photo dissociation of molecules, such as NH$_3$ and phenol [239], [240].", "In a series, Guo, Yarkony and co-workers developed invariant polynomial NNs [231], [232], [233], [234], [116], [124] to address the excited-state dynamics of NH$_3$ and H$_2$ O by fitting diabatic potential energy matrix elements.", "Absorption spectra as well as branching ratios could be obtained with high accuracy.", "The same authors further fit the diabatic 1,2$^1$ A dipole moment surfaces of NH$_3$ , which can only be fitted accurately if the topography of the PESs is reproduced correctly, validating the previously fitted diabatic potentials [119].", "Habershon, Richings and co-workers used Gaussian process regression (in their notation equal to kernel-ridge regression) to fit diabatic potentials to execute on-the-fly dynamics of the butatrien cation with variational Gaussian wavepackets [127].", "In another study, they applied an on-the-fly MCTDH scheme (DD-MCTDH) and carried out 4-mode/2-state simulations of pyrazine [120].", "By improving the ML approach with a systematic tensor decomposition of kernel ridge regression, the study of 12-mode/2-state dynamics of pyrazine was rendered possible.", "This achievement remains a huge improvement over current MCTDH simulations in terms of accuracy and efficiency [122].", "For the improvement of the diabatization by ansatz procedure, Williams et al.", "[115] applied NNs and enabled the fitting of the electronic low lying states of NO$_3$ .", "The improvement of the diabatization procedure itself is desirable [116], [117], since the generation of meaningful diabatic potentials is often a tedious task and restricts their use tremendously.", "Up to date, no rigorous diabatization procedure exists that allows the diabatization of adiabatic potentials of polyatomic systems by non-experts in this field [116], [119].", "Especially challenging for larger and more complex systems is the number of electronic states within a certain energy range that have to be considered for successful diabatization.", "An increasing computational effort to provide all relevant electronic states is the result, making diabatization further challenging [169].", "Often, more extensive approximations [115], [241], [169], e.g.", "the linear vibronic coupling model [242] are applied.", "We refer the reader to Ref.s [217], [242], [243], [50], [244] for more details on such approaches.", "Despite the advantages of diabatic potentials, due to the before-mentioned drawbacks and the fact that the direct output of a quantum chemical calculation is given in the adiabatic basis, on-the-fly NAMD in the adiabatic representation is often the method of choice for large polyatomic systems, which will be discussed in the following." ], [ "Fitting adiabatic potentials", "In order to execute NAMD simulations in the adiabatic basis, approximations have to be introduced to account for nonadiabatic transitions between different PESs.", "A good trade-off between accuracy and efficiency can be achieved with the surface hopping methodology [199], [200], which is often applied in ML-based NAMD simulations [112], [113], [114], [118].", "In surface hopping, the transitions, or so-called hops, between different states, are computed stochastically and a manifold of trajectories needs to be taken into account to analyze different reaction channels and branching ratios  [51], [245], [8] Several algorithms [245], [246], [247], [52] are frequently used to compute the hopping probability as well as its direction, with Tully's fewest switching algorithm being among the most popular ones [199], [200].", "There, the couplings between adjacent states determine the hopping probability [51].", "Other frequently applied algorithms to compute hopping probabilities are the Landau-Zener [248], [249] and the Zhu-Nakamura approximations [250], [251], [246], [247].", "Those approximations solely rely on the PESs and omit the computation of wavefunction coefficients and couplings.", "Other flavors to account for transitions exist, which have, however, not been used in ML based NAMD studies yet.", "We thus refer the reader to Ref.s [199], [250], [251], [48], [252], [245], [51], [253], [170], [254], [255], [52] for further information." ], [ "NAMD simulations with ML energies and forces", "Based on the fewest switches algorithm, one of the first ML NAMD simulation is carried out by Carbogno et al.", "[130], where the scattering of O$_2$ at Al(III) is studied [256], [130], [111].", "A set of 3768 carefully selected data points [257] allowed for interpolation of the PESs with NNs [256].", "In a first attempt, the authors include a spin-unpolarized singlet PES and a spin-polarized triplet state.", "Strictly speaking, the output of a quantum chemistry simulation for a singlet and triplet state is spin-diabatic [215], [52] and NAMD simulations ideally should carry out a diagonalization to obtain the spin-adiabatic PESs [215], [52].", "The authors took advantage of the adiabatic spin-polarized PES [258] to compute the absolute value of couplings [130].", "In this way, the transitions between the states could be approximated using surface hopping omitting the computation of wavefunction dynamics.", "This study was extended by two-state NAMD simulations of different multiple PESs arising from different spin configurations.", "Findings suggested a high probability of singlet-to-triplet conversion during scattering experiments with a non-zero probability even at low coupling values [256], [130], [111].", "Other studies using the Zhu-Nakamura method [250], [251], [246], [247] to account for nonadiabatic transitions are discussed below.", "This approximation is based solely on energies and neglects the phase of the wavefunction.", "As a drawback, PESs are always assumed to couple to each other, when they are close in energy.", "This holds true for many cases, but one must be aware that strongly and weakly coupled PESs can not be distinguished.", "Hu et al.", "[112] for example trained separate kernel ridge regression models to fit three singlet states of 6-aminopyrimidine.", "For learning, they used 65,316 data points comprising the molecular structures and energies of 6-aminopyrimidine with gradients not fitted, but computed afterwards.", "The data points were obtained from Born-Oppenheimer simulations, which were further clustered into sub-groups, from which the training points were selected randomly.", "As mentioned before, hopping geometries obtained from reference NAMD simulations were taken to find minimum conical intersections and the latter were also included in the training set.", "In contrast, Chen et al.", "[114] trained two deep NNs on 90,000 data points of two singlet states of CH$_2$ NH.", "Again, data points were obtained from Born-Oppenheimer MD simulations and NAMD simulations starting from hopping geometries.", "In both studies, the NAMD simulations of the reference method could be successfully reproduced.", "Instead of approximating the hopping probability, the NACs can also be approximated from PESs, gradients and Hessians [259], [208], [260], [261], [262], [263].", "We made use of this relation and the fact that ML Hessians can be computed efficiently, and carried out NAMD simulations with the surface hopping method for sulfur-dioxide, thioformaldehyde and the methylenimmonium cation [165].", "In addition to energies and forces, SOCs need to be fitted with ML models when states of different spin multiplicities become relevant.", "Furthermore, when approximative schemes for the computation of hopping probabilities fail, the ML models need to learn NACs.", "One of the first studies, where NACs were fitted, used 1,000 and 10,000 data points to train kernel-ridge regression models to reproduce NAMD simulations of a one-dimensional system.", "However, especially in critical regions of the PESs, the ML models could not replace quantum chemical calculations and so 13-16% electronic structure calculations were required during an NAMD simulation [113].", "The authors highlighted this as a drawback, because efficient simulations should be performed purely with ML and should not rely on intermediate quantum chemical calculations.", "Moreover, each entry of the NAC vectors was fitted by a separate kernel-ridge regression model, which turned out to be insufficiently accurate.", "As indicated before, we also aimed for reproducing NAMD simulations with ML.", "We employed multi-layer feed-forward NNs trained on 4,000 data points of 3 singlet states of CH$_2$ NH$_2^+$  [118].", "Short reference NAMD simulations based on electronic structure calculations could be reproduced.", "With the ML NAMD, long simulation times on the order of a nanosecond were successfully reached.", "Significantly different from previous ML NAMD approaches is the smaller size of the training set required to reproduce NAMD simulations.", "Further, a multi-output ML model was used to fit all NAC vectors between different states of same spin multiplicity at once.", "We term such models multi-state models.", "Per definition, kernel-ridge regression, and similar approaches such as linear regression, are single-state models.", "In order to make multi-state predictions of such models possible, the energetic state has to be encoded explicitly by using for example an additional state kernel.", "This procedure enables to model several states simultaneously.", "We studied the use of multi-state descriptors with the QML toolkit [264] for kernel-ridge regression models and showed that a multi-state description is generally superior to a single-state description in terms of accuracy [161].", "Lastly, we want to comment on the NACs as vectorial properties.", "It should be clarified that approaches relating a molecular input directly to NAC values do not provide rotational covariance.", "This drawback is independent of a single-state treatment, i.e., the use of a separate ML model for each coupling value, or a multi-state treatment, where all values are represented in one ML model.", "Very recently, Zhang et al.", "applied a symmetry-adapted high-dimensional NN  [265] and treated the couplings as derivatives of NN representations.", "In this case, electronic friction was modelled via ML and applied for MD simulations of molecules at metal surfaces to treat the electron-nuclei coupling in a rotationally covariant manner.", "For the NAC vectors, we applied a similar strategy (similar also to force-only training for potentials), and implemented them as derivatives of virtual properties (i.e., non-existent in quantum chemistry) built by a deep NN [165]." ], [ "Choosing the right descriptor", "Many of the aforementioned studies use kernel ridge regression models or NNs in combination with distance-based descriptors [114], [112], [113], , [99] such as the matrix of inverse distances or the Coulomb matrix [76].", "It is worth mentioning that the accuracy of the ML PESs also depends on the type of descriptor.", "Molecular descriptors that represent atoms in their chemical and structural environment are often superior to those who treat complete molecules [101], [154], [155].", "The symmetry functions of Behler [266], [267], their weighted counterparts [175], [268] or the FCHL (Faber-Christensen-Huang-Lilienfeld) representation [101], [155] work very well for NNs and the latter also for kernel-ridge regression and additionally provide permutation invariance.", "Further improvement can be provided by message passing neural networks [269].", "Compared to hand-crafted molecular descriptors, the representation of molecules can be seen as a part of a deep NN and, thus, is generated automatically.", "For each training set, an accurate descriptor is intrinsically designed, which accounts for the chemical and structural environment of a molecule.", "Examples for such networks are SchNet [156], [172], the DTNN [133], PhysNet [270], or HIP-NN [271].", "For the excited states, the SchNarc [165] approach offers this type of descriptor.", "To conclude, ML methods are powerful and can be used to speed up current MD approaches for the excited states.", "They have been successfully applied to circumvent existing problems due to the expenses of the underlying electronic structure methods.", "While the fitting of diabatic potentials is generally more favorable, those methods are limited by the challenges that arise in finding meaningful diabatic potentials.", "Up to date, diabatization procedures are tedious and often not feasible for large and complex systems.", "ML models have been successfully applied to improve these processes [115], [116], [117], but methods to treat large and complex polyatomic systems in the diabatic basis are still lacking.", "To investigate the photodynamics of polyatomic molecules, mixed quantum-classical MD simulations in the adiabatic basis thus often remain the method of choice.", "One advantage is that the direct output of a quantum chemical calculation is given in the adiabatic basis and so the obtained potential energies and forces can be directly fitted with an ML model.", "By applying approximations for the computation of transition probabilities from one state to another, the photodynamics can be studied efficiently with ML [112], [114], [165].", "When approaches aim for additionally fitting the coupling values between different electronic states, inconsistencies in the data need to be considered carefully.", "Those have to be either removed from the training set or the training process itself has to be adapted in order to achieve successful training.", "Both approaches have been applied and were used for NAMD simulations [113], [118], [165].", "The current challenges that have to be tackled when replacing quantum chemical calculations in photodynamics simulations of large and complex polyatomic molecules are the efficient training set generation and the accurate fitting of a manifold of energetic states, forces, and couplings between them.", "Many recent ML approaches required several thousand of data points for small polyatomic molecules and energies, forces, and couplings were often trained in separate ML models, leading to unsatisfactory accuracy.", "The development of an ML model that can treat all properties for photodynamics simulations at once is clearly desirable.", "While current studies struggle with fitting the excited states of one molecule, the transferability of ML potentials for the excited states is far from being achieved.", "Studies that try to fit more molecules than just one could give first insights into the possibility of extrapolating throughout chemical compound space with ML also for the excited states." ], [ "Acknowledgements", "This work was financially supported by the Austrian Science Fund, W 1232 (MolTag) and the uni:docs program of the University of Vienna (J.W.).", "P. M. thanks the University of Vienna for continuous support, also in the frame of the research platform ViRAPID." ] ]

[ [ "Spectral dimensions and dimension spectra of quantum spacetimes" ], [ "Abstract Different approaches to quantum gravity generally predict that the dimension of spacetime at the fundamental level is not 4.", "The principal tool to measure how the dimension changes between the IR and UV scales of the theory is the spectral dimension.", "On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterised by a discrete complex set -- the dimension spectrum.", "Here we show that these two notions complement each other and the dimension spectrum is very useful in unravelling the UV behaviour of the spectral dimension.", "We perform an extended analysis highlighting the trouble spots and illustrate the general results with two concrete examples: the quantum sphere and the $\\kappa$-Minkowski spacetime, for a few different Laplacians.", "In particular, we find out that the spectral dimensions of the former exhibit log-periodic oscillations, the amplitude of which decays rapidly as the deformation parameter tends to the classical value.", "In contrast, no such oscillations occur for either of the three considered Laplacians on the $\\kappa$-Minkowski spacetime." ], [ "Introduction", "The concept of spacetime, understood as a differentiable manifold, has proven to be extremely fruitful in modelling gravitational phenomena.", "However, it is generally expected that the smooth geometry breaks down at small scales or high energies, due to the quantum effects.", "Consequently, many of the familiar notions, such as causality, distance or dimension, have to be refined within the adopted new mathematical structure.", "An essential property of the hypothetical quantum theory of gravity, as well as a useful input for constructing it, is the ability to provide meaningful and testable predictions concerning deviations of physics from general relativity.", "The first step in this direction can be done by characterizing the structure of (static) quantum spacetime, which replaces the classical differentiable manifold, but can be seen as a certain tangible generalisation of the latter.", "This is possible only if an unambiguous notion of a spacetime could be provided.", "In some of the approaches, such a notion is preserved only at the intermediate — semiclassical — level, while in the full theory spacetime breaks down into discrete elements, determined either by the fundamental length scale or a regularization cutoff (see [1] for a conceptual discussion).", "In analogy to systems in condensed matter physics, configurations of the underlying “atoms of spacetime” may form different phases, while (classical) continuous spacetime should emerge in the limit in at least one of them.", "Other phases will naturally share some features with the classical phase.", "Let us stress in this context the distinction between the continuum limit, which is a transition from a discrete proto-spacetime to the continuous (but still quantum) spacetime, and the classical limit, in which we completely recover familiar manifolds of general relativity.", "In the recent years, calculations of the effective number of spacetime dimensions have become an ubiquitous method to characterise the quantum spacetime.", "This is one of only a few tools allowing us to find some order in the diverse landscape of quantum gravity models [2], whose predictions are notoriously difficult to compare.", "The dimension can be defined in many different ways, some of which are based on mathematical assumptions and some on physical concepts, such as thermodynamics [3].", "The most popular notion remains the so-called spectral dimension, which can be seen as determined by the mathematical properties of spectral geometry or by a physical (fictitious) diffusion process.", "The advantage of the spectral dimension in the latter context is its dependence on a parameter (auxiliary diffusion time) that can be identified with the length scale at which the geometry is probed.", "Therefore, it is naturally interpreted as a measure of how the dimension of spacetime changes with scale.", "Starting with the seminal paper [4] (updated in [5]), belonging to causal dynamical triangulations approach to quantum gravity, the spectral dimension has been calculated for Hořava-Lifshitz gravity [6], asymptotic safety scenario [7], nonlocal quantum gravity [8], spin foam models [9] (and kinematical states of loop quantum gravity in general [10]), causal sets [11], [12] and multifractional spacetimes [13].", "The almost universal prediction is the dimensional reduction at the smallest scales (the UV limit) to the value of 2, for which general relativity would actually become power-counting renormalizable.", "On the other hand, in some cases values different from 2 are obtained in the UV limit, especially in non-classical phases of models with non-trivial phase diagram.", "The situation is similar for quantum gravity in less than $3+1$ topological dimensions [14], [6], [7], [13] (see also [15]).", "Quantum spacetime often turns out to be described in terms of broadly understood noncommutative geometry, which is also sometimes treated as a stand-alone approach to quantum gravity.", "A particular example is the $\\kappa $ -Minkowski spacetime [16], associated with $\\kappa $ -Poincaré algebra [17], [18] and widely considered in doubly/deformed special relativity and relative locality.", "As it was shown in [19], the small-scale behaviour of the spectral dimension of $\\kappa $ -Minkowski spacetime (first calculated in [20]) depends on the Laplacian, which can be chosen according to several distinct principles.", "For $3+1$ -dimensional $\\kappa $ -Minkowski spacetime, there are at least three possibilities in the UV: the dimension decreasing to 3, growing to 6 or diverging.", "However, as we will discuss in Subsec.", "REF , there may be a way to reconcile these contrasting results.", "Let us also note that [21] presents an example of a noncommutative toy model (with ${\\rm U}(1) \\times {\\rm SU}(2)$ momentum space) that exhibits the dimensional reduction to 2, i.e.", "the value obtained in the approaches to quantum gravity mentioned before.", "Recently, it has also been suggested [22], [23], in the context of multifractional theories [24], that the dimension of quantum spacetimes can acquire complex values.", "These, on the other hand, result in log-periodic oscillations in various physical quantities [25] and can possibly affect the CMB spectrum [23], introduce a stochastic noise to gravitational waves [26] or modify the thermodynamics of photons [27].", "More generally, complex dimensions (or complex critical exponents) and the corresponding log-periodic oscillations can also arise in the systems with discrete scale invariance, which is observed in many contexts, including some particular cases of holography [28], as well as condensed matter physics, earthquakes and financial markets, see e.g.", "[29].", "Meanwhile, it has already been recognised by Connes and Moscovici in 1995 [30] that quantum spaces — understood as spaces determined by noncommutative algebras of observables — ought to be characterised by a discrete subset of the complex plane – the dimension spectrum, rather than a single number.", "More precisely, in noncommutative geometry the dimension spectrum is defined as the set of poles of the spectral zeta functions of geometrical provenance.", "These, on the other hand, are intimately connected with the celebrated heat trace expansion via the Mellin functional transform (cf.", "[31], [32] and also [33]).", "Both spectral zeta functions and heat trace expansions are indispensable tools in quantum field theory [34], [35], [36], also in its noncommutative version [37].", "The heat trace can be utilised to compute the one-loop effective action and allows to study the short-distance behaviour of propagators, along with quantum anomalies and some non-perturbative effects [38].", "Consequently, it is justified to expect that the structure of the entire dimension spectrum of a given noncommutative geometry is relevant for physics.", "Among the noncommutative spaces with known dimension spectra an interesting example is provided by the Podleś quantum sphere [39].", "For a particular choice of geometry [40], the dimension spectrum turns out to have surprising features [41]: Firstly, it exhibits the dimension drop (also called the dimensional reduction) from 2 to 0.", "Secondly, the associated spectral zeta-function contains poles outside the real axis, suggesting self-similarity.", "Thirdly, these poles are of second order, which is characteristic for spaces with conical singularities [42].", "Finally, the corresponding heat trace expansion turns out to be convergent – in sharp contrast to the case of smooth manifolds, where it is only asymptotic.", "As we will show, the above features are shared by two other geometries (i.e.", "other Laplacians) on the quantum sphere but the fine details of their heat trace expansions are different.", "The latter fact strengthens the observation made in [19] by one of us in the case of $\\kappa $ -Minkowski space that the spectral dimension characterises a given quantum space equipped with a specific Laplacian.", "Let us note that the effective dimensionality of spaces with the topological dimension lower than 4 is relevant not only from the perspective of toy models of lower dimensional (quantum) gravity but also in the context of the problem of entanglement entropy [43].", "The reason is that the latter can be derived from the heat trace over the boundary of some region of space [44].", "The purpose of this work is to revisit the concept of the spectral dimension from the perspective of the dimension spectrum.", "We show that the latter is a valuable rigorous tool to study the UV behaviour of the spectral dimension.", "To this end, we firstly provide, in Section , the definitions of both concepts and highlight the trouble spots.", "Then, in Section , we compute the dimension spectra and spectral dimensions of three Laplacians on the Podleś quantum sphere [39].", "We show that the dimension drop observed by Benedetti [20] has a finer structure with the square-logarithmic decay and log-periodic oscillations.", "For values of the deformation parameter $q$ close to the classical value 1 the amplitude of these oscillations becomes very small and they are invisible in numerical plots.", "On the other hand, their presence is clearly attested for any $q$ by non-real numbers in the dimension spectrum.", "Next, in Section , we study the dimension spectra of three different Laplacians on the $\\kappa $ -Minkowski spacetime in 2, 3 and 4 topological dimensions.", "We utilise these to identify the leading and subleading short-scale behaviour of the spectral dimensions obtained in [19].", "This example uncovers an ambiguity in the definition of the spectral dimension related to the order of the `Laplacian-like' operator.", "We summarise our findings in Section and discuss their consequences for model-building in quantum gravity.", "The usual perspective in quantum gravity is to introduce the spectral dimension as a characteristic of the fictitious diffusion (or random walk) process on a given configuration space.", "Let us first consider a diffusion process on a Riemannian manifold $(M,g)$ of topological dimension $d$ , which is described by the heat equation $\\frac{\\partial }{\\partial \\sigma } K(x,x_0;\\sigma ) + \\Delta K(x,x_0;\\sigma ) = 0\\,,$ with a second order differential operator $\\Delta $ in variable $x$ and an auxiliary time variable $\\sigma \\ge 0$ (playing the role of a scale parameter).", "In general, the operator $\\Delta $ does not need to be the standard Laplacian $-g^{\\mu \\nu } \\nabla _\\mu \\nabla _\\nu $ , $\\mu ,\\nu = 1,\\ldots ,d$ — it can be a Laplace-type operator [38] or even a pseudodifferential one (cf.", "[45]).", "In order to solve Eq.", "(REF ) one needs to impose appropriate initial/boundary conditions.", "Typically, one chooses the initial condition of the form $K(x,x_0; \\sigma = 0) = \\frac{\\delta ^{(d)}(x - x_0)}{\\sqrt{|\\det g(x)|}}\\,.$ In particular, in the case of $d = 4$ and the flat Euclidean metric, the solution to (REF ) can be expressed as a (inverse) Fourier transform $K(x,x_0;\\sigma ) = \\int \\frac{d^4 p}{(2\\pi )^4}\\, e^{i p_\\mu (x - x_0)^\\mu } e^{-\\sigma {\\cal L}(p)}\\,,$ where ${\\cal L}$ is the momentum space representation of $\\Delta $ .", "To characterise the diffusion process (REF ) we may use the return probability ${\\cal P}(\\sigma ) = \\frac{1}{\\mathrm {vol}\\, V} \\int _V d^4x \\sqrt{|\\det g|}\\, K(x,x;\\sigma )\\,, \\qquad \\sigma > 0\\,,$ where we integrate over a fiducial volume $V$ .", "(It factorises in the leading term and therefore can be taken to infinity if needed).", "${\\cal P}(\\sigma )$ is the probability that after the time $\\sigma $ the diffusion will return to the same point $x \\in V \\subset M$ .", "The spectral dimension is now taken to be a function of the scale parameter $\\sigma $ defined as $d_S(\\sigma ) =-2 \\, \\frac{\\partial \\log {\\cal P}(\\sigma )}{\\partial \\log \\sigma } = -\\frac{2\\sigma }{{\\cal P}(\\sigma )}\\, \\frac{\\partial {\\cal P}(\\sigma )}{\\partial \\sigma }\\,.$ In the case of $M$ being a flat Euclidean space of topological dimension $d$ , we have $d_S(\\sigma ) = d$ for all $\\sigma $ .", "Therefore, in general, if $d_S(\\sigma ) \\in {\\mathbb {N}}$ for a given value of $\\sigma $ , it can be interpreted as the effective dimension such that the ordinary diffusion process in $d_S(\\sigma )$ -dimensional Euclidean space would approximately behave as the $\\Delta $ -governed diffusion on $M$ .", "If we choose the appropriate Laplacian, small values of $\\sigma $ allow us to probe the ultraviolet structure of $M$ , while large ones correspond to its infrared geometry.", "However, for sufficiently large $\\sigma $ the function (REF ) becomes sensitive to the finite size of $M$ and the curvature of $g$ .", "The original definition (REF ) applies solely in the context of Riemannian manifolds.", "The departure from smooth geometry requires a suitable generalisation.", "Within the framework of deformed relativistic symmetries, the noncommutative geometry of spacetime is accompanied by a curved momentum space.", "Typically, the latter is a non-Abelian Lie group, equipped with an invariant Haar measure $\\mu $ .", "This suggests a natural generalisation of formula (REF ) to (cf.", "[15], [19], [21]) $K(x,x_0;\\sigma ) = \\int \\frac{d\\mu (p)}{(2\\pi )^4}\\, e^{i p_\\mu (x - x_0)^\\mu } e^{-\\sigma {\\cal L}(p)}\\,,$ representing the noncommutative Fourier transform (i.e.", "inverse of the group Fourier transform) of the function $e^{-\\sigma {\\cal L}}$ .", "More generally, one can adopt the definition of a heat operator $e^{-\\sigma T}$ , which applies for any closed, possibly unbounded, operator $T$ acting on a separable Hilbert space $\\mathcal {H}$ (cf.", "[45] and [33]).", "If $T$ is bounded from below and $e^{-\\sigma T}$ is trace-class, one defines the heat trace (or the `return probability') of an abstract operator $T$ as $\\mathcal {P}(\\sigma ) =\\operatorname{Tr}_{\\mathcal {H}} e^{-\\sigma T} = \\sum _{n=0}^{\\infty } e^{-\\sigma \\lambda _n(T)},$ where $\\lambda _n(T)$ are the eigenvalues of $T$ counted with their multiplicities.", "For a compact Riemannian manifold $M$ and $T = \\Delta $ , the trace can be computed via the standard integral kernel methods and one obtains (from now on we drop the normalization) $\\operatorname{Tr}_{L^2} e^{-\\sigma \\Delta } = \\int _M d^4x \\sqrt{|\\det g|}\\, K(x,x;\\sigma ) = {\\cal P}(\\sigma )\\,,$ for any $\\sigma > 0$ .", "Formula (REF ) allows us to extend the notion of the spectral dimension (REF ) beyond the realm of smooth manifolds.", "For an abstract operator $T$ one has $d_S(\\sigma ) = 2\\sigma \\, \\frac{\\operatorname{Tr}_{\\mathcal {H}} T e^{-\\sigma T}}{\\operatorname{Tr}_{\\mathcal {H}} e^{-\\sigma T}} = 2\\sigma \\, \\frac{\\sum _{n=0}^{\\infty } \\lambda _n(T)\\, e^{-\\sigma \\lambda _n(T)}}{\\sum _{n=0}^{\\infty } e^{-\\sigma \\lambda _n(T)}}\\,.$ Let us now point a few trouble spots with usage of (REF ) and hence the spectral dimension: Firstly, one needs to make sure that formula (REF ) is well defined.", "The trace-class property of $e^{-\\sigma T}$ for all $\\sigma > 0$ is guaranteed on general grounds if $T$ is a classical pseudodifferential operator on a compact manifold $M$ [31], [32], but it may fail, for instance, on infinite dimensional spaces [46].", "If the spacetime manifold $M$ is not compact, then $e^{-\\sigma T}$ is typically not trace-class (actually, not even compact) even if $T$ is an honest classical pseudodifferential operator.", "Consequently, to define the heat trace one needs an IR cut-off in the form of a trace-class operator $F$ $\\mathcal {P}(\\sigma ,F) =\\operatorname{Tr}_{\\mathcal {H}} F \\, e^{-\\sigma T}.$ On a manifold, one can simply take $F$ to be a function projecting on a compact fiducial volume $V$ , as done in formula (REF ).", "Then, after restoring the normalisation, $V$ can eventually be taken to infinity, showing the independence of the spectral dimension on the IR regularisation.", "On the other hand, there is no reason to assume that a similar factorisation would take place outside of the realm of manifolds.", "Although it can be demonstrated for specific examples, such as the $\\kappa $ -Minkowski spacetime which we consider in Sec.", ", in general one should expect to encounter the notorious IR/UV mixing problem [47].", "The multiplicative factor 2 in Eq.", "(REF ) originates from the fact that the Laplacian is a second order differential operator.", "This can be easily adapted if $T$ is (pseudo)differential operator of any order $\\eta > 0$ by redefining $d_S(\\sigma ) =-\\eta \\frac{\\partial \\log {\\cal P}(\\sigma )}{\\partial \\log \\sigma }.$ However, beyond the safe realm of smooth manifolds, the order of $T$ is not a priori defined and (REF ) becomes ambiguous.", "We shall illustrate this problem in Section .", "The direct computation of the spectral dimension from formula (REF ) requires full knowledge about the spectrum of $T$ , which is seldom granted.", "One could resort to asymptotic formulae for heat traces (cf.", "Formula (REF )) to unveil the small-$\\sigma $ behaviour of the spectral dimension, but the result can be very misleading if one quits the UV sector [48].", "A consistent interpretation of $d_S(\\sigma )$ as a scale-dependent dimension of spacetime requires it to reach the “classical value” in the IR sector.", "However, the latter is equal to 4 only in the very specific instance of ${\\mathbb {R}}^4$ , in which case actually $d_S(\\sigma ) = 4$ independently of the value of $\\sigma $ .", "If the classical spacetime has compact topology or non-trivial curvature, then either $d_S(\\sigma )$ tends to 0 or grows to infinity as $\\sigma \\rightarrow \\infty $ , depending on whether the operator $T$ has a trivial kernel or not (cf.", "$\\Delta ^{\\text{sc}}$ versus $\\Delta ^{\\text{sp}}$ on Fig.", "REF ).", "As for the latter, one can use the notion of the spectral variance [49] to remove the zero mode.", "In either case, in order to recover the correct dimension in the IR, one has to match the large-scale behaviour of $d_S(\\sigma )$ of the quantum model with the corresponding classical spacetime manifold, as it was done for the CDT in [14].", "Finally, in order to study the spectral dimension of spacetime, which is characterized by a Lorentzian metric, one first has to perform the Wick rotation of it, i.e.", "an analytic continuation to the Euclidean signature.", "This is a rather cumbersome procedure even in the case of curved pseudo-Riemannian manifolds and it is likely to be even more problematic in quantum spacetime models (see, however, [50]).", "We shall not explore this issue here since the considered examples are either Euclidean from the start (quantum spheres) or have the well defined Euclidean counterparts ($\\kappa $ -Minkowski momentum spaces [19])." ], [ "Dimension spectrum from asymptotic expansion", "Let us now restart with the spacetime modelled by a Riemannian manifold $M$ and turn towards the notion of the dimension spectrum.", "Abundant information about the geometry of $M$ can be learned from the celebrated heat kernel expansion [31], [32], [38]: $\\mathcal {P}(\\sigma ) \\; \\underset{\\sigma \\downarrow 0}{\\sim } \\;\\sum _{k=0}^\\infty a_k(T)\\, \\sigma ^{(k-d)/\\eta } + \\sum _{\\ell =0}^{\\infty } b_\\ell (T)\\, \\sigma ^{\\ell }\\, \\log \\sigma \\,,$ where $\\eta $ is the order of the pseudodifferential operator $T$ .", "A few comments about formula (REF ) are in order: Firstly, the infinite series in formula (REF ) are asymptotic series, which are in general divergent for any $\\sigma > 0$.", "Nevertheless, the formula has a precise meaning as an asymptotic expansion (cf.", "[51] and [33]).", "It provides accurate information about the small-$\\sigma $ asymptotic behaviour of $\\mathcal {P}(\\sigma )$ , but in general it fails for larger values of $\\sigma $ [48].", "Secondly, if $T$ is a differential operator, the coefficients $b_{\\ell }(T) = 0$ and $a_k(T)$ are locally computable quantities of geometrical origin — that is they can be expressed as $a_k(T) = \\int _{M} \\alpha _k^T(x)$ .", "On the other hand, if $T$ is only pseudodifferential, then $a_k(T)$ for $(k-d) \\in \\eta \\mathbb {N}$ are not locally computable [52].", "Thirdly, one sees that $d = \\dim M$ is encoded in Formula (REF ) as the leading small-$\\sigma $ behaviour, while $a_0(T) \\propto \\operatorname{vol}(M)$ .", "If $T$ is a scalar Laplace-type operator and $M$ has no boundary, then the odd coefficients $a_{2n+1}(T)$ vanish.", "The second coefficient reads $a_2(T) = \\tfrac{1}{6} \\, (4 \\pi )^{-d/2} \\int _M d^d x\\sqrt{g(x)} \\mathcal {R}(x)$ , where $\\mathcal {R}$ is the scalar curvature, whereas $a_{2n}(T)$ for $n \\ge 2$ involve higher order invariants constructed from the Riemann tensor.", "If $T$ acts on a vector bundle $E$ over $M$ , then $a_{n}(T)$ involve also the curvature of $E$ — see [38] for a complete catalogue.", "Finally, Formula (REF ) extends to the non-compact setting.", "As mentioned earlier, this requires an IR regularising operator — typically, a compactly supported smooth function $f$ on $M$ .", "In such a case, $\\mathcal {P}(\\sigma ,f)$ still admits an asymptotic expansion of the form (REF ) but its coefficients now depend on $f$ .", "Let us now leave the domain of smooth manifolds and trade the pseudo-differential operator for a positive unbounded operator acting on a separable Hilbert space $\\mathcal {H}$ .", "In this context, one expects a more general form of the heat trace expansion (cf.", "[33]): $\\mathcal {P}(\\sigma ) = \\operatorname{Tr}_{\\mathcal {H}} e^{-\\sigma T} \\; \\underset{\\sigma \\downarrow 0}{\\sim } \\;\\sum _{k=0}^\\infty \\sum _{m \\in {\\mathbb {Z}}} \\sum _{n=0}^{p} a_{z(k,m),n}\\, (\\log \\sigma )^n\\, \\sigma ^{-z(k,m)}\\,,$ for a (discrete, but possibly infinite) set of complex numbers $z(k,m)$ .", "We define the dimension spectrum of the operator $T$ as the collection of exponents (i.e.", "a set of numbers): $\\operatorname{Sd}(T) =\\bigcup _{k,m} z(k,m) \\subset {\\mathbb {C}}\\,,$ whence the number $(p+1)$ , capturing the maximal power of $\\log \\sigma $ terms in (REF ), is called the order of the dimension spectrum $\\operatorname{ord}\\operatorname{Sd}(T)$ .", "It is also useful to define the maximal (real) dimension in the spectrum $d_{\\operatorname{Sd}} =\\sup _{z \\in \\operatorname{Sd}}\\, {\\rm Re}(z)\\,.$ From Formula (REF ) one immediately reads out that if $T$ is a classical pseudodifferential operator, then $\\operatorname{Sd}(T) \\subset \\tfrac{1}{\\eta } (d - \\mathbb {N}) = \\lbrace (d-k)/\\eta \\, \\vert \\, k \\in \\mathbb {N}\\rbrace $ and $\\operatorname{ord}\\operatorname{Sd}(T) \\le 2$ , where $\\eta = \\eta (T)$ is the order of $T$ and $d$ is the dimension of the underlying manifold.", "In particular, if $T$ is a differential operator, then $\\operatorname{ord}\\operatorname{Sd}(T) = 1$ .", "In either case we have $d_{\\operatorname{Sd}} = d/\\eta $ , as expected.", "Dimension spectra of higher order can be found beyond the realm of classical pseudodifferential operators [53], [42], [54], [33].", "In particular, dimension spectra of order 3 were found for Fuchs-type operators on manifolds with conical singularities [42].", "Surprisingly enough, the same feature was discovered in the very different context of the quantum sphere [55], [41] (cf.", "Section ).", "Meanwhile, the presence of complex numbers in $\\operatorname{Sd}$ , typical for fractal spaces [56], is interpreted as a signature of the self-similar structure [57].", "In view of formula (REF ) this implies in turn that the heat trace exhibits oscillations [25], [27].", "The above definition of dimension spectrum is borrowed from noncommutative geometry à la Connes [58].", "The central notion of the latter is a spectral triple $(\\mathcal {A,H,D})$ consisting of a noncommutative algebra $\\mathcal {A}$ of space(time) observables represented on $\\mathcal {H}$ and an unbounded operator $\\mathcal {D}$ acting on $\\mathcal {H}$ , all tied together with a set of axioms.", "In this context, one talks about the dimension spectrum of a spectral triple $\\operatorname{Sd}(\\mathcal {A,H,D})$ [30] (cf.", "also [33]), which is the union of dimension spectra of a family of operators The precise statement is: If $(\\mathcal {A,H,D})$ is a spectral triple and $\\left|{\\mathcal {D}} \\right|$ admits an expansion of the form (REF ), then $\\operatorname{Sd}(\\left|{\\mathcal {D}} \\right|) \\subset \\operatorname{Sd}(\\mathcal {A,H,D})$ and $\\operatorname{ord}\\operatorname{Sd}(\\left|{\\mathcal {D}} \\right|) \\le \\operatorname{ord}\\operatorname{Sd}(\\mathcal {A,H,D})$ ..", "These originate from the fluctuations of the bare operator $\\mathcal {D}$ .", "On the physical side, considering fluctuated $\\mathcal {D}$ amounts to dressing it with all gauge potentials available for a given spectral triple.", "Thus, one could say that $\\operatorname{Sd}(\\mathcal {D})$ refers to the `pure gravity' scenario.", "The internal fluctuations of geometry caused by the gauge fields will in general change the dimension spectrum, including its order.", "They will not, however, change the maximal dimension $d_{\\operatorname{Sd}}$ (see [33]).", "Observe that the spectral dimension will also change in presence of other `non-gravitational' fields as the bare Laplacian will get dressed by a potential.", "It is also worth noting (see [33], [59] for the full story) that if a positive operator $T$ admits an expansion of the form (REF ), then the associated spectral zeta-function $\\zeta _T$ defined as $\\zeta _T(s) =\\operatorname{Tr}T^{-s}, \\; \\text{ for } \\; {\\rm Re}(s) \\gg 0$ admits a meromorphic extension to the whole complex plane.", "This enjoyable interplay is revealed with the help of the Mellin transform: $\\int _{0}^\\infty \\operatorname{Tr}e^{-\\sigma T} \\, \\sigma ^{s-1} \\, d\\sigma = \\Gamma (s)\\, \\zeta _T(s)\\,, \\; \\text{ for } \\; {\\rm Re}(s) \\gg 0\\,.$ Formula (REF ) allows us furthermore to retrieve the complete structure of poles of $\\zeta _T$ .", "In particular, the set $\\operatorname{Sd}(T)$ coincides with the set of poles of the function $\\Gamma \\cdot \\zeta _T$ and, moreover, $\\forall \\, z \\in \\operatorname{Sd}\\quad \\underset{s = z}{\\operatorname{Res}}(s - z)^n \\Gamma (s)\\, \\zeta _T(s) = (-1)^n n!\\, a_{z,n}\\,.$ Recall that the Gamma function has simple poles at non-positive integers with $\\operatorname{Res}_{s=-\\ell } \\Gamma (s) = (-1)^\\ell /\\ell !$ .", "In summary, a term of order $(\\log \\sigma )^{n-1} \\sigma ^{-z}$ in the small-$\\sigma $ expansion of $\\mathcal {P}(\\sigma )$ corresponds to a pole of $\\Gamma \\cdot \\zeta _T$ of order $n$ at $z \\in {\\mathbb {C}}$ .", "Let us now discuss the properties and problems of the dimension spectrum, as compared with the spectral dimension: On top of the problem of checking whether, for a given operator $T$ , Formula (REF ) is well-defined, one needs to prove the existence of an asymptotic expansion of the form (REF ).", "This is guaranteed if $T$ is a classical elliptic pseudodifferential operator [52] (cf.", "also [33]).", "On the other hand, beyond this realm it is a formidable task and no general results are available (see, however, [33], [59]).", "In the non-compact case, the dimension spectrum would suffer from the same problems with the IR/UV mixing.", "Both the coefficients $a_{z,n}$ and the set $\\operatorname{Sd}$ , as well as its order, would in general depend upon the choice of the IR regularisation (REF ).", "As in the case of the spectral dimension, the dimension spectrum depends on the order $\\eta $ of the operator at hand.", "This may result in the ambiguous interpretation of $d_{\\operatorname{Sd}}$ as the dimension of the underlying space (cf.", "Section ).", "Because the exponents $z(k,m)$ are in general complex numbers, the heat trace $\\mathcal {P}(\\sigma )$ will exhibit oscillations as $\\sigma $ tends to 0 and hence so will the spectral dimension $d_S$ .", "This oscillatory behaviour of $d_S$ in the UV may be hard to detect in the numerical plots, as we illustrate in Sec.", ".", "Within the dimension spectrum it is, however, well separated from the leading divergence rate in the UV, which is naturally given by $d_{\\operatorname{Sd}}$ .", "Moreover, we have $d_S(0) = \\eta \\, d_{\\operatorname{Sd}}$ , provided that the limit $d_S(0)$ exists.", "The dimension spectrum has no issues with the zero modes of $T$ If $\\ker T$ is non-trivial, the spectral zeta-function $\\zeta _T$ is not well defined, but this can be easily circumvented — see [33].. Also, in contrast to the spectral dimension, the curvature does not affect the exponents $z(k,m)$ contained in the spectrum but only the coefficients $a_{z,n}$ .", "As in the case of the spectral dimension, the dimension spectrum is not formally defined for spaces with Lorentzian signature (unless they are Wick-rotated).", "This is because the relevant operators are wave-operators, which are hyperbolic and not elliptic (cf.", "[38] and [33]).", "Whereas the formal heat kernel expansion (REF ) does not exist, an analogue of $a_k$ 's — called the Hadamard coefficients — can be defined (see [60]).", "In the next Section, we illustrate the (dis)similarities of the two notions of dimensionality via a careful analysis of two classes of examples." ], [ "Quantum sphere ", "A quantum sphere was first introduced by Podleś [39] as a quantum homogeneous space of the deformed group $\\mathrm {SU}_q(2)$ .", "As a topological space it is described via the complex $^*$ -algebra $\\mathcal {A}_q$ generated by $A = A^*,\\, B$ and $B^*$ subject to the relations $AB = q^2 BA\\,, && AB^* = q^{-2} B^*A\\,, && BB^* = q^{-2} A\\, (1 - A)\\,, && B^*B = A\\, (1 - q^2 A)\\,,$ for a deformation parameter $0 < q < 1$ .", "In the limit $q \\rightarrow 1$ , one recovers the classical algebra of continuous functions on the unit 2-sphere.", "This abstract algebra is faithfully represented on a Hilbert space $\\mathcal {H}_q$ spanned by orthonormal vectors $| l,m \\rangle _\\pm $ , with $m \\in \\lbrace -l,-l+1,\\ldots ,l\\rbrace $ and $l \\in {\\mathbb {N}}+ \\tfrac{1}{2}$ , mimicking the chiral spinors on $S^2$ [40].", "The geometry of quantum spheres has been extensively studied within the framework of spectral triples [61], [62], [63], [64], [40], [41].", "Among the known geometries particularly interesting is the one equivariant under the action of the Hopf algebra $\\mathcal {U}_q(\\mathfrak {su}(2))$ [40].", "Its dimension spectrum and heat trace were computed analytically in [41].", "These turned out to exhibit a number of surprising features: The maximal dimension in the dimension spectrum $d_{\\operatorname{Sd}}$ is equal to 0.", "The spectrum $\\operatorname{Sd}$ is of order 3 and the leading term in the expansion (REF ) is $\\log ^2 \\sigma $ .", "$\\operatorname{Sd}$ is a regular lattice on the complex plane, which corresponds to $\\log $ -periodic oscillations of the heat trace and suggests a self-similar structure of the quantum sphere.", "The expansion (REF ), expected to be only asymptotic, is actually convergent for all $\\sigma $ .", "The quantum sphere served in [20] as a toy example to illustrate the phenomenon of dimension drop in quantum spacetimes.", "The operator determining the geometry employed in [20] originates from the Casimir operator on the Hopf algebra $\\mathcal {U}_q(\\mathfrak {su}(2))$ .", "It could be regarded as a `scalar Laplacian', which differs slightly from the `spinor Laplacian' derived from the `Dirac operator' introduced in [40] — see Subsection REF .", "Below we present the computations of both the dimension spectrum and the spectral dimension for the above mentioned two Laplacians on the quantum sphere and a third — `simplified' — one.", "The latter allows for explicit analytic computations, while capturing the essential small-$\\sigma $ behaviour of heat traces for both scalar and spinor Laplacians." ], [ "Simplified Laplacian ", "The simplified Dirac operator $\\mathcal {D}_q^S$ on a quantum sphere was introduced in [41].", "Together with the algebra $\\mathcal {A}_q$ and the Hilbert space $\\mathcal {H}_q$ , it satisfies all of the axioms of a spectral triple.", "Its square — the simplified Laplacian — acts on basis vectors of $\\mathcal {H}_q$ as $\\Delta _q^\\text{sm}| l,m \\rangle _\\pm = u\\, q^{-(2l+1)} | l,m \\rangle _\\pm \\,,$ with $u = u(q) =(q^{-1} - q)^{-2}$ .", "The simple exponential form of eigenvalues implies a self-similarity relation $\\Delta _q^\\text{sm}=(\\mathcal {D}_q^S)^2 = u \\big \\vert \\mathcal {D}_{q^2}^S \\big \\vert $ .", "Note that $\\Delta _q^\\text{sm}$ has no zero modes and does not have a well-defined classical limit $q \\rightarrow 1$ .", "The heat trace associated with $\\Delta _q^\\text{sm}$ reads: $\\mathcal {P}_q^\\text{sm}(\\sigma ) &= \\operatorname{Tr}_{\\mathcal {H}_q} e^{-\\sigma \\Delta _q^\\text{sm}} = \\sum _{+,-} \\sum _{l \\in \\mathbb {N}+ 1/2} \\sum _{m=-l}^l {}_\\pm \\langle l,m| e^{-\\sigma \\Delta _q^\\text{sm}} | l,m \\rangle _\\pm \\\\&= 2 \\sum _{l \\in \\mathbb {N}+ 1/2} (2l+1) \\exp \\left( -\\sigma u\\, q^{-(2l+1)} \\right) = 4 \\sum _{n=1}^{\\infty } n \\exp \\left( -\\sigma u\\, q^{-2n} \\right).", "$ When $\\sigma $ tends to infinity, $\\mathcal {P}_q^\\text{sm}(\\sigma )$ decays as $e^{-\\sigma u q^{-2}}$ .", "However, the small-$\\sigma $ behaviour of $\\mathcal {P}_q^\\text{sm}(\\sigma )$ cannot be easily deduced from the doubly exponential series in Formula (REF ).", "On the other hand, the associated zeta function is a simple geometric series: $\\zeta _{\\Delta _q^\\text{sm}}(s) = \\operatorname{Tr}_{\\mathcal {H}_q} (\\Delta _q^\\text{sm})^{-s} = 4 u^{-s} \\sum _{n=1}^{\\infty } n\\, q^{2 n s} = 4 u^{-s} q^{2s} (1-q^{2s})^{-2},$ which is meromorphic on the entire complex plane.", "It has double poles located solely on the imaginary axis, for $s = \\pi i j / \\log q$ with $j \\in {\\mathbb {Z}}$ .", "In this specific case one can deduce, via (the inverse of) Formula (REF ), the explicit `non-perturbative' formula for the heat trace [41]: $\\mathcal {P}_q^\\text{sm}(\\sigma ) = \\tfrac{1}{4 \\log ^2q}\\, \\big [ 2 \\log ^2(u \\sigma ) + G\\big (\\log (u\\sigma )\\big )\\, \\log (u\\sigma ) + F\\big (\\log (u \\sigma )\\big ) \\big ] + R_{\\rm sm}(u\\sigma )\\,,$ where $F$ and $G$ are periodic bounded smooth functions on ${\\mathbb {R}}$ , defined as $G(x) &=4 \\gamma - 4 \\sum _{j \\in {\\mathbb {Z}}^*} \\Gamma (\\tfrac{\\pi i}{\\log q}\\, j)\\, e^{\\pi i j x/ \\log q}\\,, \\\\F(x) &=\\tfrac{1}{3} ( \\pi ^2 + 6 \\gamma ^2 - 4 \\log ^2 q) + 4\\sum _{j \\in {\\mathbb {Z}}^*}\\, \\Gamma (-\\tfrac{\\pi i}{\\log q} \\,j)\\, \\psi (\\tfrac{\\pi i}{\\log q}\\, j)\\, e^{\\pi i j x/ \\log q} \\, ,$ with ${\\mathbb {Z}}^* = {\\mathbb {Z}}\\setminus \\lbrace 0\\rbrace $ and $R_{\\rm sm}(x) =4\\, \\sum _{k=1}^\\infty \\tfrac{(-1)^k\\, q^{2k}}{k!", "(1 - q^{2k})^2}\\, x^k\\,.$ The symbol $\\gamma $ denotes the Euler–Mascheroni constant and $\\psi = \\Gamma ^{\\prime }/\\Gamma $ – the digamma function.", "All of the series invoked in the above formulae are absolutely convergent on ${\\mathbb {R}}$ .", "From Formula (REF ) we can quickly read out the dimension spectrum (see Fig.", "REF b)): $\\operatorname{Sd}(\\Delta _q^\\text{sm}) = \\tfrac{\\pi i}{\\log q} {\\mathbb {Z}}\\cup (- {\\mathbb {N}}) = \\lbrace \\tfrac{\\pi i}{\\log q} k\\, |\\, k \\in {\\mathbb {Z}}\\rbrace \\cup \\lbrace -n\\, |\\, n \\in {\\mathbb {N}}\\rbrace \\,, \\quad \\text{with} \\quad \\operatorname{ord}\\operatorname{Sd}= 3\\,.$ It coincides with the set of poles of the function $\\Gamma \\cdot \\zeta _{\\Delta _q^\\text{sm}}$ , as expected from general theorems discussed around Formula (REF ).", "The third order pole at $s=0$ yields the leading $\\log ^2 \\sigma $ term, the second order purely imaginary poles of $\\zeta _{\\Delta _q^\\text{sm}}$ result in the oscillating behaviour captured by $F$ and $G$ , whereas the simple poles of $\\Gamma $ at negative integers give rise to the remainder $R_{\\text{sm}}$ .", "Let us emphasize that Formula (REF ) is indeed a genuine equality valid for any $\\sigma > 0$ .", "This is in sharp contrast with a typical situation of heat trace expansion on a manifold where one has only an asymptotic formula at one's disposal.", "We can thus compute explicitly the corresponding spectral dimension: $d_S^{\\,q, \\text{sm}}(\\sigma ) = -2 \\, \\frac{\\big [ G^{\\prime }\\big (\\log (u \\sigma )\\big ) + 4 \\big ] \\log (u \\sigma ) + F^{\\prime }\\big (\\log (u \\sigma ) \\big ) + G\\big (\\log (u \\sigma ) \\big ) + u \\sigma R_{\\rm sm}^{\\prime }(u \\sigma ) }{2 \\log ^2 (u \\sigma ) + G \\big (\\log (u \\sigma ) \\big ) \\log (u \\sigma ) + F\\big (\\log (u \\sigma ) \\big ) + R_{\\rm sm}(u \\sigma )}.$ This function is plotted and analysed in Section REF below." ], [ "Spinor Laplacian ", "Let us now turn to the spinor Laplacian $\\Delta _q^\\text{sp}$ .", "It arises as the square of the Dirac operator $\\mathcal {D}_q$ , introduced in [40].", "The latter is a unique $\\mathcal {U}_q(\\mathfrak {su}(2))$ -equivariant operator, which renders a real spectral triple.", "The spinor Laplacian acts on basis vectors of $\\mathcal {H}_q$ as $\\Delta _q^\\text{sp}| l,m \\rangle _\\pm = u\\, \\left(q^{-(l+1/2)} - q^{(l+1/2)}\\right)^2 | l,m \\rangle _\\pm \\,,$ with $u = (q^{-1} - q)^{-2}$ , as previously.", "The operator $\\Delta _q^\\text{sp}$ also does not have a zero mode.", "In the limit $q \\rightarrow 1$ it tends (strongly) to the spinor Laplacian on $S^2$ (cf.", "(REF )).", "The spectral zeta function associated with $\\Delta _q^\\text{sp}$ can easily be computed (cf.", "[41] and also Appendix ): $\\zeta _{\\Delta _q^\\text{sp}}(s) &= \\operatorname{Tr}_{\\mathcal {H}_q}\\, (\\Delta _q^\\text{sp})^{-s} = \\operatorname{Tr}_{\\mathcal {H}_q} \\vert \\mathcal {D}_q \\vert ^{-2s} = \\zeta _{\\vert \\mathcal {D}_q \\vert }(2s) \\\\&= 4u^{-2s} \\sum _{k=1}^{\\infty } k \\frac{q^{2ks}}{(1 - q^{2k})^{2s}} = 4u^{-2s} q^{2s} \\sum _{n=0}^{\\infty } \\frac{\\Gamma (n + 2s)}{n!\\, \\Gamma (2s)} \\frac{q^{2n}}{(1 - q^{2(n+s)})^2}\\,.$ The last formula provides a valid meromorphic extension to the entire complex plane.", "The full asymptotic expansion of $\\mathcal {P}_q^\\text{sp}(\\sigma )$ could again be deduced from formula (REF ) via the inverse Mellin transform, along the lines of [41].", "The rather tedious computations can be bypassed by noting that $\\Delta _q^\\text{sp}$ and $\\Delta _q^\\text{sm}$ commute and differ by a bounded perturbation $\\Delta _q^\\text{sp}- \\Delta _q^\\text{sm}= - 2 + (\\Delta _q^\\text{sm})^{-1}$ .", "Consequently, we can write $\\mathcal {P}_q^\\text{sm}(\\sigma ) - \\mathcal {P}_q^\\text{sp}(\\sigma ) & = \\operatorname{Tr}\\left( e^{-\\sigma \\Delta _q^\\text{sm}} - e^{-\\sigma \\Delta _q^\\text{sp}} \\right) = \\operatorname{Tr}e^{-\\sigma \\Delta _q^\\text{sm}} \\left(1 - e^{-\\sigma (\\Delta _q^\\text{sp}- \\Delta _q^\\text{sm})} \\right) \\\\& \\le \\big \\Vert 1 - e^{-\\sigma (\\Delta _q^\\text{sp}- \\Delta _q^\\text{sm})} \\big \\Vert \\, \\mathcal {P}_q^\\text{sm}(\\sigma ) = \\mathcal {O}(\\sigma \\log ^2 \\sigma ).$ The last equality follows from an operatorial inequality $\\lim _{t\\rightarrow 0} \\tfrac{1}{t}\\left\\Vert {1- e^{-tX}} \\right\\Vert \\le \\left\\Vert {X} \\right\\Vert $ (see [41]) and Formula (REF ).", "This means that $\\mathcal {P}_q^\\text{sp}(\\sigma ) = \\tfrac{1}{4 \\log ^2q}\\, \\big [ 2 \\log ^2 (u\\sigma ) + G\\big (\\log (u\\sigma ) \\big )\\, \\log (u\\sigma ) + F\\big (\\log (u\\sigma )\\big ) \\big ] + R_{\\rm sp}(u\\sigma )\\,.$ Hence indeed the leading small-$\\sigma $ behaviour of $\\mathcal {P}_q^\\text{sp}(\\sigma )$ is captured by Formula (REF ) for the simplified Laplacian.", "This harmonises with the fact that the $n=0$ term in the last formula in (REF ) is nothing but $\\zeta _{\\Delta _q^\\text{sm}}$ .", "The structure of the remainder $R_{\\rm sp}(\\sigma )$ can be inferred from Formula (REF ) for the zeta function, in close analogy with [41].", "Observe that $\\zeta _{\\Delta _q^\\text{sp}}$ has poles located on a regular lattice in the left complex half-plane (cf.", "Figure REF c)).", "Consequently, $\\Gamma \\cdot \\zeta _{\\Delta _q^\\text{sp}}$ has third order poles at negative integers — yielding $\\sigma ^n \\log ^2\\sigma $ contribution and double poles elsewhere — giving rise to $\\sigma ^{\\pi i j / \\log q} \\sigma ^n \\log \\sigma $ oscillatory terms.", "Summa summarum, $R_{\\rm sp}(x) \\, \\; \\underset{\\sigma \\downarrow 0}{\\sim } \\;\\, \\sum _{n = 1}^{\\infty } \\left[ h_n \\log ^2 x + G_n(\\log x) \\log x + F_n(\\log x) \\right] x^{n},$ where $h_n \\in {\\mathbb {R}}$ and $F_n, G_n$ are periodic bounded functions of the form similar to that of $F$ and $G$ .", "Let us stress that the sum over $n$ need not a priori to be convergent, which is just a restatement of the fact that asymptotic expansions of heat traces are generically divergent.", "This analysis leads to the conclusion about the dimension spectrum: $\\operatorname{Sd}(\\Delta _q^\\text{sp}) = \\tfrac{\\pi i}{\\log q} {\\mathbb {Z}}- {\\mathbb {N}}= \\lbrace \\tfrac{\\pi i}{\\log q} k - n\\, |\\, k \\in {\\mathbb {Z}}, n \\in {\\mathbb {N}}\\rbrace \\,, \\quad \\text{with} \\quad \\operatorname{ord}\\operatorname{Sd}= 3\\,.$ Observe the difference with respect to (REF ), illustrated in Fig.", "REF .", "Even though we only have an asymptotic formula for $\\mathcal {P}_q^\\text{sp}$ , we can deduce the leading behaviour of the spectral dimension associated with the spinor Laplacian.", "This is because formula (REF ) grants us an explicit control on the remainder.", "We thus have $d_S^{\\,q,\\text{sp}}(\\sigma ) & = -2 \\, \\frac{\\big [ G^{\\prime }\\big (\\log (u \\sigma )\\big ) + 4 \\big ] \\log (u \\sigma ) + F^{\\prime }\\big (\\log (u \\sigma ) \\big ) + G\\big (\\log (u \\sigma ) \\big )}{2 \\log ^2 (u \\sigma ) + G \\big (\\log (u \\sigma ) \\big ) \\log (u \\sigma ) + F\\big (\\log (u \\sigma ) \\big )} + \\mathcal {O}(\\sigma ) \\\\& = d_S^{\\,q,\\text{sm}}(\\sigma ) + \\mathcal {O}(\\sigma )\\,.$ Hence, also the spectral dimensions associated with the spinor and simplified Laplacians on the quantum sphere share the same leading behaviour for small $\\sigma $ — see Figure REF ." ], [ "Scalar Laplacian ", "The `scalar Laplacian' $\\Delta _q^\\text{sc}$ , introduced in [20], originates from the Casimir operator on the Hopf algebra $\\mathcal {U}_q(\\mathfrak {su}(2))$ .", "It acts on a Hilbert space $\\mathcal {H}_q^{\\prime }$ (on which the algebra $\\mathcal {A}_q$ can also be faithfully represented) spanned by orthonormal vectors $| j,m \\rangle $ , with $m \\in \\lbrace -j,-j+1,\\ldots ,j\\rbrace $ and $j \\in {\\mathbb {N}}$ as $\\Delta _q^\\text{sc}\\, | j,m \\rangle & = \\frac{\\cosh \\big ( \\tfrac{1}{2} (2j+1) \\log q\\big ) - \\cosh \\big (\\tfrac{1}{2} \\log q \\big )}{2 \\sinh ^2 \\big (\\tfrac{1}{2} \\log q \\big )} \\, | j,m \\rangle \\\\& = u(\\sqrt{q}) \\, q^{-1/2} \\left( q^{-j} - 1 - q + q^{j+1} \\right) | j,m \\rangle \\,.", "$ In the limit $q \\rightarrow 1$ the operator $\\Delta _q^\\text{sc}$ tends (strongly) to the standard scalar Laplacian on $S^2$ (cf.", "(REF )).", "In contradistinction with $\\Delta _q^\\text{sp}$ , it does have a zero mode, which means that $\\mathcal {P}_q^\\text{sc}(\\sigma )$ tends to $\\dim \\operatorname{Ker}\\Delta _q^\\text{sc}= 1$ as $\\sigma $ goes to infinity.", "The small-$\\sigma $ behaviour of the heat trace can be deduced by singling out the unbounded part of $\\Delta _q^\\text{sc}$ , as in the case of $\\Delta _q^\\text{sp}$ .", "Namely, we have $\\mathcal {P}_q^\\text{sc}(\\sigma ) & = \\operatorname{Tr}_{\\mathcal {H}_q^{\\prime }} e^{- \\sigma \\Delta _q^\\text{sc}} = \\sum _{j=0}^{\\infty } (2j+1) \\exp \\left\\lbrace - \\sigma \\, u(\\sqrt{q}) \\, q^{-1/2} \\left( q^{-j} - 1 - q + q^{j+1} \\right) \\right\\rbrace \\\\& = 1 + \\sum _{n=1}^{\\infty } (2n+1) \\exp \\left\\lbrace - \\sigma \\, u(\\sqrt{q}) \\, q^{-1/2} \\left( q^{-n} - 1 - q + q^{n+1} \\right) \\right\\rbrace \\\\& = 1 + \\sum _{n=1}^{\\infty } (2n+1) \\exp \\left\\lbrace - \\sigma \\, u(\\sqrt{q}) \\, q^{-1/2} q^{-n} \\right\\rbrace + \\mathcal {O}(\\sigma \\log ^2 \\sigma ) \\\\& = 1 + \\tfrac{1}{2} \\mathcal {P}_{\\sqrt{q}}^\\text{sm}(\\sigma q^{-1/2}) + \\sum _{n=1}^{\\infty } \\exp \\left\\lbrace - \\sigma \\, u(\\sqrt{q}) \\, q^{-1/2} q^{-n} \\right\\rbrace + \\mathcal {O}(\\sigma \\log ^2 \\sigma ).$ The last series can be evaluated explicitly [59] using the inverse Mellin transform technique: $\\sum _{n=1}^{\\infty } e^{-x q^{-n}} = \\tfrac{1}{\\log q} \\left[ \\log x + \\gamma - \\tfrac{1}{2} \\log q + H(\\log x) \\right] + \\mathcal {O}(x)\\,,$ with $H(x) &=- \\sum _{k \\in {\\mathbb {Z}}^*} \\Gamma (-\\tfrac{2 \\pi i}{\\log q}\\, k)\\, e^{2 \\pi i k x/ \\log q}\\,.$ This yields $\\mathcal {P}_q^\\text{sc}(\\sigma ) & = \\tfrac{1}{2} \\, \\mathcal {P}_{\\sqrt{q}}^\\text{sm}(\\sigma q^{-1/2}) + \\tfrac{1}{\\log q} \\big [ \\log (\\sigma \\, u(\\sqrt{q})) + \\gamma + H\\big ( \\log (\\sigma \\, u(\\sqrt{q}) \\, q^{-1/2}) \\big ) \\big ] + \\mathcal {O}(\\sigma \\log ^2 \\sigma )\\,.$ Similarly as in the spinor case, one can unfold the structure of the remainder by an inspection of the zeta function $\\zeta _{\\Delta _q^\\text{sc}}$ — see Appendix .", "Its meromorphic structure is very similar to that of $\\zeta _{\\Delta _q^\\text{sp}}$ .", "The conclusion is that $\\Delta _q^\\text{sc}$ and $\\Delta _q^\\text{sp}$ have identical dimension spectra, both of order 3.", "Formula (REF ) allows us also to compute the spectral dimension associated with $\\zeta _{\\Delta _q^\\text{sc}}$ up to the terms of order $\\mathcal {O}(\\sigma )$ at $\\sigma = 0$ $d_S^{\\,q,\\text{sc}}(\\sigma ) & = \\frac{\\tfrac{1}{2} \\sigma q^{-1/2} \\, (\\mathcal {P}_{\\sqrt{q}}^\\text{sm})^{\\prime }(\\sigma q^{-1/2}) + \\tfrac{1}{\\log q} \\big [ 1 + H^{\\prime }\\big ( \\log (\\sigma \\, u(\\sqrt{q}) \\, q^{-1/2}) \\big ) \\big ]}{\\tfrac{1}{2} \\, \\mathcal {P}_{\\sqrt{q}}^\\text{sm}(\\sigma q^{-1/2}) + \\tfrac{1}{\\log q} \\big [ \\log (\\sigma \\, u(\\sqrt{q})) + \\gamma + H\\big ( \\log (\\sigma \\, u(\\sqrt{q}) \\, q^{-1/2}) \\big ) \\big ]} + \\mathcal {O}(\\sigma ) \\\\& = d_S^{\\sqrt{q},\\text{sm}}(q^{-1/2} \\sigma ) + \\mathcal {O}\\big ( (\\log \\sigma )^{-2} \\big )\\,, $ where the last equality follows from the exact formula (REF ).", "It shows that the leading small-$\\sigma $ behaviour of the spectral dimension for the scalar Laplacian is captured by the rescaled spectral dimension for $\\Delta _q^\\text{sm}$ ." ], [ "Comparison ", "We now compare the results obtained for the three Laplacians and contrast them with their classical counterparts.", "Recall that the spinor LaplacianIn the mathematical literature [65] the “spinor Laplacian” $\\Delta ^\\mathcal {S}$ is slightly different from $\\Delta ^\\text{sp}$ , which is the square of the Dirac operator ${\\mathcal {D}\\hspace{-6.38885pt}/\\,}^2$ .", "The two operators are related through the Schrödinger–Lichnerowicz formula ${\\mathcal {D}\\hspace{-6.38885pt}/\\,}^2 = \\Delta ^\\mathcal {S} + \\tfrac{1}{4} \\mathcal {R}$ , with $\\mathcal {R}$ being the scalar curvature.", "On the unit 2-sphere with a round metric the difference amounts to a trivial shift $\\Delta ^\\text{sp}= {\\mathcal {D}\\hspace{-6.38885pt}/\\,}^2 = \\Delta ^\\mathcal {S} + 2$ .", "acts on spinor harmonics over the unit two-sphere as follows [66], [67]: $\\Delta ^\\text{sp}| l,m \\rangle _\\pm = (l + \\tfrac{1}{2})^2 | l,m \\rangle _\\pm \\,.$ In turn, for the scalar Laplacian (a.k.a.", "the Laplace–Beltrami operator) we have [45]: $\\Delta ^\\text{sc}| j,m \\rangle = j (j+1) | j,m \\rangle \\,.$ Let us firstly have a look at the dimension spectra at Fig.", "REF .", "Both $\\Delta ^\\text{sp}$ and $\\Delta ^\\text{sc}$ are classical differential operators of second order acting over a two-dimensional manifold.", "Consequently, we have [32] (see also [48] for a direct computation): $\\operatorname{Sd}(\\Delta ^\\text{sp}) = \\operatorname{Sd}(\\Delta ^\\text{sc}) = 1 - \\mathbb {N}\\,, \\quad \\text{with} \\quad \\operatorname{ord}\\operatorname{Sd}= 1\\,.$ Figure: A comparison of dimension spectra for different Laplacians on 2-sphere Sd(Δ sp )=Sd(Δ sc )\\operatorname{Sd}(\\Delta ^\\text{sp}) = \\operatorname{Sd}(\\Delta ^\\text{sc}) (plot a)) and on quantum sphere Sd(Δ q sm )\\operatorname{Sd}(\\Delta _q^\\text{sm}) (plot b)), Sd(Δ q sp )=Sd(Δ q sc )\\operatorname{Sd}(\\Delta _q^\\text{sp}) = \\operatorname{Sd}(\\Delta _q^\\text{sc}) (plot c)).", "The symbols ×\\times , *\\ast and •\\bullet denote points in Sd\\operatorname{Sd}, corresponding to poles of the function Γ·ζ\\Gamma \\cdot \\zeta of order 1, 2 and 3 respectively, while ϕ=π/(logq)\\varphi = \\pi / (\\log q).", "An nn-th order pole of the function Γ·ζ\\Gamma \\cdot \\zeta yields a term proportional to (logσ) n-1 (\\log \\sigma )^{n-1} in the asymptotic expansion of 𝒫(σ)\\mathcal {P}(\\sigma ) (recall Eq.", "()).In the classical case we have $d_{\\operatorname{Sd}} = 1$ , in agreement with the general formula $d_{\\operatorname{Sd}} = d /\\eta $ , as $d=2$ and $\\eta =2$ .", "In contrast, for all three operators on the quantum sphere we have $d_{\\operatorname{Sd}} = 0$ .", "In order to interpret this fact as the dimension drop, i.e.", "$d=0$ , we need to argue that the `quantum Laplacians' are of order $\\eta > 0$ .", "In the framework of spectral triples, the operator $q$ verifies the so-called first-order condition [40], which mimics the demand for a classical Dirac operator to be a first order differential operator [58], [68].", "On the physical side, this condition limits the admissible fluctuations of an operator $ by gauge fields \\cite {ConnesFirst} and thus it is pertinent in building particle physics models from noncommutative geometry \\cite {WalterBook,ConnesFirstSM}.", "Since $ q$ is a first-order operator, for $ qsp= q2$ we should set $ = 2$.", "The operator $ qS$ does not meet the first-order condition \\cite {PodlesSA} and $ qsc$ does not come from a `Dirac^{\\prime } operator at all.", "Nevertheless, $ qsm$ differs from $ qsp$ by a bounded perturbation and so does $ qsc$ after a suitable rescaling and reparametrisation $ q q$.", "We can thus safely assume that $ (qsc) = (qsm) = 2$ and conclude that indeed the dimension of the quantum sphere is 0.", "The issue of the order of an operator over a quantum space is more subtle for the $$-Minkowski space, as we will see in the next section.$ The existence of non-positive numbers in the dimension spectra of the two classical Laplacians on the two-sphere certify the impact of the non-trivial Riemann tensor on the heat trace (recall Formula (REF )).", "The dimension spectra of quantum Laplacians also contain negative numbers, which suggest that quantum spheres are `curved' in some sense (see e.g.", "[46], [72], [73], [74], [75], [76] for a discussion of curvature of quantum spaces).", "On top of that, these dimension spectra contain points outside of the real axis, which hint at some kind of self-similar structure of the quantum sphere [57], [41] (see also [25], [27]).", "Note also that, excluding the simplified case of $\\Delta _q^\\text{sm}$ , the non-real points appear all over the left complex-half plane.", "This suggests that the curvature and self-similar structure of a quantum sphere are deeply interwoven.", "Finally, the dimension spectra of quantum Laplacians are of order three, which means that they are already beyond the realm of classical pseudodifferential operators, as the latter can only have $\\operatorname{ord}\\operatorname{Sd}= 2$ .", "Third order poles in the dimension spectra have been detected in the context of manifolds with conical singularities [42].", "They occur when one studies the conical singularities (and, more generally, stratified spaces) from the perspective of manifolds with boundary [77].", "Concretely, specific non-local boundary conditions related to the singularity coerce the use of Fuchs-type operators [54].", "Although the mathematical context here is very different, one might take it as a (not so surprising) indication that the geometry of the quantum sphere is not smooth.", "We now turn to the analysis of the spectral dimensions.", "Let us first have a look at the small-$\\sigma $ behaviour of $d_S(\\sigma )$ for the simplified Laplacian $\\Delta _q^\\text{sm}$ .", "From Eq.", "(REF ) one deduces the leading behaviour in the UV: $d_S^{\\,q, \\text{sm}}(\\sigma ) = \\frac{-4}{\\log \\sigma } \\left( 1 + \\frac{\\pi i}{\\log q} \\sum _{j \\in {\\mathbb {Z}}^*} j \\, \\Gamma (\\tfrac{\\pi i}{\\log q}\\, j)\\, \\sigma ^{-\\pi i j/ \\log q}\\right) + \\mathcal {O}((\\log \\sigma )^{-2})\\,.$ The function $d_S^{\\,q, \\text{sm}}$ for $q = 1/2$ is illustrated on Fig.", "(REF ).", "Figure: (LHS) The dark blue line is the spectral dimension for the Laplacian Δ q sm \\Delta _q^\\text{sm}, computed numerically from Eq.", "() up to the 100th eigenvalue.", "The leading behaviour in the UV is determined from Formula () as -4/(logσ)-4/ (\\log \\sigma ) (light green line).", "(RHS) The function d S q,sm d_S^{\\,q, \\text{sm}} after subtraction of the leading behaviour -4/(logσ)-4/ (\\log \\sigma ) clearly shows the log\\log -periodic oscillations.Formula (REF ) shows that the spectral dimension of $\\Delta _q^\\text{sm}$ drops to zero in the UV.", "Observe that the slope $-4/ (\\log \\sigma )$ does not depend on $q$ .", "On the other hand, the amplitude of oscillations exhibits strong dependence on the value of $q$ .", "Concretely, for the leading frequency ($j=1$ in the sum in Eq.", "(REF )) we have $A(q) = \\big \\vert \\tfrac{\\pi }{\\log q} \\, \\Gamma (\\tfrac{\\pi i}{\\log q}) \\big \\vert .$ The amplitude of oscillations tends to 1 as $q \\rightarrow 0$ and decays very rapidly, as $e^{\\pi ^2 /( 2(q-1))}$ , when $q$ goes to 1 — see Fig.", "REF .", "Figure: The qq-dependence of the amplitude of (leading frequency) oscillations in the spectral dimension for Δ q sp \\Delta _q^\\text{sp} (dark blue) and Δ q sc \\Delta _q^\\text{sc} (light red).", "The former is given by the function A(q)A(q) defined in (), whereas the latter is A(q)A(\\sqrt{q}), because of the relation ().In the previous sections we have shown that the leading UV behaviour of the spectral dimension for the spinor Laplacian is captured by $d_S^{\\,q, \\text{sm}}$ .", "The same is true, after suitable rescaling, also for the scalar Laplacian, though with worse precision.", "The situation is illustrated on Fig.", "REF through the numerical summation in Eq.", "(REF ) up to 100th eigenvalue.", "Figure: A comparison of the UV behaviour of the spectral dimensions for three Laplacians on the the quantum sphere for q=1/3q=1/3.", "On LHS the dark blue line corresponds to d S q,sm (σ)d_S^{\\,q, \\text{sm}}(\\sigma ) and the light green one to d S q,sp (σ)d_S^{\\,q, \\text{sp}}(\\sigma ).", "On the RHS the dark blue line is d S q,sm (σ)d_S^{\\,q, \\text{sm}}(\\sigma ), whereas the light red one shows d S q 2 ,sc (qσ)d_S^{\\,q^2, \\text{sc}}(q \\sigma ).A comparison of the spectral dimension associated with the operators $\\Delta _q^\\text{sp}, \\Delta _q^\\text{sc}$ and their classical counterparts $\\Delta ^\\text{sp}, \\Delta ^\\text{sc}$ , respectively, is presented on Figs.", "REF and REF .", "The main conclusion is that the spectral dimensions of both quantum Laplacians on the quantum sphere exhibit log-periodic oscillations in UV.", "The amplitude of these oscillations drops very rapidly when $q$ tends to the classical value 1 (cf.", "Fig.", "REF ).", "In particular, for $\\Delta _q^\\text{sc}$ the value $q = e^{-0.01} \\approx 0.99$ adopted on Fig.", "1 in [20], we have $A(\\sqrt{q}) \\sim 10^{-430}$ , which explains why the oscillations have been overlooked in that paper.", "Figure: The spectral dimension d S (σ)d_S(\\sigma ) for Δ q sc \\Delta _q^{\\rm sc} (continuous decaying curve) and Δ q sp \\Delta _q^{\\rm sp} (continuous diverging) with q=0.15q = 0.15, and for S 2 S^2 with the scalar (dashed decaying curve) and spinorial (dashed diverging) Laplacians.Figure: The spectral dimension d S (σ)d_S(\\sigma ) for Δ q sc \\Delta _q^{\\rm sc}, with q=0.1q = 0.1 (bottom continuous curve), q=0.5q = 0.5 (middle continuous) and q=0.9q = 0.9 (top continuous), and for S 2 S^2 with the scalar Laplacian (dashed)The $\\kappa $ -Minkowski space was introduced [16] in the context of quantum groups that describe deformed relativistic symmetries – it is the spacetime that is bi-covariant under the action and coaction (the former is determined by the algebraic and the latter by the coalgebraic structure) of the $\\kappa $ -Poincaré Hopf algebra.", "Both these mathematical objects can be defined in any number of dimensions [78] but $\\kappa $ -Poincaré algebra was first derived [17], [18] in the (3+1)d case, as a contraction of the quantum-deformed anti-de Sitter algebra $U_q(\\mathfrak {so}(3,2))$ , obtained by taking the limit of the (real) deformation parameter $q \\rightarrow 1$ and anti-de Sitter radius $R \\rightarrow \\infty $ , while their ratio $R\\, \\log q = \\kappa ^{-1}$ , $\\kappa > 0$ is kept fixed.", "Hence the new deformation parameter $\\kappa $ has the dimension of inverse length (in contrast to the dimensionless $q$ ), which allows for the geometrization of Planck mass $m_P = \\hbar c^{-1} \\lambda _P$ , expressed in terms of Planck length $\\lambda _P$ .", "This peculiar feature enabled application of $\\kappa $ -Poincaré algebra in the construction of models of so-called doubly special relativity, which was subsequently recast as the relative locality framework and serves as an important source for the quantum gravity phenomenology [79].", "The interest in $\\kappa $ -Poincaré and $\\kappa $ -(anti-)de Sitter algebras is also motivated by results for the 2+1-dimensional gravity, where it is known that they may arise from the structure of classical theory, at least in certain special cases (see [80] and references therein).", "The $n\\!+\\!1$ -dimensional $\\kappa $ -Minkowski noncommutative space is the dual of the subalgebra of translations of the ($n\\!+\\!1$ -dimensional) $\\kappa $ -Poincaré algebra, since the latter is naturally interpreted as the algebra of spacetime coordinates.", "The time $X_0$ and spatial coordinates $X_a$ , $a = 1,...,n$ satisfy the following commutation relations $[X_0,X_a] = \\frac{i}{\\kappa }\\, X_a\\,, \\qquad [X_a,X_b] = 0\\,.$ As a vector space, the $\\kappa $ -Minkowski space is isomorphic to the ordinary Minkowski space in $n\\!+\\!1$ dimensions, which can be recovered in the classical limit $\\kappa \\rightarrow \\infty $ .", "The Lie algebra generated by $X_0,X_a$ is usually denoted $\\mathfrak {an}(n)$ , where the notation refers to $n$ Abelian and nilpotent generators $X_a$ .", "The corresponding Lie group ${\\rm AN}(n)$ has the geometry of $n\\!+\\!1$ -dimensional elliptic de Sitter space [81], [82].", "Group elements can be written as ordered exponentials, whose ordering is equivalent to the choice of coordinates on the group manifold.", "For example, in the time-to-the-right ordering a group element has the form $g = e^{-i P^a X_a} e^{i P_0 X_0}$ and $P_0,P_a \\in {\\mathbb {R}}$ are coordinates in the so-called bicrossproduct basis.", "It is easy to notice that we can interpret ${\\rm AN}(n)$ as momentum space if such exponentials are treated as plane waves on $n\\!+\\!1$ -dimensional $\\kappa $ -Minkowski space.", "${\\rm AN}(n)$ is equipped with the structure of the algebra of translations and is related with spacetime coordinate algebra via the group Fourier transform [83], [84].", "Furthermore, the geometry of momentum space becomes evident in the classical basis (the name refers to the classical form of expression for the dispersion relation in this case), which can be introduced via the transformation $p_0 & = \\kappa \\sinh \\left(\\tfrac{P_0}{\\kappa }\\right) + \\tfrac{1}{2\\kappa } e^{P_0/\\kappa } P_aP^a\\,, \\nonumber \\\\p_a & = e^{P_0/\\kappa } P_a\\,, \\nonumber \\\\p_{-1} & = \\kappa \\cosh \\left(\\tfrac{P_0}{\\kappa }\\right) - \\tfrac{1}{2\\kappa } e^{P_0/\\kappa } P_aP^a\\,.$ The above relations lead to the constraints on $\\lbrace p_0,p_a,p_{-1}\\rbrace $ , $-p_0^2 + p_ap^a + p_{-1}^2 = \\kappa ^2$ and $p_0 + p_{-1} > 0$ , which describe the embedding of ${\\rm AN}(n)$ as half of a $(n,1)$ -hyperboloid in $(n\\!+\\!1)\\!+\\!1$ -dimensional Minkowski space.", "$p_{-1}$ is just an auxiliary coordinate, diverging in the $\\kappa \\rightarrow +\\infty $ limit.", "In order to consider the diffusion process (i.e.", "study the heat trace) determined by a Laplacian on $\\kappa $ -Minkowski space, one first has to perform the Wick rotation $(p_0 \\mapsto i p_0,p_{-1} \\mapsto i p_{-1},\\kappa \\mapsto i \\kappa ,P_0 \\mapsto i P_0)$ .", "This leads to the Euclidean momentum space, which has the hyperbolic geometry [19].", "The Euclidean version of $\\kappa $ -Minkowski space has also been studied from the perspective of spectral triples [85], [86], [87] via the star-product realisations [88], [89], [90].", "In this framework the algebra (REF ) is faithfully represented on the Hilbert space $L^2({\\mathbb {R}}^{n+1})$ through a left regular group representation.", "As mentioned in the introduction, there are several possible choices for a Laplacian on (Euclidean) $\\kappa $ -Minkowski space.", "All of them have continuous spectra, what reflects the non-compactness of the underlying space.", "Therefore, in order to define the corresponding heat traces, one firstly needs to choose an IR cut-off.", "A natural choice is a function $f$ compactly supported on ${\\mathbb {R}}^{n+1}$ and promoted to an operator on $L^2({\\mathbb {R}}^{n+1})$ through the Weyl-like quantisation (cf.", "[89], [86]).", "Fortunately, it turns out that the regularising function factors out in trace formulae and contributes just a multiplicative factor $\\left\\Vert {f} \\right\\Vert _{L^2}$ [86], [87].", "This means that the return probability does not depend on the choice of the IR regularisation and can be computed via the heat kernel formula (REF ).", "In the classical basis it reads [19] The factor $\\kappa $ in the numerator was missing in [19] but it did not affect results for the spectral dimension; here we have also rescaled the integrand by 2.: ${\\cal P}(\\sigma ) = \\int \\!", "d^{n+1}p\\ \\frac{\\kappa }{\\sqrt{p_0^2 + p_ap^a + \\kappa ^2}}\\, e^{-\\sigma {\\cal L}(p_0, \\lbrace p_a\\rbrace )},$ where ${\\cal L}$ is the (Euclidean) momentum space-representation of a Laplacian.", "In the following Subsections REF -REF we will calculate the spectral dimensions of $\\kappa $ -Minkowski space equipped with three distinct Laplacians, in 2, 3 and 4 topological dimensions for each of them.", "In this way we improve and extend the results obtained by one of us in [19].", "Furthermore, we compute the complete dimension spectra of the relevant Laplacians." ], [ "Bicovariant Laplacian", "Let us first consider the Laplacian determined by the bicovariant differential calculus on the $\\kappa $ -Minkowski space [91], [92], which we call the bicovariant Laplacian.", "In terms of bicrossproduct coordinates, the Euclidean momentum space representation of this Laplacian is given by ${\\cal L}_{\\text{cv}}(P_0,\\lbrace P_a\\rbrace ) = & \\, 4\\kappa ^2 \\sinh ^2\\left(\\tfrac{1}{2\\kappa } P_0\\right) + e^{P_0/\\kappa } P_a P^a \\nonumber \\\\&+ \\frac{1}{4\\kappa ^2} \\left(4\\kappa ^2 \\sinh ^2\\left(\\tfrac{1}{2\\kappa } P_0\\right) + e^{P_0/\\kappa } P_a P^a\\right)^2\\,,$ while in classical coordinates it acquires the familiar standard form ${\\cal L}_{\\text{cv}}(p_0, \\lbrace p_a\\rbrace ) = p_0^2 + p_ap^a.", "$ The return probability in 3+1 dimensions reads [19] ${\\cal P}_{(3+1)}(\\sigma ) = \\frac{\\pi ^2}{2\\sigma ^{3/2}} \\left(2\\kappa ^2 \\sqrt{\\sigma } - \\sqrt{\\pi }\\, e^{\\kappa ^2 \\sigma } (2\\kappa ^2 \\sigma - 1) \\left(1 - {\\rm erf}(\\kappa \\sqrt{\\sigma })\\right)\\right)\\,,$ where ${\\rm erf}(\\cdot )$ is the error function, while in 2+1 dimensions we have ${\\cal P}_{(2+1)}(\\sigma ) = \\frac{\\pi ^{3/2} \\kappa }{\\sigma }\\, U(\\tfrac{1}{2},0;\\kappa ^2 \\sigma )\\,,$ where $U(a,b;\\cdot )$ is a Tricomi confluent hypergeometric function.", "Furthermore, we may also consider the case of 1+1 dimensions (not discussed in [19]), ${\\cal P}_{(1+1)}(\\sigma ) = \\frac{\\pi ^{3/2} \\kappa }{\\sqrt{\\sigma }}\\, e^{\\kappa ^2 \\sigma } \\left(1 - {\\rm erf}(\\kappa \\sqrt{\\sigma })\\right)\\,.$ The exact formulae (REF -REF ) for heat traces can be directly developed into series around $\\sigma = 0$ ${\\cal P}_{(3+1)}(\\sigma ) &= \\frac{\\kappa \\pi ^{5/2}}{2\\sqrt{\\sigma }^3} - \\frac{\\kappa ^3 \\pi ^{5/2}}{2\\sqrt{\\sigma }} + \\frac{4\\kappa ^4 \\pi ^2}{3} - \\frac{3\\kappa ^5 \\pi ^{5/2}}{4} \\sqrt{\\sigma } + {\\cal O}(\\sigma )\\,, \\nonumber \\\\{\\cal P}_{(2+1)}(\\sigma ) &= \\frac{2\\kappa \\pi }{\\sigma } + \\kappa ^3 \\pi \\log \\sigma + \\kappa ^3 \\pi \\left(1 + \\gamma + 2 \\log \\frac{\\kappa }{2}\\right) + {\\cal O}(\\sigma )\\,, \\nonumber \\\\{\\cal P}_{(1+1)}(\\sigma ) &= \\frac{\\kappa \\pi ^{3/2}}{\\sqrt{\\sigma }} - 2\\kappa ^2 \\pi + \\kappa ^3 \\pi ^{3/2} \\sqrt{\\sigma } + {\\cal O}(\\sigma )\\,,$ where $\\gamma $ is the Euler–Mascheroni constant.", "From these expansions one can immediately read out the corresponding dimension spectra ${\\rm Sd}_{(3+1)} &= \\lbrace \\tfrac{3}{2}\\rbrace \\cup \\lbrace \\tfrac{1-n}{2} \\, \\vert \\, n \\in \\mathbb {N}\\rbrace = \\lbrace \\tfrac{3}{2},\\tfrac{1}{2},0,-\\tfrac{1}{2},-1,-\\tfrac{3}{2},\\ldots \\rbrace \\,, & \\operatorname{ord}\\operatorname{Sd}= 1\\,, \\nonumber \\\\{\\rm Sd}_{(2+1)} &= 1- \\mathbb {N} = \\lbrace 1,0,-1,-2,\\ldots \\rbrace \\,, & \\operatorname{ord}\\operatorname{Sd}= 2\\,, \\nonumber \\\\{\\rm Sd}_{(1+1)} &= \\tfrac{1}{2}(1- \\mathbb {N}) = \\lbrace \\tfrac{1}{2},0,-\\tfrac{1}{2},-1,\\ldots \\rbrace \\,, & \\operatorname{ord}\\operatorname{Sd}= 1\\,.$ Let us remind that in the 2+1 dimensional case the dimension spectrum is of the second order because of the presence of logarithmic terms in the expansion of ${\\cal P}_{(2+1)}(\\sigma )$ .", "We will now turn to results for the spectral dimensions.", "From (REF ) one finds that in 3+1 dimensions the spectral dimension is given by the expression $d_S^{(3+1)}(\\sigma ) = 3 + 2\\kappa ^2 \\sigma \\frac{2\\kappa \\sqrt{\\sigma } - \\sqrt{\\pi }\\, e^{\\kappa ^2 \\sigma } (2\\kappa ^2 \\sigma + 1) (1 - {\\rm erf}(\\kappa \\sqrt{\\sigma }))}{-2\\kappa \\sqrt{\\sigma } + \\sqrt{\\pi }\\, e^{\\kappa ^2 \\sigma } (2\\kappa ^2 \\sigma - 1) (1 - {\\rm erf}(\\kappa \\sqrt{\\sigma }))}\\,,$ which has the IR and UV limits $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow \\infty } d_S^{(3+1)}(\\sigma ) = 4$ and $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow 0} d_S^{(3+1)}(\\sigma ) = 3$ , respectively.", "Analogously, in 2+1 dimensions one uses (REF ) to get $d_S^{(2+1)}(\\sigma ) = 2 + \\frac{\\kappa ^2 \\sigma \\, U(\\frac{3}{2},1,\\kappa ^2 \\sigma )}{U(\\frac{1}{2},0,\\kappa ^2 \\sigma )}\\,,$ whose IR and UV limits are $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow \\infty } d_S^{(2+1)}(\\sigma ) = 3$ , $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow 0} d_S^{(2+1)}(\\sigma ) = 2$ , respectively.", "In 1+1 dimensions (REF ) leads to $d_S^{(1+1)}(\\sigma ) = 1 + 2 \\kappa ^2 \\sigma \\left(\\frac{1}{\\sqrt{\\pi }\\, \\kappa \\sqrt{\\sigma }} \\frac{e^{-\\kappa ^2 \\sigma }}{1 - {\\rm erf}(\\kappa \\sqrt{\\sigma })} - 1\\right)\\,,$ with the IR and UV limits $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow \\infty } d_S^{(1+1)}(\\sigma ) = 2$ , $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow 0} d_S^{(1+1)}(\\sigma ) = 1$ , respectively." ], [ "Bicrossproduct Laplacian", "Another possible Laplacian on $\\kappa $ -Minkowski space, called the bicrossproduct Laplacian, corresponds to the simplest mass Casimir of $\\kappa $ -Poincaré algebra [18] (let us stress that any function of this Casimir that has the correct classical limit is also a Casimir).", "More precisely, it is the Euclidean version of the momentum space representation of the Casimir and has the form ${\\cal L}_{\\text{cp}}(P_0,\\lbrace P_a\\rbrace ) = 4\\kappa ^2 \\sinh ^2\\left(\\tfrac{1}{2\\kappa } P_0\\right) + e^{P_0/\\kappa } P_aP^a\\,.$ In classical coordinates it becomes ${\\cal L}_{\\text{cp}}(p_0, \\lbrace p_a\\rbrace ) = 2\\kappa \\left(\\sqrt{p_0^2 + p_ap^a + \\kappa ^2} - \\kappa \\right).$ It turns out that the corresponding return probability in 3+1 dimensions is given by a simple rational function ${\\cal P}_{(3+1)}(\\sigma ) = \\pi ^2 \\frac{2\\kappa ^2 \\sigma + 1}{2\\kappa ^2 \\sigma ^3}\\,,$ while in 2+1 dimensions we obtain ([19] in this case had only numerical results) ${\\cal P}_{(2+1)}(\\sigma ) = \\frac{4\\pi \\kappa }{\\sigma }\\, e^{2\\kappa ^2 \\sigma } K_1(2\\kappa ^2 \\sigma )\\,,$ where $K_\\alpha (\\cdot )$ is a modified Bessel function of the second kind.", "Finally, in 1+1 dimensions we have just ${\\cal P}_{(1+1)}(\\sigma ) = \\frac{\\pi }{\\sigma }.$ Observe that in the latter case there is no dependence on the parameter $\\kappa $ .", "The dimension spectra can again be obtained directly from the expansions around $\\sigma = 0$ of the exact formulae (REF –REF ) (and the trivial (REF )): ${\\cal P}_{(3+1)}(\\sigma ) &= \\frac{\\pi ^2}{2\\kappa ^2 \\sigma ^3} + \\frac{\\pi ^2}{\\sigma ^2}\\,, \\nonumber \\\\{\\cal P}_{(2+1)}(\\sigma ) &= \\frac{2\\pi }{\\kappa \\sigma ^2} + \\frac{4\\kappa \\pi }{\\sigma } + 4\\kappa ^3 \\pi \\log \\sigma + 2\\kappa ^3 \\pi \\left(1 + 2\\gamma + 4\\log \\kappa \\right) + {\\cal O}(\\sigma )\\,.$ Consequently, we have ${\\rm Sd}_{(3+1)} &= \\lbrace 3,2\\rbrace \\,, & \\operatorname{ord}\\operatorname{Sd}= 1\\,,\\nonumber \\\\{\\rm Sd}_{(2+1)} &= 2-\\mathbb {N} = \\lbrace 2,1,0,-1,-2,\\ldots \\rbrace \\,, & \\operatorname{ord}\\operatorname{Sd}= 2\\,, \\nonumber \\\\{\\rm Sd}_{(1+1)} &= \\lbrace 1\\rbrace \\,, & \\operatorname{ord}\\operatorname{Sd}= 1\\,.$ If we use the standard formula (REF ), (REF ) leads to the spectral dimension $d_S^{(3+1)}(\\sigma ) = 6 - \\frac{4\\kappa ^2 \\sigma }{2\\kappa ^2 \\sigma + 1}\\,,$ with $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow \\infty } d_S^{(3+1)}(\\sigma ) = 4$ and $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow 0} d_S^{(3+1)}(\\sigma ) = 6$ ; and (REF ) leads to $d_S^{(2+1)}(\\sigma ) = 4 - 4\\kappa ^2 \\sigma \\left(1 - \\frac{K_0(2\\kappa ^2 \\sigma )}{K_1(2\\kappa ^2 \\sigma )}\\right)\\,,$ with $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow \\infty } d_S^{(2+1)}(\\sigma ) = 3$ and $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow 0} d_S^{(2+1)}(\\sigma ) = 4$ .", "In both cases we observe the dimension growing at small scales above the classical value (see, however, Subsec.", "REF ), which – from the perspective of a random walk process (REF ) – is the pattern of superdiffusion.", "Finally, the case of 1+1 dimensions is trivial – with no dimensional flow." ], [ "Relative-locality Laplacian", "The last Laplacian that is of our interest has been proposed in the framework of relative locality.", "This relative-locality Laplacian is determined by the square of the geodesic distance from the origin in momentum space [93], which has the same form for both the Lorentzian and Euclidean metric signature [19], namely $d^2(p_0,p_a) = \\kappa ^2 \\mathrm {arccosh}^2 (p_{-1}/\\kappa )$ .", "In bicrossproduct coordinates it becomes ${\\cal L}_{\\text{rl}}(P_0,\\lbrace P_a\\rbrace ) = -\\kappa ^2 \\mathrm {arccosh}^2 \\left(\\cosh \\left(\\tfrac{1}{\\kappa } P_0\\right) + \\frac{1}{2\\kappa ^2} e^{P_0/\\kappa } P_aP^a\\right)\\,,$ while in classical coordinates, ${\\cal L}_{\\text{rl}}(p_0,\\lbrace p_a\\rbrace ) = -\\kappa ^2 \\mathrm {arccosh}^2 \\left(\\tfrac{1}{\\kappa } \\sqrt{p_0^2 + p_a p^a + \\kappa ^2}\\right)\\,.$ The return probability in 3+1 dimensions is now given by ${\\cal P}_{(3+1)}(\\sigma ) = \\frac{\\pi ^{5/2} \\kappa ^3}{4\\sqrt{\\sigma }} e^{1/(4\\kappa ^2 \\sigma )} \\left(e^{2/(\\kappa ^2 \\sigma )} {\\rm erf}\\left(\\frac{3}{2\\kappa \\sqrt{\\sigma }}\\right) - 3\\, {\\rm erf}\\left(\\frac{1}{2\\kappa \\sqrt{\\sigma }}\\right)\\right)\\,,$ in 2+1 dimensions by ${\\cal P}_{(2+1)}(\\sigma ) = \\frac{\\pi ^{3/2} \\kappa ^2}{\\sqrt{\\sigma }} \\left(e^{1/(\\kappa ^2 \\sigma )} - 1\\right)$ and in 1+1 dimensions by ${\\cal P}_{(1+1)}(\\sigma ) = \\frac{\\pi ^{3/2} \\kappa }{\\sqrt{\\sigma }}\\, e^{1/(4\\kappa ^2 \\sigma )} {\\rm erf}\\left(\\frac{1}{2\\kappa \\sqrt{\\sigma }}\\right)$ ([19] had only numerical results for the Laplacian ${\\cal L}_{\\text{rl}}$ ).", "In this case, neither of the heat traces (REF –REF ) can be expanded in a series of the form (REF ).", "This is because of the factor $e^{1/\\sigma }$ , which yields an essential singularity at $\\sigma = 0$ .", "Consequently, the dimension spectrum of the relative-locality Laplacian does not exist in any of the three considered topological dimensions.", "On the other hand, given the exact formulae (REF –REF ), we can compute the corresponding spectral dimensions explicitly: $d_S^{(3+1)}(\\sigma ) = 1 + \\frac{3}{2\\kappa ^2 \\sigma } \\frac{{\\rm erf}\\left(\\frac{1}{2\\kappa \\sqrt{\\sigma }}\\right) - 3\\, e^{2/(\\kappa ^2 \\sigma )} {\\rm erf}\\left(\\frac{3}{2\\kappa \\sqrt{\\sigma }}\\right)}{3\\, {\\rm erf}\\left(\\frac{1}{2\\kappa \\sqrt{\\sigma }}\\right) - e^{2/(\\kappa ^2 \\sigma )} {\\rm erf}\\left(\\frac{3}{2\\kappa \\sqrt{\\sigma }}\\right)}\\,,$ with $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow \\infty } = 4$ , $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow 0} = \\infty $ ; similarly, in 2+1 dimensions $d_S^{(2+1)}(\\sigma ) = 1 + \\frac{2}{\\kappa ^2 \\sigma } \\frac{1}{1 - e^{-1/(\\kappa ^2 \\sigma )}}\\,,$ with $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow \\infty } = 3$ , $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow 0} = \\infty $ ; and in 1+1 dimensions $d_S^{(1+1)}(\\sigma ) = 1 + \\frac{1}{2\\kappa ^2 \\sigma } \\left(1 + \\frac{2\\kappa \\sqrt{\\sigma }\\, e^{-1/(4\\kappa ^2 \\sigma )}}{\\sqrt{\\pi }\\, {\\rm erf}\\left(\\frac{1}{2\\kappa \\sqrt{\\sigma }}\\right)}\\right)\\,,$ with $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow \\infty } = 2$ , $\\lim _{\\kappa \\sqrt{\\sigma } \\rightarrow 0} = \\infty $ .", "The UV divergence of $d_S(\\sigma )$ might be considered problematic, but such behaviour of the dimensionality (sometimes also for the Hausforff dimension) has been encountered in other models of quantum spacetime and at least in commutative geometry seems to be a consequence of the extremely high connectivity between points of space [94]." ], [ "Comparison", "For the sake of comparison, let us first recall (REF ) that the standard return probability on $\\mathbb {R}^n$ reads: $\\mathcal {P}_{\\text{class}}(\\sigma ) = (4 \\pi \\sigma )^{-n/2}.$ Hence, the dimension spectrum consists of a single element $\\lbrace n/2\\rbrace $ and is of order 1.", "The spectral dimension is a constant function equal to $n$ .", "The first observation is that for the bicovariant Laplacian the dimension spectra contain multiple elements.", "By analogy with the Riemannian geometry, one could interpret this as a signature of some sort of curved geometry of $\\kappa $ -Minkowski space.", "Furthermore, in the 2+1 dimensional case the dimension spectrum is of order 2.", "From the perspective of (pseudo)differential geometry, this means that the heat trace expansion involves non-local coefficients.", "The situation is analogous for the bicrossproduct Laplacian, apart from the 1+1 dimensional case, for which the dimension spectrum coincides with the classical one.", "The relative locality Laplacian does not have a dimension spectrum at all, which suggests that the corresponding geometry is infinite dimensional.", "In stark contrast with the quantum spheres, none of the studied Laplacians on $\\kappa $ -Minkowski space exhibits complex numbers outside of the real axis in its dimension spectrum.", "Consequently, there are no oscillations in the corresponding spectral dimensions.", "The latter is also true for the relative locality Laplacian.", "Let us now inspect the spectral dimensions closer.", "The profiles of $d_S(\\sigma )$ for different Laplacians in 3+1 and 2+1 topological dimensions are compared in Fig.", "REF (the situation in 1+1 topological dimensions is qualitatively the same apart from the case of ${\\cal L}_{\\text{cp}}$ , for which $d_S(\\sigma ) = 2$ — cf.", "Subsec.", "REF ).", "All curves in both plots exhibit the strong discrepancy in the UV, while they converge to the same IR limit, equal to the topological dimension.", "Unless we have some extra reason to claim that only one Laplacian is physically correct, the spectral dimension seems to be an ambiguous characteristic of $\\kappa $ -Minkowski space.", "Some further comments about the relations between specific UV limits visible in Fig.", "REF and various results obtained within the quantum gravity research, which could single out one of the Laplacians, can be found in [19].", "We also note that a recent analysis [95] of a static potential between two sources on 3+1-dimensional $\\kappa $ -Minkowski space brings evidence that the physical dimension in the UV is equal to 3, in agreement with the result for the bicovariant Laplacian.", "Figure: Spectral dimension d S (σ)d_S(\\sigma ) in 3+1 dimensions (left) and in 2+1 dimensions (right) for the Laplacians ℒ cv {\\cal L}_{\\text{cv}} (bottom curve), ℒ cp {\\cal L}_{\\text{cp}} (middle curve) and ℒ rl {\\cal L}_{\\text{rl}} (top curve) (we set κ=1\\kappa = 1).Let us note, however, that the spectral dimensions $d_S(\\sigma )$ in the previous subsections were calculated under an implicit assumption that all three operators ${\\cal L}$ are of order 2 — recall the discussion around Eq.", "(REF ).", "This is based on the fact that these operators are deformations of the classical Laplacian, which is a second order differential operator.", "In the case of the quantum spheres, we could provide an external argument for the order of quantum Laplacians, which is based on a rigorous first-order condition in the theory of spectral triples.", "In the case of $\\kappa $ -Minkowski space such an argument is not available, because the considered Laplacians do not originate from a Dirac operator.", "Observe that in classical coordinates the bicovariant Laplacian acquires the same form (REF ) as the standard Laplacian on ${\\mathbb {R}}^{n+1}$ .", "This justifies the assumption that its order equals 2.", "On the other hand, the bicrossproduct Laplacian in classical coordinates (REF ) looks as if was of order 1.", "Even more curiously, the relative-locality Laplacian in classical coordinates (REF ) seems to be of order 0.", "Indeed, its leading behaviour for large values of $\\vert p \\vert =\\sqrt{p_0^2 + p_a p^a}$ is $\\log \\vert p \\vert $ , which grows slower than any power of $\\vert p \\vert $ .", "The problem is that in the quantum gravity models one usually considers the quantum spacetime to be a certain deformation of the classical one (more accurately, it is the classical spacetime that is an approximation of the quantum one), with the deformation controlled by some parameter(s) related to the Planck length or mass.", "In particular, it is expected that a generalised Laplacian becomes the standard one in the classical limit, when the deformation vanishes, i.e.", "we actually have a parametrized family of operators, valid at different scales.", "In this context, Formula (REF ) allows one to track a deviation from the IR value of spacetime dimension.", "Using (REF ) with $\\eta \\ne 2$ would prevent us from recovering the correct IR limit, unless we allow the order of the operator to depend on the deformation parameter.", "If we assumed the order of the bicrossproduct Laplacian to be a continuous function of $\\kappa $ , such that $\\lim _{\\kappa \\rightarrow 0} \\eta (\\kappa ) = 1$ and $\\lim _{\\kappa \\rightarrow \\infty } \\eta (\\kappa ) = 2$ , then we would obtain the dimension $n$ in the UV and $n+1$ in the IR, as in the bicovariant case.", "In the same vein, we could set the order of the relative-locality Laplacian to be a function of $\\kappa $ , satisfying $\\lim _{\\kappa \\rightarrow \\infty } \\eta (\\kappa ) = 2$ .", "The situation in the UV limit in this case is more ambiguous since the dimension diverges, while the order should tend to zero and their ratio could yield any number.", "In particular, the behaviour of $\\eta (\\kappa )$ close to $\\kappa = 0$ might be such that $\\lim _{\\kappa \\rightarrow 0} 2/\\eta (\\kappa )\\, d_S(\\sigma ) = n$ , as for other Laplacians.", "Taking $d_S(\\sigma ) = d_S(\\kappa ^2 \\sigma )$ given by (REF -REF ) and considering the Ansatz $\\eta (\\kappa ) \\propto \\kappa ^2$ , we find that the desired limit requires $\\eta (\\kappa ) \\sim \\tfrac{4}{n} \\kappa ^2$ for $\\kappa \\rightarrow 0$ .", "Such a trick allows us to remove the discrepancy between the UV behaviour of the spectral dimensions of all three Laplacians, while maintaining the correct classical limit.", "It also agrees with the viewpoint on the operator orders suggested by the formulae in classical coordinates.", "Nevertheless, let us stress that the deformation-parameter-dependent order is only a tentative hypothesis, introduced by us here.", "Furthermore, even if this hypothesis was rigorously implemented and erased differences between spectral dimensions for the three Laplacians, the corresponding dimension spectra would still remain irreconcilably different.", "For example, the dimension spectrum in the case of 3+1 topological dimensions, depending on the Laplacian, is infinite, has only two elements or does not exist." ], [ "Summary", "Our analysis shows that the UV behaviour of the spectral dimensions of quantum spacetimes is fairly complex and not limited to a monotonic flow.", "To understand it we have adopted from noncommutative geometry à la Connes a rigorous notion of a dimension spectrum.", "The latter characterises the UV behaviour of a heat trace, from which the spectral dimension is deduced.", "The relationship between the spectral dimension and the dimension spectrum of a given Laplacian (more generally, an operator of order $\\eta $ ) is captured by the following dictionary: The dimension spectrum consisting of more than one element implies a non-constant behaviour of the spectral dimension and suggests a non-trivial geometry.", "If the UV limit $\\sigma \\rightarrow 0$ is finite, then $d_S(0) = \\eta \\, d_{\\operatorname{Sd}}$ , where $d_{\\operatorname{Sd}}$ is largest real number in the dimension spectrum (REF ).", "Note that, in general, $d_{\\operatorname{Sd}}$ need not be a natural number.", "This happens routinely in fractal geometry, where $d_{\\operatorname{Sd}}$ recovers the Hausdorff dimension [96], [56], [57], [97], [98].", "The order of the dimension spectrum determines the leading behaviour of the spectral dimension for small $\\sigma $ .", "If $\\operatorname{ord}\\operatorname{Sd}> 1$ , then this behaviour is logarithmic.", "More concretely, if $\\mathcal {P}(\\sigma ) \\sim (\\log \\sigma )^p \\sigma ^{-r}$ as $\\sigma $ tends to 0, with $p = \\operatorname{ord}\\operatorname{Sd}- 1 \\in \\mathbb {N}$ and $r \\ge 0$ , then $d_S(\\sigma ) \\sim 2r - 2p / (\\log \\sigma )$ .", "This happens on the quantum sphere with $p=2$ and $r=0$ .", "On the other hand, a sub-leading logarithmic behaviour $\\mathcal {P}(\\sigma ) \\sim \\alpha \\sigma ^{-r} + \\beta \\sigma ^{-r-1} (\\log \\sigma )^p$ translates to $d_S(\\sigma ) \\sim 2 (r-1) + 2\\alpha / (\\alpha + \\beta \\sigma (\\log \\sigma )^p)$ .", "This is exemplified on the 2+1-dimensional $\\kappa $ -Minkowski spacetime, with $p = 1$ and $r = 1$ or $r = 2$ for the bicovariant and the bicrossproduct Laplacians, respectively.", "The presence of non-real numbers in $\\operatorname{Sd}$ signifies the presence of log-periodic oscillations in the UV behaviour of the spectral dimension.", "With the help of the dimension spectrum we have detected the log-periodic oscillations of the spectral dimension for the quantum sphere.", "These were overlooked in [20], because of their very small amplitude for the deformation parameter close to the classical value.", "It is remarkable that the complex dimensions occur here in a somewhat unexpected setting, where no fractal properties or discrete scale invariance were a priori imposed.", "In contrast, such oscillations are present on $\\kappa $ -Minkowski spacetime for none of the 3 considered Laplacians, at least in the cases of 2, 3 and 4 topological dimensions that we studied.", "One might relate this to the fact the quantum sphere is compact, whereas $\\kappa $ -Minkowski spacetime is not and interpret the oscillations as a peculiar IR/UV mixing effect.", "On the other hand, from a more conservative standpoint, one could simply say that the quantum sphere is “more quantum”.", "In either case, it seems worth looking for the complex dimensions in other models of quantum spacetime.", "Finally, we would like to stress that the spectral dimension and dimension spectrum of a given quantum spacetime strongly depend on the chosen Laplacian.", "On the mathematical side, this is simply a consequence of their definitions, arising from the trace of the heat operator, which is constructed from a given Laplacian.", "As we discussed in the last subsection, the discrepancy between the UV behaviour of the spectral dimensions of $\\kappa $ -Minkowski space with different Laplacians can perhaps be removed using the concept of the deformation-dependent order of the Laplacian but even then, differences will survive in the dimension spectra.", "On the other hand, while the dimension spectra of the quantum sphere with the spinorial and scalar Laplacians are identical, their spectral dimensions do not overlap and actually behave in the opposite ways in the IR.", "Thus, the spectral dimension and dimension spectrum provide complementary information about a quantum spacetime structure.", "In terms of physics, we believe that it is of key importance to understand which aspects of dimensionality are genuine to the given quantum geometry and which are just mathematical artefacts of the chosen Laplacian.", "Our analysis suggests that such universal properties are the log-periodic oscillations for the quantum sphere, as well as the lack of these for the $\\kappa $ -Minkowski spacetime, because these features persist for all of the studied Laplacians.", "The same could be said about the appearance of third order poles in the dimension spectra on quantum sphere.", "As pointed out, this could be taken as an indication of some sort of `singular' or `non-smooth' character of this quantum spacetime.", "On the other hand, the dimension spectra of $\\kappa $ -Minkowski spacetime fit within the ones obtained in classical (pseudo)differential geometry.", "Interestingly enough, the second order poles, and hence the $\\log \\sigma $ terms in the heat trace, occur only in 2+1 topological dimensions, but for both bicovariant and bicrossproduct Laplacians.", "This suggests that the geometry of 2+1 dimensional $\\kappa $ -Minkowski spacetime departs more strongly from the classical case.", "However, further studies — possibly connected with the expected non-local nature of some coefficients of the heat trace expansion — would be needed to confirm this conclusion." ], [ "The spectral zeta function of the operator $\\Delta _q^\\text{sc}$", "In this appendix we provide an explicit meromorphic extension of the spectral zeta function associated with the operator $\\Delta _q^\\text{sc}$ defined by Formula (REF ).", "Before we start we need to take care of the zero mode, as for any $s \\in {\\mathbb {C}}$ the operator $(\\Delta _q^\\text{sc})^{-s}$ is not trace-class on the full Hilbert space $\\mathcal {H}_q^{\\prime }$ .", "We thus set $(\\ker \\Delta _q^\\text{sc})^{\\perp }$ and compute the zeta function without the zero mode.", "Note that after such a truncation, it is still possible to use the inverse Mellin transform technique for the computation of the corresponding heat trace (see [59] for the general method).", "Indeed, we have $\\mathcal {P}_q^\\text{sc}(\\sigma ) & = \\operatorname{Tr}_{\\mathcal {H}_q^{\\prime }} e^{- \\sigma \\Delta _q^\\text{sc}} = 1 + \\operatorname{Tr}_{ e^{- \\sigma \\Delta _q^\\text{sc}}}and simply use formulae (\\ref {Mell}--\\ref {poles}) with at the place of \\mathcal {H}_q^{\\prime }.$ Now, for $\\mathrm {Re}(s) > 0$ we have $\\zeta _{\\Delta _q^\\text{sc}}(s) &= \\operatorname{Tr}_{\\, (\\Delta _q^\\text{sc})^{-s} = \\sum _{j=1}^{\\infty } \\sum _{m=-j}^{j} \\langle j,m| (\\Delta _q^\\text{sc})^{-s} | j,m \\rangle \\\\& = u(\\sqrt{q})^{-s} \\sum _{j=1}^{\\infty } (2j+1) q^{s/2} \\left( q^{-j} - 1 - q + q^{j+1} \\right)^{-s} \\\\& = \\big ( u(\\sqrt{q}) q^{-3/2} \\big )^{-s} \\sum _{k=0}^{\\infty } (2k+3) \\left( q^{-k} - q - q^2 + q^{k+3} \\right)^{-s} \\\\& = \\big ( u(\\sqrt{q}) q^{-3/2} \\big )^{-s} \\sum _{k=0}^{\\infty } (2k+3) q^{ks} \\left( 1 - q^{k+1} \\right)^{-s} \\left( 1 - q^{k+2} \\right)^{-s}.", "}$ In order to construct a meromorphic extension of $\\zeta _{\\Delta _q^\\text{sc}}$ to the entire complex plane we use the standard binomial expansion formula $(1-x)^{-s} = \\sum _{n = 0}^{\\infty } \\binom{s+n-1}{n} x^n\\,$ valid for any complex number $s$ and any $x \\in {\\mathbb {C}}$ with $\\vert x \\vert < 1$ .", "The coefficients $\\binom{s+n-1}{n} = \\tfrac{\\Gamma (s+n)}{n!", "\\, \\Gamma (s)}$ are polynomials in $s$ of order $n$ .", "With the help of formula (REF ) we rewrite the zeta function as follows: $\\zeta _{\\Delta _q^\\text{sc}}(s) &= \\big ( u(\\sqrt{q}) q^{-3/2} \\big )^{-s} \\sum _{k=0}^{\\infty } (2k+3) \\sum _{\\ell = 0}^{\\infty } \\sum _{m = 0}^{\\infty } \\binom{s+\\ell -1}{\\ell } \\binom{s+m-1}{m} q^{(k+1)\\ell } q^{(k+2)m} q^{ks}.$ For $\\mathrm {Re}(s) > 0$ all of the series are absolutely convergent and we are free to change the order of summation and compute the sum over $k$ .", "We thus have $\\zeta _{\\Delta _q^\\text{sc}}(s) &= \\big ( u(\\sqrt{q}) q^{-3/2} \\big )^{-s} \\sum _{\\ell = 0}^{\\infty } \\sum _{m = 0}^{\\infty } \\binom{s+\\ell -1}{\\ell } \\binom{s+m-1}{m} \\frac{q^{\\ell + 2m} (3- q^{\\ell + m +s})}{(1-q^{\\ell +m+s})^2} \\\\&= \\big ( u(\\sqrt{q}) q^{-3/2} \\big )^{-s} \\sum _{n = 0}^{\\infty } \\sum _{m = 0}^{n} \\binom{s+n-m-1}{n-m} \\binom{s+m-1}{m} \\frac{q^{n+m} (3- q^{n +s})}{(1-q^{n+s})^2} \\\\&= \\big ( u(\\sqrt{q}) q^{-3/2} \\big )^{-s} \\sum _{n = 0}^{\\infty } \\frac{q^{n} (3- q^{n +s})}{(1-q^{n+s})^2} \\binom{s+n-1}{n} {}_2F_1(-n,s;-n+s+1;q),$ with a hypergeometric function $_2F_1$ .", "The last series over $n$ can easily be shown (cf.", "[41]) to be absolutely convergent for any complex $s$ outside of the discrete set $\\tfrac{\\pi i}{\\log q} {\\mathbb {Z}}- {\\mathbb {N}}$ .", "We have thus obtained a meromorphic extension of $\\zeta _{\\Delta _q^\\text{sc}}$ to the entire complex plane.", "It has isolated double poles precisely in the set $\\operatorname{Sd}\\Delta _q^\\text{sc}= \\operatorname{Sd}\\Delta _q^\\text{sp}= \\tfrac{\\pi i}{\\log q} {\\mathbb {Z}}- {\\mathbb {N}}$ .", "The poles of the Gamma function contribute additional poles at $-n$ , $n \\in {\\mathbb {N}}$ , hence the order of the dimension spectrum is 3.", "Formula (REF ) for the meromorphic extension of the function $\\zeta _{\\Delta _q^\\text{sp}}$ is proved along the same lines with the help of the identity (REF ).", "The two zeta functions have the same meromorphic structure, though $\\zeta _{\\Delta _q^\\text{sc}}$ has a more involved form of the coefficients.", "The work of ME was supported by the National Science Centre in Poland under the research grant Sonatina (2017/24/C/ST2/00322).", "Publication supported by the John Templeton Foundation Grant ,,Conceptual Problems in Unification Theories” (No.", "60671)." ] ]

[ [ "Global regularity for solutions of the Navier-Stokes equation\n sufficiently close to being eigenfunctions of the Laplacian" ], [ "Abstract In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier--Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian.", "This estimate improves previous regularity criteria requiring control on the $\\dot{H}^\\alpha$ norm of $u,$ with $2\\leq \\alpha<\\frac{5}{2},$ to a regularity criterion requiring control on the $\\dot{H}^\\alpha$ norm multiplied by the deficit in the interpolation inequality for the embedding of $\\dot{H}^{\\alpha-2}\\cap\\dot{H}^{\\alpha} \\hookrightarrow \\dot{H}^{\\alpha-1}.$ This regularity criterion suggests, at least heuristically, the possibility of some relationship between potential blowup solutions of the Navier--Stokes equation and the Kolmogorov-Obhukov spectrum in the theory of turbulence." ], [ "Introduction", "The Navier–Stokes equation is one of the fundamental equations of fluid mechanics.", "For an incompressible fluid, where the density of the fluid in question is constant, the Navier–Stokes equation with no external forces is given by $\\partial _t u-\\Delta u + P_{df}\\left((u\\cdot \\nabla ) u\\right)&=0, \\\\\\nabla \\cdot u&=0,$ where $u\\in \\mathbb {R}^3$ is the velocity of the fluid and $P_{df}$ is the projection onto the space of divergence free vector fields.", "We have also taken the viscosity to be $1,$ which we can do without loss of generality, because it is equivalent up to rescaling.", "In his foundational paper on the subject, Leray proved the global existence of weak solutions to the Navier–Stokes equation satisfying an energy inequality ; however, such solutions are not known to be either smooth or unique.", "This is because the bounds from the energy inequality are both supercritical with respect to scaling.", "Smooth solutions of the Navier–Stokes equation with initial data in $H^1$ must satisfy an energy equality, stating that for all $t>0,$ $ \\frac{1}{2}\\Vert u(\\cdot ,t)\\Vert _{L^2}^2+\\int _0^t \\Vert \\nabla u(\\cdot ,\\tau )\\Vert _{L^2}^2 \\mathop {}\\!\\mathrm {d}\\tau =\\frac{1}{2}\\left\\Vert u^0\\right\\Vert _{L^2}^2.$ The Navier–Stokes equation is invariant under the rescaling, $ u^\\lambda (x,t)=\\lambda u\\left(\\lambda x, \\lambda ^2 t\\right),$ for all $\\lambda >0$ , and it is a simple calculation to check that the bounds on $u$ in $L^\\infty _t L^2_x$ and $L^2_t \\dot{H}^1_x$ from the energy equality (REF ) are supercritical with respect to the rescaling (REF ).", "The only results presently available guaranteeing regularity for the Navier–Stokes equation with general, arbitrarily large initial data, require control of some scale critical quantity.", "Ladyzhenskaya , Prodi , and Serrin , proved that if a smooth solution of the Navier–Stokes equation blows up in finite time $T_{max}<+\\infty ,$ then for all $3<q\\le +\\infty , \\frac{2}{p}+\\frac{3}{q}=1,$ $\\int _0^{T_{max}}\\Vert u(\\cdot ,t)\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}t=+\\infty .$ It is straightforward to check that $L^p_t L^q_x$ is scale critical with respect to the rescaling (REF ), when $\\frac{2}{p}+\\frac{3}{q}=1.$ This result was then extended by Escauriaza, Seregin, and Šverák to the endpoint case $p=+ \\infty , q=3.$ They proved that if a smooth solution $u$ of the Navier-Stokes equation blows up in finite time $T_{max}<+\\infty $ , then $\\limsup _{t \\rightarrow T_{max}}\\Vert u(\\cdot ,t)\\Vert _{L^3(\\mathbb {R}^3)}=+\\infty .$ There have been a number of generalizations of the Ladyzhenskaya-Prodi-Serrin regularity criterion, including scale critical component reduction results involving the vorticity , two components of the vorticity , the derivative in just one direction, $\\frac{\\partial u}{\\partial x_i}$ , and involving only one component, $u_j$ Chemin1,Chemin2.", "The Ladyzhenskaya-Prodi-Serrin regularity criterion has also been generalized to endpoint Besov spaces ChenZhangBesov,KOTbesov,KozonoShimadaBesov, while the Escauriaza-Seregin-Šverák regularity criteriona has been generalized to all non-endpoint Besov spaces AlbrittonBesov,GKPbesov,AlbrittonBarker.", "In this paper, we will generalize the Ladyzhenskaya-Prodi-Serrin regularity criterion to solutions of the Navier–Stokes equation that are close to being eigenfunctions of the Laplacian.", "One tool we will use is the notion of mild solutions, which was developed by Kato and Fujita .", "Definition 1.1 Suppose $u \\in C\\left([0,T);\\dot{H}^1\\left(\\mathbb {R}^3\\right) \\right),\\nabla \\cdot u=0.$ Then $u$ is a mild solution to the Navier–Stokes equation if for all $0<t<T$ , $u(\\cdot ,t)=e^{t \\Delta }u^0+\\int _0^t e^{(t-\\tau )\\Delta }P_{df}\\left(-(u \\cdot \\nabla )u\\right)(\\cdot , \\tau )\\mathop {}\\!\\mathrm {d}\\tau ,$ where $e^{t\\Delta }$ is the heat semi-group operator given by convolution with the heat kernel; that is to say, $e^{t\\Delta }u^0$ is the solution of the heat equation after time $t,$ with initial data $u^0.$ Fujita and Kato proved the local existence in time of mild solutions to the Navier–Stokes equation for all initial data in $\\dot{H}^1\\left(\\mathbb {R}^3\\right).$ Our results for solutions of the Navier–Stokes equation sufficiently close to being eigenfunctions of the Laplacian will be proven in terms of $H^1$ mild solutions.", "We will also define, for all $\\alpha >-\\frac{3}{2},$ the homogeneous Sobolev space $\\dot{H}^\\alpha \\left(\\mathbb {R}^3\\right)$ , which is a Hilbert space for $-\\frac{3}{2}<\\alpha <\\frac{3}{2}$ , as the space with the norm $\\Vert f\\Vert _{\\dot{H}^\\alpha }^2&=\\left\\Vert (-\\Delta )^{\\frac{\\alpha }{2}}f\\right\\Vert _{L^2}^2\\\\&=\\int _{\\mathbb {R}^3} \\left( 2 \\pi |\\xi |\\right)^{2\\alpha }\\big |\\hat{f}(\\xi )\\big |^2 \\mathop {}\\!\\mathrm {d}\\xi ,$ and the inhomogeneous Hilbert space $H^\\alpha \\left(\\mathbb {R}^3\\right)$ as the space with the norm $\\Vert f\\Vert _{H^\\alpha }^2=\\int _{\\mathbb {R}^3} \\left(1+4\\pi ^2|\\xi |^2 \\right)^{\\alpha }\\big |\\hat{f}(\\xi )\\big |^2 \\mathop {}\\!\\mathrm {d}\\xi ,$ while further noting that for all $\\alpha >0,$ $H^\\alpha =L^2 \\cap \\dot{H}^\\alpha .$ For solutions of the Navier–Stokes equation, we call $\\frac{1}{2}\\Vert u(\\cdot ,t)\\Vert _{L^2}^2$ the energy and $\\frac{1}{2}\\Vert u(\\cdot ,t)\\Vert _{\\dot{H}^1}^2$ the enstrophy.", "With mild solutions and the relevant Sobolev spaces defined, we can now state the main theorem of this paper.", "Theorem 1.2 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation, and suppose $\\frac{6}{5}<q\\le 3, \\frac{2}{p}+\\frac{3}{q}=3.$ Then for all $0<t<T_{max}$ $ \\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(C_q \\int _0^t\\inf _{\\lambda (\\tau )\\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}\\tau \\right),$ where $C_q>0$ depends only on $q.$ In particular, if $T_{max}<+\\infty $ then $\\int _0^{T_{max}} \\inf _{\\lambda (t)\\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}t=+\\infty .$ An eigenfunction of the Laplacian satisfies the equation, $-\\Delta u-\\lambda u=0.$ There are no nonzero eigenfunctions of the Laplacian in $H^1\\left(\\mathbb {R}^3\\right),$ because if $-\\Delta u=\\lambda u,$ that would require its Fourier transform to be supported on a set of measure zero, specifically the set $\\left\\lbrace \\xi \\in \\mathbb {R}^3: 4\\pi ^2|\\xi |^2=\\lambda \\right\\rbrace ;$ however, the quantity $\\int _0^t \\inf _{\\lambda (\\tau ) \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}\\tau $ is nonetheless a scale critical measure of how close a solution is to being an eigenfunction of the Laplacian.", "This quantity is scale invariant because this infimum scales the same way as $-\\Delta u,$ the quantity with no parameter in the infimum, and we can see from the scale invariance (REF ), that $-\\Delta u$ has the scale invariance $-\\Delta u^\\lambda (x,t)=-\\lambda ^3\\Delta u \\left(\\lambda x, \\lambda ^2 t\\right).$ It is a simple calculation to observe that when $\\frac{2}{p}+\\frac{3}{q}=3,$ the space $L^p_t L^q_x$ is invariant under this rescaling.", "We do not have a nice expression for the quantity $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q},$ in general $L^q$ spaces, however in $L^2$ we can use the Hilbert space structure to calculate this quantity explicitly, which allows us to obtain the following result.", "Corollary 1.3 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation.", "Then for all $0<t<T_{max}$ $\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(C_2 \\int _0^t\\Vert -\\Delta u\\Vert _{L^2}^\\frac{4}{3}\\left(1-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2 \\Vert -\\Delta u\\Vert _{L^2}^2}\\right)^\\frac{2}{3} \\mathop {}\\!\\mathrm {d}\\tau \\right),$ where $C_2>0$ is taken as in Theorem REF .", "In particular, if $T_{max}<+\\infty $ then $\\int _0^{T_{max}} \\Vert -\\Delta u\\Vert _{L^2}^\\frac{4}{3}\\left(1-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2 \\Vert -\\Delta u\\Vert _{L^2}^2}\\right)^\\frac{2}{3} \\mathop {}\\!\\mathrm {d}t=+\\infty .$ More generally, we can use the Sobolev embedding of $\\dot{H}^\\beta \\hookrightarrow L^q$ to generalize Corollary REF in terms of homogeneous Sobolev spaces.", "Corollary 1.4 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation, and suppose $2\\le \\alpha \\le \\frac{5}{2},\\alpha =\\frac{1}{2}+\\frac{2}{p}.$ Then for all $0<t<T_{max}$ $\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(\\tilde{C}_\\alpha \\int _0^t\\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)^\\frac{p}{2} \\mathop {}\\!\\mathrm {d}\\tau \\right),$ where $\\tilde{C}_\\alpha >0$ depends only on $\\alpha .$ In particular, if $T_{max}<+\\infty $ then $\\int _0^{T_{max}}\\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)^\\frac{p}{2} \\mathop {}\\!\\mathrm {d}t=+\\infty .$ Note that the scaling relation between $\\alpha $ and $p$ can alternatively be expressed by $p=\\frac{2}{\\alpha -\\frac{1}{2}}$ We will note here that the $\\alpha =2$ case of Corollary REF , is precisely Corollary REF .", "For $2<\\alpha \\le \\frac{5}{2},$ Corollary REF requires that we use the fractional Sobolev inequality to bound the infimum in $L^q$ by an infimum in the appropriate homogeneous Hilbert space, which can be calculated explicitly.", "We will show that $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^2&\\le C \\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{\\dot{H}^{\\alpha -2}}^2\\\\&=C \\Vert u\\Vert _{\\dot{H}^\\alpha }^2\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right),$ where $\\alpha -2= \\frac{3}{2}-\\frac{3}{q}$ .", "We will also note that without the term $\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^2\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^2}\\right)^\\frac{p}{2},$ Corollary REF is an immediate corollary of a variant of the Ladyzhenskaya-Prodi-Serrin regularity criterion.", "Corollary REF shows that our regularity criterion for solutions of the Navier–Stokes equation sufficiently close to being eigenfunctions of the Laplacian measures the deficit in the interpolation inequality for the embedding $\\dot{H}^{\\alpha -2} \\cap \\dot{H}^{\\alpha }\\hookrightarrow \\dot{H}^{\\alpha -1},$ which states that $ \\Vert f\\Vert _{\\dot{H}^{\\alpha -1}}^2\\le \\Vert f\\Vert _{\\dot{H}^\\alpha }\\Vert f\\Vert _{\\dot{H}^{\\alpha -2}},$ where the constant 1 is sharp but not attained, because there are no nonzero eigenfunctions of the Laplacian in $\\dot{H}^{\\alpha -1}\\left(\\mathbb {R}^3\\right).$ When the inequality (REF ) is close to being saturated, then the quantity $\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)^\\frac{p}{2}$ will be small, so Corollary REF limits the extent to which blowup solutions can saturate the interpolation inequality (REF ).", "We will also prove that finite-time blowup solutions cannot concentrate on arbitrarily narrow bands in Fourier space, supported between an inner radius of $R_1(t)$ and an outer radius of $R_2(t),$ with the ratio $\\frac{R_1(t)}{R_2(t)} \\rightarrow 1$ arbitrarily quickly as $t \\rightarrow T_{max}$ relative to the size of $\\Vert u(\\cdot ,t)\\Vert _{\\dot{H}^\\alpha }^p$ .", "Corollary 1.5 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation, and suppose for all $0<t<T_{max}$ $\\operatorname{supp}{ \\hat{u}(t)} \\subset \\left\\lbrace \\xi \\in \\mathbb {R}^3:R_1(t) \\le |\\xi | \\le R_2(t) \\right\\rbrace .$ Let $2\\le \\alpha \\le \\frac{5}{2},\\alpha =\\frac{1}{2}+\\frac{2}{p}.$ Then for all $0<t<T_{max}$ $\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(\\tilde{C}_\\alpha \\int _0^t\\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{R_1(\\tau )^4}{R_2(\\tau ))^4}\\right)^\\frac{p}{2} \\mathop {}\\!\\mathrm {d}\\tau \\right),$ where $\\tilde{C}_\\alpha >0$ depends only on $\\alpha .$ In particular, if $T_{max}<+\\infty $ then $\\int _0^{T_{max}}\\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{R_1(t)^4}{R_2(t))^4}\\right)^\\frac{p}{2} \\mathop {}\\!\\mathrm {d}t=+\\infty .$ There are a number of previous results for Navier–Stokes regularity criteria involving frequency localization in Fourier space Bradshaw,Luo,Shvydkoy.", "There is not enough space to discuss these results in detail here, as stating the main theorems would require us to define a number of objects from Littlewood-Paley theory, but we will note that the regularity criterion in has an explicit connection with the Kolmogorov scaling in turbulence.", "Corollary REF can also be seen as providing a heuristic connection between the regularity criterion in Theorem REF , and the Kolmogorov phenomenological theory of turbulence.", "Corollary REF shows that solutions supported on narrow bands in Fourier space are not good candidates for finite-time blowup; this is consistent with the Kolmogorov-Obukhov phenomenology of turbulence Kolmogorov,Obukhov, which stipulates that turbulent flows cannot localize around a small number of frequencies, specifically that the energy spectrum for turbulent flows has a decay in Fourier space on the order of $|\\xi |^{-\\frac{5}{3}}$ in the inertial range." ], [ "Proofs of the results", "Before beginning the proof of the main theorem, we will first need to establish an identity for the growth of enstrophy related to eigenfunctions of the Laplacian.", "Lemma 2.1 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation.", "Then for all $0<t<T_{max},$ and for all $\\lambda (t)\\in \\mathbb {R}$ $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t} \\frac{1}{2}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2=-\\Vert \\Delta u\\Vert _{L^2}^2-\\left<-\\Delta u-\\lambda u,(u\\cdot \\nabla )u\\right>.$ It is easy to see from the Navier–Stokes equation that $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t} \\frac{1}{2}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2=-\\Vert -\\Delta u\\Vert _{L^2}^2-\\left<-\\Delta u, (u\\cdot \\nabla )u\\right>.$ We know that $u\\in H^1,$ and therefore we have sufficient regularity to integrate by parts, using the condition $\\nabla \\cdot u=0$ , to conclude that $\\left<(u\\cdot \\nabla )u,u\\right>&=-\\left<u,(u\\cdot \\nabla )u\\right>\\\\&=0.$ Because we know that for all $\\lambda \\in \\mathbb {R}$ $\\left<\\lambda u,(u\\cdot \\nabla )u\\right>=0,$ we can conclude that for all $\\lambda (t)\\in \\mathbb {R}$ , $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t} \\frac{1}{2}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2=-\\Vert \\Delta u\\Vert _{L^2}^2-\\left<-\\Delta u-\\lambda u,(u\\cdot \\nabla )u\\right>.$ This completes the proof.", "One of the other main ingredients in our proof will be the fractional Sobolev inequality, which is stated below.", "Theorem 2.2 Suppose $0<s<\\frac{3}{2},$ and $\\frac{1}{q}=\\frac{1}{2}-\\frac{s}{3}.$ Then for all $f\\in \\dot{H}^s\\left(\\mathbb {R}^3\\right),$ $\\Vert f\\Vert _{L^q}\\le C_s \\Vert f\\Vert _{\\dot{H}^s},$ where $C_s=2^{-\\frac{s}{3}}\\pi ^{-\\frac{4}{3}s}\\left(\\frac{\\Gamma \\left(\\frac{3}{2}-s\\right)}{\\Gamma \\left(\\frac{3}{2}+s\\right)}\\right)^\\frac{1}{2}$ Note that the scaling relation between the parameters $q$ and $s$ can be stated equivalently as $s=\\frac{3}{2}-\\frac{3}{q}.$ The Sobolev inequality was first proven by Sobolev in the case where $s=1$ .", "The sharp version of this inequality was proven by Talenti in the case where $s=1$ , and the general sharp version of this inequality with $0<s<\\frac{3}{2}$ was proven by Lieb .", "With these results established, we can now prove Theorem REF , which is restated here for the reader's convenience.", "Theorem 2.3 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation, and suppose $\\frac{6}{5}<q\\le 3, \\frac{2}{p}+\\frac{3}{q}=3.$ Then for all $0<t<T_{max}$ $ \\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(C_q \\int _0^t\\inf _{\\lambda (\\tau )\\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}\\tau \\right),$ where $C_q>0$ depends only on $q.$ In particular, if $T_{max}<+\\infty $ then $\\int _0^{T_{max}} \\inf _{\\lambda (t)\\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}t=+\\infty .$ We know that if $T_{max}<+\\infty ,$ then $\\lim _{t \\rightarrow T_{max}}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}=+\\infty ,$ so it suffices to prove the bound (REF ).", "We will not keep track of the value of the constants $C_q.$ It is possible to compute the value of the constant $C_q$ explicitly in terms of the sharp fractional Sobolev inequality in Theorem REF , but we will not concern ourselves with long expressions for the value of the constant that would only clutter up this paper without adding any real mathematical insight.", "First we will consider the case when $q=3, p=1.$ Using the identity for enstrophy growth in Proposition REF , and applying Hölder's inequality and the Sobolev inequality, we find that for all $0<t<T_{max},$ and for all $\\lambda (t)\\in \\mathbb {R},$ $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t} \\frac{1}{2}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2&=-\\Vert \\Delta u\\Vert _{L^2}^2-\\left<-\\Delta u-\\lambda u,(u\\cdot \\nabla )u\\right>\\\\&\\le \\Vert -\\Delta u-\\lambda u\\Vert _{L^3} \\Vert u\\Vert _{L^6}\\Vert \\nabla u\\Vert _{L^2}\\\\&\\le C \\Vert -\\Delta u-\\lambda u\\Vert _{L^3}\\Vert \\nabla u\\Vert _{L^2}^2.$ Multiplying both sides by 2 and taking the infimum over $\\lambda (t) \\in \\mathbb {R},$ we find that $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t} \\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le C_3 \\inf _{\\lambda (t)\\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^3}\\Vert \\nabla u\\Vert _{L^2}^2.$ Applying Grönwall's inequality we find that for all $0<t<T_{max}$ $\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(C_3 \\int _0^t\\inf _{\\lambda (\\tau )\\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^3} \\mathop {}\\!\\mathrm {d}\\tau \\right),$ and we are done with the case where $q=3.$ Now we will consider the case $\\frac{6}{5}<q<3.$ First we will take $6<a<+\\infty $ to be given by $ \\frac{1}{a}=\\frac{5}{18}-\\frac{1}{3q},$ and $2<b<6$ to be given by $ \\frac{1}{b}=\\frac{13}{18}-\\frac{2}{3q}.$ Note that $\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{q}=1,$ so again using the identity for enstrophy growth from Lemma REF , Hölder's inequality, and the fractional Sobolev inequality, we find that for all $0<t<T_{max}$ and for all $\\lambda (t)\\in \\mathbb {R},$ $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t}\\frac{1}{2}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2&=-\\Vert \\Delta u\\Vert _{L^2}^2-\\left<-\\Delta u-\\lambda u,(u\\cdot \\nabla )u\\right>\\\\&\\le -\\Vert \\Delta u\\Vert _{L^2}^2+\\Vert \\Delta u-\\lambda u\\Vert _{L^q}\\Vert u\\Vert _{L^a}\\Vert \\nabla u\\Vert _{L^b}\\\\&\\le -\\Vert \\Delta u\\Vert _{L^2}^2+C\\Vert \\Delta u-\\lambda u\\Vert _{L^q}\\Vert \\nabla u\\Vert _{\\dot{H}^\\alpha }\\Vert \\nabla u\\Vert _{\\dot{H}^\\beta },$ where $\\alpha =\\frac{1}{2}-\\frac{3}{a},$ and $\\beta =\\frac{3}{2}-\\frac{3}{b}.$ Plugging back into (REF ) and (REF ), we find that $\\alpha =\\frac{1}{q}-\\frac{1}{3},$ and $\\beta =\\frac{2}{q}-\\frac{2}{3}.$ It is straightforward to see that $0<\\alpha <\\frac{1}{2}$ and $0<\\beta <1,$ so we can interpolate between $L^2$ and $\\dot{H}^1$ to find that $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t}\\frac{1}{2}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2&\\le -\\Vert \\Delta u\\Vert _{L^2}^2+C\\Vert \\Delta u-\\lambda u\\Vert _{L^q}\\Vert \\nabla u\\Vert _{L^2}^{2-(\\alpha +\\beta )}\\Vert \\nabla u\\Vert _{\\dot{H}^1}^{\\alpha +\\beta }\\\\&=-\\Vert \\Delta u\\Vert _{L^2}^2+C \\Vert \\Delta u-\\lambda u\\Vert _{L^q}\\Vert \\nabla u\\Vert _{L^2}^{3-\\frac{3}{q}}\\Vert -\\Delta u\\Vert _{L^2}^{\\frac{3}{q}-1},$ where we have used the fact that $\\alpha +\\beta =\\frac{3}{q}-1.$ Recalling that $\\frac{2}{p}=3-\\frac{3}{q},$ we can see that $\\frac{6}{5}<q<3$ implies that $1<p<4.$ Take $\\frac{4}{3}<r<+\\infty $ to be the conjugate of p. $\\frac{1}{p}+\\frac{1}{r}=1.$ Observe that $\\frac{2}{r}&=2-\\frac{2}{p}\\\\&=\\frac{3}{q}-1.$ Using this to simplify and applying Young's inequality with exponents $r$ and $p,$ we find that $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t}\\frac{1}{2}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2&\\le -\\Vert \\Delta u\\Vert _{L^2}^2+C\\Vert -\\Delta u\\Vert _{L^2}^{\\frac{2}{r}}\\Vert \\Delta u-\\lambda u\\Vert _{L^q}\\Vert \\nabla u\\Vert _{L^2}^{\\frac{2}{p}}\\\\&\\le C\\Vert \\Delta u-\\lambda u\\Vert _{L^q}^p \\Vert \\nabla u\\Vert _{L^2}^2$ Multiplying both sides by 2 and taking the infimum over $\\lambda (t)\\in \\mathbb {R},$ we find that $\\frac{\\mathop {}\\!\\mathrm {d}}{\\mathop {}\\!\\mathrm {d}t}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le C_q \\inf _{\\lambda (t)\\in \\mathbb {R}}\\Vert \\Delta u-\\lambda u\\Vert _{L^q}^p \\Vert \\nabla u\\Vert _{L^2}^2$ Applying Grönwall's inequality we find that for all $0<t<T_{max}$ $\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(C_q \\int _0^t\\inf _{\\lambda (\\tau )\\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}\\tau \\right).$ This completes the proof.", "We will note here that the key element of the proof is the fact that $\\left<(u\\cdot \\nabla )u,u\\right>=0.$ Because Tao's averaged 3D Navier–Stokes model equation also has the property $\\left<\\tilde{B}(u,u),u\\right>=0,$ the regularity criterion in Theorem REF and the subsequent corollaries will also apply to Tao's model equation, for which there is finite-time blowup .", "For general $\\frac{6}{5}<q\\le 3$ we cannot compute $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}$ explicitly, but in the special case where $q=2,$ we can compute this infimum explicitly by making use of the Hilbert space structure.", "Proposition 2.4 For all $u\\in H^2\\left(\\mathbb {R}^3\\right),$ $u$ not identically zero, $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^2}^2=\\Vert -\\Delta u\\Vert _{L^2}^2\\left(1-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2 \\Vert -\\Delta u\\Vert _{L^2}^2}\\right)$ Fix $u\\in H^2.$ Define $f:\\mathbb {R} \\rightarrow \\mathbb {R},$ by $f(\\lambda )&=\\Vert -\\Delta u-\\lambda u\\Vert _{L^2}^2\\\\&=\\Vert -\\Delta u\\Vert _{L^2}^2 -2\\left<-\\Delta u, u\\right>\\lambda +\\Vert u\\Vert _{L^2}^2 \\lambda ^2\\\\&=\\Vert -\\Delta u\\Vert _{L^2}^2 -2\\Vert \\nabla u\\Vert _{L^2}^2\\lambda +\\Vert u\\Vert _{L^2}^2 \\lambda ^2$ Taking the derivative of $f$ we find that $f^{\\prime }(\\lambda )=-2\\Vert \\nabla u\\Vert _{L^2}^2+2\\Vert u\\Vert _{L^2}^2 \\lambda .$ Let $\\lambda _0=\\frac{\\Vert \\nabla u\\Vert _{L^2}^2}{\\Vert u\\Vert _{L^2}^2}.$ It is easy to see that for all $\\lambda <\\lambda _0, f^{\\prime }(\\lambda )<0,$ for all $\\lambda >\\lambda _0, f^{\\prime }(\\lambda )>0,$ and $f^{\\prime }(\\lambda _0)=0,$ so we can conclude that $f$ has a global minimum at $\\lambda _0.$ Therefore we can compute that $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^2}^2&=\\inf _{\\lambda \\in \\mathbb {R}} f(\\lambda )\\\\&=f(\\lambda _0)\\\\&=\\Vert -\\Delta u\\Vert _{L^2}^2-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2} \\\\&=\\Vert -\\Delta u\\Vert _{L^2}^2\\left(1-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2 \\Vert -\\Delta u\\Vert _{L^2}^2}\\right).$ This completes the proof.", "Using this identity for the infimum in the case where $q=2,$ we will now prove Corollary REF , which is restated here for the reader's convenience.", "Corollary 2.5 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation.", "Then for all $0<t<T_{max}$ $\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(C_2 \\int _0^t\\Vert -\\Delta u\\Vert _{L^2}^\\frac{4}{3}\\left(1-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2 \\Vert -\\Delta u\\Vert _{L^2}^2}\\right)^\\frac{2}{3} \\mathop {}\\!\\mathrm {d}\\tau \\right),$ where $C_2>0$ is taken as in Theorem REF .", "In particular, if $T_{max}<+\\infty $ then $\\int _0^{T_{max}} \\Vert -\\Delta u\\Vert _{L^2}^\\frac{4}{3}\\left(1-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2 \\Vert -\\Delta u\\Vert _{L^2}^2}\\right)^\\frac{2}{3} \\mathop {}\\!\\mathrm {d}t=+\\infty .$ We will begin by observing that when $q=2, p=\\frac{4}{3},$ then $\\frac{2}{p}+\\frac{3}{q}=3.$ Next we know from Proposition REF , that $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^2}^2=\\Vert -\\Delta u\\Vert _{L^2}^2\\left(1-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2 \\Vert -\\Delta u\\Vert _{L^2}^2}\\right).$ Taking both sides of the equation to the $\\frac{2}{3}$ power, we find that $ \\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^2}^\\frac{4}{3}=\\Vert -\\Delta u\\Vert _{L^2}^\\frac{4}{3}\\left(1-\\frac{\\Vert \\nabla u\\Vert _{L^2}^4}{\\Vert u\\Vert _{L^2}^2 \\Vert -\\Delta u\\Vert _{L^2}^2}\\right)^\\frac{2}{3},$ and then the result follows as an immediate corollary of Theorem REF .", "Now, we will note that while we cannot explicitly compute the infimum in Theorem REF , for $2<q<3,$ we can compute this infimum in the Hilbert space $\\dot{H}^\\beta \\hookrightarrow L^q,$ using the inner product structure, and this will give us an explicit, scale-critical regularity criterion, albeit one requiring a higher degree of regularity.", "Proposition 2.6 Suppose $0\\le \\beta <\\frac{3}{2}.$ For all $u\\in \\dot{H}^\\beta \\left(\\mathbb {R}^3\\right)\\cap \\dot{H}^{2+\\beta } \\left(\\mathbb {R}^3\\right),$ $u$ not identically zero, $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{\\dot{H}^\\beta }^2=\\Vert u\\Vert _{\\dot{H}^{2+\\beta }}^2\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{1+\\beta }}^4}{\\Vert u\\Vert _{\\dot{H}^\\beta }^2\\Vert u\\Vert _{\\dot{H}^{2+\\beta }}^2}\\right).$ Fix $0\\le \\beta <\\frac{3}{2},$ and $u\\in \\dot{H}^\\beta \\left(\\mathbb {R}^3\\right)\\cap H^{2+\\beta } \\left(\\mathbb {R}^3\\right).$ Define $f:\\mathbb {R} \\rightarrow \\mathbb {R},$ by $f(\\lambda )&=\\Vert -\\Delta u-\\lambda u\\Vert _{\\dot{H}^\\beta }^2\\\\&=\\Vert -\\Delta u\\Vert _{\\dot{H}^\\beta }^2 -2\\left<-\\Delta u, u\\right>_{\\dot{H}^\\beta }\\lambda +\\Vert u\\Vert _{\\dot{H}^\\beta }^2 \\lambda ^2\\\\&=\\Vert u\\Vert _{\\dot{H}^{2+\\beta }}^2-2\\Vert u\\Vert _{\\dot{H}^{1+\\beta }}^2\\lambda +\\Vert u\\Vert _{\\dot{H}^\\beta }^2 \\lambda ^2$ Taking the derivative of $f$ we find that $f^{\\prime }(\\lambda )=-2\\Vert u\\Vert _{\\dot{H}^{1+\\beta }}^2+2 \\Vert u\\Vert _{\\dot{H}^\\beta }^2 \\lambda $ Let $\\lambda _0=\\frac{\\Vert u\\Vert _{\\dot{H}^{1+\\beta }}^2}{\\Vert u\\Vert _{\\dot{H}^\\beta }^2}.$ It is easy to see that for all $\\lambda <\\lambda _0, f^{\\prime }(\\lambda )<0,$ for all $\\lambda >\\lambda _0, f^{\\prime }(\\lambda )>0,$ and $f^{\\prime }(\\lambda _0)=0,$ so we can conclude that $f$ has a global minimum at $\\lambda _0.$ Therefore we can compute that $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{\\dot{H}^\\beta }^2&=\\inf _{\\lambda \\in \\mathbb {R}} f(\\lambda )\\\\&=f(\\lambda _0)\\\\&=\\Vert u\\Vert _{\\dot{H}^{2+\\beta }}^2-\\frac{\\Vert u\\Vert _{\\dot{H}^{1+\\beta }}^4}{\\Vert u\\Vert _{\\dot{H}^\\beta }^2} \\\\&=\\Vert u\\Vert _{\\dot{H}^{2+\\beta }}^2\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{1+\\beta }}^4}{\\Vert u\\Vert _{\\dot{H}^\\beta }^2\\Vert u\\Vert _{\\dot{H}^{2+\\beta }}^2}\\right).$ This completes the proof.", "Using Proposition REF , and the Sobolev inequality corresponding to the embedding $\\dot{H}^\\beta \\hookrightarrow L^q,$ we will now prove Corollary REF , which is restated here for the reader's convenience.", "Corollary 2.7 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation, and suppose $2\\le \\alpha \\le \\frac{5}{2},\\alpha =\\frac{1}{2}+\\frac{2}{p}.$ Then for all $0<t<T_{max}$ $\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(\\tilde{C}_\\alpha \\int _0^t\\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)^\\frac{p}{2} \\mathop {}\\!\\mathrm {d}\\tau \\right),$ where $\\tilde{C}_\\alpha >0$ depends only on $\\alpha .$ In particular, if $T_{max}<+\\infty $ then $\\int _0^{T_{max}}\\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)^\\frac{p}{2} \\mathop {}\\!\\mathrm {d}t=+\\infty .$ Note that the scaling relation between $\\alpha $ and $p$ can alternatively be expressed by $p=\\frac{2}{\\alpha -\\frac{1}{2}}$ In the case where $\\alpha =2,$ this is precisely the same statement as Corollary REF , so fix $2<\\alpha \\le \\frac{5}{2}.$ Let $\\beta =\\alpha -2,$ so we have $0<\\beta \\le \\frac{1}{2}.$ Let $\\frac{1}{q}=\\frac{1}{2}-\\frac{\\beta }{3},$ so we have the Sobolev embedding $\\dot{H}^\\beta \\left(\\mathbb {R}^3\\right)\\hookrightarrow L^q\\left(\\mathbb {R}^3\\right).$ Then we can see that $\\frac{3}{q}&=\\frac{3}{2}-\\beta \\\\&= \\frac{7}{2}-\\alpha .$ Likewise we know that $\\frac{2}{p}=\\alpha -\\frac{1}{2},$ so we can conclude that $\\frac{2}{p}+\\frac{3}{q}=3.$ We know from the Sobolev inequality and from Proposition REF that $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^2&\\le C \\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{\\dot{H}^\\beta }^2\\\\&=C \\Vert u\\Vert _{\\dot{H}^{2+\\beta }}^2\\left(1-\\frac{\\Vert u\\Vert _{H^{1+\\beta }}^4}{\\Vert u\\Vert _{\\dot{H}^\\beta }^2 \\Vert u\\Vert _{\\dot{H}^{2+\\beta }}^2}\\right)\\\\&=\\Vert u\\Vert _{\\dot{H}^\\alpha }^2\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)$ Taking both sides of the equation to the power of $\\frac{p}{2}$ we find that $\\inf _{\\lambda \\in \\mathbb {R}}\\Vert -\\Delta u-\\lambda u\\Vert _{L^q}^p\\le C \\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)^\\frac{p}{2},$ and we have already shown that $\\frac{2}{p}+\\frac{3}{q}=3,$ so the result then follows as an immediate corollary of Theorem REF .", "We mentioned in the introduction that without the term $\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^2\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^2}\\right)^\\frac{p}{2},$ Corollary REF is an immediate corollary of the a variant of the Ladyzhenskaya-Prodi-Serrin regularity criterion, which states that for a smooth solution of the Navier–Stokes equation, if $T_{max}<+\\infty $ and we have $\\frac{2}{p}+\\frac{3}{q}=2, \\frac{3}{2}<q\\le +\\infty ,$ then $\\int _0^{T_{max}} \\Vert \\nabla u\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}t=+\\infty .$ This means that if $T_{max}<+\\infty ,$ then for all $1 \\le \\alpha < \\frac{5}{2}, \\alpha =\\frac{1}{2}+\\frac{2}{p},$ $\\int _0^{T_{max}} \\Vert u\\Vert _{\\dot{H}^\\alpha }^p \\mathop {}\\!\\mathrm {d}t&=\\int _0^{T_{max}} \\Vert \\nabla u\\Vert _{\\dot{H}^{\\alpha -1}}^p \\mathop {}\\!\\mathrm {d}t\\\\&\\ge C \\int _0^{T_{max}} \\Vert \\nabla u\\Vert _{L^q}^p \\mathop {}\\!\\mathrm {d}t \\\\&=+\\infty .$ What is new in Corollary REF is the term $\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)^\\frac{p}{2},$ which measures the deficit in the interpolation inequality for the embedding $\\dot{H}^{\\alpha -2} \\cap \\dot{H}^{\\alpha }\\hookrightarrow \\dot{H}^{\\alpha -1},$ stated below.", "Proposition 2.8 For all $\\alpha >\\frac{1}{2},$ we have the embedding $\\dot{H}^{\\alpha -2}\\cap \\dot{H}^\\alpha \\hookrightarrow \\dot{H}^{\\alpha -1},$ with for all $f \\in \\dot{H}^{\\alpha -2}\\cap \\dot{H}^\\alpha ,$ $\\Vert f\\Vert _{\\dot{H}^{\\alpha -1}}^2\\le \\Vert f\\Vert _{\\dot{H}^{\\alpha -2}}\\Vert f\\Vert _{\\dot{H}^{\\alpha }}$ Fix $f \\in \\dot{H}^{\\alpha -2}\\cap \\dot{H}^\\alpha .$ Using the fact that $(-\\Delta )^{\\frac{1}{2}}$ is self-adjoint, and applying Hölder's inequality we find that $\\Vert f\\Vert _{\\dot{H}^{\\alpha -1}}^2&=\\left\\Vert (-\\Delta )^{\\frac{\\alpha }{2}-\\frac{1}{2}}f\\right\\Vert _{L^2}^2 \\\\&=\\left<(-\\Delta )^{\\frac{\\alpha }{2}-1}f,(-\\Delta )^{\\frac{\\alpha }{2}}f\\right> \\\\&\\le \\left\\Vert (-\\Delta )^{\\frac{\\alpha }{2}-1}f\\right\\Vert _{L^2}\\left\\Vert (-\\Delta )^{\\frac{\\alpha }{2}}f\\right\\Vert _{L^2}\\\\&=\\Vert f\\Vert _{\\dot{H}^{\\alpha -2}}\\Vert f\\Vert _{\\dot{H}^{\\alpha }}.$ This completes the proof.", "We will note that this interpolation inequality is related to eigenfunctions of the Laplacian because the only inequality in this proof is Hölder's inequality in (REF ), which holds with equality if $(-\\Delta )^{\\frac{\\alpha }{2}}f=\\lambda (-\\Delta )^{\\frac{\\alpha }{2}-1}f,$ which in turn would imply that $-\\Delta f=\\lambda f,$ and therefore that $f$ is an eigenfunction of the Laplacian.", "This can happen on the torus when working with $f\\in \\dot{H}^{\\alpha -1}\\left(\\mathbb {T}^3\\right),$ for example the function $f(x)=\\sin (2\\pi x_1),$ is an eigenfunction of the Laplacian in $f\\in \\dot{H}^{\\alpha -1}\\left(\\mathbb {R}^3\\right),$ with $-\\Delta f= 4 \\pi ^2 f,$ but not on the whole space.", "Nonetheless, the sharp constant in Proposition REF is 1 for $\\dot{H}^{\\alpha -2}\\left(\\mathbb {R}^3\\right)\\cap \\dot{H}^\\alpha \\left(\\mathbb {R}^3\\right),$ even though, because are no eigenfunctions of the Laplacian in $\\dot{H}^{\\alpha -2}\\left(\\mathbb {R}^3\\right)\\cap \\dot{H}^\\alpha \\left(\\mathbb {R}^3\\right),$ this constant is not attained.", "We will prove this by considering functions whose Fourier transforms are supported on an annulus in $\\mathbb {R}^3.$ Proposition 2.9 Suppose $u\\in \\dot{H}^{\\alpha -2} \\cap \\dot{H}^{\\alpha },$ with $u$ not identically zero, and $\\operatorname{supp}{ \\hat{u}} \\subset \\left\\lbrace \\xi \\in \\mathbb {R}^3:R_1 \\le |\\xi | \\le R_2 \\right\\rbrace .$ Then $\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^2}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}} \\Vert u\\Vert _{\\dot{H}^\\alpha }}\\ge \\frac{R_1^2}{R_2^2}.$ This condition can be stated equivalently as $1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\le 1-\\frac{R_1^4}{R_2^4}.$ First we will observe that $ \\operatorname{supp}{ \\hat{u}} \\subset \\left\\lbrace \\xi \\in \\mathbb {R}^3:R_1 \\le |\\xi | \\le R_2 \\right\\rbrace $ implies that $\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^2&=\\int _{\\mathbb {R}^3}\\left(4 \\pi ^2 |\\xi |^2\\right)^{\\alpha -1}\\left|\\hat{u}(\\xi )\\right|^2 \\mathop {}\\!\\mathrm {d}\\xi \\\\&=\\int _{\\mathbb {R}^3}4 \\pi ^2 |\\xi |^2 \\left(4 \\pi ^2 |\\xi |^2\\right)^{\\alpha -2}\\left|\\hat{u}(\\xi )\\right|^2 \\mathop {}\\!\\mathrm {d}\\xi \\\\&\\ge 4 \\pi ^2 R_1^2 \\int _{\\mathbb {R}^3}\\left(4 \\pi ^2 |\\xi |^2\\right)^{\\alpha -2}\\left|\\hat{u}(\\xi )\\right|^2 \\mathop {}\\!\\mathrm {d}\\xi \\\\&= 4 \\pi ^2 R_1^2 \\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2.$ Likewise, the condition on the support of $\\hat{u}$ (REF ) implies that $\\Vert u\\Vert _{\\dot{H}^\\alpha }^2&=\\int _{\\mathbb {R}^3}\\left(4 \\pi ^2 |\\xi |^2\\right)^{\\alpha }\\left|\\hat{u}(\\xi )\\right|^2 \\mathop {}\\!\\mathrm {d}\\xi \\\\&=\\int _{\\mathbb {R}^3}16 \\pi ^4|\\xi |^4 \\left(4 \\pi ^2 |\\xi |^2\\right)^{\\alpha -2}\\left|\\hat{u}(\\xi )\\right|^2 \\mathop {}\\!\\mathrm {d}\\xi \\\\&\\le 16 \\pi ^4 R_2^4 \\int _{\\mathbb {R}^3}\\left(4 \\pi ^2 |\\xi |^2\\right)^{\\alpha -2}\\left|\\hat{u}(\\xi )\\right|^2 \\mathop {}\\!\\mathrm {d}\\xi \\\\&= 16 \\pi ^4 R_2^4 \\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2$ Putting together (REF ) and (REF ) we find that $\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2\\Vert u\\Vert _{\\dot{H}^{\\alpha }}^2}&\\ge \\frac{16 \\pi ^4 R_1^4 \\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^4}{16 \\pi ^4 R_2^4 \\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^4}\\\\&=\\frac{R_1^4}{R_2^4}.$ It then immediately follows that $\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^2}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}} \\Vert u\\Vert _{\\dot{H}^\\alpha }}\\ge \\frac{R_1^2}{R_2^2},$ and $1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2\\Vert u\\Vert _{\\dot{H}^{\\alpha }}^2}\\le 1-\\frac{R_1^4}{R_2^4}.$ This completes the proof.", "Proposition REF shows that the sharp constant for the interpolation inequality in Proposition REF is in fact $1,$ and so the term $\\left(1-\\frac{\\Vert u\\Vert _{\\dot{H}^{\\alpha -1}}^4}{\\Vert u\\Vert _{\\dot{H}^{\\alpha -2}}^2 \\Vert u\\Vert _{\\dot{H}^\\alpha }^2}\\right)^\\frac{p}{2}$ does indeed measure the deficit in this interpolation inequality.", "Furthermore, Corollary REF and Proposition REF show that finite-time blowup solutions cannot concentrate on arbitrarily narrow bands in Fourier space, supported between an inner radius of $R_1(t)$ and an outer radius of $R_2(t),$ with the ratio $\\frac{R_1(t)}{R_2(t)} \\rightarrow 1$ arbitrarily quickly as $t \\rightarrow T_{max}$ relative to the size of $\\Vert u(\\cdot ,t)\\Vert _{\\dot{H}^\\alpha }^p$ .", "In particular we will prove the following result, which is Corollary REF , and is restated here for the reader's convenience.", "Corollary 2.10 Suppose $u\\in C\\left(\\left[0,T_{max}\\right);H^1\\right)$ is a mild solution of the Navier–Stokes equation, and suppose for all $0<t<T_{max}$ $\\operatorname{supp}{ \\hat{u}(t)} \\subset \\left\\lbrace \\xi \\in \\mathbb {R}^3:R_1(t) \\le |\\xi | \\le R_2(t) \\right\\rbrace .$ Let $2\\le \\alpha \\le \\frac{5}{2},\\alpha =\\frac{1}{2}+\\frac{2}{p}.$ Then for all $0<t<T_{max}$ $\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}^2\\le \\left\\Vert \\nabla u^0\\right\\Vert _{L^2}^2\\exp \\left(\\tilde{C}_\\alpha \\int _0^t\\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{R_1(\\tau )^4}{R_2(\\tau ))^4}\\right)^\\frac{p}{2} \\mathop {}\\!\\mathrm {d}\\tau \\right),$ where $\\tilde{C}_\\alpha >0$ depends only on $\\alpha .$ In particular, if $T_{max}<+\\infty $ then $\\int _0^{T_{max}}\\Vert u\\Vert _{\\dot{H}^\\alpha }^p\\left(1-\\frac{R_1(t)^4}{R_2(t))^4}\\right)^\\frac{p}{2} \\mathop {}\\!\\mathrm {d}t=+\\infty .$ This corollary follows immediately from Corollary REF and Proposition REF .", "Remark 2.11 We will note that for a solution $u$ of the Navier–Stokes equation satisfying the hypotheses of Corollary REF , if $T_{max}<+\\infty ,$ then clearly $\\lim _{t\\rightarrow T_{max}} R_2(t)=+\\infty ,$ otherwise we will have $\\liminf _{t\\rightarrow T_{max}}\\Vert \\nabla u(\\cdot ,t)\\Vert _{L^2}\\le 2\\pi \\left\\Vert u^0\\right\\Vert _{L^2}\\liminf _{t\\rightarrow T_{max}}R_2(t)<+\\infty ,$ which contradicts the assumption that $T_{max}<+\\infty $ .", "This means 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[ [ "HAT: Hardware-Aware Transformers for Efficient Natural Language\n Processing" ], [ "Abstract Transformers are ubiquitous in Natural Language Processing (NLP) tasks, but they are difficult to be deployed on hardware due to the intensive computation.", "To enable low-latency inference on resource-constrained hardware platforms, we propose to design Hardware-Aware Transformers (HAT) with neural architecture search.", "We first construct a large design space with $\\textit{arbitrary encoder-decoder attention}$ and $\\textit{heterogeneous layers}$.", "Then we train a $\\textit{SuperTransformer}$ that covers all candidates in the design space, and efficiently produces many $\\textit{SubTransformers}$ with weight sharing.", "Finally, we perform an evolutionary search with a hardware latency constraint to find a specialized $\\textit{SubTransformer}$ dedicated to run fast on the target hardware.", "Extensive experiments on four machine translation tasks demonstrate that HAT can discover efficient models for different hardware (CPU, GPU, IoT device).", "When running WMT'14 translation task on Raspberry Pi-4, HAT can achieve $\\textbf{3}\\times$ speedup, $\\textbf{3.7}\\times$ smaller size over baseline Transformer; $\\textbf{2.7}\\times$ speedup, $\\textbf{3.6}\\times$ smaller size over Evolved Transformer with $\\textbf{12,041}\\times$ less search cost and no performance loss.", "HAT code is https://github.com/mit-han-lab/hardware-aware-transformers.git" ], [ "Introduction", "Transformer [34] has been widely used in natural language processing tasks.", "By stacking multiple identical encoder/decoder layers with attention modules, it provides a significant performance improvement over previous convolutional or recurrent neural network models [19].", "Figure: Framework for searching Hardware-Aware Transformers.", "We first train a SuperTransformer that contains numerous sub-networks, then conduct an evolutionary search with hardware latency feedback to find one specialized SubTransformer for each hardware.Figure: Latency of different Transformer models on different hardware.", "We find (1) FLOPs does not reflect the real measured latency; (2) Latency influencing factors of different hardware are contrasting.", "Thus we need to consider hardware latency feedback to design specialized models for different hardware.Nevertheless, it is challenging to deploy Transformers on mobile devices due to the high computation cost.", "For instance, in order to translate a sentence with only 30 words, a Transformer-Big model needs to execute 13G FLOPs and takes 20 seconds on a Raspberry Pi.", "Such long latency will hurt the user experience on edge devices.", "Thus we need hardware-efficient Transformers (Figure REF ).", "There are two common pitfalls when evaluating the efficiency of a Transformer.", "(1) FLOPs does not reflect the measured latency.", "Although FLOPs is used as an metric for efficiency in prior arts [15], [43], it is not a good latency proxy.", "As in Figure REF (Right), models with the same FLOPs can result in very different measured latencies; (2) different hardware prefers different Transformer architecture.", "As in Table REF , the Transformer model optimized on one hardware is sub-optimal for another because latency is influenced by different factors on different hardware platforms.", "For example, the embedding dimension has significant impact on the Raspberry Pi latency but hardly influences the GPU latency (Figure REF ).", "Table: BLEU score and measured inference latency of HAT on WMT'14 En-De task.", "The efficient model for GPU is not efficient for ARM CPU and vice versa.Inspired by the success of Neural Architecture Search (NAS) [3], [12], [28], [4], we propose to search for Hardware-Aware Transformers (HAT) by directly involving the latency feedback into the design loop.", "In this way, we do not need FLOPs as the latency proxy and can search specialized models for various hardware.", "We first construct a large search space with arbitrary encoder-decoder attention and heterogeneous Transformer layers.", "Traditional Transformer has an information bottleneck between the encoder and decoder.", "Arbitrary encoder-decoder attention breaks the bottleneck, allowing all decoder layers to attend to multiple and different encoder layers instead of only the last one.", "Thus low-level information from the encoder can also be used by the decoder.", "Motivated by Figure REF , we introduce heterogeneous Transformer layers to allow different layers to have different architecture adapting various hardware.", "To perform a low-cost search in such a large design space, we first train a Transformer supernet – SuperTransformer, which contains many SubTransformers sharing the weights.", "We train all SubTransformers simultaneously by optimizing the uniformly sampled SubTransformers from the SuperTransformer.", "The performance of a SubTransformer with inherited weights from the SuperTransformer can provide a good relative performance approximation for different architectures trained from-scratch.", "Unlike conventional NAS, we only need to pay the SuperTransformer training cost for once and can evaluate all the models in the design space with it.", "Finally, we conduct an evolutionary search to find the best SubTransformer under the hardware latency constraint.", "Experiments show that HAT can be naturally incorporated with model compression techniques such as quantization and knowledge distillation.", "We evaluate HAT with WMT'14 En-De, WMT'14 En-Fr, WMT'19 En-De, and IWSLT'14 De-En tasks on Raspberry Pi ARM CPU, Intel Xeon CPU, and Nvidia TITAN Xp GPU.", "Compared with previous work [34], [30], [11], [43], HAT achieves up to 3$\\times $ speedup, 3.7$\\times $ smaller size over Transformer-Big without loss of accuracy.", "With 12,041$\\times $ less search cost, HAT outperforms the Evolved Transformer with 2.7$\\times $ speedup and 3.6$\\times $ smaller size.", "It also achieves up to 1.9$\\times $ speedup over Levenshtein and Lite Transformers with no BLEU score loss.", "With 4-bit quantization, HAT can further reach 25$\\times $ model size reduction.", "Figure: HAT Overview.", "A large design space is constructed with Arbitrary Encoder-Decoder Attention and Heterogeneous Layers.", "(1) Train a weight-shared SuperTransformer by iteratively optimizing randomly sampled SubTransformers.", "It can provide a performance proxy for SubTransformers.", "(2) Collect (SubTransformer architecture, latency) data pairs on the target hardware.", "(3) Train a latency predictor for each hardware to provide fast and accurate latency feedback.", "(4) Perform an evolutionary search with hardware latency constraint to find the model with the lowest validation loss.", "(5) Finally, the searched model is trained from scratch to get the final performance.HAT has three contributions: (1) Hardware-Aware and Specialization.", "To our best knowledge, we are the first to directly involve the hardware feedback in the model design, to reduce NLP model latency for target hardware, instead of relying on proxy signals (FLOPs).", "For different hardware platforms, specialized models for low-latency inference are explored.", "(2) Low-cost Neural Architecture Search with a Large Design Space.", "We propose arbitrary encoder-decoder attention to break the information bottleneck; and heterogeneous layer to let different layers alter its capacity.", "A weight-shared SuperTransformer is trained to search for efficient models at a low cost.", "(3) Design Insights.", "Based on the search results, we reveal some design insights: Attending to multiple encoder layers is beneficial for the decoder; GPU prefers shallow and wide models while ARM CPU prefers deep and thin ones." ], [ "Proposed Approaches", "An overview of the HAT framework is shown in Figure REF .", "We firstly train a SuperTransformer with a large design space.", "Then, for a given hardware platform, we collect a dataset of (SubTransformer architecture, measured latency) pairs for different models, and train a latency predictor.", "Finally, we conduct an evolutionary search with a latency constraint to find an efficient model specialized for the target hardware." ], [ "Design Space", "We construct a large design space by breaking two conventions in the Transformer design: (1) All decoder layers only attend to the last encoder layer; (2) All the layers are identical.", "Figure: Arbitrary Encoder-Decoder Attention.", "Each encoder-decoder attention in one decoder layer can attend to the outputs from multiple encoder layers, fully leveraging the features extracted by the encoder." ], [ "Arbitrary Encoder-Decoder Attention.", "Different encoder layers extract features on different abstraction levels.", "Conventionally, all the decoder layers only attend to the last encoder layer.", "It forms an information bottleneck that forces all the decoder layers to learn solely from the high abstraction level and ignore the low-level information.", "To break the bottleneck, we propose Arbitrary Encoder-Decoder Attention to learn the most suitable connections between the encoder and the decoder.", "Each decoder layer can choose multiple encoder layers to attend.", "The key and value vectors from encoder layers are concatenated in the sentence length dimension (Figure REF ) and fed to the encoder-decoder cross attention module.", "The mechanism is efficient because it introduces no additional parameters.", "The latency overhead is also negligible.", "For example, with each decoder layer attending to two encoder layers, the latency of Transformer-Base on Nvidia TITAN Xp GPU barely increases by 0.4%.", "It improves the model capacity by allowing attention to different abstraction levels." ], [ "Heterogeneous Transformer Layers.", "Previous Transformers repeat one architecture for all layers.", "In HAT, instead, different layers are heterogeneous, with different numbers of heads, hidden dim, and embedding dim.", "In attention layers, different heads are used to capture various dependencies.", "However,  [35] shows that many heads are redundant.", "We thereby make attention head number elastic so that each attention module can decide its necessary number of heads.", "In the FFN layer, the input features are cast to a higher dimension (hidden dim), followed by an activation layer.", "Traditionally, the hidden dim is set as 2$\\times $ or 4$\\times $ of the embedding dim, but this is sub-optimal since different layers need different capacities depending on the feature extraction difficulty.", "We hence make the hidden dim elastic.", "Moreover, we also support elastic embedding dim of encoder and decoder, but it is consistent inside encoder/decoder.", "The number of encoder & decoder layers are also elastic to learn the proper level of feature encoding and decoding.", "Other design choices such as the length of $Q, K, V$ vectors in attention modules can be naturally incorporated in our framework, which we leave for future work.", "Figure: Weight Sharing of the SuperTransformer.", "All SubTransformers share the front portion of word embeddings, and weights in the fully-connected layers." ], [ "SuperTransformer", "It is critical to have a large design space in order to find high-performance models.", "However, training all the models and comparing their BLEU scores is infeasible.", "We thus propose SuperTransformer, a supernet for performance approximation, which can judge the performance of a model without fully training it.", "The SuperTransformer is the largest model in the search space with weight sharing [28], [23], [4].", "Every model in the search space (a SubTransformer) is a part of the SuperTransformer.", "All SubTransformers share the weights of their common parts.", "For elastic embedding dim, all SubTransformers share the front portion of the longest word embedding and corresponding FC layer weights.", "As in Figure REF , for elastic FFN hidden dim, the front part of the FC weights is shared.", "For elastic head number in attention modules, the whole $Q, K, V$ vectors (the lengths are fixed in our design space) are shared by dividing into $head\\_number$ parts.", "Elastic layer numbers let all SubTransformers share the first several layers.", "In the SuperTransformer training, all possible SubTransformers are uniformly sampled, and the corresponding weights are updated.", "In practice, the SuperTransformer only needs to be trained for the same steps as a baseline Transformer model, which is fast and low-cost.", "After training, we can get the performance proxy of sampled models in the design space by evaluating the corresponding SubTransformers on the validation set without training.", "Figure: The latency predictor is very accurate, with an average prediction error (RMSE) of 0.1s." ], [ "Evolutionary Search for SubTransformer", "Given a latency requirement, we perform an evolutionary search to find a satisfactory SubTransformer.", "There are two ways to evaluate the hardware latency of a SubTransformer: (1) Online measurement in which we measure the models during the search process.", "(2) Offline, where we train a latency predictor to provide the latency.", "We apply the offline method here because it is fast and accurate.", "For the online method, a single sampled SubTransformer requires hundreds of inferences to get an accurate latency, which lasts for minutes and slows down the searching.", "For the offline method, we encode the architecture of a SubTransformer into a feature vector, and predict its latency instantly with a multi-layer perceptron (MLP).", "Trained with thousands of real latency data points, the predictor yields high accuracy (Figure REF ).", "Note that the predicted latency is only used in the search process, and we report real measured latency in the experiment section.", "Compared with deducing a closed-form latency model for each hardware, the latency predictor method is more general and faster.", "We use an evolutionary algorithm to conduct the search process.", "As in Figure REF , the search engine queries the latency predictor for SubTransformer latency, and validates the loss on the validation set.", "The engine only adds SubTransformers with latency smaller than the hardware constraint to the population.", "We then train the searched models from scratch to obtain the final performance.", "Figure: Inference latency and BLEU trade-offs of WMT'14 En-De and En-Fr on three hardware platforms.", "HAT consistently outperforms the baseline Transformers and achieves up to 3×\\times faster inference speed and 3.7×\\times smaller size over Transformer-Big.", "Specific latency, BLEU and SacreBLEU   are in Appendix Table .Figure: Inference latency and BLEU trade-offs of WMT'19 and IWSLT'14 tasks on Nvidia GPU." ], [ "Datasets", "We conduct experiments on four machine translation tasks: WMT'14 En-De, WMT'14 En-Fr, WMT'19 En-De, and IWSLT'14 De-En, consisting of 4.5M, 36.3M, 43.0M, and 160K pairs of training sentences, respectively.", "For WMT'14 En-De, we apply 32K source-target BPE vocabulary, train on WMT'16, validate on newstest2013 and test on newstest2014, replicating [42]; For WMT'14 En-Fr, we use 40K source-target BPE vocabulary, validate on newstest2012&2013, and test on newstest2014, replicating [8].", "WMT'19 En-De adopts 49.6K source-target BPE vocabulary, validates on newstest2017, and tests on newstest2018, the same as [17].", "We use 10K joint BPE vocabulary in lower case for IWSLT'14 De-En [9]." ], [ "Baselines.", "Our baseline models are Transformer [34], Levenshtein Transformer [11], both with the [27] implementation, Evolved Transformer [30] and Lite Transformer [43]." ], [ "Evaluation Metrics.", "For evaluation, we use beam four and length penalty 0.6 for WMT, and beam five for IWSLT [34].", "All BLEUs are calculated with case-sensitive tokenizationhttps://github.com/moses-smt/mosesdecoder, but we also apply the compound splitting BLEUhttps://github.com/tensorflow/tensor2tensor for WMT, the same as [34].", "We test the model with the lowest validation set loss for WMT and the last ten checkpoints averaged for IWSLT.", "We test the latency of the models by measuring translation from a source sentence to a target sentence with the same length.", "The length is the average output length on the test set – 30 for WMT and 23 for IWSLT.", "For each model, we measure the latency for 300 times, remove the fastest and slowest 10% and then take the average of the rest 80%.", "We conduct experiments on three representative hardware platforms: Raspberry Pi-4 with an ARM Cortex-A72 CPU, Intel Xeon E5-2640 CPU, and Nvidia TITAN Xp GPU.", "Table: Comparisons of latency, model size, FLOPs, BLEU and training cost in terms of CO2 emissions (lbs) and cloud computing cost (USD) for Transformer, the Evolved Transformer and HAT.", "The training cost estimation is adapted from .", "The training time is for one Nvidia V100 GPU, and the latency is measured on the Raspberry Pi ARM CPU.", "The cloud computing cost is based on AWS." ], [ "SuperTransformer Setups.", "The SuperTransformer for WMT has the following design space: [512, 640] for embedding dim, [1024, 2048, 3072] for hidden dim, [4, 8] for the head number in all attention modules, [1, 2, 3, 4, 5, 6] for decoder layer number.", "Due to decoder auto-regression, encoder only accounts for less than 5% of the measured latency; thereby, we set the encoder layer number fixed as 6.", "For arbitrary encoder-decoder attention, each decoder can choose to attend to the last one, two, or three encoder layers.", "The SuperTransformer design space for IWSLT is the same as WMT except for [2048, 1024, 512] for hidden dim and [4, 2] for head number.", "We set the $Q, K, V$ vector dim fixed as 512.", "The design space contains around $10^{15}$ possible SubTransformers and covers a wide range of model size and latency (largest = 6$\\times $ smallest).", "We train the SuperTransformers of WMT for 40K steps and 50K steps for IWSLT." ], [ "Hardware-Aware Evolutionary Search Setups.", "The input of the latency predictor is a feature vector of SubTransformer architecture with ten elements: layer number, embed dim, average hidden dim, average self-attention heads, of both encoder and decoder; plus average encoder-decoder attention heads, and the average number of encoder layers each decoder layer attends.", "A dataset of 2000 (SubTransformer architecture, measured latency) samples for each hardware is collected, and split into train:valid:test=8:1:1.", "We normalize the features and latency, and train a three-layer MLP with 400 hidden dim and ReLU activation.", "We choose three-layer because it is more accurate than the one-layer model, and over three layers do not improve accuracy anymore.", "With the predictor, we conduct an evolutionary search for 30 iterations in the SuperTransformer, with population 125, parents population 25, mutation population 50 with 0.3 probability and crossover population 50." ], [ "Training Settings.", "Our training settings are in line with [42] and [43].", "For WMT, we train for 40K steps with Adam optimizer and a cosine learning rate (LR) scheduler [22], [24], where the LR is linearly warmed up from $10^{-7}$ to $10^{-3}$ , and then cosine annealed.", "For IWSLT, we train for 50K steps with inverse square root LR scheduler.", "The baseline Transformers are trained with the same settings as the searched SubTransformers for fair comparisons." ], [ "HAT Performance Comparisons", "In Figure REF , REF and Appendix Table REF , we compare HAT with Transformer baselines on four tasks.", "The embedding dims are 512 and 1024 for the Transformer-Base and Big, respectively.", "The hidden dims are $4\\times $ and $2\\times $ of the embedding dim for WMT and IWSLT.", "The IWSLT models are smaller to prevent overfitting [42].", "We obtain a series of baseline models with layer number scaling (yellow) and dimension scaling (blue).", "We set different latency constraints on three hardware to get a series of HAT models.", "HAT consistently outperforms baselines with a large gap under different latency constraints.", "On ARM CPU, HAT is 3$\\times $ faster and 3.7$\\times $ smaller than Transformer-Big with the same BLEU.", "On Intel CPU, HAT achieves over 2$\\times $ speedup.", "On Nvidia GPU, the blue dash line is nearly vertical, indicating that dimension scaling can hardly reduce the latency.", "In this case, HAT can still find models with low latency and high performance.", "We further compare various aspects of HAT with Transformer  [34] and Evolved Transformer [30] in Table REF .", "HAT achieves up to 1.6$\\times $ , 3$\\times $ , and 3.4$\\times $ speedup with up to 1.4$\\times $ , 3.7$\\times $ , and 4$\\times $ smaller size than baselines.", "We report FLOPs for translating a 23-token sentence for IWSLT and 30 for WMT.", "We show the overall GPU hours for training the SuperTransformer and the searched SubTransformer.", "We also calculate the cloud computing costs with different modes: “preemptable\" is cheaper ($0.74/h) than “on-demand\" ($2.48/h) [31].", "HAT is highly affordable since the total GPU-hour is over 12000$\\times $ smaller than the Evolved Transformer, and is even smaller than Transformer-Big by virtue of the compact model size.", "Table: Raspberry Pi ARM CPU latency and BLEU comparisons with different models on WMT'14 En-De.HAT has the lowest latency with the highest BLEU.Figure: Evolutionary search can find better SubTransformers in the SuperTransformer than random search.In Table REF , we compare HAT with other latest models.", "We scale down all models to have similar BLEU scores with Levenshtein for fair comparisons.", "We adopt the average iteration time of 2.88 for decoding [11], without limiting the length of the output sentence (12 tokens after decoding).", "HAT runs 1.3$\\times $ faster than Transformer with higher BLEU; 1.9$\\times $ faster than Levenshtein with 0.7 higher BLEU.", "Under similar latency, HAT also outperforms Lite Transformer.", "These results demonstrate HAT's effectiveness in lower latency scenarios.", "Our framework can also be adopted to speedup those models." ], [ "Design Insights.", "For all HAT WMT models in Figure REF , 10% of all decoder layers attend to three encoder layers, 40% attend to two encoder layers.", "That demonstrates the necessity of arbitrary encoder-decoder attentions.", "In Appendix Figure REF , we visualize the models specialized for different hardware mentioned in Table REF .", "We find that the GPU model is wide but shallow; the Raspberry Pi model is deep but thin.", "The phenomenon echos with our latency profiling (Figure REF ) as GPU latency is insensitive to embedding and hidden dim, but Raspberry Pi is highly sensitive.", "It guides manual designs: on GPU, we can reduce the layer number and increase dimension to reduce latency and keep high performance." ], [ "Ablation Study.", "HAT achieves higher BLEU with 1.5$\\times $ lower latency and 1.5$\\times $ smaller size compared with the largest SubTransformer (Table REF ).", "This suggests that larger models do not always provide better performance, and demonstrates the effectiveness of HAT.", "We also compare the evolutionary search with random search (Figure REF ).", "Evolutionary search can find models with lower losses than random search." ], [ "SubTransformer Performance Proxy.", "All SubTransformers inside the SuperTransformer are uniformly sampled and thus equally trained, so the performance order is well-preserved during training.", "We conduct experiments to show the effectiveness of the SubTransformer performance proxy as in Table REF and Appendix Figure REF .", "The BLEUs of SubTransformers with inherited weights and weights trained from-scratch are very close.", "More importantly, they also have the same relative performance order.", "Therefore, we can rely on the proxy to search high-performance model architecture, significantly reducing the search cost.", "Table: The performance of SubTransformers with inherited weights are close to those trained from-scratch, and have the same relative performance order.Figure: The search cost measured in pounds of CO2 emission.", "Our framework for searching HAT reduces the cost by four orders of magnitude than the Evolved Transformer ." ], [ "Low Search Cost.", "As shown in Table REF and Figure REF , the search cost of HAT is 12,041$\\times $ lower than the Evolved Transformer.", "Although both are using Evolutionary Search, the key difference is that Evolved Transformer needs to train all individual models and sort their final performance to pick top ones; on the contrary, HAT trains all models together inside SuperTransformer and sorts their performance proxy to pick top ones.", "The superior performance of HAT proves that the performance proxy is accurate enough to find good models." ], [ "Finetuning Inherited SubTransformers", "In section REF , we trained each searched SubTransformer from-scratch in order to conduct fair comparisons with baselines.", "In practice, we can also directly finetune the SubTransformers with the inherited weights from the SuperTransformer to further reduce the training cost.", "With 10K finetuning steps (1/4 of from-scratch training), the inherited SubTransformers can achieve similar or better performance than trained from-scratch ones (Table REF ).", "In this way, the training cost for a model under a new hardware constraint can be further reduced by 4$\\times $ , since the SuperTransformer training cost is amortizable among all searched models.", "Table: The SubTransformer inherited from the SuperTransformer can achieve similar or better performance than the same SubTransformer trained from-scratch.", "Training steps are saved by 4×\\times ." ], [ "Quantization Friendly.", "HAT is orthogonal to other model compression techniques such as quantization.", "We apply K-means quantization to HAT and further reduce the model size.", "We initialize centroids uniformly in the range of [min, max] of each weight matrix and run at most 300 iterations for each of them.", "Even without any finetuning, 4-bit quantization can reduce the model size by 25$\\times $ with negligible BLEU loss compared to the Transformer-Big baseline (Table REF ).", "Interestingly, the 8-bit model even has 0.1 higher BLEU than the full precision model, indicating the robustness of searched HAT.", "Compared with the Transformer-Base 4-bit quantization baseline, which has 24MB model size and 38.9 BLEU score, HAT has 2.2 higher BLEU with similar model size." ], [ "Knowledge Distillation Friendly.", "HAT is also orthogonal to knowledge distillation (KD) because HAT focuses on searching for an efficient architecture while KD focuses on better training a given architecture.", "We combine KD with HAT by distilling token-level knowledge (top-5 soft labels) from a high-performance SubTransformer to a low-performance SubTransformer on WMT'14 En-De task.", "The teacher model has a BLEU of 28.5 and 49M parameters; the student model has 30M parameters.", "KD can improve the BLEU of the student model from 25.8 to 26.1." ], [ "Transformer.", "Transformer [34] has prevailed in sequence modeling [26], [16].", "By stacking identical blocks, the model obtains a large capacity but incurs high latency.", "Recently, a research trend is to modify the Transformer to improve the performance [6], [42], [32], [38].", "Among them,  [42] introduced a convolution-based module to replace the attention; [38] proposed to train deep Transformers by propagating multiple layers together in the encoder.", "[45] and [21] also proposed AAN and SSRU to replace the attention mechanism.", "HAT is orthogonal to them and can be combined to search for efficient architecture with those new modules.", "Another trend is to apply non- or partially-autoregressive models to cut down the iteration number for decoding [11], [1], [40], [10].", "Although reducing latency, they sometimes suffer from low performance.", "[2] explored using learned linear combinations of encoder outputs as decoder inputs, while HAT concatenates the outputs without linear combinations, thus better preserving the low-level information.", "[43] investigated mobile settings for NLP tasks and proposed a multi-branch Lite Transformer.", "However, it relied on FLOPs for efficient model design, which is an inaccurate proxy for hardware latency (Figure REF ).", "There are also works [20], [18], [21], [44] using Knowledge Distillation (KD) to obtain small student models.", "Our method is orthogonal to KD and can be combined with it to improve the efficiency further.", "There are also hardware accelerators [13], [46] for attention and fully-connected layers in the Transformer to achieve efficient processing." ], [ "Neural Architecture Search.", "In the computer vision community, there has been an increasing interest in automating efficient model design with Neural Architecture Search (NAS) [47], [48], [28], [14].", "Some applied black-box optimization such as evolutionary search [39] and reinforcement learning [5], [14], [37], [36], [25]; Some leveraged backpropagation with differentiable architecture search [23].", "Some also involved hardware constraints into optimizations such as MNasNet [33], ProxylessNAS [5], FBNet [41] and APQ [39].", "To reduce the NAS cost, supernet based methods [28], [3], [12] apply a proxy for sub-network performance and adopt search algorithms to find good sub-networks.", "For NLP tasks, the benefits of the architecture search have not been fully investigated.", "Recently, [30] proposed the Evolved Transformer to search for architectures under model size constraints and surpassed the original Transformer baselines.", "However, it suffered from very high search costs (250 GPU years), making it unaffordable to search specialized models for various hardware and tasks.", "In addition, hardware latency feedback was not taken into account for better case-by-case specializations.", "Since different hardware has distinct architecture and features [7], feedback from hardware is critical for efficient NLP." ], [ "Conclusion", "We propose Hardware-Aware Transformers (HAT) framework to solve the challenge of efficient deployments of Transformer models on various hardware platforms.", "We conduct hardware-aware neural architecture search in an ample design space with an efficient weight-shared SuperTransformer, consuming four orders of magnitude less cost than the prior Evolved Transformer, and discover high-performance low-latency models.", "We hope HAT can open up an avenue towards efficient Transformer deployments for real-world applications." ], [ "Acknowledgment", "We thank NSF Career Award #1943349, MIT-IBM Watson AI Lab, Semi-conductor Research Corporation (SRC), Intel, and Facebook for supporting this research.", "Appendix for “HAT: Hardware-Aware Transformers for Efficient Natural Language Processing\" SubTransformer Performance Proxy In Figure REF , we show the relationship between the validation loss of SubTransformers directly inherited from the SuperTransformer, and the BLEU score of the SubTransformers trained from-scratch.", "We can observe that the larger the validation loss, the lower the BLEU score.", "Therefore the validation loss can be a good performance proxy.", "Figure: The validation loss of SubTransformers is a good performance proxy for BLEU of from-scratch trained SubTransformers.", "The larger the validation loss, the lower the BLEU score.", "Visualizations of Searched Models on WMT'14 En-De Task We show the HAT models searched for Raspberry Pi ARM Cortex-A72 CPU and Nvidia TITAN Xp GPU in Figure REF .", "The searched model for Raspberry Pi is deep and thin, while that for GPU is shallow and wide.", "The BLEU scores of the two models are similar: 28.10 for Raspberry Pi CPU, and 28.15 for Nvidia GPU.", "Figure: SubTransformers optimized for Raspberry Pi ARM CPU and Nvidia GPU on WMT'14 En-De task are different.", "The CPU model has BLEU 28.10, and GPU model has BLEU 28.15.", "Latency, BLEU and SacreBLEU of searched HAT models.", "In Table REF , we show the specific latency numbers, BLEU and SacreBLEU  [29] scores for searched HAT models in Figure REF and Figure REF .", "Table: Specific latency numbers, BLEU and SacreBLEU scores for searched HAT models in Figure  and Figure ." ], [ "SubTransformer Performance Proxy", "In Figure REF , we show the relationship between the validation loss of SubTransformers directly inherited from the SuperTransformer, and the BLEU score of the SubTransformers trained from-scratch.", "We can observe that the larger the validation loss, the lower the BLEU score.", "Therefore the validation loss can be a good performance proxy.", "Figure: The validation loss of SubTransformers is a good performance proxy for BLEU of from-scratch trained SubTransformers.", "The larger the validation loss, the lower the BLEU score." ], [ "Visualizations of Searched Models on WMT'14 En-De Task", "We show the HAT models searched for Raspberry Pi ARM Cortex-A72 CPU and Nvidia TITAN Xp GPU in Figure REF .", "The searched model for Raspberry Pi is deep and thin, while that for GPU is shallow and wide.", "The BLEU scores of the two models are similar: 28.10 for Raspberry Pi CPU, and 28.15 for Nvidia GPU.", "Figure: SubTransformers optimized for Raspberry Pi ARM CPU and Nvidia GPU on WMT'14 En-De task are different.", "The CPU model has BLEU 28.10, and GPU model has BLEU 28.15." ], [ "Latency, BLEU and SacreBLEU of searched HAT models.", "In Table REF , we show the specific latency numbers, BLEU and SacreBLEU  [29] scores for searched HAT models in Figure REF and Figure REF .", "Table: Specific latency numbers, BLEU and SacreBLEU scores for searched HAT models in Figure  and Figure ." ] ]

[ [ "Gluon Field Digitization via Group Space Decimation for Quantum\n Computers" ], [ "Abstract Efficient digitization is required for quantum simulations of gauge theories.", "Schemes based on discrete subgroups use fewer qubits at the cost of systematic errors.", "We systematize this approach by deriving a single plaquette action for approximating general continuous gauge groups through integrating out field fluctuations.", "This provides insight into the effectiveness of these approximations, and how they could be improved.", "We accompany the scheme by simulations of pure gauge over the largest discrete subgroup of $SU(3)$ up to the third order." ], [ "Introduction", "Large-scale quantum computers can simulate nonperturbative quantum field theories which are intractable classically [1].", "Alas, Noisy Intermediate-Scale Quantum (NISQ) era systems will be limited both in qubits and circuit depths.", "Whether any gauge theory simulations in this period are possible depends upon efficient formulations.", "The situation is similar to the early days of lattice field theory when computer memory was limited and the cost of storing $SU(3)$ elements was prohibitive.", "For fermionic fields, relatively efficient mappings to quantum registers are known [2], [3], [4], [5] evidenced by most existing quantum calculations being fermionic [6], [7], [8], [9].", "The bosonic nature of gauge fields preclude exact mappings, but many proposals exist with different costs [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].", "Digitizing reduces symmetries – either explicitly or through finite-truncations [10].", "These breakings mean a priori the original model may not be recovered in the continuum limit [29], [30], [31], [32], [33], [34].", "Further, choices of digitization may limit the use of classical resources for Euclidean simulations or state preparation [35].", "In summary, the understanding of resource costs, systematic errors, and the continuum limit for these proposals is poorly known today.", "In this work, we systematize the proposal of replacing continuous gauge groups $G$ by their discrete subgroups $H$  [11], [28] by deriving lattice actions using the group space decimation procedure of [36], [37].", "After deriving the general third order action, we will investigate the behaviour of discretizing three distinct gauge groups $U(1)$ , $SU(2)$ , and $SU(3)$ .", "We begin by reviewing the discrete subgroup approximation in Sec. .", "In Sec.", "we discuss the general aspects of the group space decimation procedure.", "Following that, in Sec.", "we derive the decimated action up to 3rd order.", "Numerical results for $\\mathbb {V}$ using the decimated actions are presented in Sec. .", "Sec.", "studies the continuous group limit, and we conclude in Sec.", "." ], [ "Discrete Subgroups", "Approximating gauge theories by replacing $G\\rightarrow H$ was explored in the early days of Euclidean lattice field theory.", "The viability of the $\\mathbb {Z}_{\\rm n}$ subgroups replacing $U(1)$ were studied in [38], [39].", "Further studies of the crystal-like discrete subgroups of $SU(N)$ were performed [40], [41], [42], including with fermions [43], [44].", "These studies met with mixed success depending on the group and action tested.", "The fundamental issue of group discretization can be understood by considering the Wilson gauge action $S[U]=-\\sum _p \\frac{\\beta }{N}\\operatorname{Re}\\operatorname{Tr}(U_p) \\,,$ where $U_p$ indicates a plaquette of continuous group gauge links $U$ (for discrete groups, $u_p$ denotes plaquettes and $u$ denotes links).", "As $\\beta \\rightarrow \\infty $ , links near the group identity $\\mathbb {1}$ dominate, i.e.", "$U\\approx \\mathbb {1}+\\varepsilon $ , where $\\varepsilon $ can be arbitrarily small.", "Therefore the gap $\\Delta S = S[\\mathbb {1}+\\varepsilon ]-S[\\mathbb {1}]$ goes to zero smoothly.", "For discrete groups, $\\varepsilon $ has a minimum given by the nearest elements $\\mathcal {N}$ to $\\mathbb {1}$ , and thus $\\Delta S=S[\\mathcal {N}]-S[\\mathbb {1}]>0$ .", "This strongly suggests a phase transition at some critical $\\beta _f=c/\\Delta S$ , where $c\\approx \\mathcal {O}(1)$ depends on spacetime dimensionality, gauge group, and entropy.", "For $U(1)\\rightarrow \\mathbb {Z}_{\\rm n}$ in 4d, $\\beta _f=\\frac{0.78}{1-\\cos (2\\pi /n)}$  [41].", "Above $\\beta _f$ , all field configurations but $u=\\mathbb {1}$ are exponentially suppressed.", "Thus, $H$ fails to approximate $G$ for $\\beta >\\beta _f$ .", "Another way to understand this behavior follows [45], where some discrete theories are shown equivalent to continuous groups coupled to a Higgs field.", "The Higgs mechanism introduces a new phase missing from the continuous gauge theory when $\\beta \\rightarrow \\infty $ .", "Both arguments suggest $H$ be viewed as an effective field theory for $G$ with a UV-cutoff at $\\Lambda _f$ .", "Provided the typical separation of scales of physics $m_{IR}\\ll \\Lambda _f$ , the approximation could be reliable up to $\\mathcal {O}(m_{IR}/\\Lambda _f)$ effects.", "In lattice calculations, one replaces $\\Lambda _f$ by a fixed lattice spacing $a=a(\\beta )$ which shrinks as $\\beta \\rightarrow \\infty $ for asymptotically free theories.", "To control errors when extrapolating to $a\\rightarrow 0$ , one should simulate in the scaling regime of $a\\ll m_{IR}^{-1}$ .", "We denote the onset of the scaling regime by $a_s,$ and $\\beta _s$ .", "For $a_f(\\beta _f)\\sim \\Lambda _f^{-1}$ , errors from the discrete group approximation would be small if $a$ can be reduced such that $m_{IR}^{-1}\\gg a\\gtrsim a_f$ i.e.", "$\\beta _s\\le \\beta _f$ .", "In the case of $U(1)$ with $\\beta _s= 1$ , $\\mathbb {Z}_{n>5}$ satisfies $\\beta _f>\\beta _s$ .", "For non-Abelian groups, only a finite set of crystal-like subgroups exist.", "$SU(2)$ has three: the binary tetrahedral $\\mathbb {BT}$ , the binary octahedral $\\mathbb {BO}$ , and the binary icosahedral $\\mathbb {BI}$ .", "While $\\mathbb {BT}$ has $\\beta _f=2.24(8)$ , $\\mathbb {BO}$ and $\\mathbb {BI}$ have $\\beta _f=3.26(8)$ and $\\beta _f=5.82(8)$ respectively [28], above $\\beta _s=2.2$ .", "Hence, $\\mathbb {BO}$ and $\\mathbb {BI}$ appear useful for $SU(2)$ .", "For the important case of $SU(3)$ with $\\beta _s=6$ , there are five crystal-like subgroups with the Valentiner group $\\mathbb {V}$ with 1080 elementsThis name is most common in the mathematical literature [46], [47].", "It has also referred to as $S(1080)$  [42], [36], [37], [28] or $\\Sigma _{3\\times 360}$  [48].. For all subgroups, $\\beta _f < \\beta _s$ , with $\\mathbb {V}$ having $\\beta _f=3.935(5)$  [28] and thus appear inadequate.", "Other work [49] has shown that extending to a subset with the midpoints between elements of $\\mathbb {V}$ raises $\\beta _f\\approx 7$ .", "However this require more qubits and – potentially more worrisome– sacrifices gauge symmetry completely which is dangerous on quantum computers [22], [50], [51].", "To decrease $a_f$ , adding additional terms to Eq.", "(REF ) was attempted  [52], [42], [39], [53], [54], [36], [37], [55], [28], although only in [42], [28] were Monte Carlo calculations undertaken for $SU(3)$ .", "Two reasons suggest this would help.", "First, additional terms which have a continuum limit $\\propto \\operatorname{Re}\\operatorname{Tr}F_{\\mu \\nu }F^{\\mu \\nu }$ , but take different values on the element of $H$ (e.g.", "$|\\operatorname{Tr}(u_p)|^2-1$ ), change $\\Delta S$ and thus $a_f$ .", "Second, new terms can reduce finite-$a$ errors as in Symanzik improvement.", "The term usually added was the adjoint trace, giving $S[u]=-\\sum _p\\left(\\frac{{\\beta }_{\\lbrace 1\\rbrace }}{3}\\operatorname{Re}\\operatorname{Tr}(u_p) +\\frac{{\\beta }_{\\lbrace 1,-1\\rbrace }}{8}|\\operatorname{Tr}(u_p)|^2\\right),$ where $u_p \\in \\mathbb {V}$ , and the first term is normalized so for $\\beta _{\\lbrace 1,-1\\rbrace }=0$ , the $S[u]$ matches the Wilson action (with $\\beta _{\\lbrace 1\\rbrace }=\\beta $ ).", "In these works, no relationship was assumed between $\\beta _{\\lbrace 1\\rbrace }$ and $\\beta _{\\lbrace 1,-1\\rbrace }$ .", "That Eq.", "(REF ) improves the viability of $\\mathbb {V}$ over the Eq.", "(REF ) will be shown in [56].", "For a different action, $S[u]=-\\sum _p \\left(\\frac{\\beta _0}{3}\\operatorname{Re}\\operatorname{Tr}(u_p) +\\beta _1\\operatorname{Re}\\operatorname{Tr}(u_p^2)\\right) \\,,$ smaller values of $a_f$ were demonstrated in [28].", "With these actions, the dimensionless product $T_c\\sqrt{t_0}$ of the pseudocritical temperature and the Wilson flow parameter were found to agree in the continuum with $SU(3)$ , allowing one to set the scale of those calculations.", "$a>0.08$ fm was achieved without the effects of $a_f$ being seen.", "This suggest that $\\mathbb {V}$ can reproduce $SU(3)$ in the scaling region with a modified action, such that practical quantum computations of $SU(3)$ could be performed.", "While promising, the choice of new terms was ad-hoc and left unclear how to systematically improve or analyze effectiveness.", "In the next section, we systematically derive lattice actions for $H$ , discovering that the terms added in these two actions are in fact the first terms generated." ], [ "Group Space Decimation", "Our ultimate goal is to approximate the path integral of group $G$ faithfully by a discrete subgroup $H$ by replacing the integration over $G$ by a summation over $H$ .", "Group space decimation can be understood in analogy to Wilsonian renormalization, where we integrate out continuous field fluctuations instead of UV modes.", "The typical method used with discrete subgroup approximations is to replace the gauge links $U\\in G$ by $u\\in H$ such that the action $S[U]\\rightarrow S[u]$ .", "This corresponds to simply regularizing a field theory.", "For strong coupling, this appears sufficient.", "As $\\beta \\rightarrow \\infty $ , correlations between gauge links increase and the average field fluctuation becomes smaller.", "When the average field fluctuations decrease below the distance between $\\mathbb {1}$ and $\\mathcal {N}$ of the discrete group, freeze-out occurs and the approximation breaks down–similar to probing a regulated theory too close to the cutoff.", "Therefore, improving this approximation and understand the systematics can be done by considering these discarded continuous field fluctuations.", "To do this, instead of performing the replacement $U\\rightarrow u$ , we will integrate out the continuous fluctuations, following the decimation formalism developed by Flyvbjerg [37], [36].", "He derived the second order decimated action for $U(1)$ , $SU(2)$ , and $SU(3)$ .", "An important general feature of the decimated action though is missing from this second order action – while new terms are generated at each order, until third order no coefficient of an existing term is modified.", "One expects such terms are critical to understanding deviations from the continuous group and therefore we compute them in Sec. .", "Figure: A schematic demonstration of Ω\\Omega (in green) of GG (a sphere) around 1\\mathbb {1} (blue point) of the discrete group (shown as points).", "𝒩\\mathcal {N} for HH are given by red points.", "We have applied the S 2 S^2 metric to obtain the Ω\\Omega .", "In groups representable in two dimensions, this region resembles a polygon while in higher dimensions, it becomes a polytope.It is natural to associate every subgroup element $u\\in H$ with an unique set, or region, $\\Omega _u$ containing all closest continuous group elements $U\\in G$ : $\\Omega _u\\equiv \\left\\lbrace U\\in G\\big | d(U,u)<d(U,u^{\\prime }), \\forall u^{\\prime }\\in H\\setminus {\\lbrace u\\rbrace }\\right\\rbrace \\, ,$ where the distance is defined as $d^2(U,u)=\\operatorname{Tr}\\big ((U-u)^\\dag (U-u)\\big )$ .", "By such a definition.", "the continuous group is fully covered, i.e., $G=\\cup _{u\\in H}\\Omega _u$ and a graphical demonstration of $\\Omega \\equiv \\Omega _{\\mathbb {1}}$ can be found in Fig.", "REF .", "Note that for any $U\\in G$ , there exist a unique $u\\in H$ and $\\epsilon \\in \\Omega $ such that $U=u\\epsilon $ , where we may treat $\\epsilon $ as the error of $u$ approximating $U$ .", "In this way, without approximation, the Euclidean path integral integrating over $G$ can be written as a summation over $H$ and integration over $\\epsilon \\in \\Omega $ : $Z=\\int _G DU \\,e^{-S[U]} = \\sum _{u\\in H}\\int _{\\Omega }D\\epsilon \\, e^{-S[u,\\epsilon ]}\\,,$ where $Z$ is a functional integral over all gauge links $U$ on the lattice, or equivalently a functional integral over $\\epsilon $ and a functional sum over $u$ .", "In this expression, $S[u,\\epsilon ]=S[U]$ is defined by replacing each gauge link $U$ by $u\\epsilon $ .", "We then expand the exponential in the path integral and integrate over $\\epsilon $ producing a moment expansion $Z &=\\sum _{u\\in H}\\int _{\\Omega }D\\epsilon \\, \\left(1-\\beta S[u,\\epsilon ]+\\frac{\\beta ^2}{2!", "}S[u,\\epsilon ]^2+\\cdots \\right)\\nonumber \\\\&=\\sum _{u\\in H}\\left(1-\\beta \\langle {S[u,\\epsilon ]}\\rangle + \\frac{\\beta ^2}{2!", "}\\langle {S[u,\\epsilon ]^2}\\rangle +\\cdots \\right) \\,,$ where we have introduced the notation $\\langle f\\rangle =\\int _{\\Omega } D\\epsilon \\, f$ with normalization $\\int _{\\Omega } D\\epsilon =1$ .", "What we are really after is an expansion for the action $S[U]$ , writing $Z$ in terms of a cumulant expansion $Z &= \\sum _{u\\in H}\\exp {\\left(-\\sum _{n=1}^\\infty \\frac{\\beta ^n}{n!", "}{S}_n[u]\\right)}\\,, $ allows us to match Eq.", "(REF ) with (REF ) to obtain an effective action.", "In this way, after integrating over $\\epsilon $ , the contributions to the action depend only on the discrete group gauge link $u$ and the effective action can be defined as $S[u]\\equiv \\sum _n \\frac{\\beta ^n}{n!", "}{S}_n[u].$ Up to $\\mathcal {O}(\\beta ^3)$ one has $&{S}_1[u]=\\langle {S[u,\\epsilon ]}\\rangle \\, ,\\\\&{S}_2[u]=-\\langle {S[u,\\epsilon ]^2}\\rangle +\\langle {S[u,\\epsilon ]}\\rangle ^2\\, ,\\\\&{S}_3[u]=\\langle {S[u,\\epsilon ]^3}\\rangle -3\\langle {S[u,\\epsilon ]}\\rangle \\langle {S[u,\\epsilon ]^2}\\rangle +2\\langle {S[u,\\epsilon ]}\\rangle ^3\\,.$ One may worry about poor convergence in the region of interest $\\beta \\ge \\beta _s\\ge 1$ .", "As will be discussed more thoroughly in Sec.", ", $\\beta ^n$ terms are suppressed by powers of the average field fluctuation.", "Thus, the size of the discrete group, which determines the size of field fluctuations integrated out, also determines the series convergence.", "Starting with the second order terms computed in Refs.", "[36], [37], the decimated action generates multi-plaquette contributions.", "Their inclusion in quantum simulations brings substantial non-locality which requires high qubit connectivity and increases circuit depth.", "Luckily these contributions will be shown to be small in Sec. .", "In the following section, we will calculate Eq.", "(REF ) to () in terms of linear combination of the group characters starting from the Wilson action of Eq.", "(REF ): $S[U]\\equiv -\\sum _p\\frac{\\beta }{N}\\operatorname{Re}\\operatorname{Tr}(U_p)=-\\sum _p\\frac{\\beta }{N}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace } \\,.", "$ Here we introduced $\\chi _r$ , the character of the group representationThrough out this work we suppress the argument of $\\chi _r$ , but it can only be $U_p$ or $u_p$ and context makes it clear which is meant.", "$r$ .", "This is the natural basis for the decimated action.", "All characters required for our $\\mathcal {\\beta }^3$ calculation are in Table REF .", "In the interest of deriving a decimated action for general gauge groups, we have chosen a nonstandard basis for $U(1)$ and $SU(2)$ .", "This allows for one general scheme for $U(N)$ and $SU(N)$ groups.", "This basis is not linearly independent and relations between representations exist.", "This dependence is typically used to write $U(1)$ and $SU(2)$ in reduced sets of representations.", "We have collated relations between the over complete basis in Appendix .", "In deriving the decimated action, integrating out the field fluctuations require us to reduce expressions of the form $\\langle {\\epsilon _{i_1j_1}\\cdots \\epsilon _{i_nj_n}}\\rangle $ .", "To simplify these, we use an identity derived in [57] for $SU(N)$ and $U(N)$ groups for any integer $n\\le N$ .", "The necessary relations for $n\\le 3$ are found in Appendix .", "From these identites, we are left with expectation values of $\\chi _r$ over $\\Omega $ $V_r\\equiv \\frac{1}{d_r}\\langle \\operatorname{Re}\\chi _r\\rangle \\,,$ where $d_r$ is the dimension of representation $r$ .", "For $U(1)\\rightarrow \\mathbb {Z}_{\\rm n}$ , there is only one representation at each order of the cumulant expansion, $V_{\\lbrace h\\rbrace }=\\langle \\epsilon ^h\\rangle $ .", "These terms can be computed analytically by a change of variables $\\epsilon =e^{i\\phi }$  [37]: $V_{\\lbrace h\\rbrace } &=\\frac{1}{V_{0}}\\int _{-\\frac{\\pi }{n}}^{\\frac{\\pi }{n}}d\\phi \\, e^{i\\phi h}= \\frac{n}{\\pi h} \\sin \\left(\\frac{\\pi h}{n}\\right)$ with $h=1,2,\\hdots $ being integers and the normalization constant $V_{0} = \\int _{\\Omega } d\\epsilon \\, \\epsilon ^0 = \\frac{2\\pi }{n}$ .", "Extending this to non-abelian groups, e.g.", "$SU(N)$ , $\\Omega $ becomes a high-dimensional polytope in $SU(N)$ space.", "In [36], the $V_r$ for $\\mathbb {BI}$ and $\\mathbb {V}$ were computed up to second order by approximating these polytopes with hyperspheres to two significant figures.", "It is crucial to remove these approximations for our purpose because the uncertainty $\\delta V_r\\sim \\mathcal {O}(1\\%)$ is magnified in the coupling constants of the decimated action.", "These couplings are combinations of powers of $V_r$ with extreme cancellations making the fraction errors grow rapidly.", "Hence we avoid the hypersphere approximation and numerically compute all the $ V_r$ necessary for the 3rd order actions to $\\mathcal {O}(0.1\\%)$ .", "(Results found in Table REF .)", "Table: The dimension, d r d_r, the character χ r \\chi _r, and V r [G→H]=d r -1 〈Reχ r 〉 V_r[G\\rightarrow H]=d_r^{-1}\\langle \\operatorname{Re}\\chi _r\\rangle of character rr for the decimations U(1)→ℤ n U(1)\\rightarrow \\mathbb {Z}_{\\rm n}, SU(2)→𝔹𝕀SU(2)\\rightarrow \\mathbb {BI}, and SU(3)→𝕍SU(3)\\rightarrow \\mathbb {V}.", "We have followed the normalizations in Table 14 of ." ], [ "Order-by-Order Decimation", "In this section, we summarize the derivation of the decimated action order-by-order.", "Further details can be found in Appendix .", "The first order is relatively straight-forward, and only contains a single plaquette term.", "Working from Eq.", "(REF ) $&\\beta {S}_1[u]= -\\frac{\\beta }{N}\\langle \\operatorname{Re}\\operatorname{Tr}{\\left(u_1\\epsilon _1u_2\\epsilon _2(u_3\\epsilon _3)^\\dag (u_4\\epsilon _4)^\\dag \\right)} \\rangle \\\\&= -\\frac{\\beta }{N}\\operatorname{Re}({u_1}_{ab}{u_2}_{cd}{u_3}_{ef}^\\dag {u_4}_{gh}^\\dag \\langle {\\epsilon _{1}}_{bc}\\rangle \\langle {\\epsilon _{2}}_{de}\\rangle \\langle {\\epsilon _{3}}_{fg}^\\dag \\rangle \\langle {\\epsilon _{4}}_{ha}^\\dag \\rangle ).$ After applying Eq.", "(REF ), ${S}_1[u]$ depends only on $u$ : $\\beta {S}_1[u]&= -V_{\\lbrace 1\\rbrace }^4\\frac{\\beta }{N}\\operatorname{Re}({u_1}_{ab}{u_2}_{cd}{u^\\dag _3}_{ef}{u^\\dag _4}_{gh})\\delta _{bc}\\delta _{de}\\delta _{fg}\\delta _{ha}\\nonumber \\\\&=-V_{\\lbrace 1\\rbrace }^4\\frac{\\beta }{N}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }\\equiv -\\beta _{\\lbrace 1\\rbrace }^{(1)} \\frac{1}{N}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }\\,,$ where $\\beta _r^{(n)}$ is the $n$ -th order term in front of $\\frac{1}{d_r}\\operatorname{Re}\\chi _r$ .", "It is comforting that at $\\mathcal {O}(\\beta )$ , no new terms are generated in $S[u]$ .", "This allows for rescaling $\\beta _{\\lbrace 1\\rbrace }^{(1)}\\rightarrow \\beta $ , recovering the procedure of directly replacing $U\\rightarrow u$ in the Wilson action.", "Although this rescaling is permitted, $V_{\\lbrace 1\\rbrace }<1$ contains content about the approximation $G\\rightarrow H$ .", "As the number of elements of $H$ increases, $\\Omega $ shrinks and $V_{\\lbrace 1\\rbrace }\\rightarrow 1$ .", "This means $V_{\\lbrace 1\\rbrace }$ quantifies how densely $H$ covers $G$ and thus the minimal fluctuation size.", "Since $\\beta _{\\lbrace 1\\rbrace }^{(1)}=V_{\\lbrace 1\\rbrace }^4\\beta $ , decreases in $V_{\\lbrace 1\\rbrace }$ signals the poorness of using Eq.", "(REF ) alone.", "This is discussed further in Sec. .", "Table: β r [G→H]\\beta _r[G\\rightarrow H] of character rr for a general group decimation.", "For completeness, we have included the 4 two-plaquette terms derived in , at second order labeled as 2r,2i,2t2r, 2i, 2t and 2u2u.We now proceed to calculate the second order decimated action while fixing a few typos in [36] along the way.", "The second order decimated action ${S}_2[u]=-\\langle {S[u,\\epsilon ]^2}\\rangle +\\langle {S[u,\\epsilon ]}\\rangle ^2$ depends upon two plaquettes $U_{p}=U_1U_2U_3^\\dag U_4^\\dag $ and $U_{q}=U_5U_6U_7^\\dag U_8^\\dag $ .", "A natural decomposition of ${S}_2[u]$ can be made into three terms based on how the two plaquettes $p$ and $q$ are related: $p=q$ (one-plaquette contribution), $p\\cap q= 1$ -link (two-plaquette contribution), and $p\\cap q=0$ -links.", "To all orders, the $p\\cap q=0$ contributions to the decimated action vanish.", "For the case of $p=q$ , we conclude that it is: ${\\frac{1}{2!", "}}\\beta ^2{S}_2[u]_{1p} =& -\\beta _{\\lbrace 0\\rbrace }^{(2)}-\\beta _{\\lbrace 2\\rbrace }^{(2)}\\frac{2}{N(N+1)}\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }\\nonumber \\\\&-\\beta _{\\lbrace 1,1\\rbrace }^{(2)}\\frac{2}{N(N-1)}\\operatorname{Re}\\chi _{\\lbrace 1,1\\rbrace }\\nonumber \\\\&-\\beta _{\\lbrace 1,-1\\rbrace }^{(2)}\\frac{1}{N^2-1}\\chi _{\\lbrace 1,-1\\rbrace }\\, ,$ where the $\\beta _r^{(2)}$ can be found in Table REF .", "Next, we calculate the case of $p\\cap q= 1$ -link for the second order decimation.", "Contracting the $\\delta $ 's in Eqs.", "(REF ) and (REF ), where unlike Eq.", "(REF ), we only identify one link as the same between the two plaquettes.", "This leads to the following expression, $&{\\frac{1}{2!", "}}\\beta ^2{S}_2[u]_{2p}\\\\&=-\\beta _{\\lbrace 2r\\rbrace }\\frac{1}{N}\\operatorname{Re}\\left[\\chi _{\\lbrace 1\\rbrace }(u_p)\\right]\\frac{1}{N}\\operatorname{Re}\\left[\\chi _{\\lbrace 1\\rbrace }(u_q)\\right]\\\\&\\quad -\\beta _{\\lbrace 2i\\rbrace }\\frac{1}{N}\\mathop {\\rm Im}\\nolimits \\left[\\chi _{\\lbrace 1\\rbrace }(u_p)\\right]\\frac{1}{N}\\mathop {\\rm Im}\\nolimits \\left[\\chi _{\\lbrace 1\\rbrace }(u_q)\\right]\\\\&\\quad -\\beta _{\\lbrace 2t\\rbrace }\\frac{1}{N}\\operatorname{Re}\\left[\\chi _{\\lbrace 1\\rbrace }(u_{p*q^\\dagger })\\right]-\\beta _{\\lbrace 2u\\rbrace }\\frac{1}{N}\\operatorname{Re}\\left[\\chi _{\\lbrace 1\\rbrace }(u_{p*q})\\right]\\,,$ where we have used the fact that all the $V_r$ 's are real due to our choice of the integration region.", "The explicit expressions for the couplings are found in Table REF .", "Note that this expression is also applicable to $U(1)$ .", "Figure: Example of two plaquettes u p u_p and u q u_q where p∩q=u 2 =u 8 p\\cap q=u_2=u_8.", "The second order contributions depend on (top) u p*q =u 1 u 5 u 6 u 7 † u 3 † u 4 † u_{p*q}=u_1u_5u_6u_7^\\dagger u_3^\\dagger u_4^\\dagger and (bottom) u p*q † =u 1 u 2 u 7 u 6 † u 5 † u 2 u 3 † u 4 † u_{p*q^\\dag }=u_1u_2u_7u_6^\\dagger u_5^\\dagger u_2 u_3^\\dagger u_4^\\dagger .We would now like to comment on how the two-plaquette – and general multiplaquette – terms contributes to the $S[u]$ .", "It would be desirable if these terms could be neglected, because they require substantial quantum resources.", "By inspecting Table REF , one observes that the two-plaquette $\\beta _{r}$ are $\\mathcal {O}(0.1)$ or smaller than the single-plaquette terms.", "The largest coupling, $\\beta _{2i}$ , multiples a term $\\mathop {\\rm Im}\\nolimits \\chi _1 \\mathop {\\rm Im}\\nolimits \\chi _1\\approx 0$ .", "Strong cancellations are expected from correlations between the remaining terms (shown in Fig.", "(REF )) as evident by the observation $\\beta _{2t}\\approx -\\beta _{2u}$ .", "It is reasonable to expect these individual reasons to persist at higher orders, suggesting that at a fixed order all multi-plaquette terms can be neglected compared to their $1-$ plaquette counterpart.", "But can we argue that the multi-plaquette terms generated at order ${\\cal O}(\\beta ^{n-1})$ are still negligible when the ${\\cal O}(\\beta ^{n})$ contribution is introduced?", "To do this, we look at the continuum limit of each term being introduced.", "In this way, we recognize that the two-plaquette terms are related to the Lüscher-Weisz action [59].", "$k$ -plaquette terms corresponds to applying $2k-2$ derivatives to $a^4\\langle \\operatorname{Tr}FF\\rangle $ and are thus $\\mathcal {O}(a^{2k+2})$ .", "Here $F$ is the field strength tensor.", "Combining this with the observation that for a coupling $\\beta _j$ generated at $\\mathcal {O}(\\beta ^n)$ has the scaling $\\beta _{j}\\approx 10^{-n}\\beta ^n$ , we estimate that $\\frac{\\langle {S}_{m}^{k-plaq}[u]\\rangle }{\\langle {S}_{n}^{\\rm 1 plaq}[u]\\rangle }\\approx \\left(\\frac{10}{\\beta }\\right)^{n-m}\\frac{a^{2k+2}\\langle D^{2k-2}(\\operatorname{Tr}FF)\\rangle }{a^4\\langle \\operatorname{Tr}FF\\rangle }\\,,$ where $D$ is a covariant derivative projected onto the lattice directions.", "The combination of higher powers of $a$ and the associated expectation values of higher-dimensional operators should be sufficient to suppress the mild $\\beta $ enhancement for $n>m$ , at least for $\\mathcal {O}(\\beta ^3)$ $S[u]$ .", "For these reasons, we will neglect higher order multi-plaquette terms.", "For the third-order terms of Eq.", "(), we, therefore, only focus on the case where three plaquettes are identical.", "Combining Eqs.", "(REF ), (REF ), and  (REF ) we arrive at the third order contribution to the single-plaquette decimated action $\\frac{\\beta ^3}{3!", "}{S}_3[u]=&-\\frac{\\beta _{\\lbrace 3\\rbrace }^{(3)}}{d_{\\lbrace 3\\rbrace }}\\operatorname{Re}\\chi _{\\lbrace 3\\rbrace }- \\frac{\\beta _{\\lbrace 2,1\\rbrace }^{(3)}}{d_{\\lbrace 2,1\\rbrace }}\\operatorname{Re}\\chi _{\\lbrace 2,1\\rbrace }\\\\ & - \\frac{\\beta _{\\lbrace 1,1,1\\rbrace }^{(3)}}{d_{\\lbrace 1,1,1\\rbrace }}\\operatorname{Re}\\chi _{\\lbrace 1,1,1\\rbrace }-\\frac{\\beta _{\\lbrace 2,-1\\rbrace }^{(3)}}{d_{\\lbrace 2,-1\\rbrace }}\\operatorname{Re}\\chi _{\\lbrace 2,-1\\rbrace }\\nonumber \\\\& -\\frac{\\beta _{\\lbrace 1,1,-1\\rbrace }^{(3)}}{d_{\\lbrace 1,1,-1\\rbrace }}\\operatorname{Re}\\chi _{\\lbrace 1,1,-1\\rbrace }- \\frac{\\beta _{\\lbrace 1\\rbrace }^{(3)}}{d_{\\lbrace 1\\rbrace }}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }\\,,$ where the overall factor of $1/3!$ has been absorbed into the definition of $\\beta _r^{(3)}$ .", "Note that, unlike the second order results where only certain decimation programs generate renormalization for existing terms, the third order $S^3[u]$ introduces corrections to $\\operatorname{Re}\\chi _1$ for all $G\\rightarrow H$ .", "Additionally, a number of the specific group identities in Eqs.", "(REF )-(REF ) also lead to renormalization.", "Putting together Eqs.", "(REF ), (REF ), and  (REF ), the single-plaquette decimated action of Eq.", "(REF ) to $\\mathcal {O}(\\beta ^3)$ for a general gauge group is, $S[u]=&\\sum _p -\\left(\\beta _{\\lbrace 1\\rbrace }^{(1)}+\\beta _{\\lbrace 1\\rbrace }^{(3)}\\right) \\frac{1}{N}\\operatorname{Re}(\\chi _{\\lbrace 1\\rbrace }) - \\beta _{\\lbrace 0\\rbrace }^{(2)}-\\beta _{\\lbrace 2\\rbrace }^{(2)}\\frac{2}{N(N+1)}\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }-\\beta _{\\lbrace 1,1\\rbrace }^{(2)}\\frac{2}{N(N-1)}\\operatorname{Re}\\chi _{\\lbrace 1,1\\rbrace }\\nonumber \\\\&\\qquad -\\beta _{\\lbrace 1,-1\\rbrace }^{(2)}\\frac{1}{N^2-1}\\chi _{\\lbrace 1,-1\\rbrace }-\\beta _{\\lbrace 3\\rbrace }^{(3)} \\frac{6}{N(N+1)(N+2)}\\operatorname{Re}\\chi _{\\lbrace 3\\rbrace }-\\beta _{\\lbrace 2,1\\rbrace }^{(3)}\\frac{3}{N(N^2-1)}\\operatorname{Re}\\chi _{\\lbrace 2,1\\rbrace }\\nonumber \\\\&\\qquad -\\beta _{\\lbrace 1,1,1\\rbrace }^{(3)}\\frac{6}{N(N-1)(N-2)}\\operatorname{Re}\\chi _{\\lbrace 1,1,1\\rbrace }-\\beta _{\\lbrace 2,-1\\rbrace }^{(3)}\\frac{2}{N(N-1)(N+2)}\\operatorname{Re}\\chi _{\\lbrace 2,-1\\rbrace }\\nonumber \\\\&\\qquad -\\beta _{\\lbrace 1,1,-1\\rbrace }^{(3)}\\frac{2}{N(N+1)(N-2)}\\operatorname{Re}\\chi _{\\lbrace 1,1,-1\\rbrace }\\,,$ where $\\beta _r$ are in Table REF .", "Note that this $S[u]$ is correct for any $G\\rightarrow H$ .", "Referring to Eqs.", "(REF ), (REF ), and (REF ), for a given $G$ simplifications occur.", "For $SU(3)$ , with $\\beta _r\\equiv \\sum _n \\frac{1}{n!", "}\\beta _r^{(n)}\\, ,$ this corresponds to: $S[u]=\\sum _p&-\\left(\\beta _{\\lbrace 1\\rbrace }+\\beta _{\\lbrace 1,1\\rbrace }\\right)\\frac{1}{3}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }-\\left(\\beta _{\\lbrace 0\\rbrace }+\\beta _{\\lbrace 1,1,1\\rbrace }\\right)\\nonumber \\\\&-\\left(\\beta _{\\lbrace 2\\rbrace }+\\beta _{\\lbrace 1,1,-1\\rbrace }\\right)\\frac{1}{6}\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }\\nonumber \\\\&-\\left(\\beta _{\\lbrace 1,-1\\rbrace }+\\beta _{\\lbrace 2,1\\rbrace }\\right)\\frac{1}{8}\\chi _{\\lbrace 1,-1\\rbrace }\\nonumber \\\\&-\\frac{\\beta _{\\lbrace 3\\rbrace }}{10}\\operatorname{Re}\\chi _{\\lbrace 3\\rbrace }-\\frac{\\beta _{\\lbrace 2,-1\\rbrace }}{15}\\operatorname{Re}\\chi _{\\lbrace 2,-1\\rbrace }\\,.$ For $U(1)$ and $SU(2)$ , we refer the reader to Appendix .", "Table: Numerical values of β r [G→H]\\beta _r[G\\rightarrow H] of character rr for the decimations U(1)→ℤ 4 U(1)\\rightarrow \\mathbb {Z}_{4}, SU(2)→𝔹𝕀SU(2)\\rightarrow \\mathbb {BI}, and SU(3)→𝕍SU(3)\\rightarrow \\mathbb {V}.", "For completeness, we have included the 4 two-plaquette terms derived in , at second order." ], [ "Results for $\\mathbb {V}$", "As a demonstration, we simulated Eq.", "(REF ) to each order in $\\beta $ for $SU(3)\\rightarrow \\mathbb {V}$ .", "For these computations 10$^2$ configurations separated by 10$^3$ sweeps were collected on a $4^4$ lattice and plotted in Fig.", "REF .", "In the figure, we compare the average energy per plaquette $\\langle E_0\\rangle $ versus the coupling, ${\\beta }_{\\lbrace 1\\rbrace }$ as defined in Eq.", "(REF ), which multiplies the Wilson term $\\operatorname{Re}\\chi _1$ .", "For $SU(3)$ , this corresponds to $\\beta $ , and $\\beta _{\\lbrace 1\\rbrace }+\\beta _{\\lbrace 1,1\\rbrace }$ for $\\mathbb {V}$ .", "Naively, $\\langle E_0\\rangle $ is monotonic in $a$ .", "Including $\\mathcal {O}(\\beta ^2)$ terms, we observe a clear reduction in $\\langle E_0\\rangle $ and thus an improvement over the Wilson action for small $a$ .", "This suggests promise in this systematic approach.", "As will be discussed in Sec.", ", it also supports the effectiveness of the previously studied ad-hoc actions.", "Instead of freezing out, the theory approaches a non-zero value of $\\langle E_0\\rangle $ .", "At $\\mathcal {O}(\\beta ^3)$ , $\\langle E_0\\rangle $ displays non-monotonic behavior due to the negative coefficient of $\\chi _1$ at $\\beta ^3$ .", "This suggests the higher order terms (4th order and beyond) required to match Eq.", "(REF ) and (REF ) are dominating the action.", "Figure: Average energy per plaquette, 〈E 0 〉=1-Re〈TrU p 〉/3\\langle E_0\\rangle = 1-\\operatorname{Re}\\langle \\operatorname{Tr}U_p\\rangle /3, vs β ˜ 1 \\tilde{\\beta }_1 on 4 4 4^4 lattice for 𝕍\\mathbb {V} action with corrections of: (fill,scale=0.4,regular polygon, regular polygon sides=4,fill=red]();) 𝒪(β 1 )\\mathcal {O}(\\beta ^1), (fill,scale=0.4,circle,fill=mycolor]();) 𝒪(β 2 )\\mathcal {O}(\\beta ^2), and (fill,scale=0.3,regular polygon, regular polygon sides=3,fill=blue!10!blue,rotate=0]();) 𝒪(β 3 )\\mathcal {O}(\\beta ^3).", "The black line is the SU(3)SU(3) result.Taken together, the $\\mathcal {O}(\\beta ^n)$ results suggests that the decimation procedure, formulated as a strong coupling expansion, converges to the continuous theory slowly.", "Sufficiently large order calculations would suppress the higher order contributions in the form $\\beta ^n/n!$ for a reasonable range of $\\beta $ .", "While an $\\mathcal {O}(\\beta ^4)$ calculation would undoubtedly be insightful to understanding this convergence, other approaches such as introducing “counter-terms\" to absorb some higher-order contributions, or instead using a character expansion may prove fruitful.", "It also may prove useful to take the expansion in Eq.", "(REF ) as an effective theory where each character is subject to field redefinitions.", "While these possibilities are interesting to investigate, they are certainly beyond the scope of this work.", "We therefore leave them for future studies." ], [ "Finite group effects", "With Eq.", "(REF ), it is possible for us to investigate systematically the effect of replacing the continuous group by its finite subgroup.", "In order to proceed, it is useful to introduce a new parameter which approximately represents the field fluctuations.", "To do this, consider the representation of a continuous group lattice gauge link in terms of the corresponding generators $\\lambda _a$ in the adjoint representation, $U=e^{i\\lambda _a A_a}$ , where a summation over color indices $a$ is implied.", "In this form, we see that the gauge fields correspond to amplitudes in each of the generators.", "For $\\epsilon \\in \\Omega $ , inserting its small parameter expansion $\\epsilon \\approx \\mathbb {1}+i \\lambda _a A_a-\\frac{1}{2}(\\lambda _a A_a)^2+\\hdots $ into Eq.", "(REF ) gives $V_r \\approx 1-\\int _\\Omega DA\\left(c_r^{(2)} \\sum _a A_a^2+\\hdots \\right)\\, ,$ where $DA$ is a measure over all $A_a$ which respects gauge symmetry and $c_r^{(n)}$ are representation and group-dependent constants.", "From this, we see that as the subgroup $H$ incorporates more elements, the size of $\\Omega $ approaches 0 and $ V_r\\rightarrow 1$ from below.", "This means that for finite $\\Omega $ the domain size of $A_a$ that grives rise to $\\Omega $ is an indicator for deviations from $G$ of $H$ .", "Flyvbjerg defines a parameter $R$ as the radius of a hypersphere with equal volume to $\\Omega $ to get a handle on the domain of $A_a$ .", "This allows him to approximate $V_{r}$ analytically [36], [37].", "Here, we can use this idea to roughly understand the scaling of $V_r$ .", "For $U(1)\\rightarrow \\mathbb {Z}_{\\rm n}$ , the hypersphere is exactly $\\Omega $ and $R$ cleanly defines $\\epsilon \\le R=\\pi /n$ .", "Beyond $U(1)$ , the connection between $\\Omega $ and a single value of $R$ is complicated because the $\\Omega $ of $H$ form polytopes in the hypervolume of their continuous partner (see Fig.", "REF for a clear demonstration).", "In this case, while one could take $\\Omega $ to be contained by a hypersphere centered at $\\mathbb {1}$ whose boundary incorporates elements of the nearest neighbors of ${\\mathbb {1}}$ in $H$ , making some element of the hypersphere not included in $\\Omega $ .", "On the other hand, there exists a largest hypersphere centered at $\\mathbb {1}$ that only contains elements in $\\Omega $ .", "In this way, we define an upper and lower bound for $R$ .", "Note, this is different from [36], [37] where the polytopes of $H$ were always approximated by hyperspheres with definite radii.", "For SU(2) with $\\mathbb {BI}$ , we find $0.09\\le R^2 \\le 0.15$ which can be compared to $R^2_{sphere}=0.12$ of  [36], [37].", "In the case of SU(3) with $\\mathbb {V}$ , $0.42\\le R^2 \\le 0.93$ compared to $R^2_{sphere}=0.62$ .", "While superficially the cumulant expansion has appeared as a strong-coupling expansion in $\\beta $ , the actual behavior is controlled by both $\\beta $ and $R$ with $R$ controlling $V_r$ .", "As pointed out in [36], [37], the leading order behavior for small $R$ for a given power of $\\beta ^\\alpha $ ($\\alpha >0$ ) is actually $\\mathcal {O}([\\beta R^{2}]^\\alpha R^{-2})$ .", "Therefore one would predict that the relative smallness of $R^2$ for $\\mathbb {BI}$ compare to $\\mathbb {V}$ signals that $\\beta _f$ should be larger for $\\mathbb {BI}$ which is indeed the case.", "For subgroups of $SU(3)$ , this scaling behavior becomes unsatisfactory because $R^2\\sim 1$ .", "It is possible to study this breakdown in $U(1)\\rightarrow \\mathbb {Z}_{\\rm n}$ where the systematic effect of decimation can be studied in detail both because errors can be made arbitrarily small for large $n$ and because $ V_r$ and $\\beta _r$ are known analytically.", "In terms of $R$ , one can expand the $\\beta _r$ for the $U(1)$ action of Eq.", "(REF ) to find: $\\beta _{\\lbrace 0\\rbrace }&\\approx \\bigg (\\frac{R^2}{3}-\\frac{19R^4}{90}+\\hdots \\bigg )\\beta ^2,$ $\\beta _{\\lbrace 1\\rbrace }+\\beta _{\\lbrace 1,1,-1\\rbrace }&\\approx \\left(1-\\frac{2R^2}{3}+\\frac{R^4}{5}+\\hdots \\right)\\beta \\nonumber \\\\&\\quad + \\left(-\\frac{17R^4}{90}+\\frac{311R^6}{945}+\\hdots \\right)\\beta ^3,$ $\\beta _{\\lbrace 2\\rbrace }&\\approx \\left(-\\frac{R^2}{3}+\\frac{53R^4}{90}+\\hdots \\right)\\beta ^2, $ $\\beta _{\\lbrace 3\\rbrace }&\\approx \\bigg (\\frac{17R^4}{90}-\\frac{1609R^6}{2835}+\\frac{46303R^8}{56700}\\nonumber \\\\&\\quad \\quad -\\frac{77603R^{10}}{103950}+\\hdots \\bigg )\\beta ^3.$ The first thing to note is that the $\\mathcal {O}([\\beta R^{2}]^\\alpha R^{-2})$ scaling found in [36], [37] continues to the third order.", "One might be tempted to use this leading behavior to estimate the $\\beta _f$ or the radius of convergence of this series, but this would be incorrect.", "Instead, it behooves one to note that for both 2nd and 3rd order contributions, the subleading terms $[R^{2}]^k R^{-2}$ with $k>\\alpha $ initially grow until a $1/k!$ factor dominates over all the other factors.", "But what is the origin of this behavior?", "For simplicity, we can understand this behavior by considering the expansion of $V_{j}^{4m}$ which form $\\beta _r$ .", "The specific combination of $V_{j}^{4m}$ dictated by the cumulant expansion ensures that orders lower than $\\mathcal {O}([\\beta R^{2}]^\\alpha R^{-2})$ cancel in $\\beta _r$ .", "The $j$ representation contributes to $\\beta _{\\lbrace r_1,\\cdots , r_k\\rbrace }$ in the form of $V_{\\lbrace j_1^1,\\cdots , j_l^1\\rbrace }^{4m_1}\\cdots V_{\\lbrace j_1^k,\\cdots ,j_n^k\\rbrace }^{4m_k}$ under the constraint $|r|=m_i\\widetilde{j}_i$ where $|r|=|r_1|+\\cdots +|r_k|$ and $\\widetilde{j}_i=|j_1^i|+\\cdots +|j_l^i|$ .", "One might worry that studying the expansion of $V_{j}^{4m}$ isn't representative, but one can verify that the scaling behavior observed below persists in $\\beta _r$ , although the numerical factors become cumbersome.", "For $V_{j}^{4m}\\equiv V_{\\lbrace j_1,\\cdots ,j_l\\rbrace }^{4m}$ , $\\widetilde{j}=|j_1|+\\cdots |j_l|$ , we have $V_{j}^{4m}\\approx 1-\\frac{2}{3}m(\\widetilde{j}R)^2+\\frac{1}{45}(10m^2-m)(\\widetilde{j}R)^4+\\mathcal {O}(m^3[\\widetilde{j}R]^6)$ from which, we see that the coefficients of the $[R^{2}]^k R^{-2}$ contributions to $\\beta _r$ are accompanied by a factor $\\propto \\frac{1}{k!", "}m^{k-1} \\widetilde{j}^{2k-2}$ .", "While the factorials ensure the series converges, $\\beta _r$ for higher representations $r$ have larger $m$ , $\\widetilde{j}$ , or both leading to higher order terms in the expansion being large for moderate $R$ .", "This helps explaining why ${\\mathbb {Z}}_4$ with the Wilson action fails to replicate $U(1)$ substantially above $\\beta =1$ – while the naive scaling would suggest $R\\lesssim \\sqrt{\\beta }^{\\frac{k}{1-k}}$ would be enough to suppress higher representations, in reality a stronger bound of $\\max \\lbrace \\frac{1}{k!", "}m^{k-1} (\\widetilde{j}R)^{2k-2}\\rbrace _{1\\le \\widetilde{j}\\le |r|}\\lesssim 1$ for $\\forall \\beta _r$ is required for all subleading terms to be small.", "Considering the range of $m$ with fixed $|r|,\\widetilde{j}$ , the bound is strictest when $|r|=\\widetilde{j}$ yielding $R\\lesssim 1/|r|^{3/2}$ in order for the lowest order contribution to dominate such that $R\\lesssim \\sqrt{\\beta }^{\\frac{\\alpha }{1-\\alpha }}$ provides a reasonable estimate for the range of $\\beta $ where the decimated action provides a reasonable approximation for its continuous partner.", "While these conditions are satisfied for $\\mathbb {BI}$ , they are violated for $\\mathbb {V}$ in which case the dominant term in the $R$ expansion isn't clear.", "Another feature observed in the $R$ expansion of the $\\mathbb {Z}_{\\rm n}$ group is that because $V_r\\propto \\sin rR$ , the sign of the $\\mathcal {O}([rR]^k)$ terms oscillate, and therefore the sign of $\\beta _r^{(n)}$ can depend sensitively on $R$ .", "Since $\\operatorname{Re}\\chi _r(\\mathbb {1})>\\operatorname{Re}\\chi _r(\\mathcal {N})$ , where ${\\cal N}$ is the nearest neighbors of ${\\mathbb {1}}$ in $H$ (see Fig.", "REF ), the overall sign of $\\beta _r^{(n)}$ determines whether or not the $r$ -th term in the action enters the frozen phase in the limit of $\\beta \\rightarrow \\infty $ .", "This behavior is observed in Fig.", "REF where $\\beta _{\\lbrace 1\\rbrace }^{(1,2)}>0$ but $\\beta _{\\lbrace 1\\rbrace }^{(3)}<0$ .", "From the behavior observed in $U(1)$ , we can improve the quantitative understanding of how well $H$ can approximate $G$ , even when $\\beta _r$ are not known analytically.", "Clearly, $V_r\\rightarrow 1$ indicates that the $R\\rightarrow 0$ , and in that limit the two actions would agree.", "Therefore, the difference between the two actions $S_G-S_H\\approx \\beta \\chi _{\\lbrace 1\\rbrace }(U)-\\beta _{\\lbrace 1\\rbrace }\\chi _{\\lbrace 1\\rbrace }(u)\\approx (1-V_{\\lbrace 1\\rbrace }^4)\\beta \\chi _{\\lbrace 1\\rbrace }(u)$ serves as an indicator of $\\beta _f$ .", "Table: Parameters of a discrete subgroups necessary to study the behavior of β f \\beta _f.Figure: β f \\beta _f as a function of (left) ΔS -1 \\Delta S^{-1}, (center) the cycle CC of 𝒩\\mathcal {N}, (right) (1-V {1} 4 ) -1 (1-V_{\\lbrace 1\\rbrace }^4)^{-1}.", "Note that for the subgroups of U(1)U(1) and SU(2)SU(2), monotonic behavior is observed for all three variables, but only for (1-V {1} 4 ) -1 (1-V_{\\lbrace 1\\rbrace }^4)^{-1} are the subgroups of SU(3)SU(3) monotonic.This proxy can be compared to others in the literature.", "The simplest estimate is $\\beta _f^{-1}\\propto \\Delta S=\\operatorname{Re}\\operatorname{Tr}(\\mathbb {1})-\\operatorname{Re}\\operatorname{Tr}(\\mathcal {N})$  [41].", "While this estimate finds monotonic behavior for discrete groups of $U(1)$ and $SU(2)$ , different $\\mathcal {O}(1)$ factors are needed.", "It also fails completely for $SU(3)$ , as seen in the left panel of Fig.", "REF .", "Observing the differing $\\mathcal {O}(1)$ factors, [41] suggested a different estimate.", "For discrete Non-abelian subgroups near $\\beta _f$ , $S[u]$ is dominated by contributions from $u_p=\\mathcal {N}$ .", "From duality arguments , the action near $\\beta _f$ could be approximately rewritten as a $\\mathbb {Z}_C$ action where $C$ is the minimal cycle such that $u^C=\\mathbb {1}$ for all $u\\subset \\mathcal {N}$ .", "Since $C=n$ for $\\mathbb {Z}_{\\rm n}$ , these arguments predict a single curve $\\beta _f\\approx 0.78/(1-\\cos (2\\pi /C))$ directly from the study of $\\beta _f$ in $\\mathbb {Z}_{\\rm n}$ for all discrete subgroups.", "The discrepancy between $SU(2)$ and $U(1)$ was reduced from $\\sim 300\\%$ to $\\sim 50\\%$ .", "The authors of [41] warned that this approximation could be poor for $SU(3)$ albeit without numerical evidence.", "Since then $\\beta _f$ for the subgroups of $SU(3)$ have been found and as anticipated, this estimator proves to be poor as presented in the center of Fig.", "REF .", "Figure: β f /β s \\beta _f/\\beta _s as a function of (1-V {1} 4 ) -1 (1-V_{\\lbrace 1\\rbrace }^4)^{-1}.In the plot on the right of Fig.", "REF , $\\beta _f$ is plotted as a function of $(1-V_{\\lbrace 1\\rbrace }^4)^{-1}$ .", "We find that monotonic, linear behavior is observed within the uncertainties for each continuous group.", "Best fit lines have been included for each group to guide the eye.", "This suggests that our estimator captures some of the non-perturbative behavior near the freezing transition better than $\\Delta S^{-1}$ or $C$ .", "Physics of the different groups differ, as signaled by their different scaling regimes.", "If we divide $\\beta _f$ by a rough estimate of $\\beta _s=[1,2.2,6]$ for $U(1),SU(2),SU(3)$ respectively, we might expect to further remove some of this group dependence.", "Doing so in Fig.", "REF , we find that $SU(2)$ and $SU(3)$ collapse onto a single line and $U(1)$ within $~25\\%$ .", "Using our higher order results, one can then gain insight into the effectiveness of the ad-hoc actions of $\\mathbb {V}$ .", "Each of these actions corresponds to terms that are generated at 2nd order in the decimated action.", "The first ad-hoc action used in [28] can be rewritten as $S[u]&=-\\sum _p \\left(\\frac{\\beta _0}{3}\\operatorname{Re}\\operatorname{Tr}(u_p) +\\beta _1\\operatorname{Re}\\operatorname{Tr}(u_p^2)\\right) \\,,\\nonumber \\\\&=-\\sum _p \\left(\\left(\\beta _0-3\\beta _1\\right)\\frac{1}{3}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace } +\\left(6\\beta _1\\right)\\frac{1}{6}\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }\\right) \\,,\\nonumber \\\\$ where we have used $\\beta _1=a\\beta _0+b$ with $a=-0.1267$ and $b=0.253$ .", "For an unpublished action of $S[u]=-\\sum _p\\left(\\frac{\\tilde{\\beta }_{\\lbrace 1\\rbrace }}{3}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace } +\\frac{\\tilde{\\beta }_{\\lbrace 1,1\\rbrace }}{8}\\operatorname{Re}\\chi _{\\lbrace 1,-1\\rbrace }\\right)$ where $\\tilde{\\beta }_{\\lbrace 1,1\\rbrace }=a\\tilde{\\beta }_{\\lbrace 1\\rbrace }+b$ with $a=-0.587$ and $b=1.80$ .", "The trajectory parameters were chosen to be parallel to the freezing point at large $\\beta _0$ by eye.", "From Fig.", "REF , we see that in both ad-hoc actions, reasonably agreement is found for intermediate $\\beta $ for the 3rd order action.", "Here $\\beta $ is the coefficient in front of $\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }$ for the ad-hoc actions.", "The ad-hoc trajectories are known to poorly reflect $G$ at low $\\beta $ , because they lack curvature to fix the known requirements at $\\beta =0$ .", "At large $\\beta $ , we expect higher order terms in the cumulant expansion to become relevant and thus disagreement is expected to occur.", "This surprising agreement in the intermediate region of $\\beta $ suggests that actions formed by neglecting terms in the cumulant expansion are optimized in their character basis by setting the couplings to results given by the resuming higher order contributions in cumulant expansion with fluctuations $G/H$ integrated out.", "Figure: β {r} \\beta _{\\lbrace r\\rbrace } trajectories of S n [u]S^n[u] to the ad-hoc actions with additional terms (left) Reχ {2} \\operatorname{Re}\\chi _{\\lbrace 2\\rbrace } of and (right) Reχ {1,-1} \\operatorname{Re}\\chi _{\\lbrace 1,-1\\rbrace } of .", "The open circles indicate the boundary between the frozen and unfrozen phases obtained on 2 4 2^4 lattices." ], [ "Conclusion", "In this work, we used the cumulant expansion to develop a systematic method for studying and improving lattice actions that replace continuous gauge groups by their discrete subgroups.", "This is a step in the ongoing trek toward developing accurate and efficient digitization on quantum computers.", "These decimated actions, through the factor $V_{\\lbrace 1\\rbrace }$ , have superior predictive power for finding the freezing transition compared to prior estimators.", "We further computed the third-order, single-plaquette contribution for the general group.", "These higher-order terms are necessary for systematizing the decimation procedure of $SU(3)\\rightarrow \\mathbb {V}$ where it has been observed that the inclusion of terms generated in the second-order cumulant expansion with ad-hoc couplings improve the approximation of $SU(3)$ .", "The most immediate work in these directions would be to compute more Euclidean observables (i.e.", "Wilson flow parameter and pseudocritical temperature) from the full decimated action of Eq.", "(REF ) and compare them to [28], [56].", "Given the large corrections from second to third order for $\\mathbb {V}$ , additional work should be devoted to computing the fourth-order contributions.", "In order to move beyond pure gauge theory, it will be necessary to consider quark fields.", "While the computational resources increase substantially for dynamical quarks, an advantage of the discrete subgroup approximations is that many standard lattice field theory techniques such as fermionic determinants and pseudofermions can be applied.", "This was demonstrated in early works on dynamical fermions where $\\mathbb {BI}$ replaced $SU(2)$  [43], [44].", "Another important step in studying the feasibility of this procedure is to explicitly construct the quantum registers and primitive gates à la [60] where smaller discrete groups were investigated.", "Together with classical lattice results, this would allows for resource counts.", "The authors would like to thank Scott Lawrence, Jesse Stryker, Justin Thaler, and Yukari Yamauchi for helpful comments on this work.", "Y.J.", "is grateful for the support of DFG, grants BR 2021/7-2 and SFB TRR 257.", "H.L.", "is supported by a Department of Energy QuantiSED grant.", "Fermilab is operated by Fermi Research Alliance, LLC under contract number DE-AC02-07CH11359 with the United States Department of Energy.", "S.Z.", "is supported by the National Science Foundation CAREER award (grant CCF-1845125)." ], [ "Creutz Identities", "A useful identity was derived in [57] for $SU(N)$ and $U(N)$ groups such that for any integer $n\\le N$ $\\langle {\\epsilon _{i_1j_1}\\cdots \\epsilon _{i_nj_n}}\\rangle &=c_1\\,\\varepsilon _{i_1A^1_1\\ldots A^1_{N-1}}\\varepsilon _{j_1A^1_1\\ldots A^1_{N-1}}\\times \\cdots \\times \\varepsilon _{i_nA^n_1\\ldots A^n_{N-1}}\\varepsilon _{j_nA^n_1\\ldots A^n_{N-1}}\\\\&+c_2\\,\\varepsilon _{i_1i_2A^1_1\\ldots A^1_{N-2}}\\varepsilon _{j_1j_2A^1_1\\ldots A^1_{N-2}}\\times \\cdots \\times \\varepsilon _{i_n A^{n-1}_1\\ldots A^{n-1}_{N-1}}\\varepsilon _{j_n A^{n-1}_1\\ldots A^{n-1}_{N-1}}+\\cdots \\\\&+c_{B_n}\\,\\varepsilon _{i_1i_2\\ldots i_n A^1_1\\ldots A^1_{N-n}}\\varepsilon _{j_1j_2\\ldots j_n A^1_1\\ldots A^1_{N-1}} \\,,$ where $\\varepsilon $ is Levi-Civita symbol, $A_i^j$ are the contracted dummy indices, and $B_n$ is the Bell number accounting for the number of ways that one can put the open indices $i_k\\,,j_l$ on $\\varepsilon $ such that no $i_k$ and $j_l$ appear in the same $\\varepsilon $ .", "In [57], Eq.", "(REF ) was derived for integrating over the entire group $G$ .", "Hence in our case, we need to determine the constants $c_i$ 's when integrating only over $\\Omega $ for $\\langle {\\epsilon _{ij}}\\rangle , \\langle {\\epsilon _{ij}\\epsilon _{k\\ell }}\\rangle $ , $\\langle {\\epsilon _{ij}\\epsilon _{k\\ell }^\\dag }\\rangle $ , $\\langle {\\epsilon _{ij}\\epsilon _{k\\ell }\\epsilon _{mn}}\\rangle $ , $\\langle {\\epsilon _{ij}\\epsilon _{k\\ell }\\epsilon _{mn}^\\dag }\\rangle $ , with $i,j,k,l,m,n\\in [N]$ .", "This is done by contracting the tensor structure on each side of Eq.", "(REF ) with products of Kronecker delta's and solving the resulting linear equations.", "At first order, only one integral is needed: $\\langle {\\epsilon _{ij}}\\rangle =&V_{\\lbrace 1\\rbrace }\\,\\delta _{ij}\\,.$ At second order, there are two relations $\\langle {\\epsilon _{ij}\\epsilon _{kl}}\\rangle =&\\frac{1}{2}\\left(V_{\\lbrace 2\\rbrace }+V_{\\lbrace 1,1\\rbrace }\\right)\\delta _{ij}\\delta _{kl}+\\frac{1}{2}\\left(V_{\\lbrace 2\\rbrace }-V_{\\lbrace 1,1\\rbrace }\\right)\\delta _{il}\\delta _{jk}\\,,$ and $\\langle {\\epsilon _{ij}\\epsilon _{kl}^\\dagger }\\rangle =&V_{\\lbrace 1,-1\\rbrace }\\delta _{ij}\\delta _{kl}+\\frac{1}{N}(1-V_{\\lbrace 1,-1\\rbrace })\\delta _{il}\\delta _{jk}\\,.$ At third order, there are four structures, but by complex conjugation one can reduce this to two unique ones: $\\langle {\\epsilon _{ij}\\epsilon _{kl}\\epsilon _{mn}}\\rangle =&\\frac{1}{6}\\left(V_{\\lbrace 3\\rbrace }+4V_{\\lbrace 2,1\\rbrace }+V_{\\lbrace 1,1,1\\rbrace }\\right)\\delta _{ij}\\delta _{kl}\\delta _{mn}+\\frac{1}{6}\\left(V_{\\lbrace 3\\rbrace }-V_{\\lbrace 1,1,1\\rbrace }\\right)(\\delta _{il}\\delta _{jk}\\delta _{mn}+\\delta _{in}\\delta _{jm}\\delta _{kl}+\\delta _{ij}\\delta _{kn}\\delta _{lm}) \\nonumber \\\\&+\\frac{1}{6}\\left(V_{\\lbrace 3\\rbrace }-2V_{\\lbrace 2,1\\rbrace }+V_{\\lbrace 1,1,1\\rbrace }\\right)(\\delta _{in}\\delta _{jk}\\delta _{lm}+\\delta _{il}\\delta _{kn}\\delta _{jm})\\,,$ $\\langle {\\epsilon _{ij}\\epsilon _{kl}\\epsilon ^\\dagger _{mn}}\\rangle =&\\frac{1}{2}\\left(V_{\\lbrace 2,-1\\rbrace }+V_{\\lbrace 1,1,-1\\rbrace }\\right)\\delta _{ij}\\delta _{kl}\\delta _{mn}+\\frac{1}{2}\\left(V_{\\lbrace 2,-1\\rbrace }-V_{\\lbrace 1,1,-1\\rbrace }\\right)\\delta _{il}\\delta _{jk}\\delta _{mn} \\nonumber \\\\&+\\left(\\frac{N}{(N-1)(N+1)}V_{\\lbrace 1\\rbrace }-\\frac{1}{2(N+1)}V_{\\lbrace 2,-1\\rbrace }-\\frac{1}{2(N-1)}V_{\\lbrace 1,1,-1\\rbrace }\\right)(\\delta _{in}\\delta _{jm}\\delta _{kl}+\\delta _{ij}\\delta _{kn}\\delta _{lm}) \\nonumber \\\\&+\\left(-\\frac{1}{(N-1)(N+1)}V_{\\lbrace 1\\rbrace }-\\frac{1}{2(N+1)}V_{\\lbrace 2,-1\\rbrace }+\\frac{1}{2(N-1)}V_{\\lbrace 1,1,-1\\rbrace }\\right)(\\delta _{in}\\delta _{jk}\\delta _{lm}+\\delta _{il}\\delta _{kn}\\delta _{jm})\\,.$" ], [ "Group Properties", "For a given group, the general basis is overcomplete.", "These leads to simplifications in our derivations for a given group.", "Here we present the related characters for three groups of relative importance: $U(1),SU(2),SU(3)$ .", "For $U(1)$ , the resulting identities are $\\chi _{\\lbrace 1\\rbrace }&=-\\chi _{\\lbrace 1,1,-1\\rbrace }\\,,\\quad \\chi _{\\lbrace 1,\\pm 1\\rbrace }=\\chi _{\\lbrace 2,\\pm 1\\rbrace }=\\chi _{\\lbrace 1,1,1\\rbrace }=0 \\,.$ For $SU(2)$ , one finds that $\\chi _{\\lbrace 1\\rbrace }&=\\chi _{\\lbrace 2,1\\rbrace }\\, ,\\quad \\chi _{\\lbrace 2\\rbrace }=\\chi _{\\lbrace 1,-1\\rbrace }\\,,\\quad \\chi _{\\lbrace 1,1\\rbrace }=1\\,,\\quad \\chi _{\\lbrace 1,1,1\\rbrace }=\\chi _{\\lbrace 1,1,-1\\rbrace }=0\\,,\\quad \\chi _{\\lbrace 3\\rbrace }=\\chi _{\\lbrace 2,-1\\rbrace }\\, ,$ and for $SU(3)$ , the set of dependent representations needed up to third order in the cumulant expansion are $\\chi _{\\lbrace 1\\rbrace }&=\\chi _{\\lbrace 1,1\\rbrace }\\, ,\\quad \\chi _{\\lbrace 2\\rbrace }=\\chi _{\\lbrace 1,1,-1\\rbrace },\\quad \\chi _{\\lbrace 1,-1\\rbrace }=\\chi _{\\lbrace 2,1\\rbrace }\\, ,\\quad \\chi _{\\lbrace 1,1,1\\rbrace }=1\\, .$ Another important set of identities are those which relate products of $\\operatorname{Re}\\chi _r$ to sum of $\\operatorname{Re}\\chi _r$ .", "They are easily enough derived, but we display a few key ones here: $ (\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace })^2&=\\frac{1}{2}\\operatorname{Re}(\\chi _{\\lbrace 2\\rbrace }+\\chi _{\\lbrace 1,1\\rbrace }+\\chi _{\\lbrace 1,-1\\rbrace }+1)\\,,\\\\\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }&=\\frac{1}{2}\\operatorname{Re}(\\chi _{\\lbrace 1\\rbrace }+\\chi _{\\lbrace 2,1\\rbrace }+\\chi _{\\lbrace 2,-1\\rbrace }+\\chi _{\\lbrace 3\\rbrace })\\,,\\\\\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }\\operatorname{Re}\\chi _{\\lbrace 1,1\\rbrace }&=\\frac{1}{2}\\operatorname{Re}(\\chi _{\\lbrace 1\\rbrace }+\\chi _{\\lbrace 1,1,1\\rbrace }+\\chi _{\\lbrace 1,1,-1\\rbrace }+\\chi _{\\lbrace 2,1\\rbrace })\\,,\\\\\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace } \\operatorname{Re}\\chi _{\\lbrace 1,-1\\rbrace }&=\\operatorname{Re}( \\chi _{\\lbrace 1\\rbrace }+\\chi _{\\lbrace 2,-1\\rbrace }+\\chi _{\\lbrace 1,1,-1\\rbrace })\\,.$ Applying all the simplifications in Eq.", "(REF ) for specific groups, we write $S[u]$ for $U(1)$ and $SU(2)$ respectively.", "For $U(1)$ : $S[u]=\\sum _p&-(\\beta _{\\lbrace 1\\rbrace }-\\beta _{\\lbrace 1,1,-1\\rbrace })\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }-\\beta _{\\lbrace 0\\rbrace }\\nonumber \\\\&-\\beta _{\\lbrace 2\\rbrace }\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }-\\beta _{\\lbrace 3\\rbrace }\\operatorname{Re}\\chi _{\\lbrace 3\\rbrace }\\,.", "$ For $SU(2)$ : $S[u]=\\sum _p&-\\left(\\beta _{\\lbrace 1\\rbrace }+\\beta _{\\lbrace 2,1\\rbrace }\\right)\\frac{1}{2}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }- \\left(\\beta _{\\lbrace 0\\rbrace }+\\beta _{\\lbrace 1,1\\rbrace }\\right)\\nonumber \\\\&-\\left(\\beta _{\\lbrace 2\\rbrace }+\\beta _{\\lbrace 1,-1\\rbrace }\\right)\\frac{1}{3}\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }\\nonumber \\\\&-(\\beta _{\\lbrace 3\\rbrace }+\\beta _{\\lbrace 2,-1\\rbrace }) \\frac{1}{4}\\operatorname{Re}\\chi _{\\lbrace 3\\rbrace }\\,.$" ], [ "Derivation of the Decimated Action", "In this appendix, we expand upon the derivation of the decimated action.", "First, for the second-order term in Eq.", "(), there are three terms which we decomposed based on the number of links that the two plaquettes $p,q$ shared.", "For case $p=q$ reads: $\\beta ^2\\langle {S[u,\\epsilon ]^2}\\rangle &= \\frac{\\beta ^2}{N^2} \\langle \\operatorname{Re}\\left(\\operatorname{Tr}\\left(u_1\\epsilon _1u_2\\epsilon _2(u_3\\epsilon _3)^\\dag (u_4\\epsilon _4)^\\dag \\right)\\right)\\operatorname{Re}\\left(\\operatorname{Tr}\\left( u_1\\epsilon _1u_2\\epsilon _2(u_3\\epsilon _3)^\\dag (u_4\\epsilon _4)^\\dag \\right)\\right) \\rangle \\\\&=\\frac{\\beta ^2}{2N^2}\\left(|V_{\\lbrace 1,1\\rbrace }|^4\\operatorname{Re}\\chi _{\\lbrace 1,1\\rbrace }+|V_{\\lbrace 2\\rbrace }|^4\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }+V_{\\lbrace 1,-1\\rbrace }^4\\chi _{\\lbrace 1,-1\\rbrace }+1\\right)\\,,$ where we have utilized Eqs.", "(REF ) and (REF ) to contract the $u$ 's after integration.", "The second term of Eq.", "() is obtained from first order action of Eq.", "(REF ) which reads, $\\beta ^2&\\langle {S[u,\\epsilon ]}\\rangle ^2 =\\left(\\frac{1}{N} \\beta ^2V_{\\lbrace 1\\rbrace }^4\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }\\right)^2= \\frac{1}{2N^2}\\beta ^2V_{\\lbrace 1\\rbrace }^8\\left( \\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }+\\operatorname{Re}\\chi _{\\lbrace 1,1\\rbrace }+\\chi _{\\lbrace 1,-1\\rbrace }+1\\right),$ where we have used Eq.", "(REF ).", "For the third-order terms of Eq.", "(), as discussed we need only consider when the three plaquettes are identical.", "This will be done term by term, where the first term is: $\\beta ^3\\langle {S[u,\\epsilon ]^3}\\rangle &= -\\frac{\\beta ^3}{N^3}\\langle \\operatorname{Re}\\operatorname{Tr}(u_1\\epsilon _1u_2\\epsilon _2(u_3\\epsilon _3)^\\dag (u_4\\epsilon _4)^\\dag )\\operatorname{Re}\\operatorname{Tr}(u_5\\epsilon _5u_6\\epsilon _6(u_7\\epsilon _7)^\\dag (u_8\\epsilon _8)^\\dag )\\operatorname{Re}\\operatorname{Tr}(u_9\\epsilon _9u_{10}\\epsilon _{10}(u_{11}\\epsilon _{11})^\\dag (u_{12}\\epsilon _{12})^\\dag )\\rangle \\nonumber \\\\&=-\\frac{\\beta ^3}{2N^3}\\bigg ( \\frac{V_{\\lbrace 3\\rbrace }^4}{2}\\operatorname{Re}\\chi _{\\lbrace 3\\rbrace } + V_{\\lbrace 2,1\\rbrace }^4\\operatorname{Re}\\chi _{\\lbrace 2,1\\rbrace }+\\frac{V_{\\lbrace 1,1,1\\rbrace }^4}{2}\\operatorname{Re}\\chi _{\\lbrace 1,1,1\\rbrace } + \\frac{3V_{\\lbrace 2,-1\\rbrace }^4}{2}\\operatorname{Re}\\chi _{\\lbrace 2,-1\\rbrace } \\nonumber \\\\&\\qquad \\qquad + \\frac{3V_{\\lbrace 1,1,-1\\rbrace }^4}{2}\\operatorname{Re}\\chi _{\\lbrace 1,1,-1\\rbrace }+3V_{\\lbrace 1\\rbrace }^4\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }\\bigg ).$ For the mixed-order term in Eq.", "(): $-&3\\beta ^3\\langle {S[u,\\epsilon ]}\\rangle \\langle {S[u,\\epsilon ]^2}\\rangle = \\frac{3\\beta ^3V_{\\lbrace 1\\rbrace }^4}{2N^3}\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }\\big [V_{\\lbrace 2\\rbrace }^4\\operatorname{Re}\\chi _{\\lbrace 2\\rbrace }+V_{\\lbrace 1,1\\rbrace }^4\\operatorname{Re}\\chi _{\\lbrace 1,1\\rbrace }+V_{\\lbrace 1,-1\\rbrace }^4\\chi _{\\lbrace 1,-1\\rbrace }+1\\big ] \\nonumber \\\\&= \\frac{3\\beta ^3 V_{\\lbrace 1\\rbrace }^4}{4N^3}\\big [(V_{\\lbrace 1,1\\rbrace }^4+2 V_{\\lbrace 1,-1\\rbrace }^4+V_{\\lbrace 2\\rbrace }^4+2)\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }+(V_{\\lbrace 1,1\\rbrace }^4+2 V_{\\lbrace 1,-1\\rbrace }^4)\\operatorname{Re}\\chi _{\\lbrace 1,1,-1\\rbrace }\\nonumber \\\\&\\hspace{62.59596pt}+(V_{\\lbrace 1,1\\rbrace }^4+V_{\\lbrace 2\\rbrace }^4)\\operatorname{Re}\\chi _{\\lbrace 2,1\\rbrace }+V_{\\lbrace 1,1\\rbrace }^4 \\operatorname{Re}\\chi _{\\lbrace 1,1,1\\rbrace }+(2V_{\\lbrace 1,-1\\rbrace }^4+V_{\\lbrace 2\\rbrace }^4)\\operatorname{Re}\\chi _{\\lbrace 2,-1\\rbrace }+V_{\\lbrace 2\\rbrace }^4 \\operatorname{Re}\\chi _{\\lbrace 3\\rbrace }\\big ]\\, , $ where the second line was simplified with the identities from Appendix .", "The final term in Eq.", "() follows from another identity: $2&\\beta ^3\\langle {S[u,\\epsilon ]}\\rangle ^3 = -2 \\frac{\\beta ^3V_{\\lbrace 1\\rbrace }^{12}}{N^3}(\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace })^3 \\nonumber \\\\&\\quad = -\\frac{\\beta ^3V_{\\lbrace 1\\rbrace }^{12}}{2N^3}\\big (\\operatorname{Re}\\chi _{\\lbrace 3\\rbrace }+2\\operatorname{Re}\\chi _{\\lbrace 2,1\\rbrace }+\\operatorname{Re}\\chi _{\\lbrace 1,1,1\\rbrace }+6\\operatorname{Re}\\chi _{\\lbrace 1\\rbrace }+3\\operatorname{Re}\\chi _{\\lbrace 2,-1\\rbrace }+3\\operatorname{Re}\\chi _{\\lbrace 1,1,-1\\rbrace }\\big ).$" ] ]

[ [ "Dp-finite fields VI: the dp-finite Shelah conjecture" ], [ "Abstract We prove the dp-finite case of the Shelah conjecture on NIP fields.", "If K is a dp-finite field, then K admits a non-trivial definable henselian valuation ring, unless K is finite, real closed, or algebraically closed.", "As a consequence, the conjectural classification of dp-finite fields holds.", "Additionally, dp-finite valued fields are henselian.", "Lastly, if K is an unstable dp-finite expansion of a field, then K admits a unique definable V-topology." ], [ "Introduction", "A structure is dp-finite if it has finite dp-rank.", "We prove the following facts about dp-finite fields and valued fields: Theorem 1.1 (dp-finite henselianity conjecture) Let $(K,v)$ be a dp-finite valued field.", "Then $v$ is henselian.", "Theorem 1.2 (dp-finite Shelah conjecture) Let $K$ be a dp-finite field.", "Then one of the following holds: $K$ is finite.", "$K$ is algebraically closed.", "$K$ is real closed.", "$K$ admits a definable non-trivial henselian valuation.", "By Theorems 3.3 and 3.11 of [4], this implies a classification of dp-finite fields up to elementary equivalence: Theorem 1.3 A field $K$ is dp-finite if and only if there is a henselian defectless valuation $v$ on $K$ such that The residue field $Kv$ is elementarily equivalent to $\\mathbb {F}_p^{alg}$ or a local field of characteristic 0, i.e., a finite extension of $\\mathbb {R}$ or $\\mathbb {Q}_p$ .", "The value group $vK$ is dp-finite as an ordered abelian group.", "If $\\operatorname{char}(K) = p$ , then the value group $vK$ is $p$ -divisible.", "If $(K,v)$ has mixed characteristic $(0,p)$ , then $[-v(p),v(p)] \\subseteq p \\cdot vK$ .", "Moreover, when the above conditions hold, the theory of $K$ is determined by the theory of $Kv$ and the theory of $vK$ (with $v(p)$ named as a constant in the mixed characteristic case).", "Remark 1.4   Dp-finite ordered abelian groups are classified in [1], [2], [3].", "Theorems 3.3 and 3.11 of [4] are phrased for strongly dependent fields, but the proof applies to dp-finite fields as well.", "All strongly dependent fields are conjectured to be dp-finite ([4], Proposition 3.9).", "The valuation $v$ is not uniquely determined by $K$ , and the map from $(\\operatorname{Th}(Kv),\\operatorname{Th}(vK))$ to $\\operatorname{Th}(K)$ is many-to-one, rather than a bijection.", "For example, if $(K,v_0) \\models \\operatorname{ACVF}$ , we could take $v$ to be either $v_0$ or the trivial valuation.", "In the first case, the value group $vK$ would be a divisible ordered abelian group; in the second case, it would be trivial.", "Theorem 1.5 Let $(K,+,\\cdot ,\\ldots )$ be an unstable dp-finite field, possibly with extra structure.", "Then $K$ admits a unique definable V-topology.", "This implies the dp-finite Shelah and henselianity conjectures, by ([8], Proposition 6.4)." ], [ "Reduction to W-topologies", "In [6], [7] we defined a “canonical topology” on any unstable dp-finite field, and proved Fact 1.6 The canonical topology is a field topology.", "If it is V-topological, then it is the unique definable V-topology.", "However, in [8], we gave an example of an expanded field $(K,+,\\cdot ,\\ldots )$ of dp-rank 2, in which the canonical topology was not a V-topology.", "In [9] we introduced a class of “finite weight” topological fields, or W-topological fields.", "Field topologies of weight 1 are exactly V-topologies.", "We proved the following: Fact 1.7   The canonical topology on an unstable dp-finite field $K$ is a definable field topology of weight at most $\\operatorname{dp-rk}(K)$ .", "If $\\tau $ is a field topology of finite weight (a W-topology), then there is at least one V-topological coarsening, i.e., a V-topology $\\sigma $ that is coarser than $\\tau $ .", "If $\\tau $ is a definable W-topology, then the V-topological coarsenings of $\\tau $ are all definable.", "If $\\tau $ is the canonical topology of an unstable dp-finite field $K$ , then the V-topological coarsenings of $\\tau $ are exactly the definable V-topologies on $K$ .", "As a corollary, we obtained the existence part of Theorem REF .", "Moreover, the uniqueness of the definable V-topology was already known in characteristic $p > 0$ , by ([6], Lemma 2.6 and [5], Proposition 3.5).", "So it remains to prove uniqueness in characteristic 0.", "In [7] (Proposition 5.17(4)), we used multiplicative infinitesimals to prove: Fact 1.8 Let $K$ be an unstable dp-finite field, viewed as a topological field via the canonical topology.", "For every neighborhood $U \\ni 1$ , the set $U^2 = \\lbrace x^2 : x \\in U\\rbrace $ is a neighborhood of 1.", "Equivalently, the squaring map $K^\\times \\rightarrow K^\\times $ is an open map.", "Thus, everything reduces to the following statement purely about W-topologies: Lemma 1.9 (= Corollary REF ) Let $(K,\\tau )$ be a W-topological field of characteristic 0.", "If the squaring map $K^\\times \\rightarrow K^\\times $ is an open map, then $\\tau $ has a unique V-topological coarsening.", "This in turn comes from the following decomposition theorem: Theorem 1.10 (= Theorems REF and REF ) Let $(K,\\tau )$ be a W-topological field.", "Then there exist W-topological coarsenings $\\tau _1, \\ldots , \\tau _n$ such that The $\\tau _i$ are jointly independent and generate $\\tau $ .", "In other words, the diagonal embedding $ (K,\\tau ) \\hookrightarrow (K,\\tau _1) \\times \\cdots \\times (K,\\tau _n)$ is a homeomorphism onto its image, and the image is dense.", "Each $\\tau _i$ has a unique V-topological coarsening, and this establishes a bijection between $\\lbrace \\tau _1,\\ldots ,\\tau _n\\rbrace $ and the set of V-topological coarsenings of $\\tau $ .", "We discuss the strategy for proving Theorem REF in §REF , after introducing some machinery in §." ], [ "Conventions", "This paper is a continuation of [9], and we use its notions of “weight,” and $W_n$ -sets, -rings, and -topologies.", "A W-topology is a topology of finite weight, i.e., a $W_n$ -topology for some $n$ .", "All rings will be commutative and unital.", "If $R$ is a ring, then $R^\\times $ will denote the group of units of $R$ , and $\\operatorname{Jac}(R)$ will denote the Jacobson radical.", "If $R$ is an integral domain, then $\\operatorname{Frac}(R)$ will denote its field of fractions, and $\\widetilde{R}$ will denote the integral closure of $R$ in $\\operatorname{Frac}(R)$ .", "We will tend to use the letter $\\mathcal {O}$ for valuation rings, and $\\mathfrak {m}$ for maximal ideals of valuation rings.", "In a list, $\\ldots ,\\widehat{x},\\ldots $ means “omit $x$ from the list.” If $R$ is a ring and $x_1, \\ldots , x_n$ are elements of an $R$ -module $M$ , then the $x_i$ are $R$ -independent if no $x_i$ lies in the $R$ -module generated by $\\lbrace x_1,\\ldots ,\\widehat{x_i},\\ldots ,x_n\\rbrace $ .", "Often $M = R$ or $M =\\operatorname{Frac}(R)$ .", "The “cube-rank” of ([9], §2.1) is the maximum length of an $R$ -independent sequence in $M$ .", "If $K$ is a field and $A, B \\subseteq K$ , we will say that $A$ is embeddable into $B$ if there is $c \\in K^\\times $ such that $c\\cdot A \\subseteq B$ .", "We say that $A$ and $B$ are co-embeddable if $A$ is embeddable into $B$ and $B$ is embeddable into $A$ .", "All topologies will be Hausdorff non-discrete locally bounded ring topologies on fields.", "We think of a topology as the filter of neighborhoods of 0, rather than the set of open sets.", "If $\\tau $ is a topology, then $\\tau ^\\perp $ will denote the ideal of bounded sets.", "By assumption, $\\tau $ always intersects $\\tau ^\\perp $ .", "We will say that $U$ “defines” or “induces” a topology $\\tau $ if $U$ is a locally bounded neighborhood in $\\tau $ , i.e., $U \\in \\tau \\cap \\tau ^\\perp $ .", "In this case, $\\lbrace c U : c \\in K^\\times \\rbrace $ is a filter basis for $\\tau $ , as well as an ideal basis for $\\tau ^\\perp $ (Lemma 2.1(e) of [10]).", "If $R$ is a proper subring of $K$ and $\\operatorname{Frac}(R) = K$ , then $R$ induces a topology $\\tau _R$ .", "The non-zero ideals of $R$ form a neighborhood basis, but we will predominantly use the basis $\\lbrace c R : c \\in K^\\times \\rbrace $ .", "The terms “local class,” “local sentence,” and “local equivalence” will be used as in [10].", "In particular, a local class is a class of topological fields defined by a set of local sentences, and two topological fields are locally equivalent if they satisfy the same local sentences.", "Local sentences allow two types of variables: Lower-case variables $a,b,c,\\ldots ,x,y,z$ , which range over the field sort.", "Upper-case variables $U, V, W, \\ldots $ , which range over $\\tau $ .", "Quantification over $\\tau $ is limited: Universal quantification $\\forall U : \\phi (U)$ is allowed only if $U$ occurs positively in $\\phi (U)$ .", "Existential quantification $\\exists U : \\phi (U)$ is allowed only if $U$ occurs negatively in $\\phi (U)$ .", "Because of this constraint, local sentences can be evaluated on a filter basis for $\\tau $ : if $\\tau _0$ is a filter basis for $\\tau $ , and $\\psi $ is a local sentence, then $ (K,\\tau ) \\models \\psi \\iff (K,\\tau _0) \\models \\psi .$ Let $K$ be a field.", "Consider the expansion of $(K,+,\\cdot )$ by all unary predicates.", "Let $\\operatorname{All}_K$ be the theory of the resulting object.", "Henceforth, an “ultrapower” of $K$ will mean a monster model of $\\operatorname{All}_K$ .", "If $K^*$ is an “ultrapower” of $K$ , and $U \\subseteq K$ , then $U^*$ will denote the corresponding subset of $K^*$ .", "In the structure $K^*$ , the $K$ -definable subsets of $K^*$ are exactly the sets $U^*$ .", "A $\\vee $ -definable set will be a complement of a type-definable set.", "Type-definable and $\\vee $ -definable sets will always be defined over small subsets of the “ultrapower.”" ], [ "Topologies and $\\vee $ -definable rings", "Fix a field $K$ and an “ultrapower” $K^*$ .", "We review the (easy) dictionary between locally bounded ring topologies on $K$ and certain $\\vee $ -definable subrings of $K^*$ .", "Proposition 2.1 For every topology $\\tau $ on $K$ , there is a ring $R = R_\\tau \\subseteq K^*$ such that $R$ is a filtered union $\\bigcup _{B \\in \\tau ^\\perp } B^*$ , i.e., the union of $B^*$ as $B$ ranges over $\\tau $ -bounded subsets of $K$ .", "$R$ is $\\vee $ -definable over $K$ .", "$R$ is a proper $K$ -subalgebra of $K^*$ .", "If $\\tau $ and $\\tau ^{\\prime }$ are two topologies, then $\\tau $ is coarser than $\\tau ^{\\prime }$ if and only if $R_{\\tau } \\supseteq R_{\\tau ^{\\prime }}$ .", "Define $R$ as in point REF .", "Then $R$ is trivially $\\vee $ -definable over $K$ .", "For the other points, we use Lemma 2.1 in [10].", "If $B_1, B_2$ are two bounded sets, then $B_1 \\cup B_2$ is bounded.", "Therefore the union is filtered.", "Any finite subset of $K$ is bounded.", "Therefore $K \\subseteq R$ .", "If $B_1, B_2$ are bounded, then $B_1 - B_2$ and $B_1 \\cdot B_2$ are bounded.", "Therefore $R$ is a $K$ -subalgebra of $K^*$ .", "The set $K$ is not itself bounded.", "By saturation of $K^*$ , it follows that $R \\ne K^*$ .", "Lastly, for point REF we prove the following chain of equivalent statements: $\\tau $ is coarser than $\\tau ^{\\prime }$ , i.e., every $\\tau $ -open is a $\\tau ^{\\prime }$ -open.", "$\\tau \\subseteq \\tau ^{\\prime }$ , i.e., every $\\tau $ -neighborhood of 0 is a $\\tau ^{\\prime }$ -neighborhood of 0.", "$(\\tau ^{\\prime })^\\perp \\subseteq \\tau ^\\perp $ , i.e., every $\\tau ^{\\prime }$ -bounded set is $\\tau $ -bounded.", "$R_{\\tau ^{\\prime }} \\subseteq R_\\tau $ .", "The equivalence (REF )$\\iff $ (REF ) is easy.", "The equivalence (REF )$\\iff $ (REF ) holds because $\\tau $ and $\\tau ^\\perp $ determine each other: $B \\in \\tau ^\\perp $ if and only if $B$ is embeddable into every $U \\in \\tau $ , in the sense that $\\exists c \\in K^\\times : cB\\subseteq U$ .", "This is Lemma 2.1(d) in [10].", "$U \\in \\tau $ if and only if every $B \\in \\tau ^\\perp $ is embeddable into $U$ .", "This holds by local boundedness of the topology.", "Finally, the equivalence (REF )$\\iff $ (REF ) follows by compactness (and the fact that the unions are filtered).", "So $R_\\tau $ is the ring of “$K$ -bounded” elements in $K^*$ .", "One could also define the group of “$K$ -infinitesimals” as the intersection $\\bigcap _{U \\in \\tau } U^*$ .", "However, the ring of $K$ -bounded elements is more useful for our purposes.", "Proposition 2.2 For every topology $\\tau $ on $K$ , there is a topology $\\tau ^*$ on $K^*$ such that $\\tau ^*$ is defined by the ring $R = R_\\tau $ of Proposition REF .", "If $U$ is any set defining $\\tau $ , then $U^*$ defines $\\tau ^*$ .", "If $\\tau _1, \\ldots , \\tau _n$ are topologies on $K$ , and $\\tau _1^*, \\ldots , \\tau _n^*$ are the corresponding topologies on $K^*$ , then there is a “local equivalence” in the sense of Prestel and Ziegler: $(K^*,\\tau _1^*,\\ldots ,\\tau _n^*) \\equiv (K,\\tau _1,\\ldots ,\\tau _n).$ Fix any $U$ defining $\\tau $ .", "Then $U$ is a bounded neighborhood of 0, so $U^* \\subseteq R_\\tau $ .", "For every bounded set $B \\in \\tau ^\\perp $ , there is $c \\in K^\\times $ such that $cB \\subseteq U$ .", "By saturation, there is $c \\in (K^*)^\\times $ such that $cB^*\\subseteq U^*$ for all $B \\in \\tau ^\\perp $ .", "Thus $cR_\\tau \\subseteq U^*$ .", "So $R_\\tau $ and $U$ are co-embeddable.", "By the proof of Lemma 2.3 in [10], $U^*$ defines a topology $\\tau ^*$ on $K^*$ , with $(K^*,\\tau ^*) \\equiv (K,\\tau )$ .", "By co-embeddability of $R_\\tau $ and $U$ , the topology $\\tau ^*$ is also defined by $R_\\tau $ .", "In particular, $\\tau ^*$ is independent of the choice of $U$ .", "Now suppose $\\tau _1, \\ldots , \\tau _n$ are fixed topologies on $K$ .", "For each $i$ , choose $U_i \\subseteq K$ defining $\\tau _i$ .", "Then there is an elementary equivalence $ (K^*,+,\\cdot ,U_1^*,\\ldots ,U_n^*) \\equiv (K,+,\\cdot ,U_1,\\ldots ,U_n).$ By Corollary 2.4 in [10], this implies the desired local equivalence, as each $U_i^*$ defines $\\tau _i^*$ .", "Proposition 2.3 Let $\\tau $ be a topology on $K$ , and let $R_\\tau $ be the corresponding $\\vee $ -definable ring.", "$\\tau $ is a $W_n$ -topology if and only if $R_\\tau $ is a $W_n$ -ring.", "In particular, the weight of $\\tau $ equals the weight of $R_\\tau $ .", "(The weight may be infinite.)", "First suppose $\\tau $ is a $W_n$ -topology.", "By definition (Definition 3.3 in [9]), there is a bounded $W_n$ -set $U\\subseteq K$ .", "Then $U^*$ is a $W_n$ -set in $K^*$ , and $U^*\\subseteq R_\\tau $ , implying that $R_\\tau $ is a $W_n$ -set and a $W_n$ -ring.", "Conversely, if $R_\\tau $ is a $W_n$ -ring, then it defines a $W_n$ -topology by Proposition 3.6 in [9].", "Therefore, $\\tau ^*$ is a $W_n$ -topology.", "By the local equivalence $(K,\\tau )\\equiv (K^*,\\tau ^*)$ , it follows that $\\tau $ is a $W_n$ -topology.", "This proves the first point.", "Then we know that $\\operatorname{wt}(\\tau ) \\le n\\iff \\operatorname{wt}(R_\\tau ) \\le n$ for all $n \\in \\mathbb {N}$ , implying that $\\operatorname{wt}(\\tau ) = \\operatorname{wt}(R_\\tau )$ .", "Lemma 2.4 Let $R_1, R_2$ be two $K$ -subalgebras of $K^*$ , both $\\vee $ -definable over $K$ .", "If $R_1$ and $R_2$ are co-embeddable, then $R_1 = R_2$ .", "More generally, if $R_1$ is embeddable into $R_2$ , then $R_1 \\subseteq R_2$ .", "Each $R_i$ is a filtered union of its $K$ -definable sets.", "Suppose $cR_1 \\subseteq R_2$ for some $c \\in (K^*)^\\times $ .", "Then for every $K$ -definable subset $U \\subseteq R_1$ , we have $cU \\subseteq R_2$ .", "By saturation, there must be some $K$ -definable subset $V \\subseteq R_2$ such that $cU \\subseteq V$ .", "The scalar $c$ is from $(K^*)^\\times $ , but $U$ and $V$ are $K$ -definable, and $K \\preceq K^*$ .", "Therefore we can find $e \\in K^\\times $ such that $eU\\subseteq V$ .", "Then $U \\subseteq e^{-1}V \\subseteq e^{-1}R_2\\subseteq R_2$ , because $e^{-1} \\in K \\subseteq R_2$ .", "As $U$ was arbitrary, $R_1 \\subseteq R_2$ .", "Proposition 2.5 Let $R$ be a subring of $K^*$ that is $\\vee $ -definable over $K$ , and satisfies $K \\subseteq R \\subsetneq \\operatorname{Frac}(R) = K^*$ .", "$R$ is of the form $R_\\tau $ for some topology on $K$ if and only if $R$ is co-embeddable with a definable set.", "$R$ is of the form $R_\\tau $ for some $W_n$ -topology on $K$ if and only if $R$ is a $W_n$ -ring.", "$R$ is of the form $R_\\tau $ for some V-topology on $K$ if and only if $R$ is a valuation ring.", "If $R = R_\\tau $ , then $R$ is co-embeddable with $U^*$ for any $U$ defining $\\tau $ , by Proposition REF (REF -REF ).", "Conversely, suppose $R$ is co-embeddable with a definable set $D$ .", "Write $D$ as $\\phi (K^*;\\vec{b})$ for some formula $\\phi (x;\\vec{y})$ (in the language of $\\operatorname{All}_K$ ), and some parameters $\\vec{b}$ from $K^*$ .", "Let $S$ be the set of $\\vec{c}$ in $K^*$ satisfying the equivalent conditions $\\phi (K^*;\\vec{c})$ is co-embeddable with $D$ $\\phi (K^*;\\vec{c})$ is co-embeddable with $R$ .", "From the first characterization, $S$ is definable.", "From the second characterization, $S$ is $\\operatorname{Aut}(K^*/K)$ -invariant.", "Therefore $S$ is $K$ -definable, and we can find $\\vec{c} \\in S(K)$ .", "Let $U = \\phi (K;\\vec{c})$ .", "Then $U^* =\\phi (K^*;\\vec{c})$ , and $U^*$ is co-embeddable with $R$ .", "Let $\\tau _R$ denote the topology on $K^*$ induced by $R$ , as in Example 1.2 of [10].", "Then $U^*$ defines $\\tau _R$ , by co-embeddability.", "The fact that $U^*$ defines a locally bounded ring-topology on $K^*$ is expressed by a first-order sentence in the structure $(K^*,+,\\cdot ,U^*)$ .", "The same sentence holds in $(K,+,\\cdot ,U)$ , and so $U$ defines a topology $\\tau $ on $K$ .", "Let $R^{\\prime } = R_\\tau $ be the corresponding $\\vee $ -definable subring on $K^*$ .", "Then $R^{\\prime }$ is co-embeddable with $U^*$ , by Proposition REF .", "Therefore $R^{\\prime }$ is co-embeddable with $R$ .", "By Lemma REF , $R^{\\prime } = R_\\tau $ .", "If $\\tau $ is a $W_n$ -topology, then $R_\\tau $ is a $W_n$ -ring by Proposition REF .", "Conversely, suppose $R$ is a $W_n$ -ring.", "By assumption, $R$ is $\\vee $ -definable.", "By Proposition 4.1 in [9], $R$ is co-embeddable with a definable set.", "By the previous point, $R = R_\\tau $ for some topology $\\tau $ .", "By Proposition REF , the fact that $R$ is a $W_n$ -ring forces $\\tau $ to be a $W_n$ -topology.", "This is the $n=1$ case of the previous point.", "Lemma 2.6 If $\\tau $ is a V-topology on $K$ and $\\mathcal {O}= R_\\tau $ is the corresonding valuation ring, then the maximal ideal $\\mathfrak {m}\\subseteq \\mathcal {O}$ is the set of $K$ -infinitesimals, i.e., $\\mathfrak {m}= \\bigcap _{U \\in \\tau } U^*$ .", "Fix a bounded neighborhood $B \\in \\tau \\cap \\tau ^\\perp $ .", "Let $C =\\lbrace x \\in K : 1/x \\notin B\\rbrace $ .", "Then $C^{-1} \\cap B = \\emptyset $ and $C \\cup B^{-1} = K$ , so that $C$ is a bounded neighborhood of 0.", "(This follows by properties of V-topologies, such as the definition of V-topologies given in §3 of [10].)", "Then $\\mathcal {O}&= \\bigcup _{U \\in \\tau ^\\perp } U^* = \\bigcup _{a \\in K^\\times } aB^*.", "\\\\\\bigcap _{U \\in \\tau } U^* &= \\bigcap _{a \\in K^\\times } aC^* = \\bigcap _{a \\in K^\\times } a^{-1}C^*.$ Therefore, the following are equivalent for $x \\in K^*$ : $x \\in \\mathfrak {m}&\\iff 1/x \\notin \\mathcal {O}\\iff \\left(\\forall a \\in K^\\times : 1/x \\notin aB^*\\right) \\\\& \\iff \\left(\\forall a \\in K^\\times : 1/(ax) \\notin B^* \\right) \\iff \\left( \\forall a \\in K^\\times : ax \\in C^*\\right) \\\\& \\iff \\left( \\forall a \\in K^\\times : x \\in a^{-1}C^*\\right) \\iff x \\in \\bigcap _{a \\in K^\\times } a^{-1}C^* \\iff x \\in \\bigcap _{U \\in \\tau } U^*.", "$ Lemma 2.7 Let $X \\subseteq K^*$ be $\\vee $ -definable over a small set.", "If $X$ has finite orbit under $\\operatorname{Aut}(K^*/K)$ , then $X$ is $\\vee $ -definable over $K$ .", "This is somewhat well-known, but we include the proof for completeness.", "We first claim that $X$ is $\\operatorname{Aut}(K^*/K)$ -invariant ($K$ -invariant).", "Recall that if $\\approx $ is a $K$ -invariant equivalence relation with boundedly-many equivalence classes, and $a \\equiv _K b$ , then $a\\approx b$ , because $K$ is a model.This is a well-known fact about Lascar strong type, and is easy to prove by using a global coheir of $\\operatorname{tp}(a/K)$ to build a sequence $c_1, c_2, \\ldots $ such that both $a,c_1,c_2,\\ldots $ and $b,c_1,c_2,\\ldots $ are $K$ -indiscernible.", "Let $a \\approx b$ indicate that $a \\in \\sigma (X) \\iff b \\in \\sigma (X)$ for all $\\sigma \\in \\operatorname{Aut}(K^*/K)$ .", "Then $\\approx $ is $K$ -invariant, with finitely many equivalence classes.", "If $X$ itself fails to be $K$ -invariant, then there are $a$ and $b$ such that $a \\equiv _K b$ but $a \\in X$ and $b \\notin X$ .", "Then $a \\lnot \\approx b$ , a contradiction.", "Thus, $X$ is $K$ -invariant.", "Now, any $\\vee $ -definable $K$ -invariant set is $\\vee $ -definable over $K$ .An equivalent, better-known statement is that a type-definable $K$ -invariant set is type-definable over $K$ .", "Indeed, if $X$ is $\\vee $ -definable over a small parameter set $B \\supseteq K$ , then $X$ corresponds to some open set $U$ in the space $S_n(B)$ of $n$ -types over $B$ .", "The $K$ -invariance means that $U$ is the preimage of some set $U^{\\prime }\\subseteq S_n(K)$ under the continuous surjection $S_n(B)\\twoheadrightarrow S_n(K)$ .", "The complement $S_n(K) \\setminus U^{\\prime }$ is the image of the closed set $S_n(B) \\setminus U$ , so $S_n(K) \\setminus U^{\\prime }$ is closed, $U^{\\prime }$ is open, and $X$ is $\\vee $ -definable over $K$ .", "Proposition 2.8 Let $\\tau $ be a W-topology on $K$ , and let $R = R_\\tau $ be the corresponding subring of $K^*$ .", "There is a unique W-topology $\\widetilde{\\tau }$ such that the corresponding ring $R_{\\widetilde{\\tau }}$ is the integral closure of $R$ .", "There are W-topologies $\\tau _1, \\ldots , \\tau _n$ such that the corresponding rings $R_{\\tau _1}, \\ldots , R_{\\tau _n}$ are exactly the localizations of $R$ at its maximal ideals.", "By Propositions 4.7 and 4.8 in [9], the integral closure and the localizations are $\\vee $ -definable.", "The integral closure is clearly $K$ -invariant, and the localizations can at most be permuted by $\\operatorname{Aut}(K^*/K)$ .", "By Lemma REF , the localizations and the integral closure are $\\vee $ -definable over $K$ .", "They are larger than $R$ , so they contain $K$ and are $K$ -algebras.", "By Lemma 2.7 in [9], they are rings of finite weight.", "By Proposition REF , they come from W-topologies on $K$ .", "Definition 2.9 Let $\\tau $ be a W-topology on $K$ .", "The integral closure of $\\tau $ is the topology $\\widetilde{\\tau }$ of Proposition REF .", "The local components of $\\tau $ are the topologies $\\tau _1, \\ldots , \\tau _n$ of Proposition REF .", "Recall from ([9], Proposition 2.12) that if $R$ is a $W_n$ -ring, then the integral closure $\\widetilde{R}$ is a multi-valuation ring, a finite intersection of valuation rings.", "$ \\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_n.$ If the $\\mathcal {O}_i$ are chosen to be pairwise incomparable, then the $\\mathcal {O}_i$ are exactly the localizations of $\\widetilde{R}$ at its maximal ideals ([7], Corollary 6.7).", "Proposition 2.10 Let $\\tau $ be a W-topology on $K$ .", "Then the V-topological coarsenings of $\\tau $ are exactly the local components of the integral closure.", "In other words, if $R \\subseteq K^*$ is the corresponding $\\vee $ -definable ring, and we write $\\widetilde{R}$ as an intersection of pairwise incomparable valuation rings $ \\widetilde{R} = \\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_n,$ then the $\\mathcal {O}_i$ are exactly the $\\vee $ -definable rings corresponding to the V-topological coarsenings of $\\tau $ .", "By Proposition REF and Proposition REF (REF ), the V-topological coarsenings of $\\tau $ correspond exactly to the valuation rings $\\mathcal {O}$ on $K$ with the following properties: $\\mathcal {O}$ is non-trivial, i.e., $\\mathcal {O}\\ne K^*$ .", "$\\mathcal {O}$ is $\\vee $ -definable over $K$ .", "$\\mathcal {O}$ contains $R$ .", "As in the proof of Proposition REF , the $\\mathcal {O}_i$ certainly have these properties.", "Let $\\mathcal {O}$ be some other valuation ring with these properties.", "Then $\\mathcal {O}\\supseteq \\widetilde{R} = \\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_n$ .", "By Corollary 6.8 in [7], there is some $i$ such that $\\mathcal {O}\\supseteq \\mathcal {O}_i$ .", "Then $\\mathcal {O}$ is a coarsening of $\\mathcal {O}_i$ .", "By non-triviality, $\\mathcal {O}$ and $\\mathcal {O}_i$ induce the same topology, so they are co-embeddable.", "By Lemma REF , $\\mathcal {O}=\\mathcal {O}_i$ ." ], [ "Local W-topologies", "Definition 2.11 A $W_n$ -topology on $K$ is local if for every bounded set $B\\subseteq K$ , there is a bounded set $C \\subseteq K$ such that $ \\forall x \\in B : (1/x \\in C \\text{ or } 1/(1-x) \\in C).$ Proposition 2.12 Let $K^*$ be an “ultrapower” of $K$ .", "Let $\\tau $ be a W-topology on $K$ , and let $R$ be the associated ring in $K^*$ .", "Then $\\tau $ is local if and only if $R$ is a local ring.", "Let $\\gamma (X,Y)$ stand for $ \\forall x \\in X : 1/x \\in Y \\text{ or } 1/(1-x) \\in Y.$ Then the following statements are equivalent: $R$ is a local ring.", "For every $x \\in R$ , at least one of $x$ or $1 -x$ is in $R^\\times $ .", "$\\gamma (R,R)$ .", "For every bounded $B \\subseteq K$ , we have $\\gamma (B^*,R)$ .", "For every bounded $B \\subseteq K$ , there is bounded $C \\subseteq K$ such that $\\gamma (B^*,C^*)$ .", "For every bounded $B \\subseteq K$ , there is bounded $C \\subseteq K$ such that $\\gamma (B,C)$ .", "$\\tau $ is a local W-topology.", "The equivalences are proven as follows: (REF )$\\iff $ (REF ): well-known commutative algebra.", "(REF )$\\iff $ (REF ): the definition of $\\gamma (-,-)$ .", "(REF )$\\iff $ (REF ): $R$ is covered by the $B^*$ .", "(REF )$\\iff $ (REF ): $R$ is a filtered union of the $C^*$ , and the structure is saturated.", "(REF )$\\iff $ (REF ): $K \\preceq K^*$ .", "(REF )$\\iff $ (REF ): the definition of “local W-topology.” For example, if $\\tau $ is a W-topology, then the local components of $\\tau $ (Definition REF ) are local W-topologies.", "Warning 2.13 A local $W_n$ -ring need not induce a local $W_n$ -topology.", "For example, let $\\mathcal {O}_1$ and $\\mathcal {O}_2$ be the two valuation rings $\\mathbb {Z}[i]_{(2+i)}$ and $\\mathbb {Z}[i]_{(2-i)}$ on the field $K = \\mathbb {Q}(i)$ .", "The residue fields are both isomorphic to $\\mathbb {Z}/(5)$ .", "Let $R$ be the set of $x \\in \\mathcal {O}_1 \\cap \\mathcal {O}_2$ such that $\\operatorname{res}_1(x) = \\operatorname{res}_2(x)$ .", "Then $R$ is a $W_2$ -ring, because it contains the valuation ring $\\mathbb {Z}_{(5)}$ on the subfield $\\mathbb {Q}$ , and $[\\mathbb {Q}(i) : \\mathbb {Q}] = 2$ .", "(See Lemma 2.7 in [9].)", "Additionally, $R$ is a local ring—the maximal ideal is the set of $x$ such that $\\operatorname{res}(x) = 0$ .", "However, $R$ is co-embeddable with $\\mathcal {O}_1 \\cap \\mathcal {O}_2$ , and induces a $V^2$ -topology, which is not local.", "Conversely, a non-local $W_n$ -ring can induce a local $W_n$ -topology.", "For example, if $\\mathcal {O}_1$ and $\\mathcal {O}_2$ are two incomparable but dependent valuation rings, then $\\mathcal {O}_1 \\cap \\mathcal {O}_2$ is a non-local $W_2$ -ring which induces the same V-topology as $\\mathcal {O}_1 \\cdot \\mathcal {O}_2$ .", "V-topologies are local.", "We shall need the fact that local $W_n$ -topologies form a local class, in the sense of [10].", "The following lemma helps translate statements about bounded sets into statements about neighborhoods.", "Lemma 2.14 Let $(K,\\tau )$ be a field with a locally bounded ring topology.", "As usual $\\tau $ denotes the filter of neighborhoods of 0 and $\\tau ^\\perp $ denotes the ideal of bounded sets.", "Let $\\phi (X)$ be a formula in $X \\subseteq K$ , depending positively on $X$ .", "Then $ \\exists B \\in \\tau ^\\perp : \\phi (B)$ is equivalent to $\\forall U \\in \\tau ~ \\exists c \\in K^\\times : \\phi (cU).", "$ Dually, if $\\phi (X)$ is a formula in which $X$ appears negatively, then $ \\forall B \\in \\tau ^\\perp : \\phi (B)$ is equivalent to $ \\exists U \\in \\tau ~ \\forall c \\in K^\\times : \\phi (cU).$ We prove the first claim; the other follows formally.", "Suppose there is a bounded set $B$ such that $\\phi (B)$ holds.", "By definition of “bounded,” for any $U \\in \\tau $ there is $e \\in K^\\times $ such that $eB \\subseteq U$ .", "Then $e^{-1} U \\supseteq B$ , and so $\\phi (e^{-1}U)$ is true.", "This proves (REF ).", "Conversely, suppose (REF ) holds.", "By local boundedness, there is $U \\in \\tau $ such that $U$ is bounded.", "By (REF ), there is $c \\in K^\\times $ such that $\\phi (cU)$ holds.", "Then $B = cU$ is bounded, and $\\phi (B)$ holds.", "Proposition 2.15 The class of local $W_n$ -topologies is cut out by a local sentence, and is therefore a local class.", "The class of $W_n$ -topologies is defined by a local sentence ([9], Remark 3.4).", "As in the proof of Proposition REF , let $\\gamma (X,Y)$ be the formula $ \\forall x \\in X : (1/x \\in Y \\text{ or } 1/(1-x) \\in Y).$ This formula is negative in $X$ and positive in $Y$ .", "Then $(K,\\tau )$ is local if and only if $ \\forall B \\in \\tau ^\\perp ~ \\exists C \\in \\tau ^\\perp : \\gamma (B,C).$ By two applications of Lemma REF , this is equivalent to $\\exists U \\in \\tau ~ \\forall c \\in K^\\times ~ \\forall V \\in \\tau ~\\exists e \\in K^\\times : \\gamma (cU,eV).$ This is a local sentence.", "In detail, it is $& \\exists U \\in \\tau ~ \\forall c \\in K^\\times ~ \\forall V \\in \\tau ~\\exists e \\in K^\\times ~ \\forall x : \\\\ & (x/c \\in U) \\rightarrow (1/(xe) \\in V \\text{ or } 1/((1-x)e) \\in V).", "$ Remark 2.16 Suppose one is only interested in locally bounded ring topologies.", "Because of Lemma REF , one could extend Prestel and Ziegler's notion of “local sentence” to allow quantification over bounded sets, i.e., over $\\tau ^\\perp $ , subject to the constraints: Universal quantification $\\forall B \\in \\tau ^\\perp : \\phi (B)$ is allowed only if $B$ occurs negatively in $\\phi (B)$ .", "Existential quantification $\\exists B \\in \\tau ^\\perp : \\phi (B)$ is allowed only if $B$ occurs positively in $\\phi (B)$ .", "These are opposite to the constraints on quantification over neighborhoods." ], [ "Where we are going", "We give an outline of the remainder of the paper, in reverse order, working backwards from our goal.", "Let $\\tau $ be a W-topology on $K$ .", "Let $\\tau _1, \\ldots , \\tau _n$ be the local components of $\\tau $ .", "For reasons discussed in the introduction, we would like to prove the following two statements: The $\\tau _i$ are jointly independent, and generate $\\tau $ .", "Each $\\tau _i$ has a unique V-topological coarsening, and this establishes a bijection between $\\lbrace \\tau _1,\\ldots ,\\tau _n\\rbrace $ and the set of V-topological coarsenings.", "One can reduce (REF ) to (REF ) using the independence criterion of ([9], Theorem 7.16), more or less.", "As for (REF ), it translates into a statement about rings.", "Fix an “ultrapower” $K^*$ , and let $R$ be the ring corresponding to $\\tau $ .", "Let $\\widetilde{R}$ be its integral closure.", "Then (REF ) translates into For any maximal ideal $\\mathfrak {p}$ of $R$ , there is a unique maximal ideal $\\mathfrak {m}$ of $\\widetilde{R}$ such that $\\widetilde{R_\\mathfrak {p}} =\\widetilde{R}_\\mathfrak {m}$ .", "This establishes a bijection between the maximal ideals of $R$ and the maximal ideals of $\\widetilde{R}$ .", "By commutative algebra, we reduce to the case where $R$ and $\\tau $ are local: If $\\tau $ is local, then $\\tau $ has a unique V-topological coarsening.", "If $R$ is local, then $\\widetilde{R}$ is a valuation ring.", "This cannot be proven purely by commutative algebra—the $W_2$ -ring of Warning REF (REF ) is local, but its integral closure is not.", "We must use the fact that $R$ is a $\\vee $ -definable ring induced by a topology.", "Assume henceforth that $R$ and $\\tau $ are local.", "By pushing the commutative algebra a little further, (REF ) reduces to...", "If $\\mathcal {O}_1, \\ldots , \\mathcal {O}_n$ are the incomparable valuation rings whose intersection is $\\widetilde{R}$ , and $\\mathfrak {m}_i$ is the maximal ideal of $\\mathcal {O}_i$ , then $ \\mathfrak {m}_i \\cap R \\subseteq \\mathfrak {m}_j \\cap R$ for $i \\ne j$ .", "Translated back into topology, this says the following If $\\tau _1$ and $\\tau _2$ are two distinct V-topological coarsenings of $\\tau $ , then the following can't happen: $\\tau _1$ induces a finer topology than $\\tau _2$ on each $\\tau $ -bounded set.", "This makes intuitive sense, as $\\tau _1$ and $\\tau _2$ are independent V-topologies.", "To make this intuition precise, we need $\\tau $ -bounded sets to be big enough: In the topology on $K^*$ induced by $\\widetilde{R}$ , the closure of $R$ includes the Jacboson radical $\\operatorname{Jac}(\\widetilde{R}) = \\mathfrak {m}_1\\cap \\cdots \\cap \\mathfrak {m}_n$ .", "In the case of DV-topologies, there was a trick to prove that $R$ is dense in its integral closure $\\mathcal {O}$ ; see §5.4 of [8].", "A variant of that method works in our setting, but we need the following configuration: For any non-zero $a$ there are $y_1, \\ldots , y_n$ such that For each $i$ , $ay_i$ is not in the $R$ -module generated by $\\lbrace y_1,\\ldots ,\\widehat{y_i},\\ldots ,y_n\\rbrace $ .", "For fixed $i$ , the elements $y_1, \\ldots , y_n$ all have the same valuation with respect to $\\mathcal {O}_i$ .", "Ignoring the factor of $a$ , we need a sequence $y_1, \\ldots , y_n$ which is $R$ -independent in the sense of §REF , but which is very far from being $\\widetilde{R}$ -independent—in fact $y_i \\in \\widetilde{R} y_j$ for all $i, j$ .", "Such a sequence can be obtained by scrambling an $R$ -independent sequence by a random matrix from $GL_n(\\mathbb {Q})$ : Let $R$ be a $W_n$ -ring, extending $\\mathbb {Q}$ for simplicity.", "Let $x_1, \\ldots , x_n$ be an $R$ -independent sequence in $R$ , and let $\\vec{y} = \\mu \\cdot \\vec{x}$ , where $\\mu $ is a “random” matrix from $GL_n(\\mathbb {Q})$ .", "If $R$ is local, then $\\vec{y}$ is an $R$ -independent sequence.", "If $R$ is a multi-valuation ring, then $y_i \\in R y_j$ for all $i, j$ .", "Thus, local $W_n$ -rings and multi-valuation rings behave differently, and this difference can be leveraged to prove (REF )." ], [ "Scrambling and density", "For the entirety of §, we will work over a fixed infinite field $K_0$ .", "All fields will extend $K_0$ , all rings will be $K_0$ -algebras, and all valuations will be trivial on $K_0$ ." ], [ "Scrambling", "Lemma 3.1 Let $(K,\\operatorname{val}_1,\\ldots ,\\operatorname{val}_m)$ be a multi-valued field.", "For any $z, w \\in K$ , there is $c \\in K_0$ such that for all $i$ , $\\operatorname{val}_i(z - cw) = \\min (\\operatorname{val}_i(z),\\operatorname{val}_i(w)).", "$ For each $i$ , let $\\operatorname{res}_i : K \\rightarrow k_i \\cup \\lbrace \\infty \\rbrace $ be the residue map of $\\operatorname{val}_i$ , extended by setting $\\operatorname{res}_i(x) = \\infty $ when $\\operatorname{val}_i(x) < 0$ .", "If $w = 0$ , then any $c$ works.", "Otherwise, take $c \\in K_0$ such that $c \\ne \\operatorname{res}_i(z/w)$ for all $i$ .", "This is possible since $K_0$ is infinite.", "If (REF ) fails, then by the strong triangle inequality $ \\operatorname{val}_i(z - cw) > \\operatorname{val}_i(z) = \\operatorname{val}_i(w).$ Then $\\operatorname{val}_i(z/w - c) = \\operatorname{val}_i(z - cw) - \\operatorname{val}_i(w) > 0$ .", "This implies $\\operatorname{res}_i(z/w) = \\operatorname{res}_i(c) = c$ , contradicting the choice of $c$ .", "Let $(K,\\operatorname{val}_1,\\ldots ,\\operatorname{val}_m)$ be a multi-valued field.", "Let $\\vec{x}$ be an $n$ -tuple in $K^n$ .", "Say that $\\vec{x}$ is scrambled if $\\operatorname{val}_1(x_1) & = \\operatorname{val}_1(x_2) = \\cdots = \\operatorname{val}_1(x_n) \\\\\\operatorname{val}_2(x_1) & = \\operatorname{val}_2(x_2) = \\cdots = \\operatorname{val}_2(x_n) \\\\\\cdots \\\\\\operatorname{val}_m(x_1) & = \\operatorname{val}_m(x_2) = \\cdots = \\operatorname{val}_m(x_n).$ If $\\mu \\in GL_n(K)$ , say that $\\mu $ scrambles $\\vec{x}$ if $\\mu \\cdot \\vec{x}$ is scrambled.", "Lemma 3.2 Let $(K,\\operatorname{val}_1,\\ldots ,\\operatorname{val}_m)$ be a multi-valued field.", "Then any $n$ -tuple $\\vec{x} \\in K^n$ can be scrambled by an element of $GL_n(K_0)$ .", "This follows by repeated applications of Lemma REF .", "In more detail, define the “discrepancy” of $\\vec{x}$ to be the size of the set $ \\lbrace (i,j) ~|~ \\exists k : \\operatorname{val}_i(x_j) > \\operatorname{val}_i(x_k)\\rbrace .$ We may assume that $\\vec{x}$ has minimal discrepancy in the coset $GL_n(K_0) \\cdot \\vec{x}$ .", "If $\\vec{x}$ has discrepancy 0, then it is scrambled and we are done.", "Otherwise, we may assume $ \\operatorname{val}_1(x_1) > \\operatorname{val}_1(x_2).$ By Lemma REF there is $c \\in K_0$ such that for all $i$ , $ \\operatorname{val}_i(x_1 - cx_2) = \\min (\\operatorname{val}_i(x_1),\\operatorname{val}_i(x_2)).$ Let $\\vec{y} = (x_1 - cx_2, x_2, x_3, \\ldots , x_n)$ .", "Then $\\vec{y}\\in GL_n(K_0) \\cdot \\vec{x}$ , and $\\vec{y}$ has lower discrepancy, a contradiction.", "Lemma 3.3 For every $n, m$ , there is a finite set $G_{n,m} \\subseteq GL_n(K_0)$ with the following property.", "Let $K$ be a field, and let $\\operatorname{val}_1, \\ldots , \\operatorname{val}_m$ be valuations on $K$ .", "Then every $n$ -tuple $\\vec{x} \\in K^n$ is scrambled by an element of $G_{n,m}$ .", "Lemma REF and compactness.", "Lemma 3.4 Let $R$ be a local integral domain.", "If $x_1, \\ldots , x_n \\in R$ is an $R$ -independent sequence in $\\operatorname{Frac}(R)$ , and $\\vec{y} \\in GL_n(K_0) \\cdot \\vec{x}$ , then $y_1, \\ldots , y_n$ is also $R$ -independent.", "Let $\\mathfrak {m}$ be the maximal ideal of $R$ , and $k$ be the residue field $R/\\mathfrak {m}$ .", "Let $M$ be the $R$ -submodule of $\\operatorname{Frac}(R)$ generated by the $x_i$ .", "Let $\\xi _i$ be the image of $x_i$ in the $k$ -vector space $M/\\mathfrak {m}M$ .", "The $\\xi _i$ generate $M/\\mathfrak {m}M$ , and are $R$ -independent by Nakayama's lemma.", "Therefore $M / \\mathfrak {m}M$ has dimension $n$ over $k$ .", "Write $\\vec{y}$ as $\\mu \\cdot \\vec{x}$ for some $\\mu \\in GL_n(K_0)$ .", "Then $y_1, \\ldots , y_n$ also generate $M$ .", "Let $\\lbrace z_1,\\ldots ,z_m\\rbrace $ be a minimal subset of $\\lbrace y_1,\\ldots ,y_n\\rbrace $ generating $M$ .", "Then $\\vec{z}$ is $R$ -independent.", "The Nakayama's lemma argument shows $m = \\dim _kM/\\mathfrak {m}M = n$ .", "Therefore $\\lbrace y_1,\\ldots ,y_n\\rbrace = \\lbrace z_1,\\ldots ,z_m\\rbrace $ , and the $\\vec{y}$ are $R$ -independent.", "Lemma 3.5 Let $R$ be an integral domain of finite weight.", "Suppose $R \\ne \\operatorname{Frac}(R) =: K$ , and the induced topology on $R$ is local of weight $n$ .", "Let $\\widetilde{R}$ be the integral closure of $R$ .", "Write $\\widetilde{R}$ as $\\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_m$ , where the $\\mathcal {O}_i$ are pairwise incomparable valuation rings.", "Let $\\operatorname{val}_i$ be the valuation associated to $i$ .", "Then for any non-zero $a \\in R$ , we can find $y_1, \\ldots , y_n \\in K$ such that The tuple $\\vec{y}$ is scrambled with respect to $(K,\\operatorname{val}_1,\\ldots ,\\operatorname{val}_n)$ , i.e., $\\operatorname{val}_i(y_j) = \\operatorname{val}_i(y_k)$ for all $i, j, k$ .", "For any $i$ , $ a \\cdot y_i \\notin R \\cdot y_1 + R \\cdot y_2 + \\cdots + \\widehat{R \\cdot y_i} + \\cdots + R \\cdot y_n.$ Let $K^*$ be an “ultrapower” of $K$ , and let $R^{\\prime }$ be the subring of $K^*$ induced by $\\tau _R$ .", "Note that $R^{\\prime }$ is a local ring of weight $n$ , and $R^{\\prime } \\supseteq R^*$ .", "Let $G_{n,m}$ be as in Lemma REF .", "Let $\\phi (x_1,\\ldots ,x_n)$ be the formula in $K^*$ expressing that for any $\\vec{y} \\in G_{n,m} \\cdot \\vec{x}$ , we have $ \\forall i : \\left(y_i \\notin \\sum _{j \\ne i} R^* \\cdot a^{-1} y_j\\right).$ Claim 3.6 Some tuple $\\vec{z} \\in (K^*)^n$ satisfies $\\phi (\\vec{x})$ .", "[] Because $R^{\\prime }$ has weight $n$ , there is an $R^{\\prime }$ -independent sequence $z_1,\\ldots ,z_n$ .", "We claim that $\\phi (\\vec{z})$ holds.", "If $\\vec{y} \\in G_{n,m} \\cdot \\vec{z} \\subseteq GL_n(K_0) \\cdot \\vec{z}$ , then $\\vec{y}$ is $R^{\\prime }$ -independent by Lemma REF .", "Therefore $ y_i \\notin \\sum _{j \\ne i} R^{\\prime } \\cdot y_j.$ But $R^* \\cup \\lbrace a^{-1}\\rbrace \\subseteq R^* \\cup K \\subseteq R^{\\prime }$ .", "Therefore $ y_i \\notin \\sum _{j \\ne i} R^* \\cdot a^{-1} \\cdot y_j.", "$ Now $\\phi (x_1,\\ldots ,x_n)$ is a formula over $K$ , so it is satisfied by some tuple $z_1,\\ldots ,z_n \\in K^n$ .", "By choice of $G_{n,m}$ in Lemma REF , there is at least one $\\vec{y} \\in G_{n,m} \\cdot \\vec{z}$ such that $\\vec{y}$ is scrambled.", "By definition of $\\phi $ , the vector $\\vec{y}$ has the desired properties." ], [ "Density in the Jacobson radical", "Lemma 3.7 Let $R$ be a local integral domain of weight $n$ , inducing a W-topology on $K = \\operatorname{Frac}(R)$ that is also local of weight $n$ .", "Let $\\widetilde{R}$ be the integral closure of $R$ .", "Then $R$ is dense in the Jacobson radical of $\\widetilde{R}$ , with respect to the topology induced by $\\widetilde{R}$ .", "Let $\\tau $ and $\\widetilde{\\tau }$ denote the topologies induced by $R$ and $\\widetilde{R}$ .", "Then $\\widetilde{\\tau }$ is coarser than $\\tau $ .", "Write $\\widetilde{R}$ as an intersection of pairwise incomparable valuation rings $\\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_m$ .", "The Jacobson radical is $\\mathfrak {m}_1 \\cap \\cdots \\cap \\mathfrak {m}_n$ , where $\\mathfrak {m}_i$ is the maximal ideal of $\\mathcal {O}_i$ .", "Let $x$ be an element of the Jacobson radical of $\\widetilde{R}$ .", "For any $U \\in \\widetilde{\\tau }$ , we must show that $x + U$ intersects $R$ .", "By taking $a$ small enough with respect to $\\tau $ , we can find non-zero $a$ such that $a \\in R$ $ax \\in R$ $a$ is in the Jacobson radical $\\mathfrak {m}_1 \\cap \\cdots \\cap \\mathfrak {m}_n$ .", "$a \\widetilde{R} \\subseteq U$ .", "Indeed, the first two requirements cut out $\\tau $ -neighborhoods of 0, the third and fourth requirements cut out $\\widetilde{\\tau }$ -neighborhoods of 0, and $\\tau $ is finer than $\\widetilde{\\tau }$ .", "By Lemma REF , there are $y_1, \\ldots , y_n \\in K$ such that For every $j \\le m$ , we have $\\operatorname{val}_j(y_1) = \\operatorname{val}_j(y_2) = \\cdots =\\operatorname{val}_j(y_n)$ .", "For every $i \\le n$ , we have $ a^2 y_i \\notin \\sum _{j \\ne i} R y_j.$ Scaling the $\\vec{y}$ 's, we may assume $y_1 = 1$ .", "Then $\\operatorname{val}_j(y_i)= 0$ for all $i, j$ .", "In particular, $y_i \\in \\widetilde{R}$ .", "Now $\\operatorname{wt}(R) < n + 1$ , so the set $\\lbrace 1,x,ay_2,ay_3,\\ldots ,ay_n\\rbrace $ cannot be $R$ -independent.", "One of three things happens: 1 is in the $R$ -module generated by $x, ay_2, ay_3, \\ldots ,ay_n$ .", "But this cannot happen, as the elements $x, ay_2, ay_3,\\ldots , ay_n$ have positive valuation with respect to any of the $\\operatorname{val}_i$ 's, and $R \\subseteq \\mathcal {O}_i$ .", "Say, $ay_2$ is in the $R$ -module generated by $1, x, ay_3,ay_4, \\ldots , ay_n$ .", "Then $ay_2 & \\in R + Rx + \\sum _{j = 3}^n R ay_j \\\\a^2 y_2 & \\in Ra + Rax + \\sum _{j = 3}^n R a^2 y_j.$ But $a, ax \\in R$ , and $a^2 \\in R$ , so that $ a^2 y_2 \\in Ra + Rax + \\sum _{j = 3}^n R a^2 y_j.", "\\subseteq R + R + \\sum _{j = 3}^n R y_j = R y_1 + \\sum _{j = 3}^n R y_j = \\sum _{j \\ne 2} R y_j,$ contradicting the choice of the $y$ 's.", "$x$ is in the $R$ -module generated by $1, ay_2, ay_3, \\ldots ,ay_n$ .", "Then there are $b, c_2, \\ldots , c_n \\in R$ such that $ x = b + ac_2y_2 + ac_3y_3 + \\cdots + ac_ny_n.$ But the $c_i \\in R \\subseteq \\widetilde{R}$ , and the $y_i \\in \\widetilde{R}$ , so we see $ b - x \\in a\\widetilde{R} \\subseteq U.$ Then $b \\in (x + U) \\cap R$ .", "Lemma 3.8 Let $R$ be a local integral domain of weight $n$ , inducing a W-topology on $K = \\operatorname{Frac}(R)$ that is also local of weight $n$ .", "Let $\\widetilde{R}$ be the integral closure of $R$ .", "Write $\\widetilde{R}$ as $\\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_m$ for incomparable valuation rings $\\mathcal {O}_i$ .", "Suppose the $\\mathcal {O}_i$ are pairwise independent.", "Then there is $a \\in K^\\times $ such that for every $b\\in K^\\times $ , $ R \\cap b\\mathcal {O}_2 \\lnot \\subseteq a\\mathcal {O}_1.$ Let $\\mathfrak {m}_i$ be the maximal ideal of $\\mathcal {O}_i$ .", "Then $\\bigcap _i \\mathfrak {m}_i$ is the Jacobson radical of $\\widetilde{R}$ .", "Take non-zero $c \\in \\bigcap _i \\mathfrak {m}_i$ .", "By the approximation theorem for V-topologies, we can find $a \\in K$ such that $\\operatorname{val}_1(a) &> \\operatorname{val}_1(c) \\\\\\operatorname{val}_i(a) &= \\operatorname{val}_i(c), \\text{ for } i > 1.$ Now, for every $b \\in K^\\times $ , there is $u \\in K^\\times $ such that $\\operatorname{val}_1(u) &= \\operatorname{val}_1(c) > 0 \\\\\\operatorname{val}_i(u) &> \\max (\\operatorname{val}_i(b),\\operatorname{val}_i(c)) > 0, \\text{ for } i > 1,$ by the approximation theorem for V-topologies.", "Then $u \\in \\bigcap _i \\mathfrak {m}_i$ .", "By Lemma REF , there are elements of $R$ arbitrarily close to $u$ with respect to the $\\widetilde{R}$ -topology.", "In particular, moving $u$ , we can take $u\\in R$ .", "Then $u \\in R$ and $u \\in b \\mathcal {O}_2$ , but $u \\notin a \\mathcal {O}_1$ , since $\\operatorname{val}_1(u) = \\operatorname{val}_1(c) < \\operatorname{val}_1(a)$ ." ], [ "The local case", "Lemma 4.1 Let $(K,\\tau )$ be a local topological field of weight $n$ .", "Let $\\tau _1, \\tau _2$ be two distinct V-topological coarsenings of $\\tau $ .", "Then for all $U \\in \\tau $ there is $V \\in \\tau _1$ such that for all $W \\in \\tau _2$ , $ U \\cap W \\lnot \\subseteq V.$ The desired condition can be expressed as a local sentence in $(K,\\tau ,\\tau _1,\\tau _2)$ .", "Let $K^*$ be an “ultrapower” of $K$ .", "Let $R, R_1, R_2$ be the rings induced by $\\tau , \\tau _1, \\tau _2$ , and let $\\tau ^*, \\tau _1^*, \\tau _2^*$ be the corresponding topologies on $K^*$ .", "Then $(K^*,\\tau ^*,\\tau _1^*,\\tau _2^*)$ is locally equivalent to $(K,\\tau ,\\tau _1,\\tau _2)$ , by Proposition REF .", "So we may work in $K^*$ instead.", "Then $\\tau ^*$ is induced by $R$ , and $\\tau ^*_i$ is induced by $R_i$ .", "Note that $R$ is a local ring of weight $n$ , inducing a local topology of weight $n$ .", "By Proposition REF , $R_i$ is one of the valuation rings whose intersection is the integral closure $\\widetilde{R}$ .", "We must show $ \\forall c \\ne 0 ~ \\exists a \\ne 0 ~ \\forall b \\ne 0 : cR \\cap bR_2 \\lnot \\subseteq aR_1.$ Up to rescaling, we may assume $c = 1$ , and then this is the content of Lemma REF (with $K$ and $K_0$ being $K^*$ and $K$ , respectively).", "Lemma 4.2 Let $(K,\\tau )$ be a local topological field of weight $n$ .", "Let $K^*$ be an “ultrapower” of $K$ .", "Let $R$ be the subring of $K^*$ induced by $\\tau $ .", "Let $\\widetilde{R}$ be the integral closure of $R$ , and let $\\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_n$ be its decomposition into incomparable valuation rings.", "Let $\\mathfrak {m}_i$ be the maximal ideal of $\\mathcal {O}_i$ .", "Then $ \\mathfrak {m}_i \\cap R \\lnot \\subseteq \\mathfrak {m}_j \\cap R,$ for $i \\ne j$ .", "Let $\\tau _i$ be the topology corresponding to $\\mathcal {O}_i$ .", "Let $U \\in \\tau $ be a bounded neighborhood.", "Let $i, j$ be given.", "By Lemma REF , there is $V \\in \\tau _j$ such that for all $W \\in \\tau _i$ , $ U \\cap W \\lnot \\subseteq V.$ By saturation, there is $\\epsilon \\in U^* \\cap \\bigcap _{W \\in \\tau _i} W^*$ with $\\epsilon \\notin V^*$ .", "Then $\\epsilon $ is a $\\tau _i$ -infinitesimal, so $\\epsilon \\in \\mathfrak {m}_1$ by Lemma REF .", "Additionally, $\\epsilon \\in U^*$ , so $\\epsilon $ is $\\tau $ -bounded, and $\\epsilon \\in R$ .", "On the other hand, $\\epsilon \\notin V^*$ , so $\\epsilon $ is not a $\\tau _j$ -infinitesimal, and $\\epsilon \\notin \\mathfrak {m}_j$ .", "Theorem 4.3 Let $(K,\\tau )$ be a local W-topological field.", "$\\tau $ has a unique V-topological coarsening.", "If $K^*$ is an “ultrapower,” if $R \\subseteq K^*$ is the ring induced by $\\tau $ , and $\\widetilde{R}$ is its integral closure, then $\\widetilde{R}$ is a valuation ring.", "The two statements are equivalent by Proposition REF ; we prove the second one.", "Take the canonical decomposition $\\widetilde{R} =\\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_n$ .", "Let $\\mathfrak {m}_i$ denote the maximal ideal of $\\mathfrak {m}$ .", "For each $i$ , the intersection $\\mathfrak {m}_i \\cap R$ is a prime ideal $\\mathfrak {p}_i \\in \\operatorname{Spec}R$ .", "By Lemma REF , the $\\mathfrak {p}_i$ are pairwise incomparable.", "Let $\\mathfrak {p}$ be the maximal ideal of the local ring $R$ .", "By Chevalley's theorem, there is a valuation ring $\\mathcal {O}$ with maximal ideal $\\mathfrak {m}$ , such that $\\mathcal {O}\\supseteq R$ and $\\mathcal {O}\\cap R = \\mathfrak {m}$ .", "Now $\\mathcal {O}\\supseteq R$ implies $\\mathcal {O}\\supseteq \\widetilde{R}$ , which in turn implies $\\mathcal {O}\\supseteq \\mathcal {O}_i$ for some $i$ ([7], Corollary 6.8).", "Then $\\mathcal {O}\\supseteq \\mathcal {O}_i \\Rightarrow \\mathfrak {m}\\subseteq \\mathfrak {m}_i \\Rightarrow \\mathfrak {m}\\cap R \\subseteq \\mathfrak {m}_i \\cap R.$ Thus $\\mathfrak {p}_i \\supseteq \\mathfrak {p}$ .", "As $\\mathfrak {p}$ is the maximal ideal, we have $\\mathfrak {p}_i = \\mathfrak {p}$ .", "Then $ \\mathfrak {p}_i = \\mathfrak {p}\\supseteq \\mathfrak {p}_j$ for all $j$ , contradicting the pairwise incomparability of the $\\mathfrak {p}_i$ —unless $n = 1$ ." ], [ "Some commutative algebra", "Let $R$ be a domain.", "We let $\\operatorname{MaxSpec}R$ denote the set of maximal ideals in $R$ .", "Definition 4.4 A key localization of $R$ is a localization $R_\\mathfrak {p}$ for some maximal ideal $\\mathfrak {p}\\in \\operatorname{MaxSpec}R$ .", "We view $R_\\mathfrak {p}$ as a subring of $K = \\operatorname{Frac}(R)$ .", "Proposition 4.5 Let $R$ be a domain.", "$R$ equals the intersection of its key localizations.", "If $\\mathfrak {p}_1, \\mathfrak {p}_2$ are two distinct maximal ideals of $R$ , then $R_{\\mathfrak {p}_1}$ is incomparable to $R_{\\mathfrak {p}_2}$ .", "If $A \\subseteq K$ is a local ring containing $R$ , then $A$ contains a key localization of $R$ .", "Therefore, the key localizations of $R$ are the minimal local subrings of $K$ containing $R$ , and they are in bijection with the maximal ideals of $R$ .", "Let $R^{\\prime }$ be the intersection of the key localizations.", "Clearly $R \\subseteq R^{\\prime }$ .", "Conversely, suppose $x \\notin R$ .", "Let $I = \\lbrace y \\in R : xy \\in R\\rbrace $ .", "Then $I$ is a proper ideal in $R$ , because $1 \\notin I$ .", "Take a maximal ideal $\\mathfrak {p}$ containing $I$ .", "We claim $x \\notin R_\\mathfrak {p}$ .", "Otherwise, $x = a/s$ for some $a \\in R$ and $s \\in R \\setminus \\mathfrak {p}$ .", "Then $s \\in I$ , contradicting $I\\subseteq \\mathfrak {p}$ .", "Thus $x \\notin R_\\mathfrak {p}$ , and $x \\notin R^{\\prime }$ .", "Suppose $\\mathfrak {p}_1, \\mathfrak {p}_2$ are distinct.", "Then $\\mathfrak {p}_1, \\mathfrak {p}_2$ are incomparable.", "Take $x \\in \\mathfrak {p}_1 \\setminus \\mathfrak {p}_2$ .", "Then $1/x \\in R_{\\mathfrak {p}_2}$ .", "If $1/x \\in R_{\\mathfrak {p}_1}$ , then $1/x = a/s$ for some $a\\in R$ and $s \\in R \\setminus \\mathfrak {p}_1$ .", "But then $s = ax \\in \\mathfrak {p}_1$ , a contradiction.", "So $1/x$ shows that $R_{\\mathfrak {p}_2} \\lnot \\subseteq R_{\\mathfrak {p}_1}$ .", "By symmetry, $R_{\\mathfrak {p}_1} \\lnot \\subseteq R_{\\mathfrak {p}_2}$ .", "Let $\\mathfrak {m}$ be the maximal ideal of $A$ .", "Then $\\mathfrak {m}\\cap R$ is a prime ideal in $R$ , so $\\mathfrak {m}\\cap R \\subseteq \\mathfrak {p}$ for some $\\mathfrak {p}\\in \\operatorname{MaxSpec}R$ .", "Then $x \\in R & \\Rightarrow x \\in A \\\\x \\in R \\setminus \\mathfrak {p}& \\Rightarrow x \\in A \\setminus \\mathfrak {m}\\Rightarrow x^{-1} \\in A.$ Therefore $R_\\mathfrak {p}\\subseteq A$ .", "Lemma 4.6 Let $R$ be a domain.", "Let $R_1, \\ldots , R_n$ be among the key localizations of $R$ .", "Let $R^{\\prime } = \\bigcap _{i = 1}^n R_i$ .", "Then the key localizations of $R^{\\prime }$ are exactly $R_1, \\ldots , R_n$ .", "Let $K = \\operatorname{Frac}(R)$ .", "By Proposition REF , it suffices to prove the following: if $A$ is a local subring of $K = \\operatorname{Frac}(R)$ , and $A\\supseteq R^{\\prime }$ , then $A \\supseteq R_i$ for some $i$ .", "Let $\\mathfrak {m}$ be the maximal ideal of $A$ .", "Then $\\mathfrak {m}\\cap R$ is a prime ideal in $R$ .", "Let $\\mathfrak {p}_i$ be the maximal ideal of $R$ whose localization is $R_i$ .", "By the prime avoidance lemma, one of two things happens: $\\mathfrak {m}\\cap R \\subseteq \\mathfrak {p}_i$ for some $i$ .", "Then $R_i\\subseteq A$ as in the proof of Proposition REF (REF ).", "$\\mathfrak {m}\\cap R \\lnot \\subseteq \\bigcup _{i = 1}^n \\mathfrak {p}_i$ .", "In this case, take $x \\in (\\mathfrak {m}\\cap R) \\setminus \\bigcup _{i = 1}^n \\mathfrak {p}_i$ .", "Then $1/x \\in R_{\\mathfrak {p}_i}$ for all $i$ , so $1/x \\in R^{\\prime } \\subseteq A$ .", "But this contradicts the fact that $x \\in \\mathfrak {m}= A \\setminus A^\\times $ .", "Lemma 4.7 Let $R$ be a local domain with maximal ideal $\\mathfrak {p}$ .", "If the integral closure $\\widetilde{R}$ is a local ring with maximal ideal $\\mathfrak {q}$ , then $\\mathfrak {q}\\cap R = \\mathfrak {p}$ .", "The intersection $\\mathfrak {q}\\cap R$ is a prime ideal in $R$ , so $\\mathfrak {q}\\cap R \\subseteq \\mathfrak {p}$ .", "If equality does not hold, take $x \\in \\mathfrak {p}\\setminus \\mathfrak {q}$ .", "Then $x \\notin \\mathfrak {q}\\Rightarrow 1/x \\in \\widetilde{R}$ , so there exist $a_0, \\ldots , a_{n-1} \\in R$ such that $ x^{-n} + a_{n-1} x^{-(n-1)} + \\cdots + a_1x^{-1} + a_0 = 0,$ or equivalently, $ -1 = a_{n-1} x + a_{n-2} x^2 + \\cdots + a_1 x^{n-1} + a_0 x^n.$ The right side is in $\\mathfrak {p}$ and the left is not, a contradiction.", "Lemma 4.8 Let $R$ be a domain with finitely many maximal ideals $\\mathfrak {p}_1,\\ldots , \\mathfrak {p}_n$ .", "Suppose that for each key localization $R_{\\mathfrak {p}_i}$ , the integral closure $\\widetilde{R_{\\mathfrak {p}_i}}$ is a valuation ring.", "Then For each key localization $R_\\mathfrak {p}$ , the integral closure $\\widetilde{R_\\mathfrak {p}}$ is a key localization of the integral closure $\\widetilde{R}$ .", "This map establishes a bijection from the key localizations of $R$ to the key localizations of $\\widetilde{R}$ .", "The integral closure $\\widetilde{R}$ is the intersection of all valuation rings on $\\operatorname{Frac}(R)$ containing $R$ .", "If $\\mathcal {O}$ is a valuation ring, the following are equivalent: $\\mathcal {O}\\supseteq R$ $\\mathcal {O}\\supseteq R_{\\mathfrak {p}_i}$ for some $i$ , by Proposition REF (REF ).", "$\\mathcal {O}\\supseteq \\widetilde{R_{\\mathfrak {p}_i}}$ , since $\\mathcal {O}$ is integrally closed.", "Therefore, the $\\widetilde{R_{\\mathfrak {p}_i}}$ are the minimal valuation rings containing $R$ , and $ \\widetilde{R} = \\bigcap _{i = 1}^n \\widetilde{R_{\\mathfrak {p}_i}}.$ Claim 4.9 The $\\widetilde{R_{\\mathfrak {p}_i}}$ are pairwise incomparable.", "[] Let $\\mathfrak {m}_i$ be the maximal ideal of the valuation ring $\\widetilde{R_{\\mathfrak {p}_i}}$ .", "By Lemma REF , $\\mathfrak {m}_i$ restricts to the maximal ideal on $R_{\\mathfrak {p}_i}$ , and then to the ideal $\\mathfrak {p}_i$ on $R$ .", "That is, $\\mathfrak {m}_i \\cap R = \\mathfrak {p}_i$ .", "Then for any $i, j$ $\\widetilde{R_{\\mathfrak {p}_i}} \\subseteq \\widetilde{R_{\\mathfrak {p}_j}} \\iff \\mathfrak {m}_i\\supseteq \\mathfrak {m}_j \\Rightarrow \\mathfrak {m}_i \\cap R \\supseteq \\mathfrak {m}_j \\cap R\\iff \\mathfrak {p}_i \\supseteq \\mathfrak {p}_j.$ The $\\mathfrak {p}_i$ are pairwise incomparable, being maximal ideals, and so the claim is proven.", "Therefore, the $\\widetilde{R_{\\mathfrak {p}_i}}$ are pairwise incomparable valuation rings, with intersection $\\widetilde{R}$ .", "By Proposition 6.2(7) in [7], the $\\widetilde{R_{\\mathfrak {p}_i}}$ are the key localizations of $\\widetilde{R}$ .", "The map $R_\\mathfrak {p}\\mapsto \\widetilde{R_\\mathfrak {p}}$ is injective by the Claim." ], [ "The non-local case", "Theorem 4.10 Let $\\tau $ be a W-topology on $K$ .", "Let $\\tau _1, \\ldots , \\tau _n$ be the local components of $\\tau $ , in the sense of Definition REF .", "Each $\\tau _i$ has a unique V-topological coarsening.", "This establishes a bijection between the local components of $\\tau $ and the V-topological coarsenings of $\\tau $ .", "The first point follows by Theorem REF .", "For the second point, take an “ultrapower” $K^*$ .", "Let $R$ and $R_i$ be the $\\vee $ -definable rings induced by $\\tau $ and $\\tau _i$ .", "By Definition REF , the $R_i$ are the key localizations of $R$ .", "By Theorem REF , each integral closure $\\widetilde{R_i}$ is a valuation ring, corresponding to the unique V-topological coarsening of $\\tau _i$ .", "By Lemma REF , the $\\widetilde{R_i}$ are pairwise distinct, and are exactly the key localizations of $\\widetilde{R}$ .", "By Proposition REF , the key localizations of $\\widetilde{R}$ correspond exactly to the the V-topological coarsenings of $\\tau $ .", "Definition 4.11 Let $\\tau , \\tau _1, \\ldots , \\tau _n$ be topologies on $K$ .", "Then $\\tau $ is an independent sum of $\\tau _1, \\ldots , \\tau _n$ if the diagonal map $ (K,\\tau ) \\rightarrow (K,\\tau _1) \\times \\cdots \\times (K,\\tau _n)$ is a homeomorphism onto its image, and the image is dense.", "More explicitly, this means that the following conditions hold: The following is a filter basis for $\\tau $ : $\\lbrace U_1 \\cap \\cdots \\cap U_n : U_1 \\in \\tau _1, ~ U_2 \\in \\tau _2,\\ldots , U_n \\in \\tau _n\\rbrace .$ If $a_i \\in K$ and $U_i \\in \\tau _i$ for all $i$ , then $\\bigcap _{i = 1}^n (a_i + U_i) \\ne \\emptyset $ .", "In other words, the $\\tau _i$ are jointly independent, and they generate $\\tau $ .", "Lemma 4.12 If $\\sigma $ is an independent sum of $\\tau _1, \\ldots , \\tau _{n-1}$ , and $\\tau $ is an independent sum of $\\sigma $ and $\\tau _n$ , then $\\tau $ is an independent sum of $\\tau _1, \\ldots , \\tau _n$ .", "Let $\\mathcal {F}$ be the class of topological embeddings with dense image.", "Then $\\mathcal {F}$ is closed under composition.", "Additionally, if $f : X \\rightarrow Y$ is in $\\mathcal {F}$ and $Z$ is another topological space, then $X \\times Z \\rightarrow Y \\times Z$ is in $\\mathcal {F}$ .", "By assumption, the diagonal maps $(K,\\sigma ) &\\rightarrow (K,\\tau _1) \\times \\cdots \\times (K,\\tau _{n-1}) \\\\(K,\\tau ) & \\rightarrow (K,\\sigma ) \\times (K,\\tau _{n-1})$ are in $\\mathcal {F}$ .", "Therefore $\\mathcal {F}$ also contains the composition $(K,\\tau ) \\rightarrow (K,\\sigma ) \\times (K,\\tau _{n}) \\rightarrow (K,\\tau _1)\\times \\cdots \\times (K,\\tau _{n-1}) \\times (K,\\tau _n),$ which is the diagonal map.", "Theorem 4.13 Let $\\tau $ be a W-topology on a field $K$ .", "Let $\\tau _1, \\ldots ,\\tau _n$ be the local components of $\\tau $ .", "Then $\\tau $ is an independent sum of the $\\tau _i$ .", "Fix an “ultrapower” $K^*$ of $K$ .", "Let $R, R_1, \\ldots , R_n$ be the $\\vee $ -definable rings corresponding to $\\tau , \\tau _1, \\ldots ,\\tau _n$ .", "The $R_i$ are the key localizations of $R$ .", "For $i \\le n$ , let $S_i = R_1 \\cap \\cdots \\cap R_i$ .", "Each $S_i$ is a $K$ -algebra that is $\\vee $ -definable over $K$ .", "Moreover, $S_i\\supseteq R$ , so each $S_i$ is a $W_n$ -ring (Lemma 2.7 in [9]).", "By Proposition REF , each $S_i$ corresponds to a W-topology $\\sigma _i$ .", "Additionally, $S_1 &= R_1 \\\\S_n &= R_1 \\cap \\cdots \\cap R_n = R,$ by Proposition REF (REF ).", "Thus $\\sigma _1 = \\tau _1$ , and $\\sigma _n = \\tau $ .", "By Lemma REF , the key localizations of $S_i$ are $R_1, \\ldots , R_i$ , and so the local components of $\\sigma _i$ are $\\tau _1, \\ldots , \\tau _i$ .", "Claim 4.14 $\\sigma _i$ is an independent sum of $\\sigma _{i-1}$ and $\\tau _i$ .", "[] We first prove independence.", "Let $\\widetilde{\\tau _j}$ denote the unique V-topological coarsening of $\\tau _j$ .", "By Theorem REF , the V-topological coarsenings of $\\sigma _{i-1}$ are $\\lbrace \\widetilde{\\tau _1}, \\ldots , \\widetilde{\\tau _{i-1}}\\rbrace $ , the V-topological coarsenings of $\\tau _i$ are $\\lbrace \\widetilde{\\tau _i}\\rbrace $ , and these two sets do not overlap.", "So $\\sigma _{i-1}$ and $\\tau _i$ have no common V-topological coarsenings.", "As $\\tau _i$ and $\\sigma _{i-1}$ are both coarsenings of the original W-topology $\\tau $ , Theorem 7.16 of [9] applies, and $\\tau _i$ and $\\sigma _{i-1}$ are independent.", "It remains to show that $\\sigma _{i-1}$ and $\\tau _i$ generate $\\sigma _i$ .", "Consider the topologies $\\sigma _{i-1}^*, \\tau _i^*,\\sigma _i^*$ on $K^*$ induced by $S_{i-1}, R_i,$ and $S_i$ , respectively.", "Note $S_i = S_{i-1} \\cap R_i$ .", "For any non-zero $a$ , there are non-zero $b, c$ such that $ b S_{i-1} \\cap c R_i \\subseteq aS_i.$ Indeed, if we take $b = c = a$ , then equality holds.", "This proves the local sentence $\\forall U \\in \\sigma _i^* ~ \\exists V \\in \\sigma _{i-1}^* ~\\exists W \\in \\tau _i^* : V \\cap W \\subseteq U.", "$ Conversely, $\\forall V \\in \\sigma _{i-1}^* ~ \\forall W \\in \\tau _i^* ~ \\exists U \\in \\sigma _i^* : U \\subseteq V \\cap W, $ because we can take $U = V \\cap W$ .", "(Both $V$ and $W$ are in $\\sigma _i^*$ , because $\\sigma _i^*$ is finer than $\\sigma _{i-1}^*$ and $\\tau _i^*$ .)", "Equations (REF ) and (REF ) are local sentences, so they hold for $\\sigma _{i-1}, \\tau _i,$ and $\\sigma _i$ , by Proposition REF (REF ).", "That is, $\\forall U \\in \\sigma _i ~ \\exists V \\in \\sigma _{i-1} ~\\exists W \\in \\tau _i &: V \\cap W \\subseteq U \\\\\\forall V \\in \\sigma _{i-1} ~ \\forall W \\in \\tau _i ~ \\exists U \\in \\sigma _i &: U \\subseteq V \\cap W.$ This expresses that $\\sigma _i$ is generated by $\\sigma _{i-1}$ and $\\tau _i$ , finishing the proof of the Claim.", "Combining the Claim with Lemma REF , we see by induction on $i$ that $\\sigma _i$ is an independent sum of $\\tau _1, \\ldots ,\\tau _i$ .", "Taking $i = n$ , we get the desired result.", "Corollary 4.15 Let $(K,\\tau )$ be a W-topological field.", "If $\\operatorname{char}(K) \\ne 2$ and the squaring map $K^\\times \\rightarrow K^\\times $ is an open map, then $\\tau $ is local and has a unique V-topological coarsening.", "If $\\operatorname{char}(K) = p > 0$ and the Artin-Schreier map $K\\rightarrow K$ is an open map, then $\\tau $ is local and has a unique V-topological coarsening.", "Let $\\tau _1, \\ldots , \\tau _n$ be the local components of $\\tau $ .", "By Theorem REF , $\\tau $ is an independent sum of $\\tau _1,\\ldots , \\tau _n$ .", "By Theorem REF , $n$ is the number of V-topological coarsenings.", "Suppose for a contradiction that $n >1$ .", "If $\\operatorname{char}(K) \\ne 2$ , then the squaring map is not open, by the proof of Claim 6.9 in [9].", "Essentially, one chooses $x$ to be infinitesimally close to 1 with respect to $\\tau _1, \\tau _3, \\tau _5, \\ldots $ , and infinitesimally close to $-1$ with respect to $\\tau _2, \\tau _4, \\tau _6, \\ldots $ .", "Then $x \\lnot \\approx 1$ and $x^2 \\approx 1$ with respect to $\\tau $ , which contradicts squaring being an open map.", "The Artin-Schreier case is similar, using $0, 1$ instead of $1, -1$ .", "Corollary 4.16   If $(K,+,\\cdot ,\\ldots )$ is an unstable dp-finite field, then the canonical topology on $K$ is a local W-topology.", "If $(K,+,\\cdot ,\\ldots )$ is an unstable dp-finite field, then $K$ admits a unique definable V-topology.", "If $(K,+,\\cdot ,v)$ is a dp-finite valued field, then $v$ is henselian.", "If $(K,+,\\cdot )$ is a dp-finite field that is neither finite, nor algebraically closed, nor real closed, then $K$ admits a non-trivial definable henselian valuation.", "The conjectural classification of dp-finite fields holds, as in Theorem 3.11 of [4].", "The canonical topology on $K$ is a W-topology by Theorem 6.3 in [9].", "By Proposition 5.17(4-5) in [7] and compactness, the canonical topology has the following properties: For every neighborhood $U \\ni 1$ there is a neighborhood $V\\ni 1$ such that $V \\subseteq \\lbrace x^2 : x \\in U\\rbrace .$ Equivalently, the squaring map $K^\\times \\rightarrow K^\\times $ is an open map.", "If $\\operatorname{char}(K) = p$ , then for every neighborhood $U\\ni 0$ there is a neighborhood $V \\ni 0$ $V \\subseteq \\lbrace x^p - x : x \\in U\\rbrace .$ Equivalently, the Artin-Schreier map $K \\rightarrow K$ is an open map.", "By Corollary REF , the canonical topology is local and has a unique V-topological coarsening.", "This proves part REF .", "By Theorem 6.6 in [9], the V-topological coarsenings of the canonical topology are exactly the definable V-topologies.", "This proves part REF .", "The remaining points then follow by (the proof of) Proposition 6.4 in [8].", "We note another characterization of local W-topologies.", "Proposition 4.17 Let $\\tau $ be a W-topology on a field $K$ .", "The following are equivalent: $\\tau $ is not local.", "$\\tau $ is an independent sum of two W-topologies.", "$\\tau $ is an independent sum of finitely many W-topologies.", "The implication (REF )$\\Rightarrow $ (REF ) follows by the Theorem REF , or rather by Claim REF in the proof.", "In the notation of the proof, $\\tau $ equals $\\sigma _n$ , which is an independent sum of $\\sigma _{n-1}$ and $\\tau _n$ .", "The implication (REF )$\\Rightarrow $ (REF ) is trivial.", "It remains to prove (REF )$\\Rightarrow $ (REF ).", "Suppose $\\tau $ is an independent sum of $\\tau _1, \\ldots , \\tau _n$ , but $\\tau $ is local.", "Claim 4.18 If $B_i$ is $\\tau _i$ -bounded for $i = 1, \\ldots , n$ , then $\\bigcap _{i = 1}^n B_i$ is $\\tau $ -bounded.", "[] By Lemma 2.1(e) in [10], a set $B$ is bounded if for every neighborhood $U$ , there is a neighborhood $V$ such that $B \\cdot V\\subseteq U$ .", "Let $U_1 \\cap \\cdots \\cap U_n$ be a basic neighborhood in $\\tau $ , so that each $U_i$ is a neighborhood in $\\tau _i$ .", "Then there are $V_i \\in \\tau _i$ such that $B_i \\cdot V_i \\subseteq U_i$ .", "Then $\\left( \\bigcap _{i = 1}^n B_i \\right) \\cdot \\left( \\bigcap _{i =1}^n V_i \\right) \\subseteq \\bigcap _{i = 1}^n U_i,$ and $\\bigcap _{i = 1}^n V_i$ is in $\\tau $ .", "For each $i$ , take a $\\tau _i$ -bounded set $B_i$ containing both 0 and 1 in its interior (i.e., $B_i \\in \\tau _i$ and $B_i - 1 \\in \\tau _i$ ).", "Let $B = B_1 \\cap \\cdots \\cap B_n$ .", "Then $B$ is $\\tau $ -bounded, by the Claim.", "Because $\\tau $ is local, there is a $\\tau $ -bounded set $C$ such that $ \\forall x \\in B : (1/x \\in C \\text{ or } 1/(1-x) \\in C).$ The fact that $\\tau $ is finer than $\\tau _i$ implies that $\\tau $ has fewer bounded sets, so $C$ is $\\tau _i$ -bounded for each $i$ .", "For each $i$ , take small enough $D_i \\in \\tau _i$ to ensure that $D_i \\cdot C$ is contained in the neighborhood $K \\setminus \\lbrace 1\\rbrace $ , i.e., $1 \\notin D_i \\cdot C$ .", "$D_i \\subseteq B_i$ and $1 - D_i \\subseteq B_i$ .", "By independence, there is $x \\in (1 - D_1) \\cap D_2 \\cap D_3 \\cap \\cdots \\cap D_n$ .", "Then $x \\in B_1 \\cap \\cdots \\cap B_n = B$ , so one of two things happens: $1/x \\in C$ .", "But $x \\in D_2$ , so then $1 = x \\cdot (1/x) \\in D_2 \\cdot C$ , a contradiction.", "$1/(1-x) \\in C$ .", "But $1 - x \\in D_1$ , so then $1 = (1-x)\\cdot (1/(1-x)) \\in D_1 \\cdot C$ , a contradiction." ], [ "Remaining questions", "We leave the following natural questions to future work.", "By Theorem REF and Proposition REF , every W-topology decomposes into an independent sum of indecomposable W-topologies.", "Is this decomposition unique?", "If a W-topology $\\tau $ decomposes as an independent sum of $\\tau _1, \\ldots , \\tau _n$ , is it true that $\\operatorname{wt}(\\tau ) = \\operatorname{wt}(\\tau _1) +\\cdots + \\operatorname{wt}(\\tau _n)$ ?", "If $\\tau _1, \\ldots , \\tau _n$ are jointly independent W-topologies, do they generate a W-topology?", "If $\\tau _1, \\tau _2$ are two W-topologies without any common V-topological coarsenings, then are $\\tau _1, \\tau _2$ necessarily independent?", "Local topologies of weight 2 on fields of characteristic 0 are exactly the “DV-topologies” of ([8], Definition 8.18).", "This follows by the classification in §8.2 of [9].", "Can this sort of classification be generalized to higher ranks, or positive characteristic?", "Let $(K,\\tau )$ be a W-topological field, and $K^*$ be an “ultrapower.” Let $R$ be the associated $\\vee $ -definable ring of $K$ -bounded elements in $K^*$ .", "Let $I$ be the type-definable ideal of $K$ -infinitesimal elements in $K^*$ .", "Is $R/I$ always an artinian ring of length equal to $\\operatorname{wt}(\\tau )$ ?", "In several places, we used the special properties of the rings $R = R_\\tau $ arising in Proposition REF .", "For example, If $\\operatorname{wt}(R) = n$ , then $R$ induces a topology of weight exactly $n$ .", "$R$ is local if and only if $R$ induces a local W-topology.", "Is there a natural algebraic condition implying these properties, and satisfied by the rings $R_\\tau $ of Proposition REF ?", "If so, there may be a more natural class of rings hidden inside the $W_n$ -rings.", "This natural class would probably include multi-valuation rings $\\mathcal {O}_1 \\cap \\cdots \\cap \\mathcal {O}_n$ for which the $\\mathcal {O}_i$ are pairwise independent, as well as the ring $R$ of §8.4 in [8].", "Do the global fields $\\mathbb {Q}$ and $\\mathbb {F}_p(t)$ support any local W-topologies other than the usual V-topologies?", "If $K$ is an unstable dp-finite field, does every heavy definable set have non-empty interior?", "How much “tame topology” can we prove?", "Do the techniques used to analyze dp-finite fields have any generalizations to finite-burden fields, or strongly dependent fields?", "Let $(K,+,\\cdot ,\\ldots )$ be an NIP field, possibly with extra structure.", "Must every definable Hausdorff non-discrete field topology on $K$ be a W-topology?", "Acknowledgments The author would like to thank Meng Chen for hosting the author at Fudan University, where this research was carried out.", "This material is based upon work supported by the National Science Foundation under Award No.", "DMS-1803120.", "Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation." ] ]

[ [ "DC$_3$N observations towards high-mass star-forming regions" ], [ "Abstract We present the study of deuteration of cyanoacetylene (HC$_3$N) towards a sample of 28 high-mass star-forming cores divided into different evolutionary stages, from starless to evolved protostellar cores.", "We report for the first time the detection of DC$_3$N towards 15 high-mass cores.", "The abundance ratios of DC$_3$N with respect HC$_3$N range in the interval 0.003$-$0.022, lower than those found in low-mas protostars and dark clouds.", "No significant trend with the evolutionary stage, or with the kinetic temperature of the region, has been found.", "We compare the level of deuteration of HC$_3$N with those of other molecules towards the same sample, finding weak correlation with species formed only or predominantly in gas phase (N$_2$H$^+$ and HNC, respectively), and no correlation with species formed only or predominantly on dust grains (CH$_3$OH and NH$_3$, respectively).", "We also present a single-dish map of DC$_3$N towards the protocluster IRAS 05358+3543, which shows that DC$_3$N traces an extended envelope ($\\sim$0.37 pc) and peaks towards two cold condensations separated from the positions of the protostars and the dust continuum.", "The observations presented in this work suggest that deuteration of HC$_3$N is produced in the gas of the cold outer parts of massive star-forming clumps, giving us an estimate of the deuteration factor prior to the formation of denser gas." ], [ "Introduction", "The deuteration level of interstellar molecules is a powerful tool to study the history of star-forming regions.", "The formation of deuterated species is significantly enhanced during the pre-stellar phases of star-forming regions due to the combination of high gas densities ($>$ 10$^{4}$ cm$^{-3}$ ) and low temperatures (T$\\le $ 20 K), which favour the depletion of CO onto interstellar dust surfaces ([10], [53]).", "This enhances the gas-phase abundances of several ions (e.g.", "H$_3^+$ ), in absence of its main destroyer, CO. Then, the reaction of H$_3^+$ with HD, the main reservoir of deuterium in molecular clouds, produces H$_{2}{\\rm D}^{+}$ through the reaction H$_{3}^{+}$ + HD $\\rightarrow $ H$_{2}$ D$^{+}$ + H$_{2}$ + 232 K, when the species involved are in the para-H$_2$ form.", "The exothermicityThe exotermicity of this reaction depends on the ortho-para ratio of the species involved, and that can vary between 84 and 256 K ( [38], [44]).", "of this reaction prevents the backwards reaction at low temperatures, thus increasing the gas-phase abundance of H$_{2}{\\rm D}^{+}$ .", "This species transfers deuterium to other molecules efficiently, increasing the deuterium fractionation in molecules by several orders of magnitude with respect to the primordial value of D/H $\\sim $ (1$-$ 2)$\\times $ 10$^{-5}$ in the solar neighborhood ([42], [34]).", "Therefore, the deuterated fraction, $D_{\\rm frac}$ , defined as the column density ratio of one species containing deuterium to its counterpart containing hydrogen, can be used as a useful probe of the evolution of star-forming cores.", "In low-mass star-forming cores, [9] found that $D_{\\rm frac}$ significantly increases during the pre-stellar phase, when the core density profile becomes more centrally peaked due to the collapse.", "As a consequence, the CO freeze-out increases, along with the abundance of deuterated molecules in the gas-phase.", "Multiple observations have shown that the value of $D_{\\rm frac}$ reaches its maximum in pre-stellar cores close to gravitational collapse (e.g.", "[17]), and decreases during the later protostellar phases ([20]).", "In high-mass star-forming regions, how $D_{\\rm frac}$ changes with the evolutionary stage is less clear.", "In previous works, we have studied the deuteration of different species towards a sample of high-mass star-forming cores at different evolutionary stages, from pre-stellar to protostellar phases.", "[23] found that enhanced values of N$_2$ D$^+$ are found towards high-mass pre-stellar cores, similarly to low-mass counterparts.", "However, other molecular species, such as HNC, NH$_3$ and CH$_3$ OH, do not show any clear trend with the evolutionary stage using the same sample of sources ([24], [25], [14]).", "In this work, we focus on the carbon-chain molecule cyanoacetylene (HC$_3$ N).", "Little is known about its deuteration, since only a few observations have been able to detect it in space.", "DC$_3$ N was first detected by [31] towards the low-mass dark cloud TMC-1, and by [27] towards several other dark clouds, finding values of $D_{\\rm frac}$ in the range 0.03-0.13.", "In the protostellar phase, DC$_3$ N has been detected only towards a handful of sources: IRAS 16293-2422 ([2]), L1527 ([46] and [3]), Chamaeleon MMS1 ([16]), and SVS13-A ([6]), with values of $D_{\\rm frac}$ $\\sim $ 0.03$-$ 0.4.", "There is still no unambiguous detection of DC$_3$ N towards high-mass star-forming regions.", "Only two tentative detections have been reported towards Orion KL and the Sgr B2 N2 hot molecular cores ([22], [4], respectively).", "These observations, along with the non detection reported towards NGC 2264 CMM3 ([16]), derived upper limits of D$_{\\rm frac}<$ 0.015, which are lower than those reported in low-mass protostars.", "There are two main questions that need to be answered about deuterium fractionation of HC$_3$ N. First, is HC$_3$ N deuterated through gas-phase reactions, like simple species such as N$_2$ H$^+$ (e.g.", "[23]), or alternatively on the surface of intersellar dust grains via H-D exchange reactions, as suggested for other carbon-chain molecule (c-C$_3$ H$_2$ , [11])?", "And second, how $D_{\\rm frac}$ changes with the evolutionary stage of the star-forming core?", "To properly answer these questions, detections of DC$_3$ N towards massive cores at different evolutionary stages are needed.", "With this aim, we present the first study of deuteration of HC$_3$ N towards a sample of massive star-forming regions spanning a wide range of evolutionary stages, from initial starless cores to more evolved phases.", "We report the detection of the DC$_3$ N $J$ =11$-$ 10 transition towards 15 regions, and present the first maps of DC$_3$ N in one of the sources of the sample, the high-mass protocluster IRAS 05358+3543.", "Table: List of the observed sources.", "Cols.", "2 and 3 give the equatorial coordinates of the sources.", "Cols.", "4 and 5 give the source distance to the Sun, and the Galactocentric distance, respectively.", "For the references on distances from the Sun see .", "In Col. 6 the 12 ^{12}C/ 13 ^{13}C ratios, derived using the galactocentric trend by , and the associated uncertainties, are given.In the last column the kinetic temperatures of the clumps derived by are listed.", "For the sources without a direct derivation of T kin T_{\\rm kin} (indicated with a ^{a}), the average value for that evolutionary stage was taken.Figure: DC 3 _3N JJ=11--10 detections towards the sample of high-mass star-forming regions studied in this work.", "The observed IRAM 30m telescope spectra are shown with a black histogram, while the best LTE fits are shown with red curves.", "Each panel is centered at the systemic velocity of each source.", "The name of each source is indicated above each panel, coloured by evolutionary groups: dark green for cold HMSCs, black for warm HMSCs, red for HMPOs and blue for UC HII regions." ], [ "Observations", "We conducted astronomical observations using the 30m-diameter telescope of the Institut de Radioastronomie Millimétrique (IRAM), located at Pico Veleta (Granada, Spain), as part of the projects 129$-$ 12 and 040-19.", "In the project 129$-$ 12, we performed single-pointing observations towards 26 high-mass star-forming cores (see details about the sample in Sect.", ").", "We used the broad-band Eight MIxer Receiver (EMIR), covering $\\sim $ 8 GHz from 89.11 to 96.89 GHz, and the fast Fourier Transform Spectrometer (FTS, [29]) in a mode that provides a channel width of 195 kHz ($\\sim $ 0.63 km s$^{-1}$ ).", "The observed setup includes the $J$ =11$-$ 10 transition of DC$_3$ N at 92.872373 GHz (spectroscopy from the recent laboratory measurements of Melosso et al.", "in prep.", "), and the $J$ =10$-$ 9 transitions of two of the three $^{13}$ C-isotopologues of HC$_3$ N, HC$^{13}$ CCN and HCC$^{13}$ CN, at 90.593059 and 90.601777 GHz, respectively ([18], [52]).", "The spatial resolution of the observations, given by the half-power beam width of the antenna (HPBW), can be calculated as HPBW()=2460/$\\nu $ (GHz), which gives 26.5at the frequecny of the DC$_3$ N $J$ =11$-$ 10 transition.", "The spectra were calibrated using the GILDASThe GILDAS software is is available at http://www.iram.fr/ IRAMFR/GILDAS software package.", "The antenna temperature units, $T_{\\rm A}^{*}$ , were converted to main beam brightness temperature, $T_{\\rm mb}$ , via the relation $T_{\\rm A}^{*}$ = $T_{\\rm mb}\\times $ (B$_{\\rm eff}$ /F$_{\\rm eff}$ ), using the corresponding telescope efficiencieshttp://www.iram.es/IRAMES/mainWiki/Iram30mEfficiencies.", "The flux density calibration uncertainty of the observations is $\\sim $ 20$\\%$ .", "More details of the observations (e.g.", "weather conditions, background subtraction, focus and pointing) are presented in [25].", "In the 040-19 project, we mapped with multiple telescope pointings the molecular emission of one of the star-forming regions of the sample, the high-mass protocluster IRAS 05358+3543 (hereafter 05358).", "We also observed the $J$ =11$-$ 10 transition of DC$_{3}$ N, and the $J$ =10$-$ 9 transitions of HC$^{13}$ CCN and HCC$^{13}$ CN, during observations performed in the periods 31st July to 5th August 2019 and 18th to 19th September 2019.", "As for the single-pointing observations, we used the 3mm receiver EMIR, and the FTS spectrometer with a frequency resolution of 195 kHz ($\\sim $ 0.63 km s$^{-1}$ ).", "The observations were made in position-switching mode.", "Pointing was checked every 1.5 h, and focus was corrected at the beginning of the observations, at dawn and every 4-6 h. The molecular datacubes were produced from On-The-Fly (OTF) mapping, covering an area of 120$\\times $ 120(corresponding to $\\sim $ 1 pc $\\times $ 1 pc, at the source distance of 1.8 kpc; [26]), with a central position of RA(J2000) = 05h:39m:13s.0, DEC(J2000)= 35°:45:51.", "The integration time for each OTF map (vertical + horizontal + calibration times) was of $\\sim $ 20 minutes.", "The antenna temperature $T_{\\rm A}^{*}$ was converted to main beam temperature $T_{\\rm mb}$ by using the same expression given above.", "Llux calibration uncertainties of 20$\\%$ are considered in the analysis.", "The GILDAS packages CLASS and GREG were used to reduce and post-process the data.", "Baselines were all fitted by constant functions or polynomials of order 1.", "We built the molecular data cube convolving the OTF map with a Gaussian kernel, using a regularly spaced grid with pixel size of 9." ], [ "Source sample", "The source sample, already used in several works ([23], [24], [25], [14], [13], [39]), includes high-mass star-forming cores spanning a wide range of evolutionary stages: High-Mass Starless Cores (HMSCs), which are not directly associated with indicators of on-going star formation, such as embedded infrared sources, outflows, or masers.", "As in previous works, we have divided this group into two subgroups: ${\\it cold}$ and ${\\it warm}$ .", "The latter includes three regions (AFGL5141-EC, 05358-mm3 and I22134-G) that have temperatures $T_{\\rm kin}\\ge $ 25 K, and that can be externally heated by nearby protostellar objects detected by high-angular resolution observations ([55], [8], [47], [15]).", "We note that our observations of these three warm HMSCs are likely contaminated by the emission of nearby protostellar sources, which fall in the IRAM 30m beam of 26.5.", "This will be discussed in Section .", "We have added to the cold HMSC group two new cores that have been identified in the DC$_3$ N maps of the 05358 region presented in this work (Sect.", "REF ): 05358-D1 and D2.", "High-Mass Protostellar Objects (HMPOs), which are associated with molecular outflows, and/or faint radio continuum emission (S$_{\\rm \\nu }$ at 3.6 cm $<$ 1 mJy) likely tracing a radio-jet, and/or infrared sources.", "Ultracompact (UC) HII regions; associated with a strong radio-continuum emission (S$_{\\rm \\nu }$ at 3.6 cm $\\ge $ 1 mJy), which likely traces photoionised gas by nascent massive stars.", "All the selected sources are located at distances $d\\le $ 5 kpc.", "The full list of the observed sources is shown in Table REF .", "More details about how the sources were selected and classified in the different evolutionary stages are given in [23].", "In summary, we observed 28 high-mass star-forming cores divided into: 9 cold HMSCs, 3 warm HMSCs, 9 HMPOs, and 7 UC HII regions.", "Figure: Differences between the velocities (upper panel) and FWHM (lower panel) of DC 3 _3N and the 13 ^{13}C-isotopologues.", "Filled(empty) circles correspond to HCC 13 ^{13}CN(HC 13 ^{13}CCN).", "The different colors indicate the different evolutionary groups, as indicated.", "The shaded area denotes the region ±\\pm 1 km s -1 ^{-1}.Figure: Values of D frac D_{\\rm frac} derived for HC 3 _3N in the sample of high-mass star-forming regions studied in this work (warm HMSC, HMPO and UC HII regions), compared with those derived for other species in previous works.", "The different colours of the circles correspond to different evolutionary groups, as indicated in each panel.", "Upper limits are denoted by triangles.", "The outputs of the Kendall τ\\tau test (τ\\tau and double-side pp value) are indicated in the upper right corner of each panel." ], [ "Single-pointing observations", "The spectra were imported into the MADCUBA packageMadrid Data Cube Analysis on ImageJ is a software developed at the Center of Astrobiology (CAB) in Madrid; http://cab.inta-csic.es/madcuba/Portada.html.", "([36]).", "The identification of the molecular lines was performed using the SLIM (Spectral Line Identification and Modeling) tool within MADCUBA.", "SLIM generates synthetic spectra of molecular species under the assumption of Local Thermodynamic Equilibrium (LTE) conditions.", "We have implemented into SLIM the new DC$_3$ N spectroscopic data recently obtained in the laboratory by Melosso et al.", "(in prep.).", "We detected the DC$_3$ N $J$ =11-10 transition towards 15 regions.", "The transitions observed, shown in Figure REF , include 2 cold HMSCs, 3 warm HMSCs, 4 HMPOs and 6 UC HII regions.", "To obtain the best fit of the observed transitions we used the MADCUBA-AUTOFIT tool that compares the observed spectra with LTE synthetic spectra, and provides the best non-linear least-squares fit using the Levenberg$-$ Marquardt (LM) algorithm (see details in [36]).", "The free parameters of the fit are: column density of the molecule ($N$ ), excitation temperature ($T_{\\rm ex}$ ), velocity (v$_{\\rm LSR}$ ) and full width at half-maximum ($FWHM$ ) of the Gaussian profiles of the lines.", "We assumed that the molecular emission fills the beam of the telescope, and thus we did not apply any beam dilution correction.", "This assumption is reasonable, since the low energies of the transitions studied (E$_{\\rm up}$ =24-27 K), which suggest that these molecules trace relatively extended gas.", "In the particular case of the 05358 region, the emission maps we present in Sect.", "REF confirm that this assumption is fulfilled.", "Since we only have one rotational transition, we fixed the value of $T_{\\rm ex}$ to the kinetic temperature $T_{\\rm kin}$ of the cores (Table REF ).", "We note that the column density of DC$_3$ N can vary by factors $<$ 1.3(1.7) depending of the $T_{\\rm ex}$ assumed in the range 15$-$ 50 K (15$-$ 100 K).", "The other parameters ($N$ , v$_{\\rm LSR}$ and $FWHM$ ) were left free whenever possible.", "Only if the LM algorithm did not converge we fixed v$_{\\rm LSR}$ and/or $FWHM$ to values that visually match the observed lines.", "The resulting fits are shown in red overplotted to the observed spectra in Figure REF , and the resulting parameters are presented in Table REF .", "In the case of non detections, upper limits for the column density of DC$_3$ N were derived with SLIM, which uses the formula 3$\\sigma \\times \\Delta $ v/$\\sqrt{N_{\\rm chan}}$ , where $\\sigma $ is the rms noise of the spectra, and $N_{\\rm chan}$ is the number of channels covered by the linewidth $\\Delta $ v. The derived upper limits for the column density of DC$_3$ N are shown in Table REF .", "To obtain the value of $D_{\\rm frac}$ , we derived the HC$_3$ N column densities towards each source with the same procedure used for DC$_3$ N. Since HC$_3$ N is an abundant molecule in high-mass star-forming regions, its emission is expected to be optically thick, which prevents a reliable accurate derivation of its column density.", "For this reason, we opted to analyze the two $^{13}$ C-isotopologues of HC$_3$ N available in the observed frequency band (HC$^{13}$ CCN and HCC$^{13}$ CN), which are expected to be optically thin.", "We have used the spectroscopic entries of the Cologne Database for Molecular Spectroscopy (CDMS, [40], [41], [21]), based on the laboratory works by [18] and [52].", "We calculated the average column density value of the $^{13}$ C-isotopologues, and assumed the $^{12}$ C/$^{13}$ C ratio as a function of the Galactocentric distance found by [37] (Table REF ) to convert to HC$_3$ N column density.", "The derived values of $D_{\\rm frac}$ are presented in the last column of Table REF .", "We note that, unlike the values of the column densities, the values of $D_{\\rm frac}$ are nearly independent of the assumed $T_{\\rm ex}$ , whenever DC$_3$ N and the $^{13}$ C-isotopologues of HC$_3$ N have a similar $T_{\\rm ex}$ .", "We found values of $D_{\\rm frac}$ in the range 0.003-0.022.", "For the sources with no DC$_3$ N detection but HC$^{13}$ CCN and HCC$^{13}$ CN detections, we give the upper limits of $D_{\\rm frac}$ .", "Only in one case, the UC HII region I19035-VLA1, DC$_3$ N was detected but not the $^{13}$ C-isotopologues.", "For this source we show in Table REF the derived lower limit of $D_{\\rm frac}$ , $\\ge $ 0.022.", "To make sure that the $^{13}$ C-isotopologues arise from similar gas than DC$_3$ N, we have checked that their velocities and linewidths are similar.", "We show in Figure REF that the differences of the velocities ($\\Delta $ v$_{\\rm LSR}$ ) and linewidths ($\\Delta FWHM$ ) between DC$_3$ N and the $^{13}$ C-isotopologues are always within a narrow range of $\\pm $ 1 km s$^{-1}$ , considering the uncertainties derived by the fits.", "This similarity in the kinematics supports our assumption that DC$_3$ N and the $^{13}$ C-isotopologues are likely tracing similar gas.", "To compare the deuteration of HC$_3$ N with that of other molecules already studied in previous works towards the same sample, we show in Figure REF the $D_{\\rm frac}$ (HC$_3$ N) versus those derived for N$_2$ H$^{+}$ ([23]), HNC ([14]), NH$_3$ ([25]) and CH$_3$ OH ([25]).", "We performed Kendall's $\\tau $ tests ([28]) to search for possible correlations between the set of values of $D_{\\rm frac}$ of the different species.", "We considered only the sources in which the deuterated species have been detected (no upper limits).", "The initial guess (null hypothesis) is that the two datasets are not correlated.", "The Kendall's $\\tau $ correlation parameter can adopt values between -1 and 1: is equal to 1 if a perfect correlation is found, 0 if there is no correlation (initial guess fulfilled), and -1 if there is perfect anticorrelation.", "As an output of the test we also give the 2-sided $p$ value, which gives the level of significance of the test between 0 and 1: 0 when it is statistically significant and 1 when it is totally not significant.", "The results of the tests are shown in the upper right corner of each panel in Figure REF .", "The deuteration of HC$_3$ N shows better correlations with N$_2$ H$^{+}$ and HNC rather than NH$_3$ and CH$_3$ OH.", "There are (weak) correlations with N$_2$ H$^{+}$ and HNC ($\\tau $ =0.47 and 0.37, respectively), which are statistically significant (6$\\%$ and 13$\\%$ probability that the initial guess is correct, i.e., no correlation).", "For NH$_3$ and CH$_3$ OH, there is no correlation ($\\tau $ =0.1 and 0.0, respectively), with 75$\\%$ and 100$\\%$ probability of no correlation, respectively.", "We discuss the implications of these results in Sect.", ".", "Figure: Left: The color scale shows the integrated emission map of the JJ=11--10 transition of DC 3 _{3}N.The red contour correspond to 0.4 times the emission peak (178 mK km s -1 ^{-1}).The gray contours show to the combined integrated emission of the JJ=10--9 transitions of HC 13 ^{13}CCN and HCC 13 ^{13}CN (0.9, 0.8, 0.7, 0.6, 0.5 and 0.4 times the emission peak, which is 265 mK km s -1 ^{-1}).The green squares indicate the positions of the continuum sources mm1, mm2, mm3 and mm4 (), and the DC 3 _{3}N peak positions (D1 and D2), respectively.", "The gray filled circle in the lower left corner indicate the beam of the DC 3 _3N observation.Right: The color scale shows the SCUBA 850 μ\\mu m continuum observations, from .", "The beam size of the SCUBA map (22.9) is indicated in the lower left corner with a blue open circle overplotted to the IRAM 30m beam (gray filled circle).", "The gray contour levels are the same as in the left panel, while the red contour levels are 0.9, 0.8, 0.7, 0.6, 0.5 and 0.4 times the peak of the integrated emission map of DC 3 _3N." ], [ "Maps of the 05358+3543 protocluster", "We produced the integrated maps of the $J$ =11$-$ 10 transition of DC$_3$ N, and the combined integrated map of the $J$ =10$-$ 9 transitions of the two $^{13}$ C-isotopologues (HC$^{13}$ CCN and HCC$^{13}$ CN), which are shown in Figure REF .", "We also indicate in the figure the positions of the continuum sources previously identified in the region with interferometric observations ([15]): mm1, mm2, mm3 and mm4.", "For comparison, we also show in the right panel of Figure REF the continuum map at 850 $\\mu $ m obtained with the Submillimetre Common User Bolometer Array (SCUBA) of the 15m-diameter James Clerk Maxwell Telescope (JCMT), from [19].", "The SCUBA observations have a HPBW of 22.9$$ , similar to our IRAM 30m observations.", "The main protostellar activity in the 05358 protocluster is located in the mm1/mm2/mm3 region, which matches with the SCUBA continuum peak, and where multiple molecular outflows have been observed ([5]).", "The source mm1 corresponds to one of the HMPOs studied in the previous section, and exhibits compact molecular emission with hot core chemistry ([32]; and Colzi, priv.", "communication).", "The source mm3 is one of the warm HMSCs of our sample, and it is considered to be pre-stellar because no compact line emission has been detected in this position.", "The evolutionary state of mm2 and the recently discovered mm4 ([15]) are still unknown.", "As shown in the Figure REF , the positions of mm1 and mm3 are separated by only 8.3, while the beam of the IRAM 30m observations is $\\sim $ 26.", "Therefore, as previously discussed, it is likely that the spectra from the warm HMSC in mm3 is contaminated by the nearby protostellar activity.", "Figure REF shows that the integrated emission of DC$_3$ N and the $^{13}$ C-isotopologues of HC$_3$ N are extended, larger than the IRAM 30m beam.", "To measure the area of the emitting regions we have considered the contour in which the emission falls to a 40$\\%$ of the peak emission.", "The emission area of DC$_3$ N and the $^{13}$ C-isotopologues are $\\sim $ 1446$^2$ and $\\sim $ 2042$^2$ , respectively.", "These emitting areas correspond to those of circular regions with diameters 0.37 pc and 0.44 pc, respectively.", "In both cases the sizes are larger than that of the IRAM 30m beam ($\\sim $ 0.26 pc).", "Therefore, the assumption of emitting region filling the telescope beam used in the previous section is fulfilled, at least, in the case of the 05358 region.", "The integrated map of the $^{13}$ C-isotopologues of HC$_3$ N peak in the region located between mm1/mm2 and mm4, slightly shifted with respect to the peak of the SCUBA continuum by $\\le $ 10.", "This shift is likely not significant, since it is less than half of the beams of the molecular and continuum observations, and considering some uncertainty of the pointings of the different telescopes.", "Interestingly, the morphology of the emission of DC$_3$ N is not fully coincident with that of the $^{13}$ C-isotopologues.", "Figure REF shows that the overall DC$_3$ N emission is shifted towards SW, with two main peaks, hereafter D1 and D2, not coincident with the peak of the $^{13}$ C-isotopologues nor with the SCUBA continuum peak.", "Since no signs of active star formation towards D1 and D2 have been reported, we considered them as cold HMSCs, and added them to the sample of sources (Table REF ).", "From the datacube, we extracted the spectra of circular regions centered at the D1 and D2 positions with diameters matching the IRAM 30m beam.", "We performed the analysis of DC$_3$ N, HC$^{13}$ CCN and HCC$^{13}$ CN using the same procedure described in the previous section.", "The values of $T_{\\rm kin}$ were derived from NH$_3$ (as for the other cores of the sample), using the VLA interferometric maps from [35], and integrating in a circular area with a diameter matching the IRAM 30 beam.", "The values ontained for D1 and D2 are 26 and 20 K, respectively, lower than that considered for mm1 and mm3 (39 K and 30 K, respectively; Table REF ), which supports their cold starless nature.", "The derived values of $D_{\\rm frac}$ are 0.020$\\pm $ 0.011 and 0.019$\\pm $ 0.011 for D1 and D2, respectively (Table REF ).", "These values are higher that those found towards mm1 and mm3, 0.005$\\pm $ 0.02 and 0.008$\\pm $ 0.03, respectively.", "Table: Total DC 3 _3N column density and D-fractionation of DC 3 _{3}N. Values without errors are fixed in the fit procedure.Figure: D frac D_{\\rm frac} of HC 3 _3N as a function of the kinetic temperature T kin T_{\\rm kin} towards the sample of high-mass star-forming regions studied in this work.", "The different colours of the circles correspond to different evolutionary groups, as indicated in the upper right corner.", "Upper limits of D frac D_{\\rm frac} are denoted by triangles pointing downwards, while the only lower limit is denoted by the triangle pointing upwards." ], [ "Discussion", "It is still not clear how DC$_3$ N is formed in star-forming regions.", "[31] suggested several ion-molecule reactions that can form it during the cold pre-stellar phase: ${\\rm H_2D^+ + HC_3N \\rightarrow HDC_3N^+ + H_2~,} \\\\{\\rm HDCN^+ + C_2H_2 \\rightarrow HDC_3N^+ + H_2~,} \\\\{\\rm H_2CN^+ + C_2HD \\rightarrow HDC_3N^+ + H_2~,}$ and ${\\rm HDC_3N^+ + e \\rightarrow DC_3N + H~. }", "$ Other possible chemical pathway might be directly linked to the formation of HC$_3$ N itself.", "Many observational works, from the study of its $^{13}$ C isotopologues, have suggested that HC$_3$ N is formed through the gas-phase neutral-neutral reaction between acetylene (C$_2$ H$_2$ ) and CN (e.g.", "[49], [50], [33], [3], [51]).", "Therefore, this pathway might also contribute to form DC$_3$ N via: ${\\rm C_2HD + CN \\rightarrow DC_3N + H~.", "}$ An alternative route would imply surface chemistry through D-H exchange reactions after the freeze-out of HC$_3$ N, as proposed by [11] for other carbon-chain species like cyclopropenylidene (c$-$ C$_3$ H$_2$ ).", "A fourth possibility was suggested by [22], based on gas-phase hot chemistry during the protostellar phase.", "We discuss in the following how our detections of DC$_3$ N towards a sample of massive cores can help us to understand how and when HC$_3$ N is deuterated during high-mass star formation.", "The comparison of $D_{\\rm frac}$ of different species in our sample (Figure REF ) shows that HC$_3$ N is better correlated with N$_2$ H$^+$ and DNC than with NH$_3$ and CH$_3$ OH.", "Interestingly, these species are formed only (the former) and predominantly (the latter) in gas phase.", "The opposite case is represented by CH$_3$ OH and NH$_3$ , which are produced efficiently on grain surfaces through hydrogenation of N and CO, respectively.", "The (weak) correlation of HC$_3$ N with the species formed in gas phase, and the null correlation with species formed on grains, supports that the deuteration of HC$_3$ N is likely produced through gas-phase chemistry rather than surface chemistry.", "We have not found a clear correlation between the presence of DC$_3$ N and the evolutionary stage, since we detected it in 2 cold HMSCs, 3 warm HMSCs, 4 HMPOs and 6 UC HII regions.", "Supporting this lack of dependence with the evolutionary phase, we do not find a correlation between $D_{\\rm frac}$ and the kinetic temperature $T_{\\rm kin}$ of the sources (Figure REF ).", "We note that although the non detections of DC$_3$ N in the sample correspond mainly to cold HMSCs (Table REF ), this does not imply necessarily that deuteration is lower in this early evolutionary stage.", "These non detections are likely due to sensitivity limits of the observations, since in most of the cases, the $^{13}$ C-isotopologues are also not detected.", "We have checked that the 6 cold HMSCs with no detection of any of the isotopologues of HC$_3$ N are those with weaker line intensities in other molecular species.", "As an example, the N$_2$ H$^+$ observations by [23] showed that these 6 sources have integrated line intensities $<$ 3 K km s$^{-1}$ , while the other HMSCs (where DC$_3$ N is detected) have 7$-$ 43 K km s$^{-1}$ .", "Another factor that might limit the detection of DC$_3$ N in cold HMSCs is spectral dilution of the line profiles.", "These cores exhibit narrow linewidths in other deuterated species, e.g.", "0.5$-$ 1.65 km s$^{-1}$ in the N$_2$ D$^+$ (2-1) transition ([23]).", "These linewidths are similar or slightly larger than the spectral resolution of the observations ($\\sim $ 0.65 km s$^{-1}$ ), which may produce some dilution of the line intensities.", "The presence of similar levels of deuteration of DC$_3$ N in cores with different evolutionary stages does not favor the hypothesis of formation on the surface of dust grains, or at high temperatures during the protostellar phase.", "In these two scenarios, the DC$_3$ N emission should be predominantly concentrated around the protostars, and higher values of $D_{\\rm frac}$ should be expected in the protostellar phase, due to thermal desorption and hot chemistry, respectively, triggered by the protostellar heating.", "However, we do not find any preference towards protostellar cores.", "Moreover, the DC$_3$ N map towards the 05358 protocluster (Figure REF ) shows that the emission is extended in the region ($\\sim $ 0.37 pc), with the main deuteration peaks (D1 and D2) shifted with respect to the dust continuum peak where the protostellar activity is on-going.", "Furthermore, hot chemistry in the hot core phase seems unlikely, since it likely too slow to significantly modify the deuteration levels after the desorption of grain mantles ([12] and [43]).", "Alternatively, the relatively constant level of HC$_3$ N deuteration during the different evolutionary phases can be naturally explained if DC$_3$ N mainly arises from material in the outer parts of the clumps, which is common to pre-stellar and protostellar cores.", "In a region with on-going star formation, only the inner gas is heated by the protostar, while the outer envelope remains cold and nearly unaltered from the pre-stellar phase (e.g.", "[1]).", "The lower values of $D_{\\rm frac}$ of HC$_3$ N with respect to other species that mainly trace the inner regions of the cores, such as N$_2$ H$^+$ or NH$_3$ (Figure REF ), points also towards an origin of DC$_3$ N in the outer envelope, where the gas is less dense, and hence deuteration is less efficient.", "To further support this hypothesis, we have compared the linewidths of DC$_3$ N with those of N$_2$ D$^+$ and o$-$ NH$_2$ D (from [23] and [25], respectively).", "We show in Figure REF that the linewidths of DC$_3$ N are in most cases larger than those of N$_2$ D$^+$ and o$-$ NH$_2$ D, as expected if the former traces the diffuse outer part of the envelope, and the two latter trace the inner compact regions.", "We note that the few cases in which N$_2$ D$^+$ or o$-$ NH$_2$ D have larger linewidths than DC$_3$ N might be explained by broadening produced by protostellar activity such as molecular outflows.", "In the external part of a star-forming region, the interstellar radiation field (ISRF) is higher, since it is less shielded by the presence of dust.", "This allows that a higher fraction of carbon is in atomic form, which starts a more rich carbon-chain chemistry.", "This effect has been detected observationally towards the L1544 pre-stellar core, where many carbon-chain molecules, including HC$_3$ N, arise from the external regions of the core, avoiding the inner dust peak ([48]).", "We have observed a similar behaviour in the 05358 protocluster, where the emission of the $^{13}$ C-isotopologues of HC$_3$ N, and mainly DC$_3$ N, are shifted with respect to the dust continuum (Figure REF ).", "Thus, all the observational evidence presented in this work suggest that DC$_3$ N might be a good tracer of deuteration fraction in star-forming regions prior the formation of the denser gas, which is an important parameter in chemical models to constrain, for instance, the ortho-para-H$_2$ ratio ([45], [30], [7]).", "New studies of deuteration of other carbon-chain species (e.g.", "HC$_5$ N, C$_2$ H or c-C$_3$ H$_2$ ) in large samples of regions like the one used in this work will serve to better understand how the deuteration of this family of species proceeds during the star formation process.", "Finally, in Figure REF we compare the $D_{\\rm frac}$ values found in our sample of high-mass cores, divided by evolutionary stage, with those reported in the literature towards other interstellar sources (low-mass dark clouds, low-mass protostars, and previous upper limits in high-mass star-forming regions; see references in FigureREF ).", "We found values in the interval 0.003$-$ 0.022 in high-mass cores, which are consistent with the previous upper limits found in Orion KL and SgrB2 N2 hot cores: 0.015 and 0.0009, respectively ([22], [4]).", "These values of $D_{\\rm frac}$ are lower than those found in protostellar low-mass stars (0.025$-$ 0.5), and pre-stellar low-mass dark clouds (0.03$-$ 0.13), as shown in Figure REF .", "This might be due to slightly higher temperatures of high-mass clumps, which would make less efficient the ion-molecule gas-phase reactions responsible of the formation of DC$_3$ N. However, we stress that the comparison between low- and high-mass cores should be done with caution, since the observations are tracing very different linear size scales.", "As previously mentioned, our observations have likely detected the DC$_3$ N arising from the external envelope of the clumps, rather than the inner cores.", "Therefore, future interferometric observations with high-angular resolution are needed to understand if and how DC$_3$ N is formed at smaller core scales ($\\sim $ 0.01 pc).", "Figure: Comparison of the FWHM values of DC 3 _3N with those of ortho--NH 2 _2D(1 1,1 -_{1,1}-1 0,1 _{0,1}) (open circles, from ) and N 2 _2D + ^+(2--1) (filled circles, from ).", "The different colors correspond to different evolutionary groups.", "The shaded area indicates the parameter space where the FWHM of DC 3 _3N is larger than those of the other deuterated species.Figure: D frac D_{\\rm frac} of HC 3 _3N in different interstellar sources: low-mass dark clouds from (orange circles); low-mass protostars from , , , , (magenta circles); high-mass star-forming regions from this work (different colors denote different evolutionary groups, as indicated above the panel; upper limits are denoted by triangles pointing downwards, while the only lower limit is denoted by the triangle pointing upwards); and previous upper limits (gray triangles) from , , ." ], [ "Conclusions", "We present the first study of DC$_3$ N toward a sample of massive cores in different evolutionary stages, from pre-stellar to protostellar phases.", "We detected the DC$_3$ N $J$ =11-10 transition towards 15 regions, which include 2 cold High Mass Starless Cores (HMSCs), 3 warm HMSCs, 4 High Mass Protostellar Objects (HMPOs) and 6 Ultra Compact HII regions.", "We found values of $D_{\\rm frac}$ (abundance ratio of DC$_3$ N with respect its HC$_3$ N) of 0.003$-$ 0.022, lower than those found in pre-stellar and protostellar low-mass star-forming regions.", "We do not find any correlation between $D_{\\rm frac}$ and the evolutionary stage of the cores, or with the kinetic temperature.", "The comparison with other deuterated species previously studied toward the same sample indicates a weak correlation of $D_{\\rm frac}$ in those species formed only or predominantly in gas phase (N$_2$ H$^+$ and HNC, respectively), and no correlation in species formed only or predominantly on dust grains (CH$_3$ OH and NH$_3$ , respectively).", "We also present the first map of DC$_3$ N in a high-mass star-forming region, the protocluster IRAS 05358+3543.", "The DC$_3$ N emission is extended ($\\sim $ 0.37 pc), and it is shifted with respect to the dust continuum peak where the protostellar activity is on-going.", "Our observational evidence indicates that DC$_3$ N is likely formed by gas-phase ion-molecule reactions in the outer and less dense part of star-forming clumps, where the interstellar radiation field keeps most of the carbon in atomic form, enhancing the formation of carbon-chain molecules such as HC$_3$ N. Thus, DC$_3$ N might be a good tracer of the deuteration level in star-forming regions prior to the formation of denser gas." ], [ "Acknowledgements", "We acknowledge the anonymous reviewer for her/his careful reading of the manuscript and her/his useful comments.", "V.M.R.", "has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 664931.", "LC acknowledges support from the Italian Ministero dell’Istruzione, Università e Ricerca through the grant Progetti Premiali 2012 - iALMA (CUP C52I13000140001).", "This work is based on observations carried out under projects number 129-12 and 040-19 with the IRAM 30m telescope.", "IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).", "This publication has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 730562 [RadioNet].", "L.C.", "and F.F.", "thank the IRAM staff for the precious help provided during the observations.", "We warmly thank Xing \"Walker\" Lu and Qizhou Zhang for sharing with us their VLA map of the 05358+3543 high-mass protocluster." ] ]

[ [ "IMDb data from Two Generations, from 1979 to 2019; Part one, Dataset\n Introduction and Preliminary Analysis" ], [ "Abstract \"IMDb\" as a user-regulating and one the most-visited portal has provided an opportunity to create an enormous database.", "Analysis of the information on Internet Movie Database - IMDb, either those related to the movie or provided by users would help to reveal the determinative factors in the route of success for each movie.", "As the lack of a comprehensive dataset was felt, we determined to do create a compendious dataset for the later analysis using the statistical methods and machine learning models; It comprises of various information provided on IMDb such as rating data, genre, cast and crew, MPAA rating certificate, parental guide details, related movie information, posters, etc, for over 79k titles which is the largest dataset by this date.", "The present paper is the first paper in a series of papers aiming at the mentioned goals, by a description of the created dataset and a preliminary analysis including some trend in data, demographic analysis of IMDb scores and their relation of genre MPAA rating certificate has been investigated." ], [ "Introduction", "Since the Web 2.0 era the internet usages has been revolutionized, more people accessing with higher bandwidth make the user from almost mere observers to creators and participants.", "Excluding private cloud storage services, the platforms created and/or owned by giants companies such as Google, Twitter, Facebook, Amazon, etc has created a gigantic data warehouse.", "To make this data usable for analysis the extracting, processing and organizing the is the very first and essential step.", "Internet Movie Database (IMDb) is an online database dedicated to all kinds of information about a wide range of motion picture contents such as films, TV and online-streaming shows, series, etc.", "The information which is presented on the IMDb portal includes cast, production crew, personal biographies, plot summaries, trivia, ratings, and fan and critical reviews and much other similar information which are mostly provided by volunteer contributors.", "To contribute, registration is required, however since no legal document is required, one is able to use an arbitrary name.", "Being a user-regulated website could be a shortcoming as it would be vulnerable to malicious attempts from a certain group to bias information.", "However, taking advantage of a large community not only overpower these attempts but also create a cornucopia of valuable data that analyzing them may shed light on many hidden factors that help movie industries and other related businesses in content production.", "There are various studies have been done on IMDb data.", "Oghina and et al.", "investigated the possibility of prediction of IMDb rating using social media contents such as tweets and YouTube comments [1].", "Otterbacher showed there is a tangible difference between men and women's review writing style using the IMDb review section [2].", "In [3] the connection between user voting data and economical characteristics of films such as budget and box office data has been investigated by Wasserman et al.", "Hsu et at., using linear combination, multiple linear regression, neural networks predicted the IMDb rating from other movie's attributes using 32968 titles [4]; and In [5] Nithin et al.", "used Logistic Regression, SVM Regression and Linear Regression to predict box office data.", "In [6], using available demographic information on IMDb Bae et al.", "created a demographic movie recommender system.", "Ramos et al.", "showed the distribution of votes showed a scale-free behavior [7].", "There are various datasets available each with a different policy.", "IMDb itself discloses a subsets their data for personal use.", "Furthermore, there are more dataset available freely on kaggle such as IMDb movies extensive dataset IMDB Dataset of 50K Movie Reviews , Also there are other which are required corresponding with the owner.", "Seeing that many datasets available online usually do not cover some important information or they are not large enough, we determined to create a dataset that covers some drawbacks that exist in the available sets.", "Still, the other datasets could be used as a complement.", "The present paper is the first paper in a series of papers aiming to create a suitable dataset, analyze it, and predict some information using those data." ], [ "Available Data", "The created dataset is based on the data available on IMDb website and some third-party datasets and resources to provide some additional information on the available data on IMDb, such as similarity of countries and languages or how much a certain actor is talked about comparing other using the number of google results.", "The data mainly extracted from IMDb Portal, IndexMundi ,Elinguistics, and Google results in a specific field of data.", "This section is dedicated to the description of gathered data from the IMDb database.", "The full description of the data is available at https://help.imdb.com.", "To learn about the gathered and processed data from IndexMundi and Elinguistics you may refer to Appendix REF .", "To access each title, we used the code which IMDb assigned uniquely to each title.", "The code started with a double t -“tt”- followed by some numbers, for example this code for the title Logan (2017) is “tt3315342”.", "Using this code one can have access to the title's main page, for example the address for the title Logan (2017) would be like https://www.imdb.com/title/tt3315342/.", "The main portion of extracted data is from the title's page and some relative addresses from that page, for instance, the rating data extracted from the relative address of /ratings of each title page ,e.g.", "https://www.imdb.com/title/tt3315342/ratings for the title Logan (2017)." ], [ "Movie Name", "The Movie name is the name which was given to each title by the producer, we found a few minor discrepancies on the titles from different part of IMDb.", "Here our reference is the name on the designated page for each title." ], [ "Poster", "There are several posters associated with each title.", "Here the main poster which has been presented on the title's page, is stored." ], [ "Alternate titles (AKAs)", "Alongside the original title, every film may have other titles or names that are known with, either in different countries and/or languages; in this case, alternate titles may be listed.", "Default alternate title is the same as the primary one [8].", "The alternate titles could be a small deviation from the original name and/or be in other languages rather than the film's language[s]; for example for the movie Logan (2017) the alternate titles are mainly are the original title plus Wolverine which is sometimes in different languages rather than its original language, English.", "In this case, the regular NLP analysis may not give any insightful results, however, the number of the alternated title could be an interesting factor.", "It could somehow show how much people and/or different nations care to give the movie their own names." ], [ "IMDb ratings and Number of votes", "Every user can vote from 1 to 10 to rate each title, there is no need to writing a review upon giving the score.", "A weighted average of the registered users will be shown as the title's rate.", "IMDb's intention is to reduce the intended attempts to change the title rating from actual worth.", "Various filters are applied for this propose and IMDb does not disclose the math [9].", "However the arithmetic mean is also available in the relative address of /ratings for each title.", "Moreover, the voting distribution histogram and demographic information of rating and number of votes are also available.", "Here demographic information contains the top 1,000 voters information, US and none-US users, and different age and genders.", "The top 1,000 voters are the top 1,000 who have voted the most titles and are unknown [10].", "For the rating section, the IMDb's rating, the arithmetic mean of rating, median, and all the demographic information about the rating (by age, sex, and information on top 1000 users, US and Non-US users) and the number of votes have been gathered." ], [ "Metacritic Score and User/Critics reviews", "Besides the rating, the metacritic score and user and professional critics reviews are available, so one could be informed of other viewers' opinions [9].", "At first glance, the semantic analysis of each review seems to be the only way to use this information.", "However, the number of reviews could be a helpful factor to validate the user's ratings.", "Despite the reviews could be biased, ignoring various downfalls of the title, especially the one written by users rather than a renown critic, the number of them could be showing how much the title worth to people dedicate their time to write about, after watching the movie.", "On the other side, the votes could be blind votes which are given by particular groups very high or low, without watching the movie as it happened for The interview (2014) which at the beginning of its release get a near-perfect score [11].", "Not only blind voting causes a problem, but also die-hard fans of some genres like Sci-Fi, ignoring major flaws, could also have very biased voting, However, after a given period of time the effect this attempt will smooth out.", "On the other hand, writing a review is less impulsive action and needs more contemplation, and of course being a fan of a genre won't be enough to write the reason why an individual liked/disliked a title." ], [ " Popularity and change ", "The popularity ranking on a title separately compares movie titles with each other [12].", "Here the popularity and its changes at the time of extraction have been stored." ], [ "Motion Picture Rating, IMDb Certificates", "To specify the appropriate audience for each title IMDb provides the Motion Picture Rating (MPAA) certificate.", "Explanations for the available entries are could be found at [13].", "Each country has its own MPAA system and/or age restriction for each title.", "Here the rating certificate given to each title within the United States has been considered as the reference.", "The information about other countries also extracted from relative url of /parentalguide for each title." ], [ "Parental Guide", "IMDb includes parental guide entry to provide the parents with additional information by describing some scenes to determine the appropriateness of each title[14].", "All the information is available in the relative address of /parentalguide of each title.", "The entries include Sex and Nudity, Violence and Gore, Profanity, Alcohol, Drugs, and Smoking, and Frightening, and Intense Scenes.", "Here just the number of scenes ( and not the description) and, if it was available, the degree of severity (Mild, Moderate, Severe) are extracted." ], [ "Genres", "There are several genres, which each title may associate with one and more.", "For the full description you may refer to [15]." ], [ "Countries and Languages", "Country is defined as the country where the production company is based.", "It is possible multiple companies are associated with each title [16].", "The languages which are spoken in each title are listed in order of frequency [17]." ], [ "Release Dates and Locations, Filming Dates and Locations ", "Release dates and locations have been gathered from the relative address /releaseinfo of each title.", "This portion of data could be an indicator of the potential popularity.", "For example, if the title released in different countries in a small time window it may be a sign for its popularity.", "Moreover, the filming dates and locations have been extracted from relative address of /locations .", "The filming locations could be a good indicator for the budget class of the movie especially when no data is available on the budget." ], [ "Box Office data - may need to add", "The extracted data here are: Budget, Opening Weekend USA Income, Opening Weekend USA, Gross USA, Cumulative Worldwide Gross." ], [ " Director, Writers, Stars ", "Director, writers, stars, and roles are also extracted.", "There is an elaborate list for each of them available but at this point, for the sake of simplicity, the first names on the main page of each title are stored.", "To machine they are some random string.", "Plus, there are not a lot of data to assign them a value or a vector with techniques such as Word2Vec.", "Some datasets are containing the number of Facebook page's likes for each actor or similar information like this dataset on kaggle.", "However the size of these datasets is limited and does not cover all the names that are needed here.", "Here we have taken another strategy and used the number of google results.", "To avoid name similarity we used the profession alongside the name to narrow down the results as much as possible; for example we searched Tom Hanks + movie star, or Steven Spielberg + director." ], [ "Production Companies", "The list of production companies has been extracted from the relative address of /companycredits of each title." ], [ "Related movies", "Up to twelve similar titles are suggested under the “More like this” entry.", "These titles are generated based on various information such as genres, country, stars, etc [18].", "Here we also extracted the IMDb rating, number of votes, and the IMDb code for each related title." ], [ "Keywords and Storyline", "There are also storyline plot and keywords available.", "This data is valuable to this extent that reveals the key and unique elements which are presented in the movie.", "The keywords are offered by users and they can vote if they are relevant or not.", "Here we gather all the keywords sorted by a relevancy score from the relative address of /keywords which is calculated by this relation: $\\text{Number of votes}\\times \\frac{\\text{Number of positive votes}}{\\text{Number of votes}}$" ], [ "Data Cleaning and Processing", "Here we briefly describe the pre-processes and labeling format that is essential to know before using the data." ], [ "Data Format", "Data is packed according to the release year of each title for better management.", "All the data are stored in a CSV file with UTF-8 encoding.", "The index of the table has been set to its unique IMDb code.", "Using the IMDb code as the index could be beneficial during the model training since it uniquely determines the title it does not contain specific information that could be used during the analysis or model training to be a part of the table.", "Moreover, there is a subdirectory for each year containing the film's poster in jpg format each with the dimension of 182$\\times $ 268, 72 DPI.", "The size of the data is around 5-25 Mb for the CSV file and 10-15 kb for each poster image.", "There are 79793 rows of data, and 67393 poster files are available in total." ], [ "Columns' names", "Since heavily relying on column numbers in the middle of analysis could be confusing, especially here which data are packed according to the release date of the titles and the number of columns may vary.", "Consequently, we introduce a specific wildcard access data columns.", "Including those patterns enable the users to search with Regular Expressions (RegEx) to narrow down the list of columns to the specific part of the table.", "Here we used capital letters at the end of each column name to distinguish them from the actual name of each column; since the multi-parted names are accompanied by underscore, using python regular expression has been made easy.", "Here we will briefly describe the wildcards' meaning *_GS GS stands for General Set, which contains general information about the title such as the name and alternate names, technical information like runtime sound mixing, the plot, keywords, related movies, filming locations and companies, etc.", "*_GENRE This wild card is related to information about the genre.", "Since Each title's genre does not necessarily fall into one category, here we created two sub-wildcard of *_SET_GENRE for a complete set of genre and *_HOTVECTOR_GENRE for their hot vector representation *_COUNTRY With this wildcard you may access the country information of each title.", "There are two sub-wildcards are also available *_SET_COUNTRY *_HOTVECTOR_COUNTRY for list of country and hot vectors of countries respectively.", "Here we included two quantized information about the country; the reference of comparison has been chosen the United States as the creator of the most titles each year.", "In *_NONGEO_DIS_COUNTRY the mean Manhattan distance between 106 parameters has been calculated, for more information about this analysis please refer to Appendix REF .", "*_GEO_DIS_COUNTRY provides information about the geographical distance by calculating the great-circle distance between the country's capital from Washinton DC using haversine formula.", "*_LANGUAGE This wildcard related to languages which are spoken in the original version of each title.", "*_SET_LANGUAGE includes list of spoken languages with descending order of usage frequency.", "*_HOTVECTOR_LANGUAGE is the hot vector of languages.", "Language comparison to English is stored in *_ENGLSIH_DIFF_LANGUAGE column.", "*_GOOGLE_RES_LANGUAGE contains the number of google search results.", "It is abundantly clear that the exact number is not a good reference but its order of magnitude would give an idea of how much a language is spoken about relative to another.", "Although the number of people who are speaking a certain language as the first and/or second language also might be a good option to assign a meaningful value to each language, however, we hadn't found any resource for all the languages.", "*_BOXOFFICE This wildcard is the data related to Boxoffice, Please note that the Currency is not converted to their today's value.", "*_DWSC This wildcard is related to Directors, Writes, Stars and their roles, and Production Companies.", "There is a comprehensive list for each field but here the list of names is restrained to the names which are appeared on the main title page.", "Also the number of google results for directors, writers and stars are included in sub-wildcard of *_GOOGLE_RES_DWS *_RATING *_RATING is associated with the voting, the rate and the number of votes.", "The general information such as the total number of votes, arithmetic mean rating, and IMDb rating and median of votes can be found using *_G_RATING wildcard.", "The sub-wildcard related to the distribution of voting are *_NUM_DIST_RATING, *_PERCENT_DIST_RATING, which are assigned to the number of specific vote and the percentage respectively.", "US and Non US voters, sore and number of vote are accessible using the wildcards of*_SCORE_GIS_RATING, and *_NUM_GIS_RATING.", "Top users score and number of votes are in Top_1000_Voters_SCORE_DEMOGRAPHIC_TOP_RATING , Top_1000_Voters_NUM_DEMOGRAPHIC_TOP_RATING columns.", "For all Ages and gender and/or separately sorted by age intervals and gender the wildcards of *_SCORE_DEMOGRAPHIC_AG_RATING, *_NUM_DEMOGRAPHIC_AG_RATING are used to access the score and number of votes respectively." ], [ "Data access", "To access the data you may contact us.", "Moreover, to have a glimpse of how data looks like, a portion of data is available at this repository." ], [ "Preliminary Analysis and Discussion", "This section aims to demonstrate an overview of data.", "Here we are going to study some trends from 1979 to 2019.", "Moreover, we are going to study the effect of other factors such as genre and parental guide information on the IMDb rating, distribution of ratings, and demographic information of ratings." ], [ "Trends ", "There are two types of information available on IMDb; one of them is related to attributes of each title such as the runtime, genre, etc, another is created by users' activities such as voting.", "An important point is that the first type could be assigned to the release year but the latter could not.", "The votes, For instance, could be cast in any year so speaking about the trends on this portion data should be interpreted as the scores that are given to the title released on a specific year, not the scores that are given within that year.", "In this study, the titles with a vote number larger than 100 have been considered.", "The number of titles with this condition grows every year.", "However, it drops after a peak in 2017, by around 300 and 1000 numbers for 2018 and 2019 respectively, Fig REF .", "Although it might be counter-intuitive, it could be an indicator that for the threshold of 100 vote numbers it takes at least three years for movies to follow the expected trend.", "The extrapolated values for 2018 and 2019 are 5007 and 5295 respectively.", "The average of the number of votes is ascending till 2010, and drop by 7000 numbers in 2019 (Fig REF ).", "This is also an indicator that vote numbers need a long time to follow the expected trends.", "The extrapolated value for 2019 is 18182; however, the changes seem to be more drastic than the number of titles.", "For example, the extrapolation polynomial prediction value for the year 1994 is 10528 which is lower than its actual value, 15122.", "The Fig REF also demostrates that the males' mean vote numbers are greater than the females'.", "Most of the votes belong to the category of males between 30-44 and female voters under 18 are the smallest category.", "There are no significant changes in trends except for the category of 18-29 males, overpassed the males over 45 in 1998.", "The under 18s are the minor portion of voters, which sound reasonable because of the restrictions for the title they can watch and using the internet.", "Another interesting trend in the data is the increasing difference between arithmetic mean and IMDb rating by 0.16 score.", "Moreover, the difference between male and female average scores has had 0.24 increment from 1979 to 2019.", "The variance of votes casted by different age category has a decreasing trend until 2009 and after that, it has an increasing trend, Fig REF ." ], [ "IMDb Scores", "The scores are given by females are higher than males Fig REF ; the average of scores given by men is 5.93 but women is 6.13.", "Their trends are different; despite the persistent decreasing trend of men's mean scores, women's votes after a descending trend has increased in from 6.02 in 2009 to 6.20 in 2019.", "This could be a sign for emerging of more politically correct content and increasing the role of women in the movie industries.", "The highest scores are given by males under 18 and the lowest scores are given by males over 45 years old.", "The score given by female voters showed fewer changes in different age categories." ], [ "Top three Languages", "The languages: English, French, Hindi, Italian, Japanese, Spanish show up as the top three languages from 1979 to 2019.", "English is always the most frequent and its number growing each year.", "However, its percentage decreased from 35.4% in 1979 to 18.6% in 2019.", "The other aforementioned languages ranked as second and third interchangeably, except for the Italian which after 1995 has not been among the top three most frequent languages.", "The top three most frequent countries during 1979-2019 were: France, Germany, India, Italy, UK, USA.", "The USA always ranked first with increasing the number of released titles each year.", "However, its share decreased from 30.1 to 17.5 percent.", "The second and third place are received by other countries interchangeably.", "The only significant trend dropping the number of titles produced by Italy from 1988, and the jump of Germany' after 1989." ], [ "Genre and IMDb Certificates", "Drama and Comedy were two first genres and Thriller, Horror, Romance, and Action are received the third place interchangeably in different years.", "Each genre has specific behavior which could be caused by many reasons such as a popular actors or directors or popular stories.", "There are some distinguished trends like increasing the percentage of documentary and short films overtime and descending trend of fantasy genre since 1994.", "The MPAA certificate of R was the most frequent.", "before 2014, PG-13, and after that, TV 14 comes as second and the third place is received by PG." ], [ "Analysis of Movie Rating", "One of the most important features in the IMDb database is the IMDb rating.", "Moreover, distribution and demographic structure of data divided by age and gender and location of voters, and being among the top 1000 voters are available.", "Here we are going to study IMDb rating alongside other parameters such as genre, rating certificate, and other parental guide information." ], [ "Demographic analysic of IMDb rating scores ", "Most of the votes, in descending order of vote numbers, are from the age category of 30-44, 18-29, over 45, and under 18.", "Each genre receives the highest rating, in descending order of scores, from males under 18, 18-29, 30-44, and over 45 and females 18-29, under 18, 30-44, and over 45.", "The distribution of voting relative to age and sex is demonstrated in the Fig REF .", "As it can be inferred from Fig REF the female voters are slightly prone to submit higher scores than males.", "Moving from younger age interval to older, the tendency of giving a very high or very low score for both genders decreases.", "However, females' changes are slightly less than the males'.", "Fig REF encapsulates the information about the correlations of scores for two age intervals (first row of each cell) and their respective percentage in the total population (second row of each cell); the least correlation for scores is between over 45 years old males and under 18 females and the most correlation is between females between 18-29 and 30-44.", "The difference between the maximum and minimum of correlations between men is 0.09 and between women is 0.15.", "An interesting point between All these data here is that either we look at the auto-correlation of each gender or cross-correlation of males and females, the most corrected part of each block belongs to age categories of 18-29 and 30-44 Which according to the second row of each cell in Fig REF , create the largest portion of the voters' populations.", "IMDb rating tends to be more correlated with Top 1000 users score than Arithmetic mean.", "In both IMDb rating and Arithmetic mean of votes are more correlated with Non-US's scores than US user, however, correlation of IMDb ratings has become more correlated with the Non-US voters than US voters Moreover, IMDb rating is more correlated with males' scores than females' scores.", "In males category, IMDb rating becomes more correlated than Arimethic mean by moving to the older age categories however in females the increments are smaller and the most changes belong to females between 30-44 years old.", "For both genders the effect of under 18-year-old voter are decreased in IMDb rating, FigREF ." ], [ "Genres and IMDb rating scores ", "The three most highly scored genres in IMDb are Drama, Comedy, and Action with the score means of 5.96, 5.91, 5.88; and the least scores are given to Musical, Western, and Sci-Fi with score means of 5.78, 5.78, 5.77.", "In each MPAA certificate, the most highly scored genres are almost the same, Drama and comedy are the most frequent genre in each certificate.", "The distribution of votes for each genre reveals a lot about the fans.", "The most frequent scores are 10, 7, and 6 without any exceptions.", "Mostly they receive 7 and after that 6.", "However, for the genre like Sci-Fi and Western and Musical, the score 10 is the most frequent.", "As males are constitute the major portion of voters the demographic gender-neutral data follows males data.", "Males' number of votes obey the same behavior similar to average behavior mentioned in the section REF for all the genre except for the Musical, and Western, males over 45 years old votes the most after the 30-44 category.", "The distribution of the scores is like the overall age distribution.", "However, females' average scores show dependence on the genre.", "The order of given scores, with descending order of mean scores, are as follows: Action, Adventure, and drama: 18-29, under 18, over 45, 30-44 years old.", "Animation, Biography, Comedy, Crime, Documentary, Family, Fantasy, Mystery, Romance, and Thriller: 18-29, over 45, under 18, 30-44 years old.", "History, Horror, Musical, Sci-Fi, Short, Sport, War, and Western: 18-29, over 45, 30-44, under 18.", "It is worth mentioning that all the order of the mean number of votes distribution is similar to the gender-neutral age categories." ], [ "MPAA rating and Paretal guide informa IMDb rating scores ", "Movies with general audience receive the highest mean IMDb rating, 6.39; and the certificates which require the minimum age of 18 years old have a minimum score of 5.6.", "Correlation between number scene containing Sex and Nudity, Violence and Gore, Profanity, Alcohol Drugs and Smoking, Frightening and Intense Scenes, Scenes show almost no correlation with the IMDb scores Fig REF (numbers are multiplied by 100).", "However, males scores are slightly more correlated." ], [ "Conclusion", "In this paper, we introduced the largest and the most comprehensive movie database created based on the IMDb dataset.", "The database contains a variety of information of over 79k titles, ranged from alternate titles, genres, MPAA rating certificates, and related movies information to demographic information on IMDb ratings.", "Other unique features that make this database special in deep learning and machine learning model training is that we tried to make it as quantize as possible.", "Alongside the name of countries, languages, actors and actresses, directors, writers, and movie companies, we assigned a number with methods that have been explained in the appendix REF .", "Moreover, the main poster of each title has been included.", "A preliminary analysis of the data has been presented.", "There were some interesting trends, for example, the difference between IMDb rating and Arithmetic mean of scores and also the difference between male and female mean scores have been increased over 1979 to 2019; in this time interval, the males' votes constitutes the major portion of voters' population.", "Males' votes average had a decreasing trend while females' started to increase after 2009; moreover, females' give a higher score than males'.", "Also, the data demonstrated that the country Italy and Italian language have not been ranked in the top three languages and countries since 1988 and 1995 respectively.", "The most frequent genres were Romance and Drama and the most frequent MPAA rating certificates were R. Analysis demographic information of IMDb rating revealed that most voters are in the age category of 30-44; and after that, the categories of 18-29, over 45 and under 18 years old are, respectively, the most voters.", "The score given by males in each category has an inverse relation by the age.", "Females also follow this pattern except the highest score comes from the category of 18-29 years old rather than under 18 categories.", "Female votes in different age categories depend on the genre but males showed a more consistent pattern.", "The most frequent votes for each genre are 7 and 6 but in some genres like sci-Fi, Western, and Musical the most frequent vote is 10.", "Study of correlation of demographic information of scores revealed that ages of 18-29 and 30-45 are the most correlated ones, either in the correlation between each gender or in the correlation of males and females.", "Two other age intervals of under 18 and over 45 show to be more inconsistent.", "The number of scenes which include Sex & nudity, Violence & Gore, Profanity, etc has an almost-zero correlation with the IMDb rating.", "The IMDb rating is calculated somehow that has become more correlated with Non-US voters, Top 1000 voters, and male voters." ], [ "Third-party data compliments", "Since part of data is in form of text, we needed to utilize an appropriate approach to turn them into numbers so we can use it in training of machine learning models.", "Despite turning them into hot-vectors might sound like the only option, we used other approaches to assign each entry a suitable value.", "Here we briefly discuss about the datasets and the process of preparation.", "As the results of some policy we are not allow to reshare some of these third-party data, thus only our results after the processing will be disclosed.", "All the data are available https://www.indexmundi.com/factbook/compare.", "Please read carefully the Term of use before using their data.", "Here we mainly used demographics information, and some information from geography and economy table.", "In total, 106 Fields of data extracted.", "All fields of data are normalized to so they are ranged from 0-1.", "Since we need to assign each country a value we calculate the geographical distance and non-geographical distance using extracted data from United States.", "The missing information was another issue; Antarctica, for instance, does not possess 96 out of 106 our data columns.", "Here we take the availability of data as similarity factor, therefore the number of missing data will increase the distance of two country.", "Here, we report the mean Manhattan distance as the non-geographical distance of countries.", "In this process the most similar, excluding geographical distance, countries was United Kingdom and the least similar was Antarctica, which sounds reasonable." ], [ "Elinguistics", "This database used to compare different languages to English.", "Despite their similarity to English, the most spoken language, could be consider as an important factor, this analogy could be misleading since two different languages from English might be highly similar.", "The reported values are from 1 to 100.", "Highly related languages, Related languages, Remotely related languages, Very remotely related languages, and No recognizable relationship receive score Between 1 and 30, Between 31 and 50, Between 51 and 70, Between 71 and 78, and Between 71 and 100 respectively.", "You may learn more about their analogy from their methodology." ], [ "Number of Google Results", "The number of google results also used to compare how much the searched key is talked about on the web.", "For the languages, stars, directors, writers, and production companies, we used number of google results.", "Figure: Number of movies with votes number larger then 100 votes, over timeFigure: Averge of votes number of male and female voters in different age categoryFigure: Trend of: difference of average of arithmetic mean and IMDb ratings, difference of male and female mean scores, variance of age categories' score over timeFigure: Average of scores given male and female voters in different age categoriesFigure: Quartiles and distributions of datings divided by age and genderFigure: Correlations of scores given by males and females in different age categoryFigure: Correlations of parental guide items with IMDb ratings, Arithmetic mean and scores given by male and female in different age category - correlation values are multiplied by 100Figure: Correlations of IMDb ratings, Arithmetic Mean and the Scores US and Non US and Top 1000 votersFigure: Correlations of US and Non-US and Top 1000 voters, Male and Females in Different age Category with IMDb ratings, Arithmetic Mean" ] ]

[ [ "Hybrid data and model driven algorithms for angular power spectrum\n estimation" ], [ "Abstract We propose two algorithms that use both models and datasets to estimate angular power spectra from channel covariance matrices in massive MIMO systems.", "The first algorithm is an iterative fixed-point method that solves a hierarchical problem.", "It uses model knowledge to narrow down candidate angular power spectra to a set that is consistent with a measured covariance matrix.", "Then, from this set, the algorithm selects the angular power spectrum with minimum distance to its expected value with respect to a Hilbertian metric learned from data.", "The second algorithm solves an alternative optimization problem with a single application of a solver for nonnegative least squares programs.", "By fusing information obtained from datasets and models, both algorithms can outperform existing approaches based on models, and they are also robust against environmental changes and small datasets." ], [ "Introduction", "Estimating the angular power spectrum (APS) of a signal impinging on an antenna array from the measured channel covariance matrix is an ill-posed problem with important applications in massive MIMO systems, including pilot decontamination [1], channel covariance matrix estimation in frequency division duplex (FDD) systems [2], [3], [4], [5], [6], [7], and localization [8], among others.", "Current approaches for APS estimation can be divided into two main groups: model based methods [2], [3], [4], [7] and data driven methods [9].", "Model-based methods are able to produce reliable estimates with little side information, no training, and potentially low computational complexity [2], [3], [4], [1].", "However, they do not exploit any information from datasets to improve the estimates or to gain robustness against measurement errors or model uncertainty, or both.", "In contrast, pure data-driven methods can provide good performance without any knowledge about physical models, but their robustness against changes in the propagation environment (i.e., the distribution of the APS) is not acceptable for many applications.", "Furthermore, even if the environment does not change, in general these methods are heuristics that do not provide any guarantees that the APS estimates are consistent with measured covariance matrices.", "In other words, using an APS estimate in the forward problem that computes the covariance matrix from the APS may not reproduce the measured covariance matrix accurately, and we note that this type of consistency is important to bound errors in some applications, such as the error of channel covariance matrix conversion in FDD massive MIMO systems [4], [10].", "Against this background, we propose algorithms that use datasets to improve the estimates obtained with model-based methods, without unduly losing robustness against environmental changes.", "To this end, we start by revisiting existing algorithms for APS estimation to establish their equivalence and to understand their limitations.", "In particular, using common assumptions in the literature, we prove that some of these algorithms solve equivalent optimization problems (Proposition REF ), in the sense that the set of solutions is the same.", "However, this set is not a singleton in general, so the performance of these existing algorithms can differ significantly because they may converge to different solutions.", "Nevertheless, we show in this study that nonuniqueness of the solution can be exploited with the paradigm of hierarchical optimization [11], [12] to improve the quality of the estimates.", "More precisely, from the set of solutions to the existing problem formulations, we select an estimate that least deviates from the expected value with respect to a Hilbertian metric learned from datasets; namely, the Mahalanobis distance.", "The unique solution to the resulting problem is then reinterpreted as a projection onto the set of fixed points of a proximal mapping, and it is computed via Haugazeau's algorithm [13].", "As an alternative to this iterative method, we also pose an optimization problem that can be solved with a single application of a solver for nonnegative least squares problems.", "Simulations show that the proposed techniques outperform previous algorithms in some scenarios, and they can be made robust against changes of the distribution of the APS, which is one of the major limitations of data driven methods, and, in particular, neural networks." ], [ "Preliminaries", "Hereafter, by $(\\cdot )^t$ , $(\\cdot )^H$ , and $(\\cdot )^\\dagger $ we denote, respectively, the transpose, the Hermitian transpose, and the pseudo-inverse.", "The set of nonnegative reals is ${\\mathbb {R}}_+$ .", "The real and imaginary components of a complex matrix ${M}\\in \\mathbb {C}^{N\\times N}$ are given by, respectively, $\\mathrm {Re}({M})\\in {\\mathbb {R}}^{N\\times N}$ and $\\mathrm {Im}({M})\\in {\\mathbb {R}}^{N\\times N}$ .", "By $(\\mathcal {H},\\left\\langle {\\cdot },{\\cdot }\\right\\rangle _\\mathcal {H})$ we denote a real Hilbert space with the inner product $\\left\\langle {\\cdot },{\\cdot }\\right\\rangle _\\mathcal {H}$ and induced norm $\\Vert x\\Vert _\\mathcal {H}:=\\sqrt{\\left\\langle {x},{x}\\right\\rangle }_\\mathcal {H}$ .", "The set of lower semicontinuous convex functions $f:\\mathcal {H}\\rightarrow {\\mathbb {R}}\\cup \\lbrace \\infty \\rbrace $ is given by $\\Gamma _0(\\mathcal {H})$ .", "The proximal mapping $\\mathrm {prox}_{ f}:\\mathcal {H}\\rightarrow \\mathcal {H}$ of $f\\in \\Gamma _0(\\mathcal {H})$ maps ${x}\\in \\mathcal {H}$ to the unique solution to: $\\mathrm {Minimize}_{{y}\\in \\mathcal {H}} f({y})+(1/2)\\Vert {x}-{y}\\Vert ^2_\\mathcal {H} $ .", "A function $f:\\mathcal {H}\\rightarrow {\\mathbb {R}}\\cup \\lbrace \\infty \\rbrace $ is said to be coercive if $\\Vert {x}\\Vert _\\mathcal {H}\\rightarrow \\infty $ implies $f({x})\\rightarrow \\infty $ .", "The projection $P_C:\\mathcal {H}\\rightarrow C$ onto a nonempty closed convex set $C\\subset \\mathcal {H}$ maps ${x}\\in \\mathcal {H}$ to the unique solution to: $\\mathrm {Minimize}_{{y}\\in C}\\Vert {x}-{y}\\Vert _\\mathcal {H}$ .", "The indicator of a set $C\\subset \\mathcal {H}$ is the function ${\\iota }_C:\\mathcal {H}\\rightarrow \\lbrace 0,\\infty \\rbrace $ given by $\\iota _C({x}) = 0$ if ${x}\\in C$ or $\\iota _C({x}) = \\infty $ otherwise.", "The norms $\\Vert \\cdot \\Vert _1$ and $\\Vert \\cdot \\Vert _2$ are, respectively, the standard $l_1$ and $l_2$ norms in Euclidean spaces.", "The set of fixed points of a mapping $T:\\mathcal {H}\\rightarrow \\mathcal {H}$ is denoted by $\\mathrm {Fix}(T):=\\lbrace x\\in \\mathcal {H}~|~T(x)=x \\rbrace $ .", "Given two real Hilbert spaces $(\\mathcal {H}^\\prime , \\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}^\\prime })$ and $(\\mathcal {H}^{\\prime \\prime },\\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}^{\\prime \\prime }})$ , the set $\\mathcal {B}(\\mathcal {H}^\\prime , \\mathcal {H}^{\\prime \\prime })$ is the set of bounded linear operators mapping vectors in $\\mathcal {H}^\\prime $ to vectors in $\\mathcal {H}^{\\prime \\prime }$ ." ], [ "System Model", "We consider the uplink of a system with one single-antenna user and one base station equipped with $N\\in {\\mathbb {N}}$ antennas.", "At time $k\\in {\\mathbb {N}}$ , the signal received at the base station spaced by multiples of the coherence interval $T_c$ in a memoryless flat fading channel is given by ${y}[k] = {h}[k]~s[k] + {n}[k] \\in \\mathbb {C}^N,$ where $s[k]\\in \\mathbb {C}$ and ${h}[k]\\in \\mathbb {C}^N$ denote, respectively, the transmitted symbol and the channel of the user; and ${n}[k]\\in \\mathbb {C}^N$ is a sample from the distribution $\\mathcal {N}_\\mathbb {C}({0},\\sigma ^2{I})$ .", "As common in the literature [14], [5], we assume that $E[|s[k]|^2] = 1$ and $E[s[k]]=0$ for every $k\\in {\\mathbb {N}}$ .", "We use the same notation for random variables and their samples.", "The meaning that should be applied is clear from the context.", "Furthermore, the transmitted symbols and noise are mutually independent, and their distributions do not change with the index $k$ in a sufficiently large time window.", "Therefore, hereafter we assume that $(\\forall k\\in {\\mathbb {N}})~E[{y}[k]{y}[k]^H] = {R} + \\sigma ^2 {I},$ where ${R} = E[{h}[k] {h}[k]^H] = {USU}^H \\in {\\mathbb {C}}^{N\\times N}$ is the channel covariance matrix, ${U}\\in {\\mathbb {C}}^{N\\times N}$ is the unitary matrix of eigenvectors of ${R}$ , and ${S}\\in {\\mathbb {C}}^{N\\times N}$ is the diagonal matrix of eigenvalues of ${R}$ .", "The channel sample ${h}[k]$ at time $k\\in {\\mathbb {N}}$ takes the form ${h}[k]={U} {S}^{1/2}{w}[k]$ , where $({w}[k])_{k\\in {\\mathbb {N}}}\\subset \\mathbb {C}^N$ are samples of i.i.d.", "random vectors with distribution $\\mathcal {N}_\\mathbb {C}({0},{I})$ .", "Hereafter, since the distribution of the random variables do not change with the time index $k$ in the memoryless channel described above, we omit this index if confusion does not arise." ], [ "The estimation problem", "Let $\\left({\\mathcal {H}_\\mathrm {1}}, \\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{{\\mathcal {H}_\\mathrm {1}}}\\right)$ be the real Hilbert space of (equivalent classes of) real square integrable functions ${\\mathcal {H}_\\mathrm {1}}=L_2(\\Omega )$ with respect to the standard Lebesgue measure $\\mu $ on a nonnull measurable set $\\Omega \\subset {\\mathbb {R}}^M$ .", "In this Hilbert space, inner products are defined by $(\\forall x\\in {\\mathcal {H}_\\mathrm {1}})(\\forall y\\in {\\mathcal {H}_\\mathrm {1}})\\left\\langle {x},{y}\\right\\rangle _{\\mathcal {H}_\\mathrm {1}}=\\int _{\\Omega } x~y~ \\mathrm {d}\\mu $ .", "Now, suppose that an array with $N\\in {\\mathbb {N}}$ antennas at a base station scans signals arriving from angles within a compact domain $\\Omega \\subset {\\mathbb {R}}^M$ , where each coordinate of $\\Omega $ corresponds to azimuth or elevation angles, possibly by also considering different antenna polarizations [3].", "Given ${\\theta }\\in \\Omega $ , we denote by $\\rho (\\theta )$ the average angular power density impinging on the array from angle $\\theta $ , and we further assume that the function $\\rho :\\Omega \\rightarrow {\\mathbb {R}}$ , hereafter called the angular power spectrum (APS), is an element of ${\\mathcal {H}_\\mathrm {1}}$ ; i.e., $\\rho \\in {\\mathcal {H}_\\mathrm {1}}$ .", "Being a power spectrum, $\\rho $ is also an element of the cone $\\mathcal {K}:=\\lbrace \\rho \\in {\\mathcal {H}_\\mathrm {1}}~|~\\mu (\\lbrace \\theta \\in \\Omega ~|~\\rho (\\theta )< 0\\rbrace ) = 0 \\rbrace $ of $\\mu $ -almost everywhere (a.e.)", "nonnegative functions.", "As shown in [2], [3], [4], [1], a common feature of realistic massive MIMO models is that the stacked version ${r} = [r_1,\\ldots ,r_{2N^2}]^t = \\phi ({R})$ of the channel covariance matrix ${R}$ in eq.contamination is related to the angular power spectrum $\\rho $ by $(\\forall n\\in \\lbrace 1,\\ldots ,2N^2\\rbrace )~r_n = \\left\\langle {\\rho },{g_n}\\right\\rangle _{\\mathcal {H}_\\mathrm {1}},$ where $(g_n)_{n\\in \\lbrace 1,\\ldots ,2N^2\\rbrace }$ are functions in ${\\mathcal {H}_\\mathrm {1}}$ defined by physical properties of the array and the propagation model, and $\\phi :{\\mathbb {C}}^{N\\times N}\\rightarrow {\\mathbb {R}}^{2N^2}:{R}\\mapsto \\mathrm {vec}\\left(\\left[\\begin{matrix}\\mathrm {Re}({R}) \\\\ \\mathrm {Im}({R})\\end{matrix}\\right]\\right)$ is the bijective mapping that vectorizes the imaginary and real components of a matrix.", "Therefore, in light of eq.innerprod, if the Hilbert space $({\\mathcal {H}_\\mathrm {2}}, \\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{{\\mathcal {H}_\\mathrm {2}}})$ denotes the standard Euclidean space ${\\mathcal {H}_\\mathrm {2}}= {\\mathbb {R}}^{2N^2}$ equipped with inner product $(\\forall {y}\\in {\\mathcal {H}_\\mathrm {2}})(\\forall {x}\\in {\\mathcal {H}_\\mathrm {2}}) \\left\\langle {{x}},{{y}}\\right\\rangle _{{\\mathcal {H}_\\mathrm {2}}}:={x}^t{y},$ then the relation between $\\rho $ and ${r}$ is given by ${r}=T\\rho $ , where $T\\in \\mathcal {B}({\\mathcal {H}_\\mathrm {1}}, {\\mathcal {H}_\\mathrm {2}})$ is the operator [1] $\\begin{array}{rcl}T:{\\mathcal {H}_\\mathrm {1}}&\\rightarrow &{\\mathcal {H}_\\mathrm {2}}\\\\\\rho &\\mapsto & [\\left\\langle {\\rho },{g_1}\\right\\rangle _{\\mathcal {H}_\\mathrm {1}},\\ldots , \\left\\langle {\\rho },{g_{2N^2}}\\right\\rangle _{\\mathcal {H}_\\mathrm {1}}]^t.\\end{array}$ Remark 1 Covariance matrices ${R}$ have structure, so we can remove many redundant equations in eq.innerprod to reduce the dimensionality of the space ${\\mathcal {H}_\\mathrm {2}}$ .", "The objective of the algorithms we propose in this study is to estimate $\\rho $ from a known (vectorized) channel covariance matrix ${r}:=\\phi ({R})=T\\rho $ .", "Note that the operator $T$ does not have an inverse in general, so this estimation problem is ill-posed.", "In particular, the null space $\\mathcal {N}(T):=\\lbrace x\\in {\\mathcal {H}_\\mathrm {1}}~|~ Tx = 0 \\rbrace $ of $T\\in \\mathcal {B}({\\mathcal {H}_\\mathrm {1}},{\\mathcal {H}_\\mathrm {2}})$ is nontrivial (i.e., $\\mathcal {N}(T)\\ne \\lbrace 0\\rbrace $ ), so there exist uncountably many functions $\\rho $ in ${\\mathcal {H}_\\mathrm {1}}$ for which $T$ maps $\\rho $ to the same vector ${r}=\\phi ({R})$ .", "Nevertheless, the studies in [2], [3], [4], [7] have shown that good estimates of $\\rho $ can be obtained with computationally efficient methods in practice.", "To improve upon these existing methods, we first need to understand their strengths and limitations, which is the topic of the next section.", "Before we proceed, we discretized all signals and operators to avoid unnecessary technical digressions.", "However, we emphasize that the results in this study can be straightforwardly extended to the infinite dimensional case described above with the tools in [2], [3], [1], [4].", "To obtain a finite dimension approximation of the estimation problem, we denote by ${\\rho }_\\mathrm {d}:=[\\rho (\\theta _1),\\ldots ,\\rho (\\theta _D)]^t \\in {\\mathbb {R}}^D$ the discrete version of true angular power spectrum $\\rho \\in {\\mathcal {H}_\\mathrm {1}}$ , where $D$ is size of the discrete grid.This approximation is somewhat heuristic because ${\\mathcal {H}_\\mathrm {1}}$ is an equivalence class of functions.", "In particular, given $\\theta \\in \\Omega $ and $\\rho \\in {\\mathcal {H}_\\mathrm {1}}$ , the value $\\rho (\\theta )$ is not well defined.", "As a result, the integrals in eq.innerprod can be approximated by $(\\forall n\\in \\lbrace 1,\\ldots ,2N^2\\rbrace )$ $\\left\\langle {\\rho },{g_n}\\right\\rangle _{\\mathcal {H}_\\mathrm {1}}= \\int _{\\Omega } \\rho ~g_n \\mathrm {d}\\mu \\approx {\\rho }_\\mathrm {d}^t {g}_{\\mathrm {d},n},$ where ${g}_{\\mathrm {d},n}=(\\mu (\\Omega )/D) [g_n(\\theta _1),\\ldots ,g_n(\\theta _D)]^t\\in {\\mathbb {R}}^D$ is a discrete approximation of the function $g_n$ of array.", "In turn, with ${A}:=[{g}_{\\mathrm {d},1}\\ldots {g}_{\\mathrm {d},2N^2}]^t,$ the operator $T_\\mathrm {d}:{\\mathbb {R}}^D\\rightarrow {\\mathbb {R}}^{2N^2}:{\\rho }\\mapsto {A}{\\rho }$ is a discrete approximation of $T$ in eq.linop, and $\\mathcal {K}_\\mathrm {d}:={\\mathbb {R}}_+^D$ is a discrete approximation of $\\mathcal {K}$ ." ], [ "Existing solutions for angular power spectrum estimation", "For simplicity, in this section we use the following assumption, which is dropped later in Sect. .", "Assumption 1 The estimated covariance matrix ${R}$ , or, equivalently, ${r}=\\phi ({R})$ , is compatible with the array, in the sense that it can be generated with one function in $\\mathcal {K}_\\mathrm {d}$ ; i.e., $\\phi ({R})={r}\\in T_\\mathrm {d}(\\mathcal {K}_\\mathrm {d}):=\\lbrace {A\\rho } \\in {\\mathbb {R}}^{2N^2}~|~ {\\rho }\\in \\mathcal {K}_\\mathrm {d}\\rbrace .$ If Assumption REF holds, we can estimate ${\\rho }_\\mathrm {d}$ from ${r}$ by solving the following set-theoretic estimation problem, which is a discrete version of one of the infinite dimensional problems posed in [2], [3], [4]: $\\text{Find }{\\rho }\\in {\\mathbb {R}}^D \\text{ such that } {\\rho }\\in V_\\mathrm {d}\\cap \\mathcal {K}_\\mathrm {d},$ where $V_\\mathrm {d}\\subset {\\mathbb {R}}^D$ is the linear variety $V_\\mathrm {d}:= \\lbrace {\\rho } \\in {\\mathbb {R}}^D~|~{A\\rho }={r}\\rbrace $ ; i.e., the set of all (not necessarily nonnegative) vectors that produce the observed channel covariance matrix ${R}=\\phi ^{-1}({r})$ .", "The idea of Problem eq.feasibility is to find an estimate that is consistent with all known information about ${\\rho }_\\mathrm {d}$ , or, more precisely, with the fact that ${\\rho }_\\mathrm {d}$ is nonnegative (i.e., ${\\rho }_\\mathrm {d}\\in \\mathcal {K}_\\mathrm {d}$ ) and it produced the observed channel covariance matrix ${R}=\\phi ^{-1}({r})$ (i.e., ${\\rho }_\\mathrm {d}\\in V_\\mathrm {d}$ ).", "In this set-theoretic paradigm, any two estimates belonging to both $V_\\mathrm {d}$ and $\\mathcal {K}_\\mathrm {d}$ are equally good because no other information about ${\\rho }_\\mathrm {d}$ is assumed to be available.", "Clearly, a necessary and sufficient condition for the convex feasibility problem in eq.feasibility to have a solution is that Assumption REF holds.", "Since the projections onto $V_\\mathrm {d}$ and $\\mathcal {K}_\\mathrm {d}$ are easy to compute in the Hilbert space $(\\mathcal {H}_2,\\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}_2})$ [15], a plethora of simple iterative projection-based algorithms with convergence guarantees are widely available [15], [16], [17].", "In particular, the variant of the Douglas-Rachford splitting method studied in [16] converges in a finite number of iterations.", "We can also reformulate Problem eq.feasibility as a standard convex program to enable us to use traditional solvers.", "For example, consider the problem below, which has been proposed in [7]: $\\text{Minimize}_{{\\rho }\\in \\mathcal {K}_\\mathrm {d}} \\Vert {A\\rho }-{r}\\Vert _2^2.$ From the definition of the linear variety $V_\\mathrm {d}$ , any estimate ${\\rho }\\in V_\\mathrm {d}$ satisfies $\\Vert {A\\rho }-{r}\\Vert _2^2=0$ , which is the global minimum of the cost function in Problem eq.nnls.", "Therefore, under Assumption REF , we verify that ${\\rho }^\\star $ solves Problem eq.feasibility if and only if ${\\rho }^\\star $ solves Problem eq.nnls.", "We emphasize that Problems eq.feasibility and eq.nnls do not have a unique solution in general.", "As a result, the quality of the estimate of ${\\rho }_\\mathrm {d}$ obtained by solving either eq.feasibility or eq.nnls depends on the choice of the iterative solver.", "Nonuniqueness of the solution provides us with additional possibilities to choose a vector in the solution set with additional desirable properties.", "For example, a common hypothesis is that ${\\rho }_\\mathrm {d}$ is a sparse vector, so, as an attempt to promote sparsity, we may select a solution to eq.feasibility with minimum $l_1$ norm (recall that the $l_1$ norm is known to promote sparsity).", "Formally, we solve the following problem: $\\text{Minimize}_{{\\rho }\\in {\\mathbb {R}}^D} \\Vert {\\rho }\\Vert _1 \\text{ subject to } {\\rho }\\in V_\\mathrm {d}\\cap \\mathcal {K}_\\mathrm {d}.$ However, as we argue below, for common array models in the literature, there is nothing to be gained by solving eq.l1 instead of eq.feasibility or the equivalent problem in eq.nnls [if Assumption assumption.range holds] because the set of solutions to Problems eq.feasibility, eq.nnls, and eq.l1 are the same.", "Some of these arrays satisfy the following assumption: Assumption 2 Let $S:=\\lbrace g_1, \\ldots , g_{2N^2}\\rbrace \\subset {\\mathcal {H}_\\mathrm {1}}$ be the set of functions of the array.", "We assume that the function $u:\\Omega \\rightarrow {\\mathbb {R}}: \\theta \\mapsto 1$ is a member of $S$ , in which case the vector $c {1}$ , where ${1}\\in {\\mathbb {R}}^D$ is the vector of ones and $c:=\\mu (\\Omega )/D$ , is a row of the matrix ${A}$ in eq.mata.", "Remark 2 Assumption REF is valid for common array models with isotropic antennas, such as uniform linear arrays and planar arrays.", "The relation among Problems eq.feasibility, eq.nnls, and eq.l1 is formally established in the next simple proposition.", "Proposition 1 Let Assumptions REF and REF be valid.", "Then set of solutions to Problems eq.feasibility, eq.nnls, and eq.l1 are the same.", "If Assumption REF holds, then $V_\\mathrm {d}\\cap \\mathcal {K}_\\mathrm {d}\\ne \\emptyset $ .", "Now, let ${\\rho }\\in V_\\mathrm {d}\\cap \\mathcal {K}_\\mathrm {d}$ be arbitrary.", "Assumption REF implies that, for $c:=\\mu (\\Omega )/D$ , there exists $k\\in \\lbrace 1,\\ldots ,2N^2\\rbrace $ such that $r_k\\overset{(\\text{a})}{=} c{1}^t{\\rho }\\overset{(\\text{b})}{=}c~\\Vert {\\rho }\\Vert _1$ , where (a) follows from ${\\rho }\\in V_\\mathrm {d}$ and (b) follows from ${\\rho }\\in \\mathcal {K}_\\mathrm {d}$ .", "Since ${\\rho }$ is arbitrary, we conclude that all vectors in $V_\\mathrm {d}\\cap \\mathcal {K}_\\mathrm {d}$ have the same $l_1$ norm, which implies that Problems eq.feasibility and eq.l1 have the same set of solutions.", "The equivalence between Problems eq.feasibility and eq.nnls has already been established, so the proof is complete.", "The practical implication of Proposition REF is that Problems eq.feasibility and eq.nnls are expected to promote sparsity implicitly, but the estimand ${\\rho }_\\mathrm {d}$ is not necessarily the sparsest vector of the solution set.", "Therefore, we need additional information in the problem formulations to improve the estimates, and in the next section we incorporate statistical information gained from datasets." ], [ "Proposed algorithms", "Given a positive definite matrix ${M}\\in {\\mathbb {R}}^{D\\times D}$ [this matrix is fixed later in eq.matm], let $(\\mathcal {H}_{{M}}, \\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}_{{M}}})$ denote the Hilbert space consisting of the vector space $\\mathcal {H}_{{M}}:={\\mathbb {R}}^D$ equipped with the inner product $(\\forall {x}\\in \\mathcal {H}_{{M}})(\\forall {y}\\in \\mathcal {H}_{{M}}) \\left\\langle {{x}},{{y}}\\right\\rangle = {x}^t{My}$ .", "By definition, the vector space $\\mathcal {H}_{{M}}={\\mathbb {R}}^D$ does not depend on ${M}$ , but the notation $\\mathcal {H}_{{M}}$ is useful to clarify the inner product defined on ${\\mathbb {R}}^D$ .", "Now, assume that a dataset $\\mathcal {M}=\\lbrace {\\rho }_{\\mathrm {d},1},\\ldots , {\\rho }_{\\mathrm {d},L}\\rbrace $ with $L$ samples of angular power spectra is available, and suppose that these samples have been independently drawn from the same distribution with mean $\\bar{{\\rho }}\\in \\mathcal {K}_\\mathrm {d}$ and covariance matrix ${C}\\in {\\mathbb {R}}^{D\\times D}$ .", "In practice, $\\bar{{\\rho }}$ and ${C}$ can be estimated from a sample average as follows (assuming $L\\gg 1$ ): $\\bar{{\\rho }} \\approx \\dfrac{1}{L} \\sum _{n=1}^L {\\rho }_{\\mathrm {d},n} \\in {\\mathbb {R}}^{D\\times D}$ and ${C} \\approx \\dfrac{1}{L-1} \\sum _{n=1}^L ({\\rho }_{\\mathrm {d},n}-\\bar{{\\rho }})({\\rho }_{\\mathrm {d},n}-\\bar{{\\rho }})^t.$ Hereafter, to exploit knowledge gained from ${C}$ and $\\bar{{\\rho }}$ , we use the Hilbert space $\\left(\\mathcal {H}_{{M}},\\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}_{{M}}}\\right)$ defined above by fixing ${M}$ to ${M}_\\alpha :=({C}+\\alpha {I})^{-1},$ where $\\alpha >0$ is a design parameter that serves two purposes: (i) it guarantees positive definiteness of ${M}_\\alpha $ , and (ii) it provides robustness against environmental changes, as discussed below.", "An important feature of the Hilbert space $(\\mathcal {H}_{{M}_\\alpha },\\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}_{{M}_\\alpha }})$ is that its induced norm $(\\forall {x}\\in \\mathcal {H}_{{M}_\\alpha }) ~ \\Vert {x}\\Vert _{\\mathcal {H}_{{M}_\\alpha }} := \\sqrt{\\left\\langle {{x}},{{x}}\\right\\rangle _{\\mathcal {H}_{{M}_\\alpha }}}$ in turn induces the Hilbertian metric $(\\forall {x}\\in \\mathcal {H}_{{M}_\\alpha })(\\forall {y}\\in \\mathcal {H}_{{M}_\\alpha }) ~ d_{\\mathcal {H}_{{M}_\\alpha }}({x},{y}):=\\Vert {x}-{y}\\Vert _{\\mathcal {H}_{{M}_\\alpha }}$ that is known as the Mahalanobis distance in statistical pattern recognition [18].", "In particular, if the design parameter $\\alpha >0$ is sufficiently small, the distance $d_{\\mathcal {H}_{{M}_\\alpha }}({x},\\bar{{\\rho }})$ between the distribution mean $\\bar{{\\rho }}$ and a given vector ${x}\\in \\mathcal {H}_{{M}_\\alpha }$ is known to provide us with a notion of distance between ${x}$ and the distribution of the dataset $\\mathcal {M}$ .", "As the parameter $\\alpha $ increases, the influence of the dataset in the metric $d_{\\mathcal {H}_{{M}_\\alpha }}$ decreases ($d_{\\mathcal {H}_{{M}_\\alpha }}$ becomes increasingly similar to a scaled version of the standard Euclidean metric), so large $\\alpha $ can be useful in scenarios in which the distribution of the angular power spectrum changes significantly over time and acquisition of datasets is difficult.", "We now propose two algorithms based on the Hilbert space $\\left(\\mathcal {H}_{{M}_\\alpha },\\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}_{{M}_\\alpha }}\\right)$ ." ], [ "Algorithm 1", "In Sect.", "we have shown that Problems eq.feasibility, eq.nnls, and eq.l1 do not have a unique solution in general, and they are equivalent if the assumptions in Proposition REF hold.", "Therefore, among all solutions, we propose to select the solution with minimum distance to the distribution of the dataset in the sense defined above; i.e., we minimize the Mahalanobis distance.", "Formally, given $\\alpha >0$ , we solve the following hierarchical problem: $\\text{Minimize}_{{{\\rho }}\\in S} \\Vert {{\\rho }}-\\bar{{\\rho }}\\Vert _{\\mathcal {H}_{{M}_\\alpha }},$ where $S:=\\operatornamewithlimits{arg \\, min}_{{\\rho }\\in \\mathcal {H}_{{M}_\\alpha }} g({\\rho }) \\subset \\mathcal {H}_{{M}_\\alpha }$ and $\\Gamma _0(\\mathcal {H}_{{M}_\\alpha })\\ni g:\\mathcal {H}_{{M}_\\alpha }\\rightarrow {\\mathbb {R}}_+:{\\rho }\\mapsto \\Vert {A\\rho }-{r}\\Vert _{2}^2+\\iota _{\\mathcal {K}_\\mathrm {d}}({\\rho }).$ Note that $S$ is the set of solutions to Problem eq.nnls, and, if the assumptions in Proposition REF hold, then $S$ is also the set of solutions to Problems eq.feasibility and eq.l1.", "However, hereafter we do not necessarily assume that the assumptions in Proposition REF hold.", "In particular, as discussed below, the proposed algorithm can deal with the case $\\mathcal {K}_\\mathrm {d}\\cap V_\\mathrm {d}=\\emptyset $ without any changes.", "One of the challenges for solving eq.generalization is that hierarchical problems are not in general canonical convex programs as defined in some well-known references [19], where constraints have to be expressed as level sets of convex functions or as equalities involving affine functions.", "Therefore, the solvers described in these references are not directly applicable.", "The proposed strategy for solving eq.generalization is to interpret its solution as the projection from $\\bar{{\\rho }}$ onto the fixed point set of a computable firmly nonexpansive mapping, which enables us to apply best approximation techniques such as those based on Haugazeau's algorithm [20].", "In more detail, recalling the definition of projections, we verify that the solution ${\\rho }^\\star $ to eq.generalization is the projection from $\\bar{{\\rho }}$ onto the closed convex set $S$ in the Hilbert space $(\\mathcal {H}_{{M}_\\alpha }, \\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}_{{M}_\\alpha }})$ ; i.e., ${\\rho }^\\star =P_S(\\bar{{\\rho }})$ .", "As a result, the solution exists and is unique provided that the set $S$ is nonempty, and we can show nonemptiness of this set even if we weaken the assumptions in Proposition REF .", "For example, let us only assume that one of the vectors $({g}_{\\mathrm {d},n})_{n\\in {\\mathbb {N}}}$ has (strictly) positive components (see Assumption REF and Remark REF ).", "In this case, we can show that $g$ is coercive, but we omit the details for brevity.", "Therefore, we have $S\\ne \\emptyset $ as an implication of [20].", "The projection onto $S$ does not have a closed-form expression in general, but it can be computed with iterative methods.", "To this end, note that the set $S$ can be equivalently expressed as the fixed point set of the mapping $\\mathrm {prox}_{\\gamma g}:\\mathcal {H}_{{M}_\\alpha }\\rightarrow \\mathcal {H}_{{M}_\\alpha }$ for every $\\gamma >0$ ; i.e., $(\\forall \\gamma >0)~\\mathrm {Fix}(\\mathrm {prox}_{\\gamma g}) = S$ .", "Therefore, given an arbitrary scalar $\\gamma >0$ , the desired solution ${\\rho }^\\star = P_S(\\bar{{\\rho }}) = P_{\\mathrm {Fix}(\\mathrm {prox}_{\\gamma g})}(\\bar{{\\rho }})$ is the limit of the sequence $({\\rho }_n)_{n\\in {\\mathbb {N}}}$ constructed with the following instance of Haugazeau's algorithm: ${\\rho }_{n+1} = Q({\\rho }_1, {\\rho }_n, \\mathrm {prox}_{\\gamma g}({\\rho }_n)),$ where ${\\rho }_1:=\\bar{{\\rho }}$ , $\\begin{array}{l}Q:\\mathcal {H}_{{M}_\\alpha }\\times \\mathcal {H}_{{M}_\\alpha }\\times \\mathcal {H}_{{M}_\\alpha }\\rightarrow {\\mathbb {R}}\\\\({x},{y},{z})\\mapsto {\\left\\lbrace \\begin{array}{ll}{z},\\text{ if }\\delta = 0 \\text{ and } \\chi \\ge 0; \\\\{x}+\\left(1+\\dfrac{\\chi }{\\nu }\\right)({z}-{y}), \\\\ \\qquad \\text{ if }\\delta > 0 \\text{ and } \\chi \\nu \\ge \\delta ;\\\\{y}+\\dfrac{\\nu }{\\delta }\\left(\\chi ({x}-{y})+\\mu ({z}-{y})\\right),\\\\ \\qquad \\text{ if }\\delta > 0 \\text{ and } \\chi \\nu < \\delta ;\\end{array}\\right.", "}\\end{array}$ $\\chi = \\left\\langle {{x}-{y}},{{y}-{z}}\\right\\rangle _{\\mathcal {H}_{{M}_\\alpha }}$ , $\\mu =\\Vert {x}-{y}\\Vert ^2_{\\mathcal {H}_{{M}_\\alpha }}$ , $\\nu = \\Vert {y}-{z}\\Vert ^2_{\\mathcal {H}_{{M}_\\alpha }}$ , and $\\delta =\\mu \\nu -\\chi ^2$ .", "The proof that the sequence $({\\rho }_n)_{n\\in {\\mathbb {N}}}$ constructed via eq.hag indeed converges to $P_S(\\bar{{\\rho }})$ is a simple application of [20].", "More precisely, recall that proximal mappings are firmly nonexpansive, so the mapping ${x}\\mapsto {x}-\\mathrm {prox}_{\\gamma g}({x})$ is demiclosed everywhere [20].", "Therefore, we fulfill all the conditions in [20] for the sequence constructed via eq.hag to converge to $P_{\\mathrm {Fix}(\\mathrm {prox}_{\\gamma g})}(\\bar{{\\rho }})=P_{S}(\\bar{{\\rho }})$ .", "Remark 3 (Computation of the proximal mapping of $g$ ) Using the definition of proximal mappings, after simple algebraic manipulations, we verify that $\\mathrm {prox}_{\\gamma g}({x})$ in the Hilbert space $(\\mathcal {H}_{{M}_\\alpha }, \\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}_{{M}_\\alpha }})$ for given ${x}\\in \\mathcal {H}_{{M}_\\alpha }$ and $\\gamma >0$ is the solution to $\\mathrm {Minimize}_{{y}\\in \\mathcal {K}_\\mathrm {d}} \\Vert {Q}^{1/2}{y}-{b}\\Vert _2^2,$ where ${Q}^{1/2}$ is the principal square root of ${Q}:={A}^t{A}+(1/(2\\gamma )){M}_\\alpha ,$ and ${b}:={Q}^{-1/2}({A}^t{r}+(1/(2\\gamma )){M}_\\alpha {x}).$ Problem eq.problemprox is a standard nonnegative least-squares program, so the proximal mapping $\\mathrm {prox}_{\\gamma g}:\\mathcal {H}_{{M}_\\alpha }\\rightarrow \\mathcal {H}_{{M}_\\alpha }$ can be computed with solvers that terminate with a finite number of steps, such as those based on the active-set method [21]." ], [ "Algorithm 2", "To derive a low-complexity alternative to Algorithm 1, we modify Problem eq.nnls by adding a regularizer based on the Mahalanobis distance as follows: $\\text{Minimize}_{{\\rho }\\in \\mathcal {H}_{{M}_\\alpha }} \\Vert {{\\rho }}-\\bar{{\\rho }}\\Vert _{\\mathcal {H}_{{M}_\\alpha }}^2+ \\mu \\Vert {A}{\\rho }-{r}\\Vert _{2}^2 + \\iota _{\\mathcal {K}_\\mathrm {d}}({\\rho }),$ where $\\mu >0$ is a design parameter that trades deviations from the set $V_\\mathrm {d}$ against the distance to the distribution of the dataset, and $\\alpha >0$ is the design parameter of the Hilbert space $(\\mathcal {H}_{{M}_\\alpha }, \\left\\langle {\\cdot },{\\cdot }\\right\\rangle _{\\mathcal {H}_{{M}_\\alpha }})$ .", "The definition of proximal mappings shows that the unique solution ${\\rho }^\\star $ to Problem eq.alg2 is ${\\rho }^\\star ={\\mathrm {prox}_{(\\mu /2)g}}(\\bar{{\\rho }})$ , where $g$ is the function defined in eq.funcg.", "As a result, in light of Remark REF , Problem eq.alg2 can be solved with a single application of the active-set method [21], unlike the algorithm in eq.hag, which uses a nonnegative least squares solver to compute the proximal mapping of $\\gamma g$ at each iteration.", "The price we pay for this reduction in computational effort is that the formulation in eq.alg2 requires knowledge of a good value for $\\mu $ because the solution to eq.alg2 depends on this parameter.", "In contrast, the parameter $\\gamma $ in eq.hag determines the path taken by the iterates, but not the vector to which the algorithm converges.", "Remark 4 Additional regularizers, such as those based on total variation techniques could also be added to eq.alg2, but we do not consider them here because of the space limitation." ], [ "Simulations and conclusions", "We assume that a base station is equipped with an uniform linear array operating with $N=16$ antennas, frequency $f=2.11$ GHz, speed of wave propagation $c=3\\cdot 10^8$ m/s, antenna spacing $d=c/(2f)$ , and the array response shown in [1].", "The samples of angular power spectra use a conventional model in the literature [2], [1].", "More precisely, each run of the simulation constructs an angular power spectrum via $\\rho :\\Omega \\rightarrow {\\mathbb {R}}_+:\\theta \\mapsto \\sum _{k=1}^Q \\alpha _k h_k(\\theta )$ , where $\\Omega :=[-\\pi /2,~\\pi /2]$ , $Q$ is uniformly drawn from $\\lbrace 1,2,3,4,5\\rbrace $ ; $h_k:\\Omega \\rightarrow {\\mathbb {R}}_+:\\theta \\mapsto ({1}/{\\sqrt{2\\pi \\Delta _k^2}})\\exp \\left({-{(\\theta -\\phi _k)^2}/{(2\\Delta _k^2)}}\\right)$ ; $\\phi _k$ , the main arriving angle of the $k$ th path, is uniformly drawn from $[0, \\pi /2]$ ; and $\\alpha _k$ is uniformly drawn from $[0,~1]$ , and it is further normalized to satisfy $\\sum _{k=1}^Q\\alpha _k=1$ .", "The discrete grid to approximate angular power spectra has $D=180$ uniformly spaced points.", "Estimates of channel covariance matrices are produced via $P_\\mathcal {T}(\\sum _{i=k}^{500} {h}[k] {h}[k]^T - \\sigma ^2 {I} )$ , where $\\sigma ^2=0.1$ is the noise variance in eq.contamination, and $P_\\mathcal {T}:{\\mathbb {C}}^{N\\times N}\\rightarrow \\mathcal {T}$ denotes the projection onto the set $\\mathcal {T}\\subset {\\mathbb {C}}^{N\\times N}$ of Toeplitz matrices with respect to the complex Hilbert space $({\\mathbb {C}}^{N\\times N}, \\left\\langle {{A}},{{B}}\\right\\rangle ={B}^H{A})$ .", "For the construction of the operator $T$ in eq.linop, we use only $2N-1$ functions because channel covariance matrices of uniform linear arrays are Toeplitz.", "The approximations in eq.rhomean and eq.C use 1,000 samples of angular power spectra, and the parameter $\\alpha $ to construct the matrix ${M}_\\alpha $ in eq.matm is set to $\\alpha = \\Vert {C}\\Vert _2/100$ , where $\\Vert {C}\\Vert _2$ denotes the spectral norm of the empirical covariance matrix ${C}$ in eq.C.", "Subsequently, we normalize the matrix ${M}_\\alpha $ to satisfy $\\Vert {M}_\\alpha \\Vert _2=1$ .", "With an abuse of notation, we use the normalized mean square error (MSE) $E\\left[{\\Vert {{\\rho }} - {\\rho }_\\mathrm {d}\\Vert _2^2}/{\\Vert {\\rho }_\\mathrm {d}\\Vert _2^2}\\right]$ as the figure of merit to compare different algorithms, where ${\\rho }$ is the estimate of ${\\rho }_\\mathrm {d}$ , and expectations are approximated with the empirical average of 200 runs of the simulation.", "Fig.", "REF shows the performance of the following algorithms: (i) the extrapolated and accelerated projection method (EAPM) used in [2], [3] operating in the standard Euclidean space ${\\mathbb {R}}^D$ with inner product $(\\forall {x}\\in {\\mathbb {R}}^D)(\\forall {y}\\in {\\mathbb {R}}^D) \\left\\langle {{x}},{{y}}\\right\\rangle :={x}^t{y}$ ; (ii) Haugazeau's algorithm in eq.hag with $\\gamma =5$ ; and (iii) the solution to the nonnegative least squares (NNLS) problem in eq.alg2, computed with SciPy NNLS solver, with $\\mu =5\\cdot 10^4$ (NNLS-1) and $\\mu =1$ (NNLS-2).", "Note that the algorithms NNLS-1 and NNLS-2 are not considered iterative methods because we assume that solvers for NNLS programs are available as a computational tool.", "Therefore, we use the convention that the estimates produced by these algorithms are the same at every iteration.", "We have also simulated a neural network similar to that in [9] with two modifications.", "First, the number of neurons in each layer was scaled by 180/128 to account for the finer grid used in this study.", "Second, the last layer based on the soft-max activation function was replaced by the rectified linear unit activation function because the desired estimand is nonnegative and the soft-max function is inappropriate for the figure of merit considered above (with the softmax activation function, simply scaling the input deteriorates the performance severely if no additional heuristics are employed).", "By carefully training this neural network with different solvers, step sizes, epochs, batch sizes, and with a training set containing 110,200 samples (which is two orders of magnitude larger than the dataset used by the proposed algorithms), we have not obtained a normalized MSE better than $6\\cdot 10^{-2}$ , which is worse than the MSE obtained with the existing EAPM algorithm in Fig.", "REF .", "Furthermore, with the scenario considered later in Figs.", "REF and REF (which uses training and test sets constructed with different distributions), the MSE increases drastically (MSE $> 2$ ).", "For these reasons, we do not show the performance of the neural network in the figures.", "Some conclusions for this first experiment are as follows: - The proposed algorithms can outperform the EAPM algorithm used in [2], [3] because statistical information obtained from a dataset is exploited, and we note that the EAPM algorithm has already been shown to outperform existing data driven methods that can cope with small datasets [2].", "- The performance gap between NNLS-1 and NNLS-2 shows that the solution to Problem eq.alg2 is sensitive to the choice of the regularization parameter $\\mu $ .", "Nevertheless, if a good value is known, which can be obtained with cross-validation techniques, then the solution to Problem eq.alg2 has performance similar to that obtained with Haugazeau's method.", "Figure: Normalized mean square error as a function of the number of iterations.A well-known limitation of data-driven methods (and, in particular, neural networks, as discussed above) is the poor generalization performance if the estimand is sampled from a distribution different from that used to construct training sets.", "As we now show, the proposed hybrid data and model driven algorithms can mitigate problems of this type.", "In Fig.", "REF , we use the proposed algorithms with the dataset in Fig.", "REF to reconstruct angular power spectra with the main angles of the paths drawn uniformly at random within the interval $[-\\pi /2, 0]$ .", "By doing so, we mimic an extreme scenario where the principal subspaces obtained from the dataset contain almost no energy of the angular power spectra being estimated.", "As seen in Fig.", "REF , the performance of the Haugazeau and NNLS-1 hybrid methods deteriorates, but the MSE does not increase to a point to render these algorithms ineffective.", "The reason is that the estimates produced by these two proposed algorithms are consistent with the measurements and the array model (i.e., they are close to the set $V_\\mathrm {d}$ ), and this fact alone may be enough to provide performance guarantees in some applications, as proved in [4], [10].", "Furthermore, the proposed algorithms have a tunable parameter to make them robust against changes in the distribution of the angular power spectrum; namely, the parameter $\\alpha $ in eq.matm.", "This fact is illustrated in Fig.", "REF , where we show the performance of the algorithms with the parameter $\\alpha $ increased to $\\alpha =\\Vert {C}\\Vert _2$ (the remaining simulation parameters are the same as those used to produce Fig.", "REF ).", "The performance of the Haugazeau and NNLS-1 algorithms in Fig.", "REF approaches the performance of the pure model-based EAPM because, by increasing $\\alpha $ , the proposed algorithms increasingly ignore the erroneous information about the distribution of the estimand, which is inferred from the dataset.", "Figure: Normalized mean square error as a function of the number of iterations.", "Angular power spectra of the dataset drawn from a distribution different from that of the estimand (α=∥C∥ 2 /100\\alpha =\\Vert {C}\\Vert _2/100).Figure: Normalized mean square error as a function of the number of iterations.", "Angular power spectra of the dataset drawn from a distribution different from that of the estimand (α=∥C∥ 2 \\alpha =\\Vert {C}\\Vert _2)." ] ]

[ [ "Exact Method for Generating Strategy-Solvable Sudoku Clues" ], [ "Abstract A Sudoku puzzle often has a regular pattern in the arrangement of initial digits and it is typically made solvable with known solving techniques, called strategies.", "In this paper, we consider the problem of generating such Sudoku instances.", "We introduce a rigorous framework to discuss solvability for Sudoku instances with respect to strategies.", "This allows us to handle not only known strategies but also general strategies under a few reasonable assumptions.", "We propose an exact method for determining Sudoku clues for a given set of clue positions that is solvable with a given set of strategies.", "This is the first exact method except for a trivial brute-force search.", "Besides the clue generation, we present an application of our method to the problem of determining the minimum number of strategy-solvable Sudoku clues.", "We conduct experiments to evaluate our method, varying the position and the number of clues at random.", "Our method terminates within $1$ minutes for many grids.", "However, as the number of clues gets closer to $20$, the running time rapidly increases and exceeds the time limit set to $600$ seconds.", "We also evaluate our method for several instances with $17$ clue positions taken from known minimum Sudokus to see the efficiency for deciding unsolvability." ], [ "Introduction", "Sudoku is a popular number-placement puzzle.", "In an ordinary Sudoku (Figure REF ), given a partially completed $9\\times 9$ grid, the goal is to fill in all empty cells with digits from 1 to 9 in such a way that each cell has a single digit, and each digit appears only once in every row, column, and $3\\times 3$ subgrid.", "Sudokus that appear in books, newspapers, etc often have the following characteristics.", "The arrangement of initial digits (called clues) forms a regular pattern.", "The level of difficulty is moderate.", "Figure: Sudoku clues (left) and its solution (right)Regarding 1, for example, the left grid of Figure REF has reflection symmetry on the two diagonal axis.", "It may also has the shape of a number, a letter, or a symbol.", "Regarding 2, completing Sudokus generally requires backtracking, which is amenable to computers.", "On the other hand, there is a set of techniques (called strategies) that humans use in solving Sudokus by hand [1].", "Typically strategies are if-then rules: If a strategy is applicable to the current grid, then it might rule out digits as candidates or might determine digits as those to be finally placed.", "A strategy-solvable Sudoku is a partially completed grid that can be completed by applying strategies repeatedly.", "Here, the strategies must be selected from a set of predetermined ones.", "Since there is no need to guess, strategy-solvable Sudokus (at least for a few basic strategies) might be referred to as Sudokus having the moderate level of difficulty for humans.", "Making new Sudoku instances having these characteristics is not an easy task.", "There is a public software The generator of Zama and Sasano [9] http://www.cs.ise.shibaura-it.ac.jp/2016-GI-35/SudokuGenerator.tar.gz , accessed May 26th, 2020., which is able to generate Sudokus from a given set of clue positions so that they are solvable with some known strategies.", "However, it is still hard to generate those with 20 or less clues and those with around 45 or more clues.", "For such instances, there might be no other choice but to rely on human intelligence involving the intuition, the inspiration and the experience of enthusiasts.", "In this paper, we consider a method for determining Sudoku clues in specified positions such that all empty cells can be filled in with a specified set of strategies (see Figure REF ).", "Figure: Clue positions (left) and clues solvable with naked singles only (right)Almost all Sudoku generators simply output proper Sudoku instances (i.e.", "partially completed grids having unique solutions).", "Clue arrangements and strategies necessary for solving are uncontrollable.", "To the best of our knowledge, the only exception is the generator of Zama and Sasano, mentioned earlier.", "Their generator is based on the generate-and-test method, which repeats the followings until the test is passed or the number of trials exceeds a predetermined limit.", "Generate clues in specified positions.", "Test whether all the other cells are completed using specified strategies only.", "The same idea is also mentioned in [4].", "Since the test step can be done quickly, the key is to devise a criterion for the generation step so that the number of trials is as small as possible.", "Since the generate-and-test method examines only a limited portion of the vast search space, it has the drawbacks listed below.", "The lower the density of solutions (i.e.", "strategy-solvable clues) over the whole search space becomes, the harder it becomes to find a solution.", "If a set of clue positions happens to be not strategy-solvable, it is unable to recognize it no matter how much time passes.", "Even for a strategy-solvable set of clue positions, there is no guarantee for being able to find a solution in a finite amount of time.", "To tackle these issues (in particular the last two), we consider it necessary to formulate the concept of strategy-solvability.", "It seems that strategy-solvability has been recognized intuitively, and no formal treatment has been given so far.", "In this paper, we introduce a rigorous framework to discuss solvability for Sudoku instances with respect to strategies and define the notion of strategy-solvable Sudoku clues.", "This allows us to handle not only known strategies but also general strategies under a few reasonable assumptions.", "We then propose an exact method for determining strategy-solvable Sudoku clues for a given set of clue positions that is solvable with a given set of strategies.", "The key is a reduction to a constraint satisfaction problem (CSP).", "Our method is able to benefit from the power of state-of-the-art CSP solvers.", "Pruning techniques of CSP solvers are expected to be effective for the first issue above.", "Moreover, as long as any complete CSP solver is utilized, it is guaranteed that if a set of clue positions is not strategy-solvable, our method eventually recognizes the unsolvability; otherwise, our method eventually finds strategy-solvable clues.", "This is the first exact method except for a trivial brute-force search.", "As strategies allowed in completing Sudokus, this paper employs naked singles, hidden singles, and locked candidates [1], which are quite basic yet enough powerful to complete many Sudokus.", "Indeed, we have confirmed that the combination of the three strategies allows for solving as many as $37,373$ of the $49,151$ minimum Sudokus (i.e., Sudokus with 17 clues) collected by Gordon F. Royle [6].", "Here we remark that our CSP formulation is almost independent of specific strategies.", "In order to allow other strategies, it is sufficient to formulate the corresponding logical constraints and add them with a minor modification of the strategy-independent part.", "Besides the clue generation, we present an application of our method to the problem of determining the minimum number of Sudoku clues that are solvable with a given set of strategies.", "We demonstrate that our method is easily customized for this problem with a small modification in the CSP constraints.", "It is well-known that there is no proper Sudoku with 16 or less clues [5].", "There are many Sudokus with 17 clues that are solvable with basic strategies such as hidden singles.", "Interestingly, this may not be the case for naked singles, arguably one of the most basic strategy, as we have confirmed that no grid solvable with only naked singles is included in the minimum Sudoku collection.", "This poses an open problem of whether there is a gap between the minimum numbers for strategy-solvable Sudokus and proper Sudokus.", "Our method will be useful in tackling this problem thanks to the ability of determining unsolvability.", "We conduct experiments to compare our method with the generator of Zama and Sasano [9], using grids varying the position and the number of clues at random.", "From the results, we observe that our method terminates within 1 minutes for many instances, showing our method being stable in terms of running time.", "However, as the number of clues gets closer to 20, the running time rapidly increases and exceeds a time limit.", "On the other hand, the generator of Zama and Sasano often can find solutions much faster even in near 20 clues, while the performance sharply deteriorates around 45 clues and exceeds the time limit for all grids with more clue positions.", "Perhaps this is due to the exponential blow-up of the search space (in other words, the exponential decline in the density of solutions).", "We also evaluate our method for several instances with 17 clue positions taken from known minimum Sudokus to see the efficiency for deciding unsolvability.", "The main contributions of the present paper are to formulate the concept of strategy-solvability, to establish an exact method for the strategy-solvable Sudoku clues problem, and to demonstrate the flexibility of the CSP-based approach.", "It remains as future work to improve our method in less clues.", "Unless otherwise noted, the size of Sudoku is fixed to $9\\times 9$ throughout the paper.", "This is simply for convenience and our method can be easily translated into general Sudokus on $n^2\\times n^2$ grid.", "The paper is organized as follows.", "Section  introduces necessary notations, terminology, and explains strategies.", "Section  formulates the concept of strategy-solvability and the strategy-solvable Sudoku clues problem.", "Section  proposes an exact method for the strategy-solvable Sudoku clues problem, and Section  presents two improvements for our method.", "Section  presents an application to the strategy-solvable minimum Sudoku problem.", "Section  presents experimental results.", "Section  concludes this paper." ], [ "Preliminaries", "In this section we introduce necessary notations and terminology, and we explain strategies." ], [ "Notations and Terminology", "For convenience, rows and columns are numbered from 0 to 8.", "A cell is denoted by the pair $\\left(i,j\\right)$ of a row index $i$ and a column index $j$ .", "The 9 subgrids of $3\\times 3$ are called blocks.", "Rows, columns, and blocks are collectively called groups.", "A group is identified with the set of all cells in the group.", "By abuse of notation, we denote by $G\\setminus \\left(i,j\\right)$ the difference of a singleton $\\lbrace \\left(i,j\\right)\\rbrace $ from a group $G$ .", "Given a partially completed grid, the goal of a Sudoku puzzle is to fill in all empty cells with digits from 1 to 9 in such a way that each cell has a single digit, and each digit appears only once in every group.", "The completed grid is called a solution.", "A Sudoku is a convenient alias for a partially completed grid given as an initial grid.", "A Sudoku is proper if it has a unique solution.", "The occurrence of a digit in an initial grid is called a clue, and Sudoku clues are the clues in an initial grid.", "A cell in an initial grid to which some digit is designated to be placed as a clue is called a clue cell or a clue position." ], [ "Strategies", "There is a set of techniques (called strategies) that humans use in solving Sudokus by hand [1].", "Typically strategies are if-then rules: If a strategy is applicable to the current grid, then it might rule out digits as candidates or might determine digits as those to be finally placed.", "Throughout the following explanation, let $n$ be a nonzero digit and $\\left(i,j\\right)$ be a cell.", "A naked single is a strategy that places $n$ in $\\left(i,j\\right)$ if no other candidate but $n$ remains at $\\left(i,j\\right)$ .", "For example, let us look at the gray cell $\\left(4,7\\right)$ in the left grid of Figure REF .", "Since all digits but 5 appear in either group having $\\left(4,7\\right)$ , these digits must be ruled out and only 5 remains.", "Hence, 5 is placed in $\\left(4,7\\right)$ .", "Starting with the left grid of Figure REF , one can reach the right grid using only naked singles, but cannot proceed any more because of two or more candidates over all empty cells.", "Figure: Starting with the partially completed grid on the left, one can reach the grid on the right using only naked singles but cannot proceed any more.A hidden single is a strategy that places $n$ in $\\left(i,j\\right)$ if there is a group $G$ having $\\left(i,j\\right)$ such that no cell in $G\\setminus \\left(i,j\\right)$ has $n$ as a candidate.", "For example, let us look at the right grid of Figure REF .", "For the row of index 1, we can observe that every cell in the row except for the gray cell $\\left(1,6\\right)$ does not have 7 as a candidate.", "Indeed, for any such cell, 7 already appears in another cell of the same column.", "Hence, a hidden single strategy determines 7 in $\\left(1,6\\right)$ .", "In this way, by applying hidden singles repeatedly, the grid can be completed.", "Let $A,B$ be groups such that $|A\\cap B|=3$ .", "A locked candidate is a strategy that rules out $n$ over all cells in one difference set $B\\setminus A$ if no cell in the other difference set $A\\setminus B$ has $n$ as a candidate.", "For example, let us look at the right grid of Figure REF .", "Let $A$ be the column of index 7, and let $B$ be the block adjacent, on the right, to the center block.", "No empty cell in $A\\setminus B$ has 3 as a candidate because for each such cell, there is another cell in the same row to which 3 is already placed.", "Hence, 3 is ruled out for all empty cells in $B\\setminus A$ .", "Because of this, 2 becomes a unique candidate at $\\left(3,6\\right)$ .", "It is determined as the digit at $\\left(3,6\\right)$ by a naked single.", "In this way, the grid also can be completed using locked candidates as well as naked singles." ], [ "Formulation", "Although some known strategies are introduced in the previous section and some solvable cases are explained using particular grids, there are still unclear points such as what is a strategy in general and what operations are allowed for placing or ruling out digits, and so on.", "In this section, we thus introduce a rigorous framework to discuss solvability for Sudoku instances.", "This allows us to handle not only known strategies but also general strategies under a few reasonable assumptions.", "This also makes it clear that, for example, it is not allowed to temporarily place digits and then cancel them when a contradiction occurs.", "Such an inference would allow us to do backtracking, which nullifies the restriction of completion methods.", "Figure: Sequence of grids in 4×44\\times 4 Sudoku, evolving from the left-most initial state to the right-most final state.Larger digits are determined ones and smaller digits are candidates.A state transition model of Sudoku is a tuple $(Q, R, I, F)$ such that $Q$ is the set of all possible states, $R$ is a state transition relation, $I$ is the set of initial states, and $F$ is the set of final states.", "We will explain each component below.", "Figure REF shows how a grid evolves as strategies are applied, which will be used as an example in the succeeding explanation.", "A state of grid (or a state in short) is a pair $\\left(f,g\\right)$ of functions.", "The function $f$ maps the set of cells to the set of digits, and $f\\left(i,j\\right)=n$ means that if $n\\ne 0$ , then $n$ is placed in $\\left(i,j\\right)$ ; otherwise, no digit is placed.", "The function $g$ maps the set of cells to the powerset of the set of nonzero digits, and $g\\left(i,j\\right)$ represents the set of all candidates at $\\left(i,j\\right)$ .", "Here, by abuse of notation, $f\\left(i,j\\right)$ and $g\\left(i,j\\right)$ denote $f\\left(\\left(i,j\\right) \\right)$ and $g\\left(\\left(i,j\\right) \\right)$ , respectively.", "All states must satisfy the conditions $S_1$ and $S_2$ below: for all cells $\\left(i,j\\right)$ and nonzero digits $n$ , $g\\left(i,j\\right)\\ne \\emptyset $ , $n\\notin g\\left(i,j\\right)$ if there is a different nonzero digit $n^{\\prime }$ from $n$ such that $f\\left(i,j\\right)=n^{\\prime }$ or there is a different cell $\\left(i^{\\prime },j^{\\prime }\\right)$ from $\\left(i,j\\right)$ with both in a common group such that $f\\left(i^{\\prime },j^{\\prime }\\right)=n$ .", "We denote by $Q$ the set of all states.", "The antecedent of $S_2$ is a condition for digits $n$ that can be safely ruled out whenever some digits are placed.", "Imposing $S_2$ ensures that all such digits are ruled out.", "This is simply to make all states having a type of normal form, and it would not deduce any wrong consequence as long as decisions for number-placements are correct such as not violating the properness.", "In Figure REF we can observe that all grids are in \"normal form\".", "A state transition relation is a binary relation $R\\subseteq Q\\times Q$ .", "The $\\left(q,q^{\\prime }\\right)\\in R$ (or $qRq^{\\prime }$ in infix notation) means that there are strategies by which $q$ is changed to $q^{\\prime }$ .", "We refer to $q$ and $q^{\\prime }$ as the source state and the target state of the transition, respectively.", "All state transitions must satisfy that for any cell $\\left(i,j\\right)$ and nonzero digit $n$ , if $f\\left(i,j\\right)=n$ holds in the source state, so does in the target state; if $n\\notin g\\left(i,j\\right)$ holds in the source state, so does in the target state.", "This simply means that once digits are placed or ruled out, then it cannot be canceled afterward.", "We put one more assumption as follows.", "For any three states $q_1=\\left(f_1,g_1\\right)$ , $q_2=\\left(f_2,g_2\\right)$ , and $q_3=\\left(f_3,g_3\\right)$ with $q_1 R q_2$ and $q_2 R q_3$ , if $g_1=g_2$ , then $q_2=q_3$ .", "We consider this assumption not so strong.", "Indeed, many strategies such as naked singles and hidden singles decide to place digits by examining only candidates.", "For such strategies, suppose that $q_2$ is obtained from $q_1$ by applying all strategies applicable to $q_1$ .", "Then, if $g_1=g_2$ , no strategy applicable to $q_2$ remains, and $q_2$ cannot be changed any more, i.e., $q_2=q_3$ .", "A state $q=\\left(f,g\\right)\\in Q$ is an initial state if it satisfies the followings: $f\\left(i,j\\right)\\ne 0$ for all clue cells; $f\\left(i,j\\right)=0$ for the other cells; $n\\notin g\\left(i,j\\right)$ if and only if the antecedent of $S_2$ holds.", "The last condition ensures that for each clue cell, all digits but the determined one are ruled out; all digits that appear as candidates in a common group to some determined digit are ruled out; (the only-if part) no other digits are ruled out.", "Notice that the only-if part is the most essential because the if part can be omitted thanks to $S_2$ .", "By the way, for non-initial states, this must not be assumed in general because of strategies for number-removals such as locked candidates.", "For example, in the left most grid of Figure REF we can observe that all digits not designated to be ruled out by the antecedent of $S_2$ remain as candidates.", "We denote by $I$ the set of all initial states.", "A final state is a state such that digits are determined for all cells, that is, $f\\left(i,j\\right)\\ne 0$ for all $\\left(i,j\\right)$ .", "We denote by $F$ the set of all final states.", "Definition 3.1 Let $(Q,R,I,F)$ be a state transition model of Sudoku.", "A state $q_0$ is strategy-solvable if there is a sequence of states $q_0,\\ldots ,q_k\\in Q$ such that $q_0\\in I$ , $q_k\\in F$ , and $q_i R q_{i+1}$ for all $i\\in \\lbrace 0,\\ldots ,k-1\\rbrace $ .", "Definition 3.2 Let $(Q,R,I,F)$ be a state transition model of Sudoku.", "The Strategy-solvable Sudoku Clues problem (or SSC in short) is to determine whether there is a strategy-solvable state of $(Q,R,I,F)$ .", "Suppose that $R$ is given as a logical formula that represents strategies and a set $S$ of clue cells is also given.", "Consider $Q$ , $I$ , and $F$ as the ones defined as described above.", "Sudoku clues are strategy-solvable if the initial state corresponding to the Sudoku clues is strategy-solvable with respect to $(Q,R,I,F)$ .", "The strategy-solvable Sudoku clues problem is defined accordingly.", "If the output of the problem is yes, in practice we will also ask for generating strategy-solvable Sudoku clues.", "Proposition 1 Let $q=\\left(f,g\\right)$ be a state of grid.", "For any cells $\\left(i,j\\right)$ , $\\left(i^{\\prime },j^{\\prime }\\right)$ in a common group, $f\\left(i,j\\right)=f\\left(i^{\\prime },j^{\\prime }\\right)$ implies $\\left(i,j\\right)=\\left(i^{\\prime },j^{\\prime }\\right)$ or $f\\left(i,j\\right)=0$ .", "Suppose that $f\\left(i,j\\right)=f\\left(i^{\\prime },j^{\\prime }\\right)$ , $\\left(i,j\\right)\\ne \\left(i^{\\prime },j^{\\prime }\\right)$ , and $f\\left(i,j\\right)\\ne 0$ .", "Let $n=f\\left(i^{\\prime },j^{\\prime }\\right)$ .", "Applying $S_2$ , we obtain $n\\notin g\\left(i,j\\right)$ .", "Since $n=f\\left(i,j\\right)$ , applying $S_2$ , we obtain $n^{\\prime }\\notin g\\left(i,j\\right)$ for all $n^{\\prime }\\ne n$ .", "From both, $g\\left(i,j\\right)=\\emptyset $ follows, and this is contradictory to $S_1$ .", "Corollary 1 Every final state represents a Sudoku solution.", "Let $q=\\left(f,g\\right)$ be a final state.", "Since $f$ is a function, each cell must have a single digit.", "From Proposition REF and $f\\left(i,j\\right)>0$ , it follows that each digit appears once in every group.", "Notice that this corollary holds no matter what state transition relation is given.", "Proposition 2 Let $R$ be a state transition relation.", "Let $q_{0}=\\left(f_0,g_0\\right),\\ldots ,q_{k}=\\left(f_k,g_k\\right)$ be a sequence of states such that $q_s R q_{s+1}$ for all $s\\in \\lbrace 0,\\ldots ,k-1\\rbrace $ .", "If $g_s\\ne g_{s+1}$ for all $s\\in \\lbrace 0,\\ldots ,k-1\\rbrace $ , then $k\\le 648$ .", "Let $m_s$ be the number of all triples $\\left(i,j,n\\right)$ such that $n\\in g_s\\left(i,j\\right)$ .", "Clearly, $729=9^3\\ge m_0>m_1>\\cdots >m_k\\ge 81$ .", "Hence $k\\le 648$ ." ], [ "Our Method", "In this section, we propose an exact method for the strategy-solvable Sudoku clues problem (SSC).", "As strategies allowed in completing Sudokus, this paper employs naked singles, hidden singles, and locked candidates, which are quite basic yet enough powerful to complete many Sudokus." ], [ "Overview", "The key of our method is to encode a given SSC instance to an equivalent constraint satisfaction problem (CSP) instance and then to solve it using a generic CSP solver.", "For the CSP encoding, we introduce variables for representing states of grid and other auxiliary variables.", "Using such variables, we represent constraints for states, state transitions, particular strategies, and so on.", "Any variable assignment that satisfies all such constraints substantially corresponds to a sequence of states $q_0,\\ldots ,q_k$ with $q_0$ strategy-solvable, as characterized in Definition REF .", "Here, the initial state $q_0$ represents an assignment of digits to clue cells and the whole sequence represents a history of evolving grids and applied strategies, which eventually reaches a Sudoku solution.", "Hence, by applying a CSP solver to the encoded constraints, we eventually obtain strategy-solvable Sudoku clues if exist; otherwise, we eventually recognize the strategy-unsolvability of the clue cells, that is, no assignment of digits to the clue cells is strategy-solvable.", "The details of our method are explained as follows.", "At first, we present constraints for a state transition framework, which is almost independent of particular strategies.", "As stated above, we introduce variables for representing states of grid and auxiliary variables.", "We then represent various constraints using such variables.", "After that, we shift to particular strategies.", "we present constraints for number-placements and constraints for number-removals.", "We finally provide the whole picture of our method as well as some remarks." ], [ "Constraints for A General State Transition Framework", "We introduce variables for representing states of grid.", "We then present constraints for initial states, constraints for state transitions, constraints for final states." ], [ "Variables", "Let $q_k=(f_k,g_k)$ be a state of grid in step $k\\ge 0$ , i.e., a state reachable from an initial state by $k$ transitions.", "Let $i$ and $j$ be a row index and a column index, respectively.", "The integer variable $X\\left(i,j,k\\right)$ encodes the value of $f_k$ at cell $\\left(i,j\\right)$ .", "In other words, $X\\left(i,j,k\\right)$ takes 0 if no digit is placed in $\\left(i,j\\right)$ , and it takes a nonzero digit $n$ if $n$ is placed in $\\left(i,j\\right)$ .", "The Boolean variable $Y\\left(i,j,n,k\\right)$ encodes whether the set of candidates, $g_k\\left(i,j\\right)$ , at cell $\\left(i,j\\right)$ includes a nonzero digit $n$ .", "Here $Y\\left(i,j,n,k\\right)$ is true if and only if $n\\in g_k\\left(i,j\\right)$ .", "We call variables of the forms $X\\left(i,j,k\\right)$ and $Y\\left(i,j,n,k\\right)$ $X$ -variables and $Y$ -variables, respectively.", "The remaining variables are Boolean variables of the form $Z\\left(m\\right)$ , which are used for auxiliary purpose.", "We call them $Z$ -variables.", "Each $Z$ -variable is introduced per particular condition for a number-placement or a number-removal.", "The parameter $m$ is simply an identifier for distinguishing between a large number of particular conditions.", "We thus number all such conditions in an arbitrary order.", "By the way, since $X$ -variables and $Y$ -variables have $k$ in their parameters, we have to fix the maximum number of steps, $K$ , in advance.", "In principle, it suffices to let $K=649$ .", "This is shown later.", "Since this is too large in practice, we will propose a practical method for doing with a smaller $K$ while keeping the exactness in Section ." ], [ "Constraints for Initial States", "The constraints for initial states are as follows.", "For all clue cells $\\left(i,j\\right)$ , $X\\left(i,j,0\\right) \\ne 0,$ and for the other cells $\\left(i,j\\right)$ , $X\\left(i,j,0\\right) = 0.$ Here we would not explicitly impose any constraint on $Y$ -variables in step 0 because such constrains can be derived from the constraints for $Y$ -variables of an arbitrary step and the constraints for number-removals." ], [ "Constraints for State Transitions", "The constraints for state transitions represent how $X$ -variables and $Y$ -variables in the current grid are determined from the previous grid.", "The constraint that allows us to determine a nonzero digit $n$ at cell $\\left(i,j\\right)$ in step $k$ is as follows.", "$X\\left(i,j,k\\right) = n \\leftrightarrow \\bigvee Z\\left(m\\right)$ Here, $Z\\left(m\\right)$ runs over all possible $Z$ -variables that can derive the left hand side.", "The only-if part means that if $X\\left(i,j,k\\right)=n$ , then there must be an evidence, i.e., some $Z$ -variable is true.", "One such $Z$ -variable is the case for which $n$ is determined at $\\left(i,j\\right)$ in the previous step.", "$Z\\left(m\\right) \\leftrightarrow X\\left(i,j,k-1\\right) = n$ As noticed before, suppose that the parameter $m$ is uniquely determined in order to distinguish the condition in the right hand side from the other particular conditions for $Z$ -variables.", "Hereafter we will follow the same implicit agreement whenever $Z\\left(m\\right)$ appears in constraints.", "The other $Z$ -variables of Formula REF will be detailed later according to particular cases.", "Notice that as discussed in Section  in order for our assumption for state transitions to be met, it is sufficient to let $Z\\left(m\\right)$ be determined by only $Y$ -variables in the previous step (except for Formula REF ).", "The constraint that allows us to rule out a nonzero digit $n$ as a candidate at cell $\\left(i,j\\right)$ in step $k$ is as follows.", "$\\lnot Y\\left(i,j,n,k\\right) \\leftrightarrow \\bigvee Z\\left(m\\right)$ In the same way as $X$ -variables, $Z\\left(m\\right)$ runs over all possible $Z$ -variables that can derive the left hand side.", "One such $Z$ -variable is the case for which $n$ is ruled out as a candidate at $\\left(i,j\\right)$ in the previous step.", "$Z\\left(m\\right) \\leftrightarrow \\lnot Y\\left(i,j,n,k-1\\right)$ The other $Z$ -variables of Formula REF will be detailed later according to particular cases." ], [ "Constraints for Final States", "The constraints for final states are those that allow us to reject the current state as soon as it turns out that there is no chance to reach final states.", "There are three cases for the current state of grid.", "All cells are completed.", "Some empty cells remain, and some candidates have been ruled out from the previous grid.", "Some empty cells remain, and no candidate has been ruled out from the previous grid.", "Clearly the first case must be accepted.", "For the third case, our assumption of state transitions (see Section ) implies that the current incomplete grid cannot be changed anymore.", "Hence, the third case must be rejected immediately.", "For the second case, since applicable strategies may remain for the current grid, the decision must be postponed to the next grid.", "The following formula is in change of this filtration.", "$\\bigwedge _{\\begin{array}{c}0\\le i,j<9\\\\ 1\\le n\\le 9\\end{array}}\\Bigl ( Y\\left(i,j,n,k-1\\right)\\leftrightarrow Y\\left(i,j,n,k\\right) \\Bigr )\\rightarrow \\bigwedge _{0\\le i,j <9} X\\left(i,j,k\\right)\\ne 0$ Suppose that the current grid is in step $k\\ (\\ge 1)$ .", "The left hand side means that no candidate has not been ruled out from the previous grid, and the right hand side means that the current grid is completed.", "It is clear that this formula rejects only the third case among the three cases.", "As announced earlier, we here show that $K=649$ is sufficient.", "Let $q_0=\\left(f_0,g_0\\right),\\ldots ,q_k=\\left(f_k,g_k\\right),\\ldots $ be any sequence of states such that $q_0$ is an initial state and $q_k$ is the next state of $q_{k-1}$ for all $k$ .", "The assumption for state transitions (see Section ) ensures that once $g_{\\bar{k}-1}=g_{\\bar{k}}$ for some $\\bar{k}$ , it follows that $q_{\\bar{k}}=q_k$ for all $k\\ge \\bar{k}$ .", "From Proposition REF , the minimum $\\bar{k}$ with $g_{\\bar{k}-1}=g_{\\bar{k}}$ is bounded from above by 649.", "Therefore it suffices to let $K=649$ .", "Notice that $\\bar{k}\\le K$ is necessary in order to decide whether $q_{\\bar{k}}$ is a final state." ], [ "Constraints for Particular Number-placements", "We explain particular cases for $Z\\left(m\\right)$ s in Formula REF that we have put off." ], [ "Naked Singles", "The right hand side of the following formula is the condition for which a naked single allows us to determine a nonzero digit $n$ at cell $\\left(i,j\\right)$ in step $k\\ (\\ge 1)$ .", "Notice that in order for this to take effect, it is necessary that the following $Z\\left(m\\right)$ is registered to Formula REF .", "$Z\\left(m\\right) \\leftrightarrow \\bigwedge _{\\begin{array}{c}n\\ne n^{\\prime }\\\\ 1\\le n^{\\prime }\\le 9\\end{array}} \\lnot Y\\left(i,j,n^{\\prime },k-1\\right)$ The right hand side means that all digits but $n$ are ruled out as candidates at cell $\\left(i,j\\right)$ in the previous step.", "There are two cases: either only $n$ remains as a candidate or no digit remains.", "Although a naked single is applicable only to the former case, there is no substantial problem.", "Because even if the latter case occurs, we obtain $X\\left(i,j,k\\right)=n$ from Formulas REF and REF , and we also obtain $X\\left(i,j,k\\right)=n^{\\prime }$ for any other digit $n^{\\prime }$ in the same way, which is a contradiction." ], [ "Hidden Singles", "The right hand side of the following formula is the condition for which a hidden single allows us to determine a nonzero digit $n$ at cell $\\left(i,j\\right)$ in step $k\\ (\\ge 1)$ .", "$Z\\left(m\\right) \\leftrightarrow \\bigwedge _{\\left(i^{\\prime },j^{\\prime }\\right)\\in G\\setminus \\left(i,j\\right)} \\lnot Y\\left(i^{\\prime },j^{\\prime },n,k-1\\right)$ Here, let $G$ be any group such that $\\left(i,j\\right)\\in G$ .", "There are three cases for $G$ , and for each case the corresponding constraint must be generated from Formula REF .", "The right hand side of Formula REF does not exclude the case for which $n$ is ruled out over all cells in $G$ , but there is no substantial problem, just like Formula REF ." ], [ "Constraints for Particular Number-removals", "We in turn explain particular cases for $Z\\left(m\\right)$ s in Formula REF that we have put off." ], [ "Conditions for States of Grid", "We have imposed the two conditions $S_1$ and $S_2$ on states of grid in order to reject states that violate the rules of a Sudoku puzzle.", "We introduce constraints corresponding to $S_1$ and $S_2$ .", "Unlike the other constraints, the constraints below determine a relation between $X$ -variables and $Y$ -variables in the same step.", "The following formula corresponds to $S_1$ , which ensures that all cells $\\left(i,j\\right)$ have at least one candidate in all steps $k\\ge 0$ .", "$\\bigvee _{1\\le n\\le 9} Y\\left(i,j,n,k\\right)$ The following formula in turn corresponds to one of the two cases in the antecedent of $S_2$ , that is, a different digit $n^{\\prime }$ from $n$ is determined at cell $\\left(i,j\\right)$ .", "$Z\\left(m\\right) \\leftrightarrow \\bigvee _{\\begin{array}{c}n\\ne n^{\\prime }\\\\ 0\\le i,j< 9\\end{array}} X\\left(i,j,k\\right)=n^{\\prime }$ Here, suppose that $Z\\left(m\\right)$ is registered in Formula REF .", "If the right hand side of Formula REF holds, then it follows from Formulas REF and REF that $n$ is ruled out as a candidate at $\\left(i,j\\right)$ in the same step $k$ , which is the consequent of $S_2$ .", "The following formula corresponds to the other case in the antecedent of $S_2$ , that is, there is a different cell $\\left(i^{\\prime },j^{\\prime }\\right)$ from $\\left(i,j\\right)$ with both in a common group $G$ such that $n$ is determined at $\\left(i^{\\prime },j^{\\prime }\\right)$ .", "$Z\\left(m\\right) \\leftrightarrow \\bigvee _{\\left(i^{\\prime },j^{\\prime }\\right)\\in G\\setminus \\left(i,j\\right)} X\\left(i^{\\prime },j^{\\prime },k\\right)=n$ Here, let $G$ be any group such that $\\left(i,j\\right)\\in G$ .", "There are three cases for $G$ , and for each case the corresponding constraint must be generated from Formula REF .", "Just like Formula REF , Formula REF together with Formula REF imply that $n$ is ruled out as a candidate at $\\left(i,j\\right)$ in the same step $k$ , which is the consequent of $S_2$ ." ], [ "Locked Candidates", "The right hand side of the following formula is the condition for which a locked candidate allows us to rule out a nonzero digit $n$ at cell $\\left(i,j\\right)$ in step $k\\ (\\ge 1)$ .", "$Z\\left(m\\right) \\leftrightarrow \\bigvee _{\\left(i^{\\prime },j^{\\prime }\\right)\\in A\\setminus B} \\lnot Y\\left(i^{\\prime },j^{\\prime },n,k-1\\right)$ Here, let $A$ and $B$ be any groups such that $\\left(i,j\\right)\\in B\\setminus A$ and $|A\\cap B|=3$ .", "For all possible combinations of $A$ and $B$ , the corresponding constraint must be generated from Formula REF .", "Notice that Formula REF can be commonly used to deduce $Y\\left(i^{\\prime \\prime },j^{\\prime \\prime },n,k\\right)$ for all $\\left(i^{\\prime \\prime },j^{\\prime \\prime }\\right)\\in B\\setminus A$ , by which the size of constraints can be largely reduced." ], [ "Remarks", "We have presented a CSP encoding for the strategy-solvable Sudoku clues problem.", "That is, given a set of cells, there are clues for the cells such that the grid can be completed using naked singles, hidden singles, and locked candidates if and only if there is an assignment of $X$ -variables, $Y$ -variables, and $Z$ -variables that satisfies all the constraints generated from Formulas REF to REF .", "A satisfying assignment substantially corresponds to a sequence of states $q_0,\\ldots ,q_k$ with $q_0$ strategy-solvable, as characterized in Definition REF .", "By applying a CSP solver to the encoded constraints, we eventually obtain strategy-solvable Sudoku clues if exist; otherwise, we eventually recognize the strategy-unsolvability of the clue cells, that is, no assignment of digits to the clue cells is strategy-solvable.", "The constraints for a general state transition framework are almost independent of particular strategies.", "The only connection is via $Z$ -variables in Formulas REF and REF .", "In order to include additional strategies, it is sufficient to model these strategies as logical formulas independently and then register new $Z$ -variables for them to Formulas REF or REF ." ], [ "Two Improvements", "In this section, we present two improvements for our method." ], [ "Reduction of Constraint Size", "The biggest issue of our method is that there are hundreds of thousands of constraints.", "A simple way for reducing a constraint size is to eliminate constraints concerning clue cells.", "For all clue cells, digits are determined in an initial state, and the determined digits do not change over all succeeding steps.", "Hence, all constraints necessary for clue cells in each step is take the same values as those in the previous step.", "In the constraints for final states, there is no need to examine the values of $X$ -variables and $Y$ -variables for clue cells.", "However, we must not ignore the constraints for clue cells in step 0 corresponding to Formulas REF , REF , and REF because otherwise, we could not reject initial states violating the rules of Sudoku puzzle." ], [ "Incremental Approach", "Our method requires to fix a maximum step size, $K$ , in advance.", "Since it has a significant impact on a constraint size, $K$ needs to be as small as possible.", "However, for a small $K$ , our method may return false clues.", "To see this, let us take a look at Figure REF .", "Suppose that the left grid is an initial state and the right grid is a state in step $K$ .", "The right grid is not yet completed, but no constraint has been violated up to step $K$ .", "Since there is no further step, our method returns the initial grid, even if only a naked single strategy is allowed.", "Notice that if the right grid was in step $K-1$ or less, our method would reject the initial grid (because the constraints for final states are violated).", "We propose a practical method for doing with a smaller $K$ while keeping the exactness.", "A basic idea is to repeatedly apply our original method while incrementing $K$ from an initial number $K_{min}$ until $K$ exceeds a sufficiently large number $K_{max}$ .", "Algorithm REF is a pseudo code for the method.", "Notice that once constraints become unsatisfiable, constraints with any lager maximum step size must be unsatisfiable.", "[t] Incremental approach for the strategy-solvable Sudoku clues (SSC) $K=K_{min}$ to $K_{max}$ Generate constraints for a given SSC instance with maximum step $K$ .", "the set of constraints is unsatisfiable return UNSAT the grid in step $K$ is completed return Sudoku clues in the initial grid." ], [ "Application", "Besides the clue generation, we present an application of our method to the problem of determining the minimum number of Sudoku clues that are solvable with a given set of strategies.", "Our method is easily customized with a small modification as follows.", "Remove Formulas REF and REF .", "Instead, introduce integer variables taking 0 or 1, $U\\left(i,j\\right)$ , for all cells $\\left(i,j\\right)$ and the following formula.", "$U\\left(i,j\\right) = 1 \\leftrightarrow X\\left(i,j,0\\right) \\ne 0$ Introduce the following formula for ensuring that the number of determined digits in step 0 is less than or equal to a threshold $\\theta $ .", "$\\sum U\\left(i,j\\right) \\le \\theta $ Here the summation runs over all variables $U\\left(i,j\\right)$ .", "In order to compute the minimum number, it is sufficient to repeatedly solve the modified constraints while decreasing $\\theta $ one by one until the constraints become unsatisfiable." ], [ "Experiments", "In this section, we conduct experiments to evaluate our method." ], [ "Common Settings and Remarks", "Our method uses Sugar version 2.3.3 [8] [7] as a CSP solver and MiniSat version 2.0 [3] [2] as a SAT solver internally invoked by Sugar.", "The maximum step size $K$ is set to 30, and our method is applied only once (i.e., no use of incremental approach) because in preliminary experiments, we have confirmed that $K=30$ is sufficient for many instances including those used in this experiments.", "The computational environment is as follows.", "OS: Ubuntu 18.04.4 LTS Main memory: 16GB CPU: Intel ® Core ™ i7-4600U 2.10GHz The implementation of our method is checked for all possible arrangements of 3 and 4 clue positions for $4\\times 4$ Sudoku.", "These instances are so small that a naive brute force search can quickly decide.", "It is confirmed that the solvability results for our method completely coincide with those for the brute force search over all instances.", "Here all strategies allowed to be used are naked singles, hidden singles, and locked candidates.", "It turns out that there is no strategy-solvable instance with 3 clue positions, and there are exactly 704 strategy-solvable instances with 4 clue positions.", "By the way, all of the 704 instances are also solvable with naked singles only.", "Since there is no proper $4\\times 4$ Sudoku with 3 clues, there is no gap between the minimum numbers of proper Sudokus and strategy-solvable Sudokus with naked singles.", "The implementation of (the CSP encoding part of) our method, tools such as the brute force search program, and all instances used in the experiments are publicly available in our websitehttp://www.disc.lab.uec.ac.jp/toda/code/scg.html , accessed in May 27th 2020.." ], [ "Running Time Comparison", "We compare our CSP-based method (CSP) with the generator of Zama and Sasano [9] (ZS).", "The time limit is set to 600 seconds.", "In both methods, only naked singles, hidden singles, and locked candidates are allowed to be used.", "Although other strategies are implemented in the generator of Zama and Sasano, we restricts the program to the three strategies by modifying their program.", "We make total 100 input instances (i.e.", "sets of cells) by repeating the following procedure: select a number $n$ from 20 to 79 at random, and generate distinct $n$ cells at random.", "Table REF shows the distribution of input instances with respect to the number of clues.", "All instances are confirmed to be strategy-solvable.", "Table: Distribution of input instances with respect to the number of cluesFigure: Comparison of running timeFigure REF shows a cactus plot of running time comparison.", "Each point is plotted so that the $x$ -coordinate is the number of instances solved within the time specified in the $y$ -coordinate.", "The points for our method (CSP) and the generator of Zama and Sasano (ZS) are indicated with $+$ and $\\times $ , respectively.", "The curve formed by the points for the same method shows an increase in the number of solved instances over time.", "Our method solves 95 of total 100 instances within the time limit, while the generator of Zama and Sasano solves 65 instances.", "The curve of our method shows a linear increase up to around 90 in the $x$ -axis.", "All instances in this range are solved within about 1 minutes.", "The curve grows rapidly after 90.", "All instances in this rage as well as unsolved instances have 25 or less cells.", "On the other hand, the generator of Zama and Sasano often can find solutions much faster even in near 20 cells as the curve of Zama-Sasano almost overlaps the $x$ -axis up to near 60.", "All instances in this range have around 45 or less cells.", "After that, the performance drops significantly around 60 as the curve sharply increases.", "All instances with 50 or more cells cannot be solved within the time limit." ], [ "Evaluation on Unsolvability", "To see the efficiency for deciding unsolvablity, we make instances which seem to be on the border between solvable cases and unsolvable cases in such a way that we randomly select 30 minimum Sudokus from the collection of Gordon F. Royle [6] and extract only clue positions, simply forgetting placed digits.", "In this experiment, all strategies allowed to be used are naked singles only.", "Notice that although no grid solvable with naked singles is included in the collection, it is possible that changing digits but in the same positions makes grids solvable.", "Within several hours, our method terminates for 14 of the 30 instances, and all of them are confirmed to be not strategy-solvable.", "Tables REF and REF in Appendix show the chosen minimum Sudokus and the running times.", "Although 10 instances take merely from 10 to 20 minutes, other instances take much more time.", "It is still unknown whether there are 17 clues solvable with naked singles only.", "It appears far from the settlement of the problem on a gap betwen the numbers of strategy-solvable Sudokus and proper Sudokus by a computer program, considering a large number of possible arrangements of 17 positions." ], [ "Conclusion", "Sudokus that appear in books, newspapers, etc often have a regular pattern in the arrangement of initial digits and they are typically made so that all empty cells can be completed using some known techniques, called strategies.", "We formally defined the problem of generating such Sudoku instances by introducing the concept of strategy-solvability, which means that all empty cell can be filled in with digits using only a given set of strategies.", "We proposed an exact method for solving this problem.", "The key is to encode a given problem instance into an equivalent CSP instance and then solve it by applying a CSP solver.", "There are a few existing researches, but all of them are based on the generate-and-test method, which repeats to generate a set of clues and then test whether it is strategy-solvable.", "There are some drawbacks such as not being able to recognize that a specified set of cells is strategy-unsolvable.", "To the best of our knowledge, our method is the first exact method except for the trivial brute-force search.", "Our method eventually can find strategy-solvable Sudoku clues if exist, and otherwise, our method eventually can recognize strategy-unsolvability.", "Besides the clue generation, we presented an application of our method to the problem of determining the minimum number of strategy-solvable Sudoku clues, demonstrating that our method is easily customized with a small modification.", "We conducted experiments to compare our method with the generator of Zama and Sasano, using grids varying the positions and the numbers of clues at random.", "From the results we observed that our method terminated within 1 minutes for many grids, showing our method being stable in terms of running time.", "However, as the number of clues got closer to 20, the running time rapidly increased and exceeded the time limit, which is set to 600 seconds.", "On the other hand, the generator of Zama and Sasano often could find solutions much faster even in near 20 clues, while the performance sharply deteriorated around 45 clues and exceeded the time limit for all grids with more clue positions.", "We also evaluated our method for several instances with 17 clue positions taken from known minimum Sudokus to see the efficiency for deciding unsolvability.", "It remains as future work to improve our method in less clues." ], [ "Acknowledgement", "This work was supported by JSPS KAKENHI Grant Number 17K17725." ] ]

[ [ "Active Brownian particle in harmonic trap: exact computation of moments,\n and re-entrant transition" ], [ "Abstract We consider an active Brownian particle in a $d$-dimensional harmonic trap, in the presence of translational diffusion.", "While the Fokker-Planck equation can not in general be solved to obtain a closed form solution of the joint distribution of positions and orientations, as we show, it can be utilized to evaluate the exact time dependence of all moments, using a Laplace transform approach.", "We present explicit calculation of several such moments at arbitrary times and their evolution to the steady state.", "In particular we compute the kurtosis of the displacement, a quantity which clearly shows the difference of the active steady state properties from the equilibrium Gaussian form.", "We find that it increases with activity to asymptotic saturation, but varies non-monotonically with the trap-stiffness, thereby capturing a recently observed active- to- passive re-entrant behavior." ], [ "Introduction", "Active matter describes systems of self-propelled particles [1], [2], [3].", "The associated energy pump and dissipation at the smallest scale maintains the system out of equilibrium, breaking the detailed balance condition and the equilibrium fluctuation-dissipation relation.", "Natural examples of active system range from molecular motors, cytoskeleton, bacteria to bird, fish and animals.", "Many situations of biological interest involves confinement, e.g., chromosomes inside the cell nucleus [4].", "Drawing inspiration from the natural examples, a large number of artificial micro- or nano- swimmers, have been fabricated [3].", "The active colloids self-propel in their instantaneous heading directions, through auto-catalytic drive utilizing ambient optical, thermal, electrical, or chemical energy.", "Such a motion can be described by the active Brownian particle (ABP) model, where each particle self-propel with a constant speed in the heading direction, which undergoes stochastic reorientation.", "Two other related models, the run and tumble particles [5], and the active Ornstein-Uhlenbeck process [6], [7] show similar dynamics at long time scales.", "Active particles in confinement show qualitatively distinct properties with respect to passive Brownian particles.", "While, a dilute gas of passive particles distribute homogeneously within a confinement, that of active particles aggregate near the boundaries and corners [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].", "In spite of the immense progress in the understanding of the collective properties of active matter, the non-equilibrium properties of non-interacting active particles is not yet completely understood.", "Even single self-propelled particles can display rich and counterintuitive physical properties [22], [23], [24], [25], [26], [27], [28], [29], [30], [31].", "Recent experiments [18], [21], [32], [10], and theoretical studies [7], [28], [33], [34], [35], [36], [37], [38] of trapped active particles have revealed hitherto unseen properties.", "These include a cross-over in the steady-state distribution of particle positions from a passive equilibrium-like Gaussian form with a peak at the trap-centre, to a strongly active non-Gaussian distribution with off-center peaks, as a function of trap stiffness and active velocity [18], [28], [35], [11], [16].", "In this paper we reconsider the problem of ABPs in a harmonic trap.", "We develop an exact analytical method to calculate all time-dependent moments of the relevant dynamical variables, the active orientation and positional displacement of the ABP, in arbitrary $d$ -dimensions.", "This is the first main contribution of the paper.", "Remarkably, the Fokker-Planck equation corresponding to the free ABP in the absence of translational noise was first studied as early as in 1952 [39], [40], in the context of the worm-like-chain model of polymers.", "Following Ref.", "[39] we develop the Laplace transform method to calculate the exact moments of the dynamical variables describing the ABP, in the presence of both the translational diffusion and a harmonic trap.", "The presence of the trapping potential ensures that the system will eventually reach a steady state.", "We show the full evolution to steady state following the first few moments of the displacement vector.", "The kurtosis of the position vector quantifies its deviation from a passive equilibrium-like Gaussian distribution peaked at the trap-center.", "With increasing active velocity, the amplitude of kurtosis grows from zero to saturate monotonically.", "In contrast, with increasing trap stiffness, it shows a non-monotonic variation with the maximal amplitude appearing at an intermediate stiffness.", "We use the kurtosis to obtain a phase diagram identifying the passive and active phases in arbitrary $d$ -dimensions, in terms of the amount of deviation from Gaussian statistics, in the activity-trapping potential plane.", "Note that there is no true phase transition in this single-particle system, this is really a crossover between active and passive behavior.", "The phase diagram shows a re-entrant crossover, from a passive to active to passive phase, with increasing trap stiffness, similar to what has recently been reported for an ABP in a two-dimensional trap [35].", "Obtaining this phase diagram in $d$ -dimensions analytically is the second main contribution of the current paper.", "In the context of the re-entrant behavior, we show that the presence of translational diffusion is crucial.", "In its absence the re-entrance disappears, and the amplitude of kurtosis grows monotonically to saturate with increasing trap stiffness.", "The plan of the paper is as follows.", "In Sec.", ", we present the model of ABP, and outline how to calculate any arbitrary moment of the relevant dynamical variables.", "In Sec.", ", we demonstrate detailed calculations of some of these moments, and analyze their time evolution.", "In Sec.", ", we describe the calculation of kurtosis, and present the phase diagram.", "Finally, in Sec.", "we conclude, summarizing our main results." ], [ "The Langevin and Fokker-Planck equations", "Let us consider an active Brownian particles (ABP) in $d$ dimensions.", "It is described by its position $ {\\bf r}=(x_1,x_2,\\ldots ,x_d)$ and orientation $ {\\hat{{u}}}=(u_1,u_2,\\ldots ,u_d)$ , where $ {\\hat{{u}}}.", "{\\hat{{u}}}=1$ .", "The orientation vector $ {\\hat{{u}}}$ performs Brownian motion on the surface of a $d$ -dimensional hypersphere and drives the motion of the active particle.", "In the presence of a translational Brownian noise and an external potential, $U( {\\bf r})$ , the particle's motion is described by the following stochastic equations (in Ito convention, see  [41], [42], [43]): $d x_i &=& \\left[ v_0 u_i - \\mu \\partial _{x_i} U \\right] dt + d{B_i^{\\rm t}}(t) \\\\d u_i &=& (\\delta _{ij} -u_i u_j)d B_j^{\\rm r} -{(d-1){D}_r} u_i dt, $ where the Gaussian noise terms $ d{B}^{\\rm t}$ and $ d{B}^{\\rm r}$ have mean zero and variances $\\langle d{B_i^{\\rm t}} d{B_j^{\\rm t}} \\rangle = 2 D \\delta _{ij} dt$ , $\\langle d{B_i^{\\rm r}} d{B_j^{\\rm r}} \\rangle = 2 {D}_r \\delta _{ij} dt$ .", "Alternatively, we can write eq:eom2 in the Stratonovich form $d u_i = (\\delta _{ij} -u_i u_j) \\circ d B_j^{\\rm r}$ .", "The form of eq:eom2 ensures the normalization $ {\\hat{{u}}}^2=1$ at all times.", "Noting that the stochastic equations, Eqs.", "(REF ,), correspond to the orientational vector $ {\\hat{{u}}}$ performing Brownian motion on the surface of a unit $d$ -dimensional hypersphere while the position vector $ {\\bf r}$ evolves via standard drift and diffusion terms, we can write the corresponding Fokker-Planck equation for the probability distribution $P( {\\bf r}, {\\hat{{u}}}, t)$ : $\\partial _t P( {\\bf r}, {\\hat{{u}}}, t) = D \\nabla ^2 P - \\nabla \\cdot [ v_0 {\\hat{{u}}}- \\mu \\nabla U] P + D_r \\nabla _u^2 P,$ where $ \\nabla = (\\partial _{x_1},\\partial _{x_2},\\ldots ,\\partial _{x_d}) $ denotes the $d$ -dimensional gradient operator, and $\\nabla _u^2$ denotes the spherical Laplacian defined on the ($d-1$ ) dimensional orientation space.", "We note that the spherical Laplacian can be expressed in terms of the cartesian coordinates ${\\bf y}$ , defined through $u_i=y_i/y$ where $y=|{\\bf y}|$ , as $\\nabla ^2_u= y^2\\sum _{i=1}^d \\partial ^2_{y_i} - [y^2\\partial ^2_y + (d-1) y \\partial _y]$ .", "In the limit of $D=0$ and $U( {\\bf r}) = 0$ , the above equation can be interpreted as that describing the probability distribution of the end-to-end separation $ {\\bf r}$ of a worm-like-chain with bending rigidity $\\kappa \\sim 1/D_r$ , interpreting the polymer contour length $L \\sim t$  [39], [44], [45], [46], [29].", "For a harmonic trap, $U ( {\\bf r}) = (1/2) k {\\bf r}^2$ , the eq:FP1 simplifies to $\\partial _t P( {\\bf r}, {\\hat{{u}}}, t) = D \\nabla ^2 P + D_r \\nabla _u^2 P - v_0\\, {\\hat{{u}}}\\cdot \\nabla P+ \\mu k {\\bf r}\\cdot \\nabla P + \\mu k d\\, P. \\nonumber $ Using the Laplace transform $\\tilde{P}( {\\bf r}, {\\hat{{u}}}, s) = \\int _0^\\infty dt e^{-s t} P( {\\bf r}, {\\hat{{u}}}, t) $ , this Fokker-Planck equation can be recast into the form, $-P( {\\bf r}, {\\hat{{u}}}, 0) + (s-\\mu k d) \\tilde{P}( {\\bf r}, {\\hat{{u}}}, s) = D \\nabla ^2 \\tilde{P} + D_r \\nabla _u^2 \\tilde{P} - v_0\\, {\\hat{{u}}}\\cdot \\nabla \\tilde{P} + \\mu k {\\bf r}\\cdot \\nabla \\tilde{P}.", "\\nonumber $ Defining the mean of an observable in the Laplace space $\\langle \\psi \\rangle _s = \\int d {\\bf r}\\, d {\\hat{{u}}}\\, \\psi ( {\\bf r}, {\\hat{{u}}}) \\tilde{P}( {\\bf r}, {\\hat{{u}}}, s)$ , and multiplying the above equation by $\\psi ( {\\bf r}, {\\hat{{u}}})$ and integrating over all possible $( {\\bf r}, {\\hat{{u}}})$ we find, $-\\langle \\psi \\rangle (0) + (s-\\mu k d) \\langle \\psi \\rangle _s = D \\langle \\nabla ^2 \\psi \\rangle _s + D_r \\langle \\nabla _u^2 \\psi \\rangle _s + v_0\\, \\langle {\\hat{{u}}}\\cdot \\nabla \\psi \\rangle _s - \\mu k \\langle \\nabla \\cdot ( {\\bf r}\\psi ) \\rangle _s.$ where the initial condition sets $\\langle \\psi \\rangle (0) = \\int d {\\bf r}\\, d {\\hat{{u}}}\\, \\psi ( {\\bf r}, {\\hat{{u}}}) P( {\\bf r}, {\\hat{{u}}}, 0)$ .", "We consider the initial position of the particle to be at $ {\\bf r}_0$ , shifted from the center of the trap, and its initial orientation of activity along $ {\\hat{{u}}}_0$ such that $P( {\\bf r}, {\\hat{{u}}}, 0) = \\delta ( {\\bf r}- {\\bf r}_0) \\delta ( {\\hat{{u}}}- {\\hat{{u}}}_0)$ .", "In the following section we show how moment2 can be used to determine various moments of interest." ], [ "Calculation of moments", "In this section, we present detailed derivation of some of the relevant moments, show their time evolution, and analyze their steady state properties." ], [ "Orientational correlation", "The confinement is not expected to change the orientational relaxation of activity in a spherically symmetric particle.", "This can be illustrated by directly computing the two time correlation function of the active orientation.", "We use $\\psi ( {\\bf r}, {\\hat{{u}}}) = {\\hat{{u}}}\\cdot {\\hat{{u}}}_0$ in moment2, to find $\\langle \\psi \\rangle (0) = 1$ , $\\langle \\nabla ^2 \\psi \\rangle _s =0$ , $\\langle \\nabla _u^2 \\psi \\rangle _s = - (d-1) {\\hat{{u}}}\\cdot {\\hat{{u}}}_0$ , and $\\langle {\\hat{{u}}}\\cdot \\nabla \\psi \\rangle _s = 0 $ , and $ \\langle \\nabla \\cdot ( {\\bf r}\\psi ) \\rangle _s = d \\langle {\\hat{{u}}}\\cdot {\\hat{{u}}}_0 \\rangle _s$ .", "This leads to the relation $\\langle {\\hat{{u}}}\\cdot {\\hat{{u}}}_0 \\rangle _s = \\frac{1}{s +(d-1)D_r},$ which, after performing an inverse Laplace transform leads to $\\langle {\\hat{{u}}}\\cdot {\\hat{{u}}}_0 \\rangle (t) = e^{- (d-1)D_r\\, t}.$ As expected, it gives an exponential decay of the two-time orientational correlation in $d$ -dimensions with a correlation time $\\tau _c = [ (d-1) D_r ]^{-1}$ , independent of the confinement.", "We use $\\tau _r = 1/D_r$ to set the unit of time in the problem.", "Along with the translational diffusion constant $D$ , this gives the unit of length $\\bar{\\ell }= \\sqrt{D/D_r}$ .", "The dimensionless active velocity $\\lambda = v_0/\\sqrt{D D_r}$ and strength of the trap $\\beta = \\mu k/D_r$ control the various properties of the ABP in harmonic trap.", "Note that the dimensionless activity $\\lambda $ is equivalent to the Péclet number defined as $v_0/\\bar{\\ell }D_r$ .", "Figure: (color online)Dependence of displacement parallel to the initial orientation 〈r ∥ ˜〉\\langle \\tilde{r_\\parallel }\\rangle on time t ˜\\tilde{t}, in d=2d=2 (aa) and d=3d=3 (bb) dimensions.We use dimensionless activity λ=1\\lambda =1.The three curves in each of these figures correspond todimensionless strengths of harmonic potentials β=0,0.5,3\\beta = 0,\\, 0.5, \\, 3.", "The initial position 𝐫 ˜ 0 {\\tilde{ {\\bf r}}}_0 was considered at the origin, coinciding the center of the trap." ], [ "Displacement vector", "Using $\\psi = {\\bf r}$ in moment2, we get for the displacement vector $\\langle {\\bf r}\\rangle _s = \\frac{v_0 \\langle {\\hat{{u}}}\\rangle _s {\\color {blue} + {\\bf r}_0}}{s + \\mu k}$ .", "The same moment2, with $\\psi = {\\hat{{u}}}$ gives $\\langle {\\hat{{u}}}\\rangle _s = \\frac{ {\\hat{{u}}}_0}{s+(d-1)D_r}.$ Thus, together, we obtain $\\langle {\\bf r}\\rangle _s = \\frac{v_0 {\\hat{{u}}}_0}{ (s+\\mu k)(s + (d-1)D_r )} { + \\frac{ {\\bf r}_0}{(s+\\mu k)} },$ which under inverse Laplace transform leads to $\\langle {\\bf r}\\rangle \\, (t) = \\frac{v_0 \\hat{u}_0}{-\\mu k + (d-1)D_r }\\left( e^{-\\mu k t} - e^{-(d-1)D_r t} \\right)+ {\\bf r}_0 e^{-\\mu k t} .", "$ The displacement components parallel and perpendicular to the initial orientation $ {\\hat{{u}}}_0$ are defined as $ {\\bf r}_\\parallel = r_\\parallel {\\hat{{u}}}_0$ with $r_\\parallel = {\\bf r}\\cdot {\\hat{{u}}}_0$ , and $ {\\bf r}_\\perp = {\\bf r}- {\\bf r}_\\parallel $ , respectively.", "By symmetry, $\\langle {\\bf r}_\\perp \\rangle = (r_0)_\\perp \\exp (-\\mu k t)$ .", "In terms of dimensionless form $\\langle {\\tilde{r}}_\\parallel \\rangle = \\langle r_\\parallel \\rangle /\\bar{\\ell }$ we find $\\langle {\\tilde{r}}_\\parallel \\rangle = \\frac{\\lambda }{-\\beta + (d-1)} \\left(e^{-\\beta \\tilde{t}} - e^{-(d-1)\\tilde{t}}\\right)+ ({\\tilde{r}}_0)_\\parallel e^{-\\beta \\tilde{t}} ,$ where, $\\tilde{t} = t \\, D_r$ .", "At short time, $\\tilde{t} \\ll 1$ , the displacement grows linearly with time $\\langle {\\tilde{r}}_\\parallel \\rangle = \\lambda \\tilde{t}$ .", "However, in the limit $\\tilde{t} \\gg 1$ , the harmonic trap ensure that the steady state displacement $\\langle {\\tilde{r}}_\\parallel \\rangle = 0$ .", "The time-scale over which the parallel component of the displacement vector vanishes is given by $\\beta ^{-1}$ , i.e., stronger the trap is the faster is the vanishing.", "This dependence for both $d=2$ and 3 are shown in Fig.", "REF ." ], [ "Second moment of displacement", "Considering the displacement squared as the dynamical variable of interest, $\\psi = {\\bf r}^2$ , and using in moment2 the related facts that $\\langle \\psi \\rangle (0) = {\\bf r}_0^2$ , $\\langle \\nabla _u^2 \\psi \\rangle _s = 0$ , $\\langle \\nabla ^2 \\psi \\rangle _s = 2d/s$ , $\\langle {\\hat{{u}}}\\cdot \\nabla \\psi \\rangle _s = 2 \\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s$ , and $\\langle \\nabla \\cdot ( {\\bf r}\\psi )\\rangle _s = (d + 2) \\langle {\\bf r}^2 \\rangle _s$ , one finds $\\langle {\\bf r}^2 \\rangle _s = \\frac{1}{s +2 \\mu k} \\left[ \\frac{2 d D}{s} + 2 v_0 \\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s + {\\bf r}_0^2 \\right].$ Further, using $\\psi = {\\hat{{u}}}\\cdot {\\bf r}$ , and the results $\\nabla ^2 ( {\\hat{{u}}}\\cdot {\\bf r}) = 0$ , $\\nabla _u^2 ( {\\hat{{u}}}\\cdot {\\bf r}) = -(d-1) ( {\\hat{{u}}}\\cdot {\\bf r})$ , $\\langle {\\hat{{u}}}\\cdot \\nabla ( {\\hat{{u}}}\\cdot {\\bf r}) \\rangle _s = \\langle {\\hat{{u}}}^2 \\rangle _s = \\langle 1 \\rangle _s = 1/s $ , and $\\nabla \\cdot [ {\\bf r}( {\\hat{{u}}}\\cdot {\\bf r})] = (d+1) ( {\\hat{{u}}}\\cdot {\\bf r})$ we obtain, $\\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s = \\frac{1}{s + \\mu k +(d-1) D_r}~\\left( \\frac{v_0}{s} + {\\hat{{u}}}_0 \\cdot {\\bf r}_0 \\right).$ Using udotr2 in rv2s one obtains, $\\langle {\\bf r}^2 \\rangle _s = \\frac{1}{s +2 \\mu k} \\left[ \\frac{2dD}{s} +\\frac{2 v_0^2}{s \\lbrace s + \\mu k +(d-1) D_r\\rbrace }{ + \\frac{2 v_0 \\, \\, {\\hat{{u}}}_0 \\cdot {\\bf r}_0}{s + \\mu k +(d-1) D_r} + {\\bf r}_0^2} \\right] .$ The inverse Laplace transform of udotr2 provides the time-dependence of the equal time cross-correlation between the active orientation and the displacement vector of the ABP $\\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle (t) = \\frac{v_0}{\\mu k+ (d-1) D_r } \\left( 1 - e^{-[ \\mu k + (d-1) D_r ] t} \\right){ + ( {\\hat{{u}}}_0 \\cdot {\\bf r}_0) \\, e^{-[\\mu k + (d-1) D_r] t}},$ which depends on the strength of the trapping potential.", "This correlation in the steady state can be directly obtained from the above relation, or using the final value theorem, $\\lim _{t \\rightarrow \\infty } \\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle (t) = \\lim _{s \\rightarrow 0_+} s\\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s = \\frac{v_0}{\\mu k +(d-1) D_r},$ which shows that orientations and positions are correlated in the steady state.", "Performing the inverse Laplace transform of rvsq2 one gets, $\\langle {\\bf r}^2(t) \\rangle = \\frac{d\\, D}{\\mu k} \\left(1 - e^{-2\\mu k t} \\right) + \\frac{v_0^2}{\\mu k[ (d-1) D_r + \\mu k ] } - \\frac{2 v_0^2 e^{-\\mu k t}}{(d-1) D_r - \\mu k } \\left[ \\frac{e^{-\\mu k t} }{2 \\mu k} - \\frac{e^{- (d-1) D_r t}}{ (d-1) D_r + \\mu k }\\right]\\nonumber \\\\{ + \\frac{2 v_0 \\, {\\hat{{u}}}_0\\cdot {\\bf r}_0}{\\mu k- (d-1)D_r} \\left( e^{-[ \\mu k + (d-1) D_r]t} - e^{- 2 \\mu k t} \\right) + {\\bf r}_0^2 e^{- 2 \\mu k t} }$ The dimensionless form, $\\langle {\\tilde{ {\\bf r}}}^2 \\rangle = \\langle {\\bf r}^2(t) \\rangle / \\bar{\\ell }^2$ , can be expressed as $\\langle {\\tilde{ {\\bf r}}}^2(\\tilde{t}) \\rangle &=& \\frac{d}{\\beta } \\left(1 - e^{-2\\beta \\tilde{t}} \\right) + \\frac{\\lambda ^2}{\\beta [ d-1+ \\beta ] }- \\frac{2 \\lambda ^2 e^{-\\beta \\tilde{t}}}{d-1 - \\beta } \\left[ \\frac{e^{-\\beta \\tilde{t}} }{2 \\beta } - \\frac{e^{- (d-1) \\tilde{t}}}{ d-1+ \\beta }\\right] \\nonumber \\\\&+& { \\frac{2 \\lambda \\, {\\hat{{u}}}_0\\cdot {\\tilde{ {\\bf r}}_0}}{\\beta - (d-1)} \\left( e^{-[ \\beta + (d-1)]\\tilde{t}} - e^{- 2 \\beta \\tilde{t}} \\right) + {\\tilde{ {\\bf r}}_0}^2 e^{- 2 \\beta \\tilde{t}} }$ Figure: (color online)Evolution of the dimensionless form of the second moment of displacement 〈𝐫 ˜ 2 (t)〉\\langle {\\tilde{ {\\bf r}}}^2(t) \\rangle , in dimensions d=2d=2 (aa) and d=3d=3 (bb) at dimensionless strengths of harmonic potentials β=0,10 -3 ,10 -2 \\beta = 0,\\, 10^{-3}, \\, 10^{-2} indicated by the three lines in the figures.The strength of activity λ=1\\lambda =1 is used.The dash-dotted lines in (aa) and (bb) denote the asymptotic steady statelim t ˜→∞ 〈𝐫 ˜ 2 〉\\lim _{\\tilde{t} \\rightarrow \\infty } \\langle {\\tilde{ {\\bf r}}}^2 \\rangle at β=10 -2 \\beta =10^{-2}, as described by Eq. .", "The initial position 𝐫 ˜ 0 {\\tilde{ {\\bf r}}}_0 was considered at the center of the trap.The second moment of displacement evolves to the steady state value ${\\tilde{ {\\bf r}}_{st}}^2 = \\lim _{\\tilde{t} \\rightarrow \\infty } \\langle {\\tilde{ {\\bf r}}}^2 \\rangle = \\frac{1}{\\bar{\\ell }^2} \\lim _{s \\rightarrow 0_+} s\\langle { {\\bf r}}^2 \\rangle _s =\\frac{d}{\\beta } + \\frac{\\lambda ^2}{\\beta ( d-1 + \\beta )}.$ It is easy to check that in the limit of vanishing trap-stiffness eq:rv2 reduces to $\\langle {\\tilde{ {\\bf r}}}^2(\\tilde{t}) \\rangle = 2d \\left( 1 + \\frac{\\lambda ^2}{d(d-1)}\\right) \\tilde{t} - \\frac{2 \\lambda ^2}{(d-1)^2} \\left( 1 - e^{- (d-1) \\tilde{t}} \\right) { + \\frac{2\\lambda \\, {\\hat{{u}}}_0 \\cdot {\\tilde{ {\\bf r}}_0}}{d-1} \\left(1 - e^{-(d-1)\\tilde{t}} \\right) + {\\tilde{ {\\bf r}}_0}^2.}", "\\nonumber \\\\$ Setting $ {\\bf r}_0$ as the origin, Eq.", "(REF ) then reduces to the known result for free ABP in $d$ -dimensions [47], [29], $\\langle {\\bf r}^2 \\rangle = 2 d D_{\\rm eff} t - \\frac{2 v_0^2}{(d-1)^2 D_r^2} \\left( 1-e^{-(d-1)D_r t}\\right),$ with the effective diffusion coefficient $D_{\\rm eff} = D + \\frac{v_0^2}{d(d-1)D_r},$ describing the long time diffusive behavior.", "This is in agreement with the $d=2$ dimensional result in Ref. [48].", "In the limiting case of $D=0$ , the expression of the second moment of displacement reduces to the well-known result of the end-to-end separation in the worm-like-chain model [44], [39], $\\langle {\\bf r}^2 \\rangle = \\frac{4\\kappa l}{d-1} - \\frac{8 \\kappa ^2(1-e^{-\\frac{(d-1)l}{2\\kappa }})}{(d-1)^2},$ where we identify the bending rigidity $\\kappa =v_0/2D_r$ and chain length $l=v_0 t$ .", "The evolutions described by eq:rv2 and eq:rv2free are shown in Fig.", "REF .", "In the absence of the harmonic potential, $\\beta = 0$ , the moment $\\langle {\\bf r}^2 \\rangle $ show two crossovers, from diffusive $\\sim t$ behavior at shortest times to ballistic $\\sim t^2$ behavior that again crossover to long time diffusion $\\sim t$  [29].", "The trapping potential brings back the displacement moment towards the steady state value described by Eq.", "(REF ).", "In Fig.", "REF , we show evolution of the scaled mean squared position $\\langle {\\tilde{ {\\bf r}}}^2 \\rangle $ for different trapping potential in 2d and 3d starting from the initial position $ {\\bf r}_0$ at the center of the trap.", "Although the steady state in the presence of the harmonic trap is independent of initial conditions, the time-evolution of individual trajectories depend on the initial position.", "In Fig.", "REF we show the evolution starting from various initial positions.", "In Fig.", "(REF a), we plot the displacements parallel to the initial orientation of activity while in Fig.", "(REF b), the second moment of displacement is plotted after taking an average over all possible initial orientations of activity.", "It is interesting to note that even if the initial value of position $\\tilde{ {\\bf r}}_0^2$ is chosen to have the steady state value $\\tilde{ {\\bf r}}_{st}^2$ , the moment shows transient deviations before evolving back to the steady state value.", "This transient deviation can be understood from Eq.", "(REF ).", "The orientational averaging renders $\\langle {\\hat{{u}}}_0 \\cdot \\tilde{ {\\bf r}}_0 \\rangle =0$ .", "The contribution from initial position $\\tilde{r}_0^2$ decays in time scale $(2\\beta )^{-1}$ , and in that same time-scale the first term in the expression of $\\langle \\tilde{r}^2 (\\tilde{t}) \\rangle $ saturates.", "The third term contributes to the transient deviation, as well, vanishing at late times.", "Figure: (color online) (aa) Dependence of 〈r ∥ ˜〉(t)\\langle \\tilde{r_\\parallel }\\rangle (t) on initial positions (r ˜ 0 ) ∥ =u ^ 0 ·𝐫 0 /ℓ ¯ ({\\tilde{r}}_0)_\\parallel = {\\hat{{u}}}_0 \\cdot {\\bf r}_0/\\bar{\\ell }, the values of which are denoted in the figure legend.", "(bb) Dependence of 〈𝐫 ˜ 2 〉(t)\\langle {\\tilde{ {\\bf r}}}^2 \\rangle (t), averaged over all possible initial orientations u ^ 0 {\\hat{{u}}}_0, on separation r ˜ 0 \\tilde{r}_0 of the initial position from the center of the trap.", "The plots are shown at d=2d=2 dimensions, using parameter values λ=1,β=0.5\\lambda =1,\\,\\beta =0.5.", "Even with a choice of initial separation r ˜ 0 =r ˜ st {\\tilde{r}}_0 = \\tilde{r}_{st} having the steady state value r ˜ st =𝐫 ˜ st 2 =2.31 \\tilde{r}_{st}=\\sqrt{\\tilde{ {\\bf r}}_{st}^2} =2.31 (Eq.", "() ), the evolution of 〈𝐫 ˜ 2 〉(t)\\langle {\\tilde{ {\\bf r}}}^2 \\rangle (t) shows transient deviations before returning to the steady state value (green dotted line)." ], [ "Fourth moment of displacement", "Again to calculate $\\langle {\\bf r}^4 \\rangle (t)$ , we consider $\\psi = {\\bf r}^4$ and use moment2.", "Note $\\langle {\\bf r}^4 \\rangle (0) = {\\bf r}_0^4$ , $\\langle \\nabla _u^2 {\\bf r}^4 \\rangle _s = 0$ .", "It is straightforward to show that $\\langle \\nabla ^2 {\\bf r}^4 \\rangle _s = 4(d+2) \\langle {\\bf r}^2 \\rangle _s$ .", "Further, $\\langle {\\hat{{u}}}\\cdot \\nabla {\\bf r}^4 \\rangle _s = 4 \\langle {\\hat{{u}}}\\cdot {\\bf r}^2 {\\bf r}\\rangle _s$ , and $\\langle \\nabla \\cdot ( {\\bf r}\\psi ) \\rangle _s = (d+4) \\langle {\\bf r}^4 \\rangle _s$ .", "Thus one gets $\\langle {\\bf r}^4 \\rangle _s = \\frac{4 (d+2) D}{s + 4\\mu k} \\langle {\\bf r}^2 \\rangle _s + \\frac{4 v_0}{s + 4 \\mu k} \\langle ( {\\hat{{u}}}\\cdot {\\bf r}) {\\bf r}^2 \\rangle _s{+ \\frac{ {\\bf r}_0^4}{s + 4\\mu k}}.$ Now we proceed to calculate $\\langle \\psi \\rangle _s$ with $\\psi = ( {\\hat{{u}}}\\cdot {\\bf r}) {\\bf r}^2$ .", "We find, $\\langle \\psi \\rangle _0 = ( {\\hat{{u}}}_0 \\cdot {\\bf r}_0) {\\bf r}_0^2$ , $\\langle \\nabla ^2 \\psi \\rangle _s = 2(d+2) \\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s$ .", "Using $\\nabla _u^2 u_i = -(d-1) u_i$ one finds $\\langle \\nabla _u^2 \\psi \\rangle _s = -(d-1) \\langle ( {\\hat{{u}}}\\cdot {\\bf r}) {\\bf r}^2\\rangle _s$ .", "The self propulsion term $\\langle {\\hat{{u}}}\\cdot \\nabla \\psi \\rangle _s = \\langle {\\bf r}^2\\rangle _s + 2 \\langle ( {\\hat{{u}}}\\cdot {\\bf r})^2 \\rangle _s$ .", "The final term $\\langle \\nabla \\cdot ( {\\bf r}\\psi )\\rangle _s = (3+d ) \\langle ( {\\hat{{u}}}\\cdot {\\bf r}) {\\bf r}^2 \\rangle _s$ .", "This leads to the relation $\\langle ( {\\hat{{u}}}\\cdot {\\bf r}) {\\bf r}^2 \\rangle _s = \\frac{2(d+2)D\\, \\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s}{s+3 \\mu k + (d-1)D_r} + \\frac{v_0 \\langle {\\bf r}^2 \\rangle _s + 2 v_0 \\langle ( {\\hat{{u}}}\\cdot {\\bf r})^2 \\rangle _s}{s+3 \\mu k + (d-1)D_r}{ + \\frac{( {\\hat{{u}}}_0 \\cdot {\\bf r}_0) {\\bf r}_0^2}{s+3 \\mu k + (d-1)D_r}}$ Note that $\\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s$ , and $\\langle {\\bf r}^2 \\rangle _s$ have already been calculated in Eq.", "(REF ), and Eq.", "(REF ), respectively.", "We are left to calculate $\\langle \\psi \\rangle _s$ with $\\psi = ( {\\hat{{u}}}\\cdot {\\bf r})^2$ , which gives $\\langle \\psi \\rangle (0) = ( {\\hat{{u}}}_0 \\cdot {\\bf r}_0)^2$ , $\\nabla ^2 \\psi = 2 {\\hat{{u}}}^2 =2$ , and thus $\\langle \\nabla ^2 \\psi \\rangle _s = 2/s$ .", "Given $\\nabla _u^2 u_m u_p = -2 d u_m u_p +2\\delta _{mp}$ we find $\\langle \\nabla _u^2 \\psi \\rangle _s = 2 \\langle {\\bf r}^2 \\rangle _s - 2 d \\langle ( {\\hat{{u}}}\\cdot {\\bf r})^2 \\rangle _s$ .", "The self propulsion term $\\langle {\\hat{{u}}}\\cdot \\nabla \\psi \\rangle _s = 2 \\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s$ .", "The final term associated with the trapping potential $\\langle \\nabla \\cdot ( {\\bf r}\\psi ) \\rangle _s = (d+2) \\langle ( {\\hat{{u}}}\\cdot {\\bf r})^2 \\rangle _s$ .", "Thus we obtain the relationship, $(s + 2 \\mu k + 2 d D_r)\\langle ( {\\hat{{u}}}\\cdot {\\bf r})^2 \\rangle _s = \\frac{2D}{s} + 2 D_r \\langle {\\bf r}^2 \\rangle _s + 2 v_0 \\langle {\\hat{{u}}}\\cdot {\\bf r}\\rangle _s{ + ( {\\hat{{u}}}_0 \\cdot {\\bf r}_0)^2} .$ Using Eqs.", "(REF ), (REF ), and (REF ) one finally obtains the expression for $\\langle {\\bf r}^4 \\rangle _s = \\frac{8 d (d+2) D^2}{s (2 \\mu k+s) (4 \\mu k+s)}+\\frac{8 (d+2) D v_0^2}{s (2 \\mu k+s) (4 \\mu k+s) ((d-1) D_r+\\mu k+s)} \\nonumber \\\\+\\frac{8 (d+2) D v_0^2 }{s (4 \\mu k+s) ((d-1) D_r+\\mu k+s) ((d-1) D_r+3 \\mu k+s)}\\nonumber \\\\+\\frac{8 d D v_0^2}{s (2 \\mu k+s) (4 \\mu k+s) ((d-1) D_r+3 \\mu k+s)}\\nonumber \\\\+\\frac{16 D v_0^2}{s (4 \\mu k+s) (2 d D_r+2 \\mu k+s) ((d-1) D_r+3 \\mu k+s)}\\nonumber \\\\+\\frac{32 d D D_r v_0^2}{s (2 \\mu k+s) (4 \\mu k+s) (2 d D_r+2 \\mu k+s) ((d-1) D_r+3 \\mu k+s)}\\nonumber \\\\+\\frac{8 v_0^4}{s (2 \\mu k+s) (4 \\mu k+s) ((d-1) D_r+\\mu k+s) ((d-1) D_r+3 \\mu k+s)}\\nonumber \\\\+\\frac{16 v_0^4}{s (4 \\mu k+s) ((d-1) D_r+\\mu k+s) (2 d D_r+2 \\mu k+s) ((d-1) D_r+3 \\mu k+s)}\\nonumber \\\\+\\frac{32 D_r v_0^4}{s (2 \\mu k+s) (4 \\mu k+s) ((d-1) D_r+\\mu k+s) (2 d D_r+2 \\mu k+s) ((d-1) D_r+3 \\mu k+s)} \\nonumber \\\\{ + \\frac{ {\\bf r}_0^4}{s+4\\mu k} + \\frac{8(d+2)D\\,\\, v_0 {\\hat{{u}}}_0\\cdot {\\bf r}_0}{(s+4\\mu k)(s+\\mu k + (d-1)D_r)} \\left( \\frac{1}{s+2\\mu k}+ \\frac{1}{s+3\\mu k +(d-1)D_r}\\right) } \\nonumber \\\\{+ \\frac{ { {\\bf r}_0}^2}{s+4\\mu k} \\left( \\frac{4(d+2)D}{s+2\\mu k} + \\frac{4 v_0( {\\hat{{u}}}_0 \\cdot {\\bf r}_0)}{s+3\\mu k+(d-1)D_r}+\\frac{8 {v_0}^2}{(s+3\\mu k +(d-1)D_r)(s+2\\mu k + 2d D_r)} \\right)} \\nonumber \\\\$ Performing inverse Laplace transform, the dimensionless fourth moment $\\langle {\\tilde{ {\\bf r}}}^4 \\rangle = \\langle {\\bf r}^4 \\rangle (t) /\\bar{\\ell }^4$ can be expressed as $\\langle {\\tilde{ {\\bf r}}}^4(\\tilde{t}) \\rangle = \\frac{d (d+2)}{\\beta ^2} + \\frac{2(d+2) \\lambda ^2}{\\beta ^2 (d-1+\\beta )} +\\frac{(2+d+3 \\beta ) \\lambda ^4}{\\beta ^2 (d-1+\\beta ) (d+\\beta ) (d-1+3 \\beta )} \\nonumber \\\\+\\frac{2 (d+2) \\left(d^2 -d (1+\\beta )+\\lambda ^2\\right) \\left(d (d-1+\\beta )+\\lambda ^2\\right)}{\\beta ^2 d \\left( \\beta ^2 - (d-1)^2 \\right)} e^{-2 \\beta \\tilde{t}}\\nonumber \\\\+\\frac{e^{-4 \\beta \\tilde{t}} }{\\beta ^2 (\\beta -d ) ( 1 +\\beta -d ) (1+3 \\beta -d )}\\left[-d (d+2) (d-1 -3 \\beta ) (d-1-\\beta ) (d -\\beta ) \\right.\\nonumber \\\\\\left.", "-2 (d+2) ( d-1 -3 \\beta ) (d-\\beta ) \\lambda ^2 - (d+2 -3 \\beta ) \\lambda ^4 \\right] \\nonumber \\\\- \\frac{4 \\lambda ^2 \\left((d+2) (d +1-\\beta ) (d-1+3 \\beta )+\\lambda ^2 (d-7+3 \\beta )\\right)}{\\beta (d-1-\\beta ) (d +1-\\beta ) (d-1+\\beta ) (d-1+ 3 \\beta )} e^{-(d -1+3 \\beta ) \\tilde{t} } \\nonumber \\\\+\\frac{4 \\lambda ^2 \\left( (d+2) (d +1+\\beta ) (d-1 -3 \\beta ) +\\lambda ^2 (d-7 -3 \\beta )\\right)}{\\beta (d-1-3 \\beta ) (d-1-\\beta ) (d-1+\\beta ) (d +1+\\beta )} e^{-( d-1 +\\beta ) \\tilde{t} } \\nonumber \\\\+\\frac{4 (d-1) \\lambda ^4 }{d (d -\\beta ) (d +1-\\beta ) (d +\\beta ) (d + 1+\\beta )} e^{-2 (d +\\beta ) \\tilde{t}}{ + {\\tilde{ {\\bf r}}}_0^4 e^{-4 \\beta \\tilde{t}} }\\nonumber \\\\{ + \\frac{4 (d+2)}{2 \\beta }\\, {\\tilde{ {\\bf r}}_0}^2 \\left( e^{-2\\beta \\tilde{t}} - e^{-4 \\beta \\tilde{t}}\\right)+ \\frac{4 \\lambda ( {\\hat{{u}}}_0\\cdot {\\tilde{ {\\bf r}}}_0) {\\tilde{ {\\bf r}}}_0^2}{\\beta - d+1} \\left( e^{-(3\\beta +d-1)\\tilde{t}} - e^{-4 \\beta \\tilde{t}}\\right)} \\nonumber \\\\{ + 8 \\lambda ^2 {\\tilde{ {\\bf r}}}_0^2 \\left[ \\frac{e^{-4 \\beta \\tilde{t}}}{(-\\beta +d-1)(-2\\beta +2d)} + \\frac{e^{-(3\\beta + d -1)\\tilde{t}} }{(\\beta -d+1)(-\\beta +d+1)} + \\frac{e^{-(2\\beta +2d)\\tilde{t}}}{(2 \\beta -2 d)(\\beta -d-1)} \\right] } \\nonumber \\\\{ + 8(d+2) \\lambda {\\hat{{u}}}_0 \\cdot {\\tilde{ {\\bf r}}}_0 \\left[ \\frac{e^{-4\\beta \\tilde{t}}}{(-3\\beta + d -1)(-2\\beta )}+ \\frac{e^{-(\\beta + d -1)\\tilde{t}}}{(3 \\beta -d+1)(\\beta - d +1) } + \\frac{e^{-2 \\beta \\tilde{t}}}{(2\\beta )(-\\beta +d-1)} \\right.}", "\\nonumber \\\\{ \\left.", "+\\frac{e^{-4 \\beta \\tilde{t}}}{(-3\\beta +d-1)(-\\beta +d-1)} + \\frac{e^{-(\\beta +d-1)\\tilde{t}}}{(3\\beta -d+1)(2\\beta )} + \\frac{e^{-(3\\beta + d-1)\\tilde{t}}}{(\\beta -d+1)(-2\\beta )} \\right] }$ Figure: (color online)Evolution of the fourth moment of displacement 〈𝐫 ˜ 4 (t)〉\\langle {\\tilde{ {\\bf r}}}^4(t)\\rangle in dimensions d=2d=2 (aa) and d=3d=3 (bb) atharmonic potential strengths β=0,10 -3 ,0.1,10\\beta = 0,\\, 10^{-3}, \\, 0.1,\\, 10 indicated by the four lines in the figures.", "In this figure 𝐫 0 {\\bf r}_0 is assumed at the origin, and we used the strength of activity λ=10\\lambda =10.", "The initial position 𝐫 ˜ 0 {\\tilde{ {\\bf r}}}_0 was considered at the center of the trap.In the limit of vanishing trap stiffness, using the initial position $ {\\bf r}_0$ at the origin, the above relation leads to $\\langle {\\tilde{ {\\bf r}}}^4(\\tilde{t}) \\rangle = \\frac{4 (d-1) \\lambda ^4 e^{-2 d \\tilde{t}}}{d^3 (d+1)^2 }-\\frac{8 \\lambda ^4 \\left(d^2 +10 d +25 \\right) e^{-(d-1) \\tilde{t}}}{(d-1)^4 (d+1)^2}+\\frac{4 \\lambda ^4 \\left(d^3 +23 d^2 -7 d +1\\right)}{(d-1)^4 d^3}\\nonumber \\\\+\\frac{8 \\tilde{t} e^{- (d-1) \\tilde{t} } \\left(d^3 \\lambda ^2+2 d^2 \\lambda ^2-d \\lambda ^2+d \\lambda ^4-2 \\lambda ^2-7 \\lambda ^4\\right)}{(d-1)^3 (d+1)}\\nonumber \\\\+\\frac{4 {\\tilde{t}}^2 \\left(d^5 -3 d^3 +2 d^3 \\lambda ^2+2 d^2 +2 d^2 \\lambda ^2-4 d \\lambda ^2+d \\lambda ^4+2 \\lambda ^4\\right)}{(d-1)^2 d }\\nonumber \\\\-\\frac{8 \\tilde{t} \\left(d^4 \\lambda ^2+d^3 \\lambda ^2-2 d^2 \\lambda ^2+d^2 \\lambda ^4+6 d \\lambda ^4-\\lambda ^4\\right)}{(d-1)^3 d^2}$ a result obtained before in Ref. [29].", "In the presence of the external trapping potential the fourth moment of displacement reaches a steady state value.", "It is relatively simple to obtain this expression using the final value theorem.", "The expression is given by, $\\lim _{t \\rightarrow \\infty } \\langle {\\tilde{ {\\bf r}}}^4 \\rangle = \\frac{1}{\\bar{\\ell }^4}\\lim _{s \\rightarrow 0_+} s \\langle {\\bf r}^4 \\rangle _s =\\frac{d (d+2)}{\\beta ^2} + \\frac{2(d+2) \\lambda ^2}{\\beta ^2 (d-1+\\beta )} +\\frac{(2+d+3 \\beta ) \\lambda ^4}{\\beta ^2 (d-1+\\beta ) (d+\\beta ) (d-1+3 \\beta )}.", "\\nonumber \\\\$ The evolution of the fourth moment with time is shown in Fig.", "REF at different strengths of the trapping potential $\\beta $ , using eq:rv4 and eq:rv4free.", "The displacement vector of a passive Brownian particle reaches an equilibrium Boltzmann distribution $P( {\\bf r}) \\sim \\exp (-U/k_B T)$ in the presence of a trapping potential $U( {\\bf r})$ and ambient temperature $T$ with $k_B$ denoting the Boltzmann constant.", "Thus in a Harmonic trap $U( {\\bf r}) = \\frac{1}{2}k {\\bf r}^2$ the displacement vector would obey the Gaussian distribution $P( {\\bf r}) = \\left(\\frac{k}{2 \\pi k_BT}\\right)^{d/2}\\exp (-k {\\bf r}^2/2 k_B T)$ .", "Such a Gaussian process with $\\langle {\\bf r}\\rangle = 0$ obeys the relation $\\langle {\\bf r}^4 \\rangle = (1+2/d) \\langle {\\bf r}^2 \\rangle ^2 $ .", "Using the expression for $\\langle {\\bf r}^2 \\rangle $ obtained for ABP in the harmonic trap, in the definition $\\mu _4 := \\left(1+ \\frac{2}{d} \\right) \\langle {\\bf r}^2 \\rangle ^2$ one can define the kurtosis ${\\cal K} = \\frac{\\langle {\\bf r}^4 \\rangle }{\\mu _4} - 1,$ which measures the deviation from the Gaussian process.", "Clearly, for a Gaussian process ${\\cal K}=0$ .", "In the steady state, the kurtosis of the ABP is given by ${\\cal K} = \\frac{2 \\mu k v_0^4 \\left[\\,(1-4 d) D_r -3 \\mu k \\right]}{(d+2) (d D_r+\\mu k) \\left[ (d-1) D_r+3 \\mu k \\right]} \\times \\frac{1}{ \\left[d^2 D D_r+d D (\\mu k-D_r)+v_0^2\\right]^2 } .$ Using the dimensionless activity $\\lambda $ and trap- stiffness $\\beta $ , the expression can be written as ${\\cal K} = \\frac{ - 2 \\beta \\lambda ^4 ( 4 d +3 \\beta -1)}{ (d+2) (d +\\beta ) (d-1+3 \\beta ) \\left(d^2 +d (\\beta -1)+\\lambda ^2\\right)^2}.$ In the limit of vanishing activity $\\lambda \\rightarrow 0$ , the ABP behaves as a particle diffusing in the harmonic trap following the Gaussian distribution of displacement.", "In this limit, the kurtosis vanishes as ${\\cal K} \\sim \\lambda ^4$ .", "As $\\lambda \\rightarrow \\infty $ the kurtosis saturates to ${\\cal K} = -\\frac{2 \\beta (4d+3\\beta -1)}{(d+2)(d+\\beta )(d-1+3\\beta )}.$ Thus with increasing $\\lambda $ , the kurtosis decreases to saturate at large $\\lambda $  (Fig.", "REF ($a$ )).", "This shows a passive to active crossover as a function of activity.", "On the other hand, with the change in trap- stiffness $\\beta $ , the kurtosis vanishes in both the limits of $\\beta \\rightarrow 0$ and $\\beta \\rightarrow \\infty $ , the two passive limits, and reaches a negative minimum for intermediate $\\beta $ values (Fig.", "REF ($b$ ) ).", "This shows the re-entrant crossover from passive to active to passive behavior with increasing trap stiffness $\\beta $ .", "In the limit of $\\beta \\rightarrow 0$ the kurtosis vanishes as ${\\cal K} \\sim -\\beta $ .", "In the other limit of $\\beta \\rightarrow \\infty $ , it vanishes as ${\\cal K} \\approx \\frac{-2 \\lambda ^4}{d^2 (d+2) \\beta ^2}.$ Figure: (color online) The variation of kurtosis 𝒦{\\cal K} with trap stiffness β\\beta , in the absence of translational diffusion (D=0D=0).", "The two graphs are for d=2,3d=2,\\,3, and the dash-dotted lines denote the asymptotic values 𝒦=-2/(d+2){\\cal K} = -2/(d+2).", "We see the monotonic dependence on β\\beta , in contrast to the non-monotonic form seen for D≠0D \\ne 0 in Fig.", "(b).Kurtosis in the absence of translational diffusion: The relation Eq.", "(REF ) simplifies to the following $\\lambda $ -independent form in the limit of $D=0$ , ${\\cal K} = \\frac{ - 2 \\beta ( 4 d +3 \\beta -1)}{ (d+2) (d +\\beta ) (d-1+3 \\beta )}.$ which is the same as Eq.", "(REF ), obtained in the limit of $\\lambda \\rightarrow \\infty $ .", "This relation describes the kurtosis in the system without translational diffusion.", "The above expression vanishes linearly ${\\cal K} \\sim - \\beta $ as $\\beta \\rightarrow 0$ .", "However, as $\\beta \\rightarrow \\infty $ , it saturates to ${\\cal K} = - 2/(d+2)$ , in contrast to the vanishing of ${\\cal K}$ at large $\\beta $ displayed by Eq.", "REF (compare Fig.REF with Fig.REF ($b$ ) ).", "Thus, in the absence of translational diffusion $D$ , we do not get the Gaussian behavior of the displacement distribution at large $\\beta $ .", "This clearly shows that the re-entrance displayed by ${\\cal K}$ as a function of $\\beta $ is possible only in the presence of translational diffusion $D \\ne 0$ .", "However, the first passive to active crossover is observed even in the absence of translational noise.", "The maximum radial extent of the trapped particle is given by $r_{ac} = v_0/\\mu k$ .", "This relation is obtained by balancing the active force with the trap force.", "A comparison of $r_{ac}$ with the persistence length of activity $\\ell _p = v_0 \\tau _c = v_0/(d-1)D_r$ can be used to understand this crossover.", "For a shallow trapping potential, $\\ell _p \\ll r_{ac}$ , the particle can undergo a large number of reorientations in the trap region, leading to effectively a simple diffusion in the trap and the Gaussian distribution of particle position as in equilibrium.", "In the other limit of stiff trapping potential, $r_{ac} \\ll \\ell _p$ , the active particle gets localized to a separation $r_{ac}$ away from the center of the trap, leading to a strongly non-Gaussian position distribution.", "Figure: (color online)Heat maps of the kurtosis at steady state 𝒦{\\cal K} as a function of the dimensionless activity λ\\lambda and potential strength β\\beta in d=2d=2 (aa) and d=3d=3 (bb) dimensions.", "The color-box shows colors corresponding to 𝒦{\\cal K} values shown.The light (yellow) and dark (black) regions denote the passive and active state, respectively.Phase diagram: We show the heat-map of the kurtosis ${\\cal K}$ as a function of trapping strength $\\beta $ and activity $\\lambda $ in Fig.", "REF , corresponding to both $d=2,\\,3$ dimensions.", "These phase diagrams show the passive to active crossovers in the presence of finite translational diffusion.", "The yellow portions of the phase space (${\\cal K} \\approx 0$ ) correspond to the passive equilibrium-like Gaussian distribution of the particle position.", "Whereas, the dark regions denote the largest amplitude of ${\\cal K}$ , capturing strong deviation from the Gaussian nature, identifying the active phase.", "The plot clearly shows a re-entrant transition from passive (Gaussian) to active (non-Gaussian) to passive (Gaussian) behavior with increasing trap stiffness $\\beta $ , e.g., in the region $10 \\lesssim \\lambda \\lesssim 100$ .", "A similar phase diagram in $d=2$ was obtained earlier by directly following the nature of the probability distribution of ABP-position in a harmonic trap [35].", "Our analytic calculation of the kurtosis admits a mathematically unique description of such phase diagrams over any range of $\\lambda $ and $\\beta $ values.", "Further, our method permits calculation of this phase diagram in arbitrary dimensions, e.g., the active-passive transition in $d=3$ is shown in Fig.", "REF ($b$ ).", "The re-entrant transition in Fig.", "REF can be understood qualitatively using the following heuristic argument [35].", "Ignoring the translational diffusion, the maximum radial extent of the trapped particle is given by $r_{ac} = v_0/\\mu k$ .", "On the other hand, the translational noise gives a broadened distribution around the origin, with an associated equilibrium length scale for the spread, $\\ell _{eq} = \\sqrt{D/\\mu k}$ .", "The off-centered peak of the ABP thus becomes insignificant for $r_{ac} \\lesssim \\ell _{eq}$ , thus giving the condition $\\beta \\gtrsim \\lambda ^2$ for a re-entrance to the passive regime.", "We note that this is consistent with Eq.", "(REF )." ], [ "Discussion", "In conclusion, we have demonstrated a method of performing exact analytical calculation of all time-dependent moments of dynamical variables describing ABPs in a harmonic trap, in the presence and absence of translational diffusion.", "The non-equilibrium activity in the trapping potential leads to a position distribution away from the equilibrium Gaussian profile [18], [35].", "In this paper, we identified such a deviation from the Gaussian distribution in terms of the kurtosis of the displacement vector.", "Our exact analytical calculation shows a re-entrant crossover from a passive Gaussian to an active non-Gaussian behavior in the activity-trapping potential plane.", "The amplitude of kurtosis grows monotonically with increasing activity to reach a saturation value.", "In contrast, with increasing trap stiffness, this measure shows non-monotonic variation, reaching a maximum for intermediate trapping strengths.", "In both the limits of vanishing and extremely stiff limits of the harmonic potential the kurtosis vanishes.", "Thereby, it identifies an passive to active to passive re-entrant crossover.", "Our analytic prediction of the phase diagram, and variation of kurtosis is amenable to direct experimental verification, e.g., using a setup similar to Ref. [18].", "The presence of translational diffusion is crucial for the observation of the re-entrant transition.", "In its absence, the amplitude of kurtosis increases monotonically to finally saturate with the increase of trap stiffness, predicting a single passive to active transition." ], [ "Acknowledgments", "D.C. thanks SERB, India for financial support through grant number MTR/2019/000750.", "A.D. acknowledges support of the Department of Atomic Energy, Government of India, under project no.12-R&D-TFR-5.10-1100.", "This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program - 7th Indian Statistical Physics Community Meeting (Code: ICTS/ispcm2020/02)." ] ]

[ [ "Minkowski decompositions for generalized associahedra of acyclic type" ], [ "Abstract We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed.", "This yields in particular a description of the Newton polytopes of the F-polynomials in terms of subword complexes as conjectured by S. Brodsky and the third author.", "We then show that the cluster complex is combinatorially isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by D. Speyer and L. Williams." ], [ "Introduction and main results", "A generalized associahedron for a cluster algebra of finite type is a simple polytope whose face lattice is dual to the cluster complex.", "Constructing such generalized associahedra has been a fruitful area of mathematical research since the introduction of cluster algebras by S. Fomin and A. Zelevinsky in the early 2000s.", "We refer to [8], [4], [11], [17], [12] in this chronological order for some of the milestones and history.", "This paper is a continuation of [3] and builds on recent results from [2], [1] and from [16].", "The paper has three major results, two of which resolve conjectures by S. Brodsky and the third author and, respectively, by D. Speyer and L. Williams.", "thm:indecompcolumns gives a self-contained combinatorial construction of the rays of the type cone of the $g$ -vector fan of a finite type cluster algebra with acyclic initial seed via subword complexes and brick polytopes.", "Using this construction together with recent results from [2], [1] and [16], cor:NewtonFpoly yields that this construction also describes the Newton polytopes of the $F$ -polynomials of the cluster algebra.", "This description was conjectured in [3].", "The appearance of the $F$ -polynomials is then as well used to derive thm:tropplusvariety showing that the totally positive part of the tropical cluster variety is, modulo its lineality space, linearly isomorphic to the $g$ -vector fan.", "As the $g$ -vector fan is combinatorially isomorphic to the cluster complex, this affirmatively answers [19] for finite type cluster algebras with principal coefficients and acyclic initial seed.", "In order to precisely state the results, let $\\Delta \\subseteq \\Phi ^+\\subseteq \\Phi _{\\ge -1}\\subseteq \\Phi $ denote a finite crystallographic root system with fundamental weights $\\nabla $ and let ${\\sf M}$ denote an initial mutation matrix with principal coefficients for a cluster algebra $\\mathcal {A}({\\sf M})$ of type $\\Phi $ with cluster variables $\\left\\lbrace u_\\beta (\\mathbf {x},\\mathbf {y}) \\;|\\; \\beta \\in \\Phi _{\\ge -1} \\right\\rbrace $ and cluster complex $S({\\sf M})$ given by the set of compatible cluster variables.", "The cluster variables have the form $u_\\beta (\\mathbf {x},\\mathbf {y}) = p(\\mathbf {x},\\mathbf {y}) / \\mathbf {x}^\\beta $ with $\\mathbf {x}= (x_1,\\dots ,x_n)$ and $\\mathbf {y}= (y_1,\\dots ,y_n)$ for $p(\\mathbf {x},\\mathbf {y}) \\in \\mathbb {N}[\\mathbf {x},\\mathbf {y}]$ and $\\mathbf {x}^\\beta = x_1^{\\beta _1} \\cdots x_n^{\\beta _n}$ with $\\beta = \\beta _1\\alpha _1+\\dots +\\beta _n\\alpha _n$ expanded in the root basis $\\Delta = \\lbrace \\alpha _1,\\dots ,\\alpha _n\\rbrace $ .", "Its $F$ -polynomials are denoted by $\\left\\lbrace F_\\beta = u_\\beta (\\mathbf {1},\\mathbf {y}) \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace $ and its $g$ -vector fan ${\\mathcal {F}_g}({\\sf M})$ is given by the cones over compatible sets of $g$ -vectors $g_\\beta = g_1\\omega _1+\\dots +g_n\\omega _n$ such that $u_\\beta (\\mathbf {x},\\mathbf {0}) = x_1^{g_1}\\cdots x_n^{g_n}$ expanded in the weight basis $\\nabla = \\lbrace \\omega _1,\\dots ,\\omega _n\\rbrace $ .", "It is well-known that the $g$ -vector fan is combinatorially isomorphic to the cluster complex $S({\\sf M})$ .", "Let $(W,\\mathcal {S})$ denote the Coxeter system generated by $\\mathcal {S}=\\left\\lbrace s_\\alpha \\;|\\; \\alpha \\in \\Delta \\right\\rbrace $ and let $c \\in W$ be a standard Coxeter element given by the product of the reflections in $\\mathcal {S}$ in some order.", "One may associate to this data an acyclic initial mutation matrix ${\\sf M}_{\\rm c}$ with principal coefficients, and as well a brick polytope $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ with normal fan given by the $g$ -vector fan ${\\mathcal {F}_g}({\\sf M}_{\\rm c})$ of $\\mathcal {A}({\\sf M}_{\\rm c})$ .", "In particular, $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ is a generalized associahedron for $\\mathcal {A}({\\sf M}_{\\rm c})$ .", "Brick polytopes for subword complexes come with natural Minkowski decompositions which in the present context may be written in the form $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}= \\sum _{\\beta \\in \\Phi ^+} \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}.$ The type cone $\\mathbb {TC}({\\mathcal {F}_g}({\\sf M}_{\\rm c}))$ of the $g$ -vector fan is the space of all its polytopal realizations.", "We thus have $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}\\in \\mathbb {TC}\\big ({\\mathcal {F}_g}({\\sf M}_{\\rm c})\\big ).", "$ While motivated by beautiful constructions in [2] and [16], the following result is entirely self-contained and only uses properties of brick polytopes developed in [17] and [3].", "Theorem 1.1 For an acyclic initial mutation matrix ${\\sf M}_{\\rm c}$ with principal coefficients, the type cone of the $g$ -vector fan ${\\mathcal {F}_g}({\\sf M}_{\\rm c})$ is the open simplicial cone generated by the natural Minkowski summands of the brick polytope $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ , $\\mathbb {TC}\\big ({\\mathcal {F}_g}({\\sf M}_{\\rm c})\\big ) \\ = \\ \\operatorname{cone}\\left\\lbrace \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace .$ Remark 1.2 This theorem and its proof are combinatorial and do not use any representation theory.", "The definition of a generalized associahedron $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ in [11], [17] extends verbatim to the noncrystallographic finite types $I_2(m)$ for $m \\notin \\lbrace 3,4,6\\rbrace $ and $H_3,H_4$ .", "The theorem also holds for noncrystallographic types when replacing the left-hand side by the type cone of weak Minkowski summands of $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ even though mutations of cluster variables, $g$ -vectors and $F$ -polynomials in these types do not behave combinatorially nicely [14].", "Combining [2] (simply-laced types) and [1] (multiply-laced types) with [16], one obtains that the rays of the type cone of the $g$ -vector fan are also equal to the Newton polytopes of the $F$ -polynomials, $\\mathbb {TC}\\big ({\\mathcal {F}_g}({\\sf M}_{\\rm c})\\big ) \\ = \\ \\operatorname{cone}\\left\\lbrace {\\operatorname{Newton}\\left(F_\\beta \\right)} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace , \\qquad \\mathrm {(\\star )}$ where the exponent vectors are written in the root basis $\\Delta $ , and in particular that $\\sum _{\\beta \\in \\Phi ^+} {\\operatorname{Newton}\\left(F_\\beta \\right)} \\ \\in \\ \\mathbb {TC}\\big ({\\mathcal {F}_g}({\\sf M}_{\\rm c})\\big ) \\qquad \\mathrm {(\\star \\star )}$ is a generalized associahedron for $\\mathcal {A}({\\sf M}_{\\rm c})$ .", "According to [16], H. Thomas announced that a future version of [2] will generalize (REF ) also to cyclic finite types.", "In this case, (REF ) was conjectured by S. Brodsky and the third author in [3].", "Combining this with thm:indecompcolumns and known properties of $F$ -polynomials, we obtain the second main result describing Newton polytopes of $F$ -polynomials for acyclic initial seeds in terms of subword complexes.", "Theorem 1.3 ([3]) Let ${\\sf M}_{\\rm c}$ be an acyclic initial mutation matrix with principal coefficients.", "For any positive root $\\beta \\in \\Phi ^+$ , we have ${\\operatorname{Newton}\\left(F_\\beta \\right)} \\ = \\ \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}.$ In [19] the authors associate to the cluster algebra $\\mathcal {A}({\\sf M})$ a polyhedral fan $\\operatorname{Trop^+}\\operatorname{Spec}\\mathcal {A}({\\sf M})$ by tropicalizing the positive part of the affine variety $\\operatorname{Spec}\\mathcal {A}({\\sf M})$ .", "Using (REF ), we finally derive the following theoremIn response to a first preprint, Thomas Lam informed us that a more general version of this theorem also follows from [1] which implies the first part of [19].. Theorem 1.4 For acyclic initial mutation matrix ${\\sf M}_{\\rm c}$ with principal coefficients, the totally positive part of the tropical variety associated to the cluster algebra $\\mathcal {A}({\\sf M}_{\\rm c})$ is, modulo its lineality space $\\mathcal {L}$ , linearly isomorphic to the $g$ -vector fan, $\\operatorname{Trop^+}\\operatorname{Spec}{\\mathcal {A}({\\sf M}_{\\rm c})}\\big / \\mathcal {L} \\ \\cong \\ {\\mathcal {F}_g}({\\sf M}_{\\rm c}).$ As the $g$ -vector fan is combinatorially isomorphic to the cluster complex, this affirmatively answers a conjecture by D. Speyer and L. Williams in this situation.", "Corollary 1.5 ([19]) In the situation of thm:tropplusvariety, the cluster complex $S({\\sf M})$ is combinatorially isomorphic to the polyhedral fan $\\operatorname{Trop^+}\\operatorname{Spec}{\\mathcal {A}({\\sf M}_{\\rm c})}$ ." ], [ "Acknowledgements", "The third author would like to thank Thomas Lam, Arnau Padrol, Markus Reineke, Raman Sanyal and Hugh Thomas for valuable discussions concerning various parts of this paper." ], [ "A natural Minkowski decomposition of generalized associahedra", "We follow the notions from [3] and refer to Section 2 therein for details." ], [ "Generalized associahedra for acyclic type", "Let $(W,\\mathcal {S})$ be a finite type Coxeter system of rank $n$ and let $\\Delta \\subseteq \\Phi ^+\\subseteq \\Phi _{\\ge -1}\\subseteq \\Phi \\subseteq V$ be a finite root system for $(W,\\mathcal {S})$ inside an Euclidean vector space $V$ , with simple roots $\\Delta = \\left\\lbrace \\alpha _s \\;|\\; s \\in \\mathcal {S} \\right\\rbrace $ , positive roots $\\Phi ^+$ and almost positive roots $\\Phi _{\\ge -1}= \\Phi ^+\\sqcup -\\Delta $ .", "Denote by $N = |\\Phi ^+|$ the number of positive roots and $n+N = |\\Phi _{\\ge -1}|$ .", "Let $C = (a_{st})_{s,t \\in \\mathcal {S}}$ denote the corresponding Cartan matrix given by $s(\\alpha _t) = \\alpha _t - a_{st} \\alpha _s$ and set $\\nabla = \\left\\lbrace \\omega _s \\;|\\; s \\in \\mathcal {S} \\right\\rbrace \\subseteq V$ to be the fundamental weights given by $\\alpha _s = \\sum _{t \\in \\mathcal {S}}a_{ts}\\omega _t.$ One then has $s(\\omega _t) = \\omega _t - \\delta _{s=t}\\alpha _s$ for $s,t \\in \\mathcal {S}$ .", "In all below considerations we consider $V \\cong \\mathbb {R}^\\Delta $ to have fixed basis $\\Delta $ , though in the examples we simultaneously consider the vector space with standard basis and standard inner product.", "We consider a fixed Coxeter element $c \\in W$ and a reduced word ${\\rm c}= {\\rm s}_1\\cdots {\\rm s}_n$ for $c$ .", "To avoid double indices we write $\\alpha _i$ for $\\alpha _{s_i}$ and $\\omega _i = \\omega _{s_i}$ .", "The initial mutation matrix ${\\sf M}_{\\rm c}= (m_{ij})$ is then obtained from the Cartan matrix by $m_{ij} = {\\left\\lbrace \\begin{array}{ll}0 &\\text{ if } i=j \\\\-a_{st} &\\text{ if } s=s_i \\text{ appears before } t=s_j \\text{ in the reduced word } {\\rm c}\\\\a_{st} &\\text{ if } s=s_i \\text{ appears after } t=s_j \\text{ in the reduced word } {\\rm c}\\end{array}\\right.", "},$ for $1 \\le i,j \\le n$ , together with an identity matrix below.", "Figure: The cluster variables with its dd-vectors, gg-vectors and FF-polynomials for the initial mutation matrices in ex:A3mutmatrix2,,ex:B2mutmatrix2.Example 2.1 ($A_3$ -example) Take $W = {\\mathfrak {S}}_4$ the symmetric group with adjacent transpositions as simple generators $\\mathcal {S}= \\big \\lbrace s_1 = (1,2),\\ s_2 = (2,3),\\ s_3 = (3,4) \\big \\rbrace $ acting on $V = \\left\\lbrace (\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4) \\in \\mathbb {R}^4 \\;|\\; \\lambda _1+\\lambda _2+\\lambda _3+\\lambda _4 = 0 \\right\\rbrace \\cong \\mathbb {R}^4 / \\mathbb {R}(1111)$ , equipped with the standard inner product, by permuting the standard basis.", "We choose $\\Delta = \\big \\lbrace \\alpha _1 = 1\\overline{1}00,~ \\alpha _2 = 01\\overline{1}0,~ \\alpha _3 = 001\\overline{1} \\big \\rbrace $ as a basis of $V$ .", "Here and below we write shorthand $\\overline{\\lambda }:= -\\lambda $ for scalars $\\lambda $ .", "We may express an element in $V$ in round brackets such as $(1,0,\\overline{1},0) = (1,1,0)_\\Delta = \\alpha _1+\\alpha _2$ where the first expression is the genuine element in $V \\subset \\mathbb {R}^4$ and the second expression is in the chosen basis $\\Delta $ in the given order.", "We obtain $\\Phi ^+&= \\big \\lbrace 1\\overline{1}00\\hspace{2.0pt}px,\\ 10\\overline{1}0\\hspace{2.0pt}px,\\ 100\\overline{1}\\hspace{2.0pt}px,\\ 01\\overline{1}0\\hspace{2.0pt}px,\\ 010\\overline{1}\\hspace{2.0pt}px,\\ 001\\overline{1}\\hspace{2.0pt}px \\big \\rbrace \\\\&= \\big \\lbrace 100_\\Delta ,\\ 110_\\Delta ,\\ 111_\\Delta ,\\ 010_\\Delta ,\\ 011_\\Delta ,\\ 001_\\Delta \\big \\rbrace ,$ and finally $\\Phi _{\\ge -1}= \\Phi ^+\\sqcup -\\Delta $ and $\\Phi = \\Phi ^+\\sqcup -\\Phi ^+$ .", "In this case, $n = |\\mathcal {S}| = 3$ and $N=|\\Phi ^+| = 6$ .", "The corresponding Cartan matrix is $C =\\begin{pmatrix}2 & \\overline{1} & 0 \\\\\\overline{1} & 2 & \\overline{1} \\\\0 & \\overline{1} & 2\\end{pmatrix}$ and the fundamental weights are $\\nabla = \\big \\lbrace \\omega _1 = 1000 = \\tfrac{1}{4}(321_\\Delta ),\\quad \\omega _2 = 1100 = \\tfrac{1}{4}(121_\\Delta ),\\quad \\omega _3 = 1110 = \\tfrac{1}{4}(123_\\Delta ) \\big \\rbrace .$ Fix the Coxeter element $c = (1,2,3,4) \\in {\\mathfrak {S}}_4$ to be the long cycle with reduced word $c = {\\rm s}_1{\\rm s}_2{\\rm s}_3$ .", "fig:examples shows cluster variables, $d$ - and $g$ -vectors and $F$ -polynomials for the initial mutation matrix ${\\sf M}_{\\rm c}=\\begin{pmatrix}0 & 1 & 0 \\\\\\overline{1} & 0 & 1 \\\\0 & \\overline{1} & 0 \\\\1 & 0 & 0 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{pmatrix}\\ .$ Example 2.2 ($B_2$ -example) Take $W = {\\mathfrak {S}}_2^B$ the group of signed permutations with simple generators $\\mathcal {S}= \\big \\lbrace s_1 = (1,2),~ s_2 = (2,\\overline{2}) \\big \\rbrace $ where $s_1$ is the usual adjacent transposition interchanging the standard basis elements $e_1$ and $e_2$ , and where $s_2$ interchanges $e_2$ and $-e_2$ .", "$W$ acts on $V = \\mathbb {R}^2$ , equipped with the standard inner product.", "We choose $\\Delta = \\big \\lbrace \\alpha _1 = 2\\overline{2},~ \\alpha _2 = 02 \\big \\rbrace $ as a basis of $V$ .", "With notation as above, we obtain $\\Phi ^+&= \\big \\lbrace 2\\overline{2}\\hspace{7.0pt}px,\\ 20\\hspace{7.0pt}px,\\ 22\\hspace{7.0pt}px,\\ 02\\hspace{7.0pt}px \\big \\rbrace \\\\&= \\big \\lbrace 10_\\Delta ,\\ 11_\\Delta ,\\ 12_\\Delta ,\\ 01_\\Delta \\big \\rbrace ,$ and finally $\\Phi _{\\ge -1}= \\Phi ^+\\sqcup -\\Delta $ and $\\Phi = \\Phi ^+\\sqcup -\\Phi ^+$ .", "In this case, $n = |\\mathcal {S}| = 2$ and $N=|\\Phi ^+|=4$ .", "The corresponding Cartan matrix is $C =\\begin{pmatrix}2 & \\overline{1} \\\\\\overline{2} & 2\\end{pmatrix}$ and the fundamental weights are $\\nabla = \\big \\lbrace \\omega _1 = 20 = 11_\\Delta ,\\quad \\omega _2 = 11 = \\tfrac{1}{2}(12_\\Delta ) \\big \\rbrace .$ Fix the Coxeter element $c = (1,2,\\overline{1}, \\overline{2}) \\in {\\mathfrak {S}}_2^B$ to be the long cycle with reduced word ${\\rm c}= {\\rm s}_1{\\rm s}_2$ .", "fig:examples shows cluster variables, $d$ - and $g$ -vectors and $F$ -polynomials for the initial mutation matrix ${\\sf M}_{\\rm c}\\ = \\ \\begin{pmatrix}0 & 1 \\\\\\overline{2} & 0 \\\\1 & 0 \\\\0 & 1\\end{pmatrix}.$ For later reference, fig:B2gfan shows the $g$ -vector fan in the weight basis and the Newton polytopes of the $F$ -polynomials in the root basis in this case.", "poly = [fill=blue!25, line width=1.2pt, opacity=0.75, draw=blue!40] segment = [draw=blue!40, line width=1.2pt, line cap=round] vertex = [fill=blue!50, draw=blue!50, line width=1.2pt] fan = [fill=red!20, opacity=0.5] ray = [draw=red!50, line width=1.4pt, line cap=round] coords = [draw=black!60,->,>=latex,line width=0.7pt, line cap=round] Figure: The gg-vector fan of type B 2 B_2 from ex:B2mutmatrix2 (left) is the common refinement of the normal fans of the Newton polytopes (blue) of the FF-polynomials.Let ${w_\\circ }\\in W$ be the unique longest element in weak order.", "For a given word ${\\rm Q}= {\\rm q}_1\\cdots {\\rm q}_m$ in the simple system $\\mathcal {S}$ define the (spherical) subword complex $\\mathcal {SC}({\\rm Q})$ as the simplicial complex of (positions of) letters in ${\\rm Q}$ whose complement contains a reduced word of ${w_\\circ }$ .", "A more general version of these complexes were introduced by A. Knutson and E. Miller in [13].", "By definition, the facets of $\\mathcal {SC}({\\rm Q})$ are subwords of ${\\rm Q}$ whose complements are reduced words for ${w_\\circ }$ .", "We consider facets as sorted lists of indices, written in set notation.", "Moreover define $I_{\\operatorname{g}}$ and $I_{\\operatorname{ag}}$ to be the lexicographically first and last facets, respectively, and call them greedy facet and antigreedy facet.", "The following notions were introduced and studied for general subword complexes in [5], [17].", "For ${\\rm Q}= {\\rm q}_1 \\cdots {\\rm q}_m$ and any facet $I \\in \\mathcal {SC}({\\rm Q})$ associate a root function ${{\\sf r}}(I,\\cdot ) : [m] \\rightarrow \\Phi = W(\\Delta ) \\subseteq V$ and a weight function ${{\\sf w}}(I,\\cdot ) : [m] \\rightarrow W(\\nabla ) \\subseteq V$ defined by ${{\\sf r}}(I,k) \\ = \\ \\Pi {{\\rm Q}}_{[k-1] \\setminus I}(\\alpha _{q_k}) \\quad \\text{and} \\quad {{\\sf w}}(I,k) \\ = \\ \\Pi {{\\rm Q}}_{[k-1] \\setminus I}(\\omega _{q_k}),$ where $\\Pi {{\\rm Q}}_{X}$ denotes the product of the simple reflections $q_x \\in {\\rm Q}$ , for $x \\in X\\subseteq [m]$ , in the order given by ${\\rm Q}$ .", "It is well known, see [13], that $\\mathcal {SC}({\\rm Q})$ is a simplicial sphere, thus for a given facet $I$ and index $i\\in I$ there exists a unique adjacent facet $J$ with $I\\setminus i = J\\setminus j$ .", "We call the transition from $I$ to $J$ the flip of $i$ in $I$ and if $i<j$ such a flip is called increasing, in which case we write $I\\prec J$ .", "This yields a poset structure on the set of facets of $\\mathcal {SC}({\\rm Q})$ with $I_{\\operatorname{g}}$ as unique minimal element and $I_{\\operatorname{ag}}$ as unique maximal element.", "Following [5], the (abstract) cluster complex $S({\\sf M}_{\\rm c})$ can be seen as a subword complex as follows.", "Denote by ${\\rm {w_\\circ }}({\\rm c})$ the Coxeter-sorting word (or ${\\rm c}$ -sorting word) of ${w_\\circ }$ , i.e., the lexicographically first subword of ${\\rm c}^N$ that is a reduced word for ${w_\\circ }$ .", "The notion of Coxeter-sorting words was introduced by N. Reading in [18] and is an essential ingredient in the combinatorial descriptions of finite type cluster algebras and, in particular, in the description of cluster complexes in terms of subword complexes.", "In this setting we get the cluster complex as $S({\\sf M}_{\\rm c}) \\cong \\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big ).", "$ Example 2.3 ($A_3$ -example) For the Coxeter element $c = s_1s_2s_3$ with fixed reduced word ${\\rm c}= {\\rm s}_1{\\rm s}_2{\\rm s}_3$ we identify the letter ${\\rm s}_i$ with its index $i$ .", "The ${\\rm c}$ -sorting word of ${w_\\circ }$ then is ${\\rm {w_\\circ }}({\\rm {\\rm c}}) = 123121$ and we obtain ${\\rm c}{\\rm {w_\\circ }}({\\rm c}) = 123123121$ for the subword complex $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ .", "The values of the root function are given by 3pt $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{341.43306pt}\\XMLaddatt {height}{106.69783pt}\\begin{tabular}{l|| c|c|c| c|c|c| c|c|c}{2}{*}{I}& \\multicolumn{9}{c}{{{\\sf r}}(I,\\cdot )}\\\\& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\\\\\hline 123 =I_{\\operatorname{g}} & 1\\overline{1}00 & 01\\overline{1}0 & 001\\overline{1}& 1\\overline{1}00 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & 010\\overline{1} & 001\\overline{1}\\\\\\hline 129 & 1\\overline{1}00 & 01\\overline{1}0 & 001\\overline{1}& 1\\overline{1}00 & 100\\overline{1} & 10\\overline{1}0& 010\\overline{1} & 01\\overline{1}0 & 00\\overline{1}1\\\\\\hline 137 & 1\\overline{1}00 & 01\\overline{1}0 & 010\\overline{1}& 10\\overline{1}0 & 1\\overline{1}00 & 100\\overline{1}& 0\\overline{1}10 & 010\\overline{1} & 001\\overline{1}\\\\\\hline 178 & 1\\overline{1}00 & 01\\overline{1}0 & 010\\overline{1}& 10\\overline{1}0 & 100\\overline{1} & 1\\overline{1}00& 001\\overline{1} & 0\\overline{1}01 & 001\\overline{1}\\\\\\hline 189 & 1\\overline{1}00 & 01\\overline{1}0 & 010\\overline{1}& 10\\overline{1}0 & 100\\overline{1} & 1\\overline{1}00& 001\\overline{1} & 01\\overline{1}0 & 001\\overline{1}\\\\\\hline 234 & 1\\overline{1}00 & 10\\overline{1}0 & 001\\overline{1}& \\overline{1}100 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & 010\\overline{1} & 001\\overline{1}\\\\\\hline 249 & 1\\overline{1}00 & 10\\overline{1}0 & 001\\overline{1}& \\overline{1}100 & 100\\overline{1} & 10\\overline{1}0& 010\\overline{1} & 01\\overline{1}0 & 001\\overline{1}\\\\\\hline 345 & 1\\overline{1}00 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & \\overline{1}010 & 100\\overline{1}& 01\\overline{1}0 & 010\\overline{1} & 001\\overline{1}\\\\\\hline 357 & 1\\overline{1}00 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & \\overline{1}100 & 100\\overline{1}& 0\\overline{1}10 & 010\\overline{1} & 001\\overline{1}\\\\\\hline 456 & 1\\overline{1}00 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & 001\\overline{1} & \\overline{1}001& 01\\overline{1}0 & 010\\overline{1} & 001\\overline{1}\\\\\\hline 469 & 1\\overline{1}00 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & 001\\overline{1} & \\overline{1}010& 010\\overline{1} & 01\\overline{1}0 & 00\\overline{1}1\\\\\\hline 567 & 1\\overline{1}00 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & 010\\overline{1} & \\overline{1}001& 0\\overline{1}10 & 010\\overline{1} & 001\\overline{1}\\\\\\hline 678 & 1\\overline{1}00 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & 010\\overline{1} & \\overline{1}100& 001\\overline{1} & 0\\overline{1}01 & 001\\overline{1}\\\\\\hline 689 = I_{\\operatorname{ag}} & 1\\overline{1}00 & 10\\overline{1}0 & 100\\overline{1}& 01\\overline{1}0 & 010\\overline{1} & \\overline{1}100& 001\\overline{1} & 0\\overline{1}10 & 00\\overline{1}1\\end{tabular}}$ and the values of the weight function are given by $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{341.43306pt}\\XMLaddatt {height}{106.69783pt}\\begin{tabular}{l|| c|c|c| c|c|c| c|c|c}{2}{*}{I}& \\multicolumn{9}{c}{{{\\sf w}}(I,\\cdot )}\\\\& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\\\\\hline 123 = I_{\\operatorname{g}} & 1000 & 1100 & 1110 & 1000 & 1100 & 1110 & 0100 & 0110 & 0010\\\\\\hline 129 & 1000 & 1100 & 1110 & 1000 & 1100 & 1101 & 0100 & 0101 & 0001\\\\\\hline 137 & 1000 & 1100 & 1110 & 1000 & 1010 & 1110 & 0010 & 0110 & 0010\\\\\\hline 178 & 1000 & 1100 & 1110 & 1000 & 1010 & 1011 & 0010 & 0011 & 0010\\\\\\hline 189 & 1000 & 1100 & 1110 & 1000 & 1010 & 1011 & 0010 & 0011 & 0001\\\\\\hline 234 & 1000 & 1100 & 1110 & 0100 & 1100 & 1110 & 0100 & 0110 & 0010\\\\\\hline 249 & 1000 & 1100 & 1110 & 0100 & 1100 & 1101 & 0100 & 0101 & 0001\\\\\\hline 345 & 1000 & 1100 & 1110 & 0100 & 0110 & 1110 & 0100 & 0110 & 0010\\\\\\hline 357 & 1000 & 1100 & 1110 & 0100 & 0110 & 1110 & 0010 & 0110 & 0010\\\\\\hline 456 & 1000 & 1100 & 1110 & 0100 & 0110 & 0111 & 0100 & 0110 & 0010\\\\\\hline 469 & 1000 & 1100 & 1110 & 0100 & 0110 & 0111 & 0100 & 0101 & 0001\\\\\\hline 567 & 1000 & 1100 & 1110 & 0100 & 0110 & 0111 & 0010 & 0110 & 0010\\\\\\hline 678 & 1000 & 1100 & 1110 & 0100 & 0110 & 0111 & 0010 & 0011 & 0010\\\\\\hline 689 = I_{\\operatorname{ag}} & 1000 & 1100 & 1110 & 0100 & 0110 & 0111 & 0010 & 0011 & 0001\\end{tabular}}$ Example 2.4 ($B_2$ -example) For the Coxeter element $c = s_1s_2$ with reduced word ${\\rm c}= {\\rm s}_1{\\rm s}_2$ we identify the letters $s_i$ with its index $i$ .", "The ${\\rm c}$ -sorting word of ${w_\\circ }$ then is ${\\rm {w_\\circ }}({\\rm c}) = 1212$ and we obtain ${\\rm c}{\\rm {w_\\circ }}({\\rm c}) = 121212$ for the subword complex $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ .", "The values of the root and weight function are given by 3pt $\\begin{tabular}{l|| c|c| c|c| c|c|| c|c| c|c| c|c}{2}{*}{I}& \\multicolumn{6}{c||}{{{\\sf r}}(I,\\cdot )}& \\multicolumn{6}{c}{{{\\sf w}}(I,\\cdot )}\\\\& 1 & 2 & 3 & 4 & 5 & 6& 1 & 2 & 3 & 4 & 5 & 6\\\\\\hline 12 = I_{\\operatorname{g}} & 2\\overline{2} & 02 & 2\\overline{2} & 20 & 22 & 02& 20 & 11 & 20 & 11 & 02 & \\overline{1}1\\\\\\hline 16 & 2\\overline{2} & 02 & 22 & 20 & 2\\overline{2} & 0\\overline{2}& 20 & 11 & 20 & 1\\overline{1} & 0\\overline{2} & \\overline{1}\\overline{1}\\\\\\hline 23 & 2\\overline{2} & 20 & \\overline{2}2 & 20 & 22 & 02& 20 & 11 & 02 & 11 & 02 & \\overline{1}1\\\\\\hline 34 & 2\\overline{2} & 20 & 22 & \\overline{2}0 & 22 & 02& 20 & 11 & 02 & \\overline{1}1 & 02 & \\overline{1}1\\\\\\hline 45 & 2\\overline{2} & 20 & 22 & 02 & \\overline{2}\\overline{2} & 02& 20 & 11 & 02 & \\overline{1}1 & \\overline{2}0 & \\overline{1}1\\\\\\hline 56 = I_{\\operatorname{ag}} & 2\\overline{2} & 20 & 22 & 02 & \\overline{2}2 & 0\\overline{2}& 20 & 11 & 02 & \\overline{1}1 & \\overline{2}0 & \\overline{1}\\overline{1}\\end{tabular}$ It was developed in [17] how one may obtain a generalized associahedron using subword complexes and brick polytopes.", "Define the brick vector of the facet $I$ of $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ as ${\\sf b}(I) \\ = \\ \\sum \\limits _{k = 1}^{N} \\big {(} {{\\sf w}}(I,n+k) - {{\\sf w}}(I_{\\operatorname{ag}},n+k) \\big {)} \\in V, $ and the brick polytope $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ in $V$ as the convex hull of all brick vectors of $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ , that is, $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}\\ = \\ \\operatorname{conv}\\left\\lbrace {\\sf b}(I) \\;\\big |\\; I \\text{ facet of } \\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big ) \\right\\rbrace .$ It was shown in [17] that $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ is a generalized associahedron.", "As explained in [3], we consider $g$ -vectors to live in the weight space.", "This is, we embed a $g$ -vector $(g_1,\\dots ,g_n)$ into the vector space $V$ as $g_1 \\omega _1 +\\dots + g_n \\omega _n \\in V$ .", "With this convention, we have the following previously known proposition.", "Proposition 2.5 The normal fan of $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ is the $g$ -vector fan.", "This is, $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}\\in \\mathbb {TC}\\big ({\\mathcal {F}_g}({\\sf M}_{\\rm c})\\big ).$ It is shown in [17] that the facet normals of all facets of $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ containing a given brick vector ${\\sf b}(I)$ for some facet $I$ of $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ are $\\left\\lbrace {{\\sf w}}(I,i) \\;|\\; i \\in I \\right\\rbrace $ .", "With the above embedding of the $g$ -vectors into $V$ , it was then shown in [3] that this set coincides with the set of $g$ -vectors inside the cluster of $\\mathcal {A}({\\sf M}_{\\rm c})$ corresponding to $I$ inside $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ under the isomorphism in (REF ) which is also explained in more detail in rem:Smutiso.", "The given definition of the brick polytope differs from the definition given in [17] by a translation and is chosen so that the brick vector ${\\sf b}(I_{\\operatorname{ag}})$ of the antigreedy facet is the origin.", "This translation corresponds to the shifted weight function as used in [3].", "Furthermore, we have for any facet $I$ of $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ that ${{\\sf w}}(I,k) = {{\\sf w}}(I_{\\operatorname{ag}},k)$ for all $1 \\le k \\le n$ .", "This clarifies why we do not consider the first $n$ weight vectors in the summation in (REF ).", "The root function of the greedy facet provides a bijection between the set of positive roots and the positions $n+1,\\ldots ,n+N$ .", "That is, $\\left\\lbrace {{\\sf r}}(I_{\\operatorname{g}},n+k) \\;|\\; 1\\le k \\le N \\right\\rbrace = \\Phi ^+$ .", "As observed in [3], we moreover have ${{\\sf r}}(I_{\\operatorname{g}},n+k) = {{\\sf w}}(I_{\\operatorname{g}},n+k) - {{\\sf w}}(I_{\\operatorname{ag}},n+k)$ for all $1 \\le k \\le N$ .", "For $\\beta ={{\\sf r}}(I_{\\operatorname{g}},n+k) \\in \\Phi ^+$ and a facet $I$ , we sometimes write ${{\\sf w}}(I,\\beta ) := {{\\sf w}}(I,n+k)$ for simplicity, and define $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}= \\operatorname{conv}\\left\\lbrace {{\\sf w}}(I,\\beta ) - {{\\sf w}}(I_{\\operatorname{ag}},\\beta ) \\;\\big |\\; I \\text{ facet of } \\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big ) \\right\\rbrace \\ .$ Remark 2.6 This identification of the positions $n+1,\\dots ,n+N$ and $\\Phi ^+$ is the same as the isomorphism in (REF ) in the following sense.", "As known since [9], sending a cluster variable $u_\\beta (\\mathbf {x},\\mathbf {y})$ to its $d$ -vector $\\beta $ is a bijection between cluster variables and almost positive roots $\\Phi _{\\ge -1}$ .", "Identifying the positions $1,\\dots ,n$ with the simple negative roots $-\\alpha _1,\\dots ,-\\alpha _n$ in this order and the above identification between positions $n+1,\\dots ,n+N$ and $\\Phi ^+$ is a bijection between cluster variables and positions $1,\\dots ,n,n+1,\\dots ,n+N$ and this bijection induces the bijection used in (REF ).", "In particular, the polytope $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ naturally correspond to the cluster variable $u_\\beta $ .", "This correspondence turns out to be a structural correspondence as discussed in sec:fpolys where we show that $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}= {\\operatorname{Newton}\\left(F_\\beta \\right)} = {\\operatorname{Newton}\\left(u_\\beta (\\mathbf {1},\\mathbf {y})\\right)}$ is the Newton polytope of the $F$ -polynomial associated to this cluster variable.", "Example 2.7 ($A_3$ -example) We display the shifted weight function for positions $n+1, \\dots , n+N$ and the brick vector in the following shifted weight table: 3pt $\\leavevmode \\xbox {resizebox}{\\XMLaddatt {width}{427.0pt}\\XMLaddatt {height}{113.81102pt}\\begin{tabular}{l|| c|c| >{{gray!20}}c|c|c|c|| c}{2}{*}{I}& \\multicolumn{6}{c||}{{{\\sf w}}(I,\\cdot ) - {{\\sf w}}(I_{\\operatorname{ag}},\\cdot )}& {2}{*}{{\\sf b}(I)}\\\\~ & 4 & 5 & 6 & 7 & 8 & 9 & ~\\\\\\hline 123 & 1\\overline{1}00 = & 10\\overline{1}0 = & 100\\overline{1} = & 01\\overline{1}0 = & 010\\overline{1} = & 001\\overline{1} = & 31\\overline{1}\\overline{3} = \\\\\\hline 129 & 1\\overline{1}00 = & 10\\overline{1}0 = & 10\\overline{1}0 = & 01\\overline{1}0 = & 01\\overline{1}0 = & 0000 = & 31\\overline{4}0 = \\\\\\hline 137 & 1\\overline{1}00 = & 1\\overline{1}00 = & 100\\overline{1} = & 0000 = & 010\\overline{1} = & 001\\overline{1} = & 3\\overline{1}1\\overline{3} = \\\\\\hline 178 & 1\\overline{1}00 = & 1\\overline{1}00 = & 1\\overline{1}00 = & 0000 = & 0000 = & 001\\overline{1} = & 3\\overline{3}1\\overline{1} = \\\\\\hline 189 & 1\\overline{1}00 = & 1\\overline{1}00 = & 1\\overline{1}00 = & 0000 = & 0000 = & 0000 = & 3\\overline{3}00 = \\\\\\hline 234 & 0000 = & 10\\overline{1}0 = & 100\\overline{1} = & 01\\overline{1}0 = & 010\\overline{1} = & 001\\overline{1} = & 22\\overline{1}\\overline{3} = \\\\\\hline 249 & 0000 = & 10\\overline{1}0 = & 10\\overline{1}0 = & 01\\overline{1}0 = & 01\\overline{1}0 = & 0000 = & 22\\overline{4}0 = \\\\\\hline 345 & 0000 = & 0000 = & 100\\overline{1} = & 01\\overline{1}0 = & 010\\overline{1} = & 001\\overline{1} = & 120\\overline{3} = \\\\\\hline 357 & 0000 = & 0000 = & 100\\overline{1} = & 0000 = & 010\\overline{1} = & 001\\overline{1} = & 111\\overline{3} = \\\\\\hline 456 & 0000 = & 0000 = & 0000 = & 01\\overline{1}0 = & 010\\overline{1} = & 001\\overline{1} = & 020\\overline{2} = \\\\\\hline 469 & 0000 = & 0000 = & 0000 = & 01\\overline{1}0 = & 01\\overline{1}0 = & 0000 = & 02\\overline{2}0 = \\\\\\hline 567 & 0000 = & 0000 = & 0000 = & 0000 = & 010\\overline{1} = & 001\\overline{1} = & 011\\overline{2} = \\\\\\hline 678 & 0000 = & 0000 = & 0000 = & 0000 = & 0000 = & 001\\overline{1} = & 001\\overline{1} = \\\\\\hline 689 & 0000 = & 0000 = & 0000 = & 0000 = & 0000 = & 0000 = & 0000 = \\end{tabular}}$ Especially, the gray marked column corresponds to the polytope ${\\sf Asso}_{111_\\Delta }({\\sf M}_{\\rm c}) = \\operatorname{conv}\\big \\lbrace 111_\\Delta ,~ 110_\\Delta ,~ 100_\\Delta ,~ 000_\\Delta \\big \\rbrace .$ Example 2.8 ($B_2$ -example) We display the shifted weight function for positions $n+1, \\dots , n+N$ and the brick vector in the following shifted weight table: 3pt $\\begin{tabular}{l|| c|c| >{{gray!20}}c|c|| c}{2}{*}{I}& \\multicolumn{4}{c||}{{{\\sf w}}(I,\\cdot ) - {{\\sf w}}(I_{\\operatorname{ag}},\\cdot )}& {2}{*}{{\\sf b}(I)}\\\\~ & 3 & 4 & 5 & 6 & ~\\\\\\hline 12 & 2\\overline{2} = & 20 = & 22 = & 02 = & 62 = \\\\\\hline 16 & 2\\overline{2} = & 2\\overline{2} = & 2\\overline{2} = & 00 = & 6\\overline{6} = \\\\\\hline 23 & 00 = & 20 = & 22 = & 02 = & 44 = \\\\\\hline 34 & 00 = & 00 = & 22 = & 02 = & 24 = \\\\\\hline 45 & 00 = & 00 = & 00 = & 02 = & 02 = \\\\\\hline 56 & 00 = & 00 = & 00 = & 00 = & 00 = \\end{tabular}$ The gray marked column corresponds to the polytope ${\\sf Asso}_{12_\\Delta }({\\sf M}_{\\rm c}) = \\operatorname{conv}\\big \\lbrace 12_\\Delta ,~ 10_\\Delta ,~ 00_\\Delta \\big \\rbrace ,$ and the brick polytope $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ can be seen in fig:B2brickpoly.", "poly = [fill=blue!25, line width=1.2pt, opacity=0.75, draw=blue!40] segment = [draw=blue!40, line width=1.2pt, line cap=round] vertex = [fill=blue!50, draw=blue!50, line width=1.2pt] fan = [fill=red!20, opacity=0.5] ray = [draw=red!50, line width=1.4pt, line cap=round] coords = [draw=black!60,->,>=latex,line width=0.7pt, line cap=round] Figure: The brick polytope error\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}} of type B 2 B_2 from ex:shiftedweightsb2 and its outer normal fan, centered at 1 2(34 Δ )\\tfrac{1}{2}(34_\\Delta ).We next collect several properties of the polytopes $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ .", "We first recall the following crucial lemma.", "Lemma 2.9 ([17]) Let $I\\setminus i = J\\setminus j$ with $i<j$ be two facets of $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ .", "For any $k\\in \\lbrace 1,\\hdots ,n+N\\rbrace $ we have ${{\\sf w}}(J,k) = {{\\sf w}}(I,k) - \\lambda {{\\sf r}}(I,i) \\text{ for some } \\lambda \\in \\mathbb {Z}_{\\ge 0} .$ Moreover for the brick vectors we obtain ${\\sf b}(J) = {\\sf b}(I) - \\lambda {{\\sf r}}(I,i) \\text{ for some } \\lambda \\in \\mathbb {Z}_{> 0}.$ For a set $X\\subseteq \\Phi ^+$ of positive roots, we set ${\\sf b}_X(I)=\\sum \\limits _{\\beta \\in X} \\big ({{\\sf w}}(I,\\beta ) - {{\\sf w}}(I_{\\operatorname{ag}},\\beta )\\big )$ and define the polytope ${\\sf Asso}_X({\\sf M}_{\\rm c}) \\subset V$ as ${\\sf Asso}_X({\\sf M}_{\\rm c}) := \\operatorname{conv}\\left\\lbrace {\\sf b}_X(I) \\;\\big |\\; I \\text{ facet of } \\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )\\ \\right\\rbrace .$ We state the following mild generalization of [17] for the present context.", "The proof given there also applies in the present generality and indeed for all root independent subword complexes as briefly defined in sec:examples below.", "Proposition 2.10 We have the Minkowski decomposition ${\\sf Asso}_X({\\sf M}_{\\rm c}) = \\sum \\limits _{\\beta \\in X} \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}.$ We may neglect the contributions of the shifts by ${{\\sf w}}(I_{\\operatorname{ag}},\\cdot )$ , as these cancel in all considerations.", "By definition we have ${\\sf Asso}_X({\\sf M}_{\\rm c}) \\subseteq \\sum \\limits _{\\beta \\in X} \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}.$ To obtain equality we show that every vertex of $\\sum _{\\beta \\in X} \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ is also a vertex of ${\\sf Asso}_X({\\sf M}_{\\rm c})$ .", "Consider a linear functional $f: V \\rightarrow \\mathbb {R}$ .", "For two adjacent facets $I \\setminus i = J \\setminus j$ of $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ and a positive root $\\beta \\in X$ we have by lem:weightsunderflips that either $f( {{\\sf w}}(I,\\beta ) ) = f( {{\\sf w}}(J,\\beta ) )$ or $f( {{\\sf w}}(I,\\beta ) ) - f( {{\\sf w}}(J,\\beta ) )$ has the same sign as $f( {\\sf b}_X(I) ) - f( {\\sf b}_X(J) )$ .", "Therefore a facet $I_f$ maximizes $f({\\sf b}_X(\\cdot ))$ among all facets if and only if it maximizes $f({{\\sf w}}(\\cdot ,\\beta ))$ for every $\\beta \\in X$ .", "Let now $v$ be a vertex of the Minkowski sum $\\sum _{\\beta \\in X} \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ and let $f: V \\rightarrow \\mathbb {R}$ be a linear functional maximized at $v$ .", "Thus, $v = \\sum _{\\beta \\in X} v_\\beta $ such that $v_\\beta $ maximizes $f$ for $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ .", "On the other hand, $f$ is also maximized by some vertex ${\\sf b}_X(I_f)$ of ${\\sf Asso}_X({\\sf M}_{\\rm c})$ .", "By the previous consideration, $f$ thus maximizes ${{\\sf w}}(I_{f},\\beta )$ for every $\\beta \\in X$ and we obtain $v_\\beta = {{\\sf w}}(I_{f},\\beta )$ .", "Hence $v = \\sum _{\\beta \\in X} {{\\sf w}}(I_{f},\\beta ) = {\\sf b}_X(I_{f})$ .", "The description of the Minkowski decomposition of the brick polytope in the previous proposition also yields the following corollary.", "Corollary 2.11 The set of vertices of ${\\sf Asso}_X({\\sf M}_{\\rm c})$ is $\\left\\lbrace {\\sf b}_X(I) \\;\\big |\\; I \\text{ facet of } \\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big ) \\right\\rbrace $ .", "Example 2.12 ($A_3$ -example) For $X = \\Phi ^+\\setminus \\lbrace 111_\\Delta \\rbrace $ the polytope ${\\sf Asso}_X({\\sf M}_{\\rm c})$ is given by $\\begin{tabular}{rl}{\\sf Asso}_X({\\sf M}_{\\rm c})& = \\ \\operatorname{conv}\\big \\lbrace 21\\overline{1}\\overline{2},~ 21\\overline{3}0,~ 2\\overline{1}1\\overline{2},~2\\overline{2}1\\overline{1},~ 2\\overline{2}00,~ 12\\overline{1}\\overline{2},~12\\overline{3}0,~ \\\\[5pt]& \\hspace{42.67912pt} 020\\overline{2},~ 011\\overline{2},~ 020\\overline{2},~02\\overline{2}0,~ 011\\overline{2},~ 001\\overline{1},~ 0000 \\big \\rbrace \\\\[10pt]\\end{tabular}& $ conv{ 232, px 230, px 212, px 201, px 200, px 132, px 130, px$\\\\[5pt]& $             022, px 012, px 022, px 020, px 012, px 001, px 000}$.$ $For later reference we note that $ r(Ig,6) = 111$ and$${\\sf b}_X( \\lbrace 345 \\rbrace ) = {\\sf b}_X( \\lbrace 456 \\rbrace ) = 022_\\Delta , \\quad {\\sf b}_X( \\lbrace 357 \\rbrace ) = {\\sf b}_X( \\lbrace 567 \\rbrace ) = 012_\\Delta \\ .$$$ Example 2.13 ($B_2$ -example) For $X = \\Phi ^+\\setminus \\lbrace 12_\\Delta \\rbrace $ the polytope ${\\sf Asso}_X({\\sf M}_{\\rm c})$ is given by $\\begin{tabular}{rl}{\\sf Asso}_X({\\sf M}_{\\rm c})& = \\ \\operatorname{conv}\\big \\lbrace 40,\\hspace{10.0pt}px4\\overline{4},\\hspace{10.0pt}px22,\\hspace{10.0pt}px02,\\hspace{10.0pt}px02,\\hspace{11.0pt}px00 \\hspace{8.0pt}\\big \\rbrace \\\\[5pt]& = \\ \\operatorname{conv}\\big \\lbrace 22_\\Delta ,~ 20_\\Delta ,~ 12_\\Delta ,~01_\\Delta ,~ 01_\\Delta ,~ 00_\\Delta \\big \\rbrace .\\end{tabular}$ For later reference we note that ${\\sf b}_X( \\lbrace 34 \\rbrace ) = {\\sf b}_X( \\lbrace 45 \\rbrace ) = 01_\\Delta $ and $12_\\Delta = {{\\sf r}}(I_{\\operatorname{g}},5)$ .", "We next introduce the following canonical long flip sequence in the subword complex $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ from the greedy to the antigreedy facet, $I_{\\operatorname{g}}= I_0 \\prec I_1 \\prec \\dots \\prec I_N = I_{\\operatorname{ag}}$ where $I_{\\ell +1}$ is obtained from $I_\\ell $ by flipping the unique index $i$ in $I_\\ell $ such that $I_{\\ell +1} \\setminus \\lbrace \\ell +1+n\\rbrace = I_\\ell \\setminus \\lbrace i\\rbrace $ .", "Indeed, up to commutation of consecutive commuting letters, the index $i$ is the smallest index that yields an increasing flip.", "Indeed, there is some flexibility in defining this sequence—any sequence of flips corresponding to source mutations in the associated cluster algebra would work.", "Example 2.14 ($A_3$ -example) For ${\\rm c}{\\rm {w_\\circ }}({\\rm c}) = 123123121$ the canonical long flip sequence is given by $I_{\\operatorname{g}}= \\lbrace 1, 2, 3\\rbrace \\prec \\lbrace 2, 3, 4\\rbrace \\prec \\lbrace 3, 4, 5\\rbrace \\prec \\lbrace 4, 5, 6\\rbrace \\prec \\lbrace 5, 6, 7\\rbrace \\prec \\lbrace 6, 7, 8\\rbrace \\prec \\lbrace 6, 8, 9\\rbrace = I_{\\operatorname{ag}}\\ .$ Example 2.15 ($B_2$ -example) For ${\\rm c}{\\rm {w_\\circ }}({\\rm c}) = 121212$ the canonical long flip sequence is given by $I_{\\operatorname{g}}= \\lbrace 1, 2\\rbrace \\prec \\lbrace 2, 3\\rbrace \\prec \\lbrace 3, 4\\rbrace \\prec \\lbrace 4, 5\\rbrace \\prec \\lbrace 5, 6\\rbrace = I_{\\operatorname{ag}}\\ .$ This flip sequence already appeared in [17] and in the proof of [3], where in particular the following property was used.", "Lemma 2.16 For every index $j \\in \\lbrace n+1, \\ldots , n+N \\rbrace $ there exists a unique pair $I_\\ell \\prec I_{\\ell +1}$ in the canonical long flip sequence and an index $i$ such that $I_\\ell \\setminus i = I_{\\ell +1} \\setminus j$ .", "Moreover, in this case the weight function ${{\\sf w}}(I_{\\ell +1},\\cdot )$ is obtained from ${{\\sf w}}(I_\\ell ,\\cdot )$ by ${{\\sf w}}(I_{\\ell +1},k) ={\\left\\lbrace \\begin{array}{ll}{{\\sf w}}(I_\\ell ,k) - {{\\sf r}}(I_\\ell ,i) & \\text{if } k = j , \\\\{{\\sf w}}(I_\\ell ,k) & \\text{otherwise} .\\end{array}\\right.", "}$ In particular, ${{\\sf w}}(I_{\\ell },\\cdot )$ and ${{\\sf w}}(I_{\\ell +1},\\cdot )$ only differ for the index $j$ .", "Up to commutations of consecutive commuting letters in the word ${\\rm c}{\\rm {w_\\circ }}({\\rm c}) = {\\rm q}_1{\\rm q}_2\\dots {\\rm q}_{n+N}$ , the facet $I_{\\ell }$ consists of the letters ${\\rm q}_{\\ell +1}\\dots {\\rm q}_{\\ell +n}$ .", "Indeed, we may assume without loss of generality that for each $0 \\le \\ell < N$ we have $I_{\\ell +1}\\setminus I_\\ell = \\lbrace \\ell +n+1\\rbrace $ .", "Moreover, $\\lbrace q_{\\ell +1},\\dots ,q_{\\ell +n}\\rbrace = \\mathcal {S}$ (this follows, for example, from [5]) and the facets $I_\\ell \\prec I_{\\ell +1}$ may be visualized inside the word ${\\rm c}{\\rm {w_\\circ }}({\\rm c})$ as $I_\\ell &= {\\rm q}_1\\dots {\\rm q}_{\\ell }\\ \\widehat{{\\rm q}}_{\\ell +1}\\widehat{{\\rm q}}_{\\ell +2}\\dots \\widehat{{\\rm q}}_{\\ell +n}\\ {\\rm q}_{\\ell +n+1}\\ {\\rm q}_{\\ell +n+2}\\ \\dots {\\rm q}_{n+N} \\\\[5pt]I_{\\ell +1} &= {\\rm q}_1\\dots {\\rm q}_{\\ell }\\ {\\rm q}_{\\ell +1}\\widehat{{\\rm q}}_{\\ell +2}\\dots \\widehat{{\\rm q}}_{\\ell +n}\\ \\widehat{{\\rm q}}_{\\ell +n+1}\\ {\\rm q}_{\\ell +n+2}\\ \\dots {\\rm q}_{n+N}$ where the letters with hats are omitted and where we assumed, again without loss of generality, that $q_{\\ell +1} = q_{\\ell +n+1}$ .", "The statement of the lemma now follows with ${{\\sf r}}(I_\\ell ,\\ell +1) = {{\\sf r}}(I_\\ell ,\\ell +n+1) = {{\\sf r}}(I_{\\ell +1},\\ell +1) = -{{\\sf r}}(I_{\\ell +1},\\ell +n+1) = q_1\\dots q_\\ell (s_{q_{\\ell +1}}).", "$ This lemma yields an interesting combinatorial property of the polytopes $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ that we do not use further below.", "Corollary 2.17 For every $\\beta \\in \\Phi ^+$ the segment connecting $\\mathbf {0}$ and $\\beta $ is an edge of $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ .", "As the brick polytope $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ realizes $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ its edges are in one-to-one correspondence to flips in $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ .", "Combining lem:weightsunderflips and cor:vertsofsubsum we obtain a similar result for $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ saying its edges are in one-to-one correspondence with flips that change the weight function ${{\\sf w}}(\\cdot ,\\beta )$ .", "Applying lem:differenceweightsequence to the canonical long flip sequence $I_{\\operatorname{g}}= I_0 \\prec I_1 \\prec \\hdots \\prec I_{N-1} \\prec I_N = I_{\\operatorname{ag}}$ we obtain for $\\beta = {{\\sf r}}(I_{\\operatorname{g}},n + i)$ that $\\begin{tabular}{lllll}{{\\sf w}}( I_{\\operatorname{g}}, \\beta ) &\\ = \\ {{\\sf w}}( I_1 , \\beta ) &\\ = \\ \\hdots &\\ = \\ {{\\sf w}}( I_{i-1} , \\beta ), &and \\\\\\end{tabular}$ w( Ii , )$ &$ = $ &$ = w( IN-1 , )$ &$ = w( Iag, )$.", "&$ $As $ w( Ig, ) - w( Iag, ) = $ we conclude the statement.$ Generators of the type cone The following definitions mostly follow [16].", "Let $\\mathcal {F}$ be an essential complete simplicial fan in $\\mathbb {R}^{d}$ .", "A polytopal realization of $\\mathcal {F}$ is a convex polytope in $\\mathbb {R}^{d}$ whose outer normal fan agrees with $\\mathcal {F}$ .", "The space of all polytopal realizations of $\\mathcal {F}$ is called the type cone of $\\mathcal {F}$ , denoted by $\\mathbb {TC}(\\mathcal {F})$ , see also [15].", "A parametrization of $\\mathbb {TC}(\\mathcal {F})$ is commonly described as follows.", "Denote by $G \\in \\mathbb {R}^{m\\times d}$ the matrix whose rows generate the rays of $\\mathcal {F}$ .", "Each height vector $h \\in \\mathbb {R}^{m}$ defines a polytope $P_h \\ = \\ \\left\\lbrace x \\in \\mathbb {R}^{d} \\mid Gx \\le h \\right\\rbrace .$ Now the type cone of $\\mathcal {F}$ can be parametrized as the open polyhedral cone $\\mathbb {TC}(\\mathcal {F}) \\ = \\ \\left\\lbrace h \\in \\mathbb {R}^{m} \\mid P_h \\text{ has normal fan } \\mathcal {F}\\right\\rbrace .$ We write $P_h \\in \\mathbb {TC}(\\mathcal {F})$ by identifying a polytope $P_h$ with its height vector $h \\in \\mathbb {R}^{m}$ .", "With this definition, $\\mathbb {TC}(\\mathcal {F})$ has $d$ -dimensional lineality space corresponding to translations in $\\mathbb {R}^{d}$ .", "More specifically, for $P_h \\in \\mathbb {TC}(\\mathcal {F})$ and a translation vector $b \\in \\mathbb {R}^{d}$ we have $P_h + b \\ = \\ P_{h+Gb} \\in \\mathbb {TC}(\\mathcal {F}) \\hspace{5.0pt},$ Thus the lineality space of $\\mathbb {TC}(\\mathcal {F})$ is given by the image of the matrix $G$ .", "We identify $\\mathbb {TC}(\\mathcal {F})$ with its pointed quotient $\\mathbb {TC}(\\mathcal {F}) / G\\mathbb {R}^{d}$ .", "The closure $\\overline{\\mathbb {TC}}(\\mathcal {F})$ is called the closed type cone.", "The faces of $\\overline{\\mathbb {TC}}(\\mathcal {F})$ correspond to (weak) Minkowki summands of $P$ with the same normal fan (which are coarsenings of $\\mathcal {F}$ ).", "In particular, the (extremal) generators of $\\overline{\\mathbb {TC}}(P)$ correspond to the indecomposable Minkowski summands of $P$ .", "We aim at the description of the type cone $\\mathbb {TC}({\\mathcal {F}_g}({\\sf M}_{\\rm c}))$ of the $g$ -vector fan ${\\mathcal {F}_g}({\\sf M}_{\\rm c})$ given in thm:indecompcolumns.", "We first state the following lemma which we then use to understand the rays of the type cone.", "Lemma 2.18 Let $C \\subset \\mathbb {R}^m$ be a full-dimensional closed polyhedral cone and let $x = x_1+\\cdots +x_m$ for $x_1,\\ldots ,x_m \\in C$ with $x$ is an interior point of $C$ and $x - x_i \\hspace{5.0pt}$ is contained in the boundary of $C$ for every $i \\in \\lbrace 1,\\ldots ,m \\rbrace $ .", "Then $C=\\operatorname{cone}\\lbrace x_1,\\ldots ,x_m\\rbrace $ .", "In particular, the cone $C$ is simplicial.", "Write $X = \\lbrace x_1,\\ldots ,x_m\\rbrace $ .", "We first show that $X$ is linearly independent.", "Assuming the contrary, one may express some $x_i$ in terms of $X \\setminus \\lbrace x_i\\rbrace $ .", "This would mean that $x = (x - x_i) + x_i$ would be in the linear span of $X \\setminus \\lbrace x_i\\rbrace $ .", "By condition (i), this point is in the interior of $C$ , while it is on the boundary by condition (ii)—a contradiction.", "It follows that $\\operatorname{cone}(X)$ is a simplicial full-dimensional cone inside $C$ .", "As condition (ii) implies that its boundary is also contained in the boundary of $C$ , we conclude the statement.", "The $g$ -vector fan ${\\mathcal {F}_g}({\\sf M}_{\\rm c})$ is an essential complete simplicial fan in $\\mathbb {R}^{n}$ with $n+N$ rays.", "Therefore, after passing to the quotient by its $n$ -dimensional lineality space, the closed type cone $\\overline{\\mathbb {TC}}({\\mathcal {F}_g}({\\sf M}_{\\rm c}))$ is an $N$ -dimensional pointed polyhedral cone.", "We aim at applying lem:simplicialcone using the points $\\left\\lbrace \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}} \\;|\\; \\beta \\in \\Phi ^+ \\right\\rbrace $ .", "We have seen in (REF ) that $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}= \\sum \\limits _{\\beta \\in \\Phi ^+} \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ is an interior point of $\\overline{\\mathbb {TC}}({\\mathcal {F}_g}({\\sf M}_{\\rm c}))$ .", "Therefore, it suffices to show that for each $\\gamma \\in \\Phi ^+$ the polytope ${\\sf Asso}_{\\Phi ^+\\setminus \\lbrace \\gamma \\rbrace }({\\sf M}_{\\rm c})$ is contained in the boundary of $\\overline{\\mathbb {TC}}({\\mathcal {F}_g}({\\sf M}_{\\rm c}))$ .", "Let $\\gamma \\in \\Phi ^+$ and let $j\\in \\lbrace n+1,\\ldots ,n+N\\rbrace $ be the unique index such that $r(I_g,j)=\\gamma $ .", "lem:differenceweightsequence ensures the existence of a unique index $\\ell $ such that $j$ is contained in $I_{\\ell +1}$ but not in $I_\\ell $ in the canonical long flip sequence $I_0 \\prec \\dots \\prec I_N$ .", "Since ${{\\sf w}}(I_{\\ell },\\cdot )$ and ${{\\sf w}}(I_{\\ell +1},\\cdot )$ only differ for the index $j$ , it follows that ${\\sf b}_{\\Phi ^+\\setminus \\lbrace \\gamma \\rbrace }(I_\\ell ) = {\\sf b}_{\\Phi ^+\\setminus \\lbrace \\gamma \\rbrace }(I_{\\ell +1}).$ cor:vertsofsubsum and the second part of lem:weightsunderflips now show that the number of vertices of ${\\sf Asso}_{\\Phi ^+\\setminus \\lbrace \\gamma \\rbrace }({\\sf M}_{\\rm c})$ is strictly less than the number of vertices of $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ .", "This means that it is a proper weak Minkowski summand and it is thus not contained in the interior of $\\overline{\\mathbb {TC}}({\\mathcal {F}_g}({\\sf M}_{\\rm c}))$ .", "Invoking lem:simplicialcone yields the proposed statement $\\overline{\\mathbb {TC}}({\\mathcal {F}_g}({\\sf M}_{\\rm c})) = \\operatorname{cone}\\left\\lbrace \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}} \\;|\\; \\beta \\in \\Phi ^+ \\right\\rbrace $ and that the type cone is in particular simplicial.", "Generators of the type cone for general spherical subword complexes We close this section with a brief discussion of properties of type cones for examples of general subword complexes.", "It turns out that the situation for cluster complexes is particularly special.", "Most importantly, the conclusion of lem:differenceweightsequence does not hold in general for spherical subword complexes.", "The complex $\\mathcal {SC}\\big ({\\rm c}{\\rm {w_\\circ }}({\\rm c})\\big )$ is known to have the following properties.", "For a word ${\\rm Q}$ , we call a spherical subword complex $\\mathcal {SC}({\\rm Q})$ root-independent if the multiset ${{\\sf R}}(I) = \\big \\lbrace \\!\\!\\big \\lbrace {{\\sf r}}(I,i) \\;\\big |\\; i \\in I \\big \\rbrace \\!\\!\\big \\rbrace $ is linearly independent for any (and thus every) facet $I$ and it is of full support if every position in ${\\rm Q}$ is contained in some facet (meaning that all elements of the ground set are indeed vertices).", "Observe that spherical subword complexes of full support are also full-dimensional, meaning that ${{\\sf R}}(I)$ generates $V$ for any facet $I$ .", "This is an immediate consequence of [17].", "We conjecture that these properties identify cluster complexes among spherical subword complexes.", "Conjecture 2.19 Let $Q$ be a word in $\\mathcal {S}$ .", "The following statements are equivalent: Up to commutations of consecutive commuting letters ${\\rm Q}= {\\rm c}{\\rm {w_\\circ }}({\\rm c})$ for some Coxeter element $c$ .", "$\\mathcal {SC}({\\rm Q})$ is root-independent and of full support.", "Remark that the first property was shown to be equivalent to the so-called SIN-property in [5].", "Furthermore they conjecture these subword complexes to maximize the number of facets a subword complex $\\mathcal {SC}({\\rm Q})$ with $|{\\rm Q}| = n+N$ can have [5].", "We next show that relaxing one of the two conditions yields examples for which the conclusion of thm:indecompcolumns does not hold.", "We denote by $P_i$ the polytope corresponding to the $i$ -th column in the shifted weight table, $P_i \\ = \\ \\operatorname{conv}\\left\\lbrace {{\\sf w}}(I,i) - {{\\sf w}}(I_{\\operatorname{ag}},i) \\;\\big |\\; I \\text{ facet of } \\mathcal {SC}({\\rm Q}) \\right\\rbrace $ for the given subword complex $\\mathcal {SC}({\\rm Q})$ .", "Example 2.20 ($B_2$ -example) For ${\\rm Q}= 1212121$ , the subword complex $\\mathcal {SC}({\\rm Q})$ is of full support but not root-independent as ${{\\sf R}}( I_{\\operatorname{g}}) = {{\\sf R}}( \\lbrace 1,2,3 \\rbrace ) =\\big \\lbrace \\!\\!\\big \\lbrace 10_\\Delta ,~ 01_\\Delta ,~ 10_\\Delta \\big \\rbrace \\!\\!\\big \\rbrace $ .", "The list of facets is $\\big \\lbrace \\lbrace 1,2,3\\rbrace ,~ \\lbrace 1,2,6\\rbrace ,~ \\lbrace 1,3,4\\rbrace ,~ \\lbrace 1,4,5\\rbrace ,~ \\lbrace 1,5,6\\rbrace ,~\\lbrace 1,6,7\\rbrace ,~ \\lbrace 2,3,7\\rbrace ,~ \\lbrace 3,4,7\\rbrace ,~ \\lbrace 4,5,7\\rbrace ,~ \\lbrace 5,6,7\\rbrace \\big \\rbrace .$ One may easily check that its brick polytope is the permutahedron of type $B_2$ whose normal fan is the Coxeter fan.", "The complete list of polytopes is $\\begin{tabular}{lll}P_1 & = \\ P_2 & = \\ \\operatorname{conv}\\lbrace 00_\\Delta \\rbrace \\\\P_3 & = \\ P_7 & = \\ \\operatorname{conv}\\lbrace 00_\\Delta ,~ 10_\\Delta \\rbrace \\\\P_4 & ~ & = \\ \\operatorname{conv}\\lbrace 00_\\Delta ,~ 10_\\Delta ,~11_\\Delta \\rbrace \\\\P_6 & ~ & = \\ \\operatorname{conv}\\lbrace 00_\\Delta ,~ 01_\\Delta ,~11_\\Delta \\rbrace \\\\P_5 & = \\ P_3 + P_{?}", "& = \\ \\operatorname{conv}\\lbrace 00_\\Delta ,~ 10_\\Delta ,~12_\\Delta ,~ 22_\\Delta \\rbrace ,\\end{tabular}$ where $P_{?}", "= \\operatorname{conv}\\lbrace 00_\\Delta ,~ 12_\\Delta \\rbrace $ is a missing generator of the type cone.", "Furthermore the sum of $P_4$ and $P_6$ can be decomposed into $P_4 + P_6 = \\operatorname{conv}\\lbrace 00_\\Delta ,~ 10_\\Delta \\rbrace +\\operatorname{conv}\\lbrace 00_\\Delta ,~ 01_\\Delta \\rbrace +\\operatorname{conv}\\lbrace 00_\\Delta ,~ 11_\\Delta \\rbrace .$ In particular, the type cone of the brick polytope is not simplicial.", "Example 2.21 ($B_2$ -example) For ${\\rm Q}= 212212$ the subword complex $\\mathcal {SC}({\\rm Q})$ is root-independent and full-dimensional but not of full support as ${{\\sf R}}( I_{\\operatorname{g}}) = {{\\sf R}}( \\lbrace 1,3 \\rbrace ) =\\big \\lbrace \\!\\!\\big \\lbrace 10_\\Delta ,~ 12_\\Delta \\big \\rbrace \\!\\!\\big \\rbrace $ and the list of facets is $\\big \\lbrace \\lbrace 1,3\\rbrace ,~ \\lbrace 1,4\\rbrace ,~ \\lbrace 3,6\\rbrace ,~ \\lbrace 4,6\\rbrace \\big \\rbrace .$ The positions 2 and 5 are not contained in any facet.", "The complete list of polytopes is $\\begin{tabular}{lll}P_1 & = \\ P_2 & = \\ \\operatorname{conv}\\lbrace 00_\\Delta \\rbrace \\\\P_3 & = \\ P_6 & = \\ \\operatorname{conv}\\lbrace 00_\\Delta ,~ 01_\\Delta \\rbrace \\\\P_5 & = \\ P_3 + P_6 & = \\ \\operatorname{conv}\\lbrace 00_\\Delta ,~ 02_\\Delta \\rbrace \\\\P_4 & = \\ P_3 + P_{?}", "& = \\ \\operatorname{conv}\\lbrace 00_\\Delta ,~ 01_\\Delta ,~11_\\Delta ,~ 12_\\Delta \\rbrace ,\\end{tabular}$ where $P_{?}", "= \\operatorname{conv}\\lbrace 00_\\Delta ,~ 11_\\Delta \\rbrace $ is the missing generator of the type cone.", "Newton polytopes of $F$ -polynomials from subword complexes Let $\\mathcal {A}({\\sf M}_{\\rm c})$ be the finite type cluster algebra with acyclic initial mutation matrix ${\\sf M}_{\\rm c}$ with principal coefficients and denote by ${\\mathcal {F}_g}({\\sf M}_{\\rm c})$ its $g$ -vector fan.", "We have seen in thm:indecompcolumns that the type cone $\\mathbb {TC}({\\mathcal {F}_g}({\\sf M}_{\\rm c}))$ is generated by the natural Minkowski summands of the brick polytope $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ , $\\mathbb {TC}\\big ( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) \\big ) \\ = \\ \\operatorname{cone}\\left\\lbrace \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace .$ A description of the generators of $\\mathbb {TC}\\big ( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) \\big )$ was also obtained by combining results from [2], [1] and [16] as follows.", "In [2] the authors provide polytopal realizations of ${\\mathcal {F}_g}( {\\sf M}_{\\rm c})$ .", "This construction produces a generalized associahedron $\\mathcal {X}_p$ for each $p \\in \\mathbb {R}_{ > 0 }^{ \\Phi ^+}$ .", "It was then shown in [2] (simply-laced types) and in [1] (multiply-laced types) that $\\mathcal {X}_p$ for $p = e_\\beta $ and $\\beta \\in \\Phi ^+$ equals the Newton polytope of the $F$ -polynomial $F_\\beta $ .", "In [16], the authors explain that within the latter constuction $\\mathbb {R}_{> 0}^{\\Phi ^+}$ can be understood as (a linear transformation of) the type cone $\\mathbb {TC}\\big ( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) \\big )$ .", "In particular, this establishes the fact that the Newton polytopes of the $F$ -polynomials generate the type cone, $\\mathbb {TC}( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) ) \\ = \\ \\operatorname{cone}\\left\\lbrace {\\operatorname{Newton}\\left(F_\\beta \\right)} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace .$ In order to prove cor:NewtonFpoly, it remains to properly identify which Newton polytope of an $F$ -polynomial corresponds to which Minkowski summand of the brick polytope.", "This is done using the following property of $F$ -polynomials.", "Proposition 3.1 ([7] (simply-laced types), [6] (multiply-laced types)) For every $\\beta \\in \\Phi ^+$ , the $F$ -polynomial $F_\\beta = F_\\beta (\\mathbf {y})$ has constant term 1 and a unique componentwise highest exponent vector given by $\\beta $ .", "In particular, $\\mathbf {0}$ and $\\beta $ are both vertices of ${\\operatorname{Newton}\\left(F_\\beta \\right)}$ .", "For simply-laced types, this is [7] and for multiply-laced types, this is [6].", "This proposition can be rechecked in types $A_3$ and $B_2$ in fig:examples.", "Now we are ready to proof our second main result.", "Since $\\mathbb {TC}\\big ( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) \\big )$ is a simplicial cone of dimension $N=| \\Phi ^+|$ we already know that the two sets of generators, $\\left\\lbrace {\\operatorname{Newton}\\left( F_\\beta \\right)} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace \\ \\text{and} \\ \\left\\lbrace \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace ,$ are non-redundant and coincide up to scalar factors.", "Let $\\beta \\in \\Phi ^+$ .", "By lem:highestandlowestexponent the unique maximal and minimal vertices of ${\\operatorname{Newton}\\left(F_\\beta \\right)}$ are $\\beta $ and $\\mathbf {0}$ , respectively.", "Since $\\beta = {\\sf b}_{\\lbrace \\beta \\rbrace }( I_{\\operatorname{g}})$ and $\\mathbf {0} = {\\sf b}_{\\lbrace \\beta \\rbrace }( I_{\\operatorname{ag}})$ , these vectors are by cor:vertsofsubsum vertices of $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ as well.", "Applying lem:weightsunderflips we see that they are the maximal and minimal vertices of $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ , respectively.", "Thus the polytopes ${\\operatorname{Newton}\\left( F_\\beta \\right)}$ and $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ coincide.", "The tropical positive cluster variety In this section, we prove thm:tropplusvariety starting from the type cone description (REF ) on page REF in terms of Newton polytopes of $F$ -polynomials.", "It is independent of the subword complex description and does not make use of it.", "We again emphasize that a more general version of thm:tropplusvariety follows from [1].", "Following [19], we start with the needed notions from tropical geometry.", "Let $E \\subset \\mathbb {Z}_{\\ge 0}^d$ be non-empty and finite and let $f=\\sum _{e \\in E} f_e \\mathbf {u}^e \\in \\mathbb {Q}[\\mathbf {u}]$ with $f_e \\ne 0$ for all $e \\in E$ be a rational polynomial supported on $E$ .", "For each weight $w\\in \\mathbb {R}^d$ we define $E(w) \\ = \\ \\underset{e \\in E}{\\arg \\max } (w\\cdot e)\\ .$ That is, $E(w)$ is the intersection of $E$ with the face of ${\\operatorname{Newton}\\left(f\\right)}=\\operatorname{conv}(E)$ that is maximized in direction $w$ .", "The tropical hypersurface $\\operatorname{Trop}(f) \\subset \\mathbb {R}^d$ is the collection of those weights $w\\in \\mathbb {R}^d$ for which $E(w)$ contains at least two elements.", "$\\operatorname{Trop}(f)$ naturally carries the structure of a polyhedral fan, whose cones are formed by those weights $w\\in \\operatorname{Trop}(f)$ that yield the same $E(w)$ .", "This fan thus agrees with the codimension-one skeleton of the normal fan of ${\\operatorname{Newton}\\left(f\\right)}$ .", "The positive part $\\operatorname{Trop^+}(f)$ of the tropical hypersurface was introduced in [19] and is defined as follows.", "Split $E=E^+_f \\sqcup E^-_f$ according to the signs of the coefficients of $f$ .", "That is, $E^+_f = \\lbrace e \\in E \\mid f_e > 0 \\rbrace , \\quad E^-_f = \\lbrace e \\in E \\mid f_e < 0 \\rbrace .$ Now $\\operatorname{Trop^+}(f)$ is defined as the subfan of $\\operatorname{Trop}(f)$ consisting of those weights for which neither $E(w)\\cap E^+_f$ nor $E(w)\\cap E^-_f$ is empty, $\\operatorname{Trop^+}(f) = \\left\\lbrace w\\in \\mathbb {R}^d \\;\\big |\\; E(w)\\cap E^+_f \\ne \\emptyset ~ \\text{ and } ~ E(w)\\cap E^-_f \\ne \\emptyset \\right\\rbrace \\ .$ For any ideal $\\mathcal {I}\\subset \\mathbb {Q}[\\mathbf {u}]$ the positive tropical variety $\\operatorname{Trop^+}(\\mathcal {I})$ is defined as the intersection of all positive tropical hypersurfaces $\\operatorname{Trop^+}(f)$ for $f\\in \\mathcal {I}$ .", "We next move to the positive tropical variety considered here.", "Let $\\mathcal {A}({\\sf M})$ be a finite type cluster algebra of rank $n$ with (not necessarily acyclic) initial mutation matrix ${\\sf M}$ with principal coefficients.", "We denote by $X_\\Delta =\\lbrace x_1,\\ldots ,x_n\\rbrace $ the set of initial cluster variables and by $X_{\\Phi ^+}=\\left\\lbrace x_\\beta \\;|\\; \\beta \\in \\Phi ^+ \\right\\rbrace $ the set of non-initial cluster variables.", "Thus the set of all cluster variables is the disjoint union $X=X_\\Delta \\sqcup X_{\\Phi ^+}$ .", "Furthermore, let $Y=\\lbrace y_1,\\ldots ,y_n\\rbrace $ be the set of principle coefficient variables.", "Recall that each non-initial cluster variable $x_\\beta \\in X_{\\Phi ^+}$ is expressed in terms of the initial seed by $x_\\beta = \\frac{p_\\beta (\\mathbf {x},\\mathbf {y})}{\\mathbf {x}^\\beta }$ where $p_\\beta (\\mathbf {x},\\mathbf {y})$ is a subtraction-free polynomial in the initial cluster and coefficient variables and $\\mathbf {x}^\\beta = x_1^{\\beta _1}\\dots x_n^{\\beta _n}$ for $\\beta = (\\beta _1,\\cdots ,\\beta _n)_\\Delta \\in \\mathbb {R}^\\Delta $ .", "Following [19] we embed $\\operatorname{Spec}\\mathcal {A}({\\sf M})$ as the affine variety $V(\\mathcal {I}_{{\\sf M}}) \\subset \\mathbb {Q}^{X\\sqcup Y}$ , where $\\mathcal {I}_{{\\sf M}}$ is the ideal generated by the non-initial cluster variables, i.e., $\\mathcal {I}_{{\\sf M}} \\ = \\ \\left\\langle x_\\beta \\cdot \\mathbf {x}^\\beta - p_\\beta (\\mathbf {x},\\mathbf {y}) ~ \\big | ~ \\beta \\in \\Phi ^+\\right\\rangle .$ Note that in this case the special form of the generators immediately yields a subtraction-free parametrization $\\Psi : (\\mathbb {Q}^\\ast )^{X_\\Delta \\sqcup Y} \\rightarrow V(\\mathcal {I}_{{\\sf M}})\\cap (\\mathbb {Q}^\\ast )^{X\\sqcup Y}$ given as the graph of the map $(\\mathbb {Q}^\\ast )^{X_\\Delta \\sqcup Y} \\ &\\longrightarrow \\ (\\mathbb {Q}^\\ast )^{X_{\\Phi ^+}} \\hspace{5.0pt}, \\\\(\\mathbf {x},\\mathbf {y}) \\ &\\longmapsto \\ \\left( \\frac{p_\\beta (\\mathbf {x},\\mathbf {y})}{\\mathbf {x}^\\beta } \\right)_{\\beta \\in \\Phi ^+}$ We denote by $\\operatorname{Trop}\\Psi : \\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow \\mathbb {R}^{X \\sqcup Y}$ the tropicalization of the map $\\Psi $ .", "This is the piecewise linear map obtained by replacing every $\\times $ in $\\Psi $ with a $+$ , every $/$ with a $−$ and every $+$ with a $\\max $ .", "The following result is an immediate consequence of [19].", "Proposition 4.1 The map $\\operatorname{Trop}\\Psi : \\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow \\mathbb {R}^{X \\sqcup Y}$ is a piecewise linear parametrization of the positive tropical variety $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ .", "It should be mentioned here that in [19] the authors are working over the field of complex Puiseux series, one of the prototype examples of a field with non-trivial valuation.", "This is standard in tropical geometry since it reveals strong connections between classical algebraic geometry and tropical geometry.", "The ideal $\\mathcal {I}_{{\\sf M}}$ in cor:paramtropplus is understood over the complex Puiseux series and for the definition of a positive tropical hypersurface for a complex Puiseux polynomial we refer to [19].", "However, the map $\\operatorname{Trop}\\Psi $ stays unchanged when working over $\\mathbb {Q}$ .", "The domains of linearity of $\\operatorname{Trop}\\Psi $ form a polyhedral fan in $\\mathbb {R}^{X_\\Delta \\sqcup Y}$ , which we denote by $\\mathcal {F}_{\\Psi }$ .", "Following [19] we equip $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ with the fan structure obtained by applying $\\operatorname{Trop}\\Psi $ to $\\mathcal {F}_{\\Psi }$ .", "Whenever we refer to $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ as a polyhedral fan we consider this fan structure.", "By cor:paramtropplus the map $\\operatorname{Trop}\\Psi : \\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow \\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ is a piecewise linear parametrization of $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ .", "Moreover, $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ is piecewise linearly isomorphic to $\\mathcal {F}_\\Psi $ by construction .", "For each $\\beta \\in \\Phi ^+$ denote by $\\mathcal {F}_\\beta $ the normal fan of ${\\operatorname{Newton}\\left(p_\\beta \\right)}$ .", "The domains of linearity of the coordinate function $\\operatorname{Trop}\\Psi _\\beta (\\mathbf {x},\\mathbf {y}) = \\operatorname{Trop}(p_\\beta (\\mathbf {x},\\mathbf {y}) / \\mathbf {x}^\\beta )$ are the maximal cones of $\\mathcal {F}_\\beta $ .", "Thus the domains of linearity of $\\operatorname{Trop}\\Psi $ , and hence the fan structure $\\mathcal {F}_\\Psi $ of the positive tropical variety $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ , are given by the common refinement of these fans $\\mathcal {F}_\\beta $ for $\\beta \\in \\Phi ^+$ .", "It follows from [10] that there exists an affine transformation $T: \\mathbb {R}^{Y} \\rightarrow \\mathbb {R}^{X_\\Delta \\sqcup Y}$ such that for all $\\beta \\in \\Phi ^+$ we have ${\\operatorname{Newton}\\left(p_\\beta \\right)} \\ = \\ T \\left( {\\operatorname{Newton}\\left(F_\\beta \\right)} \\right)\\ .$ Conversely, ${\\operatorname{Newton}\\left(F_\\beta \\right)}$ is obtained from ${\\operatorname{Newton}\\left(p_\\beta \\right)}$ by the coordinate projection $\\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow ~\\mathbb {R}^{Y}$ .", "Therefore each fan $\\mathcal {F}_\\beta $ is linearly isomorphic to the normal fan of ${\\operatorname{Newton}\\left(F_\\beta \\right)}$ .", "By cor:NewtonFpoly the common refinement of these normal fans is the $g$ -vector fan ${\\mathcal {F}_g}$ .", "This shows that $\\mathcal {F}_\\Psi $ is piecewise linearly isomorphic to ${\\mathcal {F}_g}$ .", "Example 4.2 ($B_2$ -example) We continue ex:B2mutmatrix2.", "We denote by $X_\\Delta =\\lbrace x_1,x_2\\rbrace $ the initial cluster variables and by $Y=\\lbrace y_1,y_2\\rbrace $ the principle coefficient variables.", "This yields the non-initial cluster variables $x_3 = x_{10_\\Delta } & = (x_2^2 + y_1)/{x_1} \\\\[2pt]x_4 = x_{11_\\Delta } & = (x_1 y_1 y_2 + x_2^2 + y_1)/{x_1 x_2} \\\\[2pt]x_5 = x_{12_\\Delta } & = (x_1^2 y_1 y_2^2 + 2 x_1 y_1 y_2 + x_2^2 + y_1)/{x_1 x_2^2} \\\\[2pt]x_6 = x_{01_\\Delta } & = (x_1 y_2 + 1)/{x_2}\\multicolumn{2}{l}{\\text{as given in the above example.The piecewise linear map $\\operatorname{Trop}\\Psi : \\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow \\mathbb {R}^{X \\sqcup Y}$ has non-trivial coordinate functions}}\\\\\\operatorname{Trop}\\Psi _{10_\\Delta } & = \\max (2 x_2 \\, , \\, y_1) - x_1 \\\\\\operatorname{Trop}\\Psi _{11_\\Delta } & = \\max (x_1 + y_1 + y_2 \\, , \\, 2 x_2 \\, , \\, y_1) - x_1 - x_2 \\\\\\operatorname{Trop}\\Psi _{12_\\Delta } & = \\max (2 x_1 + y_1 + 2 y_2 \\, , \\, x_1 + y_1 + y_2 \\, , \\, 2 x_2 \\, , \\, y_1) - x_1 - 2 x_2 \\\\\\operatorname{Trop}\\Psi _{01_\\Delta } & = \\max (x_1 + y_2 \\, , \\, 0) - x_2 .$ The domains of linearity of $\\operatorname{Trop}\\Psi $ define a complete four-dimensional polyhedral fan $\\mathcal {F}_{\\Psi }$ in $\\mathbb {R}^{X_\\Delta \\sqcup Y} = \\mathbb {R}^4$ with two-dimensional lineality space.", "By intersecting $\\mathcal {F}_{\\Psi }$ with the coordinate plane $\\mathbb {R}^{Y}$ by setting $x_1=x_2=0$ we obtain an essential 2-dimensional fan.", "This fan is the the common coarsening of the normal fans of the $F$ -polynomials $F_{10_\\Delta }$ , $F_{11_\\Delta }$ , $F_{12_\\Delta }$ , $F_{01_\\Delta }$ , see fig:B2gfan,,fig:B2trop.", "The normal fans are depicted in the dual basis of the root basis, known as the coweight basis, which in this case is given as $\\nabla ^\\vee =\\lbrace \\omega _1^\\vee ,\\ \\omega _2^\\vee \\rbrace =\\left\\lbrace \\tfrac{1}{2}(10),\\ \\tfrac{1}{2}(11)\\right\\rbrace .$ poly = [fill=blue!25, line width=1.2pt, opacity=0.75, draw=blue!40] segment = [draw=blue!40, line width=1.2pt, line cap=round] vertex = [fill=blue!50, draw=blue!50, line width=1.2pt] fan = [fill=red!20, opacity=0.5] ray = [draw=red!50, line width=1.4pt, line cap=round] coords = [draw=black!60,->,>=latex,line width=0.7pt, line cap=round] Figure: The gg-vector fan of type B 2 B_2 (left) is the common refinement of the domains of linearity of the coordinate functions of TropΨ\\operatorname{Trop}\\Psi from ex:B2trop after intersecting with the (y 1 ,y 2 )(y_1,y_2)-plane." ], [ "Newton polytopes of $F$ -polynomials from subword complexes", "Let $\\mathcal {A}({\\sf M}_{\\rm c})$ be the finite type cluster algebra with acyclic initial mutation matrix ${\\sf M}_{\\rm c}$ with principal coefficients and denote by ${\\mathcal {F}_g}({\\sf M}_{\\rm c})$ its $g$ -vector fan.", "We have seen in thm:indecompcolumns that the type cone $\\mathbb {TC}({\\mathcal {F}_g}({\\sf M}_{\\rm c}))$ is generated by the natural Minkowski summands of the brick polytope $\\operatorname{{{{\\sf Asso}\\left({\\sf M}_c\\right)}}}$ , $\\mathbb {TC}\\big ( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) \\big ) \\ = \\ \\operatorname{cone}\\left\\lbrace \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace .$ A description of the generators of $\\mathbb {TC}\\big ( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) \\big )$ was also obtained by combining results from [2], [1] and [16] as follows.", "In [2] the authors provide polytopal realizations of ${\\mathcal {F}_g}( {\\sf M}_{\\rm c})$ .", "This construction produces a generalized associahedron $\\mathcal {X}_p$ for each $p \\in \\mathbb {R}_{ > 0 }^{ \\Phi ^+}$ .", "It was then shown in [2] (simply-laced types) and in [1] (multiply-laced types) that $\\mathcal {X}_p$ for $p = e_\\beta $ and $\\beta \\in \\Phi ^+$ equals the Newton polytope of the $F$ -polynomial $F_\\beta $ .", "In [16], the authors explain that within the latter constuction $\\mathbb {R}_{> 0}^{\\Phi ^+}$ can be understood as (a linear transformation of) the type cone $\\mathbb {TC}\\big ( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) \\big )$ .", "In particular, this establishes the fact that the Newton polytopes of the $F$ -polynomials generate the type cone, $\\mathbb {TC}( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) ) \\ = \\ \\operatorname{cone}\\left\\lbrace {\\operatorname{Newton}\\left(F_\\beta \\right)} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace .$ In order to prove cor:NewtonFpoly, it remains to properly identify which Newton polytope of an $F$ -polynomial corresponds to which Minkowski summand of the brick polytope.", "This is done using the following property of $F$ -polynomials.", "Proposition 3.1 ([7] (simply-laced types), [6] (multiply-laced types)) For every $\\beta \\in \\Phi ^+$ , the $F$ -polynomial $F_\\beta = F_\\beta (\\mathbf {y})$ has constant term 1 and a unique componentwise highest exponent vector given by $\\beta $ .", "In particular, $\\mathbf {0}$ and $\\beta $ are both vertices of ${\\operatorname{Newton}\\left(F_\\beta \\right)}$ .", "For simply-laced types, this is [7] and for multiply-laced types, this is [6].", "This proposition can be rechecked in types $A_3$ and $B_2$ in fig:examples.", "Now we are ready to proof our second main result.", "Since $\\mathbb {TC}\\big ( {\\mathcal {F}_g}( {\\sf M}_{\\rm c}) \\big )$ is a simplicial cone of dimension $N=| \\Phi ^+|$ we already know that the two sets of generators, $\\left\\lbrace {\\operatorname{Newton}\\left( F_\\beta \\right)} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace \\ \\text{and} \\ \\left\\lbrace \\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}} \\;\\big |\\; \\beta \\in \\Phi ^+ \\right\\rbrace ,$ are non-redundant and coincide up to scalar factors.", "Let $\\beta \\in \\Phi ^+$ .", "By lem:highestandlowestexponent the unique maximal and minimal vertices of ${\\operatorname{Newton}\\left(F_\\beta \\right)}$ are $\\beta $ and $\\mathbf {0}$ , respectively.", "Since $\\beta = {\\sf b}_{\\lbrace \\beta \\rbrace }( I_{\\operatorname{g}})$ and $\\mathbf {0} = {\\sf b}_{\\lbrace \\beta \\rbrace }( I_{\\operatorname{ag}})$ , these vectors are by cor:vertsofsubsum vertices of $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ as well.", "Applying lem:weightsunderflips we see that they are the maximal and minimal vertices of $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ , respectively.", "Thus the polytopes ${\\operatorname{Newton}\\left( F_\\beta \\right)}$ and $\\operatorname{{{{\\sf Asso}_{\\beta }\\left({\\sf M}_c\\right)}}}$ coincide." ], [ "The tropical positive cluster variety", "In this section, we prove thm:tropplusvariety starting from the type cone description (REF ) on page REF in terms of Newton polytopes of $F$ -polynomials.", "It is independent of the subword complex description and does not make use of it.", "We again emphasize that a more general version of thm:tropplusvariety follows from [1].", "Following [19], we start with the needed notions from tropical geometry.", "Let $E \\subset \\mathbb {Z}_{\\ge 0}^d$ be non-empty and finite and let $f=\\sum _{e \\in E} f_e \\mathbf {u}^e \\in \\mathbb {Q}[\\mathbf {u}]$ with $f_e \\ne 0$ for all $e \\in E$ be a rational polynomial supported on $E$ .", "For each weight $w\\in \\mathbb {R}^d$ we define $E(w) \\ = \\ \\underset{e \\in E}{\\arg \\max } (w\\cdot e)\\ .$ That is, $E(w)$ is the intersection of $E$ with the face of ${\\operatorname{Newton}\\left(f\\right)}=\\operatorname{conv}(E)$ that is maximized in direction $w$ .", "The tropical hypersurface $\\operatorname{Trop}(f) \\subset \\mathbb {R}^d$ is the collection of those weights $w\\in \\mathbb {R}^d$ for which $E(w)$ contains at least two elements.", "$\\operatorname{Trop}(f)$ naturally carries the structure of a polyhedral fan, whose cones are formed by those weights $w\\in \\operatorname{Trop}(f)$ that yield the same $E(w)$ .", "This fan thus agrees with the codimension-one skeleton of the normal fan of ${\\operatorname{Newton}\\left(f\\right)}$ .", "The positive part $\\operatorname{Trop^+}(f)$ of the tropical hypersurface was introduced in [19] and is defined as follows.", "Split $E=E^+_f \\sqcup E^-_f$ according to the signs of the coefficients of $f$ .", "That is, $E^+_f = \\lbrace e \\in E \\mid f_e > 0 \\rbrace , \\quad E^-_f = \\lbrace e \\in E \\mid f_e < 0 \\rbrace .$ Now $\\operatorname{Trop^+}(f)$ is defined as the subfan of $\\operatorname{Trop}(f)$ consisting of those weights for which neither $E(w)\\cap E^+_f$ nor $E(w)\\cap E^-_f$ is empty, $\\operatorname{Trop^+}(f) = \\left\\lbrace w\\in \\mathbb {R}^d \\;\\big |\\; E(w)\\cap E^+_f \\ne \\emptyset ~ \\text{ and } ~ E(w)\\cap E^-_f \\ne \\emptyset \\right\\rbrace \\ .$ For any ideal $\\mathcal {I}\\subset \\mathbb {Q}[\\mathbf {u}]$ the positive tropical variety $\\operatorname{Trop^+}(\\mathcal {I})$ is defined as the intersection of all positive tropical hypersurfaces $\\operatorname{Trop^+}(f)$ for $f\\in \\mathcal {I}$ .", "We next move to the positive tropical variety considered here.", "Let $\\mathcal {A}({\\sf M})$ be a finite type cluster algebra of rank $n$ with (not necessarily acyclic) initial mutation matrix ${\\sf M}$ with principal coefficients.", "We denote by $X_\\Delta =\\lbrace x_1,\\ldots ,x_n\\rbrace $ the set of initial cluster variables and by $X_{\\Phi ^+}=\\left\\lbrace x_\\beta \\;|\\; \\beta \\in \\Phi ^+ \\right\\rbrace $ the set of non-initial cluster variables.", "Thus the set of all cluster variables is the disjoint union $X=X_\\Delta \\sqcup X_{\\Phi ^+}$ .", "Furthermore, let $Y=\\lbrace y_1,\\ldots ,y_n\\rbrace $ be the set of principle coefficient variables.", "Recall that each non-initial cluster variable $x_\\beta \\in X_{\\Phi ^+}$ is expressed in terms of the initial seed by $x_\\beta = \\frac{p_\\beta (\\mathbf {x},\\mathbf {y})}{\\mathbf {x}^\\beta }$ where $p_\\beta (\\mathbf {x},\\mathbf {y})$ is a subtraction-free polynomial in the initial cluster and coefficient variables and $\\mathbf {x}^\\beta = x_1^{\\beta _1}\\dots x_n^{\\beta _n}$ for $\\beta = (\\beta _1,\\cdots ,\\beta _n)_\\Delta \\in \\mathbb {R}^\\Delta $ .", "Following [19] we embed $\\operatorname{Spec}\\mathcal {A}({\\sf M})$ as the affine variety $V(\\mathcal {I}_{{\\sf M}}) \\subset \\mathbb {Q}^{X\\sqcup Y}$ , where $\\mathcal {I}_{{\\sf M}}$ is the ideal generated by the non-initial cluster variables, i.e., $\\mathcal {I}_{{\\sf M}} \\ = \\ \\left\\langle x_\\beta \\cdot \\mathbf {x}^\\beta - p_\\beta (\\mathbf {x},\\mathbf {y}) ~ \\big | ~ \\beta \\in \\Phi ^+\\right\\rangle .$ Note that in this case the special form of the generators immediately yields a subtraction-free parametrization $\\Psi : (\\mathbb {Q}^\\ast )^{X_\\Delta \\sqcup Y} \\rightarrow V(\\mathcal {I}_{{\\sf M}})\\cap (\\mathbb {Q}^\\ast )^{X\\sqcup Y}$ given as the graph of the map $(\\mathbb {Q}^\\ast )^{X_\\Delta \\sqcup Y} \\ &\\longrightarrow \\ (\\mathbb {Q}^\\ast )^{X_{\\Phi ^+}} \\hspace{5.0pt}, \\\\(\\mathbf {x},\\mathbf {y}) \\ &\\longmapsto \\ \\left( \\frac{p_\\beta (\\mathbf {x},\\mathbf {y})}{\\mathbf {x}^\\beta } \\right)_{\\beta \\in \\Phi ^+}$ We denote by $\\operatorname{Trop}\\Psi : \\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow \\mathbb {R}^{X \\sqcup Y}$ the tropicalization of the map $\\Psi $ .", "This is the piecewise linear map obtained by replacing every $\\times $ in $\\Psi $ with a $+$ , every $/$ with a $−$ and every $+$ with a $\\max $ .", "The following result is an immediate consequence of [19].", "Proposition 4.1 The map $\\operatorname{Trop}\\Psi : \\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow \\mathbb {R}^{X \\sqcup Y}$ is a piecewise linear parametrization of the positive tropical variety $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ .", "It should be mentioned here that in [19] the authors are working over the field of complex Puiseux series, one of the prototype examples of a field with non-trivial valuation.", "This is standard in tropical geometry since it reveals strong connections between classical algebraic geometry and tropical geometry.", "The ideal $\\mathcal {I}_{{\\sf M}}$ in cor:paramtropplus is understood over the complex Puiseux series and for the definition of a positive tropical hypersurface for a complex Puiseux polynomial we refer to [19].", "However, the map $\\operatorname{Trop}\\Psi $ stays unchanged when working over $\\mathbb {Q}$ .", "The domains of linearity of $\\operatorname{Trop}\\Psi $ form a polyhedral fan in $\\mathbb {R}^{X_\\Delta \\sqcup Y}$ , which we denote by $\\mathcal {F}_{\\Psi }$ .", "Following [19] we equip $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ with the fan structure obtained by applying $\\operatorname{Trop}\\Psi $ to $\\mathcal {F}_{\\Psi }$ .", "Whenever we refer to $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ as a polyhedral fan we consider this fan structure.", "By cor:paramtropplus the map $\\operatorname{Trop}\\Psi : \\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow \\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ is a piecewise linear parametrization of $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ .", "Moreover, $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ is piecewise linearly isomorphic to $\\mathcal {F}_\\Psi $ by construction .", "For each $\\beta \\in \\Phi ^+$ denote by $\\mathcal {F}_\\beta $ the normal fan of ${\\operatorname{Newton}\\left(p_\\beta \\right)}$ .", "The domains of linearity of the coordinate function $\\operatorname{Trop}\\Psi _\\beta (\\mathbf {x},\\mathbf {y}) = \\operatorname{Trop}(p_\\beta (\\mathbf {x},\\mathbf {y}) / \\mathbf {x}^\\beta )$ are the maximal cones of $\\mathcal {F}_\\beta $ .", "Thus the domains of linearity of $\\operatorname{Trop}\\Psi $ , and hence the fan structure $\\mathcal {F}_\\Psi $ of the positive tropical variety $\\operatorname{Trop^+}(\\mathcal {I}_{{\\sf M}})$ , are given by the common refinement of these fans $\\mathcal {F}_\\beta $ for $\\beta \\in \\Phi ^+$ .", "It follows from [10] that there exists an affine transformation $T: \\mathbb {R}^{Y} \\rightarrow \\mathbb {R}^{X_\\Delta \\sqcup Y}$ such that for all $\\beta \\in \\Phi ^+$ we have ${\\operatorname{Newton}\\left(p_\\beta \\right)} \\ = \\ T \\left( {\\operatorname{Newton}\\left(F_\\beta \\right)} \\right)\\ .$ Conversely, ${\\operatorname{Newton}\\left(F_\\beta \\right)}$ is obtained from ${\\operatorname{Newton}\\left(p_\\beta \\right)}$ by the coordinate projection $\\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow ~\\mathbb {R}^{Y}$ .", "Therefore each fan $\\mathcal {F}_\\beta $ is linearly isomorphic to the normal fan of ${\\operatorname{Newton}\\left(F_\\beta \\right)}$ .", "By cor:NewtonFpoly the common refinement of these normal fans is the $g$ -vector fan ${\\mathcal {F}_g}$ .", "This shows that $\\mathcal {F}_\\Psi $ is piecewise linearly isomorphic to ${\\mathcal {F}_g}$ .", "Example 4.2 ($B_2$ -example) We continue ex:B2mutmatrix2.", "We denote by $X_\\Delta =\\lbrace x_1,x_2\\rbrace $ the initial cluster variables and by $Y=\\lbrace y_1,y_2\\rbrace $ the principle coefficient variables.", "This yields the non-initial cluster variables $x_3 = x_{10_\\Delta } & = (x_2^2 + y_1)/{x_1} \\\\[2pt]x_4 = x_{11_\\Delta } & = (x_1 y_1 y_2 + x_2^2 + y_1)/{x_1 x_2} \\\\[2pt]x_5 = x_{12_\\Delta } & = (x_1^2 y_1 y_2^2 + 2 x_1 y_1 y_2 + x_2^2 + y_1)/{x_1 x_2^2} \\\\[2pt]x_6 = x_{01_\\Delta } & = (x_1 y_2 + 1)/{x_2}\\multicolumn{2}{l}{\\text{as given in the above example.The piecewise linear map $\\operatorname{Trop}\\Psi : \\mathbb {R}^{X_\\Delta \\sqcup Y} \\rightarrow \\mathbb {R}^{X \\sqcup Y}$ has non-trivial coordinate functions}}\\\\\\operatorname{Trop}\\Psi _{10_\\Delta } & = \\max (2 x_2 \\, , \\, y_1) - x_1 \\\\\\operatorname{Trop}\\Psi _{11_\\Delta } & = \\max (x_1 + y_1 + y_2 \\, , \\, 2 x_2 \\, , \\, y_1) - x_1 - x_2 \\\\\\operatorname{Trop}\\Psi _{12_\\Delta } & = \\max (2 x_1 + y_1 + 2 y_2 \\, , \\, x_1 + y_1 + y_2 \\, , \\, 2 x_2 \\, , \\, y_1) - x_1 - 2 x_2 \\\\\\operatorname{Trop}\\Psi _{01_\\Delta } & = \\max (x_1 + y_2 \\, , \\, 0) - x_2 .$ The domains of linearity of $\\operatorname{Trop}\\Psi $ define a complete four-dimensional polyhedral fan $\\mathcal {F}_{\\Psi }$ in $\\mathbb {R}^{X_\\Delta \\sqcup Y} = \\mathbb {R}^4$ with two-dimensional lineality space.", "By intersecting $\\mathcal {F}_{\\Psi }$ with the coordinate plane $\\mathbb {R}^{Y}$ by setting $x_1=x_2=0$ we obtain an essential 2-dimensional fan.", "This fan is the the common coarsening of the normal fans of the $F$ -polynomials $F_{10_\\Delta }$ , $F_{11_\\Delta }$ , $F_{12_\\Delta }$ , $F_{01_\\Delta }$ , see fig:B2gfan,,fig:B2trop.", "The normal fans are depicted in the dual basis of the root basis, known as the coweight basis, which in this case is given as $\\nabla ^\\vee =\\lbrace \\omega _1^\\vee ,\\ \\omega _2^\\vee \\rbrace =\\left\\lbrace \\tfrac{1}{2}(10),\\ \\tfrac{1}{2}(11)\\right\\rbrace .$ poly = [fill=blue!25, line width=1.2pt, opacity=0.75, draw=blue!40] segment = [draw=blue!40, line width=1.2pt, line cap=round] vertex = [fill=blue!50, draw=blue!50, line width=1.2pt] fan = [fill=red!20, opacity=0.5] ray = [draw=red!50, line width=1.4pt, line cap=round] coords = [draw=black!60,->,>=latex,line width=0.7pt, line cap=round] Figure: The gg-vector fan of type B 2 B_2 (left) is the common refinement of the domains of linearity of the coordinate functions of TropΨ\\operatorname{Trop}\\Psi from ex:B2trop after intersecting with the (y 1 ,y 2 )(y_1,y_2)-plane." ] ]

[ [ "On the index of unbalanced signed bicyclic graphs" ], [ "Abstract In this paper, we focus on the index ( largest eigenvalue) of the adjacency matrix of connected signed graphs.", "We give some general results on the index when the corresponding signed graph is perturbed.", "As applications, we determine the first five largest index among all unbalanced bicyclic graphs on n >= 36 vertices together with the corresponding extremal signed graphs whose index attain these values." ], [ "Introduction", "Given a simple graph $G=(V(G),\\ E(G))$ , let $\\sigma : E(G) \\rightarrow \\lbrace +1,\\ -1 \\rbrace $ be a mapping defined on the set $E(G)$ , then we call $\\Gamma = (G, \\sigma )$ the signed graph with underlying graph $G$ and sign function (or signature ) $\\sigma $ .", "Obviously, $G$ and $\\Gamma $ share the same set of vertices (i.e.", "$V(\\Gamma ) = V(G))$ and have equal number of edges (i.e.", "$\\mid E(\\Gamma )\\mid =\\mid E(G)\\mid $ ).", "An edge $e$ is positive (negative) if $\\sigma (e) = +1$ (resp.", "$\\sigma (e) = -1$ ).", "Actually, each concept defined for the underlying graph can be transferred with signed graphs.", "For example, the degree of a vertex $v$ in $G$ is also its degree in $\\Gamma $ .", "Furthermore, if some subgraph of the underlying graph is observed, then the sign function for the signed subgraph is the restriction of the previous one.", "Thus, if $v\\in V(G)$ , then $\\Gamma - v$ denotes the signed subgraph having $G-v$ as the underlying graph, while its signature is the restriction from $E(G)$ to $ E(G - v)$ (note, all edges incident to $v$ are deleted).", "Let $U \\subset V(G),$ then $\\Gamma [U]$ or $G(U)$ denotes the (signed) induced subgraph arising from $U$ , while $ \\Gamma - U =\\Gamma [V(G)\\backslash U]$ .", "Let $C$ be a cycle in $\\Gamma $ , the sign of $C$ is given by $\\sigma (C) =\\Pi _{e\\in C} \\sigma (e)$ .", "A cycle whose sign is $+$ (resp.$-$ ) is called positive (resp.", "negative ).", "Alternatively, we can say that a cycle is positive if it contains an even number of negative edges.", "A signed graph is balanced if all cycles are positive; otherwise it is unbalanced.", "There has been a variety of applications of balance, see [9].", "The adjacency matrix of a signed graph $\\Gamma = (G, \\sigma )$ whose vertices are $v_1, v_2,\\ldots , v_n$ is the $n\\times n$ matrix $A(\\Gamma )=(a_{ij})$ , where $a_{ij}=\\left\\lbrace \\begin{array}{lr}\\sigma (v_iv_j), & \\mbox{if }\\ v_iv_j\\in E(\\Gamma ),\\\\0, & \\mbox{otherwise}.\\end{array}\\right.$ Clearly, $A(\\Gamma )$ is real symmetric and so all its eigenvalues are real.", "The characteristic polynomial $\\det ( xI -A(\\Gamma )) $ of the adjacency matrix $A(\\Gamma )$ of a signed graph $\\Gamma $ is called the characteristic polynomial of $\\Gamma $ and is denoted by $\\phi (\\Gamma , x) $ .", "The eigenvalues of $A(\\Gamma )$ are called the eigenvalues of $\\Gamma $ .", "The largest eigenvalue is often called the index , denoted by $\\lambda (\\Gamma )$ .", "Suppose $\\theta : V(G)\\rightarrow \\lbrace +1, -1\\rbrace $ is any sign function.", "Switching by $\\theta $ means forming a new signed graph $\\Gamma ^{\\theta }=(G, \\sigma ^{\\theta })$ whose underlying graph is the same as $G$ , but whose sign function is defined on an edge $uv$ by $\\sigma ^{\\theta } (uv) = \\theta (u)\\sigma (uv)\\theta (v)$ .", "Note that switching does not change the signs or balance of the cycles of $\\Gamma $ .", "If we define a (diagonal) signature matrix $D^{\\theta }$ with $d_v=\\theta (v)$ for each $v\\in V(G),$ then $A(\\Gamma ^{\\theta }) = D^{\\theta }A(\\Gamma )D^{\\theta }$ .", "Two graphs $\\Gamma _1$ and $\\Gamma _2$ are called switching equivalent, denoted by $\\Gamma _1\\sim \\Gamma _2$ , if there exists a switching function $\\theta $ such that $\\Gamma _2=\\Gamma _1^{\\theta } $ , or equivalently $A(\\Gamma _2)=D^{\\theta }A(\\Gamma _1)D^{\\theta }$ .", "Theorem 1.1 [7] Let $\\Gamma $ be a signed graph.", "Then $\\Gamma $ is balanced if and only if $\\Gamma =(G,\\sigma )\\sim (G,+1)$ .", "Switching equivalence is a relation of equivalence, and two switching equivalent graphs have the same eigenvalues.", "In fact, the signature on bridges is not relevant, hence the edges which do not lie on some cycles are not relevant for the signature and they will be always considered as positive.", "One classical problem of graph spectra is to identify the extremal graphs with respect to the index in some given class of graphs.", "For signed graphs, since all signatures of a given tree are equivalent, the first non-trivial signature arises for unicyclic graphs, which was considered in [1].", "The authors determined signed graphs achieving the minimal or the maximal index in the class of unbalanced unicyclic graphs of order $n\\ge 3$ .", "In [5], the authors characterized the unicyclic signed graphs of order $n$ with nullity $n-2$ , $n- 3$ , $n- 4$ , $n- 5$ respectively.", "For the energy of singed graphs, see [3], [4], [6],[8],[10],[11] for details.", "Here, we will consider unbalanced bicyclic graphs, and determine the first five largest index among all unbalanced bicyclic graphs with given order $n\\ge 36$ together with the corresponding extremal signed graphs whose index attain these values.", "Here is the remainder of the paper.", "In Section 2, we study the effect of some edges moving on the index of a signed graph.", "In Section 3, we introduce the three classes of signed bicyclic graphs.", "In Section 4, we determine the first five graphs in the set of unbalanced bicyclic graphs on $n\\ge 36$ vertices, and order them according to their index in decreasing order." ], [ "Preliminaries", "The purpose of this section is to analyze how the index change when modifications are made to a signed graph.", "We start with one important tool which also works in signed graphs.", "Its general form holds for any principal submatrix of a real symmetric matrix.", "Lemma 2.1 (Interlacing theorem for signed graphs).", "Let $\\Gamma = (G, \\sigma )$ be a signed graph of order $n$ and $\\Gamma -v$ be the signed graph obtained from $\\Gamma $ by deleting the vertex $v$ .", "If $\\lambda _i$ are the (adjacency) eigenvalues, then $\\lambda _1(\\Gamma )\\ge \\lambda _1(\\Gamma -v)\\ge \\lambda _2(\\Gamma )\\ge \\lambda _2(\\Gamma -v)\\ge \\ldots \\ge \\lambda _{n-1}(\\Gamma -v)\\ge \\lambda _n(\\Gamma ).", "$ Lemma 2.2 Let $\\Gamma $ be a signed graph with cut edge $uv$ , and ${\\bf x}$ be an eigenvector corresponding to the index $\\lambda (\\Gamma )$ .", "We have $\\sigma (uv)x_{u}x_v\\ge 0$ .", "Proof.", "Without loss of generality, we assume that ${\\bf x}$ is unit and $\\sigma (uv)>0$ .", "By way of contradiction, we suppose that $ x_{u}x_v<0$ .", "Let $\\Gamma _1$ and $\\Gamma _2 $ be the two connected components of $\\Gamma -uv$ , respectively.", "Set ${\\bf x}=\\left( \\begin{matrix}\\end{matrix}{\\bf x}_1\\\\\\right.", "{\\bf x}_2$ )$, where $ x1$ and $ x2$ are the subvectors of $x$ indexed by vertices in $ 1$ and $ 2$, respectively.", "Let $y=( -x1 x2 )$, then $yTA()y-xTA()x=-4 xuxv> 0$, which contradicts to the fact that $x$ maximizes the Rayleigh quotient.\\hfill $ $\\vspace{0.0pt}$ From the above lemma, it is straightforward to derive the following result.", "Corollary 2.1 Let $T$ be a vertex induced subtree in the signed graph $\\Gamma $ , and ${\\bf x}$ be an eigenvector corresponding to the index $\\lambda (\\Gamma )$ .", "Then for any edge $uv$ of $T$ , we have $\\sigma (uv)x_{u}x_v\\ge 0$ .", "Remark 1 If $T$ is a vertex induced subtree with root $v$ in signed graph $\\Gamma $ , the above corollary implies that if $x_v\\ge 0$ we can assume that all edges in $T$ are positive and all vertices of $T$ have non-negative coordinates in ${\\bf x}$ .", "This is valid because we can prove it by using switching equivalent from the leaves of the rooted subtree.", "We proceed by considering how the index change when cut edges be moved.", "Lemma 2.3 Let $u$ , $v$ be two vertices of the signed graph $\\Gamma $ , $vv_1,\\ldots ,vv_s\\ (s\\ge 1)$ be cut edges of $\\Gamma $ , and ${\\bf x}$ be an eigenvector corresponding to $\\lambda (\\Gamma )$ .", "Let $\\Gamma ^{\\prime }=\\Gamma -vv_1-\\ldots -vv_s+uv_1+\\ldots +uv_s.$ If $x_u \\ge x_v \\ge 0$ or $x_u \\le x_v \\le 0$ , we have $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ .", "Proof.", "Without loss of generality, we assume that ${\\bf x}$ is unit.", "Due to the Rayleigh quotient, we have $\\lambda (\\Gamma ^{\\prime })-\\lambda (\\Gamma )\\ge {\\bf x}^T A(\\Gamma ^{\\prime }) {\\bf x}-{\\bf x}^T A(\\Gamma ) {\\bf x}=(x_u-x_v) \\sum \\limits _{i=1}^s \\sigma (vv_i)x_{v_i}.$ Lemma REF tell us that $\\sigma (vv_i)x_{v_i}x_{v}\\ge 0$ , one can quickly verify that $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ when $x_u \\ge x_v \\ge 0$ or $x_u \\le x_v \\le 0$ .$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ If $vv_1,\\ldots ,vv_s $ are pendant edges in the above lemma, the eigenvalue equation leads to $\\lambda (\\Gamma ) x_{v_i} =\\sigma (vv_i) x_v$ , which implies that $\\sigma (vv_i) x_v x_{v_i}>0$ when $x_v\\ne 0$ , so we can get a stronger version of the above result.", "Lemma 2.4 Let $u$ , $v$ be two vertices of signed graph $\\Gamma $ , $vv_1,\\ldots ,vv_s\\ (s\\ge 1)$ be pendant edges of $\\Gamma $ , and ${\\bf x}$ be an eigenvector corresponding to $\\lambda (\\Gamma )$ .", "Let $\\Gamma ^{\\prime }=\\Gamma -vv_1-\\ldots -vv_s+uv_1+\\ldots +uv_s.$ If $x_u \\ge x_v \\ge 0$ or $x_u \\le x_v \\le 0$ , we have $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ .", "Furthermore, if $x_u > x_v > 0$ or $x_u < x_v < 0$ , then $\\lambda (\\Gamma ^{\\prime })> \\lambda (\\Gamma )$ .", "In Lemma REF and Lemma REF , the edges be moved are all cut edges.", "Now the perturbation, $\\alpha $ -transform, described in the following can be seen in many books and many other papers, which can move non-cut edges from one vertex to another.", "Definition 2.1 Let $\\Gamma $ be a connected signed graph, $uv$ be a non-pendant edge of $\\Gamma $ which is not in any triangle.", "Let $N_{\\Gamma }(v)\\backslash \\lbrace u\\rbrace =\\lbrace v_1,\\cdots , v_d\\rbrace $ with $ d\\ge 1$ .", "The signed graph $\\Gamma ^{\\prime }=\\alpha (\\Gamma ,uv)=\\Gamma -vv_1-vv_2-\\cdots -vv_d+uv_1+uv_2+\\cdots +uv_d.$ We say that $\\Gamma ^{\\prime }$ is an $\\alpha $ -transform of $\\Gamma $ on the edge $uv$ .", "All edges retain the sign they have after $\\alpha $ -transform.", "In the next, we focus on how the index changes after $\\alpha $ -transform.", "Lemma 2.5 Let $uv$ be an edge of signed graph $\\Gamma $ , and $\\Gamma ^{\\prime }=\\alpha (\\Gamma , uv)$ be the graph obtained from $\\Gamma $ by $\\alpha $ -transform on the edge $uv$ .", "Let $\\bf {x}$ be an eigenvector corresponding to $\\lambda (\\Gamma )$ .", "If one of the following condition holds, we have $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ : if $\\sigma (uv)>0$ , and $x_v\\le x_u\\le \\lambda (\\Gamma )x_v$ , if $\\sigma (uv)<0$ and $x_u\\ge 0,\\ x_v\\ge 0$ .", "Furthermore, if one of the following can be satisfied: if $\\sigma (uv)>0$ , and $x_v< x_u< \\lambda (\\Gamma )x_v$ , if $\\sigma (uv)<0$ , and $x_u>0,\\ x_v> 0$ and $x_u\\ne x_v$ , we have $\\lambda (\\Gamma ^{\\prime })> \\lambda (\\Gamma )$ .", "Proof.", "Let $N_{\\Gamma }(u)\\backslash \\lbrace v\\rbrace =\\lbrace u_1,\\ldots ,u_r\\rbrace $ and $N_{\\Gamma }(v)\\backslash \\lbrace u\\rbrace =\\lbrace v_1,\\ldots ,v_s\\rbrace $ .", "The eigenvalue equation leads to the relation $\\lambda (\\Gamma )x_v=\\sigma (uv)x_u+\\sum \\limits _{v_i\\in N_{\\Gamma }(v)\\backslash \\lbrace u\\rbrace } \\sigma (vv_i)x_{v_i},$ $\\lambda (\\Gamma )x_u=\\sigma (uv)x_v+\\sum \\limits _{u_i\\in N_{\\Gamma }(u)\\backslash \\lbrace v\\rbrace } \\sigma (uu_i)x_{u_i}.$ These then easily imply that $ \\lambda (\\Gamma ^{\\prime })-\\lambda (\\Gamma )&\\ge {\\bf x}^T A(\\Gamma ^{\\prime }) {\\bf x}-{\\bf x}^T A(\\Gamma ) {\\bf x}=(x_u-x_v)\\sum \\limits _{v_i\\in N_G(u)\\backslash \\lbrace v\\rbrace } \\sigma (vv_i)x_{v_i}\\\\&=(x_u-x_v)(\\lambda (\\Gamma )x_v-\\sigma (uv)x_u),$ and $\\lambda (\\Gamma ^{\\prime })-\\lambda (\\Gamma )&\\ge {\\bf x}^T A(\\Gamma ^{\\prime }) {\\bf x}-{\\bf x}^T A(\\Gamma ) {\\bf x}=(x_v-x_u)\\sum \\limits _{u_i\\in N_G(v)\\backslash \\lbrace u\\rbrace } \\sigma (uu_i)x_{u_i}\\\\&= (x_v-x_u)(\\lambda (\\Gamma )x_u-\\sigma (uv)x_v).$ So that if $\\sigma (uv)>0$ , applying (3), we estimate that $\\lambda (\\Gamma ^{\\prime })-\\lambda (\\Gamma )\\ge (x_u-x_v)(\\lambda (\\Gamma )x_v-x_u).", "$ Thus, $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ when $x_v\\le x_u\\le \\lambda (\\Gamma )x_v$ , the inequality is strict when $x_v< x_u< \\lambda (\\Gamma )x_v$ .", "If $\\sigma (uv)<0$ , it seems more complicated.", "By (5), we know $\\lambda (\\Gamma ^{\\prime })-\\lambda (\\Gamma )\\ge (x_u-x_v)(\\lambda (\\Gamma )x_v+x_u).$ The symmetry tell us that we also have $\\lambda (\\Gamma ^{\\prime })-\\lambda (\\Gamma )\\ge (x_v-x_u)(\\lambda (\\Gamma )x_u+x_v).$ Therefore, if $x_u\\ge 0,\\ x_v\\ge 0$ , then $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ whenever $x_u\\ge v_v$ or $x_u<x_v$ , and the inequality is strict when $x_u>0,\\ x_v> 0$ and $x_u\\ne x_v$ .$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ In all figures, solid and dotted edges represent positive and negative edges, respectively.", "Figure: The example Γ\\Gamma in Remark 2Remark 2 The conditions in Lemma REF are necessary.", "For example, the signed graph $\\Gamma $ (as shown in Figure REF ) with index $\\lambda (\\Gamma ) \\approx 2.214$ , its positive edge $v_2v_3$ does not satisfy the condition in Lemma REF .", "If we let $\\Gamma ^{\\prime }=\\alpha (\\Gamma , v_2v_3)$ , then the index $\\lambda (\\Gamma ^{\\prime })=2$ is less than $\\lambda (\\Gamma ^{\\prime })$ .", "However, in Lemma REF , if $uv$ is a cut edge, things are easier.", "Corollary 2.2 Let $uv$ be a cut edge of signed graph $\\Gamma $ , and $\\Gamma ^{\\prime }=\\alpha (\\Gamma , uv)$ .", "We have $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ .", "Proof.", "Suppose, without loss of generality, that $uv$ is positive.", "Let ${\\bf x}$ be an unit eigenvector corresponding to $\\lambda (\\Gamma )$ .", "By Lemma REF , we can assume that $x_u\\ge x_v\\ge 0$ .", "Let $N_{\\Gamma }(v)\\setminus \\lbrace u\\rbrace =\\lbrace v_1,\\ldots ,v_s\\rbrace $ .", "The eigenvalue equation leads to the relation $\\lambda (\\Gamma )x_v= x_u+\\sum \\limits _{v_i\\in N_{\\Gamma }(v)\\setminus \\lbrace u\\rbrace } \\sigma (vv_i)x_{v_i}.$ We claim that $\\sum \\limits _{v_i\\in N_{\\Gamma }(v)\\setminus \\lbrace u\\rbrace } \\sigma (vv_i)x_{v_i}\\ge 0$ .", "Otherwise, we write the component of $\\Gamma -uv$ containing the vertex $v$ as $U$ .", "Set ${\\bf x}=\\left( \\begin{matrix}\\end{matrix}{\\bf x}_1\\\\\\right.", "{\\bf x}_2$ )$, where $ x1$ is the subvectors of $x$ indexed by vertices in $ U-v$.", "Let $y=( -x1 x2 )$, then $yTA()y-xTA()x=-4 xvviN(v){u} (vvi)xvi > 0$, which contradicts to the fact that $x$ maximizes the Rayleigh quotient.$ Since $\\sum \\limits _{v_i\\in N_{\\Gamma }(v)\\setminus \\lbrace u\\rbrace } \\sigma (vv_i)x_{v_i}\\ge 0$ , we have $x_u\\le \\lambda (\\Gamma ) x_v$ .", "By gluing together this inequality with $x_u\\ge x_v$ and Lemma REF , we get the assertion.$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ The above lemma tell us that if $T$ is a vertex induced subtree of signed graph with root $v$ , then $\\alpha $ -transform on any edge in $T$ will not decrease the index of the signed graph.", "Thus, replacing $T$ with a star ( with center $v$ and order $\\mid V(T)\\mid $ ) will not decrease the index as well.", "We recall from [2] the following Schwenk's formulas Lemma 2.6 Let $v$ be a vertex of signed graph $\\Gamma $ , $\\Phi (\\Gamma ,x)=x\\Phi (\\Gamma -v,x)-\\sum \\limits _{uv\\in E(\\Gamma )}\\Phi (\\Gamma -u-v,x)-2\\sum \\limits _{C\\in \\mathcal {C}_v}\\sigma (C)\\Phi (\\Gamma -C,x),$ where $ \\mathcal {C}_v$ is the set of signed cycles passing through $v$ , and $\\Gamma -C$ is the signed graph obtained from $\\Gamma $ by deleting $C$ ." ], [ "Three classes of signed bicyclic graphs", "A graph $G$ of order $n$ is called a bicyclic graph if $G$ is connected and the number of edges of $G$ is $n+1$ .", "A signed graph whose underlying graph is a bicyclic graph, we call it signed bicyclic graph.", "It is easy to see from the definition that $G$ is a bicyclic graph if and only if $G$ can be obtained from a tree $T$ (with the same order) by adding two new edges to $T$ .", "Let $G$ be a bicyclic graph.", "The base of bicyclic graph $G$ , denoted by $\\widehat{G}$ , is the (unique) minimal bicyclic subgraph of $G$ .", "If $\\Gamma =(G,\\sigma )$ , then we define $\\widehat{\\Gamma }=(\\widehat{G},\\sigma )$ as the base of signed bicyclic graph $\\Gamma $ .", "It is easy to see that $\\widehat{G}$ is the unique bicyclic subgraph of $G$ containing no pendant vertices, while $G$ can be obtained from $\\widehat{G}$ by attaching trees to some vertices of $\\widehat{G}$ .", "It is well-known that there are the following three types of bicyclic graphs containing no pendant vertices: Let $B(p,q)\\ (p\\ge q\\ge 3)$ be the bicyclic graph obtained from two vertex-disjoint cycles $C_p$ and $C_q$ by identifying vertices $u$ of $C_p$ and $v$ of $C_q$ (see Fig.", "2.1).", "This type of graph is also known as the infinity graph.", "Let $B(p,\\ell ,q)$ be the bicyclic graph obtained from two vertex-disjoint cycles $C_p$ and $C_q$ by joining vertices $u$ of $C_p$ and $v$ of $C_q$ by a new path $uu_1u_2\\cdots u_{\\ell -1}v$ with length $\\ell \\ (\\ell \\ge 1)$ (see Figure REF ).", "This type of graph is also known as the dumbbell graph; if the cycles are triangles, it also takes the name of hourglass graph.", "Figure: B(p,q)B(p, q) and B(p,ℓ,q)B(p, \\ell , q)Let $B(P_k,P_{\\ell },P_m)\\ (1\\le m\\le \\min \\lbrace k,\\ell \\rbrace )$ be the bicyclic graph obtained from three pairwise internal disjoint paths form a vertex $x$ to a vertex $y$ .", "These three paths are $ xv_1v_2\\cdots ,\\ v_{k-1}y$ with length $k$ , $ xu_1u_2\\cdots ,\\ u_{\\ell -1}y$ with length $\\ell $ and $xw_1w_2\\cdots ,\\ w_{m-1}y$ with length $m$ (see Figure REF ).", "This type of graph is also known as the $\\theta $ -graph.", "Figure: B(P k ,P ℓ ,P m )B(P_k,P_{\\ell },P_m)Accordingly, we denoted by $\\mathcal {B}_n$ the set of all unbalanced signed bicyclic graphs of order $n$ .", "We are now ready to describe the class of unbalanced signed bicyclic graphs.", "$\\mathcal {B}_n (p,q) =\\lbrace \\Gamma =(G,\\sigma ) \\mbox{ is \\ unbalanced} \\mid \\ \\widehat{G}=B(p,q)\\,\\,\\mbox{ for\\ some\\ } p \\ge q \\ge 3 \\rbrace , $ $ \\mathcal {B}_n (p,\\ell ,q) =\\lbrace \\Gamma =(G,\\sigma ) \\mbox{ is \\ unbalanced}\\ \\mid \\ \\widehat{G}=B(p,\\ell ,q), \\mbox{ for\\ some\\ } p \\ge 3, \\, q\\ge 3 \\,\\, \\mbox{and} \\,\\, \\ell \\ge 1\\rbrace , $ $\\mathcal {B}_n (P_k,P_{\\ell },P_m) =\\lbrace \\Gamma =(G,\\sigma ) \\mbox{ is \\ unbalanced} \\ \\mid \\ \\widehat{G}=B(P_k,P_{\\ell },P_m) \\mbox{ for\\ some\\ } 1\\le m\\le \\min \\lbrace k,l\\rbrace \\rbrace .$ It is easy to see that $\\mathcal {B}_n=\\mathcal {B}_n (p,q) \\, \\dot{\\cup } \\, \\mathcal {B}_n (p,\\ell ,q)\\, \\dot{\\cup }\\, \\mathcal {B}_n (P_k,P_{\\ell },P_m).$" ], [ "The index of unbalanced signed bicyclic graphs with given order ", "In this section, we deal with the extremal index problems for the class of unbalanced signed bicyclic graphs with order $n$ .", "We will determine the first five graphs in $\\mathcal {B}_n$ , and order them according to their index in decreasing order.", "For the unicyclic graphs, there are exactly two switching equivalent classes.", "If a unicyclic signed graph is balanced, by Theorem REF , it is switching equivalent to one with all edges positive.", "Otherwise, it is switching equivalent to one with exactly one (arbitrary) negative edge on the cycle[5].", "For unbalanced bicyclic signed graphs, we also have similar results.", "Lemma 4.1 If $\\Gamma \\in \\mathcal {B}_n (p,q) \\cup \\mathcal {B}_n (p,\\ell ,q)$ , then $\\Gamma $ is switching equivalent to one with exactly one (arbitrary) negative edge on its unbalanced cycle.", "If $\\Gamma \\in \\mathcal {B}_n (P_k,P_{\\ell },P_m)$ , then $\\Gamma $ is switching equivalent to one with exactly one (arbitrary) negative edge on its base.", "Proof.", "If $\\Gamma \\in \\mathcal {B}_n (p,q) \\cup \\mathcal {B}_n (p,\\ell ,q)$ , let $e_1$ and $e_2$ be two edges of $\\Gamma $ in different cycles, then $\\Gamma -e_1-e_2$ is a tree, which is balanced.", "So by Theorem REF , there exists a sign function $\\theta $ such that $(\\Gamma -e_1-e_2) ^{\\theta }$ consisting of positive edges.", "Returning to the graph $\\Gamma ^{\\theta }$ , the edges $e_1$ and $e_2$ must have a negative sign as switching does not change the sign of a cycle.", "If $\\Gamma \\in \\mathcal {B}_n (P_k,P_{\\ell },P_m)$ , let $e_1$ , $e_2$ and $e_3$ be the three edges of $\\Gamma $ which are incident to a common 3-degree vertex in the base.", "Similarly, $(\\Gamma -e_1-e_2) ^{\\theta }$ consisting of positive edges.", "Returning to the graph $\\Gamma ^{\\theta }$ , if exactly one of $e_1$ and $e_2$ is negative, the result follows.", "If both $e_1$ and $e_2$ are negative, then $\\Gamma $ is switching equivalent to the signed graph which has the same underlying graph as $\\Gamma $ , and just has one negative edge $e_3$ .$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ The following lemma is a starting point of our discussions.", "Lemma 4.2 Let $u_1u_2u_3u_4$ be a path in signed bicyclic graph $\\Gamma $ , and $ d_{\\hat{\\Gamma } }(u_2)=d_{\\hat{\\Gamma } }(u_3)=2$ .", "Let ${\\bf x}$ be an eigenvector corresponding to the index $\\lambda (\\Gamma )$ and $\\Gamma ^{\\prime }=\\alpha (\\Gamma ,u_2u_3)$ .", "If $x_{u_2}\\ge 0,\\ x_{u_3}\\ge 0$ , $ \\sigma (u_1u_2)x_{u_1}\\ge 0$ and $ \\sigma (u_3u_4)x_{u_4}\\ge 0$ , then $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ .", "Proof.", "From Lemma REF , it suffices to consider the case that $u_2u_3$ is a positive edge.", "If $x_{u_2}\\le x_{u_3}$ , the eigenvalue equation for the index $\\lambda (\\Gamma )$ , when restricted to the vertex $u_2$ becomes $\\lambda (\\Gamma )x_{u_2}=\\sigma (u_1u_2)x_{u_1}+\\sum \\limits _{v_i\\in N_{\\Gamma }(u_2)\\setminus \\lbrace u_1,u_3\\rbrace } \\sigma (u_2v_i)x_{v_i}+ x_{u_3}.$ The fact that $\\Gamma $ is a signed bicyclic graph and $ d_{\\hat{\\Gamma } }(u_2) =2$ imply that $u_2v_i$ is a cut edge, and then $\\sigma (u_2v_i)x_{v_i}\\ge 0$ follows from Lemma REF .", "Hence, $x_{u_3}\\le \\lambda (\\Gamma )x_{u_2} $ .", "By Lemma REF , we can get the desired result.", "Similarly, we can prove the assertion when $x_{u_2}\\ge x_{u_3}$ .$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ For convenience, we use $\\Gamma +\\widetilde{uv}$ (where $uv\\notin E(\\Gamma )$ ) to denote the signed graph obtained from $\\Gamma $ by adding a new negative edge $uv$ .", "Lemma 4.3 Let $\\Gamma $ be a $\\infty $ -type unbalanced signed bicyclic graph, and $\\hat{\\Gamma }\\notin \\mathcal {B}_n(3,3) $ , then there is some $\\infty $ -type unbalanced signed bicyclic graph $\\Gamma ^{\\prime }$ such that $|V(\\hat{\\Gamma ^{\\prime }})|<|V(\\hat{\\Gamma })|$ and $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ .", "Proof.", "By Lemma REF , we can assume that there is exactly one negative edge in an unbalanced cycle, and all edges in balanced cycle are positive.", "Let $u_1u_2\\ldots u_{g_1}$ be the unbalanced cycle of $\\Gamma $ with larger length, $u_1u_2$ be its unique negative edge, and again ${\\bf x}$ be an unit eigenvector corresponding to $\\lambda (\\Gamma )$ .", "Without loss of generality, we assume $x_{u_1}\\ge 0$ .", "If $g_1=3$ .", "Let $u_1u^{\\prime }_2\\ldots u^{\\prime }_{g_2}\\ (g_2\\ge 4)$ be another cycle of $\\Gamma $ , note that $u_1u_2u_3$ is the unbalanced cycle with larger length, and $\\hat{\\Gamma }\\notin \\mathcal {B}_n(3,3) $ , we find that $u_1u^{\\prime }_2\\ldots u^{\\prime }_{g_2}$ is balanced.", "We claim that the subvector ${\\bf x}_1$ of ${\\bf x}$ indexed by vertices in the cycle $u_1u^{\\prime }_2\\ldots u^{\\prime }_{g_2}$ is nonnegative.", "Otherwise, let ${\\bf y}$ be the vector obtained from ${\\bf x}$ by replacing all negative entries in ${\\bf x}_1$ with their absolute, then ${\\bf y}^{T}A(\\Gamma ){\\bf y}\\ge {\\bf x}^{T}A(\\Gamma ){\\bf x}$ , with equality if and only if ${\\bf y}$ is also an eigenvector of $\\lambda (\\Gamma )$ .", "Then we can either get the claim (by choosing ${\\bf x} $ as ${\\bf y}$ ) or a contradiction (contradicts to the fact that ${\\bf x}^{T}A(\\Gamma ){\\bf x}$ maximizes the Rayleigh quotient).", "Note that $g_2\\ge 4$ , we can get the desired $\\Gamma ^{\\prime }$ by using $\\alpha $ -transform on the edge $u^{\\prime }_2u^{\\prime }_3$ .", "Therefore, in the next, we assume that $g_1\\ge 4$ .", "If all non-zero elements in $\\lbrace x_{u_3},\\ x_{u_4},\\ldots , x_{u_{g_1}}\\rbrace $ have the same sign, we can get the desired unbalanced signed graph by Lemma REF .", "Now we consider the case that $\\lbrace x_{u_3},\\ldots , x_{u_{g_1}}\\rbrace $ have different signs.", "If $x_{u_2}\\ge 0,\\ x_{u_{3}}\\le 0$ , then $\\Gamma ^{\\prime }=\\Gamma -u_2u_{3}+\\widetilde{u_1u_3}$ is the desired unbalanced signed graph with unbalanced cycle $u_1u_3\\ldots u_{g_1}$ .", "If there is some edge $u_iu_{i+1}$ , where $3\\le i \\le g_1-1$ , such that $x_{u_i}\\ge 0,\\ x_{u_{i+1}}\\le 0$ , then $\\Gamma ^{\\prime }=\\Gamma -u_iu_{i+1}+u_1u_{i}$ is the desired unbalanced signed graph with unbalanced cycle $u_1u_2\\ldots u_{i}$ .", "To complete the proof, it suffices to consider the case that there is some $3\\le s \\le g_1$ such that $x_{u_2}\\le 0,\\ldots , x_{u_s}\\le 0$ and $x_{u_{s+1}}\\ge 0,\\ldots , x_{u_{g_1}}\\ge 0$ .", "If $g_1\\ge 5$ , as the larger of $s-1$ and $g-(s-1) $ is at least half of $g_1$ (which is equal to or greater than 3), so we can get the desired $\\Gamma ^{\\prime }$ by Lemma REF .", "It remains to consider the case that $g_1=4$ and $x_{u_2}\\le 0, \\ x_{u_3}\\le 0,\\ x_{u_4}\\ge 0$ .", "By using the switching equivalent, we can get a signed graph with all non-negative entries corresponding to $\\lambda (\\Gamma )$ .", "By using Lemma REF again, we can get the desired result.$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ Lemma 4.4 Let $\\Gamma $ be a $\\theta $ -type unbalanced signed bicyclic graph, and $\\hat{\\Gamma }\\notin \\mathcal {B}_n(P_1,P_2,P_2) $ , then there is some $\\theta $ -type unbalanced signed bicyclic graph $\\Gamma ^{\\prime }$ such that $|V(\\hat{\\Gamma ^{\\prime }})|<|V(\\hat{\\Gamma })|$ and $\\lambda (\\Gamma ^{\\prime })\\ge \\lambda (\\Gamma )$ .", "Proof.", "Suppose, without loss of generality, that there is just one negative edge in the base.", "Let $u_1$ be one of the 3-degree vertices of $\\hat{\\Gamma }$ , $u_1u_2$ be the unique negative edge.", "Again let ${\\bf x}$ be an unit eigenvector corresponding to $\\lambda (\\Gamma )$ with $x_{u_1}\\ge 0$ .", "If $x_{u_2}\\ge 0$ , similar to the proof of the case $g_1=3$ in Lemma REF , $\\bf {x}$ is nonnegative, we can get the desired $\\Gamma ^{\\prime }$ by using $\\alpha $ -transform.", "Consequently, if $x_{u_2}< 0$ .", "Let $u_1u_2^{\\prime }\\ldots u_p^{\\prime }u_2$ be the longest path from $u_1$ to $u_2$ .", "If there is some edge $u_i^{\\prime }u_{i+1}^{\\prime }$ such that $x_{u_i^{\\prime }}\\le 0$ , $x_{u_{i+1}^{\\prime }}\\ge 0$ , then $\\Gamma ^{\\prime }=\\Gamma -u_i^{\\prime }u_{i+1}^{\\prime }+u_2u_i^{\\prime }$ is the desired signed graph.", "If there is some edge $u_i^{\\prime }u_{i+1}^{\\prime }$ such that $x_{u_i^{\\prime }}\\ge 0$ , $x_{u_{i+1}^{\\prime }}\\le 0$ , then $\\Gamma ^{\\prime }=\\Gamma -u_i^{\\prime }u^{\\prime }_{i+1}+ \\widetilde{u_2u^{\\prime }_i}$ is the desired signed graph.", "If all non-zero entries in $x_{u^{\\prime }_2},\\ldots ,x_{u^{\\prime }_p}$ have the same sign, as before, we can set $\\Gamma ^{\\prime }=\\alpha (\\Gamma , u_2^{\\prime }u_3^{\\prime })$ .$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ Figure: Five signed graphs with maximum index in ℬ n \\mathcal { B}_nLemma 4.5 Let $\\Gamma _i \\in \\mathcal {B}_n\\ ( \\mbox{ where} \\ i=1,2,\\ldots , 5)$ be the unbalanced signed graphs as shown in Figure REF , then $\\lambda (\\Gamma _i)$ is the largest root of the equation $f_i(x)=0$ , where $f_1(x)&=x^4-nx^2+n-5, \\\\f_2(x)&=x^4-(n+1)x^2+2n-4, \\\\f_3(x)&=x^4-(n+1)x^2+4x+2n-8, \\\\f_4(x)&=x^3+x^2-(n-1)x-n+5, \\\\f_5(x)&=x^3-x^2-(n-2)x+n-4.$ Furthermore, we have $\\lambda (\\Gamma _1)>\\lambda (\\Gamma _2)>\\lambda (\\Gamma _3)>\\lambda (\\Gamma _4)>\\lambda (\\Gamma _5)$ when $n\\ge 36$ .", "Proof.", "By Lemma REF , one can get the characteristic polynomials of $\\Gamma _1, \\Gamma _2,\\Gamma _3,\\Gamma _4,\\Gamma _5$ by direct calculation, $\\Phi (\\Gamma _1,x)&=x^{n-6}(x^2-1)(x^4-nx^2+n-5), \\\\\\Phi (\\Gamma _2,x)&=x^{n-4}[x^4-(n+1)x^2+2n-4], \\\\\\Phi (\\Gamma _3,x)&=x^{n-4}[x^4-(n+1)x^2+4x+2n-8], \\\\\\Phi (\\Gamma _4,x)&=x^{n-6}(x+1)(x-1)^2[x^3+x^2-(n-1)x-n+5], \\\\\\Phi (\\Gamma _5,x)&=x^{n-5}(x+2)(x-1)[x^3-x^2-(n-2)x+n-4].$ By comparing the index of graphs and applying equations above, we have $\\Phi (\\Gamma _2,x)-\\Phi (\\Gamma _1,x)&=x^{n-6}(x^2+n-5)>0,\\\\\\Phi (\\Gamma _3,x)-\\Phi (\\Gamma _2,x)&=4x^{n-4}(x-1),\\\\\\Phi (\\Gamma _4,x)-\\Phi (\\Gamma _3,x)&=x^{n-6}(3x^2-4x-n+5).$ The Interlacing Theorem implies that $\\lambda (\\Gamma _i)>\\sqrt{n-2}>1$ for $i=2,3$ .", "It is not difficult to see that $\\Phi (\\Gamma _3,x)>\\Phi (\\Gamma _2,x)$ when $x\\ge \\lambda (\\Gamma _2)$ and $\\Phi (\\Gamma _4,x)>\\Phi (\\Gamma _3,x)$ when $x\\ge \\lambda (\\Gamma _3)$ .", "These are exactly what we need here, $\\lambda (\\Gamma _1)>\\lambda (\\Gamma _2)>\\lambda (\\Gamma _3)>\\lambda (\\Gamma _4)$ .", "To compare $\\lambda (\\Gamma _4)$ and $\\lambda (\\Gamma _5)$ , we let $f_4(x)&=x^3+x^2-(n-1)x-n+5,\\\\f_5(x)&=x^3-x^2-(n-2)x+n-4.$ Then $g(x)=f_4(x)-f_5(x)=2x^2-x-2n+9$ has the largest root $\\frac{1+\\sqrt{16n-71}}{4}$ .", "One can check directly $f_5(-\\infty )<0$ , $f_5(0)=n-4>0$ , $f_5(1)=-2<0$ and $f_5(\\frac{1+\\sqrt{16n-71}}{4})>0$ when $n\\ge 36$ .", "Hence, the largest root of $f_5(x)=0$ is less than $\\frac{1+\\sqrt{16n-71}}{4}$ , which implies that $f_4(x)<0$ when $x$ is the largest root of $f_5(x)=0$ .", "Therefore, we have $\\lambda (\\Gamma _4)>\\lambda (\\Gamma _5)$ .", "This completes the proof.$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ Figure: Signed graphs considered in the proof of Lemma and Lemma Lemma 4.6 If $\\Gamma \\in \\mathcal {B}_n$ is an $\\infty $ -type graph and is not switching equivalent to $\\Gamma _1,$ or $ \\Gamma _4$ , then $\\lambda (\\Gamma )< \\lambda (\\Gamma _5)$ .", "Proof.", "By Lemma REF , Lemma REF and Corollary REF , it is not difficult to see that, we only need to prove that if $\\Gamma \\in \\lbrace \\Gamma _4, \\ \\Gamma _6,\\ \\Gamma _7,\\ \\Gamma _8,\\ \\Gamma _1^i,\\ \\Gamma _4^j\\rbrace $ , where $1\\le i\\le 6$ and $1\\le j\\le 4$ ( as shown in Figure REF ).", "By direct computation, we can prove that $\\lambda (\\Gamma _5)>\\lambda (\\Gamma _6)=\\max \\lbrace \\lambda (\\Gamma _6),\\lambda (\\Gamma _7), \\lambda (\\Gamma _8)\\rbrace ,$ $\\lambda (\\Gamma _5)>\\lambda (\\Gamma _1^1)=\\max \\lbrace \\lambda (\\Gamma _1^1),\\ldots , \\lambda (\\Gamma _1^6)\\rbrace , $ and $\\lambda (\\Gamma _5)>\\lambda (\\Gamma _4^3)>\\lambda (\\Gamma _4^4),\\ \\lambda (\\Gamma _5)>\\lambda (\\Gamma _4^1)>\\lambda (\\Gamma _4^2).", "$ Hence, we can get the desired result.$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ Figure: Signed graphs considered in proof of Lemma Lemma 4.7 If $\\Gamma $ is a dumbbell-type unbalanced signed graph, then $\\lambda (\\Gamma )< \\lambda (\\Gamma _5)$ .", "Proof.", "Similar to the proof of Lemma REF and Lemma REF , we know that for any $\\Gamma \\in \\mathcal {B}_n(p,\\ell ,q)$ , the index of $\\lambda (\\Gamma )\\le \\max \\lbrace \\lambda (\\Gamma _{11}),\\lambda (\\Gamma _{12}),\\lambda (\\Gamma _{13})\\rbrace <\\lambda (\\Gamma _{5})$ (where $\\Gamma _{11},\\ \\Gamma _{12},\\ \\Gamma _{13}$ are the signed graphs shown as in REF ).$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ Lemma 4.8 If $\\Gamma \\in \\mathcal {B}_n$ is a $\\theta $ -type graph, and is not switching equivalent to $ \\Gamma _2$ , $\\Gamma _3$ or $\\Gamma _5$ , then $\\lambda (\\Gamma )< \\lambda (\\Gamma _5)$ .", "Proof.", "It is not difficult to see that, we only need to consider the case that $\\Gamma \\in \\lbrace \\Gamma _9, \\ \\Gamma _{10},\\ \\Gamma _2^i,\\ \\Gamma _3^j\\rbrace $ , where $1\\le i\\le 7$ and $1\\le j\\le 5$ ( as shown in Figure REF ).", "By direct computation, we can prove that $\\lambda (\\Gamma _5)>\\lambda (\\Gamma _9),\\ \\lambda (\\Gamma _5)>\\lambda (\\Gamma _{10}),$ $\\lambda (\\Gamma _5)>\\lambda (\\Gamma _2^1)=\\max \\lbrace \\lambda (\\Gamma _2^1),\\ldots , \\lambda (\\Gamma _2^4)\\rbrace ,$ $\\lambda (\\Gamma _5)>\\lambda (\\Gamma _3^1)=\\max \\lbrace \\lambda (\\Gamma _3^1),\\lambda (\\Gamma _3^3), \\lambda (\\Gamma _3^5)\\rbrace , \\ \\lambda (\\Gamma _5)>\\lambda (\\Gamma _3^2)>\\lambda (\\Gamma _3^4).", "$ Hence, we can get the desired result.$\\hbox{\\vrule height8pt depth0pt\\vbox {\\hrule width7.2pt\\vspace{7.2pt}\\hrule width7.2pt}\\vrule height8ptdepth0pt}\\smallskip $ Combining Lemma lem:inf, Lemma REF and Lemma REF , we can get the following result immediately.", "Theorem 4.1 Let $\\Gamma _i \\in \\mathcal {B}_n\\ ( \\mbox{ where} \\ i=1,2,\\ldots , 5)$ be the unbalanced signed graphs as shown in Figure REF , then the index $\\lambda (\\Gamma _i)$ is the largest root of the equation $f_i(x)=0$ , where $f_1(x)&=x^4-nx^2+n-5, \\\\f_2(x)&=x^4-(n+1)x^2+2n-4, \\\\f_3(x)&=x^4-(n+1)x^2+4x+2n-8, \\\\f_4(x)&=x^3+x^2-(n-1)x-n+5, \\\\f_5(x)&=x^3-x^2-(n-2)x+n-4,$ for $n\\ge 36$ , we have $\\lambda (\\Gamma _1)>\\lambda (\\Gamma _2)>\\lambda (\\Gamma _3)>\\lambda (\\Gamma _4)>\\lambda (\\Gamma _5)$ , if $\\displaystyle \\Gamma \\in \\mathcal { B}_n$ is not switching equivalent to $ \\Gamma _1, \\Gamma _2,\\Gamma _3,\\Gamma _4$ or $ \\Gamma _5 $ , we have $\\lambda (\\Gamma ) <\\lambda (\\Gamma _5 )$ ." ] ]

[ [ "On $r$-to-$p$ norms of random matrices with nonnegative entries:\n Asymptotic normality and $\\ell_\\infty$-bounds for the maximizer" ], [ "Abstract For an $n\\times n$ matrix $A_n$, the $r\\to p$ operator norm is defined as $$\\|A_n\\|_{r\\to p}:= \\sup_{\\boldsymbol{x} \\in \\mathbb{R}^n:\\|\\boldsymbol{x}\\|_r\\leq 1 } \\|A_n\\boldsymbol{x}\\|_p\\quad\\text{for}\\quad r,p\\geq 1.$$ For different choices of $r$ and $p$, this norm corresponds to key quantities that arise in diverse applications including matrix condition number estimation, clustering of data, and finding oblivious routing schemes in transportation networks.", "This article considers $r\\to p$ norms of symmetric random matrices with nonnegative entries, including adjacency matrices of Erd\\H{o}s-R\\'enyi random graphs, matrices with positive sub-Gaussian entries, and certain sparse matrices.", "For $1< p\\leq r< \\infty$, the asymptotic normality, as $n\\to\\infty$, of the appropriately centered and scaled norm $\\|A_n\\|_{r\\to p}$ is established.", "When $p \\geq 2$, this is shown to imply, as a corollary, asymptotic normality of the solution to the $\\ell_p$ quadratic maximization problem, also known as the $\\ell_p$ Grothendieck problem.", "Furthermore, a sharp $\\ell_\\infty$-approximation bound for the unique maximizing vector in the definition of $\\|A_n\\|_{r\\to p}$ is obtained.", "This result, which may be of independent interest, is in fact shown to hold for a broad class of deterministic sequences of matrices having certain asymptotic expansion properties.", "The results obtained can be viewed as a generalization of the seminal results of F\\\"{u}redi and Koml\\'{o}s (1981) on asymptotic normality of the largest singular value of a class of symmetric random matrices, which corresponds to the special case $r=p=2$ considered here.", "In the general case with $1< p\\leq r < \\infty$, spectral methods are no longer applicable, and so a new approach is developed, which involves a refined convergence analysis of a nonlinear power method and a perturbation bound on the maximizing vector." ], [ "Problem statement and motivation", "For any $n\\times n$ square matrix $A_n$ and $r, p \\ge 1$ , the $r\\rightarrow p$ operator norm of $A_n$ is defined as Anrp:= xr1 Anxp.", "For different values of $r$ and $p$ , the $r\\rightarrow p$ operator norm represents key quantities that arise in a broad range of disciplines.", "For example, when $p=r=2$ , this corresponds to the largest singular value of the matrix $A_n$ , which has been studied extensively for decades.", "On the other hand, when $p$ is the Hölder conjugate of $r$ , that is, $p = r/(r-1)$ , and $A_n$ has nonnegative entries and $A_n^TA_n$ is irreducible, then we will see (in Proposition REF and Section ) that this problem reduces to the famous $\\ell _r$ Grothendieck problem [33], which has inspired a vibrant line of research in the optimization community.", "Two special cases of the $\\ell _r$ Grothendieck problem, namely when $r = 2$ and $r=\\infty $ , relate to spectral partitioning [17], [22] and correlation clustering [13], respectively, and the case of general $r \\in (2,\\infty )$ can be viewed as a smooth interpolation between these two clustering criteria.", "Further, this problem is also related to finding ground states in statistical physics problems.", "Another interesting special case is when $p=r$ , which has been a classical topic; see [56], [47] for general inequalities involving the $p\\rightarrow p$ norm, [30] for applications of these norms to matrix condition number estimation, which is crucial for computing perturbations of solutions to linear equations, and [31], [9] for algorithms to approximate such norms.", "Other prime application areas are finding oblivious routing schemes in transportation networks for the $\\ell _p$ norm [19], [4], [45], [28], and data dimension reduction or sketching of these norms, with applications to the streaming model and robust regression [34].", "Understanding the computational complexity of calculating $r\\rightarrow p$ norms has generated immense recent interest in theoretical computer science.", "We refer the reader to [33] for a detailed account of the applications, approximability results, and Grothendieck-type inequalities for this norm.", "In general, this problem is NP-hard; even providing a constant-factor approximation algorithm for this problem is hard [4], [29], [6].", "However, for the case considered in this article, namely matrices with nonnegative entries and $1< p\\le r<\\infty $ , this problem can be solved in polynomial time [4], [9].", "The cases when $p=1$ and $r\\ge 1$ are equivalent to the cases $p\\le \\infty $ and $r=\\infty $  [34].", "These cases are trivial for nonnegative matrices and hence, we do not consider them in this article.", "The analysis of this norm for random matrices is motivated from a statistical point of view.", "Indeed, asymptotic results on spectral statistics and eignevectors form the bedrock of methods in high-dimensional statistics (see [53], [55], [10] for a sample of the vast literature in this area).", "Further, it is worth mentioning the seminal work of Füredi and Komlós [24], where asymptotic normality of the largest eigenvalue was first established for matrices with i.i.d. entries.", "Subsequently, this result has been extended to adjacency matrices of sparse Erdős-Rényi random graphs [20], stochastic block model [51], and rank-1 inhomogeneous random graphs [12].", "In the context of general $r\\rightarrow p$ norms for random matrices, the $p>r$ case has received much attention.", "For matrices with bounded mean-zero independent entries, asymptotic bounds on the $2\\rightarrow p$ norm was established in [3] for $2\\le p<\\infty $ .", "For $1<r\\le 2\\le p<\\infty $ and matrices having i.i.d.", "entries, $\\Vert A_n\\Vert _{r\\rightarrow p}$ is known to concentrate around its median [37].", "Furthermore, in this regime, refined bounds on the expected $r\\rightarrow p$ norm of centered Gaussian random matrices have been obtained in [27] and later extended to log-concave random matrices with dependent entries in [50].", "Another quantity of considerable interest is the maximizing vector in (REF ).", "For example, in the $p=r=2$ case, eigenvectors of adjacency matrices of graphs are known to play a pivotal role in developing efficient graph algorithms, such as spectral clustering [49], [54], spectral partitioning [17], [22], [36], [44], PageRank [43], and community detection [40], [39].", "Eigenvectors of random matrices can be viewed as perturbations of eigenvectors of the expectation matrix, in the presence of additive random noise in the entries of the latter.", "The study of eigenvector perturbation bounds can be traced back to the classical Rayleigh-Schrödinger theory [46], [48] in quantum mechanics, which gives asymptotic perturbation bounds in the $\\ell _2$ -norm, as the signal to noise ratio increases.", "Non-asymptotic perturbation bounds in the $\\ell _2$ -norm were derived later in a landmark result [16], popularly known as the Davis-Kahan $\\sin \\Theta $ theorem.", "When the perturbation is random, the above deterministic results typically yield suboptimal bounds.", "Random perturbations of low-rank matrices has recently been analyzed in [42].", "However, norms that are not unitary-invariant, such as the $\\ell _\\infty $ -norm, as considered in this paper, are typically outside the scope of the above works, although they are of significant interest in statistics and machine learning.", "The $\\ell _\\infty $ -norm bounds in the case of low-rank matrices have been studied recently in [21], [11], [18], [1], [57], [38], and  [41], [21], [1] contain extensive discussions on such perturbation bounds on eigenvectors (or singular vectors) and their numerous applications in statistics and machine learning." ], [ "Our contributions", "Fix $1 < p \\le r < \\infty $ .", "We now elaborate on the two main results of the current article, namely asymptotic normality of a suitably scaled and centered version of $\\Vert A_n\\Vert _{r\\rightarrow p}$ , and approximation of the corresponding maximizing vector." ], [ "(1) Asymptotic normality.", "Given a sequence of symmetric nonnegative random matrices $(A_n)_{n \\in \\mathbb {N}}$ , our first set of results establishes asymptotic normality of the scaled norm $\\Vert \\bar{A}_n\\Vert _{r\\rightarrow p} := n^{-(\\frac{1}{p}-\\frac{1}{r})} \\Vert A_n\\Vert _{r\\rightarrow p}$ when $1 < p \\le r < \\infty .$ Specifically, let $A_n$ have zero diagonal entries and independent and identically distributed (i.i.d.)", "off-diagonal entries that have mean $\\mu _n$ , variance $\\sigma _n^2 > 0$ .", "Under certain moment bounds on the distribution of the matrix entries, and a control on the asymptotic sparsity of the matrix sequence, expressed in terms of conditions on the (relative) rates at which $\\sigma _n^2$ and $\\mu _n$ can decay to zero, it is shown in Theorem REF that as $n \\rightarrow \\infty $ , $\\frac{1}{\\sigma _n} \\left(\\Vert \\bar{A}_n\\Vert _{r\\rightarrow p} - \\alpha _n(p,r) \\right) \\xrightarrow{}Z \\sim \\mathrm {Normal} \\big (0, 2\\big ),$ where $\\xrightarrow{}$ denotes convergence in distribution, and $\\alpha _n (p,r) := (n-1) \\mu _n + \\frac{1}{2}\\left( p-1 + \\frac{1}{r-1} \\right) \\frac{\\sigma _n^2}{\\mu _n}.$ An extension of the above result for random matrices with inhomogeneous variance profile is also provided in Theorem REF .", "In this case, however, the matrix is required to be dense.", "A result of this flavor appears to have first been established in the seminal work of Füredi and Komlós [24] for the special case $r = p = 2$ , where $\\Vert \\bar{A}_n\\Vert _{2 \\rightarrow 2} = \\Vert A_n\\Vert _{2 \\rightarrow 2}$ represents $\\lambda _1^{(n)}$ , the largest eigenvalue of $A_n$ .", "Using spectral methods, it is shown in [24] that under the assumption that $A_n$ is a symmetric $n \\times n$ random matrix with zero diagonal entries, independent, uniformly bounded off-diagonal entries having a common positive mean $\\mu >0$ and variance $\\sigma ^2>0$ (with $\\mu , \\sigma $ not depending on $n$ ), the limit (REF ) holds with $r=p=2$ , $\\sigma _n = \\sigma $ , and $\\alpha _n(2,2) = (n-1)\\mu + \\sigma ^2/\\mu ,$ which coincides with the definition in (REF ), when one sets $\\mu _n = \\mu $ and $\\sigma _n^2 = \\sigma ^2$ .", "Even for the case $p=r=2$ , our result extends the asymptotic normality result of Füredi and Komlós [24] in three directions: it allows for (a) sequences of possibly sparse matrices $(A_n)_{n\\in \\mathbb {N}}$ , that is with $\\mu _n \\rightarrow 0$ ; (b) independent and identically distributed (i.i.d.)", "off-diagonal entries satisfying suitable moment conditions, but with possibly unbounded support; (c) independent entries with possibly different variances, having a dense variance profile.", "Throughout, the assumption that the diagonal entries are identically zero is only made for simplicity of notation; the result of [24] also allows for the diagonal entries to be drawn from another independent sequence of entries with a different common positive mean and uniformly bounded support on the diagonal, and an analogous extension can also be accommodated in our setting; see Remark REF .", "It is worth mentioning two noteworthy aspects of the limit in (REF ).", "Consider the setting where $\\mu _n = \\mu > 0$ and $\\sigma _n^2 = \\sigma ^2 > 0$ , as considered in [24].", "First, note that while $\\Vert \\mathbb {E}[\\bar{A}_n]\\Vert _{r\\rightarrow p}=(n-1) \\mu $ , and $\\Vert \\bar{A}_n\\Vert _{r\\rightarrow p}/\\Vert \\mathbb {E}[\\bar{A}_n]\\Vert _{r\\rightarrow p}$ converges in probability to 1, the centering $\\alpha _n(p,r)$ is strictly larger than $(n-1) \\mu $ by a $\\Theta (1)$ asymptotically non-vanishing amount.", "Second, whereas the centering $\\alpha _n(p,r)$ for $\\Vert \\bar{A}_n\\Vert _{r\\rightarrow p}$ is $\\Theta (n)$ , the Gaussian fluctuations of $\\Vert \\bar{A}_n\\Vert _{r\\rightarrow p}$ are only $\\Theta (1)$ , having variance 2.", "Both these properties also hold for the case $r = p = 2$ analyzed in [24], and the second property can be seen as a manifestation of the rigidity phenomenon for eigenvalues of random matrices.", "This has subsequently been shown to occur in a variety of other random matrix models, but there is a priori no reason to expect this to generalize to the non-spectral setting of a general $r \\rightarrow p$ norm.", "While spectral methods can be used in the case $p=r=2$ , they are no longer applicable in the general $r \\rightarrow p$ norm setting.", "Thus, we develop a new approach, which also reveals some key reasons for these phenomena to occur, and brings to light when the shift and rigidity properties will fail when considering sparse sequences of matrices.", "(see Remark REF ).", "Our second set of results are summarized in Theorem REF , which provides an $\\ell _\\infty $ -approximation of the maximizing vector for matrices with i.i.d.", "entries, and Theorem REF , which extends this to random matrices with inhomogeneous variance profiles.", "These results rely on Proposition REF , which states an approximation result for the maximizer of the $r \\rightarrow p$ norm, for arbitrary (deterministic) sequences of symmetric matrices satisfying certain asymptotic expansion properties.", "It is not hard to see that the maximizing vector for the $r\\rightarrow p$ norm of the expectation matrix is given by $n^{-1/r}1$ , the scaled $n$ -dimensional vector of all 1's.", "Thus, the maximizing vector $v_n$ corresponding to the random matrix can be viewed as a perturbation of $n^{-1/r}1$ , and our result can be thought of as an entrywise perturbation bound of the maximizing vector for the expectation matrix.", "In contrast with the $p=r=2$ case, the unavailability of spectral methods for the general $1<p\\le r<\\infty $ case makes the problem significantly more challenging, which led us to develop a novel approach to characterize the $\\ell _\\infty $ -approximation error for arbitrary sequence of matrices having certain expansion properties." ], [ "Notation and organization", "We write $[n]$ to denote the set $\\lbrace 1,2,\\dots ,n\\rbrace $ .", "We use the standard notation of $\\xrightarrow{}$ and $\\xrightarrow{}$ to denote convergence in probability and in distribution, respectively.", "Also, we often use the Bachmann-Landau notation $O(\\cdot )$ , $o(\\cdot )$ , $\\Theta (\\cdot )$ for asymptotic comparisons.", "For two positive deterministic sequences $(f(n))_{n\\ge 1}$ and $(g(n))_{n\\ge 1}$ , we write $f(n)\\ll g(n)$ (respectively, $f(n)\\gg g(n)$ ), if $f(n) = o(g(n))$ (respectively, $f(n) = \\omega (g(n))$ ).", "For a positive deterministic sequence $(f(n))_{n\\ge 1}$ , a sequence of random variables $(X(n))_{n\\ge 1}$ is said to be $O_{\\scriptscriptstyle \\mathbb {P}}(f(n))$ and $o_{\\scriptscriptstyle \\mathbb {P}}(f(n))$ , if the sequence $(X(n)/f(n))_{n\\ge 1}$ is tight and $X(n)/f(n)\\xrightarrow{} 0$ as $n\\rightarrow \\infty $ , respectively.", "$\\mathrm {Normal}(\\mu , \\sigma ^2)$ is used to denote normal distribution with mean $\\mu $ and variance $\\sigma ^2$ .", "For two vectors $x= (x_i)_i\\in \\mathbb {R}^n$ and $y=(y_i)_i\\in \\mathbb {R}^n$ , define the `$\\circledast $ ' operation as the entrywise product given by $z= x\\circledast y= (x_iy_i)_i\\in \\mathbb {R}^n$ .", "Define 1 to be the $n$ -dimensional vector of all 1's, $J_n := 11^T$ , and $I_n$ to be the $n$ -dimensional identity matrix.", "Also, $1\\lbrace \\cdot \\rbrace $ denotes the indicator function.", "The rest of the paper is organized as follows.", "In Section  we state the main results and discuss their ramifications.", "Section  provides a high-level outline of the proofs of the main results.", "In Section  we introduce the basics of the non-linear power method, which will be a key tool for our analysis, and present some preliminary results.", "Sections  and  concern the approximation of the maximizing vector in the deterministic and random cases, respectively.", "Section  presents a two-step approximation of the $r\\rightarrow p$ norm and in particular, identifies a functional of the underlying random matrix that is `close' to the $r\\rightarrow p$ norm.", "In Section  we prove the asymptotic normality of this approximating functional.", "Finally, in Section , we end by exploring the relation between the $r\\rightarrow p$ norm and the $\\ell _p$ Grothendieck problem.", "Some of the involved but conceptually straightforward calculations are deferred to the appendix." ], [ "Main results", "In this section we present our main results.", "Section REF describes results for random matrices with i.i.d.", "entries (except possibly the diagonal entries), whereas Section REF states extension of the main results when the matrix entries can have inhomogeneity in their variances.", "Finally, in Section REF we discuss the implications of our results in two important special cases." ], [ "Matrices with i.i.d. entries", "We start by stating a general set of assumptions on the sequence of random matrices: Assumption 2.1 For each $n\\ge 1$ , let $F_n$ be a distribution supported on $[0,\\infty )$ and having finite mean $\\mu _n$ and variance $\\sigma _n^2$ .", "Let $A_n = (a_{ij}^n)_{i,j = 1}^n$ be a symmetric random matrix such that $(a_{ij}^n)_{i,j= 1,i<j}^n$ are i.i.d.", "random variables with common distribution $F_n$ .", "Also, $a_{ii}^n=0$ for all $i\\in [n]$ .", "$\\mu _n =O(1)$ , $\\mu _n = \\omega \\big (\\frac{(\\log n)^2}{n} \\big )$ , and $\\frac{\\sigma _n^2}{\\mu _n} = O(1)$ .", "$\\mathbb {P}\\left(a_{12}^n>0\\right) \\ge \\frac{(2+\\varepsilon )\\log n}{n}$ for some fixed $\\varepsilon >0$ .", "There exists $c < \\infty $ , such that $\\mathbb {E}\\big [|a_{12}^n-\\mu _n|^k\\big ]\\le \\frac{k!", "}{2} c^{k-2}\\sigma _n^2$ for all $k\\ge 3$ .", "Remark 2.2 Assumption REF  (REF ) is needed to ensure that the matrix is irreducible eventually almost surely (see Lemma REF ), which is used to get an almost sure bound on the $\\ell _\\infty $ -approximation of the maximizing vector.", "This condition can be slightly relaxed to the condition $\\mathbb {P}\\left(a_{12}^n>0\\right) \\ge \\frac{(1+\\varepsilon )\\log n}{n}\\quad \\text{ for some fixed }\\varepsilon >0,$ if one is satisfied with bounds that hold with probability tending to 1 as $n\\rightarrow \\infty $ .", "Moreover, the moment conditions in Assumption REF  (REF ) and (REF ) guarantee concentration of certain relevant polynomials of the matrix elements, which we use to approximate the operator norm.", "At first sight, Assumption REF  (REF ) might appear to be restrictive, but such conditions frequently turn up in the literature (cf.", "[35], [2]), for example, when applying Bernstein's inequality.", "Remark 2.3 Note that two important special cases that satisfy Assumption REF are (a) when $A_n$ is the adjacency matrix of an Erdős-Rényi random graph with edge probability $\\mu _n = \\omega ((\\log n)^2/n)$ , and (b) when $a_{ij}^n- \\mu _n$ has a (centered) $\\sigma _n^2$ -subgaussian upper tail." ], [ "Asymptotic normality of the $r\\rightarrow p$ norm", "Our first main result provides a central limit theorem for the $r\\rightarrow p$ norms of random matrices satisfying Assumption REF .", "Theorem REF is proved in Section REF .", "Theorem 2.4 Fix any $1< p\\le r< \\infty $ .", "Consider the sequence of random matrices $(A_n)_{n\\in \\mathbb {N}}$ satisfying Assumption REF and define $ \\bar{A}_n:= n^{-(\\frac{1}{p}-\\frac{1}{r})} A_n.$ Also assume that there exists a constant $c_0>0$ , such that $\\sigma _n \\ge n^{-\\frac{1}{2}+c_0}\\quad \\mbox{and}\\quad \\mu _n\\sigma _n^2 = \\omega \\Big (\\frac{\\log n}{n} \\Big ).$ Then, as $n\\rightarrow \\infty $ , 1n(Anrp- n(p,r)) $\\xrightarrow{}$ Z Normal(0,2), where $\\alpha _n(p,r)= (n-1)\\mu _n + \\Big (p-1+\\frac{1}{r-1}\\Big )\\frac{\\sigma _n^2}{2\\mu _n}.$ Remark 2.5 The reason why the condition in (REF ) is not included in the main set of conditions in Assumption REF is because, as we will see, (REF ) is not needed for the approximation of the maximizing vector and is only used for the asymptotic normality.", "The lower bound on $\\sigma _n$ is required when we apply existing results for the second largest eigenvalues [35] to approximate the operator norm, and the second condition in (REF ) is required only in this approximation step (see Lemma REF ).", "Remark 2.6 The assumption that $a_{ii}^n= 0$ in Theorem REF is not a strict requirement.", "In fact, if $a_{ii}^n$ 's are assumed to be independent of $a_{ij}^n$ 's and to be i.i.d.", "from some distribution $G_n$ with nonnegative support, mean $\\zeta _n = O(\\sigma _n^2)$ , and variance $\\rho _n^2 = O(\\sigma _n^2)$ , then (REF ) holds with n(p,r)=(n-1)n + n + (p-1+1r-1)n22n.", "All our proofs go through verbatim in this case, except for a minor modification that is required in Lemma REF .", "In Lemma REF we have provided a version of the result of Lemma REF in the non-zero diagonal case.", "Both Lemma REF and its proof are given in Section REF .", "However, assuming the diagonal entries to be 0 saves significant additional notational burden and computational complications in the Taylor expansion given in Lemmas REF –REF .", "For that reason, we will assume $a_{ii}^n= 0$ throughout the rest of the paper.", "Remark 2.7 As briefly mentioned in the introduction, an intriguing fact to note from Theorem REF is that although $\\Vert \\bar{A}_n\\Vert _{r\\rightarrow p}$ is concentrated around $\\Vert \\mathbb {E}[\\bar{A}_n]\\Vert _{r\\rightarrow p}$ , there is a non-trivial further $O(1)$ shift in the mean $\\alpha _n(p,r)$ on the CLT scale.", "This is consistent with [24] for the case $p=r=2$ .", "As we will see in the proof of Theorem REF in Section REF , this additional constant shift arises from a Hessian term when we perform the Taylor expansion of a suitable approximation of $\\Vert A_n\\Vert _{r\\rightarrow p}$ .", "It is also worth noting that, if $\\sigma _n^2\\ll \\mu _n$ (e.g., when $F_n$ is an exponential distribution with mean $\\mu _n\\rightarrow 0$ ), this additional shift vanishes, and thus there may be no shift for certain asymptotically sparse matrix sequences.", "Remark 2.8 There are two interesting phenomena about the asymptotic variance of $\\Vert A_n\\Vert _{r\\rightarrow p}$ that are worth pointing out.", "First, the asymptotic variance does not depend on $p,r$ beyond the scaling factor $n^{\\frac{1}{p} - \\frac{1}{r}}$ .", "Second, if $p=r$ and we are in the dense setting (i.e., $\\mu _n = \\mu >0$ and $\\sigma _n = \\sigma >0$ ), the asymptotic variance is a $\\Theta (1)$ quantity, although the mean is $\\Theta (n)$ .", "The latter is analogous to the rigidity phenomenon for eigenvalues of random matrices.", "In the $2\\rightarrow 2$ norm case when the $a_{ij}^n$ are uniformly bounded, this constant order of the asymptotic variance can be understood from the application of the bounded difference inequality (see [52], which considers the case when $a_{ij}^n$ are Bernoulli).", "However, as we see in [52], in order to bound the expected change in the operator norm after changing one entry of the matrix, the fact that $\\ell _2$ is a Hilbert space is crucial, and this method does not generalize directly for $\\ell _p$ spaces with $p\\ne 2$ .", "Nevertheless, as we have shown in Theorem REF , the variance still turns out to be $\\Theta (1)$ for the general $p=r$ case." ], [ "The maximizing vector", "The second main result is an $\\ell _\\infty $ -approximation of the maximizing vector in (REF ).", "To this end, let $\\mathbb {P}_0$ be any probability measure on $\\prod _{n} \\mathbb {R}^{n \\times n}$ , such that its projection on $\\mathbb {R}^{n\\times n}$ has the same law as $A_n$ .", "The following theorem quantifies the proximity of the maximizing vector to 1.", "Theorem REF is proved at the end of Section .", "An analogue of Theorem REF will later be proved for general deterministic sequence of matrices (see Proposition REF ).", "Theorem 2.9 Suppose Assumption REF holds.", "Also, let $v_n := \\operatornamewithlimits{arg\\,max}_{x\\in \\mathbb {R}^n: \\Vert x\\Vert _r \\le 1} \\Vert A_nx\\Vert _p $ and $K := \\max \\big \\lbrace \\limsup _{n\\rightarrow \\infty }\\frac{\\sigma _n^2}{\\mu _n}, 1\\big \\rbrace <\\infty .$ Then the following hold: For $1< p < r< \\infty $ , vn - n-1/r 1 20Kpr-p n-1rnnn,    $\\mathbb {P}$ 0 eventually almost surely, For $p = r\\in (1, \\infty )$ , vn - n-1/r 1 80K[4+ 1r-1] n-1rnnn,    $\\mathbb {P}$ 0 eventually almost surely, where 1 is the $n$ -dimensional vector of all ones." ], [ "Matrices with inhomogeneous variance profile", "We now consider random matrices having an inhomogeneous variance profile.", "In this case, to prove the asymptotic normality result we need the matrix to be dense (i.e., the matrix entries have asymptotically non-vanishing mean and variance).", "This is because our proof uses an upper bound on the second largest eigenvalue of the matrix, recently established in [2], which requires the matrix to be dense.", "The $\\ell _\\infty $ -approximation of the maximizing vector, however, still holds for analogous sparse matrices.", "We start by stating the set of assumptions on the sequence of random matrices that are needed for the $\\ell _\\infty $ -approximation of the maximizing vector.", "Assumption 2.10 For each fixed $n\\ge 1$ , let $A_n = (a_{ij}^n)_{i,j = 1}^n$ be a symmetric random matrix such that $(a_{ij}^n)_{i,j=1,i<j}^n$ is a collection of independent random variables with $a_{ij}$ having distribution $F_{ij}^n$ supported on $[0,\\infty )$ , mean $\\mu _n$ and variance $\\sigma _n^2(i,j)$ .", "Also, $a_{ii}^n=0$ for all $i\\in [n]$ .", "There exists a sequence $(\\bar{\\sigma }_n)_{n\\in \\mathbb {N}} \\subset (0,\\infty )$ , and constants $c_*,c^* \\in (0,\\infty )$ such that $c_*\\le \\liminf _{n\\rightarrow \\infty }\\min _{1\\le i<j\\le n}\\frac{\\sigma _n(i,j)}{\\bar{\\sigma }_n}\\le \\limsup _{n\\rightarrow \\infty }\\max _{1\\le i<j\\le n}\\frac{\\sigma _n(i,j)}{\\bar{\\sigma }_n} \\le c^*.$ $\\mu _n =O(1)$ , $\\mu _n = \\omega \\big (\\frac{(\\log n)^2}{n} \\big )$ , and $\\frac{\\bar{\\sigma }_n^2}{\\mu _n} = O(1)$ .", "$\\min _{1\\le i<j\\le n}\\mathbb {P}\\big (a_{ij}^n>0\\big ) \\ge \\frac{(2+\\varepsilon )\\log n}{n}$ for some fixed $\\varepsilon >0$ .", "There exists $c>0$ , such that $\\max _{1\\le i<j\\le n}\\mathbb {E}\\big [|a_{ij}^n-\\mu _n|^k\\big ]\\le \\frac{k!", "}{2} c^{k-2}\\bar{\\sigma }_n^2\\quad \\text{for all }k\\ge 3.$ Theorem 2.11 Suppose $A_n$ is a symmetric random matrix satisfying Assumption REF .", "Also, as in (REF ), recall that $v_n := \\operatornamewithlimits{arg\\,max}_{x\\in \\mathbb {R}^n: \\Vert x\\Vert _r \\le 1} \\Vert A_nx\\Vert _p $ and $K_1 := \\max \\big \\lbrace \\limsup _{n\\rightarrow \\infty }\\frac{\\bar{\\sigma }_n^2}{\\mu _n}, 1\\big \\rbrace <\\infty $ .", "Then $v_n$ satisfies the same approximations as in (REF ) and (REF ), but with $K$ replaced by $K_1$ .", "Theorem REF is proved at the end of Section .", "Next, we state the asymptotic normality result in Theorem REF below.", "Although in the statement of the theorem we assume that the sequence of matrices $(A_n)_{n\\in \\mathbb {N}}$ satisfies the conditions stated in Assumption REF , most of those conditions are trivially satisfied since the matrix sequence will be assumed to be dense.", "Theorem 2.12 Fix any $1< p\\le r< \\infty $ .", "Consider the sequence of random matrices $(A_n)_{n\\in \\mathbb {N}}$ satisfying Assumption REF and define $\\bar{A}_n:= n^{-(\\frac{1}{p}-\\frac{1}{r})} A_n.$ Also assume that $\\liminf _{n\\rightarrow \\infty } \\bar{\\sigma }_n>0.$ Then as $n\\rightarrow \\infty $ , n22i<jn2(i,j)(Anrp- n(p,r)) $\\xrightarrow{}$ Z Normal(0,2), where $\\alpha _n(p,r)= (n-1)\\mu _n + \\Big (p-1+\\frac{1}{r-1}\\Big )\\frac{\\sum _{i<j}\\sigma _n^2(i,j)}{n^2\\mu _n}.$ Theorem REF is proved in Section REF .", "Similar to Remark REF , the zero diagonal entry is not a strict requirement in Theorem REF .", "The expression of $\\alpha _n(p,r)$ in (REF ) can be suitably updated to accommodate nonnegative random diagonal entries." ], [ "Adjacency matrices of Erdős-Rényi random graphs", "As an immediate corollary to Theorems REF and REF , we obtain the asymptotic normality of $\\Vert A_n\\Vert _{r\\rightarrow p}$ and bound on the maximizing vector when $A_n$ is the adjacency matrix of an Erdős-Rényi random graph $\\mathrm {ER}_n(\\mu _n)$ with $n$ vertices and connection probability $\\mu _n$ .", "In this setting, the $\\ell _\\infty $ -vector bound for the special case $p=r=2$ was proved by Mitra [38] for $\\mu _n\\ge (\\log n)^6/n$ .", "Corollary 2.13 Fix any $1< p\\le r< \\infty $ and let $A_n$ denote the adjacency matrix of $\\mathrm {ER}_n(\\mu _n)$ .", "For $\\mu _n = \\omega ( (\\log n)^2/n)$ , the vector bounds (REF ) and (REF ) hold with $K=1$ .", "Moreover, for $\\mu _n = \\omega (\\sqrt{\\log n/n})$ , the asymptotic normality result in (REF ) holds with $\\sigma _n^2 = \\mu _n(1-\\mu _n)$ ." ], [ "Grothendieck's $\\ell _r$ -problem", "We now investigate the behavior of the $\\ell _r$ quadratic maximization problem, also known as the $\\ell _r$ Grothendieck problem.", "For any $n\\times n$ matrix $A_n$ , the $\\ell _r$ Grothendieck problem concerns the solution to the following quadratic maximization problem.", "For $r\\ge 2$ , define $M_r(A_n):= \\sup _{\\Vert x\\Vert _{r}\\le 1} x^T A_nx.$ In general, finding $M_r(A_n)$ is NP-hard [33].", "However, in the case of a matrix $A$ with nonnegative entries, for which $A^TA$ is irreducible, Proposition REF below states that the $\\ell _r$ Grothendieck problem is a special case of the $r\\rightarrow p$ norm problem.", "Proposition 2.14 Let $A^TA$ be an irreducible matrix with nonnegative entries.", "Then for any $r\\ge 2$ , $M_r(A) = \\Vert A\\Vert _{r\\rightarrow r^*},$ where $r^*= r/(r-1)$ is the Hölder conjugate of $r$ .", "Proposition REF is proved at the end of Section .", "Consequently, Theorem REF yields the limit theorem for $\\bar{A}_n:= n^{-(1-\\frac{2}{r})} A_n$ as stated in the corollary below, whose proof is immediate.", "Corollary 2.15 Let $(A_n)_{n\\in \\mathbb {N}}$ be a sequence of random matrices satisfying the assumptions of Theorem REF .", "Then for any fixed $r\\in [2,\\infty )$ , as $n\\rightarrow \\infty $ , the asymptotic normality result in (REF ) holds for $M_r(\\bar{A}_n)$ with $p = r^* = r/(r-1)$ ." ], [ "Proof outline", "The proof of Theorem REF consists of three major steps:" ], [ "Step 1: Approximating the maximizing vector.", "The first step is to find a good approximation for a maximizing vector $v_n$ for $\\Vert A_n\\Vert _{r\\rightarrow p}$ , as defined in (REF ).", "As stated in Theorem REF , we can precisely characterize the $\\ell _\\infty $ distance between $v_n$ and $n^{-1/r} 1$ , the scaled vector of all ones in $\\mathbb {R}^n$ .", "In fact we work with a general deterministic sequence of symmetric nonnegative matrices (see Proposition REF ).", "When $p<r$ , the required $\\ell _\\infty $ -bound follows whenever the row sums are approximately the same, which we call almost regularity (see Definition REF ).", "We actually have a short and elementary proof for the $p<r$ case.", "The proof for the case $p=r$ , is more complicated and requires that, given any subset $V_n\\subset [n]$ , `most' indices $i$ satisfy $\\sum _{j\\in V_n} a_{ij}^n \\approx \\mu _n |V_n|$ .", "We call the latter property, which is stated more precisely in Definition REF , well-balancedness.", "The well-balancedness property is closely related to the concept of quasi-random graphs, as elaborated in Remark REF .", "Under Assumption REF , our sequence of random matrices satisfy almost regularity and well-balancedness properties almost surely, which concludes Theorem REF .", "The next step is to construct a suitable approximation of $\\Vert A_n\\Vert _{r\\rightarrow p}$ .", "With the strong bound in Theorem REF , a natural choice would be to approximate $\\Vert A_n\\Vert _{r\\rightarrow p}$ by $\\Vert A_n n^{-1/r} 1\\Vert _{p}$ .", "However, such an approximation turns out to be insufficient on the CLT-scale.", "To this end, we use a nonlinear power iteration for finding $r\\rightarrow p$ norms, introduced by Boyd [9].", "We start the power iteration from the vector $v^{(0)}_n:=n^{-1/r}1$ .", "The rate of convergence of this power-method depends on the proximity of $v^{(0)}_n$ to $v_n$ (which we now have from Theorem REF ), and the second largest eigenvalue of $A_n$ (for which we use existing results from [35], [20], [2]).", "Our ansatz is that after only one step of Boyd's nonlinear power iteration, we arrive at a suitable approximation of $\\Vert A_n\\Vert _{r\\rightarrow p}$ .", "For any $k\\ge 1$ , $t\\in \\mathbb {R}$ , and $x= (x_1,\\ldots , x_n)$ , define $\\psi _k(t):= |t|^{k-1}\\mathrm {sgn}(t)$ , and $\\Psi _k(x)= (\\psi _k(x_i))_{i=1}^n.$ Then we show that (see Proposition REF ) the quantity Anrp(An) := Anr*(AnTp(An1))pr*(AnTp(An1))r, where $r^* := r/(r-1)$ denotes the Hölder conjugate of $r$ , provides the required approximation to $\\Vert A_n\\Vert _{r\\rightarrow p}$ .", "As in Step 1, we also first show this approximation for a deterministic sequence of matrices satisfying certain conditions, and then show that the random matrices we consider satisfy these conditions.", "The final step is to prove the asymptotic normality of the sequence $\\lbrace \\eta (A_n)\\rbrace _{n \\in \\mathbb {N}}$ .", "This is a non-linear function, and as it turns out, the state-of-the-art approaches to prove CLT do not apply directly in our case.", "For that reason, we resort to a direct approach using Taylor expansion to obtain the limit law.", "Loosely speaking, we show that $\\eta (A_n) \\approx n^{\\frac{1}{p} - \\frac{1}{r} -1}\\sum _{i,j}a_{ij}^n+ \\frac{1}{2}\\Big (p-1 + \\frac{1}{r-1}\\Big )n^{\\frac{1}{p} - \\frac{1}{r}}\\sum _{i,j}(a_{ij}^n-\\mu )^2,$ which after appropriate centering and scaling yields the CLT result as stated in Theorem REF ." ], [ "Boyd's non-linear power method", "We start by introducing the non-linear power iteration method and stating some preliminary known results, along with a rate of convergence result that will be crucial for our treatment.", "The framework for non-linear power iteration was first proposed by Boyd [9].", "It has also been used in [4] to obtain approximation algorithms for the $r\\rightarrow p$ norm of matrices with strictly positive entries.", "Henceforth, we fix $n \\in \\mathbb {N}$ , and for notational simplicity, omit the subscript $n$ , for example, using $A$ to denote $A_n$ , etc.", "Let $A$ be an $n\\times n$ matrix with nonnegative entries.", "For any $x\\ne 0$ , define the function $f(x):= \\Vert Ax\\Vert _{p}/\\Vert x\\Vert _r,$ and set $\\gamma := \\sup _{x\\ne 0}f(x)$ .", "If a vector $v$ is a local maximum (or, more generally, critical point) of the function $f$ , then the gradient of $f$ must vanish at that point.", "This critical point can further be written as the solution to a fixed point equation.", "Now, if there is a unique positive critical point, the fixed point equation can be used to construct an iteration that converges geometrically to the maximum, starting from a suitable positive vector.", "The above description is briefly formalized below.", "For $q > 1$ , $t \\in \\mathbb {R}$ and $x\\in \\mathbb {R}^n$ , define q(t):= |t|q-1sgn(t),      q(x) := (q(xi))i=1n, where $\\mathrm {sgn}(t) = -1, 1,$ and 0, for $t <0$ , $t >0$ , and $t = 0$ , respectively.", "Taking the partial derivative of $f$ with respect to $x_i$ , we obtain, for $x\\ne 0$ , $\\frac{\\partial f(x)}{\\partial x_i} = \\Vert x\\Vert _r^{-2}\\Big [\\Vert Ax\\Vert _p^{-(p-1)}\\langle \\Psi _{p}(Ax), A_i^T \\rangle \\Vert x\\Vert _r - \\Vert x\\Vert _r^{-(r-1)} \\psi _r(x_i)\\Vert Ax\\Vert _p\\Big ],$ where $A_i$ denotes the $i$ -th column of $A$ .", "Equating (REF ) to zero for $i = 1, \\ldots , n$ , yields xrrATp(Ax) = Axppr(x) .", "Now, let $u$ with $\\Vert u\\Vert _r =1$ be a (normalized) solution to (REF ) and set $\\gamma (u) := \\Vert Au\\Vert _{p}$ .", "Then straightforward algebraic manipulations show that r*(ATp(Au)) = ((u))p(r*-1)u, where recall that $r^* = r/(r-1)$ .", "We denote the operator arising on the right-hand side  of (REF ) as follows: Sx:= r*(ATp(Ax)),    Wx:= SxSxr       for x0.", "Then (REF ) implies Su= ((u))p(r*-1) u,    Wu= u, where the last equality uses the fact that $\\Vert u\\Vert _r = 1.$ Thus, any solution to (REF ) is a fixed point of the operator $W$ .", "The following lemma proves uniqueness of this fixed point among all nonnegative vectors, which can be viewed as a generalization of the classical Perron-Frobenius theorem.", "The uniqueness in Lemma REF was established for matrices with strictly positive entries in [4].", "Below we show that their proof can be adapted to matrices with nonnegative entries when $A^TA$ is irreducible.", "Lemma 4.1 Assume that $A^TA$ is irreducible.", "Then (REF ) has a unique solution $v$ among the set of all nonnegative vectors.", "Further, $v$ has all positive entries.", "Let us first show that, when $A^TA$ is irreducible, any nonnegative vector satisfying (REF ) must have strictly positive entries.", "This, in particular, will also prove that $v$ has all positive entries.", "We argue by contradiction.", "Let $x$ be a nonnegative vector satisfying (REF ) and suppose, $i \\in [n]$ be such that $x_i=0$ .", "Then, by (REF ) and (REF ) we have (Sx)i = 0 (AT (p (Ax))i = j=1n aji |k=1n ajk xk|p-1 = 0, (ATAv)i=jaji(kajkxk) =0, where the last step follows by observing that all the elements of $A$ and $x$ are nonnegative, and thus, if $\\Psi _{r^*}(A^T\\Psi _p(Ax))=0$ , then $\\Psi _{2}(A^T\\Psi _2(Ax))=0$ as well.", "Observe that (REF ) implies $x_j=0$ for all $j\\in [n]$ for which there exists $j^{\\prime }\\in [n]$ with $a_{j^{\\prime }i}>0$ and $a_{j^{\\prime }j}>0$ .", "Repeat the above argument for any such $j$ .", "Continuing this way it can be seen that since $A^TA$ is irreducible, $x_j=0$ for all $j=1,\\ldots ,n$ , which this leads to a contradiction.", "To show the uniqueness, let $u\\ne v$ be two nonnegative vectors satisfying (REF ) with $\\Vert u\\Vert _r=\\Vert v\\Vert _r=1$ .", "Further, without loss of generality, assume that $\\Vert Au\\Vert _p\\le \\Vert Av\\Vert _p$ .", "Using the above argument, both $u$ and $v$ have all positive entries.", "Let $\\theta \\in (0,1]$ be the smallest number such that $u- \\theta v$ has a zero coordinate.", "Define $U = \\lbrace k: u_k- \\theta v_k =0\\rbrace $ , so that $u_j- \\theta v_j>0$ for all $j\\in U^c$ .", "Since $\\Vert u\\Vert _r = \\Vert v\\Vert _r$ and $u\\ne v$ , we have that $U^c\\ne \\varnothing $ .", "Claim 4.2 There exists $k\\in U$ such that (Su)k > (Sv)k = p-1r-1 (S v)k. First, note that since $A^TA$ is irreducible, there exists $k_1\\in U$ , $k_2\\in [n]$ , and $k_3\\in U^c$ , such that both $a_{k_1k_2}$ and $a_{k_2k_3}$ are positive.", "Therefore, the facts that $u_{k_3}> \\theta v_{k_3}$ , $a_{k_2k_3}>0$ , and $u_i\\ge \\theta v_i$ for all $i\\in [n]$ yield (p(Au))k2>(p(A(v)))k2   and   (p(Au))i(p(A(v)))i for all i[n].", "This, together with the fact that $a_{k_1k_2}>0$ , implies $(A^T\\Psi _p(Au))_{k_1}>(A^T\\Psi _p(A(\\theta v)))_{k_1},$ and thus (REF ) holds with $k=k_1$ .", "Let us fix some $k\\in U$ satisfying (REF ).", "Then, using (REF ), (u)p = (Su)kr-1ukr-1>p-1(Sv)kr-1(vk)r-1 = p-r (v)p. Since $p\\le r$ and $\\theta \\in (0, 1]$ , this yields $\\Vert Au\\Vert _p = \\gamma (u)>\\gamma (v)=\\Vert Av\\Vert _p$ , which contradicts the initial assumption that $\\Vert Au\\Vert _p\\le \\Vert Av\\Vert _p$ .", "This proves the uniqueness.", "The (nonlinear) power iteration for finding $\\gamma $ consists of the following iterative method: Let $v^{(0)}$ be a vector with positive entries and $\\Vert v^{(0)}\\Vert _r=1$ .", "Then for $k\\ge 0$ , define v(k+1) := W v(k).", "In general, the above iteration may not converge to the global maximum $\\gamma $ .", "However, as the following result states, if in addition to having nonnegative entries, the matrix $A^TA$ is irreducible, then the iteration must converge to the unique positive fixed point.", "Proposition 4.3 ([9]) Fix any $1< p \\le r <\\infty $ .", "Let $A$ be a matrix with nonnegative entries such that $A^TA$ is irreducible.", "If $v^{(0)}$ has all positive entries, then $\\lim _{k\\rightarrow \\infty }\\Vert Av^{(k)}\\Vert _p = \\gamma $ ." ], [ "Rate of convergence", "Due to Lemma REF , henceforth we will reserve the notation $v$ to denote the unique maximizer in (REF ) having positive entries and $\\Vert v\\Vert _r=1$ .", "The notation $\\gamma = \\gamma (v) = \\Vert A v\\Vert _p$ denotes the operator norm $\\Vert A\\Vert _{r\\rightarrow p}$ .", "Next, we will study the rate of convergence of $v^{(k)}$ to $v$ .", "Specifically, we obtain a fast convergence rate once the approximating vector comes within a certain small neighborhood of the maximizing vector.", "The rate of convergence result builds on the line of arguments used in the proof of [9].", "However, as it turns out, since we are interested in the asymptotics in $n$ , the rate obtained in [9] does not suffice (see in particular, [9]), and we need the sharper result stated in Proposition REF .", "Let us consider the following linearized version of the operator $S$ at $v$ .", "Recall for any $x,y\\in \\mathbb {R}^n$ , we write $x\\circledast y= (x_iy_i)_i$ .", "Define the linear transformation Bx:=|v|2-rAT(|Av|p-2(Ax)), and the inner product [x,y]:= |v|r-2x,y.", "When $A^TA$ is irreducible, $v$ has all positive entries by Lemma REF , and thus (REF ) and (REF ) are well-defined for all $p,r\\ge 1$ .", "Observe that this inner product induces a norm, which will henceforth be referred to as the “$v$ -norm”: xv:= [x,x]1/2 = |v|r-2,|x|21/2.", "The following fact is immediate.", "Fact 4.4 The operator $B$ is symmetric and positive semi-definite with respect to the inner product in (REF ).", "Fact REF implies that the eigenspace of $B$ has $n$ orthonormal basis vectors and $n$ nonnegative eigenvalues corresponding to the Rayleigh quotient [Bx,x][x,x] = |Av|p-2,|Ax|2|v|r-2, |x|2.", "Henceforth, we will refer to (REF ) as the $v$ -Rayleigh quotient to emphasize the dependence on $v$ .", "Using (REF ), note that $B v= \\gamma ^p v$ , and hence, $\\gamma ^p$ is an eigenvalue of $B$ .", "Let $\\lambda _2\\ge \\lambda _3 \\ge \\dots \\ge \\lambda _{n}$ be the other eigenvalues.", "In fact, as shown in the proof of [9], $\\gamma ^p$ is the largest eigenvalue of $B$ and is simple.", "Now, recall that the convergence rate of the classical (linear) power iteration for the largest eigenvalue of matrices depends on the the ratio between the largest and the second largest eigenvalues.", "As it turns out, in the nonlinear case, this rate depends on the ratio of the largest and second largest eigenvalues of the operator $B$ .", "This is stated in the proposition below.", "Proposition 4.5 Let $A$ be an $n\\times n$ matrix with nonnegative entries such that $A^TA$ is irreducible and $1< p \\le r <\\infty $ .", "Also let $v^{(0)}$ have all positive entries.", "There exists $\\varepsilon _0 = \\varepsilon _0 (p,r)>0$ and $C = C(p,r) >0$ , both independent of $n$ , such that if for some $k \\ge 1$ and $\\varepsilon \\le \\varepsilon _0$ , $\\Vert v^{(k)} - v\\Vert _{\\infty } \\le \\varepsilon $ , then v(k+1) - vv (1+C)(p-1)2(r-1)p v(k) - vv.", "Remark 4.6 It is worthwhile to point out that the convergence rate of the non-linear power method is, in general, not comparable to the rate of convergence of the classical Von Mises power iteration.", "However, as we will see in Lemma REF , the $\\ell _\\infty $ -bound on the maximizing vector in the non-linear case, stated in Proposition REF , enables us to make this connection and obtain the rate of convergence result.", "[Proof of Proposition REF ] For any two fixed vectors $x,h\\in \\mathbb {R}^n$ , and a function $f$ , let us denote the directional derivative of $f$ at $x$ as $\\delta f(x;h) := \\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon } \\big (f(x+ \\varepsilon h) - f(x)\\big ),$ whenever the limit exists.", "Recall that $x\\circledast y$ denotes the vector $(x_iy_i)_i$ .", "Now, fix $1 < p \\le r < \\infty .$ First, note that for a nonnegative vector $x$ , $\\delta \\Psi _p(x;h) = (p-1) \\Psi _{p-1} (x) \\circledast h$ , and therefore, S(x; h) = (r*-1) r*-1 (ATp Ax) ( AT ((p-1)p-1(Ax) A h)) = p-1r-1 0(ATp(Ax)) SxL(x;h), where $L(x;h) := A^T (\\Psi _{p-1}(Ax) \\circledast A h)$ , and for $g(x) := \\Vert Sx\\Vert _r$ , using (REF ) and (REF ), we see that g(x;h) = 1r 1Sxrr-1 rr (Sx), S(x; h) = p-1r-1 1Sxrr-1 AT p(Ax) , 0(ATp(Ax)) SxL(x;h) = p-1r-1 1Sxrr-1Sx,L(x;h) = p-1r-1 1SxrrWx,L(x;h) .", "Now observe that since $Wx\\Vert Sx\\Vert _r = Sx$ , W(v, h) Svr + W(v)g(v; h) = S(v; h) Therefore, from (REF ) and (REF ) it follows that W(v;h) = (p-1r-1) 1Svrr-1 [|Wv|2-r L(v;h) - WvWv,L(v;h)], where we have used the fact that $v$ and $Wv$ have nonnegative entries Now, $\\delta W(v;\\cdot )$ is a linear transformation.", "Clearly, $\\delta W(v;v) = 0$ since $L(v;v) = \\Psi _r(Sv)$ .", "Further, it follows that the eigenvectors of $\\delta W(v;\\cdot )$ corresponding to the non-zero eigenvalues coincide with the eigenvectors of $B$ defined in (REF ) corresponding to $\\lambda _2,\\dots , \\lambda _n$ given by (REF ).", "This follows since $Bh= \\lambda h$ for some nonzero $\\lambda \\ne \\gamma $ implies that $L(v;h) = \\lambda |v|^{r-2} \\circledast h$ , which together with $Wv\\propto v$ yields that Wv, L(v;h) v,|v|r-2 h = [v,h] = 0.", "Thus the second term in (REF ) is zero.", "Also the first term in (REF ) is proportional to $v$ , which yields the equality of the eigenvectors.", "In fact, the eigenvalues of $\\delta W(v;\\cdot )$ are given by $\\frac{p-1}{r-1} \\gamma ^{-p}\\lambda _i$ .", "Since the Rayleigh coefficients in (REF ) are computed with respect to the $\\Vert \\cdot \\Vert _{v}$ norm, we have W(v;h)v (p-1)2(r-1)p hv.", "Now, for $t\\in [0,1]$ , define $y_t = v+t(v^{(k)} - v)$ .", "Note that $y_t$ has all positive entries, since $v$ has possitive entries, and $v^{(k)}$ has nonnegative entries whenever $v^{(0)}$ does.", "Thus, the same expression as (REF ) holds for $\\delta W(y_t;h)$ , with $v$ replace by $y_t$ .", "Now, $\\Vert y_t - v\\Vert _{\\infty } \\le \\Vert v^{(k)} - v\\Vert _{\\infty } \\le \\varepsilon $ , for any $t\\in [0,1]$ .", "Using the fact that $(1+\\varepsilon )^a = 1+O(\\varepsilon )$ , it follows that there exists a constant $C<\\infty $ and $\\varepsilon _0 > 0$ both depending only on $p,r$ , such that for all $\\varepsilon \\le \\varepsilon _0$ , W(yt;h) (1+ C )W(v;h).", "Now, observe that $\\delta W(y_t; v^{(k)}- v) = \\frac{\\mathrm {d}}{\\mathrm {d}t}(Wy_t).$ and therefore, using (REF ) and the fact that $y_0 = v$ and $y_1 = v^{(k)}$ , we obtain v(k+1) - v= W v(k) - Wv= 01 W(yt; v(k)- v) dt.", "Thus, (REF ) and (REF ) implies that v(k+1) - vv (1+ C ) (p-1)2(r-1)p v(k) - vv, and the proof follows." ], [ "An $\\ell _\\infty $ -approximation of the maximizer", "Given an $n \\times n$ nonnegative matrix $A_n = (a_{ij}^n)$ and $V\\subseteq [n]$ , we write $d_n(i,V) := \\sum _{j\\in V} a_{ij}^n, \\quad i = 1, \\ldots , n.$ Also, we simply write $d_n(i)= d_n(i,[n])$ .", "When $A_n$ is the adjacency matrix of a graph on $n$ vertices, $d_n(i)$ represents the (out)-degree of vertex $i$ .", "Definition 5.1 (Almost regular) A sequence of matrices $(A_n)_{n\\in \\mathbb {N}}$ is called $(\\varepsilon _n,\\mu _n)_{n\\in \\mathbb {N}}$ almost regular if there exists an $n_0\\ge 1$ such that for all $n\\ge n_0$ $\\max _{i\\in [n]} \\big | d_n(i)- n\\mu _n \\big |\\le n\\mu _n\\varepsilon _n.$ In order to show the proximity of the maximizing vector to $n^{-1/r}1$ for the $p=r$ case, we need another asymptotic property in addition to the almost regularity defined above.", "Definition 5.2 (Well-balanced) A sequence of matrices $(A_n)_{n\\in \\mathbb {N}}$ is called $(\\varepsilon _n,\\delta _n,\\mu _n)_{n\\in \\mathbb {N}}$ well-balanced if there exists an $n_0\\ge 1$ such that for all $n\\ge n_0$ the following holds: For any subset $V\\subseteq [n]$ there exists an exception set $V_{\\rm ex} = V_{\\rm ex}(V)$ satisfying For all $i\\in [n]\\setminus (V\\cup V_{\\rm ex})$ , $\\big |d_n(i,V) - \\mu _n|V|\\big |\\le n \\mu _n\\varepsilon _n$ , and For all $i\\in [n]$ , $d_n(i, V_{\\rm ex}) \\le n\\delta _n$ .", "In the presence of almost regularity, the well-balancedness actually implies a slightly stronger property, as stated in Claim REF below.", "However, the form in Definition REF  (a) will be useful to prove well-balancedness of random matrices since $(d_{n}(i,V))_{i\\in V^c}$ is a collection of independent random variables.", "Claim 5.3 Let $(A_n)_{n\\in \\mathbb {N}}$ be a sequence of matrices that is $(\\varepsilon _n,\\mu _n)_{n\\in \\mathbb {N}}$ almost regular and $(\\varepsilon _n,\\delta _n, \\mu _n)_{n\\in \\mathbb {N}}$ well-balanced.", "Then, the following holds for all sufficiently large $n$ : For any subset $V\\subseteq [n]$ , there exists an exception set $V_{\\rm ex} \\subset [n]$ , possibly depending on $V$ , with $d_n(i,V_{\\rm ex}) \\le 2 n\\delta _n$ , such that for any $i\\in [n]\\setminus V_{\\rm ex}$ , | dn(i,V) - n |V| |2nn n    and   |dn(i, Vc) - n |Vc| |2nn n. Indeed, since $(A_n)_{n \\in \\mathbb {N}}$ is $(\\varepsilon _n,\\delta _n,\\mu _n)_{n\\in \\mathbb {N}}$ -well-balanced, for any subset $V\\subseteq [n]$ , we can choose two exception sets $V_{\\rm ex}^{(1)}, V_{\\rm ex}^{(2)} \\subset [n]$ , such that |dn(i,V) - n |V||nn n,     i[n](VVex(1)), |dn(i,Vc) - n |Vc||nn n,     i[n](VcVex(2)), {dn(i,Vex(1)), dn(i,Vex(2))} nn ,        i[n].", "Define $V_{\\rm ex}:= V_{\\rm ex}^{(1)}\\cup V_{\\rm ex}^{(2)}$ .", "Therefore, combining almost regularity condition in Definition REF and () yields (REF ).", "Remark 5.4 (Connection to quasi-random graphs) The well-balancedness property can be understood intuitively when $A_n$ is an adjacency matrix of some graph $G_n$ on vertex set $[n]$ .", "For $\\varepsilon _n\\rightarrow 0$ and $\\delta _n\\rightarrow 0$ , this property in conjunction with almost regularity ensures that, given any $V\\subset [n]$ , most vertices have homogeneous connections with $V$ , i.e., there are approximately $\\mu _n |V|$ connections to $V$ from most vertices.", "Intuitively, this suggests that $G_n$ is `close' to an Erdős-Rényi random graph.", "Thus, it is natural to wonder how the well-balancedness is related to a somewhat related concept of quasi-randomness [15], [14].", "In fact, using Claim REF , it is straightforward to show that the well-balancedness implies condition $\\mathrm {DISC}(1)$ from [14], which states that for all $V_1,V_2 \\subset [n]$ , # edges between V1,V2 = n |V1||V2| +o (n2n).", "However, we will need the precise parameterizations in the definition of well-balancedness to work with some particular choices of $\\varepsilon _n,\\delta _n$ (see Proposition REF below).", "In the beginning of Section REF we briefly explain why this property is needed to establish the proximity of the maximizing vector to $n^{-1/r}1$ .", "We now state the main result of this section: Proposition 5.5 Let $(A_n)_{n\\in \\mathbb {N}}$ be a sequence of symmetric matrices with nonnegative entries, such that $A_n^TA_n$ is irreducible for all $n\\in \\mathbb {N}$ .", "Assume that there exists $(\\varepsilon _n)_{n \\in \\mathbb {N}} \\subset (0,\\infty )$ with $\\varepsilon _n\\rightarrow 0$ , and $(\\mu _n)_{n \\in \\mathbb {N}} \\subset (0,1)$ , such that $(A_n)_{n\\in \\mathbb {N}}$ is $(\\varepsilon _n,\\mu _n)_{n\\in \\mathbb {N}}$ almost regular.", "For each $n \\in \\mathbb {N}$ , let $v_n$ be the maximizing vector for $\\Vert A_n\\Vert _{r\\rightarrow p}$ , as defined in (REF ).", "Then there exists an $n_0\\ge 1$ , such that the following hold: For $1<p<r<\\infty $ , and for all $n\\ge n_0$ , vn - n-1/r 1 2pr-pn-1r(n + O(n2)).", "For $p=r \\in (1,\\infty )$ , further assume that there exists $\\delta _n = \\delta _n(\\varepsilon _n,\\mu _n) \\ll \\mu _n\\varepsilon _n$ such that $(A_n)_{n\\in \\mathbb {N}}$ is $(\\varepsilon _n,\\delta _n,\\mu _n)_{n\\in \\mathbb {N}}$ well-balanced.", "Then for all $n\\ge n_0$ , vn - n-1/r 1 4[4+ 1r-1] n-1r (n + o(n)).", "In Sections REF and REF , we prove Proposition REF  (a) and (b), respectively." ], [ "Maximizer for the case $p<r$", "In this section, we prove Proposition REF (a).", "Given a maximizing vector $v_n$ for $\\Vert A_n\\Vert _{r\\rightarrow p}$ as in (REF ), define $m_n:= \\min _{i= 1, \\ldots , n} v_{n,i},\\qquad \\mbox{ and } \\qquad M_n:= \\max _{i=1,\\ldots , n} v_{n,i}.$ Suppose we can show that, for all sufficiently large $n$ , and for some $c \\in (0,\\infty ),$ mnMn 1 -cn + O( 2n).", "Then, $1= \\sum _i v_{n,i}^r \\le n M_n^r$ , so that $M_n\\ge n^{-1/r}$ .", "Also, (REF ) yields $1= \\sum _{i=1}^n v_{n,i}^r \\ge n m_n^r\\ge n M_n^r(1-rc\\varepsilon _n + O(\\varepsilon _n^2)),$ Together, this shows that $\\Vert v_n - n^{-1/r} 1 \\Vert _{\\infty } \\le c n^{-\\frac{1}{r}} (\\varepsilon _n + O(\\varepsilon _n^2)).$ Thus, to show Proposition REF , it is enough to prove (REF ) for $c=\\frac{2p}{r-p}$ .", "Recall Definition REF and all the associated notation.", "Take any $i_0$ and $j_0$ such that $m_n= v_{n,i_0}$ and $M_n= v_{n,j_0}$ .", "Using (REF ), (REF ), and (REF ), together with $r^*- 1 = 1/(r-1)$ , and the fact that $A_n$ is nonnegative and symmetric, we can use (REF ) to conclude that (Svn)j0 = (r* (AnT p (An vn)))j0 = |(AnT p (An vn)))j0|1r-1 (i=1n ani j0 (Mndn(i))p-1)1r-1 (i=1n anj0 i (Mnnn (1+n) )p-1)1r-1 ((Mnnn)p-1(1+n)p-1nn (1+n) )1r-1 (Mnp-1(nn)p)1r-1(1+n)pr-1 = Mnp-1r-1(nn)pr-1(1+ pr-1n + O(n2)).", "A similar computation yields the following lower bound: (Svn)i0 mnp-1r-1(nn)pr-1(1- pr-1n + O(n2)).", "Since $v_n$ satisfies $Sv_n \\propto v_n$ , we must have $\\frac{(Sv_n)_{i_0}}{m_n}=\\frac{(Sv_n)_{j_0}}{M_n}$ , and consequently, (REF ) and (REF ) together imply that Mnp-1r-1-1(1+ pr-1n + O(n2)) mnp-1r-1-1(1- pr-1n + O(n2)), or equivalently, $ M_n^{\\frac{p-r}{r-1}} \\ge m_n^{\\frac{p-r}{r-1}}\\Big (1- \\frac{2p}{r-1}\\varepsilon _n + O(\\varepsilon _n^2)\\Big ).$ Thus, using the fact that $1 < p < r$ , we have (mnMn)r-pr-1 (1- 2pr-1n + O(n2))       mnMn (1- 2pr-pn + O(n2)).", "This completes the proof of (REF ), and hence Proposition REF (a) follows.", "$\\Box $" ], [ "Maximizer for the case $p=r$", "We now prove Proposition REF (b), which entails establishing the bound in (REF ) under both the almost-regularity and well-balanced conditions on $(A_n)_{n \\in \\mathbb {N}}$ .", "The basic idea again is to show that if a vector $v$ satisfies $Sv\\propto v$ , then the ratio of its maximum and minimum must be asymptotically vanishing in $n$ .", "However, when $p=r$ , one can see that the exponent of $M_n$ and $m_n$ in equations (REF ) and (REF ) becomes 0, and consequently, the method in Section REF fails.", "The key insight to deal with this issue is to define two sets of vertices: one, consisting of all vertex indices $i$ , such that $v_i$ is suitably large, and the other with $v_i$ 's suitably small.", "Due to the well-balancedness property, we can ensure that these two sets must share a certain number of cross-edges, unless one of them is asymptotically vanishing.", "In either case, we show that if $M_n/m_n$ is not small, then the ratio $(Sv)_i/v_i$ will be very different for the vertices in the above two sets.", "This leads to a contradiction.", "By [34] and the symmetry of $A_n$ , $\\Vert A_n\\Vert _{r\\rightarrow r} = \\Vert A_n^T\\Vert _{r^*\\rightarrow r^*}= \\Vert A_n\\Vert _{r^*\\rightarrow r^*}$ .", "Since $r\\in (1,2]$ implies $r^*\\ge 2$ , we can, without loss of generality, assume that $r\\in (1,2]$ throughout this proof.", "Let $n_0 \\in \\mathbb {N}$ be the maximum of the $n_0$ defined in the definitions of the almost-regularity and well-balanced conditions and fix $n \\ge n_0$ .", "Also, as in the proof of Proposition REF (a), we define $m_n$ and $M_n$ as in (REF ).", "First, it suffices to show that for $\\Delta _n:= (M_n-m_n)/2$ nMn [8+ 2r-1] n. Indeed, (REF ) is just a restatement of (REF ) with $c = 4(4 + \\frac{1}{r-1})$ .", "To this end, define $V := \\lbrace i: M_n-\\Delta _n\\le v_i\\le M_n\\rbrace $ .", "Denote $N_1 = |V|$ and $N_2 = n-N_1$ .", "Recall that $\\varepsilon _n$ , $\\mu _n$ , and $\\delta _n\\ll \\mu _n\\varepsilon _n$ are such that Definitions REF and REF are satisfied.", "Also suppose that Claim REF holds with the above choice of $V$ and let $V_{\\rm ex}$ be chosen accordingly.", "In the rest of the proof, we will obtain upper and lower bounds on each coordinate of $Sv= \\Psi _{r^*}(A^T\\Psi _r(Av))$ .", "Using the definition of $V$ and (REF ), we have for each $j \\in V_{\\rm ex}^c$ , (Av)j Mn(N1n + 2nnn) + (Mn-n)(N2n + 2nnn) (Av)j (Mn-n)(N1n - 2nnn)+ + mn(N2n - 2nnn)+.", "On the other hand, for each $j \\in V_{\\rm ex}$ , mnnn (1-n)(Av)j Mnnn (1+n).", "Another application of the $(\\varepsilon _n, \\delta _n, \\mu _n)_{n\\in \\mathbb {N}}$ -well-balancedness of $(A_n)_{n \\in \\mathbb {N}}$ , together with (REF ), (REF ), and the fact that $\\max _{i\\in [n]}d_n(i,V_{\\rm ex})\\le 2 n\\delta _n$ , it follows that for all vertices $i\\in [n]$ , (ATr(Av))i = j [n]Vex ajin|(Av)j|r-1 + j Vex ajin|(Av)j|r-1 [Mn(N1n + 2nnn) + (Mn-n)(N2n + 2nnn)]r-1nn (1+n)       + 2[Mnnn(1+n)]r-1 n n and, also invoking the nonnegativity of $A$ , we have (ATr(Av))i [(Mn-n)(N1n - 2nnn)+       + mn(N2n - 2nnn)+]r-1nn (1-n).", "Now, using the definitions of $S$ , $\\Psi _r$ , the identity $r^*-1= 1/(r-1)$ , and the property $Sv\\propto v$ , we see that $A^T\\Psi _r(Av)\\propto \\Psi _r(v)$ .", "Together with (REF ) and (REF ), this implies that [N1n + 2nnn + (1-nMn)(N2n + 2nnn)]r-1nn (1+n)    + 2[nn(1+n)]r-1n n [1mn(Mn-n)(N1n - 2nnn)+ + (N2n - 2nnn)+]r-1nn (1-n).", "Then, first dividing both sides of (REF ) by $(n \\mu _n)^r$ and then taking their $(\\frac{1}{r-1})$ th roots, and subsequently using the identities $N_1 + N_2 = n$ and $(M_n- \\Delta _n)/m_n = (m_n+\\Delta _n)/m_n$ , we obtain $\\Big [& 1 + 4\\varepsilon _n -\\frac{\\Delta _n}{M_n} \\left( \\frac{N_2}{n} + 2 \\varepsilon _n \\right) \\Big ] (1+\\varepsilon _n)^{\\frac{1}{r-1}} + \\varepsilon _n^{\\prime }\\\\&\\ge \\Big [\\frac{m_n+\\Delta _n}{m_n} \\left(\\frac{N_1}{n} - 2\\varepsilon _n\\right)^+ + \\left(\\frac{N_2}{n} - 2\\varepsilon _n\\right)^+\\Big ] (1-\\varepsilon _n)^{\\frac{1}{r-1}} \\\\& \\ge \\left[ 1 - 4 \\varepsilon _n + \\frac{\\Delta _n}{M_n} \\left( \\frac{N_1}{n} - 2 \\varepsilon _n \\right)^+ \\right] (1 - \\varepsilon _n)^{\\frac{1}{r-1}},$ where $\\varepsilon _n^{\\prime } := (1+\\varepsilon _n) \\left(\\frac{2\\delta _n}{\\mu _n}\\right)^{\\frac{1}{r-1}}$ is $o(\\varepsilon _n)$ due to the assumption that $\\delta _n \\ll \\mu _n\\varepsilon _n$ and $r \\in (1,2]$ .", "Thus, $\\frac{\\Delta _n}{M_n} &\\left[ (1-\\varepsilon _n)^{\\frac{1}{r-1}} \\left( \\frac{N_1}{n} - 2 \\varepsilon _n \\right)^+ + \\left( \\frac{N_2}{n} + 2 \\varepsilon _n \\right) (1 + \\varepsilon _n)^{\\frac{1}{r-1}} \\right] \\\\& \\le \\left( 1+ \\varepsilon _n\\right)^{\\frac{1}{r-1}} - \\left( 1 - \\varepsilon _n\\right)^{\\frac{1}{r-1}} +4 \\varepsilon _n \\left[ \\left( 1+ \\varepsilon _n\\right)^{\\frac{1}{r-1}} + \\left( 1 - \\varepsilon _n\\right)^{\\frac{1}{r-1}} \\right] + \\varepsilon _n^{\\prime },$ which implies $\\frac{\\Delta _n}{M_n} \\le \\left( 8 +\\frac{2}{r-1}\\right) \\varepsilon _n + o(\\varepsilon _n).$ This proves (REF ), and hence, completes the proof of Proposition REF  (b).", "$\\Box $" ], [ "Approximation of the maximizer for random matrices", "In this section, we show that the assumptions in Proposition REF are satisfied almost surely by the sequence of random matrices of interest.", "This will complete the proofs of Theorems REF and REF .", "Let $\\mathbb {P}_0$ be any probability measure on $\\prod _{n} \\mathbb {R}^{n \\times n}$ , such that its projection on $\\mathbb {R}^{n\\times n}$ has the same law as $A_n$ , as defined in Assumption REF ." ], [ "Random matrices are almost regular and well-balanced", "In Lemmas REF and REF , we verify the almost regularity and well-balancedness conditions for the homogeneous and inhomogeneous instances of the random matrix sequences, respectively.", "Lemma 6.1 Let $(A_n)_{n \\in \\mathbb {N}}$ be a sequence of random matrices that satisfies Assumption REF , and recall from Theorem REF that $K = \\max \\big \\lbrace \\limsup _{n\\rightarrow \\infty }\\sigma _n^2/\\mu _n, 1\\big \\rbrace .$ Also, suppose that $\\varepsilon _n =\\sqrt{5K\\log n/(n\\mu _n)}.$ Then $(A_n)_{n \\in \\mathbb {N}}$ is $(\\varepsilon _n, \\mu _n)_{n\\in \\mathbb {N}}$ almost regular, $\\mathbb {P}_0$ -almost surely.", "Moreover, there exists a sequence $(\\delta _n)_{n \\in \\mathbb {N}}$ with $\\delta _n\\ll \\mu _n \\varepsilon _n$ such that $(A_n)_{n \\in \\mathbb {N}}$ is also $(\\varepsilon _n, \\delta _n, \\mu _n)_{n\\in \\mathbb {N}}$ well-balanced, $\\mathbb {P}_0$ -almost surely.", "Verification of almost regularity.", "First, note that $\\sum _{j\\in [n]\\setminus \\lbrace i\\rbrace }\\mathbb {E}[(a_{ij}^n- \\mu _n)^2] \\le n\\sigma _n^2$ and Assumption REF  (REF ) provides the moment conditions required for Bernstein's inequality (see [8]).", "Therefore, using the fact that $(a_{ij}^n)_{i<j}$ are i.i.d.", "as well as the union bound, and then applying [8] for both the upper and lower tails, we conclude that for all sufficiently large $n$ , $\\mathbb {P}$ ( i: |dn(i)-nn|>nnn) n $\\mathbb {P}$ ( |dn(1)-nn|>nnn) 2 n (- n2n2n22(nn2+cnnn)) 2 (- nnn22.4K + n), where the $c$ in the second step is defined in Assumption REF  (REF ), and in the last step we have used $\\varepsilon _n\\rightarrow 0$ and $K>0$ .", "Due to the choice of $\\varepsilon _n$ , the final term in (REF ) is at most $2n^{-13/12}$ , which is summable over $n$ .", "Thus the almost regularity holds $\\mathbb {P}_0$ -almost surely using the Borel-Cantelli lemma.", "Verification of well-balancedness.", "Fix any vertex set $V\\subset [n]$ .", "Define Xn(V):= |{i[n]V: |dn(i,V) - n|V||n n n }|.", "Thus, $X_n (V)$ is the size of the exception set in Definition REF .", "Also, throughout the proof, we fix 1n = (-nn2n4.9K),    and    2n = 1nnn.", "We will show that the following two properties hold $\\mathbb {P}_0$ -eventually almost surely: $\\mbox{(i)}&~\\max _{V\\subset [n]} X_n(V) \\le n \\delta _{1n},\\\\\\mbox{(ii)}&~\\max _{U:|U| \\le n \\delta _{1n}}\\max _i d_n(i,U) \\le \\delta _{2n} n\\mu _n.", "$ Note that (REF ) ensures Condition (a) in Definition REF , where the cardinality of the exception set $V_{\\rm ex}$ is at most $n \\delta _{1n}$ .", "Further, since () holds for any $U$ with $|U| \\le n \\delta _{1n}$ , this in particular holds for $U=V_{\\rm ex}$ , thereby ensuring Condition (b) in Definition REF with $\\delta _n = \\mu _n \\delta _{2n}$ .", "Finally, the requirement that $\\delta _n \\ll \\mu _n \\varepsilon _n$ in Lemma REF follows since $\\delta _{2n}\\ll \\varepsilon _n$ .", "We first prove (REF ).", "Applying [8] again to the functional $d_n(i,V) = \\sum _{j \\in V} a_{ij}^n$ , we conclude that for $i \\notin V$ , qn := $\\mathbb {P}\\left(|d_n(i,V) - \\mu _n|V||\\ge n\\varepsilon _n \\mu _n\\right)$ 2(- n2n2n22|V|(n2+cnn)) 2(-nn2n2.4K) for large enough $n$ , where in the second inequality we have used $|V|\\le n$ .", "Note that $(d_n(i,V))_{i\\in [n]\\setminus V}$ is an independent collection, and thus the variable $X_n(V)$ defined in (REF ) clearly satisfies $X_n(V) \\sim \\mathrm {Binomial} (|[n]\\setminus V|, q_n)$ .", "Now, by our choice, $q_n \\ll \\delta _{1n}$ , using (REF ), and thus for all sufficiently large $n$ , $\\mathbb {E}$ [Xn(V)] = |[n]V| qnnqn n1n2.", "Therefore, (REF ), (REF ), and an application of [32] yields $\\mathbb {P}\\left(X_n(V)\\ge n\\delta _{1n}\\right)$ $\\mathbb {P}\\left(\\big |X_n(V) - |[n]\\setminus V| q_n\\big |\\ge \\frac{n\\delta _{1n}}{2}\\right)$ 2( - n21n212 |[n]V|qn) 2( - n1n26(nnn22.4K)).", "Let $\\mathcal {C}_n = \\big \\lbrace |X_n(V) - |[n]\\setminus V| q_n| \\ge n\\delta _{1n} \\mbox{ \\emph {for some} }V\\subset [n]\\big \\rbrace $ denote the event that the matrix $A_n$ does not satisfy (REF ).", "Then, using the union bound over the choice of the set $V$ (for which there are at most $2^n$ possibilities), we can upper bound $\\mathbb {P}$ (Cn)2 ( - n1n26(nnn22.4K) + n2).", "Again, by our choice of $\\delta _{1n}$ , and the fact that $n\\mu _n \\varepsilon _n^2 \\gg 1$ , 1n2 (n nn22.4K)1, and thus by (REF ), we get $\\sum _{n=1}^{\\infty }\\mathbb {P}(\\mathcal {C}_n)\\le 2\\sum _{n=1}^\\infty \\exp ( - n)<\\infty $ .", "An application of the Borel-Cantelli lemma proves (REF ).", "Next, we prove ().", "An identical application of the concentration inequality [8] as in (REF ), together with a union bound, yields $\\mathbb {P}$ (U:|U| n 1ni dn(i,U) > 2n nn)    n n$\\left\\lfloor n\\delta _{1n} \\right\\rfloor $ U:|U| n 1n$\\mathbb {P}$ ( dn(1,U) > 2n nn)    n $\\mathrm {e}$ n1n (1/1n) ( - n2n2 nC1n)    = n( - n 1n(2n2 nC1n2-(1/1n))), for some constant $C\\in (0,\\infty )$ , where we have used the Stirling approximation that $\\binom{n}{k} = \\mathrm {e}^{(1+o(1))k \\log (n/k)}$ , whenever $k=o(n)$ .", "Now, using (REF ), $\\mu _n(\\delta _{1n}/\\delta _{2n})^2 = (\\log n)^2 \\gg \\log (1/\\delta _{1n})$ .", "Thus, the probability in (REF ) is summable (since $n\\mu _n \\gg \\log n$ ) and hence an application of Borel-Cantelli lemma yields ().", "This completes the verification of the well-balancedness property.", "The next lemma states the version of Lemma REF in the inhomogeneous variance case.", "Lemma 6.2 Let $(A_n)_{n \\in \\mathbb {N}}$ be a sequence of random matrices that satisfies Assumption REF , and recall from Theorem REF that $K_1 = \\max \\lbrace \\limsup _{n\\rightarrow \\infty }\\bar{\\sigma }_n^2/\\mu _n, 1\\rbrace .$ Also, suppose that $\\varepsilon _n = \\sqrt{5K_1 \\log n/(n \\mu _n)}$ .", "Then $(A_n)_{n \\in \\mathbb {N}}$ is $(\\varepsilon _n, \\mu _n)_{n\\in \\mathbb {N}}$ almost regular, $\\mathbb {P}_0$ -almost surely.", "Moreover, there exists a sequence $(\\delta _n)_{n \\in \\mathbb {N}}$ with $\\delta _n\\ll \\mu _n\\varepsilon _n$ such that $\\mathbb {P}_0$ -almost surely is also $(\\varepsilon _n, \\delta _n, \\mu _n)_{n\\in \\mathbb {N}}$ well balanced.", "[Proof of Lemma REF ] The proof follows verbatim from the steps of the proof of Lemma REF by replacing $\\sigma _n$ and $K$ by $\\bar{\\sigma }_n$ and $K_1$ , respectively." ], [ "Irreducibility of $A^TA$ for random matrices", "In this section we will show that Assumption REF  (REF ) (respectively, Assumption REF  (REF )) implies that the sequence $A_n^TA_n$ must be irreducible $\\mathbb {P}_0$ -eventually almost surely.", "We start by proving a useful graph theoretic lemma.", "Lemma 6.3 Let $A$ be the adjacency matrix of a graph $G$ .", "Then $A^TA$ is irreducible if and only if $G$ is connected and non-bipartite.", "For the if part, first construct a graph $G^{\\prime }$ with adjacency matrix $A^{\\prime } = (a^{\\prime }_{ij})$ defined as follows: $a^{\\prime }_{ij} = 1$ if $\\sum _{k=1}^n a_{ik}a_{kj}>0$ .", "Note that irreducibility of $A^TA$ is equivalent to the connectedness of $G^{\\prime }$ .", "Now, since we assume that $G$ is non-bipartite, there must exist a cycle with odd number of vertices.", "Let $(x_1, x_2,\\ldots , x_{2k+1})$ be such an odd cycle, where $k\\ge 1$ , $a_{x_i, x_{i+1}}=1$ for all $i=1,\\ldots , 2k$ , and $a_{x_1, x_{2k+1}}=1$ .", "Then it is easy to see that the following constitutes a cycle in in $G^{\\prime }$ : $(x_1, x_3,\\ldots , x_{2k+1}, x_2, x_4, \\ldots , x_{2k})$ .", "Next, take any vertex $u$ outside the above odd cycle.", "We will show that $u$ must be connected to some vertex in $\\lbrace x_1, x_2,\\ldots , x_{2k+1}\\rbrace $ in $G^{\\prime }$ , thus, completing the proof.", "Since $G$ is connected, there is a path $(u_1, u_2, \\ldots , u_{2m+1})$ for some $m\\ge 1$ , such that $u_1 = u$ , $u_m = x_{i^*}$ for some $i^*\\in \\lbrace 1,2,\\ldots , 2k+1\\rbrace $ , and $a_{u_j, u_{j+1}} = 1$ for all $j =1, \\ldots , 2m$ .", "In that case, observe that the following path exists in $G^{\\prime }$ : $(u_1, u_3, u_5\\ldots , u_{2m+1})$ , and hence, $u$ is connected to $x_{i^*}$ in $G^{\\prime }$ .", "For the only if part, note that trivially, if $G$ is not connected, then $G^{\\prime }$ is also not connected, and thus $A^TA$ is not irreducible.", "Also, if $G$ is bipartite with $V_1$ and $V_2$ being the two partitions, then observe that there are no edges between $V_1$ and $V_2$ in $G^{\\prime }$ , and thus, $G^{\\prime }$ is not connected.", "This completes the proof.", "Lemma 6.4 Let $(A_n)_{n\\in \\mathbb {N}}$ be a sequence of matrices with $A_n = (a_{ij}^n)_{i,j=1}^n$ .", "Assume that for each $n\\in \\mathbb {N}$ , $(a_{ij}^n)_{i,j}$ are independent random variables supported on $[0,\\infty )$ , with $\\min _{i,j}\\mathbb {P}(a_{ij}^n>0)\\ge \\frac{(2+\\varepsilon )\\log n}{n}$ .", "Then $A_n^TA_n$ is irreducible $\\mathbb {P}_0$ -eventually almost surely.", "For each $n\\in \\mathbb {N}$ , define $\\hat{A}_n= (\\hat{a}_{ij}^n)_{i,j=1}^n$ as $\\hat{a}_{ij}^n = 1\\lbrace a_{ij}^n>0\\rbrace $ , $i, j \\in [n]$ .", "Note that $\\hat{A}_n$ represents the adjacency matrix of an Erdős-Rényi random graph $G_n$ on $n$ vertices with edge probability $p_n = \\min _{i,j}\\mathbb {P}(a_{ij}^n>0)$ .", "Thus, $A_n^TA_n$ is irreducible if and only if $\\hat{A}_n^T\\hat{A}_n$ is irreducible.", "Using Lemma REF , it suffices to show that $G_n$ is connected and non-bipartite $\\mathbb {P}_0$ -eventually almost surely.", "To see this, let $Y_i := 1\\lbrace i \\text{ is isolated}\\rbrace $ .", "A well-known result in random graph literature (cf.", "[23]) yields that for $p_n\\ge (2+\\varepsilon )\\log n/n$ $\\mathbb {P}$ (Gn is disconnected) = $\\mathbb {P}$ ( i: Yi =1) +O(n-2) = n $\\mathbb {P}$ (Y1=1) +O(n-2) n (1-pn)n-1 +O(n-2) = $\\mathrm {e}$ n - npn(1+o(1))+O(n-2) = O(n-1-), which is summable over $n$ .", "Thus, $G_n$ is connected $\\mathbb {P}_0$ -eventually almost surely.", "Note that we have used $p_n \\ge (2+\\varepsilon )\\log n/n$ instead of the connectivity threshold $p_n \\ge (1+\\varepsilon )\\log n/n$ to ensure the summability above.", "Next, we will show that $G_n$ contains a triangle $\\mathbb {P}_0$ -eventually almost surely, so that $G_n$ cannot be bipartite.", "To this end, consider the Harris coupling of the $(\\mathrm {ER}_n(p))_{p\\in [0,1]}$ , where given an independent collection of uniform random variables $(U_{ij})_{i<j}$ , we keep an edge between $i$ and $j$ in $\\mathrm {ER}_n(p)$ is $U_{ij}\\le p$ .", "We say that $(i,j,k)$ is a wedge if $\\lbrace i,j\\rbrace $ and $\\lbrace i,k\\rbrace $ are both edges.", "Let $W_n(p)$ (respectively, $E_n(p)$ ) denote the total number of wedges (respectively, edges) in $\\mathrm {ER}_n(p)$ .", "First we claim the following: Claim 6.5 If $p_n^{\\prime } = \\log n /n$ , then Wn(pn') n (n)24    and   En(pn') 7n n2   $\\mathbb {P}$ 0-eventually almost surely.", "Let $d_i$ be the degree of vertex $i$ in $\\mathrm {ER}_n(p_n^{\\prime })$ .", "Then $d_i \\sim \\mathrm {Binomial}(n-1,\\mu _n)$ .", "Note that $W_n(p_n^{\\prime }) = \\frac{1}{2} \\sum _{i\\in [n]} d_{i}(d_i-1)$ .", "Clearly, $\\mathbb {E}[W_n(p_n^{\\prime })] = n(n-1)(n-2) \\mu _n^2/2 = (1+o(1)) n(\\log n)^2/2$ .", "We now prove a concentration for $W_n(p_n^{\\prime })$ using [32].", "Note that changing the status of an edge incident to vertex $i$ changes $W_n(p_n^{\\prime })$ by at most $d_i$ .", "Thus, for any $\\delta >0$ , $\\mathbb {P}$ (|Wn(pn')-$\\mathbb {E}$ [Wn(pn')]|> n(n)2) 2(- 2n2(n)42i[n] di3).", "Now, by [32], $\\mathbb {P}(\\exists \\ i: d_i \\ge \\sqrt{n}) \\le n \\mathbb {P}(d_1\\ge \\sqrt{n}) \\le 2n \\mathrm {e}^{- \\sqrt{n}}, \\\\\\mathbb {P}\\bigg (\\sum _{i\\in [n]} d_i > 7n^2p_n^{\\prime } \\bigg ) \\le \\mathrm {e}^{-7np_n^{\\prime }} = n^{-7} .$ Both the above probabilities are summable over $n$ .", "Thus, $\\sum _{i\\in [n]} d_i^3 \\le d_{\\max }^2 \\sum _{i\\in [n]} d_i \\le 7 n^2 \\log n $ $\\mathbb {P}_0$ -eventually almost surely.", "Applying this to (REF ) yields the desired result for wedges in (REF ).", "The concentration of edges follows using (REF ).", "Under the Harris coupling, we call the edges that are present in $\\mathrm {ER}_n(2\\log n/n)$ but are not present in $\\mathrm {ER}_n(\\log n/n)$ to be the sprinkled edges.", "Conditioned on $\\mathrm {ER}_n(\\log n/n)$ , the number of sprinkled edges is stochastically lower bounded by a $\\mathrm {Binomial}\\Big (\\binom{n}{2} - \\frac{7n\\log n}{2},\\ \\frac{\\log n}{n}\\Big (1 - \\frac{\\log n}{n}\\Big )^{-1} \\Big )$ random variable and each of these edges are uniformly distributed over the complement of $\\mathrm {ER}_n(\\log n/n)$ .", "Using standard Binomial concentration, we can say that the number of sprinkled edges is at least $n\\log n/4$ , $\\mathbb {P}_0$ -eventually almost surely.", "Also, note that whenever an edge appears at an edge location $\\lbrace j,k\\rbrace $ such that $(i, j, k)$ is a wedge, a triangle is formed.", "By Claim (REF ), the probability of the latter event is at least $n(\\log n)^2/4\\binom{n}{2}$ .", "Therefore, the number of triangles in $\\mathrm {ER}_n(2\\log n/n)$ , denoted by $T_n(2\\log n/n)$ , is stochastically lower bounded by a Binomial $(n\\log n/4, (\\log n)^2/2n)$ random variable.", "Therefore, $\\mathbb {P}\\big (T_n(2\\log n/n) =0\\big )&\\le \\Big (1- \\frac{(\\log n)^2}{2n}\\Big )^{\\frac{n\\log n}{4}}\\le \\mathrm {e}^{-(\\log n)^3/16},$ which is summable in $n$ .", "This shows that $\\mathrm {ER}_n(2\\log n/n)$ contains a triangle $\\mathbb {P}_0$ -eventually almost surely, and therefore cannot be bipartite.", "This completes the proof of Lemma REF .", "[Proofs of Theorems REF and REF ] Theorems REF and REF are immediate from Proposition REF , and Lemmas REF , REF , and REF , respectively." ], [ "Approximating the $r\\rightarrow p$ norm", "The purpose of this section is to approximate $\\Vert A_n\\Vert _{r\\rightarrow p}$ .", "We use the power iteration method described in Section  starting with initial vector $v_n^{(0)} = n^{-1/r}1$ .", "Applying the power method for one step yields the following quantity n(An) = An vn(1)p = Anr*(AnTp(An1))pr*(AnTp(An1))r, which will approximate $\\Vert A_n\\Vert _{r\\rightarrow p}$ .", "We will prove the following estimate: Proposition 7.1 Let $(A_n)_{n\\in \\mathbb {N}}$ , $(\\varepsilon _n)_{n\\in \\mathbb {N}}$ , and $(\\mu _n)_{n\\in \\mathbb {N}}$ satisfy identical conditions as Proposition REF .", "Then there exists a constant $C \\in (0,\\infty )$ (possibly depending on $p$ and $r$ ) such that for all sufficiently large $n$ , $\\big |\\Vert A_nv_n \\Vert _{p} - \\eta _n(A_n)\\big |\\le C\\frac{\\Lambda _2^2 (n)\\varepsilon _n}{\\mu _n^2 n^{\\frac{3}{2}+\\frac{1}{r}}}\\Vert A_n\\Vert _{2\\rightarrow p},$ where $\\eta _n$ is defined as in () and $\\Lambda _2^2 (n) :=\\max _{x:\\langle 1, x\\rangle =0,\\ x\\ne 0} \\frac{\\Vert A_nx\\Vert _2^2}{\\Vert x\\Vert _2^2}.$ The rest of this section is organized as follows.", "First, we estimate the closeness of $v_n^{(1)}$ to $v_n$ in Proposition REF .", "In particular, we show that under the assumptions of Proposition REF (equivalently, Proposition REF ), $v_n$ can be approximated well by $v_n^{(1)}$ .", "This is then used to approximate the operator norm and complete the proof of Proposition REF .", "Proposition 7.2 Assume that the conditions of Proposition REF are satisfied.", "Recall the definition of the $v$ -norm from (REF ).", "Then there exists a constant $ C_2 < \\infty $ , possibly depending on $p, r$ , such that for all sufficiently large $n$ , $\\Vert v_n - v_n^{(1)}\\Vert _{v_n}\\le C_2\\frac{\\Lambda _2^2(n)\\varepsilon _n}{n^2\\mu _n^2},$ where $\\Lambda _2(n)$ is as defined in (REF ).", "The rest of this subsection is devoted to the proof of Proposition REF .", "The proof of Proposition REF is given in the following subsection.", "The next lemma provides key ingredients for the proof of Proposition REF .", "Lemma 7.3 Assume that $(A_n)_{n \\in \\mathbb {N}}$ satisfies the conditions of Proposition REF and $1 < p \\le r < \\infty .$ Then the following hold: $\\lim _{n\\rightarrow \\infty } \\mu _n^{-1} n^{-(1+\\frac{1}{p}-\\frac{1}{r})}\\Vert A_n\\Vert _{r\\rightarrow p} = 1$ ; $\\max _{x: \\Vert x\\Vert _{v_n}\\le 1}\\Vert A_nx\\Vert _p = (1+o(1)) n^{\\frac{1}{2}- \\frac{1}{r}}\\max _{x: \\Vert x\\Vert _{2}\\le 1}\\Vert A_nx\\Vert _p; $ Let $\\lambda _2(n)$ be the second largest eigenvalue corresponding to the $v$ -Rayleigh quotient defined in (REF ).", "Then $\\lambda _2(n)\\le 2\\mu _n^{p-2}n^{\\frac{p(r-1)}{r} - 1} \\Lambda _2^2(n).$ (a) Observe that by Proposition REF and the almost regularity condition in Definition REF , $\\Vert A_n\\Vert _{r\\rightarrow p}= \\Vert A_nv_n\\Vert _p&= \\Vert A_n 1 (n^{-1/r} + o(n^{-1/r}))\\Vert _p\\\\&= \\Vert (n\\mu _n + o(n\\mu _n)) (n^{-1/r} + o(n^{-1/r}))1\\Vert _p\\\\&= \\mu _n n^{1-1/r+1/p} + o(\\mu _n n^{1-1/r+1/p}),$ from which the claim in (a) follows.", "(b) Note that, by (REF ) and Proposition REF , we have for all sufficiently large $n$ and $x \\in \\mathbb {R}^n,$ xvn =(i=1n|vn,i|r-2|xi|2)12 = n-r-22r x2 (1+ o(1)).", "Therefore, xvn 1 Anxp = x0 Anxpxvn = x0 Anxp(1+ o(1))n-r-22r x2 = nr-22r (1+ o(1))x2 1 Anxp, which proves (b).", "(c) Recall the inner product defined in (REF ) and that $\\gamma ^p$ is the largest eigenvalue of $B$ obtained from the $v$ -Rayleigh quotient (REF ).", "Thus, by using the Courant-Fischer theorem [5], note that $\\lambda _2(n) = \\min _{u\\ne 0} \\max _{x:[u,x]=0}\\frac{[Bx,x]}{[x,x]}&\\le \\max _{x:[|v_n|^{2-r} ,x]=0}\\frac{[Bx,x]}{[x,x]}\\\\&= \\max _{x:\\langle 1, x\\rangle =0} \\frac{[Bx,x]}{\\Vert x\\Vert _{v_n}^2}\\\\&\\le n^{1-\\frac{2}{r}} \\max _{x:\\langle 1, x\\rangle =0} \\frac{\\langle |A_nv_n|^{p-2}, |A_nx|^2\\rangle }{\\Vert x\\Vert _{2}^2}\\\\&\\le 2 \\mu _n^{p-2}n^{1-\\frac{2}{r}+(1-\\frac{1}{r})(p-2)}\\max _{x:\\langle 1, x\\rangle =0} \\frac{\\Vert A_nx\\Vert _2^2}{\\Vert x\\Vert _2^2}\\\\&\\le 2\\mu _n^{p-2}n^{\\frac{p(r-1)}{r} - 1} \\Lambda _2^2(n),$ where the second inequality follows since for any $x$ , $[|v_n|^{2-r} ,x]=0$ if an only if $\\langle 1, x\\rangle =0$ , and for the third and fourth inequalities we have used () and the almost regularity.", "Now we have the ingredients to complete the proof of Proposition REF .", "[Proof of Proposition REF ] Note that for all large enough $n$ , $\\Vert v_n\\Vert _{\\infty } \\le 2 n^{-1/r}$ by Proposition REF .", "Thus, for any $x\\in \\mathbb {R}^n$ with $\\Vert x\\Vert _{\\infty }\\le 1$ , it follows that xvn = (i=1n|vn,i|r-2|xi|2)1/2 21-2r n-12+1rx2 21-2rn1r x.", "We will now use the nonlinear power iteration described in Section  with $v_n^{(0)}=n^{-1/r}\\mathbf {1}$ .", "Then by (REF ) and (REF ), $v_n^{(1)} = Wv_n^{(0)}$ .", "Thus, $\\Vert v_n-v_n^{(1)}\\Vert _{v_n}&\\le (1+o(1))\\frac{(p-1)\\lambda _2}{(r-1)\\Vert A_n\\Vert _{r\\rightarrow p}^p}\\Vert v_n - n^{-1/r}\\mathbf {1}\\Vert _{v_n} \\\\& \\le (1+o(1))\\frac{2(p-1)}{r-1}\\frac{\\Lambda _2^2(n)}{n^2\\mu _n^2} \\Vert v_n - n^{-1/r}\\mathbf {1}\\Vert _{v_n} \\\\& \\le (1+o(1)) \\frac{2^{2-\\frac{2}{r}}(p-1)}{r-1}\\frac{\\Lambda _2^2(n)}{n^{2-\\frac{1}{r}}\\mu _n^2} \\Vert v_n - n^{-1/r}\\mathbf {1}\\Vert _{\\infty }\\\\&\\le C\\frac{\\Lambda _2^2(n)\\varepsilon _n}{n^2\\mu _n^2}$ where the first inequality is due to Proposition REF and the fact that $\\Vert A_n\\Vert _{r\\rightarrow p}^p = \\gamma ^p$ , the second inequality is due to Lemma REF  (REF ) and Lemma REF  (REF ), the third inequality is due to (), and the final inequality is due to Proposition REF .", "[Proof of Proposition REF ] Again, we consider the nonlinear power iteration method described in Section  with $v_n^{(0)}=n^{-1/r}\\mathbf {1}$ and $v_n^{(1)} = W v_n^{(0)}$ by (REF )–(REF ).", "Note that then $\\Vert v_n^{(1)}\\Vert _{r} = 1$ and hence, $\\eta _n(A_n) = \\Vert A_n v_n^{(1)}\\Vert _{p},$ and $\\big |\\Vert A_nv_n\\Vert _p - \\Vert A_nv^{(1)}_n\\Vert _p\\big |&\\le \\Vert A_nv_n - A_nv_n^{(1)}\\Vert _p \\\\&\\le \\Vert v_n - v_n^{(1)}\\Vert _{v_n} \\max _{\\Vert x\\Vert _{v_n} \\le 1} \\Vert A_nx\\Vert _p\\\\&\\le \\Vert v_n - v_n^{(1)}\\Vert _{v_n} (1+o(1)) n^{\\frac{1}{2}-\\frac{1}{r}}\\max _{\\Vert x\\Vert _{2} \\le 1} \\Vert A_nx\\Vert _p\\\\&\\le \\Vert v_n - v_n^{(1)}\\Vert _{v_n} (1+o(1)) n^{\\frac{1}{2}-\\frac{1}{r}} \\Vert A_n\\Vert _{2\\rightarrow p},\\\\&\\le C\\frac{\\Lambda _2^2 (n)\\varepsilon _n}{\\mu _n^2 n^{\\frac{3}{2}+\\frac{1}{r}}}\\Vert A_n\\Vert _{2\\rightarrow p},$ where the third inequality is due to Lemma REF (REF ) and the last inequality follows from Proposition REF .", "This completes the proof." ], [ "Asymptotic normality", "In this section we establish asymptotic normality of $\\eta _n(A_n)$ when $A_n$ satisfies Assumption REF .", "We start in Section REF with some preliminary results." ], [ "Almost-sure error bound on the CLT scale", "First, recalling the definition of $\\Lambda _2(n)$ in (REF ), we prove the following lemma.", "We will need the following definition of the centered matrix: $A_n^0 := A_n - \\mu _n 11^T + \\mu _n I_n.$ Lemma 8.1 Under Assumption REF and (REF ), the following holds: 2 (n) 3n n +n,    $\\mathbb {P}$ 0 eventually almost surely.", "First observe that for all vectors $x$ with $\\langle 1, x\\rangle =0$ we can write Anx2 = (An0 + n 11T - n In )x2 An0x2 + nx2 An0x2 + nx2, where $A_n^0$ is as defined in (REF ), and the last step follows from Cauchy-Schwarz inequality.", "Therefore, 2 (n) = x:1, x=0, x0 Anx2x2 x: x0 An0x2x2 +n.", "Now, define a matrix $H_n = (h_{ij}^n)_{1\\le i,j\\le n}$ as $H_n = A_n^0/\\sqrt{n}\\sigma _n $ .", "Next we will verify that the entries of $H_n$ satisfy the conditions in [35].", "For all $i\\in [n]$ , $h^n_{ii} = 0$ , and for all $i,j\\in [n]$ with $i\\ne j$ , $\\mathbb {E}[h_{ij}^n] = 0$ , $\\mathbb {E}[(h_{ij}^n)^2] = \\frac{1}{n}$ .", "Using Assumption REF  (REF ), there exists a fixed constant $c_1 >0$ , such that for all $n\\ge 1$ and $k\\ge 3$ , $\\mathbb {E}\\big [|h_{ij}^n|^k\\big ] &\\le \\frac{\\mathbb {E}\\big [|a_{ij}^n - \\mu _n|^k\\big ]}{n^{\\frac{k}{2}}\\sigma _n^k}\\le \\frac{k!", "}{2}\\frac{c^{k-2}\\sigma _n^2}{n^{\\frac{k}{2}}\\sigma _n^k}\\le (c_1k)^{c_1k}\\frac{1}{nq_n^{k-2}},$ where $q_n = \\sqrt{n}\\sigma _n \\gg n^{-c_0},\\quad \\mbox{due to~(\\ref {eq:sigma-lower-bound})},$ and $q_n = O(\\sqrt{n})$ as $\\sigma _n^2 = O(\\mu _n) = O(1)$ .", "Therefore using [35] (also see [20]), it follows that $\\max _{x:\\ x\\ne 0} \\frac{\\Vert A_n^0x\\Vert _2}{\\Vert x\\Vert _2} \\le 3\\sqrt{n} \\sigma _n.$ This, combined with (REF ) yields (REF ).", "Below we state a general version of Lemma REF that extends the result to the non-zero diagonal entries case.", "Lemma 8.2 Under Assumption REF and (REF ), and the assumptions for non-zero diagonal entries in Remark REF , the following holds: 2 (n) 3n n +n + 2n(n2+n2),    $\\mathbb {P}$ 0 eventually almost surely.", "The proof of Lemma REF follows verbatim from the proof of Lemma REF , except the upper-bound in (REF ) will be replaced by $\\Lambda _2 (n) = \\max _{x:\\langle 1, x\\rangle =0, x\\ne 0} \\frac{\\Vert A_nx\\Vert _2}{\\Vert x\\Vert _2}\\le \\max _{x:\\ x\\ne 0} \\frac{\\Vert A_n^0x\\Vert _2}{\\Vert x\\Vert _2} +\\mu _n + \\Big (\\sum _{i=1}^n (a_{ii}^n)^2\\Big )^{\\frac{1}{2}}$ and using LLN, we can bound $\\frac{1}{n}\\sum _{i=1}^n (a_{ii}^n)^2\\le 2(\\zeta _n^2+\\rho _n^2)$ , $\\mathbb {P}_0$ -almost surely.", "Rest of the proof is omitted.", "Next, we prove a bound on the error while approximating $\\Vert A_n\\Vert _{r\\rightarrow p}$ by $\\eta _n(A_n)$ .", "Lemma 8.3 Under the conditions of Theorem REF , the following holds $\\mathbb {P}_0$ -almost surely: $\\Vert A_nv_n \\Vert _{p} = \\eta _n (A_n)+o\\big ( \\sigma _n n^{\\frac{1}{p} - \\frac{1}{r}}\\big ),$ where $\\eta _n(\\cdot )$ is defined in ().", "We will show that $\\mathbb {P}_0$ -eventually almost surely, $\\big |\\Vert A_nv_n \\Vert _{p} - \\eta _n(A_n)\\big |\\le C \\frac{\\sigma _n^2}{\\mu _n} n^{\\frac{1}{p} - \\frac{1}{r}} \\sqrt{\\frac{\\log n}{n\\mu _n}},$ for some constant $C>0$ , not depending on $n$ .", "In that case, Lemma REF is immediate from using the assumption that $\\sigma _n^2 = O(\\mu _n)$ , and the condition from (REF ) that $\\mu _n \\sigma _n^2 \\gg \\frac{\\log n}{n}$ .", "It follows from Lemma REF that under Assumption REF , and associated constants $(\\mu _n)_{n \\in \\mathbb {N}},$ $(\\sigma _n)_{n \\in \\mathbb {N}},$ the sequence $(A_n)_{n \\in \\mathbb {N}}$ is $\\mathbb {P}_0$ -almost surely $(\\varepsilon _n, \\mu _n)_{n\\in \\mathbb {N}}$ almost regular and $(\\varepsilon _n, \\delta _n, \\mu _n)_{n\\in \\mathbb {N}}$ well-balanced for $(\\varepsilon _n)_{n \\in \\mathbb {N}}, (\\delta _n)_{n \\in \\mathbb {N}}$ such that $\\varepsilon _n = \\Theta (\\sqrt{\\frac{\\log n}{n\\mu _n}})$ , and $\\delta _n \\ll \\mu _n \\varepsilon _n$ .", "In particular, the conditions of Proposition REF are satisfied and we can apply Proposition REF .", "Now we claim the following: Claim 8.4 For $p\\ge 1$ , $\\Vert A_n\\Vert _{2\\rightarrow p} = (1+o(1)) \\mu _n n^{\\frac{1}{2} + \\frac{1}{p}}$ , $\\mathbb {P}_0$ -eventually almost surely.", "For $1\\le p\\le 2$ , the claim is immediate from Lemma REF  (REF ).", "For $p>2$ , recall the centered matrix $A_n^0$ from (REF ) and note that An2p = x21 Anxp x21 ($\\mathbb {E}$ An)xp + x21 An0xp.", "For the first term on the right-hand-side of (REF ), $\\Vert \\mu _n(11^T - I_n) x\\Vert _{p } \\le \\mu _n (\\Vert x\\Vert _1 n^{1/p} + \\Vert x\\Vert _{p})$ , and thus x21 ($\\mathbb {E}$ An)xpn n12+ 1p + n, where we have used $\\max _{\\Vert x\\Vert _{2}\\le 1} \\Vert x\\Vert _1 = n^{1/2}$ , and $\\Vert x\\Vert _{p} \\le \\Vert x\\Vert _2\\le 1$ for $p>2$ .", "Further, using again the fact that $\\Vert x\\Vert _p\\le \\Vert x\\Vert _2$ for $p>2$ and any $x\\in \\mathbb {R}^n$ , we get x21 An0xpx21 An0x23n n +n, $\\mathbb {P}_0$ -eventually almost surely, where the last inequality uses (REF ).", "Applying the bounds in (REF ) and (REF ) to (REF ) completes the proof of the claim for $p> 2$ .", "Now observe that $\\big |\\Vert A_nv_n \\Vert _{p} - \\eta _n(A_n)\\big |\\le C_1\\frac{ \\Lambda _2^2(n) \\varepsilon _n}{\\mu _n n^{1-\\frac{1}{p}+\\frac{1}{r}}}\\le C_2 \\frac{\\sigma _n^2}{\\mu _n} n^{\\frac{1}{p} - \\frac{1}{r}} \\sqrt{\\frac{\\log n}{n\\mu _n}},$ $\\mathbb {P}_0$ -eventually almost surely, for constants $C_1,C_2>0$ , where the first inequality is due to Proposition REF and Claim REF , and the last step is due to Lemma REF and the choice of $\\varepsilon _n$ .", "This completes the proof of (REF ).", "Now we proceed with the proof of asymptotic normality using Taylor expansion of $\\eta _n(A_n)$ .", "Given any symmetric matrix $A= (a_{ij}^n)_{1\\le i,j\\le n}$ , we view $A$ as a vector $a\\in \\mathbb {R}^{\\binom{n}{2}}$ consisting of $(a_{ij}^n)_{1\\le i<j\\le n}$ arranged in a column vector.", "Thus, we will interchangeably write $\\eta _n(A)$ as $\\eta _n(a)$ .", "Also, let $d_{n,a}$ denote the degree vector, i.e., $d_{n,a}(i) = \\sum _{j} a_{ij}^n$ .", "Let us denote n:= {a[0,)n2: dn,a -nn1n nn}.", "First, note that $\\Omega _n$ is a convex subset of $\\mathbb {R}^{\\binom{n}{2}}$ .", "Due to the convexity of $\\Omega _n$ , we can Taylor expand $\\eta (a)$ on $\\Omega _n$ so that the intermediate points associated to the remainder terms also lies in $\\Omega _n$ , see [26].", "Specifically, Taylor expanding $\\eta (a)$ at the mean vector $\\mathbb {E}\\left[a\\right]$ , we can write (a) = ($\\mathbb {E}\\left[a\\right]$ ) + D ($\\mathbb {E}$ [a]), (a-$\\mathbb {E}\\left[a\\right]$ )+ 12(a-$\\mathbb {E}\\left[a\\right]$ )T H($\\mathbb {E}\\left[a\\right]$ ) (a-$\\mathbb {E}\\left[a\\right]$ ) +16(1) R(, )(, )(, )()(a-)(a-)(a-), where $D(\\mathbb {E}(a))$ is the gradient, $\\mathbb {H}(\\mathbb {E}\\left[a\\right])$ is the Hessian matrix, evaluated at $\\mathbb {E}\\left[a\\right]$ , $\\xi \\in \\Omega _n$ , $\\sum {^{(1)}}$ is the sum over all $\\alpha < \\beta , \\gamma < \\delta , \\kappa < \\rho $ , and $R$ is the 3-dimensional array of partial derivatives of third order evaluated at $\\xi $ , i.e., $R_{(\\alpha , \\beta )(\\gamma , \\delta )(\\kappa , \\rho )}(\\xi ) = \\frac{\\partial ^3\\eta }{\\partial a_{\\alpha \\beta }\\partial a_{\\gamma \\delta }\\partial a_{\\kappa \\rho }} (a)\\bigg |_{a= \\xi }.$ The next two lemmas identify the necessary asymptotic properties of the derivatives involved in (REF ).", "Their proofs are given in Appendix REF .", "Lemma 8.5 The following hold: As $n\\rightarrow \\infty $ , $n^{-(\\frac{1}{p} - \\frac{1}{r} -1)}D(\\mathbb {E}[a])\\big )_{(\\alpha , \\beta )} \\rightarrow 2, \\quad \\text{ uniformly in }(\\alpha , \\beta ).$ As $n\\rightarrow \\infty $ , $n^{-(\\frac{1}{p}-\\frac{1}{r}-1)}n\\mu _n (\\mathbb {H}(\\mathbb {E}\\left[a\\right]))_{(\\alpha , \\beta ), (\\alpha , \\beta )} \\rightarrow 2\\Big (p-1 + \\frac{1}{r-1}\\Big ) \\quad \\text{ uniformly in }(\\alpha , \\beta ).$ There exist positive finite constants $C_1,C_2$ (possibly depending on $p$ and $r$ ), such that $\\limsup _{n\\rightarrow \\infty }n^{-(\\frac{1}{p}-\\frac{1}{r}-1)}n\\mu _n \\big |(\\mathbb {H}(\\mathbb {E}\\left[a\\right]))_{(\\gamma , \\delta ), (\\alpha , \\gamma )} \\big |\\le C_1, \\quad \\text{ if } |\\lbrace \\alpha ,\\beta ,\\gamma ,\\delta \\rbrace | = 3,\\\\\\limsup _{n\\rightarrow \\infty }n^{-(\\frac{1}{p}-\\frac{1}{r}-2)}n\\mu _n \\big |(\\mathbb {H}(\\mathbb {E}\\left[a\\right]))_{(\\gamma , \\delta ), (\\alpha , \\gamma )} \\big |\\le C_2, \\quad \\text{ if } |\\lbrace \\alpha ,\\beta ,\\gamma ,\\delta \\rbrace | = 4.$ Lemma 8.6 Fix $1\\le \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\le n$ and $\\xi \\in \\Omega _n$ .", "Then the following hold for constants $C_1,C_2,C_3,C_4$ : If $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |\\le 3$ , then $R_{(\\alpha , \\beta )(\\gamma , \\delta )(\\kappa , \\rho )}(\\xi ) \\le C_1 (n\\mu _n)^{-4}n^{-1+\\frac{1}{p}-\\frac{1}{r}}.$ If $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace | =4$ , and if the sets $\\lbrace \\alpha ,\\beta \\rbrace , \\lbrace \\gamma ,\\delta \\rbrace , \\lbrace \\kappa ,\\rho \\rbrace $ are pairwise non-disjoint, then $R_{(\\alpha , \\beta )(\\gamma , \\delta )(\\kappa , \\rho )}(\\xi ) \\le C_2 (n\\mu _n)^{-4}n^{-1+\\frac{1}{p}-\\frac{1}{r}},$ and otherwise, $R_{(\\alpha , \\beta )(\\gamma , \\delta )(\\kappa , \\rho )} (\\xi ) \\le C_3 (n\\mu _n)^{-4}n^{-2+\\frac{1}{p}-\\frac{1}{r}}.$ For $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace | \\ge 5$ , $R_{(\\alpha , \\beta )(\\gamma , \\delta )(\\kappa , \\rho )} (\\xi ) \\le C_4 (n\\mu _n)^{-4}n^{-2+\\frac{1}{p}-\\frac{1}{r}}.$ The computation of the derivative is provided in Appendix .", "The next lemma provides asymptotics of the polynomials appearing in the Taylor expansion.", "To this end, for $k = 1, \\ldots , 6$ , let $X_{k}^n &:= \\sum _{\\begin{array}{c}\\alpha <\\beta , \\gamma < \\delta \\\\ |\\lbrace \\alpha ,\\beta ,\\gamma ,\\delta \\rbrace | = k\\end{array}} (a_{\\alpha \\beta }- \\mu _n) (a_{\\gamma \\delta }- \\mu _n), \\\\Y_{k}^n &:= \\sum _{\\begin{array}{c}\\alpha <\\beta , \\gamma < \\delta , \\kappa < \\rho \\\\ |\\lbrace \\alpha , \\beta , \\gamma , \\delta , \\kappa , \\rho \\rbrace | = k\\end{array}} (a_{\\alpha \\beta }- \\mu _n) (a_{\\gamma \\delta }- \\mu _n) (a_{\\kappa \\rho } - \\mu _n).$ Lemma 8.7 Recall that $\\sigma _n^2 = \\mathrm {Var}((a_{ij}^n)^2)$ for $i\\ne j$ .", "Then, Xkn = {ll n22 n2 + o$\\mathbb {P}$ ( nn),     if k = 2 O$\\mathbb {P}$ (n3/2 n2),     if k = 3 O$\\mathbb {P}$ (n2 n2),     if k= 4. .", "Lemma 8.8 The following hold: For $k\\le 3$ , $Y_k^n = O_{\\scriptscriptstyle \\mathbb {P}}(n^{2}\\sigma _n^{3})$ .", "For $k =4$ , let $Y_{4,1}^n $ be the partial sum of $Y_4^n$ over indices such that $\\lbrace \\alpha ,\\beta \\rbrace , \\lbrace \\gamma ,\\delta \\rbrace , \\lbrace \\kappa ,\\rho \\rbrace $ are pairwise non-empty, and let $Y_{4,2}^n$ be the rest of the sum.", "Then $Y_{4,1}^n= O_{\\scriptscriptstyle \\mathbb {P}}(n^2\\sigma _n^{3})$ and $Y_{4,2}^n = O_{\\scriptscriptstyle \\mathbb {P}}(n^3\\sigma _n^{3})$ .", "For $k \\ge 5$ , $Y_k^n = O_{\\scriptscriptstyle \\mathbb {P}}(n^3\\sigma _n^{3})$ .", "Lemmas REF  and REF are proved in Appendix ." ], [ "Proof of asymptotic normality", "We now complete the proofs of Theorem REF and Theorem REF .", "[Proof of Theorem REF ] Note that Lemma REF ensures that $\\eta _n(a^n)$ approximates $\\Vert A_n\\Vert _{r\\rightarrow p}$ on the fluctuation scale, that is, $\\big |\\Vert A_n\\Vert _{r\\rightarrow p} - \\eta _n(a^n)\\big | = o\\big ( \\sigma _n n^{\\frac{1}{p} - \\frac{1}{r}}\\big )\\quad \\mathbb {P}_0\\mbox{-almost surely}.$ Thus, it is enough to prove (REF ) when $\\Vert A_n\\Vert _{r\\rightarrow p}$ is replaced with $\\eta _n(a^n)$ .", "Now recall the Taylor expansion of $\\eta _n(a^n)$ from (REF ).", "First $\\eta _n(\\mathbb {E}[a^n]) = \\mu _n n^{1+\\frac{1}{p}-\\frac{1}{r}} $ .", "Since $a_{ij}^n$ 's are iid, we can apply the central limit theorem [7] to $\\sum _{i<j} (a_{ij}^n- \\mu _n)$ by verifying the Lyapunov condition [7].", "Let $s_n^2 = \\binom{n}{2} \\sigma _n^2$ and 1sn3 i<j $\\mathbb {E}$ [|aijn- n|3] Cn2 n2n3n3 = O(1nn), where we have used Assumption REF  (REF ).", "Now, $n\\sigma _n \\rightarrow \\infty $ by (REF ).", "This verifies that the expression on the right-hand side of (REF ) is $o(1)$ .", "Consequently, $\\frac{\\sqrt{2}\\sum _{i<j} (a_{ij}^n- \\mu _n)}{n \\sigma _n} \\xrightarrow{}\\mathrm {Normal}(0, 1),$ and thus, Lemma REF  (a) yields $\\frac{\\langle D(\\mathbb {E}[a^n]), a^n - \\mathbb {E}[a^n] \\rangle }{ \\sigma _n n^{\\frac{1}{p} - \\frac{1}{r}}} \\xrightarrow{}\\mathrm {Normal}(0, 2).", "$ Next, an application of Lemma REF  (b), (c), together with Lemma REF , shows that the Hessian term in (REF ) is equal to $&2\\Big (p-1+\\frac{1}{r-1}\\Big ) n^{\\frac{1}{p} - \\frac{1}{r}} \\frac{1}{n^2\\mu _n}\\sum _{i<j}(a_{ij}^n-\\mu _n)^2 + O_{\\scriptscriptstyle \\mathbb {P}}\\bigg (\\frac{n^{3/2} \\sigma _n^2 n^{\\frac{1}{p}- \\frac{1}{r}-1}}{ n\\mu _n}\\bigg ) + O_{\\scriptscriptstyle \\mathbb {P}}\\bigg (\\frac{n^2 \\sigma _n^2 n^{\\frac{1}{p}- \\frac{1}{r}-2}}{n\\mu _n}\\bigg )\\\\& = \\Big (p-1+\\frac{1}{r-1}\\Big ) \\frac{\\sigma _n^2}{\\mu _n} n^{\\frac{1}{p} - \\frac{1}{r}}+ n^{\\frac{1 }{p} - \\frac{1}{r}} \\sigma _n \\bigg (O_{\\scriptscriptstyle \\mathbb {P}}\\Big (\\frac{1}{n\\mu _n}\\Big )+ O_{\\scriptscriptstyle \\mathbb {P}}\\Big (\\frac{\\sigma _n}{\\sqrt{n}\\mu _n}\\Big )+ O_{\\scriptscriptstyle \\mathbb {P}}\\Big (\\frac{\\sigma _n}{n\\mu _n}\\Big )\\bigg )\\\\& = \\Big (p-1+\\frac{1}{r-1}\\Big ) \\frac{\\sigma _n^2}{\\mu _n} n^{\\frac{1}{p} - \\frac{1}{r}}+ o_{\\scriptscriptstyle \\mathbb {P}}\\big (n^{\\frac{1 }{p} - \\frac{1}{r}} \\sigma _n\\big ),$ where in the last step we have used that $n\\mu _n\\gg 1$ and $\\frac{\\sigma _n}{\\sqrt{n}\\mu _n} = \\frac{\\sigma _n^2}{\\mu _n} \\frac{1}{\\sqrt{n} \\sigma _n} = o(1),$ since $\\sigma _n^2 = O(\\mu _n)$ and $\\sigma _n \\ge n^{-\\frac{1}{2}+\\varepsilon }$ .", "On the other hand, the fourth term in (REF ) is $O_{\\scriptscriptstyle \\mathbb {P}}\\bigg (\\frac{n^2\\sigma _n^3 n^{-1+\\frac{1}{p}-\\frac{1}{r}}}{(n\\mu _n)^4}\\bigg )+ O_{\\scriptscriptstyle \\mathbb {P}}\\bigg (\\frac{n^3 \\sigma _n^3 n^{-2+\\frac{1}{p}-\\frac{1}{r}}}{(n\\mu _n)^4}\\bigg ) = O_{\\scriptscriptstyle \\mathbb {P}}\\bigg (n^{\\frac{1}{p}-\\frac{1}{r}} \\sigma _n \\frac{n\\sigma _n^2}{(n\\mu _n)^4}\\bigg ) = o_{\\scriptscriptstyle \\mathbb {P}}\\big (n^{\\frac{1}{p}-\\frac{1}{r}} \\sigma _n\\big ),$ where in the last step, we have used that $\\frac{n\\sigma _n^2}{(n\\mu _n)^4} = \\frac{\\sigma _n^2}{\\mu _n}\\frac{1}{(n\\mu _n)^3} = o(1)$ since $n\\mu _n\\gg 1$ and $\\sigma _n^2 = O(\\mu _n)$ .", "This concludes the proof of Theorem REF .", "We now turn to the proof of asymptotic normality in the dense, inhomogeneous case.", "First we will prove a version of Lemma REF in this inhomogeneous case.", "Lemma 8.9 Let $(A_n)_{n\\in \\mathbb {N}}$ be a sequence of random matrices satisfying the conditions of Theorem REF .", "Then the following holds: 2 (n) 3c*n n +n,    $\\mathbb {P}$ 0 eventually almost surely, where recall that $c^*>0$ is a constant defined in Assumption REF .", "The proof of this lemma follows similar argument as in Lemma REF , with a key difference that the bound on the $2\\rightarrow 2$ norm of the centered random matrix needs a more careful treatment.", "[Proof of Lemma REF ] Recall the centered matrix $A_n^0$ from the proof of Lemma REF .", "The proof of Lemma REF can follow verbatim from the proof of Lemma REF except the bound on $\\Vert A_n^0x\\Vert _2/\\Vert x\\Vert _2$ .", "Therefore, we will complete this proof by only providing the required bound on $\\Vert A_n^0x\\Vert _2/\\Vert x\\Vert _2$ under the conditions of Theorem REF .", "In particular, we will show that $\\frac{\\Vert A_n^0x\\Vert _2}{\\Vert x\\Vert _2}\\le 3c^* \\sqrt{n}\\bar{\\sigma }_n, \\quad \\mathbb {P}_0 \\mbox{ eventually almost surely}.$ Define a matrix $H_n = (h_{ij}^n)_{1\\le i,j\\le n}$ as $H_n = A_n^0/\\sqrt{n}\\bar{\\sigma }_n $ .", "Next we will verify that under the assumptions of Theorem REF , the entries of $H_n$ satisfy the exact conditions in [2] as stated below.", "$h^n_{ii} = 0$ for all $i\\in [n]$ and $\\mathbb {E}\\big [h_{ij}^n\\big ] = 0$ for all $i,j\\in [n]$ , $i\\ne j$ .", "By Assumption REF   (REF ), for all sufficiently large $n$ , $\\frac{c_*}{n} \\le \\min _{i,j}\\mathbb {E}\\left[(h_{ij}^n)^2\\right]\\le \\max _{i,j}\\mathbb {E}\\left[(h_{ij}^n)^2\\right] \\le \\frac{c^*}{n}$ Using (REF ), for all sufficiently large $n$ , $\\mathbb {E}\\big [|h_{ij}^n|^k\\big ] &\\le \\frac{\\mathbb {E}\\big [|a_{ij}^n - \\mu _n|^k\\big ]}{n^{\\frac{k}{2}}\\sigma _n^k}\\le \\frac{c_k}{n^{k/2}}.$ Now, using Geršgorin's circle theorem [25], observe that the largest eigenvalue of the matrix $\\big (\\mathbb {E}[(h_{ij}^n)^2]\\big )_{i,j}$ is upper-bounded by $2c^*\\bar{\\sigma }_n^2$ .", "Therefore using [2], the proof of (REF ) follows.", "The next lemma proves a version of Lemma REF in the inhomogeneous variance case.", "Lemma 8.10 Let $(A_n)_{n\\in \\mathbb {N}}$ be a sequence of random matrices satisfying the conditions of Theorem REF .", "Then the following holds $\\mathbb {P}_0$ -almost surely: $\\Vert A_nv_n \\Vert _{p} = \\Vert A_nv_n^{(1)}\\Vert _{p}+o\\big ( \\bar{\\sigma }_n n^{\\frac{1}{p} - \\frac{1}{r}}\\big ).$ The proof follows verbatim from the proof of Lemma REF , by using Lemma REF in place of Lemma REF .", "[Proof of Theorem REF ] Note that Lemma REF ensures that under the conditions of Theorem REF , $\\eta _n(a^n)$ approximates $\\Vert A_n\\Vert _{r\\rightarrow p}$ on the fluctuation scale, that is, $\\big |\\Vert A_n\\Vert _{r\\rightarrow p} - \\eta _n(a^n)\\big | = o\\big ( \\bar{\\sigma }_n n^{\\frac{1}{p} - \\frac{1}{r}}\\big )\\quad \\mathbb {P}_0\\mbox{-almost surely}.$ The rest of proof follows the exact same steps as in the proof of Theorem REF , by using $\\sum _{i<j}\\sigma _n^2(i,j)$ in place of $n^2\\sigma ^2/2$ , the upper bound $\\big (c^*\\bar{\\sigma }_n)^2$ for the variances of the entries, and using the CLT $\\frac{ \\sum _{i<j} (a_{ij}^n- \\mu _n)}{ \\sqrt{ \\sum _{i<j}\\sigma _n^2(i,j)}} \\xrightarrow{}\\mathrm {Normal}(0, 1),$ in place of (REF )." ], [ "Relation to the $\\ell _r$ Grothendieck problem", "We end this section with the proof of Proposition REF .", "[Proof of Proposition REF ] Let $x^* \\in \\mathbb {R}^n$ be a maximizer of $x^T A x$ with $\\Vert x^*\\Vert = 1$ .", "Then, using Lagrange multipliers method, there exists $\\kappa \\in \\mathbb {R}$ such that if $g: \\mathbb {R}^n \\mapsto \\mathbb {R}$ is the function given by $ g(x) = x^T A x- \\kappa \\left(\\Vert x\\Vert _r^r -1 \\right),$ then $x^*$ solves the equation $\\nabla g(x) = 2 A x- \\kappa r \\Psi _r (x) = 0,$ where recall $\\Psi _r(x) = |x|^{r-1} sgn(x).$ Taking the inner product of $x^*$ with the left-hand side of (REF ) evaluated at $x = x^*$ , and using the fact that $\\langle x^*, \\Psi _r(x^*) \\rangle =\\Vert x^*\\Vert _r = 1$ , it can be seen that $M_r (A ) = \\sup _{\\Vert x\\Vert _{r}\\le 1} x^T A x= (x^*)^T A x^* = \\frac{\\kappa r}{2}.$ Now, fix any nonnegative solution $y$ of (REF ).", "It follows that r*(ATny) = (r2)1r-1y and also, for $r\\ge 2$ and $p = r^*= r/(r-1)$ , p(A y) = (r2)p-1p(r(y)) A Tp(A y) = (r2)p-1ATp(r(y)) Sy= r*(ATp(Ay)) = (r2)p-1r-1r*(ATp(r(y))).", "Choosing $p = r^*= r/(r-1)$ , we have $\\Psi _p(\\Psi _r(y)) = y$ , and thus Sy= (r2)p-1r-1r'(ATy) = (r2)pr-1y,   due to ().", "Therefore, $Sy\\propto y$ .", "Also, note that since $r\\ge 2$ , we have $p=r^*\\le r$ .", "Thus, from Lemma REF , we know that $Sx= \\gamma ^{\\frac{p}{r-1}} x$ has a unique solution in $x$ that has all positive entries when $A$ is a symmetric matrix with nonnegative entries and $A^TA$ is irreducible (see Proposition REF ).", "Therefore, (REF ) also has a unique positive solution $x^*$ and for $p=r^*$ , Arppr-1 = (r2)pr-1 Arp = r2.", "Therefore, (REF ) yields that $M_r (A) = \\Vert A\\Vert _{r\\rightarrow r^*}$ and the proof follows.", "toc" ], [ "Proof of Lemma ", "First note that $\\mathbb {E}[X_{2}^n] = \\binom{n}{ 2} \\sigma _n^2 = \\frac{n^2\\sigma _n^2}{2} +O(n\\sigma _n^2).$ Next we recall the moment bounds in Assumption REF .", "Let $C>0$ be a generic notation for an absolute constant, whose value can change in different equations.", "Thus, Var(X2n) = Var(< (a-n)2) = < Var((a-n)2) = n2 $\\mathbb {E}$ [(a12 - n)4] C n2n2, where the final step uses Assumption REF  (REF ).", "Thus, an application of Chebyshev's inequality yields the asymptotics for $k = 2$ .", "For $k \\ge 3$ , $E[X_{k}^n] = 0$ .", "Further, when summing over $|\\lbrace \\alpha ,\\beta ,\\gamma ,\\delta \\rbrace | = 3$ , let $\\alpha = \\gamma $ .", "Thus $\\mathbb {E}$ [(X3n)2] = 9 $\\mathbb {E}$ [c1<1, 1<1 2<2, 2<2 (a11-n) (a11-n) (a22-n) (a22-n)] = C << $\\mathbb {E}$ [(a-n)2] $\\mathbb {E}$ [(a-n)2] = O(n3 n4).", "Therefore, Markov's inequality yields that $X_{3}^n = O_{\\scriptscriptstyle \\mathbb {P}}(n^{3/2}\\sigma _n^2)$ .", "Similarly, $\\mathbb {E}$ [(X4n)2] C <<<$\\mathbb {E}$ [(a - n)2]$\\mathbb {E}$ [(a - n)2] = O(n4 n4).", "Therefore, $X_{4}^n = O_{\\scriptscriptstyle \\mathbb {P}}(n^2 \\sigma _n^2)$ , and the proof follows.", "$\\Box $" ], [ "Proof of Lemma ", "The argument is similar to Lemma REF .", "We will use the moment bounds from Assumption REF  (REF ).", "First, it is always the case that $\\mathbb {E}[Y_k^n] = 0$ .", "Next, for $k = 2$ , $\\mathbb {E}$ [(Y2n)2] =< $\\mathbb {E}$ [(a-n)6] C n2 n2.", "For $k = 3$ , if we take the sum over indices with $\\rho \\ne \\beta $ , then $\\mathbb {E}$ [(Y3n)2] C $\\mathbb {E}$ [(<< (a - n)2 (a - n))2] = C << $\\mathbb {E}$ [(a - n)4] $\\mathbb {E}$ [(a - n)2]       + C 1<12< $\\mathbb {E}$ [(a1 - n)2]$\\mathbb {E}$ [(a2 - n)2]$\\mathbb {E}$ [(a - n)2] C n3 n4 + C n4 n6= O(n4n6), where the last step follows using $n\\sigma _n^2 \\gg 1$ .", "Thus the asymptotics of $Y_k^n$ for $k\\le 3$ follows from (REF ) and (REF ) and the fact that $n\\sigma _n^2 \\gg 1$ .", "For $k \\ge 5$ , the only non-zero expectation comes from product of squares.", "For this reason, it also follows that $\\mathbb {E}$ [(Ykn)2] = O(n6 n6).", "For $k = 4$ , if the index sets $\\lbrace \\alpha ,\\beta \\rbrace , \\lbrace \\gamma ,\\delta \\rbrace , \\lbrace \\kappa ,\\rho \\rbrace $ are pairwise non-empty then it must be the case that these pairwise intersections have cardinalities exactly equal to 1.", "In that case, again the only non-zero expectation comes from product of squares.", "Since one must choose only four indices $\\mathbb {E}$ [(Y4,1n)2] =O(n6 n6).", "Also with $|\\lbrace \\alpha ,\\beta ,\\gamma ,\\delta ,\\kappa ,\\rho \\rbrace | = 4$ , if $\\lbrace \\alpha ,\\beta \\rbrace \\cap \\lbrace \\kappa ,\\rho \\rbrace = \\varnothing $ , then $\\lbrace \\alpha ,\\beta \\rbrace = \\lbrace \\gamma ,\\delta \\rbrace $ .", "Thus $Y_{4,2}^n \\le C\\sum _{\\alpha < \\beta <\\kappa < \\rho } (a_{\\alpha \\beta } - \\mu _n)^2(a_{\\kappa \\rho } - \\mu _n)$ .", "Using identical arguments as (REF ), it follows that $\\mathbb {E}$ [(Y4,2n)2] =O(n4 n6) Thus the proof of Lemma REF is complete using Markov's inequality.", "$\\Box $" ], [ "Calculation of derivatives", "Define the operator $F$ Fx= AT p(Ax).", "Thus $S1 = \\Psi _{r^{\\prime }}(F1)$ .", "Let $G_{\\alpha \\beta }= (e_\\alpha e_\\beta ^T+ e_\\beta e_\\alpha ^T)$ and $e_{\\alpha \\beta }= e_{\\alpha } +e_{\\beta }$ .", "First we compute all the derivatives of $F$ .", "The derivatives of $S1$ and $AS1$ will be computed using recursion.", "Throughout this section we write generic constants $C_1,C_2,\\dots $ , whose value can be different from line to line." ], [ "First derivatives", "(,)F1 = Gp(A1) + (p-1) AT( p-1 e).", "Thus, (,)F1 A = J = (n)p-1 e+ 2(p-1) (n)p-2 1 = (n)p-2[ne+ 2(p-1) 1]." ], [ "Second derivatives", "(,)(,)F1 = (p-1)[G(p-1(A1)e)+ G(p-1(A1)e) +(p-2)AT(p-2(A1)ee)]" ], [ "Case 1: $(\\alpha , \\beta ) = (\\gamma , \\delta )$ .", "(,)(,)F1A = J= 2(p-1)[(n)p-2e+ (p-2)(n)p-31]." ], [ "Case 2: $\\alpha =\\gamma $ and {{formula:64ccc225-9742-4cdd-96a1-15f42b32dbd1}} .", "(,)(,)F1A = J= (p-1)[(n)p-2e+ (p-2)(n)p-31]." ], [ "Case 3: $\\alpha \\ne \\gamma $ and {{formula:f98792f6-fc06-4b81-bb41-d0bf9b8dc99e}} .", "(,)(,)F1A = J= 0." ], [ "Third derivative", "(,)(,)(,)F1 = (p-1)(p-2) [G(p-2(A1)ee) +G(p-2(A1)ee)+G(p-2(A1)ee) + (p-3) AT (p-3(A1)eee)]." ], [ "Case 1: $(\\alpha , \\beta ) = (\\gamma , \\delta ) = (\\kappa ,\\rho )$ .", "Since we don't care about the constants in the third derivative, we always write generic constants $C_1,C_2,\\dots $ whose value can be different from line to line.", "Also, we write $c= (c_1,c_2,\\dots ,..)$ to denote generic notation for vectors which can have at most 4 and $\\Vert c\\Vert _{\\infty } \\le 2$ .", "(,)(,)(,)F1A = J = C1 (n)p-3 c+ C2 (n)p-4 1." ], [ "Case 3: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |\\le 5$ .", "(,)(,)(,)F1A = J = C1 (n)p-3 c+ C2 (n)p-4 1." ], [ "Case 2: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |= 6$ .", "(,)(,)(,)F1A = J = 0." ], [ "First derivatives", "Recall that $S1 = \\Psi _{r^{\\prime }}(F1)$ , and therefore, (,)S1 = (r'-1) r'-1(F1)(,)F1 F1 (,)S1=(r'-1) S1 (,)F1.", "Putting the value $A = \\mu J$ at (REF ) and using (REF ) we obtain (n)p (,)S1A = J = (r'-1) (n)pr-1[(n)p-1 e + 2(p-1) (n)p-2 1] (,)S1A = J = 1r-1 (n)pr-1-2 [ne+ 2(p-1) 1]" ], [ "Second derivatives", "Differentiating (REF ), we get (,)F1 (,)S1 + F1 (,)(,)S1 = (r'-1) [(,)S1 (,)F1 + S1 (,)(,)F1], and thus, (,)(,)S1 = 0(F1)[(r'-1) [(,)S1 (,)F1               + S1 (,)(,)F1] - (,)F1 (,)S1] At $A = \\mu J$ , we evaluate the above expression." ], [ "Case 1: $(\\alpha , \\beta ) = (\\gamma , \\delta )$ .", "(,)(,)S1A = J = (n)-p[(r'-1) [1r-1 (n)pr-1+p-4 [ne+ 2(p-1) 1]2        + (n)pr-1+p-3 2(p-1)[ne+ (p-2)1]]        -(r'-1)(n)pr-1+p-4 [ne+ 2(p-1) 1]2] = (r'-1)(n)pr-1-4[ (r'-2) [ne+ 2(p-1) 1]2               + n2(p-1)[ne+ (p-2)1]] = (r'-1)(n)pr-1-4[[(2p+r'-4)(n)2 + 4(p-1)n2]e                +2(p-1)(p-2)n21 + 4(p-1)2(r'-2)21] (r'-1)(n)pr-1-3[ (2p+r'-4)ne+2(p-1)(p-2)1]" ], [ "Case 2: $\\alpha =\\gamma $ , {{formula:c8fa57db-3e5e-4c78-b460-bcba67862507}} .", "(,)(,)S1A = J = (n)-p[(r'-1) [1r-1 (n)pr-1+p-4 [ne+ 2(p-1) 1]                             [ne+ 2(p-1) 1]        + (n)pr-1+p-3 (p-1)[ne+ (p-2)1]]        -(r'-1)(n)pr-1+p-4 [ne+ 2(p-1) 1]                             [ne+ 2(p-1) 1]] = (r'-1) (n)pr-1-4[ (r'-2) [ne+ 2(p-1) 1]                             [ne+ 2(p-1) 1]        + n(p-1)[ne+ (p-2)1]] = (r'-1) (n)pr-1-4[ (r'-2) [(n)2 e+ 2(p-1)n 2(e+e) +4(p-1)221 ]        + (p-1)(n)2e+ (p-1)(p-2)n21] =(r'-1) (n)pr-1-4[(n)2[(r'-2)e + (p-1)e]               +n2[2(p-1)(r'-2)(e+e)+ (p-1)(p-2)1]+4(p-1)221] (r'-1) (n)pr-1-3[n[(r'-2)e + (p-1)e]+ (p-1)(p-2)1]" ], [ "Case 3: $|\\lbrace \\alpha ,\\gamma ,\\beta ,\\delta \\rbrace |=4$ .", "(,)(,)S1A = J = (n)-p[(r'-1) (r'-2) (n)pr-1+p-4 [ne+ 2(p-1) 1]                             [ne+ 2(p-1) 1]] = (r'-1) (r'-2) (n)pr-1-4 [ne+ 2(p-1) 1][ne+ 2(p-1) 1] (r'-1) (r'-2) (n)pr-1-4 [2(p-1)n2 (e+ e) + 4(p-1)221] = 2(p-1)(r'-1) (r'-2) (n)pr-1-4 [n2 (e+ e) + 2(p-1)21]" ], [ "Third derivatives. ", "Differentiating (REF ), we get (,)(,)F1 (,)S1+ (,)F1 (,)(,)S1               + (,)F1 (,)(,)S1 + F1 (,)(,)(,)S1 = (r'-1) [(,)(,)S1 (,)F1 + (,)S1 (,)(,)F1               + (,)S1 (,)(,)F1 + S1 (,)(,)(,)F1], and thus, (,)(,)(,)S1 =0(F1)[ (r'-1) [(,)(,)S1 (,)F1 + (,)S1 (,)(,)F1                      + (,)S1 (,)(,)F1 + S1 (,)(,)(,)F1]                   - (,)(,)F1 (,)S1 - (,)F1 (,)(,)S1                                            - (,)F1 (,)(,)S1].", "Then there exists positive constants $C_1$ and $C_2$ , such that the following hold." ], [ "Case 1: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |\\le 5$ .", "(,)(,)(,)S1A = J = C1 (n)pr-1-3 c+ C2 (n)pr-1-4 1." ], [ "Case 2: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace | = 6$ .", "(,)(,)(,)S1A = J = C1 (n)pr-1-4 2c+ C2 (n)pr-1-5 3 1." ], [ "First derivatives", "(,)S1r = 1S1rr-1 r S1, (,)S1 S1rr-1 (,)S1r = r S1, (,)S1 .", "Therefore, (,)S1r A = J = 2pr-1 (n)pr-1-1n-1+1r." ], [ "Second derivatives", "Thus (r-1)S1rr-2 (,)S1r (,)S1r + S1rr-1 (,)(,)S1r = (r-1) (r-1 S1 )(,)S1, (,)S1 + r S1, (,)(,)S1 = (r-1) r-1 S1 , (,)S1 (,)S1 + r S1, (,)(,)S1 .", "Therefore, (,)(,)S1r = -(r-1)S1r-1 (,)S1r (,)S1r + S1r-(r-1)[(r-1) r-1 S1 , (,)S1 (,)S1                      + r S1, (,)(,)S1 ]" ], [ "Case 1: $(\\alpha , \\beta ) = (\\gamma , \\delta )$ .", "(,)(,)S1r A = J = (1+o(1))2r-1(p(p-1)+1r-1)(n)pr-1-2n-1+1r" ], [ "Case 2: $\\alpha =\\gamma $ , {{formula:c6dca085-150a-4ad8-a209-83ea5bf9554e}} .", "(,)(,)S1r A = J = (1+o(1))1r-1(p(p-1)+1r-1)(n)pr-1-2n-1+1r" ], [ "Case 3: $|\\lbrace \\alpha ,\\gamma ,\\beta ,\\delta \\rbrace |=4$ .", "(,)(,)S1r A = J = - 4p2r-1(n)pr-1-2 n-2+1r(1+o(1))." ], [ "Third derivatives", "For the third derivative, we don't care about the constants.", "So we write $C_1,C_2,\\dots $ to denote generic constants (possibly depending only on $p$ and $r$ ), whose values can vary from line to line.", "Taking derivative on (REF ) yields: LHS = C1 S1rr-3 (,)S1r (,)S1r (,)S1r +C2 S1rr-2 (,)S1r (,)(,)S1r + C2 S1rr-2 (,)S1r (,)(,)S1r +C2 S1rr-2 (,)S1r (,)(,)S1r + S1rr-1 (,)(,)(,)S1r.", "RHS = C1r-2 S1, (,)S1 (,)S1(,)S1 + C2r-1 S1 , (,)S1 (,)(,)S1 + C2 r-1 S1 , (,)(,)S1 (,)S1 + C2 r-1 S1 , (,)S1(,)(,)S1 + C4 r S1 , (,)(,)(,)S1 .", "Then there exists positive constants $C_1$ and $C_2$ , such that the following hold." ], [ "Case 1: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |\\le 5$ .", "(,)(,)(,)S1rA = J C1 (n)pr-1-3n-1+1r." ], [ "Case 2: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |= 6$ .", "(,)(,)(,)S1rA = J C2 (n)pr-1-3n-2+1r." ], [ "First derivatives", "(,)(AS1) = GS1 + A(,)S1.", "Now we will compute the values of the derivatives at $A = \\mu J$ .", "From (REF ) (,)(AS1) A = J = (n)pr-1 e+ 2r-1 (n)pr-1-1 1 + 2(p-1)r-1 (n)pr-1-1 1 =(n)pr-1e+ 2pr-1 (n)pr-1-1 1" ], [ "Second derivatives ", "(,)(,)(AS1)= G(,)S1 + G(,)S1 + A(,)(,)S1 For the second order derivatives, we will consider cases." ], [ "Case 1: $(\\alpha , \\beta ) = (\\gamma , \\delta )$ .", "(,)(,)(AS1) A = J = 2G(,)S1A = J + A(,)(,)S1A = J = 2G[1r-1 (n)pr-1-1 e+ 2(p-1)r-1 (n)pr-1-2 1 ] + (r'-1)(n)pr-1-3[ 2(2p+r'-4)n21 +2(p-1)(p-2)(n)1] =2r-1 (n)pr-1-1 e+ (r'-1)(n)pr-1-2[ 2(2p+r'-4) +2(p-1)(p-2)]1 = 2r-1 (n)pr-1-1 e+ 2r-1[ p(p-1) +1r-1-1 ](n)pr-1-21 where the second equality follows using (REF ) and (REF )." ], [ "Case 2: $\\alpha =\\gamma $ , {{formula:479521e7-3cc8-4636-aed9-d07b9b95e900}} .", "There exists positive constants $c_1$ and $c_2$ , such that the following hold.", "(,)(,)(AS1) A = J = [c1 (n)pr-1-1 e+ c2(n)pr-1-21](1+o(1))." ], [ "Case 3: $|\\lbrace \\alpha ,\\beta ,,\\gamma ,\\delta \\rbrace |=4$ .", "There exists positive constant $c_1$ , such that the following hold.", "(,)(,)(AS1) A = J = c1(n)pr-1-321 (1+o(1))." ], [ "Third derivatives", "(,)(,)(,)(AS1) = G(,)(,)S1 + G(,)(,)S1               + G(,)(,)S1 + A(,)(,)(,)S1.", "Then there exists positive constants $C_1$ and $C_2$ , such that the following hold." ], [ "Case 1: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |\\le 5$ .", "(,)(,)(,)AS1A = J C1 (n)pr-1-2 c." ], [ "Case 2: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace | = 6$ .", "(,)(,)(,)AS1A = J C2 (n)pr-1-3 c." ], [ "First derivatives", "(,)AS1p = 1AS1pp-1p(AS1), (,)(AS1) , AS1pp-1(,)AS1p = p(AS1), (,)(AS1) .", "Thus, (,)AS1p A = J= 2(pr-1+1) (n)pr-1 n-1+1p ." ], [ "Second derivatives", "(p-1)AS1pp-2 (,)AS1p(,)AS1p                             + AS1pp-1 (,)(,)AS1p =(p-1)p-1(AS1)(,)(AS1), (,)(AS1)                             + p(AS1), (,)(,)(AS1) Thus, (,)(,)AS1p = - (p-1)AS1p-1(,)AS1p(,)AS1p + (p-1)AS1p-(p-1)p-1(AS1), (,)(AS1)(,)(AS1) + AS1p-(p-1)p(AS1), (,)(,)(AS1)" ], [ "Case 1: $(\\alpha , \\beta ) = (\\gamma , \\delta )$ .", "(,)(,)AS1p A = J = -(p-1) (n)pr-1 - 1 n-2+1p + (n) pr-1-3n-1+1p [(p-1) 1,(ne+2pr-11)2 + (n) 1,2r-1(ne+(p(p-1)+1r-1 - 1)1) ] = 2[p-1 + pr-1 + 1(r-1)2 + (p-1)2r-1](n)pr-1 -1 n-1+1p(1+o(1))." ], [ "Case 2: $\\alpha =\\gamma $ , {{formula:0ea0ca7b-2bb2-4b8c-8382-306c4dee3f18}} .", "There exists positive constant $c_1$ , such that $\\partial _{(\\alpha ,\\delta )(\\alpha ,\\beta )}\\Vert AS1\\Vert _p \\Big \\vert _{A = \\mu J} = c_1 (n\\mu )^{\\frac{p}{r-1} -1} n^{-1+\\frac{1}{p}} (1+o(1)).$" ], [ "Case 3: $|\\lbrace \\alpha ,\\gamma ,\\beta ,\\delta \\rbrace |=4$ .", "There exists positive constant $c_2$ , such that $\\partial _{(\\gamma ,\\delta )(\\alpha ,\\beta )}\\Vert AS1\\Vert _p \\Big \\vert _{A = \\mu J} = c_2 (n\\mu )^{\\frac{p}{r-1} -1} n^{-2+\\frac{1}{p}}(1+o(1)).$" ], [ "Third derivatives", "Taking further derivative with respect to $a_{\\kappa , \\rho }$ , we obtain from LHS= (p-1)(p-2)AS1pp-3(,)AS1p (,)AS1p(,)AS1p + (p-1)AS1pp-2 (,)(,)AS1p(,)AS1p + (p-1)AS1pp-2 (,)AS1p(,)(,)AS1p + (p-1)AS1pp-2 (,)AS1p(,)(,)AS1p +AS1pp-1 (,)(,)(,)AS1p RHS= (p-1)(p-2)p-2(AS1)(,)(AS1)(,)(AS1), (,)(AS1) +(p-1)p-1(AS1)(,)(,)(AS1), (,)(AS1) +(p-1)p-1(AS1)(,)(AS1), (,)(,)(AS1) +(p-1)p-1(AS1)(,)(AS1), (,)(,)(AS1) + p(AS1), (,)(,)(,)(AS1)" ], [ "Case 1: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |\\le 5$ .", "There exists positive constant $C_1$ , such that (,)(,)(,)AS1pA = J C1 (n)pr-1-4n-1+1p." ], [ "Case 2: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |= 6$ .", "There exists positive constant $C_2$ , such that (,)(,)(,)AS1pA = J C2 (n)pr-1-4n-2+1p." ], [ "Proofs of Lemma ", "In this appendix we will consider derivatives of $\\Vert AS1\\Vert _p/\\Vert S1\\Vert _r$ ." ], [ "First derivatives", "From the derivative calculations in Appendices REF and REF we obtain the following.", "(,)AS1pS1r A = J = 2 n1p-1r-1.", "This proves Lemma REF (a)." ], [ "Second derivatives", "From the derivative calculations in Appendices REF and REF we obtain the following.", "Let $\\alpha , \\beta , \\gamma ,$ and $\\delta $ be distinct integers in $\\lbrace 1,2,\\ldots , n\\rbrace $ .", "Then there exists positive constants $c_1, c_2$ , such that the following hold.", "$\\partial _{(\\alpha ,\\beta )(\\alpha ,\\beta )}\\frac{\\Vert AS1\\Vert _p}{\\Vert S1\\Vert _r} \\ \\bigg \\vert _{A = \\mu J} = 2 \\Big [p-1 + \\frac{1}{r-1}\\Big ]n^{\\frac{1}{p} - \\frac{1}{r} -1}\\frac{1}{n\\mu } (1+o(1))\\\\\\partial _{(\\gamma ,\\delta )(\\alpha ,\\beta )}\\frac{\\Vert AS1\\Vert _p}{\\Vert S1\\Vert _r} \\ \\bigg \\vert _{A = \\mu J} = c_1n^{\\frac{1}{p} - \\frac{1}{r} -2}\\frac{1}{n\\mu } (1+o(1))\\\\\\partial _{(\\alpha ,\\delta )(\\alpha ,\\beta )}\\frac{\\Vert AS1\\Vert _p}{\\Vert S1\\Vert _r} \\ \\bigg \\vert _{A = \\mu J} = c_2n^{\\frac{1}{p} - \\frac{1}{r} -1}\\frac{1}{n\\mu } (1+o(1)).$ This proves Lemma REF (b) and (c)." ], [ "Third derivatives", "From the derivative calculations in Appendices REF and REF we obtain the following.", "Let $\\alpha , \\beta , \\gamma ,\\delta ,\\kappa ,$ and $\\rho $ be integers in $\\lbrace 1,2,\\ldots , n\\rbrace $ .", "Then there exists positive constants $C_1$ and $C_2$ , such that the following hold." ], [ "Case 1: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |\\le 5$ .", "(,)(,)(,)AS1pS1rA = J C1 (n)-4n-1+1p-1r." ], [ "Case 2: $|\\lbrace \\alpha , \\beta ,\\gamma , \\delta ,\\kappa ,\\rho \\rbrace |= 6$ .", "(,)(,)(,)AS1pS1rA = J C2 (n)-4n-2+1p-1r.", "This completes the proof of Lemma  REF" ] ]

[ [ "Fluid dynamics on logarithmic lattices" ], [ "Abstract Open problems in fluid dynamics, such as the existence of finite-time singularities (blowup), explanation of intermittency in developed turbulence, etc., are related to multi-scale structure and symmetries of underlying equations of motion.", "Significantly simplified equations of motion, called toy-models, are traditionally employed in the analysis of such complex systems.", "In such models, equations are modified preserving just a part of the structure believed to be important.", "Here we propose a different approach for constructing simplified models, in which instead of simplifying equations one introduces a simplified configuration space: velocity fields are defined on multi-dimensional logarithmic lattices with proper algebraic operations and calculus.", "Then, the equations of motion retain their exact original form and, therefore, naturally maintain most scaling properties, symmetries and invariants of the original systems.", "Classification of such models reveals a fascinating relation with renowned mathematical constants such as the golden mean and the plastic number.", "Using both rigorous and numerical analysis, we describe various properties of solutions in these models, from the basic concepts of existence and uniqueness to the blowup development and turbulent dynamics.", "In particular, we observe strong robustness of the chaotic blowup scenario in the three-dimensional incompressible Euler equations, as well as the Fourier mode statistics of developed turbulence resembling the full three-dimensional Navier-Stokes system." ], [ "Introduction", "The theory of multi-scale nonlinear flows and, in particular, the phenomenon of hydrodynamic turbulence comprise a multitude of yet unresolved problems: the global regularity [48] and existence of finite-time singularities [53], explanation of intermittency [50] and dissipation anomaly [46], to name a few.", "Many of these problems determine the state-of-the-art in nonlinear science and open new areas in mathematics and physics.", "In these studies, toy-models employed as caricatures of complex phenomena have been proved to be indispensable as the testing ground for new ideas and theories.", "Such models retain some basic features believed to be important, while the remaining content is simplified as much as possible.", "The conventional simplifications are related to reducing the spatial dimension, e.g., the one-dimensional Burgers equation [14] or the Constantin-Lax-Majda model [31] with further generalizations [82], [28].", "The number of degrees of freedom can be drastically decreased by exploring the cascade ideas in the so-called shell models of turbulence [8].", "In these models, multi-scale properties are mimicked by geometrical progressions of scales, resulting in the popular GOY [54], [81] and Sabra models [70], the reduced wave vector set approximation (REWA) [43], [57] and tree models [5], [7], as well as more sophisticated geometric constructions [58], [59].", "Toy-models rely on the intuitive decision of what unimportant properties of the original system can be neglected.", "Of course, dealing with open problems, such decision has the risk of missing essential features of motion.", "Especially, this concerns neglected symmetries and conserved quantities, since fluid systems are known to possess highly nontrivial (infinite dimensional) symmetry groups and conservation laws [98], [76], e.g., the Kelvin Circulation Theorem.", "In the present work, we propose a different approach for constructing simplified models, in which instead of simplified equations one introduces a simplified configuration space with proper algebraic operations and calculus.", "For this purpose, we employ velocity fields defined on discrete multi-dimensional lattices with logarithmically distributed nodes.", "These lattices are designed such that the equations of motion can be used in their exact original form and, as a consequence, the symmetry groups and conservation laws automatically carry over to the new system.", "The resulting models possess much higher degree of similarity to the exact equations as compared to conventional toy-models and, at the same time, share the property of being easily accessible for numerical analysis.", "The paper is divided logically into two parts.", "The first part consists of Sections –.", "Here we classify logarithmic lattices on which the functional operations with necessary properties can be introduced.", "This classification reveals interesting connections with well-known mathematical constants, associating the two representative lattice spacings with the golden mean and the plastic number.", "We prove that the product of two fields cannot be associative on logarithmic lattices, but has the property of associativity in average.", "With this limitation, the technique is applicable to any system of partial differential equations with quadratic nonlinearities and quadratic or linear conservation laws or other integral characteristics.", "Fortunately, this is sufficient for the applicability of our approach to many fundamental models in fluid dynamics.", "The second part includes Sections ,  with the Appendix, and contains applications of the developed technique to specific equations of fluid dynamics.", "Here many properties of solutions are proved following classical derivations in fluid dynamics, as a consequence of the designed similarity of configuration spaces.", "This refers not only to the basic symmetries and inviscid invariants like, e.g., energy and helicity, but also to a number of fine properties such as incompressibility and conservation of circulation.", "In this way, we demonstrate basic properties such as local-in-time existence and uniqueness of strong solutions and the blowup criterion.", "Then, we proceed with the numerical study.", "Our central numerical result concerns the blowup problem for incompressible 3D Euler equations, where we demonstrate surprisingly strong robustness of the chaotic blowup scenario [19] with respect to the choice of the logarithmic lattice.", "We also show that, for the 3D Navier-Stokes equations at large Reynolds numbers, our models demonstrate chaotic regimes with the classical properties of developed turbulence.", "Here, velocity fields on logarithmic lattices must be considered as analogues of Fourier-transposed velocity fields in the original model.", "The paper is organized as follows.", "We classify logarithmic lattices in Section , introduce the calculus on such lattices in Section , and provide some generalizations in Section .", "Section  is devoted to the blowup problem in incompressible ideal flows, and Section  to the numerical study of developed turbulence.", "We draw conclusions in the final section.", "Appendix discusses the Burgers equation, where connections to existing shell models are given, as well as some perspectives for compressible flows." ], [ "Logarithmic lattices", "In this section, we perform a systematic study of logarithmic lattices with certain geometric properties, providing the domain on which the dynamical models shall be defined in the next sections.", "We start with one-dimensional lattices, similar to those used in shell models, and then consider the multi-dimensional case.", "Given a real number $\\lambda >1$ , the logarithmic lattice with spacing factor $\\lambda $ is the set $\\mathbb {\\Lambda } = \\lbrace \\pm \\lambda ^n\\rbrace _{n \\in \\mathbb {Z}},$ consisting of positive and negative integer powers of $\\lambda $ – see Fig.", "REF .", "This set has two properties important for applications.", "First, $\\mathbb {\\Lambda }$ is scale-invariant, i.e., $\\mathbb {\\Lambda } = k \\mathbb {\\Lambda }$ for any $k \\in \\mathbb {\\Lambda }$ .", "Secondly, the points of the lattice grow geometrically with $n$ .", "Thus, with only a few nodes we span a large range of scales.", "However, logarithmic lattices are not closed under addition as $p + q \\notin \\mathbb {\\Lambda }$ for general $p,q \\in \\mathbb {\\Lambda }$ .", "Three points $k,p,q \\in \\mathbb {\\Lambda }$ on a logarithmic lattice form a triad if $k = p + q$ .", "In this case, we say that $k$ interacts with $p$ and $q$ .", "The lattice is called nondegenerate if every two nodes interact through a finite sequence of triads.", "We are interested in a twofold task: to determine which spacings $\\lambda $ provide nondegenerate lattices, and to classify all triads of nondegenerate lattices.", "Because of the scale invariance, it is sufficient to describe the triads at unity, i.e., $1 = p + q$ .", "Table: Triads at the unity 1=p i +q i 1 = p_i + q_i for different spacing factors: TAB:DN λ=2\\lambda = 2;TAB:general λ\\lambda satisfies 1=λ b -λ a 1 = \\lambda ^b-\\lambda ^a for integers 0≤a<b0 \\le a <b.", "For example, λ=ϕ\\lambda = \\varphi is the golden mean for a=1a = 1 and b=2b = 2;TAB:extended λ=σ\\lambda = \\sigma , the plastic number.Lattices $\\mathbb {\\Lambda }$ with nontrivial triad interactions exist only for certain values of $\\lambda $ .", "Let us first present three specific nondegenerate lattices.", "The lattice with $\\lambda = 2$ has three possible types of triads described in Tab.", "REF and Fig.", "REF (a).", "For any $k \\in \\mathbb {\\Lambda }$ , these triads are $k = \\lambda k - k$ , $k = -k + \\lambda k$ and $k = \\lambda ^{-1}k + \\lambda ^{-1}k$ .", "The next example is $\\lambda = \\varphi $ , where $\\varphi = (1+\\sqrt{5})/2 \\approx 1.618$ is the golden mean.", "All triads are obtained from permutations and rescalings of the identity $1 = \\varphi ^2 - \\varphi $ , providing the richer sample of interactions in Tab.", "REF .", "In this case, each point interacts with six different neighbors – see Fig.", "REF (b).", "Another example is provided by the plastic number of Dom Van der Laan [39] $\\sigma = \\frac{\\@root 3 \\of {9+\\sqrt{69}}+\\@root 3 \\of {9-\\sqrt{69}}}{\\@root 3 \\of {18}} \\approx 1.325,$ which is the common real solution of equations $\\sigma ^3 - \\sigma - 1 = 0$ and $\\sigma ^5 - \\sigma ^4 - 1 = 0$ .", "The lattice with spacing $\\lambda = \\sigma $ has twelve types of interacting triads, enumerated in Tab.", "REF and depicted in Fig.", "REF (c).", "Because immediate neighbors are coupled, these are examples of nondegenerate lattices.", "On the other hand, if $\\lambda = \\sqrt{2}$ , the lattice is degenerate: there are no interactions that couple points $\\pm 2^n$ with $\\pm 2^n\\sqrt{2}$ .", "The main result of this section is the classification of nondegenerate logarithmic lattices with respect to their triad interactions, given by the following Theorem 1 The following three cases describe all nondegenerate lattices with spacing factors $\\lambda \\ge 1.05$ : $\\lambda = 2$ , and all triads at the unity are given in Tab.", "REF ; $\\lambda = \\sigma $ , the plastic number (REF ), and all triads at the unity are given in Tab.", "REF ; $\\lambda $ satisfies $1 = \\lambda ^b - \\lambda ^a$ , where $(a,b)$ are mutually prime integers not larger than 62, excluding also the pairs $(a,b) = (1,3)$ and $(4,5)$ .", "All triads at the unity are given in Tab.", "REF .", "Remark 2 We used the lower bound $\\lambda \\ge 1.05$ in order to make the numerically assisted proof possible.", "Still, we conjecture that Theorem REF is valid for arbitrary $\\lambda > 1$ , with no upper bound for $a$ and $b$ in the item ($iii$ ).", "A partial result in this direction is the Theorem proved in [1], which states that the plastic number is the only common root greater than unity of any two distinct polynomials $\\lambda ^a - \\lambda ^{a-1} - 1$ and $\\lambda ^b - \\lambda - 1$ with $a,b \\ge 2$ .", "Let us consider the trinomial equation $p_{a,b}(\\lambda ) = \\lambda ^b - \\lambda ^a - 1 = 0,$ with integer powers $0 \\le a < b$ .", "This equation has a single root in the interval $\\lambda > 1$ because the function $p_{a,b}(\\lambda )$ is strictly increasing in $\\lambda \\in [1,\\infty )$ with image $[-1,\\infty )$ .", "Relation (REF ) yields the three equalities $1 = \\lambda ^b - \\lambda ^a = \\lambda ^{b-a} - \\lambda ^{-a} = \\lambda ^{-b} + \\lambda ^{a-b}.$ There are six triads $1 = p + q$ corresponding to expressions (REF ) as described in Tab.", "REF .", "Let us show that the lattice is degenerate when $a$ and $b$ have a common divisor $m > 1$ .", "For the sublattice $\\mathbb {\\Lambda }^\\prime = \\lbrace \\pm \\lambda ^{mn} \\rbrace _{n \\in \\mathbb {Z}}$ to be coupled with the remaining points, the spacing $\\lambda $ should satisfy another trinomial equation (REF ) with exponents $(a^{\\prime },b^{\\prime })$ not multiples of $m$ .", "However, this is not possible, as it follows from case (b) of Lemma REF below.", "This leaves only the mutually prime pairs $(a,b)$ to our consideration.", "Now, the Theorem is a direct consequence of Lemma REF , where all triads are generated by the relations in (REF ): the case (i) corresponds to $(a,b) = (0,1)$ ; the case (ii) to $(a,b) = (1,3)$ and $(4,5)$ ; and the case (iii) to all other possibilities.", "Lemma 3 Consider two distinct trinomials (REF ) with integer powers $(a_1,b_1)$ and $(a_2,b_2)$ , where $0 \\le a_1 < a_2$ .", "These trinomials have a common root $\\lambda \\ge 1.05$ if and only if $\\lambda = \\sigma $ is the plastic number (REF ).", "In this case, $(a_1,b_1) = (1,3)$ and $(a_2,b_2) = (4,5)$ are mutually prime; $\\lambda = \\sigma ^{1/m}$ with $m = 2,\\dots ,5$ .", "In this case $(a_1,b_1) = (m,3m)$ and $(a_2,b_2) = (4m,5m)$ have the same common divisor $m$ .", "Let us denote by $\\lambda (a,b)$ the unique root of (REF ) in the interval $\\lambda >1$ .", "Note that $\\lambda (a,b) < \\lambda (a^{\\prime },b)$ if $0 \\le a^{\\prime } < a$ because the polynomials $p_{a,b}(\\lambda )$ are strictly increasing starting at $p_{a,b}(1) = p_{a^{\\prime },b}(1) = -1$ and $p_{a,b}(\\lambda ) < p_{a^{\\prime },b}(\\lambda )$ for $\\lambda >1$ .", "Therefore, if we fix the exponent $b$ of trinomial $p_{a,b}(\\lambda )$ , then $\\lambda (a,b)$ is maximized when $a = b-1$ .", "Next, $\\lambda (b-1,b)$ form a decreasing sequence with respect to $b$ , since $p_{b-1,b}(\\lambda ) < p_{b,b+1}(\\lambda )$ .", "Finally, one may check that $p_{62,63}(1.05) > 0$ , so $\\lambda (62,63)<1.05$ .", "Therefore $\\lambda (a,b) \\ge 1.05$ only if $b < 63$ .", "This bound leaves a finite number of trinomials to our consideration.", "Since the plastic number $\\sigma $ satisfies $\\sigma ^5 - \\sigma ^4 - 1 = \\sigma ^3 - \\sigma - 1 = 0$ , we obtain the two cases (a) and (b) of the Lemma.", "It remains to check that trinomials with different powers have no common root.", "This was accomplished via Validated Numerics [78], a computer assisted proof using the following strategy.", "Given two trinomials $p_{a,b}$ and $p_{a^{\\prime },b^{\\prime }}$ , we estimate their respective roots $\\lambda _1$ and $\\lambda _2$ with Newton's Method up to machine double precision.", "Next, using Symbolic Algebra [30], we evaluate exactly the product $p_{a,b}p_{a^{\\prime },b^{\\prime }}$ at the middle point $\\lambda _m = (\\lambda _1 + \\lambda _2)/2$ of their approximate roots.", "For all cases, it was verified a negative number at this point, which guarantees that $\\lambda (a,b) \\ne \\lambda (a^{\\prime },b^{\\prime })$ .", "Figure: Triad interactions on two-dimensional logarithmic lattices for different spacing factors: (a) λ=2\\lambda = 2; (b) λ=ϕ\\lambda = \\varphi , the golden mean; (c) λ=σ\\lambda = \\sigma , the plastic number.", "The red node 𝐤\\mathbf {k} can be decomposed into sums 𝐤=𝐩+𝐪\\mathbf {k} = \\mathbf {p} + \\mathbf {q} where all possible nodes 𝐩\\mathbf {p} and 𝐪\\mathbf {q} are indicated by the green lines.", "All figures are given in the same scale.", "From (a) to (c), both the density of nodes and the number of triads per each node increase.The above results for one-dimensional logarithmic lattices can be extended to higher dimensions.", "The $d$ -dimensional logarithmic lattice with spacing $\\lambda >1$ is given by the cartesian power $\\mathbb {\\Lambda }^d = \\mathbb {\\Lambda } \\times \\cdots \\times \\mathbb {\\Lambda }$ (with $d$ factors), i.e., $\\mathbf {k} = (k_1,\\dots ,k_d) \\in \\mathbb {\\Lambda }^d$ if each component $k_j \\in \\mathbb {\\Lambda }$ .", "Three points $\\mathbf {k},\\mathbf {p},\\mathbf {q} \\in \\mathbb {\\Lambda }^d$ on the lattice form a triad if $\\mathbf {k} = \\mathbf {p} + \\mathbf {q}$ .", "All nondegenerate lattices $\\mathbb {\\Lambda }^d$ are given by the spacings $\\lambda $ listed in Theorem REF and all triads are combinations of the one-dimensional triads for each component – see Fig.", "REF for the two-dimensional picture." ], [ "Calculus on logarithmic lattices", "Let us consider complex-valued functions $f(\\mathbf {k}) \\in \\mathbb {C}$ on a nondegenerate logarithmic lattice $\\mathbb {\\Lambda }^d$ , where $\\mathbf {k} \\in \\mathbb {\\Lambda }^d$ is interpreted as a wave vector in Fourier space.", "Motivated by the property of the Fourier transform of a real-valued function, we impose the reality condition $f(-\\mathbf {k}) = \\overline{f(\\mathbf {k})},$ where the bar denotes complex conjugation.", "Thus, $f(\\mathbf {k})$ is analogous to the Fourier transform of a real function, and now we are going to introduce basic operations.", "Functions $f(\\mathbf {k})$ possess a natural structure of a linear space with real scalars.", "Since we are working with Fourier-space representation, the spatial derivative $\\partial _j$ in the $j$ -th direction is defined by the Fourier factor, $\\partial _j f(\\mathbf {k}) = ik_j f(\\mathbf {k}), \\quad j =1,\\dots ,d,$ where $i$ is the imaginary unit.", "Clearly, higher order derivatives are products of such Fourier factors.", "Given two functions $f$ and $g$ , one defines their inner product naturally as $(f,g) = \\sum _{\\mathbf {k} \\in \\mathbb {\\Lambda }^d} f(\\mathbf {k})\\overline{g(\\mathbf {k})}.$ Just like the $L^2$ -inner product of real functions, expression (REF ) is real valued because of reality condition (REF ).", "The notion of differentiability on the lattice retains some important calculus identities, like the integration by parts $(\\partial _j f, g) = - (f, \\partial _j g), \\quad j =1,\\dots ,d,$ which follows from the fact that the inner product (REF ) couples $f(\\mathbf {k})$ and $\\overline{g(\\mathbf {k})} = g(-\\mathbf {k})$ .", "We next define the product of two functions on the logarithmic lattice, which in Fourier space is understood as a convolution.", "Here and below, all functions are assumed to be absolutely summable $\\sum _{\\mathbf {k} \\in \\mathbb {\\Lambda }^d} |f(\\mathbf {k})| < \\infty .$ Definition 4 A product on the logarithmic lattice $\\mathbb {\\Lambda }^d$ , denoted by $\\ast $ , is a binary operation between absolutely summable functions on $\\mathbb {\\Lambda }^d$ , which satisfies the following properties: (Reality condition) $\\!\\begin{aligned}[t](f \\ast g)(-\\mathbf {k}) = \\overline{(f \\ast g) (\\mathbf {k})};\\end{aligned}$ (Bilinearity) $\\!\\begin{aligned}[t](f + \\gamma g) \\ast h = f \\ast h + \\gamma (g \\ast h),\\end{aligned}$ for any $\\gamma \\in \\mathbb {R}$ ; (Commutativity) $\\!\\begin{aligned}[t]f \\ast g = g \\ast f;\\end{aligned}$ (Associativity in average) $\\!\\begin{aligned}[t](f \\ast g, h ) = (f, g \\ast h);\\end{aligned}$ (Leibniz rule) $\\!\\begin{aligned}[t]\\partial _j (f \\ast g) = \\partial _j f \\ast g + f \\ast \\partial _j g, \\ \\text{for } j = 1,\\dots ,d;\\end{aligned}$ Additional properties, which are related to the spatial symmetries of the lattice, may be imposed: (Scaling invariance) $\\!\\begin{aligned}[t]\\delta _{\\lambda }(f \\ast g) = \\delta _{\\lambda } f \\ast \\delta _{\\lambda } g,\\end{aligned}$ where we denoted $\\delta _\\lambda f (\\mathbf {k}) = f(\\lambda \\mathbf {k})$ , the rescaling of $f$ by the lattice spacing $\\lambda $ ; (Isotropy and parity) $\\!\\begin{aligned}[t](f\\ast g) \\circ R = (f \\circ R) \\ast (g \\circ R),\\end{aligned}$ where we denoted $(f \\circ R)(\\mathbf {k}) = f(R\\mathbf {k})$ and $R \\in \\mathsf {O_h}$ is any element of the group of cube symmetries; cf.", "[66] – it includes all transformations $(k_1,\\dots ,k_d) \\mapsto (\\pm k_{\\alpha _1},\\dots ,\\pm k_{\\alpha _d})$ , where $(\\alpha _1,\\dots ,\\alpha _d)$ are permutations of $(1,\\dots ,d)$ .", "Remark 5 Lebniz rule readily implies translation invariance on the lattice, expressed as $\\tau _{\\xi } (f \\ast g) = \\tau _{\\xi } f \\ast \\tau _{\\xi } g$ , where $\\tau _{\\xi } f (k) = e^{-ik \\cdot \\xi } f(k)$ mimics the physical-space translation (in Fourier representation) by any vector $\\xi \\in \\mathbb {R}^d$ ; The required properties for the product are chosen in order to mimic as much as possible a common pointwise product (or, equivalently, a convolution in Fourier space) of real functions defined in the Euclidean space.", "The symmetries of scaling invariance REF and isotropy REF can only be satisfied in a discrete form, because only discrete scalings and rotations are symmetries of the lattice itself.", "More importantly, we will prove shortly that the product cannot be associative.", "Nevertheless, the weaker property of associativity in average REF can be satisfied, which turns out to be sufficient for our purposes.", "We first establish the general form of the product on one-dimensional lattices.", "Later, it will be generalized to higher dimensions.", "Bilinearity REF , Leibniz rule REF and scaling invariance REF yield the following general form of the product $(f \\ast g)(k) = \\sum _{p_j + q_j = 1} c_j f(p_j k)g(q_j k), \\quad k \\in \\mathbb {\\Lambda }.$ Here, the Leibniz rule restricts the product to triad interactions, which are determined by the factors $p_j$ and $q_j$ from Tab.", "REF for each lattice of Theorem REF .", "The independence of the coefficients $c_j$ on $k$ is a consequence of the scaling invariance.", "Next, reality condition REF and parity $k \\mapsto -k$ , from REF , imply that the coefficients $c_j$ are real.", "Since the sum in (REF ) has a finite number of terms, the product of two absolutely summable functions is absolutely summable.", "As an example, consider the case $\\lambda = 2$ .", "Then, for the three triads in Tab.", "REF , formula (REF ) becomes $(f \\ast g)(k) = c_1 f(2 k)g(-k) + c_2 f(-k)g(2 k) + c_3 f(2^{-1}k)g(2^{-1}k).$ We are interested in non-trivial products (REF ), where the coefficients $c_j$ do not vanish simultaneously.", "Theorem 6 Let $\\mathbb {\\Lambda }$ be one of the logarithmic lattices (i)–(iii) described in Theorem REF .", "For the lattices (i) and (iii), the product with properties REF –REF and symmetries REF and REF is unique, up to a real prefactor which we set to unity, and has the form $(f \\ast g)(k) = \\sum _{p_j + q_j = 1} f(p_j k) g(q_j k), \\quad k \\in \\mathbb {\\Lambda },$ where the coupling factors $p_j$ and $q_j$ are given in Tab.", "REF .", "For the lattice (ii), the general form of the product is $(f \\ast g)(k) = c_1 \\sum _{\\begin{array}{c}p_j + q_j = 1 \\\\[2pt] j = 1,\\dots ,6\\end{array}}f(p_j k) g(q_j k) + c_2 \\sum _{\\begin{array}{c}p_j + q_j = 1 \\\\[2pt] j = 7,\\dots ,12\\end{array}}f(p_j k) g(q_j k), \\quad k \\in \\mathbb {\\Lambda },$ where $c_1$ and $c_2$ are arbitrary real prefactors.", "Properties REF , REF , REF and symmetries REF and REF were already used to reduce the product to the form (REF ).", "One may check that the remaining conditions REF and REF for the product can be written as linear equations with unit coefficients with respect to the variables $c_j$ .", "The system of such equations can be solved explicitly, leading to formulas (REF ) and (REF ).", "Consider, for example, the case $\\lambda = 2$ , whose product expression is given by (REF ).", "Commutativity REF requires $c_1 = c_2$ .", "On the other hand, associativity in average REF enforces all coefficients to be the same.", "Recall that the associativity condition is valid in average; see property REF in Definition REF .", "At the same time, the products cannot be associative, as it follows from Corollary 7 The non-trivial products described in Theorem REF are not associative: condition $(f \\ast g) \\ast h = f \\ast (g \\ast h)$ is not valid for all functions $f$ , $g$ and $h$ .", "Let us show that there are $p,q,r \\in \\mathbb {\\Lambda }$ such that $p+q, p+q+r \\in \\mathbb {\\Lambda }$ , but $q+r \\notin \\mathbb {\\Lambda }$ .", "From the proof of Theorem REF , there are integers $0 \\le a < b$ such that the spacing $\\lambda $ satisfies $1 = \\lambda ^b - \\lambda ^a$ .", "Take $p = \\lambda ^{2b}$ , $q = -\\lambda ^{a+b}$ and $r = -\\lambda ^{a}$ .", "Then $p+q+r = 1 \\in \\mathbb {\\Lambda }$ and $p+q = \\lambda ^b \\in \\mathbb {\\Lambda }$ .", "We claim that $q+r = -(1+\\lambda ^b)\\lambda ^a \\notin \\mathbb {\\Lambda }$ , which is equivalent to the condition $1+\\lambda ^b \\notin \\mathbb {\\Lambda }$ .", "Indeed, suppose that $1+\\lambda ^b \\in \\mathbb {\\Lambda }$ .", "In this case, $1+\\lambda ^b = \\lambda ^m$ for some integer $m>b$ .", "It follows that $\\lambda $ is a common root of trinomials (REF ) with $(a_1,b_1) = (a,b)$ and $(a_2,b_2) = (b,m)$ .", "However, such a solution is forbidden by Lemma REF , leading to a contradiction.", "Now, indicating by $\\delta _k$ the function with $\\delta _k(k) = 1$ and zero elsewhere, it follows from expression (REF ) that $(\\delta _p \\ast \\delta _q) \\ast \\delta _r = \\delta _{p+q} \\ast \\delta _r = \\delta _{p+q+r}$ , but $\\delta _p \\ast (\\delta _q \\ast \\delta _r) = \\delta _p \\ast 0 = 0$ .", "A similar argument applies to expression (REF ).", "Application of the same ideas for the two and three-dimensional cases yield similar formulas for products on these spaces, but with a larger number of free coefficients.", "For instance, the product on the three-dimensional lattice with spacing $\\lambda = \\varphi $ , the golden mean, has 10 free real coefficients.", "It is useful to give the following expression $(f \\ast g)(\\mathbf {k}) = \\sum _{\\begin{array}{c}\\mathbf {p} + \\mathbf {q} = \\mathbf {k}\\\\[2pt] \\mathbf {p},\\mathbf {q} \\in \\mathbb {\\Lambda }^d\\end{array}} f(\\mathbf {p}) g(\\mathbf {q}), \\quad \\mathbf {k} \\in \\mathbb {\\Lambda }^d,$ analogous to (REF ), which yields a product in any dimension and any lattice.", "All operations introduced in this section are implemented in LogLatt, an efficient Matlab library for the numerical calculus on logarithmic lattices [18]." ], [ "Generalized lattices and products", "In this section we discuss some generalizations of logarithmic lattices, which can be useful for applications.", "In order to mimic non-local interactions, one can add the origin to the logarithmic lattice $\\mathbb {\\Lambda } = \\lbrace 0 \\rbrace \\cup \\lbrace \\pm \\lambda ^n \\rbrace _{n \\in \\mathbb {Z}}.$ In this case, every point $k \\in \\mathbb {\\Lambda }$ interacts with the zero node: $k = k + 0 = 0 + k$ , which provides additional (non-local) terms to the products.", "The value $f(0)$ is interpreted as the mean value of $f$ in physical space, in analogy with the same value for continuous functions $\\hat{F}(0) = \\int F(x)dx$ .", "The same relations (REF )–(REF ) and Definition REF are used to define the product and other operations.", "For example, when $\\lambda = 2$ , the product (REF ) at $k \\ne 0$ generalizes to $(f \\ast g)(k) = [f(2 k)g(-k) + f(-k)g(2 k) + f(2^{-1}k)g(2^{-1}k)]+ c[f(k)g(0) + f(0)g(k)],$ with an arbitrary real parameter $c$ .", "The product $f \\ast g$ evaluated at $k = 0$ is given by $(f \\ast g)(0) = c\\sum _{k \\in \\mathbb {\\Lambda }}f(k)g(-k)$ with the same prefactor $c$ , which is the consequence of associativity in average – see REF of Definition REF .", "It is natural to set $c = 1$ , in which case expression (REF ) coincides with the inner product (REF ), i.e., $(f \\ast g)(0) = (f,g)$ .", "Furthermore, we can define generalized logarithmic lattices as arbitrary subsets $\\mathbb {\\Lambda }^{\\prime } \\subset \\mathbb {\\Lambda }^d$ of logarithmically distributed nodes.", "To ensure that functions satisfying the reality condition (REF ) can be represented in $\\mathbb {\\Lambda }^{\\prime }$ , we impose the property that if $\\mathbf {k} \\in \\mathbb {\\Lambda }^{\\prime }$ then $-\\mathbf {k} \\in \\mathbb {\\Lambda }^{\\prime }$ .", "This is the case, for example, of a truncated lattice with a finite number of points $\\mathbb {\\Lambda }^{\\prime } = \\lbrace 0, \\pm 1, \\pm \\lambda , \\dots , \\pm \\lambda ^N \\rbrace ,$ or the same subset excluding zero.", "Since a generalized lattice $\\mathbb {\\Lambda }^{\\prime }$ is not necessarily scaling invariant or isotropic, we cannot demand the corresponding product to have these symmetries.", "Therefore, a product on $\\mathbb {\\Lambda }^{\\prime }$ is an operation satisfying properties REF –REF of Definition REF .", "In the following Theorem, we provide one natural form of the product that serves for all generalized lattices.", "Theorem 8 Let $\\mathbb {\\Lambda }^{\\prime } \\subset \\mathbb {\\Lambda }^d$ be a generalized $d$ -dimensional logarithmic lattice.", "Then, operation $(f \\ast g)(\\mathbf {k}) = \\sum _{\\begin{array}{c}\\mathbf {p} + \\mathbf {q} = \\mathbf {k}\\\\[2pt] \\mathbf {p},\\mathbf {q} \\in \\mathbb {\\Lambda }^{\\prime }\\end{array}} f(\\mathbf {p}) g(\\mathbf {q}), \\quad \\mathbf {k} \\in \\mathbb {\\Lambda }^{\\prime },$ defines a product on $\\mathbb {\\Lambda }^{\\prime }$ with properties REF –REF .", "Properties REF –REF are directly verified, except for the associativity in average REF , which follows from the fact that both $(f \\ast g, h)$ and $(f,g \\ast h)$ can be written in the same form as $\\sum _{\\begin{array}{c}\\mathbf {p} + \\mathbf {q} + \\mathbf {r} = \\mathbf {0}\\\\[2pt]\\mathbf {p},\\mathbf {q},\\mathbf {r} \\in \\mathbb {\\Lambda }^{\\prime }\\end{array}} f(\\mathbf {p}) g(\\mathbf {q}) h(\\mathbf {r}).$ Note that when $\\lambda = 2$ and we let $N \\rightarrow \\infty $ , the lattice (REF ) establishes a decimation of Fourier space for $2\\pi $ -periodic functions, in the spirit of e.g.", "[51], [15], [16].", "Other examples, also for $\\lambda = 2$ are [43], [57].", "The application of lattice operations to the one-dimensional Burgers equation reproduces some well-known shell models of turbulence; see Appendix [app:A]A for the details." ], [ "Ideal incompressible flow", "In this and next sections, we make sense of incompressible hydrodynamics on logarithmic lattices by applying the operations introduced previously.", "We will consider a $d$ -dimensional logarithmic lattice $\\mathbb {\\Lambda }^d$ , for $d = 2$ or 3, where $\\mathbb {\\Lambda } = \\lbrace 0, \\pm 1, \\pm \\lambda , \\pm \\lambda ^2, \\dots \\rbrace ,$ for some $\\lambda $ from Theorem REF .", "This lattice mimics Fourier space of a system with largest integral scale $L \\sim 2 \\pi $ corresponding to $|\\mathbf {k}| \\sim 1$ .", "Our derivations below are equally valid for the case $\\mathbb {\\Lambda } = \\lbrace \\pm 1, \\pm \\lambda , \\pm \\lambda ^2, \\dots \\rbrace $ , where zero is excluded from (REF ).", "This section is subdivided as follows.", "Section REF introduces the incompressible Euler equations on the logarithmic lattice and enumerates their main properties.", "Section REF establishes rigorous results concerning the local-in-time existence and uniqueness of strong solutions and the criterion for singularity formation in this model.", "Section REF presents a numerical study of blowup in the three-dimensional equations." ], [ "Basic equations, symmetries and conservation laws", "We represent the velocity field $\\mathbf {u}(\\mathbf {k},t) = (u_1,\\dots ,u_d) \\in \\mathbb {C}^d$ as a function of the wave vector $\\mathbf {k} \\in \\mathbb {\\Lambda }^d$ and time $t \\in \\mathbb {R}$ .", "Similarly we define the scalar pressure $p(\\mathbf {k},t)$ .", "The inner product for vector fields will be understood as $(\\mathbf {u},\\mathbf {v}) = (u_1,v_1)+\\cdots +(u_d,v_d)$ with the inner product (REF ) for each scalar component.", "All functions are supposed to satisfy the reality condition (REF ).", "For the governing equations, we use the exact form of the incompressible Euler equations $\\partial _t \\mathbf {u} + \\mathbf {u} \\ast \\nabla \\mathbf {u} = -\\nabla p, \\quad \\nabla \\cdot \\mathbf {u} = 0,$ which are defined upon the logarithmic lattice $\\mathbb {\\Lambda }^d$ , with the conventional notation $(\\mathbf {u} \\ast \\nabla \\mathbf {v})_i = \\sum _{j = 1}^{d} u_j \\ast \\partial _j v_i$ for the product $\\ast $ from Theorem REF .", "Introducing the vorticity $\\omega = \\nabla \\times \\mathbf {u}$ and taking the curl of equations (REF ), we may write the Euler equations in vorticity formulation $\\partial _t \\omega + \\mathbf {u} \\ast \\nabla \\omega = \\omega \\ast \\nabla \\mathbf {u}.$ In the case of vanishing average velocity $\\mathbf {u}(\\mathbf {0}) = \\mathbf {0}$ at $\\mathbf {k} = \\mathbf {0}$ , the velocity field is recovered from the vorticity through the Biot-Savart law $\\mathbf {u}(\\mathbf {k}) = \\frac{i\\mathbf {k} \\times \\omega (\\mathbf {k})}{|\\mathbf {k}|^2} \\quad \\text{for} \\ \\mathbf {k} \\ne \\mathbf {0}; \\quad \\mathbf {u}(\\mathbf {0}) = \\mathbf {0}.$ Moreover, if we take the divergence of equation (REF ) and use the incompressibility condition, then the pressure may be obtained from the velocities by solving the Poisson equation $-\\Delta p = \\nabla \\cdot (\\mathbf {u} \\ast \\nabla \\mathbf {u}).$ The proposed model retains many properties of the continuous Euler equations, which rely only upon the structure of the equations and elementary operations on the logarithmic lattice, as described in the previous sections.", "These include the basic symmetry groups.", "Theorem 9 (Symmetry groups of the Euler equations on the logarithmic lattice) Let $\\mathbf {u}(\\mathbf {k},t)$ , $p(\\mathbf {k},t)$ be a solution of the Euler equations (REF ).", "Then the following transformations also yield solutions: (Time translations) $\\!\\begin{aligned}[t]\\mathbf {u}^\\tau (\\mathbf {k},t) = \\mathbf {u}(\\mathbf {k},t+\\tau ),\\end{aligned}$ for any $\\tau \\in \\mathbb {R}$ ; (Space translations) $\\!\\begin{aligned}[t]\\mathbf {u}^{\\xi }(\\mathbf {k},t)= e^{-i\\mathbf {k}\\cdot \\xi }\\mathbf {u}(\\mathbf {k},t),\\end{aligned}$ for any $\\xi \\in \\mathbb {R}^d$ ; (Isotropy and parity) $\\mathbf {u}^R(\\mathbf {k},t) = R^{-1}\\mathbf {u}(R\\mathbf {k},t),$ where $R \\in \\mathsf {O_h}$ is any element of the group of cube symmetries (cf.", "Definition REF ); (Scale invariance) $\\!\\begin{aligned}[t]\\mathbf {u}^{n,h}(\\mathbf {k},t) = \\lambda ^h \\mathbf {u}\\left(\\lambda ^{n} \\mathbf {k},\\lambda ^{h-n} t \\right),\\end{aligned}$ for any $h \\in \\mathbb {R}$ and $n \\in \\mathbb {Z}$ , where $\\lambda $ is the lattice spacing; (Time reversibility) $\\!\\begin{aligned}[t]\\mathbf {u}^{r}(\\mathbf {k},t) = -\\mathbf {u}\\left(\\mathbf {k},-t \\right)\\end{aligned}$ ; (Galilean invariance) $\\!\\begin{aligned}[t]\\mathbf {u}^{\\mathbf {v}}(\\mathbf {k},t) = e^{-i\\mathbf {k}\\cdot \\mathbf {v}t}\\mathbf {u}(\\mathbf {k},t) - \\widehat{\\mathbf {v}}(\\mathbf {k}),\\end{aligned}$ for any $\\mathbf {v} \\in \\mathbb {R}^d$ , where $\\widehat{\\mathbf {v}}(\\mathbf {k})$ is the constant velocity field on the lattice defined as $\\widehat{\\mathbf {v}}(\\mathbf {0}) = \\mathbf {v}$ and zero for $\\mathbf {k} \\ne \\mathbf {0}$ .", "We did not write the transformations for the pressure $p$ because it can be eliminated from the Euler equations.", "Recall that the factors $e^{-i\\mathbf {k}\\cdot \\xi }$ and $e^{-i\\mathbf {k}\\cdot \\mathbf {v}t}$ in the symmetries REF and REF are Fourier representations of physical-space translations by the vectors $\\xi $ and $\\mathbf {v}t$ .", "Thus, the listed symmetries of the Euler equations on the logarithmic lattice are the same as those for the continuous model, except that isotropy REF and scale invariance REF are given in discrete form.", "Model (REF ) also preserves the same invariants as the continuous Euler equations.", "Let us show this first for the energy and for the enstrophy or helicity, in the two or three-dimensional cases respectively.", "Here we proceed formally.", "The proofs in this section hold for strong solutions, whose existence and uniqueness for short times are established in the next Section REF .", "Theorem 10 (Conservation of energy, enstrophy, helicity) Let $\\mathbf {u}(t)$ be a solution of the three-dimensional Euler equations (REF ).", "Then the energy $E(t) = \\frac{1}{2} (\\mathbf {u}, \\mathbf {u})$ and the helicity $H(t) = (\\mathbf {u},\\omega )$ are conserved in time.", "In the the two-dimensional case, the energy (REF ) and the enstrophy $\\Omega (t) = \\frac{1}{2}(\\omega ,\\omega )$ are conserved in time.", "Taking the energy as an example, let us show how the proof can be written using the basic operations defined on the logarithmic lattice, following the standard approach of fluid dynamics.", "Using the Euler equations (REF ), we obtain $\\frac{dE}{dt} =\\displaystyle \\frac{d}{dt} \\left[ \\frac{1}{2} (\\mathbf {u}, \\mathbf {u}) \\right]= (\\mathbf {u}, \\partial _t \\mathbf {u})= -\\, (\\mathbf {u}, \\nabla p) - (\\mathbf {u}, \\mathbf {u} \\ast \\nabla \\mathbf {u}).$ The pressure term vanishes owing to the incompressibility condition as $(\\mathbf {u}, \\nabla p)= \\sum _{i = 1}^{d} (u_i,\\partial _i p)= -\\sum _{i = 1}^{d} (\\partial _i u_i, p)= - (\\nabla \\cdot \\mathbf {u}, p) = 0,$ where the second relation is obtained from the integration by parts (REF ).", "In the inertial term, using commutativity of the product REF , the associativity in average REF and the Leibniz rule REF , one obtains $(\\mathbf {u}, \\mathbf {u} \\ast \\nabla \\mathbf {u})= \\sum _{i,j = 1}^{d}(u_i, u_j \\ast \\partial _j u_i)= \\sum _{i,j = 1}^{d} (u_i \\ast \\partial _j u_i, u_j)= \\frac{1}{2} \\sum _{i,j = 1}^{d} (\\partial _j (u_i \\ast u_i), u_j).$ After integration by parts, this term vanishes due to the incompressibility condition.", "Conservation of enstrophy and helicity in their respective space dimensions can be proved following a similar line of derivations.", "One can also derive the analogue of Kelvin's Circulation Theorem for the Euler system  (REF ) on a logarithmic lattice.", "For this purpose, let us recall the relation of circulation with the cross-correlation $\\Gamma = (\\mathbf {u}, \\mathbf {h})$ for “frozen-into-fluid” divergence-free vector fields $\\mathbf {h}(\\mathbf {k},t)$ satisfying the equations [76] $\\partial _t \\mathbf {h} + \\mathbf {u} \\cdot \\nabla \\mathbf {h} - \\mathbf {h} \\cdot \\nabla \\mathbf {u} = \\mathbf {0}, \\quad \\nabla \\cdot \\mathbf {h} = 0.$ The circulation around a closed material contour $\\mathbf {C}(s,t)$ in three-dimensional physical space ($s$ is the arc length parameter) is given by the cross-correlation $\\Gamma $ with the field [98] $\\mathbf {h}(\\mathbf {x},t) = \\oint \\frac{\\partial \\mathbf {C}(s,t)}{\\partial s}\\,\\delta ^3(\\mathbf {x}-\\mathbf {C}(s,t))\\,ds,$ where $\\delta ^3$ is the 3D Dirac delta function.", "The field (REF ) satisfies equations (REF ) in the sense of distributions.", "Thus, Kelvin's Theorem follows, as a particular case, from the conservation of cross-correlation $\\Gamma $ .", "The following Theorem proves the conservation of cross-correlation in the lattice model.", "Theorem 11 (Kelvin's Theorem) Let $\\mathbf {u}(t)$ be a solution of the three-dimensional Euler equations (REF ).", "Then, for any “frozen-into-fluid” divergence-free field $\\mathbf {h}(t)$ satisfying equations $\\partial _t \\mathbf {h} + \\mathbf {u} \\ast \\nabla \\mathbf {h} - \\mathbf {h} \\ast \\nabla \\mathbf {u} = \\mathbf {0}, \\quad \\nabla \\cdot \\mathbf {h} = 0,$ the cross-correlation $\\Gamma (t) = (\\mathbf {u},\\mathbf {h})$ is conserved in time.", "Since equations (REF ) are satisfied by the vorticity field $\\omega $ , the proof for conservation of the cross-correlation follows the same steps as for conservation of helicity (REF ).", "Theorem REF provides an infinite number of circulation invariants: the cross-correlation $\\Gamma $ is conserved for any solution of system (REF ).", "For two-dimensional flows, Kelvin's Theorem can be reformulated as the conservation of flux of vorticity across surfaces moving with the fluid.", "This flux can be expressed as the inner product $\\Gamma (t) = (a,\\omega )$ of the scalar vorticity $\\omega = \\partial _1 u_2 - \\partial _2 u_1$ with a Lagrangian marker $a(\\mathbf {k},t)$  [76], which is advected by the flow and satisfies the equation $\\partial _t a + \\mathbf {u} \\cdot \\nabla a = 0$ .", "Indeed, taking the Lagrangian marker as the indicator function of a bounded surface $S_t$ carried by the flow [29], the flux of vorticity across $S_t$ yields the circulation along the contour $\\partial S_t$ , i.e., $\\Gamma (t) = \\int _{S_t} \\omega dS = \\int _{\\partial S_t} \\mathbf {u} \\cdot d\\mathbf {l}$ .", "On the logarithmic lattice, the vorticity flux is introduced similarly, as the inner product $\\Gamma (t) = (a,\\omega )$ of the scalar vorticity with a Lagrangian marker satisfying the equation $\\partial _t a + \\mathbf {u} \\ast \\nabla a = 0.$ It is straightforward to show that, given the solution $\\mathbf {u}(\\mathbf {k},t)$ of the two-dimensional Euler system (REF ), the conservation of $\\Gamma $ holds for any solution of (REF )." ], [ "Regularity of solutions", "In this section, we establish the local theory for the Euler system on the logarithmic lattice.", "Here the results are similar to those for the original model: we show local existence and uniqueness of strong solutions and the Beale-Kato-Majda (BKM) blowup criterion [4].", "Two-dimensional solutions turn out to be globally regular.", "For simplicity, we assume the vanishing average velocity $\\mathbf {u}(\\mathbf {0}) = \\mathbf {0}$ at $\\mathbf {k} = \\mathbf {0}$ and, therefore, consider only wave vectors with $|\\mathbf {k}| \\ne 0$ in the following analysis.", "For the lattice variables, we introduce the $\\ell ^2$ norm in the standard way as $||\\mathbf {u}||_{\\ell ^2} = \\left( \\sum _{\\mathbf {k} \\in \\mathbb {\\Lambda }^d} |\\mathbf {u}(\\mathbf {k})|^2 \\right)^{1/2}$ and the $\\ell ^\\infty $ norm as $||\\mathbf {u}||_{\\ell ^\\infty } = \\sup _{\\mathbf {k} \\in \\mathbb {\\Lambda }^d}|\\mathbf {u}(\\mathbf {k})|$ .", "Given a nonnegative integer $m$ , we introduce the operator $D^m$ as $D^m \\mathbf {u}(\\mathbf {k}) = |\\mathbf {k}|^m \\mathbf {u}(\\mathbf {k}),$ and define the homogeneous Sobolev spaces $h^m$ on the lattice consisting of the functions with finite norm $||\\mathbf {u}||_{h^m} = ||D^m \\mathbf {u}||_{\\ell ^2} = \\left( \\sum _{\\mathbf {k} \\in \\mathbb {\\Lambda }^d} |\\mathbf {k}|^{2m} |\\mathbf {u}(\\mathbf {k})|^2 \\right)^{1/2} <\\infty .$ Clearly, the space $h^m$ is a Hilbert space endowed with the inner product $(\\mathbf {u},\\mathbf {v})_{h^m} = (D^m \\mathbf {u}, D^m \\mathbf {v})$ , whose functions have all partial derivatives up to order $m$ in $\\ell ^2$ .", "Finally, we consider the space of divergence-free vector fields $V^m = \\lbrace \\mathbf {u} \\in h^m| \\nabla \\cdot \\mathbf {u} = 0 \\rbrace ,$ which provides the natural setting for strong solutions of the Euler equations.", "The space $V^m$ is endowed with the $h^m$ norm.", "Theorem 12 Let $\\mathbf {u}^0 \\in V^m$ for some $m \\ge 1$ .", "Then, there exists a time $T>0$ , such that the incompressible Euler equations on the logarithmic lattice (REF ) have a unique strong solution $\\mathbf {u}(t)$ in the class $\\mathbf {u} \\in C^1([0,T);V^m),$ with initial condition $\\mathbf {u}\\big |_{t=0} = \\mathbf {u}^0$ .", "This solution either exists globally in time, or there is a finite maximal time of existence $t_b$ such that $\\limsup _{t \\nearrow t_b} ||\\mathbf {u}(t)||_{h^m} = \\infty .$ The proof is similar to that in [32] for shell models of turbulence and exploits the locality of the nonlinear interactions on the logarithmic lattice, which turns the convective term into the action of a bounded operator.", "We write the Euler system (REF ) in the functional form $\\partial _t \\mathbf {u} + B(\\mathbf {u},\\mathbf {u}) = -\\nabla p,$ where we have introduced the operator $B(\\mathbf {u},\\mathbf {v}) = \\mathbf {u} \\ast \\nabla \\mathbf {v}.$ Operator $B$ is a bounded bilinear operator in $h^m$ – see the proof in Appendix [app:B]B.", "Next, in order to eliminate pressure, we project Eq.", "(REF ) onto the space of divergence-free vector fields.", "We introduce the Leray projector $\\mathbb {P}$ – cf.", "[89] – on the logarithmic lattice, explicitly given by $\\mathbb {P}_{ij}(\\mathbf {k}) = \\delta _{ij} - \\frac{k_ik_j}{|\\mathbf {k}|^2}, \\quad \\mathbf {k} \\in \\mathbb {\\Lambda }^d.$ Since $\\mathbf {u}$ is divergence free and $\\nabla p$ is a full gradient, it follows that $\\mathbb {P}\\mathbf {u} = \\mathbf {u}$ and $\\mathbb {P} \\nabla p = 0$ , and so we are reduced to the problem $\\frac{d\\mathbf {u}}{dt} = F(\\mathbf {u}), \\quad \\mathbf {u}\\big |_{t=0} = \\mathbf {u}^0,$ where $F(\\mathbf {u}) = -\\mathbb {P} B(\\mathbf {u},\\mathbf {u})$ maps functions from $V^m$ to itself.", "We claim that $F$ is locally-Lipschitz continuous.", "Since $\\mathbb {P}$ is an orthogonal projection on $h^m$ , and therefore $||\\mathbb {P}\\mathbf {v}||_{h^m} \\le ||\\mathbf {v}||_{h^m}$ , we have $\\begin{aligned}||F(\\mathbf {u}) - F(\\mathbf {v})||_{h^m} &= ||\\mathbb {P} [B(\\mathbf {u},\\mathbf {u}) - B(\\mathbf {v},\\mathbf {v})] ||_{h^m}\\\\&\\le ||B(\\mathbf {u},\\mathbf {u}) - B(\\mathbf {v},\\mathbf {v}) ||_{h^m} \\\\&\\le ||B(\\mathbf {u},\\mathbf {u}-\\mathbf {v})||_{h^m} + ||B(\\mathbf {u} - \\mathbf {v},\\mathbf {v}) ||_{h^m}.\\end{aligned}$ In the last inequality, we have applied the bilinearity of $B$ and the triangle inequality.", "Using the boundness of operator $B$ , there exists a constant $C>0$ such that $||B(\\mathbf {u},\\mathbf {u}-\\mathbf {v})||_{h^m} \\le C||\\mathbf {u}||_{h^m} ||\\mathbf {u} - \\mathbf {v}||_{h^m}.$ A similar inequality is obtained for the other term $||B(\\mathbf {u} - \\mathbf {v}, \\mathbf {v})||_{h^m}$ , which proves the Lipschitz continuity of $F$ when $||\\mathbf {u}||_{h^m}$ and $||\\mathbf {v}||_{h^m}$ are bounded by some constant.", "It follows that Eq.", "(REF ) is an ordinary differential equation with $F$ locally-Lipschitz continuous on the Banach space $V^m$ .", "In this framework, we apply the Picard Theorem on Banach spaces – see e.g.", "[20], [92] – to guarantee existence of a unique local solution in the class (REF ) and initial condition $\\mathbf {u}^0$ .", "The pressure is recovered by solving the Poisson equation (REF ).", "The blowup statement in (REF ) also follows from classical theory of ordinary differential equations [92].", "Theorem 13 (BKM blowup criterion) Let $\\mathbf {u}(t) \\in C^1([0,t_b);V^m)$ be a strong solution for the incompressible Euler equations (REF ) on the logarithmic lattice, where $t_b$ is the maximal time of existence.", "Then either $t_b = \\infty $ or $\\int _{0}^{t_b}||\\omega (t)||_{\\ell ^\\infty } dt = \\infty .$ In the later case, we have necessarily $\\limsup _{t \\nearrow t_b} ||\\omega (t)||_{\\ell ^\\infty } = \\infty .$ Let us assume that $\\int _{0}^{t_b}||\\omega (t)||_{\\ell ^\\infty } dt = M < \\infty .$ for a finite $t_b < \\infty $ .", "We are going to prove that this implies $||\\mathbf {u}(t)||_{h^m} \\le N, \\quad \\forall t<t_b,$ for some constant $N<\\infty $ , thus contradicting condition (REF ) of Theorem REF .", "To show this, we perform an energy estimate for Eq.", "(REF ).", "We set $\\mathbf {v} = D^m \\mathbf {u}$ and $q = D^m p$ and apply $D^m$ to Eq.", "(REF ) to obtain $\\partial _t \\mathbf {v} = -D^m(\\mathbf {u} \\ast \\nabla \\mathbf {u}) - \\nabla q.$ Taking the $\\ell ^2$ -inner product of Eq.", "(REF ) with $\\mathbf {v}$ yields $\\frac{1}{2} \\frac{d}{dt}||\\mathbf {v}||_{\\ell ^2}^2 = -(D^m(\\mathbf {u} \\ast \\nabla \\mathbf {u}),\\mathbf {v}) - (\\nabla q, \\mathbf {v}).$ After integrating by parts, the last term vanishes due to incompressibility as $(\\nabla q, \\mathbf {v}) = \\sum _{i = 1}^{d}(\\partial _i q, v_i) = -\\sum _{i = 1}^{d}(q,\\partial _i v_i) = -\\sum _{i = 1}^{d}(q,D^m \\partial _i u_i) = -(q,D^m \\nabla \\cdot \\mathbf {u}) = 0.$ Next, we use the following calculus inequality on logarithmic lattices $||\\mathbf {f} \\ast \\mathbf {g}||_{h^m} \\le C(||\\mathbf {f}||_{h^m} ||\\mathbf {g}||_{\\ell ^\\infty } + ||D\\mathbf {f}||_{\\ell ^\\infty } ||\\mathbf {g}||_{h^{m-1}}), \\quad \\text{for } \\mathbf {f} \\in h^m, \\mathbf {g} \\in h^{m-1}$ for some positive constant $C$ ; this inequality has a continuous analogue for Sobolev spaces $H^s$ – see e.g.", "[76] – and the lattice version (REF ) is proved in the Appendix [app:B]B.", "Then, the nonlinear term in (REF ) can be estimated using $\\mathbf {f} = \\mathbf {u}$ and $\\mathbf {g} = \\nabla u_i$ as $\\begin{array}{rcl}(D^m(\\mathbf {u} \\ast \\nabla \\mathbf {u}),\\mathbf {v}) & \\le & ||\\mathbf {u} \\ast \\nabla \\mathbf {u}||_{h^m} ||\\mathbf {v}||_{\\ell ^2}\\\\[3pt]& \\le & \\displaystyle C||\\mathbf {v}||_{\\ell ^2}\\sum _{i = 1}^d\\left(||\\mathbf {u}||_{h^m} ||\\nabla u_i||_{\\ell ^\\infty }+ ||D\\mathbf {u}||_{\\ell ^\\infty } ||\\nabla u_i||_{h^{m-1}}\\right)\\\\[15pt]& \\le & \\displaystyle 2dC||\\mathbf {v}||_{\\ell ^2} ||\\mathbf {u}||_{h^m} ||D\\mathbf {u}||_{\\ell ^\\infty }=C^{\\prime }||\\mathbf {v}||_{\\ell ^2}^2 ||D\\mathbf {u}||_{\\ell ^\\infty },\\end{array}$ where at the end we used $||\\mathbf {u}||_{h^m} = ||D^m \\mathbf {u}||_{\\ell ^2} = ||\\mathbf {v}||_{\\ell ^2}$ and set $C^{\\prime } = 2dC$ .", "Substituting relations (REF ) and (REF ) into (REF ) yields $\\frac{d}{dt}||\\mathbf {v}||_{\\ell ^2}^2 \\le 2C^{\\prime }||\\mathbf {v}||_{\\ell ^2}^2 ||D\\mathbf {u}||_{\\ell ^\\infty },$ and applying Gronwall's Inequality, we are lead to $||\\mathbf {v}(t)||_{\\ell ^2} \\le ||\\mathbf {v}(0)||_{\\ell ^2} \\exp \\left( C^{\\prime } \\int _{0}^{t} ||D\\mathbf {u}(s)||_{\\ell ^\\infty }ds \\right).$ Finally, using the estimate $||D\\mathbf {u}||_{\\ell ^\\infty } \\le ||\\omega ||_{\\ell ^\\infty },$ which follows from the Biot-Savart law (REF ), and recalling again that $||\\mathbf {v}||_{\\ell ^2} = ||\\mathbf {u}||_{h^m}$ , we obtain $||\\mathbf {u}(t)||_{h^m} \\le ||\\mathbf {u}(0)||_{h^m} \\exp \\left( C^{\\prime } \\int _{0}^{t_b} ||\\omega (s)||_{\\ell ^\\infty }ds \\right) \\le N, \\quad \\forall t \\in [0,t_b)$ for $N = ||\\mathbf {u}(0)||_{h^m} \\exp (C^{\\prime }M) < \\infty $ .", "This is the inequality (REF ), which led us to contradiction.", "Corollary 14 Strong solutions $\\mathbf {u}(t)$ of the two-dimensional incompressible Euler equations (REF ) exist globally in time.", "From Theorem REF , strong solutions of the two-dimensional Euler equations conserve the $\\ell ^2$ norm $||\\omega ||_{\\ell ^2}$ of the vorticity.", "Hence, the inequality $||\\omega ||_{\\ell ^\\infty } \\le ||\\omega ||_{\\ell ^2}$ on the lattice prevents condition (REF ) to take place." ], [ "Blowup in incompressible 3D Euler equations", "Whether three-dimensional incompressible Euler flow develops a singularity in finite time (also called blowup) remains a challenging open mathematical problem.", "According to the BKM criterion, the singularity implies a spontaneous generation of infinitely large vorticity.", "Such singularity is anticipated by Kolmogorov's theory of turbulence [50], which predicts that the vorticity diverges at small scales as $\\delta \\omega \\sim \\ell ^{-2/3}$ when energy is transferred from integral to viscous scales.", "In this context, blowup could reveal an efficient mechanism for the energy cascade and, for this reason, it is often considered a cornerstone for the theory of turbulence.", "In addition to purely mathematical approaches, see e.g.", "[22], [94] and very recent achievements [24], [44], the blowup problem was intensively investigated through Direct Numerical Simulations (DNS) [52], [56], [60].", "However, numerical results appear to be rather inconclusive, with the controversy [64], [61] only growing with the increase of resolution.", "Naturally, several simplified models have been investigated for understanding possible blowup scenarios, e.g.", "[31], [95], [42], [73].", "Despite being rather successful in the study of turbulence [8] and serving as a useful testing ground for mathematical analysis, e.g.", "[63], [26], these models fall short of reproducing basic features of Euler's blowup phenomenon: they lack important properties of Euler's flow, such as incompressibility and conservation of circulation, and often show dynamical behavior atypical for Euler solutions, such as self-similarity [21], [23].", "Note that we do not discuss here boundary effects [69], which set a different open problem.", "Unlike many previous simplified models, the Euler equations on logarithmic lattices retain most structural properties of the original equations, as we showed in Section REF .", "In the work [19], [17], we presented a numerical evidence of chaotic blowup in the three-dimensional Euler system on a golden-mean logarithmic lattice.", "Now we extend these previously reported results by testing the robustness of our conclusions on different lattices.", "For the comparison, we consider the golden mean $\\lambda = \\varphi $ and the plastic number $\\lambda = \\sigma $ , which provide two lattices $\\mathbb {\\Lambda }^3$ with increasing resolution – see Fig.", "REF ; here, $\\mathbb {\\Lambda } = \\lbrace \\pm 1, \\pm \\lambda , \\pm \\lambda ^2, \\dots \\rbrace $ is taken, with no zero component.", "We remark that the spacing factor $\\lambda = 2$ does not provide a reliable model for the blowup study, because the incompressibility condition together with a small number of triad interactions cause degeneracies in coupling of different modes.", "Numerical model.", "Aiming for the study of blowup, initial conditions are chosen to have nonzero components limited to large scales, with wavenumbers $1 \\le |k_i| \\le \\varphi ^2 = (3+\\sqrt{5})/2$ .", "This corresponds to a box of three excited modes in each direction for the golden mean and four modes for the plastic number lattice spacing.", "The velocities at these modes are explicitly given in the form $u_j(\\mathbf {k}) = \\sum _{m,n = 1}^{3}\\frac{|\\epsilon _{jmn}|}{2}k_m k_n e^{i\\theta _j(\\mathbf {k})-|\\mathbf {k}|}, \\quad \\text{for} \\quad j=1,2.$ Here $\\epsilon _{jmn}$ is the Levi-Civita permutation symbol and the phases $\\theta _j$ are given by $\\theta _j(\\mathbf {k}) = \\mathrm {sgn}(k_1)\\alpha _j + \\mathrm {sgn}(k_2)\\beta _j+\\mathrm {sgn}(k_3)\\delta _j + \\mathrm {sgn}(k_1k_2k_3)\\gamma _j,$ with the constants $(\\alpha _1,\\beta _1,\\delta _1,\\gamma _1) = (1,-7,13,-3)/4$ and $(\\alpha _2,\\beta _2,\\delta _2,\\gamma _2) = (-1,-3,11,7)/4$ .", "The third component of velocity is uniquely defined by the incompressibility condition.", "Clearly, because the nodes of different lattices do not match, it is impossible to test the same initial condition on different lattices.", "Figure: Comparison of the Euler blowup dynamics for different lattice spacings – the golden mean ϕ\\varphi in green and plastic number σ\\sigma in red.", "Owing to the lower computational cost, the simulation for the golden mean spans a larger spatial range.", "(a) Dynamic evolution of the inverse maximum vorticity 1/ω max 1/\\omega _{\\max }, reaching the blowup times t b =4.255t_b = 4.255 and 10.05210.052 for golden and plastic lattice spacings, respectively.", "(b) The maximum vorticity ω max \\omega _{\\max } in logarithmic scale fitting in average the power law ∼(t b -t) -1 \\sim (t_b - t)^{-1}.", "(c) Wave number k max k_{\\max } where the maximum vorticity occurs in logarithmic scale, following the asymptotic ∼(t b -t) -γ \\sim (t_b -t)^{-\\gamma } with γ=2.70\\gamma = 2.70.", "(d) The energy spectrum () at the final time of integration for each simulation, developing the power-law E(k)∝k -ξ E(k) \\propto k^{-\\xi }, where ξ=2.26\\xi = 2.26.Like in usual DNS, we consider the Euler equations in vorticity formulation (REF ), where the velocity field is recovered through the Biot-Savart law (REF ).", "The equations are integrated numerically with double-precision using the fourth-order Runge-Kutta-Fehlberg adaptive scheme [49].", "The time step was dynamically defined in order to keep the relative error for $\\omega _{\\max }(t) = \\max _{\\mathbf {k}}|\\omega (\\mathbf {k},t)|$ below $10^{-6}$ .", "Since only a finite number $N$ of modes in each direction can be simulated, the infinite-dimensional nature of the problem was tracked by implementing the following spatial adaptive scheme.", "At each time step, we compute the enstrophy $\\Omega (t) = \\frac{1}{2} (\\omega ,\\omega )$ due to the modes with wave vectors $|\\mathbf {k}| \\ge K_{\\max }/\\lambda $ , where $K_{\\max }$ is the largest wave number in each direction.", "This quantity estimates the enstrophy error (i.e., $\\ell ^2$ norm of vorticity) due to the mode truncation and it was kept below $10^{-15}$ during the whole simulation.", "Every time this threshold was reached we increased the number of modes in each direction by 5, i.e., multiplying $K_{\\max }$ by $\\lambda ^5$ .", "We stopped the simulation for the plastic number with $N = 95$ , thus covering a spatial range of $K_{\\max } = \\sigma ^{95} \\approx 10^{11}$ .", "Due to the higher spacing value, the golden mean allows to cover a larger spatial range with less modes.", "In this case, the simulation was stopped with $N = 70$ , which corresponds to $K_{\\max } = \\varphi ^{70} \\approx 10^{14}$ .", "For the simulations of both lattice spacings, the energy was conserved during the whole time of integration with a relative error below $10^{-6}$ .", "Results.", "Before presenting our new results, we briefly review the previous conclusions reported in [19], where a simulation of ideal flow was conducted on a golden mean logarithmic lattice.", "A large amplification of maximum vorticity $\\omega _{\\max }$ within a finite time $t_b$ was observed, demonstrating an asymptotic blowup solution $\\omega _{\\max }(t) \\sim (t_b - t)^{-1}$ , followed by a power-law development in the energy spectrum as $E(k) \\propto k^{-\\xi }$ , $\\xi \\approx 2.26$ .", "Such blowup is linked to a chaotic wave in a renormalized system, traveling with constant average speed $\\gamma \\approx 2.70$ .", "The chaotic behavior justifies the high sensitivity to perturbations, which is encountered in full DNS [56] and has theoretical foundation in developed turbulence [90].", "Instability of blowup solutions is also observed in other simplified models [95], [74], [37] and was proved recently for the full incompressible 3D Euler equations [96].", "The chaotic attractor restores the isotropy in the statistical sense, even though the solution at each particular moment is essentially anisotropic, in similarity to the recovery of isotropy in the Navier-Stokes turbulence [50], [11].", "The striking property of the attractor is its multi-scale character, covering six decades in Fourier space, which seems impossible to be reproduced by the modern numerical techniques in full DNS.", "At the respective scales, the solution of the logarithmic model displays properties that can be associated with typical coherent structures of full DNS, e.g.", "the effect of two-dimensional depletion [88], [2].", "For more details, see [19].", "Fig.", "REF compares the numerical integrations of the Euler equations for golden and plastic lattice spacings.", "Though solutions are, of course, different at earlier times, they demonstrate very close (numerically indistinguishable) asymptotic blowup dynamics, which we analyze in details now.", "Figs.", "REF (a) and REF (b) show the dynamic evolution of $\\omega _{\\max }(t)$ .", "BKM blowup criterion – see Theorem REF – requires an asymptotic growth of at least $\\omega _{\\max }(t) \\sim (t_b - t)^{-1}$ as $t$ approaches the blowup time $t_b$ .", "This is verified for both simulations by plotting the inverse value $1/\\omega _{\\max }(t)$ in Fig.", "REF (a), providing the blowup times $t_b = 4.255 \\pm 0.001$ and $t_b = 10.052 \\pm 0.001$ for the plastic number and golden mean, respectively.", "Fig.", "REF (b) shows the same results in logarithmic scale verifying the asymptotic $\\omega _{\\max }(t) \\sim (t_b - t)^{-1}$ .", "Note that a growth of five orders of magnitude is observed for the golden mean.", "The wave number $k_{\\max }$ at which the maximum vorticity occurs also grows asymptotically as $k_{\\max } \\sim (t_b - t)^{-\\gamma }$ with the same exponent $\\gamma = 2.70 \\pm 0.01$ for the two simulations, as shown in Fig.", "REF (c).", "At last, the energy spectrum $E(k) = \\frac{1}{2 \\Delta } \\sum _{k \\le |\\mathbf {k}^{\\prime }| < \\lambda k} |\\mathbf {u}(\\mathbf {k}^{\\prime })|^{2}, \\quad \\text{with } \\Delta = \\lambda k - k,$ develops a power-law $E(k) \\propto k^{-\\xi }$ .", "A dimensional argument [19] predicts the exponent $\\xi $ depending upon $\\gamma $ as $\\xi = 3 - 2/\\gamma \\approx 2.26$ , confirmed in Fig.", "REF (d).", "Chaotic attractor.", "We argued in [19] that the blowup dynamics in the Euler system on a logarithmic lattice is associated to a chaotic wave in a renormalized system.", "This chaotic behavior exemplifies the fundamental instability, which is necessary for blowup in the incompressible 3D Euler equations, as proved recently in [96].", "Here, we compare the attractors for the two simulations.", "These attractors are visualized using the renormalized variables $\\widetilde{\\omega } = (t_b-t)\\omega , \\quad \\eta = \\log |\\mathbf {k}|, \\quad \\mathbf {o} = \\mathbf {k}/|\\mathbf {k}|, \\quad \\tau = -\\log (t_b-t),$ which apply similarly in Fourier space $\\mathbb {R}^3$ and in our lattice $\\mathbb {\\Lambda }^3$ .", "The Euler equations can be rewritten as a dynamical system with respect to these new coordinates; consult [19] for more details.", "Fig.", "REF shows the time evolution, in renormalized variables, of the solutions on the two different lattices.", "For the comparison, we plot the vorticities $\\widetilde{\\omega }$ normalized with respect to their correspondent maximum values $\\widetilde{\\omega }_{\\max }$ .", "The renormalized time for the plastic number is shifted $\\tau \\mapsto \\tau + \\tau _0$ by $\\tau _0 = -1.2$ for the attractors to be aligned in space.", "Figure: Absolute value of renormalized vorticities |ω ˜||\\widetilde{\\omega }| plotted on sections of 3D Fourier space, in logarithmic scales, at three different instants τ\\tau .", "For comparison, the vorticities are normalized with respect to their maximum norm ω ˜ max \\widetilde{\\omega }_{\\max }; values below 0.010.01 are plotted in white.", "The first row shows the evolution on the golden and the second row on the plastic lattice.", "Owing to the lower computational cost, the simulation for the golden mean was integrated for longer renormalized times τ\\tau .The solutions in Fig.", "REF show convergence to chaotic waves, which travel through the main diagonal of Fourier space with the same constant speed $\\gamma $ .", "In [19], we demonstrated their chaotic nature by computing a positive largest Lyapunov exponent.", "The chaotic attractors look surprisingly similar despite the quite distinct resolutions furnished by the two lattices." ], [ "Viscous incompressible flow and turbulence", "In this section, we introduce a viscous dissipative term and a forcing $\\mathbf {f}$ into the Euler equations (REF ), leading to the incompressible 3D Navier-Stokes equations on a logarithmic lattice $\\partial _t \\mathbf {u} + \\mathbf {u} \\ast \\nabla \\mathbf {u} = -\\nabla p + \\nu \\Delta \\mathbf {u} + \\mathbf {f}, \\quad \\nabla \\cdot \\mathbf {u} = 0,$ where $\\nu \\ge 0$ is the kinematic viscosity.", "We will focus on testing some fundamental properties of hydrodynamic turbulence, when the viscous term is responsible for dissipating energy at small scales of the flow while the force injects it at large scales.", "Following the same lines of derivations as for the continuous model, we deduce the balance for the energy (REF ) as $\\frac{dE}{dt} = -2\\nu \\Omega (t) + F(t),$ where $\\Omega (t)$ is the enstrophy (REF ) and $F(t) = (\\mathbf {u},\\mathbf {f})$ is the work done by external forces.", "The term $\\varepsilon = 2\\nu \\Omega $ is the total dissipation rate of the flow." ], [ "Anomalous dissipation", "A major feature of turbulent flows is the non-vanishing energy dissipation rate $\\epsilon >0$ in the limit of large Reynolds numbers, which can also be formulated mathematically as the limit of vanishing viscosity $\\nu \\rightarrow 0$ .", "This apparently paradoxical phenomenon is known as dissipation anomaly [83], [46] and has found confirmation in many experiments [84] and numerical simulations [62].", "Dissipation anomaly is conveniently quantified by considering the evolution of energy through different scales.", "We derive from Eq.", "(REF ) the scale-by-scale energy budget equation $\\partial _t E_k = \\Pi _k -2\\nu \\Omega _k + F_k.$ Here, using the notation $(f,g)_k = \\sum _{|\\mathbf {k}^{\\prime }| \\le k} f(\\mathbf {k}^{\\prime })\\overline{g(\\mathbf {k}^{\\prime })},$ we have introduced the cumulative energy between wave number 0 and $k$ $E_k = \\frac{1}{2} (\\mathbf {u},\\mathbf {u})_k,$ the cumulative enstrophy $\\Omega _k = \\frac{1}{2} (\\omega ,\\omega )_k,$ the cumulative energy injection $F_k = (\\mathbf {u}, \\mathbf {f})_k,$ and the energy flux $\\Pi _k = -(\\mathbf {u}, \\mathbf {u} \\ast \\nabla \\mathbf {u} + \\nabla p)_k.$ Statistical steady state in a turbulent flow is achieved when $\\partial _t \\langle E_k \\rangle = 0$ .", "In this regime, the mean energy flux $\\langle \\Pi _k \\rangle $ balances with the mean energy dissipation $\\langle -2\\nu \\Omega _k \\rangle $ and the work of external forces $\\langle F_k \\rangle $ .", "Since for small viscosities it is typical to have energy injection confined to large scales and energy dissipation confined to small scales, a dissipation anomaly is related to the development of a constant energy flux in the intermediate range called the inertial interval.", "In our definition, a positive energy flux corresponds to a (direct) cascade of energy from large to small scales.", "In order to compute the energy flux, we consider the Navier-Stokes equations (REF ) on the three-dimensional logarithmic lattice of spacing $\\lambda = \\varphi $ , the golden mean.", "The energy is injected at large scales $\\varphi \\le |k_{1,2,3}| \\le \\varphi ^3$ through a constant-in-time force with randomly generated components.", "To obtain an extended inertial interval, the viscous forces $\\nu \\Delta \\mathbf {u}$ were replaced by a hyper-viscous term $-\\nu (-\\Delta )^{h}\\mathbf {u}$ with $h = 2$ .", "For models with local triad interactions, it is expected that the dynamical statistics are ultraviolet robust, i.e., does not depend on the detailed dissipation mechanism at small scales [6], [72].", "The model was integrated with double-precision using the first-order exponential time-splitting method [35]: at each time step, we first use the fourth-order Runge-Kutta method to integrate the Euler equations and next we multiply the resulting solution by the exponential factor $e^{(-\\nu |\\mathbf {k}|^{2h}\\Delta t)}$ , where $\\Delta t$ is the time step.", "Fig.", "REF shows the mean energy flux $\\langle \\Pi _k \\rangle $ along scales $k$ for different viscosities.", "The energy flux reaches the same constant positive value for all viscosities and the inertial range extends to smaller scales as the viscosity decreases, which indicates the development of a dissipation anomaly in the limit $\\nu \\rightarrow 0$ .", "Figure: Mean energy flux 〈Π k 〉\\langle \\Pi _k \\rangle along wave numbers kk, in logarithmic scale, for different values of the hyper-viscous parameter ν=10 -13 \\nu = 10^{-13}, 10 -14 10^{-14}, 10 -15 10^{-15}, 10 -16 10^{-16} and 10 -17 10^{-17}.", "In our notation, a positive energy flux corresponds to a direct cascade of energy." ], [ "Statistics of Fourier modes", "The Navier-Stokes equations on the golden mean lattice were integrated using the same numerical procedure described in Section REF , with hyper-viscous term and viscosity $\\nu = 10^{-13}$ .", "The probability distribution functions (PDF's) were numerically estimated through a histogram binning procedure using the statistics accumulated within a sample time $T$ .", "In terms of turnover time $T_0 = 1/|\\mathbf {k}_0|U_0$ , where $\\mathbf {k}_0 = (1,1,1)$ is the wave vector of integral scale and $U_0 = \\langle |\\mathbf {u}(\\mathbf {k}_0)|^2 \\rangle ^{1/2}$ , the sample time $T$ was larger than $90 T_0$ .", "The PDF's of $\\text{Re}[u_1(\\mathbf {k}_n)]$ , in units of their root-mean-square $\\langle \\text{Re}[u_1(\\mathbf {k}_n)]^2 \\rangle ^{1/2}$ values are shown in Fig.", "REF , for several wave vectors $\\mathbf {k}_n = \\lambda ^n \\mathbf {k}_0$ rescaled along the main diagonal of Fourier space.", "Figure: Normalized PDF's in logarithmic scale of the real part of xx velocity, Re[u 1 (𝐤)]\\text{Re} [u_1(\\mathbf {k})], at different wave vectors 𝐤 n =λ n 𝐤 0 \\mathbf {k}_n = \\lambda ^n \\mathbf {k}_0, 𝐤 0 =(1,1,1)\\mathbf {k_0} = (1,1,1), rescaled along the main diagonal of Fourier space.", "Scales decrease from darker to lighter colors.", "Gaussian distribution of zero mean and unit variance is shown for comparison in black dashed line.", "(a) Statistics of inertial-range wave vectors 𝐤 n \\mathbf {k}_n, with n=10,⋯,15n = 10,\\dots ,15; (b) statistics of viscous-range wave vectors 𝐤 n \\mathbf {k}_n, with n=18,⋯,22n = 18,\\dots ,22.Fig.", "REF (a) shows the statistics at inertial-range wave vectors $\\mathbf {k}_n$ , for $n = 10,\\dots ,15$ .", "The PDF's for all scales are very close to a Gaussian distribution.", "Similar Gaussian distributions for inertial-range Fourier components were observed for developed turbulence through full DNS [13] and laboratory experiments [86], [80], [27].", "For a flow of characteristic large scale $L$ and finite correlation length $\\ell $ in physical space, the univariate statistics of Fourier modes in the inertial range are normally distributed in the asymptotic limit $\\ell /L \\rightarrow 0$ , as a particular case of the Central Limit Theorem for weighted integrals [68].", "For these reasons, it is commonly argued that Fourier modes are not well suited for the study of extreme events that proportionate inertial-range turbulent intermittency.", "Large fluctuations in Fourier modes can only appear when viscous processes become important and initiate a complex interplay between nonlinearity and dissipation.", "In this regime, the velocity field exhibits strong intermittency, associated with spatial variation of large-scale motion rather than with intense small-scale structures [25].", "Unlike what occurs in the inertial range, dissipative intermittency leaves fingerprints on viscous-range Fourier components, whose statistics develop widening of tails at smaller scales [13].", "Such behavior is also reproduced by the logarithmic model.", "Fig.", "REF (b) shows the statistics at viscous-range wave vectors $\\mathbf {k}_n$ , for $n = 18,\\dots ,22$ , where we observe an increasing deviation from Gaussian distribution as we move towards finer scales of the flow.", "As presented above, there is a strong similarity between statistics of lattice variables from the logarithmic model and Fourier components of the full Navier-Stokes equations; for instance, compare Figs.", "REF (a,b) of the present paper with Figs.", "1(f,b) from the DNS results in [13].", "However, it is quite intriguing that the Gaussian behavior in our model is in sharp contrast with statistics of other simplified models, which usually present some degree of inertial-range intermittency.", "We turn now to a brief discussion about their statistical behavior.", "Shell models of turbulence exhibit chaotic intermittent dynamics in the inertial interval with statistical properties close to the Navier-Stokes developed turbulence [54], [81], [70].", "On the other hand, the reduced wave vector set approximation (REWA) model displays only weak intermittency [43], [57].", "A possible explanation for this feature was given in [13], where it is argued that REWA model can be written in a spherical model framework [79] consisting of $N$ interacting subsystems each one describing the evolution of a velocity component in a certain direction.", "In this framework, modes should have Gaussian statistics [45] and anomalous fluctuations would be destroyed in the limit $N \\rightarrow \\infty $  [87].", "A tendency towards less intermittent regime when increasing the couplings is also observed in the tree models of turbulence [7].", "In view of these results, is reasonable to relate the non-intermittent Fourier modes in our model to its rich triad couplings, although no rigorous conclusions can be made.", "How much Fourier decimation decreases intermittency in physical space is also not clear.", "This was observed for the Burgers equation with random decimation [16] and for the Navier-Stokes equations, decimating from full to REWA model [57], but not for Sabra model [75], which retains turbulent intermittent dynamics in physical space.", "We repeat that the absence of anomalous fluctuations in individual Fourier modes does not mean lack of intermittency in the flow, since this is exactly the scenario for developed turbulence, and because the intermittency is seen in the same way within the dissipative range.", "To determine whether our model mimics physical-space intermittency or not, it would be necessary to probe it properly.", "The challenging question is precisely how to capture intermittency fingerprints on Fourier variables [13], the only available quantities for our model.", "We also remark that a proper definition of integral quantities, like structure functions, should take into account lattice volumes, because these volumes vary considerably among different cells.", "For example, a straightforward way to introduce structure functions would be through powers of the energy flux, where all terms are properly balanced: $S_p(k) = \\langle |\\Pi _k|^{p/3} \\rangle $ .", "Using these definitions, our numerical simulations confirm the exact Kolmogorov scaling of structure functions in the inertial range, while the location of the dissipative range depends on $p$ in agreement with the dissipative intermittency of Fig.", "REF (b).", "One faces the same subtlety when introducing a proper analogue for the Kolmogorov energy spectrum.", "We leave the more detailed analysis of these interesting but non-trivial questions to future work." ], [ "Conclusions", "We propose a new strategy for constructing simplified models of fluid dynamics, which restricts the governing equations in their original form to a multi-dimensional logarithmic lattice in Fourier space.", "This domain receives a specially designed operational structure, which retains most of the usual calculus and algebraic properties.", "As a consequence, the resulting models preserve all symmetries (some in discrete form, namely scaling invariance and isotropy), inviscid invariants (energy and helicity, for 3D flow; energy and enstrophy, for 2D flow), and also reproduces some fine properties of Euler flow, like incompressibility and Kelvin's Circulation Theorem.", "The classification of all possible lattices supporting this construction allows us to obtain different dynamical models sharing all the above properties, and so to test the robustness and universality of the results they provide.", "Because of the strongly decimated domain, the logarithmic models can be easily simulated with great accuracy and covering a large spatial range.", "Furthermore, the solutions correlate with existing DNS at the correspondent scales [19].", "After showing rigorously that the properties of plausible finite-time singularities (blowup) for the incompressible 3D Euler equations have similar form on the logarithmic lattice, we presented the numerical evidence of blowup, characterized as a chaotic wave in a renormalized system.", "Surprisingly similar asymptotic behavior of solutions was observed for two very different lattice models, probing the robustness of our conclusions, also drawn earlier in [19].", "The multi-scale character of the attractor (ranging six decades in Fourier space) reveals the great complexity of the blowup and explains why there is a controversy around the available numerical studies, since actual computational techniques may be insufficient by far for the required resolution.", "Still, one may think of accessing the blowup through experimental measurements [91], [65], [38].", "The viscous incompressible model on a logarithmic lattice exhibits anomalous dissipation in the limit of large Reynolds numbers, similarly to hydrodynamics turbulence [50].", "This was demonstrated by measuring the mean energy flux in the inertial range for a sequence of decreasing viscosities.", "Moreover, statistics of lattice variables behave like Fourier components in the full Navier-Stokes turbulence, whose distributions are Gaussian in the inertial interval and intermittent at viscous scales.", "Such behavior contrasts with other simplified models, which usually display some degree of inertial-range intermittency.", "Though the question whether our logarithmic model reproduces a kind of physical-space intermittency was left open.", "We believe that future analysis of this model may help in better understanding the relation between physical and Fourier space representations in developed turbulence [13].", "The systematic technique we presented is applicable to any partial differential equation with quadratic nonlinearities.", "In this framework, symmetries and quadratic invariants are expected to be automatically preserved due to the designed functional structure on the lattice.", "This turns the logarithmic models into a general methodology for the study of singularities and regularity in many nonlinear systems.", "We also developed the library LogLatt [18], an efficient Matlab computational tool for the numerical calculus on logarithmic lattices.", "The proposed technique is ready-to-use in other fluid models, like natural convection [76], geostrophic motion [85], [33], porous media [34] and magnetohydrodynamics [12].", "The lower computational cost of the logarithmic models compared to full DNS may be of great use for problems in higher dimensions, like high-dimensional turbulence [55], [77], [93], [97].", "With further extensions, there is a hope to apply this technique to compressible turbulence [3], [47] and superfluids [9], [10]; see the example in Appendix [app:C]C of a possible way to model isentropic compressible flow on logarithmic lattices." ], [ "Aknowledgments", "The authors thank L. Biferale, B. Dubrulle, U. Frisch, and S. Thalabard for useful discussions and suggestions.", "The work was supported by CNPq (grants 303047/2018-6, 406431/2018-3).", "In this Appendix, we show that some well-known shell models of turbulence are equivalent to the Burgers equation on a logarithmic lattice.", "This, in particular, reinforces the idea that self-similar blowup and non-oscillatory Kolmogorov regime in shell models follow a scenario closer to Burgers' dynamics [73], [75] than to Euler's.", "The Burgers equation [14] on the one-dimensional logarithmic lattice of spacing $\\lambda $ is given by $\\partial _t u + u \\ast \\partial _x u = \\nu \\partial _x^2 u,$ where $\\nu \\ge 0$ is the viscosity.", "First, let us take $\\lambda = 2$ and consider the corresponding product (REF ) with a prefactor of 2.", "The Burgers equation (REF ) takes the form $\\partial _t u (k) = -ik\\left[ 2 u(2 k)\\overline{u(k)} + u^2 \\left( \\frac{k}{2} \\right) \\right] - \\nu k^2 u(k).$ Define the geometric progression $k_n = \\lambda ^n, \\ n \\in \\mathbb {Z}$ and consider purely imaginary solutions of type $u(\\pm k_n) = \\pm i u_n$ for $u_n \\in \\mathbb {R}$ .", "Note that this is a property of the Fourier transform for any odd function in physical space.", "Then, equation (REF ) taken at $k = k_n$ reduces to the form $\\partial _t u_n = k_n u_{n-1}^2 - k_{n+1}u_{n+1} u_n - \\nu k_n^2 u_n.$ This system is known as the Desnyansky-Novikov shell model [40], also called dyadic model.", "For our second example, we take $\\lambda = \\varphi $ , the golden mean, and consider the product (REF ) with prefactor $-\\varphi $ .", "By setting $u(k_n) = u_n$ and $u(-k_n) = \\overline{u_n}$ with $k_n = \\varphi ^n$ , the Burgers equation (REF ) is reduced to the form $\\partial _t u_n = i[k_{n+1}u_{n+2}\\overline{u_{n+1}} - (1+c)k_n u_{n+1}\\overline{u_{n-1}} - c k_{n-1}u_{n-1}u_{n-2}] - \\nu k_n^2 u_n,$ with $c = -\\varphi ^2$ .", "System (REF ) is the Sabra shell model [70].", "A third possibility is to consider $\\lambda = \\sigma $ , the plastic number (REF ), which reduces Eq.", "(REF ) to a new shell model with improved number of triad interactions.", "In this spirit, extended triads were considered in the context of helical shell models [36].", "Model (REF ) on the logarithmic lattice retains several properties of the continuous Burgers equation, like the symmetries of time translation $t \\mapsto t + t_0$ by any $t_0 \\in \\mathbb {R}$ , physical-space translation $u(k) \\mapsto e^{-ik\\xi }u(k)$ by a number $\\xi \\in \\mathbb {R}$ and, in the case of a lattice with origin, Galilean invariance $u(k,t) \\mapsto e^{-ikvt}u(k,t) - \\widehat{v}(k)$ for any $v \\in \\mathbb {R}$ , where $\\widehat{v}(0) = v$ and zero for $k \\ne 0$ .", "Inviscid ($\\nu = 0$ ) regular solutions also conserve the momentum $M(t) = u(k = 0)$ , energy $E(t) = \\frac{1}{2}(u,u)$ and the thrid-order moment $M_3(t) = (u \\ast u, u) = (u,u \\ast u)$ , which is well-defined because of associativity in average of the product – see property REF in Definition REF .", "All these conservation laws can be proved using only the operations on logarithmic lattices; see [17].", "Conservation of energy is a well-known property of shell models while the conservation of a third-order moment was revealed in the study of Hamiltonian structure in Sabra model [71].", "Unlike the continuous Burgers equation, higher-order moments are not conserved for the logarithmic models because of non-associativity on the logarithmic lattice – see Corollary REF – which turns higher powers not even well-defined.", "The non-existence of invariants of order greater than 3 was proved in [41] for the Sabra model.", "Sabra model has one more inviscid quadratic invariant of the form $I = \\sum _{n \\in \\mathbb {Z}} c^{-n} |u_n|^2$ , but this invariant do not seem to have an analogue in the Burgers equation.", "In studies of hydrodynamic turbulence, it was interpreted as the enstrophy for $c>0$ (sign definite invariant) and as helicity for $c<0$ (not sign-definite invariant).", "Our methodology not only reproduces shell models but also leads to new insights about them.", "In the spirit of Theorem REF , consider a scalar field $\\rho $ evolving as $\\partial _t \\rho + \\partial _x (\\rho \\ast u) = 0.$ This equation mimics a passive scalar advected by the flow, e.g.", "density.", "Then, the cross-correlation $\\Gamma (t) = (\\rho ,u)$ which can be seen as total momentum of the flow, is conserved in time; the proof follows similar lines as those already presented and may be found in [17].", "Since this conservation holds for all solutions $\\rho (t)$ , this provides infinitely many inviscid invariants for model (REF ), analogous to circulation in Kelvin's Theorem as described in Section .", "Up to our knowledge, this has not been shown earlier.", "Here we prove some functional inequalities and operator properties used in Section REF .", "Lemma 15 Let $\\mathbf {u} \\in h^m$ and $\\mathbf {v} \\in h^{m-1}$ , for $m \\ge 1$ .", "Then, $\\mathbf {u} \\ast \\mathbf {v} = \\sum _{i = 1}^{d}u_i \\ast v_i \\in h^m$ with $||\\mathbf {u} \\ast \\mathbf {v}||_{h^m} \\le C(||\\mathbf {u}||_{h^m} ||\\mathbf {v}||_{\\ell ^\\infty } + ||D\\mathbf {u}||_{\\ell ^\\infty } ||\\mathbf {v}||_{h^{m-1}}),$ where $C$ is a constant which does not depend on $\\mathbf {u}$ and $\\mathbf {v}$ .", "Let us prove the inequality in the one-dimensional case.", "Using elementary algebraic relations, we obtain $||u \\ast v||_{h^m}^2 &= ||D^m(u \\ast v)||_{\\ell ^2}^2= \\sum _{k \\in \\mathbb {\\Lambda }} |k|^{2m}|(u \\ast v)(k)|^2 \\\\&\\le N\\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}|k|^{2m}|u(p_jk)v(q_jk)|^2 \\\\&= N\\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}|p_jk + q_jk|^{2m}|u(p_jk)v(q_jk)|^2 \\\\&\\le 2^{2m-1}N\\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}(|p_jk|^{2m} + |q_jk|^{2m})|u(p_jk)v(q_jk)|^2 \\\\&= 2^{2m-1}N\\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}|p_jk|^{2m}|u(p_jk)v(q_jk)|^2 +2^{2m-1}N \\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}|q_jk|^{2m}|u(p_jk)v(q_jk)|^2.$ In the first term, we estimate $\\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}|p_jk|^{2m}|u(p_jk)v(q_jk)|^2 \\le ||v||_{\\ell ^\\infty }^2 \\sum _{j = 1}^{N}\\sum _{k \\in \\mathbb {\\Lambda }}|p_jk|^{2m}|u(p_jk)|^2\\le N||u||_{h^m}^2||v||_{\\ell ^\\infty }^2,$ while the sums of the second term are bounded by $\\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}|q_jk|^{2m}|u(p_jk)v(q_jk)|^2 &= \\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}|q_jk|^{2m-2}|q_jk|^{2}|u(p_jk)v(q_jk)|^2 \\\\&\\le M\\sum _{k \\in \\mathbb {\\Lambda }} \\sum _{j = 1}^{N}|q_jk|^{2m-2}|p_jk|^{2}|u(p_jk)v(q_jk)|^2 \\\\&\\le M ||Du||_{\\ell ^\\infty }^2 \\sum _{j = 1}^{N} \\sum _{k \\in \\mathbb {\\Lambda }} |q_jk|^{2m-2}|v(q_jk)|^2 \\\\&\\le MN ||Du||_{\\ell ^\\infty }^2 ||v||_{h^{m-1}}^2,$ where $M = \\max _{j = 1,\\dots ,N} |q_j|^2/|p_j|^2$ .", "In view of the estimates for the two terms, we reach to the result $|| u \\ast v ||_{h^m} \\le C\\left(||u||_{h^m}||v||_{\\ell ^\\infty } + ||Du||_{\\ell ^\\infty }||v||_{h^{m-1}}\\right)$ with the choice of $C = 2^{m-1/2}N\\max (M,1)^{1/2}$ .", "The proof extends naturally to higher dimensions, by considering multiple components.", "Lemma 16 Define the bilinear operator $B(\\mathbf {u},\\mathbf {v}) = \\mathbf {u} \\ast \\nabla \\mathbf {v},$ where $(\\mathbf {u} \\ast \\nabla \\mathbf {v})_i = \\mathbf {u} \\ast \\nabla u_i = \\sum _{j = 1}^{d} u_j \\ast \\partial _j v_i$ .", "Then, $B: h^m \\times h^m \\rightarrow h^m$ is a bounded bilinear operator, i.e., there exists a constant $C>0$ such that $||B(\\mathbf {u},\\mathbf {v}) ||_{h^m} \\le C ||\\mathbf {u}||_{h^m} ||\\mathbf {v}||_{h^m},$ for every $\\mathbf {u},\\mathbf {v} \\in h^m$ .", "Using inequality (REF ) for $\\mathbf {u}$ and $\\nabla v_i$ , we obtain $||B(\\mathbf {u},\\mathbf {v})||_{h^m}\\le \\sum _{i = 1}^d ||\\mathbf {u} \\ast \\nabla v_i||_{h^m}\\le C\\sum _{i = 1}^d\\left( ||\\mathbf {u}||_{h^m}||\\nabla v_i||_{\\ell ^\\infty } + ||D\\mathbf {u}||_{\\ell ^\\infty }||\\nabla v_i||_{h^{m-1}} \\right).$ We now use the inequalities $||\\nabla v_i||_{\\ell ^{\\infty }}\\le ||D\\mathbf {v}||_{\\ell ^\\infty }\\le ||\\mathbf {v}||_{h^1}, \\quad ||D\\mathbf {u}||_{\\ell ^\\infty } \\le ||\\mathbf {u}||_{h^1}, \\quad ||\\nabla v_i||_{h^{m-1}} \\le ||\\mathbf {v}||_{h^m}$ and the general relation $||\\mathbf {u}||_{h^1} \\le ||\\mathbf {u}||_{h^m},$ which are simple estimates from the definition of the norms on the lattice (REF ).", "This yields $||B(\\mathbf {u},\\mathbf {v})||_{h^m} \\le 2dC||\\mathbf {u}||_{h^m} ||\\mathbf {v}||_{h^m},$ which shows that $B(\\mathbf {u},\\mathbf {v}) \\in h^m$ and the boundness of operator $B$ .", "The logarithmic models presented in this paper do not extend naturally to isentropic (or general) compressible flow due to the appearance of cubic terms in the governing equations and inviscid invariants.", "Nevertheless, we present below one possible way to overcome this issue.", "The idea consists of introducing additional variables properly constrained, so the original cubic terms become quadratic with respect to the extended set of variables.", "In this formulation, the symmetries and conserved quantities are exactly those from the continuous model.", "Unfortunately, preliminary numerical simulations do not show good correspondence to dynamical features of realistic compressible flows, such as formation of shock waves.", "For this reason we restrict ourselves to the model description and its conserved quantities, leaving the numerical implementation for future analysis.", "Model.", "We introduce the scalar density $\\rho (\\mathbf {k},t)$ , the velocity field $\\mathbf {u}(\\mathbf {k},t)$ and the momentum field $\\mathbf {q}(\\mathbf {k},t)$ , defined on the lattice $\\mathbf {k} \\in \\mathbb {\\Lambda }^d$ .", "The model for ideal compressible flow consists of the continuity equation and the balance of momentum together with an algebraic constraint relating all variables, respectively given by the system (cf.", "[67]) $\\partial _t \\rho + \\nabla \\cdot \\mathbf {q} = 0,\\quad \\partial _t \\mathbf {q} = - \\nabla \\cdot \\Pi ,\\quad \\mathbf {q} = \\rho \\ast \\mathbf {u},$ where the momentum flux density tensor $\\Pi $ has its classical form $\\Pi _{ij} = p \\delta _{ij} + u_i \\ast q_j.$ In an isentropic flow, the pressure $p$ is a function of the density.", "For our logarithmic model, we consider the quadratic relation $p = A \\rho \\ast \\rho ,$ which mimics a polytropic gas $p = A \\rho ^\\gamma $ , with $\\gamma = 2$ .", "To evolve model (REF ), one needs to solve the last algebraic constraint for the velocities $\\mathbf {u}$ , i.e., express it in terms of the momentum and density.", "This is possible when the mean density $\\rho (\\mathbf {0})>0$ at $\\mathbf {k} = \\mathbf {0}$ is sufficiently larger than the sum of all other components $\\sum _{\\mathbf {k}\\ne \\mathbf {0}}|\\rho (\\mathbf {k})|$ .", "Under this condition, the density field may be interpreted as small-amplitude oscillations around a positive mean value.", "Solvability of velocities under this condition can be rigorously proved in proper functional spaces using Operator Theory.", "Conserved quantities.", "The total momentum of the flow is naturally defined as $M(t) = \\mathbf {q}(\\mathbf {0},t),$ at $\\mathbf {k} = \\mathbf {0}$ .", "The total energy $E$ decomposes into two contributions $E = K + U,$ where $K = \\frac{1}{2}(\\mathbf {q},\\mathbf {u})$ is the kinetic energy and $U = (\\rho ,e)$ is the internal energy.", "The internal energy per unit mass $e$ is defined as $e = A\\rho .$ Formula (REF ) is obtained from the pressure through the well-known (isentropic) thermodynamical relation $de = pd\\rho /\\rho ^2$ .", "System (REF ) conserves total momentum (REF ) and total energy (REF ) in time.", "Kinetic and internal energies are transferred from one another through pressure as $\\frac{dK}{dt} = -\\frac{dU}{dt} = -(\\nabla p, \\mathbf {u}).$ Viscous effects.", "Following classical derivations of fluid mechanics, viscosity is introduced in the momentum flux density tensor as $\\Pi _{ij} = p \\delta _{ij} + u_i \\ast q_j - \\sigma _{ij},$ with the viscous tensor $\\sigma $ given by $\\sigma _{ij} = \\eta \\left( \\partial _i u_j + \\partial _j u_i - \\frac{2}{3}\\nabla \\cdot \\mathbf {u}\\delta _{ij} \\right) + \\zeta \\nabla \\cdot \\mathbf {u}\\delta _{ij}.$ The constants $\\eta , \\zeta \\ge 0$ are the viscosity coefficients.", "Non-equilibrium solutions dissipate energy through the work of viscosity forces in the form $\\frac{dE}{dt} = (\\nabla \\cdot \\sigma , \\mathbf {u}).$ In this way, system (REF ) yields equations for the viscous flow." ] ]

[ [ "Matching the heavy-quark fields in QCD and HQET at four loops" ], [ "Abstract The QCD/HQET matching coefficient for the heavy-quark field is calculated up to four loops.", "It must be finite; this requirement produces analytical results for some terms in the four-loop on-shell heavy-quark field renormalization constant which were previously only known numerically.", "The effect of a non-zero lighter-flavor mass is calculated up to three loops.", "A class of on-shell integrals with two masses is analyzed in detail.", "By specifying our result to QED, we obtain the relation between the electron field and the Bloch--Nordsieck field with four-loop accuracy." ], [ "Introduction", "Some classes of QCD problems with a single heavy quark can be examined in a simpler effective theory, the so-called heavy quark effective theory (HQET, see, e. g., [1], [2], [3]).", "Let us consider QCD with a single heavy flavor $Q$ and $n_l$ light flavors ($n_f = n_l + n_h$ , $n_h = 1$ ).", "The heavy-quark momentum can be decomposed as $p = Mv + k$ , where $M$ is the on-shell $Q$ mass, and $v$ is some reference 4-velocity ($v^2 = 1$ ).", "In the case of QED, it is called Bloch–Nordsieck effective theory [4].", "In the effective theory, the heavy quark (respectively lepton) is represented by the field $h_v$ .", "The $\\overline{\\text{MS}}$ renormalized fields $Q(\\mu )$ and $h_v(\\mu )$ are related by [5] $Q(\\mu ) = e^{-iMvx} \\left[ \\sqrt{z(\\mu )} \\left( 1 + \\frac{\\unknown.", "{\\,/}D_\\bot }{2 M} \\right) h_v(\\mu ) + \\mathcal {O}\\left(\\frac{1}{M^2}\\right) \\right]\\,,$ where $D_\\bot ^\\mu = D^\\mu - v^\\mu \\,v\\cdot D$ , and the matching coefficient is given by $z(\\mu ) = \\frac{Z_h(\\alpha _s^{(n_l)}(\\mu ),\\xi ^{(n_l)}(\\mu ))\\,Z_Q^{\\text{os}}(g_0^{(n_f)},\\xi _0^{(n_f)})}{Z_Q(\\alpha _s^{(n_f)}(\\mu ),\\xi ^{(n_f)}(\\mu ))\\,Z_h^{\\text{os}}(g_0^{(n_l)},\\xi _0^{(n_l)})}\\,.$ Here $Z_Q^{\\text{os}}$ and $Z_h^{\\text{os}}$ are the on-shell field renormalization constants (they depend on the corresponding bare couplings and bare gauge-fixing parameters), and $Z_Q$ and $Z_h$ are the $\\overline{\\text{MS}}$ renormalization constants.", "The covariant-gauge fixing parameter is defined in such a way that the bare gluon propagator is given by $(g_{\\mu \\nu } - \\xi _0 p_\\mu p_\\nu /p^2)/p^2$ ; it is renormalized by the gluon-field renormalization constant: $1-\\xi _0 = Z_A(\\alpha _s(\\mu ),\\xi (\\mu )) (1-\\xi (\\mu ))$ .", "The $1/M$ correction in (REF ) is fixed by reparametrization invariance [6].", "The $\\overline{\\text{MS}}$ renormalized matching coefficient is obviously finite at $\\varepsilon \\rightarrow 0$ , because it relates the off-shell renormalized propagators in the two theories, which are both finite.", "The ultraviolet divergences cancel in the ratios $Z_Q/Z_Q^{\\text{os}}$ and $Z_h/Z_h^{\\text{os}}$ , because they relate renormalized fields; the infrared divergences cancel in $Z_Q^{\\text{os}}/Z_h^{\\text{os}}$ , because HQET is constructed to reproduce the infrared behavior of QCD; the $\\overline{\\text{MS}}$ renormalization constants $Z_Q$ and $Z_h$ (purely off-shell quantities) are infrared finite.", "If we assume that all light flavors are massless we have $Z_h^{\\text{os}} = 1$ : all loop corrections vanish because they contain no scale, ultraviolet and infrared divergences of $Z_h^{\\text{os}}$ mutually cancel.", "Taking light-quark masses $m_i$ into account produces corrections suppressed by powers of $m_i/M$ , see Sect. .", "The matching coefficient satisfies the renormalization-group equation $&&\\frac{d \\log z(\\mu )}{d \\log \\mu } =\\nonumber \\\\&&\\gamma _h(\\alpha _s^{(n_l)}(\\mu ),\\xi ^{(n_l)}(\\mu ))- \\gamma _Q(\\alpha _s^{(n_f)}(\\mu ),\\xi ^{(n_f)}(\\mu ))\\,,$ where the anomalous dimensions are defined as $\\gamma _i = d \\log Z_i / d \\log \\mu $ ($i = Q$ , $h$ ).", "It is sufficient to obtain the initial condition $z(\\mu _0)$ for some scale $\\mu _0 \\sim M$ ; $z(\\mu )$ for other renormalization scales $\\mu $ can be found by solving Eq.", "(REF ).", "We choose to present the result for $\\mu _0=M$ .", "The heavy-quark field matching coefficient $z(\\mu )$ has been calculated up to three loops [5].", "When the matching coefficient is used within a quantity containing $1/\\varepsilon $ divergences, terms with positive powers of $\\varepsilon $ in $z(\\mu )$ are needed; such terms were not given in [5].", "We present the four-loop result in Sect. .", "Power corrections due to lighter-flavor masses up to three loops are obtained in Sect. .", "The QED result, i. e. the four-loop relation between the lepton field and the Bloch–Nordsieck field, is discussed in Sect. .", "In Appendix  we provide analytic results for the decoupling coefficients for the strong coupling constant and the gluon field up to three-loop order including linear $\\varepsilon $ terms.", "Appendix  contains a detailed analysis of a class of on-shell integrals with two masses.", "It allows us, in particular, to obtain exact results for the three-loop term in the $\\overline{\\text{MS}}$ –on-shell mass relation with a closed massless and a closed lighter-flavor massive fermion loop (previously this term was only known as a truncated series in this mass ratio)." ], [ "The QCD and HQET heavy-quark fields", "If we assume that all light flavors are massless, then (REF ) gives $&&\\log z(\\mu ) = \\log Z_Q^{\\text{os}}(g_0^{(n_f)},\\xi _0^{(n_f)})\\\\&&{} - \\log Z_Q(\\alpha _s^{(n_f)}(\\mu ),\\xi ^{(n_f)}(\\mu ))+ \\log Z_h(\\alpha _s^{(n_l)}(\\mu ),\\xi ^{(n_l)}(\\mu ))\\,.\\nonumber $ The on-shell heavy-quark field renormalization constant $Z_Q^{\\text{os}}$ depends on the bare coupling $g_0^{(n_f)}$ , the bare gauge parameter $\\xi _0^{(n_f)}$ and the on-shell mass $M$ : $&&Z_Q^{\\text{os}} = 1 + \\sum _{L=1}^\\infty \\left(4 \\frac{\\bigl (g_0^{(n_f)}\\bigr )^2 M^{-2\\varepsilon }}{(4\\pi )^{d/2}} e^{-\\gamma _E\\varepsilon }\\right)^L Z_L\\,,\\nonumber \\\\&&Z_L = \\sum _{n=0}^\\infty Z_{L,n}(\\xi _0^{(n_f)}) \\varepsilon ^{n-L}\\,.$ The two-loop expression is known exactly in $\\varepsilon $  [7]; it contains a single non-trivial master integral, further terms of its $\\varepsilon $ expansion are presented in [8], [9].", "The three-loop term has been calculated in [10], [11].", "At four loops, the terms with $n_l^3$ and $n_l^2$ are known analytically [12], and the remaining ones numerically [13].", "Recently the QED-like color structures $C_F^4$ , $C_F^3 T_F n_h$ , $C_F^2 (T_F n_h)^2$ , $C_F^3 (T_F n_h)^3$ , $d_{FF} n_h$ have been calculated analytically [14].", "Here and below we use the notation $d_{FF} = \\frac{d_F^{abcd} d_F^{abcd}}{N_F}\\,,\\quad d_{FA} = \\frac{d_F^{abcd} d_A^{abcd}}{N_F}\\,,$ where $N_R = \\mathop {\\mathrm {Tr}} \\mathbf {1}_R$ (with $R=F$ ), $d_R^{abcd} = \\mathop {\\mathrm {Tr}} t_R^{(a} t_R^{b\\vphantom{(}}t_R^{c\\vphantom{)}} t_R^{d)}$ (with $R=F$ or $A$ ), and the round brackets mean symmetrization (for $SU(N_c)$ gauge group $d_{FF} = (N_c^2-1) (N_c^4-6N_c^2+18) / (96 N_c^3)$ , $d_{FA} = (N_c^2-1) (N_c^2+6) / 48$ ).", "This result contains the same master integrals as the electron $g-2$  [15], [16].", "In [15] they have been calculated numerically to 1100 digits, and analytical expressions have been reconstructed using PSLQ.", "In the case of the light-by-light contribution $d_{FF} n_h$ the results contain $\\varepsilon ^0$ terms of 6 master integrals (known numerically to 1100 digits); all the remaining constants are completely expressed via known transcendental numbers (Note that the definition of the constant $t_{63}$ is missing in the journal article [14]; it is included in the version v3 of the arXiv publication.).", "The $\\overline{\\text{MS}}$ quark-field anomalous dimension $\\gamma _q$ (and hence $\\log Z_Q$ ) is well known [17], [18], [19], [20].", "The HQET field anomalous dimension $\\gamma _h$ (and hence $\\log Z_h$ ) is known at three loops [10], [21].", "At four loops, some color structures are known analytically: $C_F (T_F n_l)^3$  [22], $C_F^2 (T_F n_l)^2$  [23], [24], $C_F C_A (T_F n_l)^2$  [13], $C_F^3 T_F n_l$  [25], $d_F^{abcd} d_F^{abcd} n_l$  [26], $C_F^2 C_A T_F n_l$ and $C_F C_A^2 T_F n_l$  [27]; $C_F C_A^3$ and $d_F^{abcd} d_A^{abcd}$ are known numerically [13].", "We need to express the three terms in (REF ) in terms of the same set of variables, for which we choose $\\alpha _s^{(n_f)}(\\mu )$ and $\\xi ^{(n_f)}(\\mu )$ .", "Expressing $g_0^{(n_f)}$ and $\\xi _0^{(n_f)}$ via these variables is straightforward, since the three-loop renormalization constants in QCD are well known.", "Expressing $\\alpha _s^{(n_l)}(\\mu )$ and $\\xi ^{(n_l)}(\\mu )$ via the $n_f$ -flavor quantities requires decoupling relations up to ${\\cal O}(\\varepsilon )$ at three loops.", "For convenience we present explicit results in Appendix .", "The resulting matching coefficient $z(M)$ must be finite at $\\varepsilon \\rightarrow 0$ .", "This requirement together with the known results for $Z_Q$ and $Z_h$ leads to analytical expressions for the four-loop coefficients $Z_{4,0}$ , $Z_{4,1}$ , and $Z_{4,2}$ in (REF ) as well as for $Z_{4,3}$ , except two color structures $C_F C_A^3$ and $d_{FA}$ where the corresponding terms in $\\gamma _h$ are not known analytically.", "The analytic results are presented in the tables REF and REF .", "We refrain from showing results for the $n_l^2$ and $n_l^3$ terms, which are already known since a few years [12].", "Furthermore, we have introduced $a_n = \\mathop {\\mathrm {Li}}\\nolimits _{n}(1/2)$ (in particular $a_1 = \\log 2$ ); $\\zeta _n$ denotes the Riemann zeta function and $\\xi _0 = \\xi _0^{(n_f)}$ .", "Analytical results for the color structures $C_F^4$ , $C_F^3 T_F n_h$ , $C_F^2 (T_F n_h)^2$ , $C_F^3 (T_F n_h)^3$ , $d_{FF} n_h$ were recently obtained [14].", "They agree with the expressions given in tables REF and REF .", "Numerical results for these coefficients are given in the tables V, VI, and VII of Ref. [13].", "Good agreement is found.", "Table: Coefficients Z 4,n Z_{4,n} of the 1/ε 4,3,2 1/\\varepsilon ^{4,3,2}terms entering the four-loop result Z 4 Z_4 in Eq.", "().Note that the color structures d FF n l d_{FF} n_l, d FF n h d_{FF} n_h, d FA d_{FA} have zero coefficients.Table: Coefficients Z 4,3 Z_{4,3} of the 1/ε1/\\varepsilon termentering the four-loop result Z 4 Z_4 in Eq.", "().Note that the color structures C F C A 3 C_F C_A^3 and d FA d_{FA} are not known analytically.Using the matching coefficient $z(\\mu )$ together with quantities which contains $1/\\varepsilon $ divergences, terms with positive powers of $\\varepsilon $ are needed.", "In order to get the finite four-loop contribution, we need the $\\alpha _s^L$ term in $z(\\mu )$ expanded up to $\\varepsilon ^{4-L}$ .", "Our result for $\\mu =M$ is given by $&&z(M) = 1 - \\frac{\\alpha _s}{\\pi } C_F\\biggl [1 + \\varepsilon \\biggl (\\frac{\\pi ^2}{16} + 2\\biggr )- \\varepsilon ^2 \\biggl (\\frac{\\zeta _3}{4} - \\frac{\\pi ^2}{12} - 4\\biggr )- \\varepsilon ^3 \\biggl (\\frac{\\zeta _3}{3} - \\frac{3}{640} \\pi ^4 - \\frac{\\pi ^2}{6} - 8\\biggr )+ \\mathcal {O}(\\varepsilon ^4) \\biggr ]\\nonumber \\\\&&{} + \\left(\\frac{\\alpha _s}{\\pi }\\right)^2 C_F\\biggl \\lbrace C_F \\biggl (\\pi ^2 a_1 - \\frac{3}{2} \\zeta _3 - \\frac{13}{16} \\pi ^2 + \\frac{241}{128}\\biggr )- \\frac{C_A}{2} \\biggl (\\pi ^2 a_1 - \\frac{3}{2} \\zeta _3 - \\frac{5}{8} \\pi ^2 + \\frac{1705}{192}\\biggr )\\nonumber \\\\&&\\qquad {}- \\frac{T_F n_h}{3} \\biggl (\\pi ^2 - \\frac{947}{96}\\biggr )+ \\frac{T_F n_l}{12} \\biggl (\\pi ^2 + \\frac{113}{8}\\biggr )\\nonumber \\\\&&\\quad {} + \\varepsilon \\biggl [- C_F \\biggl (24 a_4 + a_1^4 + 2 \\pi ^2 a_1^2 - \\frac{23}{4} \\pi ^2 a_1 + \\frac{147}{8} \\zeta _3 - \\frac{7}{20} \\pi ^4 + \\frac{347}{128} \\pi ^2 + \\frac{557}{256}\\biggr )\\nonumber \\\\&&\\qquad {}+ C_A \\biggl (12 a_4 + \\frac{a_1^4}{2} + \\pi ^2 a_1^2 - \\frac{23}{8} \\pi ^2 a_1 + \\frac{129}{16} \\zeta _3 - \\frac{7}{40} \\pi ^4 + \\frac{769}{1152} \\pi ^2 - \\frac{9907}{768}\\biggr )\\nonumber \\\\&&\\qquad {}+ T_F n_h \\biggl (2 \\pi ^2 a_1 - 7 \\zeta _3 - \\frac{445}{288} \\pi ^2 + \\frac{17971}{1728}\\biggr )+ T_F n_l \\biggl (\\zeta _3 + \\frac{127}{288} \\pi ^2 + \\frac{851}{192}\\biggr )\\biggr ]\\nonumber \\\\&&\\quad {} + \\varepsilon ^2 \\biggl [- C_F \\biggl (144 a_5 + 138 a_4 - \\frac{6}{5} a_1^5 + \\frac{23}{4} a_1^4 - 4 \\pi ^2 a_1^3 + \\frac{23}{2} \\pi ^2 a_1^2 + \\frac{13}{15} \\pi ^4 a_1 - \\frac{41}{2} \\pi ^2 a_1\\nonumber \\\\&&\\qquad \\quad {}- \\frac{609}{4} \\zeta _5 - \\frac{11}{4} \\pi ^2 \\zeta _3 + \\frac{2061}{32} \\zeta _3 - \\frac{1555}{1536} \\pi ^4 + \\frac{8947}{768} \\pi ^2 - \\frac{1817}{512}\\biggr )\\nonumber \\\\&&\\qquad {}+ C_A \\biggl (72 a_5 + 69 a_4 - \\frac{3}{5} a_1^5 + \\frac{23}{8} a_1^4 - 2 \\pi ^2 a_1^3 + \\frac{23}{4} \\pi ^2 a_1^2 + \\frac{13}{30} \\pi ^4 a_1 - \\frac{41}{4} \\pi ^2 a_1\\nonumber \\\\&&\\qquad \\quad {}- \\frac{609}{8} \\zeta _5 - \\frac{11}{8} \\pi ^2 \\zeta _3 + \\frac{7595}{288} \\zeta _3 - \\frac{14359}{23040} \\pi ^4 + \\frac{6367}{2304} \\pi ^2 - \\frac{79225}{1536}\\biggr )\\nonumber \\\\&&\\qquad {}- T_F n_h \\biggl (48 a_4 + 2 a_1^4 + 4 \\pi ^2 a_1^2 - \\frac{19}{2} \\pi ^2 a_1 + \\frac{2405}{72} \\zeta _3 - \\frac{93}{320} \\pi ^4 + \\frac{8605}{1728} \\pi ^2 - \\frac{422747}{10368}\\biggr )\\nonumber \\\\&&\\qquad {}+ \\frac{T_F n_l}{24} \\biggl (\\frac{305}{3} \\zeta _3 + \\frac{199}{80} \\pi ^4 + \\frac{853}{24} \\pi ^2 + \\frac{5753}{16}\\biggr )\\biggr ]+ \\mathcal {O}(\\varepsilon ^3)\\biggr \\rbrace \\nonumber \\\\&&{} + \\left(\\frac{\\alpha _s}{\\pi }\\right)^3 C_F\\biggl \\lbrace - C_F^2 \\biggl (28 a_4 + \\frac{7}{6} a_1^4 - \\frac{3}{2} \\pi ^2 a_1^2 - \\frac{223}{12} \\pi ^2 a_1 + \\frac{5}{16} \\zeta _5 - \\frac{\\pi ^2}{8} \\zeta _3 + \\frac{157}{8} \\zeta _3 + \\frac{19}{240} \\pi ^4 + \\frac{4801}{576} \\pi ^2 + \\frac{3023}{768}\\biggr )\\nonumber \\\\&&\\qquad {}- C_F C_A \\biggl (\\frac{a_4}{6} + \\frac{a_1^4}{144} + \\frac{181}{72} \\pi ^2 a_1^2 + \\frac{43}{9} \\pi ^2 a_1 - \\frac{145}{16} \\zeta _5 + \\frac{45}{16} \\pi ^2 \\zeta _3 + \\frac{289}{24} \\zeta _3 - \\frac{6697}{17280} \\pi ^4 - \\frac{2137}{576} \\pi ^2 - \\frac{24131}{4608}\\biggr )\\nonumber \\\\&&\\qquad {} + \\frac{C_A^2}{2} \\biggl [\\frac{1}{3} \\biggl (\\frac{85}{2} a_4 + \\frac{85}{48} a_1^4 + \\frac{127}{24} \\pi ^2 a_1^2 - \\frac{325}{24} \\pi ^2 a_1 - 37 \\zeta _5 + \\frac{127}{12} \\pi ^2 \\zeta _3 + \\frac{5857}{96} \\zeta _3 - \\frac{3419}{3840} \\pi ^4 - \\frac{4339}{576} \\pi ^2 - \\frac{1654711}{20736}\\biggr )\\nonumber \\\\&&\\qquad \\quad {}+ \\frac{\\xi }{8} \\biggl (\\frac{7}{24} \\zeta _5 + \\frac{\\pi ^2}{9} \\zeta _3 - \\frac{13}{16} \\zeta _3 + \\frac{17}{1728} \\pi ^4 - \\frac{\\pi ^2}{16} - \\frac{13}{48}\\biggr )\\biggr ]\\nonumber \\\\&&\\qquad {}+ C_F T_F n_h \\biggl (12 a_4 + \\frac{a_1^4}{2} - \\frac{\\pi ^2}{2} a_1^2 + \\frac{17}{9} \\pi ^2 a_1 + \\frac{233}{288} \\zeta _3 + \\frac{31}{720} \\pi ^4 - \\frac{553}{324} \\pi ^2 - \\frac{13571}{3456}\\biggr )\\nonumber \\\\&&\\qquad {}- C_A T_F n_h \\biggl [ \\biggl (8 a_4 + \\frac{a_1^4}{3} - \\frac{\\pi ^2}{3} a_1^2 - \\frac{80}{9} \\pi ^2 a_1 + \\frac{15}{16} \\zeta _5 - \\frac{11}{48} \\pi ^2 \\zeta _3 + \\frac{2813}{576} \\zeta _3 + \\frac{17}{360} \\pi ^4 + \\frac{9067}{1296} \\pi ^2 - \\frac{788639}{41472}\\biggr )\\nonumber \\\\&&\\qquad \\quad {}+ \\frac{\\xi }{24} \\biggl (\\zeta _3 - \\frac{2387}{576}\\biggr ) \\biggr ]\\nonumber \\\\&&\\qquad {}+ \\frac{C_F T_F n_l}{3} \\biggl (16 a_4 + \\frac{2}{3} a_1^4 + \\frac{4}{3} \\pi ^2 a_1^2 - \\frac{47}{6} \\pi ^2 a_1 + \\frac{137}{8} \\zeta _3 - \\frac{229}{720} \\pi ^4 + \\frac{113}{24} \\pi ^2 + \\frac{35}{6} \\biggr )\\nonumber \\\\&&\\qquad {}- \\frac{C_A T_F n_l}{3} \\biggl (8 a_4 + \\frac{a_1^4}{3} + \\frac{2}{3} \\pi ^2 a_1^2 - \\frac{47}{12} \\pi ^2 a_1 + \\frac{35}{24} \\zeta _3 - \\frac{19}{360} \\pi ^4 - \\frac{13}{16} \\pi ^2 - \\frac{111791}{5184}\\biggr )\\nonumber \\\\&&\\qquad {}+ \\frac{(T_F n_h)^2}{3} \\biggl (7 \\zeta _3 + \\frac{2}{15} \\pi ^2 - \\frac{8425}{864}\\biggr )+ \\frac{T_F^2 n_h n_l}{36} \\biggl (13 \\pi ^2 - \\frac{4721}{36}\\biggr )- \\frac{(T_F n_l)^2}{18} \\biggl (7 \\zeta _3 + \\frac{19}{6} \\pi ^2 + \\frac{5767}{432}\\biggr )\\nonumber \\\\&&\\quad {} + \\varepsilon \\biggl [- C_F^2 \\biggl (\\frac{440}{3} a_5 - 16 \\pi ^2 a_4 + \\frac{2444}{3} a_4 - \\frac{11}{9} a_1^5 - \\frac{2}{3} \\pi ^2 a_1^4 + \\frac{611}{18} a_1^4 + \\frac{115}{27} \\pi ^2 a_1^3 + \\frac{2}{3} \\pi ^4 a_1^2 + \\frac{2309}{36} \\pi ^2 a_1^2 - 14 \\pi ^2 \\zeta _3 a_1\\nonumber \\\\&&\\qquad \\quad {}+ \\frac{751}{432} \\pi ^4 a_1 - \\frac{367}{2} \\pi ^2 a_1 - \\frac{53}{2} \\zeta _5 - \\frac{29}{32} \\zeta _3^2 - \\frac{5861}{288} \\pi ^2 \\zeta _3 + \\frac{5119}{16} \\zeta _3 + \\frac{899}{5670} \\pi ^6 - \\frac{54467}{34560} \\pi ^4 + \\frac{74245}{2048} \\pi ^2 + \\frac{19337}{1536}\\biggr )\\nonumber \\\\&&\\qquad {}- C_F C_A \\biggl (\\frac{487}{3} a_5 - 6 \\pi ^2 a_4 - \\frac{1796}{9} a_4 - \\frac{487}{360} a_1^5 - \\frac{\\pi ^2}{4} a_1^4 - \\frac{449}{54} a_1^4 - \\frac{1135}{108} \\pi ^2 a_1^3 + \\frac{\\pi ^4}{4} a_1^2 + \\frac{7235}{216} \\pi ^2 a_1^2 - \\frac{21}{4} \\pi ^2 \\zeta _3 a_1\\nonumber \\\\&&\\qquad \\quad {}- \\frac{949}{1080} \\pi ^4 a_1 + \\frac{30803}{432} \\pi ^2 a_1 - \\frac{125473}{384} \\zeta _5 + \\frac{143}{4} \\zeta _3^2 + \\frac{2703}{128} \\pi ^2 \\zeta _3 - \\frac{16339}{288} \\zeta _3 + \\frac{27331}{181440} \\pi ^6 - \\frac{496741}{103680} \\pi ^4 - \\frac{17665}{55296} \\pi ^2\\nonumber \\\\&&\\qquad \\quad {}- \\frac{861659}{27648}\\biggr )\\nonumber \\\\&&\\qquad {} + C_A^2 \\biggl [\\frac{707}{6} a_5 - 7 \\pi ^2 a_4 + \\frac{935}{9} a_4 - \\frac{707}{720} a_1^5 - \\frac{7}{24} \\pi ^2 a_1^4 + \\frac{935}{216} a_1^4 - \\frac{905}{216} \\pi ^2 a_1^3 + \\frac{7}{24} \\pi ^4 a_1^2 + \\frac{7081}{216} \\pi ^2 a_1^2 - \\frac{49}{8} \\pi ^2 \\zeta _3 a_1\\nonumber \\\\&&\\qquad \\quad {}- \\frac{41}{8640} \\pi ^4 a_1 - \\frac{8833}{864} \\pi ^2 a_1 - \\frac{41569}{256} \\zeta _5 + \\frac{7451}{384} \\zeta _3^2 + \\frac{14915}{4608} \\pi ^2 \\zeta _3 + \\frac{67807}{3456} \\zeta _3 + \\frac{45047}{362880} \\pi ^6 - \\frac{126391}{51840} \\pi ^4 - \\frac{150229}{41472} \\pi ^2\\nonumber \\\\&&\\qquad \\quad {}- \\frac{72476083}{746496}- \\frac{\\xi }{128} \\biggl (\\frac{149}{6} \\zeta _5 - \\frac{25}{3} \\zeta _3^2 - \\frac{77}{72} \\pi ^2 \\zeta _3 + \\frac{63}{2} \\zeta _3 - \\frac{49}{405} \\pi ^6 - \\frac{383}{1080} \\pi ^4 + \\frac{35}{8} \\pi ^2 + \\frac{35}{2}\\biggr )\\biggr ]\\nonumber \\\\&&\\qquad {}+ C_F T_F n_h \\biggl (72 a_5 - \\frac{229}{6} a_4 - \\frac{3}{5} a_1^5 - \\frac{229}{144} a_1^4 + \\pi ^2 a_1^3 - \\frac{2219}{144} \\pi ^2 a_1^2 + \\frac{143}{180} \\pi ^4 a_1 + \\frac{293}{6} \\pi ^2 a_1 - \\frac{87}{8} \\zeta _5 - \\frac{81}{8} \\pi ^2 \\zeta _3\\nonumber \\\\&&\\qquad \\quad {}- \\frac{10913}{192} \\zeta _3 + \\frac{3649}{8640} \\pi ^4 - \\frac{818609}{41472} \\pi ^2 + \\frac{164069}{6912}\\biggr )\\nonumber \\\\&&\\qquad {}- C_A T_F n_h \\biggl [48 a_5 - 8 \\pi ^2 a_4 + \\frac{4247}{12} a_4 - \\frac{2}{5} a_1^5 - \\frac{\\pi ^2}{3} a_1^4 + \\frac{4247}{288} a_1^4 + \\frac{2}{3} \\pi ^2 a_1^3 + \\frac{\\pi ^4}{3} a_1^2 + \\frac{18133}{288} \\pi ^2 a_1^2 -7 \\pi ^2 \\zeta _3 a_1\\nonumber \\\\&&\\qquad \\quad {}+ \\frac{97}{180} \\pi ^4 a_1 - \\frac{775}{9} \\pi ^2 a_1 + \\frac{551}{64} \\zeta _5 - \\frac{181}{32} \\zeta _3^2 - \\frac{549}{64} \\pi ^2 \\zeta _3 + \\frac{88855}{384} \\zeta _3 + \\frac{1501}{15120} \\pi ^6 - \\frac{12607}{5760} \\pi ^4 + \\frac{286961}{13824} \\pi ^2\\nonumber \\\\&&\\qquad \\quad {}- \\frac{35801821}{248832}- \\frac{\\xi }{8} \\biggl (\\zeta _3 + \\frac{\\pi ^4}{60} - \\frac{121}{1728} \\pi ^2 - \\frac{7367}{1152}\\biggr )\\biggr ]\\nonumber \\\\&&\\qquad {}+ \\frac{C_F T_F n_l}{3} \\biggl (224 a_5 + \\frac{1028}{3} a_4 - \\frac{28}{15} a_1^5 + \\frac{257}{18} a_1^4 - \\frac{56}{9} \\pi ^2 a_1^3 + \\frac{257}{9} \\pi ^2 a_1^2 - \\frac{17}{90} \\pi ^4 a_1 - \\frac{539}{9} \\pi ^2 a_1 - \\frac{1027}{4} \\zeta _5\\nonumber \\\\&&\\qquad \\quad {}- \\frac{119}{16} \\pi ^2 \\zeta _3 + \\frac{1081}{6} \\zeta _3 - \\frac{18599}{8640} \\pi ^4 + \\frac{160081}{4608} \\pi ^2 + \\frac{3103}{72}\\biggr )\\nonumber \\\\&&\\qquad {}- C_A T_F n_l \\biggl (\\frac{112}{3} a_5 + \\frac{514}{9} a_4 - \\frac{14}{45} a_1^5 + \\frac{257}{108} a_1^4 - \\frac{28}{27} \\pi ^2 a_1^3 + \\frac{257}{54} \\pi ^2 a_1^2 - \\frac{17}{540} \\pi ^4 a_1 - \\frac{539}{54} \\pi ^2 a_1 - \\frac{859}{24} \\zeta _5\\nonumber \\\\&&\\qquad \\quad {}- \\frac{11}{16} \\pi ^2 \\zeta _3 + \\frac{1229}{432} \\zeta _3 - \\frac{3691}{6480} \\pi ^4 - \\frac{1991}{648} \\pi ^2 - \\frac{4500377}{93312}\\biggr )\\nonumber \\\\&&\\qquad {}+ \\frac{(T_F n_h)^2}{3} \\biggl (56 a_4 + \\frac{7}{3} a_1^4 - \\frac{7}{3} \\pi ^2 a_1^2 - \\frac{4}{5} \\pi ^2 a_1 + \\frac{3221}{80} \\zeta _3 - \\frac{31}{72} \\pi ^4 + \\frac{39661}{7200} \\pi ^2 - \\frac{636911}{8640}\\biggr )\\nonumber \\\\&&\\qquad {}+ T_F^2 n_h n_l \\biggl (\\frac{32}{3} a_4 + \\frac{4}{9} a_1^4 + \\frac{8}{9} \\pi ^2 a_1^2 - \\frac{35}{9} \\pi ^2 a_1 + \\frac{27}{2} \\zeta _3 + \\frac{179}{1080} \\pi ^4 + \\frac{2245}{1296} \\pi ^2 - \\frac{264817}{7776}\\biggr )\\nonumber \\\\&&\\qquad {}- \\frac{(T_F n_l)^2}{54} \\biggl (275 \\zeta _3 + \\frac{23}{5} \\pi ^4 + \\frac{1081}{16} \\pi ^2 + \\frac{253783}{864}\\biggr )\\biggr ]+ \\mathcal {O}(\\varepsilon ^2)\\biggr \\rbrace \\nonumber \\\\&&{} + \\left(\\frac{\\alpha _s}{\\pi }\\right)^4\\biggl \\lbrace C_F^4 \\biggl [L_0-\\frac{139}{2}a_5+12\\pi ^2 a_4-\\frac{9137}{16}a_4+\\frac{139}{240}a_1^5+\\frac{\\pi ^2}{2}a_1^4-\\frac{9137}{384}a_1^4-\\frac{311}{72}\\pi ^2 a_1^3-\\frac{\\pi ^4}{2}a_1^2-\\frac{8597}{192}\\pi ^2 a_1^2\\nonumber \\\\&&\\qquad {}+\\frac{21}{2}\\pi ^2 \\zeta _3 a_1-\\frac{2783}{2880}\\pi ^4 a_1+\\frac{33687}{256}\\pi ^2 a_1-\\frac{2937}{128}\\zeta _5+\\frac{87}{128}\\zeta _3^2+\\frac{2755}{192}\\pi ^2 \\zeta _3-\\frac{113181}{512}\\zeta _3-\\frac{899}{7560}\\pi ^6+\\frac{18553}{23040}\\pi ^4\\nonumber \\\\&&\\qquad {}-\\frac{24129}{1024}\\pi ^2-\\frac{90577}{8192}\\biggr ]\\nonumber \\\\&&\\quad {}+ C_F^3 C_A (14.12 \\pm 3.6)- C_F^2 C_A^2 \\bigl [8.75607 \\pm 2.9 - (0.00269 \\pm 0.0012) \\xi \\bigr ]\\nonumber \\\\&&\\quad {}- C_F C_A^3 \\bigl [142.552 \\pm 0.82 - (0.43649 \\pm 0.00076) \\xi + (0.0205278 \\pm 0.00012) \\xi ^2\\bigr ]\\nonumber \\\\&&\\quad {}+ d_{FA} \\bigl [9.4 \\pm 2.1 + (0.147 \\pm 0.013) \\xi - (0.0748 \\pm 0.0028) \\xi ^2\\bigr ]\\nonumber \\\\&&\\quad {}+ C_F^3 T_F n_h \\biggl [L_1+\\frac{46}{3}a_5+16\\pi ^2 a_4-\\frac{35189}{48}a_4-\\frac{23}{180}a_1^5+\\frac{2}{3}\\pi ^2 a_1^4-\\frac{35189}{1152}a_1^4-\\frac{703}{108}\\pi ^2 a_1^3-\\frac{2}{3}\\pi ^4 a_1^2-\\frac{77155}{1152}\\pi ^2 a_1^2\\nonumber \\\\&&\\qquad {}+14\\pi ^2 \\zeta _3 a_1-\\frac{569}{2160}\\pi ^4 a_1+\\frac{3273}{16}\\pi ^2 a_1-\\frac{3067}{32}\\zeta _5+\\frac{29}{32}\\zeta _3^2+\\frac{2981}{288}\\pi ^2 \\zeta _3-\\frac{119743}{384}\\zeta _3-\\frac{899}{5670}\\pi ^6+\\frac{65953}{69120}\\pi ^4\\nonumber \\\\&&\\qquad {}-\\frac{572525}{13824}\\pi ^2-\\frac{305411}{36864}\\biggr ]\\nonumber \\\\&&\\quad {}+ C_F^2 C_A T_F n_h \\bigl [14.893 \\pm 0.083 - (0.657352 \\pm 0.00024) \\xi \\bigr ]\\nonumber \\\\&&\\quad {}- C_F C_A^2 T_F n_h \\bigl [3.1601 \\pm 0.056 - (0.198984 \\pm 0.00013) \\xi + 0.0244254 \\xi ^2\\bigr ]\\nonumber \\\\&&\\quad {}+ C_F^2 (T_F n_h)^2 \\biggl [L_2+120a_5+\\frac{2749}{48}a_4-a_1^5+\\frac{2749}{1152}a_1^4-\\frac{\\pi ^2}{3}a_1^3-\\frac{10525}{1152}\\pi ^2 a_1^2+\\frac{43}{36}\\pi ^4 a_1+\\frac{711}{20}\\pi ^2 a_1-\\frac{493}{8}\\zeta _5\\nonumber \\\\&&\\qquad {}-\\frac{269}{24}\\pi ^2 \\zeta _3-\\frac{10127}{2560}\\zeta _3-\\frac{5513}{13824}\\pi ^4-\\frac{678719}{64800}\\pi ^2-\\frac{8452817}{414720}\\biggr ]\\nonumber \\\\&&\\quad {}- C_F C_A (T_F n_h)^2 \\bigl [0.01995 \\pm 0.0062 - 0.10436 \\xi \\bigr ]\\nonumber \\\\&&\\quad {}+ C_F (T_F n_h)^3 \\biggl [L_3+ \\frac{1}{3}\\biggl (104a_4+\\frac{13}{3}a_1^4+\\frac{5}{3}\\pi ^2 a_1^2-\\frac{103}{10}\\pi ^2 a_1+\\frac{5881}{80}\\zeta _3-\\frac{299}{360}\\pi ^4+\\frac{31451}{2700}\\pi ^2-\\frac{5981281}{51840}\\biggr )\\biggr ]\\nonumber \\\\&&\\quad {}+ d_{FF} n_h L_l- C_F^3 T_F n_l (4.92605 \\pm 0.0067)+ C_F^2 C_A T_F n_l (15.0599 \\pm 0.012)\\nonumber \\\\&&\\quad {}+ C_F C_A^2 T_F n_l \\bigl [166.421 \\pm 0.031 - 0.134051 \\xi \\bigr ]- C_F^2 T_F^2 n_h n_l (5.08715 \\pm 0.000074)\\nonumber \\\\&&\\quad {}+ C_F C_A T_F^2 n_h n_l \\bigl [0.53235 \\pm 0.0015 + 0.0910988 \\xi \\bigr ]+ 0.0138079 C_F T_F^3 n_h^2 n_l- d_{FF} n_l (2.18 \\pm 0.8)\\nonumber \\\\&&\\quad {}- C_F^2 (T_F n_l)^2 \\biggl (\\frac{32}{3} a_5 + \\frac{188}{9} a_4 - \\frac{4}{45} a_1^5 + \\frac{47}{54} a_1^4 - \\frac{8}{27} \\pi ^2 a_1^3 + \\frac{47}{27} \\pi ^2 a_1^2 - \\frac{31}{270} \\pi ^4 a_1 - \\frac{239}{54} \\pi ^2 a_1 - \\frac{601}{48} \\zeta _5 - \\frac{\\pi ^2}{2} \\zeta _3\\nonumber \\\\&&\\qquad {}+ \\frac{6925}{576} \\zeta _3 - \\frac{1181}{10368} \\pi ^4 + \\frac{1043}{384} \\pi ^2 + \\frac{3146969}{497664}\\biggr )\\nonumber \\\\&&\\quad {}+ \\frac{C_F C_A (T_F n_l)^2}{3} \\biggl (16 a_5 + \\frac{94}{3} a_4 - \\frac{2}{15} a_1^5 + \\frac{47}{36} a_1^4 - \\frac{4}{9} \\pi ^2 a_1^3 + \\frac{47}{18} \\pi ^2 a_1^2 - \\frac{31}{180} \\pi ^4 a_1 - \\frac{239}{36} \\pi ^2 a_1 - \\frac{365}{32} \\zeta _5 - \\frac{11}{12} \\pi ^2 \\zeta _3\\nonumber \\\\&&\\qquad {}- \\frac{1111}{64} \\zeta _3 - \\frac{4333}{17280} \\pi ^4 - \\frac{6815}{1152} \\pi ^2 - \\frac{4767085}{165888}\\biggr )\\nonumber \\\\&&\\quad {}+ \\frac{C_F T_F^3 n_h n_l^2}{3} \\biggl (\\frac{\\zeta _3}{16} - \\frac{4}{15} \\pi ^4 + \\frac{19}{27} \\pi ^2 + \\frac{399325}{20736}\\biggr )+ \\frac{C_F (T_F n_l)^3}{216} \\biggl (\\frac{467}{2} \\zeta _3 + \\frac{71}{20} \\pi ^4 + \\frac{167}{3} \\pi ^2 + \\frac{103933}{864}\\biggr )+ \\mathcal {O}(\\varepsilon )\\biggr \\rbrace \\nonumber \\\\&&{}+ \\mathcal {O}(\\alpha _s^5)\\,,$ where $\\alpha _s = \\alpha _s^{(n_f)}(M)$ , $\\xi = \\xi ^{(n_f)}(M)$ .", "$L_{0,l,1,2,3}$ are the $\\varepsilon ^0$ parts of the quantities $Z_2^{(4,0)}$ , $Z_2^{(4,l)}$ , $Z_2^{(4,1)}$ , $Z_2^{(4,2)}$ , $Z_2^{(4,3)}$ given in Eqs.", "(28–32) of  [14].", "Their numerical values are given in Eqs.", "(5–9) of that paper.", "The finite four-loop terms of Eq.", "(REF ) are equal to the corresponding finite four-loop terms in $Z_Q^{\\text{os}}$ plus products of lower-loop quantities which are all known analytically.", "For 14 out of 23 color structures these coefficients in $Z^{\\text{os}}_Q$ are only known numerically [13].", "We use these numerical values, together with their uncertainty estimates, from the tables V, VI, and VII of that paper.", "Note that in Ref.", "[13] $Z_Q^{\\text{os}}$ has been computed in an expansion in $\\xi $ up to the second order; 9 out of these 19 color structures are obviously gauge invariant, and 7 more seem to be either gauge-invariant or have at most linear $\\xi $ terms (though we know no explicit proof).", "The remaining 3 structures ($C_F C_A^3$ , $d_{FA}$ , $C_F C_A^2 T_F n_h$ ) may contain terms with higher powers of $\\xi $ , which are not known.", "The same is true for the corresponding terms in $z(\\mu )$ in Eq.", "(REF ).", "If we re-express $z(M)$ in Eq.", "(REF ) via $\\alpha _s^{(n_l)}(M)$ , the terms up to three loops agree with [5].", "(Note that positive powers of $\\varepsilon $ are not presented [5].)", "The $\\alpha _s^4 n_l^3$ term also agrees with [5].", "After specifying the color factors to QCD with $N_c=3$ we obtain for $\\varepsilon =0$ $&&z(M) = 1 - \\frac{4}{3} \\frac{\\alpha _s}{\\pi }- \\left(\\frac{\\alpha _s}{\\pi }\\right)^2(17.45 - 1.33 n_l)\\nonumber \\\\&&{} - \\left(\\frac{\\alpha _s}{\\pi }\\right)^3(262.42 - 0.78 \\xi - 35.81 n_l + 0.98 n_l^2)\\nonumber \\\\&&{} - \\left(\\frac{\\alpha _s}{\\pi }\\right)^4\\bigl [5137.52 - 15.67 \\xi + 1.07 \\xi ^2\\nonumber \\\\&&\\quad {}- \\bigl (1030.82 - 0.71 \\xi \\bigr ) n_l+ 60.30 n_l^2 - 1.00 n_l^3 \\bigr ]\\nonumber \\\\&&{} + \\mathcal {O}(\\alpha _s^5)\\,.$ In Landau gauge ($\\xi ^{(n_f)}=1$ ) at $n_l=4$ this gives $&&z(M) = 1 - \\frac{4}{3} \\frac{\\alpha _s}{\\pi }- 12.12 \\left(\\frac{\\alpha _s}{\\pi }\\right)^2- 134.11 \\left(\\frac{\\alpha _s}{\\pi }\\right)^3\\nonumber \\\\&&{} - 1903.22 \\left(\\frac{\\alpha _s}{\\pi }\\right)^4+ \\mathcal {O}(\\alpha _s^5)\\,,$ while the naive nonabelianization [22] (large $\\beta _0$ limit) predicts [5] $&&1 - \\frac{4}{3} \\frac{\\alpha _s}{\\pi }- 16.66 \\left(\\frac{\\alpha _s}{\\pi }\\right)^2- 153.41 \\left(\\frac{\\alpha _s}{\\pi }\\right)^3\\nonumber \\\\&&{} - 1953.40 \\left(\\frac{\\alpha _s}{\\pi }\\right)^4+ \\mathcal {O}(\\alpha _s^5)\\,.$ The comparison to Eq.", "(REF ) shows that up to four loops these predictions are rather good.", "The coefficients are all negative and grow very fast, which can be explained by the infrared renormalon at $u=1/2$  [5].", "This is the closest possible position of a renormalon singularity in the Borel plane $u$ to the origin, and it leads to the fastest possible growth of perturbative terms $(L-1)!\\,(\\beta _0/2)^L (\\alpha _s/\\pi )^L$ .", "The coefficients of powers of $\\xi $ are much smaller than the $\\xi $ -independent terms." ], [ "Effect of a lighter-flavor mass", "Now we suppose that $n_m$ light flavors have a non-zero mass $m$ , while the remaining $n_0 = n_l-n_m$ light flavors are massless.", "In practice, $n_m = 1$ , e. g. $c$ in $b$ -quark HQET.", "In this case the massless result (REF ) for the matching coefficient should be multiplied by the additional factor $z^{\\prime } &=& \\frac{Z^{\\text{os}}_Q(g_0^{(n_f)},\\xi _0^{(n_f)},m_0^{(n_f)})}{Z^{\\text{os}}_Q(g_0^{(n_f)},\\xi _0^{(n_f)},0)}\\nonumber \\\\&&{}\\times \\frac{Z^{\\text{os}}_h(g_0^{(n_l)},\\xi _0^{(n_l)},0)}{Z^{\\text{os}}_h(g_0^{(n_l)},\\xi _0^{(n_l)},m_0^{(n_l)})}\\,,$ where $Z^{\\rm os}_{Q,h}(\\ldots ,0)\\equiv Z^{\\rm os}_{Q,h}(\\ldots )$ in Eq.", "(REF ) and $Z^{\\text{os}}_h(g_0^{(n_l)},\\xi _0^{(n_l)},0) = 1$ .", "This factor does not depend on the renormalization scale $\\mu $ .", "In the expression $&&\\log z^{\\prime }= \\log Z^{\\text{os}}_Q(g_0^{(n_f)},\\xi _0^{(n_f)},m_0^{(n_f)})\\\\&&{} - \\log Z^{\\text{os}}_Q(g_0^{(n_f)},\\xi _0^{(n_f)},0)- \\log Z^{\\text{os}}_h(g_0^{(n_l)},\\xi _0^{(n_l)},m_0^{(n_l)})\\nonumber $ we re-express all terms via $\\alpha _s^{(n_f)}(M)$ , $\\xi ^{(n_f)}(M)$ and the on-shell lighter-flavor mass $m$ (it is the same in both $n_f$ and $n_l$ flavor theories).", "The result depends on the dimensionless ratio $x = \\frac{m}{M}\\,.$ If we express $z^{\\prime }$ via $\\alpha _s^{(n_f)}(\\mu )$ , $\\xi ^{(n_f)}(\\mu )$ , the coefficients will depend on $\\mu $ .", "This dependence is determined by the renormalization-group equation $\\frac{d \\log z^{\\prime }}{d \\log \\mu } = 0$ together with $&&\\frac{d \\log \\alpha _s^{(n_f)}(\\mu )}{d \\log \\mu } =- 2 \\varepsilon - 2 \\beta ^{(n_f)}(\\alpha _s^{(n_f)}(\\mu ))\\,,\\\\&&\\frac{d \\log (1 - \\xi ^{(n_f)}(\\mu ))}{d \\log \\mu } =- \\gamma _A^{(n_f)}(\\alpha _s^{(n_f)}(\\mu ),\\xi ^{(n_f)}(\\mu ))\\,.$ Ultraviolet divergences cancel in each fraction in (REF ).", "On the other hand, the on-shell wave-function renormalization factors have extra infrared divergences at $m=0$ .", "However, $z^\\prime $ in Eq.", "(REF ) has a smooth limit for $x\\rightarrow 0$ .", "In the following we illustrate the cancellation for infrared divergences at two-loop order.", "Similar mechanisms are also at work at higher loop orders.", "For dimensional reasons the two-loop corrections in Fig.", "REF a lead to $\\log Z_h^{\\text{os}}(m) \\sim g_0^4 m^{-4\\varepsilon }$ .", "Furthermore, we have $\\log Z_h^{\\text{os}}(0) = 0$ .", "Thus, the limit $x\\rightarrow 0$ is discontinuous.", "In QCD (Fig.", "REF b) we have $\\log Z_Q^{\\text{os}}(0) \\sim g_0^4 M^{-4\\varepsilon }$ for dimensional reasons.", "For $m\\ll M$ there are 3 regions (see [28], [29]): Hard (all momenta $\\sim M$ ): a regular series in $m^2$ , $\\log Z_Q^{\\text{os}}(m)\\big |_{\\text{hard}} = \\log Z_Q^{\\text{os}}(0) \\left[1 + \\mathcal {O}(x^2)\\right]$ .", "Soft-hard (momentum of one $m$ -line is $\\sim m$ , all the remaining momenta are $\\sim M$ ).", "If we take the term $m$ from the numerator $\\unknown.", "/k+m$ of the soft propagator, there is another factor $m$ in the numerator of the hard mass-$m$ propagator, and the soft-loop integral is $\\sim m^{2-2\\varepsilon }$ ; if we take $\\unknown.", "/k$ instead, we have to expand the hard subdiagram in $k$ up to the linear term, and the soft loop is $\\sim m^{4-2\\varepsilon }$ .", "We obtain $\\log Z_Q^{\\text{os}}(m)\\big |_{\\text{soft-hard}} \\sim g_0^4 M^{-2\\varepsilon } m^{-2\\varepsilon } x^4$ .", "Soft (all momenta $\\sim m$ ): the leading term is the HQET one, the Taylor series is in $x$ (not in $x^2$ ), $\\log Z_Q^{\\text{os}}(m)\\big |_{\\text{soft}} = \\log Z_h^{\\text{os}}(m) \\left[1 + \\mathcal {O}(x)\\right]$ .", "As a result, $\\log Z_Q^{\\text{os}}(m)\\big |_{\\text{hard}} - \\log Z_Q^{\\text{os}}(0)$ is smooth at $x\\rightarrow 0$ ; $\\log Z_Q^{\\text{os}}(m)\\big |_{\\text{soft-hard}}$ is subleading and hence smooth; $\\log Z_Q^{\\text{os}}(m)\\big |_{\\text{soft}}$ has the same discontinuity as $\\log Z_h^{\\text{os}}$ ; hence $\\log z^{\\prime }$  (REF ) has a smooth limit 1 at $x\\rightarrow 0$ .", "Figure: Two-loop contributions to the on-shell wave-function renormalization constants:(a) in HQET; (b) in QCD.The two-loop term in $Z^{\\text{os}}_Q(g_0^{(n_f)},\\xi _0^{(n_f)},m_0^{(n_f)})$ has been calculated up to $\\varepsilon ^0$ in [7]; the result exact in $\\varepsilon $ has been obtained in [30].", "The three-loop term has been calculated up to $\\varepsilon ^0$ in [31].", "Some master integrals are only known as truncated series in $x$ or as numerical interpolations, see [32] for detailed discussion of these master integrals.", "Exact results in $x$ for the coefficient of $C_F T_F^2 n_m n_0 \\alpha _s^3$ can be obtained using the formulas of Appendix .", "The HQET renormalization constant $Z^{\\text{os}}_h(g_0^{(n_l)},\\xi _0^{(n_l)},m_0^{(n_l)})$ at two loops has been calculated in [22], and at three loops in [33] (one of the master integrals is discussed in [34]; note that there are some typos in formulas in the journal version of [33] fixed later in arXiv).", "Altogether we are now in the position to obtain $z^{\\prime }$ up to three loops.", "The expansion of $z^\\prime $ in terms of $\\alpha _s^{(n_f)}(M)$ and its decomposition into color factors is given by $z^{\\prime } &=& 1 + C_F T_F \\left(\\frac{\\alpha _s^{(n_f)}(M)}{\\pi }\\right)^2\\left(A_0 + A_1 \\varepsilon + \\mathcal {O}(\\varepsilon ^2)\\right)\\nonumber \\\\&&{} + C_F T_F \\left(\\frac{\\alpha _s^{(n_f)}(M)}{\\pi }\\right)^3\\bigl (C_F A_F + C_A A_A+ T_F n_0 A_l + T_F n_m A_m + T_F n_h A_h+ \\mathcal {O}(\\varepsilon )\\bigr )+ \\mathcal {O}(\\alpha _s^4)\\,,$ where $A_0 &=& \\frac{1}{4} \\biggl [(1-x) (2-x-x^2-6x^3) H_{1,0}(x)- (1+x) (2+x-x^2+6x^3) H_{-1,0}(x)\\nonumber \\\\&&{} - \\frac{3}{2} \\pi ^2 x+ (4 \\log x + 7) x^2- \\frac{5}{2} \\pi ^2 x^3+ (6 \\log ^2 x + \\pi ^2) x^4\\biggr ]\\,.$ The expansion of this function in $x$ reads $A_0 &=& \\frac{1}{4} \\biggl [- \\frac{3}{2} \\pi ^2 x + 12 x^2 - \\frac{5}{2} \\pi ^2 x^3+ \\left(6 \\log ^2 x - 11 \\log x + \\pi ^2 + \\frac{125}{12}\\right) x^4+ \\sum _{n=3}^\\infty \\left(2 g(2n) \\log x + \\frac{d\\,g(2n)}{d\\,n}\\right) x^{2n}\\biggr ]\\,,\\nonumber \\\\g(x) &=& \\frac{2}{x} - \\frac{3}{x-1} - \\frac{5}{x-3} + \\frac{6}{x-4}\\,.$ Note that the only terms with odd powers of $x$ are $x^1$ and $x^3$ .", "The expansion in $x^{-1}$ is given by $A_0 = \\frac{1}{4} \\biggl [- 2 \\log ^2 x^{-1} + \\frac{19}{3} \\log x^{-1} - \\frac{\\pi ^2}{3} - \\frac{229}{36}+ \\sum _{n=1}^\\infty \\left(2 g(-2n) \\log x^{-1} + \\frac{d\\,g(-2n)}{d\\,n}\\right) x^{-2n}\\biggr ]\\,.$ For illustration we show in Fig.", "REF $A_0(x)$ for $x\\in [0,1]$ .", "The ${\\cal O}(\\varepsilon )$ term at two loops reads $A_1 &=& \\frac{1}{4} \\biggl [(1-x) (2-x-x^2-6x^3) \\bigl (2 H_{1,1,0}(x) - 4 H_{1,-1,0}(x)\\bigr )\\nonumber \\\\&&{} + (1+x) (2+x-x^2+6x^3) \\bigl (2 H_{-1,-1,0}(x) - 4 H_{-1,1,0}(x) - \\pi ^2 H_{-1}(x)\\bigr )\\nonumber \\\\&&{} + (1-x) (9-6x+6x^2-17x^3) H_{1,0}(x)- (1+x) (9+6x+6x^2+17x^3) H_{-1,0}(x)\\nonumber \\\\&&{} + 4 x (3+5x^2) \\bigl (H_{0,1,0}(x) + H_{0,-1,0}(x)\\bigr )+ 6 \\pi ^2 \\left(L + 2 a_1 - \\frac{5}{4}\\right) x+ \\left(L + 2 \\pi ^2 + \\frac{53}{2}\\right) x^2\\nonumber \\\\&&{} + 10 \\pi ^2 \\left(L + 2 a_1 - \\frac{23}{20}\\right) x^3- 12 \\left(L^3 - \\frac{17}{12} L^2 - \\zeta _3 - \\frac{17}{72} \\pi ^2\\right) x^4\\biggr ]\\nonumber \\\\&=& \\pi ^2 \\biggl (\\frac{3}{2} L + 3 a_1 - \\frac{19}{8}\\biggr ) x+ \\frac{5}{2} x^2+ \\pi ^2 \\biggl (\\frac{5}{2} L + 5 a_1 - \\frac{8}{3}\\biggr ) x^3- \\biggl (3 L^3 - \\frac{17}{4} L^2 - \\frac{3}{8} L - 3 \\zeta _3 + \\frac{2}{3} \\pi ^2 + \\frac{2827}{288}\\biggr ) x^4\\nonumber \\\\&&{} - \\frac{63}{80} \\pi ^2 x^5- \\frac{2}{15} \\biggl (\\frac{61}{5} L - 2 \\pi ^2 - \\frac{4243}{225}\\biggr ) x^6- \\frac{15}{112} \\pi ^2 x^7- \\frac{3}{56} \\biggl (\\frac{53}{35} L - \\frac{3}{2} \\pi ^2 - \\frac{5909}{1960}\\biggr ) x^8+ \\mathcal {O}(x^9)\\,,$ where $L = \\log x$ .", "At three-loop order the $C_F T_F^2 n_m n_0 \\alpha _s^3$ term is known exactly via harmonic polylogarithms of $x$ : $A_l &=& \\frac{1}{3} \\biggl [(1-x) (2-x-x^2-6 x^3) \\biggl (H_{1,-1,0}(x) + \\frac{\\pi ^2}{12} H_{1}(x)\\biggr )+ (1+x) (2+x-x^2+6 x^3) \\biggl (H_{-1,1,0}(x) + \\frac{5}{12} \\pi ^2 H_{-1}(x)\\biggr )\\nonumber \\\\&&{} - \\frac{1}{6} (1-x) (19-11 x+x^2-39 x^3) H_{1,0}(x)+ \\frac{1}{6} (1+x) (19+11 x+x^2+39 x^3) H_{-1,0}(x)\\nonumber \\\\&&{} - x (3+5 x^2) \\left(H_{0,1,0}(x) + H_{0,-1,0}(x)\\right)- \\pi ^2 \\biggl (\\frac{3}{2} L + 3 a_1 - \\frac{5}{2}\\biggr ) x- \\biggl (\\frac{17}{2} L + 2 \\pi ^2 + \\frac{91}{4}\\biggr ) \\frac{x^2}{3}- 5 \\pi ^2 \\biggl (\\frac{L}{2} + a_1 - \\frac{2}{3}\\biggr ) x^3\\nonumber \\\\&&{} + \\biggl (2 L^3 - \\frac{13}{2} L^2 - \\pi ^2 L - 9 \\zeta _3 - \\frac{13}{12} \\pi ^2\\biggr ) x^4\\biggr ]\\nonumber \\\\&=& - \\pi ^2 \\biggl (\\frac{L}{2} + a_1 - \\frac{7}{6}\\biggr ) x- \\frac{7}{3} x^2- \\frac{5}{3} \\pi ^2 \\biggl (\\frac{L}{2} + a_1 - \\frac{7}{12}\\biggr ) x^3+ \\biggl [\\frac{2}{3} L^3 - \\frac{13}{6} L^2 - \\biggl (\\frac{\\pi ^2}{3} - \\frac{3}{4}\\biggr ) L- 3 \\zeta _3 + \\frac{\\pi ^2}{4} + \\frac{1175}{432}\\biggr ] x^4\\nonumber \\\\&&{} + \\frac{21}{40} \\pi ^2 x^5+ \\frac{4}{45} \\biggl (\\frac{13}{5} L - \\frac{4}{3} \\pi ^2 - \\frac{2414}{225}\\biggr ) x^6+ \\frac{5}{56} \\pi ^2 x^7+ \\biggl (\\frac{4}{35} L - \\frac{\\pi ^2}{4} - \\frac{40489}{29400}\\biggr ) \\frac{x^8}{7}+ \\mathcal {O}(x^9)\\,,$ where after the second equality sign we show the expansion in $x$ .", "In principle, it is straightforward to obtain exact results in $x$ also the four-loop $C_F T_F^3 n_n n_0^2 \\alpha _s^4$ term.", "However, we refrain from presenting such results because the remaining four-loop color structures are not known.", "The remaining three-loop terms can be obtained in a series expansion in $x$ with the help of the result from [31].", "Including terms up to order $x^8$ gives $A_F &=& \\frac{\\pi ^2}{3} \\biggl (8 a_1 + \\frac{13}{4} \\pi - \\frac{343}{24}\\biggr ) x- \\biggl (L^2 - \\frac{67}{6} L - \\frac{17}{8} \\pi ^2 + \\frac{229}{18}\\biggr ) x^2+ \\frac{\\pi ^2}{3} \\biggl (\\frac{11}{3} L + \\frac{44}{3} a_1 + \\frac{35}{8} \\pi - \\frac{157}{8}\\biggr ) x^3\\nonumber \\\\&&{} + \\biggl [\\frac{19}{6} L^3 - \\frac{911}{120} L^2- \\biggl (3 \\pi ^2 a_1 - \\frac{3}{2} \\zeta _3 - \\frac{45}{16} \\pi ^2 - \\frac{40567}{3600}\\biggr ) L+ 20 a_4 + \\frac{5}{6} a_1^4 + \\frac{2}{3} \\pi ^2 a_1^2 + \\frac{11}{16} \\pi ^2 a_1 + \\frac{387}{32} \\zeta _3\\nonumber \\\\&&\\quad {} - \\frac{43}{144} \\pi ^4 - \\frac{155}{64} \\pi ^2 - \\frac{2534579}{216000}\\biggr ] x^4+ \\frac{7}{5} \\pi ^2 \\biggl (\\frac{3}{32} \\pi + \\frac{1}{5}\\biggr ) x^5\\nonumber \\\\&&{} + \\biggl [\\frac{1579}{70} L^2 + \\biggl (\\frac{77}{16} \\pi ^2 - \\frac{328067}{11025}\\biggr ) L- \\frac{1}{16} \\biggl (77 \\pi ^2 a_1 - \\frac{539}{2} \\zeta _3 - \\frac{83}{15} \\pi ^2 - \\frac{126231437}{1157625}\\biggr )\\biggr ] \\frac{x^6}{9}- \\frac{\\pi ^2}{28} \\biggl (\\frac{25}{16} \\pi + \\frac{1}{7}\\biggr ) x^7\\nonumber \\\\&&{} + \\biggl [\\frac{2843}{105} L^2 + \\biggl (\\frac{21}{2} \\pi ^2 - \\frac{718639}{33075}\\biggr ) \\frac{L}{2}- \\frac{1}{4} \\biggl (21 \\pi ^2 a_1 - \\frac{147}{2} \\zeta _3 - \\frac{4379}{240} \\pi ^2 + \\frac{1213332979}{83349000}\\biggr )\\biggr ] \\frac{x^8}{32}+ \\mathcal {O}(x^9)\\,,\\nonumber \\\\A_A &=& \\frac{\\pi ^2}{8} \\biggl (\\frac{25}{2} L + \\frac{313}{3} a_1 - \\frac{13}{3} \\pi - \\frac{2473}{36}\\biggr ) x\\nonumber \\\\&&{} + \\biggl [\\frac{L^2}{2} + \\biggl (\\frac{3}{2} \\zeta _3 - \\frac{31}{90} \\pi ^4 + 7 \\pi ^2 - \\frac{7}{3}\\biggr ) L- 5 \\zeta _5 - \\frac{7}{2} \\pi ^2 \\zeta _3 + \\frac{79}{4} \\zeta _3 - \\frac{17}{180} \\pi ^4 + \\frac{35}{18} \\pi ^2 + \\frac{517}{9}\\biggr ] \\frac{x^2}{4}\\nonumber \\\\&&{} + \\frac{\\pi ^2}{24} \\biggl (\\frac{269}{6} L + \\frac{1291}{3} a_1 - \\frac{35}{2} \\pi - \\frac{865}{3}\\biggr ) x^3- \\biggl [\\frac{83}{48} L^3 + \\biggl (3 \\pi ^2 - \\frac{3977}{60}\\biggr ) \\frac{L^2}{8}- \\biggl (\\frac{3}{2} \\pi ^2 a_1 - 3 \\zeta _3 - \\frac{13}{24} \\pi ^2 - \\frac{230293}{28800}\\biggr ) L\\nonumber \\\\&&\\quad {} + 10 a_4 + \\frac{5}{12} a_1^4 + \\frac{\\pi ^2}{3} a_1^2 + \\frac{11}{32} \\pi ^2 a_1 + \\frac{111}{64} \\zeta _3- \\frac{161}{1440} \\pi ^4 - \\frac{631}{1152} \\pi ^2 - \\frac{452033}{864000}\\biggr ] x^4+ \\frac{\\pi ^2}{20} \\biggl (\\frac{79}{9} L - \\frac{21}{16} \\pi - \\frac{2671}{432}\\biggr ) x^5\\nonumber \\\\&&{} + \\biggl [\\frac{5}{3} L^3 + \\frac{9911}{840} L^2 - \\biggl (\\pi ^2 + \\frac{8394157}{529200}\\biggr ) L+ \\frac{1}{12} \\biggl (77 \\pi ^2 a_1 - \\frac{509}{2} \\zeta _3 - \\frac{3607}{60} \\pi ^2 + \\frac{8471770063}{18522000}\\biggr )\\biggr ] \\frac{x^6}{24}\\nonumber \\\\&&{} + \\frac{\\pi ^2}{28} \\biggl (\\frac{57}{25} L + \\frac{25}{32} \\pi - \\frac{11549}{14000}\\biggr ) x^7+ \\biggl [\\frac{43}{27} L^3 + \\frac{209}{20} L^2 + \\biggl (125 \\pi ^2 - \\frac{12327647}{14700}\\biggr ) \\frac{L}{216}\\nonumber \\\\&&\\quad {} + \\frac{1}{8} \\biggl (21 \\pi ^2 a_1 - \\frac{1435}{18} \\zeta _3 - \\frac{1213519}{45360} \\pi ^2 + \\frac{103012097}{2058000}\\biggr )\\biggr ] \\frac{x^8}{32}+ \\mathcal {O}(x^9)\\,,\\nonumber \\\\A_h &=& - \\biggl (2 L + \\frac{13}{5}\\biggr ) \\frac{x^2}{5}+ \\frac{2}{15} \\pi ^2 x^3+ \\biggl [\\frac{3}{70} L^2 + \\biggl (\\pi ^2 - \\frac{35887}{4900}\\biggr ) \\frac{L}{3}- \\frac{1}{36} \\biggl (13 \\pi ^2 - \\frac{59985349}{514500}\\biggr )\\biggr ] x^4\\nonumber \\\\&&{} - \\biggl (244 L^2 - \\frac{92779}{315} L + \\frac{353877541}{793800}\\biggr ) \\frac{x^6}{945}- \\biggl (47 L^2 + \\frac{925823}{13860} L - \\frac{4543985839}{384199200}\\biggr ) \\frac{x^8}{770}+ \\mathcal {O}(x^9)\\,,\\nonumber \\\\A_m &=& - \\pi ^2 \\biggl (\\frac{L}{2} - \\frac{2}{15}\\biggr ) x- \\frac{7}{3} x^2- \\frac{5}{6} \\pi ^2 L x^3+ \\biggl [\\frac{2}{3} L^3 - \\frac{13}{6} L^2 - \\biggl (\\frac{\\pi ^2}{3} - \\frac{15}{4}\\biggr ) L+ \\frac{1}{4} \\biggl (\\pi ^2 + \\frac{203}{108}\\biggr )\\biggr ] x^4\\nonumber \\\\&&{} - \\biggl (\\frac{308}{5} L + \\frac{16}{3} \\pi ^2 - \\frac{13159}{225}\\biggr ) \\frac{x^6}{45}+ \\biggl (3 L^2 - \\frac{751}{70} L + \\frac{2095}{336}\\biggr ) \\frac{x^8}{14}+ \\mathcal {O}(x^9)\\,.$ Starting from three loops the individual terms in Eq.", "(REF ) are gauge parameter dependent.", "However, $\\xi $ cancels in the three-loop expression for $z^{\\prime }$ .", "It might be that $z^{\\prime }$ is gauge invariant to all orders, but we have no proof of this conjecture.", "Figure: The function A 0 (x)A_0(x)." ], [ "The QED and Bloch–Nordsieck heavy-lepton fields", "In QED the matching coefficient $z(\\mu )$ is gauge invariant to all orders in $\\alpha $  [5].", "The proof given in this paper is literally valid only for $n_f = 1$ lepton flavor, but can be easily generalized for any $n_f$ , as we demonstrate in the following.", "The QED on-shell renormalization constant $Z_\\psi ^{\\text{os}}$ is gauge invariant to all orders [35], [36], [10].", "Gauge dependence of the $\\overline{\\text{MS}}$ $Z_\\psi $ can be found using the so-called LKF transformation [37], [38] for arbitrary $n_f$ .", "In the gauge where the free photon propagator is $D^0_{\\mu \\nu }(k) = \\frac{1}{k^2} \\left(g_{\\mu \\nu } - \\frac{k_\\mu k_\\nu }{k^2}\\right) + \\Delta (k) k_\\mu k_\\nu \\,,$ the full bare lepton propagator reads $&&S(x) = S_L(x) e^{-i e_0^2 (\\tilde{\\Delta }(x) - \\tilde{\\Delta }(0))}\\,,\\nonumber \\\\&&\\tilde{\\Delta }(x) = \\int \\frac{d^d k}{(2\\pi )^d} \\Delta (k) e^{-ikx}\\,,$ where $S_L(x)$ is the Landau-gauge propagator.", "In the covariant gauge $\\Delta (k) = (1-\\xi _0)/(k^2)^2$ , and $\\tilde{\\Delta }(0) = 0$ in dimensional regularization.", "The lepton fields renormalization does not depend on their masses, so, let us assume that all $n_f$ flavors are massless.", "The propagator has a single Dirac structure $S(x) = S_0(x) e^{\\sigma (x)}\\,,$ where $S_0(x)$ is the $d$ -dimensional free propagator.", "Then $\\sigma (x) = \\sigma _L(x) + (1-\\xi _0) \\frac{e_0^2}{(4\\pi )^{d/2}} \\left(- \\frac{x^2}{4}\\right)^\\varepsilon \\Gamma (-\\varepsilon )\\,;$ re-expressing this result via the renormalized quantities, we obtain $\\log Z_\\psi (\\alpha ,\\xi ) = \\log Z_L(\\alpha ) - (1-\\xi ) \\frac{\\alpha }{4\\pi \\varepsilon }\\,.$ In QED $Z_A Z_\\alpha = 1$ due to Ward identities, hence $\\frac{d\\log ((1-\\xi (\\mu )) \\alpha (\\mu ))}{d\\log \\mu } = - 2 \\varepsilon $ exactly, and the anomalous dimension $\\gamma _\\psi (\\alpha ,\\xi ) = \\gamma _L(\\alpha ) + 2 (1-\\xi ) \\frac{\\alpha }{4\\pi }$ contains $\\xi $ only in the one-loop term.", "In the Bloch-Nordsieck EFT with $n_l$ light lepton flavors $Z_h^{\\text{os}}$ is gauge-invariant (even if some of these flavors have non-zero masses).", "Gauge dependence of the $\\overline{\\text{MS}}$ $Z_h$ can be found using exponentiation.", "The full bare propagator is $S_h(t) = S_{h0}(t) \\exp \\left(\\sum _i w_i\\right)\\,,$ where $w_i$ are webs [39], [40].", "In QED all webs have even numbers of photon legs; all webs with $>2$ legs are gauge invariant; all 2-leg webs except the trivial one (the free photon propagator) are gauge invariant, too.", "Therefore, $\\log \\frac{S_h(t)}{S_{hL}(t)} = (1-\\xi _0) \\frac{e_0^2}{(4\\pi )^{d/2}} \\left(\\frac{it}{2}\\right)^{2\\varepsilon } \\Gamma (-\\varepsilon )\\,;$ re-expressing this result via the renormalized quantities, we obtain $&&\\log Z_h(\\alpha ,\\xi ) = \\log Z_{hL}(\\alpha ) - (1-\\xi ) \\frac{\\alpha }{4\\pi \\varepsilon }\\,,\\\\&&\\gamma _h(\\alpha ,\\xi ) = \\gamma _{hL}(\\alpha ) + 2 (1-\\xi ) \\frac{\\alpha }{4\\pi }\\,.$ Finally, in the abelian case $\\zeta _\\alpha (\\mu ) = \\zeta _A(\\mu )^{-1}$ due to Ward identities, hence $(1-\\xi ^{(n_f)}(\\mu )) \\alpha ^{(n_f)}(\\mu )) = (1-\\xi ^{(n_l)}(\\mu )) \\alpha ^{(n_l)}(\\mu ))$ , and we arrive at the conclusion that $z(\\mu )$ is gauge invariant (some light flavors may be massive, this does not matter).", "Let us in the following specify $z(M)$ from Eq (REF ) to QED.", "Setting $C_F = T_F = d_{FF} = 1$ and $C_A = d_{FA} = 0$ we see that our four-loop result is indeed gauge invariant and is given by $&&z(M) = 1 - \\frac{\\alpha }{\\pi }\\biggl [1 + \\varepsilon \\biggl (\\frac{\\pi ^2}{16} + 2\\biggr )- \\varepsilon ^2 \\biggl (\\frac{\\zeta _3}{4} - \\frac{\\pi ^2}{12} - 4\\biggr )- \\varepsilon ^3 \\biggl (\\frac{\\zeta _3}{3} - \\frac{3}{640} \\pi ^4 - \\frac{\\pi ^2}{6} - 8\\biggr )+ \\mathcal {O}(\\varepsilon ^4) \\biggr ]\\nonumber \\\\&&{} + \\left(\\frac{\\alpha }{\\pi }\\right)^2\\biggl \\lbrace \\pi ^2 a_1 - \\frac{3}{2} \\zeta _3 - \\frac{55}{48} \\pi ^2 + \\frac{5957}{1152}+ \\frac{n_l}{12} \\biggl (\\pi ^2 + \\frac{113}{8}\\biggr )\\nonumber \\\\&&\\quad {} + \\varepsilon \\biggl [- 24 a_4 - a_1^4 - 2 \\pi ^2 a_1^2 + \\frac{31}{4} \\pi ^2 a_1 - \\frac{203}{8} \\zeta _3 + \\frac{7}{20} \\pi ^4 - \\frac{4903}{1152} \\pi ^2 + \\frac{56845}{6912}+ n_l \\biggl (\\zeta _3 + \\frac{127}{288} \\pi ^2 + \\frac{851}{192}\\biggr )\\biggr ]\\nonumber \\\\&&\\quad {} + \\varepsilon ^2 \\biggl [- 144 a_5 - 186 a_4 + \\frac{6}{5} a_1^5 - \\frac{31}{4} a_1^4 + 4 \\pi ^2 a_1^3 - \\frac{31}{2} \\pi ^2 a_1^2 - \\frac{13}{15} \\pi ^4 a_1 + 30 \\pi ^2 a_1 + \\frac{609}{4} \\zeta _5 + \\frac{11}{4} \\pi ^2 \\zeta _3 - \\frac{28169}{288} \\zeta _3\\nonumber \\\\&&\\qquad {} + \\frac{10007}{7680} \\pi ^4 - \\frac{114943}{6912} \\pi ^2 + \\frac{1838165}{41472}+ \\frac{n_l}{24} \\biggl (\\frac{305}{3} \\zeta _3 + \\frac{199}{80} \\pi ^4 + \\frac{853}{24} \\pi ^2 + \\frac{5753}{16}\\biggr )\\biggr ]+ \\mathcal {O}(\\varepsilon ^3)\\biggr \\rbrace \\nonumber \\\\&&{} + \\left(\\frac{\\alpha }{\\pi }\\right)^3\\biggl \\lbrace - 16 a_4 - \\frac{2}{3} a_1^4 + \\pi ^2 a_1^2 + \\frac{737}{36} \\pi ^2 a_1 - \\frac{5}{16} \\zeta _5 + \\frac{\\pi ^2}{8} \\zeta _3 - \\frac{4747}{288} \\zeta _3 - \\frac{13}{360} \\pi ^4 - \\frac{259133}{25920} \\pi ^2 -\\frac{230447}{20736}\\nonumber \\\\&&\\qquad {} + \\frac{n_l}{3} \\biggl (16 a_4 + \\frac{2}{3} a_1^4 + \\frac{4}{3} \\pi ^2 a_1^2 - \\frac{47}{6} \\pi ^2 a_1 + \\frac{137}{8} \\zeta _3 - \\frac{229}{720} \\pi ^4 + \\frac{139}{24} \\pi ^2 - \\frac{2201}{432}\\biggr )- \\frac{n_l^2}{18} \\biggl (7 \\zeta _3 + \\frac{19}{6} \\pi ^2 + \\frac{5767}{432}\\biggr )\\nonumber \\\\&&\\quad {} + \\varepsilon \\biggl [- \\frac{224}{3} a_5 + 16 \\pi ^2 a_4 - \\frac{5005}{6} a_4 + \\frac{28}{45} a_1^5 + \\frac{2}{3} \\pi ^2 a_1^4 - \\frac{5005}{144} a_1^4 - \\frac{88}{27} \\pi ^2 a_1^3 - \\frac{2}{3} \\pi ^4 a_1^2 - \\frac{11567}{144} \\pi ^2 a_1^2 + 14 \\pi ^2 \\zeta _3 a_1\\nonumber \\\\&&\\qquad \\quad {} - \\frac{2039}{2160} \\pi ^4 a_1 + \\frac{3481}{15} \\pi ^2 a_1 +\\frac{125}{8} \\zeta _5 + \\frac{29}{32} \\zeta _3^2 + \\frac{2945}{288} \\pi ^2 \\zeta _3 - \\frac{348821}{960} \\zeta _3 - \\frac{899}{5670} \\pi ^6 + \\frac{64103}{34560} \\pi ^4 - \\frac{224592113}{4147200} \\pi ^2\\nonumber \\\\&&\\qquad \\quad {} - \\frac{2783713}{207360}+ \\frac{n_l}{3} \\biggl (224 a_5 + \\frac{1124}{3} a_4 - \\frac{28}{15} a_1^5 + \\frac{281}{18} a_1^4 - \\frac{56}{9} \\pi ^2 a_1^3 + \\frac{281}{9} \\pi ^2 a_1^2 - \\frac{17}{90} \\pi ^4 a_1 - \\frac{644}{9} \\pi ^2 a_1 - \\frac{1027}{4} \\zeta _5\\nonumber \\\\&&\\qquad \\quad {}- \\frac{119}{16} \\pi ^2 \\zeta _3 + \\frac{662}{3} \\zeta _3 - \\frac{14303}{8640} \\pi ^4 + \\frac{552083}{13824} \\pi ^2 - \\frac{153109}{2592}\\biggr )\\nonumber \\\\&&\\qquad {} - \\frac{n_l^2}{54} \\biggl (275 \\zeta _3 + \\frac{23}{5} \\pi ^4 + \\frac{1081}{16} \\pi ^2 + \\frac{253783}{864}\\biggr )\\biggr ]+ \\mathcal {O}(\\varepsilon ^2)\\biggr \\rbrace \\nonumber \\\\&&{} + \\left(\\frac{\\alpha }{\\pi }\\right)^4\\biggl [L_{\\text{QED}}+\\frac{395}{6}a_5+28\\pi ^2 a_4-\\frac{58187}{48}a_4-\\frac{79}{144}a_1^5+\\frac{7}{6}\\pi ^2 a_1^4-\\frac{58187}{1152}a_1^4-\\frac{2411}{216}\\pi ^2 a_1^3-\\frac{7}{6}\\pi ^4 a_1^2\\nonumber \\\\&&\\quad {}-\\frac{69311}{576}\\pi ^2 a_1^2+\\frac{49}{2}\\pi ^2 \\zeta _3 a_1-\\frac{61}{1728}\\pi ^4 a_1+\\frac{1414153}{3840}\\pi ^2 a_1-\\frac{23093}{128}\\zeta _5+\\frac{203}{128}\\zeta _3^2+\\frac{7771}{576}\\pi ^2 \\zeta _3-\\frac{327897}{640}\\zeta _3-\\frac{899}{3240}\\pi ^6\\nonumber \\\\&&\\quad {}+\\frac{74911}{69120}\\pi ^4-\\frac{148407527}{2073600}\\pi ^2-\\frac{778181617}{9953280}- n_l (12.18 \\pm 0.8)\\nonumber \\\\&&\\quad {}- n_l^2 \\biggl (\\frac{32}{3} a_5 + \\frac{188}{9} a_4 - \\frac{4}{45} a_1^5 + \\frac{47}{54} a_1^4 - \\frac{8}{27} \\pi ^2 a_1^3 + \\frac{47}{27} \\pi ^2 a_1^2 - \\frac{31}{270} \\pi ^4 a_1 - \\frac{239}{54} \\pi ^2 a_1 - \\frac{601}{48} \\zeta _5 - \\frac{\\pi ^2}{2} \\zeta _3 + \\frac{6913}{576} \\zeta _3\\nonumber \\\\&&\\qquad {}- \\frac{1297}{51840} \\pi ^4 + \\frac{25729}{10368} \\pi ^2 - \\frac{15877}{165888}\\biggr )+ \\frac{n_l^3}{216} \\biggl (\\frac{467}{2} \\zeta _3 + \\frac{71}{20} \\pi ^4 + \\frac{167}{3} \\pi ^2 + \\frac{103933}{864}\\biggr )+ \\mathcal {O}(\\varepsilon )\\biggr ]+ \\mathcal {O}(\\alpha ^5)\\,,$ where $\\alpha = \\alpha ^{(n_f)}(M)$ ; $L_{\\text{QED}} = \\sum _{i=0,1,2,3,l} L_i$ is the $\\varepsilon ^0$ term in $Z_2^{(4)}$ of Eq.", "(26) in [14].", "Its numerical value is given in Eq.", "(15) in this paper.", "Numerically, in pure QED ($n_l=0$ ) at $\\varepsilon =0$ we have $&&z(M) = 1 - \\frac{\\alpha }{\\pi }- 1.09991 \\left(\\frac{\\alpha }{\\pi }\\right)^2+ 4.40502 \\left(\\frac{\\alpha }{\\pi }\\right)^3\\nonumber \\\\&&{} - 2.16215 \\left(\\frac{\\alpha }{\\pi }\\right)^4+ \\mathcal {O}(\\alpha ^5)\\,,$ where $\\alpha = \\alpha ^{(1)}(M)$ , the $\\overline{\\text{MS}}$ QED coupling with one active flavor at $\\mu =M$ , the on-shell electron mass.", "In contrast to the QCD case (REF ) the coefficients are numerically smaller and have different signs." ], [ "Conclusion", "We have calculated the (finite) matching coefficient between the QCD heavy-quark field $Q$ and the corresponding HQET field $h_v$ up to four loops.", "Explicit results are presented for $\\mu = M$ ; results for different values of $\\mu $ can be obtained with the help of (known) renormalization group equations.", "The effect of a non-zero light-flavor mass (e. g., $c$ in $b$ -quark HQET) is calculated up to three loops.", "We also present results for the matching constant in QED.", "As a possible application of our results we want to mention the possibility to obtain the QCD heavy-quark propagator (say, in Landau gauge) from lattice QCD results for the HQET propagator.", "A heavy-quark field can be put onto the lattice only if $M a \\ll 1$ , where $a$ is the lattice spacing.", "On the other hand, in HQET simulations there is no lattice $h_v$ field at all.", "The HQET propagator is just a straight Wilson line, i. e. a product of lattice gauge links.", "It is therefore much easier to obtain the HQET propagator from lattice simulations.", "After taking the continuum limit, one can get the continuum coordinate-space HQET propagator.", "Then the QCD heavy-quark propagator can be obtained with the help of the matching coefficient $z(\\mu )$ , provided that $1/M^n$ corrections can be neglected.", "Note that this can be done for arbitrarily heavy QCD quark, including the case when the use of the dynamic heavy-quark field on the lattice is impossible." ], [ "Acknowledgments", "We are grateful to R. N. Lee for discussions of the Appendix .", "This research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762 — TRR 257 “Particle Physics Phenomenology after the Higgs Discovery”.", "This work was supported in part by the EU TMR network SAGEX Marie Skłodowska-Curie grant agreement No.", "764850 and COST action CA16201: Unraveling new physics at the LHC through the precision frontier.", "The work of A. G. was supported by the Russian Ministry of Science and Higher Education." ], [ "The coupling and gluon-field decoupling coefficients", "The $n_l$ -flavor QCD strong coupling constant and gauge parameter are related to the corresponding quantities in the $n_f$ -flavor theory by the decoupling relations $&&\\alpha _s^{(n_l)}(\\mu ) = \\zeta _\\alpha (\\mu ) \\alpha _s^{(n_f)}(\\mu )\\,,\\\\&&1 - \\xi ^{(n_l)}(\\mu ) = \\zeta _A(\\mu ) \\left[1 - \\xi ^{(n_f)}(\\mu )\\right]\\,.\\nonumber $ The decoupling coefficients satisfy the renormalization group equations $\\frac{d \\log \\zeta _\\alpha (\\mu )}{d \\log \\mu } &=&2 \\bigl [\\beta ^{(n_f)}(\\alpha _s^{(n_f)}(\\mu )) - \\beta ^{(n_l)}(\\alpha _s^{(n_l)}(\\mu ))\\bigr ]\\,,\\nonumber \\\\\\frac{d \\log \\zeta _A(\\mu )}{d \\log \\mu } &=&\\gamma _A^{(n_f)}(\\alpha _s^{(n_f)}(\\mu ),\\xi ^{(n_f)}(\\mu ))\\nonumber \\\\&&{} - \\gamma _A^{(n_l)}(\\alpha _s^{(n_l)}(\\mu ),\\xi ^{(n_l)}(\\mu ))\\,.$ It is sufficient to have initial conditions, say, at $\\mu = M$ for solving these equations.", "For the computation of $z(M)$ we need the decoupling coefficients up to $\\alpha _s^3 \\varepsilon $ .", "Up to the order $\\alpha _s^2$ expression exact in $\\varepsilon $ can be found in [41].", "The finite three-loop results have been obtained in [42] in term of $N_c$ and in [43] for an arbitrary color group.", "The $\\alpha _s^3 \\varepsilon $ terms were derived in the course of four-loop calculations [44], [45], [43].", "However, results for an arbitrary color group, including positive powers of $\\varepsilon $ , are not explicitly presented in these publications.", "Therefore, we present them here: $&&\\zeta _\\alpha (M) = 1- \\frac{\\alpha _s}{\\pi } T_F n_h \\frac{\\varepsilon }{9}\\left( \\frac{\\pi ^2}{4} - \\zeta _3 \\varepsilon + \\frac{3}{160} \\pi ^4 \\varepsilon ^2 + \\mathcal {O}(\\varepsilon ^3) \\right)\\nonumber \\\\&&{} - \\left(\\frac{\\alpha _s}{\\pi }\\right)^2 T_F n_h\\biggl \\lbrace \\frac{15}{16} C_F - \\frac{2}{9} C_A+ \\frac{\\varepsilon }{4} \\left[ \\frac{C_F}{4} \\left( \\frac{\\pi ^2}{3} + \\frac{31}{2} \\right) + \\frac{C_A}{9} \\left( \\frac{5}{4} \\pi ^2 + \\frac{43}{3} \\right) \\right]\\nonumber \\\\&&\\quad {} - \\varepsilon ^2 \\left[ \\frac{C_F}{4} \\left( \\frac{\\zeta _3}{3} - \\frac{5}{8} \\pi ^2 - \\frac{223}{16} \\right) + \\frac{C_A}{9} \\left( \\frac{5}{4} \\zeta _3 + \\frac{\\pi ^2}{3} + \\frac{523}{72} \\right) + \\frac{\\pi ^4}{1296} T_F n_h\\right]+ \\mathcal {O}(\\varepsilon ^3) \\biggr \\rbrace \\nonumber \\\\&&{} + \\left(\\frac{\\alpha _s}{\\pi }\\right)^3 T_F n_h\\biggl \\lbrace \\frac{C_F^2}{3} \\left( \\pi ^2 a_1 - \\frac{\\zeta _3}{64} - \\frac{5}{8} \\pi ^2 - \\frac{77}{192} \\right)- \\frac{C_F C_A}{6} \\left( \\pi ^2 a_1 + \\frac{1081}{128} \\zeta _3 - \\frac{\\pi ^2}{3} + \\frac{8321}{864} \\right)\\nonumber \\\\&&\\qquad {} - \\frac{C_A^2}{768} \\left( \\frac{5}{2} \\zeta _3 - \\frac{11347}{27} \\right)\\nonumber \\\\&&\\qquad {}- C_F T_F n_h \\left( \\frac{7}{64} \\zeta _3 + \\frac{\\pi ^2}{9} - \\frac{695}{648} \\right)- \\frac{7}{64} C_A T_F n_h \\left( \\frac{\\zeta _3}{2} - \\frac{35}{81} \\right)+ \\frac{C_F T_F n_l}{18} \\left( \\pi ^2 + \\frac{311}{72} \\right)- \\frac{C_A T_F n_l}{2592}\\nonumber \\\\&&\\quad {} - \\varepsilon \\biggl [ C_F^2 \\left( \\frac{37}{12} a_4 + \\frac{37}{288} a_1^4 + \\frac{251}{288} \\pi ^2 a_1^2 - 2 \\pi ^2 a_1 + \\frac{2759}{576} \\zeta _3 - \\frac{241}{3456} \\pi ^4 + \\frac{439}{384} \\pi ^2 + \\frac{3329}{3456} \\right)\\nonumber \\\\&&\\qquad {} + C_F C_A \\left( \\frac{63}{16} a_4 + \\frac{21}{128} a_1^4 - \\frac{85}{128} \\pi ^2 a_1^2 + \\pi ^2 a_1 + \\frac{2413}{512} \\zeta _3 - \\frac{1391}{23040} \\pi ^4 - \\frac{281}{1728} \\pi ^2 + \\frac{451831}{62208} \\right)\\nonumber \\\\&&\\qquad - \\frac{C_A^2}{96} \\left( 263 a_4 + \\frac{263}{24} a_1^4 - \\frac{263}{24} \\pi ^2 a_1^2 + \\frac{27347}{288} \\zeta _3 - \\frac{1687}{1440} \\pi ^4 - \\frac{1063}{216} \\pi ^2 - \\frac{345115}{1944} \\right)\\nonumber \\\\&&\\qquad {}+ C_F T_F n_h \\left( \\frac{3}{4} a_4 + \\frac{a_1^4}{32} - \\frac{\\pi ^2}{32} a_1^2 - \\frac{2}{3} \\pi ^2 a_1 + \\frac{3353}{1152} \\zeta _3 - \\frac{17}{1920} \\pi ^4 + \\frac{407}{864} \\pi ^2 - \\frac{67037}{15552} \\right)\\nonumber \\\\&&\\qquad {}+ \\frac{C_A T_F n_h}{8} \\left( 3 a_4 + \\frac{a_1^4}{8} - \\frac{\\pi ^2}{8} a_1^2 + \\frac{1799}{864} \\zeta _3 - \\frac{17}{480} \\pi ^4 + \\frac{113}{1296} \\pi ^2 + \\frac{1165}{11664} \\right)\\nonumber \\\\&&\\qquad {}- \\frac{C_F T_F n_l}{9} \\left( \\zeta _3 + \\frac{403}{192} \\pi ^2 + \\frac{24911}{864} \\right)- \\frac{C_A T_F n_l}{27} \\left( 5 \\zeta _3 + \\frac{47}{192} \\pi ^2 - \\frac{6553}{1728} \\right)\\biggr ]+ \\mathcal {O}(\\varepsilon ^2) \\biggr \\rbrace + \\mathcal {O}(\\alpha _s^4)\\,,\\\\&&\\zeta _A(M) = 1+ \\frac{\\alpha _s}{\\pi } T_F n_h \\frac{\\varepsilon }{9}\\left( \\frac{\\pi ^2}{4} - \\zeta _3 \\varepsilon + \\frac{3}{160} \\pi ^4 \\varepsilon ^2 + \\mathcal {O}(\\varepsilon ^3) \\right)\\nonumber \\\\&&{} + \\left(\\frac{\\alpha _s}{\\pi }\\right)^2 T_F n_h\\biggl \\lbrace \\frac{1}{16} \\left( 15 C_F - \\frac{13}{12} C_A \\right)+ \\frac{\\varepsilon }{16} \\left[ C_F \\left( \\frac{\\pi ^2}{3} + \\frac{31}{2} \\right)+ \\frac{C_A}{12} \\left( 5 \\pi ^2 + \\frac{169}{6} \\right) \\right]\\nonumber \\\\&&\\qquad {} - \\frac{\\varepsilon ^2}{4}\\biggl [ C_F \\left( \\frac{\\zeta _3}{3} - \\frac{5}{8} \\pi ^2 - \\frac{223}{16} \\right) + \\frac{C_A}{12} \\left( 5 \\zeta _3 - \\frac{\\pi ^4}{48} + \\frac{13}{24} \\pi ^2 + \\frac{1765}{144} \\right) \\biggr ]+ \\mathcal {O}(\\varepsilon ^3) \\biggr \\rbrace \\nonumber \\\\&&{} - \\left(\\frac{\\alpha _s}{\\pi }\\right)^3 T_F n_h\\biggl \\lbrace \\frac{C_F^2}{3} \\left( \\pi ^2 a_1 - \\frac{\\zeta _3}{64} - \\frac{5}{8} \\pi ^2 - \\frac{77}{192} \\right)\\nonumber \\\\&&\\qquad {} - C_F C_A \\left( 2 a_4 + \\frac{a_1^4}{12} - \\frac{\\pi ^2}{12} a_1^2 + \\frac{\\pi ^2}{6} a_1 + \\frac{1765}{768} \\zeta _3 - \\frac{11}{720} \\pi ^4 - \\frac{\\pi ^2}{18} + \\frac{15977}{20736} \\right)\\nonumber \\\\&&\\qquad {} + C_A^2 \\left( a_4 + \\frac{a_1^4}{24} - \\frac{\\pi ^2}{24} a_1^2 + \\frac{1805}{4608} \\zeta _3 - \\frac{53}{5760} \\pi ^4 + \\frac{7985}{31104}- \\frac{\\xi ^{(n_f)}(M)}{48} \\left( \\zeta _3 - \\frac{677}{144} \\right) \\right)\\nonumber \\\\&&\\qquad {}- C_F T_F n_h \\left( \\frac{7}{64} \\zeta _3 + \\frac{\\pi ^2}{9} - \\frac{695}{648} \\right)- \\frac{C_A T_F n_h}{144} \\left( \\frac{287}{8} \\zeta _3 - \\frac{605}{27} \\right)+ \\frac{C_F T_F n_l}{18} \\left( \\pi ^2 + \\frac{311}{72} \\right)+ \\frac{C_A T_F n_l}{9} \\left( \\zeta _3 - \\frac{665}{432} \\right)\\nonumber \\\\&&\\quad {} - \\varepsilon \\biggl [ C_F^2 \\left( \\frac{37}{12} a_4 + \\frac{37}{288} a_1^4 + \\frac{251}{288} \\pi ^2 a_1^2 - 2 \\pi ^2 a_1 + \\frac{2759}{576} \\zeta _3 - \\frac{241}{3456} \\pi ^4 + \\frac{439}{384} \\pi ^2 + \\frac{3329}{3456} \\right)\\nonumber \\\\&&\\qquad {} + C_F C_A \\biggl ( 12 a_5 + \\frac{179}{16} a_4 - \\frac{a_1^5}{10} + \\frac{179}{384} a_1^4 + \\frac{\\pi ^2}{6} a_1^3 - \\frac{371}{384} \\pi ^2 a_1^2 + \\frac{17}{120} \\pi ^4 a_1 + \\pi ^2 a_1 - \\frac{203}{16} \\zeta _5 + \\frac{\\pi ^2}{32} \\zeta _3 + \\frac{3141}{512} \\zeta _3\\nonumber \\\\&&\\qquad \\quad {} - \\frac{1057}{7680} \\pi ^4 - \\frac{281}{1728} \\pi ^2 + \\frac{1199393}{124416} \\biggr )\\nonumber \\\\&&\\qquad {} - C_A^2 \\biggl ( 6 a_5 + \\frac{611}{96} a_4 - \\frac{a_1^5}{20} + \\frac{611}{2304} a_1^4 + \\frac{\\pi ^2}{12} a_1^3 - \\frac{611}{2304} \\pi ^2 a_1^2 + \\frac{17}{240} \\pi ^4 a_1 - \\frac{185}{32} \\zeta _5 + \\frac{3}{128} \\pi ^2 \\zeta _3 + \\frac{59395}{27648} \\zeta _3\\nonumber \\\\&&\\qquad \\quad {} - \\frac{6679}{138240} \\pi ^4 - \\frac{10181}{165888} \\pi ^2 - \\frac{886909}{373248}+ \\frac{\\xi ^{(n_f)}(M)}{96} \\left( 7 \\zeta _3 + \\frac{\\pi ^4}{10} - \\frac{233}{576} \\pi ^2 - \\frac{5737}{144} \\right) \\biggr )\\nonumber \\\\&&\\qquad {}+ C_F T_F n_h \\left( \\frac{3}{4} a_4 + \\frac{a_1^4}{32} - \\frac{\\pi ^2}{32} a_1^2 - \\frac{2}{3} \\pi ^2 a_1 + \\frac{3353}{1152} \\zeta _3 - \\frac{17}{1920} \\pi ^4 + \\frac{113}{216} \\pi ^2 - \\frac{67037}{15552} \\right)\\nonumber \\\\&&\\qquad {}+ \\frac{C_A T_F n_h}{24} \\left( 41 a_4 + \\frac{41}{24} a_1^4 - \\frac{41}{24} \\pi ^2 a_1^2 + \\frac{5551}{288} \\zeta _3 - \\frac{697}{1440} \\pi ^4 - \\frac{7}{32} \\pi ^2 - \\frac{4415}{1944} \\right)\\nonumber \\\\&&\\qquad {}- \\frac{C_F T_F n_l}{9} \\left( \\zeta _3 + \\frac{403}{192} \\pi ^2 + \\frac{24911}{864} \\right)- \\frac{C_A T_F n_l}{18} \\left( \\frac{5}{3} \\zeta _3 - \\frac{\\pi ^4}{10} + \\frac{253}{576} \\pi ^2 + \\frac{27845}{1296} \\right)\\biggr ] + \\mathcal {O}(\\varepsilon ^2) \\biggr \\rbrace + \\mathcal {O}(\\alpha _s^4)\\,,$ where $\\alpha _s = \\alpha _s^{(n_f)}(M)$ ." ], [ "On-shell diagrams with two masses", "Light-quark mass effects in the heavy-quark on-shell propagator diagrams arise for the first time at two loops, see Fig.", "REF b.", "The corresponding integral family can be defined as $&&I_{n_1 n_2 n_3 n_4} = C\\raisebox {-5mm}{\\begin{picture}(26,12)\\put (13,6){\\makebox{(}0,0){\\includegraphics {top.pdf}}}\\put (13,-1){\\makebox{(}0,0){1}}\\put (6.5,7.5){\\makebox{(}0,0){2}}\\put (20.5,8.5){\\makebox{(}0,0){3}}\\put (15,3){\\makebox{(}0,0){4}}\\end{picture}}\\nonumber \\\\&&{} = \\frac{C}{(i \\pi ^{d/2})^2}\\int \\frac{d^d k_1\\,d^d k_2}{D_1^{n_1} D_2^{n_2} D_3^{n_3} D_4^{n_4}}\\,,\\quad C = \\frac{1}{\\Gamma ^2(1+\\varepsilon )}\\,,\\nonumber \\\\&&D_1 = M^2 - (p + k_1)^2\\,,\\quad D_2 = - k_1^2\\,,\\nonumber \\\\&&D_3 = m^2 - k_2^2\\,,\\quad D_4 = m^2 - (k_1 - k_2)^2\\,,$ with $p^2 = M^2$ .", "If there are insertions to gluon lines in Fig.", "REF b containing only massless lines, such diagrams are expressed via the integrals (REF ) with $n_2 = n + l\\varepsilon $ , where $l$ is the total number of loops in these insertions and $n$ is integer ($n_{1,3,4}$ are always integer).", "These integrals have been studied in [30].", "The IBP algorithm obtained there reduces them to four master integrals $&&I_{0,l\\varepsilon ,1,1} = C\\raisebox {-6mm}{\\begin{picture}(14,14)\\put (7,7){\\makebox{(}0,0){\\includegraphics {j1.pdf}}}\\put (7,9){\\makebox{(}0,0){l\\varepsilon }}\\end{picture}}\\,,\\quad I_{1,l\\varepsilon ,1,0} = C\\raisebox {-7mm}{\\begin{picture}(20,16)\\put (10,7){\\makebox{(}0,0){\\includegraphics {j2.pdf}}}\\put (10,15){\\makebox{(}0,0){l\\varepsilon }}\\end{picture}}\\,,\\nonumber \\\\&&I_{1,l\\varepsilon ,1,1} = C\\raisebox {-5mm}{\\begin{picture}(26,12)\\put (13,6){\\makebox{(}0,0){\\includegraphics {top.pdf}}}\\put (6,7.5){\\makebox{(}0,0){l\\varepsilon }}\\end{picture}}\\,,\\nonumber \\\\&&I_{1,1+l\\varepsilon ,1,1} = C\\raisebox {-5mm}{\\begin{picture}(28,14)\\put (17,6){\\makebox{(}0,0){\\includegraphics {top.pdf}}}\\put (6.5,7.5){\\makebox{(}0,0){1+l\\varepsilon }}\\end{picture}}\\,.$ We set $M=1$ and $m=x$ .", "It is more convenient to use the column vector $j = \\bigl (I_{0,l\\varepsilon ,2,2},I_{2,l\\varepsilon ,2,0},I_{2,l\\varepsilon ,2,1},I_{1,l\\varepsilon ,2,2}\\bigr )^T$ as master integrals instead of (REF ).", "Differentiating them in $m$ and reducing the results back to $j$  [46], we obtain the differential equations $\\frac{d j}{d x} = M(\\varepsilon ,x) j\\,.$ In many cases such equations can be reduced to an $\\varepsilon $ -form [47] $j = T(\\varepsilon ,x) J\\,,\\quad \\frac{d J}{d x} = \\varepsilon M(x) J\\,.$ This makes their iterative solution to any order in $\\varepsilon $ almost trivial.", "Several terms of small-$x$ and large-$x$ expansions of these integrals (with $l=0$ ) were obtained in [48] using the method of regions (though expressed in a somewhat different language).", "Differential equations for on-shell sunsets $I_{n_1,0,n_3,n_4}$ were considered in [49], [33], but they were not in $\\varepsilon $ -form.", "Several terms of small-$x$ expansions were obtained from differential equations in [50].", "However, the easiest way to obtain any finite number of terms in the small-$x$ and large-$x$ expansions is neither the method of regions nor differential equations, but calculating the corresponding residues in the Mellin–Barnes representation [30].", "We use the Mathematica package Libra [51] which implements the algorithm of [52] to reduce the master integrals $j$ in Eq.", "(REF ) to a canonical basis $J$ : $j_1 &=& I_{0,l\\varepsilon ,2,2} = C V_{2,2,l\\varepsilon } x^{-2(l+2)\\varepsilon }= \\frac{2 (1-(l+1)\\varepsilon )}{(l+2) (1-\\varepsilon )} J_1\\,,\\nonumber \\\\j_2 &=& I_{0,l\\varepsilon ,2,0} = C V_2 M_{2,l\\varepsilon } x^{-2\\varepsilon }= \\frac{1-2(l+1)\\varepsilon }{1-(l+2)\\varepsilon } J_2\\,,\\nonumber \\\\j_3 &=& - \\frac{1}{2} \\left(J_3 + J_4\\right)\\,,\\nonumber \\\\j_4 &=& I_{1,l\\varepsilon ,2,2} = \\frac{1}{2x}\\biggl \\lbrace - \\biggl [1-2x - \\frac{2l\\varepsilon (1-x)}{1-2\\varepsilon }\\biggr ] J_3\\nonumber \\\\&&{} + \\biggl [1+2x - \\frac{2l\\varepsilon (1+x)}{1-2\\varepsilon }\\biggr ] J_4\\biggr \\rbrace \\,,$ where $&&V_{n_1} = \\raisebox {-4.5mm}{\\begin{picture}(10,11.5)\\put (5,5){\\makebox{(}0,0){\\includegraphics {v5.pdf}}}\\put (5,10.5){\\makebox{(}0,0){1}}\\end{picture}}= \\frac{\\Gamma \\bigl (\\frac{d}{2}-n_1\\bigr )}{\\Gamma (n_1)}\\,,\\\\&&V_{n_1 n_2 n_3} = \\raisebox {-8mm}{\\begin{picture}(14,17)\\put (7,8.5){\\makebox{(}0,0){\\includegraphics {v6.pdf}}}\\put (7,1){\\makebox{(}0,0){1}}\\put (7,16){\\makebox{(}0,0){2}}\\put (7,10){\\makebox{(}0,0){3}}\\end{picture}} = H_{0,n_3,n_1,n_2}\\,,\\\\&&M_{n_1 n_2}= \\raisebox {-4mm}{\\begin{picture}(22,9)\\put (11,4.5){\\makebox{(}0,0){\\includegraphics {s6.pdf}}}\\put (11,1){\\makebox{(}0,0){1}}\\put (11,8){\\makebox{(}0,0){2}}\\end{picture}}\\nonumber \\\\&&{} = \\frac{\\Gamma \\bigl (n_1+n_2-\\frac{d}{2}\\bigr ) \\Gamma (d-n_1-2n_2)}{\\Gamma (n_1) \\Gamma (d-n_1-n_2)}\\,,$ and [33] $&&H_{n_1 n_2 n_3 n_4} =\\raisebox {-5mm}{\\begin{picture}(26,12)\\put (13,6){\\makebox{(}0,0){\\includegraphics {h2.pdf}}}\\put (13,-1){\\makebox{(}0,0){1}}\\put (6.5,7.5){\\makebox{(}0,0){2}}\\put (20.5,8.5){\\makebox{(}0,0){3}}\\put (15,3){\\makebox{(}0,0){4}}\\end{picture}} =\\nonumber \\\\&&\\frac{\\Gamma (n_1/2) \\Gamma ((n_1-d)/2+n_2+n_3) \\Gamma ((n_1-d)/2+n_2+n_4)}{2 \\Gamma (n_1) \\Gamma (n_3) \\Gamma (n_4)}\\nonumber \\\\&&{}\\times \\frac{\\Gamma (n_1/2+n_2+n_3+n_4-d) \\Gamma ((d-n_1)/2-n_2)}{\\Gamma (n_1+2n_2+n_3+n_4-d) \\Gamma ((d-n_1)/2)}\\,.$ The integrals $J$ satisfy the $\\varepsilon $ -form differential equations $\\frac{d J}{d x} = \\varepsilon \\left(\\frac{M_0}{x} + \\frac{M_{+1}}{1-x} + \\frac{M_{-1}}{1+x} \\right) J\\,,$ where $&&M_0 =\\left(\\begin{array}{cccc}- 2 (l+2) & 0 & 0 & 0 \\\\0 & -2 & 0 & 0 \\\\1 & -1 & -(l+2) & l+2 \\\\1 & -1 & l+2 & -(l+2)\\end{array}\\right)\\,,\\nonumber \\\\&&M_{+1} =\\left(\\begin{array}{cccc}0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\1 & -1 & 2 & 2 (l+2) \\\\0 & 0 & 0 & 0\\end{array}\\right)\\,,\\nonumber \\\\&&M_{-1} =\\left(\\begin{array}{cccc}0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\-1 & 1 & -2 (l+2) & -2\\end{array}\\right)\\,.$ The first two are, of course, known exactly: $J_1 &=& \\frac{x^{-2 (l+2) \\varepsilon }}{(l+1) \\varepsilon ^2} \\times \\nonumber \\\\&&\\frac{\\Gamma (1 - (l+1) \\varepsilon ) \\Gamma ^2(1 + (l+1) \\varepsilon ) \\Gamma (1 + (l+2) \\varepsilon )}{\\Gamma (1-\\varepsilon ) \\Gamma ^2(1+\\varepsilon ) \\Gamma (1 + 2 (l+1) \\varepsilon )}\\,,\\nonumber \\\\J_2 &=& \\frac{x^{-2\\varepsilon }}{(l+1) \\varepsilon ^2}\\frac{\\Gamma (1 - 2 (l+1) \\varepsilon ) \\Gamma (1 + (l+1) \\varepsilon )}{\\Gamma (1 - (l+2) \\varepsilon ) \\Gamma (1+\\varepsilon )}\\,.$ The equations for $J_{3,4}$ can be solved iteratively in terms of harmonic polylogarithms [53] of $x$ .", "However, we need initial conditions.", "They can be fixed using the asymptotics of $I_{n_1 n_2 n_3 n_4}$ at $x \\rightarrow 0$ .", "It is given by contributions of three regions (Sect. )", "corresponding to residues of the Mellin–Barnes representation [30] at three series of poles: Hard: the poles $s=-n-n_3-n_4+d/2$ ($n\\ge 0$ is integer), the result is a regular series in $x^2$ .", "The leading term is $C G_{n_3 n_4} M_{n_1,n_2+n_3+n_4-d/2}$ , where $&&G_{n_1 n_2}= \\raisebox {-6mm}{\\begin{picture}(26,13)\\put (13,6.5){\\makebox{(}0,0){\\includegraphics {s5.pdf}}}\\put (13,12){\\makebox{(}0,0){2}}\\put (13,1){\\makebox{(}0,0){1}}\\end{picture}}\\nonumber \\\\&&{} = \\frac{\\Gamma \\bigl (n_1+n_2-\\frac{d}{2}\\bigr ) \\Gamma \\bigl (\\frac{d}{2}-n_1\\bigr ) \\Gamma \\bigl (\\frac{d}{2}-n_2\\bigr )}{\\Gamma (n_1) \\Gamma (n_2) \\Gamma (d-n_1-n_2)}\\,.$ Sort-hard: $s=-n-n_{3,4}$ .", "All these poles are double except the first $|n_3-n_4|$ ones (and hence, the representation of $I_{n_1 n_2 n_3 n_4}$ via hypergeometric functions of $x$ is awkward).", "We assume $n_3\\ge n_4$ , then the result is $x^{d-2n_3}$ times a regular series in $x^2$ .", "If $n_3>n_4$ then the leading term is $C V_{n_3} M_{n_1,n_2+n_4} x^{d-2n_3}$ ; if $n_3=n_4$ there is an extra factor 2 because each of the lines 3, 4 can be soft.", "Soft: $s=(n_1-d-n)/2+n_2$ , the result is $x^{2(d-n_2-n_3-n_4)-n_1}$ times a regular series in $x$ .", "The leading term is $H_{n_1 n_2 n_3 n_4} x^{2(d-n_2-n_3-n_4)-n_1}$  (REF ).", "For example, $&&I_{2,l\\varepsilon ,2,1} \\rightarrow \\frac{1}{2 \\varepsilon ^2} \\times \\nonumber \\\\&&\\biggl [ \\frac{1}{l+2}\\frac{\\Gamma ^2(1-\\varepsilon ) \\Gamma (1-2(l+2)\\varepsilon ) \\Gamma (1+(l+2)\\varepsilon )}{\\Gamma (1-2\\varepsilon ) \\Gamma (1-(l+3)\\varepsilon ) \\Gamma (1+\\varepsilon )}\\nonumber \\\\&&{} - \\frac{x^{-2\\varepsilon }}{l+1}\\frac{\\Gamma (1-2(l+1)\\varepsilon ) \\Gamma (1+(l+1)\\varepsilon )}{\\Gamma (1-(l+2)\\varepsilon ) \\Gamma (1+\\varepsilon )}\\nonumber \\\\&&{} + \\frac{x^{-2(l+2)\\varepsilon }}{(l+1) (l+2)} \\times \\nonumber \\\\&&\\frac{\\Gamma (1-(l+1)\\varepsilon ) \\Gamma ^2(1+(l+1)\\varepsilon ) \\Gamma (1+(l+2)\\varepsilon )}{\\Gamma (1-\\varepsilon ) \\Gamma ^2(1+\\varepsilon ) \\Gamma (1+2(l+1)\\varepsilon )}\\biggr ]\\,,\\nonumber \\\\$ where the 3 contributions are the hard one $C G_{21} M_{2,1+(l+1)\\varepsilon }$ (the pole $s=-1-\\varepsilon $ ), the soft-3 one $C V_2 M_{2,1+l\\varepsilon }$ (the pole $s=-1$ ), and the soft one $C H_{2,l\\varepsilon ,2,1} x^{-2(l+2)\\varepsilon }$ (the pole $s=-1-(l+1)\\varepsilon $ ).", "The leading asymptotics of $I_{1,l\\varepsilon ,2,2}$ is given by the soft contribution $C H_{1,l\\varepsilon ,2,2} x^{-1-2(l+2)\\varepsilon }$ (the pole $s=-3/2+(l+1)\\varepsilon $ ): $&&I_{1,l\\varepsilon ,2,2} \\rightarrow 2^{-1-4(l+2)\\varepsilon } \\pi ^2 x^{-1-2(l+2)\\varepsilon }\\frac{1-2(l+1)\\varepsilon }{1-2\\varepsilon }\\nonumber \\\\&&{}\\times \\frac{\\Gamma (1-\\varepsilon ) \\Gamma (1-2(l+1)\\varepsilon )}{\\Gamma (1-2\\varepsilon ) \\Gamma (1-(l+1)\\varepsilon )}\\nonumber \\\\&&{}\\times \\frac{\\Gamma (1+2(l+1)\\varepsilon ) \\Gamma (1+2(l+2)\\varepsilon )}{\\Gamma ^2(1+\\varepsilon ) \\Gamma ^2(1+(l+1)\\varepsilon ) \\Gamma (1+(l+2)\\varepsilon )}\\,.$ Now we can easily obtain any number of expansion terms of $J_{3,4}$ in $\\varepsilon $ for $x < 1$ using Libra [51]: $J_3 &=& - 2 \\biggl \\lbrace H_{1,0}(x) + H_{0,0}(x) + \\frac{\\pi ^2}{3}+ \\biggl [ - (l+2) \\bigl (2 H_{1,-1,0}(x) + H_{0,1,0}(x) + H_{0,-1,0}(x)\\bigr )+ 2 H_{1,1,0}(x)\\nonumber \\\\&&\\quad {} - 2 (l+3) H_{0,0,0}(x)- l \\frac{\\pi ^2}{6} H_{1}(x) - (l+3) \\frac{\\pi ^2}{3} H_{0}(x)+ \\frac{1}{2} (3l+2) \\zeta _3 - (l+2) \\pi ^2 a_1\\biggr ] \\varepsilon \\nonumber \\\\&&{} + 2 \\biggl [(l+2)^2 \\bigl (- 2 H_{1,-1,1,0}(x) + H_{0,1,-1,0}(x) - H_{0,-1,1,0}(x) + H_{0,0,1,0}(x) + H_{0,0,-1,0}(x)\\bigr )\\nonumber \\\\&&\\quad {} + (l+1) (l+2) \\bigl (H_{1,0,1,0}(x) + H_{1,0,-1,0}(x)\\bigr )\\nonumber \\\\&&\\quad {} + (l+2) \\bigl (- 2 H_{1,1,-1,0}(x) + 2 H_{1,-1,-1,0}(x) - H_{0,1,1,0}(x) + H_{0,-1,-1,0}(x) - 2 H_{1,0,0,0}(x)\\bigr )+ 2 H_{1,1,1,0}(x)\\nonumber \\\\&&\\quad {} + 2 (l^2+5l+7) H_{0,0,0,0}(x)+ \\frac{\\pi ^2}{12} \\bigl [- 2 l H_{1,1}(x) - (l+2) (5l+6) \\bigl (2 H_{1,-1}(x) + H_{0,-1}(x)\\bigr )\\nonumber \\\\&&\\qquad {}+ (8l^2+23l+12) H_{1,0}(x)+ l (l+2) H_{0,1}(x) + (6l^2+23l+24) H_{0,0}(x)\\bigr ]+ \\frac{1}{2} (l+3) (3l+2) \\zeta _3 H_{1}(x)\\nonumber \\\\&&\\quad {} + (l+2) \\pi ^2 a_1 \\bigl [(l+1) H_{1}(x) + (l+2) H_{0}(x)\\bigr ]+ (l+2)^2 \\pi ^2 a_1^2 + (84l^2+227l+122) \\frac{\\pi ^4}{720}\\biggr ] \\varepsilon ^2 \\biggr \\rbrace + \\mathcal {O}(\\varepsilon ^3)\\,,\\nonumber \\\\J_4 &=& - 2 \\biggl \\lbrace - H_{-1,0}(x) + H_{0,0}(x) - \\frac{\\pi ^2}{6}+ \\biggl [ - (l+2) \\bigl (2 H_{-1,1,0}(x) - H_{0,-1,0}(x) - H_{0,1,0}(x)\\bigr )+ 2 H_{-1,-1,0}(x)\\nonumber \\\\&&\\quad {} - 2 (l+3) H_{0,0,0}(x)- (5l+6) \\frac{\\pi ^2}{6} H_{-1}(x) + (2l+3) \\frac{\\pi ^2}{3} H_{0}(x)+ \\frac{1}{2} (3l+2) \\zeta _3 + (l+2) \\pi ^2 a_1\\biggr ] \\varepsilon \\nonumber \\\\&&{} + 2 \\biggl [(l+2)^2 \\bigl (2 H_{-1,1,-1,0}(x) + H_{0,-1,1,0}(x) - H_{0,1,-1,0}(x) - H_{0,0,-1,0}(x) - H_{0,0,1,0}(x)\\bigr )\\nonumber \\\\&&\\quad {} + (l+1) (l+2) \\bigl (H_{-1,0,-1,0}(x) + H_{-1,0,1,0}(x)\\bigr )\\nonumber \\\\&&\\quad {} + (l+2) \\bigl (2 H_{-1,-1,1,0}(x) - 2 H_{-1,1,1,0}(x) - H_{0,-1,-1,0}(x) + H_{0,1,1,0}(x) + 2 H_{-1,0,0,0}(x)\\bigr )- 2 H_{-1,-1,-1,0}(x)\\nonumber \\\\&&\\quad {} + 2 (l^2+5l+7) H_{0,0,0,0}(x)+ \\frac{\\pi ^2}{12} \\bigl [2 (5l+6) H_{-1,-1}(x) + l (l+2) \\bigl (2 H_{-1,1}(x) - H_{0,1}(x)\\bigr )\\nonumber \\\\&&\\qquad {}+ (4l^2+13l+12) H_{-1,0}(x)+ (l+2) (5l+6) H_{0,-1}(x) - (2l+3) (3l+8) H_{0,0}(x)\\bigr ]- \\frac{1}{2} (l+3) (3l+2) \\zeta _3 H_{-1}(x)\\nonumber \\\\&&\\quad {} + (l+2) \\pi ^2 a_1 \\bigl [(l+1) H_{-1}(x) - (l+2) H_{0}(x)\\bigr ]- (l+2)^2 \\pi ^2 a_1^2 - (36l^2+103l+58) \\frac{\\pi ^4}{720}\\biggr ] \\varepsilon ^2 \\biggr \\rbrace + \\mathcal {O}(\\varepsilon ^3)\\,.$ Up to order $\\varepsilon ^1$ all harmonic polylogarithms can be transformed to logarithms and ordinary polylogarithms up to $\\mathop {\\mathrm {Li}}\\nolimits _{3}$ , e. g., using the Mathematica package HPL [54], [55].", "Next we consider the case $x>1$ .", "We can re-write the differential equation (REF ) in the form $&&\\frac{d J}{d x^{-1}} = \\varepsilon \\times {}\\\\&&\\left( \\frac{- M_0 + M_{+1} - M_{-1}}{x^{-1}} + \\frac{M_{+1}}{1 - x^{-1}} + \\frac{M_{-1}}{1 + x^{-1}}\\right) J\\,.\\nonumber $ It can be solved in terms of harmonic polylogarithms of $x^{-1}$ , this is convenient for $x>1$ .", "We use the asymptotics $x \\rightarrow +\\infty $ for boundary conditions.", "There are 2 regions: All lines in (REF ) are hard (momenta of order $m$ ).", "This corresponds to the series of right poles in the Mellin–Barnes representation $s=n+n_1+n_2-d/2$ , i. e., to the first term in the hypergeometric representation (A1) in [30], and gives $x^{2(d-n_1-n_2-n_3-n_4)}$ times a regular series in $x^{-2}$ .", "The leading contribution to $I_{n_1 n_2 n_3 n_4}$ is $C V_{n_3,n_4,n_1+n_2} x^{2(d-n_1-n_2-n_3-n_4)}$ .", "Lines 1, 2 are soft (momenta of order $M$ ).", "This corresponds to right poles at $s=n$ , i. e., to the second hypergeometric term, and gives $x^{d-2(n_3+n_4)}$ times a regular series in $x^{-2}$ .", "The leading asymptotics is $C M_{n_1 n_2} V_{n_3+n_4} x^{d-2(n_3+n_4)}$ .", "For $I_{2,l\\varepsilon ,2,1}$ these two contributions are $\\sim x^{-2-2\\varepsilon }$ and $\\sim x^{-2-(l+2)\\varepsilon }$ ; for $I_{1,l\\varepsilon ,2,2}$ the leading contribution is hard, $\\sim x^{-2-(l+1)\\varepsilon }$ .", "This information is sufficient for solving the differential equations for $x>1$ using Libra [51]: $J_3 &=& - 2 \\biggl \\lbrace - H_{1,0}(x^{-1})+ \\biggl [- (l+4) H_{0,1,0}(x^{-1}) - 2 (l+3) H_{1,0,0}(x^{-1})+ (l+2) \\bigl (2 H_{1,-1,0}(x^{-1}) + H_{0,-1,0}(x^{-1})\\bigr )\\nonumber \\\\&&\\quad {} - 2 H_{1,1,0}(x^{-1})+ l \\frac{\\pi ^2}{6} H_{1}(x^{-1})\\biggr ] \\varepsilon \\nonumber \\\\&&{} + 2 \\biggl [(l+2)^2 \\bigl (2 H_{1,-1,1,0}(x^{-1}) + H_{0,-1,1,0}(x^{-1})\\bigr )+ (l+2) (l+5) H_{1,0,-1,0}(x^{-1})\\nonumber \\\\&&\\quad {} + (l+2) (l+4) \\bigl (H_{0,1,-1,0}(x^{-1}) + H_{0,0,-1,0}(x^{-1})\\bigr )+ (l+2) (l+3) \\bigl (2 H_{1,-1,0,0}(x^{-1}) + H_{0,-1,0,0}(x^{-1})\\bigr )\\nonumber \\\\&&\\quad {} - (l+3) (l+4) H_{0,1,0,0}(x^{-1})- (l^2+6l+10) H_{0,0,1,0}(x^{-1}) - (l^2+5l+8) H_{1,0,1,0}(x^{-1})\\nonumber \\\\&&\\quad {} - 2 (l^2+5l+7) H_{1,0,0,0}(x^{-1})- (l+4) H_{0,1,1,0}(x^{-1}) - 2 (l+3) H_{1,1,0,0}(x^{-1})\\nonumber \\\\&&\\quad {} + (l+2) \\bigl (2 H_{1,1,-1,0}(x^{-1}) - 2 H_{1,-1,-1,0}(x^{-1}) - H_{0,-1,-1,0}(x^{-1})\\bigr )- 2 H_{1,1,1,0}(x^{-1})\\nonumber \\\\&&\\quad {} + l \\frac{\\pi ^2}{12} \\bigl [(l+4) H_{0,1}(x^{-1})- (l+2) \\bigl (2 H_{1,-1}(x^{-1}) + H_{0,-1}(x^{-1})\\bigr )+ 2 H_{1,1}(x^{-1}) + H_{1,0}(x^{-1})\\bigr ]\\biggr ] \\varepsilon ^2 \\biggr \\rbrace + \\mathcal {O}(\\varepsilon ^3)\\,,\\nonumber \\\\J_4 &=& - 2 \\biggl \\lbrace H_{-1,0}(x^{-1})+ \\biggl [(l+4) H_{0,-1,0}(x^{-1}) + 2 (l+3) H_{-1,0,0}(x^{-1})+ (l+2) \\bigl (2 H_{-1,1,0}(x^{-1}) - H_{0,1,0}(x^{-1})\\bigr )\\nonumber \\\\&&\\quad {} - 2 H_{-1,-1,0}(x^{-1})- l \\frac{\\pi ^2}{6} H_{-1}(x^{-1})\\biggr ] \\varepsilon \\nonumber \\\\&&{} + 2 \\biggl [(l+2)^2 \\bigl (- 2 H_{-1,1,-1,0}(x^{-1}) + H_{0,1,-1,0}(x^{-1})\\bigr )+ (l+2) (l+5) H_{-1,0,1,0}(x^{-1})\\nonumber \\\\&&\\quad {} + (l+2) (l+4) \\bigl (H_{0,-1,1,0}(x^{-1}) - H_{0,0,1,0}(x^{-1})\\bigr )+ (l+2) (l+3) \\bigl (2 H_{-1,1,0,0}(x^{-1}) - H_{0,1,0,0}(x^{-1})\\bigr )\\nonumber \\\\&&\\quad {} + (l+3) (l+4) H_{0,-1,0,0}(x^{-1})+ (l^2+6l+10) H_{0,0,-1,0}(x^{-1}) - (l^2+5l+8) H_{-1,0,-1,0}(x^{-1})\\nonumber \\\\&&\\quad {} + 2 (l^2+5l+7) H_{-1,0,0,0}(x^{-1})- (l+4) H_{0,-1,-1,0}(x^{-1}) - 2 (l+3) H_{-1,-1,0,0}(x^{-1})\\nonumber \\\\&&\\quad {} - (l+2) \\bigl (2 H_{-1,-1,1,0}(x^{-1}) - 2 H_{-1,1,1,0}(x^{-1}) + H_{0,1,1,0}(x^{-1})\\bigr )+ 2 H_{-1,-1,-1,0}(x^{-1})\\nonumber \\\\&&\\quad {} + l \\frac{\\pi ^2}{12} \\bigl [- (l+4) H_{0,-1}(x^{-1})- (l+2) \\bigl (2 H_{-1,1}(x^{-1}) - H_{0,1}(x^{-1})\\bigr )+ 2 H_{-1,-1}(x^{-1}) - H_{-1,0}(x^{-1})\\bigr ]\\biggr ] \\varepsilon ^2 \\biggr \\rbrace \\nonumber \\\\&&{} + \\mathcal {O}(\\varepsilon ^3)\\,.$ This is, of course, the analytical continuation of (REF ) to $x>1$ .", "The same results (REF ) can be obtained if we express $J_{3,4}$ via $I_{2,l\\varepsilon ,2,1}$ and $I_{1,l\\varepsilon ,2,2}$ using (REF ) and expand the hypergeometric representations (see Eq.", "(A1) in [30]) of these two integrals in $\\varepsilon $ using the Mathematica package HypExp [56], [57].", "However, solving the differential equations (REF ) up to higher orders in $\\varepsilon $ is simpler than expanding hypergeometric functions.", "Both (REF ) and (REF ) lead to identical results at $x=1$ : $&&J_3(1) = - \\frac{\\pi ^2}{3} + \\frac{1}{2} (l+2) \\bigl (2 \\pi ^2 a_1 - 7 \\zeta _3\\bigr ) \\varepsilon \\nonumber \\\\&&{} - \\biggl [ (l+2) (l+3) \\biggl (8 a_4 + \\frac{1}{3} a_1^4 + \\frac{2}{3} \\pi ^2 a_1^2\\biggr )\\nonumber \\\\&&\\quad {} + (17l^2-36l-124) \\frac{\\pi ^4}{360}\\biggr ] \\varepsilon ^2 + \\mathcal {O}(\\varepsilon ^3)\\,,\\nonumber \\\\&&J_4(1) = \\frac{\\pi ^2}{6} - \\frac{1}{2} \\bigl (2 \\pi ^2 a_1 - 7 \\zeta _3\\bigr ) \\varepsilon \\nonumber \\\\&&{} + \\biggl [ (l+3) \\biggl (8 a_4 + \\frac{1}{3} a_1^4 + \\frac{2}{3} \\pi ^2 a_1^2\\biggr )\\nonumber \\\\&&\\quad {} + (24l^2+27l-62) \\frac{\\pi ^4}{360}\\biggr ] \\varepsilon ^2 + \\mathcal {O}(\\varepsilon ^3)\\,.$ If $l=0$ and $x=1$ , we obviously have $I_{1022}(1) = I_{2021}(1)$ , and hence $J_3(1) = - 2 J_4(1) = - 4 I_{2021}(1)\\,.$ Expanding the hypergeometric representation [30] of $I_{2021}$ (or $I_{1022}$ ) at $x=1$ in $\\varepsilon $ we get (REF ) with $l=0$ .", "Alternatively, we can use another hypergeometric representation [8], [9].", "Using integration by parts we obtain $&&I_{2021}(1) = \\frac{7}{32 \\varepsilon ^2} \\biggl [\\frac{\\Gamma (1-\\varepsilon ) \\Gamma ^2(1+2\\varepsilon ) \\Gamma (1+3\\varepsilon )}{\\Gamma ^2(1+\\varepsilon ) \\Gamma (1+4\\varepsilon )}- 1\\biggr ]\\nonumber \\\\&&{} + \\frac{2^{-2-6\\varepsilon } \\pi ^2}{3}\\frac{\\Gamma ^3(1+2\\varepsilon ) \\Gamma (1+3\\varepsilon )}{\\Gamma ^5(1+\\varepsilon ) \\Gamma ^2(1+2\\varepsilon )}+ \\frac{3}{4} \\varepsilon ^2 B_4(\\varepsilon )\\,,$ where $B_4(\\varepsilon )$ is given by the formulas (41), (43) in [9].", "This leads to the same result.", "The functions $L_\\mp (x) = - \\frac{1}{2} J_{3,4}(l=0,\\varepsilon =0)$ were used in [7], [8], [22], [30].", "In addition to the two expressions for these functions in (REF ) and (REF ), several additional representations can be found in [30].", "The results (REF ) and (REF ) are expansions in $\\varepsilon $ where the coefficients are exact functions of $x$ .", "On the other hand, it is straightforward to obtain expansions of $J_{3,4}$ in $x$ (or $x^{-1}$ ) to any finite order using residues of left (or right) poles in the Mellin–Barnes representations of the integrals $j_{3,4}$  (REF ), the coefficients being exact functions of $\\varepsilon $ .", "If we expand them in $\\varepsilon $ , they should agree with expansions of (REF ) in $x$ and of (REF ) in $x^{-1}$ .", "We have checked this up to rather high degrees of $x$ and $x^{-1}$ .", "Now we can find all contributions to $Z_j^{\\text{os}}$ ($j=M$ , $Q$ ) with the maximum number of quark loops, at most one of which is massive, to all orders exactly in $\\varepsilon $ : $&&Z_j^{\\text{os}} = 1 + C_F \\sum _l T_F^{l-1} (n_0 P)^{l-2}\\Bigl [n_0 P B_{j0}^{(l)}\\\\&&{} + (l-1) \\sum _i B_j^{(l)}(x_i)\\Bigr ]\\left(\\frac{g_0^2 M^{-2\\varepsilon }}{(4\\pi )^{d/2}} \\Gamma (\\varepsilon )\\right)^l + \\cdots \\,,\\nonumber $ where $g_0 \\equiv g_0^{(n_f)}$ , $n_0$ is the number of massless flavors, the sum runs over all massive flavors with $x_i = m_i/M$ (including the external flavor with $x=1$ ) and dots refer to other color structures.", "Here $&&P = - 4 \\frac{1-\\varepsilon }{(1-2\\varepsilon ) (3-2\\varepsilon )}\\frac{\\Gamma ^2(1-\\varepsilon )}{\\Gamma (1-2\\varepsilon )}\\,,\\nonumber \\\\&&B_{M0}^{(l)} = - 2 \\frac{(3-2\\varepsilon ) (1-l\\varepsilon )}{l (1-(l+1)\\varepsilon ) (2-(l+1)\\varepsilon )}\\frac{\\Gamma (1+l\\varepsilon ) \\Gamma (1-2l\\varepsilon )}{\\Gamma (1+\\varepsilon ) \\Gamma (1-(l+1)\\varepsilon )}\\,,\\quad B_{Q0}^{(l)} = B_{M0}^{(l)} (1+(l-1)\\varepsilon )\\,,\\nonumber \\\\&&B_M^{(l)}(x) = 2 p_0 \\biggl \\lbrace - 2 \\frac{1-\\varepsilon }{l} \\biggl [1-l\\varepsilon + l\\varepsilon \\frac{1-(l-1)\\varepsilon }{1+(l-1)\\varepsilon } x^2\\biggr ] J_1^{(l-2)}(x)+ \\biggl [p_1 + 2 \\varepsilon \\frac{1 + (2l-3)\\varepsilon - (l-1) (l+2) \\varepsilon ^2}{1+(l-1)\\varepsilon } x^2\\biggr ] J_2^{(l-2)}(x)\\nonumber \\\\&&{} - (p_1 (1+x^2) + p_2 x) (1-x)^2 J_3^{(l-2)}(x)- (p_1 (1+x^2) - p_2 x) (1+x)^2 J_4^{(l-2)}(x)\\biggr \\rbrace \\,,\\nonumber \\\\&&B_Q^{(l)}(x) = p_0 \\biggl \\lbrace - \\frac{2 \\varepsilon }{l (1-\\varepsilon ) (1+2(l-1)\\varepsilon ) (3+2(l-1)\\varepsilon )}\\nonumber \\\\&&{}\\times \\biggl [(1+(l-1)\\varepsilon )(19l-3-(11l^2+50l-11)\\varepsilon -2(4l^3-20l^2-15l+6)\\varepsilon ^2+4(4l^3-11l^2+2l+1)\\varepsilon ^3-8l(l-1)^2\\varepsilon ^4)\\nonumber \\\\&&\\quad {} + \\frac{l (1-\\varepsilon ) (1+2(l-1)\\varepsilon ) (3+2(l-1)\\varepsilon )}{1+(l-1)\\varepsilon }(4-(3l+1)\\varepsilon -(l-1)(l-7)\\varepsilon ^2-4(l-1)\\varepsilon ^3) x^2\\biggr ] J_1^{(l-2)}(x)\\nonumber \\\\&&{} + 2 \\biggl [2 (1-\\varepsilon ) (1+(l-1)\\varepsilon ) (1-l\\varepsilon )+ \\varepsilon \\frac{4+(11l-15)\\varepsilon -(l-1)(l+17)\\varepsilon ^2-2(l-1)(l^2-3)\\varepsilon ^3}{1+(l-1)\\varepsilon } x^2\\biggr ] J_2^{(l-2)}(x)\\nonumber \\\\&&{} + (p_3 + p_4 x + p_5 x^2 + p_6 x^3) (1-x) J_3^{(l-2)}(x)+ (p_3 - p_4 x + p_5 x^2 - p_6 x^3) (1+x) J_4^{(l-2)}(x) \\biggr \\rbrace \\,,$ where $&&p_0 = \\frac{2 \\varepsilon ^2}{(1-2\\varepsilon ) (1-(l+1)\\varepsilon ) (2-(l+1)\\varepsilon )}\\,,\\\\&&p_1 = 2 (1-\\varepsilon ) (1-l\\varepsilon )\\,,\\\\&&p_2 = \\frac{(1-(l+1)\\varepsilon ) (2+(l-3)\\varepsilon -2(l-1)\\varepsilon ^2)}{1+(l-1)\\varepsilon }\\,,\\\\&&p_3 = - 4 (1-\\varepsilon ) (1+(l-1)\\varepsilon ) (1-l\\varepsilon )\\,,\\\\&&p_4 = \\frac{2+(3l-5)\\varepsilon -(l-1)(5l-1)\\varepsilon ^2+4(l-1)^2\\varepsilon ^3(2-\\varepsilon )}{1+(l-1)\\varepsilon }\\,,\\\\&&p_5 = [2-(5l+13)\\varepsilon +(l^2-6l+29)\\varepsilon ^2\\\\&&\\quad {}+2(l^3-l^2+9l-13)\\varepsilon ^3-8(l-1)\\varepsilon ^4]/[1+(l-1)\\varepsilon ]\\,,\\\\&&p_6 = 4 (1-\\varepsilon ) (3-2\\varepsilon ) (1-l\\varepsilon )\\,.$ The results (REF ) at $l=2$ agree with [30] exactly in $\\varepsilon $ .", "Note that $\\lim _{x\\rightarrow 0} B_M^{(l)}(x) = B_{M0}^{(l)} P\\,,$ so that $Z_M^{\\text{os}}$ has a smooth limit $x\\rightarrow 0$ ; this is not so for $Z_Q^{\\text{os}}$ .", "The contribution of these color structures to the ratio of the $\\overline{\\text{MS}}$ mass and the on-shell one $z_m(\\mu ) = M(\\mu )/M$ can be written as $z_m(M) = z_m^{(\\beta _0)} + \\sum _i \\Delta _m(x_i) + \\cdots \\,,$ where $z_m^{(\\beta _0)}$ is the well-known large-$\\beta _0$ result [58] $&&z_m^{(\\beta _0)} = 1 + \\frac{1}{2} \\int _0^b \\frac{d b}{b}\\left(\\frac{\\gamma _m(b)}{b} - \\frac{\\gamma _{m0}}{\\beta _0}\\right)\\nonumber \\\\&&\\quad {} + \\frac{1}{\\beta _0} \\int _0^\\infty d u\\,S(u)\\,e^{-u/b}\\,,\\nonumber \\\\&&\\gamma _m(b) = \\frac{2}{3} C_F \\frac{b}{\\beta _0}\\frac{(3+2b) \\Gamma (4+2b)}{\\Gamma (3+b) \\Gamma ^2(2+b) \\Gamma (1-b)}\\,,\\nonumber \\\\&&\\gamma _{m0} = 6 C_F\\,,\\nonumber \\\\&&S(u) = - 6 C_F \\left[e^{(5/3)u} \\frac{\\Gamma (u) \\Gamma (1-2u)}{\\Gamma (3-u)} (1-u) - \\frac{1}{2u}\\right]\\,,\\nonumber \\\\&&b = \\beta _0 \\frac{\\alpha _s}{4\\pi }\\quad (\\alpha _s \\equiv \\alpha _s^{(n_f)}(M))\\,.$ Note that we first expand $S(u)$ in $u$ , then integrate term-by-term assuming $\\beta _0>0$ , and at the very end substitute $\\beta _0 \\rightarrow -(4/3) T_F n_f$ .", "$\\Delta _m(x)$ comes from the differences of diagrams with a single massive quark loop and corresponding diagrams with all quark loops being massless and is given by $&&\\Delta _m(x) = C_F T_F \\left(\\frac{\\alpha _s}{\\pi }\\right)^2 \\biggl \\lbrace \\frac{1}{2}\\biggl [(1-x)^2 (1+x+x^2) H_{1,0}(x) - (1+x)^2 (1-x+x^2) H_{-1,0}(x)+ 2 x^4 H_{0,0}(x)\\nonumber \\\\&&\\quad {} + x^2 H_{0}(x) - x (3+3x^2-x^3) \\frac{\\pi ^2}{6} + \\frac{3}{2} x^2\\biggr ]\\nonumber \\\\&&{} + T_F n_0 \\frac{\\alpha _s}{\\pi } \\frac{2}{3}\\biggl [(1-x)^2 (1+x+x^2) \\left(H_{1,-1,0}(x) + \\frac{\\pi ^2}{12} H_{1}(x)\\right)\\nonumber \\\\&&\\quad {} + (1+x)^2 (1-x+x^2) \\left(H_{-1,1,0}(x) + \\frac{5 \\pi ^2}{12} H_{-1}(x)\\right)- x (1+x^2) \\left(H_{0,1,0}(x) + H_{0,-1,0}(x) + \\pi ^2 a_1\\right)\\nonumber \\\\&&\\quad {} + x^4 \\left(2 H_{0,0,0}(x) - \\frac{13}{6} H_{0,0}(x) - \\frac{3}{2} \\zeta _3\\right)- x (3+3x^2+x^3) \\frac{\\pi ^2}{6} H_{0}(x)- \\frac{1}{12} (1-x)^2 (13+10x+13x^2) H_{1,0}(x)\\nonumber \\\\&&\\quad {} + \\frac{1}{12} (1+x)^2 (13-10x+13x^2) H_{-1,0}(x)+ x (48-12x+48x^2-13x^3) \\frac{\\pi ^2}{72}- \\frac{7}{12} x^2 H_{0}(x)- \\frac{11}{8} x^2\\biggr ]\\nonumber \\\\&&{} + \\left(T_F n_0 \\frac{\\alpha _s}{\\pi }\\right)^2 \\frac{2}{3}\\biggl [ (1-x)^2 (1+x+x^2)\\nonumber \\\\&&\\qquad {}\\times \\biggl (- 2 H_{1,-1,1,0}(x) + H_{1,0,1,0}(x) + H_{1,0,-1,0}(x)- \\frac{\\pi ^2}{6} \\left(5 H_{1,-1}(x) - 4 H_{1,0}(x)\\right)+ \\left(\\pi ^2 a_1 + \\frac{3}{2} \\zeta _3\\right) H_{1}(x)\\biggr )\\nonumber \\\\&&\\quad {} + (1+x)^2 (1-x+x^2)\\nonumber \\\\&&\\qquad {}\\times \\left(2 H_{-1,1,-1,0}(x) + H_{-1,0,-1,0}(x) + H_{-1,0,1,0}(x)+ \\frac{\\pi ^2}{6} \\left(H_{-1,1}(x) + 2 H_{-1,0}(x)\\right)+ \\left(\\pi ^2 a_1 - \\frac{3}{2} \\zeta _3\\right) H_{-1}(x)\\right)\\nonumber \\\\&&\\quad {} + 2 x (1+x^2) \\biggl (- H_{0,1,-1,0}(x) + H_{0,-1,1,0}(x) - H_{0,0,1,0}(x) - H_{0,0,-1,0}(x)+ \\frac{4}{3} \\left(H_{0,1,0}(x) + H_{0,-1,0}(x) + \\pi ^2 a_1\\right)\\nonumber \\\\&&\\qquad {} - \\frac{\\pi ^2}{12} \\left(H_{0,1}(x) - 5 H_{0,-1}(x) + 6 H_{0,0}(x)\\right)- \\pi ^2 a_1 H_{0}(x) - \\pi ^2 a_1^2\\biggr )\\nonumber \\\\&&\\quad {} + x^4 \\left(4 H_{0,0,0,0}(x) - \\frac{13}{3} H_{0,0,0}(x) + \\frac{89}{36} H_{0,0}(x)\\right)\\nonumber \\\\&&\\quad {} - \\frac{1}{6} (1-x)^2 (13+10x+13x^2) \\left(\\!H_{1,-1,0}(x) + \\frac{\\pi ^2}{12} H_{1}(x)\\right)- \\frac{1}{6} (1+x)^2 (13-10x+13x^2) \\left(\\!H_{-1,1,0}(x) + \\frac{5 \\pi ^2}{12} H_{-1}(x)\\right)\\nonumber \\\\&&\\quad {} + \\frac{1}{72} (1-x)^2 (89+68x+89x^2) H_{1,0}(x)- \\frac{1}{72} (1+x)^2 (89-68x+89x^2) H_{-1,0}(x)\\nonumber \\\\&&\\quad {} + x (48+6x+48x^2+13x^3) \\frac{\\pi ^2}{36} H_{0}(x)+ \\frac{47}{72} x^2 H_{0}(x)+ \\frac{1}{4} x^2 (6+13x^2) \\zeta _3- x (5+5x^2-2x^3) \\frac{\\pi ^4}{30}\\nonumber \\\\&&\\quad {} - x (330-192x+330x^2-89x^3) \\frac{\\pi ^2}{432}+ \\frac{33}{16} x^2 \\biggr ]+ \\mathcal {O}(\\alpha _s^3) \\biggr \\rbrace \\,.$ Note that $\\Delta _m(0) = 0$ .", "Expanding the three-loop term in $x$ we reproduce the series (up to $x^8$ ) obtained in [31].", "The three-loop coefficient exact in $x$ (Fig.", "REF ) and well as the four-loop one are new.", "The contribution of the external flavor ($m=M$ ) is given by $&&\\Delta _m(1) = C_F T_F \\left(\\frac{\\alpha _s}{\\pi }\\right)^2 \\biggl [- \\frac{\\pi ^2-3}{4}\\nonumber \\\\&&{} + T_F n_0 \\frac{\\alpha _s}{\\pi }\\left(\\zeta _3 + \\frac{13}{36} \\pi ^2 - \\frac{11}{12}\\right)\\nonumber \\\\&&{} - \\left(T_F n_0 \\frac{\\alpha _s}{\\pi }\\right)^2\\left(\\frac{13}{6} \\zeta _3 + \\frac{4}{45} \\pi ^4 + \\frac{53}{216} \\pi ^2 - \\frac{11}{8}\\right)\\nonumber \\\\&&{} + \\mathcal {O}(\\alpha _s^3) \\biggr ]\\,.$ The three- and four-loop terms here agree with [11] and [12].", "We do not present lower-loop terms of $z_m$ with positive powers of $\\varepsilon $ which may be needed when this ratio is used within calculations containing $1/\\varepsilon $ divergences; these terms can be easily obtained from Eqs.", "(REF ) and (REF ).", "Figure: The coefficient of (α s /π) 3 C F T F 2 n 0 (\\alpha _s/\\pi )^3 C_F T_F^2 n_0 in Δ m (x)\\Delta _m(x).Contributions to $Z_h^{\\text{os}}$ with the color structures $C_F T_F^{l-1} n_0^{l-2}$ (i. e., the maximum number of quark loops, one of them is massive with mass $m_i$ ) can be calculated using Eq.", "(REF ).", "The results read $Z_h^{\\text{os}} &=& 1 + C_F \\sum _{l=2}^\\infty T_F^{l-1} (n_0 P)^{l-2} (l-1) B_h^{(l)}\\nonumber \\\\&&{}\\times \\sum _i \\left(\\frac{g_0^2 m_i^{-2\\varepsilon }}{(4\\pi )^{d/2}} \\Gamma (\\varepsilon )\\right)^l+ \\cdots \\,,\\nonumber \\\\B_h^{(l)} &=&4 \\frac{(3-2\\varepsilon ) (1+(l-1)\\varepsilon ) \\Gamma ^2(1+(l-1)\\varepsilon )}{l (l-1) (3+2(l-1)\\varepsilon ) \\Gamma (2-\\varepsilon ) \\Gamma ^2(1+\\varepsilon )}\\nonumber \\\\&&{}\\times \\frac{\\Gamma (1-(l-1)\\varepsilon ) \\Gamma (1+l\\varepsilon )}{\\Gamma (2+2(l-1)\\varepsilon )}\\,,$ where $g_0 \\equiv g_0^{(n_l)}$ and dots denote other color structures.", "The ($l=2$ )-loop term agrees with [22], and the three-loop one with the corresponding color structure in [33].", "According to the regions-based argument in Sect.", ", $\\lim _{x\\rightarrow 0} \\left[B_Q^{(l)}(x) - B_{Q0}^{(l)} P - B_h^{(l)} x^{-2l\\varepsilon }\\right] = 0\\,.$" ] ]

[ [ "Delocalization transition for critical Erd\\H{o}s-R\\'enyi graphs" ], [ "Abstract We analyse the eigenvectors of the adjacency matrix of a critical Erd\\H{o}s-R\\'enyi graph $\\mathbb G(N,d/N)$, where $d$ is of order $\\log N$.", "We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices.", "In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function.", "The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent $\\gamma(\\mathbf w)$ of an eigenvector $\\mathbf w$, defined through $\\|\\mathbf w\\|_\\infty / \\|\\mathbf w\\|_2 = N^{-\\gamma(\\mathbf w)}$.", "Our results remain valid throughout the optimal regime $\\sqrt{\\log N} \\ll d \\leq O(\\log N)$." ], [ "[block]1..5em*" ], [ "0pt21" ], [ "[runin]1..4em[.]" ], [ "0pt2ex plus .1ex minus .2ex.8em" ], [ "[runin]1..3em[.]" ], [ "0pt1ex plus .1ex minus .2ex.5em" ], [ "[runin]1..3em[.]" ], [ "0pt1ex plus .1ex minus .2ex.5em arrows,decorations.pathmorphing,backgrounds,positioning,fit,petri 0.08em 0.05em 0.05em Delocalization transition for critical Erdős–Rényi graphs Johannes Alt Raphael Ducatez Antti Knowles We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi graph $\\mathbb {G}(N,d/N)$ , where $d$ is of order $\\log N$ .", "We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices.", "In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function.", "The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent $\\gamma (w̑)$ of an eigenvector $w̑$ , defined through $\\Vert w̑ \\Vert _\\infty / \\Vert w̑ \\Vert _2 = N^{-\\gamma (w̑)}$ .", "Our results remain valid throughout the optimal regime $\\sqrt{\\log N} \\ll d \\leqslant O(\\log N)$ ." ], [ "Overview", "Let $A$ be the adjacency matrix of a graph with vertex set $[N] = \\lbrace 1, \\dots , N\\rbrace $ .", "We are interested in the geometric structure of the eigenvectors of $A$ , in particular their spatial localization.", "An $\\ell ^2$ -normalized eigenvector $w̑ = (w_x)_{x \\in [N]}$ gives rise to a probability measure $\\sum _{x \\in [N]} w_x^2 \\delta _x$ on the set of vertices.", "Informally, $w̑$ is delocalized if its mass is approximately uniformly distributed throughout $[N]$ , and localized if its mass is essentially concentrated in a small number of vertices.", "There are several ways of quantifying spatial localization.", "One is the notion of concentration of mass, sometimes referred to as scarring [49], stating that there is some set $\\mathcal {B} \\subset [N]$ of small cardinality and a small $\\varepsilon > 0$ such that $\\sum _{x \\in \\mathcal {B}} w_x^2 = 1 - \\varepsilon $ .", "In this case, it is also of interest to characterize the geometric structure of the vertex set $\\mathcal {B}$ and of the eigenvector $w̑$ restricted to $\\mathcal {B}$ .", "Another convenient quantifier of spatial localization is the $\\ell ^p$ -norm $\\Vert w̑ \\Vert _p$ for $2 \\leqslant p \\leqslant \\infty $ .", "It has the following interpretation: if the mass of $w̑$ is uniformly distributed over some set $\\mathcal {B} \\subset [N]$ then $\\Vert w̑ \\Vert _p^2 = \\vert \\mathcal {B} \\vert ^{-1 + 2/p}$ .", "Focusing on the $\\ell ^\\infty $ -norm for definiteness, we define the localization exponent $\\gamma (w̑)$ through $ \\Vert w̑ \\Vert _\\infty ^2 =\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ N-(w̑) .", "Thus, $0 \\leqslant \\gamma (w̑) \\leqslant 1$ , and $\\gamma (w̑) = 0$ corresponds to localization at a single vertex while $\\gamma (w̑) = 1$ to complete delocalization.", "In this paper we address the question of spatial localization for the random Erdős-Rényi graph $\\mathbb {G}(N,d/N)$ .", "We consider the limit $N \\rightarrow \\infty $ with $d \\equiv d_N$ .", "It is well known that $\\mathbb {G}(N,d/N)$ undergoes a dramatic change in behaviour at the critical scale $d \\asymp \\log N$ , which is the scale at and below which the vertex degrees do not concentrate.", "Thus, for $d \\gg \\log N$ , with high probability all degrees are approximately equal and the graph is homogeneous.", "On the other hand, for $d \\lesssim \\log N$ , the degrees do not concentrate and the graph becomes highly inhomogeneous: it contains for instance hubs of exceptionally large degree, leaves, and isolated vertices.", "As long as $d > 1$ , the graph has with high probability a unique giant component, and we shall always restrict our attention to it.", "Here we propose the Erdős-Rényi graph at criticality as a simple and natural model on which to address the question of spatial localization of eigenvectors.", "It has the following attributes.", "Its graph structure provides an intrinsic and nontrivial notion of distance.", "Its spectrum splits into a delocalized phase and a semilocalized phase.", "The transition between the phases is sharp, in the sense of a discontinuity in the localization exponent.", "Both phases are amenable to rigorous analysis.", "Our results are summarized in the phase diagram of Figure REF , which is expressed in terms of the parameter $b$ parametrizing $d = b \\log N$ on the critical scale and the eigenvalue $\\lambda $ of $A / \\sqrt{d}$ associated with the eigenvector $w̑$ .", "To the best of our knowledge, the phase coexistence for the critical Erdős-Rényi graph established in this paper had previously not been analysed even in the physics literature.", "Figure: The phase diagram of the adjacency matrix A/dA / \\sqrt{d} of the Erdős-Rényi graph 𝔾(N,d/N)\\mathbb {G}(N,d/N) at criticality, where d=blogNd = b \\log N with bb fixed.", "The horizontal axis records the location in the spectrum and the vertical axis the sparseness parameter bb.", "The spectrum is confined to the coloured region.", "In the red region the eigenvectors are delocalized while in the blue region they are semilocalized.", "The grey regions have width o(1)o(1) and are not analysed in this paper.", "For b>b * b > b_* the spectrum is asymptotically contained in [-2,2][-2,2] and the semilocalized phase does not exist.", "For b<b * b < b_* a semilocalized phase emerges in the region (-λ max (b),-2)∪(2,λ max (b))(-\\lambda _{\\max }(b), -2) \\cup (2, \\lambda _{\\max }(b)) for some explicit λ max (b)>2\\lambda _{\\max }(b) > 2.Throughout the following, we always exclude the largest eigenvalue of $A$ , its Perron-Frobenius eigenvalue, which is an outlier separated from the rest of the spectrum.", "The delocalized phase is characterized by a localization exponent asymptotically equal to 1.", "It exists for all fixed $b > 0$ and consists asymptotically of energies in $(-2,0) \\cup (0,2)$ .", "The semilocalized phase is characterized by a localization exponent asymptotically less than 1.", "It exists only when $b < b_*$ , where $ b_* \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =122 - 1 2.59 .", "It consists asymptotically of energies in $(-\\lambda _{\\max }(b), -2) \\cup (2, \\lambda _{\\max }(b))$ , where $\\lambda _{\\max }(b) > 2$ is an explicit function of $b$ (see (REF ) below).", "The density of states at energy $\\lambda \\in \\mathbb {R}$ is equal to $N^{\\rho _b(\\lambda ) + o(1)}$ , where $\\rho _b$ is an explicit exponent defined in (REF ) below and illustrated in Figure REF .", "It has a discontinuity at 2 (and similarly at $-2$ ), jumping from $\\rho _b(2^-) = 1$ to $\\rho _b(2^+) = 1 - b / b^*$ .", "The localization exponent $\\gamma (w̑)$ from (REF ) of an eigenvector $w̑$ with associated eigenvalue $\\lambda $ satisfies with high probability $\\gamma (w̑) = 1 + o(1) \\;\\; \\text{if} \\;\\; \\vert \\lambda \\vert < 2\\,, \\qquad \\gamma (w̑) \\leqslant \\rho _b(\\lambda ) + o(1) \\;\\; \\text{if} \\;\\; \\vert \\lambda \\vert > 2\\,.$ This establishes a discontinuity, in the limit $N \\rightarrow \\infty $ , in the localization exponent $\\gamma (w̑)$ as a function of $\\lambda $ at the energies $\\pm 2$ .", "See Figure REF for an illustration; we also refer to Appendix REF for a simulation depicting the behaviour of $\\Vert w̑ \\Vert _\\infty $ throughout the spectrum.", "Moreover, in the semilocalized phase scarring occurs in the sense that a fraction $1 - o(1)$ of the mass of the eigenvectors is supported in a set of at most $N^{\\rho _b(\\lambda ) + o(1)}$ vertices.", "Figure: The behaviour of the exponents ρ b \\rho _b and γ\\gamma as a function of the energy λ\\lambda .", "The dark blue curve is the exponent ρ b (λ)\\rho _b(\\lambda ) characterizing the density of states N ρ b (λ)+o(1) N^{\\rho _b(\\lambda ) + o(1)} of the matrix A/dA / \\sqrt{d} at energy λ\\lambda .", "The entire blue region (light and dark blue) is the asymptotically allowed region of the localization exponent γ(w̑)\\gamma (w̑) of an eigenvector of A/dA / \\sqrt{d} as a function of the associated eigenvalue λ\\lambda .", "Here d=blogNd = b \\log N with b=1b = 1 and λ max (b)≈2.0737\\lambda _{\\max }(b) \\approx 2.0737.", "We only plot a neighbourhood of the threshold energy 2.", "The discontinuity at 2 of ρ b \\rho _b is from ρ b (2 - )=1\\rho _b(2^-) = 1 to ρ b (2 + )=1-b/b * =2-2log2\\rho _b(2^+) = 1 - b / b^* = 2 - 2 \\log 2.The eigenvalues in the semilocalized phase were analysed in [10], where it was proved that they arise precisely from vertices $x$ of abnormally large degree, $D_x \\geqslant 2 d$ .", "More precisely, it was proved in [10] that each vertex $x$ with $D_x \\geqslant 2 d$ gives rise to two eigenvalues of $A / \\sqrt{d}$ near $\\pm \\Lambda (D_x / d)$ , where $\\Lambda (\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =-1$.", "The same result for the $ O(1)$ largest degree vertices was independently proved in \\cite {tikhomirov2019outliers} by a different method.", "We refer also to \\cite {BBK1, BBK2} for an analysis in the supercritical and subcritical phases.$ In the current paper, we prove that the eigenvector $w̑$ associated with an eigenvalue $\\lambda $ in the semilocalized phase is highly concentrated around resonant vertices at energy $\\lambda $ , which are defined as the vertices $x$ such that $\\Lambda (D_x/d)$ is close to $\\lambda $ .", "For this reason, we also call the resonant vertices localization centres.", "With high probability, and after a small pruning of the graph, all balls $B_r(x)$ of a certain radius $r \\gg 1$ around the resonant vertices are disjoint, and within any such ball $B_r(x)$ the eigenvector $w̑$ is an approximately radial exponentially decaying function.", "The number of resonant vertices at energy $\\lambda $ is comparable to the density of states, $N^{\\rho _b(\\lambda ) + o(1)}$ , which is much less than $N$ .", "See Figure REF for a schematic illustration of the mass distribution of $w̑$ .", "Figure: A schematic representation of the geometric structure of a typical eigenvector in the semilocalized phase.", "The giant component of the graph is depicted in pale blue.", "The eigenvector's mass (depicted in dark blue) is concentrated in a small number of disjoint balls centred around resonant vertices (drawn in white), and within each ball the mass decays exponentially in the radius.", "The mass outside the balls is an asymptotically vanishing proportion of the total mass.The behaviour of the critical Erdős-Rényi graph described above has some similarities but also differences to that of the Anderson model [11].", "The Anderson model on $\\mathbb {Z}^n$ with $n \\geqslant 3$ is conjectured to exhibit a metal-insulator, or delocalization-localization, transition: for weak enough disorder, the spectrum splits into a delocalized phase in the middle of the spectrum and a localized phase near the spectral edges.", "See e.g.", "[8] for a phase diagram of its conjectured behaviour.", "So far, only the localized phase of the Anderson model has been understood rigorously, in the landmark works [39], [4], as well as contributions of many subsequent developments.", "The phase diagram for the Anderson model bears some similarity to that of Figure REF , in which one can interpret $1/b$ as the disorder strength, since smaller values of $b$ lead to stronger inhomogeneities in the graph.", "As is apparent from the proofs in [39], [4], in the localized phase the local structure of an eigenvector of the Anderson model is similar to that of the critical Erdős-Rényi graph described above: exponentially decaying around well-separated localization centres associated with resonances near the energy $\\lambda $ of the eigenvector.", "The localization centres arise from exceptionally large local averages of the potential.", "The phenomenon of localization can be heuristically understood using the following well-known rule of thumb: one expects localization around a single localization centre if the level spacing is much larger than the tunnelling amplitude between localization centres.", "It arises from perturbation theory around the block diagonal model where the complement of balls $B_r(x)$ around localization centres is set to zero.", "On a very elementary level, this rule is illustrated by the matrix $H(t) = \\bigl ( {\\begin{matrix}0 & t\\\\ t & 1\\end{matrix}}\\bigr )$ , whose eigenvectors are localized for $t = 0$ , remain essentially localized for $t \\ll 1$ , where perturbation theory around $H(0)$ is valid, and become delocalized for $t \\gtrsim 1$ , where perturbation theory around $H(0)$ fails.", "More precisely, it is a general heuristic that the tunnelling amplitude decays exponentially in the distance between the localization centres [25].", "Denoting by $\\beta (\\lambda ) > 1$ the rate of exponential decay at energy $\\lambda $ , the rule of thumb hence reads $ \\beta (\\lambda )^{-L}\\ll \\varepsilon (\\lambda )\\,,$ where $L$ is the distance between the localization centres and $\\varepsilon (\\lambda )$ the level spacing at energy $\\lambda $ .", "For the Anderson model restricted to a finite cube of $\\mathbb {Z}^n$ with side length $N^{1/n}$ , the level spacing $\\varepsilon (\\lambda )$ is of order $N^{-1}$ (see [57] and [8]) whereas the diameter of the graph is of order $N^{1/n}$ .", "Hence, the rule of thumb (REF ) becomes $\\beta (\\lambda )^{-N^{1/n}}\\ll N^{-1}\\,,$ which is satisfied and one therefore expects localization.", "For the critical Erdős-Rényi graph, the level spacing $\\varepsilon (\\lambda )$ is $N^{-\\rho (\\lambda )+o(1)}$ but the diameter of the giant component is only $\\frac{\\log N}{\\log d}$ .", "Hence, the rule of thumb (REF ) becomes $N^{-\\frac{\\log \\beta (\\lambda )}{\\log d}}\\ll N^{-\\rho (\\lambda )+o(1)}\\,,$ which is never satisfied because $\\frac{\\log \\beta (\\lambda )}{\\log d}\\rightarrow 0$ as $N \\rightarrow \\infty $ .", "Thus, the rule of thumb (REF ) is satisfied in the localized phase of the Anderson model but not in the semilocalized phase of the critical Erdős-Rényi graph.", "The underlying reason behind this difference is that the diameter of the Anderson model is polynomial in $N$ , while the diameter of the critical Erdős-Rényi graph is logarithmic in $N$ .", "Thus, the critical Erdős-Rényi graph is far more connected than the Anderson model; this property tends to push it more towards the delocalized behaviour of mean-field systems.", "As noted above, another important difference between the localized phase of the Anderson model and the semilocalized phase of the critical Erdős-Rényi graph is that the density of states is of order $N$ in the former and a fractional power of $N$ in the latter.", "Up to now we have focused on the Erdős-Rényi graph on the critical scale $d \\asymp \\log N$ .", "It is natural to ask whether this assumption can be relaxed without changing its behaviour.", "The question of the upper bound on $d$ is simple: as explained above, there is no semilocalized phase for $d > b_* \\log N$ , and the delocalized phase is completely understood up to $d \\leqslant N/2$ , thanks to Theorem REF below and [42], [35].", "The lower bound is more subtle.", "In fact, it turns out that all of our results remain valid throughout the regime $ \\sqrt{\\log N} \\ll d \\leqslant O(\\log N)\\,.$ The lower bound $\\sqrt{\\log N}$ is optimal in the sense that below it both phases are disrupted and the phase diagram from Figure REF no longer holds.", "Indeed, for $d \\lesssim \\sqrt{\\log N}$ a new family of localized states, associated with so-called tuning forks at the periphery of the graph, appear throughout the delocalized and semilocalized phases.", "We refer to Section REF below for more details.", "Previously, strong delocalization with localization exponent $\\gamma (w̑) = 1 + o(1)$ has been established for many mean-field models, such as Wigner matrices [36], [37], [34], [35], [1], supercritical Erdős-Rényi graphs [35], [42], and random regular graphs [13], [12].", "All of these models are homogeneous and only have a delocalized phase.", "Although a rigorous understanding of the metal-insulator transition for the Anderson model is still elusive, some progress has been made for random band matrices.", "Random band matrices [47], [58], [23], [40] constitute an attractive model interpolating between the Anderson model and mean-field Wigner matrices.", "They retain the $n$ -dimensional structure of the Anderson model but have proved somewhat more amenable to rigorous analysis.", "They are conjectured [40] to have a similar phase diagram as the Anderson model in dimensions $n \\geqslant 3$ .", "As for the Anderson model, dimensions $n > 1$ have so far seen little progress, but for $n = 1$ much has been understood both in the localized [50], [48] and the delocalized [52], [32], [33], [29], [28], [31], [30], [51], [22], [21], [20], [59], [43] phases.", "A simplification of band matrices is the ultrametric ensemble [41], where the Euclidean metric of $\\mathbb {Z}^n$ is replaced with an ultrametric arising from a tree structure.", "For this model, a phase transition was rigorously established in [56].", "Another modification of the $n$ -dimensional Anderson model is the Anderson model on the Bethe lattice, an infinite regular tree corresponding to the case $n = \\infty $ .", "For it, the existence of a delocalized phase was shown in [5], [38], [44].", "In [7], [6] it was shown that for unbounded random potentials the delocalized phase exists for arbitrarily weak disorder.", "It extends beyond the spectrum of the unperturbed adjacency matrix into the so-called Lifschitz tails, where the density of states is very small.", "The authors showed that, through the mechanism of resonant delocalization, the exponentially decaying tunnelling amplitudes between localization centres are counterbalanced by an exponentially large number of possible channels through which tunnelling can occur, so that the rule of thumb (REF ) for localization is violated.", "As a consequence, the eigenvectors are delocalized across many resonant localization centres.", "We remark that this analysis was made possible by the absence of cycles on the Bethe lattice.", "In contrast, the global geometry of the critical Erdős-Rényi graph is fundamentally different from that of the Bethe lattice (through the existence of a very large number of long cycles), which has a defining impact on the nature of the delocalization-semilocalization transition summarized in Figure REF .", "Transitions in the localization behaviour of eigenvectors have also been analysed in several mean-field type models.", "In [46], [45] the authors considered the sum of a Wigner matrix and a diagonal matrix with independent random entries with a large enough variance.", "They showed that the eigenvectors in the bulk are delocalized while near the edge they are partially localized at a single site.", "Their partially localized phase can be understood heuristically as a rigorous (and highly nontrivial) verification of the rule of thumb for localization, where the perturbation takes place around the diagonal matrix.", "Heavy-tailed Wigner matrices, or Lévy matrices, whose entries have $\\alpha $ -stable laws for $0 < \\alpha < 2$ , were proposed in [24] as a simple model that exhibits a transition in the localization of its eigenvectors; we refer to [3] for a summary of the predictions from [24], [53].", "In [19], [18] it was proved that for energies in a compact interval around the origin, eigenvectors are weakly delocalized, and for $0 < \\alpha < 2/3$ for energies far enough from the origin, eigenvectors are weakly localized.", "In [3], full delocalization was proved in a compact interval around the origin, and the authors even established GOE local eigenvalue statistics in the same spectral region.", "In [2], the law of the eigenvector components of Lévy matrices was computed." ], [ "Conventions", "Throughout the following, every quantity that is not explicitly constant depends on the fundamental parameter $N$ .", "We almost always omit this dependence from our notation.", "We use $C$ to denote a generic positive universal constant, and write $X = O(Y)$ to mean $\\vert X \\vert \\leqslant C Y$ .", "For $X,Y > 0$ we write $X \\asymp Y$ if $X = O(Y)$ and $Y = O(X)$ .", "We write $X \\ll Y$ or $X = o(Y)$ to mean $\\lim _{N \\rightarrow \\infty } X/Y = 0$ .", "A vector is normalized if its $\\ell ^2$ -norm is one." ], [ "Results – the semilocalized phase", "Let $\\mathbb {G} = \\mathbb {G}(N,d/N)$ be the Erdős–Rényi graph with vertex set $[N] \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={1, ..., N}$ and edge probability $ d/N$ for $ 0 d N$.", "Let $ A = (Axy)x,y [N] {0,1}NN$ be the adjacency matrix of $ G$.", "Thus, $ A =A*$, $ Axx=0$ for all $ x [N]$, and $ ( Axy x < y)$ are independent $ Bernoulli(d/N)$ random variables.$ The entrywise nonnegative matrix $A/\\sqrt{d}$ has a trivial Perron-Frobenius eigenvalue, which is its largest eigenvalue.", "In the following we only consider the other eigenvalues, which we call nontrivial.", "In the regime $d \\gg \\sqrt{\\log N / \\log \\log N}$ , which we always assume in this paper, the trivial eigenvalue is located at $\\sqrt{d} (1 + o(1))$ , and it is separated from the nontrivial ones with high probability; see [14].", "Moreover, without loss of generality in this subsection we always assume that $d \\leqslant 3 \\log N$ , for otherwise the semilocalized phase does not exist (see Section REF ).", "For $x \\in [N]$ we define the normalized degree of $x$ as $ \\alpha _x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1d y [N] Axy .", "In Theorem REF below we show that the nontrivial eigenvalues of $A / \\sqrt{d}$ outside the interval $[-2,2]$ are in two-to-one correspondence with vertices with normalized degree greater than 2: each vertex $x$ with $\\alpha _x > 2$ gives rise to two eigenvalues of $A / \\sqrt{d}$ located with high probability near $\\pm \\Lambda (\\alpha _x)$ , where we defined the bijective function $\\Lambda \\mathrel {\\hbox{.}\\hbox{.", "}}$ [2,) [2,)$ through\\begin{equation} \\Lambda (\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation}=\\frac{\\alpha }{\\sqrt{\\alpha -1}}.$ Our main result in the semilocalized phase is about the eigenvectors associated with these eigenvalues.", "To state it, we need the following notions.", "Definition 3.1 Let $\\lambda >2$ and $0 < \\delta \\leqslant \\lambda - 2$ .", "We define the set of resonant vertices at energy $\\lambda $ through $ \\mathcal {W}_{\\lambda ,\\delta } \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={x x 2, (x) - } .", "We denote by $B_r(x)$ the ball around the vertex $x$ of radius $r$ for the graph distance in $\\mathbb {G}$ .", "Define $ r_\\star = \\big \\lfloor c \\sqrt{\\log N} \\big \\rfloor \\,;$ all of our results will hold provided $c > 0$ is chosen to be a small enough universal constant.", "The quantity $r_\\star $ will play the role of a maximal radius for balls around localization centres.", "We introduce the basic control parameters $ \\xi \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =Nd d ,       u =Nd 1u , which under our assumptions will always be small (see Remark REF below).", "We now state our main result in the semilocalized phase.", "Theorem 3.2 (Semilocalized phase) For any $\\nu > 0$ there exists a constant $\\mathcal {C}$ such that the following holds.", "Suppose that $ \\mathcal {C} \\sqrt{\\log N} \\log \\log N \\leqslant d \\leqslant 3\\log N\\,.$ Let $w̑$ be a normalized eigenvector of $A/\\sqrt{d}$ with nontrivial eigenvalue $\\lambda \\geqslant 2+\\mathcal {C} \\xi ^{1/2}$ .", "Let $0<\\delta \\leqslant (\\lambda -2)/2$ .", "Then for each $x \\in \\mathcal {W}_{\\lambda , \\delta }$ there exists a normalized vector $v̑(x)$ , supported in $B_{r_\\star }(x)$ , such that the supports of $v̑(x)$ and $v̑(y)$ are disjoint for $x \\ne y$ , and $ \\sum _{x \\in \\mathcal {W}_{\\lambda ,\\delta }} \\langle v̑(x) {2mu}, w̑\\rangle ^2 \\geqslant 1 - \\mathcal {C} \\mathopen {}\\mathclose {\\left(\\frac{\\xi +\\xi _{\\lambda -2}}{\\delta }\\right)}^2$ with probability at least $1 - \\mathcal {C} N^{-\\nu }$ .", "Moreover, $v̑(x)$ decays exponentially around $x$ in the sense that for any $r \\geqslant 0$ we have $\\sum _{y \\notin B_r(x)} (v̑(x))_y^2 \\leqslant \\frac{1}{(\\alpha _x - 1)^{r+1}}\\,.$ Remark 3.3 An analogous result holds for negative eigenvalues $-\\lambda \\leqslant -2 - \\mathcal {C} \\xi ^{1/2}$ , with a different vector $v̑(x)$ .", "See Theorem REF and Remark REF below for a precise statement.", "Remark 3.4 The upper bound $d \\leqslant 3 \\log N$ in (REF ) is made for convenience and without loss of generality, because if $d > 3 \\log N$ then, as explained in Section REF , with high probability the semilocalized phase does not exist, i.e.", "eigenvalues satisfying the conditions of Theorem REF do not exist.", "Theorem REF implies that $w̑$ is almost entirely concentrated in the balls around the resonant vertices, and in each such ball $B_{r_\\star }(x)$ , $x \\in \\mathcal {W}_{\\lambda ,\\delta }$ , the vector $w̑$ is almost collinear to the vector $v̑(x)$ .", "Thus, $v̑(x)$ has the interpretation of the localization profile around the localization centre $x$.", "Since it has exponential decay, we deduce immediately from Theorem REF that the radius $r_\\star $ can be made smaller at the expense of worse error terms.", "In fact, in Definition REF and Theorem REF below, we give an explicit definition of $v̑(x)$ , which shows that it is radial in the sense that its value at a vertex $y$ depends only on the distance between $x$ and $y$ , in which it is an exponentially decaying function.", "To ensure that the supports of the vectors $v̑(x)$ for different $x$ do not overlap, $v̑(x)$ is in fact defined as the restriction of a radial function around $x$ to a subgraph of $\\mathbb {G}$ , the pruned graph, which differs from $\\mathbb {G}$ by only a small number of edges and whose balls of radius $r_\\star $ around the vertices of $\\mathcal {W}_{\\lambda ,\\delta }$ are disjoint (see Proposition REF below).", "For positive eigenvalues, the entries of $v̑(x)$ are nonnegative, while for negative eigenvalues its entries carry a sign that alternates in the distance to $x$ .", "The set of resonant vertices $\\mathcal {W}_{\\lambda ,\\delta }$ is a small fraction of the whole vertex set $[N]$ ; its size is analysed in Lemma REF below.", "Remark 3.5 Note that, by the lower bounds imposed on $d$ and $\\lambda $ in Theorem REF , we always have $\\xi , \\xi _{\\lambda - 2} \\leqslant 1/ \\mathcal {C}$ .", "Using the exponential decay of the localization profiles, it is easy to deduce from Theorem REF that a positive proportion of the eigenvector mass concentrates at the resonant vertices.", "Corollary 3.6 Under the assumptions of Theorem REF we have $\\sum _{y \\in \\mathcal {W}_{\\lambda ,\\delta }} w_y^2 = \\frac{\\sqrt{\\lambda ^2-4}}{ \\lambda + \\sqrt{\\lambda ^2-4}} +O \\bigg (\\frac{\\mathcal {C}(\\xi +\\xi _{\\lambda -2})}{\\delta }+\\frac{\\mathcal {C} \\delta }{\\lambda ^{5/2} \\sqrt{\\lambda -2}}\\bigg ) $ with probability at least $1 - \\mathcal {C} N^{-\\nu }$ .", "Next, we state a rigidity result on the eigenvalue locations in the semilocalized phase.", "It generalizes [10] by improving the error bound and extending it to the full regime (REF ) of $d$ , below which it must fail (see Section REF below).", "Its proof is a byproduct of the proof of our main result in the semilocalized phase, Theorem REF .", "We denote the ordered eigenvalues of a Hermitian matrix $M\\in {N\\times N}$ by $\\lambda _1(M) \\geqslant \\lambda _2(M) \\geqslant \\cdots \\geqslant \\lambda _N(M)$ .", "We only consider the nontrivial eigenvalues of $A / \\sqrt{d}$ , i.e.", "$\\lambda _i(A / \\sqrt{d})$ with $2 \\leqslant i \\leqslant N$ .", "For the following statements we order the normalized degrees by choosing a (random) permutation $\\sigma \\in S_N$ such that $i \\mapsto \\alpha _{\\sigma (i)}$ is nonincreasing.", "Theorem 3.7 (Eigenvalue locations in semilocalized phase) For any $\\nu > 0$ there exists a constant $\\mathcal {C}$ such that the following holds.", "Suppose that (REF ) holds.", "Let $\\mathcal {U} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ = .", "Then with probability at least $1 - \\mathcal {C} N^{-\\nu }$ , for all $1\\leqslant i\\leqslant |\\mathcal {U}|$ we have $ |\\lambda _{i + 1}(A/\\sqrt{d})-\\Lambda (\\alpha _{\\sigma (i)})| + |\\lambda _{N-i+1}(A/\\sqrt{d})+\\Lambda (\\alpha _{\\sigma (i)})| \\leqslant \\mathcal {C} (\\xi +\\xi _{\\Lambda (\\alpha _{\\sigma (i)})-2})$ and for all $\\vert \\mathcal {U} \\vert + 2 \\leqslant i \\leqslant N - \\vert \\mathcal {U} \\vert $ we have $ \\vert \\lambda _i(A/\\sqrt{d}) \\vert \\leqslant 2 + \\xi ^{1/2}\\,.$ We remark that the upper bound on $d$ from (REF ), which is necessary for the existence of a semilocalized phase, can be relaxed in Theorem REF to obtain an estimate on $\\max _{2 \\leqslant i \\leqslant N} \\vert \\lambda _i(A / \\sqrt{d}) \\vert $ in the supercritical regime $d \\geqslant 3 \\log N$ , which is sharper than the one in [10].", "The proof is the same and we do not pursue this direction here.", "We conclude this subsection with a discussion on the counting function of the normalized degrees, which we use to give estimates on the number of resonant vertices (REF ).", "For $b \\geqslant 0$ and $\\alpha \\geqslant 2$ define the exponent $ \\theta _b(\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =[1 - b (- + 1)]+ .", "Define $\\alpha _{\\max }(b) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={2 b() = 0}$.", "Thus, $ b$ is a nonincreasing function that is nonzero on $ [0, (b))$.", "Moreover, $ b(2) = [1 - b/b*]+$, so that $ (b) > 2$ if and only if $ b < b*$.From Lemma \\ref {lem:degree_distr} below it is easy to deduce that if $ d 1$ then $ (1) = (d/N) + O(/ d)$ with probability at least $ 1 - o(1)$ for any $ 1$.", "Thus, $ (d/N)$ has the interpretation of the deterministic location of the largest normalized degree.", "See Figure \\ref {fig:theta} for a plot of $ b$.$ Figure: A plot of the exponent θ b (α)\\theta _b(\\alpha ) as a function of α⩾2\\alpha \\geqslant 2 for the values b=0.3b = 0.3 (blue), b=1.3b = 1.3 (red), and b=2.3b = 2.3 (green).", "The graph hits the value 0 at α max (b)\\alpha _{\\max }(b).In Appendix REF below, we obtain estimates on the density of the normalized degrees $(\\alpha _x)_{x \\in [N]}$ and combine it with Theorem REF to deduce a lower bound on the $\\ell ^p$ -norm of eigenvectors in the semilocalized phase.", "The precise statements are given in Lemma REF and Corollary REF , which provide quantitative error bounds throughout the regime (REF ).", "Here, we summarize them, for simplicity, in simple qualitative versions in the critical regime $d \\asymp \\log N$ .", "For $b < b_*$ we abbreviate $ \\lambda _{\\max }(b) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =((b)) ,       b() ={ll b(-1()) if 2 1 if < 2 , .", "where $\\Lambda ^{-1}(\\lambda ) = \\frac{\\lambda ^2}{2}(1 + \\sqrt{1 - 4/\\lambda ^2})$ for $\\vert \\lambda \\vert \\geqslant 2$ .", "Let $d = b \\log N$ with some constant $b < b_*$ , and suppose that $2 + \\kappa \\leqslant \\lambda \\leqslant \\lambda _{\\max }(b) - \\kappa $ for some constant $\\kappa > 0$ .", "Then Lemma REF REF implies (choosing $1/d \\ll \\delta \\ll 1$ ) $ \\vert \\mathcal {W}_{\\lambda , \\delta } \\vert = N^{\\rho _b(\\lambda ) + o(1)}$ with probability $1 - o(1)$ .", "From (REF ) and Theorem REF we obtain, for any $2 \\leqslant p \\leqslant \\infty $ , $ \\Vert w̑ \\Vert _p^{2} \\geqslant N^{(2/p - 1) \\rho _b(\\lambda ) + o(1)}$ with probability $1 - o(1)$ (see Corollary REF below).", "In other words, the localization exponent $\\gamma (w̑)$ from (REF ) satisfies $\\gamma (w̑) \\leqslant \\rho _b(\\lambda ) + o(1)$ .", "See Figure REF for an illustration of the bound (REF ) for $p = \\infty $ .", "We remark that the exponent $\\rho _b(\\lambda )$ also describes the density of states at energy $\\lambda $ : under the above assumptions on $b$ and $\\lambda $ , for any interval $I$ containing $\\lambda $ and satisfying $\\xi \\ll \\vert I \\vert \\ll 1$ , the number of eigenvalues in $I$ is equal to $N^{\\rho _b(\\lambda ) + o(1)} \\vert I \\vert $ with probability $1 - o(1)$ , as can be seen from Lemma REF REF and Theorem REF ." ], [ "Results – the delocalized phase", "Let $A$ be the adjacency matrix of $\\mathbb {G}(N,d/N)$ , as in Section REF .", "For $0 < \\kappa < 1/2$ define the spectral region $ \\mathcal {S}_\\kappa \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =[-2 + , -] [, 2 - ] .", "Theorem 3.8 (Delocalized phase) For any $\\nu >0$ and $\\kappa >0$ there exists a constant $\\mathcal {C} > 0$ such that the following holds.", "Suppose that $ \\mathcal {C} \\sqrt{\\log N} \\leqslant d \\leqslant (\\log N)^{3/2}\\,.$ Let $w̑$ be a normalized eigenvector of $A / \\sqrt{d}$ with eigenvalue $\\lambda \\in \\mathcal {S}_\\kappa $ .", "Then $ \\Vert w̑ \\Vert _\\infty ^2 \\leqslant N^{-1 + \\kappa }$ with probability at least $1 - \\mathcal {C}N^{-\\nu }$ .", "In the delocalized phase, i.e.", "in $\\mathcal {S}_\\kappa $ , we also show that the spectral measure of $A / \\sqrt{d}$ at any vertex $x$ is well approximated by the spectral measure at the root of $\\mathbb {T}_{D_x,d}$ , the infinite rooted $(D_x,d)$ -regular tree, whose root has $D_x$ children and all other vertices have $d$ children.", "This approximation is a local law, valid for intervals containing down to $N^\\kappa $ eigenvalues.", "See Remark REF as well as Remark REF and Appendix REF below for details.", "Remark 3.9 In [42] it is shown that (REF ) holds with probability at least $1 - \\mathcal {C}N^{-\\nu }$ for all eigenvectors provided that $ \\mathcal {C} \\log N \\leqslant d \\leqslant N/2\\,.$ This shows that the upper bound in (REF ) is in fact not restrictive.", "Remark 3.10 (Optimality of (REF ) and (REF )) Both lower bounds in (REF ) and (REF ) are optimal (up to the value of $\\mathcal {C}$ ), in the sense that delocalization fails in each case if these lower bounds are relaxed.", "See Section REF below." ], [ "Extension to general sparse random matrices", "Our results, Theorems REF , REF , and REF , hold also for the following family of sparse Wigner matrices.", "Let $A = (A_{xy})$ be the adjacency matrix of $\\mathbb {G}(N,d/N)$ as above and $W=(W_{xy})$ be an independent Wigner matrix with bounded entries.", "That is, $W$ is Hermitian and its upper triangular entries $(W_{xy} \\mathrel {\\hbox{.}\\hbox{.", "}}$ x y)$ are independent complex-valued random variables with mean zero and variance one, $ EWxy 2 = 1$, and $ Wxy K$ almost surely for some constant $ K$.", "Then we define the \\emph {sparse Wigner matrix} $ M = (Mxy)$ as the Hadamard product of $ A$ and $ W$, with entries $ Mxy =Axy Wxy$.", "Since the entries of $ M / d$ are centred, it does not have a trivial eigenvalue like $ A / d$.$ Theorem 3.11 Let $M = (M_{xy})_{x,y \\in [N]}$ be a sparse Wigner matrix.", "Define $ \\alpha _x = \\frac{1}{d} \\sum _{y \\in [N]} \\vert M_{xy} \\vert ^2.$ Theorems REF and REF hold with (REF ) if $A$ is replaced with $M$ , and Theorem REF holds with (REF ) if $\\lambda _{i + 1}(A/\\sqrt{d})$ , $\\lambda _{N-i+1}(A/\\sqrt{d})$ , and $\\lambda _i(A/\\sqrt{d})$ are replaced with $\\lambda _{i}(M / \\sqrt{d})$ , $\\lambda _{N-i+1}(M / \\sqrt{d})$ , and $\\lambda _i(M / \\sqrt{d})$ , respectively.", "Here, the constants $\\mathcal {C}$ depend on $K$ in addition to $\\nu $ and $\\kappa $ .", "The modifications to the proofs of Theorems REF and REF required to establish Theorem REF are minor and follow along the lines of [10].", "The modification to the proof of Theorem REF is trivial, since the assumptions of the general Theorem REF below include the sparse Wigner matrix $M$ .", "We also remark that, with some extra work, one can relax the boundedness assumption on the entries of $W$ , which we shall however not do here." ], [ "The limits of sparseness and the scale $d \\asymp \\sqrt{\\log N}$", "We conclude this section with a discussion on how sparse $\\mathbb {G}$ can be for our results to remain valid.", "We show that all of our results – Theorems REF , REF , and REF – are wrong below the regime (REF ), i.e.", "if $d$ is smaller than order $\\sqrt{\\log N}$ .", "Thus, our sparseness assumptions – the lower bounds on $d$ from (REF ) and (REF ) – are optimal (up to the factor $\\log \\log N$ in (REF ) and the factor $\\mathcal {C}$ in (REF )).", "The fundamental reason for this change of behaviour will turn out to be that the ratio $\\vert S_2(x) \\vert / \\vert S_1(x) \\vert $ concentrates if and only if $d \\gg \\sqrt{\\log N}$ , where $S_i(x)$ denotes the sphere in $\\mathbb {G}$ of radius $i$ around $x$ .", "This can be easily made precise with a well-known tuning fork construction, detailed below.", "In the critical and subcritical regime $1 \\ll d = O(\\log N)$ , the graph $\\mathbb {G}$ is in general not connected, but with probability $1 - o(1)$ it has a unique giant component $\\mathbb {G}_{\\mathrm {giant}}$ with at least $N (1 - \\mathrm {e}^{- d/4})$ vertices (see Corollary REF below).", "Moreover, the spectrum of $A / \\sqrt{d}$ restricted to the complement of the giant component is contained in the $O\\bigl (\\frac{\\sqrt{\\log N}}{d}\\bigr )$ -neighbourhood of the origin (see Corollary REF below).", "Since we always assume $d \\geqslant \\mathcal {C} \\sqrt{\\log N}$ and we only consider eigenvalues in $\\mathbb {R}\\setminus [-\\kappa ,\\kappa ]$ , we conclude that all of our results listed above only pertain to the eigenvalues and eigenvectors of the giant component.", "For $D = 0,1,2,\\dots $ we introduce a starFor simplicity we only consider stars, but the same argument can be applied to arbitrary trees.", "tuning fork of degree D rooted in $\\mathbb {G}_{\\mathrm {giant}}$, or $D$ -tuning fork for short, which is obtained by taking two stars with central degree $D$ and connecting their hubs to a common base vertex in $\\mathbb {G}_{\\mathrm {giant}}$ .", "We refer to Figure REF for an illustration and Definition REF below for a precise definition.", "Figure: A star tuning fork of degree 12 rooted in a graph.", "The tuning fork is highlighted in blue.", "Its base is filled with red and its two hubs are filled with blue.It is not hard to see that every $D$ -tuning fork gives rise to two eigenvalues $\\pm \\sqrt{D/d}$ of $A / \\sqrt{d}$ restricted to $\\mathbb {G}_{\\mathrm {giant}}$ , whose associated eigenvectors are supported on the stars (see Lemma REF below).", "We denote by $\\Sigma \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =$ the spectrum of $ A / d$ restricted to $ Ggiant$ generated by the tuning forks.", "Any eigenvector associated with an eigenvalue $ D/d $ is localized on precisely $ 2D + 2$ vertices.", "Thus, $ D$-tuning forks provide a simple way of constructing localized states.", "Note that this is a very basic form of concentration of mass, supported at the periphery of the graph on special graph structures, and is unrelated to the much more subtle concentration in the semilocalized phase described in Section \\ref {sec:localized_results}.$ For $d > 0$ and $D \\in \\mathbb {N}$ we now estimate the number of $D$ -tuning forks in $\\mathbb {G}(N,d/N)$ , which we denote by $F(d,D)$ .", "The following result is proved in Appendix REF .", "Lemma 3.12 (Number of $D$ -tuning forks) Suppose that $1 \\ll d = b \\log N = O(\\log N)$ and $0 \\leqslant D \\ll \\log N / \\log \\log N$ .", "Then $F(d,D) = N^{1 - 2b - 2b D + o(1)}$ with probability $1 - o(1)$ .", "Defining $D_* \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =N2d - 1$, we immediately deduce the following result.$ Corollary 3.13 For any constant $\\varepsilon > 0$ with probability $1 - o(1)$ the following holds.", "If $D_* \\leqslant -\\varepsilon $ then $\\Sigma = \\emptyset $ .", "If $D_* \\geqslant \\varepsilon $ then $\\Sigma = $ .", "We deduce that if $d \\leqslant (1/2 - \\varepsilon ) \\log N$ then $\\Sigma \\ne \\emptyset $ and hence the delocalization for all eigenvectors from Remark REF fails.", "Hence, the lower bound (REF ) is optimal up to the value of $\\mathcal {C}$ .", "Similarly, for $d \\gg \\sqrt{\\log N}$ the set $\\Sigma $ is in general nonempty, but we always have $\\Sigma \\subset [-\\kappa , \\kappa ]$ for any fixed $\\kappa > 0$ , so that eigenvalues from $\\Sigma $ do not interfere with the statements of Theorems REF , REF , and REF .", "On the other hand, if $d = \\sqrt{\\log N} / t$ for constant $t$ , we find that $\\Sigma $ is asymptotically dense in the interval $[-t/\\sqrt{2}, t / \\sqrt{2}]$ .", "Since the conclusions of Theorems REF , REF , and REF are obviously wrong for any eigenvalue from $\\Sigma $ , they must all be wrong for large enough $t$ .", "This shows that the lower bounds $d$ from (REF ) and (REF ) are optimal (up to the factor $\\log \\log N$ in (REF ) and the factor $\\mathcal {C}$ in (REF )).", "In fact, the emergence of the tuning fork eigenvalues of order one and the failure of all of our proofs has the same underlying root cause, which singles out the scale $d \\asymp \\sqrt{\\log N}$ as the scale below which the concentration of the ratio $ \\vert S_2(x) \\vert / \\vert S_1(x) \\vert = d (1 + o(1))$ fails for vertices $x$ satisfying $D_x \\asymp d$ .", "Clearly, to have a $D$ -tuning fork with $D \\asymp d$ , (REF ) has to fail at the hubs of the stars.", "Moreover, (REF ) enters our proofs of both the semilocalized and the delocalized phase in a crucial way.", "For the former, it is linked to the validity of the local approximation by the $(D_x,d)$ -regular tree from Appendix REF , which underlies also the construction of the localization profile vectors (see e.g.", "(REF ) below).", "For the latter, in the language of Definition REF below, it is linked to the property that most neighbours of any vertex are typical (see Proposition REF REF below)." ], [ "Acknowledgements", "The authors would like to thank Simone Warzel for helpful discussions.", "The authors gratefully acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.", "715539_RandMat) and from the Swiss National Science Foundation through the NCCR SwissMAP grant.", "In this preliminary section we introduce some basic notations and definitions that are used throughout the paper, and give an overview of the proofs of Theorems REF (semilocalized phase) and REF (delocalized phase).", "These proofs are unrelated and, thus, explained separately.", "For simplicity, in this overview we only consider qualitative error terms of the form $o(1)$ , although all of our estimates are in fact quantitative." ], [ "Basic definitions", "We write $\\mathbb {N}= \\lbrace 0,1,2,\\dots \\rbrace $ .", "We set $[n] \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={1, ..., n}$for any $ n N*$ and $ [0] =$.We write $ X $ for the cardinality of a finite set $ X$.We use $ 1$ as symbol for the indicator function of the event $$.$ Vectors in $\\mathbb {R}^N$ are denoted by boldface lowercase Latin letters like $ȗ$ , $v̑$ and $w̑$ .", "We use the notation $v̑ = (v_x)_{x \\in [N]} \\in \\mathbb {R}^N$ for the entries of a vector.", "We denote by $\\operatorname{supp}v̑ \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={x [N] vx 0}$ the support of a vector $ v̑$.", "We denote by $ v̑ 2mu, w̑ = x [N] vx wx$ the Euclidean scalar product on $ RN$ and by $ v̑ = v̑ 2mu, v̑$ the induced Euclidean norm.For a matrix $ M RN N$, $ M $ is its operator norm induced by the Euclidean norm on $ RN$.For any $ x [N]$, we define the standard basis vector $ x =(xy)y [N] RN$.To any subset $ S [N]$ we assign the vector $ SRN$ given by $ S =x S x$.In particular, $ { x} = x$.$ We use blackboard bold letters to denote graphs.", "Let $\\mathbb {H} = (V(\\mathbb {H}), E(\\mathbb {H}))$ be a (simple, undirected) graph on the vertex set $V(\\mathbb {H}) = [N]$ .", "We often identify a graph $\\mathbb {H}$ with its set of edges $E(\\mathbb {H})$ .", "We denote by $A^{\\mathbb {H}} \\in \\lbrace 0,1\\rbrace ^{N \\times N}$ the adjacency matrix of $\\mathbb {H}$ .", "For $r \\in \\mathbb {N}$ and $x \\in [N]$ , we denote by $B_r^{\\mathbb {H}}(x)$ the closed ball of radius $r$ around $x$ in the graph $\\mathbb {H}$ , i.e.", "the set of vertices at distance (with respect to $\\mathbb {H}$ ) at most $r$ from the vertex $x$ .", "We denote the sphere of radius $r$ around the vertex $x$ by $S_r^{\\mathbb {H}}(x) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =BrH(x) Br - 1H(x)$.", "We denote by $ DxH$ the degree of the vertex $ x$ in the graph $ H$.", "For any subset $ V [N]$, we denote by $ H V$ the subgraph induced by $ H$ on $ V$.", "If $ H$ is a subgraph of $ G$ then we denote by $ G H$ the graph on $ [N]$ with edge set $ E(G) E(H)$.", "In the above definitions, if the graph $ H$ is the Erdős-Rényi graph $ G$, we systematically omit the superscript $ G$.$ The following notion of very high probability is a convenient shorthand used throughout the paper.", "It simplifies considerably the probabilistic statements of the kind that appear in Theorems REF , REF , and REF .", "It also introduces two special symbols, $\\nu $ and $\\mathcal {C}$ , which appear throughout the rest of the paper.", "Definition 4.1 Let $\\Xi \\equiv \\Xi _{N,\\nu }$ be a family of events parametrized by $N \\in \\mathbb {N}$ and $\\nu > 0$ .", "We say that $\\Xi $ holds with very high probability if for every $\\nu > 0$ there exists $\\mathcal {C}\\equiv \\mathcal {C}_\\nu $ such that $\\mathbb {P}(\\Xi _{N,\\nu }) \\geqslant 1 - \\mathcal {C}_\\nu N^{-\\nu }$ for all $N \\in \\mathbb {N}$ .", "Convention 4.2 In statements that hold with very high probability, we use the special symbol $\\mathcal {C} \\equiv \\mathcal {C}_\\nu $ to denote a generic positive constant depending on $\\nu $ such that the statement holds with probability at least $1 - \\mathcal {C}_\\nu N^{-\\nu }$ provided $\\mathcal {C}_\\nu $ is chosen large enough.", "Thus, the bound $\\vert X \\vert \\leqslant \\mathcal {C}Y$ with very high probability means that, for each $\\nu >0$ , there is a constant $\\mathcal {C}_\\nu >0$ , depending on $\\nu $ , such that $ \\mathbb {P}\\big ( \\vert X \\vert \\leqslant \\mathcal {C}_\\nu Y \\big ) \\geqslant 1 - \\mathcal {C}_\\nu N^{-\\nu } $ for all $N \\in \\mathbb {N}$ .", "Here, $X$ and $Y$ are allowed to depend on $N$ .", "We also write $X = \\mathcal {O}(Y)$ to mean $\\vert X \\vert \\leqslant \\mathcal {C} Y$ .", "We remark that the notion of very high probability from Definition REF survives a union bound involving $N^{O(1)}$ events.", "We shall tacitly use this fact throughout the paper.", "Moreover, throughout the paper, the constant $\\mathcal {C} \\equiv \\mathcal {C}_\\nu $ in the assumptions (REF ) and (REF ) is always assumed to be large enough." ], [ "Overview of proof in semilocalized phase", "The starting point of the proof of Theorem REF is the following simple observation.", "Suppose that $M$ is a Hermitian matrix with eigenvalue $\\lambda $ and associated eigenvector $w̑$ .", "Let $\\Pi $ be an orthogonal projection and write $\\overline{\\Pi } \\!\\,\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =I - $.", "If $$ is not an eigenvalue of $  M  $ then from $ (M - ) w̑ = 0$ we deduce\\begin{equation} \\overline{\\Pi } \\!\\,w̑ = - (\\overline{\\Pi } \\!\\,M \\overline{\\Pi } \\!\\,- \\lambda )^{-1} \\overline{\\Pi } \\!\\,M \\Pi w̑\\,.\\end{equation}If $$ is an eigenprojection of $ M$ whose range contains the eigenspace of $$ (for instance $ = w̑ w̑*$ if $$ is simple) then clearly both sides of (\\ref {intro_ev}) vanish.", "The basic idea of our proof is to apply an approximate version of this observation to $ M = A / d$, by choosing $$ appropriately, and showing that the left-hand side of (\\ref {intro_ev}) is small by estimating the right-hand side.$ In fact, we chooseThis projection $\\Pi $ is denoted by $\\Pi ^\\tau _{\\lambda ,\\delta }$ in the proof of Theorem REF below.", "$ \\Pi \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x W, v̑(x)   v̑(x)* , where $\\mathcal {W}_{\\lambda ,\\delta }$ is the set (REF ) of resonant vertices at energy $\\lambda $ , and $v̑(x)$ is the exponentially decaying localization profile from Theorem REF .", "The proof then consists of two main ingredients: $\\Vert \\overline{\\Pi } \\!\\,M \\Pi \\Vert = o(1)$ ; $\\overline{\\Pi } \\!\\,M \\overline{\\Pi } \\!\\,$ has a spectral gap around $\\lambda $ .", "Informally, REF states that $\\Pi $ is close to a spectral projection of $M$ , as $\\overline{\\Pi } \\!\\,M \\Pi = [M,\\Pi ] \\Pi $ quantifies the noncommutativity of $M$ and $\\Pi $ on the range of $\\Pi $ .", "Similarly, REF states that $\\Pi $ projects roughly onto an eigenspace of $M$ of energies near $\\lambda $ .", "Plugging REF and REF into () yields an estimate on $\\Vert \\overline{\\Pi } \\!\\,w̑ \\Vert $ from which Theorem REF follows easily.", "Thus, the main work of the proof is to establish the properties REF and REF for the specific choice of $\\Pi $ from (REF ).", "The construction of the localization profile $v̑(x)$ uses the pruned graph $\\mathbb {G}_\\tau $ from [10], a subgraph of $\\mathbb {G}$ depending on a threshold $\\tau > 1$ , which differs from $\\mathbb {G}$ by only a small number of edges and whose balls of radius $r_\\star $ around the vertices of $\\mathcal {V}_\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =$ are disjoint(see Proposition \\ref {prop:subgraph_separating_large_degrees} below).Now we set $ v̑(x) =v̑+(x)$, where, for $ = $ and $ > 1$,\\begin{equation} v̑^\\tau _\\sigma (x) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation}=\\sum _{i = 0}^{r_\\star } \\sigma ^i u_i(x) _{S_i^{\\mathbb {G}_\\tau }(x)} / \\Vert _{S_i^{\\mathbb {G}_\\tau }(x)} \\Vert \\,, \\qquad u_i(x) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x(x - 1)i/2   u0    (1 i r) .", "The motivation behind this choice is explained in Appendix REF : with high probability, the $r_\\star $ -neighbourhood of $x$ in $\\mathbb {G}_\\tau $ looks roughly like that of the root of infinite regular tree $\\mathbb {T}_{D_x, d}$ whose root has $D_x$ children and all other vertices $d$ children.", "The adjacency matrix of $\\mathbb {T}_{D_x, d}$ has the exact eigenvalues $\\pm \\sqrt{d} \\Lambda (\\alpha _x)$ with the corresponding eigenvectors given by () with $\\mathbb {G}_\\tau $ replaced with $\\mathbb {T}_{D_x, d}$ .", "The central idea of our proof is the introduction of a block diagonal approximation of the pruned graph.", "Define the orthogonal projections $\\Pi ^\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x V2 + o(1) = v̑(x) v̑(x)* ,         =I -  .", "The range of $\\Pi $ from (REF ) is a subspace of the range of $\\Pi ^\\tau $ , i.e.", "$\\Pi \\Pi ^\\tau = \\Pi $ .", "The interpretation of $\\Pi ^\\tau $ is the orthogonal projection onto all localization profiles around vertices $x$ with normalized degree at least $2 + o(1)$ , which is precisely the set of vertices around which one can define an exponentially decaying localization profile.", "Now we define the block diagonal approximation of the pruned graph as $\\widehat{H}^\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x V2 + o(1) = (x) v̑(x) v̑(x)* +  H   ; here we defined the centred and scaled adjacency matrix $H^\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =AG / d - E$, where $ E$ is a suitably chosen matrix that is close to $ EAG / d$ and preserves the locality of $ AG$ in balls around the vertices of $ V$.In the subspace spanned by the localization profiles $$, $ H$ is diagonal with eigenvalues $ (x)$.", "In the orthogonal complement, it is equal to $ H$.", "The off-diagonal blocks are zero.", "The main work of our proof consists in an analysis of $ H$.$ In terms of $\\widehat{H}^\\tau $ , abbreviating $H \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(AG - EAG) / d$, the problem of showing \\ref {item:pf_a} and \\ref {item:pf_b} reduces to showing\\begin{enumerate}[label=()]\\item \\Vert H - \\widehat{H}^\\tau \\Vert = o(1),\\item \\Vert \\overline{\\Pi } \\!\\,^\\tau H^\\tau \\overline{\\Pi } \\!\\,^\\tau \\Vert \\leqslant 2 + o(1).\\end{enumerate}Indeed, ignoring minor issues pertaining to the centring $ EAG$, we replace $ M = AG / d$ with $ H$ in \\ref {item:pf_a} and \\ref {item:pf_b}.", "Then \\ref {item:pf_a} follows immediately from \\ref {item:pf_c}, since $  H =  H + o(1) = o(1)$, as $  H= 0$ by the block structure of $ H$ and the relation $ = $.", "To show \\ref {item:pf_b}, we note that the $$-block of $ H$, $ H= x V2 + o(1) = (x) v̑(x) v̑(x)*$, trivially has a spectral gap: $  H  $ has no eigenvalues in the $$-neighbourhood of $$, simply because the projection $  $ removes the projections $ v̑(x) v̑(x)*$ with eigenvalues $ (x)$ in the $$-neighbourhood of $$.", "Moreover, the $  $-block also has such a spectral gap by \\ref {item:pf_d} and $ > 2 + o(1)$.", "Hence, by \\ref {item:pf_c}, we deduce the desired spectral gap \\ref {item:pf_b}.$ Thus, what remains is the proof of and .", "To prove , we prove $\\Vert H - H^\\tau \\Vert = o(1)$ and $\\Vert H^\\tau - \\widehat{H}^\\tau \\Vert = o(1)$ .", "The bound $\\Vert H - H^\\tau \\Vert = o(1)$ follows from a detailed analysis of the graph $\\mathbb {G} \\setminus \\mathbb {G}_\\tau $ removed from $\\mathbb {G}$ to obtain the pruned graph $\\mathbb {G}_\\tau $ , which we decompose as a union of a graph of small maximal degree and a forest, to which standard estimates of adjacency matrices of graphs can be applied (see Lemma REF below).", "To prove $\\Vert H^\\tau - \\widehat{H}^\\tau \\Vert = o(1)$ , we first prove that $v̑^\\tau _\\sigma (x)$ is an approximate eigenvector of $H^\\tau $ with approximate eigenvalue $\\sigma \\Lambda (\\alpha _x)$ (see Proposition REF below).", "Then we deduce $\\Vert H^\\tau - \\widehat{H}^\\tau \\Vert = o(1)$ using that the balls $B_{2r_\\star }(x)$ , $x \\in \\mathcal {V}_{2 + o(1)}$ , are disjoint and the locality of the operator $H^\\tau $ (see Lemma REF below).", "Thus we obtain .", "Finally, we sketch the proof of .", "The starting point is an observation going back to [15], [10]: from an estimate on the spectral radius of the nonbacktracking matrix associated with $H$ from [15] and an Ihara–Bass-type formula relating the spectra of $H$ and its nonbacktracking matrix from [15], we obtain the quadratic form inequality $\\vert H \\vert \\leqslant I + Q + o(1)$ with very high probability, where $Q = \\operatorname{diag}(\\alpha _x \\mathrel {\\hbox{.}\\hbox{.", "}}$ x [N])$, $ H $ is the absolute value of the Hermitian matrix $ H$, and $ o(1)$ is in the sense of operator norm (see Proposition \\ref {prop:operator_upper_bound} below).", "Using \\ref {item:pf_c}, we deduce the inequality\\begin{equation} \\vert \\widehat{H}^\\tau \\vert \\leqslant I + Q + o(1)\\,.\\end{equation}To estimate $  H   $, we take a normalized eigenvector $ w̑$ of $  H  $ with maximal eigenvalue $ > 0$.", "Thus, $ w̑ v̑(x)$ for all $ x V2 + o(1)$.", "We estimate $  H  $ from above (an analogous argument yields an estimate from below) using (\\ref {IB_intro}) to get\\begin{equation} \\lambda \\leqslant 1 + o(1) + \\sum _x \\alpha _x w_x^2 \\leqslant 1 + \\tau + o(1) + \\max _x \\alpha _x \\sum _{x \\in \\mathcal {V}_\\tau } w_x^2\\,.\\end{equation}Choosing $ = 1 + o(1)$, we see that \\ref {item:pf_d} follows provided that we can show that\\begin{equation} \\sum _{x\\in \\mathcal {V}_\\tau }w_{x}^{2} = o(1 / \\log N)\\,,\\end{equation}since $ x x C N$ with very high probability.$ The estimate () is a delocalization bound, in the vertex set $\\mathcal {V}_\\tau $ , for any eigenvector $w̑$ of $\\widehat{H}^\\tau $ that is orthogonal to $v̑_\\pm ^\\tau (x)$ for all $x \\in \\mathcal {V}_{2 + o(1)}$ and whose associated eigenvalue is larger than $2 \\tau + o(1)$ .", "It crucially relies on the assumption that $w̑ \\perp v̑_\\pm ^\\tau (x)$ for all $x \\in \\mathcal {V}_{2 + o(1)}$ , without which it is false (see Proposition REF below).", "The underlying principle behind its proof is the same as that of the Combes–Thomas estimate [25]: the Green function $((\\lambda - Z)^{-1})_{ij}$ of a local operator $Z$ at a spectral parameter $\\lambda $ separated from the spectrum of $Z$ decays exponentially in the distance between $i$ and $j$ , at a rate inversely proportional to the distance from $\\lambda $ to the spectrum of $Z$ .", "We in fact use a radial form of a Combes–Thomas estimate, where $Z$ is the tridiagonalization of a local restriction of $\\widehat{H}^\\tau $ around a vertex $x \\in \\mathcal {V}_\\tau $ (see Appendix REF ) and $i,j$ index radii of concentric spheres.", "The key observation is that, by the orthogonality assumption on $w̑$ , the Green function $((\\lambda - Z)^{-1})_{i r_\\star }$ , $0 \\leqslant i < r_\\star $ , and the eigenvector components in the radial basis $u_i$ , $0 \\leqslant i < r_\\star $ , satisfy the same linear difference equation.", "Thus we obtain exponential decay for the components $u_i$ , which yields $u_0^2 \\leqslant o(1/\\log N) \\sum _{i = 0}^{r_*} u_i^2$ .", "Going back to the original vertex basis, this implies that $w_x^2 \\leqslant o(1/\\log N) \\Vert w̑|_{B_{2r_\\star }^{\\mathbb {G}_\\tau }(x)}\\Vert ^2$ for all $x \\in \\mathcal {V}_\\tau $ , from which () follows since the balls $B_{2r_\\star }^{\\mathbb {G}_\\tau }(x)$ , $x \\in \\mathcal {V}_\\tau $ , are disjoint." ], [ "Overview of proof in delocalized phase", "The delocalization result of Theorem REF is an immediate consequence of a local law for the matrix $A / \\sqrt{d}$ , which controls the entries of the Green function $G \\equiv G(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(A / d - z)-1 in the form of high-probability estimates, for spectral scales $\\eta = \\operatorname{Im}z$ down to the optimal scale $1/N$ , which is the typical eigenvalue spacing.", "Such a local law was first established for $d \\gg (\\log N)^6$ in [35] and extended down to $d \\geqslant \\mathcal {C} \\log N$ in [42].", "In both of these works, the diagonal entries of $G$ are close to the Stieltjes transform of the semicircle law.", "In contrast, in the regime (REF ) the diagonal entry $G_{xx}$ is close to the Stieltjes transform of the spectral measure at the root of an infinite $(D_x,d)$ -regular tree.", "Hence, $G_{xx}$ does not concentrate around a deterministic quantity.", "The basic approach of the proof is the same as for any local law: derive an approximate self-consistent equation with very high probability, solve it using a stability analysis, and perform a bootstrapping from large to small values of $\\eta $ .", "For a set $T \\subset [N]$ denote by $A^{(T)}$ the adjacency matrix of the graph $\\mathbb {G}$ where the vertices of $T$ (and all incident edges) have been removed, and denote by $G^{(T)} = \\bigl (A^{(T)} / \\sqrt{d} - z\\bigr )^{-1}$ the associated Green function.", "In order to understand the emergence of the self-consistent equation, it is instructive to consider the toy situation where, for a given vertex $x$ , all neighbours $S_1(x)$ are in different connected components of $A^{(x)}$ .", "This is for instance the case if $\\mathbb {G}$ is a tree.", "On the global scale, where $\\eta $ is large enough, this assumption is in fact valid to a good approximation, since the neighbourhood of $x$ is with high probability a tree.", "Then a simple application of Schur's complement formula and the resolvent identity yield $ \\frac{1}{G_{xx}} = -z - \\frac{1}{d} \\sum _{y \\in S_1(x)} G_{yy}^{(x)} \\,, \\qquad G_{yy}^{(x)} - G_{yy} = (G_{yy}^{(x)})^2 \\frac{1}{d} G_{xx}\\,.$ Thus, on the global scale, using that $G$ is bounded, we obtain the self-consistent equation $ \\frac{1}{G_{xx}} = -z - \\frac{1}{d} \\sum _{y \\in S_1(x)} G_{yy} + o(1)$ with very high probability.", "It is instructive to solve the self-consistent equation (REF ) in the family $(G_{xx})_{x \\in [N]}$ on the global scale.", "To that end, we introduce the notion of typical vertices, which is roughly the set $\\mathcal {T} = $ .", "(In fact, as explained below, the actual definition for local scales has to be different; see (REF ) below.)", "A simple argument shows that with very high probability most neighbours of any vertex are typical.", "With this definition, we can try to solve (REF ) on the global scale as follows.", "From the boundedness of $G$ we obtain a self-consistent equation for the vector $(G_{xx})_{x \\in \\mathcal {T}}$ that reads $ \\frac{1}{G_{xx}} = -z - \\sum _{y \\in \\mathcal {T}} \\frac{1}{d} A_{xy} G_{yy} + \\zeta _x\\,, \\qquad \\zeta _x = o(1)\\,.$ It is not hard to see that the equation (REF ) has a unique solution, which satisfies $G_{xx} = m + o(1)$ for all $x \\in \\mathcal {T}$ .", "Here $m$ is the Stieltjes transform of the semicircle law, which satisfies $m = \\frac{1}{-z - m}$ .", "Plugging this solution back into (REF ) and using that most neighbours of any vertex are typical shows that for $x \\notin \\mathcal {T}$ we have $G_{xx} = m_{\\alpha _x} + o(1)$ , where $m_\\alpha \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1-z - m$.", "One readily finds (see Appendix \\ref {sec:mu} below) that $ mx$ is Stieltjes transform of the spectral measure of the infinite $ (Dx,d)$-regular tree at the root.$ The first main difficulty of the proof is to provide a derivation of identities of the form (REF ) (and hence a self-consistent equation of the form (REF )) on the local scale $\\eta \\ll 1$ .", "We emphasize that the above derivation of (REF ) is completely wrong on the local scale.", "Unlike on the global scale, on the local scale the behaviour of the Green function is not governed by the local geometry of the graph, and long cycles contribute to $G$ in an essential way.", "In particular, eigenvector delocalization, which follows from the local law, is a global property of the graph and cannot be addressed using local arguments; it is in fact wrong outside of the region $\\mathcal {S}_\\kappa $ , although the above derivation is insensitive to the real part of $z$ .", "We address this difficulty by replacing the identities (REF ) with the following argument, which ultimately provides an a posteriori justification of approximate versions of (REF ) with very high probability, provided we are in the region $\\mathcal {S}_\\kappa $ .", "We make an a priori assumption that the entries of $G$ are bounded with very high probability; we propagate this assumption from large to small scales using a standard bootstrapping argument and the uniform boundedness of the density of the spectral measure associated with $m_\\alpha $ .", "It is precisely this uniform boundedness requirement that imposes the restriction to $\\mathcal {S}_\\kappa $ in our local law (as explained in Remark REF , this restriction is necessary).", "The key tool that replaces the simpleminded approximation (REF ) is a series of large deviation estimates for sparse random vectors proved in [42], which, as it turns out, are effective for the full optimal regime (REF ).", "Thus, under the bootstrapping assumption that the entries of $G$ are bounded, we obtain (REF ) (and hence also (REF )), with some additional error terms, with very high probability.", "The second main difficulty of the proof is that, on the local scale and for sparse graphs, the self-consistent equation (REF ), which can be derived from (REF ) as explained above, is not stable enough to be solved in $(G_{xx})_{x \\in \\mathcal {T}}$ .", "This problem stems from the sparseness of the graphs that we are considering, and does not appear in random matrix theory for denser (or even heavy-tailed) matrices.", "Indeed, the stability estimates of (REF ) carry a logarithmic factor, which is usually of no concern in random matrix theory but is deadly for the sparse regime of this paper.", "This is a major obstacle and in fact ultimately dooms the self-consistent equation (REF ).", "To explain the issue, write the sum in (REF ) as $\\sum _y S_{xy} G_{yy}$ , where $S$ is the $\\mathcal {T} \\times \\mathcal {T}$ matrix $S_{xy} = \\frac{1}{d} A_{xy}$ .", "Writing $G_{xx} = m + \\varepsilon _x$ , plugging it into (REF ), and expanding to first order in $\\varepsilon _x$ , we obtain, using the definition of $m$ , that $\\varepsilon _x = -m^2 ((I - m^2 S)^{-1} \\zeta )_x$ .", "Thus, in order to deduce smallness of $\\varepsilon _x$ from the smallness of $\\zeta _x$ , we need an estimate on the normWe write $\\Vert \\cdot \\Vert _{p \\rightarrow p}$ for the operator norm on $\\ell ^p$ .", "$\\Vert (I - m^2 S)^{-1} \\Vert _{\\infty \\rightarrow \\infty }$ .", "In Appendix REF below we show that for typical $S$ , $\\operatorname{Re}z \\in \\mathcal {S}_\\kappa $ , and small enough $\\operatorname{Im}z$ , we have $ \\frac{\\log N}{C (\\log \\log N)^2} \\leqslant \\Vert (I - m^2 S)^{-1} \\Vert _{\\infty \\rightarrow \\infty } \\leqslant C_\\kappa \\log N$ for some universal constant $C$ and some constant $C_\\kappa $ depending on $\\kappa $ .", "In our context, where $\\zeta _x$ is small but much larger than the reciprocal of the lower bound of (REF ), such a logarithmic factor is not affordable.", "To address this difficulty, we avoid passing by the form (REF ) altogether, as it is doomed by (REF ).", "The underlying cause for the instability of (REF ) is the inhomogeneous structure of the matrix $S$ , which is essentially an adjacency matrix of a sparse graph.", "Thus, the solution is to derive a self-consistent equation of the form (REF ) but with an unstructured $S$ .", "The basic intuition is to replace the local average $\\frac{1}{d} \\sum _{y \\in S_1(x)} G_{yy}^{(x)}$ in the first identity of (REF ) with the global average $\\frac{1}{N} \\sum _{y \\ne x} G_{yy}^{(x)}$ .", "Of course, in general these two are not close, but we can include their closeness into the definition of a typical vertex.", "Thus, we define the set of typical vertices as $ \\mathcal {T} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={x [N] x = 1 + o(1)  ,  1d y S1(x) Gyy(x) = 1N y x Gyy(x) + o(1)} .", "The main work of the proof is then to prove the following facts with very high probability.", "Most vertices are typical.", "Most neighbours of any vertex are typical.", "With REF and REF at hand, we explain how to conclude the proof.", "Using REF and the approximate version of (REF ) established above, we deduce the self-consistent equation for typical vertices, $\\frac{1}{G_{xx}} = -z - \\frac{1}{\\vert \\mathcal {T} \\vert } \\sum _{y \\in \\mathcal {T}} G_{yy} + o(1)\\,, \\qquad x \\in \\mathcal {T}\\,,$ which, unlike (REF ), is stable (see Lemma REF below) and can be easily solved to show that $G_{xx} = m + o(1) = m_{\\alpha _x} + o(1)$ for all $x \\in \\mathcal {T}$ .", "Moreover, if $x \\notin \\mathcal {T}$ then we obtain from (REF ) and REF that $\\frac{1}{G_{xx}} = -z - \\frac{1}{d} \\sum _{y \\in S_1(x) \\cap \\mathcal {T}} G_{yy}^{(x)} + o(1) = -z - \\alpha _x m + o(1)\\,,$ where we used that $G_{yy} = m + o(1)$ for $y \\in \\mathcal {T}$ .", "This shows that $G_{xx} = m_{\\alpha _x} + o(1)$ for all $x \\in [N]$ with very high probability, and hence concludes the proof.", "What remains, therefore, is the proof of REF and REF ; see Proposition REF below for a precise statement.", "Using the bootstrapping assumption of boundedness of the entries of $G$ , it is not hard to estimate the probability $\\mathbb {P}(x \\in \\mathcal {T})$ , which we prove to be $1 - o(1)$ , although $\\lbrace x \\in \\mathcal {T}\\rbrace $ does not hold with very high probability (this characterizes the critical and subcritical regimes).", "Now if the events $\\lbrace x \\in \\mathcal {T}\\rbrace $ , $x \\in [N]$ , were all independent, it would then be a simple matter to deduce REF and REF .", "The most troublesome source of dependence among the events $\\lbrace x \\in \\mathcal {T}\\rbrace $ , $x \\in [N]$ , is the Green function $G_{yy}^{(x)}$ in the definition of $\\mathcal {T}$ .", "Thus, the main difficulty of the proof is a decoupling argument that allows us to obtain good decay for the probability $\\mathbb {P}(T \\subset \\mathcal {T})$ in the size of $T$ .", "This decay can only work up to a threshold in the size of $T$ , beyond which the correlations among the different events kick in.", "In fact, we essentially prove that $ \\mathbb {P}(T \\subset \\mathcal {T}) \\leqslant \\mathrm {e}^{- o(1) d \\vert T \\vert } + \\mathcal {C} N^{-\\nu } \\qquad \\text{for} \\quad \\vert T \\vert = o(d)\\,;$ see Lemma REF .", "Choosing the largest possible $T$ , $T = o(d)$ , we find that the first term on the right-hand side of (REF ) is bounded by $N^{-\\nu }$ provided that $o(1) d^2 \\geqslant \\nu \\log N$ , which corresponds precisely to the optimal lower bound in (REF ).", "Using (REF ), we may deduce REF and REF .", "To prove (REF ), we need to decouple the events $\\lbrace x \\in \\mathcal {T}\\rbrace $ , $x \\in T$ .", "We do so by replacing the Green functions $G^{(x)}$ in the definition of $\\mathcal {T}$ by $G^{(T)}$ , after which the corresponding events are essentially independent.", "The error that we incur depends on the difference $G^{(T)}_{yy} - G_{yy}$ , which we have to show is small with very high probability under the bootstrapping assumption that the entries of $G$ are bounded.", "For $T$ of fixed size, this follows easily from standard resolvent identities.", "However, for our purposes it is crucial that $T$ can have size up to $o(d)$ , which requires a more careful quantitative analysis.", "As it turns out, $G^{(T)}_{yy} - G_{yy}$ is small only up to $\\vert T \\vert = o(d)$ , which is precisely what we need to reach the optimal scale $d \\gg \\sqrt{\\log N}$ from (REF )." ], [ "The semilocalized phase", "In this section we prove the results of Section REF – Theorems REF and REF ." ], [ "The pruned graph and proof of Theorem ", "The balls $(B_r(x))_{x \\in \\mathcal {W}_{\\lambda , \\delta }}$ in Theorem REF are in general not disjoint.", "For its proof, and in order to give a precise definition of the vector $v̑(x)$ in Theorem REF , we need to make these balls disjoint by pruning the graph $\\mathbb {G}$ .", "This is an important ingredient of the proof, and will also allow us to state a more precise version of Theorem REF , which is Theorem REF below.", "This pruning was previously introduced in [10]; it is performed by cutting edges from $\\mathbb {G}$ in such a way that the balls $(B_r(x))_{x \\in \\mathcal {W}_{\\lambda , \\delta }}$ are disjoint for appropriate radii, $r = 2 r_\\star $ , by carefully cutting in the right places, thus reducing the number of cut edges.", "This ensures that the pruned graph is close to the original graph in an appropriate sense.", "The pruned graph, $\\mathbb {G}_\\tau $ , depends on a parameter $\\tau > 1$ , and its construction is the subject of the following proposition.", "To state it, we introduce the following notations.", "For a subgraph $\\mathbb {G}_\\tau $ of $\\mathbb {G}$ we abbreviate $B^\\tau _i(x) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =BGi(x) ,       Si(x) =SGi(x) .", "Moreover, we define the set of vertices with large degrees $\\mathcal {V}_\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ = .", "Proposition 5.1 (Existence of pruned graph) Let $1 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2$ and $d \\leqslant 3 \\log N$ .", "There exists a subgraph $\\mathbb {G}_\\tau $ of $\\mathbb {G}$ with the following properties.", "Any path in $\\mathbb {G}_\\tau $ connecting two different vertices in $\\mathcal {V}_\\tau $ has length at least $4 r_{\\star } +1$ .", "In particular, the balls $(B_{2 r_{\\star }}^{\\tau }(x))_{x \\in \\mathcal {V}_\\tau }$ are disjoint.", "The induced subgraph $\\mathbb {G}_\\tau |_{B_{2 r_{\\star }}^{\\tau }(x)}$ is a tree for each $x \\in \\mathcal {V}_\\tau $ .", "For each edge in $\\mathbb {G}\\setminus \\mathbb {G}_\\tau $ , there is at least one vertex in $\\mathcal {V}_\\tau $ incident to it.", "For each $x \\in \\mathcal {V}_\\tau $ and each $i \\in \\mathbb {N}$ satisfying $1 \\leqslant i \\leqslant 2 r_{\\star }$ we have $S_i^{\\tau }(x) \\subset S_i(x)$ .", "The degrees induced on $[N]$ by $\\mathbb {G}\\setminus \\mathbb {G}_\\tau $ are bounded according to $\\max _{x \\in [N]} D_x^{\\mathbb {G} \\setminus \\mathbb {G}_\\tau } \\leqslant \\mathcal {C} \\frac{\\log N}{(\\tau -1)^2d}$ with very high probability.", "Suppose that $\\sqrt{\\log N} \\leqslant d$ .", "For each $x \\in \\mathcal {V}_\\tau $ and all $2 \\leqslant i \\leqslant 2 r_\\star $ , the bound $|S_{i}(x)\\setminus S_{i}^{\\tau }(x)|\\leqslant \\mathcal {C}\\frac{\\log N}{(\\tau -1)^2}d^{i-2}$ holds with very high probability.", "The proof of Proposition REF is postponed to the end of this section, in Section REF below.", "It is essentially [10], the main difference being that REF is considerably sharper than its counterpart, [10]; this stronger bound is essential to cover the full optimal regime (REF ) (see Section REF ).", "As a guide for the reader's intuition, we recall the main idea of the pruning.", "First, for every $x \\in \\mathcal {V}_\\tau $ , we make the $2 r_\\star $ -neighbourhood of $x$ a tree by removing appropriate edges incident to $x$ .", "Second, we take all paths of length less than $4 r_\\star + 1$ connecting different vertices in $\\mathcal {V}_\\tau $ , and remove all of their edges incident to any vertex in $\\mathcal {V}_\\tau $ .", "Note that only edges incident to vertices in $\\mathcal {V}_\\tau $ are removed.", "This informal description already explains properties REF –REF .", "Properties REF and REF are probabilistic in nature, and express that with very high probability the pruning has a small impact on the graph.", "See also Lemma REF below for a statement in terms of operator norms of the adjacency matrices.", "For the detailed algorithm, we refer to the proof of [10].", "Using the pruned graph $\\mathbb {G}_\\tau $ , we can give a more precise formulation of Theorem REF , where the localization profile vector $v̑(x)$ from Theorem REF is explicit.", "For its statement, we introduce the set of vertices $ \\mathcal {V} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =V2 + 1/4 around which a localization profile can be defined.", "Definition 5.2 (Localization profile) Let $1 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2$ and $\\mathbb {G}_\\tau $ be the pruned graph from Proposition REF .", "For $x \\in \\mathcal {V}$ we introduce positive weights $u_0(x), u_1(x), \\dots , u_{r_\\star }(x)$ as follows.", "Set $u_0(x) > 0$ and define, for $i = 1, \\dots , r_\\star - 1$ , $ u_i(x) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x(x - 1)i/2   u0(x) ,       ur(x) =1(x - 1)(r- 1)/2   u0(x) .", "For $\\sigma = \\pm $ we define the radial vector $ v̑^\\tau _\\sigma (x) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =i = 0r i ui(x) Si(x)Si(x)  ,        and choose $u_0(x) > 0$ such that $v̑^\\tau _\\sigma (x)$ is normalized.", "Remark 5.3 The family $(v̑_\\sigma ^\\tau (x) \\mathrel {\\hbox{.}\\hbox{.", "}}$ x V,  = )$ is orthonormal.", "Indeed, if $ x,y V$ are distinct, then by Proposition \\ref {prop:subgraph_separating_large_degrees} \\ref {item:subgraph_paths} the vectors $ v̑(x)$ and $ v̑(y)$ are orthogonal for any $ , = $ because they are supported on disjoint sets of vertices.", "Moreover, $ v̑+(x)$ and $ v̑-(x)$ are orthogonal by the choice of $ ur(x)$ from (\\ref {ui_definition}), as can be seen by a simple computation.$ The following result restates Theorem REF by identifying $v̑(x)$ there as $v̑_+^\\tau (x)$ given in (REF ).", "It easily implies Theorem REF , and the rest of this section is devoted to its proof.", "Theorem 5.4 The following holds with very high probability.", "Suppose that $d$ satisfies (REF ).", "Let $w̑$ be a normalized eigenvector of $A/\\sqrt{d}$ with nontrivial eigenvalue $\\lambda \\geqslant 2+ \\mathcal {C} \\xi ^{1/2}$ .", "Choose $0<\\delta \\leqslant (\\lambda -2)/2$ and set $\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1 + (-2)/81$.", "Then\\begin{equation} \\sum _{x \\in \\mathcal {W}_{\\lambda ,\\delta }} \\langle v̑^\\tau _+(x) {2mu}, w̑\\rangle ^2 \\geqslant 1 -\\mathcal {C} \\mathopen {}\\mathclose {\\left(\\frac{ \\xi +\\xi _{\\tau -1}}{\\delta }\\right)}^2\\,.\\end{equation}$ Remark 5.5 An analogous result holds for negative eigenvalues $-\\lambda $ , where $\\lambda $ is as in Theorem REF and $v̑_+^\\tau (x)$ in () is replaced with $v̑_-^\\tau (x)$ .", "For the motivation behind Definition REF , we refer to the discussion in Section REF and Appendix REF .", "As explained there, if $\\mathbb {G}_\\tau $ is sufficiently close to the infinite tree $\\mathbb {T}_{D_x, d}$ in a ball of radius $r_\\star $ around $x$ , and if $r_\\star $ is large enough for $u_{r_\\star }(x)$ to be very small, we expect (REF ) to be an approximate eigenvector of $A$ .", "This will in fact turn out to be true; see Proposition REF below.", "That $r_\\star $ is in fact large enough is easy to see: the definition of $r_\\star $ in (REF ) and the bound $\\xi \\geqslant 1/d$ imply that, for $\\alpha _x\\geqslant 2+ C (\\log d)^2 / \\sqrt{\\log N}$ , we have $ (\\alpha _x - 1)^{-(r_\\star -2)/2} \\leqslant \\xi \\,.$ This means that the last element of the sequence $(u_i(x))_{i=0}^{r_\\star }$ is bounded by $\\xi $ .", "Note that the lower bound on $\\alpha _x$ imposed above always holds for $x \\in \\mathcal {V}$ , since, by (REF ), $\\frac{C (\\log d)^2}{\\sqrt{\\log N}} \\leqslant \\xi ^{1/4}\\,.$ Figure: An illustration of the three sets of vertices of increasing size that enter into the proof of Theorem .", "Each vertex xx is plotted as a dot at its normalized degree α x \\alpha _x.", "The largest set is 𝒱 τ \\mathcal {V}_{\\tau } from Proposition , where 1+ξ 1/2 ⩽τ⩽21 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2.", "It is used to define the pruned graph 𝔾 τ \\mathbb {G}_\\tau .", "The intermediate set is 𝒱≡𝒱 2+ξ 1/4 \\mathcal {V} \\equiv \\mathcal {V}_{2 + \\xi ^{1/4}} from ().", "It is the set of vertices for which we can define the localization profile vector v̑(x)v̑(x) that decays exponentially around xx.", "The smallest set 𝒲 λ,δ =Λ -1 ([λ-δ,λ+δ])\\mathcal {W}_{\\lambda ,\\delta } = \\Lambda ^{-1}([\\lambda - \\delta , \\lambda + \\delta ]) is the set of resonant vertices at energy λ\\lambda .As a guide to the reader, in Figure REF , we summarize the three main sets of vertices that are used in the proof of Theorem REF .", "We conclude this subsection by proving Theorem REF and Corollary REF using Theorem REF .", "[Proof of Theorem REF ] The first claim follows immediately from Theorem REF , with $v̑(x) = v̑^\\tau _+(x)$ .", "To verify the claim about the exponential decay of $v̑$ , we note that the graph distance in $\\mathbb {G}$ is bounded by the graph distance in $\\mathbb {G}_\\tau $ , which implies $\\sum _{y \\in B_r(x)^c} (v̑^\\tau _+(x))_y^2 \\leqslant \\sum _{y \\in B_r^\\tau (x)^c} (v̑^\\tau _+(x))_y^2 = \\sum _{i = r+1}^{r_\\star } u_i(x)^2\\,,$ from which the claim easily follows using the definition (REF ).", "[Proof of Corollary REF ] We decompose $w̑ = \\sum _{x \\in \\mathcal {W}_{\\lambda ,\\delta }} \\gamma _x v̑^\\tau _+(x) + ȇ$ , where $\\gamma _x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =v̑+(x) 2mu, w̑$ and $ ȇ$ is orthogonal to $ Span$.By Theorem \\ref {thm:general} we have $ ȇ C (+-1)$ and\\begin{equation} \\sum _{x \\in \\mathcal {W}_{\\lambda ,\\delta }} \\gamma _x^2\\geqslant 1-\\frac{\\mathcal {C} (\\xi +\\xi _{\\tau -1})}{\\delta }\\,.\\end{equation}Moreover, since $ - 2 $, we have $ W, V$, so that Proposition \\ref {prop:subgraph_separating_large_degrees} \\ref {item:subgraph_paths} implies $ (v̑+(x))y = xy u0(x)$ for $ x,y W, $.", "Thus we have\\begin{equation} \\sum _{y \\in \\mathcal {W}_{\\lambda ,\\delta }} w_y^2 = \\Vert w̑|_{\\mathcal {W}_{\\lambda ,\\delta }}\\Vert ^2 = \\Biggl \\Vert \\sum _{x \\in \\mathcal {W}_{\\lambda ,\\delta }} \\gamma _x v̑^\\tau _+(x) \\vert _{\\mathcal {W}_{\\lambda ,\\delta }} \\Biggr \\Vert ^2 + O(\\Vert ȇ \\Vert )= \\sum _{y \\in \\mathcal {W}_{\\lambda ,\\delta }} \\gamma _y^2 u_0(y)^2+\\mathcal {O}\\mathopen {}\\mathclose {\\left(\\frac{\\xi +\\xi _{\\tau -1}}{\\delta }\\right)}\\,.\\end{equation}Since $ u0(y)$ was chosen such that $ v̑+(y) $ is normalized, we find\\begin{equation*}u_0(y)^2=\\mathopen {}\\mathclose {\\left( 1 + \\sum _{i=1}^{r_\\star -1} \\frac{\\alpha _y}{(\\alpha _y-1)^i} +\\frac{1}{(\\alpha _y-1)^{r_\\star - 1}}\\right)}^{-1} = \\frac{\\alpha _y-2}{2(\\alpha _y-1)} + O \\biggl (\\frac{1}{(\\alpha _y - 1)^{r_\\star - 1}}\\biggr )\\,.\\end{equation*}Define $ =-1()$ for $ 2$.Since $ |(y)-|$ for $ yW, $, we obtain\\begin{equation*}|\\alpha _y - \\alpha | \\leqslant \\delta \\max _{t\\in [\\lambda -\\delta ,\\lambda +\\delta ]}( \\Lambda ^{-1})^{\\prime }(t) = O\\mathopen {}\\mathclose {\\left(\\delta \\lambda ^{3/2} (\\lambda -2)^{-1/2}\\right)}\\,,\\end{equation*}where we used that $ - 2 - 2$.", "Since $ dd - 22 (- 1) = 12(- 1)2 -4$, we find\\begin{equation} u_0(y)^2 = \\frac{\\alpha - 2}{2 (\\alpha - 1)} + O\\mathopen {}\\mathclose {\\left(\\frac{\\delta }{\\lambda ^{5/2} \\sqrt{\\lambda -2}} + \\frac{1}{(\\alpha _y - 1)^{r_\\star - 1}}\\right)}= \\frac{\\alpha - 2}{2 (\\alpha - 1)} + O \\mathopen {}\\mathclose {\\left( \\frac{\\delta }{\\lambda ^{5/2} \\sqrt{\\lambda -2}} + \\frac{\\xi }{\\delta }\\right)}\\,,\\end{equation}where we used (\\ref {eq:smallness_alpha_x_minus_r_star}) and the upper bound on $$ in the last step.By an elementary computation,\\begin{equation*}\\frac{\\alpha - 2}{2 (\\alpha - 1)} = \\frac{\\sqrt{\\lambda ^2 - 4}}{\\lambda + \\sqrt{\\lambda ^2 - 4}}\\,,\\end{equation*}and the claim hence follows by recalling (\\ref {eq:smallness_alpha_x_minus_r_star}) and plugging (\\ref {gamma_sum}) and (\\ref {u_0_estimate}) into (\\ref {eq:form_sum_center}).$" ], [ "Block diagonal approximation of pruned graph and proof of Theorems ", "We now introduce the adjacency matrix of $\\mathbb {G}_\\tau $ and a suitably defined centred version.", "Then we define a block diagonal approximation of this matrix, called $\\widehat{H}^\\tau $ in (REF ) below, which is the central construction of our proof.", "Definition 5.6 Let $A^\\tau $ be the adjacency matrix of $\\mathbb {G}_\\tau $ .", "Let $H \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =A   / d$ and $ H=A  / d$, where\\begin{equation}\\underline{A} \\!\\, \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation}=A - \\mathbb {E}A \\,, \\qquad \\underline{A}^\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =A- (EA) and $\\chi ^\\tau $ is the orthogonal projection onto $\\operatorname{Span}\\lbrace _y \\mathrel {\\hbox{.}\\hbox{.", "}}$ y x V B2 r(x)}$.$ The definition of $\\underline{A} \\!\\,^\\tau $ is chosen so that (i) $\\underline{A} \\!\\,^\\tau $ is close to $\\underline{A} \\!\\,$ provided that $A^\\tau $ is close to $A$ , since the kernel of $\\chi ^\\tau $ has a relatively low dimension, and (ii) when restricted to vertices at distance at most $2 r_\\star $ from $\\mathcal {V}_\\tau $ , the matrix $\\underline{A} \\!\\,^\\tau $ coincides with $A^\\tau $ .", "In fact, property (i) is made precise by the simple estimate $ \\Vert \\mathbb {E}A - \\chi ^\\tau (\\mathbb {E}A) \\chi ^\\tau \\Vert \\leqslant 2$ with very high probability (see [10] for details).", "Property (ii) means that $\\underline{A} \\!\\,^\\tau $ inherits the locality of the matrix $A$ , meaning that applying $\\underline{A} \\!\\,^\\tau $ to a vector localized in space to a small enough neighbourhood of $\\mathcal {V}_\\tau $ yields again a vector localized in space.", "This property will play a crucial role in the proof, and it can be formalized as follows.", "Remark 5.7 Let $i + j \\leqslant 2 r_\\star $ .", "Then for any $x \\in \\mathcal {V}_\\tau $ and vector $v̑$ we have $\\operatorname{supp}v̑ \\subset B_i^\\tau (x) \\quad \\Longrightarrow \\quad \\operatorname{supp}\\bigl [(H^\\tau )^j v̑\\bigr ] \\subset B_{i+j}^\\tau (x)\\,.$ The next result states that $H^\\tau $ is a small perturbation of $H$ .", "Lemma 5.8 Suppose that $d \\leqslant 3 \\log N$ .", "For any $1 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2$ we have $\\Vert H - H^\\tau \\Vert \\leqslant \\mathcal {C} \\xi _{\\tau -1}$ with very high probability.", "The next result states that $v̑_\\sigma ^\\tau (x)$ is an approximate eigenvector of $H^\\tau $ .", "Proposition 5.9 Let $d$ satisfy (REF ).", "Let $x \\in [N]$ and suppose that $1 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2$ .", "If $\\alpha _x\\geqslant 2+ C (\\log d)^2 / \\sqrt{\\log N}$ then for $\\sigma = \\pm $ we have $\\Vert (H^\\tau - \\sigma \\Lambda (\\alpha _x)) v̑^{\\tau }_\\sigma (x) \\Vert \\leqslant \\mathcal {C} \\xi $ with very high probability.", "The proofs of Lemma REF and Proposition REF are deferred to Section REF .", "The following object is the central construction in our proof.", "Definition 5.10 (Block diagonal approximation of pruned graph) Define the orthogonal projections $ \\Pi ^\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x V = v̑(x) v̑(x)* ,         =I -  , and the matrix $ \\widehat{H}^\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x V = (x) v̑(x) v̑(x)* +  H   .", "That $\\Pi ^\\tau $ and $\\overline{\\Pi } \\!\\,^\\tau $ are indeed orthogonal projections follows from Remark REF .", "Note that $\\widehat{H}^\\tau $ may be interpreted as a block diagonal approximation of $H^\\tau $.", "Indeed, completing the orthonormal family $(v̑^\\tau _\\sigma (x))_{x \\in \\mathcal {V}, \\sigma = \\pm }$ to an orthonormal basis of $\\mathbb {R}^N$ , which we write as the columns of the orthogonal matrix $R$ , we have $R^* \\widehat{H}^\\tau R =\\begin{bmatrix}\\operatorname{diag}(\\sigma \\Lambda (\\alpha _x))_{x \\in \\mathcal {V}, \\sigma = \\pm } & 0\\\\0 & [*]\\end{bmatrix}\\,.$ The following estimate states that $\\widehat{H}^\\tau $ is a small perturbation of $H^\\tau $ .", "Lemma 5.11 Let $d$ satisfy (REF ).", "If $1 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2$ then $\\Vert H^\\tau - \\widehat{H}^\\tau \\Vert \\leqslant \\mathcal {C}\\xi $ with very high probability.", "The proof of Lemma REF is deferred to Section REF .", "The following result is the key estimate of our proof; it states that on the range of $\\overline{\\Pi } \\!\\,^\\tau $ the matrix $H^\\tau $ is bounded by $2\\tau + o(1)$ .", "Proposition 5.12 Let $d$ satisfy (REF ).", "If $1 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2$ then $\\Vert \\overline{\\Pi } \\!\\,^\\tau H^\\tau \\overline{\\Pi } \\!\\,^\\tau \\Vert \\leqslant 2\\tau + \\mathcal {C} (\\xi +\\xi _{\\tau -1})$ with very high probability.", "The proof of Proposition REF is deferred to Section REF .", "We now use Lemma REF and Proposition REF to conclude Theorems REF and REF .", "[Proof of Theorem REF ] Define the orthogonal projections $\\Pi ^\\tau _{\\lambda ,\\delta } \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x W, v̑+(x)   v̑+(x)*  ,               , =I - ,  .", "By definition, the orthogonal projections $\\Pi ^\\tau $ and $\\Pi ^\\tau _{\\lambda ,\\delta }$ commute.", "Moreover, under the assumptions of Theorem REF we have the inclusion property $ \\Pi ^\\tau \\Pi ^\\tau _{\\lambda ,\\delta } = \\Pi ^\\tau _{\\lambda ,\\delta }\\,.$ See also Figure REF .", "To show (REF ), we note that the condition on $\\delta $ and the lower bound on $\\lambda $ in Theorem REF imply $\\lambda - \\delta \\geqslant 2 + \\mathcal {C} \\xi ^{1/2}$ .", "Using $\\Lambda (2 + x) - 2 \\asymp x^2 \\wedge x^{1/2}$ for $x \\geqslant 0$ we conclude that for any $\\alpha \\geqslant 2$ we have the implication $\\Lambda (\\alpha ) \\geqslant \\lambda - \\delta \\; \\Rightarrow \\; \\alpha \\geqslant 2 + \\xi ^{1/4}$ , which implies (REF ).", "Next, we abbreviate $E^\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(EA / d) $ and note that $ E= 0$ because $ = 0$ by construction of $ v̑(x)$.", "From (\\ref {incl_Pi}) we obtain $  , =  , +  $, which yields\\begin{equation} \\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta } (\\widehat{H}^\\tau + E^\\tau ) \\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta } = \\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta } \\Pi ^\\tau \\widehat{H}^\\tau \\Pi ^\\tau \\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta } + \\bigl (\\overline{\\Pi } \\!\\,^\\tau \\widehat{H}^\\tau \\overline{\\Pi } \\!\\,^\\tau + E^\\tau \\bigr )\\,.\\end{equation}$ The core of our proof is the spectral gap $ \\operatorname{spec}\\Bigl (\\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta } (\\widehat{H}^\\tau + E^\\tau ) \\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta }\\Bigr ) \\subset \\mathbb {R}\\setminus [\\lambda - \\delta , \\lambda + \\delta ]\\,.$ To establish (REF ), it suffices to establish the same spectral gap for each term on the right-hand side of () separately, since the right-hand side of () is a block decomposition of its left-hand side.", "The first term on the right-hand side of () is explicit: $\\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta } \\Pi ^\\tau \\widehat{H}^\\tau \\Pi ^\\tau \\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta } = \\sum _{x \\in \\mathcal {V}} \\sum _{\\sigma = \\pm } \\sigma \\Lambda (\\alpha _x) \\, \\mathbb {1}_{\\vert \\sigma \\Lambda (\\alpha _x) - \\lambda \\vert > \\delta } \\, v̑^\\tau _\\sigma (x) v̑^\\tau _\\sigma (x)^*\\,,$ which trivially has no eigenvalues in $[\\lambda - \\delta , \\lambda + \\delta ]$ .", "In order to establish the spectral gap for the second term of (), we begin by remarking that $E^\\tau $ has rank one and, by (REF ), its unique nonzero eigenvalue is $\\sqrt{d} + O(1/\\sqrt{d})$ .", "Hence, by rank-one interlacing and Proposition REF , we find $ \\operatorname{spec}\\bigl (\\overline{\\Pi } \\!\\,^\\tau (H^\\tau + E^\\tau ) \\overline{\\Pi } \\!\\,^\\tau \\bigr ) \\subset \\bigl [-2\\tau -\\mathcal {C} (\\xi +\\xi _{\\tau -1}) \\,, 2\\tau +\\mathcal {C} (\\xi +\\xi _{\\tau -1})\\bigr ] \\cup \\bigl \\lbrace \\mu \\bigr \\rbrace $ for some simple eigenvalue $\\mu = \\sqrt{d} + O(1)$ .", "Thus, to conclude the proof of the spectral gap for the second term of (), it suffices to show that $ \\lambda - \\delta &> 2\\tau +\\mathcal {C} (\\xi +\\xi _{\\tau -1})\\\\ \\lambda + \\delta &< \\mu \\,.$ To prove (REF ), we suppose that $\\lambda \\geqslant 2 + 8 \\mathcal {C} \\xi ^{1/2}$ and, recalling the condition on $\\delta $ and the choice of $\\tau $ in Theorem REF , obtain $ \\lambda - \\delta \\geqslant 2 + \\frac{\\lambda - 2}{2} \\geqslant 2\\tau + 2 \\mathcal {C} \\xi ^{1/2} > 2\\tau + \\mathcal {C}(\\xi + \\xi _{\\tau -1})\\,,$ where in the last step we used that $\\xi _{\\tau -1} < \\xi ^{1/2}$ by our choice of $\\tau $ and the lower bound on $\\lambda $ .", "This is (REF ).", "For the following arguments, we compare $A / \\sqrt{d}$ with $\\widehat{H}^\\tau + E^\\tau $ using the estimate $ \\Vert A / \\sqrt{d} - (\\widehat{H}^\\tau + E^\\tau ) \\Vert \\leqslant \\Vert (H^\\tau - \\widehat{H}^\\tau ) + (H - H^\\tau ) + (\\mathbb {E}A / \\sqrt{d} - E^\\tau )\\Vert \\leqslant \\mathcal {C} (\\xi +\\xi _{\\tau -1})$ with very high probability, which follows from Lemma REF , Lemma REF , (REF ) and $d^{-1/2} \\leqslant \\mathcal {C} \\xi $ .", "Next, we use (REF ) to conclude the proof of ().", "The only nonzero eigenvalue of $E^\\tau $ is $\\sqrt{d}(1 + O(1/d))$ , and from Proposition REF and Remark REF we have $\\Vert \\widehat{H}^\\tau \\Vert \\leqslant \\Lambda (\\max _{x \\in \\mathcal {V}} \\alpha _x) + O(1)$ with very high probability, so that Lemma REF and the assumption (REF ) yield $\\Vert \\widehat{H}^\\tau \\Vert \\leqslant \\mathcal {C} \\sqrt{\\frac{\\log N}{d}}$ with very high probability.", "Hence, (REF ) and (REF ) imply that $A/\\sqrt{d}$ has one eigenvalue bigger than $\\sqrt{d} - O(1)$ and all other eigenvalues are at most $\\mathcal {C} \\sqrt{\\frac{\\log N}{d}}$ .", "Since $\\lambda $ is nontrivial, we conclude that $\\lambda \\leqslant \\mathcal {C} \\sqrt{\\frac{\\log N}{d}}$ .", "By the upper bound $\\delta \\leqslant (\\lambda - 2)/2$ and the lower bound on $d$ in (REF ), this concludes the proof of () and, thus, the one of the spectral gap (REF ).", "Next, from (REF ), and (REF ), we conclude the spectral gap for the full adjacency matrix $ \\operatorname{spec}\\Bigl (\\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta } (A / \\sqrt{d}) \\overline{\\Pi } \\!\\,^\\tau _{\\lambda ,\\delta }\\Bigr ) \\subset \\mathbb {R}\\setminus \\bigl [\\lambda -\\delta +\\mathcal {C} (\\xi +\\xi _{\\tau -1}),\\lambda +\\delta -\\mathcal {C} (\\xi +\\xi _{\\tau -1})\\bigr ]\\,.$ Using (REF ) we may conclude the proof.", "The eigenvalue-eigenvector equation $(A/ \\sqrt{d} - \\lambda ) w̑ = 0$ yields $ \\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau w̑ = - \\Bigl (\\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau (A/\\sqrt{d}) \\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau -\\lambda \\Bigr )^{-1} \\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau (A/\\sqrt{d}) \\Pi _{\\lambda ,\\delta }^\\tau w̑\\,.$ Assuming that $\\delta > \\mathcal {C} (\\xi + \\xi _{\\tau - 1})$ , from (REF ) we get $ \\Bigl \\Vert \\Bigl (\\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau (A/\\sqrt{d}) \\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau -\\lambda \\Bigr )^{-1} \\Bigr \\Vert \\leqslant \\frac{1}{\\delta -\\mathcal {C} (\\xi +\\xi _{\\tau -1})}\\,.$ Moreover, since $\\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau \\widehat{H}^\\tau \\Pi _{\\lambda ,\\delta }^\\tau = 0$ and $E^\\tau \\Pi _{\\lambda ,\\delta }^\\tau =0$ , we deduce from (REF ) that $\\Vert \\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau (A/\\sqrt{d}) \\Pi _{\\lambda ,\\delta }^\\tau \\Vert \\leqslant \\mathcal {C} (\\xi + \\xi _{\\tau -1})\\,.$ Plugging (REF ) and (REF ) into (REF ) yields $\\Vert \\overline{\\Pi } \\!\\,_{\\lambda ,\\delta }^\\tau w̑ \\Vert \\leqslant \\frac{\\mathcal {C} (\\xi +\\xi _{\\tau -1})}{\\delta - \\mathcal {C}(\\xi + \\xi _{\\tau -1})}\\wedge 1 \\leqslant \\frac{2\\mathcal {C} (\\xi +\\xi _{\\tau -1})}{\\delta }\\,,$ since $w̑$ is normalized.", "This concludes the proof if $\\delta > \\mathcal {C} (\\xi + \\xi _{\\tau - 1})$ , and otherwise the claim is trivial.", "Proposition REF is also the main tool to prove Theorem REF .", "[Proof of Theorem REF ] The proof uses Proposition REF , Lemma REF , and Lemma REF for $\\tau \\in [1 + \\xi ^{1/2}/3,2]$ .", "Note that the lower bound $1 + \\xi ^{1/2}/3$ is smaller than the lower bound $1 + \\xi ^{1/2}$ imposed in these results, but their proofs hold verbatim also in this regime of $\\tau $ .", "We set $E^\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(EA/d)$ with $$ from Definition~\\ref {def:underline_A_underline_A_tau_chi_tau}.We now compare $ A/d$ and $ H+ E$, as in the proof of Theorem~\\ref {thm:general},and use some estimates from its proof.For any $ [1 + 1/2/3,2]$, we have\\begin{equation} \\operatorname{spec}(\\widehat{H}^\\tau + E^\\tau ) = \\lbrace \\pm \\Lambda (\\alpha _x)\\mathrel {\\hbox{.}\\hbox{.", "}}\\end{equation}x \\in \\mathcal {U} \\rbrace \\cup \\operatorname{spec}\\big (\\overline{\\Pi } \\!\\,^\\tau ( H^\\tau + E^\\tau ) \\overline{\\Pi } \\!\\,^\\tau \\big )\\,,$ since as $\\Pi ^\\tau \\chi ^\\tau =0$ .", "By perturbation theory and the choice $\\tau =2$ , we get from (), (REF ) and (REF ) that $\\lambda _1(A/\\sqrt{d}) = \\mu + O(\\xi ) = \\sqrt{d} + O(1)$ and $\\lambda _1(A/\\sqrt{d})$ is well separated from the other eigenvalues of $A/\\sqrt{d}$ (see the proof of Theorem REF ).", "Combining (), (REF ), and (REF ), choosing $\\tau = 1 + \\xi ^{1/2}/3$ as well as using $\\mathcal {C}(\\xi + \\xi _{\\tau - 1}) \\leqslant \\xi ^{1/2}/3$ for this choice of $\\tau $ imply (REF ).", "Moreover, we apply perturbation theory to () using (REF ) and (REF ), and obtain $|\\lambda _{i + 1}(A/\\sqrt{d}) - \\Lambda (\\alpha _{\\sigma (i)})| + |\\lambda _{N-i+1}(A/\\sqrt{d}) + \\Lambda (\\alpha _{\\sigma (i)})| \\leqslant \\mathcal {C}(\\xi + \\xi _{\\tau - 1})$ with very high probability for all $\\tau \\in [1+\\xi ^{1/2}/3,2]$ and all $i \\in [\\vert \\mathcal {U} \\vert ]$ satisfying $ 2 (\\tau -1) + \\mathcal {C} ( \\xi + \\xi _{\\tau - 1}) < \\Lambda (\\alpha _{\\sigma (i)})-2.$ What remains is choosing $\\tau \\equiv \\tau _i$ , depending on $i \\in [\\vert \\mathcal {U} \\vert ]$ , such that the condition (REF ) is satisfied and the error estimate from (REF ) transforms into the form of (REF ).", "Both are achieved by setting $ \\tau = 1 + \\frac{1}{3}\\big [ (\\Lambda (\\alpha _{\\sigma (i)}) -2 ) \\wedge 3 \\big ].$ Note that $\\tau \\in [1 + \\xi ^{1/2}/3,2]$ as $\\sigma (i) \\in \\mathcal {U}$ .", "From $\\Lambda (\\alpha _{\\sigma (i)})-2 \\geqslant 3(\\tau -1) $ due to (REF ) and $\\Lambda (\\alpha _{\\sigma (i)})-2 \\geqslant \\xi ^{1/2}$ by the definition of $\\mathcal {U}$ , we conclude that $ \\Lambda (\\alpha _{\\sigma (i)})-2 \\geqslant \\frac{5}{2} (\\tau - 1) + \\frac{1}{6} \\xi ^{1/2} \\geqslant 2 (\\tau - 1) + \\mathcal {C} (\\xi _{\\tau - 1} + \\xi ), $ where we used $\\tau - 1 \\geqslant 3 \\xi _{\\tau - 1} \\log d$ as $\\tau -1 \\geqslant \\xi ^{1/2}/3$ .", "This proves (REF ) and, thus, (REF ) for any $\\sigma (i) \\in \\mathcal {U}$ with the choice of $\\tau $ from (REF ).", "In order to show that the right-hand side of (REF ) is controlled by the one in (REF ), we now distinguish the two cases, $\\Lambda (\\alpha _{\\sigma (i)}) -2 \\leqslant 3$ and $\\Lambda (\\alpha _{\\sigma (i)}) -2 > 3$ .", "In the latter case, $\\tau = 2$ by (REF ) and (REF ) follows immediately from (REF ) as $\\xi _1 \\leqslant \\xi $ .", "If $\\Lambda (\\alpha _{\\sigma (i)}) -2 \\leqslant 3$ then $\\tau - 1 = (\\Lambda (\\alpha _{\\sigma (i)}) -2)/3$ and, thus, $\\xi _{\\tau - 1} = 3 \\xi _{\\Lambda (\\alpha _{\\sigma (i)}) -2}$ .", "Hence, (REF ) implies (REF ).", "This concludes the proof of Theorem REF ." ], [ "Proof of Lemma ", "[Proof of Lemma REF ] To begin with, we reduce the problem to the adjacency matrices by using the estimate (REF ).", "Hence, with very high probability, $\\sqrt{d} \\Vert H - H^\\tau \\Vert \\leqslant \\Vert \\mathbb {E}A - \\chi ^\\tau (\\mathbb {E}A) \\chi ^\\tau \\Vert + \\Vert A - A^\\tau \\Vert \\leqslant 2 + \\Vert A^{\\mathbb {D}_\\tau } \\Vert \\,,$ where $A^{\\mathbb {D}_\\tau }$ is the adjacency matrix of the graph $\\mathbb {D}_\\tau \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =G G$.", "Hence, since $ d-1/2 C -1 $ by $ d 3 N$ and the definition (\\ref {eq:main_error_term}), it suffices to show that$ AD C -1 d$.$ We know from Proposition REF REF and REF that with very high probability $\\mathbb {D}_\\tau $ consists of (possibly overlapping) starsA star around a vertex $x$ is a set of edges incident to $x$ .", "around vertices $x \\in \\mathcal {V}_\\tau $ of central degree $D_x^{\\mathbb {D}_\\tau } \\leqslant \\mathcal {C} d \\xi _{\\tau -1}^2$ .", "Moreover, with very high probability, any ball $B_{2 r_\\star }(x)$ around $x \\in \\mathcal {V}_\\tau $ has at most $\\mathcal {C}$ cycles; any ball $B_{2 r_\\star }(x)$ around $x \\in \\mathcal {V}_\\tau $ contains at most $\\mathcal {C} d \\xi _{\\tau -1}^2$ vertices in $\\mathcal {V}_\\tau $ .", "Claim REF follows from [10], the definition (REF ), and Lemma REF .", "Claim REF follows from [10] and $h((\\tau -1)/2) \\asymp (\\tau -1)^2$ for $1 \\leqslant \\tau \\leqslant 2$ .", "Let $x \\in \\mathcal {V}_\\tau $ .", "We claim that we can remove at most $\\mathcal {C}$ edges of $\\mathbb {D}_\\tau $ incident to $x$ so that no cycle passes through $x$ .", "Indeed, if there were more than $\\mathcal {C}$ cycles in $\\mathbb {D}_\\tau $ passing through $x$ , then at least one such cycle would have to leave $B_{2 r_\\star }(x)$ (by REF ), which would imply that $B_{2 r_\\star }(x)$ has at least $r_\\star $ vertices in $\\mathcal {V}_\\tau $ , which, by REF , is impossible since $r_\\star \\geqslant 2 \\mathcal {C} d \\xi _{\\tau -1}^2$ by $\\tau \\geqslant 1 + \\xi ^{1/2}$ .", "See Figure REF for an illustration of $\\mathbb {D}_\\tau $ .", "Figure: An illustration of a connected component of 𝔻 τ \\mathbb {D}_\\tau .", "Vertices of 𝒱 τ \\mathcal {V}_\\tau are drawn in white and the other vertices in black.", "The ball B 2r ☆ (x)B_{2 r_\\star }(x) around a chosen white vertex xx is drawn in grey, where 2r ☆ =42 r_\\star = 4.", "The illustrated component of 𝔻 τ \\mathbb {D}_\\tau has three cycles, two of which are in B 2r ☆ (x)B_{2 r_\\star }(x).", "The blue and red cycles pass through xx.", "The purple edge is removed from the blue cycle, i.e.", "it is put into the graph 𝕌 τ \\mathbb {U}_\\tau .", "With very high probability, the red cycle cannot appear, because it leaves the ball B 2r ☆ (x)B_{2 r_\\star }(x) and therefore contains more white vertices in B 2r ☆ (x)B_{2 r_\\star }(x) than allowed by property .Thus, we can remove a graph $\\mathbb {U}_\\tau $ from $\\mathbb {D}_\\tau $ such that $\\mathbb {U}_\\tau $ has maximal degree $\\mathcal {C}$ and $\\mathbb {D}_\\tau \\setminus \\mathbb {U}_\\tau $ is a forest of maximal degree $\\mathcal {C}d \\xi _{\\tau -1}^2$ (by REF ).", "The claim now follows from Lemma REF .", "[Proof of Proposition REF ] We focus on the case $\\sigma = +$ ; trivial modifications yield (REF ) for $\\sigma = -$ .", "The basic strategy is to decompose $(H^\\tau - \\Lambda (\\alpha _x))v̑_+^\\tau (x)$ into several error terms that are estimated separately.", "A similar argument was applied in [10] to the original graph $\\mathbb {G}$ instead of $\\mathbb {G}^\\tau $ , which however does not yield sharp enough estimates to reach the optimal scale $d \\gg \\sqrt{\\log N}$ (see Section REF ).", "We omit $x$ from the notation in this proof and write $u_i$ , $v̑_+^\\tau $ and $S_i^\\tau $ instead of $u_i(x)$ , $v̑^\\tau _+(x)$ and $S_i^\\tau (x)$ .", "We define $ s̑^\\tau _i \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =SiSi ,             Ni(y) =S1(y) Si  .", "$Note that $ s̑i)i=02r$ form an orthonormal system.", "Defining the vectors\\begin{equation} \\begin{aligned}w̑_2 & \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{aligned}=\\sum _{i=2}^{{r_\\star }} \\frac{u_i}{\\sqrt{d \\vert S^\\tau _i \\vert }} \\sum _{y \\in S^\\tau _{i-1}} \\bigg ( N_i^\\tau (y) - \\frac{\\vert S^\\tau _i \\vert }{\\vert S^\\tau _{i-1} \\vert } \\bigg ) _y, \\\\w̑_3 & \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation}=u_2 \\mathopen {}\\mathclose {\\left( \\frac{\\sqrt{\\vert S^\\tau _2 \\vert }}{\\sqrt{d\\vert S^\\tau _1 \\vert }} - 1 \\right)} s̑^\\tau _1 + \\sum _{i=2}^{{r_\\star } - 1} \\mathopen {}\\mathclose {\\left[ u_{i+1} \\mathopen {}\\mathclose {\\left(\\frac{\\sqrt{\\vert S^\\tau _{i+1} \\vert }}{\\sqrt{d\\vert S^\\tau _i \\vert }} -1 \\right)}+ u_{i-1} \\mathopen {}\\mathclose {\\left( \\frac{\\sqrt{\\vert S^\\tau _i \\vert }}{\\sqrt{d \\vert S^\\tau _{i-1} \\vert }} - 1 \\right)} \\right]} s̑^\\tau _i , \\\\w̑_4 & \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =ur (1 - 1x ) s̑r-1+ ur-1 ( Sr d Sr-1 - 1x - 1 ) s̑r + ur Sr+1 dSr   s̑r+1 , a straightforward computation using the definition of $v̑^\\tau _+$ yields $ (H^\\tau - \\Lambda (\\alpha _x)) v̑_+^\\tau = w̑_2 + w̑_3 + w̑_4.$ For a detailed proof of (REF ) in a similar setup, we refer the reader to [10] (note that in the analogous calculation of [10] the left-hand side of (REF ) is multiplied by $\\sqrt{d}$ ).", "The terms in (REF ) analogous to $w̑_0$ and $w̑_1$ in [10] vanish, respectively, because the projection $\\chi ^\\tau $ is included in () and because $\\mathbb {G}_\\tau |_{B_{2r_\\star }^\\tau }$ is a tree by Proposition REF REF .", "The vector $w̑_4$ from () differs from the one in [10] due to the special choice of $u_{r_\\star }$ in (REF ).", "We now complete the proof of (REF ) by showing that each term on the right-hand side of (REF ) is bounded in norm by $\\mathcal {C} \\xi $ with very high probability.", "We start with $w̑_3$ by first proving the concentration bound $ \\mathopen {}\\mathclose {\\left|\\frac{|S_{i+1}^\\tau |}{d|S_i^\\tau |}-1\\right|}=\\mathcal {O}\\mathopen {}\\mathclose {\\left({\\frac{\\sqrt{\\log N}}{d}}\\right)}$ with very high probability, for $i = 1, \\ldots , r_\\star $ .", "To prove this, we use Proposition REF REF and REF , as well as [10], to obtain $ \\frac{|S_i^\\tau |}{|S_i|}=1-\\frac{|S_{i}\\setminus S_{i}^\\tau |}{|S_i|}\\geqslant 1-\\mathcal {C}\\frac{\\log N}{(\\tau - 1)^2 d^2}$ with very high probability, where we used that $\\alpha _x \\geqslant 1$ , and the assumption [10] is satisfied by the definition (REF ).", "Therefore, invoking [10] in the following expansion yields $\\frac{\\vert S_{i+1}^\\tau \\vert }{d \\vert S_i^\\tau \\vert } = \\frac{\\vert S_{i+1} \\vert }{d \\vert S_i \\vert }\\, \\frac{\\vert S_{i} \\vert }{\\vert S_i^\\tau \\vert }\\, \\frac{\\vert S_{i+1}^\\tau \\vert }{\\vert S_{i+1} \\vert } = \\biggl (1 + \\mathcal {O} \\biggl ({\\frac{\\sqrt{\\log N}}{d}}\\biggr )\\biggr ) \\biggl (1 + \\mathcal {O} \\biggl (\\frac{\\log N}{d^2 (\\tau - 1)^2}\\biggr )\\biggr )$ with very high probability.", "Hence, recalling the lower bound $\\tau \\geqslant 1 + \\xi ^{1/2}$ , we obtain (REF ).", "We take the norm in the definition of $w̑_3$ , use the orthonormality of $(s̑_i^\\tau )_{i=0}^{r_\\star }$ , and end up with $\\Vert w̑_3 \\Vert ^2 \\leqslant \\mathopen {}\\mathclose {\\left[ \\mathopen {}\\mathclose {\\left( \\frac{\\sqrt{\\vert S^\\tau _2 \\vert }}{\\sqrt{d \\vert S^\\tau _1 \\vert }} -1 \\right)}^2 u_2^2 + 2 \\sum _{i=2}^{{r_\\star }-1} \\mathopen {}\\mathclose {\\left( \\mathopen {}\\mathclose {\\left(\\frac{\\sqrt{\\vert S^\\tau _{i+1} \\vert }}{\\sqrt{d \\vert S^\\tau _i \\vert }} - 1\\right)}^2 u_{i+1}^2 + \\mathopen {}\\mathclose {\\left( \\frac{\\sqrt{\\vert S^\\tau _i \\vert }}{\\sqrt{d \\vert S^\\tau _{i-1} \\vert }} - 1 \\right)}^2 u_{i-1}^2 \\right)} \\right]}.$ Consequently, (REF ) and $\\sum _{i=0}^{r_\\star } u_i^2 =1$ yield the desired bound on $\\Vert w̑_3 \\Vert $ .", "In order to estimate $\\Vert w̑_2 \\Vert $ , we use the definitions $ N_i(y) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =S1(y) Si ,       Yi =1Si-1 y Si-1 ( Ni(y) - E[Ni(y) | Bi-1] )2 $and the Pythagorean theorem to obtain{\\begin{@align}{1}{-1}\\Vert w̑_2\\Vert ^2 & = \\sum _{i=2}^{{r_\\star }} \\frac{u_i^2}{d \\vert S^\\tau _i \\vert } \\sum _{y \\in S^\\tau _{i-1}} \\bigg ( N^\\tau _i(y) - \\frac{\\vert S^\\tau _i \\vert }{\\vert S^\\tau _{i-1} \\vert } \\bigg )^2 \\nonumber \\\\ & \\leqslant 4 \\sum _{i=2}^{{r_\\star }} \\frac{u_i^2}{d \\vert S^\\tau _i \\vert } \\sum _{y \\in S^\\tau _{i-1}} \\bigg [ \\big ( N_i(y) - \\mathbb {E}[N_i(y) | B_{i-1}]\\big )^2 + \\big (\\mathbb {E}[N_i(y) | B_{i-1}] - d \\big )^2\\nonumber \\\\ & \\qquad \\qquad \\qquad \\qquad +\\bigg (d- \\frac{\\vert S^\\tau _i \\vert }{\\vert S^\\tau _{i-1} \\vert } \\bigg )^2+(N_i^{\\tau }(y)-N_i(y))^2 \\bigg ] \\nonumber \\\\ & \\leqslant 4 \\max _{2\\leqslant i\\leqslant r_\\star } \\frac{\\vert S^{\\tau }_{i-1} \\vert }{d \\vert S^\\tau _i \\vert } \\Big [ Y_i+ \\mathcal {C} \\log N+\\big (\\max _y D_y^{\\mathbb {G} \\setminus \\mathbb {G}_\\tau }\\big )^2 \\Big ] \\end{@align}}with very high probability.Here, in the last step,we used (\\ref {S_tau_S_1}), $ i=0r ui2 = 1$ and $ d - E[Ni(y) | Bi-1] = d Bi-1 /N C$ with very high probability due to\\cite [Eq.~(5.12b)]{ADK19} and Lemma~\\ref {lem:upper_bound_degrees}.$ Next, we claim that $ Y_i \\leqslant \\mathcal {C} \\log N \\log d$ with very high probability, for $i = 2, \\ldots , r_\\star $ .", "The proof of (REF ) is based on a dyadic decomposition analogous to the one used in the proof of [10].", "We distinguish two regimes and estimate $Y_i &\\leqslant d + \\frac{1}{\\vert S_{i-1}^{\\tau } \\vert }\\sum _{y \\in S_{i-1}^\\tau } \\mathbb {1}_{|N_i(y) - \\mathbb {E}[N_i(y) | B_{i-1}] |>d^{1/2}} \\mathopen {}\\mathclose {\\left( N_i(y) - \\mathbb {E}[N_i(y) | B_{i-1}] \\right)}^2\\\\ &\\leqslant d + \\frac{1}{\\vert S_{i-1}^{\\tau } \\vert }\\sum _{k=k_{\\min }}^{0} d^2 \\mathrm {e}^{k+1} \\vert \\mathcal {N}^\\tau _{i,k} \\vert $ with very high probability, where we introduced $k_{\\min }\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =-d  ,      Ni,k ={ySi-1d2 ek< (Ni(y)-E[Ni(y) | Bi-1])2 d2 ek+1} .", "$In (\\ref {eq:decomposition_Z_i}), we used that, with very high probability, $ (Ni(y)-E[Ni(y) | Bi-1])2 d2 ((- 1/2)2 1) d2 e$, because$ ySi-1$ implies the conditions $ 0Ni(y)Dyd $due to Proposition \\ref {prop:subgraph_separating_large_degrees} \\ref {item:subgraph_paths} and $ d/2 E[Ni(y) |Bi-1] d$with very high probability.By Proposition~\\ref {prop:subgraph_separating_large_degrees} \\ref {item:subgraph_inclusion_S_i}, we have $ Ni,kNi-1k$,where $ Ni-1k$ is defined as in the proof of \\cite [Eq.~(5.26)]{ADK19}.", "(Note that, in the notation of \\cite {ADK19}, there is a one-to-one mapping between $ A( Bi-1)$ and $ Bi$.", ")In this proof it is shown that, with very high probability,$$|\\mathcal {N}^{i-1}_k| \\leqslant \\ell _k, \\qquad \\qquad \\ell _k \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =Cd(|Si-1|+N) e-k. $Using (\\ref {eq:lower_bound_S_i_tau_over_S_i}) and (\\ref {eq:ratio_S_i_tau_S_i}), and then plugging the resulting bound into (\\ref {eq:decomposition_Z_i}) concludes the proof of (\\ref {eq:bound_Z_i}).$ Thus, we obtain $\\Vert w̑_2 \\Vert \\leqslant \\mathcal {C} \\xi $ with very high probability, by starting from (REF ) and using (REF ), (REF ) and Proposition REF REF as well as the assumption $1 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2$ .", "Finally, we estimate $w̑_4$ .", "Since $\\alpha _x \\geqslant 2$ and $u_0 \\leqslant 1$ we have that $u_{r_\\star } + u_{r_\\star -1} \\leqslant 3 (\\alpha _x - 1)^{-(r_\\star -2)/2}$ .", "The other coefficients of $s̑_{r_\\star -1}^\\tau $ , $s̑_{r_\\star }^\\tau $ and $s̑_{r_\\star +1}^\\tau $ are bounded by $\\mathcal {C}$ with very high probability, due to $\\alpha _x \\geqslant 2$ and (REF ), respectively.", "Therefore, (REF ) implies $\\Vert w̑_4 \\Vert \\leqslant \\mathcal {C} \\xi $ .", "This concludes the proof of Proposition REF .", "[Proof of Lemma REF ] We have to estimate the norm of $ H^\\tau -\\widehat{H}^\\tau = \\Pi ^\\tau H^\\tau \\Pi ^\\tau -\\sum _{x \\in \\mathcal {V}} \\sum _{\\sigma = \\pm } \\sigma \\Lambda (\\alpha _x) v̑^\\tau _\\sigma (x) v̑^\\tau _\\sigma (x)^* +\\overline{\\Pi } \\!\\,^\\tau H^\\tau \\Pi ^\\tau + (\\overline{\\Pi } \\!\\,^\\tau H^\\tau \\Pi ^\\tau )^*.$ Each $x \\in \\mathcal {V}$ satisfies the condition of Proposition REF since $\\xi ^{1/4} \\geqslant C (\\log d)^2 / \\sqrt{\\log N}$ (see (REF )).", "Hence, for any $x \\in \\mathcal {V}$ and $\\sigma = \\pm $ , Proposition REF yields $H^\\tau v̑_\\sigma ^\\tau (x)= \\sigma \\Lambda (\\alpha _x) v̑_\\sigma ^\\tau (x) + ȇ_\\sigma ^\\tau (x)\\,,\\qquad \\operatorname{supp}ȇ_\\sigma ^\\tau (x) \\subset B_{r_{\\star }+1}^{\\tau }(x) \\,, \\qquad \\Vert ȇ_\\sigma ^\\tau (x) \\Vert \\leqslant \\mathcal {C} \\xi $ with very high probability, where the second statement follows from the first, the definition (REF ) of $v̑_\\sigma ^\\tau (x)$ , and Remark REF .", "By Proposition REF REF , the balls $B_{2r_\\star }^\\tau (x)$ and $B_{2r_\\star }^\\tau (y)$ are disjoint for $x, y \\in \\mathcal {V}_\\tau $ with $x \\ne y$ .", "Hence, in this case, $v̑_\\sigma ^\\tau (x),ȇ_\\sigma ^\\tau (x) \\perp v̑_{\\sigma ^{\\prime }}^\\tau (y),ȇ_{\\sigma ^{\\prime }}^\\tau (y)$ .", "For any $ȃ = \\sum _{x\\in \\mathcal {V}} \\sum _{\\sigma = \\pm } a_{x,\\sigma } v̑_\\sigma ^\\tau (x)$ , we obtain $\\overline{\\Pi } \\!\\,^\\tau H^\\tau \\Pi ^\\tau ȃ = \\sum _{x \\in \\mathcal {V}} \\sum _{\\sigma = \\pm } a_{x,\\sigma } \\overline{\\Pi } \\!\\,^\\tau H^\\tau v̑_\\sigma ^\\tau (x) = \\overline{\\Pi } \\!\\,^\\tau \\sum _{x \\in \\mathcal {V}} \\sum _{\\sigma = \\pm } a_{x,\\sigma } ȇ_\\sigma ^\\tau (x)\\,.$ Thus, with very high probability, $\\Vert \\overline{\\Pi } \\!\\,^\\tau H^\\tau \\Pi ^\\tau ȃ \\Vert ^2 \\leqslant \\sum _{x \\in \\mathcal {V}} \\Vert \\sum _{\\sigma = \\pm } a_{x,\\sigma } ȇ_\\sigma ^\\tau (x)\\Vert ^2 \\leqslant 4 \\mathcal {C}^2\\sum _{x\\in \\mathcal {V}} \\sum _{\\sigma = \\pm } a_{x,\\sigma }^2 \\xi ^2 = 4 \\mathcal {C}^2 \\xi ^2 \\Vert ȃ \\Vert ^2$ by orthogonality.", "Therefore, $\\Vert \\overline{\\Pi } \\!\\,^\\tau H^\\tau \\Pi ^\\tau \\Vert \\leqslant \\mathcal {C} \\xi $ with very high probability.", "Similarly, the representation $\\mathopen {}\\mathclose {\\left(\\Pi ^\\tau H^\\tau \\Pi ^\\tau -\\sum _{x \\in \\mathcal {V}} \\sum _{\\sigma = \\pm } \\sigma \\Lambda (\\alpha _{x}) v̑^\\tau _\\sigma (x) v̑^\\tau _\\sigma (x)^*\\right)}ȃ = \\Pi ^\\tau \\sum _{x \\in \\mathcal {V}} \\sum _{\\sigma = \\pm } a_{x,\\sigma } ȇ_\\sigma ^\\tau (x)$ yields the desired estimate on the sum of the two first terms on the right-hand side of (REF )." ], [ "Proof of Proposition ", "In this section we prove Proposition REF .", "Its proof relies on two fundamental tools.", "The first tool is a quadratic form estimate, which estimates $H$ in terms of the diagonal matrix of the vertex degrees.", "It is an improvement of [10].", "To state it, for two Hermitian matrices $X$ and $Y$ we use the notation $X \\leqslant Y$ to mean that $Y - X$ is a nonnegative matrix, and $\\vert X \\vert $ is the absolute value function applied to the matrix $X$ .", "Proposition 5.13 Let $4 \\leqslant d \\leqslant 3 \\log N$ .", "Then, with very high probability, we have $\\vert H \\vert \\leqslant I+(1+ 2d^{-1/2}) Q +\\mathcal {C}\\frac{\\log N}{d^{2}}\\vee d^{-1/2},$ where $Q$ is the diagonal matrix with diagonal $(\\alpha _x)_{x \\in [N]}$ .", "The second tool is a delocalization estimate for an eigenvector $w̑$ of $\\widehat{H}^\\tau $ associated with an eigenvalue $\\lambda > 2$ .", "Essentially, it says that $w_x$ is small at any $x \\in \\mathcal {V}_\\tau $ unless $w̑$ happens to be the specific eigenvector $v̑^\\tau _\\pm (x)$ of $\\widehat{H}^\\tau $ , which is by definition localized around $x$ .", "Thus, in any ball $B_{2 r_\\star }^\\tau (x)$ around $x \\in \\mathcal {V}_\\tau $ , all eigenvectors except $v̑^\\tau _\\pm (x)$ are locally delocalized in the sense that their magnitudes at $x$ are small.", "Using that the balls $(B_{2 r_\\star }^\\tau (x))_{x \\in \\mathcal {V}_\\tau }$ are disjoint, this implies that eigenvectors of $\\overline{\\Pi } \\!\\,^\\tau H^\\tau \\overline{\\Pi } \\!\\,^\\tau $ have negligible mass on the set $\\mathcal {V}$ .", "Proposition 5.14 Let $d$ satisfy (REF ).", "If $1 + \\xi ^{1/2} \\leqslant \\tau \\leqslant 2$ then the following holds with very high probability.", "Let $\\lambda $ be an eigenvalue of $\\widehat{H}^{\\tau }$ with $\\lambda >2\\tau +\\mathcal {C} \\xi $ and $w̑=(w_x)_{x \\in [N]}$ its corresponding eigenvector.", "If $x \\in \\mathcal {V}$ and $v̑_\\pm ^{\\tau }(x)\\perp w̑$ or if $x \\in \\mathcal {V}_\\tau \\setminus \\mathcal {V}$ then $\\frac{|w_{x}|}{\\Vert w̑|_{B_{2r_\\star }^\\tau (x)}\\Vert }\\leqslant \\frac{\\lambda ^2}{(\\lambda -2 \\tau - \\mathcal {C}\\xi )^{2}}\\bigg (\\frac{2 \\tau +\\mathcal {C} \\xi }{\\lambda }\\bigg )^{r_{\\star }}\\,.$ Let $w̑$ be normalized.", "If $v̑_\\pm ^{\\tau }(x)\\perp w̑$ for all $x \\in \\mathcal {V}$ then $\\sum _{x\\in \\mathcal {V}_\\tau }w_{x}^{2}\\leqslant \\frac{\\lambda ^4}{(\\lambda -2 \\tau -\\mathcal {C} \\xi )^{4}}\\bigg (\\frac{2 \\tau +\\mathcal {C} \\xi }{\\lambda }\\bigg )^{2r_{\\star }}\\,.$ Analogous results hold for $\\lambda < -2 \\tau - \\mathcal {C} \\xi $ .", "We may now conclude the proof of Proposition REF .", "[Proof of Proposition REF ] By Proposition REF , Lemma REF , and Lemma REF we have $\\begin{aligned}\\widehat{H}^\\tau & \\leqslant I+(1+2 d^{-1/2}) Q +\\mathcal {C}\\frac{\\log N}{d^{2}}\\vee d^{-1/2}+\\Vert H-H^\\tau \\Vert +\\Vert H^\\tau - \\widehat{H}^\\tau \\Vert \\\\ & \\leqslant I+(1+2 d^{-1/2}) Q +\\mathcal {C} (\\xi +\\xi _{\\tau -1})\\end{aligned}$ with very high probability, where we used $\\frac{\\log N}{d^{2}}\\vee d^{-1/2} \\leqslant (\\xi +\\xi _{\\tau -1})$ .", "Arguing by contradiction, we assume that there exists an eigenvalue $\\lambda > 2\\tau + \\mathcal {C}^{\\prime } (\\xi +\\xi _{\\tau -1})$ of $\\overline{\\Pi } \\!\\,^\\tau H^\\tau \\overline{\\Pi } \\!\\,^\\tau $ for some $\\mathcal {C}^{\\prime } \\geqslant 2 \\mathcal {C}$ to be chosen later.", "By the lower bound in (REF ), we may assume that $\\mathcal {C}^{\\prime } \\xi \\leqslant 1$ .", "Thus, by the definition of $\\widehat{H}^\\tau $ , there is an eigenvector $w̑$ of $\\widehat{H}^\\tau $ corresponding to $\\lambda $ , which is orthogonal to $v̑_\\pm ^\\tau (x)$ for all $x \\in \\mathcal {V}$ .", "From (REF ), we conclude $ \\lambda = \\langle w̑, \\widehat{H}^\\tau w̑ \\rangle \\leqslant 1+(1+2 d^{-1/2}) \\sum _{x \\notin \\mathcal {V}_\\tau }w_x^{2} \\tau +(1+2 d^{-1/2}) \\sum _{x \\in \\mathcal {V}_\\tau }w_x^{2} \\max _{y\\in [N]} \\alpha _y +\\mathcal {C} (\\xi +\\xi _{\\tau -1}).$ It remains to estimate the two sum on right-hand side of (REF ).", "Since $w̑ \\perp v̑^\\tau _\\pm (x)$ for all $x \\in \\mathcal {V}$ , we can apply Proposition REF REF .", "We find $ 2r_\\star \\log \\mathopen {}\\mathclose {\\left( \\frac{2 \\tau +\\mathcal {C} \\xi }{\\lambda }\\right)}\\leqslant 2r_\\star \\log \\mathopen {}\\mathclose {\\left( \\frac{2\\tau +\\mathcal {C} \\xi }{2\\tau +\\mathcal {C}^{\\prime } \\xi }\\right)}\\leqslant - 2 r_\\star \\frac{(\\mathcal {C}^{\\prime } - \\mathcal {C}) \\xi }{2\\tau + \\mathcal {C}^{\\prime } \\xi }\\leqslant - \\frac{c (\\mathcal {C}^{\\prime } - \\mathcal {C})}{3} \\sqrt{\\log N} \\, \\xi \\,,$ where in the last step we recalled the definition (REF ) and used that $\\tau \\leqslant 2$ and $\\mathcal {C}^{\\prime } \\xi \\leqslant 1$ .", "Using the estimate $\\frac{\\lambda ^4}{(\\lambda -2\\tau -\\mathcal {C} \\xi )^{4}} \\leqslant \\frac{C}{(\\mathcal {C}^{\\prime } - \\mathcal {C})^4 \\xi ^4}\\,,$ combined with Proposition REF REF , (REF ) and Lemma REF , yields $\\frac{1}{\\xi } \\sum _{x \\in \\mathcal {V}_\\tau }w_x^{2} \\max _{y\\in [N]} \\alpha _y &\\leqslant \\frac{C \\log N}{(\\mathcal {C}^{\\prime } - \\mathcal {C})^4 \\xi ^5}\\exp \\biggl (- \\frac{c (\\mathcal {C}^{\\prime } - \\mathcal {C})}{3} \\sqrt{\\log N} \\, \\xi \\biggr )\\\\&\\leqslant \\frac{C d^5 \\log N}{(\\mathcal {C}^{\\prime } - \\mathcal {C})^4}\\exp \\biggl (- \\frac{c (\\mathcal {C}^{\\prime } - \\mathcal {C})}{3} \\frac{\\log N}{d} \\log d\\biggr )\\\\&\\leqslant \\frac{C d^5 \\log N}{(\\mathcal {C}^{\\prime } - \\mathcal {C})^4} \\frac{1}{d^8} \\leqslant 1\\,,$ where the third step follows by choosing $\\mathcal {C}^{\\prime }$ large enough, depending on $\\mathcal {C}$ .", "Plugging this estimate into (REF ) and using $\\sum _x w_x^2\\leqslant 1$ to estimate the first sum in (REF ), we obtain $\\lambda \\leqslant 2\\tau + 2 \\mathcal {C} (\\xi +\\xi _{\\tau -1})$ .", "This is a contradiction to the assumption $\\lambda > 2\\tau + \\mathcal {C}^{\\prime } (\\xi +\\xi _{\\tau -1})$ .", "The proof of Proposition REF is therefore complete.", "[Proof of Proposition REF ] We only establish an upper bound on $H$ .", "The proof of the same upper bound on $-H$ is identical and, therefore, omitted.", "We introduce the matrices $H(t) = (H_{xy}(t))_{x, y \\in [N]}$ and $M(t) = (\\delta _{xy} m_x(t))_{x,y \\in [N]}$ with entries $H_{xy}(t) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =tHxyt2 -Hxy2,   mx(t)=1+y Hxy2t2-Hxy2 $$ By the estimate on the spectral radius of the nonbacktracking matrix associated with $H$ in [15] and the Ihara–Bass-type formula in [15] we have, with very high probability, $\\det (M(t)-H(t))\\ne 0$ for all $t\\geqslant 1+\\mathcal {C} d^{-1/2}$ .", "Because $(M(t)-H(t))\\rightarrow I$ as $t\\rightarrow \\infty $ , the matrix $M(t)-H(t)$ is positive definite for large enough $t$ .", "By continuity of the eigenvalues, we conclude that all eigenvalues of $M(t)-H(t)$ stay positive for $t\\geqslant 1+\\mathcal {C} d^{-1/2}$ , and hence $H(t)\\leqslant M(t) $ for all $t\\geqslant 1+\\mathcal {C} d^{-1/2}$ with very high probability.", "We now define the matrix $\\Delta = (\\Delta _{xy})_{x,y \\in [N]}$ with $ \\Delta _{xy}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={ll Hxy(t)-t-1Hxy if xy y' |Hxy'(t)-t-1Hxy'| if x=y .", ".", "$It is easy to check that $ $ is a nonnegative matrix.We also have$$\\sum _{y^{\\prime }} |H_{xy^{\\prime }}(t)-t^{-1}H_{xy^{\\prime }}|\\leqslant \\sum _{y^{\\prime }} \\frac{|H_{xy^{\\prime }}|^3}{t(t^2 -H_{xy^{\\prime }}^2)}\\leqslant \\frac{2}{t^3d^{1/2}}\\bigg (\\alpha _x+\\frac{1}{d} \\bigg )\\,,$$where we used that $ Hxy d-1/2$ and $ y' Hx y'2 = x + O (NN)$ with very high probability, by Lemma~\\ref {lem:upper_bound_degrees}.We use this to estimate the diagonal entries of $$ and obtain\\begin{equation} 0\\leqslant \\Delta \\leqslant H(t)-t^{-1}H+\\frac{2}{t^3\\sqrt{d}}Q+\\frac{2}{t^3 d^{3/2}}.\\end{equation}On the other hand, for the diagonal matrix $ M(t)$, we have the trivial upper bound\\begin{equation}M(t)\\leqslant I +t^{-2}Q+\\mathcal {C}\\frac{\\log N}{d^2}\\end{equation}since $ x C (N)/d$ with very high probability due to Lemma~\\ref {lem:upper_bound_degrees}.Finally, combining (\\ref {eq:IharaBassInequality}), (\\ref {eq:errorAtA}) and (\\ref {eq:errorDM}) yields$$t^{-1} H\\leqslant I +\\biggl (t^{-2}+\\frac{2}{t^3\\sqrt{d}}\\biggr )Q+\\mathcal {C}\\frac{\\log N}{d^2}$$and Proposition \\ref {prop:operator_upper_bound} follows by choosing $ t= 1+C d-1/2$.$ What remains is the proof of Proposition REF .", "The underlying principle behind the proof is the same as that of the Combes–Thomas estimate [25]: the Green function $((\\lambda - Z)^{-1})_{ij}$ of a local operator $Z$ at a spectral parameter $\\lambda $ separated from the spectrum of $Z$ decays exponentially in the distance between $i$ and $j$ , at a rate inversely proportional to the distance from $\\lambda $ to the spectrum of $Z$ .", "Here local means that $Z_{ij}$ vanishes if the distance between $i$ and $j$ is larger than 1.", "Since a graph is equipped with a natural notion of distance and the adjacency matrix is a local operator, a Combes–Thomas estimate would be applicable directly on the level of the graph, at least for the matrix $H^\\tau $ .", "For our purposes, however, we need a radial version of a Combes–Thomas estimate, obtained by first tridiagonalizing (a modification of) $\\widehat{H}^\\tau $ around a vertex $x \\in \\mathcal {V}_\\tau $ (see Appendix REF ).", "In this formulation, the indices $i$ and $j$ have the interpretation of radii around the vertex $x$ , and the notion of distance is simply that of $\\mathbb {N}$ on the set of radii.", "Since $Z$ is tridiagonal, the locality of $Z$ is trivial, although the matrix $\\widehat{H}^\\tau $ (or its appropriate modification) is not a local operator on the graph $\\mathbb {G}_\\tau $ .", "To ensure the separation of $\\lambda > 2\\tau + o(1)$ and the spectrum of $Z$ , we cannot choose $Z$ to be the tridiagonalization of $\\widehat{H}^\\tau $ , since $\\lambda $ is an eigenvalue of $\\widehat{H}^\\tau $ .", "In fact, $Z$ is the tridiagonalization of a new matrix $\\widehat{H}^{\\tau ,x}$ , obtained by restricting $\\widehat{H}^\\tau $ to the ball $B^\\tau _{2 r_\\star }(x)$ and possibly subtracting a suitably chosen rank-two matrix, which allows us to show $\\Vert \\widehat{H}^{\\tau , x} \\Vert \\leqslant 2 \\tau + o(1)$ .", "By the orthogonality assumption on $w̑$ , we then find that the Green function $((\\lambda - Z)^{-1})_{i r_\\star }$ , $0 \\leqslant i < r_\\star $ , and the eigenvector components in the radial basis $u_i$ , $0 \\leqslant i < r_\\star $ , satisfy the same linear difference equation.", "The exponential decay of $((\\lambda - Z)^{-1})_{i r_\\star }$ in $r_\\star - i$ then implies that, for each $x \\in \\mathcal {V}_\\tau $ , $u_0^2 \\leqslant o(1/\\log N) \\sum _{i = 0}^{r_*} u_i^2$ .", "Going back to the original vertex basis, this implies that $w_x^2 \\leqslant o(1/\\log N) \\Vert w̑|_{B_{2r_\\star }^\\tau (x)}\\Vert ^2$ for all $x \\in \\mathcal {V}_\\tau $ , from which Proposition REF follows since the balls $B_{2r_\\star }^\\tau (x)$ , $x \\in \\mathcal {V}_\\tau $ , are disjoint.", "[Proof of Proposition REF ] For a matrix $M \\in \\mathbb {R}^{N \\times N}$ and a set $V \\subset [N]$ , we use the notation $(M \\vert _V)_{xy} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1x,y V Mxy$.$ We begin with part REF .", "We first treat the case $x\\in \\mathcal {V}$ .", "To that end, we introduce the matrix $ \\widehat{H}^{\\tau ,x} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =H|B2r(x)-(x) v̑+(x)v̑+(x)*+(x) v̑-(x)v̑-(x)* .", "We claim that, with very high probability, $ \\Vert \\widehat{H}^{\\tau ,x} \\Vert \\leqslant 2 \\tau +\\mathcal {C}\\xi \\,.$ To show (REF ), we begin by noting that, by Proposition REF REF and REF , $\\mathbb {G}_{\\tau }$ restricted to $B_{2r_{\\star }}^{\\tau }(x)$ is a tree whose root $x$ has $\\alpha _x d$ children and all other vertices have at most $\\tau d$ children.", "Hence, Lemma REF yields $\\bigl \\Vert H^{\\tau }|_{B_{2r_{\\star }}^{\\tau }(x)} \\bigr \\Vert \\leqslant \\sqrt{\\tau } \\Lambda (\\alpha _x /\\tau \\vee 2)$ .", "Using Lemma REF we find $ \\Vert \\widehat{H}^{\\tau }|_{B_{2r_{\\star }}^\\tau (x)}-H^{\\tau }|_{B_{2r_{\\star }}(x)}\\Vert \\leqslant \\mathcal {C} \\xi $ with very high probability, and since $v̑_\\pm ^\\tau (x)$ is an eigenvector of $\\widehat{H}^\\tau |_{B_{2r_{\\star }}^{\\tau }(x)}$ with eigenvalue $\\pm \\Lambda (\\alpha _x)$ , we conclude $ \\Vert \\widehat{H}^{\\tau ,x} \\Vert \\leqslant \\sqrt{\\tau } \\Lambda (\\alpha _x /\\tau \\vee 2) + \\mathcal {C} \\xi $ with very high probability.", "The estimate (REF ) is rough in the sense that the subtraction of the two last terms of (REF ) is not needed for its validity (since $\\Lambda (\\alpha _x) \\leqslant \\sqrt{\\tau } \\Lambda (\\alpha _x/\\tau \\vee 2)$ ).", "Nevertheless, it is sufficient to establish (REF ) in the following cases, which may be considered degenerate.", "If $\\alpha _x \\leqslant 2 \\tau $ then (REF ) immediately implies (REF ), since $\\sqrt{\\tau } \\leqslant \\tau $ .", "Moreover, if $\\alpha _x > 2 \\tau $ and $\\Lambda (\\alpha _x) \\leqslant 2 \\sqrt{\\tau } + \\mathcal {C} \\xi $ , then (REF ) implies $\\Vert \\widehat{H}^{\\tau ,x} \\Vert \\leqslant \\sqrt{\\tau } \\Lambda (\\alpha _x /\\tau ) + \\mathcal {C} \\xi \\leqslant \\sqrt{\\tau } \\Lambda (\\alpha _x) + \\mathcal {C} \\xi \\leqslant 2 \\tau + 3 \\mathcal {C} \\xi \\,,$ which is (REF ) after renaming the constant $\\mathcal {C}$ .", "Hence, to prove (REF ), it suffices to consider the case $\\Lambda (\\alpha _x) > 2 \\sqrt{\\tau } + \\mathcal {C} \\xi $ .", "By Proposition REF REF and REF , $\\mathbb {G}_{\\tau }$ restricted to $B_{2r_{\\star }}^{\\tau }(x) \\setminus \\lbrace x\\rbrace $ is a forest of maximal degree at most $\\tau d$ .", "Lemma REF therefore yields $\\Vert H^{\\tau }|_{B_{2r_{\\star }}^\\tau (x) \\setminus \\lbrace x\\rbrace }\\Vert \\leqslant 2\\sqrt{\\tau }$ .", "Moreover, the adjacency matrix of the star graph consisting of all edges of $\\mathbb {G}_\\tau $ incident to $x$ has precisely two nonzero eigenvalues, $\\pm \\sqrt{d \\alpha _x}$ .", "By eigenvalue perturbation theory, we therefore conclude that $H^{\\tau }|_{B_{2r_{\\star }}^\\tau (x)}$ has at most one eigenvalue strictly larger than $2\\sqrt{\\tau }$ and at most one strictly smaller than $-2\\sqrt{\\tau }$ .", "Using (REF ) we conclude that $\\widehat{H}^{\\tau }|_{B_{2r_{\\star }}^\\tau (x)}$ has at most one eigenvalue strictly larger than $2\\sqrt{\\tau }+\\mathcal {C} \\xi $ and at most one strictly smaller than $-2\\sqrt{\\tau }-\\mathcal {C} \\xi $ .", "Since $v̑_+^{\\tau }(x)$ (respectively $v̑_-^{\\tau }(x)$ ) is an eigenvector of $\\widehat{H}^{\\tau }|_{B_{2r_{\\star }}^\\tau (x)}$ with eigenvalue $\\Lambda (\\alpha _{x})$ (respectively $-\\Lambda (\\alpha _{x})$ ), and since $\\Lambda (\\alpha _x) > 2 \\sqrt{\\tau } + \\mathcal {C} \\xi $ , we conclude (REF ).", "Next, let $(g̑_i)_{i=0}^{r_\\star }$ be the Gram–Schmidt orthonormalization of the vectors $((\\widehat{H}^{\\tau ,x})^i _x)_{i=0}^{r_\\star }$ .", "We claim that $ \\operatorname{supp}g̑_i \\subset B_{r_\\star +i}^\\tau (x)\\,.$ for $i = 0, \\ldots , r_\\star $ .", "The proof proceeds by induction.", "The base case for $i =0$ holds trivially.", "For the induction step, it suffices to prove for $0 \\leqslant i < r_\\star $ that if $\\operatorname{supp}g̑_i \\subset B_{r_\\star +i}^\\tau (x)$ then $ \\operatorname{supp}(\\widehat{H}^{\\tau ,x} g̑_i) \\subset B_{r_\\star +i+1}^\\tau (x)$ To that end, we note that by Proposition REF REF we have $\\widehat{H}^{\\tau ,x} = \\bigl (\\overline{\\Pi } \\!\\,^{\\tau } H^\\tau \\overline{\\Pi } \\!\\,^\\tau \\bigr ) |_{B_{2r_{\\star }}^{\\tau }(x)}$ .", "Hence, by induction assumption, Proposition REF REF , and Remark REF , $\\widehat{H}^{\\tau ,x} g̑_i = \\biggl (I - \\sum _{\\sigma = \\pm } v̑^\\tau _\\sigma (x) v̑^\\tau _\\sigma (x)^*\\biggr ) H^\\tau \\biggl (I - \\sum _{\\sigma = \\pm } v̑^\\tau _\\sigma (x) v̑^\\tau _\\sigma (x)^*\\biggr ) g̑_i\\,,$ and we conclude (REF ), as $\\operatorname{supp}v̑_\\sigma ^\\tau (x) \\subset B_{r_\\star }^\\tau (x)$ .", "Let $Z = (Z_{ij})_{i,j=0}^{r_\\star }$ , $Z_{ij} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =g̑i 2mu, H,x g̑j$, be the tridiagonal representation of $ H,x$ up to radius $ r$ (see Appendix \\ref {sec:mu} below).Owing to (\\ref {eq:norm_wh_H_tau_x}), we have\\begin{equation} \\Vert Z\\Vert \\leqslant 2 \\tau +\\mathcal {C} \\xi .\\end{equation}$ We set $u_{i}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =g̑i,w̑$ for any $ 0 i r$.", "Because $ w̑$ is an eigenvector of $ H$ that is orthogonal to $ v̑(x)$, for any $ i<r$,(\\ref {eq:b_g_i_support}) implies\\begin{equation} \\begin{aligned}\\lambda u_i & = \\mathopen {}\\mathclose {\\left\\langle g̑_i,\\mathopen {}\\mathclose {\\left(\\widehat{H}^\\tau -\\Lambda (\\alpha _{x}) v̑_+^{\\tau }(x)v̑_+^{\\tau }(x)^{*}+\\Lambda (\\alpha _{x}) v̑_{-}^{\\tau }(x)v̑_{-}^{\\tau }(x)^{*}\\right)}w̑\\right\\rangle }\\\\ &=\\mathopen {}\\mathclose {\\left\\langle \\widehat{H}^{\\tau ,x}g̑_i, w̑ \\right\\rangle }\\\\& = \\langle Z_{ii}g̑_{i}+Z_{i\\,i+1}g̑_{i+1}+Z_{i\\,i-1}g̑_{i-1}, w̑\\rangle \\\\& =Z_{ii}u_{i}+Z_{i\\,i+1}u_{i+1}+Z_{i\\,i-1}u_{i-1}\\,\\end{aligned}\\end{equation}with the conventions $ u-1=0$ and $ Z0,-1=0$.Let $ G() =(- Z)-1$be the resolvent of $ Z$ at $$.", "Note that $ -Z$ is invertible since $ > Z $ by assumption and (\\ref {eq:norm_wh_M}).Since $ ((- Z) G())i   r = 0$ for $ i<r$, we find$$\\lambda G_{i r_{\\star }}(\\lambda )=Z _{ii}G_{i r_{\\star }}(\\lambda )+Z _{i\\,i+1}G_{i +1 \\, r_{\\star }}(\\lambda )+Z _{i\\,i-1}G_{i -1\\, r_{\\star }}(\\lambda ).$$Therefore $ (Gi r())ir$ and $ (ui)ir$ satisfy the same linearrecursive equation (cf.\\ (\\ref {eq:wh_lambda_u_i})); solving them recursively from $ i = 0$ to $ i = r$ yields\\begin{equation} \\frac{G_{i r_{\\star }}(\\lambda )}{G_{r_\\star r_{\\star }}(\\lambda )}=\\frac{u_{i}}{u_{r_{\\star }}}\\end{equation}for all $ ir$.", "Moreover, as $ >Z$ by assumption and (\\ref {eq:norm_wh_M}), we have the convergent Neumann series $ G()= 1k0(Z / )k$.Thus, the offdiagonal entries of the resolvent satisfy{\\begin{@align*}{1}{-1}G_{0 r_{\\star }}(\\lambda ) =\\frac{1}{\\lambda }\\sum _{k \\geqslant 0} \\bigl ((Z / \\lambda )^{k}\\bigr )_{0 r_\\star }\\end{@align*}}Since $ Z$ is tridiagonal, we deduce that $ ((Z / )k)0 r = 0$ if $ k < r$, so that, by (\\ref {eq:norm_wh_M}),\\begin{equation} |G_{0 r_{\\star }}(\\lambda )| \\leqslant \\bigg (\\frac{2 \\tau +\\mathcal {C} \\xi }{\\lambda }\\bigg )^{r_{\\star }}\\frac{1}{\\lambda -2\\tau -\\mathcal {C} \\xi }\\,.\\end{equation}On the other hand, for the diagonal entries of the resolvent, we get, by splitting the summation over $ k$ into even and odd values,\\begin{multline} G_{r_\\star r_\\star }(\\lambda ) = \\frac{1}{\\lambda } \\sum _{k \\geqslant 0} \\bigl ((Z/\\lambda )^k\\bigr )_{r_\\star r_\\star }= \\frac{1}{\\lambda } \\sum _{k \\geqslant 0} \\Bigl ((Z/\\lambda )^{k} (I + Z/\\lambda ) (Z/\\lambda )^{k}\\Bigr )_{r_\\star r_\\star }\\\\\\geqslant \\frac{1}{\\lambda } (I + Z/\\lambda )_{r_\\star r_\\star }\\geqslant \\frac{1}{\\lambda }\\bigg (1-\\frac{2\\tau +\\mathcal {C} \\xi }{\\lambda }\\bigg )\\,,\\end{multline}where in the thid step we discarded the terms $ k > 0$ to obtain a lower bound using that $ I + Z/0$ by (\\ref {eq:norm_wh_M}), and in the last step we used (\\ref {eq:norm_wh_M}).Hence, the definition of $ ui$ and (\\ref {eq:b_g_i_support}) imply$$\\frac{|w_{x}|}{\\Vert w̑|_{B^\\tau _{2r_{\\star }}(x)}\\Vert }\\leqslant \\frac{|u_{0}|}{\\mathopen {}\\mathclose {\\left(\\sum _{i=0}^{r_\\star } u_{i}^2\\right)}^{1/2}}\\leqslant \\frac{|u_{0}|}{|u_{r_{\\star }}|}=\\frac{\\vert G_{0r_{\\star }}(\\lambda ) \\vert }{G_{r_{\\star } r_{\\star }}(\\lambda )}\\leqslant \\frac{\\lambda ^2}{(\\lambda -2\\tau -\\mathcal {C} \\xi )^{2}}\\bigg (\\frac{2\\tau +\\mathcal {C} \\xi }{\\lambda }\\bigg )^{r_{\\star }}.$$Here, we used (\\ref {eq:G_entries_u_i_s}) in third step and (\\ref {eq:G_diagonal}) as well as (\\ref {eq:G_offdigonal}) in the last step.This concludes the proof of \\ref {item:deloc1} for $ x V$.$ In the case $x\\in \\mathcal {V}_\\tau \\setminus \\mathcal {V}$ , we set $\\widehat{H}^{\\tau ,x} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =H |B2r(x)$.", "We claim that (\\ref {eq:norm_wh_H_tau_x}) holds.", "To see that, we use Proposition \\ref {prop:subgraph_separating_large_degrees} \\ref {item:subgraph_paths} and \\ref {item:subgraph_tree} as well as Lemma \\ref {lem:normTree} with $ p = d(2 + 1/4)$ and $ q = d $ to obtain$$ \\Vert H^\\tau |_{B_{2r_{\\star }}^{\\tau }(x)} \\Vert \\leqslant \\sqrt{\\tau } \\Lambda ((2 + \\xi ^{1/4})/\\tau \\vee 2) \\leqslant 2 \\tau .", "$$Here, the last step is trivial if $ 1 + 1/4/2$ and, if $ [1 + 1/2, 1 + 1/4/2]$, we used that $ f() =((2 + 1/4) / )/(2)$ ismonotonically decreasing on this interval and $ f(1 + 1/2) 1$, as can be seen by an explicit analysis of the function $ f$.Now we may take over the previous argument verbatim to prove \\ref {item:deloc1} for $ x VV$.$ Finally, we prove REF .", "By REF we have $\\sum _{x\\in \\mathcal {V}_\\tau }w_{x}^2\\leqslant \\sum _{x \\in \\mathcal {V}_\\tau } \\Vert w̑|_{B^\\tau _{2r_{\\star }}(x)}\\Vert ^2 \\frac{\\lambda ^4}{(\\lambda -2\\tau -\\mathcal {C} \\xi )^{4}}\\bigg (\\frac{2\\tau +\\mathcal {C} \\xi }{\\lambda }\\bigg )^{2 r_{\\star }}\\leqslant \\frac{\\lambda ^4}{(\\lambda -2\\tau -\\mathcal {C} \\xi )^{4}}\\bigg (\\frac{2\\tau +\\mathcal {C} \\xi }{\\lambda }\\bigg )^{2 r_{\\star }}\\,,$ where we used that the the balls $\\lbrace B^\\tau _{2r_\\star }(x) \\mathrel {\\hbox{.}\\hbox{.", "}}$ x V}$ are disjoint, which implies $ 1=w̑2 xV w̑|B2r(x)2$.$" ], [ "Proof of Proposition ", "We conclude this section with the proof of Proposition REF .", "[Proof of Proposition REF ] Parts REF –REF follow immediately from parts (i)–(iv) and (vi) of [10].", "To see this, we remark that the function $h$ from [10] satisfies $h((\\tau - 1)/2) \\asymp (\\tau - 1)^2$ for $1 < \\tau \\leqslant 2$ .", "Moreover, by Lemma REF and the upper bound on $d$ , we have $\\max _x D_x \\leqslant \\mathcal {C} \\log N$ with very high probability.", "Hence, choosing the universal constant $c$ small enough in (REF ) and recalling the lower bound on $\\tau - 1$ , in the notation of [10] we obtain for any $x \\in \\mathcal {V}_\\tau $ the inequality $2 r_\\star \\leqslant (\\frac{1}{4} r_x ) \\wedge (\\frac{1}{2} r(\\tau ))$ with very high probability.", "This yields parts REF –REF .", "It remains to prove REF , which is the content of the rest of this proof.", "From now on we systematically omit the argument $x$ from our notation.", "Part REF already implies the bound $ |S_{1} \\setminus S_{1}^{\\tau }|=D_x^{\\mathbb {G} \\setminus \\mathbb {G}_\\tau }\\leqslant \\mathcal {C}\\frac{\\log N}{(\\tau -1)^2 d}$ with very high probability, which is (REF ) for $i=1$ .", "From [10] we find $\\vert S_i \\setminus S_i^\\tau \\vert \\leqslant \\sum _{y \\in S_1 \\setminus S_1^\\tau } \\vert S_{i-1}(y) \\vert \\,.$ (As a guide to the reader, this estimate follows from the construction of $\\mathbb {G}_\\tau $ given in [10], which ensures that if a vertex $z \\in S_i$ is not in $S_i^\\tau $ then any path in $\\mathbb {G}$ of length $i$ connecting $z$ to $x$ is cut in $\\mathbb {G}_\\tau $ at its edge incident to $x$ .)", "Hence, in order to show REF for $i \\geqslant 2$ , it suffices to prove $\\sum _{y \\in S_1 \\setminus S_1^\\tau } \\vert S_{i-1}(y) \\vert \\leqslant \\mathcal {C}\\frac{\\log N}{(\\tau -1)^2}d^{i-2}$ with very high probability, for all $2 \\leqslant i \\leqslant 2 r_\\star $ .", "We start with the case $i=2$ .", "We shall use the relation $ \\sum _{y \\in S_1 \\setminus S_1^\\tau } \\vert S_{1}(y) \\vert = \\sum _{y \\in S_1 \\setminus S_1^\\tau } N_2(y) + \\sum _{y \\in S_1\\setminus S_1^\\tau } \\vert S_1(y) \\cap S_1 \\vert + \\vert S_1 \\setminus S_1^\\tau \\vert \\,,$ where, for $y \\in S_1$ , we introduced $N_2(y) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =S1(y) S2 $.Note that $ N2(y)$ is the number of vertices in $ S2$ connected to $ x$ via a path of minimal length passing through $ y$.The identity (\\ref {eq:relation_S_1_and_N_2}) is a direct consequence of $ S1(y) = S1(y) S2 + S1(y) S1 + S1(y) S0 $ using the definition of $ N2$ and$ S1(y) S0 = S1(y) {x } = 1$.$ The second and third terms of (REF ) are smaller than the right-hand side of (REF ) for $i=2$ due to [10] and (REF ), respectively.", "Hence, it remains to estimate the first term on the right-hand side of (REF ) in order to prove (REF ) for $i =2$ .", "To that end, we condition on the ball $B_1$ and abbreviate $\\mathbb {P}_{B_1}(\\cdot ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =P(  B1)$.", "Since\\begin{equation}N_2(y) = \\sum _{z \\in [N] \\setminus B_1} A_{yz}\\,,\\end{equation}we find that conditioned on $ B1$ the random variables $ (N2(y))y S1$ are independent $ Binom(N - B1 , d/N)$ random variables.", "We abbreviate $ =N(-1)2$.", "For given $ C, C'$, we set $ C” =C' + 2 C$ and estimate{\\begin{@align}{1}{-1}& \\mathbb {P}_{B_1}\\Biggl (\\sum _{y\\in S_1\\setminus S_1^\\tau } N_2(y)\\geqslant \\mathcal {C}^{\\prime \\prime } \\Gamma \\Biggr )\\\\ & \\leqslant \\mathbb {P}_{B_1}\\Biggl (\\sum _{y\\in S_1\\setminus S_1^\\tau } \\mathbb {1}_{N_2(y) \\geqslant 2 d}N_2(y)\\geqslant (\\mathcal {C}^{\\prime \\prime } - 2 \\mathcal {C} )\\Gamma \\Biggr )+\\mathbb {P}_{B_1}\\Biggl (\\sum _{y\\in S_1\\setminus S_1^\\tau } \\mathbb {1}_{N_2(y)< 2 d}N_2(y)\\geqslant 2 \\mathcal {C} \\Gamma \\Biggr )\\\\ & \\leqslant \\mathbb {P}_{B_1}\\Biggl (\\sum _{y\\in S_1} \\mathbb {1}_{2d \\leqslant N_2(y) \\leqslant N^{1/4}}N_2(y)\\geqslant \\mathcal {C}^{\\prime }\\Gamma \\Biggr )+ \\sum _{y \\in S_1} \\mathbb {P}_{B_1}\\bigl (N_2(y) \\geqslant N^{1/4}\\bigr )+\\mathbb {P}_{B_1}\\bigl ( |S_1\\setminus S_1^\\tau |\\geqslant \\mathcal {C} \\Gamma d^{-1}\\bigr ).\\end{@align}}In order to estimate the first term on the right-hand side of (\\ref {P_B_1}), we shall prove that if $ B1 N1/4$ then\\begin{equation}\\mathbb {E}_{B_1} \\Bigl [\\exp \\Bigl (\\mathbb {1}_{2d \\leqslant N_2(y) \\leqslant N^{1/4}} N_2(y)t\\Bigr )\\Bigr ] \\leqslant 2\\end{equation}for all $ yS1$ and $ t 1/8$.", "To that end, we estimate\\begin{equation*}\\mathbb {E}_{B_1} \\Bigl [\\exp \\Bigl (\\mathbb {1}_{2d \\leqslant N_2(y) \\leqslant N^{1/4}} N_2(y)t\\Bigr )\\Bigr ] \\leqslant 1 + \\mathbb {E}_{B_1} \\Bigl [\\mathbb {1}_{2d \\leqslant N_2(y) \\leqslant N^{1/4}} \\mathrm {e}^{N_2(y)t}\\Bigr ]\\,.\\end{equation*}With Poisson approximation, Lemma \\ref {lem:binomial_estimate} below, we obtain (assuming that $ 2d$ is an integer to simplify notation){\\begin{@align*}{1}{-1}\\mathbb {E}_{B_1} \\Bigl [\\mathbb {1}_{2d \\leqslant N_2(y) \\leqslant N^{1/4}} \\mathrm {e}^{N_2(y)t}\\Bigr ]&= \\sum _{2d \\leqslant k \\leqslant N^{1/4}}\\frac{(d - d \\vert B_1 \\vert /N)^k \\mathrm {e}^{tk}}{k!", "}\\mathrm {e}^{-d + d \\vert B_1 \\vert /N} \\bigl (1+O(N^{-1/2})\\bigr )\\\\&{-100mu}\\leqslant \\sum _{k\\geqslant 2 d}\\frac{d^k \\mathrm {e}^{tk}}{k!", "}\\mathrm {e}^{-d} \\bigl (1+O(N^{-1/2})\\bigr )=\\frac{d^{2d} \\mathrm {e}^{2td}}{(2d)!", "}\\mathrm {e}^{-d}\\sum _{i\\geqslant 0}\\frac{d^i \\mathrm {e}^{t i}}{\\prod _{j=2 d+1}^{2d + i}j} \\bigl (1+O(N^{-1/2})\\bigr )\\\\&{-100mu} \\leqslant \\frac{d^{2d} \\mathrm {e}^{2td}}{(2d)!", "}\\mathrm {e}^{-d} \\sum _{i\\geqslant 0}\\frac{2d^i \\mathrm {e}^{t i}}{(2d)^i}= \\frac{d^{2d} \\mathrm {e}^{2td}}{(2d)!", "}\\mathrm {e}^{-d}\\frac{2}{(1-e^t/2)}.\\end{@align*}}By Stirling^{\\prime }s approximation we get{\\begin{@align*}{1}{-1}\\log \\mathopen {}\\mathclose {\\left(\\frac{d^{2d} \\mathrm {e}^{2td}}{(2d)!", "}\\mathrm {e}^{-d}\\right)}&= d\\mathopen {}\\mathclose {\\left(2t- 2 \\log 2+1 \\right)}-\\frac{1}{2}\\log (4 \\pi d)+ \\mathrm {o}(1).\\end{@align*}}The term in the parentheses on the right-hand side is negative for $ t 1/8$, and hence\\begin{equation*}\\mathbb {E}_{B_1} \\Bigl [\\mathbb {1}_{2d \\leqslant N_2(y) \\leqslant N^{1/4}} \\mathrm {e}^{N_2(y)t}\\Bigr ] \\leqslant 1\\end{equation*}for large enough $ d$, which gives (\\ref {eq:bound_markov_geqtau}).", "Since the family $ (N2(y))y S1$ is independent conditioned on $ B1$, we can now use Chebyshev^{\\prime }s inequality to obtain, for $ 0 t 1/8$,{\\begin{@align*}{1}{-1}\\mathbb {P}_{B_1}\\Biggl (\\sum _{y\\in S_1} \\mathbb {1}_{2d \\leqslant N_2(y) \\leqslant N^{1/4}} N_2(y)\\geqslant \\mathcal {C}^{\\prime }\\Gamma \\Biggr )& \\leqslant \\frac{\\max _{y \\in S_1}\\mathopen {}\\mathclose {\\left( \\mathbb {E}_{B_1} \\exp \\Bigl (\\mathbb {1}_{2d \\leqslant N_2(y) \\leqslant N^{1/4}} N_2(y)t\\Bigr )\\right)}^{|S_1|}}{\\mathrm {e}^{t \\mathcal {C}^{\\prime }\\Gamma }} \\\\ & \\leqslant \\exp \\mathopen {}\\mathclose {\\left( \\vert S_1 \\vert \\log 2 - \\mathcal {C}^{\\prime }\\frac{t}{(\\tau -1)^2}\\log N \\right)}\\,.\\end{@align*}}Now we set $ t = 1/8$, recall the bound $ 2$, plug this estimate back into (\\ref {P_B_1}), and take the expectation.", "We use Lemma \\ref {lem:upper_bound_degrees} to estimate $ S1 $, which in particular implies that $ B1 N1/4$ with very high probability; this concludes the estimate of the expectation of the first term of (\\ref {P_B_1}) by choosing $ C'$ large enough.", "Next, the expectation of the second term is easily estimated by Lemma \\ref {lem:upper_bound_degrees} since $ N2(y)$ has law $ Binom(N - B1 , d/N)$ when conditioned on $ B1$.", "Finally, the expectation of the last term of (\\ref {P_B_1}) is estimated by (\\ref {S1_est1}) by choosing $ C$ large enough.", "This concludes the proof of (\\ref {eq:sumN_i}) for $ i = 2$.$ We now prove (REF ) for $i + 1$ with $i\\geqslant 2$ by induction.", "Using [10] combined with Lemma REF , we deduce that $\\vert S_{i}(y) \\vert \\leqslant d \\vert S_{i-1}(y) \\vert +\\mathcal {C} \\sqrt{d \\vert S_{i-1}(y) \\vert \\log N}$ with very high probability for all $y\\in S_1\\setminus S_1^{\\tau }$ and all $i \\leqslant r_\\star $ .", "Therefore, using the induction assumption, i.e.", "(REF ) for $i$ , we obtain $\\sum _{y\\in S_1\\setminus S_1^\\tau } \\vert S_{i}(y) \\vert & \\leqslant \\mathcal {C}\\frac{\\log N}{(\\tau -1)^2}d^{i-1}+\\mathcal {C} \\sqrt{d \\log N}\\sum _{y\\in S_1\\setminus S_1^\\tau }\\sqrt{\\vert S_{i-1}(y) \\vert }\\\\& \\leqslant \\mathcal {C}\\frac{\\log N}{(\\tau -1)^2}d^{i-1}+\\mathcal {C} \\sqrt{d \\log N} |S_1\\setminus S_1^\\tau | \\Biggl ( \\sum _{y\\in S_1\\setminus S_1^\\tau }\\frac{\\vert S_{i-1}(y) \\vert }{|S_1\\setminus S_1^\\tau |}\\Biggr )^{1/2}\\\\& \\leqslant \\mathcal {C}\\frac{\\log N}{(\\tau -1)^2}d^{i-1}+\\mathcal {C} \\sqrt{d \\log N} \\frac{\\log N}{d (\\tau -1)^2} \\sqrt{d^{i-1}}$ with very high probability, where we used the concavity of $\\sqrt{\\,\\cdot \\,}$ in the second step, (REF ) and (REF ) for $i$ in the last step.", "Since $\\sqrt{d^i \\log N}\\leqslant d^{i/2+1}\\leqslant d^i$ for $i\\geqslant 2 $ and the sequence $(d^{1-i/2})_{i \\in \\mathbb {N}}$ is summable, this proves (REF ) for $i+1$ with a constant $\\mathcal {C}$ independent of $i$ .", "This concludes the proof of Proposition REF ." ], [ "The delocalized phase", "In this section we prove Theorem REF .", "In fact, we state and prove a more general result, Theorem REF below, which immediately implies Theorem REF ." ], [ "Local law", "Theorem REF is a local law for a general class of sparse random matrices of the form $ M = H + f ȇ ȇ^*\\,,$ where $f \\geqslant 0$ and $ȇ \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =N-1/2(1,1,...,1)*$.", "Here $ H$ is a Hermitian random matrix satisfying the following definition.$ Definition 6.1 Let $0 < d < N$ .", "A sparse matrix is a complex Hermitian $N\\times N$ matrix $H=H^* \\in \\mathbb {C}^{N \\times N}$ whose entries $H_{ij}$ satisfy the following conditions.", "The upper-triangular entries ($H_{ij}\\mathrel {\\hbox{.}\\hbox{.", "}}$ 1 i jN$) are independent.\\item [(ii)] We have $ E Hij=0$ and $ E Hij 2=(1 + O(ij))/N$ for all $ i,j$.\\item [(iii)] For any $ k3$, we have$ E|Hij|k 1/(Ndk-22)$for all $ i,j$.$ It is easy to check that the set of matrices $M$ defined as in (REF ) and Definition REF contains those from Theorem REF (see the proof of Theorem REF below).", "The local law for the matrix $M$ established in Theorem REF below provides control of the entries of the Green function $ G(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =( M - z)-1 for $z$ in the spectral domain $ \\mathbf {S} \\equiv S̑_{\\kappa , L, N} = \\mathcal {S}_\\kappa \\times [N^{-1 + \\kappa }, L]$ for some constant $L \\geqslant 1$ .", "We also define the Stieltjes transform $g$ of the empirical spectral measure of $M$ given by $ g(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1N i =1N 1i(M) - z = 1N TrG(z) .", "The limiting behaviour of $G$ and $g$ is governed by the following deterministic quantities.", "Denote by $+ \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={z Imz > 0}$ the complex upper half-plane.For $ z +$ we define $ m(z)$ as the Stieltjes transform of the semicircle law $ 1$,\\begin{equation} m(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation}=\\int \\frac{\\mu _1(\\mathrm {d}u)}{u - z} \\,, \\qquad \\mu _1(\\mathrm {d}u) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =12 (4 - u2)+   du .", "An elementary argument shows that $m(z)$ can be characterized as the unique solution $m$ in $+$ of the equation $ \\frac{1}{m(z)} = -z - m(z)\\,.$ For $\\alpha \\geqslant 0$ and $z \\in +$ we define $ m_\\alpha (z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =- 1 z + m(z) , so that $m_1 = m$ by (REF ).", "In Lemma REF below we show that $m_\\alpha $ is bounded in the domain $S̑$ , with a bound depending only on $\\kappa $ .", "For $x \\in [N]$ we denote the square Euclidean norm of the $x$ th row of $H$ by $ \\beta _x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =y Hxy 2 , which should be thought of as the normalized degree of $x$ ; see Remark REF below.", "Theorem 6.2 (Local law for $M$ ) Fix $0 < \\kappa \\leqslant 1/2$ and $L \\geqslant 1$ .", "Let $H$ be a sparse matrix as in Definition REF , define $M$ as in (REF ) for some $0 \\leqslant f \\leqslant N^{\\kappa /6}$ , and define $G$ and $g$ as in (REF ) and (REF ) respectively.", "Then with very high probability, for $d$ satisfying (REF ), for all $z \\in S̑$ we have $ \\max _{x,y \\in [N]} \\bigl \\vert G_{xy}(z) - \\delta _{xy} m_{\\beta _x}(z) \\bigr \\vert &\\leqslant \\mathcal {C}\\bigg ( \\frac{\\log N}{d^2} \\bigg )^{1/3}\\,,\\\\\\bigl \\vert g(z) - m(z) \\bigr \\vert &\\leqslant \\mathcal {C}\\Bigg ( \\frac{\\log N}{d^2} \\bigg )^{1/3}\\,.$ [Proof of Theorem REF ] Under the assumptions of Theorem REF we find that $M \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =A / d$ is of the form (\\ref {eq:def_M}) for some $ H$ and $ f$ satisfying the assumptions of Theorem \\ref {thm:local_law}.Now Theorem~\\ref {thm:delocalization} is a well-known consequence of Theorem \\ref {thm:local_law} and the boundedness of $ m(z)$ in (\\ref {eq:m_alpha_bounded}) below.", "For the reader^{\\prime }s convenience, we give the short proof.", "Denoting the eigenvalues of $ M$ by $ (i(M))i [N]$ and the associated eigenvectors by $ (w̑i(M))i [N]$, setting $ z = + i$ with $ = N-1 + $, by (\\ref {eq:local_law_entrywise}) and (\\ref {eq:m_alpha_bounded}) we have with very high probability\\begin{equation*}\\mathcal {C} \\geqslant \\operatorname{Im}G_{xx}(z) = \\sum _{i \\in [N]} \\frac{\\eta \\vert \\langle _x {2mu}, w̑_i(M)\\rangle \\vert ^2}{\\eta ^2 + (\\lambda - \\lambda _i(M))^2} \\geqslant \\frac{1}{\\eta } \\, \\vert \\langle _x {2mu}, w̑\\rangle \\vert ^2\\,,\\end{equation*}where in the last step we omitted all terms except $ i$ satisfying $ i(M) = $.", "The claim follows by renaming $ / 2$.", "(Here we used that Theorem \\ref {thm:local_law} holds also for random $ z S̑$, as follows form a standard net argument; see e.g.\\ \\cite [Remark 2.7]{BenyachKnowles2017}.", ")$ Remark 6.3 (Relation between $\\alpha _x$ and $\\beta _x$ ) In the special case $M = d^{-1/2} A$ with $A$ the adjacency matrix of $\\mathbb {G}(N,d/N)$ , we have $\\beta _x = \\frac{1}{d} \\sum _{y} \\bigg (A_{xy} - \\frac{d}{N}\\bigg )^2 = \\alpha _x + O \\biggl (\\frac{d (1 + \\alpha _x)}{N}\\biggr ) = \\alpha _x + \\mathcal {O} \\biggl (\\frac{d + \\log N}{N}\\biggr )$ with very high probability, by Lemma REF .", "By definition, $m_\\alpha (z) \\in +$ for $z \\in +$ , i.e.", "$m_\\alpha $ is a Nevanlinna function, and $\\lim _{z \\rightarrow \\infty } z m_\\alpha (z) = -1$ .", "By the integral representation theorem for Nevanlinna functions, we conclude that $m_\\alpha $ is the Stieltjes transform of a Borel probability measure $\\mu _\\alpha $ on $\\mathbb {R}$ , $ m_\\alpha (z) = \\int \\frac{\\mu _\\alpha (\\mathrm {d}u)}{u - z}\\,.$ Theorem REF implies that the spectral measure of $M$ at a vertex $x$ is approximately $\\mu _{\\beta _x}$ with very high probability.", "Inverting the Stieltjes transform (REF ) and using the definitions () and (REF ), we find after a short calculation $ \\mu _\\alpha (\\mathrm {d}u) = g_\\alpha (u) \\, \\mathrm {d}u + h_\\alpha \\delta _{s_\\alpha }(\\mathrm {d}u)+ h_\\alpha \\delta _{-s_\\alpha }(\\mathrm {d}u)\\,,$ where $g_\\alpha (u) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1u < 22 4-u2(1-)u2 + 2 ,       h=1> 2 - 22 - 2 + 1= 02 ,       s=1> 2 () .", "The family $(\\mu _\\alpha )_{\\alpha \\geqslant 0}$ contains the semicircle law ($\\alpha = 1$ ), the Kesten-McKay law of parameter $d$ ($\\alpha = d / (d - 1)$ ), and the arcsine law ($\\alpha = 2$ ).", "For rational $\\alpha = p/q$ , the measure $\\mu _{p/q}$ can be interpreted as the spectral measure at the root of the infinite rooted $(p,q)$ -regular tree, whose root has $p$ children and all other vertices have $q$ children.", "We refer to Appendix REF for more details.", "See Figure REF for an illustration of the measure $\\mu _\\alpha $ .", "Remark 6.4 Using a standard application the Helffer-Sjöstrand formula (see e.g.", "[16]), we deduce from Theorem REF the following local law for the spectral measure.", "Denote by $\\varrho _x$ the spectral measure of $M$ at vertex $x$ .", "Under the assumptions of Theorem REF , with very high probability, for any inverval $I \\subset \\mathcal {S}_\\kappa $ , we have $\\varrho _x(I) = \\mu _{\\beta _x}(I) + \\mathcal {O} \\biggl (\\vert I \\vert \\bigg ( \\frac{\\log N}{d^2} \\bigg )^{1/3} + N^{\\kappa - 1}\\biggr )\\,.$ The error is smaller than the left-hand side provided that $\\vert I \\vert \\geqslant \\mathcal {C} N^{\\kappa - 1}$ .", "Figure: An illustration of the probability measure μ α \\mu _\\alpha for various values of α\\alpha .", "For α>2\\alpha > 2, μ α \\mu _\\alpha has two atoms which we draw using vertical lines.", "The measure μ α \\mu _\\alpha is the semicircle law for α=1\\alpha = 1, the arcsine law for α=2\\alpha = 2, and the Kesten-McKay law with d=α α-1d = \\frac{\\alpha }{\\alpha - 1} for 1<α<21 < \\alpha < 2.", "Note that the density of μ α \\mu _\\alpha is bounded in 𝒮 κ \\mathcal {S}_\\kappa , uniformly in α\\alpha .", "The divergence of the density near 0 is caused by values of α\\alpha close to 0, and the divergence of the density near ±2\\pm 2 by values of α\\alpha close to 2.The remainder of this section is devoted to the proof of Theorem REF .", "For the rest of this section, we assume that $M$ is as in Theorem REF .", "To simplify notation, we consistently omit the $z$ -dependence from our notation in quantities that depend on $z \\in S̑$ .", "Unless mentioned otherwise, from now on all statements are uniform in $z \\in S̑$ .", "For the proof of Theorem REF , it will be convenient to single out the generic constant $\\mathcal {C}$ from (REF ) by introducing a new constant $\\mathcal {D}$ and replacing (REF ) with $ \\mathcal {D} \\sqrt{\\log N} \\leqslant d \\leqslant (\\log N)^{3/2}\\,.$ Our proof will always assume that $\\mathcal {C} \\equiv \\mathcal {C}_\\nu $ and $\\mathcal {D} \\equiv \\mathcal {D}_\\nu $ are large enough, and the constant $\\mathcal {C}$ in (REF ) can be taken to be $\\mathcal {C} \\vee \\mathcal {D}$ .", "For the rest of this section we assume that $d$ satisfies (REF ) for some large enough $\\mathcal {D}$ , depending on $\\kappa $ and $\\nu $ .", "To guide the reader through the proof, in Figure REF we include a diagram of the dependencies of the various quantities appearing throughout this section.", "Figure: The dependency graph of the various quantities appearing in the proof of Theorem .", "An arrow from xx to yy means that yy is chosen as a function of xx.", "The independent parameters, κ\\kappa and ν\\nu , are highlighted in blue.Typical vertices We start by introducing the key tool in the proof of Theorem REF , a decomposition of vertices into typical vertices and the complementary atypical vertices.", "Heuristically, a typical vertex $x$ has close to $d$ neighbours and the spectral measure of $M$ at $x$ is well approximated by the semicircle law.", "In fact, in order to be applicable to the proof of Proposition REF below, the notion of a typical vertex is somewhat more complicated, and when counting the number of neighbours of a vertex $x$ we also need to weight the neighbours with diagonal entries of a Green function, so that the notion of typical vertex also depends on the spectral parameter $z \\in S̑$ .", "It is precisely defined using the parameters $\\Phi _x$ and $\\Psi _x$ from (REF ) below.", "The main result of this subsection is Proposition REF below, which states, in the language of graphs when $M = d^{-1/2} A$ with $A$ the adjacency matrix of $\\mathbb {G}(N,d/N)$ , that most vertices are typical and most neighbours of any vertex are typical.", "To state it, we introduce some notation.", "Definition 6.5 For any subset $T \\subset [N]$ , we define the minor $M^{(T)}$ with indices in $T$ as the $(N-\\vert T \\vert ) \\times (N-\\vert T \\vert )$ -matrix $ M^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(Mxy)x,y [N] T. If $T$ consists only of one or two elements, $T = \\lbrace x\\rbrace $ or $T=\\lbrace x,y\\rbrace $ , then we abbreviate $M^{(x)}$ and $M^{(xy)}$ for $M^{(\\lbrace x\\rbrace )}$ and $M^{(\\lbrace x,y\\rbrace )}$ .", "We also abbreviate $M^{(Tx)}$ for $M^{(T \\cup \\lbrace x\\rbrace )}$ .", "The Green function of $M^{(T)}$ is denoted by $ G^{(T)}(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(M(T) - z)-1.", "We use the notation $ \\sum _{x}^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x [N]T .", "Definition 6.6 (Typical vertices) Let $\\mathfrak {a}> 0$ be a constant, and define the set of typical vertices $ \\mathcal {T}_\\mathfrak {a}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ = ,       a=a( Nd2 )1/3 , where $ \\Phi _x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =y(x) (Hxy 2 - 1N) ,       x =y(x) (Hxy 2 - 1N) Gyy(x) .", "Note that this notion depends on the spectral parameter $z$ , i.e.", "$\\mathcal {T}_\\mathfrak {a}\\equiv \\mathcal {T}_\\mathfrak {a}(z)$ .", "The constant $\\mathfrak {a}$ will depend only on $\\nu $ and $\\kappa $ .", "It will be fixed in () below.", "The constant $\\mathcal {D} \\geqslant \\mathfrak {a}^{3/2}$ from (REF ) is always chosen large enough so that $\\varphi _{\\mathfrak {a}} \\leqslant 1$ .", "The following proposition holds on the event $\\lbrace \\theta = 1\\rbrace $ , where we introduce the indicator function $ \\theta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1x,y Gxy depending on some deterministic constant $\\Gamma \\geqslant 1$ .", "In (REF ) below, we shall choose a constant $\\Gamma \\equiv \\Gamma _\\kappa $ , depending only on $\\kappa $ , such that the condition $\\theta = 1$ can be justified by a bootstrapping argument along the proof of Theorem REF in Section REF below.", "Throughout the sequel we use the following generalization of Definition REF .", "Definition 6.7 An event $\\Xi $ holds with very high probability on an event $\\Omega $ if for all $\\nu > 0$ there exists $\\mathcal {C} > 0$ such that $\\mathbb {P}(\\Xi \\cap \\Omega ) \\geqslant \\mathbb {P}(\\Omega ) - \\mathcal {C} N^{-\\nu }$ for all $N \\in \\mathbb {N}$ .", "We now state the main result of this subsection.", "Proposition 6.8 There are constants $0 < q \\leqslant 1$ , depending only on $\\Gamma $ , and $\\mathfrak {a}> 0$ , depending only on $\\nu $ and $q$ , such that, on the event $\\lbrace \\theta = 1\\rbrace $ , the following holds with very high probability.", "Most vertices are typical: $ \\vert \\mathcal {T}_\\mathfrak {a}^c \\vert \\leqslant \\exp ( q \\varphi _\\mathfrak {a}^2 d ) + N \\exp ( - 2 q \\varphi _\\mathfrak {a}^2 d).", "$ Most neighbours of any vertex are typical: $ \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}^c}^{(x)}\\vert H_{xy} \\vert ^2 \\leqslant \\mathcal {C} \\varphi _\\mathfrak {a}+ \\mathcal {C}d^4 \\exp (- q \\varphi _\\mathfrak {a}^2 d ) $ uniformly for $x \\in [N]$ .", "For the interpretation of Proposition REF REF , one should think of the motivating example $M = d^{-1/2} A$ , for which $d \\sum _{y \\in \\mathcal {T}^c_\\mathfrak {a}}^{(x)}\\vert H_{xy} \\vert ^2$ is the number of atypical neighbours of $x$ , up to an error term $\\mathcal {O}\\bigl (\\frac{d^2 + d \\log N}{N}\\bigr )$ by Remark REF .", "The remainder of Section REF is devoted to the proof of Proposition REF .", "We need the following version of $\\mathcal {T}_\\mathfrak {a}$ defined in terms of $H^{(T)}$ instead of $H$ .", "Definition 6.9 For any $x \\in [N]$ and $T \\subset [N]$ , we define $ \\Phi _x^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =y(Tx) (Hxy 2 - 1N) ,       x(T) =y(Tx) (Hxy 2 - 1N) Gyy(Tx)  $and\\begin{equation*}\\mathcal {T}^{(T)} _\\mathfrak {a}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\,.$ Note that $\\Phi _x^{(\\emptyset )} = \\Phi _x$ and $\\Psi _x^{(\\emptyset )} = \\Psi _x$ with the definitions from (REF ), and hence $\\mathcal {T}_\\mathfrak {a}^{(\\emptyset )} = \\mathcal {T}_\\mathfrak {a}$ .", "The proof of Proposition REF relies on the two following lemmas.", "Lemma 6.10 There are constants $0 < q \\leqslant 1$ , depending only on $\\Gamma $ , and $\\mathfrak {a}> 0$ , depending only on $\\nu $ and $q$ , such that, for any deterministic $X \\subset [N]$ , the following holds with very high probability on the event $\\lbrace \\theta = 1\\rbrace $ .", "$\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/2}^c \\vert \\leqslant \\exp ( q \\varphi _\\mathfrak {a}^2 d) + \\vert X \\vert \\exp (- 2 q \\varphi _\\mathfrak {a}^2 d)$ .", "If $\\vert X \\vert \\leqslant \\exp ( 2 q \\varphi _\\mathfrak {a}^2 d)$ then $\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/2}^c \\vert \\leqslant \\mathcal {\\varphi }_\\mathfrak {a}d$ .", "For any deterministic $x \\in [N]$ , the same estimates hold for $\\big (\\mathcal {T}^{(x)}_{\\mathfrak {a}/ 2}\\big )^c$ instead of $\\mathcal {T}^c_{\\mathfrak {a}/2}$ and a random set $X \\subset [N] \\setminus \\lbrace x\\rbrace $ that is independent of $H^{(x)}$ .", "Lemma 6.11 With very high probability, for any constant $\\mathfrak {a}> 0$ we have $ \\theta \\vert \\Phi _y - \\Phi _y^{(x)} \\vert \\leqslant \\varphi _{\\mathfrak {a}/2}, \\qquad \\theta \\vert \\Psi _y - \\Psi _y^{(x)} \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2} $ for all $x,y \\in [N]$ .", "Before proving Lemmas REF and REF , we use them to establish Proposition REF .", "As a preparation, we record the following simple consequence of Definition REF and Bennett's inequality.", "Lemma 6.12 We have $\\vert H_{xy} \\vert ^2 \\leqslant 1/d$ almost surely, and $\\sum _y \\vert H_{xy} \\vert ^2 \\leqslant \\mathcal {C} d$ with very high probability.", "[Proof of Proposition REF ] For REF , we choose $X = [N]$ in Lemma REF REF , using that $\\mathcal {T}_{\\mathfrak {a}/2} \\subset \\mathcal {T}_\\mathfrak {a}$ .", "We now turn to the proof of REF .", "By Lemma REF , on the event $\\lbrace \\theta = 1\\rbrace $ we have $\\mathcal {T}^c_\\mathfrak {a}\\subset \\big (\\mathcal {T}^{(x)}_{\\mathfrak {a}/2}\\big )^c$ with very high probability and hence $\\theta \\sum ^{(x)}_{y \\in \\mathcal {T}_\\mathfrak {a}^c} \\vert H_{xy} \\vert ^2 \\leqslant \\theta \\sum ^{(x)}_{y \\in (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c} \\vert H_{xy} \\vert ^2$ with very high probability.", "Since $\\vert H_{xy} \\vert ^2 \\leqslant 1 / d$ by Lemma REF , we obtain the decomposition $ \\begin{aligned}\\sum ^{(x)}_{y \\in (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c} \\vert H_{xy} \\vert ^2 &\\leqslant \\sum _{k = 0}^{\\log N} \\sum ^{(x)}_{y \\in (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c} \\vert H_{xy} \\vert ^2 \\mathbb {1}_{d^{-k-2} \\leqslant \\vert H_{xy} \\vert ^2 \\leqslant d^{-k - 1}} + \\frac{1}{N}\\\\&\\leqslant \\sum _{k = 0}^{\\log N} \\sum ^{(x)}_{y \\in (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c} d^{-k - 1} \\mathbb {1}_{\\vert H_{xy} \\vert ^2 \\geqslant d^{-k-2}} + \\frac{1}{N}\\\\&= \\sum _{k = 0}^{\\log N} d^{-k - 1} \\vert X_k \\cap (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c \\vert + \\frac{1}{N}\\,,\\end{aligned}$ where we defined $X_k \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={y x Hxy 2 d-k - 2} .", "Since $\\sum ^{(x)}_y \\vert H_{xy} \\vert ^2 \\leqslant \\mathcal {C} d$ with very high probability by Lemma REF , we conclude that $ \\vert X_k \\vert \\leqslant \\mathcal {C} d^{k + 3}$ with very high probability.", "We shall apply Lemma REF to the sets $X = X_k$ and $(\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c$ .", "To that end, note that $X_k \\subset [N] \\setminus \\lbrace x\\rbrace $ is a measurable function of the family $(H_{xy})_{y \\in [N]}$ , and hence independent of $H^{(x)}$ .", "Thus, we may apply Lemma REF .", "We define $K \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={k 0 C dk + 3 e 2qa2 d}$ and decompose the sum on the right-hand side of (\\ref {eq:sum_a_y_in_Tc}) into{\\begin{@align*}{1}{-1}\\sum _{k = 0}^{\\log N} d^{-k - 1} \\vert X_k \\cap \\big (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}\\big )^c \\vert &= \\sum _{k = 0}^{K} d^{-k - 1} \\vert X_k \\cap \\big (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}\\big )^c \\vert + \\sum _{k = K+1}^{\\log N} d^{-k - 1} \\vert X_k \\cap \\big (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}\\big )^c \\vert \\\\&\\leqslant \\sum _{k = 0}^{K} d^{-k - 1} \\varphi _\\mathfrak {a}d + \\sum _{k = K+1}^{\\log N} d^{-k - 1} \\bigl (\\mathrm {e}^{q \\varphi _\\mathfrak {a}^2 d} + \\mathcal {C} d^{k+3} \\mathrm {e}^{- 2q \\varphi _\\mathfrak {a}^2 d}\\bigr )\\\\&\\leqslant 2 \\varphi _\\mathfrak {a}+ \\mathcal {C} d^2 \\mathrm {e}^{-q \\varphi _\\mathfrak {a}^2 d}\\log N\\,\\end{@align*}}with very high probability.Here, we used Lemma \\ref {lem:XcapTc} \\ref {item:XcapTc_X_small} to estimate the summands if $ k K$ and Lemma~\\ref {lem:XcapTc} \\ref {item:XcapTc_general} and (\\ref {X_k_estimate}) for the other summands.", "Since $ N d2$, this concludes the proof of \\ref {item:a2}.$ The rest of this subsection is devoted to the proofs of Lemmas REF and REF .", "Let $\\theta $ be defined as in (REF ) for some constant $\\Gamma \\geqslant 1$ .", "For any subset $T \\subset [N]$ , we define the indicator function $\\theta ^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1a,b T Gab(T) 2  .", "Lemma REF is a direct consequence of the following two lemmas.", "The first one, Lemma REF , is mainly a decoupling argument for the random variables $(\\Psi _x)_{x \\in [N]}$ .", "Indeed, the probability that any fixed vertex $x$ is atypical is only small, $o(1)$ , and not very small, $N^{-\\nu }$ ; see (REF ) below.", "If the events of different vertices being atypical were independent, we could deduce that the probability that a sufficiently large set of vertices are atypical is very small.", "However, these events are not independent.", "The most serious breach of independence arises from the Green function $G^{(x)}_{yy}$ in the definition of $\\Psi _x$ .", "In order to make this argument work, we have to replace the parameters $\\Phi _x$ and $\\Psi _x$ with their decoupled versions $\\Phi _x^{(T)}$ and $\\Psi _x^{(T)}$ from Definition REF .", "To that end, we have to estimate the error involved, $\\vert \\Phi _x - \\Phi _x^{(T)} \\vert $ and $\\vert \\Psi _x - \\Psi _x^{(T)} \\vert $ .", "Unfortunately the error bound on the latter is proportional to $\\beta _x$ (see (REF )), which is not affordable for vertices of large degree.", "The solution to this issue involves the observation that if $\\beta _x$ is too large then the vertex is atypical by the condition on $\\Phi _x$ , which allows us to disregard the size of $\\Psi _x$ .", "The details are given in the proof of Lemma REF below.", "The second one, Lemma REF , gives a priori bounds on the entries of the Green function $G^{(T)}$ , which shows that if the entries of $G$ are bounded then so are those of $G^{(T)}$ for $\\vert T \\vert o(d)$ .", "For $T$ of fixed size, this fact is a standard application of the resolvent identities from Lemma REF .", "For our purposes, it is crucial that $T$ can have size up to $o(d)$ , and such a quantitative estimate requires slightly more care.", "Lemma 6.13 There is a constant $0 < q \\leqslant 1$ , depending only on $\\Gamma $ , such that, for any $\\nu >0$ , there is $\\mathcal {C}>0$ such that the following holds for any fixed $\\mathfrak {a}> 0$ .", "If $x \\notin T \\subset [N]$ are deterministic with $\\vert T \\vert \\leqslant \\varphi _\\mathfrak {a}d /\\mathcal {C}$ then $\\mathbb {P}\\big ( T \\subset \\mathcal {T}_{\\mathfrak {a}/ 2}^c,\\, \\theta = 1 \\big ) & \\leqslant \\mathrm {e}^{- 4 q \\varphi _\\mathfrak {a}^2 d \\vert T \\vert } + \\mathcal {C}N^{-\\nu } , \\\\\\mathbb {P}\\big ( T \\subset \\big (\\mathcal {T}_{\\mathfrak {a}/2}^{(x)}\\big )^c, \\theta ^{(x)} =1 \\big ) &\\leqslant \\mathrm {e}^{-4 q \\varphi _\\mathfrak {a}^2 d \\vert T \\vert } + \\mathcal {C}N^{-\\nu }\\,.$ Lemma 6.14 For any subset $T \\subset [N]$ satisfying $\\vert T \\vert \\leqslant \\frac{d}{\\mathcal {C} \\Gamma ^2}$ we have $\\theta \\leqslant \\theta ^{(T)}$ with very high probability.", "Before proving Lemma REF and Lemma REF , we use them to show Lemma REF .", "[Proof of Lemma REF ] Throughout the proof we abbreviate $\\mathbb {P}_\\theta (\\Xi ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =P({ = 1})$.", "Let $ C$ be the constant from Lemma \\ref {lem:decoupling}, and set\\begin{equation} \\mathfrak {a}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation}=\\biggl (\\frac{\\mathcal {C} \\nu }{4 q}\\biggr )^{1/3}\\,.$ For the proof of REF , we choose $k = \\varphi _\\mathfrak {a}d /\\mathcal {C}$ and estimate $\\mathbb {P}_\\theta (\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/ 2}^c \\vert \\geqslant k) \\leqslant \\sum _{Y \\subset X : \\vert Y \\vert = k} \\mathbb {P}_\\theta (Y \\subset \\mathcal {T}_{\\mathfrak {a}/2}^c)\\leqslant \\binom{\\vert X \\vert }{k} \\Big ( \\mathrm {e}^{-4 q \\varphi _\\mathfrak {a}^2 d k} + \\mathcal {C}N^{-\\nu } \\Big )\\\\\\leqslant \\big (\\vert X \\vert \\mathrm {e}^{- 4 q \\varphi _\\mathfrak {a}^2 d}\\big )^k + \\mathcal {C}\\vert X \\vert ^k N^{-\\nu }\\leqslant \\mathrm {e}^{- 2 q \\varphi _\\mathfrak {a}^2 d k} + \\mathcal {C}\\mathrm {e}^{2 q \\varphi _\\mathfrak {a}^2 d k} N^{-\\nu } = N^{-2q\\mathfrak {a}^3/\\mathcal {C}} + \\mathcal {C} N^{2q\\mathfrak {a}^3/\\mathcal {C} - \\nu }\\,.$ where in the second step we used (REF ).", "Thus, by our choice of $\\mathfrak {a}$ , we have $\\mathbb {P}_\\theta (\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/ 2}^c \\vert \\geqslant k) \\leqslant (\\mathcal {C} + 1) N^{-\\nu /2}$ , from which REF follows after renaming $\\nu $ and $\\mathcal {C}$ .", "To prove REF we estimate, for $t>0$ and $l \\in \\mathbb {N}$ , $\\mathbb {P}_\\theta (\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/ 2}^c \\vert \\geqslant t) \\leqslant \\frac{1}{t^l} \\mathbb {E}\\Biggl (\\sum _{x \\in X} \\mathbb {1}_{x \\in \\mathcal {T}_{\\mathfrak {a}/ 2}^c}\\theta \\Biggr )^l = \\frac{1}{t^l} \\sum _{x_1, \\dots , x_l \\in X} \\mathbb {P}_\\theta (x_1 \\in \\mathcal {T}_{\\mathfrak {a}/ 2}^c, \\dots , x_l \\in \\mathcal {T}_{\\mathfrak {a}/ 2}^c)\\,.$ Choosing $l = \\varphi _\\mathfrak {a}d/\\mathcal {C}$ , regrouping the summation according to the partition of coincidences, and using Lemma REF yield $\\mathbb {P}_\\theta (\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/ 2}^c \\vert \\geqslant t) \\leqslant \\frac{1}{t^l} \\sum _{\\pi \\in \\mathfrak {P}_l} \\vert X \\vert ^{\\vert \\pi \\vert } \\big ( \\mathrm {e}^{- 4 q \\varphi _\\mathfrak {a}^2 d \\vert \\pi \\vert } + \\mathcal {C}N^{- \\nu } \\big )\\\\\\leqslant \\frac{1}{t^l} \\sum _{k = 0}^l \\binom{l}{k} l^{l - k} \\vert X \\vert ^k \\big ( \\mathrm {e}^{-4 q \\varphi _\\mathfrak {a}^2 dk} + \\mathcal {C}N^{- \\nu } \\big )= \\frac{(l + \\vert X \\vert \\mathrm {e}^{- 4 q \\varphi _\\mathfrak {a}^2 d})^l + \\mathcal {C}N^{-\\nu } (l + \\vert X \\vert )^l}{t^l}\\,.$ Here, $\\mathfrak {P}_l$ denotes the set of partitions of $[l]$ , and we denote by $k = \\vert \\pi \\vert $ the number of blocks in the partition $\\pi \\in \\mathfrak {P}_l$ .", "We also used that the number of partitions of $l$ elements consisting of $k$ blocks is bounded by $\\binom{l}{k} l^{l - k}$ .", "The last step follows from the binomial theorem.", "Therefore, using $l = \\varphi _\\mathfrak {a}d/\\mathcal {C}$ and choosing $t = \\mathrm {e}^{q \\varphi _\\mathfrak {a}^2 d} + \\vert X \\vert \\mathrm {e}^{- 2 q \\varphi _\\mathfrak {a}^2 d}$ as well as $\\mathcal {C}$ and $\\nu $ sufficiently large imply the bound in Lemma REF REF with very high probability, after renaming $\\mathcal {C}$ and $\\nu $ .", "Here we used (REF ).", "To obtain the same statements for $\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}$ instead of $\\mathcal {T}_{\\mathfrak {a}/ 2}$ , we estimate $ \\mathbb {P}_\\theta \\Big ( \\vert X \\cap (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c \\vert \\geqslant t\\Big ) \\leqslant \\mathbb {E}\\Big [ \\mathbb {P}\\Bigl ( \\vert X \\cap (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c \\vert \\geqslant t, \\theta ^{(x)} = 1 \\Big \\vert X \\Bigr ) \\Big ] + \\mathbb {P}\\big ( \\theta ^{(x)} = 0 , \\theta = 1\\big ).", "$ For both parts, REF and REF , the conditional probability $\\mathbb {P}\\bigl ( \\vert X \\cap (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c \\vert \\geqslant t, \\theta ^{(x)} = 1 \\big \\vert X \\bigr )$ can be bounded as before using () instead of (REF ) since, by assumption on $X$ , the set $\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}$ and the indicator function $\\theta ^{(x)}$ are independent of $X$ .", "The smallness of $\\mathbb {P}(\\theta ^{(x)} = 0, \\theta = 1) \\leqslant \\mathbb {P}(\\theta ^{(x)} < \\theta )$ is a consequence of Lemma REF .", "This concludes the proof of Lemma REF .", "The rest of this subsection is devoted to the proofs of Lemmas REF , REF , and REF .", "Lemma 6.15 There is $\\mathfrak {c}\\equiv \\mathfrak {c}_\\nu >0$ , depending on $\\nu $ and $\\kappa $ , such that for any deterministic $T \\subset [N]$ satisfying $\\vert T \\vert \\leqslant \\mathfrak {c}d / \\Gamma ^2$ we have with very high probability $ \\theta \\max _{x,y \\notin T} \\bigl \\vert G_{xy}^{(T)} \\bigr \\vert \\leqslant 2 \\Gamma \\,.$ Moreover, under the same assumptions on $T$ and for any $u \\in [N] \\setminus T$ , we have $ \\theta \\max _{x,y \\notin T \\cup \\lbrace u\\rbrace } \\bigl \\vert G_{xy}^{(Tu)} - G_{xy}^{(T)} \\bigr \\vert \\leqslant \\mathcal {C}d^{-1}$ with very high probability.", "Before proving Lemma REF , we use it to conclude the proof of Lemma REF .", "[Proof of Lemma REF ] The bound in (REF ) of Lemma REF implies that $\\theta = \\theta \\theta ^{(T)}$ with very high probability.", "Since $\\theta \\leqslant 1$ , the proof is complete.", "[Proof of Lemma REF ] Throughout the proof we work on the event $\\lbrace \\theta = 1\\rbrace $ exclusively.", "After a relabelling of the vertices $[N]$ , we can suppose that $T = [k]$ with $k \\leqslant cd/\\Gamma ^2$ .", "For $k \\in [N]$ , we set $ \\Gamma _k \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1 x,y [k] Gxy([k])  .", "$Note that $ 0 $ by definition of $$.$ We now show by induction on $k$ that there is $\\mathcal {C}>0$ such that $ \\Gamma _k \\leqslant \\Gamma _0 \\bigg (1 + \\frac{16 \\mathcal {C}\\Gamma ^2}{d} \\bigg )^k$ for all $k \\in \\mathbb {N}$ satisfying $k \\leqslant \\frac{d}{32 \\, \\mathcal {C}\\Gamma ^2}$ .", "Since $1 + x \\leqslant \\mathrm {e}^x$ , (REF ) implies that $\\Gamma _k \\leqslant \\mathrm {e}^{1/2} \\Gamma _0 \\leqslant 2 \\Gamma $ .", "This directly implies (REF ) by the definition of $\\theta $ .", "The initial step with $k = 0$ is trivially correct.", "For the induction step $k \\rightarrow k+1$ , we set $T = [k]$ and $u = k + 1$ .", "The algebraic starting point for the induction step is the identities (REF ) and (REF ).", "We shall need the following two estimates.", "First, abbreviating $\\eta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =Imz$, from Lemma \\ref {lem:Ward} and Cauchy--Schwarz, we get\\begin{equation} \\frac{f}{N}\\biggl \\vert G_{uy}^{(T)} \\sum _a^{(Tu)} G_{xa}^{(Tu)} \\biggr \\vert \\leqslant \\frac{f}{N} \\Gamma _k \\sqrt{\\frac{N}{\\eta }} \\Gamma _{k+1} \\leqslant N^{-\\kappa /3} \\Gamma _k \\Gamma _{k+1}\\,,\\end{equation}where we used that $ k+1 1$, $ f N/6$, and $ N-1 + $.Second, the first estimate of (\\ref {eq:LDB_wvhp}) in Corollary~\\ref {cor:large_deviation_very_high_probabilty} with $ = k+1/d$ and $ = k+1/(N)$, Lemma \\ref {lem:Ward}, and $ k+1 1$ imply\\begin{equation} \\biggl \\vert \\sum _{a}^{(Tu)} G_{xa}^{(Tu)} H_{au} \\biggr \\vert \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}} \\Gamma _{k+1}\\end{equation}with very high probability.$ Hence, owing to (REF ) and (REF ) with $T = [k]$ and $u = k + 1$ , we get, respectively, $ \\Gamma _{k+1} \\leqslant \\Gamma _k + \\frac{\\mathcal {C}}{\\sqrt{d}} \\Gamma _k \\Gamma _{k+1}, \\qquad \\qquad \\Gamma _{k+1} \\leqslant \\Gamma _k + \\frac{\\mathcal {C}}{d} \\Gamma _k \\Gamma _{k+1}^2$ with very high probability.", "By the induction assumption (REF ) we have $\\mathcal {C} \\Gamma _k / \\sqrt{d} \\leqslant 2 \\mathcal {C} \\Gamma / \\sqrt{d} \\leqslant 1/2$ , so that the first inequality in (REF ) implies the rough a priori bound $ \\Gamma _{k+1} \\leqslant 2 \\Gamma _k$ with very high probability.", "From the second inequality in (REF ) and (REF ), we deduce that $\\Gamma _{k+1} \\leqslant \\Gamma _k \\biggl (1 + \\frac{4 \\mathcal {C}}{d} \\Gamma _k^2\\biggr ) \\leqslant \\Gamma _k \\biggl (1 + \\frac{16 \\mathcal {C} \\Gamma ^2}{d}\\biggr )\\,,$ where in the second step we used $\\Gamma _k \\leqslant 2 \\Gamma $ , by the induction assumption (REF ).", "This concludes the proof of (REF ), and, hence, of (REF ).", "For the proof of (REF ), we start from (REF ) and use (), () as well as (REF ).", "This concludes the proof of Lemma REF .", "The next result provides concentration estimates for the parameters $\\Phi _x$ and $\\Psi _x$ .", "Lemma 6.16 There is a constant $0 < q \\leqslant 1$ , depending only on $\\Gamma $ , such that the following holds.", "Let $\\mathfrak {c}>0$ be as in Lemma REF , and let $x \\in [N]$ and $T \\subset [N]$ be deterministic and satisfy $\\vert T \\vert \\leqslant \\mathfrak {c}d / \\Gamma ^2$ .", "Then for any $0 < \\varepsilon \\leqslant 1$ we have $\\theta ^{(T)} \\mathbb {P}\\big ( \\vert \\Phi _x^{(T)} \\vert > \\varepsilon \\bigm \\vert H^{(T)} \\big ) \\leqslant \\mathrm {e}^{- 32 q \\varepsilon ^2 d}\\,, \\qquad \\theta ^{(T)} \\mathbb {P}\\big (\\vert \\Psi _x^{(T)} \\vert > \\varepsilon \\bigm \\vert H^{(T)} \\big ) \\leqslant \\mathrm {e}^{ - 32 q \\varepsilon ^2 d}\\,,$ and, for any $u \\notin T$ , $ \\Phi _x^{(Tu)} - \\Phi _x^{(T)} = O\\biggl (\\frac{1}{d}\\biggr )\\,,\\qquad \\theta ^{(T)} \\bigl (\\Psi _x^{(Tu)} - \\Psi _x^{(T)}\\bigr ) = \\mathcal {O}\\biggl (\\frac{1 + \\beta _x}{d}\\biggr )$ with very high probability.", "Before proving Lemma REF , we use it conclude the proof of Lemma REF .", "[Proof of Lemma REF ] Using (), we find that $\\beta _x \\leqslant \\mathcal {C} (1 + \\frac{\\log N}{d})$ with very high probability.", "The claim now follows from (REF ) with $T = \\emptyset $ and the definition of $\\varphi _{\\mathfrak {a}}$ , choosing the constant $\\mathcal {D}$ in (REF ) large enough.", "[Proof of Lemma REF ] Set $q \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1211(e)2$.We get, using (\\ref {eq:LDB_linear_noncentered}) with $ r =32 q 2 d d$, $ EHxy 2 = 1/N$, and Chebyshev^{\\prime }s inequality,\\begin{multline*}\\theta ^{(T)} \\mathbb {P}\\Big ( \\vert \\Psi _x^{(T)} \\vert > \\varepsilon \\Bigm \\vert H^{(T)} \\Big )= \\mathbb {P}\\Bigg (\\theta ^{(T)} \\Biggl \\vert \\sum _y^{(Tx)} (\\vert H_{xy} \\vert ^2 - \\mathbb {E}\\vert H_{xy} \\vert ^2) G_{yy}^{(T)} \\Biggr \\vert > \\varepsilon \\biggm \\vert H^{(T)} \\Bigg )\\\\\\leqslant \\biggl (\\frac{8 \\Gamma }{\\varepsilon } \\sqrt{\\frac{r}{d}}\\biggr )^r= \\mathrm {e}^{ -32 q \\varepsilon ^2 d}\\end{multline*}with very high probability for any $ 0 < 1$.", "This proves the estimate on $ x(T)$ in (\\ref {eq:bound_Phi_Psi_x}), and the estimate for $ x(T)$ is proved similarly.$ We now turn to the proof of (REF ).", "If $x = u$ then the statement is trivial.", "Thus, we assume $x \\ne u$ .", "In this case we have $\\Phi _x^{(Tu)} - \\Phi _x^{(T)} = - \\bigg (\\vert H_{xu} \\vert ^2 - \\frac{1}{N} \\bigg )$ and the claim for follows from Lemma REF .", "Next, $ \\Psi _x^{(Tu)} - \\Psi _x^{(T)} = \\sum _{y}^{(Tux)} \\bigg ( \\vert H_{xy} \\vert ^2 - \\frac{1}{N} \\bigg ) \\Big ( G_{yy}^{(Tux)} - G_{yy}^{(Tx)} \\Big ) - \\bigg (\\vert H_{xu} \\vert ^2 - \\frac{1}{N} \\bigg ) G_{uu}^{(Tx)}\\,.", "$ The last term multiplied by $\\theta ^{(T)}$ is estimated by $O(\\Gamma / d)$ by Lemma REF and the fact that $\\theta ^{(T)} \\vert G_{uu}^{(Tx)} \\vert \\leqslant 4 \\Gamma $ by (REF ).", "We estimate the first term using (REF ) in Lemma REF , which yields $\\theta ^{(T)} \\bigl \\vert \\Psi _x^{(Tu)} - \\Psi _x^{(T)} \\bigr \\vert \\leqslant \\sum _{y}^{(Tux)} \\vert H_{xy} \\vert ^2 \\frac{\\mathcal {C}}{d} + \\frac{1}{N} \\sum _{y}^{(Tux)} \\frac{\\mathcal {C}}{d} + O \\biggl ( \\frac{\\Gamma }{d}\\biggr ) = \\mathcal {O} \\biggl (\\frac{1 + \\beta _x}{d}\\biggr )$ with very high probability.", "This concludes the proof of Lemma REF .", "[Proof of Lemma REF ] Throughout the proof we abbreviate $\\mathbb {P}_\\theta (\\Xi ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =P({ = 1})$.", "We have\\begin{equation*}\\mathbb {P}\\big ( T \\subset \\mathcal {T}_{\\mathfrak {a}/ 2}^c,\\, \\theta = 1 \\big ) = \\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x\\Biggr )\\,,\\end{equation*}where we defined the event\\begin{equation*}\\Omega _x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\bigl \\lbrace \\vert \\Phi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace \\cup \\bigl \\lbrace \\vert \\Psi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace = \\bigl \\lbrace \\vert \\Phi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace \\cup \\bigl \\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}, \\vert \\Psi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace \\,.$ We have the inclusions $\\bigl \\lbrace \\vert \\Phi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace &\\subset \\bigl \\lbrace \\vert \\Phi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr \\rbrace \\cup \\bigl \\lbrace \\vert \\Phi _x - \\Phi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr \\rbrace \\,,\\\\\\bigl \\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}, \\vert \\Psi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace &\\subset \\bigl \\lbrace \\vert \\Psi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr \\rbrace \\cup \\bigl \\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}, \\vert \\Psi _x - \\Psi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr \\rbrace \\,.$ Defining the event $\\Omega _x^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={x(T) > a/ 4} {x(T) > a/ 4} , we therefore deduce by a union bound that $ \\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x\\Biggr ) \\leqslant \\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x^{(T)}\\Biggr ) + \\sum _{x \\in T} \\mathbb {P}_\\theta \\bigl (\\vert \\Phi _x - \\Phi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr )\\\\+ \\sum _{x \\in T} \\mathbb {P}_\\theta \\bigl (\\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}, \\vert \\Psi _x - \\Psi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr )\\,.$ We begin by estimating the first term of (REF ).", "To that end, we observe that, conditioned on $H^{(T)}$ , the family $(\\Omega _x^{(T)})_{x \\in T}$ is independent.", "Using Lemma REF we therefore get $\\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x^{(T)}\\Biggr ) \\leqslant \\mathbb {E}\\Biggl [ \\theta ^{(T)} \\mathbb {P}\\Biggl (\\bigcap _{x \\in T}\\Omega _x^{(T)} \\biggm | H^{(T)}\\Biggr )\\Biggr ] + \\mathcal {C} N^{-\\nu } = \\mathbb {E}\\biggl [\\theta ^{(T)} \\prod _{x \\in T} \\mathbb {P}(\\Omega _x^{(T)} | H^{(T)})\\biggr ] + \\mathcal {C} N^{-\\nu }\\,,$ and we estimate each factor using (REF ) from Lemma REF as $\\theta ^{(T)} \\mathbb {P}(\\Omega _x^{(T)} | H^{(T)}) \\leqslant \\theta ^{(T)} \\mathbb {P}\\big ( \\vert \\Phi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/4} \\bigm \\vert H^{(T)} \\big ) + \\theta ^{(T)} \\mathbb {P}\\big ( \\vert \\Psi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4} \\bigm \\vert H^{(T)} \\big )\\\\\\leqslant 2 \\mathrm {e}^{-8 q \\varphi _{\\mathfrak {a}}^2 d} \\leqslant \\mathrm {e}^{-4 q \\varphi _{\\mathfrak {a}}^2 d}\\,,$ where in the last step we used that $\\mathrm {e}^{-4 q \\varphi _{\\mathfrak {a}}^2 d} \\leqslant 1/2$ .", "We conclude that $\\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x^{(T)}\\Biggr ) \\leqslant \\mathrm {e}^{-4 q \\varphi _{\\mathfrak {a}}^2 d \\vert T \\vert } + \\mathcal {C} N^{-\\nu }\\,.$ Next, we estimate the second term of (REF ).", "After renaming the vertices, we may assume that $T = [k]$ with $k \\leqslant \\varphi _\\mathfrak {a}d / \\mathcal {C}$ , so that we get from (REF ) from Lemma REF (using that $\\varphi _\\mathfrak {a}d / \\mathcal {C} \\leqslant \\mathfrak {c}d / \\Gamma ^2$ provided that $\\mathcal {D}$ in (REF ) is chosen large enough, depending on $\\mathfrak {a}$ ), by telescoping and recalling Lemma REF , $ \\vert \\Phi _x - \\Phi _x^{(T)} \\vert \\leqslant \\sum _{i = 0}^{k-1} \\bigl \\vert \\Phi _x^{([i])} - \\Phi _x^{([i+1])} \\bigr \\vert \\leqslant O \\biggl (\\frac{k}{d}\\biggr ) \\leqslant \\varphi _{\\mathfrak {a}/ 4}$ with very high probability on the event $\\lbrace \\theta = 1\\rbrace $ , if the constant $\\mathcal {C}$ in the upper bound $\\varphi _\\mathfrak {a}d / \\mathcal {C}$ on $k$ is large enough.", "The last term of (REF ) is estimated analogously, with the additional observation that, by definition of $\\Phi _x$ and since $\\varphi _{\\mathfrak {a}/2} \\leqslant 1/2$ , on the event $\\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}\\rbrace $ we have $\\beta _x \\leqslant 2$ .", "Thus, on the event $\\lbrace \\theta = 1\\rbrace \\cap \\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}\\rbrace $ we have, by Lemma REF , $ \\vert \\Psi _x - \\Psi _x^{(T)} \\vert \\leqslant \\sum _{i = 0}^{k-1} \\bigl \\vert \\Psi _x^{([i])} - \\Psi _x^{([i+1])} \\bigr \\vert \\leqslant \\mathcal {O} \\biggl (\\frac{k(1 + \\beta _x)}{d}\\biggr ) \\leqslant \\varphi _{\\mathfrak {a}/ 4}$ with very high probability, for large enough $\\mathcal {C}$ in the upper bound on $k$ .", "We conclude that the two last terms of (REF ) are bounded by $\\mathcal {C} N^{-\\nu }$ , and the proof of (REF ) is therefore complete.", "The proof of () is identical, replacing the matrix $M$ with the matrix $M^{(x)}$ .", "Self-consistent equation and proof of Theorem  REF In this subsection, we derive an approximate self-consistent equation for the Green function $G$ , and use it to prove Theorem REF .", "The key ingredient is Proposition REF below, which provides a bootstrapping bound stating that if $G_{xx} - m_{\\beta _x}$ is smaller than some constant then it is in fact bounded by $\\varphi _\\mathfrak {a}$ with very high probability.", "It is proved by first deriving and solving a self-consistent equation for the entries $G_{xx}$ indexed by typical vertices $x \\in \\mathcal {T}_\\mathfrak {a}$ , and using the obtained bounds to analyse $G_{xx}$ for atypical vertices $x \\in \\mathcal {T}^c_\\mathfrak {a}$ .", "We begin with a simple algebraic observation.", "Lemma 6.17 (Approximate self-consistent equation) For any $x \\in [N]$ , we have $ \\frac{1}{G_{xx}} = - z - \\sum _{y}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} + Y_x\\,, $ where we introduced the error term $ Y_x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =Hxx + fN - a b(x) Hxa Gab(x) Hbx - a,b(x) ( fN ( HxaGab(x) + Gab(x) Hbx ) + f2N2 Gab(x)) .", "The lemma follows directly from (REF ) and the definition (REF ).", "Let $\\theta $ be defined as in (REF ) with some $\\Gamma \\geqslant 1$ .", "The following lemma provides a priori bounds on the error terms appearing in the self-consistent equation.", "Lemma 6.18 For all $z \\in \\mathbf {S}$ , with very high probability, $\\theta \\max _x \\vert Y_x \\vert & \\leqslant \\mathcal {C}d^{-1/2}, \\\\\\theta \\max _{x \\ne y} \\vert G_{xy} \\vert & \\leqslant \\mathcal {C}d^{-1/2}, \\\\\\theta \\max _{x \\ne a \\ne y} \\vert G_{xy} - G_{xy}^{(a)} \\vert & \\leqslant \\mathcal {C}d^{-1}.", "$ We first estimate $Y_x$ .", "From Lemma REF , the upper bound on $f$ , and (REF ), we conclude that $\\vert H_{xx} \\vert + f / N = O(d^{-1/2})$ almost surely.", "Moreover, the Cauchy–Schwarz inequality, Lemma REF , (REF ) and the upper bound on $f$ imply $ \\theta \\frac{f^2}{N^2} \\biggl \\vert \\sum _{a,b}^{(x)} G_{ab}^{(x)} \\biggr \\vert \\leqslant C_\\kappa \\frac{f^2}{\\sqrt{N\\eta }} \\leqslant C_\\kappa N^{-\\kappa /6} \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}}\\,, $ for some constant $C_\\kappa $ depending only on $\\kappa $ .", "Next, we use the first estimate of (REF ), Lemma REF , and the upper bound on $f$ to conclude that $ \\frac{f}{N} \\theta \\biggl \\vert \\sum _{a,b}^{(x)} H_{xa} G_{ab}^{(x)} \\biggr \\vert + \\frac{f}{N} \\theta \\biggl \\vert \\sum _{a,b}^{(x)} G_{ab}^{(x)} H_{bx} \\biggr \\vert \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}} \\frac{f}{\\sqrt{N\\eta }} \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}} N^{-\\kappa /3} \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}} $ with very high probability (compare the proof of ()).", "Moreover, from Lemma REF and the second estimate of (REF ) we deduce that remaining term in (REF ) is $\\mathcal {O}(d^{-1}) = \\mathcal {O}(d^{-1/2})$ .", "This concludes the proof of (REF ).", "For the proof of (), we start from (REF ) and use $M_{xa} = H_{xa} + f/ N $ to obtain $ G_{xy} = - G_{xx} \\sum _{a}^{(x)} H_{xa} G_{ay}^{(x)} - G_{xx} H_{xy} G_{yy}^{(x)} - \\frac{f}{N} G_{xx} \\sum _a^{(x)} G_{ay}^{(x)}.", "$ Similar arguments as in () and () show that the first and third term, respectively, are bounded by $\\mathcal {C}d^{-1/2}$ with very high probability.", "The same bound for the second term follows from Lemma REF and (REF ) in Lemma REF .", "This proves ().", "Finally, () follows directly from (REF ).", "Proposition REF below is the main tool behind the proof of Theorem REF .", "To formulate it, we introduce the $z$ -dependent random control parameters $\\Lambda _{\\mathrm {d}} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x Gxx - mx  ,       o =x y Gxy  ,       =d o , and, for some constant $\\lambda \\leqslant 1$ , the indicator function $ \\phi \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1 .", "Proposition REF below provides a strong bound on $\\Lambda $ provided the a priori condition $\\phi = 1$ is satisfied.", "Each step of its proof is valid provided $\\lambda $ is chosen small enough depending on $\\kappa $ .", "Note that, owing to (REF ), there is a deterministic constant $\\Gamma $ , depending only on $\\kappa $ , such that, for all $z \\in \\mathbf {S}$ , we have $ \\phi \\max _{x,y} \\vert G_{xy} \\vert \\leqslant \\Gamma \\,.$ In particular, if $\\Gamma $ in the definition (REF ) of $\\theta $ is chosen as in (REF ) then $ \\phi \\leqslant \\theta \\,.$ Proposition 6.19 There exists $\\lambda > 0$ , depending only on $\\kappa $ , such that, for all $z \\in \\mathbf {S}$ , with very high probability, $ \\phi \\Lambda \\leqslant \\mathcal {C}\\varphi _\\mathfrak {a}\\,.", "$ For the proof of Proposition REF , we employ the results of the previous subsections to show that the diagonal entries $(G_{xx})_{x \\in \\mathcal {T}_\\mathfrak {a}}$ of the Green function of $M$ at the typical vertices satisfy the approximate self-consistent equation (REF ) below.", "This is a perturbed version of the relation (REF ) for the Stieltjes transform $m$ of the semicircle law, which holds for all $z \\in +$ .", "The stability estimate, (REF ) below, then implies that $G_{xx}$ and $m$ are close for all $x \\in \\mathcal {T}_\\mathfrak {a}$ .", "From this we shall, in a second step, deduce that $G_{xx}$ is close to $m_{\\beta _x}$ for all $x$ ; this steps includes also the atypical vertices.", "The next lemma is a relatively standard stability estimate of self-consistent equations in random matrix theory (compare e.g.", "to [27]).", "It is proved in Appendix REF .", "Lemma 6.20 (Stability of the self-consistent equation for $m$ ) Let $\\mathcal {X}$ be a finite set, $\\kappa >0$ , and $z \\in +$ satisfy $\\vert \\mathrm {Re}\\,z \\vert \\leqslant 2- \\kappa $ .", "We assume that, for two vectors $(g_x)_{x \\in \\mathcal {X}}$ , $(\\varepsilon _x)_{x \\in \\mathcal {X}} \\in {\\mathcal {X}}$ , the identities $ \\frac{1}{g_x} = -z - \\frac{1}{\\vert \\mathcal {X} \\vert }\\sum _{y \\in \\mathcal {X}} g_y + \\varepsilon _x$ hold for all $x \\in \\mathcal {X}$ .", "Then there are constants $b, C \\in (0,\\infty )$ , depending only on $\\kappa $ , such that if $\\max _{x \\in \\mathcal {X}} \\vert g_x -m(z) \\vert \\leqslant b$ then $ \\max _{x \\in \\mathcal {X}} \\vert g_x - m(z) \\vert \\leqslant C \\max _{x \\in \\mathcal {X}} \\vert \\varepsilon _x \\vert ,$ where $m(z)$ satisfies (REF ).", "[Proof of Proposition REF ] Throughout the proof, we work on the event $\\lbrace \\phi = 1\\rbrace $ , which, by (REF ), is contained in the event $\\lbrace \\theta = 1\\rbrace $ .", "Fix $\\mathfrak {a}$ as in Lemma REF .", "Throughout the proof we use that $d^{-1/2} \\leqslant \\varphi _\\mathfrak {a}$ by the upper bound in (REF ).", "Owing to (), it suffices to estimate $\\Lambda _{\\mathrm {d}}$ .", "Let $b$ be chosen as in Lemma REF , and set $\\lambda \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =b/2$ in the definition (\\ref {eq:def_vartheta}) of $$.$ For the analysis of $G_{xx}$ we distinguish the two cases $x \\in \\mathcal {T}_\\mathfrak {a}$ and $x \\notin \\mathcal {T}_\\mathfrak {a}$ .", "If $x \\in \\mathcal {T}_\\mathfrak {a}$ then we write using Lemma REF and the definition (REF ) of $\\Psi _x$ that $\\frac{1}{G_{xx}} = -z - \\sum _y^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} + Y_x = -z - \\frac{1}{N} \\sum _y^{(x)} G_{yy}^{(x)} + Y_x - \\Psi _x = -z - \\frac{1}{\\vert \\mathcal {T}_\\mathfrak {a} \\vert } \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}} G_{yy} + \\varepsilon _x\\,,$ where the error term $\\varepsilon _x$ satisfies $ \\vert \\varepsilon _x \\vert = \\mathcal {O} \\biggl (d^{-1/2} + \\frac{1}{N} \\exp ( q \\varphi _\\mathfrak {a}^2 d) + \\exp (-2 q \\varphi _\\mathfrak {a}^2 d ) + \\varphi _\\mathfrak {a}\\biggr ) = \\mathcal {O}(\\varphi _\\mathfrak {a})$ with very high probability.", "Here, in the first step of (REF ) we used (REF ), (), Proposition REF REF , and the bound on $\\Psi _x$ in the definition (REF ) of $\\mathcal {T}_\\mathfrak {a}$ , and in the second step of (REF ) we used that $\\varphi _\\mathfrak {a}^2 d = \\mathfrak {a}^2 (\\log N)^{2/3} d^{-1/3}$ and (REF ) imply $(\\log N)^{1/6} / \\mathcal {C} \\leqslant \\varphi ^2_\\mathfrak {a}d \\leqslant \\mathcal {C} (\\log N)^{1/2}$ , which yields $ \\frac{1}{N} \\exp ( q \\varphi _\\mathfrak {a}^2 d) + \\exp (-2 q \\varphi _\\mathfrak {a}^2 d ) \\leqslant \\mathcal {C} d^{-10} \\leqslant \\varphi _\\mathfrak {a}\\,.$ Thus, for $(G_{xx})_{x \\in \\mathcal {T}_\\mathfrak {a}}$ we get the self-consistent equation in (REF ) with $g_x = G_{xx}$ and $\\mathcal {X} = \\mathcal {T}_\\mathfrak {a}$ .", "Moreover, by the bound on $\\Phi _x$ in the definition (REF ) of $\\mathcal {T}_\\mathfrak {a}$ , we have $\\beta _x = 1 + \\mathcal {O}(\\varphi _\\mathfrak {a})$ .", "Hence, by (), the assumption $\\phi = 1$ and $d \\geqslant \\mathcal {C}\\sqrt{\\log N}$ , we find that $\\vert G_{xx} - m \\vert \\leqslant \\vert G_{xx} - m_{\\beta _x} \\vert + \\vert m_{\\beta _x} - m \\vert \\leqslant b\\,,$ choosing the constant $\\mathcal {D}$ in (REF ) large enough that the right-hand side of (), i.e.", "$C \\vert \\beta _x - 1 \\vert $ , is bounded by $b/2$ .", "Hence Lemma REF is applicable and we obtain $\\vert G_{xx} - m \\vert = O(\\max _{y \\in \\mathcal {T}_\\mathfrak {a}} \\vert \\varepsilon _y \\vert )$ .", "Therefore, we obtain $ \\vert G_{xx} - m_{\\beta _x} \\vert \\leqslant \\vert G_{xx} - m \\vert + \\vert m - m_{\\beta _x} \\vert \\leqslant \\mathcal {C}\\varphi _\\mathfrak {a}$ with very high probability.", "This concludes the proof in the case $x \\in \\mathcal {T}_\\mathfrak {a}$ .", "What remains is the case $x \\notin \\mathcal {T}_\\mathfrak {a}$ .", "In that case, we obtain from Lemma REF that $ \\frac{1}{G_{xx}} = -z - \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} - \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}^c}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} + Y_x = -z - \\beta _x m + \\varepsilon _x\\,,$ where the error term $\\varepsilon _x$ satisfies $\\varepsilon _x = \\mathcal {O} ((1 + \\beta _x) \\varphi _\\mathfrak {a})$ with very high probability.", "Here we used (REF ) as well as (), (REF ), (REF ) and Proposition REF REF twice to conclude that $\\sum _{y \\in \\mathcal {T}_\\mathfrak {a}}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} = \\beta _x m + \\mathcal {O} (\\beta _x \\varphi _\\mathfrak {a}) \\,, \\qquad \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}^c}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} = \\mathcal {O}\\big (\\varphi _\\mathfrak {a}+ d^4 \\exp (-q \\varphi _\\mathfrak {a}^2 d )\\big )= \\mathcal {O}(\\varphi _\\mathfrak {a}) $ with very high probability.", "From (REF ) and (REF ) we therefore get $ G_{xx} - m_{\\beta _x} = - m_{\\beta _x} \\, \\frac{1}{-z - \\beta _x m + \\varepsilon _x} \\, \\varepsilon _x\\,.$ To estimate the right-hand side of (REF ), we consider the cases $\\beta _x \\leqslant 1$ and $\\beta _x > 1$ separately.", "If $\\beta _x \\leqslant 1$ then, by (REF ), the first factor of (REF ) is bounded by $C$ .", "Thus, by (REF ), the second factor is bounded by $2 C$ provided that $\\vert \\varepsilon _x \\vert \\leqslant 1/{2C}$ by choosing $\\mathcal {D}$ in (REF ) large enough, and the third factor is bounded by $\\mathcal {C} \\varphi _\\mathfrak {a}$ .", "This yields the claim.", "If $\\beta _x > 1$ , we use that $\\operatorname{Im}m \\geqslant c$ for some constant $c > 0$ depending only on $\\kappa $ and $L$ .", "Thus, the right-hand side of (REF ) is bounded in absolute value, again using (REF ), by $C \\frac{1}{\\beta _x c/2} \\mathcal {C} \\beta _x \\varphi _\\mathfrak {a}$ , provided that $\\mathcal {D}$ in (REF ) is chosen large enough.", "This yields the claim.", "[Proof of Theorem REF ] After possibly increasing $L$ , we can assume that $L$ in the definition of $\\mathbf {S}$ in (REF ) satisfies $L \\geqslant 2/\\lambda + 1$ , where $\\lambda $ is chosen as in Proposition REF .", "We first show that () follows from (REF ).", "Indeed, averaging the estimate on $\\vert G_{xx} - m_{\\beta _x} \\vert $ in (REF ) over $x \\in [N]$ , using that $m_{\\beta _x} = m + O(\\varphi _\\mathfrak {a})$ for $x \\in \\mathcal {T}_\\mathfrak {a}$ by () and estimating the summands in $\\mathcal {T}_\\mathfrak {a}^c$ by Proposition REF REF and (REF ) yield () due to (REF ).", "What remains is the proof of (REF ).", "Let $z_0 \\in \\mathbf {S}$ , set $J \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={ j N0 Im z0 + j N-3 2 / }$,and define $ zj =z0 + ij N-3$ for $ j [J]$.We shall prove the bound in (\\ref {eq:local_law_entrywise}) at $ z = zj$ by induction on $ j$, starting from $ j = J$ and going down to $ j = 0$.Since $ Gxy(z) (Im z)-1$ and $ mx(z) (Im z)-1$ for all $ x,y [N]$,we have $ x Gxx(zJ) - mx(zJ) $ and $ (zJ) = 1$.$ For the induction step $j \\rightarrow j - 1$ , suppose that $\\phi (z_j) = 1$ with very high probability.", "Then, by Proposition REF , we deduce that $\\Lambda (z_j) \\leqslant \\mathcal {C} \\varphi _\\mathfrak {a}$ with very high probability.", "Since $G_{xy}$ and $m_{\\beta _x}$ are Lipschitz-continuous on $\\mathbf {S}$ with constant $N^2$ , we conclude that $\\Lambda (z_{j-1}) \\leqslant \\mathcal {C}\\varphi _\\mathfrak {a}+ N^{-1}$ with very high probability.", "If $N$ is sufficiently large and $\\varphi _\\mathfrak {a}$ is sufficiently small, obtained by choosing $\\mathcal {D}$ in (REF ) large enough, then we deduce that $\\Lambda (z_{j-1}) \\leqslant \\lambda $ with very high probability and hence $\\phi (z_{j - 1}) = 1$ with very high probability.", "Using Proposition REF , this concludes the induction step, and hence establishes $\\Lambda (z_0) \\leqslant \\mathcal {C} \\varphi _\\mathfrak {a}$ with very high probability.", "Here we used that the intersection of $J$ events of very high probability is an event of very high probability, since $J \\leqslant C N^3$ , where $C$ depends on $\\kappa $ .", "Appendices toc In the following appendices we collect various tools and explanations used throughout the paper.", "Simulation of the $\\ell ^\\infty $ -norms of eigenvectors In Figure REF we depict a simulation of the $\\ell ^\\infty $ -norms of the eigenvectors of the adjacency matrix $A / \\sqrt{d}$ of the Erdős-Rényi graph $\\mathbb {G}(N,d/N)$ restricted to its giant component.", "We take $d = b \\log N$ with $N = 10^{\\prime }000$ and $b = 0.6$ .", "The eigenvalues and eigenvectors are drawn using a scatter plot, where the horizontal coordinate is the eigenvalue and the vertical coordinate the $\\ell ^\\infty $ -norm of the associated eigenvector.", "The higher a dot is located, the more localized the associated eigenvector is.", "Complete delocalization corresponds to a vertical coordinate $\\approx 0.01$ , and localization at a single site to a vertical coordinate 1.", "Note the semilocalization near the origin and outside of $[-2,2]$ .", "The two semilocalized blips around $\\pm 0.4$ are a finite-$N$ effect and tend to 0 as $N$ is increased.", "The Perron-Frobenius eigenvalue is an outlier near $2.8$ with delocalized eigenvector.", "Figure: A scatter plot of (λ,∥w̑∥ ∞ )(\\lambda , \\Vert w̑ \\Vert _\\infty ) for all eigenvalue-eigenvector pairs (λ,w̑)(\\lambda , w̑) of the adjacency matrix A/dA / \\sqrt{d} of the critical Erdős-Rényi graph restricted to its giant component, where N=10 ' 000N = 10^{\\prime }000 and d=0.6logNd = 0.6 \\log N. Spectral analysis of the infinite rooted $(p,q)$ -regular tree In this appendix we describe the spectrum, eigenvectors, and spectral measure of the following simple graph.", "Definition 7.1 For $p,q \\in \\mathbb {N}^*$ we define $\\mathbb {T}_{p,q}$ as the infinite rooted $(p,q)$ -regular tree, whose root has $p$ children and all other vertices have $q$ children.", "A convenient way to analyse the adjacency matrix of $\\mathbb {T}_{p,q}$ is by tridiagonalizing it around its root.", "To that end, we first review the tridiagonalizationThe tridiagonalization algorithm that we use is the Lanczos algorithm.", "Tridiagonalizing matrices in numerical analysis and random matrix theory [26], [55] is usually performed using the numerically more stable Householder algorithm.", "However, when applied to the adjacency matrix $X = A$ of a graph, the Lanczos algorithm is more convenient because it can exploit the sparseness and local geometry of $A$ .", "of a general symmetric matrix $X \\in \\mathbb {R}^{N \\times N}$ around a vertex $x \\in [N]$ ; we refer to [10] for details.", "Let $r \\in \\mathbb {N}$ and $x \\in [N]$ .", "Suppose that the vectors $_x, X _x, X^2 _x, \\dots , X^r _x$ are linearly independent, and denote by $g̑_0, g̑_1, g̑_2, \\dots , g̑_r$ the associated orthonormalized sequence.", "Then the tridiagonalization of $X$ around $x$ up to radius $r$ is the $(r + 1) \\times (r+1)$ matrix $Z = (Z_{ij})_{i,j = 0}^r$ with $Z_{ij} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =g̑i 2mu, Z g̑j$.", "By construction, $ Z$ is tridiagonal and conjugate to $ X$ restricted to the subspace $ Span{g̑0, g̑1, ..., g̑r}$.$ Let now $X = A \\equiv A^{\\mathbb {T}_{p,q}}$ be the adjacency matrix of $\\mathbb {T}_{p,q}$ , whose root we denote by $o$ .", "Then it is easy to see that $g̑_i = _{S_i(o)} / \\Vert _{S_i(o)} \\Vert $ and the tridiagonalization of $A$ around the root up to radius $\\infty $ is the infinite matrix $\\sqrt{q} Z(p/q)$ , where $ Z(\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ = 0 0 1 1 0 1 1 0  .", "If $\\alpha > 2$ , a transfer matrix analysis (see [10]) shows that $Z(\\alpha )$ has precisely two eigenvalues in $\\mathbb {R}\\setminus [-2,2]$ , which are $\\pm \\Lambda (\\alpha )$ .", "The associated eigenvectors are $((\\pm )^i u_i)_{i \\in \\mathbb {N}}$ , where $u_0 > 0$ and $u_i \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(- 1)i/2   u0$ for $ i 1$.", "Note that the eigenvector components are exponentially decaying since $ > 2$, and hence $ u0$ can be chosen so that the eigenvectors are normalized.", "Going back to the original vertex basis of $ Tp,q$, setting $ = p/q$, we conclude that the adjacency matrix $ A$ has eigenvalues $ q ()$ with associated eigenvectors $ i N ()i ui Si(o) / Si(o) $.$ Next, we show that the measure $\\mu _\\alpha $ from (REF ) is the spectral measure at the root of $A^{\\mathbb {T}_{p,q}} / \\sqrt{d}$ and the spectral measure at 0 of (REF ).", "Lemma 7.2 For any $\\alpha \\geqslant 0$ the measure $\\mu _\\alpha $ is the spectral measure of $Z(\\alpha )$ at 0.", "For any $p,q \\in \\mathbb {N}^*$ the measure $\\mu _{p/q}$ is the spectral measure of the normalized adjacency operator $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ at the root.", "For REF , define the vector $ȇ_0 = (1,0,0,\\dots ) \\in \\ell ^2(\\mathbb {N})$ .", "The spectral measure of $Z(\\alpha )$ with respect to $ȇ_0$ is characterized by its Stieltjes transform $ \\bigl \\langle ȇ_0 {2mu}, (Z(\\alpha ) - z)^{-1} ȇ_0\\bigr \\rangle = \\frac{1}{- z - \\alpha \\bigl \\langle ȇ_0 {2mu}, (Z(1) - z)^{-1} ȇ_0\\bigr \\rangle }\\,.$ Here, we used Schur's complement formula on the Green function $(Z(\\alpha ) - z)^{-1}$ , observing that the minor of $Z(\\alpha )$ obtained by removing the zeroth row and column is $Z(1)$ .", "Setting $\\alpha = 1$ in (REF ) and recalling the defining relation (REF ) of the Stieltjes transform $m$ of the semicircle law, we conclude that $\\bigl \\langle ȇ_0 {2mu}, (Z(1) - z)^{-1} ȇ_0\\bigr \\rangle = m(z)$ and hence from (REF ) and (REF ) we get $\\bigl \\langle ȇ_0 {2mu}, (Z(\\alpha ) - z)^{-1} ȇ_0\\bigr \\rangle = m_\\alpha (z)$ , as desired.", "The proof of REF is analogous.", "Denote the root of $\\mathbb {T}_{p,q}$ by $o$ .", "Again using Schur's complement formula to remove the $o$ th row and column of $H = A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ , we deduce that $ \\bigl \\langle _o {2mu}, \\bigl (A^{\\mathbb {T}_{p,q}} / \\sqrt{q} - z\\bigr )^{-1} _o\\bigr \\rangle = \\biggl (-z - \\frac{p}{q} \\bigl \\langle _o {2mu}, \\bigl (A^{\\mathbb {T}_{q,q}} / \\sqrt{q} - z\\bigr )^{-1} _o\\bigr \\rangle \\biggr )^{-1}\\,,$ where we used that $\\mathbb {T}_{p,q}$ from which $o$ has been removed consists of $p$ disconnected copies of $\\mathbb {T}_{q,q}$ .", "Setting $p = q$ in (REF ) and comparing to (REF ) implies that the left-hand side of (REF ) is equal to $m(z)$ if $p = q$ , and hence REF for general $p$ follows from (REF ).", "Finally, we remark that the equality of the spectral measures of $Z(p/q)$ and $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ can also be seen directly, by noting that $Z(p/q)$ is the tridiagonalization of $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ around the root $o$ .", "We conclude with some basic estimates for the Stieltjes transform $m_\\alpha $ of $\\mu _\\alpha $ used in Section .", "Lemma 7.3 For each $\\kappa >0$ there is a constant $C>0$ depending only on $\\kappa $ such that for all $z \\in S̑$ and all $\\alpha \\geqslant 0$ we have $ \\vert m_\\alpha (z) \\vert & \\leqslant C\\,, \\\\\\vert m_\\alpha (z) - m(z) \\vert & \\leqslant C \\vert \\alpha - 1 \\vert \\,.$ The simple facts follow directly from the corresponding properties of the semicircle law and its Stieltjes transform $m$ (see e.g.", "[16]).", "We leave the details to the reader.", "Bounds on adjacency matrices of trees In this appendix we derive estimates on the operator norm of a tree.", "We start with a standard estimate on the operator norm of a graph.", "Lemma 7.4 Let $\\mathbb {T}$ be a graph whose vertices have degree at most $q+1$ for some $q \\geqslant 1$ .", "Then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant q+1$ and if in addition $\\mathbb {T}$ is a tree then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant 2 \\sqrt{q}$ .", "The first claim is obvious by the Schur test for the operator norm.", "To prove the second claim, choose a root $o$ and denote by $C_x$ the set of children of the vertex $x$ .", "Then for any vector $w̑ = (w_x)$ we have $\\bigl \\vert \\bigl \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\bigr \\rangle \\bigr \\vert = \\Biggl \\vert \\sum _{x,y} w_x A_{xy}^{\\mathbb {T}} w_y \\Biggr \\vert = 2 \\Biggl \\vert \\sum _x \\sum _{y \\in C_x} w_x w_y \\Biggr \\vert \\leqslant \\sum _{x} \\sum _{y \\in C_x} \\biggl (\\frac{1}{\\sqrt{q}} w_x^2 + \\sqrt{q} w_y^2\\biggr )\\\\\\leqslant \\frac{q+1}{\\sqrt{q}} w_o^2 + \\sum _{x \\ne o} \\biggl (\\frac{q}{\\sqrt{q}} w_x^2 + \\sqrt{q} w_x^2\\biggr ) \\leqslant 2 \\sqrt{q} \\sum _x w_x^2\\,,$ where in third step we used Young's inequality and in the fourth step that each vertex in the sum appears once as a child and at most $q$ times as a parent.", "This concludes the proof.", "The same proof shows that if $\\mathbb {T}$ is a rooted tree whose root has at most $p$ children and all other vertices at most $q$ children, then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant \\sqrt{q} (p/q \\vee 2)$ .", "This bound is sharp for $p \\leqslant 2q$ but not for $p > 2q$ .", "The sharp bound in the latter case is established in the following result.", "Lemma 7.5 Let $p,q \\in \\mathbb {N}^*$ .", "Let $\\mathbb {T}$ be a tree whose root has $p$ children and all the other vertices have at most $q$ children.", "Then the adjacency matrix $A^{\\mathbb {T}}$ of $\\mathbb {T}$ satisfies $\\Vert A^{\\mathbb {T}}\\Vert \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2)$ .", "Let $r \\in \\mathbb {N}$ and denote by $\\mathbb {T}_{p,q}(r)$ the rooted $(p,q)$ -regular tree of depth $r$ , whose root $x$ has $p$ children, all vertices at distance $1 \\leqslant i \\leqslant r$ from $x$ have $q$ children, and all vertices at distance $r+1$ from $x$ are leaves.", "For large enough $r$ , we can exhibit $\\mathbb {T}$ as a subgraph of $\\mathbb {T}_{p,q}(r)$ .", "By the Perron-Frobenius theorem, $ \\Vert A^{\\mathbb {T}} \\Vert = \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\rangle $ for the some normalized eigenvector $w̑$ whose entries are nonnegative.", "We extend $w̑$ to a vector indexed by the vertex set of $\\mathbb {T}_{p,q}(r)$ by setting $w_y = 0$ for $y$ not in the vertex set of $\\mathbb {T}$ .", "Clearly, $ \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\rangle \\leqslant \\langle w̑ {2mu}, A^{\\mathbb {T}_{p,q}(r)} w̑\\rangle \\,.$ Abbreviating $A \\equiv A^{\\mathbb {T}_{p,q}(r)}$ , it therefore remains to estimate the right-hand side of (REF ) for large enough $r$ .", "To that end, we define $Z$ as the tridiagonalization of $A$ around the root up to radius $r$ (see Appendix REF ).", "The associated orthonormal set $g̑_0, g̑_1, \\dots , g̑_r$ is given by $g̑_i = _{S_i(x)}/\\Vert _{S_i(x)}\\Vert $ , and $Z = \\sqrt{q} Z_r(p/q)$ , where $Z_r(\\alpha )$ is the upper-left $(r+1) \\times (r+1)$ block of (REF ).", "We introduce the orthogonal projections $P_0 \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =g̑0 g̑0*$ and $ P =i = 0r g̑i g̑i*$.", "Clearly, $ P0 P = P0$ and hence $ (1 - P) (1 - P0) = 1 - P$.", "For large enough $ r$ the vectors $ g̑r$ and $ w̑$ have disjoint support, and hence $ (1 - P) A P w̑ = (1 - P) A i = 0r - 1 g̑i g̑i 2mu, w̑ = 0$,since $ A g̑i Span{g̑i-i, g̑i+1}$ for $ i < r$.", "Thus we have{\\begin{@align}{1}{-1}\\langle w̑ {2mu}, A w̑\\rangle &= \\langle w̑ {2mu}, PAP w̑\\rangle + \\langle w̑ {2mu}, (1 - P) A (1 - P) w̑\\rangle \\\\ &= \\langle w̑ {2mu}, PAP w̑\\rangle + \\langle w̑ {2mu}, (1 - P) (1 - P_0) A (1 - P_0) (1 - P) w̑\\rangle \\,.\\end{@align}}From \\cite [Appendices B and C]{ADK19} we find\\begin{equation}\\lim _{r \\rightarrow \\infty } \\Vert P A P \\Vert = \\lim _{r \\rightarrow \\infty } \\Vert Z \\Vert = \\sqrt{q} \\Lambda (p/q \\vee 2)\\,.\\end{equation}Moreover, the operator $ (1 - P0) A (1 - P0)$ is the adjacency matrix of a forest whose vertices have degree at most $ q$.", "By Lemma \\ref {lem:forest_bound}, we therefore obtain $ (1 - P0) A (1 - P0) 2 q$.From (\\ref {w_quad_est}) we therefore get\\begin{equation*}\\limsup _{r \\rightarrow \\infty } \\langle w̑ {2mu}, A w̑\\rangle \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2) \\Vert P w̑ \\Vert ^2 + 2 \\sqrt{q} \\Vert (1 - P) w̑ \\Vert ^2 \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2) \\Vert w̑ \\Vert ^2\\,.\\end{equation*}By (\\ref {PF_quad1}) and (\\ref {PF_quad2}), the proof is complete.$ Degree distribution and number of resonant vertices In this appendix we record some basic facts about the distribution of degrees of the graph $\\mathbb {G}(N,d/N)$ , and use them to estimate the number of resonant vertices $\\mathcal {W}_{\\lambda , \\delta }$ .", "The following is a quantitative version of the Poisson approximation of a binomial random variable.", "Lemma 7.6 (Poisson approximation) If $D$ is a random variable with law $\\operatorname{Binom}(n,p)$ then for $k\\leqslant \\sqrt{n}$ and $p \\leqslant 1 / \\sqrt{n}$ we have $\\mathbb {P}(D = k) = \\frac{(pn)^k}{k!}", "\\mathrm {e}^{-pn} \\biggl (1+O\\biggl (\\frac{k^2}{n} + p^2 n\\biggr )\\biggr )\\,.$ Plugging the estimates $(1-p)^{n-k}= \\mathrm {e}^{(n-k)\\log (1-p)} = \\mathrm {e}^{-np + O(pk + p^2n)}$ and $\\frac{n!}{(n-k)!}", "= n^k\\prod _{i=0}^{k-1}\\biggl (1-\\frac{i}{n}\\biggr )=n^k \\mathrm {e}^{\\sum _{i=0}^{k-1} \\log \\bigl (1-\\frac{i}{n}\\bigr )}= n^k \\mathrm {e}^{O \\bigl (\\frac{k^2}{n}\\bigr )}\\,,$ into $\\mathbb {P}(D_x = k) = \\frac{n!}{k!", "(n-k)!}", "p^k (1-p)^{n-k}$ yields the claim, since $pk \\leqslant k^2/n + p^2 n$ .", "Lemma 7.7 For $\\mathbb {G}(N,d/N)$ we have $\\alpha _x \\leqslant \\mathcal {C}\\bigl (1 + \\frac{ \\log N}{d}\\bigr )$ with very high probability.", "This is a simple application of Bennett's inequality; see [10] for details.", "Next, we recall some standard facts about the distribution of the degrees.", "Define the function $f_d : [1,\\infty ) \\rightarrow \\big [\\frac{1}{2} \\log (2 \\pi d), \\infty \\big )$ through $ f_d(\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =d( - + 1) + 12 (2 d)  , which is bijective and increasing.", "For its interpretation, we note that if $Y \\overset{\\mathrm {d}}{=}\\operatorname{Poisson}(d)$ then by Stirling's formula we have $\\mathbb {P}(Y = k) = \\exp \\bigl (-f_d(k/d) + O \\bigl (\\frac{1}{k}\\bigr )\\bigr )$ for any $k \\in \\mathbb {N}$ .", "There is a universal constant $C > 0$ such that for $1 \\leqslant l \\leqslant \\frac{N}{C \\sqrt{d}}$ the equation $f_d(\\beta ) = \\log (N/l)$ has a unique solution $\\beta \\equiv \\beta _l(d)$ .", "The interpretation of $\\beta _l(d)$ is the typical location of $\\alpha _{\\sigma (l)}$ .", "By the implicit function theorem, we find that $d \\mapsto \\beta _l(d)$ on the interval $\\bigl (0, \\frac{N^2}{C l^2}\\bigr ]$ is a decreasing bijective function.", "Definition 7.8 An event $\\Xi \\equiv \\Xi _N$ holds with high probability if $\\mathbb {P}(\\Xi ) = 1 - o(1)$ .", "The following result is a slight generalization of [10], which can be established with the same proof.", "We note that the qualitative notion of high probability can be made stronger and quantitative with some extra effort, which we however refrain from doing here.", "Lemma 7.9 If $d \\geqslant 1$ and $l \\geqslant 1$ satisfies $\\beta _l(d) \\geqslant 3/2$ then $ \\vert \\alpha _{\\sigma (l)} - \\beta _l(d) \\vert \\leqslant \\frac{1 \\vee (\\zeta / \\log \\beta _l(d))}{d}$ with high probability, where $\\zeta $ is any sequence tending to infinity with $N$ .", "The following resultThe assumption $d \\gg \\log \\log N$ in Lemma REF is tailored so that it covers the entire range $\\alpha \\geqslant 2$ , which is what we need in this paper.", "The assumption on $d$ could also be removed at the expense of introducing a nontrivial lower bound on $\\alpha $ .", "gives bounds on the counting function of the normalized degrees $(\\alpha _x)_{x \\in [N]}$ .", "Lemma 7.10 Suppose that $\\zeta $ satisfies $ 1 \\ll \\zeta \\leqslant \\frac{d}{C \\log \\log N}$ for some large enough universal constant $C$ .", "Then for any $\\alpha \\geqslant 2$ we have with high probability $ \\big \\lfloor (N \\mathrm {e}^{-f_d(\\alpha )} - 1) (\\log N)^{-2 \\zeta } \\big \\rfloor \\leqslant \\vert \\vert \\leqslant \\big \\lceil (N \\mathrm {e}^{-f_d(\\alpha )} + 1) (\\log N)^{2 \\zeta } \\big \\rceil \\,.$ If $d > 3 \\log N$ , then an elementary analysis using Bennett's inequality shows that $\\vert \\vert = 0$ with high probability.", "Since $N \\mathrm {e}^{-f_d(\\alpha )} \\leqslant 1$ for $\\alpha \\geqslant 2$ , the claim follows.", "Thus, for the following we assume that $d \\leqslant 3 \\log N$ .", "Abbreviate $\\Upsilon \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =32 d$, which is an upper bound for the right-hand side of (\\ref {deg_est_1}).For the following we adopt the convention that $ 0(d) = $.", "Choose $ l 0$ such that\\begin{equation} \\beta _{l+1}(d) < \\alpha \\leqslant \\beta _l(d)\\,,\\end{equation}and define\\begin{equation*}\\underline{k} \\!\\, \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\big \\lfloor l (\\log N)^{-2\\zeta } \\big \\rfloor \\,, \\qquad \\overline{k} \\!\\, \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(l+1) (N)2  .", "We shall show that $ \\beta _{\\underline{k} \\!\\,}(d) - \\Upsilon \\geqslant \\beta _l(d)$ for $\\underline{k} \\!\\, \\geqslant 1$ , $ \\beta _{\\overline{k} \\!\\,}(d) + \\Upsilon \\leqslant \\beta _{l+1}(d)\\,,$ and $ \\overline{k} \\!\\, \\leqslant N \\mathrm {e}^{-f_d(3/2)}\\,.$ Thus $\\beta _{\\overline{k} \\!\\,}(d) \\geqslant 3/2$ and, assuming $\\underline{k} \\!\\, \\geqslant 1$ , Lemma REF is applicable to the indices $\\overline{k} \\!\\,$ and $\\underline{k} \\!\\,$ .", "We obtain, with high probability, $\\alpha _{\\sigma (\\overline{k} \\!\\,)} \\leqslant \\beta _{\\overline{k} \\!\\,}(d) + \\Upsilon \\leqslant \\beta _{l+1}(d) \\leqslant \\alpha \\leqslant \\beta _l(d) \\leqslant \\beta _{\\underline{k} \\!\\,}(d) - \\Upsilon \\leqslant \\alpha _{\\sigma (\\underline{k} \\!\\,)}\\,,$ from which we deduce that $ \\underline{k} \\!\\, \\leqslant \\vert \\vert \\leqslant \\overline{k} \\!\\,\\,,$ which also holds trivially also for the case $\\underline{k} \\!\\, = 0$ .", "By applying the function $f_d$ to () we obtain $l \\leqslant N \\mathrm {e}^{-f_d(\\alpha )} \\leqslant l+1$ , so that (REF ) yields (REF ).", "Next, we verify (REF ).", "We consider the cases $l = 0$ and $l \\geqslant 1$ separately.", "If $l = 0$ then, by the definition of $\\beta _{\\overline{k} \\!\\,}(d)$ , for (REF ) we require $(\\log N)^{2 \\zeta } + 1 \\leqslant N \\mathrm {e}^{-f_d(3/2)}$ , which holds by the assumption $d \\leqslant 3 \\log N$ and the upper bound on $\\zeta $ .", "Let us therefore suppose that $l \\geqslant 1$ .", "By (), $\\alpha \\geqslant 2$ , and the definition of $\\beta _l(d)$ , we have $l \\leqslant N \\mathrm {e}^{-f_d(2)}$ , and we have to ensure that $(l+2) (\\log N)^{2 \\zeta } \\leqslant N \\mathrm {e}^{-f_d(3/2)}$ .", "Since $l \\geqslant 1$ , this is satisfied provided that $3 \\mathrm {e}^{-f_d(2)} (\\log N)^{2 \\zeta } \\leqslant \\mathrm {e}^{-f_d(3/2)}$ , which holds provided that $f_d(2) - f_d(3/2) \\geqslant 3 \\zeta \\log \\log N$ .", "This inequality is true because $f_d(2) - f_d(3/2) \\geqslant f^{\\prime }_d(3/2) /2 \\geqslant d/C$ , where we used that $f_d^{\\prime }(\\alpha ) = d \\log \\alpha + \\frac{1}{2 \\alpha }$ .", "What remains, therefore, is the proof of (REF ) and (REF ).", "We begin with the proof of (REF ).", "We get from the mean value theorem that $ \\beta _{\\underline{k} \\!\\,}(d) - \\beta _l(d) = f_d^{-1}\\biggl (\\log \\biggl (\\frac{N}{\\underline{k} \\!\\,}\\biggr )\\biggr ) - f_d^{-1}\\biggl (\\log \\biggl (\\frac{N}{l}\\biggr )\\biggr )\\geqslant \\frac{3}{4 d \\log \\beta _{\\underline{k} \\!\\,}(d)} \\log \\biggl (\\frac{l}{\\underline{k} \\!\\,}\\biggr )\\,.$ The right-hand side of (REF ) is bounded from below by $\\Upsilon $ provided that $ \\log \\biggl (\\frac{l}{\\underline{k} \\!\\,}\\biggr ) \\geqslant 2 \\zeta \\log \\beta _{\\underline{k} \\!\\,}(d)\\,.$ We estimate $\\beta _{\\underline{k} \\!\\,}(d) \\leqslant \\beta _1(d)$ using the elementary bound $f_d(\\beta ) \\geqslant \\frac{d}{10} \\beta $ for $\\beta \\geqslant 2$ , which yields $\\log N = f_d(\\beta _1(d)) \\geqslant \\frac{d}{10} \\beta _1(d)$ .", "By assumption on $d$ we therefore get $ \\beta _1(d) \\leqslant \\log N\\,.$ Thus, (REF ) holds by $\\underline{k} \\!\\, \\leqslant l / (\\log N)^{2 \\zeta }$ .", "This concludes the proof of (REF ).", "Next, we prove (REF ).", "As in (REF ), we find $ \\beta _{l + 1}(d) - \\beta _{\\overline{k} \\!\\,}(d) = f_{d}^{-1}\\biggl (\\log \\biggl (\\frac{N}{l+1}\\biggr )\\biggr ) - f_{d}^{-1}\\biggl (\\log \\biggl (\\frac{N}{\\overline{k} \\!\\,}\\biggr )\\biggr )\\geqslant \\frac{3}{4 d \\log \\beta _{l+1}(d)} \\log \\biggl (\\frac{\\overline{k} \\!\\,}{l+1}\\biggr )\\,.$ Together with $\\beta _{l+1}(d) \\leqslant \\beta _1(d) \\leqslant \\log N$ from (REF ), we deduce that the right-hand side of (REF ) is bounded from below by $\\Upsilon $ provided that $\\log \\bigl (\\frac{\\overline{k} \\!\\,}{l+1}\\bigr ) \\geqslant 2 \\zeta \\log \\log N$ , which is true by definition of $\\overline{k} \\!\\,$ .", "This concludes the proof of (REF ).", "The following result follows easily from Lemma REF .", "Recall the definition (REF ) of the exponent $\\theta _b(\\alpha )$ .", "Corollary 7.11 Suppose that $\\zeta $ satisfies (REF ).", "Write $d = b \\log N$ .", "Then for any $\\alpha \\geqslant 2$ we have $\\vert \\vert \\vee 1 = N^{\\theta _b(\\alpha ) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )$ with high probability.", "Using the exponent $\\theta _b(\\alpha )$ from (REF ) and $\\alpha _{\\max }(b)$ defined below it, we may state the following estimate on the density of the normalized degrees and the number of resonant vertices.", "Lemma 7.12 The following holds for a large enough universal constant $C$ .", "Suppose that $\\zeta $ satisfies (REF ).", "Write $d = b \\log N$ .", "For $2 \\leqslant \\alpha < \\beta \\leqslant \\alpha _{\\max }(b)$ satisfying $\\beta - \\alpha \\geqslant C \\frac{\\zeta \\log \\log N}{d \\log \\alpha }$ , with high probability we have $ \\vert \\vert = N^{\\theta _b(\\alpha ) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )\\,.$ For $\\delta \\geqslant C \\frac{\\zeta \\log \\log N}{d}$ and $2 + \\delta \\leqslant \\lambda \\leqslant \\Lambda (\\alpha _{\\max }(b))$ , with high probability we have $\\vert \\mathcal {W}_{\\lambda ,\\delta } \\vert = N^{\\theta _b(\\Lambda ^{-1}(\\lambda - \\delta )) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )\\,.$ Note that, since $\\xi \\geqslant d^{-1/2}$ , if the conclusion of Theorem REF is nontrivial then $\\delta \\geqslant d^{-1/2}$ , and hence the assumption on $\\delta $ in Lemma REF REF is automatically satisfied for suitably chosen $\\zeta $ .", "[Proof of Lemma REF ] Part REF follows Corollary REF below by noting that the assumption on $\\beta $ implies $\\theta _b(\\alpha ) - \\theta _b(\\beta ) \\geqslant C \\frac{\\zeta \\log \\log N}{\\log N}$ by the mean value theorem.", "Part REF follows from Part REF , using that $\\log (\\lambda - \\delta ) \\geqslant \\log 2$ , that $\\Lambda ^{\\prime }$ is bounded on $[2,\\infty )$ , and the mean value theorem.", "Corollary 7.13 The following holds for large enough universal constants $C, \\mathcal {C}$ .", "Suppose that (REF ) holds.", "Write $d = b \\log N$ .", "Let $w̑ = (w_x)_{x \\in [N]}$ be a normalized eigenvector of $A/\\sqrt{d}$ with nontrivial eigenvalue $2+\\mathcal {C} \\xi ^{1/2} \\leqslant \\lambda \\leqslant \\Lambda (\\alpha _{\\max }(b))$ .", "Then with high probability for any $2 \\leqslant p \\leqslant \\infty $ we have $\\Vert w̑ \\Vert _p^{2} \\geqslant N^{(2/p - 1)\\theta _b(\\Lambda ^{-1}(\\lambda )) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\Biggl [ \\frac{\\log \\log N}{\\sqrt{\\log N}} + b (\\log \\lambda ) \\biggl (\\lambda + \\frac{1}{\\sqrt{\\lambda - 2}}\\biggr ) (\\xi + \\xi _{\\lambda - 2})\\Biggr ]\\,.$ We choose $\\delta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =C (+ - 2)$.", "Then by assumption on $$ we have $ (- 2)/2$, and hence Theorem \\ref {thm:localisation} yields, using that $ v̑(x)$ is supported in $ Br(x)$, $ x W, y Br(x) wy2 12$with high probability.", "Thus, by Hölder^{\\prime }s inequality,\\begin{equation} \\Vert w̑ \\Vert _{p}^{2} \\geqslant \\frac{1}{2} \\Biggl (\\sum _{x \\in \\mathcal {W}_{\\lambda ,\\delta }} \\vert B_{r_{\\star }}(x) \\vert \\Biggr )^{2/p - 1} \\geqslant \\frac{1}{2} \\Bigl (\\vert \\mathcal {W}_{\\lambda ,\\delta } \\vert N^{C \\log \\log N / \\sqrt{\\log N}}\\Bigr )^{2/p - 1}\\end{equation}with high probability, where we used Lemma \\ref {lem:upper_bound_degrees} to estimate $ x [N]Br(x) NC N / N$ with high probability.$ Next, using the mean value theorem and elementary estimates on the derivatives of $\\theta _b$ and $\\Lambda ^{-1}$ , we estimate $\\theta _b(\\Lambda ^{-1}(\\lambda - \\delta )) - \\theta _b(\\Lambda ^{-1}(\\lambda )) \\leqslant C b (\\log \\lambda ) \\biggl (\\lambda + \\frac{1}{\\sqrt{\\lambda - 2}}\\biggr ) \\delta \\,.$ Invoking Lemma REF REF with $\\zeta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =N$, and recalling (\\ref {w_infty_lower_est}), therefore yields the claim.$ Connected components of $\\mathbb {G}(N,d/N)$ In this appendix we give some basic estimates on the sizes of connected components of $\\mathbb {G}(N,d/N)$ .", "These are needed for the analysis of the tuning forks in Appendix REF below.", "The arguments are standard and are tailored to work well in the regime $1 \\ll d \\leqslant \\log N$ that we are interested in.", "For smaller values of $d$ , see e.g.", "[17].", "Lemma 7.14 Let $W_k$ be the number of connected components that have $k$ vertices and $\\widehat{W}_k$ the number of connected components that have $k$ vertices and are not a tree.", "Then for $k \\leqslant N/2$ we have $\\mathbb {E}[W_k] \\leqslant N \\mathrm {e}^{-k (d/2 - \\log d - 1)}\\,, \\qquad \\mathbb {E}[\\widehat{W}_k] \\leqslant \\mathrm {e}^{-k (d/2 - \\log d - 1)}\\,.$ For a set $X \\subset [N]$ , denote by $\\mathcal {T}(X)$ the set of spanning trees of $X$ .", "If $X$ is a connected component of $\\mathbb {G}$ then there exists $\\mathbb {T} \\in \\mathcal {T}(X)$ a subgraph of $\\mathbb {G}$ such that no vertex of $X$ is connected to a vertex of $[N] \\setminus X$ .", "Hence, $W_k \\leqslant \\sum _{X \\subset [N]} \\mathbb {1}_{\\vert X \\vert = k} \\sum _{\\mathbb {T} \\in \\mathcal {T}(X)} \\mathbb {1}_{\\mathbb {T} \\subset \\mathbb {G}} \\prod _{x \\in X} \\prod _{y \\in [N] \\setminus X} (1 - A_{xy})\\,.$ Taking the expectation now easily yields the claim, using $\\vert \\mathcal {T}(X) \\vert = \\vert X \\vert ^{\\vert X \\vert - 2}$ by Cayley's theorem, that a tree on $k$ vertices has $k - 1$ edges, Stirling's approximation, and $1 - x \\leqslant \\mathrm {e}^{-x}$ .", "The argument to estimate $\\widehat{W}_k$ is similar, noting that in addition to a spanning tree $\\mathbb {T}$ of $X$ , we also have to have at least one edge not in $\\mathbb {T}$ connecting two vertices of $X$ .", "Thus, $\\widehat{W}_k \\leqslant \\sum _{X \\subset [N]} \\mathbb {1}_{\\vert X \\vert = k} \\sum _{\\mathbb {T} \\in \\mathcal {T}(X)} \\mathbb {1}_{\\mathbb {T} \\subset \\mathbb {G}} \\prod _{x \\in X} \\prod _{y \\in [N] \\setminus X} (1 - A_{xy}) \\sum _{\\lbrace u,v\\rbrace \\in X^2 \\setminus E(\\mathbb {T})} A_{uv}\\,,$ and we may estimate the expectation as before.", "We call a connected component of $\\mathbb {G}$ small if it is not the giant component.", "For the following statement we recall the definition of high probability from Definition REF .", "Corollary 7.15 Suppose that $d \\gg 1$ .", "All small components of $\\mathbb {G}$ have at most $O\\bigl (\\frac{\\log N}{d}\\bigr )$ vertices with very high probability.", "All small components of $\\mathbb {G}$ are trees with high probability.", "The giant component of $\\mathbb {G}$ has at least $N (1 - \\mathrm {e}^{-d/4})$ vertices with high probability.", "Any small component has at most $N/2$ vertices.", "Using Lemma REF we therefore get that the probability that there exists a small component with at least $K$ vertices is bounded by $\\mathbb {P}(\\exists k \\in [K,N/2] \\,, W_k \\geqslant 1) \\leqslant \\sum _{k = K}^{N/2} \\mathbb {E}[W_k] \\leqslant 2 N \\mathrm {e}^{-K (d/2 - \\log d - 1)}\\,,$ by summing the geometric series.", "Since $d/2 - \\log d - 1 \\geqslant c d$ for some universal constant $c$ , we obtain the first claim.", "To obtain the second claim, we use Lemma REF to estimate the probability that there exists a small component that is not a tree by $\\sum _{k = 1}^{N/2} \\mathbb {E}{\\widehat{W}_k} \\leqslant \\mathrm {e}^{-d/3}$ .", "To obtain the last claim, we estimate the expected number of vertices in small components by $\\mathbb {E}\\bigl [\\sum _{k = 1}^{N/2} k W_k\\bigr ] \\leqslant N \\sum _{k = 1}^\\infty k \\mathrm {e}^{-k (d/2 - \\log d - 1)} \\leqslant C N \\mathrm {e}^{- d/3}$ using Lemma REF , and the third claim follows from Chebyshev's inequality.", "We may now estimate the adjacency matrix on the small components of $\\mathbb {G}(N,d/N)$ .", "The following result follows immediately from Corollary REF and Lemma REF .", "Corollary 7.16 Suppose that $d \\gg 1$ .", "Then the operator norm of $A / \\sqrt{d}$ restricted to the small components of $\\mathbb {G}$ is bounded by $O\\bigl (\\frac{\\sqrt{\\log N}}{d}\\bigr )$ with high probability.", "Corollary REF makes it explicit that Theorem REF excludes all eigenvectors on small components of $\\mathbb {G}$ , whose eigenvalues lie outside $\\mathcal {S}_\\kappa $ precisely under the lower bound from (REF ).", "Tuning forks and proof of Lemma REF In this appendix we give a precise definition of the $D$ -tuning forks from Section REF and prove Lemma REF .", "Definition 7.17 A star of degree $D \\in \\mathbb {N}$ consists of a vertex, the hub, and $D$ leaves adjacent to the hub, the spokes.", "A star tuning fork of degree $D$ is obtained by taking two disjoint stars of degree $D$ along with an additional vertex, the base, and connecting both hubs to the base.", "We say that a star tuning fork is rooted in a graph $\\mathbb {H}$ if it is a subgraph of $\\mathbb {H}$ in which both hubs have degree $D+1$ and all spokes are leaves.", "Lemma 7.18 If a star tuning fork of degree $D$ is rooted in some graph $\\mathbb {H}$ , then the adjacency matrix of $\\mathbb {H}$ has eigenvalues $\\pm \\sqrt{D}$ with corresponding eigenvectors supported on the stars of the tuning fork, i.e.", "on $2D + 2$ vertices.", "Suppose first that $D \\geqslant 1$ .", "Note first that the adjacency matrix of a star of degree $D$ has rank two and has the two nonzero eigenvalues $\\pm \\sqrt{D}$ , with associated eigenvector equal to $\\pm \\sqrt{D}$ at the hub and 1 at the spokes.", "Now take a star tuning fork of degree $D$ rooted in a graph $\\mathbb {H}$ .", "Define a vector on the vertex set of $\\mathbb {H}$ by setting it to be $\\pm \\sqrt{D}$ at the hub of the first star, 1 at the spokes of the first star, $\\mp \\sqrt{D}$ at the hub of the second star, $-1$ at the spokes of the second star, and 0 everywhere else.", "Then it is easy to check that this vector is an eigenvector of the adjacency matrix of $\\mathbb {H}$ with eigenvalue $\\pm \\sqrt{D}$ .", "If $D = 0$ the construction is analogous, defining the vector to be $+1$ at one hub and $-1$ at the other.", "We recall from Section REF that $F(d,D)$ denotes the number of star tuning forks of degree $D$ rooted in $\\mathbb {G}_{\\mathrm {giant}}$ .", "Lemma 7.19 Suppose that $1 \\ll d \\ll \\sqrt{N}$ and $0 \\leqslant D \\ll \\sqrt{N}$ .", "Then $ \\mathbb {E}[F(d,D)] = \\frac{N d^2 \\mathrm {e}^{-2d}}{2 D!^2} (d \\mathrm {e}^{-d + 1})^{2D} (1 + o(1))$ and $\\mathbb {E}[F(d,D)^2] \\leqslant \\mathbb {E}[F(d,D)]^2 (1 + o(1))$ .", "[Proof of Lemma REF ] From Lemma REF we deduce that if $1 \\ll d = b \\log N = O(\\log N)$ and $D \\ll \\log N / \\log \\log N$ , then $\\mathbb {E}[F(d,D)] = N^{1 - 2b - 2b D + o(1)}$ .", "The claim then follows from the second moment estimate in Lemma REF and Chebyshev's inequality.", "[Proof of Lemma REF ] Let $x_1,x_2 \\in [N]$ be distinct vertices and $R_1, R_2 \\subset [N] \\setminus \\lbrace x_1,x_2\\rbrace $ be disjoint subsets of size $D$ .", "We abbreviate $U = (x_1, x_2, R_1, R_2)$ and sometimes identify $U$ with $\\lbrace x_1, x_2\\rbrace \\cup R_1 \\cup R_2$ .", "The family $U$ and a vertex $o \\in [N] \\setminus U$ define a star tuning fork of degree $D$ with base $o$ , hubs $x_1$ and $x_2$ , and associated spokes $R_1$ and $R_2$ .", "Let $C_k(\\mathbb {H})$ denote the vertex set of the $k$ th largest connected component of the graph $\\mathbb {H}$ .", "Then $F(d,D) = \\frac{1}{2}\\sum _U \\sum _{o \\in [N] \\setminus U} \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}$ , where $S_{o,U} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =i = 12 (u Ri {o} Axi u u [N] (Ri {o}) (1 - Axi u) u Ri v [N] {xi} (1 - Auv)).", "The factor $\\frac{1}{2}$ corrects the overcounting from the labelling of the two stars.", "For disjoint deterministic $U$ , we split the random variables $A = (A^{\\prime }, A^{\\prime \\prime })$ into two independent families, where $A^{\\prime } \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(Auv u U or v U)$ and $ A” =(Auv u,v [N] U)$.", "Note that $ So,U$ is $ A'$-measurable.", "We define the event\\begin{equation*}\\Xi \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\,,$ which is $A^{\\prime \\prime }$ -measurable.", "By Corollary REF and the assumption on $D$ , the event $\\Xi $ holds with high probability.", "Moreover, we have $\\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U} = \\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})} S_{o,U}$ , since the component of $o$ in $\\mathbb {G}$ and $\\mathbb {G} \\vert _{[N] \\setminus U}$ differ by $2D + 2$ vertices.", "Thus, for fixed $o \\in [N] \\setminus U$ , using the independence of $A^{\\prime }$ and $A^{\\prime \\prime }$ , we get $\\mathbb {E}[\\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}] &= \\mathbb {E}[\\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})} S_{o,U}] + \\mathbb {E}[\\mathbb {1}_{\\Xi ^c} \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}]\\\\&= \\mathbb {E}[S_{o,U}] \\bigl [\\mathbb {P}\\bigl (o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})\\bigr ) + O\\bigl (\\mathbb {P}(\\Xi ^c)\\bigr )\\bigr ]\\,.$ We have $\\mathbb {P}(\\Xi ^c) = o(1)$ and $\\mathbb {P}\\bigl (o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})\\bigr ) = 1 - o(1)$ by Corollary REF and the assumption on $D$ .", "Computing $\\mathbb {E}[S_{o,U}]$ and performing the sum over $o$ and $U$ , we therefore conclude that $\\mathbb {E}[F(d,D)] = \\frac{N (N - 1) \\cdots (N - 2D - 3 + 1)}{2 D!^2} \\biggl (\\frac{d}{N}\\biggr )^{2D + 2} \\biggl (1 - \\frac{d}{N}\\biggr )^{2 (N - D - 1) + 2 D (N - 1)} (1 + o(1))\\,,$ from which (REF ) follows.", "The estimate of the second moment is similar; one can even disregard the restriction to the giant component by estimating $\\mathbb {E}[F(d,D)^2] \\leqslant \\frac{1}{4} \\sum _{U, \\tilde{U}} \\sum _{o,\\tilde{o} \\in [N]} \\mathbb {E}[S_{o,U} S_{\\tilde{o}, \\tilde{U}}]$ ; we omit the details.", "Multilinear large deviation bounds for sparse random vectors In this appendix we collect basic large deviation bounds for multilinear functions of sparse random vectors, which are proved in [42].", "The following result is proved in Propositions 3.1, 3.2, and 3.5 of [42].", "We denote by $\\Vert X \\Vert _r \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(EX r)1/r$ the $ Lr$-norm of a random variable $ X$.$ Proposition 7.20 Let $r$ be even and $1 \\leqslant d \\leqslant N$ .", "Let $X_1, \\ldots , X_N$ be independent random variables satisfying $ \\mathbb {E}X_i = 0, \\qquad \\mathbb {E}\\vert X_i \\vert ^k \\leqslant \\frac{1}{N d^{(k-2)/2}}$ for all $i \\in [N]$ and $2 \\leqslant k \\leqslant r$ .", "Let $a_i \\in and $ bij be deterministic for all $i,j \\in [N]$ .", "Suppose that $\\bigg (\\frac{1}{N} \\sum _i \\vert a_i \\vert ^2 \\bigg )^{1/2} \\leqslant \\gamma \\,, \\qquad \\frac{\\max _i \\vert a_i \\vert }{\\sqrt{d}} \\leqslant \\psi ,$ and $\\bigg ( \\max _i \\frac{1}{N} \\sum _{j} \\vert b_{ij} \\vert ^2 \\bigg )^{1/2} \\vee \\bigg ( \\max _j \\frac{1}{N}\\sum _i \\vert b_{ij} \\vert ^2 \\bigg )^{1/2} \\leqslant \\gamma , \\qquad \\frac{\\max _{i,j} \\vert b_{ij} \\vert }{d} \\leqslant \\psi $ for some $\\gamma , \\psi \\geqslant 0$ .", "Then $\\biggl \\Vert \\sum _i a_i X_i \\biggr \\Vert _r & \\leqslant \\bigg ( \\frac{ 2r}{1 + 2 (\\log (\\psi /\\gamma ))_+} \\vee 2 \\bigg ) \\big ( \\gamma \\vee \\psi \\big ), \\\\\\biggl \\Vert \\sum _i a_i \\big ( \\vert X_i \\vert ^2 - \\mathbb {E}\\vert X_i \\vert ^2 \\big ) \\biggr \\Vert _r & \\leqslant 2 \\bigg ( 1 + \\frac{2d}{N} \\bigg ) \\max _i \\vert a_i \\vert \\bigg ( \\frac{r}{d} \\vee \\sqrt{\\frac{r}{d}} \\bigg ), \\\\\\biggl \\Vert \\sum _{i\\ne j} b_{ij} X_iX_j \\biggr \\Vert _r & \\leqslant \\bigg ( \\frac{ 4r}{1 + (\\log (\\psi /\\gamma ))_+} \\vee 4 \\bigg ) ^2\\big ( \\gamma \\vee \\psi \\big ).", "$ The $L^r$ -norm bounds in Proposition REF induce bounds that hold with very high probability.", "Corollary 7.21 Fix $\\kappa \\in (0,1)$ .", "Let the assumptions of Proposition REF be satisfied.", "If $\\psi /\\gamma \\geqslant N^{\\kappa /4}$ then with very high probability $ \\biggl \\vert \\sum _i a_i X_i \\biggr \\vert \\leqslant \\mathcal {C}\\psi \\,, \\qquad \\biggl \\vert \\sum _{i \\ne j} b_{ij} X_iX_j \\biggr \\vert \\leqslant \\mathcal {C}\\psi \\,.$ Remark 7.22 Our proof of Corollary REF shows that $\\mathcal {C}$ can be chosen as a linear function of $\\nu $ for the first estimate of (REF ) and as a quadratic function of $\\nu $ for the second estimate of (REF ).", "Fix $\\nu \\geqslant 1$ .", "We choose $r= \\nu \\log N$ in (REF ) of Proposition REF and obtain from Cheybshev's inequality that $ \\mathbb {P}\\bigg ( \\biggl \\vert \\sum _i a_i X_i \\biggr \\vert > \\mathcal {C}\\psi \\bigg ) \\leqslant N^{-\\nu }, \\qquad \\mathcal {C}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =4e $as $ (0,1)$.Similarly, choosing $ r = 1 2 N$ in(\\ref {eq:LDB_quadratic}) yields$$\\mathbb {P}\\bigg ( \\biggl \\vert \\sum _{i \\ne j} b_{ij} X_i X_j \\biggr \\vert > 4 \\mathcal {C}\\psi \\bigg ) \\leqslant N^{-\\nu },\\qquad \\mathcal {C}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =16e22 2 .", "$$ Resolvent identities In this appendix we record some well-known identities for the Green function (REF ) and its minors from Definition REF .", "Lemma 7.23 (Ward identity) For $x \\notin T \\subset [N]$ we have, with the notation $\\eta = \\operatorname{Im}z$ , $\\sum _y^{(T)} \\vert G_{xy}^{(T)} \\vert ^2 = \\frac{1}{\\eta } \\operatorname{Im}G_{xx}^{(T)} \\,.$ This is a standard identity for resolvents, see e.g.", "[16].", "Lemma 7.24 Let $T \\subset [N]$ .", "For $x, y \\notin T$ and $x \\ne y$ , we have $ G_{xy}^{(T)} = - G_{yy}^{(T)} \\sum _{a}^{(Ty)} G_{xa}^{(T y)} {M_{ay}} = - G_{xx}^{(T)} \\sum _{b}^{(Tx)} {M_{xb}} G_{by}^{(Tx)} .$ For $x, y, a \\notin T$ and $x \\ne a \\ne y$ , we have $ G_{xy}^{(Ta)} = G_{xy}^{(T)} - \\frac{G_{xa}^{(T)} G_{ay}^{(T)}}{G_{aa}^{(T)}} .$ For any $x \\in [N]$ , we have $ \\frac{1}{G_{xx}} = M_{xx} - z- \\sum _{a,b}^{(x)} M_{xa} G_{ab}^{(x)} M_{bx}.$ All identities are standard and proved e.g.", "in [16]: (REF ) in [16], (REF ) in [16] and (REF ) in [16].", "We recall (REF ) and derive two expansions used in Section .", "For any $T \\subset [N]$ and $x,y, u \\notin T$ , $x \\ne u \\ne y$ , we have $G_{xy}^{(Tu)} = G_{xy}^{(T)} + \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} G_{uy}^{(T)} + \\frac{f}{N} G_{uy}^{(T)} \\sum _{a}^{(Tu)} G_{xa}^{(Tu)} ,$ which follows from (REF ) and (REF ).", "Under the same assumptions, applying (REF ) to (REF ) yields $\\begin{aligned}G_{xy}^{(Tu)} = \\, & \\phantom{-} G_{xy}^{(T)} -G_{uu}^{(T)} \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} \\sum _{b}^{(Tu)} H_{ub} G_{by}^{(Tu)} \\\\& - \\frac{f}{N} G_{uu}^{(T)} \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} \\sum _b^{(Tu)} G_{by}^{(Tu)} + \\frac{f}{N} G_{uy}^{(T)} \\sum _{a}^{(Tu)} G_{xa}^{(Tu)}.", "\\end{aligned}$ Stability estimate – proof of Lemma  REF In this appendix we prove Lemma REF .", "The estimate in [27] corresponding to (REF ) has logarithmic factors, which are not affordable for our purposes: they have to be replaced with constants.", "The following proof of Lemma REF is analogous to that of the more complicated bulk stability estimate from [9].", "[Proof of Lemma REF ] We introduce the vectors $g̑ \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(gx)x X$ and $ =(x)x X$.", "Moreover, with the abbreviation $ m =m(z)$ we introduce the constant vectors $ m̑ = (m)x X$ and $ ȇ =X -1/2 (1)x X$.", "We regard all vectors as column vectors.A simple computation starting from the difference of (\\ref {m_quadr}) and (\\ref {eq:self_consistent_eq_perturbed})reveals that\\begin{equation} B(g̑ - m̑) = m (g̑ - m̑) \\bigl (\\mathbf {e} \\mathbf {e}^* (g̑ - m̑)\\bigr ) - (g̑ - m̑) m - m^2 ,\\end{equation}where $ B =1 - m2 ȇ ȇ*$, and column vectors are multiplied entrywise.The inverse of $ B$ is$$ B^{-1} = 1 + \\frac{m^2}{1- m^2} \\mathbf {e} \\mathbf {e}^*.", "$$For a matrix $ R X X$, we write $ R $ for the operator norm induced by the norm $ ȓ = x X rx $ on $ X$.It is easy to see that there is $ c>0$, depending only on $$, such that $ 1- m(w)2 c$ for all $ w +$ satisfying $ Re w 2 - $.Hence, owing to $ ȇ ȇ* = 1$,we obtain$ B-1 1 + 1- m2 -1 1+ c-1$.Therefore, inverting $ B$ in (\\ref {eq:stability_equation}) and choosing $ b$, depending only on $$,sufficiently small to absorb the term quadratic in $ g̑ - m̑$ into the left-hand side of the resulting boundyields (\\ref {eq:stability_estimate}) for some sufficiently large $ C>0$, depending only on $$.This concludes the proof of Lemma~\\ref {lem:stability}.$ Instability estimate – proof of (REF ) In this appendix we prove (REF ), which shows that the self-consistent equation (REF ) is unstable with a logarithmic factor, which renders it useless for the analysis of sparse random graphs.", "More precisely, we show that the norm $\\Vert (I - m^2 S)^{-1} \\Vert _{\\infty \\rightarrow \\infty }$ is ill-behaved precisely in the situation where we need it.", "For simplicity, we replace $m^2$ with a phase $\\alpha ^{-1} \\in S^1$ separated from $\\pm 1$ , since for $\\operatorname{Re}z \\in \\mathcal {S}_\\kappa $ we have $ \\vert m(z) \\vert ^2 = 1 - O(\\operatorname{Im}z) \\,, \\qquad \\operatorname{Im}m(z) \\asymp 1\\,,$ by [33].", "Moreover, for definiteness, recalling that with very high probability most of the $d (1 + o(1))$ neighbours of any vertex in $\\mathcal {T}$ are again in $\\mathcal {T}$ , we assume that $S$ is the adjacency matrix of a $d$ -regular graph on $\\mathcal {T}$ divided by $d$ .", "By the spectral theorem and because $S$ is Hermitian, $\\Vert (\\alpha - S)^{-1} \\Vert _{2 \\rightarrow 2}$ is bounded, but, as we now show, the same does not apply to $\\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty }$ .", "Indeed, the upper bound of (REF ) follows from [34], and the lower bound from the following result.", "Lemma 7.25 (Instability of (REF )) Let $S$ be $1/d$ times the adjacency matrix of a graph whose restriction to the ball of radius $r \\in \\mathbb {N}^*$ around some distinguished vertex is a $d$ -regular tree.", "Let $\\alpha \\in S^1$ be an arbitrary phase.", "Then $ \\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty } \\geqslant c \\biggl (\\frac{r}{\\log r} \\wedge d\\biggr )$ for some universal constant $c > 0$ .", "In particular, denoting by $N$ the number of vertices in the tree (which may be completed to a $d$ -regular graph by connecting the leaves to each other), for $d \\asymp \\log N$ and $r \\asymp \\frac{\\log N}{\\log d}$ we find $ \\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty } \\geqslant \\frac{c \\log N}{(\\log \\log N)^2}\\,,$ which is the lower bound of (REF ).", "[Proof of Lemma REF ] After making $r$ smaller if needed, we may assume that $\\frac{r}{\\log r} \\leqslant d$ .", "We shall construct a vector $ȗ$ satisfying $\\Vert ȗ \\Vert _\\infty = 1$ and $\\Vert (\\alpha - S) ȗ \\Vert _\\infty = O\\bigl (\\frac{\\log r}{r}\\bigr )$ , from which (REF ) will follow.", "To that end, we construct the sequence $a_0, a_1, \\dots , a_r$ by setting $a_0 \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1 ,       a1 = ,       ak+1 =dd - 1 ak - 1d - 1 ak - 1    for    1 k r - 1 .", "A short transfer matrix analysis shows that $\\vert a_k \\vert \\leqslant \\mathrm {e}^{C_1 k /d}$ for some constant $C_1$ .", "Now choose $\\mu \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =C2 rr$ with $ C2 =2 2 C1$, and define $ bk =e-k ak$.", "Calling $ o$ the distinguished vertex, we define $ ux =bk$ if $ k = dist(o,x) r$ and $ ux = 0$ otherwise.", "It is now easy to check that $ (- S) ȗ = O(rr)$, by considering the cases $ k = 0$, $ 1 k r - 1$, and $ k r$ separately.", "The basic idea of the construction is that if $$ were zero, then $ (- S) ȗ$ would vanish exactly on $ Br - 1(o)$, but it would be large on the boundary $ Sr(o)$.", "The factor $ e-k$ introduces exponential decay in the radius which dampens the contribution of the boundary $ Sr(o)$ at the expense of introducing errors in the interior $ Br - 1(o)$.$ Johannes Alt (johannes.alt@unige.ch) Raphaël Ducatez (raphael.ducatez@unige.ch) Antti Knowles (antti.knowles@unige.ch) University of Geneva, Section of Mathematics, 2-4 Rue du Lièvre, 1211 Genève 4, Switzerland." ], [ "The delocalized phase", "In this section we prove Theorem REF .", "In fact, we state and prove a more general result, Theorem REF below, which immediately implies Theorem REF ." ], [ "Local law", "Theorem REF is a local law for a general class of sparse random matrices of the form $ M = H + f ȇ ȇ^*\\,,$ where $f \\geqslant 0$ and $ȇ \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =N-1/2(1,1,...,1)*$.", "Here $ H$ is a Hermitian random matrix satisfying the following definition.$ Definition 6.1 Let $0 < d < N$ .", "A sparse matrix is a complex Hermitian $N\\times N$ matrix $H=H^* \\in \\mathbb {C}^{N \\times N}$ whose entries $H_{ij}$ satisfy the following conditions.", "The upper-triangular entries ($H_{ij}\\mathrel {\\hbox{.}\\hbox{.", "}}$ 1 i jN$) are independent.\\item [(ii)] We have $ E Hij=0$ and $ E Hij 2=(1 + O(ij))/N$ for all $ i,j$.\\item [(iii)] For any $ k3$, we have$ E|Hij|k 1/(Ndk-22)$for all $ i,j$.$ It is easy to check that the set of matrices $M$ defined as in (REF ) and Definition REF contains those from Theorem REF (see the proof of Theorem REF below).", "The local law for the matrix $M$ established in Theorem REF below provides control of the entries of the Green function $ G(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =( M - z)-1 for $z$ in the spectral domain $ \\mathbf {S} \\equiv S̑_{\\kappa , L, N} = \\mathcal {S}_\\kappa \\times [N^{-1 + \\kappa }, L]$ for some constant $L \\geqslant 1$ .", "We also define the Stieltjes transform $g$ of the empirical spectral measure of $M$ given by $ g(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1N i =1N 1i(M) - z = 1N TrG(z) .", "The limiting behaviour of $G$ and $g$ is governed by the following deterministic quantities.", "Denote by $+ \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={z Imz > 0}$ the complex upper half-plane.For $ z +$ we define $ m(z)$ as the Stieltjes transform of the semicircle law $ 1$,\\begin{equation} m(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation}=\\int \\frac{\\mu _1(\\mathrm {d}u)}{u - z} \\,, \\qquad \\mu _1(\\mathrm {d}u) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =12 (4 - u2)+   du .", "An elementary argument shows that $m(z)$ can be characterized as the unique solution $m$ in $+$ of the equation $ \\frac{1}{m(z)} = -z - m(z)\\,.$ For $\\alpha \\geqslant 0$ and $z \\in +$ we define $ m_\\alpha (z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =- 1 z + m(z) , so that $m_1 = m$ by (REF ).", "In Lemma REF below we show that $m_\\alpha $ is bounded in the domain $S̑$ , with a bound depending only on $\\kappa $ .", "For $x \\in [N]$ we denote the square Euclidean norm of the $x$ th row of $H$ by $ \\beta _x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =y Hxy 2 , which should be thought of as the normalized degree of $x$ ; see Remark REF below.", "Theorem 6.2 (Local law for $M$ ) Fix $0 < \\kappa \\leqslant 1/2$ and $L \\geqslant 1$ .", "Let $H$ be a sparse matrix as in Definition REF , define $M$ as in (REF ) for some $0 \\leqslant f \\leqslant N^{\\kappa /6}$ , and define $G$ and $g$ as in (REF ) and (REF ) respectively.", "Then with very high probability, for $d$ satisfying (REF ), for all $z \\in S̑$ we have $ \\max _{x,y \\in [N]} \\bigl \\vert G_{xy}(z) - \\delta _{xy} m_{\\beta _x}(z) \\bigr \\vert &\\leqslant \\mathcal {C}\\bigg ( \\frac{\\log N}{d^2} \\bigg )^{1/3}\\,,\\\\\\bigl \\vert g(z) - m(z) \\bigr \\vert &\\leqslant \\mathcal {C}\\Bigg ( \\frac{\\log N}{d^2} \\bigg )^{1/3}\\,.$ [Proof of Theorem REF ] Under the assumptions of Theorem REF we find that $M \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =A / d$ is of the form (\\ref {eq:def_M}) for some $ H$ and $ f$ satisfying the assumptions of Theorem \\ref {thm:local_law}.Now Theorem~\\ref {thm:delocalization} is a well-known consequence of Theorem \\ref {thm:local_law} and the boundedness of $ m(z)$ in (\\ref {eq:m_alpha_bounded}) below.", "For the reader^{\\prime }s convenience, we give the short proof.", "Denoting the eigenvalues of $ M$ by $ (i(M))i [N]$ and the associated eigenvectors by $ (w̑i(M))i [N]$, setting $ z = + i$ with $ = N-1 + $, by (\\ref {eq:local_law_entrywise}) and (\\ref {eq:m_alpha_bounded}) we have with very high probability\\begin{equation*}\\mathcal {C} \\geqslant \\operatorname{Im}G_{xx}(z) = \\sum _{i \\in [N]} \\frac{\\eta \\vert \\langle _x {2mu}, w̑_i(M)\\rangle \\vert ^2}{\\eta ^2 + (\\lambda - \\lambda _i(M))^2} \\geqslant \\frac{1}{\\eta } \\, \\vert \\langle _x {2mu}, w̑\\rangle \\vert ^2\\,,\\end{equation*}where in the last step we omitted all terms except $ i$ satisfying $ i(M) = $.", "The claim follows by renaming $ / 2$.", "(Here we used that Theorem \\ref {thm:local_law} holds also for random $ z S̑$, as follows form a standard net argument; see e.g.\\ \\cite [Remark 2.7]{BenyachKnowles2017}.", ")$ Remark 6.3 (Relation between $\\alpha _x$ and $\\beta _x$ ) In the special case $M = d^{-1/2} A$ with $A$ the adjacency matrix of $\\mathbb {G}(N,d/N)$ , we have $\\beta _x = \\frac{1}{d} \\sum _{y} \\bigg (A_{xy} - \\frac{d}{N}\\bigg )^2 = \\alpha _x + O \\biggl (\\frac{d (1 + \\alpha _x)}{N}\\biggr ) = \\alpha _x + \\mathcal {O} \\biggl (\\frac{d + \\log N}{N}\\biggr )$ with very high probability, by Lemma REF .", "By definition, $m_\\alpha (z) \\in +$ for $z \\in +$ , i.e.", "$m_\\alpha $ is a Nevanlinna function, and $\\lim _{z \\rightarrow \\infty } z m_\\alpha (z) = -1$ .", "By the integral representation theorem for Nevanlinna functions, we conclude that $m_\\alpha $ is the Stieltjes transform of a Borel probability measure $\\mu _\\alpha $ on $\\mathbb {R}$ , $ m_\\alpha (z) = \\int \\frac{\\mu _\\alpha (\\mathrm {d}u)}{u - z}\\,.$ Theorem REF implies that the spectral measure of $M$ at a vertex $x$ is approximately $\\mu _{\\beta _x}$ with very high probability.", "Inverting the Stieltjes transform (REF ) and using the definitions () and (REF ), we find after a short calculation $ \\mu _\\alpha (\\mathrm {d}u) = g_\\alpha (u) \\, \\mathrm {d}u + h_\\alpha \\delta _{s_\\alpha }(\\mathrm {d}u)+ h_\\alpha \\delta _{-s_\\alpha }(\\mathrm {d}u)\\,,$ where $g_\\alpha (u) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1u < 22 4-u2(1-)u2 + 2 ,       h=1> 2 - 22 - 2 + 1= 02 ,       s=1> 2 () .", "The family $(\\mu _\\alpha )_{\\alpha \\geqslant 0}$ contains the semicircle law ($\\alpha = 1$ ), the Kesten-McKay law of parameter $d$ ($\\alpha = d / (d - 1)$ ), and the arcsine law ($\\alpha = 2$ ).", "For rational $\\alpha = p/q$ , the measure $\\mu _{p/q}$ can be interpreted as the spectral measure at the root of the infinite rooted $(p,q)$ -regular tree, whose root has $p$ children and all other vertices have $q$ children.", "We refer to Appendix REF for more details.", "See Figure REF for an illustration of the measure $\\mu _\\alpha $ .", "Remark 6.4 Using a standard application the Helffer-Sjöstrand formula (see e.g.", "[16]), we deduce from Theorem REF the following local law for the spectral measure.", "Denote by $\\varrho _x$ the spectral measure of $M$ at vertex $x$ .", "Under the assumptions of Theorem REF , with very high probability, for any inverval $I \\subset \\mathcal {S}_\\kappa $ , we have $\\varrho _x(I) = \\mu _{\\beta _x}(I) + \\mathcal {O} \\biggl (\\vert I \\vert \\bigg ( \\frac{\\log N}{d^2} \\bigg )^{1/3} + N^{\\kappa - 1}\\biggr )\\,.$ The error is smaller than the left-hand side provided that $\\vert I \\vert \\geqslant \\mathcal {C} N^{\\kappa - 1}$ .", "Figure: An illustration of the probability measure μ α \\mu _\\alpha for various values of α\\alpha .", "For α>2\\alpha > 2, μ α \\mu _\\alpha has two atoms which we draw using vertical lines.", "The measure μ α \\mu _\\alpha is the semicircle law for α=1\\alpha = 1, the arcsine law for α=2\\alpha = 2, and the Kesten-McKay law with d=α α-1d = \\frac{\\alpha }{\\alpha - 1} for 1<α<21 < \\alpha < 2.", "Note that the density of μ α \\mu _\\alpha is bounded in 𝒮 κ \\mathcal {S}_\\kappa , uniformly in α\\alpha .", "The divergence of the density near 0 is caused by values of α\\alpha close to 0, and the divergence of the density near ±2\\pm 2 by values of α\\alpha close to 2.The remainder of this section is devoted to the proof of Theorem REF .", "For the rest of this section, we assume that $M$ is as in Theorem REF .", "To simplify notation, we consistently omit the $z$ -dependence from our notation in quantities that depend on $z \\in S̑$ .", "Unless mentioned otherwise, from now on all statements are uniform in $z \\in S̑$ .", "For the proof of Theorem REF , it will be convenient to single out the generic constant $\\mathcal {C}$ from (REF ) by introducing a new constant $\\mathcal {D}$ and replacing (REF ) with $ \\mathcal {D} \\sqrt{\\log N} \\leqslant d \\leqslant (\\log N)^{3/2}\\,.$ Our proof will always assume that $\\mathcal {C} \\equiv \\mathcal {C}_\\nu $ and $\\mathcal {D} \\equiv \\mathcal {D}_\\nu $ are large enough, and the constant $\\mathcal {C}$ in (REF ) can be taken to be $\\mathcal {C} \\vee \\mathcal {D}$ .", "For the rest of this section we assume that $d$ satisfies (REF ) for some large enough $\\mathcal {D}$ , depending on $\\kappa $ and $\\nu $ .", "To guide the reader through the proof, in Figure REF we include a diagram of the dependencies of the various quantities appearing throughout this section.", "Figure: The dependency graph of the various quantities appearing in the proof of Theorem .", "An arrow from xx to yy means that yy is chosen as a function of xx.", "The independent parameters, κ\\kappa and ν\\nu , are highlighted in blue.Typical vertices We start by introducing the key tool in the proof of Theorem REF , a decomposition of vertices into typical vertices and the complementary atypical vertices.", "Heuristically, a typical vertex $x$ has close to $d$ neighbours and the spectral measure of $M$ at $x$ is well approximated by the semicircle law.", "In fact, in order to be applicable to the proof of Proposition REF below, the notion of a typical vertex is somewhat more complicated, and when counting the number of neighbours of a vertex $x$ we also need to weight the neighbours with diagonal entries of a Green function, so that the notion of typical vertex also depends on the spectral parameter $z \\in S̑$ .", "It is precisely defined using the parameters $\\Phi _x$ and $\\Psi _x$ from (REF ) below.", "The main result of this subsection is Proposition REF below, which states, in the language of graphs when $M = d^{-1/2} A$ with $A$ the adjacency matrix of $\\mathbb {G}(N,d/N)$ , that most vertices are typical and most neighbours of any vertex are typical.", "To state it, we introduce some notation.", "Definition 6.5 For any subset $T \\subset [N]$ , we define the minor $M^{(T)}$ with indices in $T$ as the $(N-\\vert T \\vert ) \\times (N-\\vert T \\vert )$ -matrix $ M^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(Mxy)x,y [N] T. If $T$ consists only of one or two elements, $T = \\lbrace x\\rbrace $ or $T=\\lbrace x,y\\rbrace $ , then we abbreviate $M^{(x)}$ and $M^{(xy)}$ for $M^{(\\lbrace x\\rbrace )}$ and $M^{(\\lbrace x,y\\rbrace )}$ .", "We also abbreviate $M^{(Tx)}$ for $M^{(T \\cup \\lbrace x\\rbrace )}$ .", "The Green function of $M^{(T)}$ is denoted by $ G^{(T)}(z) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(M(T) - z)-1.", "We use the notation $ \\sum _{x}^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x [N]T .", "Definition 6.6 (Typical vertices) Let $\\mathfrak {a}> 0$ be a constant, and define the set of typical vertices $ \\mathcal {T}_\\mathfrak {a}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ = ,       a=a( Nd2 )1/3 , where $ \\Phi _x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =y(x) (Hxy 2 - 1N) ,       x =y(x) (Hxy 2 - 1N) Gyy(x) .", "Note that this notion depends on the spectral parameter $z$ , i.e.", "$\\mathcal {T}_\\mathfrak {a}\\equiv \\mathcal {T}_\\mathfrak {a}(z)$ .", "The constant $\\mathfrak {a}$ will depend only on $\\nu $ and $\\kappa $ .", "It will be fixed in () below.", "The constant $\\mathcal {D} \\geqslant \\mathfrak {a}^{3/2}$ from (REF ) is always chosen large enough so that $\\varphi _{\\mathfrak {a}} \\leqslant 1$ .", "The following proposition holds on the event $\\lbrace \\theta = 1\\rbrace $ , where we introduce the indicator function $ \\theta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1x,y Gxy depending on some deterministic constant $\\Gamma \\geqslant 1$ .", "In (REF ) below, we shall choose a constant $\\Gamma \\equiv \\Gamma _\\kappa $ , depending only on $\\kappa $ , such that the condition $\\theta = 1$ can be justified by a bootstrapping argument along the proof of Theorem REF in Section REF below.", "Throughout the sequel we use the following generalization of Definition REF .", "Definition 6.7 An event $\\Xi $ holds with very high probability on an event $\\Omega $ if for all $\\nu > 0$ there exists $\\mathcal {C} > 0$ such that $\\mathbb {P}(\\Xi \\cap \\Omega ) \\geqslant \\mathbb {P}(\\Omega ) - \\mathcal {C} N^{-\\nu }$ for all $N \\in \\mathbb {N}$ .", "We now state the main result of this subsection.", "Proposition 6.8 There are constants $0 < q \\leqslant 1$ , depending only on $\\Gamma $ , and $\\mathfrak {a}> 0$ , depending only on $\\nu $ and $q$ , such that, on the event $\\lbrace \\theta = 1\\rbrace $ , the following holds with very high probability.", "Most vertices are typical: $ \\vert \\mathcal {T}_\\mathfrak {a}^c \\vert \\leqslant \\exp ( q \\varphi _\\mathfrak {a}^2 d ) + N \\exp ( - 2 q \\varphi _\\mathfrak {a}^2 d).", "$ Most neighbours of any vertex are typical: $ \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}^c}^{(x)}\\vert H_{xy} \\vert ^2 \\leqslant \\mathcal {C} \\varphi _\\mathfrak {a}+ \\mathcal {C}d^4 \\exp (- q \\varphi _\\mathfrak {a}^2 d ) $ uniformly for $x \\in [N]$ .", "For the interpretation of Proposition REF REF , one should think of the motivating example $M = d^{-1/2} A$ , for which $d \\sum _{y \\in \\mathcal {T}^c_\\mathfrak {a}}^{(x)}\\vert H_{xy} \\vert ^2$ is the number of atypical neighbours of $x$ , up to an error term $\\mathcal {O}\\bigl (\\frac{d^2 + d \\log N}{N}\\bigr )$ by Remark REF .", "The remainder of Section REF is devoted to the proof of Proposition REF .", "We need the following version of $\\mathcal {T}_\\mathfrak {a}$ defined in terms of $H^{(T)}$ instead of $H$ .", "Definition 6.9 For any $x \\in [N]$ and $T \\subset [N]$ , we define $ \\Phi _x^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =y(Tx) (Hxy 2 - 1N) ,       x(T) =y(Tx) (Hxy 2 - 1N) Gyy(Tx)  $and\\begin{equation*}\\mathcal {T}^{(T)} _\\mathfrak {a}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\,.$ Note that $\\Phi _x^{(\\emptyset )} = \\Phi _x$ and $\\Psi _x^{(\\emptyset )} = \\Psi _x$ with the definitions from (REF ), and hence $\\mathcal {T}_\\mathfrak {a}^{(\\emptyset )} = \\mathcal {T}_\\mathfrak {a}$ .", "The proof of Proposition REF relies on the two following lemmas.", "Lemma 6.10 There are constants $0 < q \\leqslant 1$ , depending only on $\\Gamma $ , and $\\mathfrak {a}> 0$ , depending only on $\\nu $ and $q$ , such that, for any deterministic $X \\subset [N]$ , the following holds with very high probability on the event $\\lbrace \\theta = 1\\rbrace $ .", "$\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/2}^c \\vert \\leqslant \\exp ( q \\varphi _\\mathfrak {a}^2 d) + \\vert X \\vert \\exp (- 2 q \\varphi _\\mathfrak {a}^2 d)$ .", "If $\\vert X \\vert \\leqslant \\exp ( 2 q \\varphi _\\mathfrak {a}^2 d)$ then $\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/2}^c \\vert \\leqslant \\mathcal {\\varphi }_\\mathfrak {a}d$ .", "For any deterministic $x \\in [N]$ , the same estimates hold for $\\big (\\mathcal {T}^{(x)}_{\\mathfrak {a}/ 2}\\big )^c$ instead of $\\mathcal {T}^c_{\\mathfrak {a}/2}$ and a random set $X \\subset [N] \\setminus \\lbrace x\\rbrace $ that is independent of $H^{(x)}$ .", "Lemma 6.11 With very high probability, for any constant $\\mathfrak {a}> 0$ we have $ \\theta \\vert \\Phi _y - \\Phi _y^{(x)} \\vert \\leqslant \\varphi _{\\mathfrak {a}/2}, \\qquad \\theta \\vert \\Psi _y - \\Psi _y^{(x)} \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2} $ for all $x,y \\in [N]$ .", "Before proving Lemmas REF and REF , we use them to establish Proposition REF .", "As a preparation, we record the following simple consequence of Definition REF and Bennett's inequality.", "Lemma 6.12 We have $\\vert H_{xy} \\vert ^2 \\leqslant 1/d$ almost surely, and $\\sum _y \\vert H_{xy} \\vert ^2 \\leqslant \\mathcal {C} d$ with very high probability.", "[Proof of Proposition REF ] For REF , we choose $X = [N]$ in Lemma REF REF , using that $\\mathcal {T}_{\\mathfrak {a}/2} \\subset \\mathcal {T}_\\mathfrak {a}$ .", "We now turn to the proof of REF .", "By Lemma REF , on the event $\\lbrace \\theta = 1\\rbrace $ we have $\\mathcal {T}^c_\\mathfrak {a}\\subset \\big (\\mathcal {T}^{(x)}_{\\mathfrak {a}/2}\\big )^c$ with very high probability and hence $\\theta \\sum ^{(x)}_{y \\in \\mathcal {T}_\\mathfrak {a}^c} \\vert H_{xy} \\vert ^2 \\leqslant \\theta \\sum ^{(x)}_{y \\in (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c} \\vert H_{xy} \\vert ^2$ with very high probability.", "Since $\\vert H_{xy} \\vert ^2 \\leqslant 1 / d$ by Lemma REF , we obtain the decomposition $ \\begin{aligned}\\sum ^{(x)}_{y \\in (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c} \\vert H_{xy} \\vert ^2 &\\leqslant \\sum _{k = 0}^{\\log N} \\sum ^{(x)}_{y \\in (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c} \\vert H_{xy} \\vert ^2 \\mathbb {1}_{d^{-k-2} \\leqslant \\vert H_{xy} \\vert ^2 \\leqslant d^{-k - 1}} + \\frac{1}{N}\\\\&\\leqslant \\sum _{k = 0}^{\\log N} \\sum ^{(x)}_{y \\in (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c} d^{-k - 1} \\mathbb {1}_{\\vert H_{xy} \\vert ^2 \\geqslant d^{-k-2}} + \\frac{1}{N}\\\\&= \\sum _{k = 0}^{\\log N} d^{-k - 1} \\vert X_k \\cap (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c \\vert + \\frac{1}{N}\\,,\\end{aligned}$ where we defined $X_k \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={y x Hxy 2 d-k - 2} .", "Since $\\sum ^{(x)}_y \\vert H_{xy} \\vert ^2 \\leqslant \\mathcal {C} d$ with very high probability by Lemma REF , we conclude that $ \\vert X_k \\vert \\leqslant \\mathcal {C} d^{k + 3}$ with very high probability.", "We shall apply Lemma REF to the sets $X = X_k$ and $(\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c$ .", "To that end, note that $X_k \\subset [N] \\setminus \\lbrace x\\rbrace $ is a measurable function of the family $(H_{xy})_{y \\in [N]}$ , and hence independent of $H^{(x)}$ .", "Thus, we may apply Lemma REF .", "We define $K \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={k 0 C dk + 3 e 2qa2 d}$ and decompose the sum on the right-hand side of (\\ref {eq:sum_a_y_in_Tc}) into{\\begin{@align*}{1}{-1}\\sum _{k = 0}^{\\log N} d^{-k - 1} \\vert X_k \\cap \\big (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}\\big )^c \\vert &= \\sum _{k = 0}^{K} d^{-k - 1} \\vert X_k \\cap \\big (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}\\big )^c \\vert + \\sum _{k = K+1}^{\\log N} d^{-k - 1} \\vert X_k \\cap \\big (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}\\big )^c \\vert \\\\&\\leqslant \\sum _{k = 0}^{K} d^{-k - 1} \\varphi _\\mathfrak {a}d + \\sum _{k = K+1}^{\\log N} d^{-k - 1} \\bigl (\\mathrm {e}^{q \\varphi _\\mathfrak {a}^2 d} + \\mathcal {C} d^{k+3} \\mathrm {e}^{- 2q \\varphi _\\mathfrak {a}^2 d}\\bigr )\\\\&\\leqslant 2 \\varphi _\\mathfrak {a}+ \\mathcal {C} d^2 \\mathrm {e}^{-q \\varphi _\\mathfrak {a}^2 d}\\log N\\,\\end{@align*}}with very high probability.Here, we used Lemma \\ref {lem:XcapTc} \\ref {item:XcapTc_X_small} to estimate the summands if $ k K$ and Lemma~\\ref {lem:XcapTc} \\ref {item:XcapTc_general} and (\\ref {X_k_estimate}) for the other summands.", "Since $ N d2$, this concludes the proof of \\ref {item:a2}.$ The rest of this subsection is devoted to the proofs of Lemmas REF and REF .", "Let $\\theta $ be defined as in (REF ) for some constant $\\Gamma \\geqslant 1$ .", "For any subset $T \\subset [N]$ , we define the indicator function $\\theta ^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1a,b T Gab(T) 2  .", "Lemma REF is a direct consequence of the following two lemmas.", "The first one, Lemma REF , is mainly a decoupling argument for the random variables $(\\Psi _x)_{x \\in [N]}$ .", "Indeed, the probability that any fixed vertex $x$ is atypical is only small, $o(1)$ , and not very small, $N^{-\\nu }$ ; see (REF ) below.", "If the events of different vertices being atypical were independent, we could deduce that the probability that a sufficiently large set of vertices are atypical is very small.", "However, these events are not independent.", "The most serious breach of independence arises from the Green function $G^{(x)}_{yy}$ in the definition of $\\Psi _x$ .", "In order to make this argument work, we have to replace the parameters $\\Phi _x$ and $\\Psi _x$ with their decoupled versions $\\Phi _x^{(T)}$ and $\\Psi _x^{(T)}$ from Definition REF .", "To that end, we have to estimate the error involved, $\\vert \\Phi _x - \\Phi _x^{(T)} \\vert $ and $\\vert \\Psi _x - \\Psi _x^{(T)} \\vert $ .", "Unfortunately the error bound on the latter is proportional to $\\beta _x$ (see (REF )), which is not affordable for vertices of large degree.", "The solution to this issue involves the observation that if $\\beta _x$ is too large then the vertex is atypical by the condition on $\\Phi _x$ , which allows us to disregard the size of $\\Psi _x$ .", "The details are given in the proof of Lemma REF below.", "The second one, Lemma REF , gives a priori bounds on the entries of the Green function $G^{(T)}$ , which shows that if the entries of $G$ are bounded then so are those of $G^{(T)}$ for $\\vert T \\vert o(d)$ .", "For $T$ of fixed size, this fact is a standard application of the resolvent identities from Lemma REF .", "For our purposes, it is crucial that $T$ can have size up to $o(d)$ , and such a quantitative estimate requires slightly more care.", "Lemma 6.13 There is a constant $0 < q \\leqslant 1$ , depending only on $\\Gamma $ , such that, for any $\\nu >0$ , there is $\\mathcal {C}>0$ such that the following holds for any fixed $\\mathfrak {a}> 0$ .", "If $x \\notin T \\subset [N]$ are deterministic with $\\vert T \\vert \\leqslant \\varphi _\\mathfrak {a}d /\\mathcal {C}$ then $\\mathbb {P}\\big ( T \\subset \\mathcal {T}_{\\mathfrak {a}/ 2}^c,\\, \\theta = 1 \\big ) & \\leqslant \\mathrm {e}^{- 4 q \\varphi _\\mathfrak {a}^2 d \\vert T \\vert } + \\mathcal {C}N^{-\\nu } , \\\\\\mathbb {P}\\big ( T \\subset \\big (\\mathcal {T}_{\\mathfrak {a}/2}^{(x)}\\big )^c, \\theta ^{(x)} =1 \\big ) &\\leqslant \\mathrm {e}^{-4 q \\varphi _\\mathfrak {a}^2 d \\vert T \\vert } + \\mathcal {C}N^{-\\nu }\\,.$ Lemma 6.14 For any subset $T \\subset [N]$ satisfying $\\vert T \\vert \\leqslant \\frac{d}{\\mathcal {C} \\Gamma ^2}$ we have $\\theta \\leqslant \\theta ^{(T)}$ with very high probability.", "Before proving Lemma REF and Lemma REF , we use them to show Lemma REF .", "[Proof of Lemma REF ] Throughout the proof we abbreviate $\\mathbb {P}_\\theta (\\Xi ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =P({ = 1})$.", "Let $ C$ be the constant from Lemma \\ref {lem:decoupling}, and set\\begin{equation} \\mathfrak {a}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation}=\\biggl (\\frac{\\mathcal {C} \\nu }{4 q}\\biggr )^{1/3}\\,.$ For the proof of REF , we choose $k = \\varphi _\\mathfrak {a}d /\\mathcal {C}$ and estimate $\\mathbb {P}_\\theta (\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/ 2}^c \\vert \\geqslant k) \\leqslant \\sum _{Y \\subset X : \\vert Y \\vert = k} \\mathbb {P}_\\theta (Y \\subset \\mathcal {T}_{\\mathfrak {a}/2}^c)\\leqslant \\binom{\\vert X \\vert }{k} \\Big ( \\mathrm {e}^{-4 q \\varphi _\\mathfrak {a}^2 d k} + \\mathcal {C}N^{-\\nu } \\Big )\\\\\\leqslant \\big (\\vert X \\vert \\mathrm {e}^{- 4 q \\varphi _\\mathfrak {a}^2 d}\\big )^k + \\mathcal {C}\\vert X \\vert ^k N^{-\\nu }\\leqslant \\mathrm {e}^{- 2 q \\varphi _\\mathfrak {a}^2 d k} + \\mathcal {C}\\mathrm {e}^{2 q \\varphi _\\mathfrak {a}^2 d k} N^{-\\nu } = N^{-2q\\mathfrak {a}^3/\\mathcal {C}} + \\mathcal {C} N^{2q\\mathfrak {a}^3/\\mathcal {C} - \\nu }\\,.$ where in the second step we used (REF ).", "Thus, by our choice of $\\mathfrak {a}$ , we have $\\mathbb {P}_\\theta (\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/ 2}^c \\vert \\geqslant k) \\leqslant (\\mathcal {C} + 1) N^{-\\nu /2}$ , from which REF follows after renaming $\\nu $ and $\\mathcal {C}$ .", "To prove REF we estimate, for $t>0$ and $l \\in \\mathbb {N}$ , $\\mathbb {P}_\\theta (\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/ 2}^c \\vert \\geqslant t) \\leqslant \\frac{1}{t^l} \\mathbb {E}\\Biggl (\\sum _{x \\in X} \\mathbb {1}_{x \\in \\mathcal {T}_{\\mathfrak {a}/ 2}^c}\\theta \\Biggr )^l = \\frac{1}{t^l} \\sum _{x_1, \\dots , x_l \\in X} \\mathbb {P}_\\theta (x_1 \\in \\mathcal {T}_{\\mathfrak {a}/ 2}^c, \\dots , x_l \\in \\mathcal {T}_{\\mathfrak {a}/ 2}^c)\\,.$ Choosing $l = \\varphi _\\mathfrak {a}d/\\mathcal {C}$ , regrouping the summation according to the partition of coincidences, and using Lemma REF yield $\\mathbb {P}_\\theta (\\vert X \\cap \\mathcal {T}_{\\mathfrak {a}/ 2}^c \\vert \\geqslant t) \\leqslant \\frac{1}{t^l} \\sum _{\\pi \\in \\mathfrak {P}_l} \\vert X \\vert ^{\\vert \\pi \\vert } \\big ( \\mathrm {e}^{- 4 q \\varphi _\\mathfrak {a}^2 d \\vert \\pi \\vert } + \\mathcal {C}N^{- \\nu } \\big )\\\\\\leqslant \\frac{1}{t^l} \\sum _{k = 0}^l \\binom{l}{k} l^{l - k} \\vert X \\vert ^k \\big ( \\mathrm {e}^{-4 q \\varphi _\\mathfrak {a}^2 dk} + \\mathcal {C}N^{- \\nu } \\big )= \\frac{(l + \\vert X \\vert \\mathrm {e}^{- 4 q \\varphi _\\mathfrak {a}^2 d})^l + \\mathcal {C}N^{-\\nu } (l + \\vert X \\vert )^l}{t^l}\\,.$ Here, $\\mathfrak {P}_l$ denotes the set of partitions of $[l]$ , and we denote by $k = \\vert \\pi \\vert $ the number of blocks in the partition $\\pi \\in \\mathfrak {P}_l$ .", "We also used that the number of partitions of $l$ elements consisting of $k$ blocks is bounded by $\\binom{l}{k} l^{l - k}$ .", "The last step follows from the binomial theorem.", "Therefore, using $l = \\varphi _\\mathfrak {a}d/\\mathcal {C}$ and choosing $t = \\mathrm {e}^{q \\varphi _\\mathfrak {a}^2 d} + \\vert X \\vert \\mathrm {e}^{- 2 q \\varphi _\\mathfrak {a}^2 d}$ as well as $\\mathcal {C}$ and $\\nu $ sufficiently large imply the bound in Lemma REF REF with very high probability, after renaming $\\mathcal {C}$ and $\\nu $ .", "Here we used (REF ).", "To obtain the same statements for $\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}$ instead of $\\mathcal {T}_{\\mathfrak {a}/ 2}$ , we estimate $ \\mathbb {P}_\\theta \\Big ( \\vert X \\cap (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c \\vert \\geqslant t\\Big ) \\leqslant \\mathbb {E}\\Big [ \\mathbb {P}\\Bigl ( \\vert X \\cap (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c \\vert \\geqslant t, \\theta ^{(x)} = 1 \\Big \\vert X \\Bigr ) \\Big ] + \\mathbb {P}\\big ( \\theta ^{(x)} = 0 , \\theta = 1\\big ).", "$ For both parts, REF and REF , the conditional probability $\\mathbb {P}\\bigl ( \\vert X \\cap (\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)})^c \\vert \\geqslant t, \\theta ^{(x)} = 1 \\big \\vert X \\bigr )$ can be bounded as before using () instead of (REF ) since, by assumption on $X$ , the set $\\mathcal {T}_{\\mathfrak {a}/ 2}^{(x)}$ and the indicator function $\\theta ^{(x)}$ are independent of $X$ .", "The smallness of $\\mathbb {P}(\\theta ^{(x)} = 0, \\theta = 1) \\leqslant \\mathbb {P}(\\theta ^{(x)} < \\theta )$ is a consequence of Lemma REF .", "This concludes the proof of Lemma REF .", "The rest of this subsection is devoted to the proofs of Lemmas REF , REF , and REF .", "Lemma 6.15 There is $\\mathfrak {c}\\equiv \\mathfrak {c}_\\nu >0$ , depending on $\\nu $ and $\\kappa $ , such that for any deterministic $T \\subset [N]$ satisfying $\\vert T \\vert \\leqslant \\mathfrak {c}d / \\Gamma ^2$ we have with very high probability $ \\theta \\max _{x,y \\notin T} \\bigl \\vert G_{xy}^{(T)} \\bigr \\vert \\leqslant 2 \\Gamma \\,.$ Moreover, under the same assumptions on $T$ and for any $u \\in [N] \\setminus T$ , we have $ \\theta \\max _{x,y \\notin T \\cup \\lbrace u\\rbrace } \\bigl \\vert G_{xy}^{(Tu)} - G_{xy}^{(T)} \\bigr \\vert \\leqslant \\mathcal {C}d^{-1}$ with very high probability.", "Before proving Lemma REF , we use it to conclude the proof of Lemma REF .", "[Proof of Lemma REF ] The bound in (REF ) of Lemma REF implies that $\\theta = \\theta \\theta ^{(T)}$ with very high probability.", "Since $\\theta \\leqslant 1$ , the proof is complete.", "[Proof of Lemma REF ] Throughout the proof we work on the event $\\lbrace \\theta = 1\\rbrace $ exclusively.", "After a relabelling of the vertices $[N]$ , we can suppose that $T = [k]$ with $k \\leqslant cd/\\Gamma ^2$ .", "For $k \\in [N]$ , we set $ \\Gamma _k \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1 x,y [k] Gxy([k])  .", "$Note that $ 0 $ by definition of $$.$ We now show by induction on $k$ that there is $\\mathcal {C}>0$ such that $ \\Gamma _k \\leqslant \\Gamma _0 \\bigg (1 + \\frac{16 \\mathcal {C}\\Gamma ^2}{d} \\bigg )^k$ for all $k \\in \\mathbb {N}$ satisfying $k \\leqslant \\frac{d}{32 \\, \\mathcal {C}\\Gamma ^2}$ .", "Since $1 + x \\leqslant \\mathrm {e}^x$ , (REF ) implies that $\\Gamma _k \\leqslant \\mathrm {e}^{1/2} \\Gamma _0 \\leqslant 2 \\Gamma $ .", "This directly implies (REF ) by the definition of $\\theta $ .", "The initial step with $k = 0$ is trivially correct.", "For the induction step $k \\rightarrow k+1$ , we set $T = [k]$ and $u = k + 1$ .", "The algebraic starting point for the induction step is the identities (REF ) and (REF ).", "We shall need the following two estimates.", "First, abbreviating $\\eta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =Imz$, from Lemma \\ref {lem:Ward} and Cauchy--Schwarz, we get\\begin{equation} \\frac{f}{N}\\biggl \\vert G_{uy}^{(T)} \\sum _a^{(Tu)} G_{xa}^{(Tu)} \\biggr \\vert \\leqslant \\frac{f}{N} \\Gamma _k \\sqrt{\\frac{N}{\\eta }} \\Gamma _{k+1} \\leqslant N^{-\\kappa /3} \\Gamma _k \\Gamma _{k+1}\\,,\\end{equation}where we used that $ k+1 1$, $ f N/6$, and $ N-1 + $.Second, the first estimate of (\\ref {eq:LDB_wvhp}) in Corollary~\\ref {cor:large_deviation_very_high_probabilty} with $ = k+1/d$ and $ = k+1/(N)$, Lemma \\ref {lem:Ward}, and $ k+1 1$ imply\\begin{equation} \\biggl \\vert \\sum _{a}^{(Tu)} G_{xa}^{(Tu)} H_{au} \\biggr \\vert \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}} \\Gamma _{k+1}\\end{equation}with very high probability.$ Hence, owing to (REF ) and (REF ) with $T = [k]$ and $u = k + 1$ , we get, respectively, $ \\Gamma _{k+1} \\leqslant \\Gamma _k + \\frac{\\mathcal {C}}{\\sqrt{d}} \\Gamma _k \\Gamma _{k+1}, \\qquad \\qquad \\Gamma _{k+1} \\leqslant \\Gamma _k + \\frac{\\mathcal {C}}{d} \\Gamma _k \\Gamma _{k+1}^2$ with very high probability.", "By the induction assumption (REF ) we have $\\mathcal {C} \\Gamma _k / \\sqrt{d} \\leqslant 2 \\mathcal {C} \\Gamma / \\sqrt{d} \\leqslant 1/2$ , so that the first inequality in (REF ) implies the rough a priori bound $ \\Gamma _{k+1} \\leqslant 2 \\Gamma _k$ with very high probability.", "From the second inequality in (REF ) and (REF ), we deduce that $\\Gamma _{k+1} \\leqslant \\Gamma _k \\biggl (1 + \\frac{4 \\mathcal {C}}{d} \\Gamma _k^2\\biggr ) \\leqslant \\Gamma _k \\biggl (1 + \\frac{16 \\mathcal {C} \\Gamma ^2}{d}\\biggr )\\,,$ where in the second step we used $\\Gamma _k \\leqslant 2 \\Gamma $ , by the induction assumption (REF ).", "This concludes the proof of (REF ), and, hence, of (REF ).", "For the proof of (REF ), we start from (REF ) and use (), () as well as (REF ).", "This concludes the proof of Lemma REF .", "The next result provides concentration estimates for the parameters $\\Phi _x$ and $\\Psi _x$ .", "Lemma 6.16 There is a constant $0 < q \\leqslant 1$ , depending only on $\\Gamma $ , such that the following holds.", "Let $\\mathfrak {c}>0$ be as in Lemma REF , and let $x \\in [N]$ and $T \\subset [N]$ be deterministic and satisfy $\\vert T \\vert \\leqslant \\mathfrak {c}d / \\Gamma ^2$ .", "Then for any $0 < \\varepsilon \\leqslant 1$ we have $\\theta ^{(T)} \\mathbb {P}\\big ( \\vert \\Phi _x^{(T)} \\vert > \\varepsilon \\bigm \\vert H^{(T)} \\big ) \\leqslant \\mathrm {e}^{- 32 q \\varepsilon ^2 d}\\,, \\qquad \\theta ^{(T)} \\mathbb {P}\\big (\\vert \\Psi _x^{(T)} \\vert > \\varepsilon \\bigm \\vert H^{(T)} \\big ) \\leqslant \\mathrm {e}^{ - 32 q \\varepsilon ^2 d}\\,,$ and, for any $u \\notin T$ , $ \\Phi _x^{(Tu)} - \\Phi _x^{(T)} = O\\biggl (\\frac{1}{d}\\biggr )\\,,\\qquad \\theta ^{(T)} \\bigl (\\Psi _x^{(Tu)} - \\Psi _x^{(T)}\\bigr ) = \\mathcal {O}\\biggl (\\frac{1 + \\beta _x}{d}\\biggr )$ with very high probability.", "Before proving Lemma REF , we use it conclude the proof of Lemma REF .", "[Proof of Lemma REF ] Using (), we find that $\\beta _x \\leqslant \\mathcal {C} (1 + \\frac{\\log N}{d})$ with very high probability.", "The claim now follows from (REF ) with $T = \\emptyset $ and the definition of $\\varphi _{\\mathfrak {a}}$ , choosing the constant $\\mathcal {D}$ in (REF ) large enough.", "[Proof of Lemma REF ] Set $q \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1211(e)2$.We get, using (\\ref {eq:LDB_linear_noncentered}) with $ r =32 q 2 d d$, $ EHxy 2 = 1/N$, and Chebyshev^{\\prime }s inequality,\\begin{multline*}\\theta ^{(T)} \\mathbb {P}\\Big ( \\vert \\Psi _x^{(T)} \\vert > \\varepsilon \\Bigm \\vert H^{(T)} \\Big )= \\mathbb {P}\\Bigg (\\theta ^{(T)} \\Biggl \\vert \\sum _y^{(Tx)} (\\vert H_{xy} \\vert ^2 - \\mathbb {E}\\vert H_{xy} \\vert ^2) G_{yy}^{(T)} \\Biggr \\vert > \\varepsilon \\biggm \\vert H^{(T)} \\Bigg )\\\\\\leqslant \\biggl (\\frac{8 \\Gamma }{\\varepsilon } \\sqrt{\\frac{r}{d}}\\biggr )^r= \\mathrm {e}^{ -32 q \\varepsilon ^2 d}\\end{multline*}with very high probability for any $ 0 < 1$.", "This proves the estimate on $ x(T)$ in (\\ref {eq:bound_Phi_Psi_x}), and the estimate for $ x(T)$ is proved similarly.$ We now turn to the proof of (REF ).", "If $x = u$ then the statement is trivial.", "Thus, we assume $x \\ne u$ .", "In this case we have $\\Phi _x^{(Tu)} - \\Phi _x^{(T)} = - \\bigg (\\vert H_{xu} \\vert ^2 - \\frac{1}{N} \\bigg )$ and the claim for follows from Lemma REF .", "Next, $ \\Psi _x^{(Tu)} - \\Psi _x^{(T)} = \\sum _{y}^{(Tux)} \\bigg ( \\vert H_{xy} \\vert ^2 - \\frac{1}{N} \\bigg ) \\Big ( G_{yy}^{(Tux)} - G_{yy}^{(Tx)} \\Big ) - \\bigg (\\vert H_{xu} \\vert ^2 - \\frac{1}{N} \\bigg ) G_{uu}^{(Tx)}\\,.", "$ The last term multiplied by $\\theta ^{(T)}$ is estimated by $O(\\Gamma / d)$ by Lemma REF and the fact that $\\theta ^{(T)} \\vert G_{uu}^{(Tx)} \\vert \\leqslant 4 \\Gamma $ by (REF ).", "We estimate the first term using (REF ) in Lemma REF , which yields $\\theta ^{(T)} \\bigl \\vert \\Psi _x^{(Tu)} - \\Psi _x^{(T)} \\bigr \\vert \\leqslant \\sum _{y}^{(Tux)} \\vert H_{xy} \\vert ^2 \\frac{\\mathcal {C}}{d} + \\frac{1}{N} \\sum _{y}^{(Tux)} \\frac{\\mathcal {C}}{d} + O \\biggl ( \\frac{\\Gamma }{d}\\biggr ) = \\mathcal {O} \\biggl (\\frac{1 + \\beta _x}{d}\\biggr )$ with very high probability.", "This concludes the proof of Lemma REF .", "[Proof of Lemma REF ] Throughout the proof we abbreviate $\\mathbb {P}_\\theta (\\Xi ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =P({ = 1})$.", "We have\\begin{equation*}\\mathbb {P}\\big ( T \\subset \\mathcal {T}_{\\mathfrak {a}/ 2}^c,\\, \\theta = 1 \\big ) = \\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x\\Biggr )\\,,\\end{equation*}where we defined the event\\begin{equation*}\\Omega _x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\bigl \\lbrace \\vert \\Phi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace \\cup \\bigl \\lbrace \\vert \\Psi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace = \\bigl \\lbrace \\vert \\Phi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace \\cup \\bigl \\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}, \\vert \\Psi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace \\,.$ We have the inclusions $\\bigl \\lbrace \\vert \\Phi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace &\\subset \\bigl \\lbrace \\vert \\Phi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr \\rbrace \\cup \\bigl \\lbrace \\vert \\Phi _x - \\Phi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr \\rbrace \\,,\\\\\\bigl \\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}, \\vert \\Psi _x \\vert > \\varphi _{\\mathfrak {a}/ 2}\\bigr \\rbrace &\\subset \\bigl \\lbrace \\vert \\Psi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr \\rbrace \\cup \\bigl \\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}, \\vert \\Psi _x - \\Psi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr \\rbrace \\,.$ Defining the event $\\Omega _x^{(T)} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={x(T) > a/ 4} {x(T) > a/ 4} , we therefore deduce by a union bound that $ \\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x\\Biggr ) \\leqslant \\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x^{(T)}\\Biggr ) + \\sum _{x \\in T} \\mathbb {P}_\\theta \\bigl (\\vert \\Phi _x - \\Phi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr )\\\\+ \\sum _{x \\in T} \\mathbb {P}_\\theta \\bigl (\\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}, \\vert \\Psi _x - \\Psi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4}\\bigr )\\,.$ We begin by estimating the first term of (REF ).", "To that end, we observe that, conditioned on $H^{(T)}$ , the family $(\\Omega _x^{(T)})_{x \\in T}$ is independent.", "Using Lemma REF we therefore get $\\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x^{(T)}\\Biggr ) \\leqslant \\mathbb {E}\\Biggl [ \\theta ^{(T)} \\mathbb {P}\\Biggl (\\bigcap _{x \\in T}\\Omega _x^{(T)} \\biggm | H^{(T)}\\Biggr )\\Biggr ] + \\mathcal {C} N^{-\\nu } = \\mathbb {E}\\biggl [\\theta ^{(T)} \\prod _{x \\in T} \\mathbb {P}(\\Omega _x^{(T)} | H^{(T)})\\biggr ] + \\mathcal {C} N^{-\\nu }\\,,$ and we estimate each factor using (REF ) from Lemma REF as $\\theta ^{(T)} \\mathbb {P}(\\Omega _x^{(T)} | H^{(T)}) \\leqslant \\theta ^{(T)} \\mathbb {P}\\big ( \\vert \\Phi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/4} \\bigm \\vert H^{(T)} \\big ) + \\theta ^{(T)} \\mathbb {P}\\big ( \\vert \\Psi _x^{(T)} \\vert > \\varphi _{\\mathfrak {a}/ 4} \\bigm \\vert H^{(T)} \\big )\\\\\\leqslant 2 \\mathrm {e}^{-8 q \\varphi _{\\mathfrak {a}}^2 d} \\leqslant \\mathrm {e}^{-4 q \\varphi _{\\mathfrak {a}}^2 d}\\,,$ where in the last step we used that $\\mathrm {e}^{-4 q \\varphi _{\\mathfrak {a}}^2 d} \\leqslant 1/2$ .", "We conclude that $\\mathbb {P}_\\theta \\Biggl (\\bigcap _{x \\in T}\\Omega _x^{(T)}\\Biggr ) \\leqslant \\mathrm {e}^{-4 q \\varphi _{\\mathfrak {a}}^2 d \\vert T \\vert } + \\mathcal {C} N^{-\\nu }\\,.$ Next, we estimate the second term of (REF ).", "After renaming the vertices, we may assume that $T = [k]$ with $k \\leqslant \\varphi _\\mathfrak {a}d / \\mathcal {C}$ , so that we get from (REF ) from Lemma REF (using that $\\varphi _\\mathfrak {a}d / \\mathcal {C} \\leqslant \\mathfrak {c}d / \\Gamma ^2$ provided that $\\mathcal {D}$ in (REF ) is chosen large enough, depending on $\\mathfrak {a}$ ), by telescoping and recalling Lemma REF , $ \\vert \\Phi _x - \\Phi _x^{(T)} \\vert \\leqslant \\sum _{i = 0}^{k-1} \\bigl \\vert \\Phi _x^{([i])} - \\Phi _x^{([i+1])} \\bigr \\vert \\leqslant O \\biggl (\\frac{k}{d}\\biggr ) \\leqslant \\varphi _{\\mathfrak {a}/ 4}$ with very high probability on the event $\\lbrace \\theta = 1\\rbrace $ , if the constant $\\mathcal {C}$ in the upper bound $\\varphi _\\mathfrak {a}d / \\mathcal {C}$ on $k$ is large enough.", "The last term of (REF ) is estimated analogously, with the additional observation that, by definition of $\\Phi _x$ and since $\\varphi _{\\mathfrak {a}/2} \\leqslant 1/2$ , on the event $\\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}\\rbrace $ we have $\\beta _x \\leqslant 2$ .", "Thus, on the event $\\lbrace \\theta = 1\\rbrace \\cap \\lbrace \\vert \\Phi _x \\vert \\leqslant \\varphi _{\\mathfrak {a}/ 2}\\rbrace $ we have, by Lemma REF , $ \\vert \\Psi _x - \\Psi _x^{(T)} \\vert \\leqslant \\sum _{i = 0}^{k-1} \\bigl \\vert \\Psi _x^{([i])} - \\Psi _x^{([i+1])} \\bigr \\vert \\leqslant \\mathcal {O} \\biggl (\\frac{k(1 + \\beta _x)}{d}\\biggr ) \\leqslant \\varphi _{\\mathfrak {a}/ 4}$ with very high probability, for large enough $\\mathcal {C}$ in the upper bound on $k$ .", "We conclude that the two last terms of (REF ) are bounded by $\\mathcal {C} N^{-\\nu }$ , and the proof of (REF ) is therefore complete.", "The proof of () is identical, replacing the matrix $M$ with the matrix $M^{(x)}$ .", "Self-consistent equation and proof of Theorem  REF In this subsection, we derive an approximate self-consistent equation for the Green function $G$ , and use it to prove Theorem REF .", "The key ingredient is Proposition REF below, which provides a bootstrapping bound stating that if $G_{xx} - m_{\\beta _x}$ is smaller than some constant then it is in fact bounded by $\\varphi _\\mathfrak {a}$ with very high probability.", "It is proved by first deriving and solving a self-consistent equation for the entries $G_{xx}$ indexed by typical vertices $x \\in \\mathcal {T}_\\mathfrak {a}$ , and using the obtained bounds to analyse $G_{xx}$ for atypical vertices $x \\in \\mathcal {T}^c_\\mathfrak {a}$ .", "We begin with a simple algebraic observation.", "Lemma 6.17 (Approximate self-consistent equation) For any $x \\in [N]$ , we have $ \\frac{1}{G_{xx}} = - z - \\sum _{y}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} + Y_x\\,, $ where we introduced the error term $ Y_x \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =Hxx + fN - a b(x) Hxa Gab(x) Hbx - a,b(x) ( fN ( HxaGab(x) + Gab(x) Hbx ) + f2N2 Gab(x)) .", "The lemma follows directly from (REF ) and the definition (REF ).", "Let $\\theta $ be defined as in (REF ) with some $\\Gamma \\geqslant 1$ .", "The following lemma provides a priori bounds on the error terms appearing in the self-consistent equation.", "Lemma 6.18 For all $z \\in \\mathbf {S}$ , with very high probability, $\\theta \\max _x \\vert Y_x \\vert & \\leqslant \\mathcal {C}d^{-1/2}, \\\\\\theta \\max _{x \\ne y} \\vert G_{xy} \\vert & \\leqslant \\mathcal {C}d^{-1/2}, \\\\\\theta \\max _{x \\ne a \\ne y} \\vert G_{xy} - G_{xy}^{(a)} \\vert & \\leqslant \\mathcal {C}d^{-1}.", "$ We first estimate $Y_x$ .", "From Lemma REF , the upper bound on $f$ , and (REF ), we conclude that $\\vert H_{xx} \\vert + f / N = O(d^{-1/2})$ almost surely.", "Moreover, the Cauchy–Schwarz inequality, Lemma REF , (REF ) and the upper bound on $f$ imply $ \\theta \\frac{f^2}{N^2} \\biggl \\vert \\sum _{a,b}^{(x)} G_{ab}^{(x)} \\biggr \\vert \\leqslant C_\\kappa \\frac{f^2}{\\sqrt{N\\eta }} \\leqslant C_\\kappa N^{-\\kappa /6} \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}}\\,, $ for some constant $C_\\kappa $ depending only on $\\kappa $ .", "Next, we use the first estimate of (REF ), Lemma REF , and the upper bound on $f$ to conclude that $ \\frac{f}{N} \\theta \\biggl \\vert \\sum _{a,b}^{(x)} H_{xa} G_{ab}^{(x)} \\biggr \\vert + \\frac{f}{N} \\theta \\biggl \\vert \\sum _{a,b}^{(x)} G_{ab}^{(x)} H_{bx} \\biggr \\vert \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}} \\frac{f}{\\sqrt{N\\eta }} \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}} N^{-\\kappa /3} \\leqslant \\frac{\\mathcal {C}}{\\sqrt{d}} $ with very high probability (compare the proof of ()).", "Moreover, from Lemma REF and the second estimate of (REF ) we deduce that remaining term in (REF ) is $\\mathcal {O}(d^{-1}) = \\mathcal {O}(d^{-1/2})$ .", "This concludes the proof of (REF ).", "For the proof of (), we start from (REF ) and use $M_{xa} = H_{xa} + f/ N $ to obtain $ G_{xy} = - G_{xx} \\sum _{a}^{(x)} H_{xa} G_{ay}^{(x)} - G_{xx} H_{xy} G_{yy}^{(x)} - \\frac{f}{N} G_{xx} \\sum _a^{(x)} G_{ay}^{(x)}.", "$ Similar arguments as in () and () show that the first and third term, respectively, are bounded by $\\mathcal {C}d^{-1/2}$ with very high probability.", "The same bound for the second term follows from Lemma REF and (REF ) in Lemma REF .", "This proves ().", "Finally, () follows directly from (REF ).", "Proposition REF below is the main tool behind the proof of Theorem REF .", "To formulate it, we introduce the $z$ -dependent random control parameters $\\Lambda _{\\mathrm {d}} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =x Gxx - mx  ,       o =x y Gxy  ,       =d o , and, for some constant $\\lambda \\leqslant 1$ , the indicator function $ \\phi \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1 .", "Proposition REF below provides a strong bound on $\\Lambda $ provided the a priori condition $\\phi = 1$ is satisfied.", "Each step of its proof is valid provided $\\lambda $ is chosen small enough depending on $\\kappa $ .", "Note that, owing to (REF ), there is a deterministic constant $\\Gamma $ , depending only on $\\kappa $ , such that, for all $z \\in \\mathbf {S}$ , we have $ \\phi \\max _{x,y} \\vert G_{xy} \\vert \\leqslant \\Gamma \\,.$ In particular, if $\\Gamma $ in the definition (REF ) of $\\theta $ is chosen as in (REF ) then $ \\phi \\leqslant \\theta \\,.$ Proposition 6.19 There exists $\\lambda > 0$ , depending only on $\\kappa $ , such that, for all $z \\in \\mathbf {S}$ , with very high probability, $ \\phi \\Lambda \\leqslant \\mathcal {C}\\varphi _\\mathfrak {a}\\,.", "$ For the proof of Proposition REF , we employ the results of the previous subsections to show that the diagonal entries $(G_{xx})_{x \\in \\mathcal {T}_\\mathfrak {a}}$ of the Green function of $M$ at the typical vertices satisfy the approximate self-consistent equation (REF ) below.", "This is a perturbed version of the relation (REF ) for the Stieltjes transform $m$ of the semicircle law, which holds for all $z \\in +$ .", "The stability estimate, (REF ) below, then implies that $G_{xx}$ and $m$ are close for all $x \\in \\mathcal {T}_\\mathfrak {a}$ .", "From this we shall, in a second step, deduce that $G_{xx}$ is close to $m_{\\beta _x}$ for all $x$ ; this steps includes also the atypical vertices.", "The next lemma is a relatively standard stability estimate of self-consistent equations in random matrix theory (compare e.g.", "to [27]).", "It is proved in Appendix REF .", "Lemma 6.20 (Stability of the self-consistent equation for $m$ ) Let $\\mathcal {X}$ be a finite set, $\\kappa >0$ , and $z \\in +$ satisfy $\\vert \\mathrm {Re}\\,z \\vert \\leqslant 2- \\kappa $ .", "We assume that, for two vectors $(g_x)_{x \\in \\mathcal {X}}$ , $(\\varepsilon _x)_{x \\in \\mathcal {X}} \\in {\\mathcal {X}}$ , the identities $ \\frac{1}{g_x} = -z - \\frac{1}{\\vert \\mathcal {X} \\vert }\\sum _{y \\in \\mathcal {X}} g_y + \\varepsilon _x$ hold for all $x \\in \\mathcal {X}$ .", "Then there are constants $b, C \\in (0,\\infty )$ , depending only on $\\kappa $ , such that if $\\max _{x \\in \\mathcal {X}} \\vert g_x -m(z) \\vert \\leqslant b$ then $ \\max _{x \\in \\mathcal {X}} \\vert g_x - m(z) \\vert \\leqslant C \\max _{x \\in \\mathcal {X}} \\vert \\varepsilon _x \\vert ,$ where $m(z)$ satisfies (REF ).", "[Proof of Proposition REF ] Throughout the proof, we work on the event $\\lbrace \\phi = 1\\rbrace $ , which, by (REF ), is contained in the event $\\lbrace \\theta = 1\\rbrace $ .", "Fix $\\mathfrak {a}$ as in Lemma REF .", "Throughout the proof we use that $d^{-1/2} \\leqslant \\varphi _\\mathfrak {a}$ by the upper bound in (REF ).", "Owing to (), it suffices to estimate $\\Lambda _{\\mathrm {d}}$ .", "Let $b$ be chosen as in Lemma REF , and set $\\lambda \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =b/2$ in the definition (\\ref {eq:def_vartheta}) of $$.$ For the analysis of $G_{xx}$ we distinguish the two cases $x \\in \\mathcal {T}_\\mathfrak {a}$ and $x \\notin \\mathcal {T}_\\mathfrak {a}$ .", "If $x \\in \\mathcal {T}_\\mathfrak {a}$ then we write using Lemma REF and the definition (REF ) of $\\Psi _x$ that $\\frac{1}{G_{xx}} = -z - \\sum _y^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} + Y_x = -z - \\frac{1}{N} \\sum _y^{(x)} G_{yy}^{(x)} + Y_x - \\Psi _x = -z - \\frac{1}{\\vert \\mathcal {T}_\\mathfrak {a} \\vert } \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}} G_{yy} + \\varepsilon _x\\,,$ where the error term $\\varepsilon _x$ satisfies $ \\vert \\varepsilon _x \\vert = \\mathcal {O} \\biggl (d^{-1/2} + \\frac{1}{N} \\exp ( q \\varphi _\\mathfrak {a}^2 d) + \\exp (-2 q \\varphi _\\mathfrak {a}^2 d ) + \\varphi _\\mathfrak {a}\\biggr ) = \\mathcal {O}(\\varphi _\\mathfrak {a})$ with very high probability.", "Here, in the first step of (REF ) we used (REF ), (), Proposition REF REF , and the bound on $\\Psi _x$ in the definition (REF ) of $\\mathcal {T}_\\mathfrak {a}$ , and in the second step of (REF ) we used that $\\varphi _\\mathfrak {a}^2 d = \\mathfrak {a}^2 (\\log N)^{2/3} d^{-1/3}$ and (REF ) imply $(\\log N)^{1/6} / \\mathcal {C} \\leqslant \\varphi ^2_\\mathfrak {a}d \\leqslant \\mathcal {C} (\\log N)^{1/2}$ , which yields $ \\frac{1}{N} \\exp ( q \\varphi _\\mathfrak {a}^2 d) + \\exp (-2 q \\varphi _\\mathfrak {a}^2 d ) \\leqslant \\mathcal {C} d^{-10} \\leqslant \\varphi _\\mathfrak {a}\\,.$ Thus, for $(G_{xx})_{x \\in \\mathcal {T}_\\mathfrak {a}}$ we get the self-consistent equation in (REF ) with $g_x = G_{xx}$ and $\\mathcal {X} = \\mathcal {T}_\\mathfrak {a}$ .", "Moreover, by the bound on $\\Phi _x$ in the definition (REF ) of $\\mathcal {T}_\\mathfrak {a}$ , we have $\\beta _x = 1 + \\mathcal {O}(\\varphi _\\mathfrak {a})$ .", "Hence, by (), the assumption $\\phi = 1$ and $d \\geqslant \\mathcal {C}\\sqrt{\\log N}$ , we find that $\\vert G_{xx} - m \\vert \\leqslant \\vert G_{xx} - m_{\\beta _x} \\vert + \\vert m_{\\beta _x} - m \\vert \\leqslant b\\,,$ choosing the constant $\\mathcal {D}$ in (REF ) large enough that the right-hand side of (), i.e.", "$C \\vert \\beta _x - 1 \\vert $ , is bounded by $b/2$ .", "Hence Lemma REF is applicable and we obtain $\\vert G_{xx} - m \\vert = O(\\max _{y \\in \\mathcal {T}_\\mathfrak {a}} \\vert \\varepsilon _y \\vert )$ .", "Therefore, we obtain $ \\vert G_{xx} - m_{\\beta _x} \\vert \\leqslant \\vert G_{xx} - m \\vert + \\vert m - m_{\\beta _x} \\vert \\leqslant \\mathcal {C}\\varphi _\\mathfrak {a}$ with very high probability.", "This concludes the proof in the case $x \\in \\mathcal {T}_\\mathfrak {a}$ .", "What remains is the case $x \\notin \\mathcal {T}_\\mathfrak {a}$ .", "In that case, we obtain from Lemma REF that $ \\frac{1}{G_{xx}} = -z - \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} - \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}^c}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} + Y_x = -z - \\beta _x m + \\varepsilon _x\\,,$ where the error term $\\varepsilon _x$ satisfies $\\varepsilon _x = \\mathcal {O} ((1 + \\beta _x) \\varphi _\\mathfrak {a})$ with very high probability.", "Here we used (REF ) as well as (), (REF ), (REF ) and Proposition REF REF twice to conclude that $\\sum _{y \\in \\mathcal {T}_\\mathfrak {a}}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} = \\beta _x m + \\mathcal {O} (\\beta _x \\varphi _\\mathfrak {a}) \\,, \\qquad \\sum _{y \\in \\mathcal {T}_\\mathfrak {a}^c}^{(x)} \\vert H_{xy} \\vert ^2 G_{yy}^{(x)} = \\mathcal {O}\\big (\\varphi _\\mathfrak {a}+ d^4 \\exp (-q \\varphi _\\mathfrak {a}^2 d )\\big )= \\mathcal {O}(\\varphi _\\mathfrak {a}) $ with very high probability.", "From (REF ) and (REF ) we therefore get $ G_{xx} - m_{\\beta _x} = - m_{\\beta _x} \\, \\frac{1}{-z - \\beta _x m + \\varepsilon _x} \\, \\varepsilon _x\\,.$ To estimate the right-hand side of (REF ), we consider the cases $\\beta _x \\leqslant 1$ and $\\beta _x > 1$ separately.", "If $\\beta _x \\leqslant 1$ then, by (REF ), the first factor of (REF ) is bounded by $C$ .", "Thus, by (REF ), the second factor is bounded by $2 C$ provided that $\\vert \\varepsilon _x \\vert \\leqslant 1/{2C}$ by choosing $\\mathcal {D}$ in (REF ) large enough, and the third factor is bounded by $\\mathcal {C} \\varphi _\\mathfrak {a}$ .", "This yields the claim.", "If $\\beta _x > 1$ , we use that $\\operatorname{Im}m \\geqslant c$ for some constant $c > 0$ depending only on $\\kappa $ and $L$ .", "Thus, the right-hand side of (REF ) is bounded in absolute value, again using (REF ), by $C \\frac{1}{\\beta _x c/2} \\mathcal {C} \\beta _x \\varphi _\\mathfrak {a}$ , provided that $\\mathcal {D}$ in (REF ) is chosen large enough.", "This yields the claim.", "[Proof of Theorem REF ] After possibly increasing $L$ , we can assume that $L$ in the definition of $\\mathbf {S}$ in (REF ) satisfies $L \\geqslant 2/\\lambda + 1$ , where $\\lambda $ is chosen as in Proposition REF .", "We first show that () follows from (REF ).", "Indeed, averaging the estimate on $\\vert G_{xx} - m_{\\beta _x} \\vert $ in (REF ) over $x \\in [N]$ , using that $m_{\\beta _x} = m + O(\\varphi _\\mathfrak {a})$ for $x \\in \\mathcal {T}_\\mathfrak {a}$ by () and estimating the summands in $\\mathcal {T}_\\mathfrak {a}^c$ by Proposition REF REF and (REF ) yield () due to (REF ).", "What remains is the proof of (REF ).", "Let $z_0 \\in \\mathbf {S}$ , set $J \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ ={ j N0 Im z0 + j N-3 2 / }$,and define $ zj =z0 + ij N-3$ for $ j [J]$.We shall prove the bound in (\\ref {eq:local_law_entrywise}) at $ z = zj$ by induction on $ j$, starting from $ j = J$ and going down to $ j = 0$.Since $ Gxy(z) (Im z)-1$ and $ mx(z) (Im z)-1$ for all $ x,y [N]$,we have $ x Gxx(zJ) - mx(zJ) $ and $ (zJ) = 1$.$ For the induction step $j \\rightarrow j - 1$ , suppose that $\\phi (z_j) = 1$ with very high probability.", "Then, by Proposition REF , we deduce that $\\Lambda (z_j) \\leqslant \\mathcal {C} \\varphi _\\mathfrak {a}$ with very high probability.", "Since $G_{xy}$ and $m_{\\beta _x}$ are Lipschitz-continuous on $\\mathbf {S}$ with constant $N^2$ , we conclude that $\\Lambda (z_{j-1}) \\leqslant \\mathcal {C}\\varphi _\\mathfrak {a}+ N^{-1}$ with very high probability.", "If $N$ is sufficiently large and $\\varphi _\\mathfrak {a}$ is sufficiently small, obtained by choosing $\\mathcal {D}$ in (REF ) large enough, then we deduce that $\\Lambda (z_{j-1}) \\leqslant \\lambda $ with very high probability and hence $\\phi (z_{j - 1}) = 1$ with very high probability.", "Using Proposition REF , this concludes the induction step, and hence establishes $\\Lambda (z_0) \\leqslant \\mathcal {C} \\varphi _\\mathfrak {a}$ with very high probability.", "Here we used that the intersection of $J$ events of very high probability is an event of very high probability, since $J \\leqslant C N^3$ , where $C$ depends on $\\kappa $ .", "Appendices toc In the following appendices we collect various tools and explanations used throughout the paper.", "Simulation of the $\\ell ^\\infty $ -norms of eigenvectors In Figure REF we depict a simulation of the $\\ell ^\\infty $ -norms of the eigenvectors of the adjacency matrix $A / \\sqrt{d}$ of the Erdős-Rényi graph $\\mathbb {G}(N,d/N)$ restricted to its giant component.", "We take $d = b \\log N$ with $N = 10^{\\prime }000$ and $b = 0.6$ .", "The eigenvalues and eigenvectors are drawn using a scatter plot, where the horizontal coordinate is the eigenvalue and the vertical coordinate the $\\ell ^\\infty $ -norm of the associated eigenvector.", "The higher a dot is located, the more localized the associated eigenvector is.", "Complete delocalization corresponds to a vertical coordinate $\\approx 0.01$ , and localization at a single site to a vertical coordinate 1.", "Note the semilocalization near the origin and outside of $[-2,2]$ .", "The two semilocalized blips around $\\pm 0.4$ are a finite-$N$ effect and tend to 0 as $N$ is increased.", "The Perron-Frobenius eigenvalue is an outlier near $2.8$ with delocalized eigenvector.", "Figure: A scatter plot of (λ,∥w̑∥ ∞ )(\\lambda , \\Vert w̑ \\Vert _\\infty ) for all eigenvalue-eigenvector pairs (λ,w̑)(\\lambda , w̑) of the adjacency matrix A/dA / \\sqrt{d} of the critical Erdős-Rényi graph restricted to its giant component, where N=10 ' 000N = 10^{\\prime }000 and d=0.6logNd = 0.6 \\log N. Spectral analysis of the infinite rooted $(p,q)$ -regular tree In this appendix we describe the spectrum, eigenvectors, and spectral measure of the following simple graph.", "Definition 7.1 For $p,q \\in \\mathbb {N}^*$ we define $\\mathbb {T}_{p,q}$ as the infinite rooted $(p,q)$ -regular tree, whose root has $p$ children and all other vertices have $q$ children.", "A convenient way to analyse the adjacency matrix of $\\mathbb {T}_{p,q}$ is by tridiagonalizing it around its root.", "To that end, we first review the tridiagonalizationThe tridiagonalization algorithm that we use is the Lanczos algorithm.", "Tridiagonalizing matrices in numerical analysis and random matrix theory [26], [55] is usually performed using the numerically more stable Householder algorithm.", "However, when applied to the adjacency matrix $X = A$ of a graph, the Lanczos algorithm is more convenient because it can exploit the sparseness and local geometry of $A$ .", "of a general symmetric matrix $X \\in \\mathbb {R}^{N \\times N}$ around a vertex $x \\in [N]$ ; we refer to [10] for details.", "Let $r \\in \\mathbb {N}$ and $x \\in [N]$ .", "Suppose that the vectors $_x, X _x, X^2 _x, \\dots , X^r _x$ are linearly independent, and denote by $g̑_0, g̑_1, g̑_2, \\dots , g̑_r$ the associated orthonormalized sequence.", "Then the tridiagonalization of $X$ around $x$ up to radius $r$ is the $(r + 1) \\times (r+1)$ matrix $Z = (Z_{ij})_{i,j = 0}^r$ with $Z_{ij} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =g̑i 2mu, Z g̑j$.", "By construction, $ Z$ is tridiagonal and conjugate to $ X$ restricted to the subspace $ Span{g̑0, g̑1, ..., g̑r}$.$ Let now $X = A \\equiv A^{\\mathbb {T}_{p,q}}$ be the adjacency matrix of $\\mathbb {T}_{p,q}$ , whose root we denote by $o$ .", "Then it is easy to see that $g̑_i = _{S_i(o)} / \\Vert _{S_i(o)} \\Vert $ and the tridiagonalization of $A$ around the root up to radius $\\infty $ is the infinite matrix $\\sqrt{q} Z(p/q)$ , where $ Z(\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ = 0 0 1 1 0 1 1 0  .", "If $\\alpha > 2$ , a transfer matrix analysis (see [10]) shows that $Z(\\alpha )$ has precisely two eigenvalues in $\\mathbb {R}\\setminus [-2,2]$ , which are $\\pm \\Lambda (\\alpha )$ .", "The associated eigenvectors are $((\\pm )^i u_i)_{i \\in \\mathbb {N}}$ , where $u_0 > 0$ and $u_i \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(- 1)i/2   u0$ for $ i 1$.", "Note that the eigenvector components are exponentially decaying since $ > 2$, and hence $ u0$ can be chosen so that the eigenvectors are normalized.", "Going back to the original vertex basis of $ Tp,q$, setting $ = p/q$, we conclude that the adjacency matrix $ A$ has eigenvalues $ q ()$ with associated eigenvectors $ i N ()i ui Si(o) / Si(o) $.$ Next, we show that the measure $\\mu _\\alpha $ from (REF ) is the spectral measure at the root of $A^{\\mathbb {T}_{p,q}} / \\sqrt{d}$ and the spectral measure at 0 of (REF ).", "Lemma 7.2 For any $\\alpha \\geqslant 0$ the measure $\\mu _\\alpha $ is the spectral measure of $Z(\\alpha )$ at 0.", "For any $p,q \\in \\mathbb {N}^*$ the measure $\\mu _{p/q}$ is the spectral measure of the normalized adjacency operator $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ at the root.", "For REF , define the vector $ȇ_0 = (1,0,0,\\dots ) \\in \\ell ^2(\\mathbb {N})$ .", "The spectral measure of $Z(\\alpha )$ with respect to $ȇ_0$ is characterized by its Stieltjes transform $ \\bigl \\langle ȇ_0 {2mu}, (Z(\\alpha ) - z)^{-1} ȇ_0\\bigr \\rangle = \\frac{1}{- z - \\alpha \\bigl \\langle ȇ_0 {2mu}, (Z(1) - z)^{-1} ȇ_0\\bigr \\rangle }\\,.$ Here, we used Schur's complement formula on the Green function $(Z(\\alpha ) - z)^{-1}$ , observing that the minor of $Z(\\alpha )$ obtained by removing the zeroth row and column is $Z(1)$ .", "Setting $\\alpha = 1$ in (REF ) and recalling the defining relation (REF ) of the Stieltjes transform $m$ of the semicircle law, we conclude that $\\bigl \\langle ȇ_0 {2mu}, (Z(1) - z)^{-1} ȇ_0\\bigr \\rangle = m(z)$ and hence from (REF ) and (REF ) we get $\\bigl \\langle ȇ_0 {2mu}, (Z(\\alpha ) - z)^{-1} ȇ_0\\bigr \\rangle = m_\\alpha (z)$ , as desired.", "The proof of REF is analogous.", "Denote the root of $\\mathbb {T}_{p,q}$ by $o$ .", "Again using Schur's complement formula to remove the $o$ th row and column of $H = A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ , we deduce that $ \\bigl \\langle _o {2mu}, \\bigl (A^{\\mathbb {T}_{p,q}} / \\sqrt{q} - z\\bigr )^{-1} _o\\bigr \\rangle = \\biggl (-z - \\frac{p}{q} \\bigl \\langle _o {2mu}, \\bigl (A^{\\mathbb {T}_{q,q}} / \\sqrt{q} - z\\bigr )^{-1} _o\\bigr \\rangle \\biggr )^{-1}\\,,$ where we used that $\\mathbb {T}_{p,q}$ from which $o$ has been removed consists of $p$ disconnected copies of $\\mathbb {T}_{q,q}$ .", "Setting $p = q$ in (REF ) and comparing to (REF ) implies that the left-hand side of (REF ) is equal to $m(z)$ if $p = q$ , and hence REF for general $p$ follows from (REF ).", "Finally, we remark that the equality of the spectral measures of $Z(p/q)$ and $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ can also be seen directly, by noting that $Z(p/q)$ is the tridiagonalization of $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ around the root $o$ .", "We conclude with some basic estimates for the Stieltjes transform $m_\\alpha $ of $\\mu _\\alpha $ used in Section .", "Lemma 7.3 For each $\\kappa >0$ there is a constant $C>0$ depending only on $\\kappa $ such that for all $z \\in S̑$ and all $\\alpha \\geqslant 0$ we have $ \\vert m_\\alpha (z) \\vert & \\leqslant C\\,, \\\\\\vert m_\\alpha (z) - m(z) \\vert & \\leqslant C \\vert \\alpha - 1 \\vert \\,.$ The simple facts follow directly from the corresponding properties of the semicircle law and its Stieltjes transform $m$ (see e.g.", "[16]).", "We leave the details to the reader.", "Bounds on adjacency matrices of trees In this appendix we derive estimates on the operator norm of a tree.", "We start with a standard estimate on the operator norm of a graph.", "Lemma 7.4 Let $\\mathbb {T}$ be a graph whose vertices have degree at most $q+1$ for some $q \\geqslant 1$ .", "Then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant q+1$ and if in addition $\\mathbb {T}$ is a tree then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant 2 \\sqrt{q}$ .", "The first claim is obvious by the Schur test for the operator norm.", "To prove the second claim, choose a root $o$ and denote by $C_x$ the set of children of the vertex $x$ .", "Then for any vector $w̑ = (w_x)$ we have $\\bigl \\vert \\bigl \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\bigr \\rangle \\bigr \\vert = \\Biggl \\vert \\sum _{x,y} w_x A_{xy}^{\\mathbb {T}} w_y \\Biggr \\vert = 2 \\Biggl \\vert \\sum _x \\sum _{y \\in C_x} w_x w_y \\Biggr \\vert \\leqslant \\sum _{x} \\sum _{y \\in C_x} \\biggl (\\frac{1}{\\sqrt{q}} w_x^2 + \\sqrt{q} w_y^2\\biggr )\\\\\\leqslant \\frac{q+1}{\\sqrt{q}} w_o^2 + \\sum _{x \\ne o} \\biggl (\\frac{q}{\\sqrt{q}} w_x^2 + \\sqrt{q} w_x^2\\biggr ) \\leqslant 2 \\sqrt{q} \\sum _x w_x^2\\,,$ where in third step we used Young's inequality and in the fourth step that each vertex in the sum appears once as a child and at most $q$ times as a parent.", "This concludes the proof.", "The same proof shows that if $\\mathbb {T}$ is a rooted tree whose root has at most $p$ children and all other vertices at most $q$ children, then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant \\sqrt{q} (p/q \\vee 2)$ .", "This bound is sharp for $p \\leqslant 2q$ but not for $p > 2q$ .", "The sharp bound in the latter case is established in the following result.", "Lemma 7.5 Let $p,q \\in \\mathbb {N}^*$ .", "Let $\\mathbb {T}$ be a tree whose root has $p$ children and all the other vertices have at most $q$ children.", "Then the adjacency matrix $A^{\\mathbb {T}}$ of $\\mathbb {T}$ satisfies $\\Vert A^{\\mathbb {T}}\\Vert \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2)$ .", "Let $r \\in \\mathbb {N}$ and denote by $\\mathbb {T}_{p,q}(r)$ the rooted $(p,q)$ -regular tree of depth $r$ , whose root $x$ has $p$ children, all vertices at distance $1 \\leqslant i \\leqslant r$ from $x$ have $q$ children, and all vertices at distance $r+1$ from $x$ are leaves.", "For large enough $r$ , we can exhibit $\\mathbb {T}$ as a subgraph of $\\mathbb {T}_{p,q}(r)$ .", "By the Perron-Frobenius theorem, $ \\Vert A^{\\mathbb {T}} \\Vert = \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\rangle $ for the some normalized eigenvector $w̑$ whose entries are nonnegative.", "We extend $w̑$ to a vector indexed by the vertex set of $\\mathbb {T}_{p,q}(r)$ by setting $w_y = 0$ for $y$ not in the vertex set of $\\mathbb {T}$ .", "Clearly, $ \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\rangle \\leqslant \\langle w̑ {2mu}, A^{\\mathbb {T}_{p,q}(r)} w̑\\rangle \\,.$ Abbreviating $A \\equiv A^{\\mathbb {T}_{p,q}(r)}$ , it therefore remains to estimate the right-hand side of (REF ) for large enough $r$ .", "To that end, we define $Z$ as the tridiagonalization of $A$ around the root up to radius $r$ (see Appendix REF ).", "The associated orthonormal set $g̑_0, g̑_1, \\dots , g̑_r$ is given by $g̑_i = _{S_i(x)}/\\Vert _{S_i(x)}\\Vert $ , and $Z = \\sqrt{q} Z_r(p/q)$ , where $Z_r(\\alpha )$ is the upper-left $(r+1) \\times (r+1)$ block of (REF ).", "We introduce the orthogonal projections $P_0 \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =g̑0 g̑0*$ and $ P =i = 0r g̑i g̑i*$.", "Clearly, $ P0 P = P0$ and hence $ (1 - P) (1 - P0) = 1 - P$.", "For large enough $ r$ the vectors $ g̑r$ and $ w̑$ have disjoint support, and hence $ (1 - P) A P w̑ = (1 - P) A i = 0r - 1 g̑i g̑i 2mu, w̑ = 0$,since $ A g̑i Span{g̑i-i, g̑i+1}$ for $ i < r$.", "Thus we have{\\begin{@align}{1}{-1}\\langle w̑ {2mu}, A w̑\\rangle &= \\langle w̑ {2mu}, PAP w̑\\rangle + \\langle w̑ {2mu}, (1 - P) A (1 - P) w̑\\rangle \\\\ &= \\langle w̑ {2mu}, PAP w̑\\rangle + \\langle w̑ {2mu}, (1 - P) (1 - P_0) A (1 - P_0) (1 - P) w̑\\rangle \\,.\\end{@align}}From \\cite [Appendices B and C]{ADK19} we find\\begin{equation}\\lim _{r \\rightarrow \\infty } \\Vert P A P \\Vert = \\lim _{r \\rightarrow \\infty } \\Vert Z \\Vert = \\sqrt{q} \\Lambda (p/q \\vee 2)\\,.\\end{equation}Moreover, the operator $ (1 - P0) A (1 - P0)$ is the adjacency matrix of a forest whose vertices have degree at most $ q$.", "By Lemma \\ref {lem:forest_bound}, we therefore obtain $ (1 - P0) A (1 - P0) 2 q$.From (\\ref {w_quad_est}) we therefore get\\begin{equation*}\\limsup _{r \\rightarrow \\infty } \\langle w̑ {2mu}, A w̑\\rangle \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2) \\Vert P w̑ \\Vert ^2 + 2 \\sqrt{q} \\Vert (1 - P) w̑ \\Vert ^2 \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2) \\Vert w̑ \\Vert ^2\\,.\\end{equation*}By (\\ref {PF_quad1}) and (\\ref {PF_quad2}), the proof is complete.$ Degree distribution and number of resonant vertices In this appendix we record some basic facts about the distribution of degrees of the graph $\\mathbb {G}(N,d/N)$ , and use them to estimate the number of resonant vertices $\\mathcal {W}_{\\lambda , \\delta }$ .", "The following is a quantitative version of the Poisson approximation of a binomial random variable.", "Lemma 7.6 (Poisson approximation) If $D$ is a random variable with law $\\operatorname{Binom}(n,p)$ then for $k\\leqslant \\sqrt{n}$ and $p \\leqslant 1 / \\sqrt{n}$ we have $\\mathbb {P}(D = k) = \\frac{(pn)^k}{k!}", "\\mathrm {e}^{-pn} \\biggl (1+O\\biggl (\\frac{k^2}{n} + p^2 n\\biggr )\\biggr )\\,.$ Plugging the estimates $(1-p)^{n-k}= \\mathrm {e}^{(n-k)\\log (1-p)} = \\mathrm {e}^{-np + O(pk + p^2n)}$ and $\\frac{n!}{(n-k)!}", "= n^k\\prod _{i=0}^{k-1}\\biggl (1-\\frac{i}{n}\\biggr )=n^k \\mathrm {e}^{\\sum _{i=0}^{k-1} \\log \\bigl (1-\\frac{i}{n}\\bigr )}= n^k \\mathrm {e}^{O \\bigl (\\frac{k^2}{n}\\bigr )}\\,,$ into $\\mathbb {P}(D_x = k) = \\frac{n!}{k!", "(n-k)!}", "p^k (1-p)^{n-k}$ yields the claim, since $pk \\leqslant k^2/n + p^2 n$ .", "Lemma 7.7 For $\\mathbb {G}(N,d/N)$ we have $\\alpha _x \\leqslant \\mathcal {C}\\bigl (1 + \\frac{ \\log N}{d}\\bigr )$ with very high probability.", "This is a simple application of Bennett's inequality; see [10] for details.", "Next, we recall some standard facts about the distribution of the degrees.", "Define the function $f_d : [1,\\infty ) \\rightarrow \\big [\\frac{1}{2} \\log (2 \\pi d), \\infty \\big )$ through $ f_d(\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =d( - + 1) + 12 (2 d)  , which is bijective and increasing.", "For its interpretation, we note that if $Y \\overset{\\mathrm {d}}{=}\\operatorname{Poisson}(d)$ then by Stirling's formula we have $\\mathbb {P}(Y = k) = \\exp \\bigl (-f_d(k/d) + O \\bigl (\\frac{1}{k}\\bigr )\\bigr )$ for any $k \\in \\mathbb {N}$ .", "There is a universal constant $C > 0$ such that for $1 \\leqslant l \\leqslant \\frac{N}{C \\sqrt{d}}$ the equation $f_d(\\beta ) = \\log (N/l)$ has a unique solution $\\beta \\equiv \\beta _l(d)$ .", "The interpretation of $\\beta _l(d)$ is the typical location of $\\alpha _{\\sigma (l)}$ .", "By the implicit function theorem, we find that $d \\mapsto \\beta _l(d)$ on the interval $\\bigl (0, \\frac{N^2}{C l^2}\\bigr ]$ is a decreasing bijective function.", "Definition 7.8 An event $\\Xi \\equiv \\Xi _N$ holds with high probability if $\\mathbb {P}(\\Xi ) = 1 - o(1)$ .", "The following result is a slight generalization of [10], which can be established with the same proof.", "We note that the qualitative notion of high probability can be made stronger and quantitative with some extra effort, which we however refrain from doing here.", "Lemma 7.9 If $d \\geqslant 1$ and $l \\geqslant 1$ satisfies $\\beta _l(d) \\geqslant 3/2$ then $ \\vert \\alpha _{\\sigma (l)} - \\beta _l(d) \\vert \\leqslant \\frac{1 \\vee (\\zeta / \\log \\beta _l(d))}{d}$ with high probability, where $\\zeta $ is any sequence tending to infinity with $N$ .", "The following resultThe assumption $d \\gg \\log \\log N$ in Lemma REF is tailored so that it covers the entire range $\\alpha \\geqslant 2$ , which is what we need in this paper.", "The assumption on $d$ could also be removed at the expense of introducing a nontrivial lower bound on $\\alpha $ .", "gives bounds on the counting function of the normalized degrees $(\\alpha _x)_{x \\in [N]}$ .", "Lemma 7.10 Suppose that $\\zeta $ satisfies $ 1 \\ll \\zeta \\leqslant \\frac{d}{C \\log \\log N}$ for some large enough universal constant $C$ .", "Then for any $\\alpha \\geqslant 2$ we have with high probability $ \\big \\lfloor (N \\mathrm {e}^{-f_d(\\alpha )} - 1) (\\log N)^{-2 \\zeta } \\big \\rfloor \\leqslant \\vert \\vert \\leqslant \\big \\lceil (N \\mathrm {e}^{-f_d(\\alpha )} + 1) (\\log N)^{2 \\zeta } \\big \\rceil \\,.$ If $d > 3 \\log N$ , then an elementary analysis using Bennett's inequality shows that $\\vert \\vert = 0$ with high probability.", "Since $N \\mathrm {e}^{-f_d(\\alpha )} \\leqslant 1$ for $\\alpha \\geqslant 2$ , the claim follows.", "Thus, for the following we assume that $d \\leqslant 3 \\log N$ .", "Abbreviate $\\Upsilon \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =32 d$, which is an upper bound for the right-hand side of (\\ref {deg_est_1}).For the following we adopt the convention that $ 0(d) = $.", "Choose $ l 0$ such that\\begin{equation} \\beta _{l+1}(d) < \\alpha \\leqslant \\beta _l(d)\\,,\\end{equation}and define\\begin{equation*}\\underline{k} \\!\\, \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\big \\lfloor l (\\log N)^{-2\\zeta } \\big \\rfloor \\,, \\qquad \\overline{k} \\!\\, \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(l+1) (N)2  .", "We shall show that $ \\beta _{\\underline{k} \\!\\,}(d) - \\Upsilon \\geqslant \\beta _l(d)$ for $\\underline{k} \\!\\, \\geqslant 1$ , $ \\beta _{\\overline{k} \\!\\,}(d) + \\Upsilon \\leqslant \\beta _{l+1}(d)\\,,$ and $ \\overline{k} \\!\\, \\leqslant N \\mathrm {e}^{-f_d(3/2)}\\,.$ Thus $\\beta _{\\overline{k} \\!\\,}(d) \\geqslant 3/2$ and, assuming $\\underline{k} \\!\\, \\geqslant 1$ , Lemma REF is applicable to the indices $\\overline{k} \\!\\,$ and $\\underline{k} \\!\\,$ .", "We obtain, with high probability, $\\alpha _{\\sigma (\\overline{k} \\!\\,)} \\leqslant \\beta _{\\overline{k} \\!\\,}(d) + \\Upsilon \\leqslant \\beta _{l+1}(d) \\leqslant \\alpha \\leqslant \\beta _l(d) \\leqslant \\beta _{\\underline{k} \\!\\,}(d) - \\Upsilon \\leqslant \\alpha _{\\sigma (\\underline{k} \\!\\,)}\\,,$ from which we deduce that $ \\underline{k} \\!\\, \\leqslant \\vert \\vert \\leqslant \\overline{k} \\!\\,\\,,$ which also holds trivially also for the case $\\underline{k} \\!\\, = 0$ .", "By applying the function $f_d$ to () we obtain $l \\leqslant N \\mathrm {e}^{-f_d(\\alpha )} \\leqslant l+1$ , so that (REF ) yields (REF ).", "Next, we verify (REF ).", "We consider the cases $l = 0$ and $l \\geqslant 1$ separately.", "If $l = 0$ then, by the definition of $\\beta _{\\overline{k} \\!\\,}(d)$ , for (REF ) we require $(\\log N)^{2 \\zeta } + 1 \\leqslant N \\mathrm {e}^{-f_d(3/2)}$ , which holds by the assumption $d \\leqslant 3 \\log N$ and the upper bound on $\\zeta $ .", "Let us therefore suppose that $l \\geqslant 1$ .", "By (), $\\alpha \\geqslant 2$ , and the definition of $\\beta _l(d)$ , we have $l \\leqslant N \\mathrm {e}^{-f_d(2)}$ , and we have to ensure that $(l+2) (\\log N)^{2 \\zeta } \\leqslant N \\mathrm {e}^{-f_d(3/2)}$ .", "Since $l \\geqslant 1$ , this is satisfied provided that $3 \\mathrm {e}^{-f_d(2)} (\\log N)^{2 \\zeta } \\leqslant \\mathrm {e}^{-f_d(3/2)}$ , which holds provided that $f_d(2) - f_d(3/2) \\geqslant 3 \\zeta \\log \\log N$ .", "This inequality is true because $f_d(2) - f_d(3/2) \\geqslant f^{\\prime }_d(3/2) /2 \\geqslant d/C$ , where we used that $f_d^{\\prime }(\\alpha ) = d \\log \\alpha + \\frac{1}{2 \\alpha }$ .", "What remains, therefore, is the proof of (REF ) and (REF ).", "We begin with the proof of (REF ).", "We get from the mean value theorem that $ \\beta _{\\underline{k} \\!\\,}(d) - \\beta _l(d) = f_d^{-1}\\biggl (\\log \\biggl (\\frac{N}{\\underline{k} \\!\\,}\\biggr )\\biggr ) - f_d^{-1}\\biggl (\\log \\biggl (\\frac{N}{l}\\biggr )\\biggr )\\geqslant \\frac{3}{4 d \\log \\beta _{\\underline{k} \\!\\,}(d)} \\log \\biggl (\\frac{l}{\\underline{k} \\!\\,}\\biggr )\\,.$ The right-hand side of (REF ) is bounded from below by $\\Upsilon $ provided that $ \\log \\biggl (\\frac{l}{\\underline{k} \\!\\,}\\biggr ) \\geqslant 2 \\zeta \\log \\beta _{\\underline{k} \\!\\,}(d)\\,.$ We estimate $\\beta _{\\underline{k} \\!\\,}(d) \\leqslant \\beta _1(d)$ using the elementary bound $f_d(\\beta ) \\geqslant \\frac{d}{10} \\beta $ for $\\beta \\geqslant 2$ , which yields $\\log N = f_d(\\beta _1(d)) \\geqslant \\frac{d}{10} \\beta _1(d)$ .", "By assumption on $d$ we therefore get $ \\beta _1(d) \\leqslant \\log N\\,.$ Thus, (REF ) holds by $\\underline{k} \\!\\, \\leqslant l / (\\log N)^{2 \\zeta }$ .", "This concludes the proof of (REF ).", "Next, we prove (REF ).", "As in (REF ), we find $ \\beta _{l + 1}(d) - \\beta _{\\overline{k} \\!\\,}(d) = f_{d}^{-1}\\biggl (\\log \\biggl (\\frac{N}{l+1}\\biggr )\\biggr ) - f_{d}^{-1}\\biggl (\\log \\biggl (\\frac{N}{\\overline{k} \\!\\,}\\biggr )\\biggr )\\geqslant \\frac{3}{4 d \\log \\beta _{l+1}(d)} \\log \\biggl (\\frac{\\overline{k} \\!\\,}{l+1}\\biggr )\\,.$ Together with $\\beta _{l+1}(d) \\leqslant \\beta _1(d) \\leqslant \\log N$ from (REF ), we deduce that the right-hand side of (REF ) is bounded from below by $\\Upsilon $ provided that $\\log \\bigl (\\frac{\\overline{k} \\!\\,}{l+1}\\bigr ) \\geqslant 2 \\zeta \\log \\log N$ , which is true by definition of $\\overline{k} \\!\\,$ .", "This concludes the proof of (REF ).", "The following result follows easily from Lemma REF .", "Recall the definition (REF ) of the exponent $\\theta _b(\\alpha )$ .", "Corollary 7.11 Suppose that $\\zeta $ satisfies (REF ).", "Write $d = b \\log N$ .", "Then for any $\\alpha \\geqslant 2$ we have $\\vert \\vert \\vee 1 = N^{\\theta _b(\\alpha ) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )$ with high probability.", "Using the exponent $\\theta _b(\\alpha )$ from (REF ) and $\\alpha _{\\max }(b)$ defined below it, we may state the following estimate on the density of the normalized degrees and the number of resonant vertices.", "Lemma 7.12 The following holds for a large enough universal constant $C$ .", "Suppose that $\\zeta $ satisfies (REF ).", "Write $d = b \\log N$ .", "For $2 \\leqslant \\alpha < \\beta \\leqslant \\alpha _{\\max }(b)$ satisfying $\\beta - \\alpha \\geqslant C \\frac{\\zeta \\log \\log N}{d \\log \\alpha }$ , with high probability we have $ \\vert \\vert = N^{\\theta _b(\\alpha ) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )\\,.$ For $\\delta \\geqslant C \\frac{\\zeta \\log \\log N}{d}$ and $2 + \\delta \\leqslant \\lambda \\leqslant \\Lambda (\\alpha _{\\max }(b))$ , with high probability we have $\\vert \\mathcal {W}_{\\lambda ,\\delta } \\vert = N^{\\theta _b(\\Lambda ^{-1}(\\lambda - \\delta )) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )\\,.$ Note that, since $\\xi \\geqslant d^{-1/2}$ , if the conclusion of Theorem REF is nontrivial then $\\delta \\geqslant d^{-1/2}$ , and hence the assumption on $\\delta $ in Lemma REF REF is automatically satisfied for suitably chosen $\\zeta $ .", "[Proof of Lemma REF ] Part REF follows Corollary REF below by noting that the assumption on $\\beta $ implies $\\theta _b(\\alpha ) - \\theta _b(\\beta ) \\geqslant C \\frac{\\zeta \\log \\log N}{\\log N}$ by the mean value theorem.", "Part REF follows from Part REF , using that $\\log (\\lambda - \\delta ) \\geqslant \\log 2$ , that $\\Lambda ^{\\prime }$ is bounded on $[2,\\infty )$ , and the mean value theorem.", "Corollary 7.13 The following holds for large enough universal constants $C, \\mathcal {C}$ .", "Suppose that (REF ) holds.", "Write $d = b \\log N$ .", "Let $w̑ = (w_x)_{x \\in [N]}$ be a normalized eigenvector of $A/\\sqrt{d}$ with nontrivial eigenvalue $2+\\mathcal {C} \\xi ^{1/2} \\leqslant \\lambda \\leqslant \\Lambda (\\alpha _{\\max }(b))$ .", "Then with high probability for any $2 \\leqslant p \\leqslant \\infty $ we have $\\Vert w̑ \\Vert _p^{2} \\geqslant N^{(2/p - 1)\\theta _b(\\Lambda ^{-1}(\\lambda )) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\Biggl [ \\frac{\\log \\log N}{\\sqrt{\\log N}} + b (\\log \\lambda ) \\biggl (\\lambda + \\frac{1}{\\sqrt{\\lambda - 2}}\\biggr ) (\\xi + \\xi _{\\lambda - 2})\\Biggr ]\\,.$ We choose $\\delta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =C (+ - 2)$.", "Then by assumption on $$ we have $ (- 2)/2$, and hence Theorem \\ref {thm:localisation} yields, using that $ v̑(x)$ is supported in $ Br(x)$, $ x W, y Br(x) wy2 12$with high probability.", "Thus, by Hölder^{\\prime }s inequality,\\begin{equation} \\Vert w̑ \\Vert _{p}^{2} \\geqslant \\frac{1}{2} \\Biggl (\\sum _{x \\in \\mathcal {W}_{\\lambda ,\\delta }} \\vert B_{r_{\\star }}(x) \\vert \\Biggr )^{2/p - 1} \\geqslant \\frac{1}{2} \\Bigl (\\vert \\mathcal {W}_{\\lambda ,\\delta } \\vert N^{C \\log \\log N / \\sqrt{\\log N}}\\Bigr )^{2/p - 1}\\end{equation}with high probability, where we used Lemma \\ref {lem:upper_bound_degrees} to estimate $ x [N]Br(x) NC N / N$ with high probability.$ Next, using the mean value theorem and elementary estimates on the derivatives of $\\theta _b$ and $\\Lambda ^{-1}$ , we estimate $\\theta _b(\\Lambda ^{-1}(\\lambda - \\delta )) - \\theta _b(\\Lambda ^{-1}(\\lambda )) \\leqslant C b (\\log \\lambda ) \\biggl (\\lambda + \\frac{1}{\\sqrt{\\lambda - 2}}\\biggr ) \\delta \\,.$ Invoking Lemma REF REF with $\\zeta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =N$, and recalling (\\ref {w_infty_lower_est}), therefore yields the claim.$ Connected components of $\\mathbb {G}(N,d/N)$ In this appendix we give some basic estimates on the sizes of connected components of $\\mathbb {G}(N,d/N)$ .", "These are needed for the analysis of the tuning forks in Appendix REF below.", "The arguments are standard and are tailored to work well in the regime $1 \\ll d \\leqslant \\log N$ that we are interested in.", "For smaller values of $d$ , see e.g.", "[17].", "Lemma 7.14 Let $W_k$ be the number of connected components that have $k$ vertices and $\\widehat{W}_k$ the number of connected components that have $k$ vertices and are not a tree.", "Then for $k \\leqslant N/2$ we have $\\mathbb {E}[W_k] \\leqslant N \\mathrm {e}^{-k (d/2 - \\log d - 1)}\\,, \\qquad \\mathbb {E}[\\widehat{W}_k] \\leqslant \\mathrm {e}^{-k (d/2 - \\log d - 1)}\\,.$ For a set $X \\subset [N]$ , denote by $\\mathcal {T}(X)$ the set of spanning trees of $X$ .", "If $X$ is a connected component of $\\mathbb {G}$ then there exists $\\mathbb {T} \\in \\mathcal {T}(X)$ a subgraph of $\\mathbb {G}$ such that no vertex of $X$ is connected to a vertex of $[N] \\setminus X$ .", "Hence, $W_k \\leqslant \\sum _{X \\subset [N]} \\mathbb {1}_{\\vert X \\vert = k} \\sum _{\\mathbb {T} \\in \\mathcal {T}(X)} \\mathbb {1}_{\\mathbb {T} \\subset \\mathbb {G}} \\prod _{x \\in X} \\prod _{y \\in [N] \\setminus X} (1 - A_{xy})\\,.$ Taking the expectation now easily yields the claim, using $\\vert \\mathcal {T}(X) \\vert = \\vert X \\vert ^{\\vert X \\vert - 2}$ by Cayley's theorem, that a tree on $k$ vertices has $k - 1$ edges, Stirling's approximation, and $1 - x \\leqslant \\mathrm {e}^{-x}$ .", "The argument to estimate $\\widehat{W}_k$ is similar, noting that in addition to a spanning tree $\\mathbb {T}$ of $X$ , we also have to have at least one edge not in $\\mathbb {T}$ connecting two vertices of $X$ .", "Thus, $\\widehat{W}_k \\leqslant \\sum _{X \\subset [N]} \\mathbb {1}_{\\vert X \\vert = k} \\sum _{\\mathbb {T} \\in \\mathcal {T}(X)} \\mathbb {1}_{\\mathbb {T} \\subset \\mathbb {G}} \\prod _{x \\in X} \\prod _{y \\in [N] \\setminus X} (1 - A_{xy}) \\sum _{\\lbrace u,v\\rbrace \\in X^2 \\setminus E(\\mathbb {T})} A_{uv}\\,,$ and we may estimate the expectation as before.", "We call a connected component of $\\mathbb {G}$ small if it is not the giant component.", "For the following statement we recall the definition of high probability from Definition REF .", "Corollary 7.15 Suppose that $d \\gg 1$ .", "All small components of $\\mathbb {G}$ have at most $O\\bigl (\\frac{\\log N}{d}\\bigr )$ vertices with very high probability.", "All small components of $\\mathbb {G}$ are trees with high probability.", "The giant component of $\\mathbb {G}$ has at least $N (1 - \\mathrm {e}^{-d/4})$ vertices with high probability.", "Any small component has at most $N/2$ vertices.", "Using Lemma REF we therefore get that the probability that there exists a small component with at least $K$ vertices is bounded by $\\mathbb {P}(\\exists k \\in [K,N/2] \\,, W_k \\geqslant 1) \\leqslant \\sum _{k = K}^{N/2} \\mathbb {E}[W_k] \\leqslant 2 N \\mathrm {e}^{-K (d/2 - \\log d - 1)}\\,,$ by summing the geometric series.", "Since $d/2 - \\log d - 1 \\geqslant c d$ for some universal constant $c$ , we obtain the first claim.", "To obtain the second claim, we use Lemma REF to estimate the probability that there exists a small component that is not a tree by $\\sum _{k = 1}^{N/2} \\mathbb {E}{\\widehat{W}_k} \\leqslant \\mathrm {e}^{-d/3}$ .", "To obtain the last claim, we estimate the expected number of vertices in small components by $\\mathbb {E}\\bigl [\\sum _{k = 1}^{N/2} k W_k\\bigr ] \\leqslant N \\sum _{k = 1}^\\infty k \\mathrm {e}^{-k (d/2 - \\log d - 1)} \\leqslant C N \\mathrm {e}^{- d/3}$ using Lemma REF , and the third claim follows from Chebyshev's inequality.", "We may now estimate the adjacency matrix on the small components of $\\mathbb {G}(N,d/N)$ .", "The following result follows immediately from Corollary REF and Lemma REF .", "Corollary 7.16 Suppose that $d \\gg 1$ .", "Then the operator norm of $A / \\sqrt{d}$ restricted to the small components of $\\mathbb {G}$ is bounded by $O\\bigl (\\frac{\\sqrt{\\log N}}{d}\\bigr )$ with high probability.", "Corollary REF makes it explicit that Theorem REF excludes all eigenvectors on small components of $\\mathbb {G}$ , whose eigenvalues lie outside $\\mathcal {S}_\\kappa $ precisely under the lower bound from (REF ).", "Tuning forks and proof of Lemma REF In this appendix we give a precise definition of the $D$ -tuning forks from Section REF and prove Lemma REF .", "Definition 7.17 A star of degree $D \\in \\mathbb {N}$ consists of a vertex, the hub, and $D$ leaves adjacent to the hub, the spokes.", "A star tuning fork of degree $D$ is obtained by taking two disjoint stars of degree $D$ along with an additional vertex, the base, and connecting both hubs to the base.", "We say that a star tuning fork is rooted in a graph $\\mathbb {H}$ if it is a subgraph of $\\mathbb {H}$ in which both hubs have degree $D+1$ and all spokes are leaves.", "Lemma 7.18 If a star tuning fork of degree $D$ is rooted in some graph $\\mathbb {H}$ , then the adjacency matrix of $\\mathbb {H}$ has eigenvalues $\\pm \\sqrt{D}$ with corresponding eigenvectors supported on the stars of the tuning fork, i.e.", "on $2D + 2$ vertices.", "Suppose first that $D \\geqslant 1$ .", "Note first that the adjacency matrix of a star of degree $D$ has rank two and has the two nonzero eigenvalues $\\pm \\sqrt{D}$ , with associated eigenvector equal to $\\pm \\sqrt{D}$ at the hub and 1 at the spokes.", "Now take a star tuning fork of degree $D$ rooted in a graph $\\mathbb {H}$ .", "Define a vector on the vertex set of $\\mathbb {H}$ by setting it to be $\\pm \\sqrt{D}$ at the hub of the first star, 1 at the spokes of the first star, $\\mp \\sqrt{D}$ at the hub of the second star, $-1$ at the spokes of the second star, and 0 everywhere else.", "Then it is easy to check that this vector is an eigenvector of the adjacency matrix of $\\mathbb {H}$ with eigenvalue $\\pm \\sqrt{D}$ .", "If $D = 0$ the construction is analogous, defining the vector to be $+1$ at one hub and $-1$ at the other.", "We recall from Section REF that $F(d,D)$ denotes the number of star tuning forks of degree $D$ rooted in $\\mathbb {G}_{\\mathrm {giant}}$ .", "Lemma 7.19 Suppose that $1 \\ll d \\ll \\sqrt{N}$ and $0 \\leqslant D \\ll \\sqrt{N}$ .", "Then $ \\mathbb {E}[F(d,D)] = \\frac{N d^2 \\mathrm {e}^{-2d}}{2 D!^2} (d \\mathrm {e}^{-d + 1})^{2D} (1 + o(1))$ and $\\mathbb {E}[F(d,D)^2] \\leqslant \\mathbb {E}[F(d,D)]^2 (1 + o(1))$ .", "[Proof of Lemma REF ] From Lemma REF we deduce that if $1 \\ll d = b \\log N = O(\\log N)$ and $D \\ll \\log N / \\log \\log N$ , then $\\mathbb {E}[F(d,D)] = N^{1 - 2b - 2b D + o(1)}$ .", "The claim then follows from the second moment estimate in Lemma REF and Chebyshev's inequality.", "[Proof of Lemma REF ] Let $x_1,x_2 \\in [N]$ be distinct vertices and $R_1, R_2 \\subset [N] \\setminus \\lbrace x_1,x_2\\rbrace $ be disjoint subsets of size $D$ .", "We abbreviate $U = (x_1, x_2, R_1, R_2)$ and sometimes identify $U$ with $\\lbrace x_1, x_2\\rbrace \\cup R_1 \\cup R_2$ .", "The family $U$ and a vertex $o \\in [N] \\setminus U$ define a star tuning fork of degree $D$ with base $o$ , hubs $x_1$ and $x_2$ , and associated spokes $R_1$ and $R_2$ .", "Let $C_k(\\mathbb {H})$ denote the vertex set of the $k$ th largest connected component of the graph $\\mathbb {H}$ .", "Then $F(d,D) = \\frac{1}{2}\\sum _U \\sum _{o \\in [N] \\setminus U} \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}$ , where $S_{o,U} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =i = 12 (u Ri {o} Axi u u [N] (Ri {o}) (1 - Axi u) u Ri v [N] {xi} (1 - Auv)).", "The factor $\\frac{1}{2}$ corrects the overcounting from the labelling of the two stars.", "For disjoint deterministic $U$ , we split the random variables $A = (A^{\\prime }, A^{\\prime \\prime })$ into two independent families, where $A^{\\prime } \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(Auv u U or v U)$ and $ A” =(Auv u,v [N] U)$.", "Note that $ So,U$ is $ A'$-measurable.", "We define the event\\begin{equation*}\\Xi \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\,,$ which is $A^{\\prime \\prime }$ -measurable.", "By Corollary REF and the assumption on $D$ , the event $\\Xi $ holds with high probability.", "Moreover, we have $\\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U} = \\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})} S_{o,U}$ , since the component of $o$ in $\\mathbb {G}$ and $\\mathbb {G} \\vert _{[N] \\setminus U}$ differ by $2D + 2$ vertices.", "Thus, for fixed $o \\in [N] \\setminus U$ , using the independence of $A^{\\prime }$ and $A^{\\prime \\prime }$ , we get $\\mathbb {E}[\\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}] &= \\mathbb {E}[\\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})} S_{o,U}] + \\mathbb {E}[\\mathbb {1}_{\\Xi ^c} \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}]\\\\&= \\mathbb {E}[S_{o,U}] \\bigl [\\mathbb {P}\\bigl (o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})\\bigr ) + O\\bigl (\\mathbb {P}(\\Xi ^c)\\bigr )\\bigr ]\\,.$ We have $\\mathbb {P}(\\Xi ^c) = o(1)$ and $\\mathbb {P}\\bigl (o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})\\bigr ) = 1 - o(1)$ by Corollary REF and the assumption on $D$ .", "Computing $\\mathbb {E}[S_{o,U}]$ and performing the sum over $o$ and $U$ , we therefore conclude that $\\mathbb {E}[F(d,D)] = \\frac{N (N - 1) \\cdots (N - 2D - 3 + 1)}{2 D!^2} \\biggl (\\frac{d}{N}\\biggr )^{2D + 2} \\biggl (1 - \\frac{d}{N}\\biggr )^{2 (N - D - 1) + 2 D (N - 1)} (1 + o(1))\\,,$ from which (REF ) follows.", "The estimate of the second moment is similar; one can even disregard the restriction to the giant component by estimating $\\mathbb {E}[F(d,D)^2] \\leqslant \\frac{1}{4} \\sum _{U, \\tilde{U}} \\sum _{o,\\tilde{o} \\in [N]} \\mathbb {E}[S_{o,U} S_{\\tilde{o}, \\tilde{U}}]$ ; we omit the details.", "Multilinear large deviation bounds for sparse random vectors In this appendix we collect basic large deviation bounds for multilinear functions of sparse random vectors, which are proved in [42].", "The following result is proved in Propositions 3.1, 3.2, and 3.5 of [42].", "We denote by $\\Vert X \\Vert _r \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(EX r)1/r$ the $ Lr$-norm of a random variable $ X$.$ Proposition 7.20 Let $r$ be even and $1 \\leqslant d \\leqslant N$ .", "Let $X_1, \\ldots , X_N$ be independent random variables satisfying $ \\mathbb {E}X_i = 0, \\qquad \\mathbb {E}\\vert X_i \\vert ^k \\leqslant \\frac{1}{N d^{(k-2)/2}}$ for all $i \\in [N]$ and $2 \\leqslant k \\leqslant r$ .", "Let $a_i \\in and $ bij be deterministic for all $i,j \\in [N]$ .", "Suppose that $\\bigg (\\frac{1}{N} \\sum _i \\vert a_i \\vert ^2 \\bigg )^{1/2} \\leqslant \\gamma \\,, \\qquad \\frac{\\max _i \\vert a_i \\vert }{\\sqrt{d}} \\leqslant \\psi ,$ and $\\bigg ( \\max _i \\frac{1}{N} \\sum _{j} \\vert b_{ij} \\vert ^2 \\bigg )^{1/2} \\vee \\bigg ( \\max _j \\frac{1}{N}\\sum _i \\vert b_{ij} \\vert ^2 \\bigg )^{1/2} \\leqslant \\gamma , \\qquad \\frac{\\max _{i,j} \\vert b_{ij} \\vert }{d} \\leqslant \\psi $ for some $\\gamma , \\psi \\geqslant 0$ .", "Then $\\biggl \\Vert \\sum _i a_i X_i \\biggr \\Vert _r & \\leqslant \\bigg ( \\frac{ 2r}{1 + 2 (\\log (\\psi /\\gamma ))_+} \\vee 2 \\bigg ) \\big ( \\gamma \\vee \\psi \\big ), \\\\\\biggl \\Vert \\sum _i a_i \\big ( \\vert X_i \\vert ^2 - \\mathbb {E}\\vert X_i \\vert ^2 \\big ) \\biggr \\Vert _r & \\leqslant 2 \\bigg ( 1 + \\frac{2d}{N} \\bigg ) \\max _i \\vert a_i \\vert \\bigg ( \\frac{r}{d} \\vee \\sqrt{\\frac{r}{d}} \\bigg ), \\\\\\biggl \\Vert \\sum _{i\\ne j} b_{ij} X_iX_j \\biggr \\Vert _r & \\leqslant \\bigg ( \\frac{ 4r}{1 + (\\log (\\psi /\\gamma ))_+} \\vee 4 \\bigg ) ^2\\big ( \\gamma \\vee \\psi \\big ).", "$ The $L^r$ -norm bounds in Proposition REF induce bounds that hold with very high probability.", "Corollary 7.21 Fix $\\kappa \\in (0,1)$ .", "Let the assumptions of Proposition REF be satisfied.", "If $\\psi /\\gamma \\geqslant N^{\\kappa /4}$ then with very high probability $ \\biggl \\vert \\sum _i a_i X_i \\biggr \\vert \\leqslant \\mathcal {C}\\psi \\,, \\qquad \\biggl \\vert \\sum _{i \\ne j} b_{ij} X_iX_j \\biggr \\vert \\leqslant \\mathcal {C}\\psi \\,.$ Remark 7.22 Our proof of Corollary REF shows that $\\mathcal {C}$ can be chosen as a linear function of $\\nu $ for the first estimate of (REF ) and as a quadratic function of $\\nu $ for the second estimate of (REF ).", "Fix $\\nu \\geqslant 1$ .", "We choose $r= \\nu \\log N$ in (REF ) of Proposition REF and obtain from Cheybshev's inequality that $ \\mathbb {P}\\bigg ( \\biggl \\vert \\sum _i a_i X_i \\biggr \\vert > \\mathcal {C}\\psi \\bigg ) \\leqslant N^{-\\nu }, \\qquad \\mathcal {C}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =4e $as $ (0,1)$.Similarly, choosing $ r = 1 2 N$ in(\\ref {eq:LDB_quadratic}) yields$$\\mathbb {P}\\bigg ( \\biggl \\vert \\sum _{i \\ne j} b_{ij} X_i X_j \\biggr \\vert > 4 \\mathcal {C}\\psi \\bigg ) \\leqslant N^{-\\nu },\\qquad \\mathcal {C}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =16e22 2 .", "$$ Resolvent identities In this appendix we record some well-known identities for the Green function (REF ) and its minors from Definition REF .", "Lemma 7.23 (Ward identity) For $x \\notin T \\subset [N]$ we have, with the notation $\\eta = \\operatorname{Im}z$ , $\\sum _y^{(T)} \\vert G_{xy}^{(T)} \\vert ^2 = \\frac{1}{\\eta } \\operatorname{Im}G_{xx}^{(T)} \\,.$ This is a standard identity for resolvents, see e.g.", "[16].", "Lemma 7.24 Let $T \\subset [N]$ .", "For $x, y \\notin T$ and $x \\ne y$ , we have $ G_{xy}^{(T)} = - G_{yy}^{(T)} \\sum _{a}^{(Ty)} G_{xa}^{(T y)} {M_{ay}} = - G_{xx}^{(T)} \\sum _{b}^{(Tx)} {M_{xb}} G_{by}^{(Tx)} .$ For $x, y, a \\notin T$ and $x \\ne a \\ne y$ , we have $ G_{xy}^{(Ta)} = G_{xy}^{(T)} - \\frac{G_{xa}^{(T)} G_{ay}^{(T)}}{G_{aa}^{(T)}} .$ For any $x \\in [N]$ , we have $ \\frac{1}{G_{xx}} = M_{xx} - z- \\sum _{a,b}^{(x)} M_{xa} G_{ab}^{(x)} M_{bx}.$ All identities are standard and proved e.g.", "in [16]: (REF ) in [16], (REF ) in [16] and (REF ) in [16].", "We recall (REF ) and derive two expansions used in Section .", "For any $T \\subset [N]$ and $x,y, u \\notin T$ , $x \\ne u \\ne y$ , we have $G_{xy}^{(Tu)} = G_{xy}^{(T)} + \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} G_{uy}^{(T)} + \\frac{f}{N} G_{uy}^{(T)} \\sum _{a}^{(Tu)} G_{xa}^{(Tu)} ,$ which follows from (REF ) and (REF ).", "Under the same assumptions, applying (REF ) to (REF ) yields $\\begin{aligned}G_{xy}^{(Tu)} = \\, & \\phantom{-} G_{xy}^{(T)} -G_{uu}^{(T)} \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} \\sum _{b}^{(Tu)} H_{ub} G_{by}^{(Tu)} \\\\& - \\frac{f}{N} G_{uu}^{(T)} \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} \\sum _b^{(Tu)} G_{by}^{(Tu)} + \\frac{f}{N} G_{uy}^{(T)} \\sum _{a}^{(Tu)} G_{xa}^{(Tu)}.", "\\end{aligned}$ Stability estimate – proof of Lemma  REF In this appendix we prove Lemma REF .", "The estimate in [27] corresponding to (REF ) has logarithmic factors, which are not affordable for our purposes: they have to be replaced with constants.", "The following proof of Lemma REF is analogous to that of the more complicated bulk stability estimate from [9].", "[Proof of Lemma REF ] We introduce the vectors $g̑ \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(gx)x X$ and $ =(x)x X$.", "Moreover, with the abbreviation $ m =m(z)$ we introduce the constant vectors $ m̑ = (m)x X$ and $ ȇ =X -1/2 (1)x X$.", "We regard all vectors as column vectors.A simple computation starting from the difference of (\\ref {m_quadr}) and (\\ref {eq:self_consistent_eq_perturbed})reveals that\\begin{equation} B(g̑ - m̑) = m (g̑ - m̑) \\bigl (\\mathbf {e} \\mathbf {e}^* (g̑ - m̑)\\bigr ) - (g̑ - m̑) m - m^2 ,\\end{equation}where $ B =1 - m2 ȇ ȇ*$, and column vectors are multiplied entrywise.The inverse of $ B$ is$$ B^{-1} = 1 + \\frac{m^2}{1- m^2} \\mathbf {e} \\mathbf {e}^*.", "$$For a matrix $ R X X$, we write $ R $ for the operator norm induced by the norm $ ȓ = x X rx $ on $ X$.It is easy to see that there is $ c>0$, depending only on $$, such that $ 1- m(w)2 c$ for all $ w +$ satisfying $ Re w 2 - $.Hence, owing to $ ȇ ȇ* = 1$,we obtain$ B-1 1 + 1- m2 -1 1+ c-1$.Therefore, inverting $ B$ in (\\ref {eq:stability_equation}) and choosing $ b$, depending only on $$,sufficiently small to absorb the term quadratic in $ g̑ - m̑$ into the left-hand side of the resulting boundyields (\\ref {eq:stability_estimate}) for some sufficiently large $ C>0$, depending only on $$.This concludes the proof of Lemma~\\ref {lem:stability}.$ Instability estimate – proof of (REF ) In this appendix we prove (REF ), which shows that the self-consistent equation (REF ) is unstable with a logarithmic factor, which renders it useless for the analysis of sparse random graphs.", "More precisely, we show that the norm $\\Vert (I - m^2 S)^{-1} \\Vert _{\\infty \\rightarrow \\infty }$ is ill-behaved precisely in the situation where we need it.", "For simplicity, we replace $m^2$ with a phase $\\alpha ^{-1} \\in S^1$ separated from $\\pm 1$ , since for $\\operatorname{Re}z \\in \\mathcal {S}_\\kappa $ we have $ \\vert m(z) \\vert ^2 = 1 - O(\\operatorname{Im}z) \\,, \\qquad \\operatorname{Im}m(z) \\asymp 1\\,,$ by [33].", "Moreover, for definiteness, recalling that with very high probability most of the $d (1 + o(1))$ neighbours of any vertex in $\\mathcal {T}$ are again in $\\mathcal {T}$ , we assume that $S$ is the adjacency matrix of a $d$ -regular graph on $\\mathcal {T}$ divided by $d$ .", "By the spectral theorem and because $S$ is Hermitian, $\\Vert (\\alpha - S)^{-1} \\Vert _{2 \\rightarrow 2}$ is bounded, but, as we now show, the same does not apply to $\\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty }$ .", "Indeed, the upper bound of (REF ) follows from [34], and the lower bound from the following result.", "Lemma 7.25 (Instability of (REF )) Let $S$ be $1/d$ times the adjacency matrix of a graph whose restriction to the ball of radius $r \\in \\mathbb {N}^*$ around some distinguished vertex is a $d$ -regular tree.", "Let $\\alpha \\in S^1$ be an arbitrary phase.", "Then $ \\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty } \\geqslant c \\biggl (\\frac{r}{\\log r} \\wedge d\\biggr )$ for some universal constant $c > 0$ .", "In particular, denoting by $N$ the number of vertices in the tree (which may be completed to a $d$ -regular graph by connecting the leaves to each other), for $d \\asymp \\log N$ and $r \\asymp \\frac{\\log N}{\\log d}$ we find $ \\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty } \\geqslant \\frac{c \\log N}{(\\log \\log N)^2}\\,,$ which is the lower bound of (REF ).", "[Proof of Lemma REF ] After making $r$ smaller if needed, we may assume that $\\frac{r}{\\log r} \\leqslant d$ .", "We shall construct a vector $ȗ$ satisfying $\\Vert ȗ \\Vert _\\infty = 1$ and $\\Vert (\\alpha - S) ȗ \\Vert _\\infty = O\\bigl (\\frac{\\log r}{r}\\bigr )$ , from which (REF ) will follow.", "To that end, we construct the sequence $a_0, a_1, \\dots , a_r$ by setting $a_0 \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1 ,       a1 = ,       ak+1 =dd - 1 ak - 1d - 1 ak - 1    for    1 k r - 1 .", "A short transfer matrix analysis shows that $\\vert a_k \\vert \\leqslant \\mathrm {e}^{C_1 k /d}$ for some constant $C_1$ .", "Now choose $\\mu \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =C2 rr$ with $ C2 =2 2 C1$, and define $ bk =e-k ak$.", "Calling $ o$ the distinguished vertex, we define $ ux =bk$ if $ k = dist(o,x) r$ and $ ux = 0$ otherwise.", "It is now easy to check that $ (- S) ȗ = O(rr)$, by considering the cases $ k = 0$, $ 1 k r - 1$, and $ k r$ separately.", "The basic idea of the construction is that if $$ were zero, then $ (- S) ȗ$ would vanish exactly on $ Br - 1(o)$, but it would be large on the boundary $ Sr(o)$.", "The factor $ e-k$ introduces exponential decay in the radius which dampens the contribution of the boundary $ Sr(o)$ at the expense of introducing errors in the interior $ Br - 1(o)$.$ Johannes Alt (johannes.alt@unige.ch) Raphaël Ducatez (raphael.ducatez@unige.ch) Antti Knowles (antti.knowles@unige.ch) University of Geneva, Section of Mathematics, 2-4 Rue du Lièvre, 1211 Genève 4, Switzerland." ], [ "Appendices", "toc In the following appendices we collect various tools and explanations used throughout the paper." ], [ "Simulation of the $\\ell ^\\infty $ -norms of eigenvectors", "In Figure REF we depict a simulation of the $\\ell ^\\infty $ -norms of the eigenvectors of the adjacency matrix $A / \\sqrt{d}$ of the Erdős-Rényi graph $\\mathbb {G}(N,d/N)$ restricted to its giant component.", "We take $d = b \\log N$ with $N = 10^{\\prime }000$ and $b = 0.6$ .", "The eigenvalues and eigenvectors are drawn using a scatter plot, where the horizontal coordinate is the eigenvalue and the vertical coordinate the $\\ell ^\\infty $ -norm of the associated eigenvector.", "The higher a dot is located, the more localized the associated eigenvector is.", "Complete delocalization corresponds to a vertical coordinate $\\approx 0.01$ , and localization at a single site to a vertical coordinate 1.", "Note the semilocalization near the origin and outside of $[-2,2]$ .", "The two semilocalized blips around $\\pm 0.4$ are a finite-$N$ effect and tend to 0 as $N$ is increased.", "The Perron-Frobenius eigenvalue is an outlier near $2.8$ with delocalized eigenvector.", "Figure: A scatter plot of (λ,∥w̑∥ ∞ )(\\lambda , \\Vert w̑ \\Vert _\\infty ) for all eigenvalue-eigenvector pairs (λ,w̑)(\\lambda , w̑) of the adjacency matrix A/dA / \\sqrt{d} of the critical Erdős-Rényi graph restricted to its giant component, where N=10 ' 000N = 10^{\\prime }000 and d=0.6logNd = 0.6 \\log N." ], [ "Spectral analysis of the infinite rooted $(p,q)$ -regular tree", "In this appendix we describe the spectrum, eigenvectors, and spectral measure of the following simple graph.", "Definition 7.1 For $p,q \\in \\mathbb {N}^*$ we define $\\mathbb {T}_{p,q}$ as the infinite rooted $(p,q)$ -regular tree, whose root has $p$ children and all other vertices have $q$ children.", "A convenient way to analyse the adjacency matrix of $\\mathbb {T}_{p,q}$ is by tridiagonalizing it around its root.", "To that end, we first review the tridiagonalizationThe tridiagonalization algorithm that we use is the Lanczos algorithm.", "Tridiagonalizing matrices in numerical analysis and random matrix theory [26], [55] is usually performed using the numerically more stable Householder algorithm.", "However, when applied to the adjacency matrix $X = A$ of a graph, the Lanczos algorithm is more convenient because it can exploit the sparseness and local geometry of $A$ .", "of a general symmetric matrix $X \\in \\mathbb {R}^{N \\times N}$ around a vertex $x \\in [N]$ ; we refer to [10] for details.", "Let $r \\in \\mathbb {N}$ and $x \\in [N]$ .", "Suppose that the vectors $_x, X _x, X^2 _x, \\dots , X^r _x$ are linearly independent, and denote by $g̑_0, g̑_1, g̑_2, \\dots , g̑_r$ the associated orthonormalized sequence.", "Then the tridiagonalization of $X$ around $x$ up to radius $r$ is the $(r + 1) \\times (r+1)$ matrix $Z = (Z_{ij})_{i,j = 0}^r$ with $Z_{ij} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =g̑i 2mu, Z g̑j$.", "By construction, $ Z$ is tridiagonal and conjugate to $ X$ restricted to the subspace $ Span{g̑0, g̑1, ..., g̑r}$.$ Let now $X = A \\equiv A^{\\mathbb {T}_{p,q}}$ be the adjacency matrix of $\\mathbb {T}_{p,q}$ , whose root we denote by $o$ .", "Then it is easy to see that $g̑_i = _{S_i(o)} / \\Vert _{S_i(o)} \\Vert $ and the tridiagonalization of $A$ around the root up to radius $\\infty $ is the infinite matrix $\\sqrt{q} Z(p/q)$ , where $ Z(\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ = 0 0 1 1 0 1 1 0  .", "If $\\alpha > 2$ , a transfer matrix analysis (see [10]) shows that $Z(\\alpha )$ has precisely two eigenvalues in $\\mathbb {R}\\setminus [-2,2]$ , which are $\\pm \\Lambda (\\alpha )$ .", "The associated eigenvectors are $((\\pm )^i u_i)_{i \\in \\mathbb {N}}$ , where $u_0 > 0$ and $u_i \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(- 1)i/2   u0$ for $ i 1$.", "Note that the eigenvector components are exponentially decaying since $ > 2$, and hence $ u0$ can be chosen so that the eigenvectors are normalized.", "Going back to the original vertex basis of $ Tp,q$, setting $ = p/q$, we conclude that the adjacency matrix $ A$ has eigenvalues $ q ()$ with associated eigenvectors $ i N ()i ui Si(o) / Si(o) $.$ Next, we show that the measure $\\mu _\\alpha $ from (REF ) is the spectral measure at the root of $A^{\\mathbb {T}_{p,q}} / \\sqrt{d}$ and the spectral measure at 0 of (REF ).", "Lemma 7.2 For any $\\alpha \\geqslant 0$ the measure $\\mu _\\alpha $ is the spectral measure of $Z(\\alpha )$ at 0.", "For any $p,q \\in \\mathbb {N}^*$ the measure $\\mu _{p/q}$ is the spectral measure of the normalized adjacency operator $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ at the root.", "For REF , define the vector $ȇ_0 = (1,0,0,\\dots ) \\in \\ell ^2(\\mathbb {N})$ .", "The spectral measure of $Z(\\alpha )$ with respect to $ȇ_0$ is characterized by its Stieltjes transform $ \\bigl \\langle ȇ_0 {2mu}, (Z(\\alpha ) - z)^{-1} ȇ_0\\bigr \\rangle = \\frac{1}{- z - \\alpha \\bigl \\langle ȇ_0 {2mu}, (Z(1) - z)^{-1} ȇ_0\\bigr \\rangle }\\,.$ Here, we used Schur's complement formula on the Green function $(Z(\\alpha ) - z)^{-1}$ , observing that the minor of $Z(\\alpha )$ obtained by removing the zeroth row and column is $Z(1)$ .", "Setting $\\alpha = 1$ in (REF ) and recalling the defining relation (REF ) of the Stieltjes transform $m$ of the semicircle law, we conclude that $\\bigl \\langle ȇ_0 {2mu}, (Z(1) - z)^{-1} ȇ_0\\bigr \\rangle = m(z)$ and hence from (REF ) and (REF ) we get $\\bigl \\langle ȇ_0 {2mu}, (Z(\\alpha ) - z)^{-1} ȇ_0\\bigr \\rangle = m_\\alpha (z)$ , as desired.", "The proof of REF is analogous.", "Denote the root of $\\mathbb {T}_{p,q}$ by $o$ .", "Again using Schur's complement formula to remove the $o$ th row and column of $H = A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ , we deduce that $ \\bigl \\langle _o {2mu}, \\bigl (A^{\\mathbb {T}_{p,q}} / \\sqrt{q} - z\\bigr )^{-1} _o\\bigr \\rangle = \\biggl (-z - \\frac{p}{q} \\bigl \\langle _o {2mu}, \\bigl (A^{\\mathbb {T}_{q,q}} / \\sqrt{q} - z\\bigr )^{-1} _o\\bigr \\rangle \\biggr )^{-1}\\,,$ where we used that $\\mathbb {T}_{p,q}$ from which $o$ has been removed consists of $p$ disconnected copies of $\\mathbb {T}_{q,q}$ .", "Setting $p = q$ in (REF ) and comparing to (REF ) implies that the left-hand side of (REF ) is equal to $m(z)$ if $p = q$ , and hence REF for general $p$ follows from (REF ).", "Finally, we remark that the equality of the spectral measures of $Z(p/q)$ and $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ can also be seen directly, by noting that $Z(p/q)$ is the tridiagonalization of $A^{\\mathbb {T}_{p,q}} / \\sqrt{q}$ around the root $o$ .", "We conclude with some basic estimates for the Stieltjes transform $m_\\alpha $ of $\\mu _\\alpha $ used in Section .", "Lemma 7.3 For each $\\kappa >0$ there is a constant $C>0$ depending only on $\\kappa $ such that for all $z \\in S̑$ and all $\\alpha \\geqslant 0$ we have $ \\vert m_\\alpha (z) \\vert & \\leqslant C\\,, \\\\\\vert m_\\alpha (z) - m(z) \\vert & \\leqslant C \\vert \\alpha - 1 \\vert \\,.$ The simple facts follow directly from the corresponding properties of the semicircle law and its Stieltjes transform $m$ (see e.g.", "[16]).", "We leave the details to the reader." ], [ "Bounds on adjacency matrices of trees", "In this appendix we derive estimates on the operator norm of a tree.", "We start with a standard estimate on the operator norm of a graph.", "Lemma 7.4 Let $\\mathbb {T}$ be a graph whose vertices have degree at most $q+1$ for some $q \\geqslant 1$ .", "Then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant q+1$ and if in addition $\\mathbb {T}$ is a tree then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant 2 \\sqrt{q}$ .", "The first claim is obvious by the Schur test for the operator norm.", "To prove the second claim, choose a root $o$ and denote by $C_x$ the set of children of the vertex $x$ .", "Then for any vector $w̑ = (w_x)$ we have $\\bigl \\vert \\bigl \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\bigr \\rangle \\bigr \\vert = \\Biggl \\vert \\sum _{x,y} w_x A_{xy}^{\\mathbb {T}} w_y \\Biggr \\vert = 2 \\Biggl \\vert \\sum _x \\sum _{y \\in C_x} w_x w_y \\Biggr \\vert \\leqslant \\sum _{x} \\sum _{y \\in C_x} \\biggl (\\frac{1}{\\sqrt{q}} w_x^2 + \\sqrt{q} w_y^2\\biggr )\\\\\\leqslant \\frac{q+1}{\\sqrt{q}} w_o^2 + \\sum _{x \\ne o} \\biggl (\\frac{q}{\\sqrt{q}} w_x^2 + \\sqrt{q} w_x^2\\biggr ) \\leqslant 2 \\sqrt{q} \\sum _x w_x^2\\,,$ where in third step we used Young's inequality and in the fourth step that each vertex in the sum appears once as a child and at most $q$ times as a parent.", "This concludes the proof.", "The same proof shows that if $\\mathbb {T}$ is a rooted tree whose root has at most $p$ children and all other vertices at most $q$ children, then $\\Vert A^{\\mathbb {T}} \\Vert \\leqslant \\sqrt{q} (p/q \\vee 2)$ .", "This bound is sharp for $p \\leqslant 2q$ but not for $p > 2q$ .", "The sharp bound in the latter case is established in the following result.", "Lemma 7.5 Let $p,q \\in \\mathbb {N}^*$ .", "Let $\\mathbb {T}$ be a tree whose root has $p$ children and all the other vertices have at most $q$ children.", "Then the adjacency matrix $A^{\\mathbb {T}}$ of $\\mathbb {T}$ satisfies $\\Vert A^{\\mathbb {T}}\\Vert \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2)$ .", "Let $r \\in \\mathbb {N}$ and denote by $\\mathbb {T}_{p,q}(r)$ the rooted $(p,q)$ -regular tree of depth $r$ , whose root $x$ has $p$ children, all vertices at distance $1 \\leqslant i \\leqslant r$ from $x$ have $q$ children, and all vertices at distance $r+1$ from $x$ are leaves.", "For large enough $r$ , we can exhibit $\\mathbb {T}$ as a subgraph of $\\mathbb {T}_{p,q}(r)$ .", "By the Perron-Frobenius theorem, $ \\Vert A^{\\mathbb {T}} \\Vert = \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\rangle $ for the some normalized eigenvector $w̑$ whose entries are nonnegative.", "We extend $w̑$ to a vector indexed by the vertex set of $\\mathbb {T}_{p,q}(r)$ by setting $w_y = 0$ for $y$ not in the vertex set of $\\mathbb {T}$ .", "Clearly, $ \\langle w̑ {2mu}, A^{\\mathbb {T}} w̑\\rangle \\leqslant \\langle w̑ {2mu}, A^{\\mathbb {T}_{p,q}(r)} w̑\\rangle \\,.$ Abbreviating $A \\equiv A^{\\mathbb {T}_{p,q}(r)}$ , it therefore remains to estimate the right-hand side of (REF ) for large enough $r$ .", "To that end, we define $Z$ as the tridiagonalization of $A$ around the root up to radius $r$ (see Appendix REF ).", "The associated orthonormal set $g̑_0, g̑_1, \\dots , g̑_r$ is given by $g̑_i = _{S_i(x)}/\\Vert _{S_i(x)}\\Vert $ , and $Z = \\sqrt{q} Z_r(p/q)$ , where $Z_r(\\alpha )$ is the upper-left $(r+1) \\times (r+1)$ block of (REF ).", "We introduce the orthogonal projections $P_0 \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =g̑0 g̑0*$ and $ P =i = 0r g̑i g̑i*$.", "Clearly, $ P0 P = P0$ and hence $ (1 - P) (1 - P0) = 1 - P$.", "For large enough $ r$ the vectors $ g̑r$ and $ w̑$ have disjoint support, and hence $ (1 - P) A P w̑ = (1 - P) A i = 0r - 1 g̑i g̑i 2mu, w̑ = 0$,since $ A g̑i Span{g̑i-i, g̑i+1}$ for $ i < r$.", "Thus we have{\\begin{@align}{1}{-1}\\langle w̑ {2mu}, A w̑\\rangle &= \\langle w̑ {2mu}, PAP w̑\\rangle + \\langle w̑ {2mu}, (1 - P) A (1 - P) w̑\\rangle \\\\ &= \\langle w̑ {2mu}, PAP w̑\\rangle + \\langle w̑ {2mu}, (1 - P) (1 - P_0) A (1 - P_0) (1 - P) w̑\\rangle \\,.\\end{@align}}From \\cite [Appendices B and C]{ADK19} we find\\begin{equation}\\lim _{r \\rightarrow \\infty } \\Vert P A P \\Vert = \\lim _{r \\rightarrow \\infty } \\Vert Z \\Vert = \\sqrt{q} \\Lambda (p/q \\vee 2)\\,.\\end{equation}Moreover, the operator $ (1 - P0) A (1 - P0)$ is the adjacency matrix of a forest whose vertices have degree at most $ q$.", "By Lemma \\ref {lem:forest_bound}, we therefore obtain $ (1 - P0) A (1 - P0) 2 q$.From (\\ref {w_quad_est}) we therefore get\\begin{equation*}\\limsup _{r \\rightarrow \\infty } \\langle w̑ {2mu}, A w̑\\rangle \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2) \\Vert P w̑ \\Vert ^2 + 2 \\sqrt{q} \\Vert (1 - P) w̑ \\Vert ^2 \\leqslant \\sqrt{q} \\Lambda (p/q \\vee 2) \\Vert w̑ \\Vert ^2\\,.\\end{equation*}By (\\ref {PF_quad1}) and (\\ref {PF_quad2}), the proof is complete.$" ], [ "Degree distribution and number of resonant vertices", "In this appendix we record some basic facts about the distribution of degrees of the graph $\\mathbb {G}(N,d/N)$ , and use them to estimate the number of resonant vertices $\\mathcal {W}_{\\lambda , \\delta }$ .", "The following is a quantitative version of the Poisson approximation of a binomial random variable.", "Lemma 7.6 (Poisson approximation) If $D$ is a random variable with law $\\operatorname{Binom}(n,p)$ then for $k\\leqslant \\sqrt{n}$ and $p \\leqslant 1 / \\sqrt{n}$ we have $\\mathbb {P}(D = k) = \\frac{(pn)^k}{k!}", "\\mathrm {e}^{-pn} \\biggl (1+O\\biggl (\\frac{k^2}{n} + p^2 n\\biggr )\\biggr )\\,.$ Plugging the estimates $(1-p)^{n-k}= \\mathrm {e}^{(n-k)\\log (1-p)} = \\mathrm {e}^{-np + O(pk + p^2n)}$ and $\\frac{n!}{(n-k)!}", "= n^k\\prod _{i=0}^{k-1}\\biggl (1-\\frac{i}{n}\\biggr )=n^k \\mathrm {e}^{\\sum _{i=0}^{k-1} \\log \\bigl (1-\\frac{i}{n}\\bigr )}= n^k \\mathrm {e}^{O \\bigl (\\frac{k^2}{n}\\bigr )}\\,,$ into $\\mathbb {P}(D_x = k) = \\frac{n!}{k!", "(n-k)!}", "p^k (1-p)^{n-k}$ yields the claim, since $pk \\leqslant k^2/n + p^2 n$ .", "Lemma 7.7 For $\\mathbb {G}(N,d/N)$ we have $\\alpha _x \\leqslant \\mathcal {C}\\bigl (1 + \\frac{ \\log N}{d}\\bigr )$ with very high probability.", "This is a simple application of Bennett's inequality; see [10] for details.", "Next, we recall some standard facts about the distribution of the degrees.", "Define the function $f_d : [1,\\infty ) \\rightarrow \\big [\\frac{1}{2} \\log (2 \\pi d), \\infty \\big )$ through $ f_d(\\alpha ) \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =d( - + 1) + 12 (2 d)  , which is bijective and increasing.", "For its interpretation, we note that if $Y \\overset{\\mathrm {d}}{=}\\operatorname{Poisson}(d)$ then by Stirling's formula we have $\\mathbb {P}(Y = k) = \\exp \\bigl (-f_d(k/d) + O \\bigl (\\frac{1}{k}\\bigr )\\bigr )$ for any $k \\in \\mathbb {N}$ .", "There is a universal constant $C > 0$ such that for $1 \\leqslant l \\leqslant \\frac{N}{C \\sqrt{d}}$ the equation $f_d(\\beta ) = \\log (N/l)$ has a unique solution $\\beta \\equiv \\beta _l(d)$ .", "The interpretation of $\\beta _l(d)$ is the typical location of $\\alpha _{\\sigma (l)}$ .", "By the implicit function theorem, we find that $d \\mapsto \\beta _l(d)$ on the interval $\\bigl (0, \\frac{N^2}{C l^2}\\bigr ]$ is a decreasing bijective function.", "Definition 7.8 An event $\\Xi \\equiv \\Xi _N$ holds with high probability if $\\mathbb {P}(\\Xi ) = 1 - o(1)$ .", "The following result is a slight generalization of [10], which can be established with the same proof.", "We note that the qualitative notion of high probability can be made stronger and quantitative with some extra effort, which we however refrain from doing here.", "Lemma 7.9 If $d \\geqslant 1$ and $l \\geqslant 1$ satisfies $\\beta _l(d) \\geqslant 3/2$ then $ \\vert \\alpha _{\\sigma (l)} - \\beta _l(d) \\vert \\leqslant \\frac{1 \\vee (\\zeta / \\log \\beta _l(d))}{d}$ with high probability, where $\\zeta $ is any sequence tending to infinity with $N$ .", "The following resultThe assumption $d \\gg \\log \\log N$ in Lemma REF is tailored so that it covers the entire range $\\alpha \\geqslant 2$ , which is what we need in this paper.", "The assumption on $d$ could also be removed at the expense of introducing a nontrivial lower bound on $\\alpha $ .", "gives bounds on the counting function of the normalized degrees $(\\alpha _x)_{x \\in [N]}$ .", "Lemma 7.10 Suppose that $\\zeta $ satisfies $ 1 \\ll \\zeta \\leqslant \\frac{d}{C \\log \\log N}$ for some large enough universal constant $C$ .", "Then for any $\\alpha \\geqslant 2$ we have with high probability $ \\big \\lfloor (N \\mathrm {e}^{-f_d(\\alpha )} - 1) (\\log N)^{-2 \\zeta } \\big \\rfloor \\leqslant \\vert \\vert \\leqslant \\big \\lceil (N \\mathrm {e}^{-f_d(\\alpha )} + 1) (\\log N)^{2 \\zeta } \\big \\rceil \\,.$ If $d > 3 \\log N$ , then an elementary analysis using Bennett's inequality shows that $\\vert \\vert = 0$ with high probability.", "Since $N \\mathrm {e}^{-f_d(\\alpha )} \\leqslant 1$ for $\\alpha \\geqslant 2$ , the claim follows.", "Thus, for the following we assume that $d \\leqslant 3 \\log N$ .", "Abbreviate $\\Upsilon \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =32 d$, which is an upper bound for the right-hand side of (\\ref {deg_est_1}).For the following we adopt the convention that $ 0(d) = $.", "Choose $ l 0$ such that\\begin{equation} \\beta _{l+1}(d) < \\alpha \\leqslant \\beta _l(d)\\,,\\end{equation}and define\\begin{equation*}\\underline{k} \\!\\, \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\big \\lfloor l (\\log N)^{-2\\zeta } \\big \\rfloor \\,, \\qquad \\overline{k} \\!\\, \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(l+1) (N)2  .", "We shall show that $ \\beta _{\\underline{k} \\!\\,}(d) - \\Upsilon \\geqslant \\beta _l(d)$ for $\\underline{k} \\!\\, \\geqslant 1$ , $ \\beta _{\\overline{k} \\!\\,}(d) + \\Upsilon \\leqslant \\beta _{l+1}(d)\\,,$ and $ \\overline{k} \\!\\, \\leqslant N \\mathrm {e}^{-f_d(3/2)}\\,.$ Thus $\\beta _{\\overline{k} \\!\\,}(d) \\geqslant 3/2$ and, assuming $\\underline{k} \\!\\, \\geqslant 1$ , Lemma REF is applicable to the indices $\\overline{k} \\!\\,$ and $\\underline{k} \\!\\,$ .", "We obtain, with high probability, $\\alpha _{\\sigma (\\overline{k} \\!\\,)} \\leqslant \\beta _{\\overline{k} \\!\\,}(d) + \\Upsilon \\leqslant \\beta _{l+1}(d) \\leqslant \\alpha \\leqslant \\beta _l(d) \\leqslant \\beta _{\\underline{k} \\!\\,}(d) - \\Upsilon \\leqslant \\alpha _{\\sigma (\\underline{k} \\!\\,)}\\,,$ from which we deduce that $ \\underline{k} \\!\\, \\leqslant \\vert \\vert \\leqslant \\overline{k} \\!\\,\\,,$ which also holds trivially also for the case $\\underline{k} \\!\\, = 0$ .", "By applying the function $f_d$ to () we obtain $l \\leqslant N \\mathrm {e}^{-f_d(\\alpha )} \\leqslant l+1$ , so that (REF ) yields (REF ).", "Next, we verify (REF ).", "We consider the cases $l = 0$ and $l \\geqslant 1$ separately.", "If $l = 0$ then, by the definition of $\\beta _{\\overline{k} \\!\\,}(d)$ , for (REF ) we require $(\\log N)^{2 \\zeta } + 1 \\leqslant N \\mathrm {e}^{-f_d(3/2)}$ , which holds by the assumption $d \\leqslant 3 \\log N$ and the upper bound on $\\zeta $ .", "Let us therefore suppose that $l \\geqslant 1$ .", "By (), $\\alpha \\geqslant 2$ , and the definition of $\\beta _l(d)$ , we have $l \\leqslant N \\mathrm {e}^{-f_d(2)}$ , and we have to ensure that $(l+2) (\\log N)^{2 \\zeta } \\leqslant N \\mathrm {e}^{-f_d(3/2)}$ .", "Since $l \\geqslant 1$ , this is satisfied provided that $3 \\mathrm {e}^{-f_d(2)} (\\log N)^{2 \\zeta } \\leqslant \\mathrm {e}^{-f_d(3/2)}$ , which holds provided that $f_d(2) - f_d(3/2) \\geqslant 3 \\zeta \\log \\log N$ .", "This inequality is true because $f_d(2) - f_d(3/2) \\geqslant f^{\\prime }_d(3/2) /2 \\geqslant d/C$ , where we used that $f_d^{\\prime }(\\alpha ) = d \\log \\alpha + \\frac{1}{2 \\alpha }$ .", "What remains, therefore, is the proof of (REF ) and (REF ).", "We begin with the proof of (REF ).", "We get from the mean value theorem that $ \\beta _{\\underline{k} \\!\\,}(d) - \\beta _l(d) = f_d^{-1}\\biggl (\\log \\biggl (\\frac{N}{\\underline{k} \\!\\,}\\biggr )\\biggr ) - f_d^{-1}\\biggl (\\log \\biggl (\\frac{N}{l}\\biggr )\\biggr )\\geqslant \\frac{3}{4 d \\log \\beta _{\\underline{k} \\!\\,}(d)} \\log \\biggl (\\frac{l}{\\underline{k} \\!\\,}\\biggr )\\,.$ The right-hand side of (REF ) is bounded from below by $\\Upsilon $ provided that $ \\log \\biggl (\\frac{l}{\\underline{k} \\!\\,}\\biggr ) \\geqslant 2 \\zeta \\log \\beta _{\\underline{k} \\!\\,}(d)\\,.$ We estimate $\\beta _{\\underline{k} \\!\\,}(d) \\leqslant \\beta _1(d)$ using the elementary bound $f_d(\\beta ) \\geqslant \\frac{d}{10} \\beta $ for $\\beta \\geqslant 2$ , which yields $\\log N = f_d(\\beta _1(d)) \\geqslant \\frac{d}{10} \\beta _1(d)$ .", "By assumption on $d$ we therefore get $ \\beta _1(d) \\leqslant \\log N\\,.$ Thus, (REF ) holds by $\\underline{k} \\!\\, \\leqslant l / (\\log N)^{2 \\zeta }$ .", "This concludes the proof of (REF ).", "Next, we prove (REF ).", "As in (REF ), we find $ \\beta _{l + 1}(d) - \\beta _{\\overline{k} \\!\\,}(d) = f_{d}^{-1}\\biggl (\\log \\biggl (\\frac{N}{l+1}\\biggr )\\biggr ) - f_{d}^{-1}\\biggl (\\log \\biggl (\\frac{N}{\\overline{k} \\!\\,}\\biggr )\\biggr )\\geqslant \\frac{3}{4 d \\log \\beta _{l+1}(d)} \\log \\biggl (\\frac{\\overline{k} \\!\\,}{l+1}\\biggr )\\,.$ Together with $\\beta _{l+1}(d) \\leqslant \\beta _1(d) \\leqslant \\log N$ from (REF ), we deduce that the right-hand side of (REF ) is bounded from below by $\\Upsilon $ provided that $\\log \\bigl (\\frac{\\overline{k} \\!\\,}{l+1}\\bigr ) \\geqslant 2 \\zeta \\log \\log N$ , which is true by definition of $\\overline{k} \\!\\,$ .", "This concludes the proof of (REF ).", "The following result follows easily from Lemma REF .", "Recall the definition (REF ) of the exponent $\\theta _b(\\alpha )$ .", "Corollary 7.11 Suppose that $\\zeta $ satisfies (REF ).", "Write $d = b \\log N$ .", "Then for any $\\alpha \\geqslant 2$ we have $\\vert \\vert \\vee 1 = N^{\\theta _b(\\alpha ) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )$ with high probability.", "Using the exponent $\\theta _b(\\alpha )$ from (REF ) and $\\alpha _{\\max }(b)$ defined below it, we may state the following estimate on the density of the normalized degrees and the number of resonant vertices.", "Lemma 7.12 The following holds for a large enough universal constant $C$ .", "Suppose that $\\zeta $ satisfies (REF ).", "Write $d = b \\log N$ .", "For $2 \\leqslant \\alpha < \\beta \\leqslant \\alpha _{\\max }(b)$ satisfying $\\beta - \\alpha \\geqslant C \\frac{\\zeta \\log \\log N}{d \\log \\alpha }$ , with high probability we have $ \\vert \\vert = N^{\\theta _b(\\alpha ) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )\\,.$ For $\\delta \\geqslant C \\frac{\\zeta \\log \\log N}{d}$ and $2 + \\delta \\leqslant \\lambda \\leqslant \\Lambda (\\alpha _{\\max }(b))$ , with high probability we have $\\vert \\mathcal {W}_{\\lambda ,\\delta } \\vert = N^{\\theta _b(\\Lambda ^{-1}(\\lambda - \\delta )) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\biggl (\\frac{\\zeta \\log \\log N}{\\log N}\\biggr )\\,.$ Note that, since $\\xi \\geqslant d^{-1/2}$ , if the conclusion of Theorem REF is nontrivial then $\\delta \\geqslant d^{-1/2}$ , and hence the assumption on $\\delta $ in Lemma REF REF is automatically satisfied for suitably chosen $\\zeta $ .", "[Proof of Lemma REF ] Part REF follows Corollary REF below by noting that the assumption on $\\beta $ implies $\\theta _b(\\alpha ) - \\theta _b(\\beta ) \\geqslant C \\frac{\\zeta \\log \\log N}{\\log N}$ by the mean value theorem.", "Part REF follows from Part REF , using that $\\log (\\lambda - \\delta ) \\geqslant \\log 2$ , that $\\Lambda ^{\\prime }$ is bounded on $[2,\\infty )$ , and the mean value theorem.", "Corollary 7.13 The following holds for large enough universal constants $C, \\mathcal {C}$ .", "Suppose that (REF ) holds.", "Write $d = b \\log N$ .", "Let $w̑ = (w_x)_{x \\in [N]}$ be a normalized eigenvector of $A/\\sqrt{d}$ with nontrivial eigenvalue $2+\\mathcal {C} \\xi ^{1/2} \\leqslant \\lambda \\leqslant \\Lambda (\\alpha _{\\max }(b))$ .", "Then with high probability for any $2 \\leqslant p \\leqslant \\infty $ we have $\\Vert w̑ \\Vert _p^{2} \\geqslant N^{(2/p - 1)\\theta _b(\\Lambda ^{-1}(\\lambda )) + \\varepsilon }\\,, \\qquad \\varepsilon = O \\Biggl [ \\frac{\\log \\log N}{\\sqrt{\\log N}} + b (\\log \\lambda ) \\biggl (\\lambda + \\frac{1}{\\sqrt{\\lambda - 2}}\\biggr ) (\\xi + \\xi _{\\lambda - 2})\\Biggr ]\\,.$ We choose $\\delta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =C (+ - 2)$.", "Then by assumption on $$ we have $ (- 2)/2$, and hence Theorem \\ref {thm:localisation} yields, using that $ v̑(x)$ is supported in $ Br(x)$, $ x W, y Br(x) wy2 12$with high probability.", "Thus, by Hölder^{\\prime }s inequality,\\begin{equation} \\Vert w̑ \\Vert _{p}^{2} \\geqslant \\frac{1}{2} \\Biggl (\\sum _{x \\in \\mathcal {W}_{\\lambda ,\\delta }} \\vert B_{r_{\\star }}(x) \\vert \\Biggr )^{2/p - 1} \\geqslant \\frac{1}{2} \\Bigl (\\vert \\mathcal {W}_{\\lambda ,\\delta } \\vert N^{C \\log \\log N / \\sqrt{\\log N}}\\Bigr )^{2/p - 1}\\end{equation}with high probability, where we used Lemma \\ref {lem:upper_bound_degrees} to estimate $ x [N]Br(x) NC N / N$ with high probability.$ Next, using the mean value theorem and elementary estimates on the derivatives of $\\theta _b$ and $\\Lambda ^{-1}$ , we estimate $\\theta _b(\\Lambda ^{-1}(\\lambda - \\delta )) - \\theta _b(\\Lambda ^{-1}(\\lambda )) \\leqslant C b (\\log \\lambda ) \\biggl (\\lambda + \\frac{1}{\\sqrt{\\lambda - 2}}\\biggr ) \\delta \\,.$ Invoking Lemma REF REF with $\\zeta \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =N$, and recalling (\\ref {w_infty_lower_est}), therefore yields the claim.$" ], [ "Connected components of $\\mathbb {G}(N,d/N)$", "In this appendix we give some basic estimates on the sizes of connected components of $\\mathbb {G}(N,d/N)$ .", "These are needed for the analysis of the tuning forks in Appendix REF below.", "The arguments are standard and are tailored to work well in the regime $1 \\ll d \\leqslant \\log N$ that we are interested in.", "For smaller values of $d$ , see e.g.", "[17].", "Lemma 7.14 Let $W_k$ be the number of connected components that have $k$ vertices and $\\widehat{W}_k$ the number of connected components that have $k$ vertices and are not a tree.", "Then for $k \\leqslant N/2$ we have $\\mathbb {E}[W_k] \\leqslant N \\mathrm {e}^{-k (d/2 - \\log d - 1)}\\,, \\qquad \\mathbb {E}[\\widehat{W}_k] \\leqslant \\mathrm {e}^{-k (d/2 - \\log d - 1)}\\,.$ For a set $X \\subset [N]$ , denote by $\\mathcal {T}(X)$ the set of spanning trees of $X$ .", "If $X$ is a connected component of $\\mathbb {G}$ then there exists $\\mathbb {T} \\in \\mathcal {T}(X)$ a subgraph of $\\mathbb {G}$ such that no vertex of $X$ is connected to a vertex of $[N] \\setminus X$ .", "Hence, $W_k \\leqslant \\sum _{X \\subset [N]} \\mathbb {1}_{\\vert X \\vert = k} \\sum _{\\mathbb {T} \\in \\mathcal {T}(X)} \\mathbb {1}_{\\mathbb {T} \\subset \\mathbb {G}} \\prod _{x \\in X} \\prod _{y \\in [N] \\setminus X} (1 - A_{xy})\\,.$ Taking the expectation now easily yields the claim, using $\\vert \\mathcal {T}(X) \\vert = \\vert X \\vert ^{\\vert X \\vert - 2}$ by Cayley's theorem, that a tree on $k$ vertices has $k - 1$ edges, Stirling's approximation, and $1 - x \\leqslant \\mathrm {e}^{-x}$ .", "The argument to estimate $\\widehat{W}_k$ is similar, noting that in addition to a spanning tree $\\mathbb {T}$ of $X$ , we also have to have at least one edge not in $\\mathbb {T}$ connecting two vertices of $X$ .", "Thus, $\\widehat{W}_k \\leqslant \\sum _{X \\subset [N]} \\mathbb {1}_{\\vert X \\vert = k} \\sum _{\\mathbb {T} \\in \\mathcal {T}(X)} \\mathbb {1}_{\\mathbb {T} \\subset \\mathbb {G}} \\prod _{x \\in X} \\prod _{y \\in [N] \\setminus X} (1 - A_{xy}) \\sum _{\\lbrace u,v\\rbrace \\in X^2 \\setminus E(\\mathbb {T})} A_{uv}\\,,$ and we may estimate the expectation as before.", "We call a connected component of $\\mathbb {G}$ small if it is not the giant component.", "For the following statement we recall the definition of high probability from Definition REF .", "Corollary 7.15 Suppose that $d \\gg 1$ .", "All small components of $\\mathbb {G}$ have at most $O\\bigl (\\frac{\\log N}{d}\\bigr )$ vertices with very high probability.", "All small components of $\\mathbb {G}$ are trees with high probability.", "The giant component of $\\mathbb {G}$ has at least $N (1 - \\mathrm {e}^{-d/4})$ vertices with high probability.", "Any small component has at most $N/2$ vertices.", "Using Lemma REF we therefore get that the probability that there exists a small component with at least $K$ vertices is bounded by $\\mathbb {P}(\\exists k \\in [K,N/2] \\,, W_k \\geqslant 1) \\leqslant \\sum _{k = K}^{N/2} \\mathbb {E}[W_k] \\leqslant 2 N \\mathrm {e}^{-K (d/2 - \\log d - 1)}\\,,$ by summing the geometric series.", "Since $d/2 - \\log d - 1 \\geqslant c d$ for some universal constant $c$ , we obtain the first claim.", "To obtain the second claim, we use Lemma REF to estimate the probability that there exists a small component that is not a tree by $\\sum _{k = 1}^{N/2} \\mathbb {E}{\\widehat{W}_k} \\leqslant \\mathrm {e}^{-d/3}$ .", "To obtain the last claim, we estimate the expected number of vertices in small components by $\\mathbb {E}\\bigl [\\sum _{k = 1}^{N/2} k W_k\\bigr ] \\leqslant N \\sum _{k = 1}^\\infty k \\mathrm {e}^{-k (d/2 - \\log d - 1)} \\leqslant C N \\mathrm {e}^{- d/3}$ using Lemma REF , and the third claim follows from Chebyshev's inequality.", "We may now estimate the adjacency matrix on the small components of $\\mathbb {G}(N,d/N)$ .", "The following result follows immediately from Corollary REF and Lemma REF .", "Corollary 7.16 Suppose that $d \\gg 1$ .", "Then the operator norm of $A / \\sqrt{d}$ restricted to the small components of $\\mathbb {G}$ is bounded by $O\\bigl (\\frac{\\sqrt{\\log N}}{d}\\bigr )$ with high probability.", "Corollary REF makes it explicit that Theorem REF excludes all eigenvectors on small components of $\\mathbb {G}$ , whose eigenvalues lie outside $\\mathcal {S}_\\kappa $ precisely under the lower bound from (REF )." ], [ "Tuning forks and proof of Lemma ", "In this appendix we give a precise definition of the $D$ -tuning forks from Section REF and prove Lemma REF .", "Definition 7.17 A star of degree $D \\in \\mathbb {N}$ consists of a vertex, the hub, and $D$ leaves adjacent to the hub, the spokes.", "A star tuning fork of degree $D$ is obtained by taking two disjoint stars of degree $D$ along with an additional vertex, the base, and connecting both hubs to the base.", "We say that a star tuning fork is rooted in a graph $\\mathbb {H}$ if it is a subgraph of $\\mathbb {H}$ in which both hubs have degree $D+1$ and all spokes are leaves.", "Lemma 7.18 If a star tuning fork of degree $D$ is rooted in some graph $\\mathbb {H}$ , then the adjacency matrix of $\\mathbb {H}$ has eigenvalues $\\pm \\sqrt{D}$ with corresponding eigenvectors supported on the stars of the tuning fork, i.e.", "on $2D + 2$ vertices.", "Suppose first that $D \\geqslant 1$ .", "Note first that the adjacency matrix of a star of degree $D$ has rank two and has the two nonzero eigenvalues $\\pm \\sqrt{D}$ , with associated eigenvector equal to $\\pm \\sqrt{D}$ at the hub and 1 at the spokes.", "Now take a star tuning fork of degree $D$ rooted in a graph $\\mathbb {H}$ .", "Define a vector on the vertex set of $\\mathbb {H}$ by setting it to be $\\pm \\sqrt{D}$ at the hub of the first star, 1 at the spokes of the first star, $\\mp \\sqrt{D}$ at the hub of the second star, $-1$ at the spokes of the second star, and 0 everywhere else.", "Then it is easy to check that this vector is an eigenvector of the adjacency matrix of $\\mathbb {H}$ with eigenvalue $\\pm \\sqrt{D}$ .", "If $D = 0$ the construction is analogous, defining the vector to be $+1$ at one hub and $-1$ at the other.", "We recall from Section REF that $F(d,D)$ denotes the number of star tuning forks of degree $D$ rooted in $\\mathbb {G}_{\\mathrm {giant}}$ .", "Lemma 7.19 Suppose that $1 \\ll d \\ll \\sqrt{N}$ and $0 \\leqslant D \\ll \\sqrt{N}$ .", "Then $ \\mathbb {E}[F(d,D)] = \\frac{N d^2 \\mathrm {e}^{-2d}}{2 D!^2} (d \\mathrm {e}^{-d + 1})^{2D} (1 + o(1))$ and $\\mathbb {E}[F(d,D)^2] \\leqslant \\mathbb {E}[F(d,D)]^2 (1 + o(1))$ .", "[Proof of Lemma REF ] From Lemma REF we deduce that if $1 \\ll d = b \\log N = O(\\log N)$ and $D \\ll \\log N / \\log \\log N$ , then $\\mathbb {E}[F(d,D)] = N^{1 - 2b - 2b D + o(1)}$ .", "The claim then follows from the second moment estimate in Lemma REF and Chebyshev's inequality.", "[Proof of Lemma REF ] Let $x_1,x_2 \\in [N]$ be distinct vertices and $R_1, R_2 \\subset [N] \\setminus \\lbrace x_1,x_2\\rbrace $ be disjoint subsets of size $D$ .", "We abbreviate $U = (x_1, x_2, R_1, R_2)$ and sometimes identify $U$ with $\\lbrace x_1, x_2\\rbrace \\cup R_1 \\cup R_2$ .", "The family $U$ and a vertex $o \\in [N] \\setminus U$ define a star tuning fork of degree $D$ with base $o$ , hubs $x_1$ and $x_2$ , and associated spokes $R_1$ and $R_2$ .", "Let $C_k(\\mathbb {H})$ denote the vertex set of the $k$ th largest connected component of the graph $\\mathbb {H}$ .", "Then $F(d,D) = \\frac{1}{2}\\sum _U \\sum _{o \\in [N] \\setminus U} \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}$ , where $S_{o,U} \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =i = 12 (u Ri {o} Axi u u [N] (Ri {o}) (1 - Axi u) u Ri v [N] {xi} (1 - Auv)).", "The factor $\\frac{1}{2}$ corrects the overcounting from the labelling of the two stars.", "For disjoint deterministic $U$ , we split the random variables $A = (A^{\\prime }, A^{\\prime \\prime })$ into two independent families, where $A^{\\prime } \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(Auv u U or v U)$ and $ A” =(Auv u,v [N] U)$.", "Note that $ So,U$ is $ A'$-measurable.", "We define the event\\begin{equation*}\\Xi \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}\\end{equation*}=\\,,$ which is $A^{\\prime \\prime }$ -measurable.", "By Corollary REF and the assumption on $D$ , the event $\\Xi $ holds with high probability.", "Moreover, we have $\\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U} = \\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})} S_{o,U}$ , since the component of $o$ in $\\mathbb {G}$ and $\\mathbb {G} \\vert _{[N] \\setminus U}$ differ by $2D + 2$ vertices.", "Thus, for fixed $o \\in [N] \\setminus U$ , using the independence of $A^{\\prime }$ and $A^{\\prime \\prime }$ , we get $\\mathbb {E}[\\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}] &= \\mathbb {E}[\\mathbb {1}_{\\Xi } \\mathbb {1}_{o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})} S_{o,U}] + \\mathbb {E}[\\mathbb {1}_{\\Xi ^c} \\mathbb {1}_{o \\in C_1(\\mathbb {G})} S_{o,U}]\\\\&= \\mathbb {E}[S_{o,U}] \\bigl [\\mathbb {P}\\bigl (o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})\\bigr ) + O\\bigl (\\mathbb {P}(\\Xi ^c)\\bigr )\\bigr ]\\,.$ We have $\\mathbb {P}(\\Xi ^c) = o(1)$ and $\\mathbb {P}\\bigl (o \\in C_1(\\mathbb {G} \\vert _{[N] \\setminus U})\\bigr ) = 1 - o(1)$ by Corollary REF and the assumption on $D$ .", "Computing $\\mathbb {E}[S_{o,U}]$ and performing the sum over $o$ and $U$ , we therefore conclude that $\\mathbb {E}[F(d,D)] = \\frac{N (N - 1) \\cdots (N - 2D - 3 + 1)}{2 D!^2} \\biggl (\\frac{d}{N}\\biggr )^{2D + 2} \\biggl (1 - \\frac{d}{N}\\biggr )^{2 (N - D - 1) + 2 D (N - 1)} (1 + o(1))\\,,$ from which (REF ) follows.", "The estimate of the second moment is similar; one can even disregard the restriction to the giant component by estimating $\\mathbb {E}[F(d,D)^2] \\leqslant \\frac{1}{4} \\sum _{U, \\tilde{U}} \\sum _{o,\\tilde{o} \\in [N]} \\mathbb {E}[S_{o,U} S_{\\tilde{o}, \\tilde{U}}]$ ; we omit the details." ], [ "Multilinear large deviation bounds for sparse random vectors", "In this appendix we collect basic large deviation bounds for multilinear functions of sparse random vectors, which are proved in [42].", "The following result is proved in Propositions 3.1, 3.2, and 3.5 of [42].", "We denote by $\\Vert X \\Vert _r \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(EX r)1/r$ the $ Lr$-norm of a random variable $ X$.$ Proposition 7.20 Let $r$ be even and $1 \\leqslant d \\leqslant N$ .", "Let $X_1, \\ldots , X_N$ be independent random variables satisfying $ \\mathbb {E}X_i = 0, \\qquad \\mathbb {E}\\vert X_i \\vert ^k \\leqslant \\frac{1}{N d^{(k-2)/2}}$ for all $i \\in [N]$ and $2 \\leqslant k \\leqslant r$ .", "Let $a_i \\in and $ bij be deterministic for all $i,j \\in [N]$ .", "Suppose that $\\bigg (\\frac{1}{N} \\sum _i \\vert a_i \\vert ^2 \\bigg )^{1/2} \\leqslant \\gamma \\,, \\qquad \\frac{\\max _i \\vert a_i \\vert }{\\sqrt{d}} \\leqslant \\psi ,$ and $\\bigg ( \\max _i \\frac{1}{N} \\sum _{j} \\vert b_{ij} \\vert ^2 \\bigg )^{1/2} \\vee \\bigg ( \\max _j \\frac{1}{N}\\sum _i \\vert b_{ij} \\vert ^2 \\bigg )^{1/2} \\leqslant \\gamma , \\qquad \\frac{\\max _{i,j} \\vert b_{ij} \\vert }{d} \\leqslant \\psi $ for some $\\gamma , \\psi \\geqslant 0$ .", "Then $\\biggl \\Vert \\sum _i a_i X_i \\biggr \\Vert _r & \\leqslant \\bigg ( \\frac{ 2r}{1 + 2 (\\log (\\psi /\\gamma ))_+} \\vee 2 \\bigg ) \\big ( \\gamma \\vee \\psi \\big ), \\\\\\biggl \\Vert \\sum _i a_i \\big ( \\vert X_i \\vert ^2 - \\mathbb {E}\\vert X_i \\vert ^2 \\big ) \\biggr \\Vert _r & \\leqslant 2 \\bigg ( 1 + \\frac{2d}{N} \\bigg ) \\max _i \\vert a_i \\vert \\bigg ( \\frac{r}{d} \\vee \\sqrt{\\frac{r}{d}} \\bigg ), \\\\\\biggl \\Vert \\sum _{i\\ne j} b_{ij} X_iX_j \\biggr \\Vert _r & \\leqslant \\bigg ( \\frac{ 4r}{1 + (\\log (\\psi /\\gamma ))_+} \\vee 4 \\bigg ) ^2\\big ( \\gamma \\vee \\psi \\big ).", "$ The $L^r$ -norm bounds in Proposition REF induce bounds that hold with very high probability.", "Corollary 7.21 Fix $\\kappa \\in (0,1)$ .", "Let the assumptions of Proposition REF be satisfied.", "If $\\psi /\\gamma \\geqslant N^{\\kappa /4}$ then with very high probability $ \\biggl \\vert \\sum _i a_i X_i \\biggr \\vert \\leqslant \\mathcal {C}\\psi \\,, \\qquad \\biggl \\vert \\sum _{i \\ne j} b_{ij} X_iX_j \\biggr \\vert \\leqslant \\mathcal {C}\\psi \\,.$ Remark 7.22 Our proof of Corollary REF shows that $\\mathcal {C}$ can be chosen as a linear function of $\\nu $ for the first estimate of (REF ) and as a quadratic function of $\\nu $ for the second estimate of (REF ).", "Fix $\\nu \\geqslant 1$ .", "We choose $r= \\nu \\log N$ in (REF ) of Proposition REF and obtain from Cheybshev's inequality that $ \\mathbb {P}\\bigg ( \\biggl \\vert \\sum _i a_i X_i \\biggr \\vert > \\mathcal {C}\\psi \\bigg ) \\leqslant N^{-\\nu }, \\qquad \\mathcal {C}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =4e $as $ (0,1)$.Similarly, choosing $ r = 1 2 N$ in(\\ref {eq:LDB_quadratic}) yields$$\\mathbb {P}\\bigg ( \\biggl \\vert \\sum _{i \\ne j} b_{ij} X_i X_j \\biggr \\vert > 4 \\mathcal {C}\\psi \\bigg ) \\leqslant N^{-\\nu },\\qquad \\mathcal {C}\\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =16e22 2 .", "$$" ], [ "Resolvent identities", "In this appendix we record some well-known identities for the Green function (REF ) and its minors from Definition REF .", "Lemma 7.23 (Ward identity) For $x \\notin T \\subset [N]$ we have, with the notation $\\eta = \\operatorname{Im}z$ , $\\sum _y^{(T)} \\vert G_{xy}^{(T)} \\vert ^2 = \\frac{1}{\\eta } \\operatorname{Im}G_{xx}^{(T)} \\,.$ This is a standard identity for resolvents, see e.g.", "[16].", "Lemma 7.24 Let $T \\subset [N]$ .", "For $x, y \\notin T$ and $x \\ne y$ , we have $ G_{xy}^{(T)} = - G_{yy}^{(T)} \\sum _{a}^{(Ty)} G_{xa}^{(T y)} {M_{ay}} = - G_{xx}^{(T)} \\sum _{b}^{(Tx)} {M_{xb}} G_{by}^{(Tx)} .$ For $x, y, a \\notin T$ and $x \\ne a \\ne y$ , we have $ G_{xy}^{(Ta)} = G_{xy}^{(T)} - \\frac{G_{xa}^{(T)} G_{ay}^{(T)}}{G_{aa}^{(T)}} .$ For any $x \\in [N]$ , we have $ \\frac{1}{G_{xx}} = M_{xx} - z- \\sum _{a,b}^{(x)} M_{xa} G_{ab}^{(x)} M_{bx}.$ All identities are standard and proved e.g.", "in [16]: (REF ) in [16], (REF ) in [16] and (REF ) in [16].", "We recall (REF ) and derive two expansions used in Section .", "For any $T \\subset [N]$ and $x,y, u \\notin T$ , $x \\ne u \\ne y$ , we have $G_{xy}^{(Tu)} = G_{xy}^{(T)} + \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} G_{uy}^{(T)} + \\frac{f}{N} G_{uy}^{(T)} \\sum _{a}^{(Tu)} G_{xa}^{(Tu)} ,$ which follows from (REF ) and (REF ).", "Under the same assumptions, applying (REF ) to (REF ) yields $\\begin{aligned}G_{xy}^{(Tu)} = \\, & \\phantom{-} G_{xy}^{(T)} -G_{uu}^{(T)} \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} \\sum _{b}^{(Tu)} H_{ub} G_{by}^{(Tu)} \\\\& - \\frac{f}{N} G_{uu}^{(T)} \\sum _a^{(Tu)} G_{xa}^{(Tu)} H_{au} \\sum _b^{(Tu)} G_{by}^{(Tu)} + \\frac{f}{N} G_{uy}^{(T)} \\sum _{a}^{(Tu)} G_{xa}^{(Tu)}.", "\\end{aligned}$" ], [ "Stability estimate – proof of Lemma ", "In this appendix we prove Lemma REF .", "The estimate in [27] corresponding to (REF ) has logarithmic factors, which are not affordable for our purposes: they have to be replaced with constants.", "The following proof of Lemma REF is analogous to that of the more complicated bulk stability estimate from [9].", "[Proof of Lemma REF ] We introduce the vectors $g̑ \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =(gx)x X$ and $ =(x)x X$.", "Moreover, with the abbreviation $ m =m(z)$ we introduce the constant vectors $ m̑ = (m)x X$ and $ ȇ =X -1/2 (1)x X$.", "We regard all vectors as column vectors.A simple computation starting from the difference of (\\ref {m_quadr}) and (\\ref {eq:self_consistent_eq_perturbed})reveals that\\begin{equation} B(g̑ - m̑) = m (g̑ - m̑) \\bigl (\\mathbf {e} \\mathbf {e}^* (g̑ - m̑)\\bigr ) - (g̑ - m̑) m - m^2 ,\\end{equation}where $ B =1 - m2 ȇ ȇ*$, and column vectors are multiplied entrywise.The inverse of $ B$ is$$ B^{-1} = 1 + \\frac{m^2}{1- m^2} \\mathbf {e} \\mathbf {e}^*.", "$$For a matrix $ R X X$, we write $ R $ for the operator norm induced by the norm $ ȓ = x X rx $ on $ X$.It is easy to see that there is $ c>0$, depending only on $$, such that $ 1- m(w)2 c$ for all $ w +$ satisfying $ Re w 2 - $.Hence, owing to $ ȇ ȇ* = 1$,we obtain$ B-1 1 + 1- m2 -1 1+ c-1$.Therefore, inverting $ B$ in (\\ref {eq:stability_equation}) and choosing $ b$, depending only on $$,sufficiently small to absorb the term quadratic in $ g̑ - m̑$ into the left-hand side of the resulting boundyields (\\ref {eq:stability_estimate}) for some sufficiently large $ C>0$, depending only on $$.This concludes the proof of Lemma~\\ref {lem:stability}.$" ], [ "Instability estimate – proof of (", "In this appendix we prove (REF ), which shows that the self-consistent equation (REF ) is unstable with a logarithmic factor, which renders it useless for the analysis of sparse random graphs.", "More precisely, we show that the norm $\\Vert (I - m^2 S)^{-1} \\Vert _{\\infty \\rightarrow \\infty }$ is ill-behaved precisely in the situation where we need it.", "For simplicity, we replace $m^2$ with a phase $\\alpha ^{-1} \\in S^1$ separated from $\\pm 1$ , since for $\\operatorname{Re}z \\in \\mathcal {S}_\\kappa $ we have $ \\vert m(z) \\vert ^2 = 1 - O(\\operatorname{Im}z) \\,, \\qquad \\operatorname{Im}m(z) \\asymp 1\\,,$ by [33].", "Moreover, for definiteness, recalling that with very high probability most of the $d (1 + o(1))$ neighbours of any vertex in $\\mathcal {T}$ are again in $\\mathcal {T}$ , we assume that $S$ is the adjacency matrix of a $d$ -regular graph on $\\mathcal {T}$ divided by $d$ .", "By the spectral theorem and because $S$ is Hermitian, $\\Vert (\\alpha - S)^{-1} \\Vert _{2 \\rightarrow 2}$ is bounded, but, as we now show, the same does not apply to $\\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty }$ .", "Indeed, the upper bound of (REF ) follows from [34], and the lower bound from the following result.", "Lemma 7.25 (Instability of (REF )) Let $S$ be $1/d$ times the adjacency matrix of a graph whose restriction to the ball of radius $r \\in \\mathbb {N}^*$ around some distinguished vertex is a $d$ -regular tree.", "Let $\\alpha \\in S^1$ be an arbitrary phase.", "Then $ \\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty } \\geqslant c \\biggl (\\frac{r}{\\log r} \\wedge d\\biggr )$ for some universal constant $c > 0$ .", "In particular, denoting by $N$ the number of vertices in the tree (which may be completed to a $d$ -regular graph by connecting the leaves to each other), for $d \\asymp \\log N$ and $r \\asymp \\frac{\\log N}{\\log d}$ we find $ \\Vert (\\alpha - S)^{-1} \\Vert _{\\infty \\rightarrow \\infty } \\geqslant \\frac{c \\log N}{(\\log \\log N)^2}\\,,$ which is the lower bound of (REF ).", "[Proof of Lemma REF ] After making $r$ smaller if needed, we may assume that $\\frac{r}{\\log r} \\leqslant d$ .", "We shall construct a vector $ȗ$ satisfying $\\Vert ȗ \\Vert _\\infty = 1$ and $\\Vert (\\alpha - S) ȗ \\Vert _\\infty = O\\bigl (\\frac{\\log r}{r}\\bigr )$ , from which (REF ) will follow.", "To that end, we construct the sequence $a_0, a_1, \\dots , a_r$ by setting $a_0 \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =1 ,       a1 = ,       ak+1 =dd - 1 ak - 1d - 1 ak - 1    for    1 k r - 1 .", "A short transfer matrix analysis shows that $\\vert a_k \\vert \\leqslant \\mathrm {e}^{C_1 k /d}$ for some constant $C_1$ .", "Now choose $\\mu \\mathrel {\\hbox{\\scriptsize .", "}\\hbox{\\scriptsize .", "}}$ =C2 rr$ with $ C2 =2 2 C1$, and define $ bk =e-k ak$.", "Calling $ o$ the distinguished vertex, we define $ ux =bk$ if $ k = dist(o,x) r$ and $ ux = 0$ otherwise.", "It is now easy to check that $ (- S) ȗ = O(rr)$, by considering the cases $ k = 0$, $ 1 k r - 1$, and $ k r$ separately.", "The basic idea of the construction is that if $$ were zero, then $ (- S) ȗ$ would vanish exactly on $ Br - 1(o)$, but it would be large on the boundary $ Sr(o)$.", "The factor $ e-k$ introduces exponential decay in the radius which dampens the contribution of the boundary $ Sr(o)$ at the expense of introducing errors in the interior $ Br - 1(o)$.$ Johannes Alt (johannes.alt@unige.ch) Raphaël Ducatez (raphael.ducatez@unige.ch) Antti Knowles (antti.knowles@unige.ch) University of Geneva, Section of Mathematics, 2-4 Rue du Lièvre, 1211 Genève 4, Switzerland." ] ]

[ [ "Vertex-Faithful Regular Polyhedra" ], [ "Abstract We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a \"vertex-faithful\" polyhedron with the same number of vertices.", "We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set.", "In particular, we classify all regular polyhedra where the number of vertices is prime or twice a prime.", "We also construct the smallest regular polyhedra with a prime squared number of vertices." ], [ "Introduction", "In recent years, the study of maps on compact surfaces has seen much attention.", "Often maps are investigated by considering how their automorphism groups act on faces of a certain dimension.", "For instance, maps where the automorphism group acts transitively on the edges (edge-transitive maps) have been classified by Graver, Širáň, Tucker, and Watkins [13], [22].", "Regular maps — those with maximal symmetry, whose automorphism groups act transitively on flags (incident triples of vertices, edges, and facets) — have received the most consideration.", "A regular map that satisfies one more condition (called the diamond property) can be seen as an abstract regular polyhedron, and the study of abstract regular polytopes has a rich history of its own [19].", "In this paper, we will restrict our study to regular abstract polyhedra, and will be interested in both the action of the automorphism group on the vertices, and the prime factorization of the number of vertices.", "Regular maps with specific actions on their vertices have been considered before; for instance, in [15], where regular maps with quasiprimitive automorphism groups on vertices are discussed, and in [17], where regular maps with automorphism groups that do not act faithfully on their vertices, edges, or faces are analyzed.", "Similarly, the prime factorization of the number of vertices (or faces of another dimension) has also been utilized in previous works.", "In [2], regular maps with a prime number of faces are examined, and in [11] regular maps with simple underlying graphs whose order is the product of two primes are classified.", "Many of the maps in these classifications do not correspond to abstract polyhedra, and we find that the classification of regular polyhedra with a given number of vertices is somewhat tamer than corresponding classifications of maps.", "When examining polyhedra where the size of the vertex set has few prime factors, it is useful to understand the smallest examples fully.", "Thankfully, much is known about the small regular polyhedra, as those with up to 4000 flags have been enumerated [5].", "Using these data, we observed that there appeared to be only two regular polyhedra with $b$ vertices for each prime $b \\ge 5$ ; an observation which inspired this whole project.", "This paper is structured as follows.", "In Section  we provide some definitions and basic results about regular maps and regular abstract polyhedra.", "In particular, in Section  we will recall various results about flat regular polyhedra, and provide some basic results about regular polyhedra where the automorphism group acts faithfully on the vertices.", "In Section  we consider such polyhedra with few vertices.", "Then, utilizing these results, in Sections , , and , we consider regular polyhedra where the number of vertices is prime, twice a prime, or a prime squared, respectively.", "Specifically, we classify all regular polyhedra with either a prime number of vertices or twice a prime number of vertices.", "We also construct the smallest regular polyhedra with a prime squared number of vertices." ], [ "Abstract Polyhedra", "In what follows, we recall some definitions and results from the theory of abstract regular polytopes (see [19]).", "From here on out, we will omit the word abstract, and simply refer to polytopes and polyhedra.", "A polyhedron is a ranked poset $\\mathcal {P}$ with the following properties.", "The elements of $\\mathcal {P}$ are called faces, with ranks in $\\lbrace -1, \\ldots , 3\\rbrace $ .", "The poset $\\mathcal {P}$ has a unique face of rank -1, a unique face of rank 3, and the faces of ranks 0, 1, and 2 are called vertices, edges, and facets respectively.", "The maximal totally ordered subsets of $\\mathcal {P}$ are called flags, and each flag has exactly 5 faces, one for each rank.", "We say that two faces are incident if they are on the same flag, and we say that two flags are adjacent if they differ by exactly one face.", "In particular, we say that $\\Psi $ and $\\Phi $ are $i$ -adjacent if they differ exactly in a face of rank $i$ .", "Given any two flags $\\Psi $ and $\\Phi $ of $\\mathcal {P}$ , there is a sequence of flags $\\Psi = \\Psi _0, \\Psi _1, \\ldots , \\Psi _k = \\Phi $ , so that each flag in the sequence contains $\\Psi \\cap \\Phi $ and any two successive flags are adjacent.", "Additionally, whenever $F < G$ , and $\\textrm {rank}(F) = \\textrm {rank}(G)-2$ , there are exactly two faces $H$ such that $F < H < G$ ; this is called the diamond property.", "Given a vertex $v$ of a polyhedron, the collection of faces $F$ so that $v \\le F$ is incident to $v$ is called the vertex figure at $v$.", "The automorphism group $\\Gamma (\\mathcal {P})$ of a polyhedron is the group of rank-preserving automorphisms of the partially ordered set.", "When the size of the automorphism group is the same as the number of flags, all the flags will be in the same orbit, and the polyhedron is called regular.", "In fact, each regular polyhedron can be thought of as a regular map.", "Informally, a map $\\mathcal {P}$ is a family of finite polygons with the following four properties.", "Any two polygons of the map meet in a common edge or vertex, or do not meet at all.", "Each edge of the map belongs to exactly two polygons.", "The set of polygons containing a given vertex form a single cycle of polygons, where adjacent polygons in the cycle share a common edge.", "Finally, between any two polygons is a chain of adjacent polygons.", "Much is known about the automorphism groups of regular polyhedra.", "If we fix a base flag $\\Psi $ of $\\mathcal {P}$ , then the automorphism group $\\Gamma (\\mathcal {P})$ of a regular polyhedron $\\mathcal {P}$ is generated by the three involutions $\\rho _i$ (with $i \\in \\lbrace 0,1,2\\rbrace $ ), where $\\rho _i$ sends $\\Psi $ to the adjacent flag $\\Psi ^i$ differing in a face of rank $i$ .", "Thus, for example, the generator $\\rho _0$ sends the base vertex to the other vertex incident to the base edge and keeps the base face fixed.", "Any regular polyhedron $\\mathcal {P}$ has a (Schläfli) type $\\lbrace p,q\\rbrace $ where each facet is a polygon with $p$ vertices, and each vertex is incident to $q$ edges.", "In this case the automorphism group $\\Gamma (\\mathcal {P})$ is a smooth quotient of the string Coxeter group $[p,q]$ , and is called a string C-group of rank 3, where $[p,q] := \\langle a, b, c \\mid a^2 = b^2 = c^2 = (ab)^p = (bc)^q = (ac)^2 = 1 \\rangle .$ Each string C-group satisfies a particular intersection condition inherited from the Coxeter group.", "Namely, let $\\Gamma = \\langle \\rho _0, \\rho _1, \\rho _2 \\rangle $ be the automorphism group of a regular polyhedron, and let $\\Gamma _0 := \\langle \\rho _1, \\rho _2 \\rangle $ be the stabilizer of the base vertex, and $\\Gamma _2 := \\langle \\rho _0, \\rho _1 \\rangle $ be the stabilizer of the base facet.", "Then the intersection condition implies that $\\Gamma _0 \\cap \\Gamma _2 \\cong \\langle \\rho _1 \\rangle $ .", "In fact, every rank 3 string C-group is the automorphism group of a unique regular polyhedron (see Theorem.", "2E11 of [19] ).", "Due to this, we can refer to the (possibly infinite) universal regular polytope of type $\\lbrace p,q\\rbrace $ which has automorphism group the Coxeter group $[p,q]$ .", "Let $\\mathcal {P}$ and $\\mathcal {Q}$ be regular polyhedra.", "We say that $\\mathcal {P}$ covers $\\mathcal {Q}$ if there a surjective function $\\psi $ from $\\mathcal {P}$ to $\\mathcal {Q}$ that preserves incidence, rank, and has the property that if two flags of $\\mathcal {P}$ are $i$ -adjacent then so are their images under $\\psi $ .", "An isomorphism from a regular polyhedron $\\mathcal {P}$ to a regular polyhedron $\\mathcal {Q}$ is a bijection that preserves incidence and rank.", "If $\\mathcal {P}$ covers $\\mathcal {Q}$ , and $\\mathcal {P}$ is not isomorphic to $\\mathcal {Q}$ , we say that $\\mathcal {P}$ is a proper cover of $\\mathcal {Q}$ .", "Furthermore, $\\mathcal {P}$ covers $\\mathcal {Q}$ if and only if there is an epimorphism from $\\Gamma (\\mathcal {P})=\\langle \\rho _0, \\rho _1, \\rho _2 \\rangle $ to $\\Gamma (\\mathcal {Q})=\\langle r_0, r_1, r_2 \\rangle $ sending each $\\rho _i$ to $r_i$ .", "Let $\\Gamma (\\mathcal {P}) = \\langle \\rho _0, \\rho _1, \\rho _2 \\rangle $ be the automorphism group of a regular polyhedron.", "It will be useful to define the abstract rotations $\\sigma _1$ and $\\sigma _2$ , where $\\sigma _1 := \\rho _0 \\rho _1$ and $\\sigma _2 := \\rho _1 \\rho _2$ .", "The index of $\\langle \\sigma _1, \\sigma _2 \\rangle $ in $\\Gamma (\\mathcal {P})$ is at most 2.", "When the index is exactly 2, we say that $\\mathcal {P}$ is orientably regular; otherwise, when the index is 1, we say that $\\mathcal {P}$ is non-orientably regular.", "A regular polyhedron $\\mathcal {P}$ is orientably regular (or simply orientable) if and only if all identity words $w$ in terms of the generators $\\rho _i$ have even length.", "Thus $\\mathcal {P}$ is non-orientable when there is a trivial word $w$ in terms of the generators $\\rho _i$ which has odd length.", "To each polyhedron, we may associate its dual polyhedron $\\mathcal {P}^\\delta $ , which is constructed by reversing the partial order of $\\mathcal {P}$ .", "If $\\mathcal {P}^\\delta $ and $\\mathcal {P}$ are isomorphic, then we say that $\\mathcal {P}$ is self-dual.", "Similarly, to each regular polyhedron $\\mathcal {P}$ we may associate its Petrie dual (or more briefly Petrial) $\\mathcal {P}^\\pi $ .", "The Petrial has the same vertices and edges as the original polyhedron.", "The facets of $\\mathcal {P}^\\pi $ are the Petrie polygons of $\\mathcal {P}$ , defined so that exactly two successive edges of a Petrie polygon are edges of a facet of $\\mathcal {P}$ .", "Note that the Petrie dual of a polyhedron may fail to be a polyhedron.", "If $\\mathcal {P}$ is isomorphic to $\\mathcal {P}^\\pi $ , then $\\mathcal {P}$ is said to be self-Petrie.", "For a regular polyhedron, the length $r$ of a Petrie polygon is the order of the element $\\rho _0 \\rho _1 \\rho _2$ in its automorphism group.", "We refer to the (universal) regular polytope of type $\\lbrace p,q\\rbrace _r$ which has an automorphism group with the same presentation as the Coxeter group $[p,q]$ plus the additional relator $(\\rho _0 \\rho _1 \\rho _2)^r$ , denoted $[p,q]_r$ .", "For instance the (universal) polyhedron $\\lbrace 4,4\\rbrace _6$ is the regular polyhedron that has automorphism group with the following presentation $ \\langle a, b, c \\mid a^2 = b^2 = c^2 = (ab)^4 = (bc)^4 = (ac)^2 = (abc)^6= 1 \\rangle .$ We will later utilize the fact that the polytope $\\lbrace 4,4\\rbrace _{2s}$ is isomorphic to the toroidal map $\\lbrace 4,4\\rbrace _{(s,s)}$ (see [7] Section 8.6).", "Furthermore, the dual of the Petrial of the toroidal map $\\lbrace 4,4\\rbrace _{(s,0)}$ will play an important role, and it will be denoted $(\\lbrace 4,4\\rbrace _{(s,0)})^{\\pi \\delta }$ .", "Note that the Petrie polygons of $\\lbrace 4,4\\rbrace _{(s,0)}$ visit each vertex at most once, and so by Lemma 7B3 of [19], the Petrial of $\\lbrace 4, 4\\rbrace _{(s,0)}$ (and thus its dual) is in fact a polyhedron.", "Given a regular polyhedron $\\mathcal {P}$ of type $\\lbrace p,q\\rbrace $ , it is easy to see that $| \\Gamma (\\mathcal {P}) | = 4e = 2vq = 2fp$ where $e$ is the number of edges, $v$ is the number of vertices, and $f$ is the number of facets.", "Furthermore, $\\mathcal {P}$ has at least $p$ vertices, and so it has at least $2pq$ flags." ], [ "Flat regular polyhedra", "A regular polyhedron is said to be flat if every face is incident to every vertex.", "If $\\mathcal {P}$ is a regular polyhedron of type $\\lbrace p, q\\rbrace $ , then it is flat if and only if it is tight, which means that it has the minimum possible number of flags ($2pq$ ).", "Flat (tight) regular polyhedra were classified in [9].", "The non-orientable ones fall into a small number of infinite families.", "The orientable ones are somewhat more diverse.", "From Theorem 3.3 of [9], we know that every flat orientably regular polyhedron has an automorphism group of the form $ \\Lambda (p,q)_{i,j} = [p,q] / (\\sigma _2^{-1} \\sigma _1 = \\sigma _1^i \\sigma _2^j), $ with $i$ and $j$ taken modulo $p$ and $q$ , respectively.", "If we fix the number of vertices (which is equal to $p$ ), then we would like to know what values of $q$ , $i$ , and $j$ actually yield a flat polyhedron of type $\\lbrace p, q\\rbrace $ .", "Here we summarize the results from [9] that allow us to do so.", "Proposition 2.1 ([9] Proposition 4.2) Suppose $p,q \\ge 3$ and that $0 \\le i \\le p-1$ and $0 \\le j \\le q-1$ .", "Then $\\Lambda (p,q)_{i,j}$ is the automorphism group of a flat orientably regular polyhedron of type $\\lbrace p, q\\rbrace $ if and only if there are values $p^{\\prime }$ and $q^{\\prime }$ such that: $p^{\\prime }$ divides $p$ and $i+1$ , $q^{\\prime }$ divides $q$ and $j-1$ , The group $\\Lambda (p,q^{\\prime })_{i,1}$ is the automorphism group of a flat orientably regular polyhedron of type $\\lbrace p, q^{\\prime }\\rbrace $ , with $\\langle \\sigma _2 \\rangle $ core-free in $\\Lambda (p,q^{\\prime })_{i,1}$ , and The group $\\Lambda (p^{\\prime },q)_{-1,j}$ is the automorphism group of a flat orientably regular polyhedron of type $\\lbrace p^{\\prime }, q\\rbrace $ , with $\\langle \\sigma _1 \\rangle $ core-free in $\\Lambda (p^{\\prime },q)_{-1,j}$ .", "Proposition 2.2 The group $\\Lambda (p,q^{\\prime })_{i,1}$ is the automorphism group of a flat orientably regular polyhedron $\\mathcal {P}$ of type $\\lbrace p, q^{\\prime }\\rbrace $ , with $\\langle \\sigma _2 \\rangle $ core-free in $\\Lambda (p,q^{\\prime })_{i,1}$ , if and only if one of the following is true: $q^{\\prime } = 2$ and $i = -1$ , or $q^{\\prime }$ is odd, $p = 2q^{\\prime }$ , and $i = 3$ , or $q^{\\prime }$ is an even divisor of $p$ such that $\\gcd (p/q^{\\prime }, q^{\\prime })$ is a power of 2 If the maximal power of 2 that divides $p$ is $2^{\\alpha }$ , then the maximal power of 2 that divides $q^{\\prime }$ is either 2, 4, or $2^{\\alpha -1}$ , and it is only 4 is $\\alpha \\ge 3$ , and $i$ satisfies a particular system of congruences.", "In particular, if we write $p = 2^{\\alpha } p_1 p_2$ , where $p_1$ is coprime with $q^{\\prime }$ and $p_2$ divides $q^{\\prime }$ , and if $2^{\\beta }$ is the maximal power of 2 that divides $q^{\\prime }$ , then: $\\frac{1-i}{2} \\equiv -1$ (mod $p_2$ ), $\\frac{1-i}{2} \\equiv 1$ (mod $p_1$ ), If $\\beta = 1$ , then $\\frac{1-i}{2} \\equiv 1$ (mod $2^{\\alpha -1}$ ) If $\\beta = 2$ , then $\\frac{1-i}{2} \\equiv 2^{\\alpha -2}+1$ (mod $2^{\\alpha -1}$ ) If $\\beta = \\alpha -1$ , then $\\frac{1-i}{2} \\equiv 2^{\\alpha -2}-1$ or $-1$ (mod $2^{\\alpha -1}$ ) The proof of the previous proposition is contained in the contents of Section 4.1 of [9].", "Proposition REF also has a dual version, with the roles of $p$ and $q$ reversed and with the roles of $i$ and $-j$ reversed.", "We note that for tight orientable regular polyhedra, $\\langle \\sigma _2 \\rangle $ is core-free in $\\Lambda (p,q^{\\prime })_{i,1}$ if and only if $\\mathcal {P}$ has no multiple edges (that is, if no pair of vertices is incident to two or more edges)." ], [ "Group Actions and Permutation Representation Graphs", "In much of this paper we will be concerned with how the automorphism group of a regular polyhedron acts on its vertices.", "Here we give some of the required concepts about group actions and permutation representation graphs that we will use in our proofs.", "We will follow [10] for the results on permutation groups, and [20] for the notation about permutation representation graphs.", "Let $\\Gamma $ be the automorphism group of a finite regular polytope $\\mathcal {P}$ and let $\\Omega $ be a set of faces or flags of $\\mathcal {P}$ .", "Any homomorphism $f$ of $\\Gamma $ into the symmetric group $\\textrm {Sym}(\\Omega )$ is called a (permutation) representation of $\\Gamma $ on $\\Omega $ , and we will say that $\\Gamma $ acts on $\\Omega $ .", "The kernel of $f$ is called the kernel of the representation, and the representation is called faithful when the kernel is trivial; in that case, the image of $\\Gamma $ under $f$ is isomorphic to $\\Gamma $ .", "For each element $i \\in \\Omega $ and each element $g \\in \\Gamma $ , we denote the image of $i$ under $g$ by $i^g$ or $(i)g$ .", "(The former notation is standard, but since our group elements $g$ often involve superscripts, the latter notation will be more convenient for us later.)", "We can extend this notion to subgroups, and denote the orbit of $i$ under $H$ as $i^H : = \\lbrace i ^ g \\mid g \\in H \\rbrace $ , where $H \\le \\Gamma $ .", "When the orbit of $i$ under $H$ is all of $\\Omega $ , $H$ is said to act transitively.", "Additionally, we can extend this notion to subsets of $\\Omega $ , denoting $S^g := \\lbrace i^g \\mid i \\in S\\rbrace $ , where $S \\subseteq \\Omega $ .", "Finally, we say a nonempty subset $B$ of $\\Omega $ is a block for $\\Gamma $ if for each $g \\in \\Gamma $ either $B^g = B$ or $B^g \\cap B = \\emptyset $ .", "For each $i \\in \\Omega $ , the set $\\lbrace i\\rbrace $ forms a trivial block for $\\Gamma $ , as does $\\Omega $ itself.", "If $\\Gamma $ acts transitively on $\\Omega $ and $B$ is a block for $\\Gamma $ , then the set $\\Sigma = \\lbrace B^g \\mid g \\in \\Gamma \\rbrace $ is a partition of $\\Omega $ and each element of the partition is a block for $\\Gamma $ .", "The set $\\Sigma $ is called a system of blocks (or a block system) for $\\Gamma $ .", "The group $\\Gamma $ is said to be primitive if it has no nontrivial blocks on $\\Omega $ .", "If there is a nontrivial partition of $\\Omega $ by blocks of $\\Gamma $ , then $\\Gamma $ is said to be imprimitive.", "As each element of the partition will have the same size (say $k$ ), then $|\\Omega | = km$ , where there are $m$ blocks each of size $k$ .", "Let $\\Gamma $ act transitively on $\\Omega $ .", "There are two common types of block system we will use: If $H$ is a normal subgroup of $\\Gamma $ , then the orbits of $H$ form a block system for $\\Gamma $ .", "If $\\textrm {Stab}(\\Gamma ,i)$ is the stabilizer of some point $i \\in \\Omega $ , then the set of fixed points of $\\textrm {Stab}(\\Gamma ,i)$ form a block for $\\Gamma $ .", "We will also need Burnside's Theorem for transitive groups of prime degree: Theorem 2.3 (Burnside, [3]) A transitive permutation group of prime degree $b$ is doubly transitive or has a normal Sylow $b$ -subgroup.", "Given a regular polyhedron $\\mathcal {P}$ , and thus a string C-group $\\Gamma (\\mathcal {P})$ , when $\\Gamma (\\mathcal {P})$ acts faithfully on a set $\\Omega $ , we can form a CPR graph for this representation, which stands for “C-group Permutation Representation\" graph.", "This concept extends naturally to any group generated by involutions, and is defined as follows.", "Let $f$ be an embedding of $\\Gamma (\\mathcal {P}) = \\langle \\rho _0, \\rho _1, \\rho _2 \\rangle $ into the symmetric group $\\textrm {Sym}(\\Omega )$ .", "The CPR graph $X$ of $\\mathcal {P}$ given by $f$ is a 3-edge-labeled multigraph with nodes $\\Omega $ such that for any $i,j \\in \\Omega $ with $i \\ne j$ , there is a single $ij$ edge of $X$ of label $k$ if and only if the image of $\\rho _k$ under $f$ sends $i$ to $j$ .", "When $\\Gamma (\\mathcal {P})$ acts faithfully on its vertex set, we can thus construct a vertex CPR graph for the regular polyhedron $\\mathcal {P}$ , where the vertices of the graph $X$ are themselves vertices of $\\mathcal {P}$ .", "Given a CPR graph $X$ , and any subset $I$ of $\\lbrace 0,1,2\\rbrace $ , we can construct the subgraph $X_I$ which denotes the spanning subgraph of $X$ , including all the vertices of $X$ , whose edge set consists of the edges with labels $k \\in I$ .", "Much is known about the structure of these subgraphs $X_I$ .", "For instance, if $I = \\lbrace 0 ,2 \\rbrace $ then every connected component of $X_I$ is either a single vertex, a single edge, a double edge, or an alternating square.", "The type of the polyhedron also can easily be determined from the CPR graph.", "Let $X$ be a CPR graph for a regular polyhedron of type $\\lbrace p,q\\rbrace $ .", "Then, if $I = \\lbrace 0,1\\rbrace $ (or dually $\\lbrace 1,2\\rbrace $ ) then the connected components of $X_I$ are single vertices, double edges, alternating paths, or alternating cycles, and $p$ (or dually $q$ ) is the least common multiple of the number of vertices in each alternating path and half the number of vertices in each alternating cycle." ], [ "Vertex-faithful polyhedra", "We say that a regular polyhedron is vertex-faithful if its automorphism group acts faithfully on its vertices; in other words, if the only automorphism that fixes every vertex is the identity.", "We note that if a polyhedron is vertex-describable, meaning that each face is completely determined by its vertex-set, then it must also be vertex-faithful.", "This is because, if an automorphism fixes every vertex of a vertex-describable polyhedron, it must fix every face and thus must be the identity (since the automorphism group of a polyhedron acts freely on the flags).", "The converse is not true; the hemi-cube $\\lbrace 4,3\\rbrace _3$ is an example of a vertex-faithful polyhedron that is not vertex-describable.", "If $\\mathcal {P}$ is a regular polyhedron with $\\Gamma (\\mathcal {P}) = \\langle \\rho _0, \\rho _1, \\rho _2 \\rangle $ and with base vertex $u$ , then the stabilizer of $u$ is $\\langle \\rho _1, \\rho _2 \\rangle $ , and the stabilizer of an arbitrary vertex $u \\varphi $ is $\\varphi ^{-1} \\langle \\rho _1, \\rho _2 \\rangle \\varphi $ , where $\\varphi \\in \\Gamma (\\mathcal {P})$ .", "Then the kernel of the action of $\\Gamma (\\mathcal {P})$ on the vertices is $ \\bigcap _{\\varphi \\in \\Gamma (\\mathcal {P})} \\varphi ^{-1} \\langle \\rho _1, \\rho _2 \\rangle \\varphi , $ which is the normal core of $\\langle \\rho _1, \\rho _2 \\rangle $ in $\\Gamma (\\mathcal {P})$ (also called simply the core of $\\langle \\rho _1, \\rho _2 \\rangle $ in $\\Gamma (\\mathcal {P})$ ).", "Thus, $\\mathcal {P}$ is vertex-faithful if and only if $\\langle \\rho _1, \\rho _2 \\rangle $ is core-free in $\\Gamma (\\mathcal {P})$ .", "In order to analyze regular polyhedra with a fixed number of vertices, we will treat flat and non-flat polyhedra separately.", "Proposition REF below gives one of the main reasons why.", "Proposition 2.4 (Proposition 2.2 of [9]) Let $\\Gamma = \\langle \\rho _0, \\rho _1, \\rho _2 \\rangle $ be a string C-group.", "Let $N = \\langle (\\rho _1 \\rho _2)^k \\rangle $ for some $k \\ge 2$ .", "If $N$ is normal in $\\Gamma $ , then $\\Gamma /N$ is a string C-group.", "Proposition 2.5 Suppose that $\\mathcal {P}$ is a non-flat regular polyhedron of type $\\lbrace p, q\\rbrace $ with $v$ vertices, where $v$ is finite.", "Let $N$ be the normal core of $\\langle \\rho _1, \\rho _2 \\rangle $ in $\\Gamma (\\mathcal {P})$ .", "Then: $N = \\langle (\\rho _1 \\rho _2)^{q^{\\prime }} \\rangle $ for some $q^{\\prime } \\ge 3$ where $q^{\\prime }$ divides $q$ .", "$\\Gamma (\\mathcal {P})/N$ is the automorphism group of a non-flat, vertex-faithful regular polyhedron of type $\\lbrace p, q^{\\prime }\\rbrace $ with $v$ vertices.", "Let $\\varphi \\in N$ .", "If $\\varphi $ is a reflection (that is, the product of an odd number of generators), then $\\varphi $ is conjugate to either $\\rho _1$ or $\\rho _2$ .", "Since $N$ is normal, that implies that $\\rho _1$ or $\\rho _2$ is in $N$ and thus fixes all vertices.", "If $\\rho _1 \\in N$ , then $\\rho _0 \\rho _1 \\rho _0 \\in N$ , and by the intersection condition, $\\rho _0 \\rho _1 \\rho _0 \\in \\langle \\rho _1 \\rangle $ .", "This forces $(\\rho _0 \\rho _1)^2 = 1$ , and so $p = 2$ , which would make $\\mathcal {P}$ flat.", "So, suppose $\\rho _2 \\in N$ .", "Then $\\Gamma (\\mathcal {P}) / N$ is a quotient of the dihedral group $\\langle \\rho _0, \\rho _1 \\rangle $ , and so $[\\Gamma (\\mathcal {P}) : N] \\le 2p$ .", "(Note that $p$ must be finite since $v$ is finite.)", "Now, since $N$ is a subgroup of a dihedral group of order $2q$ and does not contain $\\rho _1$ , the index of $N$ in $\\langle \\rho _1, \\rho _2 \\rangle $ is at least 2.", "Furthermore, the index of $\\langle \\rho _1, \\rho _2 \\rangle $ in $\\Gamma (\\mathcal {P})$ is equal to $v$ which is at least as large as $p$ .", "So $ 2p \\le 2v \\le [\\Gamma (\\mathcal {P}) : \\langle \\rho _1, \\rho _2 \\rangle ] [\\langle \\rho _1, \\rho _2 \\rangle : N] = [\\Gamma (\\mathcal {P}) : N] \\le 2p.", "$ So in this case we get $v = p$ , meaning that $\\mathcal {P}$ is flat.", "So when $\\mathcal {P}$ is non-flat, neither $\\rho _1$ nor $\\rho _2$ can lie in $N$ .", "Thus $N$ cannot contain any reflections, and so $N = \\langle (\\rho _1 \\rho _2)^{q^{\\prime }} \\rangle $ for some $q^{\\prime }$ dividing $q$ .", "Next, Proposition REF shows that $\\Gamma (\\mathcal {P})/N$ is the automorphism group of a regular polyhedron $\\mathcal {Q}$ , which will have type $\\lbrace p, q^{\\prime }\\rbrace $ .", "Clearly the image of $\\langle \\rho _1, \\rho _2 \\rangle $ will be core-free, so that $\\mathcal {Q}$ is vertex-faithful.", "Furthermore, since $N$ was the kernel of the action on the vertices, $\\mathcal {Q}$ will have the same number of vertices as $\\mathcal {P}$ .", "Since $\\mathcal {P}$ is not flat, that implies that $v > p$ , and thus $\\mathcal {Q}$ is also not flat.", "Finally, this implies that $q^{\\prime } \\ge 3$ , since if $q^{\\prime } = 2$ then $\\mathcal {Q}$ has type $\\lbrace p, 2\\rbrace $ which would imply that it is flat.", "Corollary 2.6 If there are no non-flat, vertex-faithful regular polyhedra with $v$ vertices, then there are no non-flat regular polyhedra whatsoever with $v$ vertices.", "We will see later that there is an infinite family of flat regular polyhedra with no vertex-faithful quotients, so the assumption that $\\mathcal {P}$ was non-flat was essential.", "Given a vertex-faithful regular polyhedron $\\mathcal {Q}$ , there may be infinitely many regular polyhedra $\\mathcal {P}$ that cover $\\mathcal {Q}$ and have the same number of vertices.", "However, it is sometimes possible to bound the size of $\\mathcal {P}$ .", "Proposition 2.7 Suppose that $\\mathcal {Q}$ is a vertex-faithful regular polyhedron of type $\\lbrace p, q^{\\prime }\\rbrace $ and that $\\mathcal {P}$ is a regular polyhedron of type $\\lbrace p, q\\rbrace $ that properly covers $\\mathcal {Q}$ and has the same number of vertices.", "Let $\\psi : \\Gamma (\\mathcal {P}) \\rightarrow \\Gamma (\\mathcal {Q})$ be the canonical covering that sends generators to generators.", "If $\\mathcal {Q}$ is non-orientable, then $\\mathcal {P}$ is non-orientable.", "In $\\Gamma (\\mathcal {P})$ , $\\rho _0 \\sigma _2^{q^{\\prime }} \\rho _0 = \\sigma _2^{aq^{\\prime }}$ for some $a$ satisfying $a^2 \\equiv 1$ (mod $q/q^{\\prime }$ ).", "If $\\alpha $ is a word in $\\Gamma (\\mathcal {P})$ such that $\\alpha \\psi = 1$ , then $\\alpha $ commutes with $\\sigma _2^{q^{\\prime }}$ .", "Suppose that $\\alpha \\in \\Gamma (\\mathcal {P})$ such that $\\alpha \\psi = 1$ .", "Let $s$ be the number of occurrences (mod 2) of $\\rho _0$ in $\\alpha $ , and let $t$ be the total number of occurrences (mod 2) of $\\rho _1$ and $\\rho _2$ in $\\alpha $ .", "Then $\\sigma _2^{q^{\\prime }} = \\sigma _2^{a^s (-1)^t q^{\\prime }}$ .", "In particular, if $\\alpha $ has odd length and $\\rho _0$ occurs an even number of times, then $q = 2q^{\\prime }$ , and if $p$ is odd, then $\\rho _0$ inverts $\\sigma _2^{q^{\\prime }}$ .", "Let $N = \\ker \\psi = \\langle \\sigma _2^{q^{\\prime }} \\rangle $ .", "If $\\mathcal {Q}$ is non-orientable, that means that there is a word $\\alpha $ of odd length in $\\Gamma (\\mathcal {P})$ such that $\\alpha \\in N$ .", "But the elements of $N$ all have even length, and so $\\mathcal {P}$ is non-orientable.", "That proves part (a).", "For part (b), the first part is obvious since $N$ is normal.", "Then since $\\rho _2$ inverts $\\sigma _2$ , $ \\sigma _2^{q^{\\prime }} = (\\rho _0 \\rho _2)^2 \\sigma _2^{q^{\\prime }} (\\rho _2 \\rho _0)^2 = \\sigma _2^{a^2 q^{\\prime }}, $ proving the second part.", "Part (c) is clear since in this case $\\alpha \\in N$ .", "To prove part (d), first recall that $\\rho _1$ and $\\rho _2$ both invert $\\sigma _2^{q^{\\prime }}$ .", "The result then follows immediately from parts (b) and (c) considering the equation $\\alpha ^{-1} \\sigma _2^{q^{\\prime }} \\alpha = \\sigma _2^{q^{\\prime }}$ and expanding the left-hand side.", "We can say more about the covering of vertex-faithful regular polyhedra depending on the lengths of their $j-$holes and $j-$zigzags (see Section 7B of [19] for a combinatorial description).", "Algebraically, the length of a 1-hole, $p$ , is the order of the element $\\rho _0 \\rho _1$ , where the length of a 2-hole is the order of the element $h=\\rho _0 \\rho _1 \\rho _2 \\rho _1$ .", "The length of a 1-zigzag is the size of the Petrie polygon of the map, which is the order of the element $z_1=\\rho _0 \\rho _1 \\rho _2$ .", "Finally, the order of the element $z_2=\\rho _0 \\rho _1 \\rho _2 \\rho _1 \\rho _2$ gives the length of the 2-zigzags of $\\mathcal {P}$ .", "Corollary 2.8 Suppose that $\\mathcal {Q}$ is a vertex-faithful regular polyhedron of type $\\lbrace p, q^{\\prime }\\rbrace $ and that $\\mathcal {P}$ is a regular polyhedron of type $\\lbrace p, q\\rbrace $ that properly covers $\\mathcal {Q}$ and has the same number of vertices.", "Let $h$ , $z_1$ , and $z_2$ be the elements of $\\Gamma (\\mathcal {Q})$ as described above.", "If either $p$ or $|h|$ is odd and either $|z_1|$ or $|z_2|$ is odd, then $q = 2q^{\\prime }$ .", "In particular, $|\\Gamma (\\mathcal {P})| = 2|\\Gamma (\\mathcal {Q})|$ .", "In this case, part (d) says that $\\rho _0$ commutes with $\\sigma _2^{q^{\\prime }}$ and inverts $\\sigma _2^{q^{\\prime }}$ .", "It follows that $\\sigma _2^{-q^{\\prime }} = \\sigma _2^{q^{\\prime }}$ , and so $q = 2q^{\\prime }$ .", "Then $|\\Gamma (\\mathcal {P})| = 2qv = 4q^{\\prime }v = 2|\\Gamma (\\mathcal {Q})|$ .", "The following proposition is an adaptation of [23].", "Proposition 2.9 Let $\\mathcal {P}$ be a regular polyhedron of type $\\lbrace p, q\\rbrace $ .", "For every $q^{\\prime } < q/2$ such that $q^{\\prime }$ divides $q$ , the number of vertices fixed by $(\\rho _1 \\rho _2)^{q^{\\prime }}$ is a divisor of the total number of vertices.", "Let $u$ be the base vertex, with stabilizer $\\langle \\rho _1, \\rho _2 \\rangle $ , and fix a $q^{\\prime } < q/2$ such that $q^{\\prime }$ divides $q$ .", "If $v$ is an arbitrary vertex of $\\mathcal {P}$ , then we may write $v = u \\alpha $ for some $\\alpha \\in \\Gamma (\\mathcal {P})$ .", "Then $(\\rho _1 \\rho _2)^{q^{\\prime }}$ fixes $v$ if and only if $\\sigma := \\alpha (\\rho _1 \\rho _2)^{q^{\\prime }} \\alpha ^{-1}$ fixes $u$ , which is true if and only if $\\sigma \\in \\langle \\rho _1, \\rho _2 \\rangle $ .", "Now, since $q^{\\prime } < q/2$ , it follows that $(\\rho _1 \\rho _2)^{q^{\\prime }}$ has order 3 or more, and thus the same is true of $\\sigma $ .", "Then since $\\sigma \\in \\langle \\rho _1, \\rho _2 \\rangle $ , which is dihedral, we find that $\\sigma \\in \\langle (\\rho _1 \\rho _2)^{q^{\\prime }} \\rangle $ .", "Therefore, $(\\rho _1 \\rho _2)^{q^{\\prime }}$ fixes $u \\alpha $ if and only if $\\alpha $ normalizes $\\langle (\\rho _1 \\rho _2)^{q^{\\prime }} \\rangle $ .", "Now, let $N$ be the normalizer of $\\langle (\\rho _1 \\rho _2)^{q^{\\prime }} \\rangle $ in $\\Gamma (\\mathcal {P})$ , and let $H = \\langle \\rho _1, \\rho _2 \\rangle $ .", "The stabilizer of $u$ is $H$ , and so $(\\rho _1 \\rho _2)^{q^{\\prime }}$ fixes $u \\alpha $ if and only if it fixes $u \\beta \\alpha $ for every $\\beta \\in H$ .", "By the previous paragraph, this means that $\\alpha \\in N$ if and only if $\\beta \\alpha \\in N$ for every $\\beta \\in H$ .", "It follows that the number of vertices fixed by $(\\rho _1 \\rho _2)^{q^{\\prime }}$ is the index of $H$ in $N$ .", "Since the total number of vertices is the index of $H$ in $\\Gamma (\\mathcal {P})$ , the result follows.", "All the calculations in this paper were verified in GAP [12] and Magma [1].", "When these programs were used for a calculation, we will say that they were done on a Computer Algebra System (CAS)." ], [ "Polyhedra with few vertices", "In this section we consider vertex-faithful regular polyhedra with few vertices.", "If a regular polyhedron has vertex figures of size $q$ , this means that the base vertex is incident to $q$ edges.", "We will refer to the other vertices incident to these $q$ edges as the vertices on the base vertex figure.", "For a fixed size of vertex figure $q$ , we will show that the number of vertices $v$ is bounded below by $q$ , and we will consider what happens when $v$ obtains this minimum.", "We note that the same bound does not apply for arbitrary regular polyhedra; for example the regular map $\\lbrace 3,6\\rbrace _{(2,0)}$ is a polyhedron with only four vertices.", "In this section we also classify the vertex-faithful regular polyhedra with fewer than sixteen vertices.", "This classification will be leveraged in the later sections.", "Proposition 3.1 Suppose $\\mathcal {P}$ is a vertex-faithful regular polyhedron of type $\\lbrace p, q\\rbrace $ .", "If $\\mathcal {P}$ is orientable, then $q < v$ ; if $\\mathcal {P}$ is non-orientable, then $q \\le v$ .", "Let $\\Gamma (\\mathcal {P}) = \\langle \\rho _0, \\rho _1, \\rho _2 \\rangle $ , and let $H = \\langle \\rho _1, \\rho _2 \\rangle $ .", "The vertices of $\\mathcal {P}$ correspond to cosets $H \\varphi $ , with the base vertex corresponding to the coset $H = H1$ (where 1 is the identity of $\\Gamma (\\mathcal {P})$ ).", "We may assume that $v \\ge 4$ .", "Otherwise, if $\\mathcal {P}$ is vertex-faithful, then $\\Gamma (\\mathcal {P})$ embeds into the symmetric group acting on three elements, and thus has size less than or equal to six.", "It can be checked (for instance in [14]) that there is no regular polyhedron with an automorphism group this small.", "Let $\\sigma _2 = \\rho _1 \\rho _2$ , and consider the action of $\\sigma _2$ on the vertex $H \\rho _0$ , which is the other vertex incident to the base edge.", "Assume that $q \\ge v$ .", "There are at most $v-1$ distinct vertices on the base vertex figure, and thus the stabilizer of $H \\rho _0$ in $\\langle \\sigma _2 \\rangle $ is non-trivial and generated by $\\sigma _2^a$ for some smallest positive integer $a$ .", "Now, to say that $\\sigma _2^a$ fixes $H \\rho _0$ is to say that $\\rho _0 \\sigma _2^a \\rho _0$ lies in $H = \\langle \\rho _1, \\rho _2 \\rangle $ .", "If $\\mathcal {P}$ is orientable, or if $\\sigma _2^a$ has order 3 or more, then $\\rho _0 \\sigma _2^a \\rho _0$ must be a power of $\\sigma _2^a$ .", "In this case, $\\langle \\sigma _2^a \\rangle $ is normal.", "Then for any vertex $H \\varphi $ , we have that $H \\varphi \\sigma _2^a = H \\sigma _2^{ak} \\varphi = H \\varphi $ , and so $\\sigma _2^a$ fixes every vertex of $\\mathcal {P}$ , and thus $\\mathcal {P}$ is not vertex-faithful.", "Thus in particular, if $\\mathcal {P}$ is orientable and vertex-faithful, then $q < v$ .", "The remaining case to consider is when $\\mathcal {P}$ is non-orientable, $a = q/2$ , and $\\rho _0 \\sigma _2^{q/2} \\rho _0$ is not in $\\langle \\sigma _2^{q/2} \\rangle $ .", "Then $\\rho _0 \\sigma _2^{q/2} \\rho _0 = \\alpha \\in \\langle \\rho _1, \\rho _2 \\rangle $ , where $\\alpha $ has odd length.", "We note that $\\rho _2$ commutes with the left side, so we have $(\\rho _2 \\alpha )^2 = 1$ .", "Then since $\\rho _2 \\alpha $ has even length and order 2, it follows that $\\rho _2 \\alpha = \\sigma _2^{q/2}$ , and so $\\alpha = \\rho _2 \\sigma _2^{q/2}$ .", "In other words, the following relation holds in $\\Gamma (\\mathcal {P})$ : $ \\rho _0 \\sigma _2^{q/2} \\rho _0 = \\rho _2 \\sigma _2^{q/2}$ It follows that $\\sigma _2^{q/2} \\rho _0 \\sigma _2^{q/2} = \\rho _0 \\rho _2$ .", "Furthermore, $\\sigma _2^{q/2} \\rho _0 \\rho _1 \\sigma _2^{q/2} = \\rho _0 \\rho _2 \\rho _1$ .", "This means that $\\mathcal {P}$ is internally self-Petrie (see [8]).", "This also implies that $(\\mathcal {P}^{\\delta })^{\\pi }$ is internally self-dual.", "Consider what Relation REF means for the $\\frac{q}{2}$ vertices on the base vertex figure, where $\\sigma _2^\\frac{q}{2}$ acts trivially.", "It implies that if any neighbor of the base vertex is also fixed by $\\rho _0$ , then it is fixed by $\\rho _2$ .", "Thus $\\rho _0$ has at most one fixed point on the base vertex figure (with zero fixed points when $\\frac{q}{2}$ is odd).", "See Figure REF .", "Figure: Subgraphs of vertex CPR graphs induced by the base vertex v 0 v_0 and the q 2\\frac{q}{2} vertices on the base vertex figure, for q 2\\frac{q}{2} even and odd respectively.Let $X$ be the vertex CPR graph for $\\mathcal {P}$ , and let $w$ be any vertex on the base vertex figure not fixed by $\\rho _2$ , and thus not fixed by $\\rho _0$ .", "Since $\\rho _0$ and $\\rho _2$ commute, in $X$ the vertex $w$ must be part of either an alternating square with edges of labels 0 and 2, or an end point of a double edge of labels 0 and 2.", "Since $\\sigma _2^{q/2}$ fixes $w$ , Relation REF shows that the double edge case is not possible, and thus $w$ and $(w) \\rho _2$ are part of an alternating square with two other vertices $x$ and $(x) \\rho _2$ .", "Similarly, Relation REF implies that $x$ and $(x) \\rho _2$ are interchanged by $\\sigma _2^{q/2}$ , and so they are not on the base vertex figure.", "Therefore for every pair of vertices $w$ and $(w) \\rho _2$ on the base vertex figure, there exists a pair of vertices $x$ and $(x) \\rho _2$ not on the base vertex figure.", "Counting up the vertices, we have the base vertex, the $\\frac{q}{2}$ vertices on the base vertex figure, and the at least $\\frac{q}{2}-2$ (or $\\frac{q}{2} -1$ if $\\frac{q}{2}$ is odd) vertices that are sent to the base vertex figure by $\\rho _0$ .", "Thus $v \\ge 1+\\frac{q}{2}+\\frac{q}{2}-2 = q-1$ (or $v\\ge q$ when $\\frac{q}{2}$ is odd).", "If $q = v+1$ , then $\\frac{q}{2}$ is even.", "Furthermore, if $\\frac{q}{2}$ is even, then exactly $\\frac{q}{2}-2$ vertices are moved by $\\sigma _2^{q/2}$ and sent to the base vertex figure by $\\rho _0$ .", "We will now rule out the possibility that $q \\ge v$ when $\\frac{q}{2}$ is even, which will then prove that $q \\le v$ .", "Consider the action of $\\sigma _2^{q/4}$ on the $\\frac{q}{2}-2$ vertices that are moved by $\\sigma _2^{q/2}$ .", "Since $\\sigma _2^{q/2}$ interchanges these vertices in pairs, $\\sigma _2^{q/4}$ must act as 4-cycles on all of these vertices.", "Thus $\\frac{q}{2}-2$ is divisible by 4, which implies that $q \\equiv 4$ (mod 8).", "So we may write $q = 4q^{\\prime }$ where $q^{\\prime }$ is odd.", "In order for $\\sigma _2$ to have order $q$ , there must be some orbit of size $4k$ .", "Furthermore, since the $q/2$ vertices on the base vertex figure form a single orbit of size $2q^{\\prime }$ , the orbit of size $4k$ must be among vertices that are not fixed by $\\sigma _2^{q/2}$ , all of which are sent to the base vertex figure by $\\rho _0$ .", "Call this orbit $T$ .", "In the subgraph $X_{\\lbrace 1,2\\rbrace }$ the vertices of $T$ form either an alternating path or cycle.", "In either case you can label the vertices of $T$ as $(0,1,2,3,\\ldots ,4k-1)$ based on the order in $X_{\\lbrace 1,2\\rbrace }$ , where vertex $i$ is adjacent to $i+1$ for $0 \\le i \\le 4k-2$ .", "Figure: Possible subgraphs of the vertex CPR graph for vertices in TT.Using the left hand side of Relation REF , each vertex in $T$ is fixed by $\\rho _0 \\sigma _2^{\\frac{q}{2}} \\rho _0$ .", "On the other hand, for all vertices $i$ , $(i) \\sigma _2^{\\frac{q}{2}} = 4k-1-i$ when $T$ is an alternating path, and $(i) \\sigma _2^{\\frac{q}{2}} = i \\pm 2k$ when $T$ is an alternating cycle.", "Furthermore, in both cases $(i) \\rho _2 \\equiv i \\pm 1 \\pmod {4k}$ .", "In either case, $\\rho _2 \\sigma _2^{\\frac{q}{2}}$ cannot fix all vertices of $T$ , and so Relation REF would not hold.", "Therefore, $q \\ge v$ and $\\frac{q}{2}$ even is not possible for any vertex-faithful regular polyhedron.", "In particular, for any vertex-faithful regular polyhedron $q \\le v$ and $q = v$ only if $\\frac{q}{2}$ is odd and $\\mathcal {P}$ is internally self-Petrie and non-orientable.", "We saw that if a vertex-faithful regular polyhedron has the property that $q=v$ , then $v$ is even, $\\frac{v}{2}$ is odd, $\\mathcal {P}$ is non-orientable, self-Petrie, and satisfies Relation REF .", "This is extremely restrictive, and will allow us to fully understand the structure of $\\mathcal {P}$ up to isomorphism.", "In this case, the orbits of the vertices under the action of $\\langle \\rho _1 \\rho _2 \\rangle $ are of size 1 in the case of the base vertex, size $\\frac{v}{2}$ for the vertices on the base vertex figure.", "Additionally, every vertex not on the base vertex figure, is sent to the base vertex figure by $\\rho _0$ .", "Let $T$ be any $\\langle \\rho _1, \\rho _2 \\rangle $ -orbit of a vertex not on the base vertex figure (other than the base vertex).", "We will show that $T$ always consists of two vertices connected by a single edge labeled 2.", "Assume to the contrary that $T$ has at least 3 vertices.", "Since $\\rho _2$ does not fix any of these vertices, there must be an even number $2j$ of vertices in $T$ , with $j > 1$ .", "Similar to in the previous proof, each vertex of $T$ is fixed by $\\rho _0 \\sigma _2^{\\frac{q}{2}} \\rho _0$ .", "On the other hand, for all vertices $i$ , $(i) \\sigma _2^{\\frac{q}{2}} = 2j-1-i$ when $T$ is an alternating path, and $(i) \\sigma _2^{\\frac{q}{2}} = i \\pm j$ when $T$ is an alternating cycle.", "Furthermore, in both cases $(i) \\rho _2 \\equiv i \\pm 1 \\pmod {2j}$ .", "In either case, $\\rho _2 \\sigma _2^{\\frac{q}{2}}$ cannot fix all vertices of $T$ , and so Relation REF would not hold if $T$ had at least 3 vertices.", "So $T$ has two vertices that are connected by an edge labeled 2.", "If they were also connected by an edge labeled 1, then $\\rho _2 \\sigma _2^{\\frac{q}{2}}$ would move both vertices whereas $\\rho _0 \\sigma _2^{\\frac{q}{2}} \\rho _0$ would fix both vertices, violating Relation REF .", "Thus $T$ consists of a single edge labeled 2.", "The vertex-set of the base facet consists of the $\\langle \\rho _0, \\rho _1 \\rangle $ -orbit of the base vertex.", "We have that $\\rho _0$ sends the base vertex to one of its neighbors; then $\\rho _1$ sends that vertex to another neighbor of the base vertex, and then $\\rho _0$ sends that vertex to a vertex not on the base vertex figure.", "The above argument demonstrates that $\\rho _1$ fixes this vertex, and so there are exactly 4 vertices incident to the base facet, and thus $\\mathcal {P}$ is of type $\\lbrace 4,v\\rbrace $ .", "This gives a unique possible regular polyhedron for each $v \\equiv 2$ (mod 4).", "The vertex CPR graphs for $v=6$ and $v=10$ are shown below.", "Using Theorem 4.4 of [20] we see that these are in fact polyhedra, but it remains to show that this permutation representation is the action on the vertices.", "$\\includegraphics [scale=.5]{Type4,v}$ Corollary 3.2 If $\\mathcal {P}$ is a vertex-faithful regular polyhedron of type $\\lbrace p, q\\rbrace $ with $q = v$ , then $\\mathcal {P}$ is $(\\lbrace 4,4\\rbrace _{(q/2,0)})^{\\pi \\delta }$ , with $q \\ge 6$ and $q/2$ odd.", "We have shown above that there is only one possibility for such a vertex-faithful regular polyhedron of type $\\lbrace p,q\\rbrace $ with $q=v$ to exist.", "Furthermore, we have seen that if it does exist, then $p=4$ and $v \\equiv 2$ (mod 4).", "It remains to show that such a polyhedron does exist for these cases.", "The result can be checked for $v \\le 5$ using a CAS.", "Fix $q$ such that $q \\ge 6$ and $q \\equiv 2$ (mod 4), and let $\\mathcal {P}$ be the regular polyhedron with the vertex CPR graph above.", "We have established that $\\mathcal {P}$ is a polyhedron of type $\\lbrace 4, q\\rbrace $ that satisfies Relation REF .", "In particular, this means that $\\mathcal {P}$ is self-Petrie and thus a quotient of the universal polyhedron $\\mathcal {Q}= \\lbrace 4, q\\rbrace _4$ .", "Now, $\\mathcal {Q}$ is the dual of the Petrial of $\\lbrace 4, 4\\rbrace _q$ , the latter of which is isomorphic to the toroidal map $\\lbrace 4, 4\\rbrace _{(q/2, q/2)}$ and which has an automorphism group of order $4q^2$ .", "Since $\\mathcal {Q}$ is orientable whereas $\\mathcal {P}$ is non-orientable, it follows that $\\mathcal {P}$ is a proper quotient of $\\mathcal {Q}$ .", "Then the Petrial of the dual of $\\mathcal {P}$ is a proper smooth quotient of $\\lbrace 4, 4\\rbrace _q$ that still has petrie polygons of length $q$ .", "The only possibility is $\\mathcal {T} = \\lbrace 4, 4\\rbrace _{(q/2, 0)}$ , and so $\\mathcal {P}$ is the dual of the Petrial of $\\mathcal {T}$ .", "Now let us consider non-vertex-faithful regular polyhedra whose vertex-faithful quotient is $(\\lbrace 4,4\\rbrace _{(q/2,0)})^{\\pi \\delta }$ .", "Lemma 3.3 Suppose $\\mathcal {P}$ is a finite non-flat vertex-faithful regular polyhedron of type $\\lbrace p,q\\rbrace $ which satisfies Relation REF , with $q$ even.", "If $\\mathcal {Q}$ is a regular polyhedron that covers $\\mathcal {P}$ and $\\mathcal {Q}$ has the same number of vertices as $\\mathcal {P}$ , then $\\mathcal {Q}= \\mathcal {P}$ .", "In particular, there are no non-vertex-faithful regular polyhedra whose vertex-faithful quotient is $(\\lbrace 4,4\\rbrace _{(q/2,0)})^{\\pi \\delta }$ with $q \\ge 6$ and $q/2$ odd.", "Let $\\mathcal {P}$ and $\\mathcal {Q}$ satisfy the conditions of the lemma.", "Let $\\langle \\rho _0, \\rho _1, \\rho _2 \\rangle = \\Gamma (\\mathcal {Q})$ , and let $\\sigma _2 = \\rho _1 \\rho _2$ .", "By Proposition REF , $N = \\langle \\sigma _2^q \\rangle $ , and thus lifting Relation REF from $\\Gamma (\\mathcal {P})$ to $\\Gamma (\\mathcal {Q})$ we get $\\rho _0 \\sigma _2^{\\frac{q}{2}} \\rho _0 \\sigma _2^{-\\frac{q}{2}} \\rho _2 = \\sigma _2^{kq}$ for some integer $k$ .", "Therefore, in $\\Gamma (\\mathcal {Q})$ $ \\rho _0 \\sigma _2^{\\frac{q}{2}} \\rho _0 = \\sigma _2^{kq} \\rho _2 \\sigma _2^{\\frac{q}{2}}.$ The right hand side of this equation has order 2 (being an element of odd length in a dihedral group), and thus $\\sigma _2^{\\frac{q}{2}}$ also has order 2.", "Therefore $\\mathcal {Q}$ is of type $\\lbrace p,q\\rbrace $ , and since $\\mathcal {P}$ was also of this type we know that $N$ was trivial.", "Thus $\\mathcal {P}= \\mathcal {Q}$ .", "The rest follows immediately.", "Motivated by the preceding results, we briefly consider vertex-faithful regular polyhedra with $q=v-1$ .", "We leave the full details of this case as an open question.", "Up to 2000 flags, there are only 7 regular polyhedra with $q = v-1$ , of types $\\lbrace 3, 3\\rbrace $ , $\\lbrace 4, 3\\rbrace $ , $\\lbrace 3, 5\\rbrace $ , $\\lbrace 5, 5\\rbrace $ , $\\lbrace 4, 15\\rbrace $ , $\\lbrace 8, 15\\rbrace $ , and $\\lbrace 12, 15\\rbrace $ .", "Only the first 4 are vertex-faithful.", "Suppose $\\mathcal {P}$ is a vertex-faithful regular polyhedron with $q = v-1$ .", "If $\\mathcal {P}$ is orientably regular, then the vertices on the base vertex figure are all distinct.", "This implies that the edge graph of $\\mathcal {P}$ is the complete graph on $v$ vertices, (which is to say, $\\mathcal {P}$ is a neighborly polyhedron), as the base vertex is adjacent to every other vertex.", "Then, one can use the fact that the regular imbeddings of the complete graphs in orientable surfaces have been classified in [16].", "Finally, we mention one further corollary of Proposition REF that helps with searching for vertex-faithful regular polyhedra.", "Corollary 3.4 A vertex-faithful polyhedron with $v$ vertices has at most $2v^2$ flags." ], [ "Vertex-faithful polyhedra with few vertices ", "Here we will give details of all the vertex-faithful polyhedra with fewer than sixteen vertices.", "The results can be achieved by analyzing the classification of all regular polytopes with up to 4000 flags [5], or by classifying regular polytopes for transitive groups up to degree 15.", "The details of our classification are found in Table REF and Table REF .", "There are no vertex-faithful regular polyhedra where the number of vertices is a prime number less than 16.", "We will see in Section  that this continues to hold for all primes.", "Additionally, we point out that there are exactly two vertex-faithful regular polyhedra with $v=14$ vertices; one of them is flat and will be described in the second row of Table REF , the other is $(\\lbrace 4,4\\rbrace _{(v,0)})^{\\pi \\delta }$ , the dual of the Petrial of $\\lbrace 4,4\\rbrace _{(v,0)}$ .", "We will see in Section  that this classification remains whenever $v$ is twice any prime at least 7.", "In our tables, we provide the “Atlas Canonical Name” of the polyhedron from Hartley's Atlas [14], as well as a description of the automorphism group of the polyhedron, where each one is a quotient of the string Coxeter group of type $\\lbrace p,q\\rbrace $ , with $p$ and $q$ found in the Atlas Canonical Name.", "For each group, we give the order of the Petrie element $z_1=\\rho _0 \\rho _1 \\rho _2$ , which gives the length of a 1-zigzag of $\\mathcal {P}$ .", "We also give the orders of the elements $h=\\rho _0 \\rho _1 \\rho _2 \\rho _1$ which describes the sizes of the 2-holes in the map, and $z_2=\\rho _0 \\rho _1 \\rho _2 \\rho _1 \\rho _2$ which gives the length of the 2-zigzag of $\\mathcal {P}$ .", "In Problem 7 of [21], it is asked to what extent a regular polyhedron of fixed type is determined by the lengths of its $j-$ holes and the lengths of its $j-$ zigzags.", "Considering this question, we also note whether the polyhedron is “universal” with respect to the length of its 1-zigzags, 2-zigzags, and 2-holes; otherwise further relators are needed to form a presentation for the automorphism group.", "Table: The vertex-faithful regular polyhedra with 10 or fewer vertices.Table: The vertex-faithful regular polyhedra with 12, 14, or 15 vertices." ], [ "Polyhedra with a prime number of vertices", "In this section, we will fully classify the regular polyhedra with a prime number of vertices." ], [ "Flat polyhedra", "If $\\mathcal {P}$ is a flat regular polyhedron with $b$ vertices, then it has type $\\lbrace b, q\\rbrace $ for some $q$ .", "When $b = 2$ , the polyhedron $\\lbrace 2, q\\rbrace $ is a flat polyhedron with 2 vertices for every $q \\ge 2$ (including $q = \\infty $ ).", "If $b$ is an odd prime, then there are exactly two flat orientably regular polyhedra with $b$ vertices [6].", "The first is the universal polyhedron of type $\\lbrace b, 2\\rbrace $ .", "The second has type $\\lbrace b, 2b\\rbrace $ , and the automorphism group is the quotient of $[b, 2b]$ by the extra relation $(\\rho _0 \\rho _1 \\rho _2 \\rho _1 \\rho _2)^2 = 1$ .", "This is equivalent to the relation $\\sigma _1 \\sigma _2^{-1} \\sigma _1 \\sigma _2^3 = 1$ , which yields $\\sigma _2^{-1} \\sigma _1 = \\sigma _1^{-1} \\sigma _2^{-3}$ .", "Thus the flat regular polyhedron of type $\\lbrace b, 2b\\rbrace $ has automorphism group $\\Lambda (b,2b)_{-1,-3}$ (see Section REF ).", "Additionally, the only flat non-orientably regular polyhedron with a prime number of vertices is the hemi-octahedron $\\lbrace 3,4\\rbrace _3$  [9].", "All of the flat regular polyhedra with a prime number of vertices are summarized in Table REF .", "We list the relations needed to define the automorphism group as a quotient of the string Coxeter group $[p, q]$ that is indicated by the type.", "Note that none of the flat regular polyhedra with a prime number of vertices is vertex-faithful.", "Indeed, in every case, $\\rho _2$ fixes every vertex.", "Table: Flat regular polyhedra with a prime number of vertices bb" ], [ "Non-flat polyhedra", "To classify non-flat regular polyhedra with $b$ vertices, we start by looking for vertex-faithful polyhedra (bearing in mind Corollary REF ).", "Proposition 4.1 Suppose that $\\mathcal {P}$ is a non-flat, vertex-faithful regular polyhedron with a prime number of vertices $b$ .", "Then: $q \\le b-1$ .", "$\\Gamma (\\mathcal {P})$ does not have a normal Sylow $b$ -subgroup.", "$\\Gamma (\\mathcal {P})$ acts doubly transitively on the vertices.", "By Proposition REF , $q \\le b$ .", "Since $|\\Gamma (\\mathcal {P})| = 2qb = 4e$ , it follows that $q$ is even and thus $q \\le b-1$ .", "This proves part (a).", "Since $b$ does not divide $q$ , any Sylow $b$ -subgroup of $\\Gamma (\\mathcal {P})$ is cyclic and of order $b$ .", "Let $\\pi $ be a $b$ -cycle, and suppose that $\\langle \\pi \\rangle $ is normal.", "Then $\\rho _1 \\pi \\rho _1= \\pi ^j$ and $\\rho _2 \\pi \\rho _2 =\\pi ^k$ where $k^2=j^2 \\equiv 1$ (mod $b)$ .", "Since $b$ is prime, $j$ and $k$ are each $\\pm 1$ , which is to say that conjugating $\\pi $ by $\\rho _1$ or $\\rho _2$ either fixes $\\pi $ or inverts $\\pi $ .", "Label the vertices so that $\\pi $ acts on them as a cycle $(0,1,2,3, \\ldots , b-1)$ where 0 is the base vertex.", "Suppose that for $i=1$ or $i=2$ that $\\rho _i \\pi \\rho _i = \\pi $ .", "Then since $\\rho _i$ fixes 0, it follows that for all $j \\in \\lbrace 0,\\ldots , b-1\\rbrace $ , $ j \\rho _i = 0 \\pi ^j \\rho _i = 0 \\rho _i \\pi ^j = 0 \\pi ^j = j.$ Thus $\\rho _i$ would fix all vertices, which would imply that $\\mathcal {P}$ is not vertex-faithful.", "So $\\rho _1 \\pi \\rho _1 = \\rho _2 \\pi \\rho _2 = \\pi ^{-1}$ .", "But then $(\\rho _1 \\rho _2) \\pi (\\rho _2 \\rho _1) = \\pi $ , and by the same argument as for $\\rho _1$ and $\\rho _2$ , this implies that $\\rho _1 \\rho _2$ fixes every vertex and that $\\mathcal {P}$ is not vertex-faithful.", "So $\\langle \\pi \\rangle $ must not be normal.", "Part (c) follows from part (b) and Theorem REF .", "Lemma 4.2 There are no non-flat, vertex-faithful regular polyhedra with a prime number of vertices.", "Suppose $\\mathcal {P}$ is a non-flat, vertex-faithful regular polyhedron with a prime number of vertices $b$ .", "Let $\\mathcal {P}$ be of type $\\lbrace p, q\\rbrace $ and thus $| \\Gamma (\\mathcal {P}) | = 2bq$ .", "Since $\\mathcal {P}$ is not flat, $b \\ge 3$ .", "By Proposition REF (c), $\\Gamma (\\mathcal {P})$ acts doubly transitively on the vertices, and so $\\langle \\rho _1, \\rho _2 \\rangle $ (which is the stabilizer of the base vertex) acts transitively on the remaining $b-1$ vertices; in particular, $b-1$ divides $2q$ .", "Let $n$ be the number of Sylow $b$ -subgroups.", "By Proposition REF (b), $n \\ne 1$ , and by the Sylow Theorems, $n \\equiv 1$ (mod $b$ ) and $n$ divides $2q$ .", "Since $q \\le b-1$ by Proposition REF (a), this implies that $n = b+1 = 2q$ .", "But then $b-1$ divides $b+1$ , which is only true if $b = 3$ .", "In that case we get $q = 2$ , which yields a flat polyhedron, and indeed such a polyhedron is not vertex-faithful after all.", "Theorem 4.3 Every regular polyhedron with a prime number of vertices $b$ is one of the flat polyhedra of type $\\lbrace b, q\\rbrace $ included in Table REF .", "In particular, for each prime $b \\ge 5$ , up to isomorphism there are exactly two regular polyhedra with $b$ vertices.", "From Lemma REF and Corollary REF we conclude that there are no non-flat regular polyhedra with a prime number of vertices.", "Table REF describes all such flat regular polyhedra." ], [ "Polyhedra with twice a prime number of vertices ", "We start with a useful result about vertex-faithful regular polyhedra with $2b$ vertices.", "Next, we will consider the flat regular polyhedra.", "The vertex-faithful regular polyhedra will be broken down by the action of the automorphism groups on the vertices.", "Finally, we conclude this section with a classification of the regular polyhedra with twice a prime number of vertices.", "Proposition 5.1 Suppose that $\\mathcal {P}$ is a vertex-faithful regular polyhedron of type $\\lbrace p, q\\rbrace $ with $2b$ vertices, with $b$ a prime at least 5, and with $q < 2b$ .", "Then $q = b$ if and only if $\\Gamma (\\mathcal {P})$ has a normal $b$ -subgroup.", "Furthermore, if $\\mathcal {P}$ is non-flat, then $q \\ne b$ and $\\Gamma (\\mathcal {P})$ does not have a normal $b$ -subgroup.", "Suppose that $q = b$ , and let $n$ be the number of Sylow $b$ -subgroups.", "Then $|\\Gamma (\\mathcal {P})| = 4b^2$ , and so $n$ divides 4 and $n \\equiv 1$ (mod $b$ ).", "If $b \\ge 5$ , then $n = 1$ and the Sylow $b$ -subgroup is normal.", "Conversely, suppose that $\\Gamma (\\mathcal {P})$ has a normal $b$ -subgroup $S$ .", "Since $|\\Gamma (\\mathcal {P})| = 4qb$ , if $S$ has order greater than $b$ , then $b$ divides $q$ and since $q < 2b$ that means $q = b$ .", "So let us assume that $S = \\langle \\pi \\rangle $ with $\\pi $ an element of order $b$ .", "The orbits of $\\langle \\pi \\rangle $ form a block system on the $2b$ vertices, and since $\\mathcal {P}$ is vertex-faithful, there must be two blocks of size $b$ .", "Let $B_1$ be the block containing the base vertex, and $B_2$ the other block.", "Then we can argue similarly to the proof of Proposition REF as follows.", "Since $S$ is normal, $\\rho _1$ and $\\rho _2$ each either commute with $\\pi $ or invert it.", "Note that since the blocks have odd size, both $\\rho _1$ and $\\rho _2$ must fix a vertex in each block.", "Then if either $\\rho _i$ commutes with $\\pi $ , it follows that $\\rho _i = \\pi ^{-k} \\rho _i \\pi ^k$ for all $k$ , which would demonstrate that $\\rho _i$ fixes all vertices, violating faithfulness of the action.", "So $\\rho _1$ and $\\rho _2$ both invert $\\pi $ , implying that $\\rho _1 \\rho _2$ commutes with $\\pi $ .", "Since $\\rho _1 \\rho _2$ fixes one of the vertices in $B_1$ , it must fix $B_1$ pointwise (by the same argument as for $\\rho _i$ ).", "The only way to maintain a faithful action is if $\\rho _1 \\rho _2$ does not fix any point of $B_2$ .", "Then if $\\rho _1 \\rho _2$ sends some vertex $w$ to $w \\pi ^k$ , it must send every $w \\pi ^i$ to $w \\pi ^{i+k}$ .", "Thus $\\rho _1 \\rho _2 = \\pi ^{k-1}$ of order $b$ , and so $q = b$ .", "Thus $q = b$ if and only if $\\Gamma (\\mathcal {P})$ has a normal $b$ -subgroup.", "Assume now that $\\mathcal {P}$ is non-flat and that $q=b$ .", "We have just seen that this implies $\\Gamma (\\mathcal {P})$ has a normal $b$ -subgroup.", "This normal subgroup acts on the vertices with two blocks of size $b$ .", "Since $\\rho _0$ interchanges these blocks, this implies that every relation in $\\Gamma (\\mathcal {P})$ has an even number of occurrences of $\\rho _0$ .", "In particular, $p$ is even.", "The order of $\\Gamma (\\mathcal {P})$ is $2qv = 4b^2$ , and so $p$ is a proper even divisor of $4b^2$ .", "If $\\mathcal {P}$ is not flat, then $2 < p < 2b$ , and so the only possible value of $p$ is 4.", "However, the order of $\\Gamma (\\mathcal {P})$ is also $2pf$ where $f$ is the number of facets of $\\mathcal {P}$ , and thus $pf = 2b^2$ , and so this is also impossible." ], [ "Flat polyhedra", "Using Proposition REF and Proposition REF , we can classify the flat orientably regular polyhedra with $2b$ vertices (that is, of type $\\lbrace 2b, q\\rbrace $ ) where $b$ is prime.", "If $b$ is an odd prime, then there are only two possibilities for $\\Lambda (2b,q^{\\prime })_{i,1}$ with $\\langle \\sigma _2 \\rangle $ core-free: $\\Lambda (2b, 2)_{-1,1}$ ($i = -1$ ) and $\\Lambda (2b, b)_{3,1}$ ($i = 3$ ).", "The three possibilities for $\\Lambda (p^{\\prime },q)_{-1,j}$ where $p^{\\prime }$ divides $p$ and $\\langle \\sigma _1 \\rangle $ is core-free are: $\\Lambda (2, q)_{-1,1}$ ($j = 1$ ), $\\Lambda (b, 2b)_{-1,-3}$ .", "($q = 2b$ by [9], $j = -3$ ), and $\\Lambda (2b, q)_{-1,j}$ .", "($i = -1$ , $j$ depends on $q$ ).", "Putting these together, we find the following possible automorphism groups of flat regular polyhedra with $2b$ vertices: $\\Lambda (2b, q)_{-1,1}$ , with $q$ even, $\\Lambda (2b, q)_{3,1}$ , with $q$ divisible by $b$ , $\\Lambda (2b, 2b)_{-1,-3}$ , and $\\Lambda (2b, q)_{-1,j}$ , with $q$ divisible by $2b$ but not $2b^2$ , and where $j$ is the unique positive integer with $1 \\le j \\le q-1$ that satisfies $\\frac{j+1}{2} \\equiv -1$ (mod $b$ ) and $\\frac{j+1}{2} \\equiv 1$ (mod $q/2b$ ).", "Of these, only $\\Lambda (2b, b)_{3,1}$ yields a vertex-faithful polyhedron.", "Now suppose that $b = 2$ (so that $\\mathcal {P}$ has 4 vertices).", "The only possibility for $\\Lambda (4,q^{\\prime })_{i,1}$ is $\\Lambda (4,2)_{-1,1}$ .", "There are two possibilities for $\\Lambda (p^{\\prime }, q)_{-1,j}$ ; either $\\Lambda (2, q)_{-1,1}$ or $\\Lambda (4, q)_{-1,j}$ , and in the second case $q$ is divisible by 8.", "Thus there are two families; one with group $\\Lambda (4,q)_{-1,1}$ with $q$ even, and one with group $\\Lambda (4,q)_{-1,j}$ with $q$ divisible by 8 and $j$ determined by $q$ .", "Now we consider the flat non-orientably regular polyhedra.", "By [9], the only such polyhedra with $2b$ vertices for an odd prime $b$ have type $\\lbrace 6, 4r\\rbrace $ , with $r \\ge 1$ and odd.", "Using the dual of [9], we find the following groups: The quotient of $[6, 4]$ by the relations $\\sigma _2^{-1} \\sigma _1 = \\sigma _1^{-1} \\rho _1 \\sigma _2^2$ and $\\sigma _2^{-1} \\sigma _1^2 =\\sigma _1^{-2} \\sigma _2$ .", "In fact, we can verify with a CAS that the second relation is uneccessary.", "The quotient of $[6, 4r]$ by the relations $\\sigma _2^{-1} \\sigma _1 = \\sigma _1^2 \\rho _1 \\sigma _2^{r+1}$ and $\\sigma _2^{-1} \\sigma _1^2 = \\sigma _1^{-2} \\sigma _2^{2r-1}$ , for $r \\equiv 1$ (mod 4).", "The quotient of $[6, 4r]$ by the relations $\\sigma _2^{-1} \\sigma _1 = \\sigma _1^2 \\rho _1 \\sigma _2^{-r+1}$ and $\\sigma _2^{-1} \\sigma _1^2 = \\sigma _1^{-2} \\sigma _2^{2r-1}$ , for $r \\equiv 3$ (mod 4).", "In cases (b) and (c), [9] shows that $\\langle \\sigma _2^4 \\rangle $ is normal, and so for $r > 1$ the corresponding polyhedra are not vertex-faithful.", "We can check with a CAS that $\\langle \\sigma _2 \\rangle $ is core-free when $r = 1$ for the group in case (b).", "Finally, there are two families of flat non-orientably regular polyhedra with 4 vertices (of type $\\lbrace 4, q\\rbrace $ ), described in [9].", "The first is the quotient of $[4, 3k]$ by the relations $\\sigma _2^{-1} \\sigma _1 = \\sigma _1^2 \\rho _1 \\sigma _2$ and $\\sigma _2^{-2} \\sigma _1 = \\sigma _1^{-1} \\sigma _2^2$ , and the second is the quotient of $[4, 6k]$ by the relations $\\sigma _2^{-1} \\sigma _1 = \\sigma _1^2 \\rho _1 \\sigma _2^{1+3k}$ and $\\sigma _2^{-2} \\sigma _1 = \\sigma _1^{-1} \\sigma _2^2$ .", "In fact, we can show that the second relation is unnecessary to define these groups.", "By [9], the quotient of $[4, 3k]$ by the single relation $\\sigma _2^{-1} \\sigma _1 = \\sigma _1^2 \\rho _1 \\sigma _2$ already causes $\\langle \\sigma _2^3 \\rangle $ to be normal, and taking the quotient by this normal subgroup yields the group of the hemicube, where the second relation is unnecessary.", "(This also demonstrates that if $k > 1$ then this polyhedron is not vertex-faithful since $\\langle \\sigma _2 \\rangle $ has a nontrivial core.)", "A similar argument holds for the quotient of $[4, 6k]$ by the single relation $\\sigma _2^{-1} \\sigma _1 = \\sigma _1^2 \\rho _1 \\sigma _2^{1+3k}$ , where $\\langle \\sigma _2^6 \\rangle $ is normal.", "We summarize all of the flat regular polyhedra with $2b$ vertices in Table REF .", "Table: Flat regular polyhedra with 4 or 2b2b vertices, where bb is an odd prime" ], [ "Non-flat polyhedra – the primitive case", "Now we will consider non-flat vertex-faithful regular polyhedra whose automorphism groups have a primitive action on their $2b$ vertices, where $b$ is prime.", "When $b \\in \\lbrace 2, 3, 5\\rbrace $ there do exist non-flat vertex-faithful regular polyhedra whose automorphism groups act primitively on their $2b$ vertices, for instance $\\lbrace 3,3\\rbrace $ , $\\lbrace 5,5\\rbrace _3$ , and $\\lbrace 5,3\\rbrace _5$ respectively.", "On the other hand, when $b=7$ , there are only two vertex-faithful regular polyhedra; both have automorphism groups which act imprimitively with blocks of 7 vertices.", "When $b = 11$ , we can use similar techniques as in Section REF to see that there are two vertex-faithful regular polyhedra with 22 vertices, both of which have automorphism groups acting imprimitively with blocks of size 11.", "When $b > 11$ , we will use the following result which relies on the classification of finite simple groups and the work of Liebeck and Saxl [18].", "Theorem 5.2 ([4], Theorem 1.1) Let $b$ be a prime number.", "A primitive permutation group of degree $2b$ is doubly transitive provided that $b \\ne 5$ .", "The results above then lead us to the following theorem.", "Theorem 5.3 When $b \\ge 7$ is prime, there are no non-flat vertex-faithful regular polyhedra with automorphism groups that act primitively on their $2b$ vertices.", "Assume to the contrary that there is a non-flat vertex-faithful regular polyhedron with automorphism group $\\Gamma $ acting primitively on $2b$ vertices where $b$ is prime.", "The discussion above shows that we may assume $b \\ge 13$ and that $\\Gamma $ acts doubly transitively.", "Since $\\Gamma $ acts doubly transitively on a set of size $2b$ , the stabilizer of the base vertex acts transitively on the remaining $2b-1$ vertices; in particular $| \\Gamma | = (k)(2b)(2b-1)$ .", "Additionally $| \\Gamma | = 4e$ , where $e$ is the number of edges of the polyhedron.", "Thus $k$ must be even.", "Also $| \\Gamma | = (2)(q)(2b)$ , and thus $k(2b-1) = 2q$ .", "When $k > 2$ (and thus $k >3$ ), this implies that $q > 2b =v$ , which contradicts Proposition REF .", "Thus $k=2$ , $q=2b-1$ , and $| \\Gamma | = 4b(2b-1)$ .", "Let $n$ be the number of Sylow $b$ -subgroups of $\\Gamma $ .", "We know that $n = 1 + tb$ for some integer $t$ , and that $n$ divides $8b-4$ .", "Thus $8b-4 = r(1+tb)$ for some integer $r$ .", "Solving this for $b$ we get $b=\\frac{r+4}{8-rt}$ .", "Furthermore, by Proposition REF , we know that $t \\ne 0$ .", "Thus, $rt \\le 7$ , and in particular $b \\le r +4 \\le 11$ ." ], [ "Non-flat polyhedra – the imprimitive case", "Now we will classify the non-flat vertex-faithful regular polyhedra with twice a prime number of vertices whose automorphism groups act imprimitively on the vertices.", "We again assume that $b$ is an odd prime with $b \\ge 11$ .", "Corollary REF provides the unique vertex-faithful regular polyhedron of type $\\lbrace p, 2b\\rbrace $ , so by Proposition REF , we may assume for the remainder of this section that $q < 2b$ ." ], [ "$b$ blocks of 2 vertices", "First we will show that there are no such regular polyhedra $\\mathcal {P}$ where the action of $\\Gamma (\\mathcal {P})$ on the vertices is imprimitive with blocks of size 2.", "Let $\\Gamma = \\Gamma (\\mathcal {P})$ and let $B_1$ be the block containing the base vertex of a regular polyhedron of type $\\lbrace p,q\\rbrace $ .", "In this case, $|\\textrm {Stab}(\\Gamma , B_1)| = 4q$ .", "Since $q < 2b$ , Proposition REF implies that $b$ does not divide $q$ .", "Thus the Sylow $b$ -subgroup of $\\Gamma $ is cyclic of order $b$ , generated by an element $\\pi $ that permutes the $b$ blocks cyclically.", "Note that, since the blocks have size 2 and $\\rho _1$ and $\\rho _2$ both fix the base vertex, they in fact both fix $B_1$ pointwise.", "Then by Theorem REF and Proposition REF , $\\Gamma $ acts doubly transitively on the blocks.", "In particular, $\\textrm {Stab}(\\Gamma , B_1)$ acts transitively on the $b-1$ remaining blocks.", "Then $ |\\textrm {Stab}(\\Gamma , B_1)| = |\\textrm {Stab}( \\textrm {Stab}(\\Gamma , B_1),B_2)| \\cdot (b-1), $ and since $\\rho _2 \\in \\textrm {Stab}( \\textrm {Stab}(\\Gamma , B_1),B_2)$ , it follows that $2b-2$ divides $4q$ .", "Let $n$ be the number of Sylow $b$ -subgroups.", "Since $\\langle \\pi \\rangle $ is not normal (by Proposition REF ) we have $n \\ne 1$ .", "Thus, we need for $4q$ to be a multiple of $2b-2$ and of $n$ , where $n = kb+1$ for some $k \\ge 1$ .", "Since $q \\le v-1$ , we have $4q \\le 8b-4$ .", "Then $4q = (2b-2)r$ with $1 \\le r \\le 4$ .", "Similarly $4q = (kb+1)s$ with $1 \\le k \\le 8$ and $1 \\le s \\le 8$ .", "With a CAS, we find that the only solutions this system of equations (with odd prime $b$ ) occur when $b \\le 11$ .", "We know that there are only two vertex-faithful polyhedra with 22 vertices: the one of type $\\lbrace 4, 22\\rbrace $ ($(\\lbrace 4,4\\rbrace _{(11,0)})^{\\pi \\delta }$ ) and the flat one of type $\\lbrace 22, 11\\rbrace $ described by the second line of Table REF .", "Therefore, for all primes $b \\ge 5$ , there is no non-flat vertex-faithful regular polyhedron with $2b$ vertices where the action of the automorphism group has blocks of size 2.", "Additionally it can be checked that when $b=2$ , this result also holds.", "On the other hand, the octahedron has an automorphism group that acts with 3 blocks of 2 vertices." ], [ "2 blocks of $b$ vertices", "Here, for all primes at least seven, we show that the non-flat vertex-faithful regular polyhedra from Corollary REF with twice a prime number of vertices is unique.", "Proposition 5.4 For all primes $b \\ge 7$ , if $\\mathcal {P}$ is a non-flat vertex-faithful regular polyhedron with $2b$ vertices such that the action on vertices is imprimitive with two blocks of size $b$ , then $q = 2b$ and $\\mathcal {P}$ is $(\\lbrace 4,4\\rbrace _{(b,0)})^{\\pi \\delta }$ .", "Let $\\Gamma = \\Gamma (\\mathcal {P})$ and suppose that $\\Gamma $ acts faithfully and imprimitively on the $2b$ vertices with two blocks each with $b$ vertices.", "Let the base vertex $u$ be in the block $B_1$ .", "The set of fixed points of $\\Gamma _0:=\\langle \\rho _1, \\rho _2 \\rangle = \\textrm {Stab}(\\Gamma ,u)$ forms a block for $\\Gamma $ , so the number of fixed points of $\\Gamma _0$ is either 1, 2, $b$ , or $2b$ .", "The number of fixed points for $\\Gamma _0$ cannot be $2b$ or $\\Gamma $ would not act faithfully.", "If the number of of fixed points is 2, then there is a block system for $\\Gamma $ consisting of $b$ blocks each of size 2; this was ruled out in subsection REF .", "Let $q^{\\prime }$ be a divisor of $q$ and let $H = \\langle \\sigma _2^{q^{\\prime }} \\rangle $ .", "Recall from the proof of Proposition REF that if $q^{\\prime } < q/2$ , then $ \\langle \\sigma _2^{q^{\\prime }} \\rangle $ fixes a vertex $u \\alpha $ if and only if $\\alpha $ normalizes $H$ .", "Suppose that $H$ fixes a vertex $u \\alpha $ in $B_2$ .", "Then $H \\alpha = \\alpha H$ , which implies that $H$ fixes a vertex $w$ if and only if it fixes $w \\alpha $ .", "Since $u$ is in $B_1$ and $u \\alpha $ is in $B_2$ , that means that $\\alpha $ interchanges $B_1$ with $B_2$ , and so the number of vertices fixed by $H$ must be even.", "By Proposition REF and the fact that $\\mathcal {P}$ is vertex-faithful, this implies that either $H$ fixes exactly two vertices, or that $q^{\\prime } = q/2$ .", "Consider what this implies when $q^{\\prime }=1$ and $H = \\langle \\sigma _2 \\rangle $ .", "If $\\sigma _2$ fixes a vertex $v$ , then either both $\\rho _1$ and $\\rho _2$ fix $v$ , or there is a distinct vertex $w = (v) \\rho _1 = (v) \\rho _2$ .", "If the number of fixed points of $H$ is 2, then since $\\Gamma _0$ fixes the base vertex $u$ , then $\\Gamma _0$ also fixes the other fixed point of $H$ .", "However, we have already ruled out the case where $\\Gamma _0$ has 2 fixed points, so the number of fixed points of $H$ cannot be 2, and thus $\\sigma _2$ cannot fix any vertex in $B_2$ Therefore, the orbits of vertices in $B_2$ under $ \\langle \\sigma _2 \\rangle $ all have size at least 2.", "Suppose that there are two orbits of coprime size $2 \\le m < n$ , where $m$ and $n$ are divisors of $q$ .", "Note that $m \\ne q/2$ since if $m = q/2$ , then $n > m$ would force $n = q$ , in which case $m$ and $n$ are not coprime.", "The above argument shows that $\\langle \\sigma _2^m \\rangle $ fixes exactly two vertices.", "However, in addition to fixing the base vertex u, the group $\\langle \\sigma _2^m \\rangle $ fixes at least $m$ vertices of $B_2$ that are in the same $ \\langle \\sigma _2 \\rangle $ orbit.", "Therefore, the orbit sizes in $B_2$ all have a common factor.", "Since there are no orbits of size 1 and the block has prime size $b$ , it follows that the orbit must be the entire block $B_2$ .", "Therefore, $b$ divides $q$ , and by Proposition REF , $q \\ne b$ .", "Then Proposition REF and Corollary REF imply the rest of the claim." ], [ "Representation of a flat polyhedron", "Let us also examine what happens to the above argument when $q=b$ and thus $\\mathcal {P}$ is flat.", "Consider the vertex CPR graph of $\\Gamma $ .", "Since $\\sigma _2$ has prime order $b$ and it fixes one vertex of $B_1$ , all the orbits in $B_1$ must have size 1, which is to say that $\\sigma _2$ fixes $B_1$ pointwise.", "Thus, the $b-1$ vertices in $B_1$ other than the base vertex are connected in pairs by parallel edges labeled 1 and 2.", "This completely determines the induced subgraph on vertices in $B_1$ .", "By the argument of Proposition REF , the vertices of $B_2$ form a single $\\langle \\sigma _2 \\rangle $ orbit, and so the induced subgraph on vertices in $B_2$ is an alternating path of length $b$ .", "This leaves only 1 possible vertex CPR graph for $\\Gamma $ : $\\includegraphics [height=6cm]{PossiblePiNormal.pdf}$ In this case $p$ is equal to $2b$ and $\\Gamma $ is the automorphism group of the flat vertex-faithful regular polytope of type $\\lbrace 2b, b\\rbrace $ (see Table REF )." ], [ "Classification of all regular polyhedra with twice a prime number of vertices", "We have seen the classification of the regular flat polyhedra with $2b$ vertices, and we have constructed the unique non-flat vertex-faithful regular polyhedron with $2b$ vertices for each prime $b \\ge 7$ .", "Now we will classify the non-flat regular polyhedra with 4, 6, and 10 vertices.", "Any such polyhedron covers one of the vertex-faithful polyhedra in Table REF .", "Lemma 5.5 Suppose that $\\mathcal {P}$ is a non-flat regular polyhedron of type $\\lbrace p, q\\rbrace $ that is not vertex-faithful, and let $\\mathcal {Q}$ be the vertex-faithful regular polyhedron of type $\\lbrace p, q^{\\prime }\\rbrace $ such that $\\mathcal {P}$ covers $\\mathcal {Q}$ .", "If $\\mathcal {Q}= \\lbrace 3, 3\\rbrace $ , then $q = 6$ or 12.", "If $\\mathcal {Q}= \\lbrace 3, 4\\rbrace $ , then $q = 8, 12$ , or 24.", "If $\\mathcal {Q}$ is one of the following, then $q = 2q^{\\prime }$ : $\\lbrace 3, 5\\rbrace *60, \\lbrace 5,5\\rbrace *60, \\lbrace 4,6\\rbrace *120, \\lbrace 5,3\\rbrace *60, \\lbrace 5,6\\rbrace *120a,\\lbrace 6,6\\rbrace *120$ .", "For parts (a) and (b), Proposition REF (d) says that $\\rho _0$ inverts $\\sigma _2^{q^{\\prime }}$ in $\\Gamma (\\mathcal {P})$ .", "Adding the relation $\\rho _0 \\sigma _2^{q^{\\prime }} \\rho _0 = \\sigma _2^{-q^{\\prime }}$ to the group $[3, \\infty ]$ with $q^{\\prime } = 3$ causes a collapse to a quotient of $[3, 12]$ , and when $q^{\\prime } = 4$ we get a collapse to a quotient of $[3, 24]$ .", "For part (c), we can appeal to Corollary REF , using the information in Table REF .", "Given this bound on $q$ , and thus on the size of the group, we can use [14] to finish the classification of non-vertex-faithful regular polyhedra.", "Table REF summarizes such polyhedra with 4, 6, or 10 vertices.", "Table: The non-flat, non-vertex-faithful regular polyhedra with 4, 6, or 10 vertices.With the small cases fully understood, this leads us to the following theorem.", "Theorem 5.6 A regular polyhedron with $2b$ vertices (where $b$ is prime) is one of the following: A vertex-faithful polyhedron with $b \\le 5$ , described in Table REF , A flat polyhedron described in Table REF , A non-vertex-faithful polyhedron with $b \\le 5$ , described in Table REF , or $(\\lbrace 4,4\\rbrace _{(b,0)})^{\\pi \\delta }$ , which is the unique non-flat vertex-faithful regular polyhedron of type $\\lbrace 4,2b \\rbrace $ with $b \\ge 3$ .", "The analysis in this section proves that the only vertex-faithful non-flat regular polyhedron with $2b$ vertices and $b \\ge 7$ is $(\\lbrace 4,4\\rbrace _{(b,0)})^{\\pi \\delta }$ .", "Lemma REF then says that no other (non-vertex-faithful) polyhedra with $2b$ vertices cover this polyhedron." ], [ "Flat polyhedra", "The flat regular polyhedra with 4 vertices were covered in Section  and the corresponding polyhedra are in Table REF .", "So we will assume that $b$ is an odd prime.", "Again, we use Proposition REF and Proposition REF to determine the flat orientably regular polyhedra of type $\\lbrace b^2, q\\rbrace $ .", "The only possibility for $\\Lambda (b^2,q^{\\prime })_{i,1}$ with $\\langle \\sigma _2 \\rangle $ core-free is $\\Lambda (b^2, 2)_{-1,1}$ .", "There are two possibilities for $\\Lambda (p^{\\prime }, q)_{-1,j}$ with $p^{\\prime }$ dividing $b^2$ and $\\langle \\sigma _1 \\rangle $ core-free: either $\\Lambda (b^2, 2b^2)_{-1,-3}$ or $\\Lambda (b, 2b)_{-1,-3}$ .", "Putting these together, we find two flat orientably regular polyhedra with $b^2$ vertices: one of type $\\lbrace b^2, 2b^2\\rbrace $ with group $\\Lambda (b^2, 2b^2)_{-1,-3}$ , and one of type $\\lbrace b^2, 2b\\rbrace $ with group $\\Lambda (b^2, 2b)_{-1,-3}$ .", "Moving on to non-orientably regular flat polyhedra, [9] implies that there are no non-orientably regular flat polyhedra of type $\\lbrace b^2, q\\rbrace $ for any odd prime $b \\ge 5$ .", "When $b = 3$ , there is a single non-orientably regular flat polyhedron, which has type $\\lbrace 9, 4\\rbrace $ .", "The group of this polyhedron is the quotient of $[9, 4]$ by the extra relation $(\\rho _0 \\rho _1 \\rho _2 \\rho _1)^2 \\rho _2 = 1$ [14].", "We summarize the flat regular polyhedra with $b^2$ vertices in Table REF .", "Table: Flat regular polyhedra with b 2 b^2 vertices, where bb is an odd prime" ], [ "Non-flat polyhedra", "Fully classifying the non-flat regular polyhedra with $b^2$ vertices appears to be somewhat more difficult than the problems we have considered so far.", "Instead, we will consider only the smallest non-flat regular polyhedra with $b^2$ vertices (where by “smallest” we mean having the fewest flags).", "Since the number of flags is $2qv = 2qb^2$ , finding the smallest polyhedra amounts to finding the smallest value of $q$ .", "Furthermore, since the number of flags is also $4e$ , we find that $q$ must be even if $b$ is odd.", "If $\\mathcal {P}$ is not flat, then $q \\ne 2$ .", "Then the smallest possible value for $q$ is $q = 4$ , and there is at least one non-flat regular polyhedron with $b^2$ vertices: the toroidal map $\\lbrace 4, 4\\rbrace _{(b,0)}$ .", "Let us classify the non-flat regular polyhedra of type $\\lbrace p, 4\\rbrace $ with $b^2$ vertices.", "Lemma 6.1 Suppose that $b$ is an odd prime and that $\\mathcal {P}$ is a non-flat regular polyhedron of type $\\lbrace p, 4\\rbrace $ with $b^2$ vertices.", "Then: $\\Gamma (\\mathcal {P})$ has a normal Sylow $b$ -subgroup.", "$p \\ne b$ .", "First, let $n$ be the number of Sylow $b$ -subgroups of $\\Gamma (\\mathcal {P})$ .", "We have $|\\Gamma (\\mathcal {P})| = 2qv = 8b^2$ .", "Then by the Sylow theorems, $n$ divides 8 and $n \\equiv 1 ( \\textrm {mod } b)$ .", "Clearly if $b \\ge 11$ then $n = 1$ .", "For $3 \\le b \\le 7$ , we can verify the claim using [14] and a CAS.", "Now, suppose that $p = b$ and let $S$ be the normal Sylow $b$ -subgroup.", "Since $|\\Gamma (\\mathcal {P})| = 8b^2 = 2pf$ , the polyhedron $\\mathcal {P}$ has $4b$ faces.", "Consider the action of $\\Gamma (\\mathcal {P})$ on the faces.", "Since $S$ is normal, the orbits of the faces under $S$ form a system of blocks for $\\Gamma $ .", "Let $B_1$ be the block containing the base face and let $B_2 = (B_1) \\rho _2$ .", "Then $\\rho _0$ stabilizes $B_1$ and thus $B_2$ (since $\\rho _0$ commutes with $\\rho _2$ ), and $\\rho _1$ stabilizes $B_1$ .", "Furthermore, since $\\rho _0 \\rho _1$ has order $b$ , it must lie in $S$ , and so it stabilizes $B_2$ .", "Thus $\\rho _1$ also stabilizes $B_2$ , and so the block system consists of these two blocks only.", "The size of each block divides $b^2$ , and so the total number of faces must divide $2b^2$ , contradicting that there are $4b$ faces.", "We end with a classification of the smallest regular non-flat polyhedra with a prime squared number of vertices.", "Theorem 6.2 For $b \\ge 3$ , up to isomorphism, there are exactly two smallest regular non-flat polyhedra with $b^2$ vertices: the toroidal map $\\lbrace 4, 4\\rbrace _{(b,0)}$ and its Petrial of type $\\lbrace 2b, 4\\rbrace $ .", "Suppose that $\\mathcal {P}$ is a smallest regular non-flat polyhedron with $b^2$ vertices.", "We have already established that $\\mathcal {P}$ must have type $\\lbrace p, 4\\rbrace $ for some $p$ .", "Furthermore, the toroidal map $\\lbrace 4,4\\rbrace _{(b,0)}$ and its Petrial of type $\\lbrace 2b, 4\\rbrace $ both have $b^2$ vertices.", "First, let us show that there is no other possible value of $p$ .", "We have that $|\\Gamma (\\mathcal {P})| = 8b^2 = 2pf$ .", "In order for $\\mathcal {P}$ to be non-flat, we need $2 < p < b^2$ and for $p$ to properly divide $4b^2$ .", "Lemma REF rules out the case $p = b$ , and if $p = 4b$ , then $f = b$ , and the dual of Theorem REF implies that $\\mathcal {P}$ must be flat.", "So $p = 4$ or $p = 2b$ .", "Now, if $p = 4$ , then $\\mathcal {P}$ is a regular polyhedron of type $\\lbrace 4, 4\\rbrace $ , and so it must be either the toroidal map $\\lbrace 4, 4\\rbrace _{(a,0)}$ , which has $a^2$ vertices, or the toroidal map $\\lbrace 4,4\\rbrace _{(a,a)}$ , which has $2a^2$ vertices.", "So $\\mathcal {P}$ must be $\\lbrace 4,4\\rbrace _{(b,0)}$ .", "If instead $p = 2b$ , then $|\\Gamma (\\mathcal {P})| = 8b^2 = 2pf$ implies that $f = 2b$ .", "Then $\\mathcal {P}^{\\delta }$ is a non-flat regular polyhedron of type $\\lbrace 4, 2b\\rbrace $ with $2b$ vertices.", "By Theorem REF , $\\mathcal {P}^{\\delta }$ must be $(\\lbrace 4,4\\rbrace _{(b,0)})^{\\pi \\delta }$ , proving the claim.", "Acknowledgement: We would like to thank the referees for many useful comments." ] ]

[ [ "Chaos, Extremism and Optimism: Volume Analysis of Learning in Games" ], [ "Abstract We present volume analyses of Multiplicative Weights Updates (MWU) and Optimistic Multiplicative Weights Updates (OMWU) in zero-sum as well as coordination games.", "Such analyses provide new insights into these game dynamical systems, which seem hard to achieve via the classical techniques within Computer Science and Machine Learning.", "The first step is to examine these dynamics not in their original space (simplex of actions) but in a dual space (aggregate payoff space of actions).", "The second step is to explore how the volume of a set of initial conditions evolves over time when it is pushed forward according to the algorithm.", "This is reminiscent of approaches in Evolutionary Game Theory where replicator dynamics, the continuous-time analogue of MWU, is known to always preserve volume in all games.", "Interestingly, when we examine discrete-time dynamics, both the choice of the game and the choice of the algorithm play a critical role.", "So whereas MWU expands volume in zero-sum games and is thus Lyapunov chaotic, we show that OMWU contracts volume, providing an alternative understanding for its known convergent behavior.", "However, we also prove a no-free-lunch type of theorem, in the sense that when examining coordination games the roles are reversed: OMWU expands volume exponentially fast, whereas MWU contracts.", "Using these tools, we prove two novel, rather negative properties of MWU in zero-sum games: (1) Extremism: even in games with unique fully mixed Nash equilibrium, the system recurrently gets stuck near pure-strategy profiles, despite them being clearly unstable from game theoretic perspective.", "(2) Unavoidability: given any set of good points (with your own interpretation of \"good\"), the system cannot avoid bad points indefinitely." ], [ "Introduction", "In recent years, fuelled by AI applications such as Generative Adversarial Networks (GANs), there has a been a strong push towards a more detailed understanding of the behavior of online learning dynamics in zero-sum games and beyond.", "Even when focusing on the canonical case of bilinear zero-sum games, the emergent behavior depends critically on the choice of the training algorithms.", "Results can macroscopically be grouped in three distinct categories: convergent, divergent and cyclic/recurrent.", "Specifically, most standard regret minimizing dynamics and online optimization dynamics, such as Multiplicative Weights Updates (MWU) or gradient descent [6], although their time average converges [12], their day-to-day behavior diverges away from Nash equilibria [3], [7].", "On the other hand, some game-theoretically inspired dynamics, such as Optimistic Multiplicative Weights Updates (OMWU) converge [10], [9].", "(Numerous other convergent heuristics have also been recently analyzed, e.g.", "[18], [14], [15], [4], [1].)", "Finally, if we simplify learning into continuous-time ordinary differential equations (ODEs), such as Replicator Dynamics, the continuous time analogue of MWU, the emergent behavior becomes almost periodic (Poincaré recurrence) [20], [19], [5].", "This level of complex case-by-case analysis just to understand bilinear zero-sum games seems daunting.", "Can we find a more principled approach behind these results that is also applicable to more general games?", "One candidate is volume analysis, a commonly used tool in the area of Dynamical Systems.", "Briefly speaking, what it does is to consider a set of starting points with positive volume (Lebesgue measure), and analyze how the volume changes as the set evolves forward in time.", "As we shall see, an advantage of volume analysis is its general applicability, for it can be used to analyze not just ODEs but different discrete-time algorithms such as MWU and OMWU in different types of games.", "In Evolutionary Game Theory, volume analysis has been applied to continuous-time dynamical systems/ODEs (see [16] and [21]).", "Eshel and Akin [11] showed that RD in any multi-player matrix game is volume preserving in the dual (aggregate payoff) space.", "This result is in fact a critical step in the proof of Poincaré recurrence in zero-sum games.", "Loosely speaking, if we think of the set of initial conditions as our uncertainty about where is the starting point of the system, since uncertainty does not decrease (convergence) or increase (divergence) we end up cycling in space.", "Cheung and Piliouras [8] recently applied volume analysis to discrete-time numerical algorithms in a series of games, including two-person zero-sum games, graphical constant-sum games, generalized Rock-Paper-Scissors games and general $2\\times 2$ bimatrix games.", "Among other results, they showed that MWU in zero-sum games is Lyapunov chaotic in the dual space.", "This is done by showing that the volume of any set is expanding exponentially fast.", "Lyapunov chaos is one of the most classical notions in the area of Dynamical Systems that captures instability and unpredictability.", "More precisely, it captures the following type of butterfly effect: when the starting point of a dynamical system is slightly perturbed, the resulting trajectories and final outcomes diverge quickly.", "Lyapunov chaos means that such system is very sensitive to round-off errors in computer simulations; thus, the result of Cheung and Piliouras provides a rigorous mathematical explanation to the numerical instability of MWU experiments.", "Our Contributions, and Roadmap for This Paper.", "Our contributions can be summarized into two categories, both stemming from volume analyses.", "First, besides the numerical instability and unpredictability already mentioned, we discover two novel and negative properties of MWU in zero-sum games in this paper, which are consequences of exponential volume expansion.", "We call them unavoidability and extremism.", "We have given informal descriptions of these two properties in the abstract; we will give more details about them below.", "Second, we carry out volume analysis on OMWU and discover that its volume-change behavior is in stark contrast with MWU.", "To understand why we should be interested in such an analysis, we first point out that in the study of game dynamics, a primary target is to seek algorithms that behave well in as broad family of games as possible.", "Recently, OMWU was shown to achieve stability in zero-sum games, despite its strong similarity with MWU (which is chaotic in zero-sum games).", "It is natural to ask how these stability behaviors generalize to other games.", "We provide a negative answer, by proving that OMWU is volume-expanding and Lyapunov chaotic in coordination games; see Figure REF for a summary of this no-free-lunch phenomena.", "We show that the volume is exponentially decreasing for OMWU in zero-sum game, mirroring with the recent stability results [9], [10] in the original (primal) space (simplex of actions) about these game dynamics.", "The details are presented in Section .", "As unavoidability and extremism are shown via volume expansion argument, it is easy to generalize and show that these two properties also appear in OMWU in coordination games.", "On a technical note, the volume analysis on OMWU is more involved than that on MWU.", "Along the process, we propose an ODE system which is the continuous-time analogue of OMWU in games, and the volume analysis relies crucially on the fact that discrete-time OMWU is an online Euler discretization of the ODE system; see Section  for details.", "Figure: How volume changes in the dual space.", "“++” denotes exponential volume expansion, unavoidability and extremism, while “--” denotes exponential volume contraction.See Figures  and  for graphical illuminations.In Section REF , we discuss how volume analyses can be carried out on learning algorithms that are gradual, i.e.", "controlled by a step-size $\\epsilon $ .", "We demonstrate that volume analyses can often be boiled down to analyzing a polynomial of $\\epsilon $ that arises from the expansion of the determinant of some Jacobian matrix.", "This convincingly indicates that volume analyses can be readily applicable to a broad family of learning algorithms.", "In the rest of this introduction, we discuss extremism and unavoidability with more details.", "Section  contains the necessary background for this work.", "All missing proofs can be found in the appendix.", "Extremism (Section ).", "The more formal statement of our extremism theorem (Theorem REF ) is: given any zero-sum game that satisfies mild regularity condition, there is a dense set of starting points from which MWU will lead to a state where both players concentrate their game-plays on only one strategy.", "More precisely, let $\\mathbf {x},\\mathbf {y}$ denote the mixed strategies of the two players.", "For any $\\delta > 0$ , there is a dense set of starting points $(\\mathbf {x}^0,\\mathbf {y}^0)$ , from which MWU with a suitably small step-size leads to $(\\mathbf {x}^t,\\mathbf {y}^t)$ for some time $t$ , where there exists a strategy $j$ of Player 1 with $x^t_j \\ge 1-\\delta $ , and a strategy $k$ of Player 2 with $y^t_k \\ge 1-\\delta $ .", "To understand how bizarre extremism is, consider the classical Rock-Paper-Scissors game, which is a zero-sum game with a unique fully-mixed Nash equilibrium, where each strategy is chosen with equal probability $1/3$ .", "The extremism theorem indicates that there exists a starting point arbitrarily close to the Nash equilibrium, which will eventually lead to a situation where each player essentially sticks with one strategy for a long period of timeWhen $x_j^t < \\delta $ , it takes at least $\\Omega \\left( \\frac{1}{\\epsilon }\\ln \\frac{1}{\\delta } \\right)$ time before $x_j$ can possibly resume a “normal” value, say above $1/20$ .. As no pure Nash equilibrium exists, the trajectory will recurrently approach and then escape such extremal points infinitely often (Theorem REF ), demonstrating that the dynamic is very unstable.", "Unavoidability (Section ).", "The extremism theorem is actually an indirect consequence of an unavoidability theorem of MWU in zero-sum games.", "Unavoidability is a notion first introduced in a topic of (automatic) control theory called “avoidance control” [17], which addresses the following type of problems: for dynamical/automatic systems, analyze whether they can always avoid reaching certain bad states, e.g.", "collisions of robots/cars, or places with severe weather conditions.", "To explain unavoidability of MWU in general games, we need another notion of uncontrollability.", "Let $U$ be a region which is in the strict interior of the primal simplex, and let $V$ be the correspondence set of $U$ in the dual space.", "Informally, we say a region $U$ is uncontrollable if any subset of $V$ is exponentially volume expanding in the dual space.", "As the volume is expanding quickly, it is impossible for $V$ to contain the evolved set after a sufficiently long period of time, which, when converting back to the primal space, implies that the dynamic escapes from $U$ (Theorem REF ).", "When $U$ is thought as a set of good points and its complements are the set of bad points, the above discussions can be summarized by a punchline: When a good set is uncontrollable, the bad set is unavoidable.", "Note that the above discussion concerns general games.", "When we narrow down to zero-sum games, the results of Cheung and Piliouras [8] indicate that under mild regularity condition, any set $U$ in the the strict interior of the primal simplex is uncontrollable.", "Thus, for MWU in zero-sum game, you can have whatsoever interpretation of “good”, but the corresponding bad set is unavoidable.", "Some ideas behind the proof of unavoidability come from Cheung and Piliouras [8], who demonstrated several negative properties of MWU in special games, including generalized Rock-Paper-Scissors games.", "Our key contribution here is to formulate and prove the fully generalized statement about this property.", "In the proof of the extremism theorem, the first step uses unavoidability to show that extremism appears for one player.", "But to show that extremism appears for both players simultaneously, we need some substantially novel arguments in the subsequent steps." ], [ "Games", "In this paper, we focus on two-person general normal-form games.", "The strategy set of Player $i$ is $S_i$ .", "Let $n=|S_1|$ and $m=|S_2|$ .", "We assume $n,m\\ge 2$ throughout this paper.", "Let $\\mathbf {A}$ and $\\mathbf {B}$ be two $S_1\\times S_2$ matrices, which represent the payoffs to Players 1 and 2 respectively.", "We assume all payoffs are bounded within the interval $[-1,1]$ .", "Let $\\Delta ^d := \\left\\lbrace (z_1,z_2,\\cdots ,z_d)\\in \\mathbb {R}^d ~\\big |~ \\sum _{j=1}^d z_j = 1,~~\\text{and}~~\\forall ~j,~z_j\\ge 0\\right\\rbrace $ .", "We call $\\Delta := \\Delta ^n \\times \\Delta ^m$ the primal simplex or primal space of the game, which contains the set of all mixed strategy profiles of the two players.", "We use $\\mathbf {x}\\in \\Delta ^n$ and $\\mathbf {y}\\in \\Delta ^m$ to denote strategy vectors of Players 1 and 2 respectively.", "When a zero-sum game is concerned, only the matrix $\\mathbf {A}$ needs to be specified, as $\\mathbf {B}= -\\mathbf {A}$ .", "Definition 1 A zero-sum game $(\\mathbf {A},-\\mathbf {A})$ is trivial if there exist real numbers $a_1,a_2,\\cdots ,a_n$ and $b_1,b_2,\\cdots ,b_m$ such that $A_{jk} = a_j + b_k$ .", "A trivial game is not interesting as each player has a clear dominant strategy; for Player 1 it is $\\operatornamewithlimits{arg\\,max}_{j\\in S_1} a_j$ , while for Player 2 it is $\\operatornamewithlimits{arg\\,min}_{k\\in S_2} b_k$ .", "Following [8], we measure the distance of a zero-sum game $\\mathbf {A}$ from triviality by $c(\\mathbf {A}) ~:=~ \\min _{\\begin{array}{c}a_1,\\cdots ,a_n\\in \\mathbb {R}\\\\b_1,\\cdots ,b_m\\in \\mathbb {R}\\end{array}}~~\\left[\\max _{\\begin{array}{c}j\\in S_1\\\\k\\in S_2\\end{array}}~ \\left(A_{jk} - a_j - b_k\\right)~~-~~ \\min _{\\begin{array}{c}j\\in S_1\\\\k\\in S_2\\end{array}}~ \\left(A_{jk} - a_j - b_k\\right)\\right].$ Observe that if $\\mathbf {A}^{\\prime }$ is a sub-matrix of $\\mathbf {A}$ , then $c(\\mathbf {A}^{\\prime }) \\le c(\\mathbf {A})$ .", "If one of the two dimensions of $\\mathbf {A}$ is one, then $c(\\mathbf {A}) = 0$ .", "By setting all $a_j,b_k$ to zero, we have the trivial bound $c(\\mathbf {A}) \\le 2$ .", "For a coordination game, i.e.", "a game with payoff matrices in the form of $(\\mathbf {A},\\mathbf {A})$ , we also measure its distance from triviality using Equation (REF )." ], [ "MWU and OMWU Update Rules in Dual and Primal Spaces", "As is well-known, MWU and OMWU can be implemented either in the primal space, or in a dual space.", "The dual space is $\\mathcal {D}:= \\mathbb {R}^n\\times \\mathbb {R}^m$ , in which MWU with positive step size $\\epsilon $ generates a sequence of updates $(\\mathbf {p}^0,\\mathbf {q}^0),(\\mathbf {p}^1,\\mathbf {q}^1),$ $(\\mathbf {p}^2,\\mathbf {q}^2),\\cdots $ , where $p_j^t - p_j^0$ is $\\epsilon $ times the cumulative payoff to Player 1's strategy $j$ up to time $t$ , and $q_k^t - q_k^0$ is $\\epsilon $ times the cumulative payoff to Player 2's strategy $k$ up to time $t$ .", "At each time step, a point $(\\mathbf {p}^t,\\mathbf {q}^t)\\in \\mathcal {D}$ is converted to a point $(\\mathbf {x}^t,\\mathbf {y}^t)=(\\mathbf {x}(\\mathbf {p}^t),\\mathbf {y}(\\mathbf {q}^t))\\in \\Delta $ by the following rules: $x_j^t = x_j(\\mathbf {p}^t) = \\exp (p_j^t)\\left\\bad.\\left(\\sum _{\\ell \\in S_1} \\exp (p_\\ell ^t)\\right)\\right.~~~\\text{and}~~~y_k^t = y_k(\\mathbf {q}^t) = \\exp (q_k^t)\\left\\bad.\\left(\\sum _{\\ell \\in S_2} \\exp (q_\\ell ^t)\\right)\\right..$ For convenience, we let $\\mathsf {G}$ denote the function that converts a dual point to a primal point, i.e.", "$\\mathsf {G}(\\mathbf {p},\\mathbf {q}) = (\\mathbf {x}(\\mathbf {p}),\\mathbf {y}(\\mathbf {q}))$ .", "For MWU in a general-sum game, the payoffs to Player 1's all strategies in round $(t+1)$ can then be represented by the vector $\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^t)$ , while the payoffs to Player 2's all strategies in round $(t+1)$ can be represented by the vector $\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p}^t)$ .", "Accordingly, the MWU update rule in the dual space can be written as $\\mathbf {p}^{t+1} ~=~ \\mathbf {p}^t + \\epsilon \\cdot \\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^t)~~~~~~\\text{and}~~~~~~\\mathbf {q}^{t+1} ~=~ \\mathbf {q}^t + \\epsilon \\cdot \\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p}^t).$ The above update rule in the dual space is equivalent to the following MWU update rule in the primal space with starting point $\\mathsf {G}(\\mathbf {p}^0,\\mathbf {q}^0)$ , which some readers might be more familiar with: $x_j^{t+1} ~=~ \\frac{x_j^t \\cdot \\exp (\\epsilon \\cdot [\\mathbf {A}\\cdot \\mathbf {y}^t]_j)}{\\sum _{\\ell \\in S_1} x_\\ell ^t \\cdot \\exp (\\epsilon \\cdot [\\mathbf {A}\\cdot \\mathbf {y}^t]_\\ell )}~~~~~~\\text{and}~~~~~~y_k^{t+1} ~=~ \\frac{y_k^t \\cdot \\exp (\\epsilon \\cdot [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}^t]_k)}{\\sum _{\\ell \\in S_2} y_\\ell ^t \\cdot \\exp (\\epsilon \\cdot [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}^t]_\\ell )}.$ For OMWU in a general-sum game with step-size $\\epsilon $ , its update rule in the dual space starts with $(\\mathbf {p}^0,\\mathbf {q}^0)=(\\mathbf {p}^1,\\mathbf {q}^1)$ , and for $t\\ge 1$ , $\\mathbf {p}^{t+1} = \\mathbf {p}^t + \\epsilon \\cdot \\left[2\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^t)-\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^{t-1})\\right]~~\\text{and}~~\\mathbf {q}^{t+1} = \\mathbf {q}^t + \\epsilon \\cdot \\left[2\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p}^t) - \\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p}^{t-1})\\right],$ where $\\mathbf {x}(\\mathbf {p}^t),\\mathbf {y}(\\mathbf {q}^t)$ are as defined in (REF ).", "The above update rule in the dual space has an equivalent update rule in the primal space, which is the same as (REF ), except we replace $\\mathbf {A}\\cdot \\mathbf {y}^t$ there by $2\\mathbf {A}\\cdot \\mathbf {y}^t-\\mathbf {A}\\cdot \\mathbf {y}^{t-1}$ , and replace $\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}^t$ there by $2\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}^t - \\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}^{t-1}$ .", "Note that for the update rule (REF ), for $t\\ge 2$ , we have $\\mathbf {p}^t-\\mathbf {p}^0 = \\epsilon (\\sum _{\\tau =1}^{t-2} \\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^\\tau ) ~+~ 2\\cdot \\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^{t-1}))$ , which can be viewed as $\\epsilon $ times the cumulative payoff to strategy $j$ from time 2 up to time $t$ , but with a double weight on the last-iterate payoff." ], [ "Relationships between Primal and Dual Spaces", "Here, we clarify some facts about primal and dual spaces and their relationships.", "Equation (REF ) provides a conversion from a point in $\\mathcal {D}$ to a point in the interior of the primal space, i.e., $\\mathsf {int}(\\Delta )$ .", "It is not hard to see that there exist multiple points in $\\mathcal {D}$ which convert to the same point in $\\mathsf {int}(\\Delta )$ .", "Precisely, by [8], if $(\\mathbf {p},\\mathbf {q}),(\\mathbf {p}^{\\prime },\\mathbf {q}^{\\prime })\\in \\mathcal {D}$ , then $(\\mathbf {x}(\\mathbf {p}),\\mathbf {y}(\\mathbf {q})) = (\\mathbf {x}(\\mathbf {p}^{\\prime }),\\mathbf {y}(\\mathbf {q}^{\\prime }))$ if and only if $\\mathbf {p}-\\mathbf {p}^{\\prime } = c_1 \\cdot \\mathbf {1}$ and $\\mathbf {q}-\\mathbf {q}^{\\prime } = c_2 \\cdot \\mathbf {1}$ for some $c_1,c_2\\in \\mathbb {R}$ .", "For any $S\\subset \\mathsf {int}(\\Delta )$ , we let $\\mathsf {G}^{-1}(S)$ denote the set of points $(\\mathbf {p},\\mathbf {q})$ in the dual space $\\mathcal {D}$ such that $\\mathsf {G}(\\mathbf {p},\\mathbf {q})\\in S$ .", "Since the primal and dual spaces are not in one-one correspondence, some readers might argue that the reduced dual space used by Eshel and Akin [11] (in which its $(n+m-2)$ dual variables denote the quantities $p_1-p_n,p_2-p_n,\\cdots ,p_{n-1}-p_n,q_1-q_m,q_2-q_m,\\cdots ,q_{m-1}-q_m$ ) is a better choice.", "Our reason for choosing $\\mathcal {D}$ as the dual space to work with is simply because we are unable to establish the same type of results (like Lemma REF below) for the reduced dual space.", "The following proposition shows that volume expansion in the dual space implies large diameter in the primal space, if the corresponding primal set is bounded away from the simplex boundary.", "Proposition 2 Let $S$ be a set in the dual space with Lebesgue volume $v$ .", "Suppose there exists $j\\in S_1$ and $k\\in S_2$ such that $\\max _{(\\mathbf {p},\\mathbf {q})\\in S} p_j - \\min _{(\\mathbf {p},\\mathbf {q})\\in S} p_j \\le R_j$ and $\\max _{(\\mathbf {p},\\mathbf {q})\\in S} q_k - \\min _{(\\mathbf {p},\\mathbf {q})\\in S} q_k \\le R_k$ .", "Also, suppose that for some $\\kappa > 0$ , there exists a point $(\\mathbf {x},\\mathbf {y})\\in \\mathsf {G}(S)$ such that either every entry of $\\mathbf {x}$ is at least $\\kappa $ or every entry of $\\mathbf {y}$ is at least $\\kappa $ .", "Then the diameter of $\\mathsf {G}(S)$ is at least $\\left[ 1 - \\exp \\left(-\\frac{1}{4} \\cdot \\left( \\frac{v}{R_j R_k} \\right)^{1/(n+m-2)}\\right) \\right] \\cdot \\kappa .$ We point out that while we use volume as the mean for analyses, when measuring instability what we really care is the diameter of the set $S$ or its corresponding primal set.", "Indeed, volume is not an ideal benchmark, as we present concrete examples in Appendix  to show that (A) volume contraction in the dual space does not necessarily imply stability in either the dual or the primal space; (B) volume expansion in the dual space does not necessarily imply instability in the primal space if the primal set is near the simplex boundary.", "We show that (B) remains true in the reduced dual space." ], [ "Dynamical System, Jacobian, and Volume of Flow", "A dynamical system is typically described by a system of ordinary differential equations (ODE) over time in $\\mathbb {R}^d$ , governed by $d$ differential equations on the variables $s_1,s_2,\\cdots ,s_d$ , which are of the form $\\forall j\\in [d],~~\\frac{\\mathsf {d} s_j}{\\mathsf {d} t} = F_j(s_1,s_2,\\cdots ,s_d).$ Given a starting point $(s_1^\\circ ,s_2^\\circ ,\\cdots ,s_d^\\circ )$ , the values of the variables at any time $t\\ge 0$ are typically uniquely determined; precisely, given the starting point, for each $j\\in [d]$ , there is a function $s_j:\\mathbb {R}^+\\rightarrow \\mathbb {R}$ such that altogether they satisfy the ODE system, with $(s_1(0),s_2(0),\\cdots ,s_d(0))$ being the given starting point.", "The collection of the functions $s_1,s_2,\\cdots ,s_d$ is called the trajectory of the given starting point.", "The flow of a given starting point at time $t$ is simply $(s_1(t),s_2(t),\\cdots ,s_d(t))$ .", "In this paper, we assume that $F_j$ is smooth everywhere.", "Given a measurable set $S$ and an ODE system, the flow of $S$ at time $t$ is simply the collection of the flows of all starting points in $S$ at time $t$ ; when the underlying ODE system is clear from context, we denote it by $S(t)$ .", "Let $\\mathsf {vol}(S)$ denote the Lebesgue volume of set $S$ .", "The Jacobian of the ODE system at $\\mathbf {s}=(s_1,s_2,\\cdots ,s_d)$ is the $d\\times d$ -matrix $\\mathbf {J}(\\mathbf {s})$ : $\\mathbf {J}(\\mathbf {s}) ~=~\\begin{bmatrix}\\frac{\\partial }{\\partial s_1}F_1(\\mathbf {s}) & \\frac{\\partial }{\\partial s_2}F_1(\\mathbf {s}) & \\cdots & \\frac{\\partial }{\\partial s_d}F_1(\\mathbf {s})\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\frac{\\partial }{\\partial s_1}F_d(\\mathbf {s}) & \\frac{\\partial }{\\partial s_2}F_d(\\mathbf {s}) & \\cdots & \\frac{\\partial }{\\partial s_d}F_d(\\mathbf {s})\\end{bmatrix}.$ The Liouville's theorem states that if $S(0)\\subset \\mathbb {R}^d$ is a bounded and measurable set, then $\\frac{\\mathsf {d} }{\\mathsf {d} t}\\mathsf {vol}(S(t)) = \\int _{\\mathbf {s}\\in S} \\mathsf {trace}(\\mathbf {J}(\\mathbf {s}))\\,\\mathsf {d}V.$ The Liouville's theorem is indeed the continuous analogue of integration by substitution for multi-variables, which applies for calculating volume changes of discrete-time update rules.", "We present a simplified version of it which suffices for our purposes.", "For a gradual update rule $\\mathbf {s}_{t+1} = \\mathbf {s}_t + \\epsilon \\cdot F(\\mathbf {s}_t),$ where $F:\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ is a smooth function and step-size $\\epsilon >0$ , if $S\\subset \\mathbb {R}^d$ is a bounded and measurable set, and if the discrete flow in one time step maps $S$ to $S^{\\prime }$ injectively, then $\\mathsf {vol}(S^{\\prime }) ~=~ \\int _{\\mathbf {s}\\in S} \\det \\left( \\mathbf {I}+ \\epsilon \\cdot \\mathbf {J}(\\mathbf {s}) \\right) \\,\\mathsf {d}V,$ where $\\mathbf {J}(\\mathbf {s})$ is as defined in (REF ), and $\\mathbf {I}$ is the identity matrix.", "Clearly, analyzing the determinant in the integrand is crucial in volume analysis; we call it the volume integrand in this paper.", "When the determinant is expanded using the Leibniz formula, it becomes a polynomial of $\\epsilon $ , in the form of $1 + C(\\mathbf {s}) \\cdot \\epsilon ^h + \\mathcal {O}(\\epsilon ^{h+1})$ for some integer $h\\ge 1$ .", "Thus, when $\\epsilon $ is sufficiently small, the sign of $C(\\mathbf {s})$ dictates on whether the volume expands or contracts.", "In our case, $\\mathbf {s}$ refers to a cumulative payoff vector $(\\mathbf {p},\\mathbf {q})$ .", "Cheung and Piliouras [8] showed that for the MWU update rule (REF ) in the dual space, the volume integrand can be written as $1 + C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q}) \\cdot \\epsilon ^2 + \\mathcal {O}(\\epsilon ^4)$ , where $C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q}) ~=~ -\\sum _{j\\in S_1}~\\sum _{k\\in S_2}~x_j(\\mathbf {p}) \\cdot y_k(\\mathbf {q}) \\cdot (A_{jk} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j) \\cdot (B_{jk} - [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k).$ Note that $C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q})$ depends on the primal variables $\\mathbf {x}(\\mathbf {p}),\\mathbf {y}(\\mathbf {q})$ but not explicitly on $\\mathbf {p},\\mathbf {q}$ .", "Thus, it is legitimate to refer to this value using the primal variables as input parameters to $C_{(\\mathbf {A},\\mathbf {B})}$ , i.e., we can refer to its value by $C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {x},\\mathbf {y})$ too.", "Cheung and Piliouras [8] showed the following lemma.", "Lemma 3 [8] The following hold: When $\\epsilon \\le 1/4$ , the update rule (REF ) in the dual space is injective.", "In any two-person zero-sum game $(\\mathbf {A},-\\mathbf {A})$ , at any point $(\\mathbf {x},\\mathbf {y})\\in \\Delta $ , $C_{(\\mathbf {A},-\\mathbf {A})}(\\mathbf {x},\\mathbf {y}) \\ge 0$ .", "Indeed, $C_{(\\mathbf {A},-\\mathbf {A})}(\\mathbf {x},\\mathbf {y})$ equals to the variance of the random variable $X$ such that $X = (A_{jk} - [\\mathbf {A}\\mathbf {y}]_j - [\\mathbf {A}^{\\mathsf {T}}\\mathbf {x}]_k)$ with probability $x_j y_k$ , for all $(j,k)\\in S_1\\times S_2$ .", "When $\\epsilon < \\min \\left\\lbrace 1/(32n^2m^2) , C(\\mathbf {p},\\mathbf {q}) \\right\\rbrace $ , the volume integrand at point $(\\mathbf {p},\\mathbf {q})$ is lower bounded by $1+(C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q})-\\epsilon )\\epsilon ^2$ .", "Thus, in (REF ), if $\\overline{C} := \\min _{(\\mathbf {p},\\mathbf {q})\\in S} C_{\\mathbf {A},\\mathbf {B}}(\\mathbf {p},\\mathbf {q}) > 0$ , then for all $0 < \\epsilon \\le \\overline{C}$ , $\\mathsf {vol}(S^{\\prime }) ~\\ge ~ \\left[ 1 + \\left( \\overline{C} - \\epsilon \\right) \\epsilon ^2 \\right] \\cdot \\mathsf {vol}(S).$ By the definition of $C_{(\\mathbf {A},\\mathbf {B})}$ , it is straight-forward to see that $C_{(\\mathbf {A},\\mathbf {A})}(\\mathbf {p},\\mathbf {q}) ~=~ -C_{(\\mathbf {A},-\\mathbf {A})}(\\mathbf {p},\\mathbf {q}).$ Thus, for any coordination game $(\\mathbf {A},\\mathbf {A})$ , and for any $(\\mathbf {p},\\mathbf {q})\\in \\mathcal {D}$ , $C_{(\\mathbf {A},\\mathbf {A})}(\\mathbf {p},\\mathbf {q})\\le 0$ due to Lemma REF Part (2)." ], [ "Lyapunov Chaos", "In the study of dynamical systems, Lyapunov chaos generally refers to following phenomenon in some systems: a tiny difference in the starting points can yield widely diverging outcomes quickly.", "A classical measure of chaos is Lyapunov time, defined as: when the starting point is perturbed by a distance of tiny $\\gamma $ , for how long will the trajectories of the two starting points remain within a distance of at most $2\\gamma $ .", "Cheung and Piliouras [8] showed that if the volume of a set increases at a rate of $\\Omega ((1+\\beta )^t)$ , its diameter increases at a rate of at least $\\Omega ((1+\\beta /d)^t)$ , where $d$ is the dimension of the system, thus indicating that the Lyapunov time is at most $\\mathcal {O}(d/\\beta )$ ." ], [ "Unavoidability of MWU in Games", "The result in this section holds for MWU in general-sum games.", "Recall the definition of $C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q})$ in Equation (REF ), and the discussion on extending the definition of the function $C$ to the primal space (i.e.", "$C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {x},\\mathbf {y})$ ) below Equation (REF ).", "To avoid cluster, when the underlying game $(\\mathbf {A},\\mathbf {B})$ is clear from context, we write $C(\\cdot )$ for $C_{(\\mathbf {A},\\mathbf {B})}(\\cdot )$ .", "Definition 4 A set $U\\subset \\mathsf {int}(\\Delta )$ is called a primal open set if there is an open set $U^{\\prime }$ in $\\mathbb {R}^{n+m}$ , such that $U = U^{\\prime }\\cap \\Delta $ .", "A primal open set $U$ is uncontrollable if $\\inf _{(\\mathbf {x},\\mathbf {y})\\in U} C(\\mathbf {x},\\mathbf {y}) > 0$ .", "The following is the unavoidability theorem, the main theorem in this section.", "Theorem 5 Let $U$ be an uncontrollable primal open set with $\\inf _{(\\mathbf {x},\\mathbf {y})\\in U} C(\\mathbf {x},\\mathbf {y}) \\ge \\overline{C}> 0$ .", "If the step-size $\\epsilon $ in the update rule (REF ) satisfies $\\epsilon < \\min \\left\\lbrace \\frac{1}{32n^2m^2},\\overline{C}\\right\\rbrace $ , then there exists a dense subset of points in $U$ such that the flow of each such point must eventually reach a point outside $U$ .", "Recall that one perspective to think about the unavoidability theorem is to consider $U$ as a collection of good points, while $\\Delta \\setminus U$ is the set of bad points that we want to avoid.", "We desire the game dynamic to stay within $U$ forever, so long as the starting point is in $U$ .", "The theorem then presents a negative property, which states that if $U$ is uncontrollable, then there is a dense set of points in $U$ such that the game dynamic must eventually reach a point that we want to avoid.", "In particular, when the underlying game is a zero-sum game, due to Lemma REF , $\\inf _{(\\mathbf {x},\\mathbf {y})\\in U} C(\\mathbf {x},\\mathbf {y}) \\ge 0$ for any $U$ .", "With some mild assumptions on $U$ and the underlying game, it is foreseeable that the infimum will become strictly positive, for which Theorem REF is applicable.", "For instance, if the zero-sum game is not trivial (see Definition REF ), and $U$ collects all primal points $(\\mathbf {x},\\mathbf {y})$ such that all $x_j,y_k\\ge \\delta $ for some fixed $\\delta > 0$ , then the infimum is strictly positive due to Lemma REF Part (2); see [8] for a detailed explanation.", "Informally speaking, for quite general scenarios, MWU in zero-sum game cannot avoid bad states, regardless of what “good” or “bad” really mean." ], [ "Proof of Theorem ", "In Definition REF , we have defined uncontrollability of a set in the primal space.", "In the dual space, the definition of uncontrollability is similar: an open set $V$ in the dual space is uncontrollable if $\\inf _{(\\mathbf {p},\\mathbf {q})\\in V} C(\\mathbf {p},\\mathbf {q}) > 0$ .", "The following is the key lemma to proving Theorem REF .", "Lemma 6 Let $V$ be an uncontrollable open set in the dual space, with $\\inf _{(\\mathbf {p},\\mathbf {q})\\in V} C(\\mathbf {p},\\mathbf {q}) \\ge \\overline{C}> 0$ .", "Assume that the step-size $\\epsilon $ in the update rule (REF ) satisfies $0< \\epsilon < \\min \\left\\lbrace \\frac{1}{32n^2m^2},\\overline{C}\\right\\rbrace $ .", "Let $S\\subset V$ be a measurable set with positive volume, and let $S(t)$ be the flow of $S$ at time $t$ .", "Also, let $d(S) ~:=~ \\max \\left\\lbrace \\max _{j\\in S_1} \\left\\lbrace \\max _{(\\mathbf {p},\\mathbf {q})\\in S} p_j ~-~ \\min _{(\\mathbf {p},\\mathbf {q})\\in S} p_j \\right\\rbrace ~,~\\max _{k\\in S_2} \\left\\lbrace \\max _{(\\mathbf {p},\\mathbf {q})\\in S} q_k ~-~ \\min _{(\\mathbf {p},\\mathbf {q})\\in S} q_k \\right\\rbrace \\right\\rbrace .$ Then there exists a time $\\tau $ with $\\tau ~\\le ~ \\max \\left\\lbrace \\frac{d(S)}{2\\epsilon } ~,~ \\frac{8(n+m)}{(\\overline{C}-\\epsilon )\\epsilon ^2} \\ln \\frac{4(n+m)}{(\\overline{C}-\\epsilon )\\epsilon ^2} ~,~ \\frac{4}{(\\overline{C}-\\epsilon )\\epsilon ^2} \\ln \\frac{1}{\\mathsf {vol}(S)} \\right\\rbrace ,$ such that $S(\\tau )$ contains a point which is not in $V$ .", "[Proof: ] We suppose the contrary, i.e., for all $\\tau \\le T$ , $S(\\tau ) \\subset V$ , where $T$ will be specified later.", "We analyze how the volume of $S(t)$ changes with $t$ using formula (REF ).", "We rewrite it here: $\\mathsf {vol}(S(t+1)) ~=~ \\int _{(\\mathbf {p},\\mathbf {q})\\in S(t)} \\det \\left( \\mathbf {I}+ \\epsilon \\cdot \\mathbf {J}(\\mathbf {p},\\mathbf {q}) \\right) \\,\\mathsf {d}V.$ By Lemma REF , if $S(t)\\subset V$ , then the above inequality yields $\\mathsf {vol}(S(t+1)) ~\\ge ~ \\mathsf {vol}(S(t)) \\cdot \\left( 1+(\\overline{C}-\\epsilon )\\epsilon ^2 \\right)$ , and hence $\\forall t\\le T+1,~~~~\\mathsf {vol}(S(t)) ~\\ge ~ \\mathsf {vol}(S) \\cdot \\left( 1+(\\overline{C}-\\epsilon )\\epsilon ^2 \\right)^t.$ On the other hand, observe that in the update rule (REF ), each variable is changed by a value in the interval $[-\\epsilon ,\\epsilon ]$ per time step, since every entry in $\\mathbf {A},\\mathbf {B}$ is in the interval $[-1,1]$ .", "Consequently, the range of possible values for each variable in $S(t)$ lies within an interval of length at most $d(S) + 2\\epsilon t$ , and hence $S(t)$ is a subset of a hypercube with side length $d(S) + 2\\epsilon t$ .", "Therefore, $\\forall t\\le T+1,~~~~\\mathsf {vol}(S(t)) ~\\le ~ \\left( d(S) + 2\\epsilon t \\right)^{n+m}.$ Note that the lower bound in (REF ) is exponential in $t$ , while the upper bound in (REF ) is polynomial in $t$ .", "Intuitively, it is clear that the two bounds cannot be compatible for some large enough $T$ , and hence a contradiction.", "The rest of this proof is to derive how large $T$ should be.", "Precisely, we seek $T$ such that $\\left( d(S) + 2\\epsilon T \\right)^{n+m} ~<~ \\mathsf {vol}(S) \\cdot \\left( 1+(\\overline{C}-\\epsilon )\\epsilon ^2 \\right)^T.$ First, we impose that $T\\ge d(S)/(2\\epsilon )=:T_1$ .", "Taking logarithm on both sides, to satisfy the above inequality, it suffices that $(n+m) \\ln (4\\epsilon T) ~<~ \\frac{T \\cdot (\\overline{C}-\\epsilon )\\epsilon ^2}{2} + \\ln (\\mathsf {vol}(S)).$ Since $4\\epsilon \\le 1$ , it suffices that $(\\overline{C}-\\epsilon )\\epsilon ^2T - 2(n+m) \\ln T ~>~ 2\\cdot \\ln \\frac{1}{\\mathsf {vol}(S)}.$ Next, observe that when $T\\ge \\frac{8(n+m)}{(\\overline{C}-\\epsilon )\\epsilon ^2} \\ln \\frac{4(n+m)}{(\\overline{C}-\\epsilon )\\epsilon ^2} =: T_2$ , we have $(\\overline{C}-\\epsilon )\\epsilon ^2T - 2(n+m) \\ln T \\ge (\\overline{C}-\\epsilon )\\epsilon ^2T/2$ .", "(We will explain why in the next paragraph.)", "Then it is easy to see that $T\\ge \\frac{4}{(\\overline{C}-\\epsilon )\\epsilon ^2} \\ln \\frac{1}{\\mathsf {vol}(S)} =: T_3$ suffices.", "Overall, we need $T = \\max \\lbrace T_1,T_2,T_3\\rbrace $ .", "Lastly, we explain why the inequality in the last paragraph holds.", "Observe that it is equivalent to $\\frac{T}{\\ln T} \\ge \\frac{4(n+m)}{(\\overline{C}-\\epsilon )\\epsilon ^2} =: \\gamma $ .", "Then it suffices to know that $\\frac{T}{\\ln T}$ is an increasing function of $T$ when $T\\ge 3$ , and $\\frac{T_2}{\\ln T_2} ~=~ \\frac{2\\gamma \\ln \\gamma }{\\ln 2 + \\ln \\gamma + \\ln \\ln \\gamma } ~\\ge ~ \\frac{2\\gamma \\ln \\gamma }{2\\ln \\gamma } ~=~ \\gamma ,$ where the only inequality sign in the above expression holds because $\\ln \\gamma \\ge \\ln \\ln \\gamma + \\ln 2 > 0$ when $\\gamma \\ge 3$ .", "The following proposition is straight-forward.", "Proposition 7 If $U$ is a primal open set, then $\\mathsf {G}^{-1}(U)$ is an open and unbounded subset in $\\mathcal {D}$ .", "[Proof of Theorem REF : ] Let $U^{\\prime }$ denote the set of points in $U$ which, when taken as a starting point, will eventually reach a point outside $U$ .", "Suppose the theorem does not hold, i.e., $U^{\\prime }$ is not dense.", "Then we can find a primal open set $B\\subset U$ such that its flow must stay in $U$ forever.", "Let $V := \\mathsf {G}^{-1}(U)$ and $S^{\\prime } := \\mathsf {G}^{-1}(B)$ .", "Due to the discussion immediately after Equation (REF ) and the assumption that $U$ is uncontrollable in the primal space, $V$ is uncontrollable in the dual space.", "On the other hand, $S^{\\prime }$ is open and unbounded due to Proposition REF .", "But it is easy to find a subset $S\\subset S^{\\prime }$ which is open and bounded.", "Thus, $S$ has positive and finite volume.", "We apply Lemma REF with the sets $V,S$ given above, to show that using update rule (REF ), the flow of $S$ at some time $\\tau $ contains a point $(\\mathbf {p}^\\tau ,\\mathbf {q}^\\tau )\\notin V$ .", "By definition of $V$ , $\\mathsf {G}(\\mathbf {p}^\\tau ,\\mathbf {q}^\\tau )\\notin U$ .", "Let $(\\mathbf {p}^0,\\mathbf {q}^0)$ denote a point in $S$ such that its flow at time $\\tau $ is $(\\mathbf {p}^\\tau ,\\mathbf {q}^\\tau )$ .", "Since $S$ is a subset of $S^{\\prime }$ , $\\mathsf {G}(\\mathbf {p}^0,\\mathbf {q}^0)\\in B$ .", "Due to the equivalence between the primal update rule (REF ) and the dual update rule (REF ), we can conclude that when $\\mathsf {G}(\\mathbf {p}^0,\\mathbf {q}^0) \\in B$ is used as the starting point of the primal update rule (REF ), at time $\\tau $ its flow is $\\mathsf {G}(\\mathbf {p}^\\tau ,\\mathbf {q}^\\tau )$ which is not in $U$ , a contradiction." ], [ "Extremism of MWU in Zero-Sum Games", "Here, we focus on MWU in zero-sum game.", "[3] and [7] showed that the dynamic converges to the boundary of $\\Delta $ and fluctuates bizarrely near the boundary by using a potential function argument.", "However, the potential function has value $+\\infty $ at every point on the boundary so it cannot be distinctive there, and hence it cannot provide any useful insight on how the dynamic behaves near the boundary.", "In general, the behaviors near boundary can be highly unpredictable, as suggested by the “chaotic switching” phenomenon found by [2], although more regular (yet still surprising) patterns were found in lower-dimensional systems [13].", "In [3], [7], a central discouraging message is convergence towards boundary of $\\Delta $ is inevitable even when the underlying zero-sum game has a fully-mixed Nash equilibrium.", "What can we still hope for after this?", "Will $(\\mathbf {x}^t,\\mathbf {y}^t)$ remain within a somewhat reasonable range around the Nash equilibrium forever?", "We answer the latter question with a strikingly general negative answer for almost all zero-sum games, with the two theorems below.", "Definition 8 The extremal domain with threshold $\\delta $ consists of all points $(\\mathbf {x},\\mathbf {y})$ such that each of $\\mathbf {x},\\mathbf {y}$ has exactly one entry of value at least $1-\\delta $ .", "Theorem 9 Let $(\\mathbf {A},-\\mathbf {A})$ be a two-person zero-sum game.", "Suppose the following: Every $2\\times 2$ sub-matrix of $\\mathbf {A}$ is non-trivial.", "Let $\\alpha _1 > 0$ denote the minimum distance from triviality of all $2\\times 2$ sub-matrices of $\\mathbf {A}$ .", "(Recall the distance measure (REF ).)", "No two entries in the same row or the same column have exactly the same value.", "Let $\\alpha _2 > 0$ be the minimum difference between any two entries of $\\mathbf {A}$ in the same row or the same column.", "Let $N := \\max \\lbrace n,m\\rbrace $ .", "For any $0 < \\delta < \\alpha _2/4$ , if both players use MWU with step-size $\\epsilon $ satisfying $0< \\epsilon < \\min \\left\\lbrace \\frac{1}{32n^2 m^2} ~,~ \\frac{(\\alpha _1)^2}{18} \\cdot \\left(\\frac{\\delta }{N-1}\\right)^{8(N-1)/(\\alpha _2-4\\delta )+2} \\right\\rbrace ,$ then there exists a dense subset of points in $\\mathsf {int}(\\Delta )$ , such that the flow of each such point must eventually reach the extremal domain with threshold $\\delta $ .", "Theorem 10 Let $v$ denote the game value of the zero-sum game $(\\mathbf {A},-\\mathbf {A})$ .", "In addition to the conditions required in Theorem REF , if (i) $\\min _{j\\in S_1,k\\in S_2} |A_{jk} - v| \\ge r > 0$ , and (ii) $6\\epsilon + 4\\delta \\le r$ , then there exists a dense subset of points in $\\mathsf {int}(\\Delta )$ , such that the flow of each such point visits and leaves extremal domain with threshold $\\delta $ infinitely often.", "To see the power of the Theorem REF , consider a zero-sum game with a fully-mixed Nash Equilibrium.", "The theorem implies that in any arbitrarily small open neighbourhood of the Nash equilibrium, there exists a starting point such that its flow will eventually reach a point where each player concentrates her game-play on only one strategy.", "We call this extremism of game-play, since both players are single-minded at this point: they are concentrating on one strategy and essentially ignoring all the other available options.", "There are two assumptions on the matrix $\\mathbf {A}$ .", "If the matrix is to be drawn uniformly randomly from the space $[-1,+1]^{n\\times m}$ , the random matrix satisfies assumptions (A) and (B) almost surely.", "Unfortunately, the classical Rock-Paper-Scissors game is a zero-sum game which does not satisfy assumption (A) in Theorem REF , and thus the theorem is not applicable.", "In Appendix REF , we provide a separate proof which shows similar result to Theorem REF for this specific game." ], [ "Proof Sketch of Theorem ", "The full proofs of the two theorems are deferred to Appendix .", "Here, we give high-level description of the proof of Theorem REF .", "We first define a family of primal open sets in $\\mathsf {int}(\\Delta )$ .", "Let $\\mathcal {E}^{\\delta }_{a,b}$ be the collection of all points $(\\mathbf {x},\\mathbf {y})$ , such that at least $a$ entries in $\\mathbf {x}$ are larger than $\\delta $ , and at least $b$ entries in $\\mathbf {y}$ are larger than $\\delta $ .", "The first step is to use condition (A) to show that for any $1/3 > \\kappa > 0$ , $\\mathcal {E}^\\kappa _{2,2}~\\text{is an uncontrollable primal set with}~~~\\inf _{(\\mathbf {x},\\mathbf {y})\\in \\mathcal {E}^\\kappa _{2,2}} C(\\mathbf {x},\\mathbf {y}) \\ge \\kappa ^2 (\\alpha _1)^2 / 2.$ Then we can apply Theorem REF to show that for any sufficiently small step-size $\\epsilon $ , there exists a dense subset of points in $\\mathcal {E}^\\kappa _{2,2}$ such that the flow of each such point must eventually reach a point outside $\\mathcal {E}^\\kappa _{2,2}$ .", "Let $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ denote the reached point outside $\\mathcal {E}^\\kappa _{2,2}$ .", "At $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ , one of the two players, which we assume to be Player 1 without loss of generality, concentrates her game-play on only one strategy, which we denote by strategy $\\hat{j}$ .", "We have: for any $j\\ne \\hat{j}$ , $\\hat{x}_j \\le \\kappa $ , and hence $\\sum _{j\\in S_1\\setminus \\lbrace \\hat{j}\\rbrace } \\hat{x}_j \\le (N-1)\\kappa $ .", "When we pick $(N-1)\\kappa \\ll \\delta $ , where $\\delta $ is the quantity specified in Theorem REF , after the dynamic reaches $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ , the total probability of choosing any strategy other than $\\hat{j}$ by Player 1 will be at most $\\delta $ for a long period of time — this is true because that total probability can increase by a factor of at most $\\exp (2\\epsilon )$ per time step.", "In other words, we may think that the game essentially becomes an $1\\times m$ sub-game of $(\\mathbf {A},-\\mathbf {A})$ during this long period of time.", "We then show that during the long period of time, no matter what $\\hat{\\mathbf {y}}$ is, the game-play of Player 2 must become concentrating on one strategy too.", "Naively, one might think that this strategy ought to be $k$ which maximizes $-A_{\\hat{j}k}$ , which is the dominant strategy of Player 2 in the $1\\times m$ sub-game.", "However, if $\\hat{y}_k$ is tiny, this might not be true.", "To reach the conclusion, we have to use a technical lemma, Lemma REF in Appendix ." ], [ "Continuous Analogue of OMWU", "As the update rule (REF ) at time $t+1$ depends on the past updates at times $t$ and $t-1$ , at first sight it might seem necessary to perform volume analysis in the product space $\\Delta \\times \\Delta $ that contains $((\\mathbf {p}_t,\\mathbf {q}_t),(\\mathbf {p}_{t-1},\\mathbf {q}_{t-1}))$ .", "However, this raises a number of technical difficulties.", "First, since the initialization sets $(\\mathbf {p}_1,\\mathbf {q}_1)$ as a function of $(\\mathbf {p}_0,\\mathbf {q}_0)$ , the initial set has to lie in a proper manifold in $\\Delta \\times \\Delta $ , thus it has zero Lebesgue measure w.r.t.", "$\\Delta \\times \\Delta $ , making volume analysis useless, as the volume must remain zero when the initial set is of measure zero.", "Second, even if we permit $\\mathbf {p}_1,\\mathbf {q}_1$ to be unrelated to $\\mathbf {p}_0,\\mathbf {q}_0$ so that we can permit an initial set with positive measure, the OMWU update rule is not of the same type that is presumed by the formula (REF ).", "We will need to use the more general form of integration by substitution, and the volume integrand there will not be of the form $\\mathbf {I}+ \\epsilon \\cdot \\mathbf {J}$ , hence the determinant is not a polynomial of $\\epsilon $ with constant term 1.", "This imposes huge difficulty in analysis, forbidding us to present a clean volume analysis as was done in [8].", "To bypass the issues, we first derive a continuous analogue of OMWU in games as an ODE system, which will permit us to have a clean volume analysis." ], [ "Continuous Analogue of OMWU in General Contexts", "We focus on Player 1 who uses the OMWU update rule (REF ) in the dual space.", "To set up for the most general context, we replace $\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^t)$ by $\\mathbf {u}(t)$ , which represents the utility (or payoff) vector at time $t$ .", "We assume $\\mathbf {u}(t)$ is $C^2$ -differentiable.", "We rewrite the rule as below: $\\frac{\\mathbf {p}^{t+1} ~-~ \\mathbf {p}^t}{\\epsilon } ~=~ \\mathbf {u}(t) ~+~ \\epsilon \\cdot \\frac{\\mathbf {u}(t) - \\mathbf {u}(t-1)}{\\epsilon }.$ Recall that for any smooth function $f:\\mathbb {R}\\rightarrow \\mathbb {R}$ , its first derivative is $\\lim _{\\epsilon \\rightarrow 0} (f(x+\\epsilon )-f(x))/\\epsilon $ .", "For readers familiar with Euler discretization and finite-difference methods, the above discrete-time rule naturally motivates the following differential equation, where for any variable $v$ , $\\dot{v} \\equiv \\frac{\\mathsf {d} v}{\\mathsf {d} t}$ : $\\dot{\\mathbf {p}}~=~ \\mathbf {u}~+~ \\epsilon \\cdot \\dot{\\mathbf {u}}.$ To numerically simulate (REF ), we should take into account various informational constraints: [leftmargin=0.2in] If the function $\\mathbf {u}$ is explicitly given and it is a simple function of time (e.g.", "a polynomial), the function $\\dot{\\mathbf {u}}$ can be explicitly computed.", "Euler method with step-size $\\Delta t = \\epsilon $ is the update rule $\\mathbf {p}(t+\\epsilon ) = \\mathbf {p}(t) + \\epsilon \\cdot \\mathbf {u}(t) + \\epsilon ^2 \\cdot \\dot{\\mathbf {u}}(t).$ However, in some scenarios, $\\mathbf {u}$ is a rather complicated function of $t$ , so computing explicit formula for $\\dot{\\mathbf {u}}$ might not be easy.", "Yet, we have full knowledge of values of $\\mathbf {u}(0),\\mathbf {u}(\\Delta t),\\mathbf {u}(2\\cdot \\Delta t),\\mathbf {u}(3\\cdot \\Delta t),\\cdots $ .", "Then a common approach to approximately compute $\\dot{\\mathbf {u}}(N\\cdot \\Delta t)$ is to use the central finite-difference method: $\\dot{\\mathbf {u}}(N\\cdot \\Delta t) ~=~ \\frac{\\mathbf {u}((N+1)\\cdot \\Delta t) - \\mathbf {u}((N-1)\\cdot \\Delta t)}{2\\cdot \\Delta t} ~+~ \\mathcal {O}((\\Delta t)^2).$ Euler method with step-size $\\Delta t = \\epsilon $ which makes use of the above approximation gives the update rule $\\mathbf {p}(t+\\epsilon ) = \\mathbf {p}(t) + \\epsilon \\cdot \\mathbf {u}(t) + \\epsilon \\cdot \\frac{\\mathbf {u}(t+\\epsilon ) - \\mathbf {u}(t-\\epsilon )}{2}.$ Even worse, in the context of online learning or game dynamics, at time $N\\cdot \\Delta t$ , the players have only observed $\\mathbf {u}(0),\\mathbf {u}(\\Delta t),\\mathbf {u}(2\\cdot \\Delta t),\\cdots ,\\mathbf {u}(N\\cdot \\Delta t)$ , but they do not have any knowledge on the future values of $\\mathbf {u}$ .", "Due to the more severe constraint on information, we have to settle with the backward finite-difference method to approximately compute $\\dot{\\mathbf {u}}(N\\cdot \\Delta t)$ : $\\dot{\\mathbf {u}}(N\\cdot \\Delta t) ~=~ \\frac{\\mathbf {u}(N\\cdot \\Delta t) - \\mathbf {u}((N-1)\\cdot \\Delta t)}{\\Delta t} ~+~ \\mathcal {O}(\\Delta t),$ which has a higher-order error when compared with the central finite-difference method.", "Euler method with step-size $\\Delta t = \\epsilon $ which makes use of the above approximation gives the rule (REF ), by identifying $\\mathbf {p}(t+\\epsilon )$ as $\\mathbf {p}^{t+1}$ .", "Due to an error that occurs when we approximate $\\dot{\\mathbf {u}}$ as above, $\\epsilon \\cdot \\mathbf {u}(t) + \\epsilon \\cdot (\\mathbf {u}(t)-\\mathbf {u}(t-1)) ~=~ \\epsilon \\left[\\mathbf {u}(t) + \\epsilon \\cdot \\dot{\\mathbf {u}}(t)\\right] + \\mathcal {O}(\\epsilon ^3),$ where the LHS is the quantity $\\mathbf {p}^{t+1} - \\mathbf {p}^t$ in the OMWU update rule (REF ), and the first term in the RHS is the standard Euler discretization of (REF ).", "Proposition 11 From differential equation (REF ), when only online value oracle for a $C^2$ -differentiable function $\\mathbf {u}$ is given, the discrete-time update rule (REF ) is obtained by first using backward finite-difference method with step-size $\\epsilon $ to approximate $\\dot{\\mathbf {u}}$ , and then applying the Euler discretization method with step-size $\\epsilon $ .", "Also, Equation (REF ) holds.", "While we heuristically derived the ODE system (REF ) from OMWU with $\\epsilon $ being the step-size, but after it is derived, $\\epsilon $ becomes a parameter of a parametrized family of learning dynamics.", "When this parameter $\\epsilon $ is zero, system (REF ) recovers the Replicator Dynamics.", "When $\\epsilon > 0$ , it reduces the increment of $p_j$ if $\\dot{u}_j$ is negative.", "It can be interpreted as a common learning behavior (e.g.", "in financial markets), which is mainly depending on the payoffs, but also having a tuning which depends on the trend of payoffs.", "There is nothing to stop us from having a negative $\\epsilon $ , although it is not clear in what contexts such learning dynamics are motivated." ], [ "Continuous Analogue of OMWU in General-Sum Games", "Next, we use (REF ) to derive a system of differential equations for OMWU in general-sum games.", "In these and also many other learning contexts, $\\mathbf {u},\\dot{\\mathbf {u}}$ depend on the driving variables $\\mathbf {p},\\mathbf {q}$ .", "In (REF ), for Player 1, we replace $\\mathbf {u}(t)$ by $\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^t)$ .", "By the chain rule, $\\frac{\\mathsf {d} p_j}{\\mathsf {d} t} ~=~ [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j + \\epsilon \\cdot \\frac{\\mathsf {d} [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j}{\\mathsf {d} t} ~=~ [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j + \\epsilon \\cdot \\sum _{k\\in S_2} \\frac{\\partial [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j}{\\partial q_k} \\cdot \\frac{\\mathsf {d} q_k}{\\mathsf {d} t}.$ Recall from [8] that $\\frac{\\partial [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j}{\\partial q_k} = y_k(\\mathbf {q}) \\cdot (A_{jk} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j)$ .", "Thus, $\\frac{\\mathsf {d} p_j}{\\mathsf {d} t} &= [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j + \\epsilon \\sum _{k\\in S_2} y_k(\\mathbf {q}) \\cdot (A_{jk} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j) \\cdot \\frac{\\mathsf {d} q_k}{\\mathsf {d} t}.\\\\\\text{Analogously,}~~~~\\frac{\\mathsf {d} q_k}{\\mathsf {d} t}&= [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k + \\epsilon \\sum _{j\\in S_1} x_j(\\mathbf {p}) \\cdot (B_{jk} - [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k) \\cdot \\frac{\\mathsf {d} p_j}{\\mathsf {d} t}.~~~~~~~~~~~~~~~~~$ Formally, the above two formulae, which are in a recurrence format, have not yet formed an ODE system.", "To settle this issue, in Appendix , we show that when $\\epsilon $ is small enough, they can be reduced to a standard ODE system of the form $\\left( \\frac{\\mathsf {d} \\mathbf {p}}{\\mathsf {d} t} , \\frac{\\mathsf {d} \\mathbf {q}}{\\mathsf {d} t} \\right)^{\\mathsf {T}}= \\left[ \\mathbf {I}- \\epsilon \\mathbf {M}(\\mathbf {p},\\mathbf {q}) \\right]^{-1}\\cdot \\mathbf {v}(\\mathbf {p},\\mathbf {q})$ for some matrix $\\mathbf {M}(\\mathbf {p},\\mathbf {q})$ and vector $\\mathbf {v}(\\mathbf {p},\\mathbf {q})$ .", "This then formally permits us to use (REF ) and () in the analysis below, as is standard in formal power series when dealing with generating functions." ], [ "Volume Analysis of OMWU in Games", "Iterating the recurrence (REF ) and () yields the following system.", "$\\frac{\\mathsf {d} p_j}{\\mathsf {d} t} &~=~ [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j ~+~ \\epsilon \\sum _{k\\in S_2} y_k(\\mathbf {q})\\cdot (A_{jk} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j) \\cdot [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k ~+~ \\mathcal {O}(\\epsilon ^2); \\nonumber \\\\\\frac{\\mathsf {d} q_k}{\\mathsf {d} t} &~=~ [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k ~+~ \\epsilon \\sum _{j\\in S_1} x_j(\\mathbf {p}) \\cdot (B_{jk} - [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k) \\cdot [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j ~+~ \\mathcal {O}(\\epsilon ^2).$ Proposition REF establishes that in general contexts, (REF ) is the online Euler discretization of the differential equation (REF ).", "As a special case in games, (REF ) is the online Euler discretization of the recurrence system (REF ) and ().", "Via Equations (REF ) and (REF ), we rewrite (REF ) as $p_j^{t+1} &=~ p_j^t + \\epsilon [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^t)]_j ~+~ \\epsilon ^2 \\sum _{k\\in S_2} y_k(\\mathbf {q}^t)\\cdot (A_{jk} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^t)]_j) \\cdot [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p}^t)]_k + \\mathcal {O}(\\epsilon ^3); \\nonumber \\\\q_k^{t+1} &=~ q_k^t + \\epsilon [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p}^t)]_k ~+~ \\epsilon ^2 \\sum _{j\\in S_1} x_j(\\mathbf {p}^t) \\cdot (B_{jk} - [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p}^t)]_k) \\cdot [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q}^t)]_j + \\mathcal {O}(\\epsilon ^3).$ Update rule (REF ) can be implemented easily by the players in distributed manner, but it is hard to be used for volume analysis.", "The above update rule cannot be implemented by the players in distributed manner, since Player 1 does not know the values of $y_k$ and $[\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k$ .", "However, it will permit us to perform a clean volume analysis, since its RHS involves only $\\mathbf {p}^t,\\mathbf {q}^t$ but not $\\mathbf {p}^{t-1},\\mathbf {q}^{t-1}$ .", "We will show that when we ignore the $\\mathcal {O}(\\epsilon ^3)$ terms and perform volume analysis as described in Section REF , the volume integrand is of the format $1+C^{\\prime }\\epsilon ^2+ \\mathcal {O}(\\epsilon ^3)$ .", "Thus, taking the ignored terms into account does not affect the crucial $C^{\\prime }\\epsilon ^2$ term which dictates volume change.", "For the moment, we ignore the $\\mathcal {O}(\\epsilon ^3)$ terms.", "To use (REF ) for computing volume change, we need to derive $\\epsilon \\cdot \\mathbf {J}(\\mathbf {p},\\mathbf {q})$ in the volume integrand: $\\forall j_1,j_2\\in S_1,~~~~~~& \\epsilon J_{j_1 j_2} ~=~ \\epsilon ^2 \\sum _{k\\in S_2} y_k(\\mathbf {q}) \\cdot (A_{j_1k} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_{j_1}) \\cdot x_{j_2}(\\mathbf {p}) \\cdot (B_{j_2k} - [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k)~;\\\\\\forall k_1,k_2\\in S_2,~~~~~~& \\epsilon J_{k_1 k_2} ~=~ \\epsilon ^2 \\sum _{j\\in S_1} x_j(\\mathbf {p}) \\cdot (B_{jk_1} - [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_{k_1}) \\cdot y_{k_2}(\\mathbf {q}) \\cdot (A_{jk_2} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j)~;\\\\\\forall j\\in S_1, k\\in S_2,~~& ~\\epsilon J_{jk}~~=~ \\epsilon \\cdot y_k(\\mathbf {q}) \\cdot (A_{jk} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j) ~+~ \\mathcal {O}(\\epsilon ^2)~;\\\\\\forall k\\in S_2, j\\in S_1,~~& ~\\epsilon J_{kj}~~=~ \\epsilon \\cdot x_j(\\mathbf {p}) \\cdot (B_{jk} - [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k) ~+~ \\mathcal {O}(\\epsilon ^2)~.$ With the above formulae, we expand $\\det (\\mathbf {I}+ \\epsilon \\cdot \\mathbf {J}(\\mathbf {p},\\mathbf {q}))$ via the Leibniz formula.", "The determinant is of the form $1+C^{\\prime }(\\mathbf {p},\\mathbf {q})\\cdot \\epsilon ^2 + \\mathcal {O}(\\epsilon ^3)$ , where $C^{\\prime }(\\mathbf {p},\\mathbf {q})$ is the coefficient of $\\epsilon ^2$ in the expression $\\sum _{j\\in S_1} \\epsilon J_{jj} ~+~ \\sum _{k\\in S_2} \\epsilon J_{kk} ~-~ \\sum _{\\begin{array}{c}j\\in S_1\\\\k\\in S_2\\end{array}}~(\\epsilon J_{jk})(\\epsilon J_{kj}).$ A straight-forward arithmetic shows the above expression equals to $-\\epsilon ^2 \\cdot C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q}) + \\mathcal {O}(\\epsilon ^3)$ , and hence $\\det (\\mathbf {I}+ \\epsilon \\cdot \\mathbf {J}(\\mathbf {p},\\mathbf {q})) ~=~ 1 ~-~ C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q}) \\cdot \\epsilon ^2 ~+~ \\mathcal {O}(\\epsilon ^3).$" ], [ "OMWU in Coordination Games is Lyapunov Chaotic in the Dual Space", "At this point, it is important to address the similarity of MWU in zero-sum games $(\\mathbf {A},-\\mathbf {A})$ and OMWU in coordination games $(\\mathbf {A},\\mathbf {A})$ .", "Recall from [8] that the volume integrand for the former case is $1+C_{(\\mathbf {A},-\\mathbf {A})}(\\mathbf {p},\\mathbf {q})\\cdot \\epsilon ^2 + \\mathcal {O}(\\epsilon ^4),$ while by (REF ) and (REF ), the volume integrand for the latter case is $1-C_{(\\mathbf {A},\\mathbf {A})}(\\mathbf {p},\\mathbf {q})\\cdot \\epsilon ^2 + \\mathcal {O}(\\epsilon ^3) = 1+C_{(\\mathbf {A},-\\mathbf {A})}(\\mathbf {p},\\mathbf {q})\\cdot \\epsilon ^2 + \\mathcal {O}(\\epsilon ^3).$ When $\\epsilon $ is the sufficiently small, their volume-change behaviors are almost identical.", "Using (REF ), we can deduce all the Lyapunov chaos, unavoidability and extremism results in Section  for OMWU in coordination games.", "We also have volume contraction results for OMWU in zero-sum game and MWU in coordination game, which are stated formally in Theorems REF and REF in Appendix .", "Theorem 12 Suppose the underlying game is a non-trivial coordination game $(\\mathbf {A},\\mathbf {A})$ and the parameter $\\alpha _1$ as defined in Theorem REF is strictly positive.", "For any $1/2 > \\delta > 0$ , for any sufficiently small $0<\\epsilon \\le \\bar{\\epsilon }$ where the upper bound depends on $\\delta $ , and for any set $S=S(0)\\subset \\mathsf {G}^{-1}(\\mathcal {E}^{\\delta }_{2,2})$ in the dual space, if $S$ is evolved by the OMWU update rule (REF ) and if its flow remains a subset of $\\mathsf {G}^{-1}(\\mathcal {E}^{\\delta }_{2,2})$ for all $t\\le T-1$ , then $\\mathsf {vol}(S(T)) ~\\ge ~ \\left( 1 + \\frac{\\epsilon ^2 \\delta ^2 (\\alpha _1)^2}{4} \\right)^T \\cdot \\mathsf {vol}(S).$ Consequently, the system is Lyapunov chaotic within $\\mathsf {G}^{-1}(\\mathcal {E}^{\\delta }_{2,2})$ of the dual space, with Lyapunov time $\\mathcal {O}((n+m)/(\\epsilon ^2 \\delta ^2 (\\alpha _1)^2))$ ." ], [ "Negative Consequences of Volume Expansion of OMWU in Coordination Game", "In Sections  and , the unavoidability and extremism theorems are proved largely due to volume expansion; For the extremism theorems, it requires some additional arguments that seem specific to MWU, which comprise of Lemma REF and Step 3 in the proof of Theorem REF , both in Appendix ).", "But a careful examination of the proof of Lemma REF and the Step 3 finds these additional arguments work for OMWU too (with very minor modifications).", "Thus, the unavoidability and extremism theorems hold for OMWU too, after suitably modifying the condition needed for volume expansion, and the upper bounds on the step-sizes.", "Suppose a coordination game has a non-pure Nash equilibrium (i.e.", "a Nash equilibrium $(\\mathbf {x}^*,\\mathbf {y}^*)$ in which the supports of $\\mathbf {x}^*,\\mathbf {y}^*$ are both of size at least 2).", "By Theorem REF (the OMWU analogue), for any tiny open ball $B$ around the equilibrium, there is a dense subset of points in $\\mathsf {int}(\\Delta )\\cap B$ such that the flow of this point eventually reaches close to an extremal point.", "In other words, there are points arbitrarily close to the equilibrium with their flows reaching extremal points, i.e.", "the flows not only move away from the equilibrium locally, but they move away for a big distance.", "This kind of global instability result can be applied quite broadly, as many coordination games have non-pure Nash equilibrium.", "In the standard coordination game $(\\mathbf {A},\\mathbf {A})$ where $\\mathbf {A}= \\left[{\\begin{matrix}1 & 0 \\\\0 & 1\\end{matrix}}\\right]$ ., the game has three Nash equilibria, namely $((1,0),(1,0))$ , $((0,1),(0,1))$ and $((1/2,1/2),(1/2,1/2))$ .", "The latest one is a non-pure Nash equilibrium.", "Another example is the following generalization.", "Consider a two-player coordination game where each player has $n$ strategies.", "Suppose that when both players choose strategy $i$ , they both earn $$A_i$ , and otherwise they both lose $$Z$ , where $A_i>0$ and $Z\\ge 0$ .", "Then the game has a non-pure Nash equilibrium $(\\mathbf {x}^*,\\mathbf {x}^*)$ , where $x^*_i = \\frac{1}{A_i+Z} \\left\\bad.", "\\left(\\sum _j \\frac{1}{A_j+Z}\\right) \\right.$ , which is strictly positive for all $i$ ." ], [ "Acknowledgments", "We thank several anonymous reviewers for their suggestions, which help to improve the readability of this paper from its earlier version.", "Yun Kuen Cheung and Georgios Piliouras acknowledge AcRF Tier 2 grant 2016-T2-1-170, grant PIE-SGP-AI-2018-01, NRF2019-NRF-ANR095 ALIAS grant and NRF 2018 Fellowship NRF-NRFF2018-07." ], [ "Missing Examples and Proof in Section ", "In Section , we pointed out two facts: volume contraction in the dual space does not necessarily imply stability in either the dual or the primal space; volume expansion in the dual space does not necessarily imply instability in the primal space.", "To see why (A) is true, consider the following parametrized rectangular set $S(z)$ around the origin in the dual space: $S(z) ~:=~ \\lbrace (\\mathbf {p},\\mathbf {q})\\in \\mathbb {R}^2\\times \\mathbb {R}^2 ~\\Big |~ |p_1|,|q_1|\\le 1/z,~|p_2|,|q_2|\\le \\sqrt{z} \\rbrace .$ As $z$ increases, the volume of $S(z) = 1/z$ decreases, but its diameter and the quantities $\\max \\lbrace p_2-p_1\\rbrace $ , $\\max \\lbrace q_2-q_1\\rbrace $ are $\\Theta (\\sqrt{z})$ which increase with $z$ .", "Also, since $S$ contains the points $((0,\\sqrt{z}),(0,\\sqrt{z})), ((0,-\\sqrt{z}),(0,-\\sqrt{z})),$ when the set $S(z)$ is converted to the primal space, $\\mathsf {G}(S)$ contains points close to $((0,1),(0,1)),((1,0),(1,0))$ as $z\\rightarrow \\infty $ , so the diameter of $\\mathsf {G}(S)$ increases to 2 as $z\\rightarrow \\infty $ .", "To see why (B) is true, consider the following parametrized set $S(z)$ in the dual space: $S(z) ~:=~ \\lbrace (\\mathbf {p},\\mathbf {q})\\in \\mathbb {R}^2\\times \\mathbb {R}^2 ~\\Big |~ p_2 \\ge p_1 + z \\text{~and~} q_2\\ge q_1 + z, \\text{~and~} 0\\le p_1,p_2,q_1,q_2\\le 3z \\rbrace .$ It is not hard to compute its volume $4z^4$ which increases with $z$ , but its primal counterpart contracts and converges to a single point $((0,1),(0,1))$ .", "We also note that (B) remains true in the dual space used by Eshel and Akin.", "An example is $S(z) = \\lbrace ((p_1-p_3,p_2-p_3),(q_1-q_3,q_2-q_3)) ~\\big |~ z\\le p_1-p_3,q_1-q_3\\le 2z ~\\text{and}~ -2z\\le p_2-p_3,q_2-q_3\\le -z \\rbrace .$ The volume of $S(z)$ is $z^4$ which increases with $z$ , but its primal counterpart converges to the primal point $((1,0,0),(1,0,0))$ as $z\\rightarrow \\infty $ .", "Proposition REF follows directly from the following proposition.", "Proposition 13 Let $S$ be a set in the dual space with Lebesgue volume $v$ .", "Also, suppose there exists $j\\in S_1$ and $k\\in S_2$ such that $\\max _{(\\mathbf {p},\\mathbf {q})\\in S} p_j - \\min _{(\\mathbf {p},\\mathbf {q})\\in S} p_j \\le R_j$ and $\\max _{(\\mathbf {p},\\mathbf {q})\\in S} q_k - \\min _{(\\mathbf {p},\\mathbf {q})\\in S} q_k \\le R_k$ .", "Then for $\\beta := \\exp \\left(\\left( \\frac{v}{R_j R_k} \\right)^{1/(n+m-2)}\\right)$ , at least one of the followings holds: There exists $j^{\\prime }\\in S_1$ such that $\\left(\\max _{(\\mathbf {p},\\mathbf {q})\\in S} \\frac{x_{j^{\\prime }}(\\mathbf {p})}{x_j(\\mathbf {p})}\\right) \\Big / \\left(\\min _{(\\mathbf {p},\\mathbf {q})\\in S} \\frac{x_{j^{\\prime }}(\\mathbf {p})}{x_j(\\mathbf {p})}\\right) \\ge \\beta $ .", "Furthermore, if there exists $(\\mathbf {p}^\\#,\\mathbf {q}^\\#) \\in S$ such that $x_j(\\mathbf {p}^\\#),x_{j^{\\prime }}(\\mathbf {p}^\\#) \\ge \\kappa > 0$ , then the diameter of $\\mathsf {G}(S)$ w.r.t.", "$\\ell _2$ norm is at least $\\left( 1 - \\beta ^{-1/4} \\right) \\kappa $ .", "There exists $k^{\\prime }\\in S_2$ such that $\\left(\\max _{(\\mathbf {p},\\mathbf {q})\\in S} \\frac{y_{k^{\\prime }}(\\mathbf {q})}{y_k(\\mathbf {q})}\\right) \\Big / \\left(\\min _{(\\mathbf {p},\\mathbf {q})\\in S} \\frac{y_{k^{\\prime }}(\\mathbf {q})}{y_{k}(\\mathbf {q})}\\right) \\ge \\beta $ .", "Furthermore, if there exists $(\\mathbf {p}^\\#,\\mathbf {q}^\\#) \\in S$ such that $y_k(\\mathbf {q}^\\#),y_{k^{\\prime }}(\\mathbf {q}^\\#) \\ge \\kappa > 0$ , then the diameter of $\\mathsf {G}(S)$ w.r.t.", "$\\ell _2$ norm is at least $\\left( 1 - \\beta ^{-1/4} \\right) \\kappa $ .", "[Proof: ]Without loss of generality, we assume that $j=1$ and $k=1$ .", "Consider the mapping: $((p_1,p_2,\\cdots ,p_n) ~,~ (q_1,q_2,\\cdots ,q_m)) ~~\\rightarrow ~~ ((p_1,p_2-p_1,\\cdots ,p_n-p_1) ~,~ (q_1,q_2-q_1,\\cdots ,q_m-q_1)).$ This is a linear mapping, and it is easy to verify that the determinant of the matrix that describes this linear mapping has determinant 1, so the mapping is volume-preserving.", "Suppose that each of the quantities $p_{j^{\\prime }}-p_1$ and $q_{k^{\\prime }}-q_1$ is bounded by an interval of length at most $R$ within the set $S$ , for a value of $R$ to be specified later.", "Then $S$ is a subset of a rectangular box in $\\mathbb {R}^{n+m}$ , with $n+m-2$ sides of lengths at most $R$ , and the remaining two sides of lengths at most $R_j$ and $R_k$ .", "Thus, the volume of $S$ after the above linear mapping is at most $R^{n+m-2} R_j R_k$ .", "When $R < ( \\frac{v}{R_j R_k} )^{1/(n+m-2)}$ , this is a contradiction.", "Thus, there exists one quantity $p_{j^{\\prime }}-p_1$ or $q_{k^{\\prime }}-q_1$ which is not bounded by an interval of length at most $( \\frac{v}{R_j R_k} )^{1/(n+m-2)}$ .", "Then we are done by recalling that $\\frac{x_{j^{\\prime }}(\\mathbf {p})}{x_1(\\mathbf {p})} = \\exp (p_{j^{\\prime }}-p_1)$ and $\\frac{y_{k^{\\prime }}(\\mathbf {q})}{y_1(\\mathbf {q})} = \\exp (q_{k^{\\prime }}-q_1)$ .", "If furthermore, there exists $(\\mathbf {p}^\\#,\\mathbf {q}^\\#) \\in S$ such that $x_j(\\mathbf {p}^\\#),x_{j^{\\prime }}(\\mathbf {p}^\\#) \\ge \\kappa > 0$ , then there exists $(\\mathbf {p}^*,\\mathbf {q}^*) \\in S$ such that either $\\frac{x_j(\\mathbf {p}^*)}{x_{j^{\\prime }}(\\mathbf {p}^*)} \\left\\bad.", "\\frac{x_j(\\mathbf {p}^\\#)}{x_{j^{\\prime }}(\\mathbf {p}^\\#)} \\right.", "\\ge \\beta ^{1/2}~~~~~~\\text{or}~~~~~~\\frac{x_j(\\mathbf {p}^*)}{x_{j^{\\prime }}(\\mathbf {p}^*)} \\left\\bad.", "\\frac{x_j(\\mathbf {p}^\\#)}{x_{j^{\\prime }}(\\mathbf {p}^\\#)} \\right.", "\\le \\beta ^{-1/2}.$ We focus on the former case, as the latter case is similar.", "We have $x_j(\\mathbf {p}^*) - x_j(\\mathbf {p}^\\#) ~\\ge ~ x_j(\\mathbf {p}^\\#) \\cdot \\left( \\frac{x_{j^{\\prime }}(\\mathbf {p}^*)}{x_{j^{\\prime }}(\\mathbf {p}^\\#)} \\cdot \\beta ^{1/2}-1 \\right)$ .", "If $\\frac{x_{j^{\\prime }}(\\mathbf {p}^*)}{x_{j^{\\prime }}(\\mathbf {p}^\\#)} \\ge \\beta ^{-1/4}$ , we have $x_j(\\mathbf {p}^*) - x_j(\\mathbf {p}^\\#) \\ge \\kappa (\\beta ^{1/4}-1) \\ge \\kappa (1-\\beta ^{-1/4})$ .", "Otherwise, $\\frac{x_{j^{\\prime }}(\\mathbf {p}^*)}{x_{j^{\\prime }}(\\mathbf {p}^\\#)} < \\beta ^{-1/4}$ , and hence $x_{j^{\\prime }}(\\mathbf {p}^\\#) - x_{j^{\\prime }}(\\mathbf {p}^*) > x_{j^{\\prime }}(\\mathbf {p}^\\#) \\cdot \\left( 1 - \\beta ^{-1/4} \\right) \\ge \\kappa (1-\\beta ^{-1/4})$ ." ], [ "Extremism of MWU in Zero-Sum Games", "Lemma 14 Suppose an agent has $m$ options which she use MWU with step-size $\\epsilon $ to decide the mixed strategy $\\mathbf {y}^t = (y_1^t,\\cdots ,y_m^t)$ in each time step.", "Suppose at each round $t$ , the payoff to each option $k$ is $a_k + \\delta _k^t$ , where each $a_k\\in [-1,1]$ ; there exists a positive number $\\alpha _2>0$ , such that for any $2\\le k \\le m$ , $a_{k-1} - a_{k} \\ge \\alpha _2$ ; there exists a positive number $\\delta \\le \\alpha _2/8$ , such that $\\delta _k^t\\in [-2\\delta ,2\\delta ]$ .", "Let $\\hat{k}(t)$ denote the strategy $\\min \\lbrace k\\in [m] ~|~ y_k^t > \\delta /(m-1)\\rbrace $ .", "Then for $T:=\\left\\lceil \\frac{2}{\\epsilon (\\alpha _2 - 4\\delta )}\\cdot \\ln \\frac{m-1}{\\delta } \\right\\rceil $ , (i) if $\\mathbf {y}^{\\tau +T}$ has more than one entries larger than $\\delta /(m-1)$ for some $\\tau \\ge 0$ , then $\\hat{k}(\\tau +T) \\le \\hat{k}(\\tau )-1$ , and (ii) for some $t\\le (m-1)T$ , $\\mathbf {y}^t$ has an entry which is larger than or equal to $1-\\delta $ .", "For part (i), we prove the contrapositive statement instead: if $\\hat{k}(\\tau +T) \\ge \\hat{k}(\\tau )$ , then $\\mathbf {y}^{\\tau +T}$ has exactly one entry larger than $\\delta /(m-1)$ .", "Let $k = \\hat{k}(\\tau )$ .", "For any $\\ell > k$ , due to the definition of the MWU update rule (REF ) and our assumptions, for $t\\ge \\tau $ , $\\frac{y_\\ell ^{t+1}}{y_k^{t+1}} ~=~ \\frac{y_\\ell ^{t}}{y_k^{t}} \\cdot \\exp \\left( \\epsilon (a_\\ell + \\delta _\\ell ^t - a_k - \\delta _k^t) \\right)~\\le ~ \\frac{y_\\ell ^{t}}{y_k^{t}} \\cdot \\exp \\left( -\\epsilon (\\alpha _2 - 4\\delta ) \\right).$ Since $k = \\hat{k}(\\tau )$ , we have $y_k^\\tau > \\delta /(m-1)$ .", "Also, $y_k^t,y_\\ell ^\\tau \\le 1$ trivially.", "Thus, for any $t\\ge \\tau $ , $y_\\ell ^{t} ~\\le ~ y_k^{t} \\cdot \\frac{y_\\ell ^{\\tau }}{y_k^{\\tau }} \\cdot \\exp \\left( -\\epsilon (\\alpha _2 - 4\\delta ) (t-\\tau ) \\right)~<~ \\frac{m-1}{\\delta } \\cdot \\exp \\left( -\\epsilon (\\alpha _2 - 4\\delta ) (t-\\tau ) \\right).$ When $\\exp \\left( -\\epsilon (\\alpha _2 - 4\\delta ) (t-\\tau ) \\right)\\le \\delta ^2 / (m-1)^2$ , or equivalently $t \\ge \\tau + \\left\\lceil \\frac{2}{\\epsilon (\\alpha _2 - 4\\delta )}\\cdot \\ln \\frac{m-1}{\\delta } \\right\\rceil = \\tau + T$ , we have $y_\\ell ^t \\le \\delta /(m-1)$ .", "Due to the conclusion of the last paragraph, we have $\\hat{k}(\\tau +T) \\le k$ .", "But we also have the assumption $\\hat{k}(\\tau +T) \\ge \\hat{k}(\\tau ) = k$ .", "Thus, $\\hat{k}(\\tau +T) = k$ , and hence for any $k^{\\prime }<k$ , $y_{k^{\\prime }}^{\\tau +T} \\le \\delta /(m-1)$ .", "This, together with the conclusion of the last paragraph, shows that $y_k^{\\tau +T}$ is the only entry in $\\mathbf {y}^{\\tau +T}$ which is larger than $\\delta /(m-1)$ .", "This completes the proof of part (i).", "We prove part (ii) by contradiction.", "Suppose that for all $t\\le (m-1)T$ , $\\mathbf {y}^t$ has more than one entries larger than $\\delta /(m-1)$ .", "First of all, $\\hat{k}(0)\\ne m$ , for otherwise $y_m^0$ is the only entry in $\\mathbf {y}^0$ which is larger than $\\delta /(m-1)$ .", "Next, we apply part (i) for $(m-1)$ times to yield that $\\hat{k}((m-1)T) \\le \\hat{k}(0) - (m-1) \\le 0$ , a contradiction.", "Thus, for some $\\mathbf {y}^t$ with $t\\le (m-1)T$ , it has exactly one entry which is larger than $\\delta /(m-1)$ .", "The entry has to be larger than or equal to $1-(m-1)(\\delta /(m-1)) = 1-\\delta $ .", "[Proof of Theorem REF : ] The proof comprises of three steps.", "Step 1.", "We show that for any $\\kappa > 0$ , $\\mathcal {E}^\\kappa _{2,2}$ is an uncontrollable primal set with $\\inf _{(\\mathbf {x},\\mathbf {y})\\in \\mathcal {E}^\\kappa _{2,2}} C(\\mathbf {x},\\mathbf {y}) \\ge \\kappa ^2 (\\alpha _1)^2 / 2$ .", "Recall Lemma REF that $C(\\mathbf {x},\\mathbf {y})$ is the variance of a random variable $X$ , which is equal to $\\mathbb {E}\\left[ (X-\\mathbb {E}\\left[ X \\right])^2 \\right]$ .", "For any point $(\\mathbf {x},\\mathbf {y})\\in \\mathcal {E}^\\kappa _{2,2}$ , each of $\\mathbf {x},\\mathbf {y}$ has at least two entries larger than $\\kappa $ .", "Suppose $x_{j_1},x_{j_2},y_{k_1},y_{k_2} > \\kappa $ .", "Then $C(\\mathbf {x},\\mathbf {y}) ~\\ge ~ \\sum _{j\\in \\lbrace j_1,j_2\\rbrace } ~\\sum _{k\\in \\lbrace k_1,k_2\\rbrace }~ \\kappa ^2 \\left[ \\underbrace{\\left(A_{jk} - [\\mathbf {A}\\mathbf {y}]_j - [\\mathbf {A}^{\\mathsf {T}}\\mathbf {x}]_k\\right) - \\mathbb {E}\\left[ X \\right]}_{A^{\\prime }_{jk}} \\right]^2.$ Due to Condition (A) and Equation (REF ), we are guaranteed that among the four possible values of $A^{\\prime }_{jk}$ , the maximum and minimum values differ by at least $\\alpha _1$ , for otherwise we can choose $a_j = [\\mathbf {A}\\mathbf {y}]_j + \\mathbb {E}\\left[ X \\right]$ and $b_k = -[\\mathbf {A}^{\\mathsf {T}}\\mathbf {x}]_k$ in (REF ) to show that the $2\\times 2$ sub-matrix of $\\mathbf {A}$ corresponding to strategies $\\lbrace j_1,j_2\\rbrace \\times \\lbrace k_1,k_2\\rbrace $ has distance from triviality strictly less than $\\alpha _1$ .", "Consequently, $C(\\mathbf {x},\\mathbf {y}) \\ge \\kappa ^2 (\\alpha _1/2)^2 \\cdot 2 = \\kappa ^2 (\\alpha _1)^2/2$ .", "Step 2.", "Then we apply Theorem REF to show that for any step-size $\\epsilon < \\min \\left\\lbrace \\frac{1}{32n^2 m^2}~,~\\frac{\\kappa ^2 (\\alpha _1)^2}{2} \\right\\rbrace $ , there exists a dense subset of points in $\\mathsf {int}(\\Delta )$ such that the flow of each such point must eventually reach a point outside $\\mathcal {E}^\\kappa _{2,2}$ .", "Let $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ denote the point outside $\\mathcal {E}^\\kappa _{2,2}$ .", "At $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ , one of the two players, which we assume to be Player 1 without loss of generality, concentrates her game-play on only one strategy, which we denote by strategy $\\hat{j}$ .", "Precisely, for any $j\\ne \\hat{j}$ , $\\hat{x}_j \\le \\kappa $ , and hence $\\sum _{j\\in S_1\\setminus \\lbrace \\hat{j}\\rbrace } \\hat{x}_j \\le (N-1)\\kappa $ .", "Step 3.", "Now, we consider the flow starting from $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ .", "Since $x_j^{t+1} / x_j^t \\le \\exp (2\\epsilon )$ always, we are sure that for the next $T_1 := \\left\\lfloor \\frac{1}{2\\epsilon }\\ln \\frac{\\delta }{(N-1)\\kappa } \\right\\rfloor $ time steps, $\\sum _{j\\in S_1\\setminus \\lbrace \\hat{j}\\rbrace } x_j^t \\le \\delta $ .", "Thus, within this time period, the payoff to strategy $k$ of Player 2 in each time step is $-A_{\\hat{j}k}$ plus a perturbation term in the interval $[-2\\delta ,2\\delta ]$ .", "Then by Lemma REF part (ii) (a sanity check on the conditions required by the lemma is easy and thus skipped), if $(N-1)\\cdot \\left\\lceil \\frac{2}{\\epsilon (\\alpha _2-4\\delta )}\\cdot \\ln \\frac{N-1}{\\delta } \\right\\rceil \\le T_1$ , we are done.", "A direct arithmetic shows that this inequality holds if $\\kappa \\le (\\delta /(N-1))^{4(N-1)/(\\alpha _2-4\\delta )+1}/3$ .", "[Proof of Theorem REF : ] By Theorem REF , we are guaranteed that there exists a dense set of starting points such that the flow of each of them must eventually reach the extremal domain with threshold $\\delta $ .", "When we apply Theorem REF , This is our starting point to prove Theorem REF .", "Step 1.", "We show that: for each such starting point $y$ , we prove that its flow cannot remain in the extremal domain forever.", "First, observe that the extremal domain is the union of small neighbourhoods of extremal points, and each such neighbourhood is far from the other neighbourhoods.", "Suppose the contrary that there exists a starting point such that its flow remains in the extremal domain forever.", "Due to the above observation, its flow must remain in the small neighbourhood of one extremal point forever.", "Suppose the utility values at this extremal point is $(u,-u)$ ; recall that by assumption, $|u-v| \\ge r$ .", "Since the flow remains near this extremal point, in the long run, the average utility gained by Player 1 must lie in the interval $(1-\\delta )u \\pm \\delta $ , which is a subset of the interval $u \\pm 2\\delta $ .", "On the other hand, due to a well-known regret bound of MWU (see, for instance, [7]), in the long run, the average utility gained by Player 1 must lie in the interval $v\\pm 3\\epsilon $ .", "When $3\\epsilon + 2\\delta \\le r/2$ , this is incompatible with the interval derived in the previous paragraph, thus a contradiction.", "Step 2.", "Indeed, we have a stronger version of the result in Step 1.", "Recall that the complement of the extremal domain is an open set.", "Since the MWU update rule is a continuous mapping, it preserves openness, and hence we not only one point $y$ that visits and leaves the extremal domain, but we have an open neighbourhood $\\mathcal {O}_1$ around $y$ , such that the flow of $\\mathcal {O}_1$ visits and leaves the extremal domain.", "Let $\\mathcal {O}^{\\prime }$ denote the flow of $\\mathcal {O}_1$ at the moment when the flow leaves the extremal domain.", "$\\mathcal {O}^{\\prime }$ is open, and hence has positive Lebesgue measure.", "Then we construct a closed subset $\\mathcal {C}_1 \\subset \\mathcal {O}_1$ with positive Lebesgue measure.", "This is easy as follows.", "First, we take an arbitrary point $z \\in \\mathcal {O}^{\\prime }$ .", "Since $\\mathcal {O}^{\\prime }$ is open, there exists an open ball around $z$ with some radius $r>0$ which is contained in $\\mathcal {O}^{\\prime }$ .", "Since the MWU update rule is a continuous mapping, its inverse for arbitrary finite time preserves closeness, the inverse (back to the starting time) of the closed ball around $z$ with radius $r/2$ is a closed set, which we take as $\\mathcal {C}_1$ ; $\\mathcal {C}_1\\subset \\mathcal {O}_1$ since the closed ball around $z$ with radius $r/2$ is a subset of $\\mathcal {O}^{\\prime }$ , and the inverse (back to the starting time) of $\\mathcal {O}^{\\prime }$ is $\\mathcal {O}_1$ .", "Step 3.", "Since $\\mathcal {C}_1$ has positive Lebesgue measure, we can reiterate the arguments in Steps 1 and 2, and construct open set $\\mathcal {O}_2\\subset \\mathcal {C}_1$ and closed set $\\mathcal {C}_2\\subset \\mathcal {O}_2$ that visit and leave the extremal domain again.", "By iterating these arguments repeatedly, we get a sequence of closed (and indeed compact) sets $\\mathcal {C}_1 \\supset \\mathcal {C}_2 \\supset \\mathcal {C}_3 \\supset \\cdots $ .", "By the Cantor's intersection theorem, the intersection of this sequence of closed sets must be non-empty.", "Then any point in this intersection is a starting point that visits and leaves the extremal domain infinitely often." ], [ "Classical Rock-Paper-Scissors Game", "The standard Rock-Paper-Scissors game is the zero-sum game $(\\mathbf {A},-\\mathbf {A})$ with the following payoff matrix: $\\mathbf {A}=\\left[{\\begin{matrix}0 & -1 & 1\\\\1 & 0 & -1\\\\-1 & 1 & 0\\end{matrix}}\\right]$ .", "There are two types of $2\\times 2$ sub-matrices of $\\mathbf {A}$ .", "Consider such a sub-matrix which corresponds to strategy set $Q_i\\subset \\lbrace R,P,S\\rbrace $ for Players $i=1,2$ .", "The first type is when $Q_1 = Q_2$ , then the sub-matrix is $\\mathbf {A}^{\\prime } = \\left[{\\begin{matrix} 0 & -1\\\\ 1 & 0 \\end{matrix}}\\right]$ , which is trivial, i.e., $c(\\mathbf {A}^{\\prime }) = 0$ .", "The second type is when $|Q_1\\cap Q_2| = 1$ , then the sub-matrix is $\\mathbf {A}^{\\prime \\prime } = \\left[{\\begin{matrix} 0 & 1\\\\ 1 & -1 \\end{matrix}}\\right]$ ; it is easy to show that $c(\\mathbf {A}^{\\prime \\prime }) = 3/2$ .", "Due to the existence of the first type of sub-matrices, Theorem REF cannot be applied.", "We provide a separate proof to show that the same conclusion of Theorem REF holds for this specific game.", "Theorem 15 Suppose the underlying game is the standard Rock-Paper-Scissors game.", "For any $0<\\delta < 1/20$ , if both players use MWU with step-size $\\epsilon $ satisfying $\\epsilon < \\delta ^{22}/(34\\times 10^6)$ , then there exists a dense subset of points in $\\mathsf {int}(\\Delta )$ , such that the flow of each such point must eventually reach a point $(\\mathbf {x},\\mathbf {y})$ where each of $\\mathbf {x},\\mathbf {y}$ has exactly one entry larger than or equal to $1-\\delta $ .", "To start, we define a new family of primal set $\\mathcal {E}^\\kappa $ .", "To define it, let $(\\mathbf {x},\\mathbf {y})$ be a point in $\\mathsf {int}(\\Delta )$ , and let $Q_i$ denote the set of strategies of Player 1 with probability density larger than $\\kappa $ .", "Then $(\\mathbf {x},\\mathbf {y}) \\in \\mathcal {E}^\\kappa $ if and only if $|Q_1|,|Q_2|\\ge 2$ , and furthermore, there exists $Q^{\\prime }_1\\subset Q_1$ , $Q^{\\prime }_2\\subset Q_2$ such that $|Q^{\\prime }_1|,|Q^{\\prime }_2|=2$ and $|Q^{\\prime }_1\\cap Q^{\\prime }_2| = 1$ .", "The definition of $\\mathcal {E}^\\kappa $ deliberately avoids us from deriving a lower bound of $C(\\mathbf {x},\\mathbf {y})$ in the manner of (REF ) when $\\lbrace j_1,j_2\\rbrace = \\lbrace k_1,k_2\\rbrace $ , which corresponds to a trivial sub-matrix.", "Then by following Step 1 in the proof of Theorem REF , we have $\\inf _{\\mathbf {x},\\mathbf {y}\\in \\mathcal {E}^\\kappa } \\ge \\kappa ^2 c(\\mathbf {A}^{\\prime \\prime })^2/2 = 9\\kappa ^2 / 8$ .", "By following Step 2 in the proof of Theorem REF , when $\\epsilon < \\min \\left\\lbrace \\frac{1}{32n^2 m^2}~,~\\frac{9\\kappa ^2}{8} \\right\\rbrace $ , there exists a dense set of points in $\\mathsf {int}(\\Delta )$ such that the flow of each such point must reach a point $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ outside $\\mathcal {E}^\\kappa $ .", "Below, we assume the time is reset to zero with starting point $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ .", "We proceed on a case analysis below.", "Case 1: either $|Q_1|=1$ or $|Q_2|=1$ .", "For this case, we can simply follow Step 3 in the proof of Theorem REF .", "$\\kappa \\le \\delta ^{11}/6144$ suffices.", "Case 2: $Q_1=Q_2$ , and $|Q_1|=2$ .", "Without loss of generality, we assume $Q_1=Q_2=\\lbrace R,P\\rbrace $ .", "In the sub-game corresponding to $Q_1\\times Q_2$ , each player has a strictly dominant strategy, namely $P$ .", "Intuitively, the probability of choosing strategy $P$ must strictly increase with time (when we ignore the tiny effect of strategy $S$ ).", "More formally, starting from time zero, for the next $T_1 := \\left\\lfloor \\frac{1}{2\\epsilon }\\ln \\frac{\\delta }{2\\kappa } \\right\\rfloor $ time steps, $x_S^t,y_S^t \\le \\delta /2$ , and hence $x_P^t+x_R^t,y_P^t+y_R^t \\ge 1-\\delta /2$ .", "Then $&{\\small (\\text{the payoff to strategy }P\\text{ of Player 1 in round }t) - (\\text{the payoff to strategy }R\\text{ of Player 1 in round }t)}\\\\=~&\\left[y_P^t \\cdot 0 + y_R^t \\cdot 1 + y_S^t \\cdot (-1)\\right] - \\left[y_P^t \\cdot (-1) + y_R^t \\cdot 0 + y_S^t \\cdot 1\\right]\\\\\\ge ~&y_P^t + y_R^t -\\delta ~\\ge ~ 1-2\\delta .$ Thus, $\\frac{x_P^{t+1}}{x_R^{t+1}} ~\\ge ~ \\frac{x_P^t}{x_R^t}\\cdot \\exp \\left( \\epsilon (1-2\\delta ) \\right)$ , and hence $\\frac{x_P^t}{x_R^t} ~\\ge ~ \\hat{x}_P \\cdot \\exp \\left( \\epsilon (1-2\\delta )t \\right).$ The above inequality holds also when all $x$ 's are replaced by $y$ 's.", "Case 2(a): at $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ , each of the two players have one strategy with probability larger than or equal to $1-\\delta $ .", "Then we are done.", "Case 2(b): at $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ , each of the two players have all strategies with probability less than $1-\\delta $ .", "Then we know that $\\hat{x}_P,\\hat{y}_P\\ge 1-(1-\\delta )-\\delta /2 = \\delta /2$ .", "By (REF ), when $\\exp \\left( \\epsilon (1-2\\delta )t \\right)\\ge 4/\\delta ^2$ , we have $x^t_P/x^t_R , y^t_P/y^t_R \\ge 2/\\delta $ .", "And since we still have $x^t_S,y^t_S\\le \\delta /2$ , it is easy to show that $x^t_P,y^t_P \\ge 1-\\delta $ .", "Case 2(c): at $(\\hat{\\mathbf {x}},\\hat{\\mathbf {y}})$ , exactly one of the two players have one strategy with probability larger than or equal to $1-\\delta $ .", "Without loss of generality, we assume the player is Player 2.", "Then we know that $\\hat{x}_P,\\hat{x}_R\\ge \\delta /2$ .", "Similar to the argument for Case 2(b), when $\\exp \\left( \\epsilon (1-2\\delta )t \\right)\\ge 4/\\delta ^2$ , we have $x^t_P \\ge 1-\\delta $ .", "If at this time $t$ , we have either $y^t_P\\ge 1-\\delta $ or $y^t_R\\ge 1-\\delta $ , we are done.", "Otherwise, we have $y^t_P\\ge \\delta /2$ .", "Thus, after another period of time $t^{\\prime }$ such that $\\exp \\left( \\epsilon (1-2\\delta )t^{\\prime } \\right)\\ge 4/\\delta ^2$ , we have $y^{t+t^{\\prime }}_P\\ge 1-\\delta $ , while $x^{t+t^{\\prime }}_P\\ge 1-\\delta $ still.", "For the arguments for Cases 2(b),(c) to hold, we need $2\\cdot \\left\\lceil \\frac{1}{(1-2\\delta )\\epsilon }\\ln \\frac{4}{\\delta ^2} \\right\\rceil ~~\\le ~~ T_1,$ A direct arithmetic shows that $\\kappa \\le \\delta ^{10}/2845$ suffices." ], [ "Continuous Analogue of OMWU in General-Sum Games", "In equations (REF ) and (), observe that each $\\frac{\\mathsf {d} p_j}{\\mathsf {d} t}$ is expressed as an affine combination of various $\\frac{\\mathsf {d} q_k}{\\mathsf {d} t}$ , while each $\\frac{\\mathsf {d} q_k}{\\mathsf {d} t}$ is expressed as an affine combination of various $\\frac{\\mathsf {d} p_j}{\\mathsf {d} t}$ .", "Thus, we may rewrite all these expressions into a matrix-form differential equation.", "Let $\\mathbf {v}(\\mathbf {p},\\mathbf {q})$ denote the following vector in $\\mathbb {R}^{n+m}$ : $\\mathbf {v}(\\mathbf {p},\\mathbf {q}) = ([\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_1,\\cdots ,[\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_n~,~[\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_1,\\cdots ,[\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_m)^{\\mathsf {T}}~,$ and let $\\mathbf {M}(\\mathbf {p},\\mathbf {q})$ denote the $(S_1\\cup S_2)\\times (S_1\\cup S_2)$ matrix $\\left[{\\begin{matrix} \\mathbf {0} & \\mathbf {M}^1\\\\ \\mathbf {M}^2 & \\mathbf {0} \\end{matrix}}\\right]$ , where $\\mathbf {M}^1\\equiv \\mathbf {M}^1(\\mathbf {p},\\mathbf {q})$ is a $S_1\\times S_2$ sub-matrix and $\\mathbf {M}^2 \\equiv \\mathbf {M}^2(\\mathbf {p},\\mathbf {q})$ is a $S_2\\times S_1$ sub-matrix defined as below: $M^1_{jk} = y_k(\\mathbf {q}) \\cdot (A_{jk} - [\\mathbf {A}\\cdot \\mathbf {y}(\\mathbf {q})]_j)~~~~\\text{and}~~~~M^2_{kj} = x_j(\\mathbf {p}) \\cdot (B_{jk} - [\\mathbf {B}^{\\mathsf {T}}\\cdot \\mathbf {x}(\\mathbf {p})]_k).$ Then we can rewrite the recurrence system (REF ) and () as $\\left( \\frac{\\mathsf {d} \\mathbf {p}}{\\mathsf {d} t} ~,~ \\frac{\\mathsf {d} \\mathbf {q}}{\\mathsf {d} t} \\right)^{\\mathsf {T}}= \\mathbf {v}(\\mathbf {p},\\mathbf {q}) + \\epsilon \\cdot \\mathbf {M}(\\mathbf {p},\\mathbf {q}) \\cdot \\left( \\frac{\\mathsf {d} \\mathbf {p}}{\\mathsf {d} t} ~,~ \\frac{\\mathsf {d} \\mathbf {q}}{\\mathsf {d} t} \\right)^{\\mathsf {T}}$ .", "This can be easily solved to a standard (non-recurring) system of ODE: $\\left( \\frac{\\mathsf {d} \\mathbf {p}}{\\mathsf {d} t} ~,~ \\frac{\\mathsf {d} \\mathbf {q}}{\\mathsf {d} t} \\right)^{\\mathsf {T}}~=~ \\left( \\mathbf {I}- \\epsilon \\cdot \\mathbf {M}(\\mathbf {p},\\mathbf {q}) \\right)^{-1} \\cdot \\mathbf {v}(\\mathbf {p},\\mathbf {q}),$ if the inverse of the matrix $(\\mathbf {I}- \\epsilon \\cdot \\mathbf {M}(\\mathbf {p},\\mathbf {q}))$ exists.", "We proceed by using the following identity: if a square matrix $\\mathbf {R}$ satisfies $\\sup _{\\Vert \\mathbf {z}\\Vert =1} \\Vert \\mathbf {R}\\mathbf {z}\\Vert < 1$ , then $(\\mathbf {I}- \\mathbf {R})^{-1} = \\mathbf {I}+ \\sum _{\\ell =1}^{\\infty } \\mathbf {R}^\\ell $ .", "In our case, we desire $\\sup _{\\Vert \\mathbf {z}\\Vert =1} \\Vert \\epsilon \\cdot \\mathbf {M}(\\mathbf {p},\\mathbf {q}) \\cdot \\mathbf {z}\\Vert ~<~ 1$ .", "Observe that for each row of $\\mathbf {M}(\\mathbf {p},\\mathbf {q})$ , its $\\ell _2$ -norm is at most $2\\Vert \\mathbf {x}\\Vert $ or $2\\Vert \\mathbf {y}\\Vert $ , which are upper bounded by 2.", "Thus, each entry in $\\epsilon \\cdot \\mathbf {M}(\\mathbf {p},\\mathbf {q}) \\cdot \\mathbf {z}$ is absolutely bounded by $2\\epsilon $ , and hence $\\Vert \\epsilon \\cdot \\mathbf {M}(\\mathbf {p},\\mathbf {q}) \\cdot \\mathbf {z}\\Vert \\le 2\\epsilon \\sqrt{n+m}$ .", "Consequently, $\\epsilon < 1/(2\\sqrt{n+m})$ suffices to guarantee that the inverse of $(\\mathbf {I}- \\epsilon \\cdot \\mathbf {M}(\\mathbf {p},\\mathbf {q}))$ exists, and the identity mentioned above holds for its inverse: $\\left( \\frac{\\mathsf {d} \\mathbf {p}}{\\mathsf {d} t} ~,~ \\frac{\\mathsf {d} \\mathbf {q}}{\\mathsf {d} t} \\right)^{\\mathsf {T}}~=~ \\left( \\mathbf {I}+ \\sum _{\\ell =1}^\\infty \\epsilon ^\\ell \\cdot \\mathbf {M}(\\mathbf {p},\\mathbf {q})^\\ell \\right) \\cdot \\mathbf {v}(\\mathbf {p},\\mathbf {q}).$" ], [ "Volume Analysis of Discrete-Time OMWU", "Recall from [8] that the volume integrand for MWU is $1 ~+~ C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q}) \\cdot \\epsilon ^2 ~+~ \\mathcal {O}(\\epsilon ^4),$ while by (REF ), the volume integrand for OMWU is $1 ~-~ C_{(\\mathbf {A},\\mathbf {B})}(\\mathbf {p},\\mathbf {q}) \\cdot \\epsilon ^2 ~+~ \\mathcal {O}(\\epsilon ^3).$ By (REF ), within $\\mathsf {G}^{-1}(\\mathcal {E}^\\delta _{2,2})$ , $C_{(\\mathbf {A},-\\mathbf {A})}(\\mathbf {p},\\mathbf {q}) \\ge \\delta ^2 (\\alpha _1)^2 / 2$ , thus $C_{(\\mathbf {A},\\mathbf {A})}(\\mathbf {p},\\mathbf {q}) ~=~ -C_{(\\mathbf {A},-\\mathbf {A})}(\\mathbf {p},\\mathbf {q})$ $\\le -\\delta ^2 (\\alpha _1)^2 / 2$ .", "Therefore, when $\\epsilon $ is sufficiently small, the volume integrands for MWU in coordination game and OMWU in zero-sum game are both at most $1- \\epsilon ^2 \\delta ^2 (\\alpha _1)^2 / 4$ .", "Theorem 16 Suppose the underlying game is a non-trivial zero-sum game $(\\mathbf {A},-\\mathbf {A})$ and the parameter $\\alpha _1$ as defined in Theorem REF is strictly positive.", "For any $1/2 > \\delta > 0$ , for any sufficiently small $0<\\epsilon \\le \\bar{\\epsilon }$ where the upper bound depends on $\\delta $ , and for any set $S=S(0)\\subset \\mathsf {G}^{-1}(\\mathcal {E}^{\\delta }_{2,2})$ in the dual space, if $S$ is evolved by the OMWU update rule (REF ) and if its flow remains a subset of $\\mathsf {G}^{-1}(\\mathcal {E}^{\\delta }_{2,2})$ for all $t\\le T-1$ , then $\\mathsf {vol}(S(T)) ~\\le ~ \\left( 1 - \\frac{\\epsilon ^2 \\delta ^2 (\\alpha _1)^2}{4} \\right)^T \\cdot \\mathsf {vol}(S).$ Theorem 17 Suppose the underlying game is a non-trivial coordination game $(\\mathbf {A},\\mathbf {A})$ and the parameter $\\alpha _1$ as defined in Theorem REF is strictly positive.", "For any $1/2 > \\delta > 0$ , for any sufficiently small $0<\\epsilon \\le \\bar{\\epsilon }$ where the upper bound depends on $\\delta $ , and for any set $S=S(0)\\subset \\mathsf {G}^{-1}(\\mathcal {E}^{\\delta }_{2,2})$ in the dual space, if $S$ is evolved by the MWU update rule (REF ) and if its flow remains a subset of $\\mathsf {G}^{-1}(\\mathcal {E}^{\\delta }_{2,2})$ for all $t\\le T-1$ , then $\\mathsf {vol}(S(T)) ~\\le ~ \\left( 1 - \\frac{\\epsilon ^2 \\delta ^2 (\\alpha _1)^2}{4} \\right)^T \\cdot \\mathsf {vol}(S).$" ] ]

[ [ "BPM Systems: A brief Introduction to Beam Position Monitoring" ], [ "Abstract This introduction on beam position monitors (BPM) summarizes the fundamental parts of the tutorial presented at the CAS 2018 on beam instrumentation.", "The focus is on the signal detection and normalization, and on the principle of operation of commonly used broadband pickups, i.e.\\ button and stripline BPMs.", "Other BPM types, such as split-plane and cavity BPMs are also discussed, as well as the detection of low-$\\beta$ beams.", "Finally, a note on BPM signal processing techniques is given." ], [ "Introduction", "The beam position monitor (BPM) system is one of the most utilized and powerful beam instrumentation tools in a particle accelerator or accelerator beam-line.", "A BPM system consists out of many beam position monitors distributed along the accelerator beam-line, see Fig.", "REF for a ring accelerator, monitoring the passing beam.", "Each beam position monitor consists our of: a BPM pickup, which is part of the vacuum chamber and consists out of two or four symmetrically arranged electrodes which couple to the the electromagnetic fields of the beam and generate electrical signals at their output ports.", "Typically, a BPM pickup is located near each quadrupole magnet along the beam-line.", "the read-out electronics, which is used to condition and process the signals from the pickup electrodes to provide the beam position information in a digital data format, such that it can be acquired and further processed by the accelerator control system.", "Figure: A BPM systemPopular BPM pickup styles are of button or stripline type, have broadband characteristics, and generate a pulse-like output signal for each passing bunch.", "This enables bunch-by-bunch BPM signal processing possibilities, but requires the synchronization of the BPM data of all monitors in the system.", "In a ring accelerator a synchronized turn-by-turn monitoring of the beam position is of interest to follow the response of the beam to a kick or chirp excitation, e.g.", "for beam optic studies and analysis.", "Averaging the BPM data over many turns (ring accelerator) or over many beam pulses (linac) at each BPM provides a high resolution measurement of the beam orbit or beam trajectory, used for the alignment of the beam, for an orbit feedback, etc.", "This summary of the CAS 2018 tutorial on “BPM Systems” gives a short introduction on the principles and techniques of a beam position monitor.", "The principle of operation of broadband BPM pickups are covered in more detail, as they are widely used, other types of BPM pickups are mentioned briefly.", "Some information on the BPM signal processing is also provided, however, as this part is subject to the advances in electronics technologies and certainly will change, or will be outdated soon, therefore the discussion on read-out electronics is kept brief." ], [ "Principle of Operation of a Beam Position Monitor", "This introduction on beam position monitoring covers BPM systems based on non-invasive, electromagnetic-type beam pickups, however it should be noticed, there are other methods and ways to detect the position of the beam.", "Figure REF illustrates a typical pickup, in this example the so-called “button”-style BPM, which consists out of four round, metallic, coin-like electrodes (the “buttons”), which are arranged symmetrically along the horizontal and vertical axes in a vacuum chamber of circular cross-section.", "The horizontal ($x$ ) and vertical ($y$ ) offset of the beam trajectory wrt.", "the center of the vacuum chamber is the beam position or beam displacement, and has to be monitored with high resolution, accuracy and repeatability, this is the goal of the beam position measurement.", "Figure: Operation principle of a BPM pickup (courtesy O.R.", "Jones).Broadband BPM pickups operate on the principle of the image currents (or image charges), each charged particle of the beam is “compensated” by an image charge of opposite sign in the metallic vacuum chamber.", "Figure REF shows the induced image charges on a pair of electrostatic BPM electrodes and the resulting bunch response voltage signal for the upper electrode on a load resistor.", "The time-domain response of a BPM electrode to a bunched beam appears as a differentiated pulse as the image charges are induced as a displacement current through the capacitive button electrode.", "Figure REF shows the equivalent circuit and the frequency-domain response of this capacitive coupling BPM electrode, which is equivalent to that of a simple, 1$^{\\mathrm {st}}$ -order high-pass filter.", "There is no “DC-coupling”, and therefore broadband BPM pickups cannot operated with debunched, direct-current (DC) beams.", "Figure: The beam (bunch) intensity effect on the difference signal of a BPM pickup.As illustrated in Fig.", "REF , the pulse-like signal waveforms out of the symmetric pair of BPM electrodes are always identical, but as for each electrode $v_{\\mathrm {elec}}\\propto \\mathrm {pos}\\times \\mathrm {int}$ the signal amplitude is proportional to the distance between BPM electrode and beam, i.e.", "the beam position ($\\mathrm {pos}$ ) and the intensity ($\\mathrm {int}$ ) of the beam, it is obvious to subtract the two voltage signals of the symmetrically arranged BPM electrodes, lets call them $A$ and $B$ , to get a signal proportional to the beam position ($\\mathrm {pos}$ ): $v_{\\Delta } = (v_A-v_b) \\propto \\mathrm {pos}$ While this is true, the signal of each BPM electrode remains proportional to the beam intensity $v_{\\mathrm {elec}} \\propto \\mathrm {int}$ , and due to this so-called common-mode signal the difference signal $v_{\\Delta }$ of the BPM electrodes still contains the beam intensity: $v_{\\Delta } = (v_A-v_B) \\propto \\mathrm {pos} \\times \\mathrm {int}$ A normalization procedure is required to extract a pure, beam intensity independent position signal from the BPM electrodes: $\\frac{v_{\\Delta }}{v_{\\mathrm {int}}} \\propto \\frac{\\mathrm {pos} \\times \\mathrm {int}}{\\mathrm {int}} = \\mathrm {pos}$ Typically, the intensity signal required for the normalization is derived from the BPM electrodes as well, e.g.", "$v_{\\mathrm {int}}=v_{\\Sigma }=v_A+v_B$ , which leads to the well-known method to normalize the beam position signal: $\\frac{v_{\\Delta }}{v_{\\Sigma }}=\\frac{v_A-v_B}{v_A+v_B} \\propto \\mathrm {pos}$ however, many other methods for the beam intensity normalization also exist." ], [ "The BPM Pickup", "A BPM pickup is a passive, linear electromagnetic coupler, each of its electrodes delivers a signal (here defined in the frequency-domain): $V_{\\mathrm {elec}}(x,y,\\omega ) = s(x,y)Z(\\omega )I_b(\\omega )$ with $Z(\\omega )$ being the transfer impedance of a BPM pickup electrode (unit: $\\Omega $ ), $I_b(\\omega )$ being the beam or bunch current, i.e.", "the frequency-domain equivalent of the time-domain envelope function of the longitudinal particle distribution.", "Often the bunch current $I_{\\mathrm {bunch}}(\\omega )$ is assumed as a Gaussian envelope function, which is a good approximation for electron bunches, but may have some limitations in case of bunched proton or ion beams.", "$s(x,y)$ is a sensitivity function reflecting the cross-section geometry of the BPM pickup, to describe the transverse position characteristic, i.e.", "the strength of the coupling between beam and BPM electrode as function of the transverse beam position $(x,y)$ .", "Hidden in Eq.", "(REF ) is an electromagnetic coverage factor $\\phi =\\frac{\\int J_w dA_{\\mathrm {elec}}}{ \\int J_w dA_{\\mathrm {BPM}}}$ which is defined as ratio of the integrated wall current $J_w$ between the surface of the pickup electrode $A_{\\mathrm {elec}}$ and the total surface area of the BPM pickup $A_{\\mathrm {BPM}}$ , expressed for a centered beam.", "Figure: Electromagnetic coverage of button BPM electrodes.Figure REF illustrates the coverage factor on the longitudinal section of a button-style BPM pickup: Fig.", "REF a indicates the surface areas $A_{\\mathrm {elec}}$ and $A_{\\mathrm {BPM}}$ .", "Fig.", "REF b shows a “snap-shot” of the image (wall) current density $J_w$ for a Gaussian bunch as it is passing the BPM pickup, and Fig.", "REF c shows the E-field local in the BPM volume.", "While similar, the geometric and the electromagnetic coverage are not identical, $\\frac{\\int J_w dA_{\\mathrm {elec}}}{ \\int J_w dA_{\\mathrm {BPM}}} \\ne \\frac{A_{\\mathrm {elec}}}{ A_{\\mathrm {BPM}}}$ which is illustrated for the cross-section of three different button BPM arrangements in Fig.", "REF .", "Figure: E-field for different BPM electrode arrangements.Evidently, for the the curved, flush BPM electrodes, Fig.", "REF (left), geometric and electromagnetic coverage are very similar, but this is not true for the other examples with flat BPM electrodes, arranged flush with the beam pipe, Fig.", "REF (center), or raised, Fig.", "REF (right).", "As Eq.", "(REF ) expresses the coupling ratio between beam and BPM electrode for a centered beam, it is equivalent to $s(x=0,y=0)=\\phi $ .", "But while it is an intrinsic part of the BPM position characteristic, there is no practical meaning linked as it vanishes through the normalization procedure.", "The coverage factor $\\phi $ is based on size and shape of the BPM electrode, which also defines the frequency depending beam coupling characteristic, therefore it also appears in the the discussion of the transfer impedance $Z(\\omega )$ for broadband BPM pickup electrodes, relevant to compute the signal levels of the BPM electrodes.", "Furthermore, please notice that Eq.", "(REF ) applies only to broadband BPM pickups, where the transverse position sensitivity $s \\protect {\\protect \\mathrel {\\unknown.", "{\\perp }\\hspace{1.111pt}{\\perp }}\\omega is frequency independentfor relativistic beams (\\beta \\rightarrow 1).For low-\\beta beams s(x,y)is a function of the beam velocity v=\\beta c_0, and is discussed at the end of this section.", "}Equation~(\\ref {eq:transfer}) separates the frequency dependent, but beam position independentresponse function $ Z()$ from the frequency independent beam position characteristic $ s(x,y)$, which is basedon the image charge model (for broadband BPM pickups).This methodology simplifies the understanding and analysis of a BPM pickup.The characteristics of a resonant (cavity) BPMs does not simply match to Eq.~(\\ref {eq:transfer}),and is discussed separately.$" ], [ "Image charges", "From electrostatic principles (Gauss law, Poisson equation) we find for a point charge in the rest frame of the charge: $\\nabla \\vec{E} =-\\nabla ^2 \\phi = \\frac{\\rho }{\\epsilon } \\Rightarrow \\phi ^{\\prime }=\\frac{q}{4\\pi \\epsilon _0} \\frac{1}{r^{\\prime }}$ the potential $\\phi ^{\\prime }$ follows a uniformly distributed static electric field, see also Fig.", "REF $\\vec{E^{\\prime }}=\\frac{q}{4\\pi \\epsilon _0} \\frac{\\vec{r^{\\prime }}}{r^{\\prime 3}}$ where $|\\vec{r}|=\\sqrt{x^2+y^2+z^2}$ is the modulus of the vector between the charge and the observer.", "From the Lorentz transformation for the electromagnetic fields we find the $E$ -field components for a point charge moving with constant velocity $v=\\beta c_0$ along the $z$ -axis: $E_x=\\gamma E^{\\prime }_x$ , $E_y=\\gamma E^{\\prime }_y$ , and $E_z=E^{\\prime }_z$ , with $\\gamma =1/\\sqrt{1-\\beta ^2}$ .", "The transformation $r^{\\prime }=\\sqrt{x^2+y^2+\\gamma ^2 (z-vt)^2}$ results at $t=0$ in the electric field $\\vec{E}=\\frac{q}{4\\pi \\epsilon _0}\\frac{\\gamma }{(x^2+y^2+\\gamma ^2 z^2)^{3/2}}\\vec{r}$ where $\\vec{r}=(x,y,z-vt)$ is the vector from the current position of the particle to the observer.", "The magnetic field is found by the relation $\\vec{B}=1/c_0^2 (\\vec{v}\\times \\vec{E})$ .", "Figure REF and REF show how the electric field gets “compressed” as the velocity of the a point charge increases.", "The opening angle of the EM-field is $\\propto 1/(\\beta \\gamma )$ .", "Figure: TEM field of a point charge moving with v≃c 0 v\\simeq c_0 in a conductive beam pipe.At the relativistic limit $\\beta \\rightarrow 1$ the electromagnetic field of the point charge is purely transverse, a so-called transverse electro-magnetic (TEM) field (“pancake”-like field), which also applies if the charge moves with relativistic velocity in a conducting vacuum chamber, see Fig.", "REF .", "In this case the point charge $q$ is compensated by image (wall) charges $q_w$ which are distributed at the inner surface of the beam pipe.", "$q$ and $q_w$ are always linked through the electromagnetic field." ], [ "Position characteristic in a circular vacuum chamber", "A charged particle beam $I_{\\mathrm {beam}}$ with infinite small transverse dimension, a so-called pencil beam which is equivalent to a line charge, traveling with relativistic velocity ($\\beta \\rightarrow 1$ ) in a beam pipe with circular cross-section of radius $R$ has only transverse electromagnetic field components (TEM field), and is equivalent to a line charge.", "Figure REF illustrates the equivalent 2-dimensional electrostatic image charge problem for a beam charge at the position $(x=r\\cos \\varphi ,y=r\\cos \\varphi )$ .", "The solution for the wall current distribution can be found in form of a series expansion: Jw(R, , r, )= -Ibeam2R[ 1+n=1 (rR )n n (-) ] or as a closed-form expression: Jw(R, , r, )= -Ibeam2R R2-r2R2+r2-2Rr(-) A wall current related fraction $I_{\\mathrm {elec}}$ is induced on a BPM electrode covering an arc $\\alpha $ : $I_{\\mathrm {elec}}=R\\int _{-\\alpha /2}^{+\\alpha /2} J_w(R, \\Phi , r, \\varphi ) \\mathrm {d}\\Phi $ For the two horizontal arranged electrodes $A$ and $B$ of Fig, REF we find: $I_{\\mathrm {elec}}=-\\frac{I_{beam}}{2\\pi } s_{\\mathrm {elec}} \\left( r/R,\\varphi , \\alpha \\right)$ with the sensitivity functions for the $A$ and $B$ electrodes based on (REF ): sA ( r/R,, ) = + 4 n=1 1n ( rR )n ( n) ( n2 ) sB ( r/R,, ) = + 4 n=1 1n ( rR )n ( n) [n ( +2 ) ] Following the intensity normalization concept of Eq.", "(REF ) for our two symmetric, horizontal electrodes $A$ and $B$ , using (REF ), (REF ), (REF ), (REF ), we can approximate the horizontal position characteristic for a circular beam pipe of radius $R$ as: $\\frac{\\Delta }{\\Sigma } = \\frac{A-B}{A+B} \\simeq \\frac{4\\sin (\\alpha /2)}{\\alpha } \\frac{x}{R}+ \\mathrm {higher\\: order \\: terms} = \\mathrm {hor.", "\\: position}$ with $x/R$ being the normalized horizontal beam position (normalized to the beam pipe radius $R$ ), and $\\alpha $ being the coverage angle of the electrodes, intercepting some fraction of the image current $J_w$ .", "In general, the normalized position characteristic is non-linear, however, for narrow electrodes $\\sin (\\alpha /2)\\approx \\alpha /2$ and small beam displacements $x^2+y^2 \\ll R^2$ the normalized beam position response follows approximatively: $\\mathrm {hor.", "\\: position} \\approx \\frac{2}{R}x$ with $2/R=k_{\\mathrm {PU}}$ sometimes called the monitor constant.", "From (REF ) and REF ) we can also find a closed form expression for the normalized position characteristic in Cartesian coordinates: $\\frac{\\Delta }{\\Sigma } = \\frac{A-B}{A+B} =\\frac{s(x,y,R,\\alpha )-s(-x,y,R,\\alpha )}{s(x,y,R,\\alpha )+s(-x,y,R,\\alpha )} = \\mathrm {hor.", "\\: position}$ with $\\begin{aligned}s(x,y,R,\\alpha ) &= \\pi \\int _{-\\alpha /2}^{+\\alpha /2} J_w(R, \\Phi ) \\mathrm {d} \\Phi \\\\&= \\arctan \\frac{[\\left(R+x \\right)^2+y^2 ] \\tan \\left( \\frac{\\alpha }{4} \\right) - 2Ry}{x^2+y^2-R^2} +\\arctan \\frac{[\\left(R+x \\right)^2+y^2 ] \\tan \\left( \\frac{\\alpha }{4} \\right) + 2Ry}{x^2+y^2-R^2}\\end{aligned}$ being the sensitivity function.", "Figure: Horizontal Δ/Σ\\Delta / \\Sigma position characteristic in a circular beam pipe of radius R=12.5mmR=12.5~mm,with electrodes each covering an arc of α=30 ∘ \\alpha =30^{\\circ }.An example of the horizontal position characteristic based on (REF ), (REF ) is illustrated in Figure REF , on the left side (Fig.", "REF ) as parametric plot with contours of constant $\\Delta /\\Sigma $ , on the right side (Fig.", "REF ) as function of the horizontal beam position.", "The small covering angle ($\\alpha =30^{\\circ }$ ) of the electrodes lead to high non-linear effects in the position characteristic, with substantial cross-coupling to the vertical plane.", "From that perspective a covering angle of $\\alpha \\approx 60^{\\circ }$ is preferable, however, other aspects like a minimum beam coupling impedance often have to be considered and call for small BPM electrodes.", "For accelerators covering a large range of beam intensities, a pair of logarithmic detectors in the BPM read-out electronics can be beneficial.", "In this case the intensity normalization is performed by: hor.", "position 20 10 (vA) - 20 10 (vB) = 2010 ( AB ) 2010 (4(/2) xR + higher order terms) = 2010 s(x,y,R,)s(-z,y,R,) As Eq.", "(REF ) indicates, the signals of the BPM electrodes $A$ and $B$ are detected with separate logarithmic amplifiers and their detected output signals are subtracted, resulting in a normalized beam position signal.", "Each logarithmic detector follows the characteristic $v_{out}=20\\log _{10} (v_{in})$ over a large input signal range, typically $40\\ldots 60~dB$ .", "Figure: Horizontal position log-ratio characteristic in a circular beam pipe of radius R=12.5mmR=12.5~mm,with electrodes each covering an arc of α=30 ∘ \\alpha =30^{\\circ }.Figure REF shows the position characteristic of the log-ratio normalization for the same cross-section geometry of the BPM pickup as Fig.", "REF , which appears to be more linear compared to the $\\Delta /\\Sigma $ method." ], [ "BPM cross-section with “rotated” electrodes", "In some cases the BPM electrodes cannot be located along the horizontal and vertical planes, e.g.", "in electron storage rings due to the synchrotron light fan which can have unwanted effects on the horizontal electrodes, or due to collision debris at BPM pickups near the interaction point of a particle collider, or just due to real-estate restrictions.", "Intense photons, charged primary or secondary particles on the BPM electrodes have to be strictly avoided, they would alter the signal response in a non-predictable way.", "For those situations, the BPM pickup electrodes can be arranged in a different way, but still symmetric to the horizontal and vertical plane.", "For a simple circular cross-section we can still use Eq.", "(REF ), applying: x = x1 - x2 y = x1 + x2 for the rotated, not necessarily orthogonal coordinates ($\\tilde{x}$ , $\\tilde{y}$ ) of the BPM electrodes.", "Most popular is an arrangement of all electrodes rotated by $\\theta _1=\\theta _2=\\theta =45^{\\circ }$ , see Fig.", "REF .", "We used the same cross-section dimensions ($R=12.5~mm$ , $\\alpha =30^{\\circ }$ ), but by comparing Fig.", "REF with Fig.", "REF (or Fig.", "REF with Fig.", "REF ) it is evident that the position non-linearities increase substantially for a BPM with rotated electrodes." ], [ "Numerical analysis of the position characteristic", "For the general case of an arbitrary shape of the BPM pickup and its electrodes the analytical analysis has limitations, we just may not find an analytical expression for the position characteristic $s(x,y)$ .", "In this case the BPM pickup must by characterized in a different way, e.g.", ": by measurements with a stretched wire, which requires a BPM test-stand and specific lab equipment.", "by numerical computation, either by solving the Laplace equation in two-dimensions (2D) to find the electrostatic potential, or by solving the Maxwell equations in three-dimansions (3D) to find coupled electrode signals through the EM fields excited by a transient beam stimulus.", "While the stretched wire measurement method is useful as verification of the final prototype, the numerical computation is already helpful during the construction phase, enabling the optimization of shapes towards specific goals and the study of the effect of manufacturing tolerances." ], [ "Electrostatic analysis in 2D", "For a relativistic beam with sufficiently long bunches compared to the beam pipe aperture ($\\sigma _l \\gg R$ ) and large values of $\\gamma $ ($1/\\gamma ^2 \\ll (\\sigma _l/R)^2$ ), the analysis of a BPM electrode configuration can be reduced to a simple two-dimansional electrostatic problem: $ \\nabla _{\\perp }^{2}\\mathbf {\\Phi _{elec}(r)} = \\frac{\\rho }{\\phi } \\mathbf {\\delta (r-r_0)},$ with the Fourier expansion of line charge density $\\rho $ and potential $\\phi $ being a constant factor ($\\rho \\propto \\phi $ ) of no influence.", "Instead of solving Eq.", "REF for many positions $\\mathbf {r} \\in (x,y)$ of the beam equivalent line charge density $\\rho $ , we apply Green's reciprocity theorem and solve the Laplace equation numerically for one of the BPM electrodes, see Fig.", "REF : $ \\nabla _{\\perp }^{2}\\mathbf {\\Phi _{elec}(r)} = 0 \\rightarrow \\Phi _{elec}(x,y).$ As of the symmetries, the potentials of the other electrodes are simply found by coordinate rotation or mirroring, e.g.", "$\\Phi _{B}(x,y)=\\Phi _{A}(-x,y)$ , etc.", "We now combine the electrode potentials $\\Phi _{A}$ and $\\Phi _{B}$ , respectively $\\Phi _{C}$ and $\\Phi _{D}$ to compute the normalized H and V potential ratios, which are equivalent to the raw, normalized $\\Delta / \\Sigma $ beam position (see also Eq.", "REF ) $ \\hat{\\Phi }_x (x,y) = \\frac{\\Phi _{A}-\\Phi _{B}}{\\Phi _{A}+\\Phi _{B}} \\;\\;\\;\\;\\text{ and }\\;\\;\\;\\;\\hat{\\Phi }_y (x,y) = \\frac{\\Phi _{C}-\\Phi _{D}}{\\Phi _{C}+\\Phi _{D}}.$ Figure: Comparison of analytical and numerical analysis of a horizontal BPM pickup.Figure REF compares the 2D numerical result for an “ideal” horizontal BPM, having circular beam pipe of $R = 10$  mm with electrodes spanning $\\alpha = 30^\\circ $ with the analytical result.", "The error reaches up to 5% at some large beam displacements, but stays within $<1$ % for typical beam positions (see Fig.", "REF ).", "This 2D approach basically gives the position sensitivity $s(x,y)$ of an arbitrary BPM cross-section, thus cannot distinguish between, e.g.", "button and stripline configuration, and neglects fringe fields and any other 3D effects.", "The advantage lies in a very efficient computation of the BPM position characteristic, the beam-to-electrode coupling, and it also can be used to determine the characteristic impedance of a stripline electrode." ], [ "3D electromagnetic analysis", "Different options exists for a three-dimensional electromagnetic analysis of the BPM pickup, among other simulation results, they all allow in some way to extract the position characteristic of a BPM pickup with arbitrary shape of BPM body and electrodes: S-parameter The definition of a set of coaxial ports, one for each BPM electrode, plus two ports for the upstream and downstream beam pipe enables the extraction of the scattering parameters based on a 3D electromagnetic analysis.", "In this numerical simulation a thin metal conductor (“wire”) in $z$ -direction, scanned in the $x$ -$y$ plane similar to the stretched-wire measurement method, replaces the beam.", "As this coaxial stimulus arrangement excites TEM-fields, the result of this method is equivalent to that of a relativistic beam ($\\beta =1$ ) as stimulus signal.", "For the position characteristic the S-parameter between the upstream beam port and the pickup electrodes are relevant.", "The analysis can be performed either in the frequency-domain, or in time-domain, the results may be valuable to be compared with a practical, stretched-wire bench measurement setup.", "Wakefield A wakefield simulation calculates the EM fields in the time-domain, driven by a line-current signal following a single bunch envelope function (often Gaussian), moving at a given, constant velocity with a specified transverse ($x$ -$y$ ) displacement through the specified structure (here: a BPM pickup) along the $z$ -direction.", "While the primary goal of the wakefield simulation is the estimation of the wake-potential, which is proportional to the integrated force acting on the trailing beam particles due to EM interaction with the surrounding structure by the leading particles, the beam-like longitudinal excitation signal can also be used to study other characteristics of a BPM pickup, e.g.", "the position characteristic (requires many runs with different beam offsets), the transfer impedance of the BPM electrodes in time- and frequency domain, unwanted beam-excited RF resonances, detailed field analysis, etc.", "Therefore, waveguide ports, voltage signal and field monitors have to be added.", "Particle-in-cell (PIC) The PIC method offers the most flexible time-domain simulation of the response of a BPM pickup to a beam which is defined in terms of macro-particles, defined by a particle emission model As the computational demands and times are substantially higher compared to the above 3D numerical methods, PIC and other particle tracking methods are rarely used for the characterization of a BPM pick, except for e.g.", "for low-$\\beta $ beams or other exotic beam conditions.", "Usually the wakefield is the preferred numerical 3D method to analyze the BPM pickup characteristic, Fig.", "REF gives an example for a simplified button BPM geometry in CST Studio.", "The black and orange traces indicate the beam trajectory, respectively the wakefield integration path through the BPM pickup structure, little arrows at the end of each coaxial port indicate the voltage signal monitors.", "To analyze the position characteristic, the “beam” stimulus has to be scanned in reasonable steps, typically $<1/10$ of the aperture, in the $x$ -$y$ plane while recording the voltage signals at the four buttons.", "However, as of the transverse symmetry in this particular case it would be sufficient to scan $1/8$ of the area.", "Beside the position characteristic the wakefield simulation will give other relevant information, like bunch response function of the BPM electrodes in time and frequency domain transfer impedance), wake-potential and beam coupling impedance, etc." ], [ "Correction of the position non-linearities of a broadband BPM pickup", "As analyzed in the previous chapter, all types of broadband (non-resonating), 2-dimensional BPM pickups with circular or similar beam-pipe cross-section are based on the wall current principle, and have a non-linear position behavior, regardless of their normalization principle (Figures REF and REF ) or electrode orientation (Fig.", "REF ).", "This non-linear position behavior originates in Eq.", "(REF ), and is minimum for horizontal / vertical oriented electrodes spanning an arc of $\\alpha \\approx 60^\\circ $ .", "But also other two-dimensional cross-section geometries, e.g.", "elliptical or rectangular vacuum chambers with flat or curved electrodes do have a non-linear position behavior, the only exception is the split-plane – sometimes called “shoe-box” – type BPM pickup, which consists out of a pair capacitive electrodes which is “sliced” in a three-dimensional way and discussed later.", "However, all the popular button-style, or stripline BPMs have a non-linear position behavior that usually needs to be corrected, in particular if large beam displacements have to be monitored.", "As the non-linearities of the BPM pickup are known and as the raw position data (uncorrected and uncalibrated) is made available in a digital format, a correction can be applied to linearize the pickup response.", "Two linearization principles exists: a look-up table an equation-based linearization based on a 1D or 2D polynomial fit of the normalized position characteristic A look-up table based procedure directly corrects the digitized raw data, i.e.", "for each quantized value (raw ADC counts) the table responds with a corrected value based on the equations or a numerical analysis as discussed in the previous chapter.", "E.g., for a digitizer with an 8-bit analog-digital converter (ADC) the BPM pickup signals would be quantized into 256 values, thus requiring a 256-value look-up table.", "If each of the four BPM electrodes is digitized individually, it requires four 256-entry look-up tables, each with a correction following the analytical Eq.", "(REF ) or numerical derived position characteristic Eq.", "(REF ) of the BPM electrode.", "In practice however, things are a bit more complicated.", "Any non-linearity, or change in the analog signal processing – which also maps the dynamic signal range of the BPM electrode to the full-scale range of the ADC – needs to be included in the correction for this procedure, thus the look-up table linearization combines the BPM pickup response with the transfer characteristic of the analog signal conditioning section.", "This “mixing” makes this concept more complicated in terms of maintenance and housekeeping aspects for large scale BPM systems, and becomes even more complicated if the two opposite electrodes signals are combined before the digitalization.", "A polynomial fit f the BPM pickup characteristic allows a clear separation between the pickup non-linearities and the following analog signal conditioning section.", "The following example explains the procedure for the normalized $\\Delta /\\Sigma $ position response on the simple example of a BPM pickup with circular cross-section:" ], [ "Calibration with linear correction", "The linear calibration does not perform any correction of the non-linearities, it just scales the normalized position characteristic of a pair of BPM electrodes, e.g.", "for the $\\Delta /\\Sigma $ normalization: $x_{\\mathrm {raw}} = \\frac{A-B}{A+B} \\;\\;\\;\\mbox{ or } \\;\\;\\; y_{\\mathrm {raw}} = \\frac{C-D}{C+D}$ with $A$ , $B$ , $C$ , and $D$ being the measured signal levels of the four BPM electrodes, usually in ADC counts.", "The derived normalized “raw” horizontal and vertical positions range -1...+1, and are unit-less.", "A simple, linear calibration is then: $x_{\\mathrm {bpm}} = k_x x_{\\mathrm {raw}} \\approx x \\;\\;\\;\\mbox{ or } \\;\\;\\;y_{\\mathrm {bpm}} = k_y y_{\\mathrm {raw}} \\approx y$ with $k_{x,y}=R/2$ for a BPM with circular cross-section, see also Eq.", "REF .", "As Figure REF shows, this linear “fit” can only be used for small beam displacements, at large beam displacements it underestimates the true beam position." ], [ "Calibration with non-linear correction", "To better account for the non-linearities of the normalized position behavior a calibration with higher-order polynomials can be applied.", "Consider a pair of horizontal oriented electrodes as BPM pickup and beam displacements only in the horizontal plane, $x\\ne 0$ , $y=0$ .", "The function $x_{\\mathrm {raw}}=f(x)$ between the true horizontal beam position $x$ and the measured value $x_{\\mathrm {raw}}$ can be estimated analytical or numerically as described in the previous section.", "By its nature we can assume $f$ to be smooth and invertible $x=f^{-1}(x_{\\mathrm {raw}})$ The coefficients $c_i$ of a one-dimensional polynomial of power $p$ can be calculated to fit $f^{-1}$ and find an approximate horizontal beam position: $ x_{\\mathrm {bpm}}^{1D} = \\sum _{i=0}^{p}c_i x_{\\mathrm {raw}}^i = U_p(x_{\\mathrm {raw}}) \\approx x .$ In case of a double symmetry of the BPM, e.g.", "circular cross-section, the polynomial for of the vertical axis $y_{\\mathrm {bpm}}^{1D}=U_p(y_{\\mathrm {raw}})$ will be identical to $x_{\\mathrm {bpm}}^{1D}$ .", "In other cases, e.g.", "elliptical or rectangular BPMs, a different set of polynomial coefficients $c_i$ for $y_{\\mathrm {bpm}}^{1D}\\ne x_{\\mathrm {bpm}}^{1D}$ has to be computed.", "As of the symmetry, all even coefficients $c_{i=0,2,4,\\ldots }=0$ , and don't have to be considered.", "However, $c_0$ is an offset which comes to play in case of alignment errors or as “electronics” offset.", "Typically, polynomials of order $p=3\\ldots 7$ give a sufficient correction of the non-linearities.", "Figure REF shows the remaining error of this 1D polynomial calibration procedure for the example of a CERN LHC stripline BPM with circular cross-section.", "The polynomial for was limited to $\\mathbb {R}=68\\%$ of the cross-section area, which seemed to be a good compromise between remaining errors in the area of interest ($\\sim $ 1/3 of the aperture) and reasonable order (here: $p=5$ ) of the polynomials.", "While this calibration with 1D polynomials for the non-linear correction does a good job along the horizontal and vertical axis, the position errors remain rather high for off-axis beam displacements ($x=r\\sin \\varphi $ , $x=r\\sin \\varphi $ ) for $\\varphi \\ne n\\pi /2$ with $n\\in (0,1,2,3)$ .", "Figure: Remaining position error after applying 1D correction polynomials U 5 (x)U_5(x), U 5 (y)U_5(y)for an area of ℝ=68%\\mathbb {R}=68\\% (courtesy A. Nosych).Consider an arbitrary beam offset $x,y \\ne 0$ within the BPM aperture.", "The relation between raw and the true beam position can be linked by the functions: ${\\left\\lbrace \\begin{array}{ll}x = f(x_{\\mathrm {raw}},y_{\\mathrm {raw}})\\\\y = g(x_{\\mathrm {raw}},y_{\\textrm {raw}})\\end{array}\\right.", "}$ where, in case of circular beam pipe and symmetries, $f=g$ and $y = f(y_{\\mathrm {raw}},x_{\\mathrm {raw}})$ (note the variable swap).", "By mapping and fitting $f(x,y)$ by a two-dimensional surface polynomial with $p$ and $q$ being the maximum powers for $x$ and $y$ respectively, a coupled relationship between the original, true beam position in each plane and its “raw” response is obtained for both planes: $ {\\left\\lbrace \\begin{array}{ll}x_{\\mathrm {bpm}}^{\\mathrm {2D}} = \\sum _{i,j=0}^{p,q}(c_{ij} x_{\\mathrm {raw}}^i y_{\\mathrm {raw}}^j ) =Q_{p,q}(x_{\\mathrm {raw}},y_{\\mathrm {raw}}) \\approx x \\\\y_{\\mathrm {bpm}}^{\\mathrm {2D}} = \\sum _{i,j=0}^{p,q}(c_{ij} y_{\\mathrm {raw}}^i x_{\\mathrm {raw}}^j ) =Q_{p,q}(y_{\\mathrm {raw}},x_{\\mathrm {raw}}) \\approx y\\end{array}\\right.", "}$ Figure: Pin-cushion effects and remaining errors for a calibration with 2D correction polynomials(courtesy A. Nosych).The 2D polynomial fit of the position characteristic returns a matrix of order $p,q$ with the polynomial coefficients $c_{ij}$ .", "In practice not only the even coefficients of the main rows are zero due to the symmetry conditions, $c_{i=0,2,4,\\ldots ,j} = 0$ , also many of the cross-coupling coefficients appear as very small value and can be neglected, e.g.", "for the $5^{\\mathrm {th}}$ order 2D polynomial fit of the LHC stripline BPM $ Q_{5,4} \\approx Q_{5,5} =\\begin{pmatrix}\\colorbox {green!50}{\\displaystyle c_{10}} & c_{11} & \\colorbox {green!50}{\\displaystyle c_{12}} & c_{13} & \\colorbox {green!50}{\\displaystyle c_{14}} & c_{15} \\\\\\cdots & & & & & \\\\\\colorbox {green!50}{\\displaystyle c_{30}} & c_{31} & \\colorbox {green!50}{\\displaystyle c_{32}} & c_{33} & c_{34} & c_{35} \\\\\\cdots & & & & & \\\\\\colorbox {green!50}{\\displaystyle c_{50}} & c_{51} & c_{52} & c_{53} & c_{54} & c_{55}\\end{pmatrix}$ only the highlighted coefficients need to be considered.", "Figure REF visualizes the results of a 2D polynomial fit applied in the area $\\mathbb {R}=40\\%$ on the CERN LHC stripline BPM, using the reduced, minimum number of cross-coupling coefficients." ], [ "Higher-order moments", "Until now the charged particle beam was treated as a point charge traveling with constant velocity through the BPM pickup, or as line charge.", "In practice, the beam consists out of many particles, and as the EM field of each particle is linear and time-invariant, we can apply the superposition principle to evaluate the effect of many particles passing the BPM pickup.", "Figure REF illustrates the concept for our well known, ideal BPM with circular cross-section.", "For most practical beam position monitoring applications we can ignore the transverse expansion of the beam, and replace it by a single point charge of $Q=eN$ , where $N$ is the number of particles in the beam and $e\\approx 1.6\\cdot 10^{-19}C$ the elementary charge.", "The measured transverse position $\\vec{r}=(\\bar{x}$ , $\\bar{y})$ of $Q$ represents the center-of-charge of the beam: $\\vec{r}=\\frac{1}{eN}\\sum _{i=1}^N q_i \\vec{\\rho }_i$ with $\\vec{\\rho }_i=(x_i, y_i)$ being the transverse position of the $i^{th}$ -particle $q_i=e$ .", "Strictly speaking Eq.", "(REF ) is only valid as long as the transverse dimensions of the beam are small compared to the aperture of the BPM pickup, $\\sigma _x, \\sigma _y\\ll R$ .", "The effect of the beam size on the signal of a BPM electrode can be studied by applying the superposition principle on Eq.", "(REF ), from which for the signal on electrode $A$ in Fig.", "REF follows: $I_{\\mathrm {A}}=-\\frac{I_{beam}}{2\\pi } \\left[\\alpha + 4 \\sum _{i=1}^N \\sum _{n=1}^{\\infty } \\frac{1}{n} \\left( \\frac{r_i}{R} \\right)^n\\cos ( n\\varphi _i ) \\sin \\left( \\frac{n\\alpha }{2} \\right) \\right]$ with the position of the $i^{th}$ -particle given in cylindrical coordinates ($x_i=r_i \\cos \\varphi _i$ , $y_i=r_i \\sin \\varphi _i$ ).", "Applying $Q\\sum _{i=1}^N x_i^2/N=\\sigma _x^2+\\bar{x}^2$ , respectively $Q\\sum _{i=1}^N y_i^2/N=\\sigma _y^2+\\bar{y}^2$ Eq.", "(REF ) can be rewritten in terms of moments: $I_{\\mathrm {A}}=-\\frac{I_{beam}}{\\pi } \\bigg [ \\underbrace{\\frac{\\alpha }{2}}_{\\text{monopole}}+\\underbrace{\\frac{2}{R}\\sin \\left(\\frac{\\alpha }{2}\\right) \\bar{x}}_{\\text{dipole}} +\\underbrace{\\frac{1}{R^2}\\sin ( \\alpha ) \\left( \\sigma _x^2-\\sigma _y^2+\\bar{x}^2-\\bar{y}^2 \\right) }_{\\text{quadrupole}}+ \\ldots \\bigg ]$ Monople moment The first term in Eq.", "(REF ) is the monopole moment, which is only proportional to the beam intensity.", "It often is called the common mode, as it appears in all BPM electrodes.", "Dipole moment The second term is of interest for the beam position measurement, it is proportional to the mean value of the beam displacement (here: $\\bar{x}$ ) times the beam intensity.", "Quadrupolar moment and other higher-order moments can be defined as of the transverse expansion of the beam.", "The quadrupolar moment includes information of the transverse beam dimensions $\\sigma _x$ , $\\sigma _y$ , and is $\\propto (\\Delta \\mathrm {size}+\\Delta \\mathrm {pos})\\times \\mathrm {int}$ .", "The higher-order moments exist because of the non-linear position behavior of a point charge in a BPM with circular cross-section, a BPM pickup with linear position characteristic would contain only monopole and dipolar moments.", "The constants between the moments scale with $\\sim 1/R$ , which in practice is more than an order of magnitude, thus, the dipolar moment is typically $10\\ldots 100$ smaller than the common mode, while the quadrupolar moment is smaller by a similar amount compared to the dipolar moment, which makes the detection of the quadrupolar moment very challenging.", "To extract information about the transverse beam size, two or more BPM pickups need to be located along the beam-line, preferable with no magnetic elements in between, at a location with $\\sigma _x\\ne \\sigma _y$ and where $\\bar{x}-\\bar{y}=0$ , aligned with a very high accuracy $\\ll 1/R^2$ .", "Imperfections of the symmetry of the BPM electrodes have to be in the same oder of magnitude, a BPM with a small aperture $R$ wrt.", "the beam size dimension is preferable.", "To insure $\\bar{x}-\\bar{y}=0$ the BPM pickups can be mounted on remote controlled translation stages, which allows to center the beam in the BPM pickups without moving the beam." ], [ "Bunched beam signals", "For the analysis of the position characteristic of a BPM pickup based the sensitivity function $s(x,y)$ the point charge approach is sufficient.", "However, to estimate the waveform and signal power out of a BPM pickup electrode in Eq.", "(REF ), the bunch current signal $i_{\\mathrm {bunch}}(t)$ or $I_{\\mathrm {bunch}}(\\omega )$ needs to be known – as discussed in this section – along with the transfer impedance $Z(\\omega )$ of the pickup electrode – discussed in the next section.", "A point-like charge $q=zeN$ , with $z$ being the charge state (for ions), $e\\approx 1.6\\cdot 10^{-19}C$ the elementary charge and $N$ the number of particles, traveling with relativistic velocity $\\beta =v/c_0=1$ in a perfectly conducting vacuum chamber (Fig.", "REF ) has a bunch current: $i_{\\mathrm {bunch}}(t)=q\\cdot \\delta (t)= -i_w(t)$ which is cancelled by the wall current $i_w(t)$ , originated by the image charges $q_w=-q$ distributed around the azimuth of the beam pipe wall.", "In some cases, e.g.", "very short bunches, this $\\delta $ -signal approach for the bunch current is an acceptable approximation, with the Dirac ($\\delta $ ) function defined as $\\int _{-\\infty }^{+\\infty }\\delta (t)=1$ .", "In most other cases the longitudinal distribution of the particles in the RF bucket is approximated by an analytic function to describe the longitudinal bunch current distribution vs. time, e.g.", "$i_{\\mathrm {bunch}}(t)=\\frac{zeN}{\\sqrt{2\\pi }\\sigma _t}e^{-\\frac{t^2}{2\\sigma _t^2}}$ for a Gaussian particle distribution of length $\\sigma _t$ (in time), see also Fig.", "REF .", "Figure: Bunch and beam current signals in the time-domain.In a ring accelerator of circumference $l_c=vT_{\\mathrm {rev}}=v/f_{\\mathrm {rev}}$ , RF buckets can be filled with charged particles at equidistance times intervals Table: Harmonic amplitude factors for various bunch shapes.$T=\\frac{2\\pi }{\\omega }=\\frac{h}{f_{RF}}$ where $h=f_{RF}/f_{\\mathrm {rev}}$ is the harmonic number.", "The resulting beam current is composed out of charged particle bunches, in the ideal case of equal intensity and shape, spaced by an equidistant time $T$ , and can be written in terms of a Fourier series, see also Fig.", "REF : $i_{\\mathrm {beam}}(t)=\\langle I_{DC}\\rangle + 2 \\langle I_{DC}\\rangle \\sum _{m=1}^{\\infty } A_m \\cos (m\\omega t)$ with the average beam current $\\langle I_{DC}\\rangle =zeN/T$ and frequency harmonics spaced by $\\omega = 2\\pi f$ .", "The bunch shape in Eq.", "(REF ) is defined by a harmonic amplitude factor.", "Table REF lists the harmonic amplitude factor $A_m$ for some typical bunch shape functions, with $t_b$ being the bunch length (in time) at the base, and a normalization of $A_m\\rightarrow 1$ for $\\omega \\rightarrow 0$ .", "Figure: Bunch signals with Gaussian and cos 2 ^2 shape in time and frequency domain.$\\begin{aligned}&\\text{time-domain} &\\qquad \\text{freq}&\\text{uency-domain} \\\\i_{Gauss}(t)&=\\frac{zeN}{\\sqrt{2\\pi }\\sigma _t}e^{-\\frac{t^2}{2\\sigma _t^2}} &\\qquad I_{Gauss}(f )&=eNe^{-2(\\pi f\\sigma _t)^2} \\\\[1em]i_{cos^2}(t)&=\\smash{\\left\\lbrace \\begin{array}{c@{}c@{}}\\frac{N}{t_b} \\left( 1+\\cos \\frac{2\\pi t}{t_b} \\right), \\;\\; -t_b/2<t<t_b/2 \\\\ []0, \\qquad \\qquad \\text{elsewhere}\\end{array}\\right.", "}&\\qquad I_{cos^2}(f)&=\\frac{eN\\sin \\pi f t_b}{\\pi f t_b \\left[1-(f t_b)^2\\right]}\\end{aligned}$ For many practical cases however, it is more convenient to use the Fourier transformation instead of a Fourier series expansion with infinite sums.", "Figure REF illustrates the two examples given in Eq.", "(REF ) of a bunch shape following a Gaussian and a raised-cosine (cos$^2$ ) function, having similar parameters.", "The differences between the two bunch shapes become more obvious compering them in the frequency-domain, at higher frequencies on a logarithmic scale (Fig.", "REF ).", "For illustration, bunch harmonics spaced by $f=100$  MHz have been indicated in the logarithmic plot, demonstrating how this concept of the Fourier transformation also covers the line spectrum of bunches, e.g.", "in a ring accelerator.", "Figure: Examples of bunched beam formats.However, in practice a beam of equidistant bunches of same intensity rarely exists, instead beam-free gaps for the injection / extraction or other purposes have to be provided, the beam in a linac is usually pulsed, and furthermore it is challenging to ensure all bunches in the accelerator to have the same intensity.", "Examples for different beam formats are shown in Fig.", "REF , which will cause the related beam spectrum to be more complicated.", "However, for the evaluation of the response of a BPM pickup, characterized by the transfer impedance $Z(\\omega )$ or $z(t)$ , a single bunch excitation signal $I_{\\mathrm {bunch}}(\\omega )$ or $i_{\\mathrm {bunch}}(t)$ with the bunch-length of relevance is sufficient.", "As the BPM pickup is a linear, time-invariant system the response to other beam formats can be simply evaluated by superposition of the response signal $v_{\\mathrm {elec}}(t)$ to a single bunch with the appropriate time-delays and intensity factors." ], [ "The “button” BPM", "The button-style BPM is the most popular type of BPM pickup because of its simplicity, robustness and compact design, and has well defined, reproducible characteristics at relatively moderate costs.", "In most cases it consists out of four symmetrically arranged circular electrodes, the “buttons”, each located at the end of a coaxial transmission-line connector of usually $Z_0=50\\;\\Omega $ characteristic impedance, similar as illustrated in Fig.", "REF .", "In practice, button electrode, coaxial connector and vacuum feedthrough barrier are integrated in a single element, the button BPM feedthrough, as shown in Fig.", "REF and Fig.", "REF .", "Different styles are available, often custom designs, for flange mounting or welding, with shape and dimensions optimized for the application.", "Figure REF shows a pair of vertical oriented button electrodes with the variables for the most relevant mechanical dimensions defined.", "To avoid confusion with the electrical symbols, we define $D$ as the diameter of the beam-pipe (assuming a circular cross-section), $r$ as the radius of the metallic button electrode, $t$ being the thickness of the button, $g$ as the width of the gap between the rim of the button electrode and the beam-pipe, and $d$ as the distance between button electrode and beam-pipe.", "Figure: Equivalent circuit of a button electrode.Electrically, the button electrode couples capacitively to the beam, it is dominated by its capacitance $C_b$ to ground (the beam-pipe), and forms a high-pass filter with the load impedance $R_l$ .", "Figure REF shows the equivalent circuit of a button electrode, where $v_{\\mathrm {beam}}(t)\\simeq -\\frac{1}{C_b} \\overbrace{i_{\\mathrm {beam}}(t)\\underbrace{\\frac{r}{2D}}_{\\phi }}^{\\Delta i_{\\mathrm {beam}}(t)}\\underbrace{\\frac{2r}{c_0}}_{\\Delta t}=-i_{\\mathrm {beam}}(t) \\frac{r^2}{Dc_o C_b}$ is the beam related generator voltage at the button, no load impedance assumed.", "This induced voltage is proportional to the interception area of the button, given by the coverage factor $\\phi \\approx r/(2D)$ (see also Eq.", "(REF )), the transit time $\\Delta t= 2r/c_0$ , and inverse proportional to the button capacitance $C_b$ .", "Eq.", "(REF ) holds for relativistic beams $v\\simeq c_0$ and a beam current signal $i_{\\mathrm {beam}}(t)$ that resembles a bunch length longer then the button diameter $\\sigma _s \\gg 2r$ .", "For a centered beam ($x=y=0$ ) the transfer impedance of a button electrode follows as: $Z_b(\\omega )=\\frac{V_b(\\omega )}{I_{\\mathrm {beam}}(\\omega )}=\\phi R_l \\frac{\\omega _1}{\\omega _2}\\frac{j\\omega /\\omega _1}{1+\\omega /\\omega _1}$ with: $\\text{1/time constant: } \\omega _1=\\frac{1}{R_l C_b}\\; , \\qquad \\text{1/transit time: } \\omega _2=\\frac{c_0}{2r}\\; , \\qquad \\text{coverage factor: } \\phi \\approx \\frac{r}{2D}$ The button-to-ground capacitance can be estimated as $C_b \\approx \\frac{\\pi \\epsilon t_b}{\\ln \\left( \\frac{r+g}{r} \\right) } + \\frac{\\pi \\epsilon r^2 }{d} + C_{\\mathrm {fringe}}$ but in practice a measurement or a numerical computation of $C_b$ will give better, more accurate results, the same is true for the coverage factor $\\phi $ .", "Figure REF show some examples of $Z_b(\\omega )$ for different dimensions of the button ($r$ , $t$ ) and the gap ($g$ ), as well as for a variation of the load impedance ($R_l$ ).", "Typical values of the transfer impedance for high frequencies are around $Z\\approx 1\\;\\Omega \\;@\\; f\\gtrsim 1$  GHz, the 3 dB cutoff-frequency is rather high, typically $f_{3dB} \\approx 500\\ldots 2000$  MHz.", "A lower cutoff frequency and increase of $Z_b$ could be achieved by increasing the load impedance $R_l$ , however, usually $R_l$ is fixed to 50 $\\Omega $ to stay compatible with the impedance standard in RF technology.", "Increasing the diameter $2r$ of the button will increase $Z_b$ and result in a higher coupling (sensitivity) to the beam, and – as of the increase of $C_b$ – will also lower $f_{3dB}$ , but simultaneously will lower the position sensitivity.", "Moreover, a larger button size will also result in lowering the frequencies of trapped eigenmodes, which can lead to an increase of unwanted wake field effects.", "An increase of the button thickness can reduce the beam coupling (wake) impedance, but increases the weight of the button, which has to be handled by the thin pin / vacuum feedthrough construction.", "Lowering the gap is even more beneficial to reduce the beam coupling impedance, the limit is set by manufacturing tolerances.", "Figure: Transfer impedance of a button BPM electrode." ], [ "The stripline BPM", "Similar to the button BPM, the stripline BPM is a broadband coupler.", "Two or four stripline-like electrodes of length $\\ell $ and width $w$ are arranged symmetrically in the beam-pipe and form the stripline BPM pickup, see Fig.", "REF .", "Each stripline is a TEM transmission-line of characteristic impedance $Z_0$ , usually matched to the 50 $\\Omega $ impedance of the coaxial feedthrough ports, located at both ends.", "Unlike a button, which couples capacitively to the beam, the stripline electrode is an electro-magnetic coupler, Fig.", "REF illustrates how stripline and beam-pipe form a “loop” that also couples to the magnetic field components of the beam.", "Electrically, stripline electrode and beam form a 4-port network, similar to that of a directional coupler, but in case of a stripline BPM, two of the ports are beam signal related waveguide ports.", "An intuitive explanation of the principle of operation is illustrated in the time-domain in Fig.", "REF , assuming the signal velocity along the stripline and the velocity of a point charge being the same, close to the speed-of-light, $v_s=v\\simeq c_0$ .", "Figure: Operational principle of a stripline BPM electrode.$\\mathbf {t=0}$ A positive charged beam particle enters the upstream portion of the stripline electrode.", "The corresponding image charges (wall current) induces a point charge on the upstream end of the stripline, causing a positive, pulse-like signal with an intensity that depends on the distance (beam position) between charged particle and electrode, the width $w$ of the electrode, and the intensity $q$ of the beam particle.", "As the induced image charge “sees” a transmission-line of same characteristic impedance $Z_0$ in both directions – upstream through the vacuum feedthrough as coaxial transmission-line and downstream as stripline electrode – it splits in two parts of equal amplitude, traveling with $v_s\\simeq c_0$ in opposite directions.", "$\\mathbf {t=\\ell / c_0}$ The beam particle exits the downstream portion of the stripline electrode.", "Likewise to $t=0$ , an image charge is induced on the downstream end of the stripline, now with opposite sign, causing a negative, pulse-like signal which also splits off into two equal parts traveling in opposite directions.", "At the same time the positive image charge signal that is linked to the beam charge arrives at the downstream end of the stripline, its signal has the same intensity but opposite sign as the freshly induced signal portion.", "Both signal portions compensate each other and no signal appears at the termination resistor outside at the downstream port.", "$\\mathbf {t=2\\ell /c_0}$ The negative signal from the downstream induction exits the upstream port, following the positive signal pulse at a distance of $2\\ell $ , which will generate a “doublet”-pulse voltage signal with a time delay between positive and negative parts of $\\Delta t=2\\ell / c_0$ , which is then observable in a load resistor $R_l=Z_0$ (not shown).", "As of the symmetry, a beam charge entering from the opposite end will cause the same signal pattern in the other port: The stripline BPM has directivity, in the ideal case the beam signal appears only at the upstream port.", "The time-domain transfer impedance of an ideal, lossless stripline electrode is given as a $\\delta $ -doublet signal (see also Fig.", "REF ): Figure: Transfer characteristic of a stripline BPM electrode.$z(t)=\\phi \\frac{Z_0}{2} \\left[ \\delta (t)-\\delta \\left( t-2\\frac{\\ell }{c_0} \\right) \\right]$ with $\\phi =\\frac{1}{\\pi }\\arcsin (w/D)$ being the geometric coverage factor and $Z_0$ the characteristic impedance of the stripline.", "The frequency-domain transfer impedance of an ideal stripline electrode is then the Fourier transformation of Eq.", "(REF ): $Z(\\omega )=\\phi \\frac{Z_0}{2} \\left( 1-e^{-j2\\omega \\frac{\\ell }{c_0}} \\right)=j\\phi Z_0 e^{-j\\omega \\frac{\\ell }{c_0}}\\sin \\left( \\omega \\frac{\\ell }{c_0} \\right)$ The modulus of Eq.", "(REF ), $|Z(f)|$ is visualized in Fig.", "REF for $\\ell =100$  mm, $\\phi =1$ , and $Z_0=50\\;\\Omega $ as parameters of the stripline electrode.", "The lobes of $|Z(\\omega )|$ peak at $f_c=(2n-1)c_0/(4\\ell )$ , each has a bandwidth $f_{BW}=f_{hi}-f_{lo}=f_{c1}$ with $f_{lo}=f_{c1}/2$ , $f_{hi}=3f_{lo}$ , and $f_{c1}=c_0/(4\\ell )$ .", "In most applications the length $\\ell $ of the stripline is chosen to operate within the first lobe.", "Figure: Equivalent circuit of a stripline electrode.Figure REF shows the equivalent circuit for a stripline BPM electrode.", "If the stripline electrodes are rather wide, their coupling between each other cannot be neglected.", "In that case, and depending if the BPM has two (single-plane) or four (dual-plane) electrodes the characteristic line impedance $Z_0$ needs to be replace by $Z_0=\\sqrt{Z_{0e} Z_{0o}}$ for two electrodes, or by $Z_0=\\sqrt{Z_{0s} Z_{0q}}$ for four electrodes.", "The concept of odd and even mode impedances $Z_{0e}$ , $Z_{0o}$ for two coupled TEM transmission-lines is well known, it just requires a simple electrostatic analysis of the cross-section geometry, similar to that of a single electrode.", "In case of four electrodes, $Z_{0s}$ represents the sum mode, equivalent to the even mode of two electrodes but now for all four electrodes, and $Z_{0q}$ is a quadrupolar mode with always two opposite electrodes in even mode but one pair in odd mode wrt.", "the other pair.", "To ensure the annihilation of the signals at the downstream port, the signal velocity on the stripline $v=\\beta _s c_0$ and the velocity of the beam $v=\\beta c_0$ must be identical, $v_s=v$ .", "If the strip-electrode includes dielectric material $e_r>1$ , e.g.", "ceramic supports, $v_s\\ne v$ , the directivity is broken, and some parts of the upstream beam signal will also appear at the downstream port, therefore: zdownstream(t)= Z02 [ (t-s c0)-( t-c0 ) ] while the upstream port couples: zupstream(t)= Z02 [ (t)-( t-s c0-c0 ) ] The frequency-domain equivalent of the above equations is then: Zdownstream() =Z02 [ e-js c0-e-jc0 ] =-jZ0 e-j2s c0 e-j2c0) [ 2(s c0-c0) ] Zupstream() =Z02 [ 1-e-j(s c0+c0) ] =jZ0 e-j2(s c0+c0) [ 2(s c0+c0) ] The stripline BPM is a TEM coupler, in the ideal case electric and magnetic fields have only transverse components, and the coupler operates for a beam of relativistic velocity.", "In practice, due of the transition from a coaxial-line to the stripline, in and near the vacuum feedthroughs at both ports, those discontinuities “disturb” the TEM field propagation, causing a perturbation of the characteristic impedance and therefore signal reflections, which degrade the performance of the BPM, in particular limits its directivity at higher frequencies.", "In a very simplistic approach the width $w$ of the stripline causes a reflection near both ports with a time delay $\\Delta t=w / c_0$ , which will alter the frequency-domain transfer characteristic: $Z_{RF}(\\omega )=j\\phi Z_0 e^{-j\\omega \\left(\\frac{\\ell }{c_0}+\\Delta t\\right)} \\frac{\\sin (\\omega \\Delta t)}{\\omega \\Delta t}\\sin \\left( \\omega \\frac{\\ell }{c_0} \\right)$ This RF effect of Eq.", "(REF ) is visualized in Fig.", "REF for $\\Delta t=100$  ps, i.e.", "stripline width $w=30$  mm, in comparison with the ideal response (REF ).", "As the stripline basically is a lossless coupler, the difference between $Z(\\omega )$ and $Z_{RF}(\\omega )$ will appear as unwanted signal at the downstream port and limit its directivity.", "The TEM operation of the stripline BPM, as basis of the theory presented here, can only be ensured for $\\ell \\gg w$ .", "If $\\ell \\approx w$ the stripline converges into an electrostatic or button electrode, and the TEM-based theory becomes invalid.", "Compared to a button BPM, the stripline BPM is a more complicated, fragile and costly pickup, it also requires more real estate.", "Still, the stripline offers several advantages, e.g.", ": The matched line-impedance terminated at the downstream port acts as almost perfect 50 $\\Omega $ signal source, and minimizes reflections between the BPM pickup and the read-out electronics.", "In case of hadron beams, the length of the stripline can be tuned to match the bunch length, thus improving the signal levels, therefore the signal-to-noise ratio (S/N) and the resolution.", "The downstream port can be utilized to feed test or calibration signals.", "A stripline BPM with good directivity can be used in particle colliders near the interaction point (IP) to provide separate beam position signals for the counterrotating beams.", "Stripline BPM are also popular in hadron linacs, often with downstream ports short circuited.", "This safes 2 or 4 vacuum feedthroughs and simplifies the mechanical construction, while stiffening the stripline mechanics." ], [ "The split-plane BPM", "Consider a hollow metallic cylinder of length $2\\ell $ and diameter $2R$ floating in the beam-pipe as shown in Fig.", "REF .", "The cylinder has a diagonal cut, which forms two electrodes $A$ and $B$ , each connected to a coaxial vacuum feedthrough to deliver the corresponding beam signal.", "Based of Eq.", "(REF ), a beam current $I_{\\mathrm {beam}}(r,\\varphi )$ induces a wall current $I_{\\mathrm {elec}}$ on the split-plane electrode: $I_{\\mathrm {elec}} =-\\frac{I_{\\mathrm {beam}}}{2\\pi } \\frac{\\ell }{R} \\int _0^{2\\pi }\\frac{(1+\\cos \\Phi )\\left(R^2-r^2\\right)}{R^2+r^2-2Rr\\cos (\\Phi -\\varphi )} d\\Phi $ for a beam position $x=r\\cos \\varphi $ , $y=r\\sin \\varphi $ , where $\\ell (1+\\cos \\Phi )$ describes the variation of the electrode length wrt.", "the integration path $d\\Phi $ .", "The solution of Eq.", "(REF ) results in: $I_{\\mathrm {elec}} =-\\frac{I_{\\mathrm {beam}}}{2\\pi } \\frac{\\ell }{R} \\left(1+\\frac{r\\cos \\varphi }{R}\\right)=-\\frac{I_{\\mathrm {beam}}}{2\\pi } \\frac{\\ell }{R} \\left(1+\\frac{x}{R}\\right)$ offering a perfectly linear position characteristic.", "The linear position behavior is reflected in the normalization of the two horizontal electrodes $A$ and $B$ : $\\frac{\\Delta }{\\Sigma }=\\frac{A-B}{A+B}=\\frac{1}{R}x=\\mathrm {hor.\\; position}$ The position sensitivity (REF ) of the split-plane BPM is half of the sensitivity of two-dimensional BPM electrodes, e.g.", "buttons or striplines, approximated by Eq.", "REF .", "Figure: The “shoe-box” BPM without, and with ground-guards (courtesy P. Kowina).Figure REF shows a variant of the split-plane BPM, the arrangement of split electrodes in a rectangular cross-section of the beam-pipe, the so-called “shoe-box'`BPM.", "This configuration, as any other linear cut electrode configuration, offers the same linear position characteristic.", "However, numerical studies show the coupling along the cut between the electrodes to be an error source, limiting the linearity in particular at higher frequencies, see Fig.", "REF (left).", "A narrow ground-guard between the electrodes reduces the cross-coupling and improves the linearity, also at higher frequencies, see Fig.", "REF (right).", "Split-plane BPMs were popular in the early days – particular in hardon accelerators – when the BPM read-out electronics was entirely based on analog signal-processing, and a correction of a non-linear position behavior was difficult to implement.", "The electrode dimensions are large, in the order of the beam-pipe diameter, which results in a high electrode-to-ground capacitance and limits the usable frequency range.", "At higher frequencies the electrodes develop eigenmodes which results in an unwanted, high beam coupling impedance, therefore this type of BPM should not be used in accelerators with short bunches." ], [ "The cavity BPM", "The types of BPM pickups discussed so far were based on beam image currents, induced into the pickup electrodes, and have a broadband characteristic: Their transfer impedance $Z(\\omega )$ covers a wide frequency range with a position behavior independent of the operation frequency.", "For a pair of stripline electrodes, the maximum achievable value of the transfer impedance is limited to half of the characteristic impedance of the electrodes ($Z_0/2$ ), but in practice the transfer impedance of a broadband BPM rarely exceeds a few-$\\Omega $ (see Fig.", "REF and Fig.", "REF ).", "The BPM transfer impedance, together with the intensity and spectrum of the bunched beam, defines the achievable output signal level at the ports (vacuum feedthroughs) of the BPM, therefore the signal-to-noise ratio, and finally the resolution of the beam position measurement.", "Figure: Beam excited eigenmodes in a cylindrical cavity (courtesy D. Lipka).A beam-excited, passive resonant cavity can be used a beam detector, and offers, similar to an accelerating structure, a high shunt impedance $R_s$ (typically $k\\Omega /mm$ ) at the resonant frequency.", "Figure REF shows the first three beam excited eigenmodes of a cylindrical resonator, the so-called “pillbox” cavity.", "For beam position measurements the TM110 dipole mode is of relevance, $v_{\\mathrm {TM110}}\\propto i_{\\mathrm {beam}}\\cdot r$ it returns a signal which is proportional to the beam intensity and the beam displacement $|\\vec{r}|=\\sqrt{x^2+y^2}$ wrt.", "the center of the resonator.", "In contrast to the signals from broadband BPM electrodes, a dipole-like mode signal from a cavity resonator has no common-mode signal contribution, for a centered beam $r=0$ the TM110-mode signal vanishes, see Eq.", "(REF ).", "For broadband BPM pickups the difference signal of Eq.", "(REF ) has to be arranged externally, with help of the read-out electronics, in case of the cylindrical cavity the TM110 mode already is a $\\Delta $ -signal.", "Figure: Operation principle of a “pillbox” cavity BPM.A single bunch or a train of bunches with a sufficient wide spectrum is be able to excite the TM110 eigenmode, but it also may excite unwanted resonances like the TM010 and TM020 monopole modes and other higher-order modes, as illustrated in Fig.", "REF .", "The principle of operation of a cylindrical “pillbox”cavity BPM of radius $R$ and length (height) $h$ is sketched in Fig.", "REF , with a beam passing the resonator with some displacement $dx$ away from the $z$ symmetry axis.", "The velocity vector of the beam points dominantly to the $z$ -direction, therefore it couples to the eigenmodes that have longitudinal E-field components, the so-called E or TM modes.", "As the E-field always has to be zero at the conductive boundary wall of the lateral area, i.e.", "the cavity rim, the diameter $2R$ defines many (but not all) of the possible E-field mode patterns, the E-field of the fundamental TM010 monopole mode and the TM110 dipole mode are indicated in the longitudinal section sketch of Fig.", "REF .", "The E-field of the dipole mode (TM110) is zero at the center, along the $z$ symmetry axis, where the E-field of the monopole mode (TM010) has its maximum.", "Therefore, a centered beam will only excite the TM010, plus some other higher-order modes, a displaced beam $dx$ will also excite the TM110 dipole mode, which changes its sign as the beam crosses the center.", "The beam excited fields are coupled to a coaxial port, usually loaded with $R_l=50\\;\\Omega $ , by means of a pin (capacitive coupling to the E-field), or loop antenna (inductive coupling to the H-field).", "For symmetry reasons a symmetric arrangement of coupling antennas is preferred, Fig.", "REF shows two pin-antennas which will detect the same mode signal, except the TM110 dipole mode signals will have opposite sign.", "The parameters of the cavity resonances (frequencies and Q-values) are defined by its geometric shape, the dimensions and the conductivity of the metallic material.", "For simple geometries, e.g.", "a cuboid (brick-shape) or a cylinder (“pillbox”), the Laplace equation of the vector potential $\\Delta \\Psi +k_0^2\\epsilon _r\\mu _r\\Psi =0$ can be solved analytically in form an orthonormal series expansion.", "$k_0=2\\pi /\\lambda _0$ is the free space wave number in (REF ), and is $k_0^2=\\omega ^2\\epsilon _0\\mu _0$ , while $\\lambda _0=2\\pi c_0/\\omega $ is the free space wave length.", "Without going into the details, lets summarize the theory first for a hollow cuboid in Cartesian coordinates of dimensions $0<x<a$ , $0<y<b$ , $0<z<c$ as boundary conditions.", "A product “Ansatz” for the vector potential: $\\Psi = X(x)Y(y)Z(z)$ with the separation condition $k_x^2+k_y^2+k_z^2=k_0^2\\epsilon _r\\mu _r$ finds a general solution for the vector potential as orthonormal series expansion: ${0.93}{!", "}{\\psi =\\left\\lbrace \\begin{array}{cc} A\\cos \\left(k_x x\\right)+B\\sin \\left(k_x x\\right) \\\\\\grave{A}e^{-jk_x x}+\\grave{B}e^{-jk_x x}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cc} C\\cos \\left(k_y y\\right)+D\\sin \\left(k_y y\\right) \\\\\\grave{C}e^{-jk_y y}+\\grave{D}e^{-jk_y y}\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cc} E\\cos \\left(k_z z\\right)+F\\sin \\left(z_z z\\right) \\\\\\grave{E}e^{-jk_z z}+\\grave{F}e^{-jk_z z}\\end{array}\\right\\rbrace \\begin{array}{ll}\\rightarrow \\text{standing waves} \\\\\\rightarrow \\text{traveling waves}\\end{array}}$ with $k_x=m\\pi /a$ , $k_y=n\\pi /b$ , and $k_z=p\\pi /c$ .", "In our case the standing wave solution of Eq.", "(REF ) will give the results for the fields and for the frequencies of the eigenmodes: $f_{mnp}=\\frac{c_0}{2\\pi \\epsilon _r\\mu _r}\\sqrt{\\left(\\frac{m\\pi }{a}\\right)^2+\\left(\\frac{n\\pi }{b}\\right)^2+\\left(\\frac{p\\pi }{c}\\right)^2}$ While there are a few examples using a cuboid as high resolution cavity BPM, often a cylinder is preferred, e.g.", "for manufacturing advantages on a precision lathe.", "The orthonormal series expansion procedure of (REF ) for a cylinder is the same as for the cuboid, but as of the cylindrical coordinates, now cylindrical functions come into play.", "For a hollow cylinder its dimensions $0<\\rho <R$ and $0<z<h$ , as well as the angle $0<\\varphi <2\\pi $ will set the boundary conditions for the product approach: $\\Psi =R(\\rho )F(\\varphi )Z(z)$ This results in a general solution for the vector potential, again as orthonormal series expansion, but now in cylindrical coordinates: ${0.93}{!", "}{\\psi =\\left\\lbrace \\begin{array}{cc} AJ_m\\left(k_{\\rho }\\rho \\right)+BY_m\\left(k_{\\rho }\\rho \\right) \\\\\\grave{A}H_m^{(2)}\\left(k_{\\rho }\\rho \\right)+\\grave{B}H_m^{(1)}\\left(k_{\\rho }\\rho \\right)\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cc} C\\cos (m\\varphi )+D\\sin (m\\varphi ) \\\\\\grave{C}e^{-jm\\varphi }+\\grave{D}e^{-jm\\varphi }\\end{array}\\right\\rbrace \\left\\lbrace \\begin{array}{cc} E\\cos \\left(k_z z\\right)+F\\sin \\left(k_z z\\right) \\\\\\grave{E}e^{-jk_z z}+\\grave{F}e^{-jk_z z}\\end{array}\\right\\rbrace \\begin{array}{ll}\\rightarrow \\text{standing waves} \\\\\\rightarrow \\text{traveling waves}\\end{array}}$ Here the cylindrical function are: $J_m$ Bessel function of first kind of order $m$ .", "$Y_m$ Bessel function of second kind of order $m$ (also called Weber function or Neumann function).", "$H_m^{(1)}$ Hankel function of first kind of order $m$ (for outward traveling waves, also called Bessel function of third kind).", "$H_m^{(2)}$ Hankel function of second kind of order $m$ (for inward traveling waves, also called Bessel function of third kind).", "Here, two separation conditions are used: $\\left(\\frac{j_{mn}}{R}\\right)^2+k_z^2=k_0^2 \\qquad \\qquad \\left(\\frac{j^{\\prime }_{mn}}{R}\\right)^2+k_z^2=k_0^2$ with $k_z=p\\pi /h$ , and $j_{mn}$ being the $n^{th}$ root of $J_m(x)$ and $j^{\\prime }_{mn}$ being the $n^{th}$ root of $J^{\\prime }_m(x)$ , with $J^{\\prime }_m$ being the derivative of the Bessel function $J_m$ .", "Eq.", "(REF ) can be expressed in form of TM$_{mnp}$ and TE$_{mnp}$ eigenmodes, with the frequencies found by the separation conditions: $f_{\\mathrm {TM}mnp}&=\\frac{c_0}{2\\pi \\epsilon _r\\mu _r}\\sqrt{\\left(\\frac{j_{mn}}{R}\\right)^2+\\left(\\frac{p\\pi }{h}\\right)^2} \\\\f_{\\mathrm {TE}mnp}&=\\frac{c_0}{2\\pi \\epsilon _r\\mu _r}\\sqrt{\\left(\\frac{j^{\\prime }_{mn}}{R}\\right)^2+\\left(\\frac{p\\pi }{h}\\right)^2}$ The analytical approach gives a rough estimation of the cavity dimensions and the related eigen-frequencies for a given shape.", "The ports of the beam-pipe act as waveguides, therefore most of the cavity eigemodes with $f_{\\mathrm {cavity}}>f_{\\mathrm {TE}11}$ will not be trapped, assuming a beam-pipe with circular cross-section.", "As the waveguide ports alter the boundary conditions, a so-called “mode matching” approach is required to correctly solve the cavity/beam-pipe geometry in an analytical way.", "In practice a numerical analysis often is more convenient, which also allows to include the coaxial port antennas and other details, and provides additional information, e.g.", "due to the finite conductive of the cavity walls.", "Figure: Equivalent circuit of a resonant mode of a cavity BPM.Electrically, each mode can be described as $RLC$ equivalent circuit (Fig.", "REF ), the circuit parameters can be derived from analytical formulas, from a numerical analysis, or from RF measurements in the laboratory.", "The resonant frequency is given by $\\omega _{\\mathrm {mode}}=1/\\sqrt{LC}$ , the losses are represented by $R$ , and the geometry (or shape) of the resonator is reflected by $R/Q=\\omega _{\\mathrm {mode}}L=1/(\\omega _{\\mathrm {mode}}C)=\\sqrt{L/C}$ , which is an equivalent to the characteristic impedance of a transmission-line.", "The output power delivered at a load impedance $R_l$ is maximum if $R$ is matched to $R_l$ , i.e.", "at critical coupling (no reflections).", "This can be achieved by tuning the coupling to a value $k=\\sqrt{R/R_l}$ .", "Figure: A common-mode free cavity BPM.As Fig.", "REF suggests, the TM010 monopole mode is very strong and always present, and causes a major limitation to detect a weak TM110 dipole mode signal, despite the fact that it is located at a higher frequency, The reason lies in the finite Q-value of the resonant modes, with the “tail” of $f_{\\mathrm {TM}010}$ leaking into $f_{\\mathrm {TM}110}$ .", "The TM010 mode is equivalent the common-mode (intensity) signal contribution in a broadband BPM, except in case of the cavity BPM there is some benefit due to the frequency separation.", "Figure REF shows a pillbox cavity BPM with two slot-coupled rectangular waveguides attached, oriented in the horizontal and vertical plane.", "The waveguides of width $a$ act as very efficient high-pass filter, and are dimensioned such that the cutoff frequency of the TE10 waveguide mode lies between the monopole and dipole-mode frequency of the cavity resonator, $f_{\\mathrm {TM}010}<f_{\\mathrm {TE}10\\mathrm {wg}}=1/(2a\\sqrt{\\epsilon \\mu })<f_{\\mathrm {TM}010}$ .", "To minimize the monopole-mode leakage of this so-called “common-mode free” cavity BPM, the rectangular coupling slot between cavity resonator and waveguide has to be made reasonable narrow, the coupling mechanism is shown qualitatively in Fig.", "REF .", "A cavity BPM based on a cylindrical pillbox resonator requires the two TM110 mode polarizations to be perfectly orthogonal, aligned to the horizontal and vertical plane.", "Usually this is achieved by precise manufacturing of the coupling slots or antennas, which cause an “imperfection” for the cylinder and gives the TM110 fields a boundary to align the polarization axis.", "In practice, a residual cross-coupling between the planes of $<-40$  dB can be achieved, while the frequencies of the two TM110 polarization are almost identical.", "Alternatively two cuboid (brick-style) resonators with different resonant frequencies can be arranged to detect horizontal and vertical beam displacements separately with minimum cross-talk.", "While the operational principle seems to be simple, the realization of a BPM system based on resonant cavities is non-trivial.", "Along with the high transfer impedance goes a high beam coupling impedance that can have negative effects on the beam quality, e.g.", "beam instabilities, beam breakup, etc., therefore cavity BPMs are not used in storage rings.", "Beside the precision manufacturing and calibration of the cavity BPM pickup, the RF front-end and signal processing is more demanding compared to read-out electronics for broadband BPMs, moreover, a separate TM010 monopole mode resonator is required as reference for the beam intensity normalization and as beam phase reference to detect the phase of the dipole mode." ], [ "Other types of BPM pickups", "The presented BPM pickup types, button-style, stripline, split-plane and cavity BPM are the most common used BPM pickups.", "However, there exists a larger variety of other BPM pickups, e.g.", "Exponentially tapered stripline BPM This modified stripline BPM alters the transfer function such that the lobes of $|Z(\\omega )|$ have a wider bandwidth.", "Similar techniques are used in the RF technology, known as multi-element direction coupler.", "Re-entrant BPM is a “hybrid” between broadband, stripline-style and resonant, cavity-style BPM pickup.", "Resonant button or stripline BPM An external reactive circuit element is used to change the broadband behavior of a traditional capacitive pickup, e.g.", "button BPM, or a stripline BPM into a resonant circuit.", "While the beam position characteristic stays unchanged, the transfer function is altered towards a band-pass characteristic, with a higher $|Z(\\omega _{\\mathrm {res}})|$ at the resonance frequency.", "Inductive BPM The wall current is detected by a symmetric arrangement of inductive couplers.", "The list is incomplete, but in common of all type of BPM pickups is the symmetric arrangement of beam coupling elements, and the better the symmetry is realized in practice, the better is the performance potential in terms of accuracy and achievable resolution.", "Other, more ambitious electromagnetic BPM detectors proposed will operate in domain of optical wavelengths, but require a short bunch length.", "The beam field could be observed by a symmetric arrangement of electro-optical crystals, similar to an electro-optical modulator, or could generate diffraction or Cherenkov radiation with radiators and detectors in a symmetrical setup to extract the beam position information." ], [ "Broadband BPM pickup response to a ", "The response of a broadband BPM pickup electrode, button or stripline, to a single bunch in terms of time-domain output signal waveform and level is of fundamental relevance, as this is the signal to be detected and processed by the read-out electronics.", "From the single bunch response it is straightforward – applying the superposition principle – to analyze more complex beam formats, like multi-bunch trains with or without equidistant bunch spacing, also with varying bunch intensities.", "Figure: Stripline BPM response to a single Gaussian bunchin the frequency domain (magnitude).The frequency-domain output signal for the upstream (coupled) port of an ideal, air-dielectric stripline electrode, i.e.", "$\\beta _s=\\beta $ , neglecting high frequency and fringe field effects, to a relativistic ($\\beta =1$ ) Gaussian beam bunch is given by the multiplication of the frequency-domain Gaussian bunch stimulus in Eq.", "(REF ), and the transfer impedance of the stripline electrode Eq.", "(REF ): $V_{\\mathrm {strip}_{\\mathrm {upstream}}}(\\omega ) = j\\phi Z_0 e Ne^{-\\frac{(\\omega \\sigma _t)^2}{2}}e^{-j\\omega \\frac{\\ell }{c_0}}\\sin \\left( \\omega \\frac{\\ell }{c_0} \\right)$ and is illustrated in Fig.", "REF .", "Figure: Stripline BPM response to a single Gaussian bunchin the time domain.The time-domain output signal at the upstream port of a stripline electrode – with the impulse response according to Eq.", "(REF ) – to a Gaussian bunch following Eq.", "(REF ) is simply: $v_{\\mathrm {strip}_{\\mathrm {upstream}}}&=\\phi \\frac{Z_0}{2}\\frac{eN}{\\sqrt{2\\pi }\\sigma _t}\\left[e^{-\\frac{\\left(t+\\tau \\right)^2}{2\\sigma _t^2}}-e^{-\\frac{\\left(t-\\tau \\right)^2}{2\\sigma _t^2}}\\right] \\\\\\text{with: } \\tau &=\\frac{\\ell }{2c_0}\\left(\\frac{1}{\\beta }+\\frac{1}{\\beta _s}\\right)$ For relativistic beams ($\\beta =1$ ) and air-stripline electrodes ($\\epsilon _r=1\\rightarrow \\beta _s=1$ ) $\\tau $ in Eq.", "(REF ) simplifies $\\beta =\\beta _s=1\\rightarrow \\tau =\\ell /c_0$ .", "Figure REF and REF show the output signal waveforms of a stripline electrode of length $\\ell =100$  mm for a Gaussian bunch ($\\beta =1$ ) of $\\sigma =150$  mm and 75 mm length, respectively, positive and negative parts of the doublet impulse response waveform are overlapping.", "A shorter bunch of $\\sigma =15$  mm length for the same stripline length separates the those parts, see Fig.", "REF , and in the general case $t_{\\mathrm {bunch}}<\\ell /2$ , with $t_{\\mathrm {bunch}}$ being the total bunch length, the stripline BPM can also be used as detector to monitor the longitudinal bunch profile.", "Please note, the instantaneous signal voltages out of a stripline BPM electrode for short bunches can be high, e.g.", "several 100 volts, in our example computed for $N=10^{11}$ charges per bunch.", "Figure: Button BPM response to a single Gaussian bunchin the frequency domain (magnitude).Figure: Button BPM signal in the frequency-domain (upper row),and the time-domain (lower row)to a single Gaussian bunch.Likewise for the stripline BPM, the frequency domain response of a button BPM electrode is computed by: $V_{\\mathrm {button}}(f) = Z_{\\mathrm {button}}(f) I_{\\mathrm {bunch}}(f)$ with $I_{\\mathrm {bunch}}(f)$ as e.g.", "Gaussian bunch according to Eq.", "REF , and $Z_{\\mathrm {button}}(f)$ given by Eq.", "REF , examples see Fig.", "REF .", "For the time-domain output signal of a button BPM electrode to a Gaussian bunch there unfortunately is no closed-form analytical expression.", "However, if the frequency spectrum of the bunch stays below the high-pass cut-off frequency of the button electrode, $f_1=\\omega _1/(2\\pi )$ in Eq.", "(REF ), the button is purely differentiating the bunch signal, so that: $v_{\\mathrm {button}}&\\approx \\frac{A}{\\pi D} \\frac{R_l}{\\beta c_0}\\frac{di_{\\mathrm {bunch}(t)}}{dt} \\\\&=\\frac{r^2 R_l}{D\\beta c_0}\\frac{eN}{\\sqrt{2\\pi }\\sigma _t^3}te^{-\\frac{t^2}{2\\sigma _t^2}} \\quad \\text{for: }f_{\\text{3dB}} \\ll f_1$ with $A=r^2\\pi $ being the surface are of the button electrode, and all other parameters as defined in section REF .", "The 3 dB cut-off frequency for a Gaussian bunch spectrum, used in Eq.", "(REF ), is found from Eq.", "(REF ): $f_{\\text{3dB, Gauss}}=\\frac{\\sqrt{\\frac{\\ln \\sqrt{2}}{2}}}{\\pi } \\frac{1}{\\sigma _t} \\approx \\frac{0.1325}{\\sigma _t}$ Figure REF shows a button output signal with same bunch parameters as for the response of a $\\ell =100$  mm long stripline electrode in Fig.", "REF .", "As Figure REF indicates, the transfer function of the $r=4$  mm button is not well matched to the spectrum of the Gaussian bunch of length $\\sigma =150$  mm, therefore the output voltage is substantially lower compered to that of the stripline electrode, see Fig.", "REFTo better visualize bunch current and button voltage in Fig.", "REF , the button output signal voltage was magnified by $\\times $ 10, i.e.", "the voltage scale has to be divided by 10!.", "Short bunches in a button BPM are better matched to the higher frequency range of the button transfer impedance, including and beyond $f_1$ , as Fig.", "REF indicates.", "To include that regime we have to evaluate Eq.", "(REF ) and numerically compute the time-domain button output signal by applying the inverse discrete Fourier transformation (iDFT or iFFT).", "For a Gaussian bunch the button output signal is of form: $v_{\\mathrm {button}}(t)\\propto \\int _{-\\infty }^{+\\infty }\\frac{jfe^{-2\\left(\\pi f\\sigma _t\\right)^2}}{f_1+jf}e^{-j2\\pi f t}df$ Fig.", "REF shows the response in time and frequency-domain for three different cases of the bunch length, based on Eq.", "(REF ) and (REF ).", "Figure: Broadband BPM signals as response to ato a single Gaussian bunch (eN=100eN=100 pC, σ=25\\sigma =25 mm, β=1\\beta =1, beam displacement: 1 mm)as result of a 3D numerical analysis.Evidently, the computation of the output signal of the BPM electrodes based on analytical expressions has some limitations, it also misses many details like high frequency effects at the vacuum feedthrough transition, geometrical details of the BPM pickup, effects of low-$\\beta $ beams, losses, etc.", "A numerical analysis of the electromagnetic problem, using a good approximation of the exact BPM pickup shape, and a beam field as stimulus signal allows a more realistic estimation of the electrode signals.", "Figure REF shows the results of a so-called wakefield simulation for the button and stripline BPMs as illustrated in Fig.", "REF and Fig.", "REF , for both cases the beam-pipe diameter is $D=25$  mm and the coverage factor is $\\phi \\approx 0.1$ , the length of the stripline electrode is $\\ell =100$  mm.", "As a result, the time-domain output signal waveforms for both pickup electrodes are computed (Fig.", "REF and Fig.", "REF ), as well as the frequency domain response (Fig.", "REF ), here normalized to the peak value of the upper stripline electrode.", "Fig.", "REF also demonstrates the broadband behavior of the position characteristic, in this example both BPM pickups have a sensitivity of 2.7 dB/mm." ], [ "Beam position monitoring of low-$\\beta $ beams", "Until now we considered the beam moving at a relativistic velocity $v=\\beta c_0$ , with $\\beta \\simeq 1$ .", "In that situation all electromagnetic field components of the charged particles are purely transverse, the so-called TEM field configuration, see Fig.", "REF .", "As a consequence, the longitudinal wall current density $J_w(s)=dq_w/ds$ is always a replica of the longitudinal particle distribution, the line-current $i_{\\mathrm {beam}}(t)$ , with $t=s/(\\beta c_0)$ .", "At lower beam velocities $\\beta \\ll 1$ , the EM-field of a charged particle also develops longitudinal field components, Fig.", "REF shows the effect for a moving point charge in free space.", "In Figure REF the electric field lines and the related – for the beam monitoring more interesting – longitudinal wall current density $J_w(s)$ are analyzed for a point charge, moving with constant velocity $v=\\beta c_0$ in a circular beam-pipe of radius $R$ at a displacement $|\\vec{r}|=\\sqrt{x^2+y^2}$ .", "While there is no closed-form analytical expression for $J_w(s, r)$ the RMS-value of the current distribution can be expressed simply by: $\\sigma _s=\\frac{d}{\\sqrt{2 \\gamma }}$ where $d$ is the distance between the point charge and the beam-pipe wall, which is the difference $\\vec{r}-\\vec{R}$ between the beam at the transverse position ($r$ , $\\varphi $ ) and a location on beam-pipe surface ($R$ , $\\Phi $ ), see also Fig.", "REF .", "As of the additional longitudinal component of the wall current distribution, the electrostatic problem of low-$\\beta $ beams now requires to solve the Laplace equation in three-dimension.", "As a result, for the configuration of Fig.", "REF , the wall current density around the azimuth is: $J_w(\\omega , R, \\Phi , r, \\varphi )=-\\frac{I_{\\mathrm {beam}}(\\omega )}{2\\pi R}\\left\\lbrace \\frac{I_0(gr)}{I_0(gR)}+2\\sum _{n=1}^{\\infty } \\frac{I_n(gr)}{I_n(gR)} \\cos \\left[ n (\\Phi -\\varphi ) \\right] \\right\\rbrace $ where $I_{\\mathrm {beam}}(\\omega )=I_{\\mathrm {beam}}(m\\omega _0)=\\sqrt{2}\\langle I_{DC}\\rangle A_m$ is expressed as the RMS amplitude at a specific frequency harmonic $m$ .", "$I_n$ is a modified Bessel function of order $n$ , and $g(\\omega )=\\frac{2\\pi }{\\gamma \\lambda }=\\frac{m\\omega _0}{\\beta \\gamma c_0}=\\frac{\\omega }{\\beta \\gamma c_0}$ is a frequency depending function.", "Eq.", "REF is similar to Eq.", "REF for relativistic beams.", "Again, by integration over the electrode surface defined by the angle $\\alpha $ , see Fig.", "REF , we find the beam position characteristic for a pair of BPM electrodes $A$ and $B$ in a beam-pipe with circular cross-section of radius $R$ , now for non-relativistic beams: $I_{\\mathrm {elec}}=R\\int _{-\\alpha /2}^{+\\alpha /2} J_w(\\omega , R, \\Phi , r, \\varphi ) \\mathrm {d}\\Phi $ For the two horizontal arranged electrodes $A$ and $B$ follows: $I_{\\mathrm {elec}}=-\\frac{I_{beam}}{2\\pi } s_{\\mathrm {elec}} \\left( \\omega , r/R,\\varphi , \\alpha \\right)$ with the sensitivity functions for the $A$ and $B$ electrodes based on (REF ): sA ( , r/R,, ) = I0(gr)I0(gR)+ 4 n=1 1n Im(gr)Im(gR) [n ( 2-) ] sB ( , r/R,, ) = I0(gr)I0(gR)+ 4 n=1 1n Im(gr)Im(gR) [n ( +2 -) ] Figure: Button BPM position characteristic for low-β\\beta beams, β∈\\beta \\in (0.1, 0.3),observed at frequencies f∈f\\in (325, 650, 975 MHz) (courtesy P. Kowina).Similar to section REF for a relativistic beam, we can combine (REF ) and (REF ) in terms of a $\\Delta /\\Sigma $ or log-ratio normalization and find the normalized position characteristic.", "However, unlike for the relativistic case Eq.", "(REF ), a closed form expression has never been worked out for the wall current distribution in the non-relativistic case.", "The main difference for the BPM position characteristic between relativistic and non-relativistic beams is the dependence on the frequency in the latter, given by the term Eq.", "(REF ).", "Figure REF shows a numerical study of the effects for low-$\\beta $ beams on the position characteristic wrt.", "operation frequency of the read-out electronics, which qualitatively verifies the analytical analysis discussed above.", "Still, the situation may become more complicated if the read-out system operates over a broad range of frequencies, e.g.", "for single bunch signal processing, as the impulse response of the position signal, i.e.", "its shape, is a function of the frequency content modulated by the beam velocity." ], [ "BPM Signal Processing", "The signals from the BPM electrodes need to be conditioned and processed to extract the beam intensity independent beam position information.", "This is realized by a set of electronics hardware (analog, RF and digital), digital gateware and real-time software that follows the discussed normalization procedure, typically the $\\Delta /\\Sigma $ -normalization.", "The exact concept, design, layout, and implementation of the BPM signal processing system depends on a variety of factors, e.g.", "type of accelerator, beam parameters and formatting, performance requirements of the beam position measurement, environmental conditions, etc., but also on less technical factors, like budget and manpower resources, infrastructure, as well as on laboratory standards, rules and regulations.", "Figure: Symmetry concept of a BPM system.As Fig.", "REF illustrates, the BPM signal processing is a continuation of the symmetry concept of the BPM pickup, used to detect a small asymmetry, i.e.", "the beam displacement with a perfect symmetric system in presence of a high common-mode signal.", "The more perfect the symmetry of the entire BPM system – pickup and signal processing – the higher is its performance potential, BPM resolution and accuracy." ], [ "Read-out electronics", "Figure REF shows the building blocks of a typical BPM read-out electronics.", "Each BPM pickup is equipped with this so-called front-end electronics, processing one or both (horizontal and vertical) planes.", "In most cases it is preferred to locate the BPM electronics outside the accelerator tunnel to avoid damage of the semiconductors and other elements from the ionizing radiation during machine operation.", "Figure: Read-out (front-end) electronics for a BPM pickup.This requires rather long coaxial cables, TEM transmission-lines, between the BPM pickup in the accelerator tunnel and the read-out electronics located in a gallery or service building, often installed in temperature stabilized racks.", "Sometimes, e.g.", "to detect the position of beam with very low intensity, some signal conditioning elements, e.g.", "pre-amplifiers, RF filters, hybrids, etc.", "have to be installed in the tunnel in close proximity to the BPM pickup, or a special impedance-matching network requires to be located directly on the BPM feedthrough.", "Figure: BPM analog signal processing concepts (courtesy G. Vismara).There is a very large variety of BPM signal processing concepts and post-processing schemas, impossible to cover even a few of them in this tutorial.", "Moreover, many of the different BPM signal processing concepts reflect the present state-of-the-art in electronics and telecommunication technologies, but will be quickly outdated as these techniques advance.", "Figure REF shows an overview of the various BPM analog signal processing concepts which were popular until the last millennium; Fig.", "REF shows a more “modern” concept, utilizing a digital based signal processor which was developed in the beginning of the millinium, and is based on I/Q demodulation techniques known from the telecom industry.", "The latter technique requires four separate processing channels, one for each BPM electrode.", "Figure: BPM signal processing with analog and digital electronics.For the BPM read-out concepts, the technology advances of analog-to-digital (ADC) converters have the greatest influence on selecting a particular signal processing schema.", "Clearly, the trend goes to digital signal processing techniques, which prevents imperfections and drift effects in analog and RF circuit sections due to the aging of electronics components, their tolerances, and the variation of their nominal values with the ambient temperature.", "The digital hardware also enables more complex arithmetics on the BPM signals, typically implemented as gateware in a field-progamable-gate-array (FPGA).", "Still, some minimalistic conditioning of the BPM electrode signals with classical analog and/or RF techniques remains, to adapt the signal levels and modify the waveforms to the optimal ADC operational conditions.", "Also, the generation of test or calibration signals, necessary to ensure long term reliability and performance of the BPM processor, requires some analog and RF electronics." ], [ "BPM resolution", "The resolution of a beam position monitor, i.e.", "the smallest detectable change of the beam position is – beside accuracy and long term stability (reproducibility) – the most important performance parameter of a BPM system.", "Both, BPM pickup and the following signal processor define the achievable BPM resolution.", "Consider a broadband BPM, e.g.", "a pair of button electrodes $A$ and $B$ .", "For small beam displacements we consider only the linear term of the normalization function, the beam position is detected as: pos=kPUmeasmeas =kPUmeas+noisemeas+meas kPUPUPU+ kPUnoisePUresolution kPUnoisePU=resolution =kPUAnoise-BnoiseAPU+BPU with $k_{\\mathrm {PU}}\\approx 2/R$ , see Eq.", "(REF ).", "Assuming small beam displacements: APUBPU = S AnoiseBnoise = N and $A_{\\mathrm {noise}}$ , $B_{\\mathrm {noise}}$ being uncorrelated, we find: $\\text{resolution}=k_{\\mathrm {PU}}\\frac{\\sqrt{2}N}{2S} = \\frac{k_{\\mathrm {PU}}}{\\sqrt{2}}\\left(\\frac{S}{N}\\right)^{-1}$ The noise $N$ is contributed by the signal processor, and has a minimum level of: $v_{\\mathrm {noise}}=\\sqrt{4kTR\\Delta f}$ with $k=1.38\\cdot 10^{-23}$  J/K (Bolzmann constant), $T=300$  K (typical operating temperature), and $R=50\\;\\Omega $ (typical load impedance of the signal source, here: BPM electrode).", "In practice the thermal noise level of Eq.", "REF is the lower limit of $v_{\\mathrm {noise}}$ , passive components including cables (counting as insertion loss), and the so-called noise figure of gain stages (amplifiers) will always results in a higher value of $v_{\\mathrm {noise}}$ .", "$\\Delta f$ is the 3 dB overall bandwidth of the BPM processing electronics, this includes all filter, averaging, and other bandwidth limiting elements on the BPM signal or data (analog, digital, and software).", "The signal level $S$ can be estimated from the discussion and equations in section REF , selecting a frequency band as defined by the input filters of the front-end electronics." ], [ "Summary", "This summary of the tutorial on BPM systems, presented at the CAS2018 on beam instrumentation, covered the more elementary topics and aspects on beam position monitoring.", "The presented material is of common knowledge, to be found in textbooks, papers and conference contributions, see the bibliography below.", "Typos and errors found in the CAS presentation have been corrected, including some “clean-up” of the used symbols, also some topics required clarifications and enhancements.", "As this introduction on BPM systems was set in frame of a school tutorial, many technology aspects have been omitted, and the focus was set on the physics and engineering basics of the beam position detectors, the beam signals, and a few, basic aspects of the BPM signal processing.", "The equations and mathematical expression have not been derived from the fundamental field theory and principles, instead known basic solutions have been introduced as is, on which the formalism for the beam position monitoring was build upon.", "With the presented material the reader should be able to design a BPM system, in particular to select a BPM pickup for given beam conditions and parameters, to evaluate its characteristics, and to estimate the performance of the BPM in terms of resolution potential.", "For more details on this subject I recommend the literature below and the contributions to the workshops and conference series of the BIW, DIPAC and IBIC." ], [ "Bibliography", "J. Cuperus, Monitoring of Particle Beams at High Frequencies, Nuclear Instruments and Methods, 145 (1977) 219, CERN/PS/LIN-76-7 D. A. Goldberg and G. R. Lambertson, Dynamic Devices – A Primer on Pickups and Kickers, AIP Conference Proceedings, 249, (1992), pp.", "537-600 R. E. Shafer, Beam Position Monitoring, AIP Conference Proceedings, 249, (1992), pp.", "601-636 R. E. Shafer, Beam Position Monitor Sensitivity for Low-$\\beta $ Beams, AIP Conference Proceedings, 319, (1994), pp.", "303-308 D. P. McGinnis, The Design of Beam Pickup and Kickers, AIP Conference Proceedings, 333, (1995), pp.", "64-85 F. Marcellini, M. Serio, M. Zobov, DA$\\mathit {\\Phi }$ NE Broad-Band Button Electrodes, INFN-LNF, Accelerator Division, DA$\\Phi $ NE Technical Note CD-6, Frascati, Italy, Jan. 1996" ] ]

[ [ "Impact of eccentricity on the gravitational wave searches for binary\n black holes: High mass case" ], [ "Abstract The possible formation of stellar-mass binary black holes through dynamical interactions in dense stellar environments predicts the existence of binaries with non-negligible eccentricity in the frequency band of ground-based gravitational wave detectors; the detection of binary black hole mergers with measurable orbital eccentricity would validate the existence of this formation channel.", "Waveform templates currently used in the matched-filter gravitational-wave searches of LIGO-Virgo data neglect effects of eccentricity which is expected to reduce their efficiency to detect eccentric binary black holes.", "Meanwhile, the sensitivity of coherent unmodeled gravitational-wave searches (with minimal assumptions about the signal model) have been shown to be largely unaffected by the presence of even sizable orbital eccentricity.", "In this paper, we compare the performance of two state-of-the-art search algorithms recently used by LIGO and Virgo to search for binary black holes in the second Observing Run (O2), quantifying their search sensitivity by injecting numerical-relativity simulations of inspiral-merger-ringdown eccentric waveforms into O2 LIGO data.", "Our results show that the matched-filter search PyCBC performs better than the unmodeled search cWB for the high chirp mass ($>20 M_{\\odot}$) and low eccentricity region ($e_{30 Hz} < 0.3$) of parameter space.", "For moderate eccentricities and low chirp mass, on the other hand, the unmodeled search is more sensitive than the modeled search." ], [ "Introduction", "The number of detections of gravitational wave (GW) signals has steeply increased from the first and second Observing runs (O1/O2) of Advanced LIGO and Advanced Virgo [1] to the third Observing run (O3), where tens of GW candidates have already been recorded [2], [3], [4], [5].", "So far, all GW detections of binary black holes (BBHs) are consistent with signals emitted from quasicircular binaries [6], [7].", "Generally, two main scenarios can be considered regarding possible formation channels for BBH mergers: 1) isolated binary evolution [8], [9], [10], [11], during which BBHs shed their formation eccentricity through GW emission and have circularized by the time they enter the frequency band of the ground-based detectors [12], [13]; 2) binaries dynamically formed in dense stellar environments like globular clusters and active galactic nuclei [14], [15], [16], [17], [18], which may still retain a significant eccentricity by the time they enter the frequency band of the Advanced LIGO [19] and Advanced Virgo [20] detectors.", "Although both formation channels (and their different astrophysical scenarios) predict BBH mergers with distinct distributions of masses and spins [21], [22], [23], [24], the model uncertainties —as well as the low statistics due to the limited number of GW detections —do not permit to set tight constraints on BBH formation scenarios from the mass and spin distributions alone.", "Dynamical BBH formation, however, is distinctly characterized by the potential existence of binaries with non-negligible eccentricity in the frequency band of the ground-based detectors, which were formed through dynamical capture at very close separations (without time to circularize before merger) or through a dynamical process that increased the eccentricity of the binary (e.g.", "Kozai-Lidov oscillations [25], [26]).", "The detection of a GW signal with an unambiguous signature of non-negligible orbital eccentricity would therefore confirm the dynamical formation channel for BBHs and provide information about possible formation mechanisms and the astrophysical environments of such sources.", "In order to be able to confidently detect eccentric binary black hole signals it is necessary to assess the sensitivity of the pipelines used to search for such signals.", "As a consequence several studies have analysed the sensitivity of different search pipelines to eccentric compact binary mergers over data from O1 and O2 Advanced LIGO and Advanced Virgo observing runs [27], [28], [29].", "In this paper we quantify the sensitivity of two different gravitational-wave search pipelines to eccentric inspiral-merger-ringdown (IMR) signals calculated from numerical relativity (NR) simulations.", "The two search pipelines are: 1) the template-based PyCBC algorithm [30], [31], and 2) the unmodeled coherent WaveBurst (cWB) algorithm [32], [33].", "We study the sensitivity of the pipelines with increasing eccentricity of the signal for three different mass ratios $q=1,2,4$ , with $q=m_1/m_2>1$ and $m_1$ , $m_2$ the component masses of the binary.", "Furthermore, for mass ratio $q=1$ we inject eccentric simulations with increasing dimensionless component spins $|\\vec{\\chi }_i| \\le 0.75$ (aligned with the orbital angular momentum of the system), where $\\vec{\\chi }_i= \\vec{S}_i/m^2_i$ and $\\vec{S}_i$ the spin vector of the i-component, with $i=1,2$ .", "Due to the restricted length of the NR simulations the waveforms are injected at a start frequency of 30Hz, and the eccentricity is consistently defined at that frequency according to the procedure detailed in Sec.", ".", "The paper is organised as follows: In Sec.", "we provide details about the IMR NR eccentric waveforms used in this work.", "In Sec.", "we briefly summarize the two search algorithms considered in this study, the template-based search PyCBC and the un-modeled search, cWB.", "We present in Sec.", "the results of the sensitivity estimates of both studied pipelines.", "We conclude in Sec.", "discussing the results obtained and reporting our conclusions." ], [ "Eccentric binary black holes", "The gravitational wave signals emitted from generic binary black holes are described by 17 parameters [34].", "The parameters of a binary can be separated into 10 intrinsic parameters, i.e.", "properties of the emitting source, and 7 extrinsic parameters, describing the position of the source in the detector sky.", "The intrinsic parameters are the two component masses $m_i$ , the six dimensionless spin vectors $\\vec{\\chi }_i=\\vec{S}_i/m^2_i$ , the eccentricity parameter $e$ , and the argument of the periapsis $\\Omega $ .", "Another useful mass parameter in gravitational wave data analysis is the chirp mass $\\mathcal {M}$ of a binary with masses $m_1$ and $m_2$ , which is defined as $\\mathcal {M}\\equiv (m_{1}m_{2})^{3/5}(m_1+m_2)^{-1/5}$ .", "The extrinsic parameters are the luminosity distance $d_L$ , the azimuthal angle $\\varphi $ , the inclination $\\iota $ , the time of coalescence $t_c$ , the polarization angle $\\psi $ , the right ascension $\\phi $ and the declination $\\theta $ .", "The strain induced in a gravitational wave detector can be written in terms of these parameters as [35], [36] $\\begin{split}h(t,\\zeta ,\\Theta ) & = F_+ (\\theta ,\\phi ,\\psi ) \\, h_+(t-t_c;\\iota ,\\varphi , \\zeta ) \\\\& + F_\\times (\\theta ,\\phi ,\\psi ) \\, h_\\times (t-t_c;\\iota ,\\varphi , \\zeta ) ,\\end{split}$ where $F_+$ , $F_\\times $ are the antenna pattern functions, and $\\Theta =\\lbrace t_c, r, \\theta , \\varphi ,\\psi ,\\iota , \\varphi \\rbrace $ and $\\zeta =\\lbrace m_1, m_2, \\vec{S}_1, \\vec{S}_2, e, \\Omega \\rbrace $ represent the sets of extrinsic and intrinsic parameters, respectively.", "The gravitational wave polarizations $(h_+,h_\\times )$ appearing in the detector response can be expressed as a complex waveform strain $h(t)=h_+ - i h_\\times = \\sum _{l=2}^{\\infty }\\sum _{m=-l}^{l} Y^{-2}_{lm}(\\iota , \\varphi ) h_{lm} (t-t_c;\\zeta ),$ where $h_{lm}$ are the $(l,m)$ waveform modes and $Y^{-2}_{lm}(\\iota , \\varphi )$ the spherical harmonics of spin-weight $-2$ ." ], [ "Numerical Relativity data set", "In this work we inject eccentric NR waveforms produced with the open-source EinsteinToolkit (ET) code [37], [38] and the SpEc code [39].", "The ET waveforms were presented in [40], and the SXS ones in [41].", "The injected waveforms are displayed in Table REF , where we show for each simulation its identifier (ID, an integer number), the simulation name, mass ratio, z-components of the dimensionless spin vectors $(\\chi _{1,z},\\chi _{2,z})$ and the initial eccentricity measured with the method developed in [40].", "Table: Summary of the injected NR simulations.", "The first column denotes the identifier of the simulation, the second column indicates the name of the simulation as presented in , .", "Next columns show the mass ratio, z-component of the dimensionless spin vectors and the initial NR eccentricity as measured using the procedure detailed in .The injected data set is chosen with the following criteria: simulations with IDs $1-4$ are equal mass non-spinning cases which serve as control cases because eccentric equal mass non-spinning binaries have already been studied in the literature [27], while simulations with IDs $5-10$ extend the equal mass case to the spinning sector.", "Finally, simulation sets $11-14$ and $15-17$ allow to test the efficiency of the pipelines at higher mass ratios without including spin effects.", "The eccentricity parameter describes the ellipticity of the binary's orbit, values close to 0 indicate a quasi-circular evolution while values close to 1 represent an almost head-on collision.", "In general relativity the eccentricity is a gauge dependent quantity.", "As a consequence, a plethora of eccentricity estimators have been developed to measure the eccentricity in numerical relativity simulations [42], [43], [44], [45], [46], [47], [48].", "Eccentricity estimators are combinations of dynamical or wave quantities, like the orbital frequency of the binary, the orbital separation, the gravitational wave frequency of the $(2,2)$ mode, etc., measuring the relative oscillations in those quantities due to eccentricity.", "In this work we measure the eccentricity from the gravitational wave frequency of the $h_{22}$ mode, $\\omega _{22}$ , following the procedures of [40].", "We remark that the eccentricities presented in Table REF are measured from the gravitational wave frequency and their values differ from those presented in [40] as they were calculated there using the orbital frequency computed from the trajectories of the black holes.", "In the top panel of Fig.", "REF we show the time evolution of the eccentricity of the simulation with ID 17 in Table REF .", "Moreover, we choose the end of the inspiral given by the minimum energy circular orbit (MECO) [49], and explicitly set the eccentricity to zero from the MECO time onwards as at that point the eccentricity is so small which is practically zero.", "In this study we are interested in injecting the waveforms presented in Table REF at a certain detector frequency and for a certain total mass distribution.", "The modification of the total mass of the system implies a change in length of the waveform within the frequency band of the detector, as a consequence different total masses imply also different initial eccentricities, as one can appreciate from the top panel of Fig.", "REF , which shows the eccentricity as a monotonically decaying function as the binary evolves.", "One possible solution might be to express the eccentricity measured from the NR simulation as a function of gravitational wave frequency of the 22-mode scaled by the total mass of the system, $ 2 \\pi M f_{22}= M \\omega _{22} $ , approximate the value of the injection frequency by the frequency of the 22-mode, $f_{22} \\approx f_{\\text{GW}}$ , and construct a function $e(M f_{\\text{GW}})$ which would provide the value of the eccentricity at a certain total mass for a given injection frequency.", "However, in the eccentric case the gravitational wave frequency is a non-monotonic function due to the asymmetric gravitational interaction along the orbit of the binary as one can observe in the mid panel of Fig.", "REF , where the time domain frequency of the 22-mode for the eccentric simulation with ID 17 from Table REF and the frequency of the quasicircular IMRPhenomT [50] waveform model for the same configuration are displayed.", "We note that after the MECO time both curves converge indicating circularization of the eccentric system at merger.", "Figure: Top panel: Time domain evolution of the eccentricity estimated from the eccentric NR simulation with ID 17 in Table .", "Mid panel: Time domain 22-mode gravitational wave frequencies of the eccentric case with ID 17 from Table and of the quasicircular IMRPhenomT waveform model, highlighted in blue and green colors respectively.", "Bottom panel: Eccentricity as a function of the gravitational frequency of the (2,2)(2,2) mode for the same configuration as in the upper panel.", "With vertical lines in the top and bottom plots we have highlighted the MECO time and frequency, respectively.One possibility for the definition of the eccentricity as a function of a monotonically increasing frequency is to consider the post-Newtonian (PN) approximation, and use the Radiation Reaction (RR) equations [51] for the PN parameter, x, which can be written in terms of the orbital frequency, $x= \\omega ^{2/3}$ , and the temporal eccentricityWe recall that within the quasi-Keplerian parametrization [52], [53] one defines three eccentricities, $e_t$ , $e_r$ and $e_\\phi $ , which can be related to each other by PN expressions.", "We refer the reader to [51] for details.", "$e_t$ .", "The RR equations are ordinary differential equations for the temporal evolutions of $x$ and $e_t$ , derived from the angular momentum and gravitational wave energy fluxes [51].", "In practice, in the RR equation for $e_t$ one could replace it by the eccentricity measured from the NR simulation and solve the differential equation for $\\dot{x}$ .", "However, we find that this procedure does not work satisfactorily, as we have checked that the RR equations show a divergent behavior before the MECO time in some cases, indicating the breakdown of the post-Newtonian approximation.", "Therefore, we decide to take the gravitational wave frequency of IMRPhenomT and combine it with the eccentricity measured from the simulation to construct the function $e_{NR}(Mf_{22})$ .", "The outcome of such a calculation for the simulation with ID 17 in Table REF is shown in the bottom plot of Fig.", "REF .", "Hence, given an injection with total mass $M_T$ and an injection frequency of $f_{GW}$ , we can compute the eccentricity at that frequency and total mass as $e_\\text{inj}= e_{NR}(M_T f_{GW}).$ We note that we focus only on the eccentricity parameter as the initial argument of the periapsisAlso called initial mean anomaly in the quasi-Keplerian parametrization [52], [53].", "in the non-precessing case acts as an initial phase during the inspiral.", "Its main impact is in the morphology of the waveform at plunge, whose detailed study would require going beyond the maximum total mass considered in this communication ($M_T>100 M_\\odot $ ).", "We leave for future work analyzing such high total mass regime.", "Figure: Temporal evolution of the GW polarization state h x (t)h_x(t) for non-spinning, eccentric stellar-mass binary black holes with total mass M T =50M ⊙ M_T = 50 M_\\odot and mass ratio q=2q = 2 , provided by numerical-relativity simulations.", "The characteristic orbital eccentricity of the system —defined at a reference frequency of 30Hz30 Hz —is estimated to be 0.050.05 (in blue, simulation ID 11) and 0.230.23 (in orange, simulation ID 14), respectively.Finally, in Fig.", "REF we plot the time evolution of the GW polarization state $h_x(t)$ for non-spinning, eccentric stellar-mass binary black holes with total mass $M_T = 50 M_\\odot $ and mass ratio $q = 2 $ , provided by numerical-relativity simulations.", "The characteristic orbital eccentricity of the system —defined at a reference frequency of $30 Hz$ —is estimated to be $0.05$ (in blue, simulation ID 11) and $0.23$ (in orange, simulation ID 14), respectively.", "The time-domain waveforms clearly demonstrate the effects of increasing initial orbital eccentricity: rapid dephasing, as well as pronounced amplitude modulations due to the advance of the periastron.", "The data set used to conduct this study is part of the O2 Data Release through the Gravitational Wave Open Science Center [54].", "This covers approximately $\\approx 5$ days of the coincident data between LIGO Livingston and LIGO Hanford between UTC Interval 2017-02-28 16:30:00 - 2017-03-10 13:35:00.", "Times with significant instrumental disturbances have been removed from the time period considered here [55], [56].", "We consider two search algorithms, PyCBC and cWB, which are described in the following sections." ], [ "PyCBC : The matched filter algorithm", "PyCBC is a search pipeline devised to detect GWs from compact binary coalescences using the PyCBC software package [57].", "In this work we have employed the PyCBC search algorithm in a similar configuration as was used for the first catalogue of gravitational waves transients GWTC-1 [1].", "For details of the algorithm see [57], [58], [30], [59], [31].", "In the PyCBC analysis presented here the template bank described in [60] is used.", "This bank covers binary systems with a total mass between $2 M_{\\odot }$ and $500 M_{\\odot }$ and mass ratios $q < 98$ .", "Binary components with masses below $2 M_{\\odot }$ are assumed to be neutron stars with a maximum dimensionless spin magnitude of $0.05$ ; otherwise the maximum dimensionless spin magnitude is $0.998$ .", "This template bank includes no effects of eccentric orbits.", "In a previous study it has been found that a quasicircular bank does not provide a good match for searching binaries with eccentric orbits [61].", "Furthermore, it is known that the signal morphology of the eccentric BBH is orthogonal to the aligned-spin quasicircular BBH [62].", "As a consequence the template bank, which is restricted to the dominant harmonic of quasicircular non-precessing waveforms, becomes ineffective for searching eccentric BBH with high eccentricities.", "The way eccentricity affects the matched-filter search by a quasicircular template bank is twofold, first the collection of matched filtered signal-to-noise ratio (SNR) is reduced as a function of eccentricity (this can be quantified by studying the overlap of eccentric and quasicircular waveforms), second the signal-based $\\chi ^2$ veto [58] used for weighting the single detector SNR to compute the rank also penalizes the final detection statistics of the search." ], [ "cWB : The un-modeled search algorithm", "The cWB search pipeline [32], [27], [33] is designed to detect and reconstruct short-lived signals which are weakly modeled or unmodeled using a network of GW detectors [33], but is also effective for signals with a known morphology, as is the case of BBH events reported in GWTC-1 [1].", "The configuration of cWB used in this work is the same as used in the GWTC-1 catalog.", "We refer the reader to [32], [27], [28], [33], [63] for details of the detection process in cWB.", "The lack of a template bank for binary black holes in eccentric orbits, which could be used by matched filter pipelines, motivated the use of cWB as a robust tool to search eccentric BBH signals during the first and second observing runs of the LIGO and Virgo detectors [28].", "The cWB search pipeline performs worse than matched-filter pipelines in the case the signal is well recovered by the template bank.", "In this case matched-filtering would indeed be the optimal method for Gaussian and stationary noise (these simplifications do however not apply to actual LIGO noise).", "The sensitivity of cWB significantly improves in parts of the parameter space where the template bank does not faithfully reproduce the incoming signal.", "In an earlier version of cWB, its search sensitivity was found to have almost no dependency on eccentricity [27].", "This was also confirmed in the latest results for observing runs O1 and O2 of the LIGO and Virgo detectors [28].", "As a weakly modelled search cWB is more affected by the background noise and hence has a lower sensitivity as compared to matched filter searches for known signals.", "Nevertheless cWB has been found to provide a valuable complementarity to matched filter searches to detect signals which are outside the template bank, as in the case of eccentric BBH signals [27] or intermediate mass BBH signals [64].", "Figure: Top row: The left and right panel show the sensitivity range for the cWB and PyCBC search pipelines for various chirp mass bins as a function of eccentricity defined at 30Hz at an IFAR >> 10 years, respectively.", "Bottom row: The left and right panels display the sensitivity range for the cWB and PyCBC search pipeline for various chirp mass bins as a function of eccentricity defined at 30Hz at an IFAR >> 100 years, respectively.", "The plot markers are placed in the center of the eccentricity bins." ], [ "Visible range", "The visible range for a given source parameters is calculated by injecting simulated waveforms into the data [31], [59], [32].", "The False Alarm Rate (FAR) is a statistic measuring the frequency with which the search would rank non-astrophysical events with a detection statistic comparable to the one of a candidate event.", "In practice, each recovered signal is assigned an inverse false alarm rate (IFAR=1/FAR) according to its detection statistic.", "Then, one can compute for each bin of the source parameters the visible volume over a certain IFAR threshold.", "For a generic binary, the sensitive volume $V$ of a network of detectors with a given sensitivity can be defined as $V(\\xi ) = \\int _0^{\\infty } f(z|\\xi ) \\frac{\\mathrm {d} V_c}{\\mathrm {d}z} \\frac{1}{1+z} \\ dz,$ where $f(z|\\xi )$ is the detection probability of a binary with a given parameter set $\\xi $ at redshift $z$ , averaged over the extrinsic binary orientation parameters [65].", "In Eq.", "(REF ) the sensitivity is assumed to be constant over the observing time, $T_\\mathrm {obs}$ , which is why we have chosen the specific chunk of O2 data where sensitivity was almost uniform.", "Given a population with parameters $\\theta $ , the total observed volume can be computed as $V_{\\theta } = \\int _{\\xi } p(\\xi | \\theta ) V(\\xi ) \\ d\\xi ,$ where $p(\\xi | \\theta )$ describes the underlying distribution of the intrinsic parameters.", "The visible range can be then estimated as the radius of the visible volume.", "The sensitivity of GW searches is a strongly dependent function of the binary chirp mass and distance, and it also varies with spin.", "We also note that the eccentricity can be a relevant factor depending on the pipeline used to conduct the search.", "Thus, we have mainly chosen chirp mass binning to study the impact of eccentricity on visible range as it shows more clearly the dependence of the search sensitivity than other parameters, like the total mass." ], [ "Injection set", "The injection set used in this study is composed of the NR waveforms listed in Table REF .", "As a consequence, injections have fixed spin vectors and mass ratio values corresponding to those of the NR waveforms.", "Nonetheless, the total mass of the system acts as a scale parameter which can be freely specified, subject only to consistency with the length of the NR waveforms such that the injected signals start at the specified starting frequency in the band of the detectors.", "Due to the length limitations of the NR waveforms we set the starting frequency of the injection set to $30Hz$ , which is also the frequency at which the value of eccentricity is specified.", "We have chosen the number of performed injections to limit the computational cost necessary to run the search pipelines.", "We note that PyCBC is more expensive than cWB, however it also allows to achieve higher IFAR values.", "The largest subset of injected waveforms corresponds to $q=1$ nonspinning with 17317 injections distributed among cases with IDs 1, 2, 3 and 4 in Table REF .", "While the number of injections for the rest of waveforms has been decreased substantially due to the limited computational resources available to 3591 for $\\chi _{\\text{eff}}<0$ cases (IDs 5, 6 and 7), 4723 for $\\chi _{\\text{eff}}>0$ configurations (IDs 8, 9 and 10), 6416 for $q=2$ simulations and 4458 for $q=4$ simulations (IDs 15, 16 and 17).", "As a consequence, the equal mass non-spinning eccentric case provides better statistics and permits to clearly identify the behavior of the sensitivity of both pipelines for specific values of chirp mass and eccentricity, as shown in Sec.", "REF .", "The injection set is constructed using a uniform distribution in distance scaled by the chirp mass [66].", "The total mass values are uniformly distributed from a minimum value consistent with the length of the NR waveforms, between $[30-50] M_\\odot $ for our dataset, to a maximum total mass of $100 M_\\odot $ .", "The orbital eccentricity of the individual injections, defined at a reference frequency of $30 Hz$ is estimated through Eq.", "(REF ).", "We note that with this method the maximum eccentricity at $30 Hz$ of a given injected NR waveform is given by the values of the last column of Table REF , as these values are measured at the start of the NR waveforms.", "The moderate values of eccentricity considered here are well-suited for a first study of the sensitivity of gravitational waves searches to full IMR signals.", "Furthermore, many astrophysical models for eccentric binary black hole coalescences in the frequency band of ground-based detectors predict similar eccentricity values as those used here [67], [68], [69], [15]." ], [ "Effect of eccentricity on search sensitivities", "We now turn to discussing the visible range at IFAR thresholds of 10 and 100 years for both search pipelines and the same injection set.", "Although matched-filter searches are an optimal method to search for signals of known morphologies, in the case of eccentric BBHs computationally efficient waveform models describing the full GW signal of eccentric BBH coalescences have not yet been developed.", "For this reason it is expected that the quasicircular template bank used by PyCBC will not be able to detect eccentric BBH events with orbital eccentricities beyond a certain threshold.", "On the other hand, cWB does not require signal models for detection and thus its sensitivity to eccentric BBH signals is expected to only vary significantly as a function of signal strength, but only weakly in terms of other parameters like eccentricity.", "It should be noted, however, that cWB is not an optimal method to detect BBH merger events and thus has lower sensitivity than PyCBC for regions of parameter space which are either explicitly covered by the PyCBC template bank or where the signal is otherwise `mimicked' by templates in the bank.", "Figure: The upper (lower) panel shows the sensitivity range for the cWB and PyCBC search pipelines for equal mass non-spinning, positive spins and negative spins as a function of eccentricity defined at 30Hz at an IFAR >> 10 years (IFAR >> 100 years).", "The plot markers are placed in the center of the eccentricity bins.In Figure REF we exhibit the visible ranges of the PyCBC and cWB pipelines binned in chirp mass and eccentricity for all the injected signals.", "The results show a reduction in visible range of PyCBC with increasing eccentricity.", "The steepness of the reduction of visible range becomes more apparent when one goes to lower chirp masses; this is due to the fact that for high chirp masses the number of cycles visible in the sensitivity band of the LIGO detectors (and hence the inspiral part of the signal where eccentricity effects are pronounced) is rather short.", "One can conclude that for high chirp mass events with moderate to low eccentricities the PyCBC search and its quasicircular template bank does not lose much visible range.", "This behaviour is contrary to the low chirp mass case with moderate eccentricities, where the loss in visible range is substantial.", "Table: Summary of the injected signals in Fig.", ".", "The first column denotes the identifier of the simulation.", "The next columns indicate the total mass, M T M_T, the luminosity distance, D L D_L, the eccentricity, e inj e_{\\text{inj}} , of the injection, the match between the injected signal and the recovered one by cWB, ℳ Ecc \\mathcal {M}^{\\text{Ecc}}, and the match between the injected signal and the QC template with the same injected parameters, ℳ QC \\mathcal {M}^{\\text{QC}}.Regarding cWB, previous work [28] found that the search pipeline is almost independent of eccentricity for a given chirp mass bin.", "However, the waveforms used in that investigation [70] were based on geodesics in Kerr spacetime and the quadrupole formula for energy loss, and significantly less accurate than the NR simulations used here.", "We note an interesting feature in the dependency of the range as a function of eccentricity for cWB for the lowest chirp mass bin at IFAR$>$ 10 years.", "The range increases slightly as a function of eccentricity.", "This might probably be attributed to the power content in higher harmonics in eccentric BBH signals which is enhanced when the eccentricity increases.", "cWB captures the total excess power in the network of detectors and therefore can observe eccentric BBH events at larger range.", "However, we note that this particular small increase in sensitivity is also compatible with a constant sensitive distance as the values are within the statistical error bars.", "We find that our results are robust when changing the IFAR threshold from 10 to 100 years: sensitivity results for the higher IFAR choice are shown in the lower panels of Fig.", "REF .", "One observes the expected overall decrease of the sensitive distance of both pipelines with increasing IFAR.", "Moreover, it can be noted that the dependence of the visible range on eccentricity retains the same features as at IFAR$>10$ years for both pipelines.", "Figure: In the upper and lower panels the whitened detector response of LIGO Hanford, H1, in time domain are displayed for an injected signal corresponding to the waveform with ID 9 and 12 from Table , respectively.", "Some of the parameters of the injected signals are shown in Table .", "In both panels the injected and reconstructed signals are represented with blue and orange colors, respectively.In our NR simulations we only have waveforms with moderate eccentricities ($e_{30 Hz}$ $<$ 0.3).", "The subset of spinning waveforms is even more restricted in eccentricity values ($e_{30 Hz} < 0.12 $ ).", "As a consequence, a study of the impact of the effect of eccentricity and spins is more difficult.", "In Fig.", "REF we show the sensitive distance for the equal mass spinning and non-spinning eccentric waveforms as a function of eccentricity for the chirp mass bin $[13,30]$ $M_\\odot $ .", "As expected, one observes that PyCBC has larger sensitivity for positive spins than for negative spins, as for positive spins the matched filter pipeline can collect more SNR than for negative spins.", "There is also a drop in sensitivity for PyCBC with increasing eccentricity, while for cWB the small drop in sensitivity is consistent with statistical error bars.", "We note here that the results for the spinning waveforms are computed over a smaller number of injections compared to the nonspinning case, as explained in Sec.", "REF .", "In addition the small range in eccentricity of the spinning simulation does not allow to identify specific trends for the sensitivity as a function of the eccentricity.", "We point out that more insightful results could be obtained by increasing the number of injections and the range of values of initial eccentricity of the waveforms, and we leave the study of a large parameter space of the eccentric non-precessing spin sector, as well as eccentric spin-precessing systems, to future work.", "Finally, we illustrate an example of the robust waveform reconstruction procedure of cWB [71] applied to eccentric signals.", "In Fig.", "REF we display the whitened detector response of LIGO Hanford (H1) to two eccentric injected signals and the corresponding reconstructed waveforms by cWB, specifically for injections corresponding to the cases with ID 9 and 12 from Table REF , with injection parameters specified in Table REF .", "We have also calculated the match, defined as the noise weighted inner product [72], between the injected and recovered signal with cWB obtaining high agreement between both.", "The results of the match are also reported for a quasicircular template waveform computed against the same injected signals.", "One observes that the match against the quasicircular template, calculated using the SEOBNRv4 waveform model [73], decreases significantly with increasing eccentricity, from $\\mathcal {M}=0.93$ for ID 12 to $\\mathcal {M}=0.85$ for ID 9, while the drop in the match against the cWB reconstructed waveform goes from $\\mathcal {M}=0.98$ for ID 12 to $\\mathcal {M}=0.95$ for ID 9.", "For cWB the match decreases because the signal with ID 9 is longer than for ID 12, and the reconstruction is expected to degrade the longer the signal is, while for the quasicircular template the increase of eccentricity decreases substantially the match due to the inability to resemble eccentric features in the injected signal.", "It should also be remarked that for cWB as an unmodelled search algorithm a high match between reconstructed and injected signal does not directly translate into having a high sensitivity of the pipeline as it is not a matched filter search pipeline.", "However, from a waveform modelling perspective it is still relevant to observe the ability of cWB to reconstruct the eccentric signal and the inability of the quasicircular template to resemble the injected signal with increasing eccentricity.", "As expected the reconstructed signal degrades after the waveform peak, thus, the ringdown is poorly reproduced due to the decrease in power of the signal.", "With these examples we want to illustrate the capability of cWB to recover features of eccentric signals, and we leave a thorough analysis of the reconstruction procedure of cWB applied to eccentric BBH signals for future work." ], [ "Comparisons of search sensitivities and astrophysical implications", "In Figure REF we show the comparison of the visible volumes of PyCBC and cWB at IFAR thresholds of 10 and 100 years.", "Within our injection set PyCBC almost always performs better than or similar to cWB in terms of visible volume.", "In the case of low chirp mass and high eccentricity the situation is reversed: PyCBC loses sensitivity and cWB becomes more sensitive, specially at IFAR $>$ 10 years.", "Figure: In the upper (lower) panel the relative difference in sensitivity volume between the search sensitivity of pyCBC and cWB for various chirp mass bins is presented as function of eccentricity at an IFAR >> 10 (IFAR >> 100) years.", "The plot markers are placed in the center of the eccentricity bins.In the lower panel of Fig.", "REF the comparison in sensitive volume between both pipelines at IFAR $>$ 100 years is shown.", "One observes that the trends in relative volume between both pipelines are similar to the case at an IFAR threshold of 10 years, although the error estimates are larger due to the decrease in sensitivity of both pipelines with increasing IFAR values.", "The results show that PyCBC has a larger sensitive volume than cWB, even for the low chirp mass bin.", "However, with increasing eccentricity the decrease in sensitive volume of PyCBC and the constancy of the cWB one, make the relative volume to be an increasing function, which for higher values of eccentricity is expected to cross zero, as it is the case at IFAR $>$ 10 years.", "These comparison results can also be viewed in the light of coalescence rate.", "Suppose the coalescence rate of eccentric BBH mergers with eccentricities between $(0,0.3)$ at 30 Hz is $R_{eBBH}$ ; then the number of visible events will be simply $N_{events} = R_{eBBH} \\times V_{iFAR} \\times T_{obs}$ .", "The relative difference in the number of detected events will be the same as the relative difference between the visible volume for the two search algorithms that we have considered.", "From this, we can conclude that at IFAR$>10$ years cWB will see $\\sim $ 10 % more events than PyCBC if the chirp mass is between $[13 M_{\\odot }, 20 M_{\\odot }]$ and the eccentricity at 30 Hz is between [0.25-0.3]." ], [ "Conclusions", "In this paper we have quantified for the first time the sensitivities of GW search algorithms to eccentric BBH signals, using NR simulations of eccentric BBH mergers.", "The effect of eccentricity on matched filtered searches has only been studied for inspiral-only waveforms until now [61]; we have extended those studies to complete IMR signals.", "The search range of unmodeled searches for eccentric signals has been previously investigated with a particular IMR waveform model [28]; however, that waveform model is far less accurate than the NR simulations used here.", "We have employed two different gravitational wave searches for BBHs to compare the search sensitivity in terms of visible volume.", "The matched filter search PyCBC performs better than the unmodeled search cWB in most parts of the limited parameter space that we have considered.", "Only in the parameter space region of low chirp mass and high eccentricity does cWB perform better than PyCBC.", "It should also be noted that the parameter space that is covered by our NR injections is rather small.", "Due to the restricted length of the NR simulations, the parameter space of low chirp mass ($\\mathcal {M}_c < $ 13) and high eccentricity $e_{30 Hz} > 0.3 $ is not yet probed in this work.", "This, however, is the most interesting part of parameter space for eccentric BBHs, with waveform morphologies that are substantially different than those of quasicircular BBHs.", "We plan to investigate this part of parameter space in subsequent work, with eccentric hybrid waveforms that combine NR data with an analytic description of the inspiral, or with future waveform models for the full IMR signal.", "The two search pipelines used here —very different algorithms as described in Sec.", "—offer a complementary way to search for BBH mergers in different parts of the source parameter space.", "Constructing a template-based search for eccentric BBH will be challenging as the rate of background triggers increases with the increase of template bank parameters.", "In the light of astrophysical considerations [14], [15], most of the BBH events observable by LIGO and Virgo are expected to have eccentricities lower than $0.2$ at 30 Hz; this region of parameter space has been demonstrated to be well-covered by the PyCBC search, even with a quasi-circular template bank.", "Certain astrophysical scenarios suggest LIGO-Virgo relevant BBH events with higher eccentricities: for such sources the cWB search provides decent coverage.", "With the expected availability of computationally efficient and accurate eccentric IMR BBH waveforms models (and/or eccentric hybrids) in the near future it will be interesting to probe the low chirp mass and high eccentricity part of the parameter space, where the modelled search is penalized due to substantial dephasing between the quasicircular template bank and the signal.", "With future upgrades [74], the detectors' low-frequency sensitivity (in the range $24-100$ Hz) is expected to improve significantly; this will in turn allow a significant gain in SNR during the inspiral even for BBH systems with relatively high masses, adding more prominence to detectable inspiral features like eccentricity and penalizing the matched-filter searches for eccentric BBH even further.", "With future improvements at low frequencies, the role of un-modeled searches is therefore expected to become important also for the part of parameter space which is well-covered by matched-filter searches at current detector configuration." ], [ "Acknowledgements", "We would like to thank Debnandini Mukherjee for useful comments about the manuscript.", "This work was supported by the Spanish Ministry of Education and Professional Formation grants FPU15/03344.", "The author also acknowledges the support by the Govern de les Illes Balears through the Vicepresidència i Conselleria d’Innovació, Recerca i Turisme and the Direcció General de Política Universitària i Recerca with funds from the Tourist Stay Tax Law ITS 2017-006 (PRD2018/24), the European Union FEDER funds and EU COST Actions CA18108, CA17137, CA16214, and CA16104, the Ministry of Science, Innovation and Universities and the Spanish Agencia Estatal de Investigación grants FPA2016-76821-P, RED2018-102661-T, RED2018-102573-E, FPA2017-90566-REDC, FPA2017-90687-REDC, and the Generalitat Valenciana (PROMETEO/2019/071).", "The author thankfully acknowledges the computer resources at MareNostrum and the technical support provided by Barcelona Supercomputing Center (BSC) through Grants No.", "AECT-2020-1-0025, AECT-2019-3-0020, AECT-2019-2-0010, AECT-2019-1-0022, AECT-2018-3-0017, AECT-2018-2-0022, AECT-2018-1-0009, AECT-2017-3-0013, AECT-2017-2-0017, AECT2017-1-0017, AECT-2016-3-0014, AECT2016-2-0009, from the Red Española de Supercomputación (RES) and PRACE (Grant No.", "2015133131).", "ET simulations were carried out on the BSC MareNostrum computer center under PRACE and RES allocations and on the FONER cluster at the University of the Balearic Islands.", "The authors are grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.", "MH acknowledges support from Swiss National Science Foundation (SNSF) grant IZCOZ0-177057.", "ARB is grateful to the Pauli Center for Theoretical Studies at ETHZ which provided valuable travel support during stages of this work.", "S.T.", "is supported by Forschungskredit Nr.", "FK-19-114 and Swiss National Science Foundation grant number 200020 182047.", "This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration.", "LIGO is funded by the U.S. National Science Foundation.", "Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.", "The authors gratefully acknowledge the support of the NSF CIT cluster for the provision of computational resources for pyCBC and cWB runs." ] ]

[ [ "Self-averaging in many-body quantum systems out of equilibrium: Time\n dependence of distributions" ], [ "Abstract In a disordered system, a quantity is self-averaging when the ratio between its variance for disorder realizations and the square of its mean decreases as the system size increases.", "Here, we consider a chaotic disordered many-body quantum system and search for a relationship between self-averaging behavior and the properties of the distributions over disorder realizations of various quantities and at different timescales.", "An exponential distribution, as found for the survival probability at long times, explains its lack of self-averaging, since the mean and the dispersion are equal.", "Gaussian distributions, however, are obtained for both self-averaging and non-self-averaging quantities.", "Our studies show also that one can make conclusions about the self-averaging behavior of one quantity based on the distribution of another related quantity.", "This strategy allows for semianalytical results, and thus circumvents the limitations of numerical scaling analysis, which are restricted to few system sizes." ], [ "Introduction", "Experimental advances with cold atoms [1], ion traps [2], superconducting devices [3], and nuclear magnetic resonance platforms [4], [5] allow for the high level of control and long coherence times of many-body quantum systems.", "This has invigorated experimental and theoretical studies of the long-time evolution of these systems.", "Common questions include the viability of thermalization [6], [7], [8], [9], the description of the dynamics [10], [11], and the time to reach equilibrium [12], [13].", "Much less explored is the question of self-averaging [14], [15], [16].", "A quantity of a disordered system is self-averaging when its relative variance — the ratio between its variance for disorder realizations and the square of its mean — decreases as the system size increases.", "If self-averaging holds, as the system size increases, then one can decrease the number of samples used in theoretical and experimental analyses.", "In this case, the properties of the system do not depend on the specific realization selected.", "Lack of self-averaging, however, makes the study of disordered systems more challenging.", "Take as an example the scaling analysis of many-body quantum systems.", "The problem is already hard, because the many-body Hilbert space grows exponentially with system size.", "If in addition to this, one cannot decrease the number of disorder realizations as the system size grows, the problem becomes intractable.", "Non-self-averaging behavior is often associated with disordered many-body quantum systems at the transition between the delocalized and the localized phase [17] and systems at a critical point in general [18], [19], [20], [21], [22], [23], [24], [25], [26], [27].", "This sort of studies have mostly been done at equilibrium [20].", "Recently, however, the analysis has been extended to systems out of equilibrium close to the localization transition point [15], [16] and also in the chaotic regime [14].", "It has been shown that self-averaging is not directly related with quantum chaos [14], [28], [29], [30], as one might naively expect.", "Quantum chaos refers to specific properties of the eigenvalues and eigenstates of systems that are chaotic in the classical limit.", "The eigenvalues are correlated [31], [32], [33] and the eigenstates are close to the random vectors [34], [35], [8] of full random matrices.", "If the system shows these properties, then it is usual to refer to it as chaotic even if its classical limit is not well defined.", "In Ref.", "[14], the analysis of self-averaging was done for both a disordered spin model in the chaotic regime and a model consisting of full random matrices of a Gaussian orthogonal ensemble (GOE).", "It was shown numerically and analytically that the survival probability (the probability for finding the system in its initial state at a later time) is non-self-averaging at any timescale.", "Other quantities considered include the inverse participation ratio, which measures the spread of the initial state in the many-body Hilbert space, and observables measured in experiments with cold atoms and ion traps, namely the spin autocorrelation function and the connected spin-spin correlation function.", "The self-averaging behavior of the inverse participation ratio and spin autocorrelation function varies in time, while the connected spin-spin correlation function is self-averaging at all times.", "Motivated by the results in Ref.", "[14], we now study numerically and analytically the distributions over disorder realizations of those same quantities throughout their evolution to equilibrium using again both the GOE and the disordered spin model.", "In addition, to avoid the negative values that can be reached with the spin autocorrelation function, we consider also the absolute value and the square of the spin autocorrelation function.", "Our goal is to understand how the shape and overall properties of the distributions depend on time, observables, and models, and whether they can help us determine when self-averaging holds.", "We find that at short times, the distributions are model dependent.", "Due to the locality of the spin model Hamiltonian, the distributions of the quantities considered here exhibit a fragmented structure with peaks at different energy windows, while the distributions are Gaussian for the GOE model.", "At long times, the distributions become similar for both models, but they differ depending on the quantity.", "The survival probability, for example, shows an exponential distribution [36], [28], [29], [30] as soon as the correlations between the eigenvalues get manifested in the dynamics.", "This distribution, where mean and standard deviation coincide, explains the lack of self-averaging of this quantity at long times.", "For the other quantities, the distribution is either Gaussian or related to a normal distribution.", "Gaussian distributions are found for both self-averaging and non-self-averaging quantities.", "A useful outcome of these studies is the realization that the shape of the distribution for one quantity can assist with the analysis of the self-averaging behavior of another related quantity.", "As an example, we discuss the case of the spin autocorrelation function, $I(t)$ , and its absolute value, $|I(t)|$ .", "The numerical analysis of the self-averaging behavior of $|I(t)|$ at long times are inconclusive, due to the limited system sizes available.", "However, in hands of the Gaussian distribution for $I(t)$ , we find analytically the dependence on system size of the relative variance of $|I(t)|$ .", "With this strategy, we are able to deduce that $|I(t)|$ is non-self-averaging at long times.", "The paper is organized as follows.", "Section  contains the necessary background for the following sections.", "It presents the model, initial states, quantities, and our previous results about the dynamics and self-averaging properties of the survival probability and inverse participation ratio.", "In Secs.", ", , and , we proceed with the analysis of the distributions of these global quantities.", "This study is separated by time intervals: short times in Sec.", ", long times in Sec.", ", and intermediate times in Sec. .", "The analysis of the local quantities and how to use the distribution of one quantity to describe the self-averaging behavior of another one is explained in Sec. .", "Conclusions are presented in Sec.", "." ], [ "Models, Quantities, and Time Scales", "We study two models described by Hamiltonians of the form $H=H_0+ V,$ where $H_0$ is the unperturbed part of the total Hamiltonian and $V$ is a strong perturbation that takes the system into the chaotic regime.", "The notation adopted is the following: $|n\\rangle $ stands for the eigenstates of $H_0$ , $|\\alpha \\rangle $ for the eigenstates of $H$ , and $E_\\alpha $ for the eigenvalues of $H$ .", "One model consists of random matrices from a GOE and the other is a many-body spin-1/2 system." ], [ "GOE model", "For the GOE model, $H_0$ is the diagonal part of a full random matrix of dimension $D$ and $V$ contains the off-diagonal elements.", "The entries are all real random numbers from a Gaussian distribution with mean value $\\left<H_{ij}\\right>=0$ and variance $\\left<H_{ij}^2\\right>=\\left\\lbrace \\begin{array}{ll}1 & i=j, \\\\1/2 & i\\ne j.\\end{array}\\right.$ The Hamiltonian matrix $H$ can be generated by creating a matrix $M$ with random numbers from a Gaussian distribution with mean 0 and variance 1 and then adding $M$ to its transpose as $H=(M+M^{T})/2$  [37].", "The eigenvalues of this model are highly correlated [31], [32], [33] and the eigenstates are normalized random vectors [35].", "There are no realistic systems described by this model, but it allows for analytical derivations not only for static properties [38], [31], [33], but also for the dynamics [39], [40], [12], [14]." ], [ "Disordered spin model", "We consider a one-dimensional chaotic spin-$1/2$ model of great experimental interest [41] and often used in studies of many-body localization [42], [43], [44], [45], [46], [47].", "It has onsite disorder and nearest neighboring couplings [48], $H_0 &=& J \\sum _{k=1}^L (h_k S_k^z + S_k^z S_{k+1}^z), \\nonumber \\\\V &=& J\\sum _{k=1}^L (S_k^x S_{k+1}^x + S_k^y S_{k+1}^y).$ Above, $\\hbar =1$ , $J=1$ is the coupling strength, $S_k^{x,y,z}$ are spin operators on site $k$ , $L$ is the size of the chain, which is even throughout this work, and periodic boundary conditions are used.", "The Zeeman splittings $h_i$ are random numbers uniformly distributed in $[-h,h]$ .", "The total magnetization in the $z$ direction is conserved, so we take the largest subspace, where the total $z$ magnetization is zero and the dimension is $D=L!/(L/2)!^2$ .", "We use disorder strength $h=0.75$ , which places the system in the chaotic regime.", "The level statistics and the structure of the eigenstates away from the borders of the spectrum are comparable to those of the GOE model." ], [ "Initial state", "The initial state $\\left| {\\rm ini} \\right> = \\left| \\Psi (0)\\right> $ is an eigenstate $|n\\rangle $ of $H_0$ .", "We take $\\left|\\Psi (0)\\right>$ with energy close to the middle of the spectrum, where the eigenstates are chaotic [49], $E_{\\rm ini} = \\langle \\Psi (0)|H|\\Psi (0)\\rangle = \\sum _\\alpha \\left|C_\\alpha ^{\\rm ini}\\right|^2 E_\\alpha \\sim 0.$ In the equation above, $C_\\alpha ^{\\rm ini}=\\left<\\alpha |\\Psi (0)\\right>$ are real components, since the Hamiltonian matrices treated in this work are real and symmetric.", "For the spin model, the initial states are product states in the $z$ direction, where on each site the spin either points up or down in the $z$ direction, such as $|\\uparrow \\downarrow \\uparrow \\downarrow \\downarrow \\uparrow \\ldots \\rangle $ .", "They are often referred to as site-basis vectors or computational basis vectors." ], [ "Quantities", "We analyze in detail the distributions over disorder realizations of the survival probability and the inverse participation ratio.", "Both are nonlocal quantities in real space.", "We also present results for the spin autocorrelation function, its absolute value and its square value, and for the connected spin-spin correlation function.", "These four quantities are local in space.", "Our studies of the survival probability and the inverse participation ratio are presented for the GOE model and the chaotic spin model.", "For the local quantities, this is done only for the spin model, since the notion of locality does not exist in full random matrices.", "The survival probability is the squared overlap of the initial state and its evolved counterpart, $P_S(t)&=&\\left|\\left<\\Psi (0)\\right|e^{-iHt}\\left|\\Psi (0)\\right>\\right|^2 =\\left| \\sum _{\\alpha } \\left|C_\\alpha ^{\\rm ini}\\right|^2 e^{-i E_{\\alpha } t} \\right|^2 \\nonumber \\\\&=&\\left| \\int dE e^{-iEt} \\rho _{\\rm ini}(E) \\right|^2,$ where $\\rho _{\\rm ini}(E)=\\sum _{\\alpha }\\left|C_\\alpha ^{\\rm ini}\\right|^2\\delta (E-E_\\alpha )$ is the energy distribution of the initial state.", "$\\rho _{\\rm ini}(E)$ is usually referred to as local density of states (LDOS) or strength function.", "The width $\\Gamma $ of this distribution depends on the number of states $|n\\rangle $ that are directly coupled with $\\left|\\Psi (0)\\right>$ , $\\Gamma ^2&=& \\sum _{\\alpha } \\left| C_\\alpha ^{\\rm ini} \\right|^2 E_{\\alpha }^2 - \\left( \\sum _{\\alpha } \\left| C_\\alpha ^{\\rm ini} \\right|^2 E_{\\alpha } \\right)^2 \\nonumber \\\\&=& \\langle \\Psi (0) |H H |\\Psi (0) \\rangle - \\langle \\Psi (0) |H|\\Psi (0) \\rangle ^2 \\nonumber \\\\&=& \\sum _{n } \\langle \\Psi (0) |H |n\\rangle \\langle n| H| \\Psi (0) \\rangle - \\langle \\Psi (0) |H |\\Psi (0) \\rangle ^2 \\nonumber \\\\&=& \\sum _{n \\ne {\\rm ini}} | \\langle n |H| \\Psi (0) \\rangle |^2.$ The survival probability is a quantity of great theoretical and experimental [50] relevance.", "It has been used in studies of the quantum speed limit [51], [52], onset of exponential [53], [54] and power-law [55], [56], [57], [58], [59] decays, quench dynamics [60], [61], [62], [63], [64], [65], ground-state and excited-state quantum phase transitions [66], [67], quantum scars [68], [69], multifractality in disordered systems [70], [71], [72], [73], and emergence of the correlation hole [74], [75], [76], [77], [78], [79], [80], [81], [82].", "The inverse participation ratio measures the degree of delocalization of a state in a certain basis [83], [84], [71].", "Here, we study a dynamical version of it [85], [86], [87], which accounts for the spreading in time of the initial many-body state in the basis of unperturbed many-body states $\\left|n\\right>$ .", "It is defined as ${\\rm {IPR}}(t)=\\sum _n\\left|\\left<n\\right|e^{-iHt}\\left|\\Psi (0)\\right>\\right|^4 .$ At $t=0$ , when $|\\Psi (0)\\rangle $ is one of the states $|n\\rangle $ , ${\\rm {IPR}}(t)=1$ .", "As $|\\Psi (0)\\rangle $ spreads into other states $|n\\rangle $ , ${\\rm {IPR}}(t)$ decays.", "For chaotic systems perturbed far from equilibrium, it reaches very small values.", "The spin autocorrelation function measures the proximity of a spin $k$ at time $t$ to its orientation at $t=0$ and it is averaged over all sites, $I(t)=\\frac{4}{L}\\sum _{k=1}^L \\left<\\Psi (0) \\right|S^z_k e^{iHt} S^z_k e^{-iHt}\\left|\\Psi (0) \\right>.$ This quantity is equivalent to the density imbalance between even and odd sites measured in experiments with cold atoms [41], as can be seen by mapping the spins into hardcore bosons.", "The self-averaging behavior of this quantity was studied in Refs.", "[14], [15].", "Here, we analyze also $|I(t)|$ and $I^2(t)$ .", "This is done because at long times, $I(t)$ can reach negative values and the oscillations between negative and positive values may complicate the analysis of self-averaging, which is avoided with the other two quantities.", "The connected spin-spin correlation function is given by $C(t)&=&\\frac{4}{L}\\sum _k\\left[\\left<\\Psi (t)\\right|S_k^zS_{k+1}^z\\left|\\Psi (t)\\right>\\right.\\\\&-&\\left.\\left<\\Psi (t)\\right|S_k^z\\left|\\Psi (t)\\right>\\left<\\Psi (t)\\right|S_{k+1}^z\\left|\\Psi (t)\\right>\\right]\\nonumber $ and is measured in experiments with ion traps [88]." ], [ "Self-Averaging and Timescales", "The results presented in this subsection have already appeared in Refs.", "[12], [14].", "The purpose of this summary is to serve as a reference for the discussions in the next sections.", "We show first the evolution of the mean survival probability.", "The various timescales involved in the relaxation process of this quantity are the ones used in the analysis of the distributions of all quantities in the next sections.", "We also describe here the time-dependence of the relative variance of the survival probability and of the inverse participation ratio, whose distributions are the subjects of Secs.", ", , and  .", "A quantity $O$ is self-averaging when its relative variance ${\\cal R}_{O}(t)= \\frac{\\sigma _{O}^2 (t) }{\\left<O(t)\\right>^2} = \\frac{\\left<O^2(t)\\right>-\\left<O(t)\\right>^2}{\\left<O(t)\\right>^2}$ decreases as the system size increases.", "The notation $\\left< \\cdot \\right>$ indicates in our case the average over disorder realizations and also initial states.", "We consider $0.01D$ initial states and at least $10^4/(0.01D)$ disorder realizations, so that each point for the curves of $\\left<O(t)\\right>$ and ${\\cal R}_{O}(t)$ is an average over $10^4$ data." ], [ "Survival probability", "The top panels of Fig.", "REF show the survival probability for the GOE model [Fig.", "REF  (a)] and the spin model [Fig.", "REF  (b)].", "The shape and bounds of the LDOS [Eq.", "(REF )] determine the initial decay of the survival probability.", "The LDOS for the GOE model is semicircular.", "The square of the Fourier transform of a semicircle gives $\\mathcal {J}^2_1(2 \\Gamma t)/(\\Gamma ^2 t^2)$ , where ${\\cal J}_1$ indicates the Bessel function of the first kind [89].", "This implies that after a very rapid initial decay, $\\langle P_S(t) \\rangle $ shows oscillations that decay according to a power law $\\propto t^{-3}$ [58], [59], [79], as seen in Fig.", "REF  (a).", "The LDOS for the spin model is Gaussian [60], [61], as found in many-body quantum systems with two-body couplings and perturbed far from equilibrium [35], [90], [91], [85], [92], [93].", "The square of the Fourier transform of a bounded Gaussian gives $\\exp (-\\Gamma ^2 t^2) {\\cal F}(t)/(4 {\\cal N}^2)$ , where ${\\cal F}(t)$ involves error functions and ${\\cal N}$ is a normalization constant (see the appendices in Refs.", "[59], [12]).", "This implies that after an initial Gaussian decay [60], [61], $\\langle P_S(t) \\rangle $ shows a power-law behavior $\\propto t^{-2}$ [58], [59], [94], as observed in Fig.", "REF  (b).", "The origin of the power-law decay of the survival probability in bounded spectra has been discussed at least since the 1950's  [55], [95], [96], [97], [98], [99] and more recently in Ref. [100].", "The experimental detection of algebraic decay at long times has been reported in Ref.", "[101], and evidence of slower relaxation for the density imbalance in the context of many-body localization of one- and two-dimensional quasiperiodic systems was presented in Refs.", "[102], [103].", "Figure: Evolution of the mean of the survival probability (a, b), of the relative variance of the survival probability (c, d), and of the relative variance of the inverse participation ratio (e, f) for the GOE model (left panels) and the chaotic disordered spin model (right panels).", "The time intervals for the fast initial decay, power-law behavior, correlation hole, and saturation are indicated in panels (a) and (b).", "The horizontal dashed line marks the saturation value of P S P_S for the largest size.", "System sizes: D=252,924,3432,12870D=252, 924, 3\\,432, 12\\,870 (L=10,12,14,16L=10,12, 14, 16).", "For the spin model, L=18L=18 is also shown.", "In all panels: 0.01D0.01D disorder realizations and 10 4 /(0.01D)10^4/(0.01D) initial states.The power-law decays in Figs.", "REF  (a) and REF  (b) persist up to a time denoted by $t_{\\rm {Th}}$  [12], where $\\langle P_S(t) \\rangle $ reaches its minimum value.", "Beyond this point, the survival probability increases until the dynamics saturates for $t>t_{\\rm {R}}$ , where $t_{\\rm {R}}$ is the relaxation time.", "At this point, $\\langle P_S (t>t_{\\rm {R}})\\rangle $ fluctuates around the infinite-time average $ \\langle \\sum _{\\alpha } \\left|C_\\alpha ^{\\rm ini}\\right|^4 \\rangle $ .", "The dip below the saturation point is known as correlation hole [74], [75], [76] and it appears only in systems where the eigenvalues are correlated, reflecting short- and long-range correlations [104].", "The four time intervals for the distinct behaviors of $\\langle P_S(t) \\rangle $ – fast initial decay, power-law behavior, correlation hole, and saturation – are indicated in Figs.", "REF  (a) and REF  (b).", "These are the timescales that we consider in the next sections to investigate the distributions of the survival probability and of the other quantities as well.", "In Figs.", "REF  (c) and REF  (d), we show the results for the relative variance ${\\cal R}_{P_S}(t)$ for different system sizes.", "The survival probability is non-self-averaging at any timescale, as shown analytically in Ref. [14].", "Initially, ${\\cal R}_{P_S}(t)$ grows with system size, while for $t>t_{\\rm {Th}}$ , it reaches a constant value, ${\\cal R}_{P_S}(t) \\sim 1$ .", "There is no noticeable difference between the value of ${\\cal R}_{P_S}(t) $ in the interval $[t_{\\rm {Th}}, t_{\\rm {R}}]$ and for $t>t_{\\rm {R}}$ ." ], [ "Inverse participation ratio", "Plots for the mean of the inverse participation ratio can be seen in Ref. [14].", "There are two different behaviors for $\\langle {\\rm {IPR}}(t) \\rangle $ at short times.", "The decay is initially very fast and then it either oscillates in the case of the GOE model or it slows down for the spin model.", "These two timescales coincide with the intervals for the fast decay and the power-law behavior of $\\langle P_S(t) \\rangle $ .", "Beyond this point, however, a correlation hole is not visible for $\\langle {\\rm {IPR}}(t) \\rangle $ .", "It exists, but it is extremely small [12] and, contrary to what we find for the survival probability, the ratio between the saturation point of $\\langle {\\rm {IPR}}(t) \\rangle $ and its minimum value at the correlation hole decreases as the system size increases.", "The evolution of the relative variance of $\\rm {IPR}$ is seen in Figs.", "REF  (e) and REF  (f).", "It shows that the inverse participation ratio is non-self-averaging at short times, which is understandable, since for small times,$\\langle {\\rm {IPR}}(t) \\rangle \\sim \\langle P_S^2(t) \\rangle $ .", "But for times $t>t_{\\rm Th}$ , the inverse participation ratio becomes “super” self-averaging, by which we mean that ${\\cal R}_{\\rm IPR}(t) \\propto 1/D$ instead of $\\propto 1/L$ ." ], [ "Distributions at short times", "In Fig.", "REF , we show the distributions of the survival probability [Figs.", "REF  (a) and REF  (b)] and of the inverse participation ratio [Figs.", "REF  (c) and REF  (d)] for the GOE model [Figs.", "REF  (a) and REF  (c)] and the spin model [Figs.", "REF  (b) and REF  (d)] at short times, $t < \\Gamma ^{-1}$ , when the decays of $\\langle P_S(t) \\rangle $ and $\\langle {\\rm IPR}(t) \\rangle $ are very fast.", "The distributions are similar for both quantities, but differ between the models.", "At short times, the main contribution for $\\langle {\\rm IPR}(t) \\rangle $ is the square of the survival probability, $\\langle {\\rm IPR}(t\\ll \\Gamma ^{-1}) \\rangle \\sim \\left|\\left< \\Psi (0) \\right| e^{-iHt}\\left|\\Psi (0)\\right>\\right|^4$ , which explains why the distributions for both quantities are so similar.", "Compare Fig.", "REF  (a) with Fig.", "REF  (c), and Fig.", "REF  (b) with Fig.", "REF  (d).", "Therefore, it suffices to describe below the distributions for the survival probability.", "Figure: Distributions of the survival probability (a, b) and inverse participation ratio (c, d) for the GOE (a, c) and the spin (b, d) model at very short times: t=0.01t=0.01 and t=0.5t=0.5, respectively.", "Solid line in (a) is the theoretical Gaussian distribution with mean from Eq.", "() and variance from Eq.", "(), and dashed line in (c) is the Gaussian with the numerical values for the mean and variance." ], [ "Survival probability", "At short times, the decay of the survival probability is controlled by the short-time expansion of $\\mathcal {J}^2_1(2 \\Gamma t)/(\\Gamma ^2 t^2)$ for the GOE model and of $\\exp (-\\Gamma ^2 t^2)$ for the spin model.", "The distribution of $P_S(t)$ at a fixed time $t< \\Gamma ^{-1}$ reflects then the distribution of the square of the width of the LDOS, $\\Gamma ^2$ , and its higher powers.", "For the GOE model, the expansion gives $\\frac{\\mathcal {J}^2_1(2 \\Gamma t)}{\\Gamma ^2 t^2}= 1 - \\Gamma ^2 t^2 + \\frac{5}{12}\\Gamma ^4 t^4 - \\frac{7}{72}\\Gamma ^6 t^6 + \\frac{7}{480} \\Gamma ^8 t^8 \\dots $ As we saw in Eq.", "(REF ), $\\Gamma ^2$ is the sum of the square of the off-diagonal elements contained in the row of the Hamiltonian matrix where the initial state lies.", "For the GOE model, this means the sum of the square of $D-1$ Gaussian random numbers with $\\langle H_{ij}\\rangle =0$ and $\\langle H_{ij}^2\\rangle =1/2$ , which gives a $\\chi ^2$ -distribution with $D-1$ degrees of freedom.", "This is approximately a Gaussian distribution with mean $\\mu _{\\Gamma ^2 }=(D-1)/2$ and variance $\\sigma _{\\Gamma ^2 }^2 =(D-1)/2$ .", "Using $g_n$ as a notation for the moments of $\\Gamma ^2$ , that is, $g_n= \\frac{1}{ \\sqrt{2 \\pi \\sigma _{\\Gamma ^2 }^2}}\\int (\\Gamma ^2)^n \\exp \\left[- \\frac{(\\Gamma ^2 -\\mu _{\\Gamma ^2 })^2}{2 \\sigma _{\\Gamma ^2 }^2}\\right]d\\Gamma ^2,$ and keeping terms up to 8th order in time we see that $\\langle P_S (t) \\rangle \\approx 1 - g_1 t^2 + \\frac{5}{12}g_2 t^4 - \\frac{7}{72}g_3 t^6 + \\frac{7}{480}g_4 t^8,$ and the variance $&& \\sigma _{P_S(t)}^2 \\\\&& =(g_2 - g_1^2) t^4- \\frac{5}{6} (g_3 - g_1 g_2) t^6 \\nonumber \\\\&& + \\left[ \\frac{25}{144} (g_4 - g_2^2) +\\frac{7}{36} (g_4 - g_1 g_3) \\right]t^8 \\nonumber \\\\&& - \\left[ \\frac{7}{240} (g_5 - g_1 g_4) + \\frac{35}{432} (g_5 - g_2 g_3) \\right] t^{10}\\nonumber \\\\&& +\\left[ \\frac{49}{5184} (g_6 \\!-\\!", "g_3^2) + \\frac{11}{3600} (g_6 \\!-\\!", "g_1 g_5) + \\frac{7}{576} (g_6 \\!-\\!", "g_2 g_4)\\right] t^{12} .", "\\nonumber $ For a fixed $t=0.01 $ and $D=12\\,870$ , $\\langle P_S (0.01) \\rangle \\sim 0.505$ and $\\sigma _{P_S(0.01)}^2 \\sim 2.1 \\times 10^{-5} $ , which are the values used in the Gaussian indicated with a solid line in Fig.", "REF  (a)." ], [ "Survival probability: Spin model", "For the spin model, the energy $E_{\\text{ini}}$ [Eq.", "(REF )] of the initial state depends on the disorder strength and on the number $n_p$ of neighboring pairs of up-spins as determined by the Ising interaction, $\\sum _k S_k^z S_{k+1}^z$ .", "Focusing only on the Ising interaction, one can see that it leads to $L/2$ energy bands that go from the band of lowest energy with no pairs of up-spins, which has only the two Néel states $|\\uparrow \\downarrow \\uparrow \\downarrow \\ldots \\rangle $ and $|\\downarrow \\uparrow \\downarrow \\uparrow \\ldots \\rangle $ , to the band of highest energy with $n_p=L/2-1$ neighboring pairs of up-spins, which has $L$ states [105].", "The number of states in a band grows as we approach the middle of the spectrum.", "The most populated band for chain sizes that are multiple of 4 is centered at energy zero, and for the chains of other even sizes, it is centered at $-1/2$ .", "The fragmented distribution in Fig.", "REF  (b) reflects the bands created by the Ising interaction.", "Each state in a band with $n_p$ pairs of neighboring up-spins couples with $(L-2n_p)$ other states, so according to Eq.", "(REF ), $\\Gamma ^2=(L-2n_p)/4$ .", "For the $L=16$ case shown in Fig.", "REF  (b), the states in the most populated band at energy zero has $n_p=4$ and $\\Gamma ^2=2$ , so $P_S(t<\\Gamma ^{-1}) \\sim \\exp (-\\Gamma ^2 t^2)$ gives $\\sim 0.61$ for $t=0.5$ , which is indeed the center of the highest peak in Fig.", "REF  (b).", "The two other highest peaks correspond to the Ising band at $-1$ with $n_p=3$ and $P_S(0.5)\\sim 0.54$ and the band at 1 with $n_p=5$ and $P_S(0.5) \\sim 0.69$ .", "As discussed in Ref.", "[14], both the survival probability and in the inverse participation ratio are non-self-averaging at short times.", "This can be understood from the expansion of the survival probability at the lowest order in $t$ , $P_S(t) \\sim 1 - \\Gamma ^2 t^2,$ which gives ${\\cal R}_{P_S}(t) &\\sim &\\frac{\\left< (1 - \\Gamma ^2 t^2)^2 \\right>-\\left<1 - \\Gamma ^2 t^2\\right>^2}{\\left<1 - \\Gamma ^2 t^2\\right>^2} \\nonumber \\\\&=& \\sigma _{\\Gamma ^2 }^2 t^4 .$ The lack of self-averaging happens because $\\sigma _{\\Gamma ^2 }^2 $ grows with $L$ for the spin model and with $D$ for the GOE model, having no relationship with the shape of the distributions." ], [ "Distributions after Saturation", "In Fig.", "REF , we show the distributions of $P_S(t)$ and ${\\rm IPR}(t)$ after the saturation of the dynamics, for a fixed time $t>t_{\\rm R}$ .", "In contrast with the behavior at short times, the distributions for both models are now similar, while they differ between quantities.", "In realistic chaotic systems, properties similar to those of random matrices manifest themselves at long times." ], [ "Survival probability", "The distribution of $P_S(t)$ for the GOE and the spin model for $t>t_{\\rm R}$ is exponential, as shown in Figs.", "REF  (a) and REF  (b).", "Since the mean and the dispersion of exponential distributions are equal, ${\\cal R}_{P_S}(t>t_{\\rm R}) \\sim 1$ , as indeed found numerically in Figs.", "REF  (c) and REF  (d).", "This justifies the lack of self-averaging of the survival probability for $t>t_{\\rm R}$ .", "Figure: Distributions of the survival probability (a, b) and inverse participation ratio (c, d) for the GOE model (a, c) at t=10 3 t=10^3 and for the spin model (b, d) at t=5×10 4 t=5\\times 10^4 .", "Solid lines in (a) are the exponential distribution with rate parameter D/3D/3, and in (c) they are the Gaussian distribution with mean and variance from Eqs.", "() and ().", "Dashed lines in (b) are the exponential distribution with rate parameter 1/〈∑ α |C α ini | 4 〉1/\\langle \\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4 \\rangle , and in (d) they are the Gaussian curve with the numerical values for 〈IPR(t)〉\\langle \\text{IPR} (t) \\rangle and σ IPR (t)\\sigma _{\\text{IPR}} (t) at t=5×10 4 t=5\\times 10^4.The rate parameter of an exponential distribution is the reciprocal of the mean.", "For the distribution of $P_S(t>t_{\\rm R})$ , the rate parameter is $1/\\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4$ .", "This can be understood by writing the survival probability as $P_S (t) = \\sum _{\\alpha < \\beta } 2 |C_{\\alpha }^{\\rm ini}|^2 |C_{\\beta }^{\\rm ini}|^2 \\cos \\left[ (E_{\\alpha } - E_{\\beta }) t \\right] + \\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4 .$ On average, the first term on the right hand side of the equation above cancels out, so $\\langle P_S(t>t_{\\text{R}}) \\rangle \\sim \\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4$ .", "The eigenstates of the GOE model are random vectors, so $C_{\\alpha }^{\\rm ini}$ 's are random numbers from a Gaussian distribution satisfying the constraint $\\sum _{\\alpha =1}^D |C_{\\alpha }^{\\rm ini}|^2 =1$ .", "Using ${\\cal P}(C) = \\sqrt{D/(2\\pi )} e^{-DC^2/2}$ for the components, we have $\\langle C \\rangle $ =0, $\\langle C^2 \\rangle =1/D$ , and $\\langle C^4 \\rangle =3/D^2$ , so $\\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4=\\sum _{\\alpha } (3/D^2) = 3/D$ .", "For the chaotic spin model, the eigenstates away from the edges of the spectrum are also approximately random vectors, so $\\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4$ is close to $3/D$ , although slightly larger.", "This discrepancy becomes particularly evident if one fits the numerical distribution in Fig.", "REF  (b) with a single parameter.", "The fact that we get a value slightly larger than $3/D$ indicates some remaining degree of correlations between the components of the initial state.", "A simple justification for the exponential shape of the distribution for $P_S(t)$ can be given by substituting $\\left| \\sum _{\\alpha } \\left|C_\\alpha ^{\\rm ini}\\right|^2 e^{-i E_{\\alpha } t} \\right|^2,$ with $\\frac{1}{D^2} \\left\\lbrace \\left[ \\sum _{\\alpha } \\cos (E_{\\alpha } t) \\right]^2 + \\left[ \\sum _{\\alpha } \\sin (E_{\\alpha } t) \\right]^2\\right\\rbrace .$ The sum of the cosines and the sum of the sines are Gaussian random variables, as discussed in Ref.", "[29] for full random matrices.", "The distribution of the sum of the square of two Gaussian random numbers is exponential, which explains the shape seen in Figs.", "REF  (a) and REF  (b).", "Notice, however, that this simplification gives $1/D$ as the mean value for $P_S(t)$ , which differs from the correct value by a factor of 3.", "Furthermore, we verified numerically that for $t>t_\\text{R}$ , the sum of the cosines and the sum of the sines are Gaussian random variables also when $E_{\\alpha }$ are random numbers from a Gaussian distribution, which indicates that for such long times, the correlations between the eigenvalues are not essential for the onset of the exponential shape of the distribution for $P_S(t)$ .", "This means that even for an integrable model with uncorrelated eigenvalues, the distribution of $P_S(t>t_{\\rm R})$ is exponential.", "The proper derivation of the exponential distribution for $P_S (t)$ involves the convolution of the distribution for the components of the initial state with the distribution for $e^{-i E_{\\alpha } t}$ , as done in Ref.", "[30] for random matrices.", "The result for $t>t_\\text{R}$ is ${\\cal P}(P_S) =\\frac{1}{\\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4} \\exp \\left[- \\frac{P_S(t) }{\\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4} \\right].$ The agreement between this theoretical curve and the numerical distribution of $P_S (t)$ for the GOE and also for the spin model is excellent, as seen in Figs.", "REF  (a) and REF  (b)." ], [ "Inverse participation ratio", "The distribution of the inverse participation ratio for the GOE and the spin model at a fixed time $t>t_{\\text{R}}$ is Gaussian, as evident in Figs.", "REF  (c) and REF  (d).", "Following the steps described in Ref.", "[30], it should be possible to formally derive the Gaussian distribution by doing the convolutions between the distributions for the components $C_{\\alpha }^n$ and $C_{\\alpha }^{\\rm ini} $ , which are nearly Gaussian random numbers, and for $e^{- E_{\\alpha } t}$ .", "Taking into account the sum over all basis vectors $|n\\rangle $ in ${\\rm IPR}(t) = \\sum _n \\left| \\sum _{\\alpha } C_{\\alpha }^n C_{\\alpha }^{\\rm ini} e^{- E_{\\alpha } t}\\right|^4,$ which is a large sum, one should arrive at the Gaussian shape.", "The mean of the distribution of $ {\\rm IPR}(t)$ is obtained by realizing that the only terms in $&& {\\rm IPR}(t) \\nonumber \\\\&& \\!\\!", "= \\sum _n \\!\\!\\!", "\\sum _{\\alpha , \\beta , \\gamma , \\delta } \\!\\!\\!", "C_{\\alpha }^n C_{\\alpha }^{\\rm ini}C_{\\beta }^n C_{\\beta }^{\\rm ini}C_{\\gamma }^n C_{\\gamma }^{\\rm ini}C_{\\delta }^n C_{\\delta }^{\\rm ini} e^{- (E_{\\alpha } - E_{\\beta } + E_{\\gamma } - E_{\\delta })t}\\nonumber $ that do not average out at long times are those where $\\alpha = \\beta $ , $\\gamma = \\delta $ , with $\\alpha \\ne \\delta $ ; $\\alpha = \\delta $ , $\\beta =\\gamma $ , with $\\alpha \\ne \\beta $ ; and $\\alpha = \\beta =\\gamma = \\delta $ , which gives $2 \\sum _{n} \\left( \\sum _{\\alpha } |C_{\\alpha }^{n}|^2 |C_{\\alpha }^{\\rm ini}|^2 \\right)^2 -\\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4 \\left( \\sum _{n} |C_{\\alpha }^{n}|^4 \\right).$ Since for the random matrices, $|C_{\\alpha }^{n}|^2 \\sim 1/D$ , we have $\\frac{2}{D} - \\frac{9}{D^2}.$ To compute the variance of the distribution, we need the dominant terms of $&& {\\rm IPR}^2 (t) \\nonumber \\\\&& =\\sum _{n} \\sum _{\\alpha , \\beta , \\gamma , \\delta } \\sum _{n^{\\prime }} \\sum _{\\alpha ^{\\prime }, \\beta ^{\\prime }, \\gamma ^{\\prime }, \\delta ^{\\prime }} \\nonumber \\\\&& \\times C_{\\alpha }^n C_{\\alpha }^{\\rm ini}C_{\\beta }^n C_{\\beta }^{\\rm ini}C_{\\gamma }^n C_{\\gamma }^{\\rm ini}C_{\\delta }^n C_{\\delta }^{\\rm ini}e^{- (E_{\\alpha } - E_{\\beta } + E_{\\gamma } - E_{\\delta })t} \\nonumber \\\\&& \\times C_{\\alpha ^{\\prime }}^{n^{\\prime }} C_{\\alpha ^{\\prime }}^{\\rm ini}C_{\\beta ^{\\prime }}^{n^{\\prime }} C_{\\beta ^{\\prime }}^{\\rm ini}C_{\\gamma ^{\\prime }}^{n^{\\prime }} C_{\\gamma ^{\\prime }}^{\\rm ini}C_{\\delta ^{\\prime }}^{n^{\\prime }} C_{\\delta ^{\\prime }}^{\\rm ini}e^{- (E_{\\alpha ^{\\prime }} - E_{\\beta ^{\\prime }} + E_{\\gamma ^{\\prime }} - E_{\\delta ^{\\prime }})t} .", "\\nonumber $ There are four terms similar to the one with $\\alpha =\\beta $ , $\\alpha ^{\\prime }=\\beta ^{\\prime }$ , $\\gamma = \\delta $ , $\\gamma ^{\\prime } = \\delta ^{\\prime }$ , which gives $4 \\sum _{n} \\left( \\sum _{\\alpha } |C_{\\alpha }^{n}|^2 |C_{\\alpha }^{\\rm ini}|^2 \\right)^2 - 4 \\sum _{\\alpha } |C_{\\alpha }^{\\rm ini}|^4 \\left( \\sum _{n} |C_{\\alpha }^{n}|^4 \\right) $ , and they cancel the dominant terms of $\\langle {\\rm IPR} (t>t_{\\rm R}) \\rangle ^2$ , so they do not contribute to the variance.", "But there are also four terms similar to the one with $\\alpha =\\delta $ , $\\alpha ^{\\prime }=\\delta ^{\\prime }$ , $\\beta =\\gamma $ , $\\beta ^{\\prime }=\\gamma ^{\\prime } $ , which for $n=n^{\\prime }$ gives $4 \\sum _{n} \\!\\!", "\\sum _{\\alpha , \\beta , \\gamma , \\delta } \\!\\!", "|C_{\\alpha }^n|^2 |C_{\\alpha }^{\\rm ini} |^2| C_{\\beta }^n |^2 |C_{\\beta }^{\\rm ini} |^2| C_{\\gamma }^n |^2 |C_{\\gamma }^{\\rm ini} |^2| C_{\\delta }^n |^2 |C_{\\delta }^{\\rm ini}|^2,$ so the variance of the distribution of ${\\rm IPR} (t)$ for a fixed $t>t_{\\rm R}$ is $\\sigma ^2_{\\rm IPR} \\sim \\frac{4}{D^3}.$ The Gaussian distribution with the mean from Eq.", "(REF ) and the variance from Eq.", "(REF ) matches very well the histogram for the GOE model in Fig.", "REF  (c).", "Furthermore, our numerical analysis of the distributions obtained for random matrices of different sizes shows that the skewness $\\rightarrow 0$ and the kurtosis $\\rightarrow 3$ as the dimension of the matrices increases, just as we would expect for a symmetric Gaussian distribution.", "For the spin model, the dashed line in Fig.", "REF  (d) is a Gaussian curve with the numerical values obtained for $\\langle \\text{IPR} (t) \\rangle $ and $\\sigma ^2_{\\text{IPR}} (t)$ for a fixed $t>t_{\\text{R}}$ .", "The mean and variance for this curve are slightly larger than the values in Eqs.", "(REF ) and (REF ), indicating again some degree of correlation between the components of the eigenstates of the realistic model.", "We might expect the results to approach those for the GOE model as $L$ increases, although our numerical analysis of the distributions for $L=10, 12, 14, 16, 18$ indicates that the skewness $\\rightarrow 1$ and the kurtosis $\\rightarrow 4$ as the system size increases.", "These values indicate a nonsymmetric distribution with heavier tails than a Gaussian distribution.", "The results for the mean and variance of ${\\rm IPR}(t)$ in Eqs.", "(REF ) and (REF ) make it clear that ${\\cal R}_{\\rm IPR}(t)$ decreases as $1/D$ and therefore, the inverse participation ratio becomes self-averaging at long times.", "The dependence of ${\\cal R}_{\\rm IPR}(t>t_{\\rm R})$ on the dimension of the Hamiltonian matrix instead of the system size $L$ is characteristic of interacting many-body quantum systems.", "This is related to the fact that the spread of the initial state takes place in the many-body Hilbert space instead of the real space." ], [ "Distributions at intermediate times", "As time grows from zero, the distributions for the various quantities studied here gradually change their shapes from those observed at short times (Fig.", "REF ) to those at long times (Fig.", "REF ).", "Illustrations of the distributions for $P_S(t)$ and $\\text{IPR}(t)$ for the spin model at intermediate times are shown in Fig.", "REF and discussed below." ], [ "Survival probability", "As time increases, the Gaussian distribution that $P_S(t)$ shows for the GOE model at short times becomes gradually more skewed until an exponential distribution emerges.", "For the spin model, the bands found in the distribution at short times [Fig.", "REF  (a)] broaden and simultaneously become more skewed [Fig.", "REF  (b)] until the distribution becomes exponential as well [Figs.", "REF  (d) and REF  (e)].", "Figure: Distributions of the survival probability (left) and inverse participation ratio (right) for the spin model at times indicated in the panels.", "Dashed lines in (d) and (e) are the exponential distribution with rate parameter given by 1/〈P S (t)〉1/\\langle P_S(t) \\rangle , and in (i) and (j) they are the Gaussian distribution with the mean and variance obtained numerically.Notice that for both models, the exponential distribution is seen even before $t_{\\rm R}$ .", "It starts taking shape already in the interval of the power-law decay [Fig.", "REF  (c)] and it becomes clearly exponential at $t\\sim t_{\\rm {Th}}$ when the spectral correlations get manifested in the dynamics and the correlation hole develops [Figs.", "REF  (d) and REF  (e)].", "For $t\\ge t_{\\rm Th}$ , the rate parameter of the exponential distribution is given by $1/\\langle P_S(t) \\rangle $ , as shown with a dashed line in Figs.", "REF  (d) and REF  (e).", "It is only for $t>t_{\\rm R}$ that $1/\\langle P_S(t) \\rangle \\sim 1/\\sum _{\\alpha } |C_{\\alpha }^{\\text{ini}}|^4$ and we recover the curve from Fig.", "REF  (b).", "The fact that we have an exponential distribution for $P_S(t)$ , with mean equal to the dispersion during the entire duration of the correlation hole $( t_{\\rm Th} \\le t \\le t_{\\rm R})$ implies that both $\\langle P_S(t) \\rangle $ and $\\sigma _{P_S}(t)$ decrease below their saturation values and that ${\\cal R}_{P_S}(t) \\sim 1$ for any time $t \\ge t_{\\rm Th}$ , as we indeed see numerically in Figs.", "REF  (c) and REF  (d).", "We notice that for an integrable model, where the correlation hole does not exist, one should not expect an exponential distribution for $P_S(t)$ before saturation, that is, for $t_{\\rm Th} \\le t \\le t_{\\rm R}$ .", "However, as discussed below Eq.", "(REF ), it should emerge for $t>t_{\\rm R}$ .", "The analysis of how the distribution of the survival probability may serve as an indicator of quantum chaos is a subject worth further studies." ], [ "Inverse participation ratio", "The distribution of ${\\rm IPR}(t)$ for the GOE model is throughout Gaussian, although some level of skewness and kurtosis larger than 3 are seen for times where $\\langle {\\rm IPR}(t) \\rangle $ oscillates, which corresponds to the power-law region of the survival probability.", "The width of the distribution depends on the dimension of the GOE matrix.", "At short times, the variance is related with the distribution of $\\Gamma ^2$ , so it increases as the matrix grows, while at long times, the variance is related with the distributions of the components $C_{\\alpha }^{n}$ , so it decreases as $D$ grows.", "We therefore have a Gaussian distribution that shrinks as time grows.", "The fact that the distribution is Gaussian at short and long times, but self-averaging holds at long times only, reiterates our claim that there is not a one to one correspondence between the shape of the distribution and the presence of self-averaging.", "The distribution for ${\\rm IPR}(t)$ for the spin model is hybrid.", "It starts similar to the distribution for the survival probability of the spin model [Figs.", "REF  (f) and REF  (g)], but it later acquires a shape equivalent to the distribution of ${\\rm IPR}(t)$ for the GOE model [Figs.", "REF  (i) and REF  (j)]." ], [ "Inferring self-averaging behaviors from distributions", "The main purpose of this section is to show how we can use the distribution of one quantity to assist determining the self-averaging behavior of another related quantity.", "But before that, we summarize the self-averaging behavior and the shapes of the distributions of the two experimental local quantities, the connected spin-spin correlation function and the spin autocorrelation function evolved under the spin model." ], [ "Distributions of local quantities", "Both quantities $C$ and $I$ are self-averaging up to the correlation hole.", "The connected spin-spin correlation function does not detect the hole and remains self-averaging at all times [14].", "In contrast, the spin autocorrelation function exhibits a correlation hole and stops being self-averaging beyond its minimum value.", "Similarly to the survival probability and the inverse participation ratio, the distributions of the values of the two local quantities at short times also reflect the distribution of $\\Gamma ^2$ .", "They exhibit fragmented structures similar to those in Figs.", "REF  (b) and REF  (d).", "However, the main difference between the global and local quantities at short times is that $P_S$ and ${\\rm IPR}$ are not self-averaging, while the local quantities are, because they have an explicit dependence on the system size in the denominator [14], $I(t) \\sim 1 - 4\\frac{\\Gamma ^2 t^2}{L} ,$ so ${\\cal R}_{I}(t) \\sim 16\\frac{\\sigma _{\\Gamma ^2 }^2 t^4}{L^2} ,$ which decreases with $L$ .", "As time grows, the distributions for the connected spin-spin correlation function and for the spin autocorrelation function progress in a way similar to the distribution for the inverse participation ratio shown in Fig.", "REF , that is, from a fragmented structure at short times to a Gaussian shape at long times.", "Even though both quantities show Gaussian distributions at long times, $C$ is strongly self-averaging, with ${\\cal R}_{\\rm C}(t>t_{\\rm R})$ decreasing exponentially as $L$ increases [14], while $I$ is non-self-averaging at long times.", "This is what we observe by studying system sizes with $L\\le 18$ , although one cannot rule out the possibility that this behavior might change for much larger $L$ 's.", "Based on the results at hand, the fact that both quantities exhibit a Gaussian distribution makes us conclude that there is no direct connection between self-averaging for $t>t_{\\text{R}}$ and a Gaussian distribution." ], [ "Semianalytical results for self-averaging", "The spin autocorrelation function can reach negative values at long times, which could suggest that ${\\cal R}_{I}(t)$ increases with $L$ just because $\\langle I(t) \\rangle $ gets very close to zero.", "This motivates us to study also the self-averaging behavior of $|I(t)|$ and $I^2(t)$ .", "In Figs.", "REF  (a) and REF  (b), we compare the results for the mean of the spin autocorrelation function and for the mean of its absolute value.", "The correlation hole is less evident for $\\langle |I(t)| \\rangle $ and for $\\langle I^2(t) \\rangle $ (this one is not shown) than for $\\langle I(t) \\rangle $ , but it is still present.", "For the three quantities, however, the ratio between the saturation point and the minimum of the hole decreases as $L$ increases, which contrasts with the survival probability, where the ratio is constant.", "Figure: Evolution of the mean of the spin autocorrelation function (a) and its relative variance (c), the mean of the absolute value of the spin autocorrelation function (b) and its relative variance (d), the relative variance of the square of the spin autocorrelation function (e), and the relative variance of the three quantities for t>t R t>t_{\\text{R}} vs. LL (f).", "All panels are for the chaotic disordered spin model.", "In panels (a, b) the four time intervals identified in the evolution of the survival probability are indicated, and the horizontal dashed line marks the saturation value for L=16L=16.", "In all panels: Average over 10 4 10^4 data and (f) includes also an average for 100 different instants of times.As expected for local quantities, the three observables are self-averaging at short times, with ${\\cal R}_{I, |I|,I^2}(t)$ decreasing as $L$ increases [see Figs.", "REF  (c), REF  (d) and REF  (e)].", "For $t \\sim t_{\\rm Th}$ , the curves cross.", "Beyond this point, for $t > t_{\\rm Th}$ , the behavior of ${\\cal R}_{I}(t)$ , ${\\cal R}_{|I|}(t)$ , and ${\\cal R}_{I^2}(t)$ differ.", "${\\cal R}_{I}(t)$ increases with system size, confirming the non-self-averaging behavior mentioned above, while the curves for ${\\cal R}_{I^2}(t)$ cross once again, recovering self-averaging at very long times.", "The results for ${\\cal R}_{|I|}(t)$ , however, are much less conclusive.", "Excluding $L=10$ , which is very small, the curves for $t> t_{\\text{Th}}$ seem to reach a nearly constant value independent of $L$ , as shown also in the scaling analysis in Fig.", "REF  (f).", "This suggests lack of self-averaging, but how can we better convinced of it with the system sizes that we have access to?", "Our strategy to circumvent the limited system sizes available is to use the numerical results for $I(t)$ to infer the self-averaging behavior of $|I(t)|$ , as we explain next.", "We verified that distribution for $|I(t>t_{\\rm R})|$ is a folded Gaussian, which further supports that the distribution for $I(t>t_{\\rm R})$ is indeed Gaussian.", "Both the standard deviation and the mean of $I(t)$ for $t>t_{\\rm R}$ decrease as the system size increases.", "The exponents $s$ and $m$ in $\\sigma _I(t>t_{\\rm R}) \\propto L^{-s}$ and $\\langle I(t>t_{\\rm R}) \\rangle \\propto L^{-m}$ can be obtained numerically.", "We find that $m>s$ .", "With this information, we can compute the mean and the variance of the folded Gaussian distribution for $|I(t)|$ using $\\langle |I| \\rangle &=& \\sqrt{\\frac{2}{\\pi }} \\,\\sigma _I \\exp \\left( -\\frac{\\langle I \\rangle ^2}{2 \\sigma ^2_I} \\right)+ \\langle I \\rangle \\,\\text{erf} \\left( \\frac{\\langle I \\rangle }{\\sigma _I} \\right),\\nonumber \\\\\\sigma ^2_{|I|} &=& \\langle I \\rangle ^2+\\sigma ^2_{I} - \\langle |I| \\rangle ^2.\\nonumber $ For large $L$ , we find that $\\langle |I| \\rangle &\\rightarrow & L^{-s} \\sqrt{\\frac{2}{\\pi }} ,\\\\\\sigma ^2_{|I|} &\\rightarrow &L^{-2s} \\left( 1-\\frac{2}{\\pi } \\right) ,$ which implies that the relative variance goes asymptotically to a constant, ${\\cal R}_{| I |}(t>t_{\\rm R}) \\rightarrow \\frac{\\pi - 2 }{2} \\sim 0.57.$ This value is indeed very close to what we have in Fig.", "REF  (f), but the semianalytical strategy described above provides a much stronger evidence that $| I (t)|$ is non-self-averaging at long times than what we can conclude from the numerical results in Fig.", "REF  (f)." ], [ "Conclusions", "We investigated the distributions over disorder realizations of different quantities and at various timescales of the evolution of a realistic chaotic spin model, from very short times up to equilibration.", "We compared these distributions with the quantities' self-averaging properties.", "The distributions for the global quantities — the survival probability and the inverse participation ratio — were contrasted also with those for the GOE model.", "The results for the two models are comparable at long times, but not at short times.", "At long times, the distribution of the survival probability for the GOE and for the chaotic spin model is exponential, which accounts for the lack of self-averaging of this quantity.", "The exponential shape emerges as soon as the dynamics detect the spectral correlations typical of chaotic systems.", "At long times, the distribution of the inverse participation ratio and also of the local quantities — the spin-spin correlation function and the spin autocorrelation function — are Gaussian.", "The fact that the first two are self-averaging, while the spin autocorrelation function is not, demonstrates that there is no direct relationship between the presence of self-averaging and the onset of a Gaussian distribution.", "We also studied the absolute value and the square of the spin autocorrelation function, $|I(t)|$ and $I^2(t)$ .", "The evolution of their mean values shows features similar to those observed for $\\langle I(t) \\rangle $ , but their self-averaging behaviors differ.", "Based on the system sizes available, we conclude that at long times the spin autocorrelation function is non-self-averaging, while $I^2(t)$ is.", "The numerical scaling analysis of the relative variance of $|I(t)|$ is less conclusive.", "A main result of this work is to show that knowledge of the distribution of one quantity may be used to uncover the self-averaging behavior of another related quantity.", "This is what we achieved using $I(t)$ and $|I(t)|$ as an example.", "Starting with the Gaussian distribution and non-self-averaging behavior of $I(t)$ at long times, we showed semianalytically that the relative variance of $|I(t)|$ for times $t>t_{\\rm {R}}$ goes asymptotically to a constant as $L$ increases, concluding in this way that $|I(t)|$ is non-self-averaging at long times.", "This strategy circumvents the limitations of the numerical scaling analysis, for which few system sizes can be accessed.", "We are grateful to Mauro Schiulaz for various discussions during the beginning of this project.", "E.J.T.-H. and I.V.-F. acknowledge funding from VIEP-BUAP (Grant Nos.", "MEBJ-EXC19-G and No.", "LUAGEXC19-G), Mexico.", "They are also grateful to LNS-BUAP for allowing use of their supercomputing facility.", "L.F.S.", "is supported by the NSF Grant No.", "DMR-1936006." ] ]

[ [ "HetPipe: Enabling Large DNN Training on (Whimpy) Heterogeneous GPU\n Clusters through Integration of Pipelined Model Parallelism and Data\n Parallelism" ], [ "Abstract Deep Neural Network (DNN) models have continuously been growing in size in order to improve the accuracy and quality of the models.", "Moreover, for training of large DNN models, the use of heterogeneous GPUs is inevitable due to the short release cycle of new GPU architectures.", "In this paper, we investigate how to enable training of large DNN models on a heterogeneous GPU cluster that possibly includes whimpy GPUs that, as a standalone, could not be used for training.", "We present a DNN training system, HetPipe (Heterogeneous Pipeline), that integrates pipelined model parallelism (PMP) with data parallelism (DP).", "In HetPipe, a group of multiple GPUs, called a virtual worker, processes minibatches in a pipelined manner, and multiple such virtual workers employ data parallelism for higher performance.", "We also propose a novel parameter synchronization model, which we refer to as Wave Synchronous Parallel (WSP) to accommodate both PMP and DP for virtual workers, and provide convergence proof of WSP.", "Our experimental results on a given heterogeneous setting show that with HetPipe, DNN models converge up to 49% faster compared to the state-of-the-art DP technique." ], [ "Introduction", "We have, in our facilities, four systems each with a different set of GPUs.", "Each, at the time of purchase, was (close to) state-of-the-art affordable with what budget we could muster.", "With technology advancing in such rapid pace, these systems have become outdated.", "Furthermore, the world is requiring us to run larger and larger Deep Neural Networks (DNNs) models.", "What we have is a bunch of (now) old technologies, individually unable to run these large models and funds depleted, unable to purchase the high-priced state-of-the-art systems.", "Our boss is asking what happened to those machines bought in the past years, why those cannot be used, unaware that this war is about purchasing power.", "Except for those exceptionals (you know who you are), this is a typical scenario we are faced with.", "Deep Neural Networks have been popularly used to solve various problems such as image classification [14], [26], speech recognition [15], topic modeling [3], and text processing [8].", "The size of DNN models (i.e., the number of parameters) have continuously been increasing in order to improve the accuracy and quality of models and to deal with complex features of data [17], [42], [48], [49].", "The size of input data and batches used for training have also increased to achieve higher accuracy and throughput [17], [23].", "For training large DNN models, data parallelism [4], [28], [45], [29], which employs multiple workers using parameter servers or AllReduce communication, and model parallelism [10], [25], [27], which divides the network layers of a DNN model into multiple partitions and assigns each partition to a different GPU, have commonly been leveraged.", "Furthermore, to mitigate the critical issue of low GPU utilization of naive model parallelism, pipelined model parallelism, where minibatches are continuously fed to the GPUs one after the other and processed in a pipelined manner, has recently been proposed [34], [17].", "For training DNN models, the use of GPU clusters is now commonplace.", "In such an environment, the use of heterogeneous GPUs is inevitable due to the short release cycle of new GPU architectures [21].", "Moreover, several types of GPUs targeted for high-end servers, workstations, and desktops are being released for purchase [35], [36], [37], [38].", "Table REF shows the hardware specifications for four different types of GPUs that we have purchased in our institution in the short span of the last three years, with each type determined by the year of purchase and the expenses available at the time.", "Due to their cost-effectiveness, less expensive GPUs targeted for desktops and workstations, rather than high-end servers are also commonly used for machine learning training, especially for small and medium size clusters [12], [19], [44], [50], [51], [53].", "Due to the same reason, spot instances with different types of GPUs that are offered by cloud service providers are being used [2], [33], [21].", "Table: Heterogeneous GPUsThere are benefits to enabling DNN training with heterogeneous resources.", "First, it allows for large model training with lower-class GPUs.", "While unable to train individually due to their limited resources, aggregated together, they may be used for training.", "These GPUs, which likely would have been retired, become usable, possibly used to create (virtual) workers that show similar performance as high-class GPUs.", "Second, low-class GPUs can be used to improve the performance of even high-class GPUs by incrementally adding on the resources of the (old) lower class systems to the (new) high-class systems.", "We call a group of aggregated GPUs that could satisfy the resource constraint and be used for training a virtual worker.", "Internally, such a virtual worker could leverage pipelined model parallelism (PMP) to process a minibatch, while externally, a number of virtual workers could leverage data parallelism (DP) for higher performance.", "In this paper, we explore the integration of PMP and DP to maximize the parallelism of DNN model training.", "In particular, we investigate a DNN model training system, which employs both PMP and DP, for a heterogeneous GPU cluster that possibly includes whimpy GPUs that, as a standalone, could not be used for training large models.", "There are numerous technical challenges that need to be overcome to realize a truly ideal solution of PMP and DP based DNN training systems for heterogeneous GPU clusters: How are the heterogeneous GPUs to be divided and allocated into a virtual worker?", "How do we reduce virtual worker stragglers when we consider DP?", "How do we partition the model to maximize the performance of PMP using heterogeneous GPUs?", "How are the weights synchronized in this setting?", "That is, what version of parameters is used for a next minibatch while previous minibatches are still executing in a pipelined manner within each virtual worker?", "How do multiple virtual workers synchronize their parameters?", "Can we guarantee convergence?", "While DP [4], [28], [45], [29], PMP [17], [34], and heterogeneity [22], [21], [30] for training have been considered separately, to the best of our knowledge, this is the first paper that tackles these issues together in attempting to answer some of the aforementioned questions.", "In this work, we design a DNN training system, HetPipe (Heterogeneous Pipeline), that integrates PMP of a virtual worker, which is composed of multiple (possibly whimpy) heterogeneous GPUs, with DP of virtual workers using parameter servers to enable and also speed up training of large models.", "HetPipe can aggregate heterogeneous resources from multiple GPUs to form a virtual worker such that the performance of each virtual worker is similar to each other, reducing the straggler problem.", "For HetPipe, we propose a novel parameter synchronization model, which we refer to as Wave Synchronous Parallel (WSP).", "WSP is adapted from the Stale Synchronous Parallel (SSP) model [16] to accommodate both PMP and DP for virtual workers composed of heterogeneous GPUs.", "We also prove the convergence of WSP.", "Note that while HetPipe would work in a homogeneous GPU cluster, with the rapid turnaround of newer GPU architectures, it is more likely that one will end up with a cluster of heterogeneous GPUs.", "This is the environment that we target.", "We implement HetPipe by modifying TensorFlow, a commonly used machine learning training system.", "We evaluate the performance of HetPipe for two DNN models using a heterogeneous GPU cluster composed of four different types of GPUs.", "Our experimental results demonstrate that the performance of HetPipe is better than that of the state-of-the-art DP via Horovod [45] that uses AllReduce communication [40].", "Compared to Horovod, the convergence of VGG-19 with a large parameter set to a desired accuracy becomes 49% faster, and that of ResNet-152 which is too big to be loaded in four whimpy GPUs in our cluster becomes 39% faster by using all the GPUs (including whimpy ones).", "Strategies to leverage PMP have been explored in previous studies [6], [34], [17], [24].", "Compared to these, our study makes forward strides in three aspects.", "First, we generalize PMP of a virtual worker to be used together with DP of virtual workers, increasing the parallelism of DNN model training.", "Consequently, this results in speeding up training.", "Second, we consider a heterogeneous GPU cluster, which allows the use of GPUs, which otherwise, could not be used for training.", "Finally, we present a parameter synchronization model that guarantees convergence, of which we provide a proof.", "We provide a more in-depth comparative discussion on these studies in Section REF ." ], [ "DNN Training", "The goal of training of a DNN model composed of multiple layers is to find the parameters (or weights) $w$ of the model that minimizes the sum of a loss function for the training dataset that consists of training samples and their labels.", "In a popularly used training method, stochastic gradient descent (SGD), it computes the weight updates, i.e., gradients on a subset of training samples, called a minibatch, and updates weights $w$ .", "The training process consists of a forward pass and then a backward pass.", "In the forward pass, the model first predicts the label for each of the samples in a minibatch.", "Each layer computes activations for the next layer using the given input data and the current parameters.", "Finally, the last layer of the model computes loss based on the predicted and actual labels.", "In the backward pass, the loss is backpropagated over all the layers of the model where each layer computes gradients using the gradients computed by the upper layer and activations previously computed in the forward pass." ], [ "Data Parallelism", " Data parallelism (DP) utilizes multiple workers to speed up training of a DNN model.", "It divides the training dataset into subsets and assigns each worker a different subset.", "Each worker has a replica of the DNN model and processes each minibatch in the subset, thereby computing the weight updates.", "Therefore, if a DNN model cannot be loaded into the memory of a single GPU, DP cannot be used.", "Among the multiple workers, the parameters are synchronized using parameter servers [28] or AllReduce communications [45], [29].", "For Bulk Synchronous Parallel (BSP) [32], [1], each worker must wait for all other workers to finish the current minibatch $p$ before it starts to process the next minibatch $p+1$ so that it can use an updated version of the weights for minibatch $p+1$ .", "For Asynchronous Parallel (ASP) [43], [1], each worker need not wait for other workers to finish minibatch $p$ , possibly using a stale version of the weights.", "With BSP, which is possible for both the parameter servers and AllReduce communications, the system may suffer from high synchronization overhead, especially in a heterogeneous GPU cluster where each worker with a different GPU provides different training performance [30].", "On the other hand, while ASP, which is possible for the parameter servers, has no synchronization overhead, it is known that ASP does not ensure convergence [43], [52].", "Figure: Pipeline execution of minibatches where M p,k M_{p,k} indicates the execution of a minibatch pp in partition kk, which is executed in GPU k _k and the yellow and green colors indicate the forward and backward passes, respectively.A method that takes the middle ground between BSP and ASP is Stale Synchronous Parallel (SSP) [16].", "With SSP, each worker is allowed to proceed the training of minibatches using a stale version of the weights that may not reflect the most recent updates computed by other workers.", "Thus, workers need not synchronize with other workers whenever it finishes the processing of a minibatch.", "As such, parameter staleness can occur.", "However, this staleness is bounded as defined by the user and referred to as the staleness threshold.", "As SSP is beneficial when worker performance is varied, it has been explored especially in the context of heterogeneous systems [21].", "In SSP, each worker periodically pushes the weight updates to the parameter server.", "This synchronization interval is called a clock.", "Thus, each worker increases its local clock by one for every iteration, which is the training period of a minibatch.", "For a given staleness threshold $s$ where $s \\ge 0$ , each worker with clock $c$ is allowed to use a stale version of the weights, which includes all the updates from iteration 0 to $c-s-1$ and, possibly, more recent updates past iteration $c-s-1$ .", "That is, a worker can continue training of the next minibatch with parameters whose updates may be missing from up to the $s$ most recent minibatches." ], [ "Model Parallelism and Pipeline Execution", " Model parallelism (MP) is typically exploited for large DNN models that are too large to be loaded into memory of a single GPU.", "In particular, a DNN model composed of multiple layers is divided into $k$ partitions and each partition is assigned to a different GPU.", "Each GPU executes both the forward and backward passes for the layers of the assigned partition.", "Note that it is important to execute the forward and backward passes of a partition on the same GPU as the activation result computed for the minibatch during the forward pass needs to be kept in the GPU memory until the backward pass of the same minibatch for efficient convergence, as similarly discussed by Narayanan and others [34].", "Otherwise, considerable extra overhead will incur for managing the activation through either recomputation or memory management.", "In the basic form of MP, $k$ GPUs, individually, act as one virtual worker to process a minibatch as follows: For each minibatch, execution of the forward pass starts from GPU$_1$ up to GPU$_k$ .", "When each GPU$_i$ , where $1 \\le i < k$ , completes the forward pass of the assigned partition, it sends the computed activations of only the last layer in its partition to GPU$_{i+1}$ .", "Once GPU$_k$ finishes the forward pass of its partition, the backward pass of the minibatch is executed from GPU$_k$ down to GPU$_1$ .", "When each GPU$_{i^{\\prime }}$ , where $1 < i^{\\prime } \\le k$ , finishes the backward pass, it sends the computed local gradient of only the first layer in its assigned partition to GPU$_{i^{\\prime }-1}$ .", "This basic form of MP results in low GPU utilization as only one GPU is actively executing either the forward or backward pass.", "Nonetheless, MP allows execution of large DNN models that are too large for a single GPU.", "To improve utilization of the GPUs in a virtual worker, minibatches can be processed in a pipelined manner.", "The subsequent minibatches are fed into the first GPU in MP (i.e., GPU$_1$ ) one by one once the GPU completes the processing of the previous minibatch.", "This allows for multiple GPUs to simultaneously execute either the forward or backward pass of their assigned layers for different minibatches.", "This is referred to as Pipelined Model Parallelism (PMP).", "Table: Comparison of HetPipe with GPipe and PipeDreamThis PMP strategy has been investigated in previous studies [34], [17].", "PipeDream exploits PMP of a single virtual worker to avoid the parameter communication overhead of DP [34].", "Considering only homogeneous GPUs, when PipeDream partitions a model into stages to maximize pipeline performance, it does not take into account the memory requirement of each stage.", "Thus, PipeDream processes a limited number of minibatches, which is large enough to saturate the pipeline, to reduce memory overhead.", "PipeDream also provides a form of DP, but it considers DP within a virtual worker to speed up the execution of lagging layers.", "No proof of single pipeline convergence is provided in PipeDream.", "GPipe is a scheme that leverages PMP of a single virtual worker to support large DNN models, also in a homogeneous GPU cluster [17].", "In GPipe, a minibatch is divided into multiple microbatches that are injected into the pipeline.", "Using the same weights, GPipe executes the forward passes for all the microbatches, and then executes the backward passes for them.", "When the backward pass of the last microbatch is done, it updates the weights all together for the minibatch.", "GPipe incurs frequent pipeline flushes, possibly resulting in low GPU utilization [34].", "In GPipe, DP of multiple virtual workers can be done using existing synchronization schemes like BSP as a virtual worker processes one minibatch at a time.", "GPipe saves on GPU memory by recomputing the activations again in the backward pass instead of keeping the activations computed in the forward pass in memory.", "We do not use this optimization though there are no fundamental reasons forbidding it.", "A comparison of HetPipe with previous studies is given in Table REF ." ], [ "System Overview", "The system that we propose focuses on training a large DNN model in a heterogeneous GPU cluster composed of various types of GPUs that have different computation capability and memory capacity.", "Except for the exceptional few who always have the luxury to be provided with the most advanced systems, most DNN practitioners will inevitably find themselves in this type of environment as systems evolve and the demand for larger DNN models continues.", "In such settings, for some types of GPUs in the cluster, the DNN model of interest may be too large to be loaded into the memory of a single GPU.", "The system that we propose in this paper leverages both pipelined model parallelism (PMP) and data parallelism (DP) to enable training of such large DNN models and, in the process, enhance performance as well as the utilization of the heterogeneous GPU resources of the cluster.", "Figure REF shows the architecture of the proposed cluster system composed of $H$ nodes.", "Each node comprises a homogeneous set of GPUs, but the GPUs (and memory capacity) of the nodes themselves can be heterogeneous.", "Two key novelties exist in this architecture.", "First, DP is supported through a notion of a virtual worker (VW), which consists of $k$ , possibly heterogeneous, GPUs, and encapsulates the notion of a worker in typical DNN systems.", "That is, a virtual worker is used to train the DNN model.", "In Figure REF , note that there are $N$ virtual workers with 4 GPUs each, that is, $k= 4$ , and that the GPUs comprising the virtual worker may be different for each virtual worker.", "While in this paper we consider $k$ to be constant for each virtual worker, our design does not restrain it to be so; this is simply a choice we make for simplicity.", "The key aspect here is that a virtual worker allows DP by aggregating GPUs possibly even when individual GPUs may be resource limited.", "Figure: System architecture (VW: Virtual Worker)The second novelty is that each virtual worker processes each minibatch based on model parallelism, in a pipelined manner, to fully utilize the GPU resources, as shown in Figure REF , to accommodate large DNN models.", "While PMP has been proposed before (which we compare in Section REF ), to the best of our knowledge, we are the first to present PMP in a heterogeneous setting.", "We refer to our system as HetPipe as it is heterogeneous, in GPUs, across and, possibly, within virtual workers and makes use of pipelining in virtual workers for resource efficiency.", "To train DNN models based on pipelined model parallelism in virtual workers, the resource allocator first assigns $k$ GPUs to each virtual worker based on a resource allocation policy (which will be discussed in Section ).", "Note that for allocating the heterogeneous GPUs to the virtual workers, the resource allocation policy must consider several factors such as the performance of individual GPUs as well as the communication overhead caused by sending activations and gradients within a virtual worker, and synchronizing the weights among the virtual workers and the parameter server.", "Then, for the given DNN model and allocated $k$ GPUs, the model partitioner divides the model into $k$ partitions for the virtual worker such that the performance of the pipeline executed in the virtual worker can be maximized.", "As any typical DP, multiple virtual workers must periodically synchronize the global parameters via parameter servers or AllReduce communication; in HetPipe, parameter servers are used to maintain the global weights.", "Each virtual worker has a local copy of the global weights and periodically synchronizes the weights with the parameter server.", "Evidently, when managing the weights within a virtual worker and across virtual workers, two types of staleness, local staleness and global staleness, need to be permitted to improve the performance of DNN training.", "Local staleness refers to staleness within a virtual worker.", "As each virtual worker processes minibatches in a pipelined manner, there are multiple minibatches that are being processed in parallel.", "Thus, staleness is inevitable as weights seen by a minibatch may not reflect the updates of all of its previous minibatches.", "Global staleness, on the other hand, is similar to the staleness notion introduced by Ho et al. [16].", "That is, the system needs to reduce communication overhead between the parameter server and (virtual) workers, and, in our case, also mitigate the synchronization overhead caused by possibly heterogeneous virtual workers.", "Therefore, similarly to SSP [16], each virtual worker should be allowed to proceed training without querying the global weights for every minibatch, unless its local copy is so old such that there are too many missing recent updates made by other virtual workers.", "Note that such staleness condition is set by the user [16].", "For our system, we propose the Wave Synchronous Parallel (WSP) model to synchronize the weights.", "A wave is a sequence of minibatches that are processed concurrently in a virtual worker.", "Let the number of minibatches in a wave be $N_m$ .", "Within a wave, processing of the $i$ -th minibatch is allowed to proceed without waiting for the preceding minibatchs $i^{\\prime }$ to be completed, where $1 < i \\le N_m$ and $1 \\le i^{\\prime } < i$ .", "As the virtual worker does not enforce the updates even from the first minibatch in a wave to be reflected in the weights used by the last minibatch, the local staleness threshold in WSP is $N_m - 1$ .", "Moreover, each virtual worker only pushes the aggregated updates from all the minibatches in a wave, instead of for every minibatch, to the parameter server.", "This results in considerable reduction in communication overhead.", "As it is important that the results generated through our proposed system configuration are correct [16], [21], [54], we show the convergence of our methodology in Section .", "Note that HetPipe uses parameter servers, which may incur synchronization and communication overhead.", "However, HetPipe mitigates such overhead by permitting global staleness among virtual workers and executing the pipeline in each virtual worker such that it continues to process minibatches that have already been injected while waiting for the parameter update.", "We believe HetPipe can be further optimized by taking decentralized approaches, but leave this for future work." ], [ "Pipelined Model Parallelism Within a VW", "Number of Minibatches in the Pipeline: In our system, each virtual worker processes up to $N_m$ minibatches concurrently in a pipeline manner so that the executions of the minibatches can overlap.", "Given a DNN model and $k$ GPUs, the maximum number of minibatches executed concurrently in the virtual worker, $Max_m$ , is basically determined by the memory requirement for training the model.", "For a model that requires a huge amount of memory for output activations and weights, $Max_m$ may be less than $k$ .", "Note that in such cases, the utilization of each GPU is unlikely to be high.", "$N_m$ , the actual number of minibatches in the pipeline will be $N_m \\le Max_m$ and basically determined by considering the throughput of the pipeline.", "Note that $N_m$ must be the same in every virtual worker, and thus, $N_m$ is set to the minimum $Max_m$ among all the virtual workers.", "$N_m$ will affect the local staleness that we discuss later in this section.", "Model Partitioning: To train a DNN model, a set of $k$ GPUs is allocated to a virtual worker by a resource allocation policy, which we discuss in Section .", "For now, let us assume that $k$ , the number of possibly heterogeneous GPUs, and $N_m$ are given.", "Then, a partitioning algorithm is employed to divide multiple layers of the model into $k$ partitions, assigning them to the $k$ different GPUs.", "The goal of the partitioning algorithm is to maximize the performance of the pipeline, while satisfying the memory requirement of each partition to process $N_m$ minibatches.", "In particular, in this study, for memory, we consider the fact that the actual memory requirement will vary depending on the stage of the pipeline that the GPU is used for.", "For example, contrast GPU$_4$ and GPU$_1$ in Figure REF .", "GPU$_4$ , the GPU that handles the last stage of the pipeline, handles only one minibatch at a time and is immediately done with the minibatch as exemplified by the yellow (forward pass) and green (backward pass) M$_{i,4}$ pairs for $i = 1, 2, ...$ , that are side-by-side.", "In contrast, for GPU$_1$ , the yellow and green M$_{i,1}$ pairs are far apart, meaning that the forward pass M$_{i,1}$ needs to hold up memory until the backward pass M$_{i,1}$ is finished with its execution.", "Thus, with GPU$_1$ , the memory requirement is high as it needs to hold on to the results of the forward pass for all stages of the pipeline.", "This variance in memory requirement is considered in partitioning the layers.", "Execution time must also be considered when partitioning the layers.", "To do so, we calculate the execution time of a partition to be the sum of the computation time of all the layers in the partition and the communication time needed for receiving the activations (in the forward pass) and local gradients (in the backward pass).", "Our partitioning algorithm attempts to minimize the maximum execution time of the partitions within the bounds of satisfying the memory requirement.", "Partition Scheduling: Once the partition is set, the partitions need to be scheduled for each of the GPUs.", "Each GPU$_q$ responsible for partition $q$ may have multiple forward pass and backward pass tasks to schedule at a time.", "Each GPU schedules a task by enforcing the following conditions: A forward pass task for a minibatch $p$ will be executed only after a forward pass task for every minibatch $p^{\\prime }$ is done where $1 \\le p^{\\prime } < p$ .", "Similarly, a backward pass task for a minibatch $p$ will be executed only after a backward pass task for every minibatch $p^{\\prime }$ is done where $1 \\le p^{\\prime } < p$ .", "Among multiple forward and backward pass tasks, a FIFO scheduling policy is used.", "Note that in the last partition, for a minibatch, processing a forward pass immediately followed by a backward pass is executed as a single task.", "Considering Staleness: Given the description of pipelining, the question of staleness of weights used needs to be considered.", "That is, as a minibatch is scheduled, it may be that the layers are not using the most up-to-date weights.", "For example, in Figure REF , when the forward pass M$_{2,1}$ , the second minibatch, begins to be processed, it must use stale weights as the first minibatch has not completed and hence, the changes in the weights due to the first minibatch have not yet been appropriately reflected, which is in contrast with typical processing where minibatches are processed one at a time.", "We now discuss how this staleness issue is considered.", "Let local staleness be the maximum number of missing updates from the most recent minibatches that is allowed for a minibatch to proceed in a virtual worker.", "As training with $N_m$ minibatches can proceed in parallel in a virtual worker, the local staleness threshold, $s_{local}$ , is determined as $N_m -1$ , where $1 \\le N_m \\le Max_m$ .", "If $N_m =1$ , the behavior is exactly the same as naive model parallelism.", "Larger $N_m$ improves the performance (i.e., throughput) of the pipeline as a larger number of concurrent minibatches are executed, but local staleness increases, possibly affecting the convergence of training.", "In a real setting, typically, $N_m$ will not be large enough to affect convergence as it will be bounded by the total amount of GPU memory of a virtual worker.", "Such local staleness also exists in PipeDream [34].", "As PipeDream basically employs weight stashing that uses the latest version of weights available on each partition to execute the forward pass of a minibatch, a different version of weights is used across partitions for the same minibatch.", "Unfortunately, PipeDream only shows empirical evidence of convergence when weight stashing is used.", "Note that PipeDream also discusses vertical sync, which is similar to HetPipe, but it excludes vertical sync in its evaluations [34].", "Now let $w_p$ be the weights used by minibatch $p$ .", "Then, initially, we can assume that $w_0$ , the initial version of weights, is given to the virtual worker.", "Then, the first ($s_{local}+1$ ) minibatches are processed in a pipelined manner with $w_0 = w_1 = \\cdots = w_{s_{local}} = w_{s_{local}+1}$ .", "To accommodate staleness in our system, when processing of minibatch $p$ completes, the virtual worker updates the local version of the weights, $w_{local}$ as $w_{local}= w_{local}+ u_p$ , where $u_p$ is the updates computed by processing minibatch $p$ .", "When the virtual worker starts to process a new minibatch, it makes use of the lastest value of $w_{local}$ without waiting for the other minibatches to update their weights.", "For example, once the virtual worker is done for minibatch 1 and updates $w_{local}$ with $u_1$ , it will start to process minibatch $s_{local}+2$ by using the updated weights without waiting for minibatches 2 up to $s_{local}+1$ to be completed.", "Similarly, when the virtual worker is done with minibatch $s_{local}+1$ and updates $w_{local}$ with $u_{s_{local}+1}$ , it will start to process minibatch $2 \\times (s_{local}+1)$ without waiting for the previous most recent $s_{local}$ minibatches to be completed.", "Therefore, except for the initial minibatches 1 to $s_{local}+1$ , for minibatch $p$ the virtual worker will use the version of the weights that reflects (at least) all the local updates from minibatches 1 to $p-(s_{local}+1)$ .", "Note that for every minibatch $p$ , $w_p$ must be kept in GPU memory until the backward pass for $p$ is executed.", "Note that staleness in SSP is caused by the different processing speed of minibatches among multiple workers.", "Thus, in SSP, staleness is used as a means to reduce the synchronization and communication overhead.", "However, local staleness in HetPipe is caused inherently as minibatches are processed in a pipelined manner within a virtual worker." ], [ "Data Parallelism with Multiple VWs", "In this section, we discuss data parallelism (DP) with virtual workers.", "The first and foremost observation of DP being supported with virtual workers is that the virtual workers may be composed of (whimpy) heterogeneous GPUs.", "While it is well known that DP helps expedite DNN execution, DP, in typical systems, is not possible if individual GPUs, that is, workers, do not have sufficient resources to handle the DNN model, in particular, large DNNs.", "By allowing a virtual worker to be composed of multiple GPUs that are lacking in resources, our system allows DP even with whimpy GPUs.", "The other key observation in properly supporting DP with virtual workers is that each virtual worker now retains local staleness as discussed in Section .", "Making sure that, despite such individual staleness, we understand and show that the results obtained from DP among virtual workers (globally) converges is an important issue that must be addressed.", "The rest of the section elaborates on this matter.", "Workings of WSP: As stated in the system overview, HetPipe uses parameter servers.", "We assume that such synchronization occurs in clock units, a notion taken from SSP [16].", "Precisely, a clock unit is defined as the progress of completing one wave.", "Recall from Section  (and Figure REF ) that a wave is a sequence of $s_{local}+1$ minibatches concurrently executed such that a virtual worker is allowed to process a later minibatch in a wave without updates from an earlier minibatch in the same wave.", "Similarly to SSP, each virtual worker maintains a local clock $c_{local}$ , while the parameter server maintains a global clock $c_{global}$ , which holds the minimum $c_{local}$ value of all the virtual workers.", "Initially, the local clocks and the global clock are 0.", "At the end of every clock $c$ , each virtual worker completes the execution of all the minibatches in wave $c$ .", "At this point, the virtual worker computes the aggregated updates from minibatch $c \\times (s_{local}+1)+1$ to minibatch $(c+1) \\times (s_{local}+1)$ and pushes the updates ũ to the parameter server.", "We see that, similar to in SSP [16], ũ is synchronized with a clock value $c$ .", "For example, as shown in Figure REF where $s_{local}=3$ , at the end of clock 0, the virtual worker pushes the aggregated updates of wave 0, and at the end of clock 1, the aggregated updates of wave 1, which is composed of minibatches from 5 to 8, and so on.", "It is important to note that in WSP, the virtual worker pushes ũ to the parameter server for every wave, instead of pushing ũ for every minibatch, which will significantly reduce the communication overhead.", "When the parameter server receives the updates ũ from the virtual worker, the parameter server updates the global version of the weights as $ w_{global}= w_{global}+$ ũ.", "Note that the parameter server updates its $c_{global}$ to $c+1$ only after every virtual worker has pushed the aggregated updates of wave $c$ .", "In WSP, each virtual worker is allowed to proceed training without retrieving the global weights for every wave.", "Thus, the virtual worker may use a weight version that, from a global standpoint, may be stale, as the most recent updates received by the parameter servers may not be reflected in its local version of the weights.", "We discuss how global staleness among the virtual workers is bounded.", "Global Staleness Bound: Let clock distance be the maximum difference between the values of $c_{local}$ of any two virtual workers in the system.", "That is, the clock distance is the difference in $c_{local}$ between the fastest and slowest virtual workers.", "In WSP, the maximum clock distance must be at most $D$ , where $D$ is a threshold set by the user.", "Therefore, a virtual worker with local clock $c$ , where $c \\ge D+1$ , must use a version of the weights that includes all the (aggregated) updates from wave 0 up to $c - D - 1$ or beyond.", "That is, using weights that exclude any updates from waves between 0 up to $c - D - 1$ is not permitted.", "Thus, a virtual worker can proceed training of the next minibatch without updates from up to $D$ most recent waves.", "When a virtual worker pulls the global weights at the end of clock $c$ to maintain this distance, it may need to wait for other virtual workers to push their updates upon completion of wave $c-D$ .", "Note, however, that while a virtual worker waits for other virtual workers to possibly catch up at the end of clock $c$ , local processing is allowed to proceed with $s_{local}$ minibatches of wave $c+1$ as the minibatches are executed in a pipelined manner.", "Take, for example, the case when $D=0$ in Figure REF .", "As the virtual worker completes minibatch 4, it computes the aggregated updates ũ for wave 0 (composed of minibatches 1 to 4) and pushes ũ to the parameter server.", "This virtual worker now waits for the other virtual workers to complete wave 0 before proceeding with minibatch 8.", "However, note that as shown in the figure, this virtual worker has already started to process minibatches 5, 6 and 7, which belong to wave 1.", "Similarly, once it completes minibatch 8, it pushes the aggregated updates ũ for wave 1 (composed of minibatches 5 to 8) to the parameter server; in the meantime, it has already started processing minibatches 9, 10, and 11, which belong to wave 2.", "Note that this processing of local minibatches in the virtual worker does not violate the local staleness bound.", "Note also that when $D=0$ , each virtual worker must wait for each other at the end of every clock to synchronize the weights for every wave, which is BSP-like behavior with pipelined execution in each virtual worker.", "Now let us define the global staleness bound, $s_{global}$ , to be the maximum number of missing updates from the most recent minibatches, globally computed by all the other virtual workers in the system, that is allowed for a minibatch to proceed in a virtual worker.", "We want to identify $s_{global}$ based on our discussion so far.", "This will allow each virtual worker to determine whether it can proceed with its current minibatch.", "Initially, all virtual workers start processing the first $(D+1)$ waves without querying the global weights from the parameter server.", "Furthermore, they can start to process up to $s_{local}$ minibatches of the next wave before receiving the global weights that include the recent updates as discussed above.", "Therefore, for those initial minibatches, the virtual worker uses $w_0$ or a weight version that may include some recent local updates.", "For any minibatch $p$ thereafter, that is, where $p > (D+1) \\times (s_{local}+1) + s_{local}$ , $p$ must use a weight version that reflects, at the very least, all the global updates from all the other virtual workers from minibatch 1 to minibatch $p - (s_{global}+1)$ , where ${s_{global}} = (D+1) \\times (s_{local}+1)+s_{local}-1$ .", "The first term of this equation is due to the fact that a virtual worker is allowed to proceed with the next $(D+1)$ waves (i.e., $(D+1) \\times (s_{local}+1)$ minibatches), and the second term is due to the additional $s_{local}$ minibatches that can be started because of pipelined execution.", "Continuing with the example above, where $D =0$ and $s_{local}= 3$ , the virtual worker proceeds the training of minibatch 11 without the global and/or local updates from wave 1 (minibatches 5 to 8) or the two local updates from minibatches 9 and 10.", "However, it must have a version of the weights that includes all the global updates from minibatches 1 to 4." ], [ "Convergence Analysis", "In this section, we discuss the convergence property of the WSP model.", "Let $N$ be the number of virtual workers and $u_{n,p}$ be the update of worker $n$ at minibatch execution $p$ .", "Given $s_g = s_{global}$ , $s_l = s_{local} + 1$ for abbreviations and following the analysis of [16], the noisy weight parameter $\\tilde{w}_{n,p}$ (which is a defined term for a way of updating the weights in our proof), for worker $n$ at minibatch execution $p$ , is decomposed into $\\tilde{w}_{n,p} = w_0 + \\left[\\sum _{n^{\\prime } = 1}^N \\sum _{p^{\\prime } = 1}^{p - s_g - 1} u_{n^{\\prime },p^{\\prime }}\\right] + \\left[\\sum _{p^{\\prime } \\in \\mathcal {C}_{n,p}} u_{n,p^{\\prime }}\\right] \\\\ + \\left[\\sum _{\\left(n^{\\prime },p^{\\prime }\\right) \\in \\mathcal {E}_{n,p}} u_{n^{\\prime },p^{\\prime }}\\right].$ Here $w_0$ refers to the initial parameter.", "The noisy weight has three terms which respectively include 1. updates of all workers (guaranteed to be included) to process minibatch execution $p$ , 2.", "$\\mathcal {C}_{n,p} \\subseteq [p - s_g, p - 1]$ : the index set of latest updates of the querying worker $n$ in the range of current global staleness bound, and 3.", "$\\mathcal {E}_{n,p} \\subseteq \\left([1,N]\\backslash \\lbrace n\\rbrace \\right) \\times \\left[p - s_g, p + s_g + s_l\\right]$ : the index set of extra updates of other workers in the range of current global staleness bound.", "When the execution $p$ is not at synchronization point, $\\mathcal {E}_{n,p} = \\emptyset $ .", "We define $\\lbrace u_t\\rbrace $ as the sequence of updates of each virtual worker after processing each minibatch and $w_t = w_0 + \\sum _{t^{\\prime } = 0}^{t - s_l N} u_{t^{\\prime }}$ as the reference sequence of weights, where $u_t u_{t \\bmod {N}, \\lfloor t/N \\rfloor + \\, t \\bmod {s_l}},$ in which we loop over the workers ($t \\bmod {N}$ ) and over each update after a minibatch execution inside a worker ($\\lfloor t/N \\rfloor + \\, t \\bmod {s_l}$ ).", "(Here $s_l N$ ($= s_l \\times N$ ) is the number of total minibatch updates in one wave from all virtual workers.)", "Since a virtual worker uses a version of the weights that reflects all the local updates from minibatch 1 to $p - s_l$ for worker $p$ , the reference and noisy sequences at iteration $t$ are updated up to $t - s_lN$ .", "The set $\\mathcal {E}_t$ and the noisy sequence $\\tilde{w}_t$ are defined similarly and the difference between $w_t$ and $\\tilde{w}_t$ is $\\tilde{w}_t = w_t - \\left[\\sum _{i \\in \\mathcal {R}_t} u_i\\right] + \\left[\\sum _{i \\in \\mathcal {Q}_t} u_i\\right]$ where $\\mathcal {R}_t$ is the index set of missing updates in the reference weights but not in noisy weights, and $\\mathcal {Q}_t$ is the index set of extra updates in the noisy weights but not in reference weights.", "After $T$ updates, When we represent the target function as $f(w) \\frac{1}{T} \\sum _{t = 1}^T f_t(w)$ , the regret of two functions with $\\tilde{w}_t$ , the parameter learned from the noisy update, and $w^*$ , the parameter learned from the synchronized update is $R[W] \\frac{1}{T}\\sum _{t = 1}^T f_t\\left(\\tilde{w}_t\\right) - f\\left(w^*\\right).$ Thus, when we bound the regret of the two functions, we can bound the error of the noisy updates incurred by the distributed pipeline staleness gradient descent.", "We first bound the cardinality of $\\mathcal {R}_t$ and $\\mathcal {Q}_t$ in the following lemma.", "Lemma 1 The following two inequalities, $\\vert \\mathcal {R}_t \\vert + \\vert \\mathcal {Q}_t \\vert \\le (2s_g + s_l)(N - 1)$ and $\\min \\left(\\mathcal {R}_t \\cup \\mathcal {Q}_t\\right) \\ge \\max (1, t - (s_g + s_l)N)$ , hold.", "Since $\\mathcal {Q}_t \\subseteq \\mathcal {E}_t$ and $\\mathcal {R}_t \\subseteq \\mathcal {E}_t \\backslash \\mathcal {Q}_t$ , $ \\vert \\mathcal {R}_t \\vert + \\vert \\mathcal {Q}_t \\vert \\le \\vert \\mathcal {E}_t \\vert \\le (2s_g + s_l)(N - 1)$ .", "The second claim follows from $\\mathcal {E}_t \\supseteq \\mathcal {R}_t \\cup \\mathcal {Q}_t$ .", "To prove the convergence, we have the following two assumptions and leave the proof to the Appendix , which generally follows Qirong et al. [16].", "Assumption 1 ($L$ -Lipschitz components) For all $t$ , the component function $f_t$ is convex and has bounded subdifferential $\\left\\Vert \\nabla f_t(w) \\right\\Vert \\le L$ , in which $L > 0$ is a constant.", "Assumption 2 (Bounded distances) For all $w, w^{\\prime }$ , the distance between them is bounded $D(w \\Vert w^{\\prime }) \\le M$ , in which $M > 0$ is a constant.", "We also denote $\\frac{1}{2} \\Vert w - w^{\\prime } \\Vert ^2$ as $D\\left(w\\Vert w^{\\prime }\\right)$ .", "Then, we can bound the regret of the function trained with our noisy distributed, pipeline update as in Theorem REF .", "Theorem 1 Suppose $w^*$ is the minimizer of $f(w)$ .", "Let $\\displaystyle u_t - \\eta _t \\nabla f_t\\left(\\tilde{w}_t\\right)$ where $\\eta _t = \\frac{\\sigma }{\\sqrt{t}}$ with $\\sigma = \\frac{M}{L\\sqrt{(2s_g + s_l)N}}$ , in which $M, L$ are the constants defined in the assumptions.", "Then the regret is bounded as $R[W] \\le 4ML\\sqrt{\\frac{(2s_g + s_l)N}{T}}.", "$ Our theoretical results are similar with existing work on non pipelined version of staleness update [16], [21].", "However, we reflect the new characteristics of distributed pipeline staleness update in Lemma REF , and thus in Theorem REF ." ], [ "Partitioning Algorithm", "Recall that the goal of our partitioning algorithm is to minimize the maximum execution time of the partitions within the bounds of satisfying the memory requirement.", "To obtain a performance model to predict the execution time of each layer of a model in a heterogeneous GPU, we first profile the DNN model on each of the different types of GPUs in a cluster, where we measure the computation time of each layer of the model.", "For GPU memory usage, we measure the usage of each layer (by using the logging feature of TensorFlow) on only one GPU type (as it is roughly the same for all GPU types).", "To compute the memory requirement for a given partition, we take into account the total memory usage to store the data to process the layers as well as the maximum number of minibatches concurrently assigned to the partition.", "For communication time between layers in the model, we first derive the amount of input data for each layer in the forward and backward pass from the model graph.", "For the given data size, we predict intra-node communication based on the PCI-e bandwidth, then multiply it by a scaling-down constant (which is similarly done in Paleo [41]), since in practice, it is not possible to utilize the peak bandwidth.", "The scaling-down constant is derived by running a synthetic model that sends various sizes of data from one GPU to another GPU in the same node.", "For inter-node communication (via Infiniband), we use linear regression to estimate the communication time for the given data size.", "To build a prediction model, we collect 27 samples by training two DNN models, used in our experiments, with arbitrary partitions.", "Note that in this work, the heterogeneity of network performance such as slow network links is not considered (as in [30]).", "However, for such cases, we can extend our partitioning algorithm to consider different network performance between two nodes when estimating the communication time.", "Also, a model that estimates the memory requirement for each stage more accurately will be helpful in partitioning a DNN model in a more balanced manner.", "To find the best partitions of a DNN model, we make use of CPLEX, which is an optimizer for solving linear programming problems [18].", "Memory requirements for each partition on the pipeline to support $N_m$ concurrent minibatches are provided as constraints to the optimizer.", "We first investigate the performance of the 7 different individual virtual workers that are possible according to the allocation schemes in Table .", "Figure  shows the throughput over various values of $N_m$ , which is the number of minibatches executed concurrently, in the virtual worker normalized to that of when $N_m = 1$ and the maximum average GPU utilization among the four partitions for ResNet-152 and VGG-19.", "The numbers shown (in the box) along with the allocation policy are the absolute throughput (images/sec) when $N_m = 1$ .", "Note that some results for larger $N_m$ are not shown.", "This is because the GPU memory cannot accommodate such situations and hence, cannot be run.", "From the results, we can see that as $N_m$ increases, normalized throughput of a virtual worker as well as the maximum GPU utilization generally increases.", "However, depending on the resource allocation scheme (which results in different partitions of a model) as well as the DNN model, the effect of having larger $N_m$ varies.", "When a virtual worker is configured with homogeneous GPUs, the average GPU utilization of each partition is similar to each other.", "However, when it is configured with heterogeneous GPUs, there is a tendency that the GPU utilization of the first or last partition is higher than those of the other partitions.", "For this configuration, different computation capabilities and memory capacity of the GPUs are considered when partitioning a model.", "As it is possible that only a small number of layers are assigned to some GPUs, the overall GPU utilization may turn out to be low." ], [ "Performance of multiple virtual workers", " Figure: VGG-19Figure REF shows the throughput of training each model with the three resource allocation policies, where “Horovod” indicates the state-of-the-art DP via Horovod that uses AllReduce communication.", "In these experiments, for each resource allocation policy, $N_m$ is set such that performance is maximized while every virtual worker uses the same value of $N_m$ as this is the assumption behind HetPipe.", "For ResNet-152, the whole model is too large to be loaded into a single GPU with G type, and thus, Horovod uses only 12 GPUs.", "The results in Figure REF show that the performance of DNN training is strongly affected by how heterogeneous GPUs are allocated to virtual workers.", "From the results, we can make the following observations: For VGG-19 whose parameter size is 548MB, the performance of Horovod, which reduces communication overhead for parameter synchronization, is better than those of NP, ED, and HD.", "However, for ResNet-152 whose parameter size is 230MB, ED and HD, which utilize virtual workers with similar performance, show similar performance to Horovod (with 12 GPUs).", "With NP, training performance of ResNet-152 and VGG-19 is low as $N_m$ is bounded by the virtual worker with the smallest GPU memory.", "With the local placement policy, intra-communication occurs between each GPU and the parameter server, significantly reducing communication overhead across the nodes, especially for VGG-19, the model with a large parameter set.", "For VGG-19, the amount of data transferred across the nodes with ED-local (i.e., 103MB) is much smaller than that with Horovod (i.e., 515MB).", "Thus, the performance of ED-local (which also mitigates the straggler problem) is 1.8 times higher than Horovod.", "For ResNet-152, the amount of data transferred with ED-local (i.e., 298MB) is larger than that with Horovod (i.e., 211MB) because the sizes of output activations to be sent between partitions are large, even though the parameter size is relatively small.", "However, the throughput of ED-local is still 40% higher than Horovod.", "This is because Hetpipe allows each virtual worker to process a large number of minibatches concurrently.", "Table: Performance improvement of adding whimpy GPUs(The number in parenthesis presents the totalnumber of concurrent minibatches in HetPipe)Next, we investigate how the throughput is improved when whimpy GPUs are additionally used for training.", "Table REF shows the throughput of VGG-19 and ResNet-152 when DP via Horovod and HetPipe with ED-local are used over different sets of heterogeneous GPUs.", "For these experiments, HetPipe is configured to use four virtual workers, except for V4 where a single virtual worker is used.", "In the table, the number and type of GPUs used for each experiment are also given.", "From the results, we can see that the performance of both Horovod and HetPipe increases when additional whimpy GPUs are used for training.", "With additional GPUs, HetPipe can increase the total number of concurrent minibatches processed, having up to 2.3 times speedup.", "This scenario can be thought of as an answer to when new, higher end nodes are purchased, but one does not know what to do with existing nodes.", "For example, one can imagine node V to be the most recent purchase, with earlier systems R, Q, and G. The results show that making use of the earlier whimpy systems allows for faster training of larger models.", "Figure: ResNet-152 top-1 accuracy" ], [ "Convergence", "Our HetPipe based on the WSP model is guaranteed to converge as proven in Section .", "In this section, we analyze the convergence performance of HetPipe with ED-local using ResNet-152 and VGG-19.", "For our experiments, the desired target accuracy of ResNet-152 and VGG-19 is 74% and 67%, respectively.", "Figure REF shows the top-1 accuracy of ResNet-152 with Horovod (12 GPUs), HetPipe (12 GPUs), and HetPipe (16 GPUs), where $D$ is set to 0 for HetPipe.", "For the experiments with 12 GPUs, the 4 G type GPUs are not used.", "When the same set of GPUs are used, convergence with HetPipe is 35% faster than that of Horovod by reducing the straggler problem in a heterogeneous environment and exploiting both PMP and DP.", "Furthermore, by adding four more whimpy G GPUs, HetPipe improves training performance even more, converging faster than Horovod by 39%.", "Figure REF shows the top-1 accuracy of VGG-19 with Horovod and HetPipe as we vary $D$ to 0, 4, and 32.", "For the experiments, all 16 GPUs are used.", "The figure shows that convergence with the BSP-like configuration (i.e., $D=0$ ) of HetPipe is roughly 29% faster than that with Horovod.", "As we increase $D$ to 4, the straggler effect is mitigated and the communication overhead due to parameter synchronization is reduced.", "Thus, convergence is faster by 28% and 49% compared to $D=0$ and Horovod, respectively.", "In this experiment with ED-local (where the training speed of each virtual worker is similar), when $D$ becomes very large (i.e., 32), the throughput remains similar but the convergence performance becomes degraded by 4.7%, compared to $D=4$ .", "This is because it is unlikely that the clock distance between the fastest and slowest virtual workers becomes large as 32, but higher global staleness can degrade the convergence performance (similarly discussed in [16]).", "Note that though not shown, using larger $D$ has a greater effect for HetPipe with NP, ED and HD resource allocation, and the different resource allocations only affect the set of heterogeneous GPUs used for each virtual worker and do not affect the convergence behavior.", "We also analyze the synchronization overhead as $D$ is varied.", "We find that as $D$ increases, the waiting time of a virtual worker to receive the updated global weight decreases.", "In our experiments, the average waiting time with $D=4$ is found to be 62% of that with $D=0$ .", "Furthermore, the actual idle time is only 18% of the waiting time as the virtual worker can continue to proceed in the pipeline while waiting.", "Figure: VGG-19 top-1 accuracy" ], [ "Related Work", " Pipelining has been leveraged to improve the performance of machine learning systems [5], [6], [34], [17], [29].", "A pipelining scheme is employed to handle expensive backpropagation [6].", "Pipe-SGD pipelines the processing of a minibatch to hide communication time in AllReduce based systems [29].", "A weight prediction technique is proposed to address the staleness issue in pipelined model parallelism [5].", "Detailed comparisons of HetPipe with PipeDream [34] and GPipe [17] are provided in Section REF .", "Note that the feature of overlapping computation and communication, presented in PipeDream [34], will also improve the performance of our system.", "PipeDream employs the one-forward-one-backward scheduling algorithm for pipeline execution where the minimum number of minibatches that is large enough to saturate the pipeline are admitted.", "Sophisticated schedulers that consider various factors such as heterogeneous configurations, the number of partitions, and the number of concurrent minibatches within a virtual worker, can potentially improve the performance of HetPipe.", "Techniques to optimize learning rates have been studied [13], which can also be applied to HetPipe to help converge faster.", "Decentralized training systems that consider heterogeneous environments have also been studied [31], [30].", "However, these techniques do not consider integration of DP with PMP, which allows support for large models that do not fit into single GPU memory.", "In AD-PSGD, once a mini-batch is processed, a worker updates the parameters by averaging them with only one neighbor which is randomly selected [30].", "This is done asynchronously, allowing faster workers to continue.", "In theory, the convergence rate of AD-PSGD is the same as SGD.", "In principle, the contribution of AD-PSGD is orthogonal with the contributions of HetPipe in that we can extend our HetPipe further by adapting the idea of asynchronous decentralized update in AD-PSGD when there is a bottleneck in the parameter server.", "When it comes to the experimental evaluations, the performance of AD-PSGD is evaluated for DNN models whose sizes are 1MB, 60MB, and 100MB, which are smaller than the models we consider in HetPipe.", "For a decentralized training system, Hop [31] considers the bounded staleness and backup workers, and uses CIFAR-10 for performance evaluation on a CNN model.", "There have been earlier efforts to employ DP and/or MP for model training.", "Project Adam uses both DP and MP to train machine learning models on CPUs [7].", "Pal et al.", "combine DP and MP in a similar way as our system, but do not consider pipelining nor heterogeneous GPUs [39].", "STRADS leverages MP to address the issues of uneven convergence of parameters and parameter dependencies [24].", "FlexFlow considers utilizing parallelism in various dimensions such as sample, operator, attribute and parameters to maximize parallelization performance [20].", "Bounded staleness has been explored where Jiang et al.", "present heterogeneity-aware parameter synchronization algorithms that are based on the SSP model [21], while Cui et al.", "analyze the effects of bounded staleness [9]." ], [ "Conclusion", "In this paper, we presented a DNN training system, HetPipe, that integrates pipelined model parallelism with data parallelism.", "Leveraging multiple virtual workers, each of which consists of multiple, possibly whimpy, heterogeneous GPUs, HetPipe makes it possible to efficiently train large DNN models.", "We proved that HetPipe converges and presented results showing the fast convergence of DNN models with HetPipe." ], [ "Appendix", "The analysis follows Qirong Ho et al.", "[16], except the addition of global staleness $s_g = s_{global}$ and local staleness $s_l = s_{local} + 1$ .", "Theorem 1.", "Suppose $w^*$ is the minimizer of convex function $f(w)$ .", "Let $\\displaystyle u_t - \\eta _t \\nabla f_t\\left(\\tilde{w}_t\\right)$ where $\\eta _t = \\frac{\\sigma }{\\sqrt{t}}$ with $\\sigma = \\frac{M}{L\\sqrt{(2s_g + s_l)N}}$ , in which $M, L$ are the constants defined in the assumptions.", "We assume that the components $f_t$ are also convex.", "Then after $T$ iterations, the regret is bounded as $R[W] \\le 4ML\\sqrt{\\frac{(2s_g + s_l)N}{T}}, $ with $f(w) \\frac{1}{T} \\sum _{t = 1}^T f_t(w), \\qquad \\text{and} \\qquad R[W] \\frac{1}{T}\\sum _{t = 1}^T f_t\\left(\\tilde{w}_t\\right) - f\\left(w^*\\right).$ Since $f_t$ are convex $T\\cdot R[W] \\le \\sum _{t=1}^T \\left\\langle \\nabla f_t\\left(\\tilde{w}_t\\right), \\tilde{w}_t - w^* \\right\\rangle = \\sum _{t=1}^T \\left\\langle \\tilde{g}_t, \\tilde{w}_t - w^* \\right\\rangle ,$ where $\\tilde{g}_t \\nabla f_t(\\tilde{w}_t)$ .", "If $T \\cdot R[W] \\le \\mathcal {O}(\\sqrt{T})$ , we will have $\\mathbb {E}_t \\lbrace f_t(\\tilde{w}_t) - f_t(w^*)\\rbrace \\rightarrow 0$ and thus convergence.", "First, we need this lemma.", "Note that this is not Lemma REF in the paper: Lemma 2 With $\\tilde{w}_t = w_t - \\left[\\sum _{i \\in \\mathcal {R}_t} u_i\\right] + \\left[\\sum _{i \\in \\mathcal {Q}_t} u_i\\right]$ , for all $w^* \\in \\mathbb {R}^n$ and $t > 0$ , we have $\\left\\langle \\tilde{w}_t - w^*, \\tilde{g}_t \\right\\rangle = \\frac{\\eta _t}{2} \\Vert \\tilde{g}_t \\Vert ^2 + \\frac{D\\left(w^* \\Vert w_{t+1}\\right) - D\\left(w^* \\Vert w_{t}\\right)}{\\eta _t}+ \\left[\\sum _{i \\in \\mathcal {R}_t} \\eta _i \\left\\langle \\tilde{g}_i, \\tilde{g}_t \\right\\rangle - \\sum _{i \\in \\mathcal {Q}_i} \\eta _i \\left\\langle \\tilde{g}_i, \\tilde{g}_t \\right\\rangle \\right],$ with $D\\left(w\\Vert w^{\\prime }\\right) = \\frac{1}{2} \\Vert w - w^{\\prime } \\Vert ^2$ .", "We have $D\\left(w^* \\Vert w_{t+1}\\right) - D\\left(w^* \\Vert w_{t}\\right) = \\frac{1}{2} \\eta _t^2 \\left\\Vert \\tilde{g}_t \\right\\Vert ^2 - \\eta _t \\left\\langle \\tilde{w}_t - w^*, \\tilde{g}_t \\right\\rangle - \\eta _t \\left\\langle w_t - \\tilde{w}_t, \\tilde{g}_t \\right\\rangle ,$ with the last term is $\\left\\langle w_t - \\tilde{w}_t, \\tilde{g}_t \\right\\rangle = - \\sum _{i \\in \\mathcal {R}_t}\\eta _i \\left\\langle \\tilde{g}_i, \\tilde{g}_t \\right\\rangle + \\sum _{i \\in \\mathcal {Q}_t} \\eta _i \\left\\langle \\tilde{g}_i, \\tilde{g}_t \\right\\rangle .$ Therefore, $\\left\\langle \\tilde{w}_t - w^*, \\tilde{g}_t \\right\\rangle = \\frac{\\eta _t}{2} \\Vert \\tilde{g}_t \\Vert ^2 + \\frac{D\\left(w^* \\Vert w_{t+1}\\right) - D\\left(w^* \\Vert w_{t}\\right)}{\\eta _t}+ \\left[\\sum _{i \\in \\mathcal {R}_t}\\eta _i \\left\\langle \\tilde{g}_i, \\tilde{g}_t \\right\\rangle - \\sum _{i \\in \\mathcal {Q}_t} \\eta _i \\left\\langle \\tilde{g}_i, \\tilde{g}_t \\right\\rangle \\right].$ We use the above Lemma to find the upper bound of each term in the regret $R[W]$ : $\\sum _{t = 1}^T \\frac{\\eta _t}{2} \\Vert \\tilde{g}_t \\Vert ^2 & \\le \\sum _{t = 1}^T \\frac{\\eta _t}{2} L^2 \\quad \\text{ ($L$-Lipschitz assumption)} \\nonumber \\\\& = \\frac{1}{2}\\sum _{t = 1}^T \\frac{\\sigma }{\\sqrt{t}} L^2 \\le \\sigma L^2 \\sqrt{T} \\quad \\left(\\text{since } \\sum _{k=a}^b \\frac{1}{2\\sqrt{k}} \\le \\sqrt{b - a + 1}\\right), \\nonumber $ and $& \\sum _{t = 1}^T \\frac{D\\left(w^* \\Vert w_{t+1}\\right) - D\\left(w^* \\Vert w_{t}\\right)}{\\eta _t} \\\\= \\quad & \\frac{D(w^* \\Vert w_1)}{\\eta _1} - \\frac{D(x^* \\Vert w_{T+1})}{\\eta _t} + \\sum _{t=2}^{T}\\left[D(w^* \\Vert w_t) \\left(\\frac{1}{\\eta _t} - \\frac{1}{\\eta _{t-1}} \\right)\\right] \\\\\\le \\quad & \\frac{M^2}{\\sigma } + 0 + \\frac{M^2}{\\sigma } \\sum _{t=2}^T \\left[\\sqrt{t} - \\sqrt{t - 1}\\right] \\quad \\text{(Bounded distances assumption)} \\\\= \\quad & \\frac{M^2}{\\sigma }\\sqrt{T},$ and $& \\sum _{t = 1}^T \\left[\\sum _{i \\in \\mathcal {R}_t}\\eta _i \\left\\langle \\tilde{g}_i, \\tilde{g}_t \\right\\rangle - \\sum _{i \\in \\mathcal {Q}_t} \\eta _i \\left\\langle \\tilde{g}_i, \\tilde{g}_t \\right\\rangle \\right] \\nonumber \\\\\\le \\quad & \\sum _{t = 1}^T \\left(\\left|\\mathcal {R}_t \\right|+ \\left|\\mathcal {Q}_t \\right|\\right) \\eta _{\\max (1, t - (s_g + s_l)N)} L^2 \\nonumber \\\\\\quad & \\qquad \\qquad \\qquad (\\text{from paper's lemma: } \\min \\left(\\mathcal {R}_t \\cup \\mathcal {Q}_t\\right) \\ge \\max (1, t - (s_g + s_l)N)) \\nonumber \\\\= \\quad & L^2 \\left[\\sum _{t=1}^{(s_g + s_l)N} \\left(\\left|\\mathcal {R}_t \\right|+ \\left|\\mathcal {Q}_t \\right|\\right) \\eta _1 + \\sum _{t=(s_g + s_l)N + 1}^T \\left(\\left|\\mathcal {R}_t \\right|+ \\left|\\mathcal {Q}_t \\right|\\right) \\eta _{t - (s_g + s_l)N}\\right] \\nonumber \\\\\\quad & \\qquad \\qquad \\qquad (\\text{split the sum and use decreasing sequence property of } \\lbrace \\eta _t\\rbrace ) \\nonumber \\\\= \\quad & L^2 \\left[\\sum _{t=1}^{(s_g + s_l)N} \\left(\\left|\\mathcal {R}_t \\right|+ \\left|\\mathcal {Q}_t \\right|\\right) \\sigma + \\sum _{t=(s_g + s_l)N + 1}^T \\left(\\left|\\mathcal {R}_t \\right|+ \\left|\\mathcal {Q}_t \\right|\\right) \\frac{\\sigma }{\\sqrt{t - (s_g + s_l)N}}\\right] \\nonumber \\\\\\le \\quad & \\sigma L^2 (2s_g + s_l)(N - 1) \\left[(s_g + s_l)N + \\sum _{t=(s_g + s_l)N + 1}^T \\frac{1}{\\sqrt{t - (s_g + s_l)N}}\\right] \\nonumber \\\\\\quad & \\qquad \\qquad \\qquad (\\text{from paper's lemma: } \\vert \\mathcal {R}_t \\vert + \\vert \\mathcal {Q}_t \\vert \\le (2s_g + s_l)(N - 1)) \\nonumber \\\\\\le \\quad & \\sigma L^2 (2s_g + s_l)N \\left[(s_g + s_l)N + 2\\sqrt{T - (s_g + s_l)N}\\right] \\qquad \\left(\\text{since } \\sum _{k=a}^b \\frac{1}{2\\sqrt{k}} \\le \\sqrt{b - a + 1}\\right) \\nonumber \\\\\\le \\quad & \\sigma L^2 (2s_g + s_l)(s_g + s_l)N^2 + 2 \\sigma L^2 (2s_g + s_l)N \\sqrt{T}$ Therefore, $T \\cdot R[W] \\le \\sigma L^2 \\sqrt{T} + \\frac{M^2}{\\sigma }\\sqrt{T} + \\sigma L^2 (2s_g + s_l)(s_g + s_l)N^2+ 2 \\sigma L^2 (2s_g + s_l)N \\sqrt{T}$ Let the initial $\\sigma = \\frac{M}{L\\sqrt{(2s_g + s_l)N}}$ , then $T \\cdot R[W] & \\le \\frac{ML\\sqrt{T}}{\\sqrt{(2s_g + s_l)N}} + ML\\sqrt{(2s_g + s_l)NT} + ML(s_g + Fs_l)N\\sqrt{(2s_g + s_l)N} + 2ML \\sqrt{(2s_g + s_l)NT} \\nonumber \\\\& = ML \\sqrt{(2s_g + s_l)NT} \\left[\\frac{1}{(2s_g + s_l)N} + 1 + \\frac{(s_g + s_l)N}{\\sqrt{T}} + 2\\right].$ We have $\\frac{1}{(2s_g + s_l)N} + \\frac{(s_g + s_l)N}{\\sqrt{T}} \\le 1$ when $T$ is large enough.", "Therefore we get $T \\cdot R[W] \\le 4ML\\sqrt{(2s_g + s_l)NT},$ or $R[W] \\le 4ML\\sqrt{\\frac{(2s_g + s_l)N}{T}}$" ] ]

[ [ "Microscopic approach to the macrodynamics of matter with broken\n symmetries" ], [ "Abstract A unified set of hydrodynamic equations describing condensed phases of matter with broken continuous symmetries is derived using a generalization of the statistical-mechanical approach based on the local equilibrium distribution.", "The dissipativeless and dissipative parts of the current densities and the entropy production are systematically deduced in this approach by expanding in powers of the gradients of the macrofields.", "Green-Kubo formulas are obtained for all the transport coefficients.", "The results apply to both crystalline solids and liquid crystals.", "The consequences of microreversibility and spatial symmetries are investigated, leading to the prediction of cross effects resulting from Onsager-Casimir reciprocal relations." ], [ "Introduction", "The spontaneous breaking of continuous symmetries is a ubiquitous phenomenon in Nature.", "It manifests itself in the quantum vacuum of spacetime [1], [2], [3], in condensed phases of matter at equilibrium [4], or in dissipative structures existing far from equilibrium [5], [6].", "In nonrelativistic condensed matter at equilibrium, different kinds of continuous symmetries may be broken including internal gauge symmetries in superfluids or superconductors [7], [8], [9], [10], and spatial symmetries of translations or rotations in crystals, liquid crystals, or magnetic materials [11], [12].", "The breaking of continuous symmetries generates long-range order, rigidity (i.e., the possibility to drag the whole condensed phase from its boundaries), as well as Nambu-Goldstone modes [8], [9] beside the hydrodynamic modes resulting from the five fundamental conservation laws for mass, energy, and linear momentum.", "All these soft modes have frequencies vanishing with their wave number, since they represent perturbations with respect to equilibrium becoming slower and slower as their wave length increases.", "These modes may be propagative (e.g., the sound modes) or diffusive (e.g., the heat mode).", "In any case, they are damped because of energy dissipation coming from thermal fluctuations at positive temperature.", "All these effects can be described in terms of macroscopic equations ruling the time evolution of these modes.", "The macroscale formulation of hydrodynamics in matter characterized by broken symmetries has been achieved, in particular, for crystalline solids and liquid crystals [13], [14].", "A basic issue is to deduce the macroscopic equations from the underlying microscopic dynamics of atoms and molecules composing matter.", "For this purpose, statistical mechanics is required, not only for matter at equilibrium, but also away from equilibrium to obtain the time-dependent properties including the transport coefficients associated with energy dissipation [15], [16], [17], [18], [19], [20], [21].", "In the regime of relaxation towards global equilibrium, linear response theory combined with projection-operator method has been much developed to deduce the transport properties of condensed phases with broken symmetries [22], [11].", "This approach has also been formulated for crystalline solids in Refs.", "[23], [24].", "However, nonlinear effects often arise because of advection induced by the velocity of the system, as it is the case in fluid turbulence.", "In order to deduce the Eulerian terms in hydrodynamics, a more systematic approach consists of using local equilibrium probability distributions instead of global equilibrium distributions.", "This method, which has been developed since the sixties [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], provides not only the microscopic expressions for the dissipativeless fluxes of Eulerian type, but also local thermodynamics and the statistical-mechanical expression for entropy production in terms of the dissipative fluxes.", "Furthermore, these latter fluxes can be obtained at leading order in the gradients of the macrofields together with the transport coefficients given by Green-Kubo formulas.", "In this framework, the consequences of microreversibility and spatial symmetries can also be investigated.", "However, this method has been defined and used only for normal fluids and its generalization to a system with broken symmetry is still lacking.", "Our aim is here to use this systematic approach to derive a unified set of macroscopic equations applicable to both crystalline solids and liquid crystals.", "The full derivation relies on the identification of the hydrodynamic variables and local order parameters associated with the broken continuous symmetries at the microscopic level of description.", "These variables obey balance equations in the form of local conservation laws, which can be obtained from the microscopic Hamiltonian dynamics, as presented in Sec. .", "On this basis, the local equilibrium probability distribution is introduced and its time evolution can be investigated using the microscopic Hamiltonian dynamics.", "In this way, the entropy functional can be defined, as well as the Massieu functional given by its Legendre transform.", "The microscopic expression for entropy production is deduced, allowing us to identify the dissipativeless and dissipative fluxes (also called current densities) as summarized in Sec. .", "The calculations are carried out by expanding the macrofields in powers of their gradients around local equilibrium.", "The local thermodynamic relations for matter with broken symmetries are obtained at leading order in the gradients in Sec. .", "The dissipativeless current densities are determined at next order in Sec.", ", including the extra contributions arising from the broken symmetries.", "In Sec.", ", the dissipative current densities are derived and the Green-Kubo formulas are found giving all the possible transport coefficients.", "Even if the microscopic expressions of the quantities of interest are essential for a concrete evaluation of the dissipative coefficients, the full derivation can be performed without their explicit knowledge.", "The applicability to crystalline solids is discussed in Sec.", "and the case of liquid crystals in Sec. .", "Finally, our concluding remarks are presented in Sec. .", "Notations and conventions: Latin letters $a, b, c, ... = x, y, z$ correspond to spatial coordinates and Greek letters $\\alpha , \\beta , \\gamma , \\dots $ label the hydrodynamic variables and the order parameters.", "The indices $i,j,k,\\dots = 1, 2,\\dots , N$ label the particles, where $N$ is the total number of particles.", "We take, for simplicity, a single species of particles.", "The microscopic expression of a field such as a density $\\hat{c}^{\\alpha }$ or a current density $\\hat{J}^a_{c^{\\alpha }}$ is denoted with a hat to make the distinction with respect to its expectation value denoted without the hat.", "Einstein's convention of summation over repeated indices is adopted.", "We consider a system of $N$ particles of mass $m$ , all supposed of the same species.", "The particle $i$ has the position ${\\bf r}_i$ , the velocity ${\\bf \\dot{r}}_i=d{\\bf r}_i/dt$ , and the momentum ${\\bf p}_i=m{\\bf \\dot{r}}_i$ .", "Their microscopic dynamics is ruled by the following Hamiltonian function $H = \\sum _i \\frac{{\\bf p}_i^2}{2m} + \\frac{1}{2}\\sum _{i\\ne j} V(r_{ij}) \\, ,$ where $V$ is the interaction potential energy and $r_{ij} = ||\\mathbf {r}_i - \\mathbf {r}_j||$ the distance between the particles $i$ and $j$ (with $i,j=1,2,\\dots ,N$ ).", "The positions and the momenta are three-dimensional vectors with the components ${\\bf r}_i=(r_i^a)$ and ${\\bf p}_i=(p_i^a)$ (with $a=x,y,z$ ).", "The time evolution of this system is represented in the phase space $\\Gamma =({\\bf r}_i,{\\bf p}_i)_{i=1}^N$ of dimension $6N$ , where the trajectories $\\Gamma _t=\\Gamma _t(\\Gamma _0)$ are uniquely determined by their initial condition $\\Gamma _0$ .", "Any observable function $A(\\Gamma )$ defined in phase space is thus evolving in time according to $dA/dt= \\lbrace A,H\\rbrace +\\partial _t A$ , where $\\lbrace A,B\\rbrace \\equiv \\sum _i(\\partial _{{\\bf r}_i}A\\cdot \\partial _{{\\bf p}_i}B - \\partial _{{\\bf p}_i}A\\cdot \\partial _{{\\bf r}_i}B)$ is the Poisson bracket between the phase-space functions $A$ and $B$ .", "For time-independent observable functions such that $\\partial _tA=0$ , time integration gives $A_t=A(\\Gamma _t)$ , since $d\\Gamma _t/dt=\\lbrace \\Gamma _t,H\\rbrace $ .", "This time evolution can be expressed in terms of the Liouvillian operator ${\\mathcal {L}}A\\equiv \\lbrace A,H\\rbrace $ according to $d\\Gamma _t/dt={\\mathcal {L}} \\Gamma $ .", "The observable quantities may be global, i.e., extensive, such as the total energy given by the Hamiltonian function, or local, i.e., intensive, such as densities.", "In particular, the densities of energy $\\hat{e}$ , mass $\\hat{\\rho }$ , and momentum $\\hat{\\bf g}=(\\hat{g}^a)$ are the variables associated with five fundamental conservation laws in the system.", "The time variations of these densities can be expressed in terms of the divergence of a current density or flux, leading to slow modes called hydrodynamic modes.", "Beyond, there may exist other densities, such as the densities of kinetic energy and potential energy, which are not directly associated with conservation laws.", "Their time variations are ruled by a rate, instead of a flux divergence, so that they most often generate fast modes, called kinetic modes.", "Nevertheless, upon continuous symmetry breaking, some of these fast local quantities may turn into slow modes, called Nambu-Goldstone modes [8], [9], which thus behave as hydrodynamic modes.", "According to the Goldstone theorem [9], there are as many such slow modes as continuous symmetries that are broken.", "Therefore, the total number of hydrodynamic variables is the sum of the five conservation laws and the number of broken continuous symmetries (one, two or three depending on the type of system in consideration, i.e., a liquid crystal or a crystalline solid).", "For the rest of this section, we introduce the microscopic definitions of the hydrodynamic variables and their current densities." ], [ "Conserved quantities", "The Hamiltonian system ruled by Eq.", "(REF ) has several conserved quantities of fundamental origin.", "Since the Hamiltonian function is time independent $\\partial _tH=0$ , this function is itself a conserved quantity because $\\lbrace H,H\\rbrace =0$ , which represents the total energy $E=H$ .", "Besides, the total mass $M=mN$ is conserved.", "Moreover, the Hamiltonian function is invariant under spatial translations ${\\bf r}_i\\rightarrow {\\bf r}_i+{\\bf a}$ with ${\\bf a}\\in {\\mathbb {R}}^3$ , so that the total momentum ${\\bf P}=\\sum _i{\\bf p}_i$ is conserved.", "Because of the symmetry of the Hamiltonian function (REF ) under rotations $\\mathsf {O}\\in $  SO(3), the angular momentum ${\\bf L}=\\sum _i {\\bf r}_i\\times {\\bf p}_i$ is also conserved.", "In order to obtain the local conservation laws, we introduce the densities associated with the total mass, energy, and linear momentum according to $&&\\text{mass density:} \\\\&&\\hat{\\rho }(\\mathbf {r};\\Gamma ) \\equiv \\sum _i m\\, \\delta (\\mathbf {r} - \\mathbf {r}_i) = m\\, \\hat{n}(\\mathbf {r};\\Gamma ) \\, , \\nonumber \\\\&&\\text{energy density:} \\\\&&\\hat{e}(\\mathbf {r};\\Gamma ) \\equiv \\sum _i \\left[ \\frac{{\\bf p}_i^2}{2m} + \\frac{1}{2}\\sum _{j(\\ne i)} V(r_{ij})\\right] \\delta (\\mathbf {r} - \\mathbf {r}_i)\\, , \\nonumber \\\\&&\\text{momentum density:} \\\\&&\\hat{ {g}}^a(\\mathbf {r};\\Gamma ) \\equiv \\sum _i {p}^a_i \\, \\delta (\\mathbf {r} - \\mathbf {r}_i)\\, , \\nonumber $ where $\\hat{n}$ denotes the particle density.", "The extensive quantities are given by integrating these densities over space: $M=\\int \\hat{\\rho }\\, d{\\bf r}$ , $E=\\int \\hat{e}\\, d{\\bf r}$ , and ${\\bf P}=\\int \\hat{\\bf g}\\, d{\\bf r}$ .", "We note that a density of angular momentum could be introduced similarly.", "However, the density of angular momentum is not strictly local since it is defined with respect to a coordinate origin, so that its status is different from the five aforedefined densities [11].", "It is known [19] that the densities (REF )-() obey the following local conservation equations, $&\\partial _t\\hat{\\rho }(\\mathbf {r};\\Gamma _t) + \\nabla ^a \\hat{J}_\\rho ^{a}(\\mathbf {r};\\Gamma _t) = 0\\;, \\\\&\\partial _t\\hat{e}(\\mathbf {r};\\Gamma _t) + \\nabla ^a \\hat{J}_e^{a}(\\mathbf {r};\\Gamma _t) = 0\\;, \\\\&\\partial _t\\hat{{g}}^b(\\mathbf {r};\\Gamma _t) + \\nabla ^a \\hat{J}_{g^b}^{a}(\\mathbf {r};\\Gamma _t) = 0\\;, $ given in terms of the following current densities or fluxes, $&&\\hat{{J}}_\\rho ^{a}(\\mathbf {r};\\Gamma ) \\equiv \\hat{g}^a(\\mathbf {r};\\Gamma ) \\; , \\\\&&\\hat{{J}}_e^{a}(\\mathbf {r};\\Gamma ) \\equiv \\sum _i \\left[ \\frac{{\\bf p}_i^2}{2m} + \\frac{1}{2}\\sum _{j(\\ne i)} V(r_{ij}) \\right] \\frac{p_i^a}{m}\\, \\delta (\\mathbf {r} - \\mathbf {r}_i) \\nonumber \\\\&&\\quad + \\frac{1}{2}\\sum _{i<j}({r}_i^a - r_j^a)\\, \\frac{{p}^b_i + {p}^b_j}{m}\\, {F}^b_{ij}\\, D(\\mathbf {r};\\mathbf {r}_i,\\mathbf {r}_j) \\; , \\\\&&\\hat{{J}}_{g^b}^{a}(\\mathbf {r};\\Gamma ) \\equiv \\sum _{i}\\frac{p_i^ap_i^b}{m}\\, \\delta (\\mathbf {r} - \\mathbf {r}_i) \\nonumber \\\\&&\\qquad \\qquad + \\sum _{i<j} ({r}_i^a - r_j^a) \\, {F}^b_{ij} \\, D(\\mathbf {r};\\mathbf {r}_i,\\mathbf {r}_j) \\; ,$ where $F^b_{ij}\\equiv - \\frac{\\partial V(r_{ij})}{\\partial r_i^b}\\;,$ is the force exerted on the particle $i$ by the particle $j$ , and $D(\\mathbf {r};\\mathbf {r}_i,\\mathbf {r}_j) \\equiv \\int _{0}^1 d\\xi \\ \\delta \\left[\\mathbf {r} - \\mathbf {r}_i + (\\mathbf {r}_i - \\mathbf {r}_j)\\xi \\right]\\;,$ is a uniform linear density distributed on the straight line joining the positions of the particles $i$ and $j$ [35].", "As expected, there are five hydrodynamic variables coming from the conservation laws." ], [ "Breaking of continuous symmetries", "A low enough temperature, phase transitions happen from normal fluids to liquid crystals or crystalline solids.", "In these new phases, continuous symmetries are broken in the structure of matter at equilibrium.", "In nematic liquid crystals, continuous rotational symmetry is broken by the emergence of a special orientation of the molecules, while the continuous translational symmetry is broken in crystals, where only discrete translational symmetry remains.", "In these phases of matter, the continuous symmetries of uniform and isotropic normal fluids are thus broken.", "Such phenomena are not described by Gibbsian statistical distributions based on the Hamiltonian function (REF ) since this latter is invariant under continuous translations and rotations.", "Therefore, an external potential energy should be added to the Hamiltonian in order to break explicitly the symmetries, $H_{\\epsilon } &\\equiv & H +\\epsilon \\sum _i V^{\\rm (ext)}({\\bf r}_i) \\nonumber \\\\&=& H +\\epsilon \\int V^{\\rm (ext)}({\\bf r}) \\, \\hat{n}({\\bf r};\\Gamma )\\, d{\\bf r} \\, .$ The symmetric Hamiltonian function (REF ) is recovered in the limit $\\epsilon \\rightarrow 0$ .", "At the inverse temperature $\\beta =(k_{\\rm B}T)^{-1}$ and chemical potential $\\mu $ , the equilibrium phase of matter can be described by the following Gibbsian grand canonical probability distribution $p_{\\rm eq}(\\Gamma ) = \\frac{1}{\\Xi \\, \\Delta \\Gamma } \\, {\\rm e}^{-\\beta (H_{\\epsilon }-\\mu M) } \\, ,$ expressed in terms of the total mass $M=mN$ and depending on the random variables $\\Gamma =(\\Gamma _N,N)$ since the particle number $N$ is also a random variable.", "In Eq.", "(REF ), $\\Xi $ is the partition function given by the normalization condition $\\int p(\\Gamma )\\, d\\Gamma = \\sum _{N=0}^{\\infty }\\frac{1}{N!", "}\\int _{{\\mathbb {R}}^{6N}} p(\\Gamma _N,N)\\, d\\Gamma _N = 1 \\, ,$ and $\\Delta \\Gamma =h^{3N}$ is the elementary phase-space volume, where $h$ is Planck's constant.", "In the following, Boltzmann's constant is set equal to unity, $k_{\\rm B}=1$ , except if it is explicitly written.", "The statistical average with respect to the probability distribution is denoted $\\langle A\\rangle \\equiv \\int A(\\Gamma )\\, p(\\Gamma )\\, d\\Gamma $ .", "Because of the external potential $V^{\\rm (ext)}$ , continuous symmetries are explicitly broken.", "For instance in crystals, the particle density can now have an equilibrium mean value $n_{\\rm eq}({\\bf r})=\\langle \\hat{n}({\\bf r};\\Gamma )\\rangle _{\\rm eq}$ that is a periodic function in space, which is invariant under the discrete crystalline group of symmetry, but no longer under the continuous group of three-dimensional translations and rotations.", "We note that the system can undergo a phenomenon of spontaneous symmetry breaking without the help of the external potential, i.e., for the Hamiltonian function (REF ) in the limit $\\epsilon \\rightarrow 0$ .", "For $\\epsilon =0$ , all the continuous symmetries are restored for the probability distribution (REF ).", "However, the system manifests long-range order.", "For instance, in crystals, the particle density is periodic with respect to the center of mass of the crystal.", "Accordingly, the symmetric Hamiltonian function (REF ) should be split into the part describing the motion of the center of mass and the other part ruling the dynamics in the frame moving with the center of mass.", "This latter part is no longer symmetric under continuous translations and can itself define a grand canonical probability distribution with the broken symmetry.", "This reasoning shows that including the center of mass in the Hamiltonian function restores the continuous symmetry in Gibbsian equilibrium probability distributions.", "Similarly, in nematic liquid crystals, the part of the Hamiltonian ruling the rotation of the system around the orientation selected by spontaneous symmetry breaking should be separated from the rest of the Hamiltonian function in order to define equilibrium probability distributions describing the properties of the phase with broken symmetry.", "Phases with broken symmetries can be characterized by local order parameters denoted $\\hat{x}^\\alpha $ , where the index $\\alpha $ runs over the subset of variables originating from symmetry breaking.", "The decay of these variables is given by a rate $\\hat{J}_{x^\\alpha }$ such that $\\partial _t\\hat{x}^\\alpha (\\mathbf {r};\\Gamma _t) + \\hat{J}_{x^\\alpha }(\\mathbf {r};\\Gamma _t) &= 0\\; .$ We may also introduce the variables $\\hat{u}^{b\\alpha }\\equiv \\nabla ^b\\hat{x}^\\alpha $ that obey the following equations similar to the local conservation equations (REF )-(), $\\partial _t\\hat{u}^{b\\alpha }(\\mathbf {r};\\Gamma _t) + \\nabla ^a\\hat{J}^a_{u^{b\\alpha }}(\\mathbf {r};\\Gamma _t) &= 0\\;,$ where the corresponding current density is defined as $\\hat{J}^a_{u^{b\\alpha }} & \\equiv \\delta ^{ab} \\hat{J}_{x^\\alpha }\\;.$ The microscopic expression for $\\hat{x}^\\alpha $ depends on the phase in consideration, as discussed in Secs.", "and  for crystalline solids and liquid crystals.", "However, it is possible to proceed with a general derivation of the macroscopic equations without using any explicit microscopic expression for $\\hat{x}^\\alpha $ .", "Because of long-range order, the equilibrium correlation functions of the variables $\\hat{x}^\\alpha $ decay in space as $\\langle \\delta \\hat{x}^\\alpha ({\\bf r}) \\, \\delta \\hat{x}^\\beta ({\\bf r^{\\prime }}) \\rangle _{\\rm eq} \\sim \\Vert {\\bf r}-{\\bf r^{\\prime }}\\Vert ^{-1} \\, ,$ or for their Fourier transforms, $\\delta \\hat{\\tilde{x}}^\\alpha ({\\bf q})\\equiv \\int \\exp (-\\imath {\\bf q}\\cdot {\\bf r})\\, \\delta \\hat{x}^\\alpha ({\\bf r})\\, d{\\bf r} \\, ,$ according to $\\langle \\delta \\hat{\\tilde{x}}^\\alpha ({\\bf q}) \\, \\delta \\hat{\\tilde{x}}^\\beta ({\\bf -q}) \\rangle _{\\rm eq} \\sim \\Vert {\\bf q}\\Vert ^{-2} \\, .$ In this regard, these order parameters have a singular behavior, contrary to regular conserved quantities that have correlations of medium or short range.", "Nevertheless, the gradients $\\hat{u}^{a\\alpha }\\equiv \\nabla ^a\\hat{x}^\\alpha $ of the order parameters have medium- or short-ranged correlations because $\\langle \\delta \\hat{\\tilde{u}}^{a\\alpha }({\\bf q}) \\, \\delta \\hat{\\tilde{u}}^{b\\beta }({\\bf -q}) \\rangle _{\\rm eq} \\sim \\Vert {\\bf q}\\Vert ^{0} \\, ,$ so that they are regular as for the conserved variables." ], [ "Nambu-Goldstone modes", "As a consequence of the emergence of long-range order, there exist Nambu-Goldstone modes behaving as the conserved modes with vanishing dispersion relations for long enough wave length [8], [9], [6].", "The equations ruling all the conserved and kinetic modes $\\psi ^{\\beta }=\\langle \\hat{\\psi }^{\\beta }\\rangle $ could be written in the following form $\\partial _t \\, \\psi ^{\\beta }+ F^{\\beta }(\\psi ^{\\gamma },\\nabla ^c\\psi ^{\\gamma },\\dots ) = 0\\, .$ Let us suppose that there exists an equilibrium solution $\\psi ^{\\beta }_{\\rm eq}({\\bf r};x^{\\alpha }) \\, ,$ depending on the uniform equilibrium values of the order parameters $x^{\\alpha }$ , such as the global displacement vector in crystals, or the global director in nematic liquid crystals.", "Since Eq.", "(REF ) is a stationary solution of Eq.", "(REF ), we have that $F^{\\beta }(\\psi ^{\\gamma }_{\\rm eq},\\nabla ^c\\psi ^{\\gamma }_{\\rm eq},\\dots ) = 0\\, .$ Now, we may consider small perturbations of different kinds with respect to this equilibrium solution.", "On the one hand, an additive perturbation may be considered giving solutions of the following form, $\\psi ^{\\beta }({\\bf r},t) = \\psi ^{\\beta }_{\\rm eq}({\\bf r};x^{\\alpha }) + \\delta \\psi ^{\\beta }({\\bf r},t) \\, .$ Substituting into Eq.", "(REF ) and linearizing, we obtain the following evolution equations, $&& \\partial _t \\, \\delta \\psi ^{\\beta } + \\underbrace{\\left(F^{\\beta }\\right)_{\\rm eq}}_{=\\, 0} + \\left( \\frac{\\partial F^{\\beta }}{\\partial \\psi ^{\\gamma }}\\right)_{\\rm eq} \\delta \\psi ^{\\gamma } \\nonumber \\\\&&\\ \\ + \\left( \\frac{\\partial F^{\\beta }}{\\partial \\nabla ^c\\psi ^{\\gamma }}\\right)_{\\rm eq} \\nabla ^c\\delta \\psi ^{\\gamma } + \\cdots = 0\\, .$ Such a mode is decaying exponentially in time with a rate that is not vanishing for long enough wave length, because $\\left(\\partial F^{\\beta }/\\partial \\psi ^{\\gamma }\\right)_{\\rm eq}$ defines a matrix with non-vanishing elements in general.", "On the other hand, another perturbation may be considered where the order parameters are locally varying in space and time as $\\psi ^{\\beta }({\\bf r},t) = \\psi ^{\\beta }_{\\rm eq}\\left[{\\bf r};x^{\\alpha }({\\bf r},t)\\right] \\, .$ For this solution, we have that $\\partial _t \\psi ^{\\beta } &=& \\frac{\\partial \\psi ^{\\beta }_{\\rm eq}}{\\partial x^{\\alpha }}\\, \\partial _t x^{\\alpha } \\, , \\\\\\nabla ^c\\psi ^{\\gamma } &=& \\nabla ^c\\psi ^{\\gamma }_{\\rm eq} + \\frac{\\partial \\psi ^{\\gamma }_{\\rm eq}}{\\partial x^{\\alpha }}\\, \\nabla ^c x^{\\alpha } \\, .$ Now, substituting into Eq.", "(REF ) and linearizing, we find $&&\\frac{\\partial \\psi ^{\\beta }_{\\rm eq}}{\\partial x^{\\alpha }}\\, \\partial _t x^{\\alpha } + \\underbrace{\\left(F^{\\beta }\\right)_{\\rm eq}}_{=\\, 0} \\nonumber \\\\&& \\ \\ + \\left( \\frac{\\partial F^{\\beta }}{\\partial \\nabla ^c\\psi ^{\\gamma }}\\right)_{\\rm eq} \\frac{\\partial \\psi ^{\\gamma }_{\\rm eq}}{\\partial x^{\\alpha }}\\, \\nabla ^c x^{\\alpha } + \\cdots = 0\\, , \\qquad $ where the term with the non-vanishing coefficients $\\left(\\partial F^{\\beta }/\\partial \\psi ^{\\gamma }\\right)_{\\rm eq}$ no longer appears.", "If we multiply Eq.", "(REF ) by $\\partial \\psi ^{\\beta }_{\\rm eq}/\\partial x^{\\gamma }$ , sum over $\\beta $ , integrate over the volume $V$ of the system to average out the local variations of the symmetry-breaking equilibrium solution, and relabel the quantities, we get ${\\mathcal {N}}^{\\alpha \\beta } \\, \\partial _t\\, x^{\\beta } + {\\mathcal {M}}^{\\alpha b\\beta } \\, \\nabla ^b x^{\\beta } + \\cdots = 0 \\, ,$ where ${\\mathcal {N}}^{\\alpha \\beta } &\\equiv & \\frac{1}{V} \\int _V \\frac{\\partial \\psi ^{\\gamma }_{\\rm eq}}{\\partial x^{\\alpha }}\\, \\frac{\\partial \\psi ^{\\gamma }_{\\rm eq}}{\\partial x^{\\beta }} \\, d{\\bf r} \\, , \\\\{\\mathcal {M}}^{\\alpha b\\beta } &\\equiv & \\frac{1}{V} \\int _V \\frac{\\partial \\psi ^{\\gamma }_{\\rm eq}}{\\partial x^{\\alpha }} \\left( \\frac{\\partial F^{\\gamma }}{\\partial \\nabla ^b\\psi ^{\\delta }}\\right)_{\\rm eq} \\frac{\\partial \\psi ^{\\delta }_{\\rm eq}}{\\partial x^{\\beta }} \\, d{\\bf r} \\, .", "\\quad \\qquad $ In Eq.", "(REF ), the dots denote terms with higher spatial derivatives for $x^{\\beta }({\\bf r},t)$ .", "Since the solution (REF ) is breaking continuous symmetries, we have that $(\\partial \\psi ^{\\beta }_{\\rm eq}/\\partial x^{\\alpha })\\ne 0$ and thus the matrix $\\mathcal {N}=({\\mathcal {N}}^{\\alpha \\beta })$ is non vanishing and can be inverted to define the quantity $\\mathcal {V}=\\mathcal {N}^{-1}\\cdot \\mathcal {M}$ .", "If we suppose that the local order parameters behave as $x({\\bf r},t)\\sim \\exp (\\imath {\\bf q}\\cdot {\\bf r}-\\imath \\omega t)$ , the frequency $\\omega $ is related to the wave vector $\\bf q$ according to $\\left[ \\omega \\, \\mathsf {1} - \\mathcal {V}\\cdot {\\bf q} + O({\\bf q}^2)\\right]\\cdot x = 0 \\, ,$ showing that the dispersion relations of these solutions are vanishing with the wave number as $\\lim _{{\\bf q}\\rightarrow 0}\\omega ({\\bf q})=0$ , as in the case of locally conserved quantities.", "The eigenvalues of the matrix $ \\mathcal {V}\\cdot {\\bf q}$ are giving the propagation speed of the modes.", "The mode is diffusive if its propagation speed is equal to zero.", "The existence of the Nambu-Goldstone modes is thus a consequence of continuous symmetry breaking.", "There are as many such modes as components of the vector $x=(x^{\\alpha })$ , i.e., as continuous symmetries that are broken, which is the statement of the Goldstone theorem [9], [6].", "In order to investigate the effects of the Nambu-Goldstone modes, we should thus consider Eq.", "(REF ) on the same footing as the equations (REF )-() for the locally conserved quantities." ], [ "Nonequilibrium statistical mechanics", "In this section, we extend the formalism introduced for normal fluids [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35] to phases with broken continuous symmetries." ], [ "Time evolution", "On the one hand, Eqs.", "(REF )-() for the locally conserved quantities, as well Eq.", "(REF ) for the gradients of the order parameters can all be written as $\\partial _t\\, \\hat{c}^{\\alpha }(\\mathbf {r},t) + \\nabla ^a \\hat{J}^{a}_{c^\\alpha }(\\mathbf {r},t) & = 0\\;, $ where $(\\hat{c}^{\\alpha }) &=& (\\hat{e},\\hat{\\rho },\\hat{g}^b,\\hat{u}^{b\\beta }) \\qquad \\mbox{and} \\\\(\\hat{J}^{a}_{c^\\alpha }) &=& (\\hat{J}^{a}_{e},\\hat{J}^{a}_{\\rho },\\hat{J}^{a}_{g^b},\\hat{J}^{a}_{u^{b\\beta }})$ are respectively the densities and the corresponding current densities or fluxes.", "At time $t$ , the densities are given in terms of the Liouvillian operator $\\mathcal {L}$ or the trajectories $\\Gamma _t$ by $\\hat{c}^{\\alpha }(\\mathbf {r},t) \\equiv {\\rm e}^{{\\mathcal {L}} t} \\, \\hat{c}^{\\alpha }(\\mathbf {r};\\Gamma ) = \\hat{c}^{\\alpha }(\\mathbf {r};\\Gamma _t)$ with similar expressions for the current densities.", "On the other hand, any phase-space probability density $p_t(\\Gamma )$ at time $t$ is given by $p_t(\\Gamma ) = {\\rm e}^{-{\\mathcal {L}} t} \\, p_0(\\Gamma ) = p_0(\\Gamma _{-t})$ in terms of the initial probability density $p_0(\\Gamma )$ and the reversed trajectory $\\Gamma _{-t}$ going from the current phase-space point $\\Gamma $ back to the initial conditions $\\Gamma _0$ of the trajectory.", "Consequently, the macroscopic densities can be obtained by taking the mean value of the time-independent densities over the time-evolved probability distribution $p_t(\\Gamma )$ or, equivalently, the mean values of the time-dependent densities over the initial probability distribution $p_0(\\Gamma )$ , $\\langle \\hat{c}^{\\alpha }(\\mathbf {r};\\Gamma )\\rangle _t &\\equiv & \\int d\\Gamma \\, p_t(\\Gamma ) \\,\\hat{c}^{\\alpha }(\\mathbf {r};\\Gamma ) \\nonumber \\\\&=& \\int d\\Gamma _0 \\, p_0(\\Gamma _0) \\, \\hat{c}^{\\alpha }(\\mathbf {r};\\Gamma _t) \\, , \\qquad $ because of Liouville's theorem $d\\Gamma _0=d\\Gamma _t$ , the expectation value with respect to $p_{t}(\\Gamma ) $ being denoted as $\\langle \\cdot \\rangle _{{t}}$ .", "Similar results hold for the current densities and other fields." ], [ "Local equilibrium distribution", "The key assumption of the formalism is the initial condition being the local equilibrium distribution $p_{\\rm leq}(\\Gamma ;\\lambda ) & = \\frac{1}{\\Delta \\Gamma } \\, \\exp \\left[ - \\lambda ^{\\alpha }\\ast \\hat{c}^{\\alpha } (\\Gamma )- \\Omega (\\lambda ) \\right] \\;,$ where $\\lambda =(\\lambda ^{\\alpha })=(\\lambda _{c^{\\alpha }})$ are inhomogeneous fields conjugated to the density fields $\\hat{\\bf c}=(\\hat{c}^{\\alpha })$ , and the asterisk $\\ast $ corresponds to the integration over space $f\\ast g & \\equiv \\int d\\mathbf {r}\\, f(\\mathbf {r}) \\, g(\\mathbf {r})\\, .$ The normalization condition (REF ) for the local equilibrium distribution (REF ) gives the functional $\\Omega (\\lambda ) & = \\ln \\int \\frac{d\\Gamma }{\\Delta \\Gamma }\\, \\exp \\left[ - \\lambda ^{\\alpha }\\ast \\hat{c}^{\\alpha }(\\Gamma )\\right] \\, .$ The expectation value with respect to the local equilibrium distribution (REF ) is denoted by $\\langle \\cdot \\rangle _{{\\rm leq},\\lambda }$ .", "In this formalism, the expectation values of the densities can be obtained by taking the functional derivative of the functional (REF ) with respect to the conjugated fields as follows, $c^{\\alpha }({\\bf r}) =- \\frac{\\delta \\Omega (\\lambda )}{\\delta \\lambda ^{\\alpha }({\\bf r})} \\, , \\quad \\mbox{where}\\quad c^{\\alpha }({\\bf r})\\equiv \\langle \\hat{c}^{\\alpha }(\\mathbf {r};\\Gamma )\\rangle _{{\\rm leq},\\lambda } \\, .$ The entropy is defined as $S \\equiv -\\int p(\\Gamma ) \\, \\ln \\left[ p(\\Gamma ) \\Delta \\Gamma \\right] d\\Gamma \\, ,$ leading for the local equilibrium distribution (REF ) to the entropy functional $S({\\bf c}) = \\inf _{\\lambda }\\left[\\lambda ^\\alpha \\ast {c}^{\\alpha }+\\Omega (\\lambda ) \\right] ,$ which is the Legendre transform of the previously introduced functional (REF ).", "The conjugated fields are thus given by the following functional derivatives, $\\lambda ^{\\alpha }({\\bf r}) = \\frac{\\delta S({\\bf c})}{\\delta c^{\\alpha }({\\bf r})} \\, ,$ this relation being called the second identity in Ref. [35].", "Vice versa, the Legendre transform of the entropy functional (REF ) gives back the functional (REF ).", "We note that the equilibrium grand canonical distribution (REF ) with $\\epsilon =0$ is recovered if the conjugated fields $\\lambda $ are uniform with $\\lambda _e=\\beta $ , $\\lambda _\\rho = -\\beta \\mu $ , $\\lambda _{g^a}=0$ , $\\lambda _{u^{a\\alpha }}=0$ , and $\\Omega =\\ln \\Xi $ ." ], [ "Time evolution of the local equilibrium distribution", "The basic idea of the formalism is that the time-evolved probability density (REF ) should remain close to the local equilibrium distribution (REF ) with the conjugated fields $\\lambda _t$ considered at time $t$ .", "In this regard, these latter should be determined by the conditions $\\langle \\hat{c}^{\\alpha }(\\mathbf {r};\\Gamma )\\rangle _t = \\langle \\hat{c}^{\\alpha }(\\mathbf {r};\\Gamma )\\rangle _{{\\rm leq},\\lambda _t} \\equiv c^{\\alpha }(\\mathbf {r},t) \\, ,$ according to which the expectation values (REF ) of the densities with respect to the probability distribution $p_t(\\Gamma )$ are equal to their expectation values with respect to the local equilibrium distribution with the conjugated fields $\\lambda _t$ at time $t$ .", "The conditions (REF ) are thus defining the macroscopic densities $c^{\\alpha }(\\mathbf {r},t)$ , which are given by the functional derivatives (REF ) of the functional (REF ) with respect to the conjugated fields $\\lambda =\\lambda _t$ at time $t$ .", "Now, we consider the time evolution of the probability density starting from the initial condition given by the local equilibrium distribution (REF ) with $\\lambda =\\lambda _0$ , $p_{t}(\\Gamma ) &=& {\\rm e}^{-{\\mathcal {L}} t} p_{\\rm leq}(\\Gamma ;\\lambda _0) \\\\&=& \\frac{1}{\\Delta \\Gamma } \\, \\exp \\left[ - \\lambda _0^{\\alpha }\\ast \\hat{c}^{\\alpha } (\\Gamma _{-t})- \\Omega (\\lambda _0)\\right] .", "\\qquad \\nonumber $ Since the normalization of this probability distribution should be preserved during the time evolution, we must have that $(d/dt)\\int p_t(\\Gamma )\\, d\\Gamma =0$ .", "The calculation using Eq.", "(REF ) leads to the following relation, which should hold for any conjugated field $\\lambda _0$ that may thus be replaced by $\\lambda $ to get $\\nabla ^a\\lambda ^\\alpha \\ast \\langle \\hat{J}^{a}_{c^{\\alpha }}\\rangle _{{\\rm leq},{\\lambda }} = 0\\;.", "$ This relation has been obtained notably in Refs.", "[31], [35] and is called the first identity in Ref. [35].", "Remarkably, we have that $p_t(\\Gamma ) = p_{\\rm leq}(\\Gamma ;\\lambda _t) \\, {\\rm e}^{\\Sigma _t(\\Gamma )}$ with the quantity $\\Sigma _t(\\Gamma ) & \\equiv \\int _0^t d\\tau \\, \\partial _\\tau \\left[{\\lambda }_\\tau ^\\alpha \\ast \\hat{c}^\\alpha (\\Gamma _{\\tau - t}) + \\Omega (\\lambda _\\tau )\\right] , $ as shown in Refs.", "[25], [35].", "Therefore, the expectation value of any observable $A(\\Gamma )$ with respect to the time-evolved probability distribution $p_t(\\Gamma )$ is thus given in terms of the expectation value with respect to the local equilibrium distribution according to $\\langle A(\\Gamma ) \\rangle _t = \\langle A(\\Gamma )\\, {\\rm e}^{\\Sigma _t(\\Gamma )}\\rangle _{{\\rm leq},{\\lambda }_t}\\;,$ which is called the third identity in Ref. [35].", "In particular, the conditions (REF ) are equivalent to the following relations, $\\langle \\hat{c}^{\\alpha }({\\bf r};\\Gamma ) \\big [ {\\rm e}^{\\Sigma _t(\\Gamma )}-1\\big ]\\rangle _{{\\rm leq},\\lambda _t} = 0 \\, .$" ], [ "Entropy production and dissipative current densities", "The identity (REF ) is a universal relation, which is reminiscent of the nonequilibrium work and integral fluctuation theorems [35], [36], [37], [38], [39], [40], [41], [42], [43].", "Choosing $A(\\Gamma ) = {\\rm e}^{-\\Sigma _t(\\Gamma )}$ in this third identity gives [35] $\\langle {\\rm e}^{-\\Sigma _t(\\Gamma )} \\rangle _t & = 1\\;,$ which implies by Jensen's inequality [44] that $\\langle \\Sigma _t(\\Gamma ) \\rangle _t & = S({\\bf c}_t)- S({\\bf c}_0) \\ge 0\\; .$ In open systems, the entropy $S$ changes in time due to the exchanges $d_{\\rm e}S$ with the environment and its production $d_{\\rm i}S$ inside the system: $dS=d_{\\rm e}S+d_{\\rm i}S$ .", "Since the system is here isolated, there is no exchange with the environment $d_{\\rm e}S=0$ , so that the change in time of the entropy is equal to the entropy production $dS=d_{\\rm i}S$ .", "In this regard, the result (REF ) may be interpreted as the non-negativity of the entropy production, in agreement with the second law of thermodynamics.", "Using Eqs.", "(REF ) and (REF ), we have that $\\frac{d}{dt}\\, \\Omega (\\lambda _t) = -\\partial _t\\lambda _t^\\alpha \\ast \\langle \\hat{c}^\\alpha \\rangle _{{\\rm leq},{\\lambda }_t}$ and $\\frac{d}{dt}\\lambda _t^\\alpha \\ast \\langle \\hat{c}^\\alpha \\rangle _{{\\rm leq},{\\lambda }_t}=\\partial _t\\lambda _t^\\alpha \\ast \\langle \\hat{c}^\\alpha \\rangle _{{\\rm leq},{\\lambda }_t}+\\lambda _t^\\alpha \\ast \\partial _t\\langle \\hat{c}^\\alpha \\rangle _{{\\rm leq},{\\lambda }_t} \\;.$ According to the definition (REF ) of the entropy functional, the relation (REF ), Eq.", "(REF ), and integrations by parts, the entropy production is thus given by $&&\\frac{d_{\\rm i}S}{dt} = \\frac{dS}{dt} = \\frac{d}{dt} \\left[\\lambda _t^\\alpha \\ast \\langle \\hat{c}^\\alpha \\rangle _{{\\rm leq},{\\lambda }_t}+\\Omega (\\lambda _t)\\right] \\\\&&= \\lambda _t^\\alpha \\ast \\partial _t\\langle \\hat{c}^\\alpha \\rangle _{t}= - \\lambda ^{\\alpha }_t\\ast \\nabla ^a\\langle \\hat{J}^a_{c^{\\alpha }}\\rangle _{t}= \\nabla ^a\\lambda ^{\\alpha }_t\\ast \\langle \\hat{J}^a_{c^{\\alpha }}\\rangle _{t} \\;.", "\\nonumber $ Now, using the identity (REF ) with $A$ taken as $\\hat{J}^a_{c^\\alpha }$ , the expectation values of the current densities with respect to the phase-space probability distribution (REF ) can be decomposed as [25] $J^{a}_{c^\\alpha }(\\mathbf {r},t) & \\equiv \\langle \\hat{J}^{a}_{{c}^\\alpha }(\\mathbf {r};\\Gamma ) \\rangle _t\\\\& = \\langle \\hat{J}^{a}_{{c}^\\alpha }(\\mathbf {r};\\Gamma )\\left\\lbrace 1 + \\big [{\\rm e}^{\\Sigma _t(\\Gamma )} - 1\\big ]\\right\\rbrace \\rangle _{{\\rm leq},{\\lambda }_t}\\\\& = \\bar{J}^{a}_{c^\\alpha }(\\mathbf {r},t) + \\mathcal {J}^{a}_{c^\\alpha }(\\mathbf {r},t) $ into $\\bar{J}^{a}_{c^{\\alpha }}(\\mathbf {r},t) & \\equiv \\langle \\hat{J}^a_{c^{\\alpha }}(\\mathbf {r};\\Gamma )\\rangle _{{\\rm leq},\\lambda _t} $ and ${\\mathcal {J}}^a_{c^{\\alpha }}(\\mathbf {r},t) & \\equiv \\langle \\hat{J}^a_{c^{\\alpha }}(\\mathbf {r};\\Gamma )\\big [{\\rm e}^{\\Sigma _t(\\Gamma )}-1\\big ]\\rangle _{{\\rm leq},\\lambda _t}\\;.$ The entropy production is thus given by $\\frac{d_{\\rm i}S}{dt} = \\underbrace{\\nabla ^a\\lambda ^{\\alpha }_t\\ast \\langle \\hat{J}^a_{c^{\\alpha }}\\rangle _{{\\rm leq},\\lambda _t}}_{=\\, 0} + \\nabla ^a\\lambda ^{\\alpha }_t\\ast {\\mathcal {J}}^a_{c^{\\alpha }}(t)\\;, $ where the first term vanishes because of the identity (REF ).", "This term is thus expressing the conservation of entropy in adiabatic (isoentropic) processes induced by the dissipativeless current densities defined by Eqs.", "(REF ).", "The second term in Eq.", "(REF ) is in general non vanishing and related to the production of entropy, leading to the definition of the dissipative current densities by Eqs.", "(REF ).", "Accordingly, the entropy production can be expressed as $\\frac{d_{\\rm i}S}{dt} = \\nabla ^a\\lambda ^{\\alpha }_t\\ast {\\mathcal {J}}^a_{c^{\\alpha }}(t) \\ge 0$ in terms of the dissipative current densities (REF ) and the gradients of the conjugated fields, which play the role of thermodynamic forces, also called the affinities, which is in accordance with macroscopic nonequilibrium thermodynamics [45], [46], [47], [48], [49].", "As we will show explicitly below, the three identities (REF ), (REF ), and (REF ) allow us to fully deduce the macroscopic equations and identify the dissipative coefficients.", "The local conservation equations for the mean values (REF ) can indeed be obtained from the expectation values of Eq.", "(REF ), giving $\\partial _t\\, c^{\\alpha } + \\nabla ^a\\left(\\bar{J}^{a}_{c^\\alpha }+{\\mathcal {J}}^{a}_{c^\\alpha }\\right) = 0$ in terms of the dissipativeless (REF ) and dissipative (REF ) current densities.", "In summary, the method is carried out as follows using expansions in powers of the gradients: First, the slow modes of the system are identified and the local thermodynamic relations between these variables are established at leading order in the gradients.", "Once the hydrodynamic variables are identified, the identity (REF ) is used to obtain the conjugate fields $\\lambda $ .", "The dissipativeless current densities are computed by taking the expectation values of the microscopic current densities over the local equilibrium distribution, according to Eq.", "(REF ).", "The dissipative current densities arise from Eq.", "(REF ) as a direct consequence of the third identity (REF ).", "The Green-Kubo relations giving the linear response coefficients between the dissipative current densities and the gradients of the conjugated fields can thus be obtained.", "We now proceed with the explicit computation for systems with broken symmetries, such as crystalline solids and liquid crystals." ], [ "The local Euler, Gibbs, and Gibbs-Duhem relations", "The identity (REF ) shows that the dissipativeless current densities (REF ) are leaving the entropy constant in time.", "All the processes involved by the dissipativeless current densities may thus be considered as reversible (i.e., adiabatic or isoentropic).", "The approach of nonequilibrium statistical mechanics based on the local equilibrium distribution (REF ) is therefore providing the local thermodynamic relations in every element of matter, as shown here below.", "At leading order in the expansion in the gradients, the functionals (REF ) and (REF ) may be supposed of the forms $\\Omega (\\lambda ) &=& \\int \\omega (\\lambda ) \\, d{\\bf r} + O(\\nabla ^2) \\qquad \\mbox{and}\\nonumber \\\\S({\\bf c}) &=& \\int s({\\bf c}) \\, d{\\bf r} + O(\\nabla ^2) \\, ,$ defined by introducing the densities $\\omega (\\lambda )$ and $s({\\bf c})$ , which are respectively functions of the conjugated fields and mean densities.", "Since both functionals are interrelated by Legendre transforms, we have that $c^{\\alpha } = - \\frac{\\partial \\omega }{\\partial \\lambda ^{\\alpha }} \\qquad \\mbox{and}\\qquad \\lambda ^{\\alpha } = \\frac{\\partial s}{\\partial c^{\\alpha }} \\, ,$ giving the local relations $s = \\lambda ^{\\alpha } \\, c^{\\alpha } +\\omega \\, , \\quad ds =\\lambda ^{\\alpha } \\, dc^{\\alpha } \\, , \\quad d\\omega = - c^{\\alpha } \\, d\\lambda ^{\\alpha } \\, ,$ up to terms of second order in the gradients.", "In Eq.", "(REF ), the first relation can be identified as the local Euler relation, the second as the local Gibbs relation for the entropy density, and the third as the associated Gibbs-Duhem relation.", "At this stage, a comparison becomes possible with previous works on the thermodynamics of matter with broken symmetry [13], [14], where the relevant thermodynamic relations are given in the laboratory frame by $&\\text{Euler relation:} \\\\ &\\quad e = Ts + \\mu \\rho + {v}^a {g}^a + {\\phi }^{a\\alpha }u^{a\\alpha } - p\\;,\\\\&\\text{Gibbs relation:} \\\\&\\quad de = Tds + \\mu d\\rho + {v}^a d{g}^a + {\\phi }^{a\\alpha }du^{a\\alpha }\\;,\\\\&\\text{Gibbs-Duhem relation:}\\\\&\\quad dp = sdT + \\rho d\\mu + {g}^a d{v}^a + u^{a\\alpha }d{\\phi }^{a\\alpha }\\;,$ where $e\\equiv E/V$ is the mean energy density, $T$ the temperature, $s\\equiv S/V$ the entropy density, $\\mu $ the chemical potential, $\\rho \\equiv M/V$ the mean mass density, $v^a$ the velocity, $g^a=\\rho v^a$ the mean momentum density, $u^{a\\alpha }$ the gradients of the mean order parameters $x^{\\alpha }$ , $\\phi ^{a\\alpha }$ the fields thermodynamically conjugated to $u^{a\\alpha }$ , and $p$ the hydrostatic pressure." ], [ "The conjugated fields", "The Euler relation (REF ) can be written to give the entropy density instead of the energy density.", "After substitution into the entropy functional (REF ) and taking the functional derivatives (REF ), the conjugated fields are obtained as $\\lambda _e(\\mathbf {r},t) & \\equiv \\frac{\\delta S({\\bf c})}{\\delta e({\\bf r},t)} = \\beta (\\mathbf {r},t) + O(\\nabla ^2)\\;, \\\\\\lambda _\\rho (\\mathbf {r},t) & \\equiv \\frac{\\delta S({\\bf c})}{\\delta \\rho ({\\bf r},t)} = - \\beta (\\mathbf {r},t)\\,\\mu (\\mathbf {r},t) + O(\\nabla ^2)\\;, \\\\\\lambda _{g^a}(\\mathbf {r},t) & \\equiv \\frac{\\delta S({\\bf c})}{\\delta g^a({\\bf r},t)} = - \\beta (\\mathbf {r},t)\\, v^a(\\mathbf {r},t) + O(\\nabla ^2)\\;, \\\\\\lambda _{u^{a\\alpha }}(\\mathbf {r},t) & \\equiv \\frac{\\delta S({\\bf c})}{\\delta u^{a\\alpha }({\\bf r},t)} = - \\beta (\\mathbf {r},t)\\, \\phi ^{a\\alpha }(\\mathbf {r},t) + O(\\nabla ^2)\\;, $ in terms of the local inverse temperature $\\beta =(k_{\\rm B}T)^{-1}$ .", "Furthermore, the comparison between the phenomenological Euler relation (REF ) and the theoretical expression given by the third relation in Eq.", "(REF ) allows us to identify the Legendre transform of the entropy density as the local thermodynamic potential $\\omega =\\beta p$ , proportional to the hydrostatic pressure $p$ and referred to as a Massieu function (here, per unit volume) [49].", "Now, the Gibbs relation () written for the entropy density gives the relations $&&\\frac{1}{T} =\\left(\\frac{\\partial s}{\\partial e}\\right)_{\\mathbf {g},\\rho ,\\mathsf {u}},\\qquad - \\frac{\\mu }{T} = \\left(\\frac{\\partial s}{\\partial \\rho }\\right)_{e,\\mathbf {g},\\mathsf {u}}, \\\\&& - \\frac{v^a}{T} = \\left(\\frac{\\partial s}{\\partial g^a}\\right)_{e,\\rho ,\\mathsf {u}},\\qquad - \\frac{\\phi ^{a\\alpha }}{T}=\\left(\\frac{\\partial s}{\\partial u^{a\\alpha }}\\right)_{e,\\mathbf {g},\\rho },\\nonumber $ holding in the laboratory frame where the center of mass of the matter element moves at the velocity ${\\bf v}=(v^x,v^y,v^z)$ .", "In the frame moving with the element where ${\\bf v}=0$ , the energy density and the chemical potential are given by $e_0 = e-\\frac{1}{2} \\, \\rho \\, {\\bf v}^2 \\, , \\qquad \\mu _0 = \\mu + \\frac{1}{2} \\, {\\bf v}^2 \\, ,$ and the relations (REF ) become $&&\\beta = \\left(\\frac{\\partial s}{\\partial e_0}\\right)_{\\rho ,\\mathsf {u}},\\qquad \\beta \\mu _0= - \\left(\\frac{\\partial s}{\\partial \\rho }\\right)_{e_0,\\mathsf {u}},\\nonumber \\\\&&\\beta \\phi ^{a\\alpha }= - \\left(\\frac{\\partial s}{\\partial u^{a\\alpha }}\\right)_{e_0,\\rho } .", "$" ], [ "Maxwell relations", "Using Eq.", "(REF ), the following Maxwell relations [49] are obtained $\\left(\\frac{\\partial \\beta }{\\partial \\rho }\\right)_{e_0,\\mathsf {u}} & = - \\left[\\frac{\\partial ( \\beta \\mu _0)}{\\partial e_0}\\right]_{\\rho ,\\mathsf {u}} ,\\\\\\left(\\frac{\\partial \\beta }{\\partial u^{a\\alpha }}\\right)_{e_0,\\rho } & = - \\left[\\frac{\\partial (\\beta \\phi ^{a\\alpha })}{\\partial e_0}\\right]_{\\rho ,\\mathsf {u}} ,\\\\\\left[\\frac{\\partial (\\beta \\mu _0)}{\\partial u^{a\\alpha }}\\right]_{e_0,\\rho } & = \\left[\\frac{\\partial (\\beta \\phi ^{a\\alpha })}{\\partial \\rho }\\right]_{e_0,\\mathsf {u}} ,\\\\\\left[\\frac{\\partial (\\beta \\phi ^{c\\gamma })}{\\partial u^{a\\alpha }}\\right]_{e_0,\\rho } & = \\left[\\frac{\\partial (\\beta \\phi ^{a\\alpha })}{\\partial u^{c\\gamma }}\\right]_{e_0,\\rho } .$" ], [ "Gibbs-Duhem relation and consequences", "The Massieu density $\\omega =\\beta p$ obeys the following Gibbs-Duhem relation $d\\omega = -e \\, d\\beta + \\rho \\, d(\\beta \\mu ) + g^a\\, d(\\beta v^a) + u^{a\\alpha }\\, d(\\beta \\phi ^{a\\alpha }) \\, ,$ which is equivalent to Eq. ().", "From this latter Gibbs-Duhem relation, we find $&& \\left(\\frac{\\partial p}{\\partial \\beta }\\right)_{\\beta \\mathbf {v},\\beta \\mu ,\\beta \\phi } = - \\frac{e + p}{\\beta }\\, ,\\ \\left[\\frac{\\partial p}{\\partial (\\beta \\mu )}\\right]_{\\beta ,\\beta \\mathbf {v},\\beta \\phi } = \\frac{\\rho }{\\beta }\\, , \\nonumber \\\\&& \\left[\\frac{\\partial p}{\\partial (\\beta v^a)}\\right]_{\\beta ,\\beta \\mu ,\\beta \\phi } = \\frac{g^a}{\\beta }\\, ,\\ \\left[\\frac{\\partial p}{\\partial (\\beta \\phi ^{a\\alpha })}\\right]_{\\beta ,\\beta \\mathbf {v},\\beta \\mu }=\\frac{u^{a\\alpha }}{\\beta }\\, .", "\\nonumber \\\\&&$ In the frame moving with the element where $\\mathbf {v}=0$ , we get $&&\\left(\\frac{\\partial p}{\\partial \\beta }\\right)_{\\beta \\mu _0,\\beta \\phi }= - \\frac{e_0 + p}{\\beta }\\, ,\\quad \\left[\\frac{\\partial p}{\\partial (\\beta \\mu _0)}\\right]_{\\beta ,\\beta \\phi }=\\frac{\\rho }{\\beta }\\, ,\\nonumber \\\\&& \\left[\\frac{\\partial p}{\\partial (\\beta \\phi ^{a\\alpha })}\\right]_{\\beta ,\\beta \\mu _0}=\\frac{u^{a\\alpha }}{\\beta }\\, ,$ where the subscript 0 denote the rest-frame quantities (REF )." ], [ "Dissipativeless current densities", "The expression (REF ) for the entropy production is showing that entropy is conserved if the dissipative parts of the current densities are vanishing.", "This is the case for the leading parts of the current densities giving the dissipativeless current densities (REF ).", "In normal fluids, these parts lead to Euler's equations.", "Our purpose is now to obtain their expressions in phases with broken symmetries.", "With this purpose, we first consider Galilean transformations from the laboratory frame to the frame moving at the velocity $\\bf v$ of the system.", "In this way, the expectation values of the current densities with respect to the local equilibrium distribution (REF ) can be calculated according to the definition  (REF ).", "In phases with broken symmetries, extra contributions are expected, which need to be determined, in particular, using the microscopic expressions of the corresponding decay rates $\\hat{J}_{x^{\\alpha }}$ introduced in Eq.", "(REF ).", "These microscopic expressions will be given in Secs.", "and  for crystals and liquid crystals.", "However, the most general form of their expectation value is known on the basis of time-reversal symmetry [13].", "Once this general form is fixed, the contributions of broken symmetry to the dissipativeless current densities of momentum and energy can be obtained using the identity (REF ) expressing the conservation of entropy, which finds its origin in the adiabaticity of the reversible processes ruled by the dissipativeless current densities (REF )." ], [ "Galilean transformation", "The particle momenta ${\\bf p}_i$ in the laboratory frame and those ${\\bf p}_{i0}$ in the frame moving at the local velocity ${\\bf v}({\\bf r})$ are related to each other according to ${\\bf p}_i={\\bf p}_{i0}+m{\\bf v}({\\bf r}_i)$ by the principle of Galilean relativity.", "Carrying out this change of variables in the microscopic expressions for the energy density $\\hat{e}$ , the momentum density $\\hat{g}^a$ , and their current densities, we find $\\hat{e} &=& \\hat{e}_0 + \\hat{g}^a_0 v^a + \\frac{1}{2}\\, \\hat{\\rho } \\, {\\bf v}^2\\, , \\\\\\hat{g}^a &=& \\hat{g}^a_0 + \\hat{\\rho }\\, v^a \\, ,\\\\\\hat{J}^a_{e} &=& \\hat{J}^a_{e 0} + \\hat{e}_0\\, v^a +\\hat{J}^a_{g^b 0} v^b + \\hat{g}^b_0\\, v^b\\, v^a \\nonumber \\\\&&+ \\frac{1}{2}\\, {\\bf v}^2 (\\hat{g}^a_0+\\hat{\\rho }\\, v^a) - \\hat{\\Delta }^a \\, , \\\\\\hat{J}^a_{g^b} &=& \\hat{J}^a_{g^b 0} + \\hat{g}^a_0\\,v^b + v^a\\,\\hat{g}^b_0+\\hat{\\rho }\\, v^b\\, v^a \\, , $ where the quantities with the subscript 0 are those with the momenta ${\\bf p}_{i0}$ replacing ${\\bf p}_{i}$ , and [35] $&&\\hat{\\Delta }^a({\\bf r};\\Gamma ) \\equiv \\frac{1}{2} \\sum _{i\\ne j} (r_i^a - r_j^a)\\, F_{ij}^b \\\\&&\\times \\left[v^b({\\bf r})-\\frac{v^b({\\bf r}_i)+v^b({\\bf r}_j)}{2}\\right] D({\\bf r};{\\bf r}_i,{\\bf r}_j) \\, .", "\\nonumber $ Similar expressions hold for the current densities $\\hat{J}^a_{u^{b\\alpha }}$ of the gradients of order parameters.", "In Eq.", "(), the contribution (REF ) would vanish if the velocity field $\\bf v$ was uniform.", "Furthermore, we note that $\\hat{\\Delta }^a$ goes as the square of gradients, in particular, the square of the gradients of the velocity field, so that this term can be dropped if the corrections of $O(\\nabla ^2)$ are neglected.", "Now, the terms with odd powers of the momenta ${\\bf p}_{i0}$ are vanishing after averaging over local equilibrium in the frame moving with the element of matter.", "Consequently, we get $\\langle \\hat{g}^a_0\\rangle _{\\rm leq} =0$ , $\\langle \\hat{J}^a_{e 0}\\rangle _{\\rm leq}=0$ , and $\\langle \\hat{\\rho }\\rangle _{\\rm leq}=\\rho $ .", "Moreover, the internal energy density is given by $\\langle \\hat{e}_0\\rangle _{\\rm leq}=e_0$ .", "The velocity field is thus defined as the ratio of the momentum to the mass densities: $v^a\\equiv \\langle \\hat{g}^a\\rangle _{\\rm leq}/\\langle \\hat{\\rho }\\rangle _{\\rm leq}$ .", "As aforementioned, we need to consider the possibility that symmetry breaking may introduce extra contributions in the local equilibrium expectation values of the currents densities of energy, momentum, and the gradients $u^{a\\alpha }$ of the order parameter, so that we obtain the following forms for these expectation values, $&&\\langle \\hat{e}\\rangle _{\\rm leq} = e\\, ,\\\\&&\\langle \\hat{g}^a\\rangle _{\\rm leq} = \\rho \\, v^a \\, ,\\\\&&\\langle \\hat{J}^a_{e}\\rangle _{\\rm leq} = (e+p)\\, v^a + \\bar{J}^a_{e}\\big \\vert _{\\rm BS} +O(\\nabla ^2) \\, , \\\\&&\\langle \\hat{J}^a_{g^b}\\rangle _{\\rm leq} = \\rho \\, v^b \\, v^a + p\\, \\delta ^{ab} + \\bar{J}^a_{g^b}\\big \\vert _{\\rm BS} +O(\\nabla ^2) \\, , \\ \\qquad \\\\&&\\langle \\hat{J}^a_{u^{b\\alpha }}\\rangle _{\\rm leq} = u^{b\\alpha }\\, v^a + \\bar{J}^a_{u^{b\\alpha }}\\big \\vert _{\\rm BS} +O(\\nabla ^2) \\, , $ where $e= e_0+\\rho {\\bf v}^2/2$ , the hydrostatic pressure $p$ is separated from the contributions $\\bar{J}^a_{e}\\big \\vert _{\\rm BS}$ and $\\bar{J}^a_{g^b}\\big \\vert _{\\rm BS}$ of broken symmetries, and $\\bar{J}^a_{u^{b\\alpha }}\\big \\vert _{\\rm BS} = \\delta ^{ab}\\, \\bar{J}_{x^{\\alpha }} \\, .$ Since the order parameters $\\hat{x}^{\\alpha }$ are usually even under time-reversal symmetry, their rate $\\hat{J}_{x^{\\alpha }}$ should be odd.", "Therefore, the most general form of the dissipativeless mean rates is given by $\\bar{J}_{x^{\\alpha }} = -A^{a\\alpha } \\, v^a - B^{ab\\alpha } \\, \\nabla ^a v^b +O(\\nabla ^2) \\, ,$ where the coefficients $A^{a\\alpha }$ and $B^{ab\\alpha }$ are defined with the sign convention of Ref. [13].", "This form will be justified on the basis of the microscopic dynamics in Secs.", "and .", "Here, we note that $B^{ab\\alpha }=0$ in crystals and $A^{a\\alpha }=0$ in nematic liquid crystals.", "In Eqs.", "(REF )-(), the terms vanishing with the velocity are due to the advection of the corresponding quantity by the motion of the element of matter at the velocity ${\\bf v}=(v^a)$ .", "The next issue is to determine the contributions $\\bar{J}^a_{e}\\big \\vert _{\\rm BS}$ and $\\bar{J}^a_{g^b}\\big \\vert _{\\rm BS}$ to the dissipativeless current densities of energy and momentum by using the identity (REF )." ], [ "Dissipativeless current densities of energy and momentum", "The dissipativeless currents are satisfying the identity (REF ).", "As shown in Eq.", "(REF ), this identity is equivalent to the requirement that the dissipativeless terms conserve the entropy.", "This identity can be used to deduce the energy and momentum current densities from the expression (REF ), as follows.", "Using the leading-order contributions of the conjugate fields $\\lambda ^\\alpha $ , derived in Eqs.", "(REF )-(), and writing the dissipativeless currents as $\\bar{J}^{a}_{c^{\\alpha }} \\equiv \\langle \\hat{J}^{a}_{c^{\\alpha }}\\rangle _{{\\rm leq},\\lambda }$ , the identity (REF ) is giving $&& \\nabla ^a\\beta \\ast \\bar{J}^{a}_e - \\nabla ^a\\left(\\beta \\mu \\right)\\ast \\bar{J}^{a}_{\\rho } - \\nabla ^a\\left(\\beta v^b\\right)\\ast \\bar{J}^{a}_{g^b}\\nonumber \\\\&& - \\nabla ^a\\left(\\beta \\phi ^{b\\alpha }\\right)\\ast \\bar{J}^{a}_{u^{b\\alpha }} =0\\; ,$ up to higher-order corrections.", "We note that $\\bar{J}^{a}_{\\rho } =\\rho v^a$ because of Eq. ().", "Using the Gibbs-Duhem relation (REF ) with the differential $d$ replaced by the gradient $\\nabla ^a$ , we get $\\rho \\, \\nabla ^a(\\beta \\mu ) &=& \\nabla ^a(\\beta p) + e\\, \\nabla ^a\\beta - g^b\\, \\nabla ^a(\\beta v^b)\\nonumber \\\\&& -u^{b\\alpha } \\, \\nabla ^a(\\beta \\phi ^{b\\alpha }) \\, .$ Besides, the scalar product of the term $\\nabla ^a(\\beta p)$ with the velocity $v^a$ can be transformed according to $v^a \\, \\nabla ^a(\\beta p) = \\nabla ^a(\\beta p \\, v^a) + p \\, v^a \\nabla ^a\\beta - p \\nabla ^a(\\beta v^a) \\, .$ Substituting the relations (REF ) and (REF ) into Eq.", "(REF ), we obtain $&& \\nabla ^a\\beta \\ast \\left(\\bar{J}^{a}_e - e v^a - p v^a\\right) \\nonumber \\\\&& - \\nabla ^a\\left(\\beta v^b\\right)\\ast \\left(\\bar{J}^{a}_{g^b} - g^b v^a -p \\, \\delta ^{ab}\\right) \\nonumber \\\\&& - \\nabla ^a\\left(\\beta \\phi ^{b\\alpha }\\right)\\ast \\left(\\bar{J}^{a}_{u^{b\\alpha }} -u^{b\\alpha } v^a\\right)=0\\; ,\\qquad $ because the integral (REF ) of the divergence $\\nabla ^a(\\beta p \\, v^a)$ is vanishing.", "Replacing the dissipativeless current densities by their expressions ()-() with $g^b=\\rho v^b$ given by Eq.", "(), we find that the contributions of broken symmetry to the dissipativeless current densities should satisfy the following identity $&& \\nabla ^a\\beta \\ast \\bar{J}^{a}_e\\big \\vert _{\\rm BS} - \\nabla ^a\\left(\\beta v^b\\right)\\ast \\bar{J}^{a}_{g^b}\\big \\vert _{\\rm BS} \\nonumber \\\\&&\\quad - \\nabla ^a\\left(\\beta \\phi ^{b\\alpha }\\right)\\ast \\bar{J}^{a}_{u^{b\\alpha }}\\big \\vert _{\\rm BS} =0\\; ,$ or equivalently $&& \\nabla ^a\\beta \\ast \\left(\\bar{J}^{a}_e\\big \\vert _{\\rm BS} - v^b \\bar{J}^{a}_{g^b}\\big \\vert _{\\rm BS} - \\phi ^{a\\alpha } \\bar{J}_{x^{\\alpha }}\\right) \\\\&& - (\\beta \\, \\nabla ^a v^b)\\ast \\bar{J}^{a}_{g^b}\\big \\vert _{\\rm BS} - (\\beta \\, \\nabla ^a\\phi ^{a\\alpha })\\ast \\bar{J}_{x^\\alpha } =0\\; , \\qquad \\nonumber $ after using Eq.", "(REF ) with the mean decay rate of the order parameter given by Eq.", "(REF ).", "In order to satisfy this identity, we first require that $\\nabla ^a v^b\\bar{J}^{a}_{g^b}\\big \\vert _{\\rm BS}+\\nabla ^a\\phi ^{a\\alpha }\\bar{J}_{x^\\alpha }$ can be expressed as a divergence, which leads to $\\bar{J}^{a}_{g^b}\\big \\vert _{\\rm BS} = -\\phi ^{a\\alpha } \\, A^{b\\alpha } + B^{ab\\alpha } \\, \\nabla ^c \\phi ^{c\\alpha } +O(\\nabla ^2)$ and the conditions $\\nabla ^a A^{b\\alpha } = 0 \\, .$ Now, the divergence $\\nabla ^a(\\phi ^{a\\alpha } A^{b\\alpha } v^b)$ resulting from the last two terms in Eq.", "(REF ) is combined with the factor $\\beta $ in front of them to give an extra contribution to the first term in $\\nabla ^a\\beta $ , so that we finally obtain $&&\\bar{J}^a_e\\big \\vert _{\\rm BS} \\nonumber \\\\&&= v^b \\bar{J}^{a}_{g^b}\\big \\vert _{\\rm BS} + \\phi ^{b\\alpha } \\bar{J}^{a}_{u^{b\\alpha }}\\big \\vert _{\\rm BS} + \\phi ^{a\\alpha } A^{b\\alpha } v^b + O(\\nabla ^2)\\nonumber \\\\&&= - \\phi ^{a\\alpha } A^{b\\alpha } v^b + B^{ab\\alpha } \\, v^b \\, \\nabla ^c \\phi ^{c\\alpha } - \\phi ^{a\\alpha } B^{bc\\alpha } \\, \\nabla ^b v^c \\nonumber \\\\&&\\quad + O(\\nabla ^2) \\, .$ The dissipativeless parts of the momentum and energy current densities are thus determined as $\\bar{J}^a_{g^b} &=& \\rho v^a v^b -\\sigma ^{ab} +O(\\nabla ^2) \\, , \\\\\\bar{J}^a_{e} &=& e\\, v^a - \\sigma ^{ab} v^b - \\phi ^{a\\alpha } B^{bc\\alpha } \\nabla ^b v^c +O(\\nabla ^2) \\, , \\nonumber \\\\&& $ with the reversible stress tensor defined by $\\sigma ^{ab} \\equiv -p\\, \\delta ^{ab} +\\phi ^{a\\alpha }\\, A^{b\\alpha } -B^{ab\\alpha } \\nabla ^c \\phi ^{c\\alpha } +O(\\nabla ^2) \\, .$ These results are consistent with those obtained in Refs.", "[13], [14].", "To summarize, the dissipativeless current densities are obtained from the condition that they should conserve the entropy, which is equivalent to the identity (REF ) of the formalism.", "Galilean invariance and the behavior of the variables under time-reversal symmetry provide the identification of the most general form of the dissipativeless current densities.", "This approach, based on the conservation of entropy, does not require the knowledge of the explicit microscopic expressions for the variables $\\hat{x}^\\alpha $ ." ], [ "Dissipativeless local conservation equations", "Now, the expressions obtained for the dissipativeless current densities can be substituted back into the local conservation equations (REF ), here neglecting the dissipative part of the current densities.", "We introduce the total time derivative of any field $f({\\bf r},t)$ along a streamline of matter advected by the velocity ${\\bf v}=(v^a)$ according to $\\frac{df}{dt} \\equiv \\partial _t f + v^a \\, \\nabla ^a f \\, .$ Expanding the Eulerian local conservation equations, we obtain their following Lagrangian forms, $&& \\frac{d\\rho }{dt} = -\\rho \\, \\nabla ^a v^a \\, , \\\\&& \\frac{d e_0}{dt} = -e_0\\, \\nabla ^a v^a + \\sigma ^{ab} \\, \\nabla ^a v^b + \\nabla ^a(\\phi ^{a\\alpha } B^{bc\\alpha } \\nabla ^b v^c) \\, , \\nonumber \\\\ && \\\\&& \\rho \\, \\frac{dv^a}{dt} = \\nabla ^b \\sigma ^{ba} \\, , \\\\&& \\frac{d u^{a\\alpha }}{dt} = - u^{a\\alpha } \\, \\nabla ^b v^b + \\nabla ^a\\left( A^{b\\alpha } v^b + B^{bc\\alpha } \\nabla ^b v^c \\right) \\, , \\nonumber \\\\ && $ in terms of the reversible stress tensor (REF ) and up to higher-order corrections.", "These equations of motion thus describe adiabatic processes leaving constant the entropy." ], [ "Dissipativeless equations for the conjugated fields", "Since the conjugate fields $\\beta $ , $\\beta \\mu _0$ , and $\\beta \\phi ^{a\\alpha }$ are functions of the variables $e_0$ , $\\rho $ , and $u^{a\\alpha }$ , their time evolution can be obtained from the dissipativeless equations (REF )-().", "As shown in detail in App.", "using the Maxwell relations (REF )-(), they obey the following equations, $\\frac{d\\beta }{dt} &=& -\\beta \\left(\\frac{\\partial \\sigma ^{ab}}{\\partial e_0}\\right)_{\\rho , \\mathsf {u}} \\nabla ^av^b \\, , \\\\\\frac{d(\\beta \\mu _0)}{dt} &=& \\beta \\left(\\frac{\\partial \\sigma ^{ab}}{\\partial \\rho }\\right)_{e_0, \\mathsf {u}} \\nabla ^av^b \\, , \\\\\\frac{d(\\beta \\phi ^{a\\alpha })}{dt} &=& \\beta \\left(\\frac{\\partial \\sigma ^{bc}}{\\partial u^{a\\alpha }}\\right)_{e_0,\\rho } \\nabla ^bv^c \\, , $ up to corrections of order $\\nabla ^2$ .", "Furthermore, we also have $\\rho \\, \\frac{dv^a}{dt} &=& \\frac{e_0 + p}{\\beta }\\, \\nabla ^a\\beta - \\frac{\\rho }{\\beta }\\, \\nabla ^a(\\beta \\mu _0) + A^{a\\alpha }\\, \\nabla ^b\\phi ^{b\\alpha } \\nonumber \\\\&&+ O(\\nabla ^2) \\, ,$ which is obtained from Eq.", "() with the reversible stress tensor (REF ), using Eq.", "(REF ) and the fact that $u^{a\\alpha }=\\nabla ^a x^{\\alpha } = O(\\nabla )$ by definition." ], [ "Heat current density", "Once the dissipativeless current densities are identified, we turn to the derivation of the dissipative current densities defined by Eq.", "(REF ) and contributing to the entropy production (REF ).", "This latter can be expressed in terms of the gradients of the conjugated field given by Eqs.", "(REF )-() at leading order in the gradients.", "Using the fact that the dissipative current density of mass is equal to zero $\\mathcal {J}^a_{\\rho }=0$ , the relation $\\mathcal {J}^a_{u^{b\\alpha }}=\\delta ^{ab} \\mathcal {J}_{x^{\\alpha }}$ , and the expansions $\\nabla ^a(\\beta f)=f\\nabla ^a\\beta +\\beta \\nabla ^a f$ for any field $f$ , we obtain $\\frac{d_{\\rm i}S}{dt} &=& \\int d{\\bf r}\\, \\Big ( \\nabla ^a\\beta \\, \\mathcal {J}^a_q - \\beta \\, \\nabla ^a v^b\\, \\mathcal {J}^a_{g^b} \\nonumber \\\\&&\\ \\ \\qquad - \\beta \\, \\nabla ^a \\phi ^{a\\alpha }\\, \\mathcal {J}_{x^{\\alpha }} \\Big )\\ge 0 $ in terms of the heat current density defined as $\\mathcal {J}^a_q \\equiv \\mathcal {J}^a_e - v^b \\mathcal {J}^a_{g^b} -\\phi ^{a\\alpha } \\mathcal {J}_{x^\\alpha } \\, .$ This result shows that the dissipative current density of energy is reduced to the heat current density under the conditions where $v^a=0$ and $\\phi ^{a\\alpha }=0$ , i.e., in the frame moving with matter and in the absence of the contribution from $\\phi ^{a\\alpha }$ to the stress." ], [ "Deduction at leading order", "According to Eq.", "(REF ), the leading-order term in the gradient expansion of the dissipative current densities is given by $\\mathcal {J}^{a}_{c^\\alpha }(\\mathbf {r},t) & = \\langle \\hat{J}^{a}_{{c}^\\alpha }(\\mathbf {r};\\Gamma )\\,\\Sigma _t(\\Gamma )\\rangle _{{\\rm leq},\\lambda _t} + O(\\Sigma _t^2)\\;.$ Therefore, their derivation relies on the evaluation of the quantity (REF ), which can be expressed as $\\Sigma _t(\\Gamma ) &=& \\int _0^t d\\tau \\, \\Big [\\partial _\\tau {\\lambda }^{\\alpha }_{\\tau }\\ast \\delta \\hat{c}^\\alpha (\\Gamma _{\\tau - t}) \\nonumber \\\\&&\\qquad + \\nabla ^a{\\lambda }^{\\alpha }_{\\tau }\\ast \\delta \\hat{J}^{a}_{{c}^\\alpha }(\\Gamma _{\\tau - t})\\Big ] $ in terms of $&& \\delta \\hat{c}^\\alpha (\\mathbf {r};\\Gamma _{\\tau - t}) \\equiv \\hat{c}^\\alpha (\\mathbf {r};\\Gamma _{\\tau - t}) - \\langle \\hat{c}^\\alpha (\\mathbf {r};\\Gamma )\\rangle _{{\\rm leq},\\lambda _{\\tau }} \\, ,\\nonumber \\\\ && \\\\&& \\delta \\hat{J}^{a}_{{c}^\\alpha }(\\mathbf {r};\\Gamma _{\\tau - t}) \\equiv \\hat{J}^{a}_{{c}^\\alpha }(\\mathbf {r};\\Gamma _{\\tau - t}) - \\langle \\hat{J}^{a}_{{c}^\\alpha }(\\mathbf {r};\\Gamma )\\rangle _{{\\rm leq},\\lambda _{\\tau }} \\, .", "\\nonumber \\\\ && $ The detailed calculations are carried out in App. .", "Under the conditions $v^a=0$ and $\\phi ^{a\\alpha }=0$ where the dissipative current density of energy coincides with the heat current density, we find at first order in the gradients that $&& \\partial _\\tau {\\lambda }^{\\alpha }\\ast \\delta \\hat{c}^\\alpha + \\nabla ^a{\\lambda }^{\\alpha }\\ast \\delta \\hat{J}^{a}_{{c}^\\alpha } \\\\&& = \\nabla ^a\\beta \\ast \\delta \\hat{J}^{\\prime a}_{e}- (\\beta \\, \\nabla ^a v^b)\\ast \\delta \\hat{J}^{\\prime a}_{g^b} - (\\beta \\, \\nabla ^a\\phi ^{a\\alpha })\\ast \\delta \\hat{J}^{\\prime }_{x^{\\alpha }}\\;, \\nonumber $ with the following definitions, $\\delta \\hat{J}^{\\prime a}_{e} & \\equiv \\delta \\hat{J}^a_{e} -\\rho ^{-1}(e_0+p)\\, \\delta \\hat{g}^a \\, , \\\\\\delta \\hat{J}^{\\prime a}_{g^b} & \\equiv \\delta \\hat{J}_{g^b}^a +\\left(\\frac{\\partial \\sigma ^{ab}}{\\partial e_0}\\right)_{\\rho , \\mathsf {u}} \\delta \\hat{e} + \\left(\\frac{\\partial \\sigma ^{ab}}{\\partial \\rho }\\right)_{e_0, \\mathsf {u}} \\delta \\hat{\\rho } \\, , \\\\\\delta \\hat{J}^{\\prime }_{x^{\\alpha }} & \\equiv \\delta \\hat{J}_{x^\\alpha } + \\rho ^{-1} A^{b\\alpha }\\, \\delta \\hat{g}^b\\; .$ Substituting Eq.", "(REF ) back into Eq.", "(REF ), the dissipative current densities are given at leading order by Eq.", "(REF ), giving $&&\\mathcal {J}^a_{{c}^\\alpha }(\\mathbf {r},t)= \\\\&&\\int _0^t d\\tau \\int d\\mathbf {r}^{\\prime } \\, \\langle \\delta \\hat{J}^a_{c^\\alpha }(\\mathbf {r},0) \\, \\delta \\hat{J}^{\\prime b}_{e}(\\mathbf {r}^{\\prime },\\tau -t)\\rangle _{\\text{leq}, t} \\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\times \\nabla ^{\\prime b}\\beta (\\mathbf {r}^{\\prime },\\tau ) \\nonumber \\\\&-& \\int _0^t d\\tau \\int d\\mathbf {r}^{\\prime } \\, \\langle \\delta \\hat{J}^a_{c^\\alpha }(\\mathbf {r},0) \\, \\delta \\hat{J}^{\\prime b}_{g^c}(\\mathbf {r}^{\\prime },\\tau -t) \\rangle _{\\text{leq}, t} \\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\times \\beta (\\mathbf {r}^{\\prime },\\tau ) \\, \\nabla ^{\\prime b} v^c(\\mathbf {r}^{\\prime },\\tau ) \\nonumber \\\\&-& \\int _0^t d\\tau \\int d\\mathbf {r}^{\\prime } \\, \\langle \\delta \\hat{J}^a_{c^\\alpha }(\\mathbf {r},0) \\, \\delta \\hat{J}^{\\prime }_{x^{\\gamma }}(\\mathbf {r}^{\\prime },\\tau -t) \\rangle _{\\text{leq}, t} \\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\times \\beta (\\mathbf {r}^{\\prime },\\tau ) \\, \\nabla ^{\\prime b}\\phi ^{b\\gamma }(\\mathbf {r}^{\\prime },\\tau )\\, ,\\nonumber $ up to higher-order corrections.", "Assuming that the characteristic length and time scales of the conjugated fields $\\lambda $ are much larger than the correlation length and time of the current densities (REF )-(), we can replace $\\nabla ^{\\prime c}\\lambda ^{\\gamma }(\\mathbf {r}^{\\prime },\\tau )$ by $\\nabla ^c\\lambda ^{\\gamma }(\\mathbf {r},t)$ in the previous equation to find $&&\\mathcal {J}^a_{{c}^\\alpha }(\\mathbf {r},t) = \\nabla ^{b}\\beta (\\mathbf {r},t) \\\\&&\\times \\int _0^t d\\tau \\int d\\mathbf {r}^{\\prime } \\, \\langle \\delta \\hat{J}^a_{c^\\alpha }(\\mathbf {r},0) \\, \\delta \\hat{J}^{\\prime b}_{e}(\\mathbf {r}^{\\prime },\\tau -t)\\rangle _{\\text{leq}, t} \\nonumber \\\\&-& \\beta (\\mathbf {r},t) \\, \\nabla ^{b} v^c(\\mathbf {r},t) \\nonumber \\\\&&\\times \\int _0^t d\\tau \\int d\\mathbf {r}^{\\prime } \\, \\langle \\delta \\hat{J}^a_{c^\\alpha }(\\mathbf {r},0) \\, \\delta \\hat{J}^{\\prime b}_{g^c}(\\mathbf {r}^{\\prime },\\tau -t) \\rangle _{\\text{leq}, t} \\nonumber \\\\&-& \\beta (\\mathbf {r},t) \\, \\nabla ^{b}\\phi ^{b\\gamma }(\\mathbf {r},t) \\nonumber \\\\&&\\times \\int _0^t d\\tau \\int d\\mathbf {r}^{\\prime } \\, \\langle \\delta \\hat{J}^a_{c^\\alpha }(\\mathbf {r},0) \\, \\delta \\hat{J}^{\\prime }_{x^{\\gamma }}(\\mathbf {r}^{\\prime },\\tau -t) \\rangle _{\\text{leq}, t}\\, .", "\\nonumber $ Since the conjugated fields evolve in time on a longer time scale than the correlation time of the current densities, the local equilibrium distribution at time $t$ may be considered as the equilibrium distribution at the local values of the conjugated fields in the frame where $v^a=0$ and $\\phi ^{a\\alpha }=0$ , i.e., for given local values of temperature and chemical potential.", "Since the equilibrium distribution is stationary, we have that $\\langle \\delta \\hat{a}({\\bf r},0)\\,\\delta \\hat{b}({\\bf r}^{\\prime },\\tau -t)\\rangle =\\langle \\delta \\hat{a}({\\bf r},t-\\tau )\\, \\delta \\hat{b}({\\bf r}^{\\prime },0)\\rangle $ .", "Replacing $t-\\tau $ by $\\tau $ , the integral over time becomes $\\int _0^t d\\tau \\, \\langle \\delta \\hat{a}({\\bf r},0)\\,\\delta \\hat{b}({\\bf r}^{\\prime },\\tau -t)\\rangle =\\int _0^t d\\tau \\,\\langle \\delta \\hat{a}({\\bf r},\\tau )\\, \\delta \\hat{b}({\\bf r}^{\\prime },0)\\rangle $ , where the limit $t\\rightarrow \\infty $ can be taken since the time scale $t$ of the conjugated fields is longer than the correlation time of the current densities.", "Furthermore, the material properties over spatial scales larger than the microscopic structure of the phase can be defined by averaging over space.", "The microscopic currents are thus introduced as $\\hat{\\mathbb {J}}^a_{c^{\\alpha }}(t) \\equiv \\int _V \\hat{J}^a_{c^{\\alpha }}({\\bf r},t) \\, d{\\bf r}$ by integrating over the volume $V$ of the system.", "Therefore, Eqs.", "(REF )-() give $\\delta \\hat{\\mathbb {J}}^{\\prime a}_{e}(t) & = \\delta \\hat{\\mathbb {J}}^a_{e}(t) -\\rho ^{-1}(e_0+p)\\, \\delta \\hat{P}^a(t) \\, ,\\\\\\delta \\hat{\\mathbb {J}}^{\\prime a}_{g^b}(t) & = \\delta \\hat{\\mathbb {J}}_{g^b}^a(t) +\\left(\\frac{\\partial \\sigma ^{ab}}{\\partial e_0}\\right)_{\\rho , \\mathsf {u}} \\delta \\hat{E}(t) \\\\&\\qquad \\qquad \\ + \\left(\\frac{\\partial \\sigma ^{ab}}{\\partial \\rho }\\right)_{e_0, \\mathsf {u}} \\delta \\hat{M}(t) \\, , \\\\\\delta \\hat{\\mathbb {J}}^{\\prime }_{x^{\\alpha }}(t) & = \\delta \\hat{\\mathbb {J}}_{x^\\alpha }(t) + \\rho ^{-1} A^{b\\alpha }\\, \\delta \\hat{P}^b(t)\\; ,$ where $\\delta \\hat{P}^a=\\int \\delta \\hat{g}^a d{\\bf r}$ , $\\delta \\hat{E}=\\int \\delta \\hat{e}\\, d{\\bf r}$ , and $\\delta \\hat{M}=\\int \\delta \\hat{\\rho }\\, d{\\bf r}$ are the deviations in the total momentum, energy, and mass with respect to their local equilibrium value.", "Since the total momentum, energy, and mass are conserved, they are constants of motion [i.e., they do not fluctuate in time, but they are still random variables because the initial conditions in phase space may give different values to these constants of motion, for instance, in the grand canonical ensemble of distribution (REF )].", "They may thus be added into the equilibrium time correlation functions because $\\langle \\delta \\hat{\\mathbb {J}}^a_{c^{\\alpha }}\\rangle _{\\rm eq}=0$ .", "Consequently, the unprime quantities can be replaced by the prime ones in the time correlation functions.", "Therefore, Eq.", "(REF ) becomes $&&\\mathcal {J}^a_{{c}^\\alpha }(\\mathbf {r},t) = \\\\&&\\ \\ \\frac{1}{V} \\, \\nabla ^{b}\\beta (\\mathbf {r},t) \\int _0^{\\infty } d\\tau \\, \\langle \\delta \\hat{\\mathbb {J}}^{\\prime a}_{c^\\alpha }(\\tau ) \\, \\delta \\hat{\\mathbb {J}}^{\\prime b}_{e}(0)\\rangle _{\\text{leq}, t} \\nonumber \\\\&& - \\frac{1}{V} \\, \\beta (\\mathbf {r},t) \\, \\nabla ^{b} v^c(\\mathbf {r},t) \\int _0^{\\infty } d\\tau \\, \\langle \\delta \\hat{\\mathbb {J}}^{\\prime a}_{c^\\alpha }(\\tau ) \\, \\delta \\hat{\\mathbb {J}}^{\\prime b}_{g^c}(0) \\rangle _{\\text{leq}, t} \\nonumber \\\\&& - \\frac{1}{V} \\, \\beta (\\mathbf {r},t) \\, \\nabla ^{b}\\phi ^{b\\gamma }(\\mathbf {r},t) \\int _0^{\\infty } d\\tau \\, \\langle \\delta \\hat{\\mathbb {J}}^{\\prime a}_{c^\\alpha }(\\tau ) \\, \\delta \\hat{\\mathbb {J}}^{\\prime }_{x^{\\gamma }}(0) \\rangle _{\\text{leq}, t}\\, ,\\nonumber $ which holds for the dissipative current densities $(\\mathcal {J}^a_{{c}^\\alpha })=(\\mathcal {J}^a_{e},\\mathcal {J}^a_{g^b},\\mathcal {J}_{x^\\beta })$ under the conditions $v^a=0$ and $\\phi ^{a\\alpha }=0$ , where the dissipative energy current density is equal to the heat current density $\\mathcal {J}^a_{e}=\\mathcal {J}^a_{q}$ ." ], [ "Green-Kubo formulas for the transport coefficients", "As a consequence, Eq.", "(REF ) leads to the following expressions for the dissipative current densities of heat, momentum, and order parameters, $\\mathcal {J}^a_{q} &=& -\\kappa ^{ab}\\, \\nabla ^bT -\\chi ^{abc}\\, \\nabla ^b v^c - \\xi ^{a\\alpha }\\, \\nabla ^b\\phi ^{b\\alpha }\\, , \\nonumber \\\\&& \\\\\\mathcal {J}^a_{g^b} &=& \\chi ^{cab}\\, \\frac{\\nabla ^c T}{T} -\\eta ^{abcd} \\, \\nabla ^c v^d- \\theta ^{ab\\alpha }\\nabla ^c\\phi ^{c \\alpha }\\, , \\nonumber \\\\&&\\\\\\mathcal {J}_{x^\\alpha } &=& -\\xi ^{a\\alpha }\\, \\frac{\\nabla ^aT}{T} +\\theta ^{ab\\alpha }\\, \\nabla ^a v^b - \\zeta ^{\\alpha \\beta }\\, \\nabla ^a\\phi ^{a\\beta }\\, , \\nonumber \\\\&&$ with the transport coefficients given by the following Green-Kubo formulas, $\\kappa ^{ab} & \\equiv \\lim _{V\\rightarrow \\infty } \\frac{1}{k_{\\rm B} T^2V}\\int _0^{\\infty }dt \\, \\langle \\delta \\hat{\\mathbb {J}}^{\\prime a}_{e}(t)\\, \\delta \\hat{\\mathbb {J}}^{\\prime b}_{e}(0)\\rangle _{\\text{eq}}\\;, \\\\\\eta ^{abcd} & \\equiv \\lim _{V\\rightarrow \\infty } \\frac{1}{k_{\\rm B} T V} \\int _0^{\\infty }dt \\, \\langle \\delta \\hat{\\mathbb {J}}^{\\prime a}_{g^b}(t)\\, \\delta \\hat{\\mathbb {J}}^{\\prime c}_{g^d}(0)\\rangle _{\\text{eq}}\\;, \\\\\\xi ^{a\\alpha } &\\equiv \\lim _{V\\rightarrow \\infty } \\frac{1}{k_{\\rm B} T V}\\int _0^{\\infty }dt \\, \\langle \\delta \\hat{\\mathbb {J}}^{\\prime a}_{e}(t)\\, \\delta \\hat{\\mathbb {J}}^{\\prime }_{x^{\\alpha }}(0)\\rangle _{\\text{eq}}\\;, \\\\\\zeta ^{\\alpha \\beta } & \\equiv \\lim _{V\\rightarrow \\infty } \\frac{1}{k_{\\rm B} T V}\\int _0^{\\infty }dt \\, \\langle \\delta \\hat{\\mathbb {J}}^{\\prime }_{{x}^{\\alpha }}(t) \\, \\delta \\hat{\\mathbb {J}}^{\\prime }_{x^{\\beta }}(0)\\rangle _{\\text{eq}}\\;, \\\\\\chi ^{abc} & \\equiv \\lim _{V\\rightarrow \\infty } \\frac{1}{k_{\\rm B} T V} \\int _0^{\\infty }dt \\, \\langle \\delta \\hat{\\mathbb {J}}^{\\prime a}_{e}(t)\\, \\delta \\hat{\\mathbb {J}}^{\\prime b}_{g^c}(0)\\rangle _{\\text{eq}}\\;, \\\\\\theta ^{ab\\alpha } & \\equiv \\lim _{V\\rightarrow \\infty } \\frac{1}{k_{\\rm B} T V}\\int _0^{\\infty }dt \\, \\langle \\hat{\\mathbb {J}}^{\\prime a}_{g^b}(t)\\, \\delta \\hat{\\mathbb {J}}^{\\prime }_{x^{\\alpha }}(0)\\rangle _{\\text{eq}}\\;.", "$ Here, the limit $V\\rightarrow \\infty $ is taken at constant chemical potential, in order to define the bulk transport properties in arbitrarily large systems, removing in this way possible finite-size effects encountered in molecular dynamics simulation [50].", "We note that, in isotropic phases of matter, the rank-three tensors $\\chi ^{abc}$ and $\\theta ^{ab\\alpha }$ are vanishing, so that they are expected to be small in general." ], [ "Time-reversal symmetry", "Since the Hamiltonian function (REF ) has the symmetry $H(\\Theta \\Gamma )=H(\\Gamma )$ under the time-reversal transformation $\\Theta ({\\bf r}_i,{\\bf p}_i)=({\\bf r}_i,-{\\bf p}_i)$ , the equilibrium probability distribution (REF ) is also symmetric and we have the Onsager-Casimir reciprocal relations [51], [52], [53] $&& \\int _0^{\\infty } dt \\, \\langle \\delta \\hat{\\mathbb {J}}_{\\alpha }(t) \\, \\delta \\hat{\\mathbb {J}}_{\\beta }(0)\\rangle _{\\text{eq}}\\\\&=& \\epsilon _{\\alpha } \\, \\epsilon _{\\beta } \\int _0^{\\infty } dt \\, \\langle \\delta \\hat{\\mathbb {J}}_{\\beta }(t)\\, \\delta \\hat{\\mathbb {J}}_{\\alpha }(0)\\rangle _{\\text{eq}} \\, , \\nonumber $ where $\\epsilon _\\alpha =\\pm 1$ if the current $\\delta \\hat{\\mathbb {J}}_{\\alpha }$ is even or odd under time reversal (and there is no Einstein summation here).", "Since $\\hat{e}$ and $\\hat{x}^{\\alpha }$ are even, their currents $\\hat{\\mathbb {J}}^a_e$ and $\\hat{\\mathbb {J}}_{x^{\\alpha }}$ are odd.", "Besides, the currents $\\hat{\\mathbb {J}}^a_{g^b}$ are even because $\\hat{g}^b$ is odd.", "Consequently, we have the Onsager-Casimir reciprocal relations $\\kappa ^{ab}=\\kappa ^{ba}$ , $\\eta ^{abcd}=\\eta ^{cdab}$ , $\\xi ^{a\\alpha }=\\xi ^{\\alpha a}$ , and $\\zeta ^{\\alpha \\beta }=\\zeta ^{\\beta \\alpha }$ .", "This latter relation explains how the coefficient in front of $\\nabla ^b\\phi ^{b\\alpha }$ in Eq.", "(REF ) is the same as the one in front of $\\nabla ^aT/T$ in Eq. ().", "Moreover, we have the Onsager-Casimir reciprocal relations $\\chi ^{abc}=-\\chi ^{bca}$ and $\\theta ^{ab\\alpha }=-\\theta ^{\\alpha ab}$ , which explains the changes of sign in front of $\\nabla ^cT/T$ in Eq.", "() and in front of $\\nabla ^av^b$ in Eq.", "() with respect to the corresponding terms with the same coefficients.", "The coefficients $\\chi ^{abc}$ generate a coupling between momentum transport and temperature gradients in a similar way as at the interface between two phases [54], which is the mechanism inducing the phenomenon of thermophoresis [55]." ], [ "Entropy production", "Because of the antisymmetry of the coefficients $\\chi ^{abc}$ and $\\theta ^{ab\\alpha }$ , the associated terms do not contribute to entropy production and thus dissipation.", "In order to confirm this result, the entropy production (REF ) is calculated from Eqs.", "(REF )-(), giving $&& \\frac{d_{\\rm i}S}{dt} = \\int d{\\bf r} \\, \\frac{1}{T}\\bigg (\\eta ^{abcd} \\, \\nabla ^a v^b \\, \\nabla ^c v^d+ \\frac{\\kappa ^{ab}}{T} \\, \\nabla ^aT \\, \\nabla ^b T \\nonumber \\\\&& + 2\\, \\frac{\\xi ^{a\\alpha }}{T} \\, \\nabla ^a T \\, \\nabla ^b \\phi ^{b\\alpha } + \\zeta ^{\\alpha \\beta } \\, \\nabla ^a \\phi ^{a\\alpha } \\, \\nabla ^b \\phi ^{b\\beta } \\bigg )\\ge 0 \\, , \\nonumber \\\\ &&$ where, indeed, the terms with the coefficients $\\chi ^{abc}$ and $\\theta ^{ab\\alpha }$ do not appear.", "In this regard, these terms may be considered as dissipativeless contributions to the current densities.", "In particular, the terms with the coefficients $\\theta ^{ab\\alpha }$ in Eq.", "() are similar to those with the coefficients $B^{ab\\alpha }$ in Eq.", "(REF ).", "The non-negativity of the entropy production results in particular from the conditions $\\eta ^{abab}\\ge 0$ , $\\kappa ^{ab}\\ge 0$ , $\\zeta ^{\\alpha \\alpha }\\ge 0$ , and $\\kappa ^{aa}\\zeta ^{\\alpha \\alpha } \\ge (\\xi ^{a\\alpha })^2/T$ [47]." ], [ "Macroscopic equations", "Finally, the macroscopic equations read $\\partial _t \\rho +\\nabla ^a(\\rho v^a) &=& 0 \\, ,\\\\\\partial _t e+\\nabla ^a( \\bar{J}^a_e + {\\mathcal {J}}^a_e) &=& 0 \\, ,\\\\\\partial _t (\\rho v^b) +\\nabla ^a( \\bar{J}^a_{g^b} + {\\mathcal {J}}^a_{g^b}) &=& 0 \\, ,\\\\\\partial _t x^\\alpha + \\bar{J}_{x^{\\alpha }} + {\\mathcal {J}}_{x^{\\alpha }} &=& 0 \\, ,$ with the dissipativeless and dissipative current densities and rates given by Eqs.", "(REF ), (REF ), (), (REF ), (REF ), (), and (), as obtained with the expansion in powers of the gradients.", "The system of macroscopic equations can be closed using the thermodynamic relations.", "Keeping the terms that are linear in the gradients and the velocity, we find $&& \\partial _t \\rho \\simeq - \\rho \\, \\nabla ^a v^a \\, , \\\\&& \\partial _t e_0 \\simeq -(e_0+p) \\nabla ^a v^a + \\chi ^{abc} \\nabla ^a\\nabla ^b v^c \\nonumber \\\\&&\\qquad \\qquad + \\kappa ^{ab} \\nabla ^a\\nabla ^b T + \\xi ^{a\\alpha } \\nabla ^a\\nabla ^b\\phi ^{b\\alpha } \\, , \\\\&& \\rho \\, \\partial _t v^b \\simeq - \\nabla ^b p + A^{b\\alpha } \\nabla ^a \\phi ^{a\\alpha }\\nonumber \\\\&&\\quad \\qquad - (B^{ab\\alpha }-\\theta ^{ab\\alpha }) \\nabla ^a\\nabla ^c \\phi ^{c\\alpha } - \\frac{\\chi ^{cab}}{T} \\, \\nabla ^a\\nabla ^c T \\nonumber \\\\&&\\quad \\qquad + \\eta ^{abcd} \\nabla ^a\\nabla ^c v^d \\, , \\\\&& \\partial _t x^\\alpha \\simeq A^{a\\alpha } v^a +(B^{ab\\alpha }-\\theta ^{ab\\alpha }) \\nabla ^a v^b +\\frac{\\xi ^{a\\alpha }}{T} \\, \\nabla ^a T \\nonumber \\\\&&\\quad \\qquad + \\zeta ^{\\alpha \\beta } \\nabla ^a \\phi ^{a\\beta } \\, , $ which is exactly Eq.", "(6.1) of Ref.", "[13] in the case where $\\chi ^{abc}=0$ and $\\theta ^{ab\\alpha }=0$ .", "The coefficients $\\kappa ^{ab}$ can be interpreted as the heat conductivities and $\\eta ^{abcd}$ as the viscosities.", "Finally, note that the microscopic expressions for the order parameters and its associated current densities were not used in the derivation.", "However, these expressions are essential in practice and in order to evaluate the transport coefficients with the Green-Kubo formulas." ], [ "Crystalline Solids", "The method applies in particular to crystals where the continuous symmetry under the group of spatial translations is broken into the discrete symmetry of one of the 230 crystallographic space groups.", "Accordingly, crystals have eight hydrodynamic modes with dispersion relations vanishing with the wave number according to the Goldstone theorem.", "These modes are the two longitudinal sound modes, the four transverse sound modes, the mode of heat conduction, and the mode of vacancy diffusion.", "These modes are damped because of energy dissipation and the present results give the Green-Kubo formulas for the coefficients ruling their damping." ], [ "Order parameters", "For crystals, the variables $\\hat{x}^{\\alpha }$ associated with continuous symmetry breaking are the components $\\hat{u}^a$ of the displacement field.", "Since the broken symmetry is a spatial symmetry in three dimensions, we can replace the Greek index $\\alpha $ by a Latin index $a$ .", "The microscopic expression for the displacement field is known [23], [24], [56], [57].", "In cubic crystals, it reads $\\hat{{u}}^a(\\mathbf {r};\\Gamma ) & \\equiv - \\frac{1}{\\mathcal {N}} \\int _{\\rm BZ} \\frac{d\\mathbf {k}}{(2\\pi )^3}\\ {\\rm e}^{\\imath \\mathbf {k}\\cdot \\mathbf {r}} \\\\&\\times \\int d\\mathbf {r}^{\\prime }\\ {\\rm e}^{ - \\imath \\mathbf {k}\\cdot \\mathbf {r}^{\\prime }} \\,\\frac{\\partial n_{\\text{eq}}(\\mathbf {r}^{\\prime })}{\\partial {r}^{\\prime a}}\\, \\hat{n}(\\mathbf {r}^{\\prime };\\Gamma )\\;,$ given in terms of the particle density $\\hat{n}=\\hat{\\rho }/m$ , the equilibrium particle density $n_{\\rm eq}({\\bf r})$ , an integral over the Brillouin zone (BZ), and a normalization factor $\\cal N$ .", "The equilibrium particle density is a periodic function of space, which has the symmetry of the crystallographic group.", "In a uniform phase where $\\partial n_{\\text{eq}}/\\partial r^a=0$ , the displacement field would be vanishing, as expected for order parameters.", "The associated rate defined by Eq.", "(REF ) is thus given by $\\hat{J}_{{{u}}^a}(\\mathbf {r};\\Gamma ) & = - \\frac{1}{m\\mathcal {N}} \\int _{\\rm BZ} \\frac{d\\mathbf {k}}{(2\\pi )^3}\\ {\\rm e}^{\\imath \\mathbf {k}\\cdot \\mathbf {r}} \\\\& \\times \\int d\\mathbf {r}^{\\prime }\\ {\\rm e}^{ - \\imath \\mathbf {k}\\cdot \\mathbf {r}^{\\prime }}\\frac{\\partial n_{\\text{eq}}(\\mathbf {r}^{\\prime })}{\\partial {r}^{\\prime a}} \\, \\frac{\\partial \\hat{g}^b(\\mathbf {r}^{\\prime };\\Gamma )}{\\partial r^{\\prime b}}\\;.$ In crystals, the microscopic strain tensor is defined as the symmetric rank-two tensor $\\hat{u}^{ab} & \\equiv \\frac{1}{2}\\left(\\nabla ^a \\hat{u}^b + \\nabla ^b \\hat{u}^a\\right) .$ Accordingly, the associated current density introduced in Eq.", "(REF ) is here given by $\\hat{J}^c_{u^{ab}}(\\mathbf {r};\\Gamma ) & = \\frac{1}{2}\\left(\\delta ^{ac}\\delta ^{bd}+\\delta ^{ad}\\delta ^{cb}\\right)\\hat{J}_{{{u}}^d}(\\mathbf {r};\\Gamma )\\; .$" ], [ "Dissipativeless current densities", "According to Eq.", "(REF ), the dissipativeless part of the rate can be obtained by computing the expectation value of the microscopic expression (REF ) over the local equilibrium distribution.", "For conjugated fields slowly varying in space over length scales much larger than the size of the lattice unit cell, we have that $\\langle \\hat{J}_{u^{a}}(\\mathbf {r};\\Gamma )\\rangle _{{\\rm leq},\\lambda _t} & = - v^a(\\mathbf {r},t)\\;.$ Consequently, the comparison with Eq.", "(REF ) gives $A^{ab} = \\delta ^{ab} \\qquad \\mbox{and} \\qquad B^{abc} =0 \\;.", "$ We note that the reversible stress tensor (REF ) is thus also symmetric.", "In a crystal, it is not possible to assign a particle to a lattice site due to the presence of defects, which are vacancies and interstitials.", "As a consequence, there is an associated phenomenon of diffusion and a corresponding mode [14], [23], [24], [56], [57].", "To describe this phenomenon, the density of vacancies is defined as $\\hat{c} \\equiv -\\hat{n} - n_{{\\rm eq},0} \\nabla ^a \\hat{u}^a\\;,$ where $n_{\\text{eq},0} & \\equiv \\frac{1}{v}\\int _v d\\mathbf {r}\\ n_{\\text{eq}}(\\mathbf {r})\\;, $ is the mean equilibrium density at equilibrium, $v$ being the volume of the unit cell.", "In this way, the eight hydrodynamic modes of crystals can be obtained with the methods of Ref.", "[14]." ], [ "Green-Kubo formulas", "As shown in Sec.", ", the dissipative current densities are given by Eqs.", "(REF )-() with the coefficients given by the Green-Kubo formulas (REF )-().", "The formulas for the heat conductivities $\\kappa ^{ab}$ , the viscosities $\\eta ^{abcd}$ , the coefficients $\\zeta ^{ab}$ related to vacancy diffusion, and $\\xi ^{ab}$ to the cross effect of vacancy thermal diffusion [14] are consistent with the results of Ref. [24].", "Moreover, there also exist dissipativeless cross effects described by the rank-three tensors $\\chi ^{abc}=\\chi ^{acb}$ and $\\theta ^{abc}=\\theta ^{bac}$ .", "In isotropic media, such rank-three tensors are known to vanish according to Curie's principle based on space rotational symmetries.", "Such rank-three tensors may be non-vanishing only for 20 among the 32 crystallographic structures.", "These 20 crystallographic structures are the same as those selected to allow the possibility of a non-vanishing piezoelectric tensor, which is also of rank three [58].", "Nevertheless, the cross effects described by the coefficients $\\chi ^{abc}$ and $\\theta ^{abc}$ often play a negligible role.", "Finally, the hydrodynamic modes can be obtained from the macroscopic equations (REF )-(), which are consistent with earlier results [13], [14], [23], [24]." ], [ "Liquid crystals", "Liquid crystals are composed of nonspherical molecules, which interact with different types of intermolecular forces.", "Rotational or translational symmetries may be broken in liquid crystals, because of the emergence of a privileged orientation, e.g., in nematics, or two-dimensional columnar order, e.g., in some phases of discotic liquid crystals [12], [59].", "For rotational symmetry breaking, we note that the total angular momentum can be decomposed as ${\\bf L}={\\bf L}_0+\\sum _{k} {\\bf L}_{k}$ in terms of the angular momentum ${\\bf L}_0$ with respect to the origin of the laboratory frame and the angular momenta ${\\bf L}_{k}$ of the molecules $k$ with respect to their center of mass (or any other property).", "Such angular momenta ${\\bf L}_{k}$ allow us to carry out local rotations in the system.", "The corresponding Nambu-Goldstone modes can be defined as soft modes associated with such local rotations, as discussed in Subsec.", "REF ." ], [ "Order parameters", "For apolar nematogens (i.e., nematic molecules), an external electric field ${\\cal E}^a({\\bf r})$ will explicitly break the rotation symmetry.", "In this case, the total external potential energy in the Hamiltonian function (REF ) is given by $V_{\\rm tot}^{\\rm (ext)} = -\\frac{1}{2} \\int {\\cal E}^a({\\bf r})\\, \\hat{q}^{ab}({\\bf r})\\, {\\cal E}^b({\\bf r})\\, d{\\bf r}$ with the local traceless polarizability tensor $q^{ab}({\\bf r})$ .", "In general, this local order parameter can be taken as the quadrupolar contribution to the density of some property associated with the nematogens $&&\\hat{q}^{ab}({\\bf r}) = \\sum _{k} \\sum _{i\\in {k}} q_{i}\\bigg [(r_i^a-r_k^a)(r_i^b-r_k^b)\\nonumber \\\\&&\\ \\ \\qquad -\\frac{1}{3} \\, ({\\bf r}_i-{\\bf r}_k)^2\\, \\delta ^{ab}\\bigg ] \\delta ({\\bf r}-{\\bf r}_k) \\, , \\quad $ where $q_i$ is the relevant property attached to the atom $i\\in {k}$ in the molecule $k$ , and ${\\bf r}_k=(r_k^a)$ is a position at the center of the molecule $k$ [12]." ], [ "Dissipativeless current densities", "If the variables (REF ) are taken as the order parameters $\\hat{x}^{\\alpha }$ , the corresponding microscopic rates are given by Eq.", "(REF ).", "Carrying out the change of variables ${\\bf p}_i={\\bf p}_{i0}+m_i{\\bf v}({\\bf r}_i)$ where ${\\bf v}({\\bf r})$ is the velocity field, the expectation values of those rates over the local equilibrium distribution give the dissipativeless parts $\\langle \\hat{J}_{q^{ab}}\\rangle _{\\rm leq} = -A^{abc}\\, v^c - B^{abcd}\\, \\nabla ^c \\, v^d+O(\\nabla ^2) \\, ,$ where $A^{abc} &\\equiv & -\\nabla ^c \\langle \\hat{q}^{ab}\\rangle _{\\rm leq} \\, , \\\\B^{abcd} &\\equiv & \\langle \\hat{y}^{abcd}\\rangle _{\\rm leq} - \\langle \\hat{q}^{ab}\\rangle _{\\rm leq}\\, \\delta ^{cd} \\, ,$ with $&& \\hat{y}^{abcd}({\\bf r}) \\equiv \\sum _{k} \\sum _{i\\in {k}} q_{i}\\Big [(r_i^b-r_k^b)(r_i^c-r_k^c)\\,\\delta ^{ad} \\nonumber \\\\&& +(r_i^a-r_k^a)(r_i^c-r_k^c)\\,\\delta ^{bd} \\nonumber \\\\&& -\\frac{2}{3} \\, (r_i^c-r_k^c)(r_i^d-r_k^d)\\,\\delta ^{ab}\\Big ] \\delta ({\\bf r}-{\\bf r}_{k}) \\, .$ For nematics, we have that $A^{abc} = -\\nabla ^c \\langle \\hat{q}^{ab}\\rangle _{\\rm leq} = 0$ , since they are uniform at equilibrium.", "Therefore, Eq.", "(REF ) can be also justified for such liquid crystals on the basis of the microscopic approach." ], [ "Green-Kubo formulas", "Again, the transport coefficients will be given by the Green-Kubo formulas (REF )-().", "The coefficients $\\chi ^{abc}$ and $\\theta ^{ab\\alpha }$ describing the dissipativeless cross effects may be expected to be equal to zero in most phases of liquid crystals.", "However, since piezoelectricity also exists in some liquid crystals [60], [61], it is possible that such cross effects exist here as well, although being small." ], [ "Conclusion", "In this paper, we have shown that the macroscopic equations ruling the time evolution of matter with broken continuous symmetries can be derived in a unified microscopic approach based on the local equilibrium distribution, extending to crystalline solids and liquid crystals results previously obtained for normal fluids.", "In the presence of broken continuous symmetries, the description should be extended to include the microscopic expressions of the order parameters beside the microscopic densities of mass, energy, and momentum, which are locally conserved.", "The time evolution of these variables is generated at the fundamental level of description by the underlying Hamiltonian microdynamics.", "The manifestation of spontaneous symmetry breaking is the emergence of as many Nambu-Goldstone modes as there are broken continuous symmetries.", "Those modes have frequencies $\\omega ({\\bf q})$ vanishing with their wave number $\\bf q$ , as for the hydrodynamic modes associated with the local conservation laws.", "All these modes are damped because of their interaction with the other degrees of freedom.", "This damping is determined by the transport coefficients responsible for energy dissipation and entropy production.", "Here, we have deduced these properties using the microscopic approach based on the local equilibrium distribution and its time evolution ruled by the Hamiltonian microdynamics.", "For every density, we have systematically obtained the dissipativeless and dissipative parts of the corresponding current density using expansions in powers of the gradients of the macrofields.", "With this approach, the dissipativeless part of each current density can be inferred including their nonlinear dependence on the velocity field, and their dissipative part can be identified in direct relation with entropy production.", "In this way, Green-Kubo formulas have been derived for all the possible transport coefficients, giving them a microscopic foundation.", "The symmetries under time reversal and the point group of unbroken spatial rotations have been used to reduce the number of transport coefficients.", "Time-reversal symmetry leads to the Onsager-Casimir reciprocal relations between the coefficients coupling different transport processes.", "These latter may contribute to entropy production if their coupling is symmetric under time reversal, or be dissipativeless if their coupling is antisymmetric.", "Among the former, there are the heat conductivities, the viscosities, the coefficients associated with the order parameters, and the cross effects between heat transport and the order parameters.", "Among the latter, cross effects are furthermore predicted between heat and momentum transport, and between momentum transport and the order parameters.", "In isotropic phases of matter, these latter cross effects are absent according to Curie's principle based on the full continuous group of three-dimensional rotations.", "However, such cross effects become possible in the presence of anisotropies, such as planar interfaces between two isotropic phases [54].", "Here, we have found that, in some classes of crystalline solids and liquid crystals, properties may be coupled together with a rank-three tensor of non-vanishing coefficients because of anisotropy, as it is the case for piezoelectricity [58].", "The microscopic expressions of all these transport coefficients are here given by Green-Kubo formulas.", "For crystalline solids, previously obtained results are recovered [23], [24].", "Moreover, the unified approach also provides the microscopic expressions of transport properties in liquid crystals, depending on their broken continuous symmetries.", "We note that the approach can be extended to quantum systems [17], [18], [27], [28], [30], [26].", "In future work, we hope to use the methods developed in the present paper, in particular, to investigate the hydrodynamic and Nambu-Goldstone modes and their damping in crystalline solids and liquid crystals." ], [ "Acknowledgements", "Financial support from the Université Libre de Bruxelles (ULB) and the Fonds de la Recherche Scientifique - FNRS under the Grant PDR T.0094.16 for the project \"SYMSTATPHYS\" is acknowledged.", "Joël Mabillard: orcid.org/0000-0001-6810-3709 Pierre Gaspard: orcid.org/0000-0003-3804-2110" ], [ "Deduction of dissipativeless equations for conjugated fields", "In this appendix, the method is presented for deducing the dissipativeless equations (REF )-() of the conjugated fields.", "Since the conjugated fields are functions of the fields $e_0$ , $\\rho $ , and $u^{a\\alpha }$ , we have in particular that $\\frac{d\\beta }{dt} = \\left(\\frac{\\partial \\beta }{\\partial e_0}\\right)_{\\rho ,\\mathsf {u}} \\frac{de_0}{dt} + \\left(\\frac{\\partial \\beta }{\\partial \\rho }\\right)_{e_0,\\mathsf {u}} \\frac{d\\rho }{dt} + \\left(\\frac{\\partial \\beta }{\\partial u^{a\\alpha }}\\right)_{e_0,\\rho } \\frac{d u^{a\\alpha }}{dt} \\, ,$ where the time derivatives in the right-hand side are given by the Lagrangian equations (REF )-().", "Using Eq.", "(REF ), we get $\\frac{d\\beta }{dt} &=& \\left(\\frac{\\partial \\beta }{\\partial e_0}\\right)_{\\rho ,\\mathsf {u}} \\left[\\beta \\left(\\frac{\\partial p}{\\partial \\beta }\\right)_{\\beta \\mu _0,\\beta \\phi } \\nabla ^a v^a +\\phi ^{a\\alpha }\\, A^{b\\alpha }\\, \\nabla ^a v^b -B^{ab\\alpha } \\nabla ^c \\phi ^{c\\alpha } \\, \\nabla ^a v^b + \\nabla ^a(\\phi ^{a\\alpha } B^{bc\\alpha } \\nabla ^b v^c)\\right] \\nonumber \\\\&& + \\left(\\frac{\\partial \\beta }{\\partial \\rho }\\right)_{e_0,\\mathsf {u}} \\left\\lbrace -\\beta \\left[\\frac{\\partial p}{\\partial (\\beta \\mu _0)}\\right]_{\\beta ,\\beta \\phi }\\nabla ^a v^a\\right\\rbrace \\nonumber \\\\&& + \\left(\\frac{\\partial \\beta }{\\partial u^{a\\alpha }}\\right)_{e_0,\\rho } \\left\\lbrace -\\beta \\left[\\frac{\\partial p}{\\partial (\\beta \\phi ^{a\\alpha })}\\right]_{\\beta ,\\beta \\mu _0}\\nabla ^b v^b + \\nabla ^a\\left( A^{b\\alpha } v^b + B^{bc\\alpha } \\nabla ^b v^c \\right)\\right\\rbrace .$ Gathering together all the terms with the divergence $\\nabla \\cdot {\\bf v}$ and using the Maxwell relations (REF )-(), we find $&&\\frac{d\\beta }{dt} = \\beta \\left\\lbrace \\left(\\frac{\\partial p}{\\partial \\beta }\\right)_{\\beta \\mu _0,\\beta \\phi } \\left(\\frac{\\partial \\beta }{\\partial e_0}\\right)_{\\rho ,\\mathsf {u}} +\\left[\\frac{\\partial p}{\\partial (\\beta \\mu _0)}\\right]_{\\beta ,\\beta \\phi } \\left[\\frac{\\partial ( \\beta \\mu _0)}{\\partial e_0}\\right]_{\\rho ,\\mathsf {u}} + \\left[\\frac{\\partial p}{\\partial (\\beta \\phi ^{a\\alpha })}\\right]_{\\beta ,\\beta \\mu _0} \\left[\\frac{\\partial (\\beta \\phi ^{a\\alpha })}{\\partial e_0}\\right]_{\\rho ,\\mathsf {u}} \\right\\rbrace \\nabla ^b v^b \\nonumber \\\\&&\\qquad + \\left(\\frac{\\partial \\beta }{\\partial e_0}\\right)_{\\rho ,\\mathsf {u}} \\left[ \\phi ^{a\\alpha }\\, A^{b\\alpha }\\, \\nabla ^a v^b -B^{ab\\alpha } \\nabla ^c \\phi ^{c\\alpha } \\, \\nabla ^a v^b + \\nabla ^a(\\phi ^{a\\alpha } B^{bc\\alpha } \\nabla ^b v^c)\\right] \\nonumber \\\\&&\\qquad -\\left[\\frac{\\partial (\\beta \\phi ^{a\\alpha })}{\\partial e_0}\\right]_{\\rho ,\\mathsf {u}} \\nabla ^a\\left( A^{b\\alpha } v^b + B^{bc\\alpha } \\nabla ^b v^c \\right) \\, .$ The coefficient of the first term is given by $(\\partial p/\\partial e_0)_{\\rho ,\\mathsf {u}}$ .", "Neglecting the terms of $O(\\nabla ^2)$ , we find $\\frac{d\\beta }{dt} = \\beta \\left(\\frac{\\partial p}{\\partial e_0}\\right)_{\\rho ,\\mathsf {u}} \\nabla ^a v^a -\\beta \\left(\\frac{\\partial \\phi ^{a\\alpha }}{\\partial e_0}\\right)_{\\rho ,\\mathsf {u}} A^{b\\alpha } \\, \\nabla ^a v^b + O(\\nabla ^2)\\, .$ We thus find Eq.", "(REF ) as a consequence of the definition (REF ) of the reversible stress tensor and the condition (REF ).", "Similar deductions can be carried out for Eqs.", "() and ()." ], [ "Deduction of ${\\Sigma _t}$ at first order", "Here, we deduce the leading contribution to the quantity $\\Sigma _t(\\Gamma )$ given in Eq.", "(REF ).", "The two terms in the integrand are computed separately using $\\partial _\\tau (\\beta v^a) = v^a\\partial _\\tau \\beta + \\beta \\partial _\\tau v^a$ and a similar expression with $\\nabla ^a$ replacing $\\partial _\\tau $ , together with the chemical potential (REF ) in the frame moving with matter.", "For the first and second series of terms, we respectively obtain $\\partial _\\tau {\\lambda }^{\\alpha } \\ast \\delta \\hat{c}^\\alpha = \\partial _\\tau \\beta \\ast \\left(\\delta \\hat{e} - v^a\\, \\delta \\hat{g}^a + \\frac{{\\bf v}^2}{2}\\, \\delta \\hat{\\rho }\\right) - (\\beta \\, \\partial _\\tau v^a)\\ast \\left(\\delta \\hat{g}^a - v^a\\delta \\hat{\\rho }\\right) - \\partial _\\tau (\\beta \\mu _0)\\ast \\delta \\hat{\\rho } - \\partial _\\tau (\\beta \\phi ^{a \\alpha })\\ast \\delta \\hat{u}^{a\\alpha }$ and $\\nabla ^a{\\lambda }^{\\alpha } \\ast \\delta \\hat{J}^{a}_{{c}^\\alpha } = \\nabla ^a\\beta \\ast \\left(\\delta \\hat{J}^a_e - v^b\\delta \\hat{J}_{g^b}^a + \\frac{{\\bf v}^2}{2}\\, \\delta \\hat{J}^a_\\rho \\right) - (\\beta \\,\\nabla ^a v^b)\\ast \\left(\\delta \\hat{J}_{g^b}^a - v^b\\delta \\hat{J}^a_\\rho \\right) - \\nabla ^a(\\beta \\mu _0)\\ast \\delta \\hat{J}^a_\\rho - \\nabla ^a(\\beta \\phi ^{a\\alpha })\\ast \\delta \\hat{J}_{x^\\alpha }$ with $\\delta \\hat{J}^a_\\rho = \\delta \\hat{g}^a$ .", "Next, Eq.", "(REF ) for the total time derivative along the stream lines is used to obtain the partial time derivatives of the conjugated fields as $\\partial _{\\tau }\\lambda ^{\\alpha }=d\\lambda ^{\\alpha }/d\\tau -v^a\\nabla ^a\\lambda ^{\\alpha }$ from the dissipativeless equations (REF )-(REF ).", "In this way, Eq.", "(REF ) is transformed into $&&\\partial _\\tau {\\lambda }^{\\alpha } \\ast \\delta \\hat{c}^\\alpha = \\nabla ^a\\beta \\ast \\left\\lbrace -v^a\\left(\\delta \\hat{e} - v^b\\, \\delta \\hat{g}^b + \\frac{{\\bf v}^2}{2}\\, \\delta \\hat{\\rho }\\right) -\\rho ^{-1}\\left[(e_0+p)\\, \\delta ^{ab} - \\phi ^{a\\alpha } A^{b\\alpha }\\right]\\left(\\delta \\hat{g}^b-v^b\\delta \\hat{\\rho }\\right)\\right\\rbrace \\nonumber \\\\&& - (\\beta \\, \\nabla ^a v^b)\\ast \\left[- v^a \\left(\\delta \\hat{g}^b - v^b\\delta \\hat{\\rho }\\right)+\\left(\\frac{\\partial \\sigma ^{ab}}{\\partial e_0}\\right)_{\\rho , \\mathsf {u}} \\left(\\delta \\hat{e} - v^c\\, \\delta \\hat{g}^c + \\frac{{\\bf v}^2}{2}\\, \\delta \\hat{\\rho }\\right) + \\left(\\frac{\\partial \\sigma ^{ab}}{\\partial \\rho }\\right)_{e_0, \\mathsf {u}} \\delta \\hat{\\rho } + \\left(\\frac{\\partial \\sigma ^{ab}}{\\partial u^{c\\gamma }}\\right)_{e_0,\\rho } \\delta \\hat{u}^{c\\gamma }\\right] \\nonumber \\\\&&+\\nabla ^a(\\beta \\mu _0)\\ast \\delta \\hat{g}^a - \\nabla ^a(\\beta \\phi ^{b \\alpha })\\ast \\left[\\rho ^{-1} \\delta ^{ab} \\, A^{c\\alpha } \\left(\\delta \\hat{g}^c-v^c\\delta \\hat{\\rho }\\right) - v^a\\, \\delta \\hat{u}^{b\\alpha }\\right]+ O(\\nabla ^2) \\, .$ Summing Eqs.", "(REF ) and (REF ), we find $&&\\partial _\\tau {\\lambda }^{\\alpha } \\ast \\delta \\hat{c}^\\alpha + \\nabla ^a{\\lambda }^{\\alpha } \\ast \\delta \\hat{J}^{a}_{{c}^\\alpha } \\nonumber \\\\&&=\\nabla ^a\\beta \\ast \\left\\lbrace \\delta \\hat{J}^a_e - v^b\\delta \\hat{J}_{g^b}^a + \\frac{{\\bf v}^2}{2}\\, \\delta \\hat{g}^a -v^a\\left(\\delta \\hat{e} - v^b\\, \\delta \\hat{g}^b + \\frac{{\\bf v}^2}{2}\\, \\delta \\hat{\\rho }\\right) -\\rho ^{-1}\\left[(e_0+p)\\, \\delta ^{ab} - \\phi ^{a\\alpha } A^{b\\alpha }\\right]\\left(\\delta \\hat{g}^b-v^b\\delta \\hat{\\rho }\\right)\\right\\rbrace \\nonumber \\\\&& -(\\beta \\, \\nabla ^a v^b)\\ast \\bigg [\\delta \\hat{J}_{g^b}^a - v^b\\delta \\hat{g}^a - v^a\\delta \\hat{g}^b + v^a v^b\\delta \\hat{\\rho } \\nonumber \\\\&&\\qquad \\qquad \\qquad +\\left(\\frac{\\partial \\sigma ^{ab}}{\\partial e_0}\\right)_{\\rho , \\mathsf {u}} \\left(\\delta \\hat{e}- v^c\\, \\delta \\hat{g}^c + \\frac{{\\bf v}^2}{2}\\, \\delta \\hat{\\rho }\\right) + \\left(\\frac{\\partial \\sigma ^{ab}}{\\partial \\rho }\\right)_{e_0, \\mathsf {u}} \\delta \\hat{\\rho } + \\left(\\frac{\\partial \\sigma ^{ab}}{\\partial u^{c\\gamma }}\\right)_{e_0,\\rho } \\delta \\hat{u}^{c\\gamma }\\bigg ] \\nonumber \\\\&&-\\nabla ^a(\\beta \\phi ^{b \\alpha })\\ast \\left[\\delta ^{ab}\\, \\delta \\hat{J}_{x^\\alpha } + \\rho ^{-1} \\delta ^{ab} \\, A^{c\\alpha } \\left(\\delta \\hat{g}^c-v^c\\delta \\hat{\\rho }\\right) - v^a\\, \\delta \\hat{u}^{b\\alpha }\\right]+ O(\\nabla ^2) \\, .", "$ We note that $\\delta \\hat{u}^{c\\gamma }=\\nabla ^c\\delta \\hat{x}^{\\gamma }$ is of $O(\\nabla )$ , so that the terms involving this quantity multiplied by another gradient contribute to the corrections $O(\\nabla ^2)$ in Eq.", "(REF ).", "Finally, setting $v^a=0$ and $\\phi ^{a\\alpha }=0$ in Eq.", "(REF ), we obtain the expression (REF ), where the deviations of the current densities are given by Eqs.", "(REF )-()." ] ]

[ [ "Adversarial Attacks and Defense on Texts: A Survey" ], [ "Abstract Deep learning models have been used widely for various purposes in recent years in object recognition, self-driving cars, face recognition, speech recognition, sentiment analysis, and many others.", "However, in recent years it has been shown that these models possess weakness to noises which force the model to misclassify.", "This issue has been studied profoundly in the image and audio domain.", "Very little has been studied on this issue concerning textual data.", "Even less survey on this topic has been performed to understand different types of attacks and defense techniques.", "In this manuscript, we accumulated and analyzed different attacking techniques and various defense models to provide a more comprehensive idea.", "Later we point out some of the interesting findings of all papers and challenges that need to be overcome to move forward in this field." ], [ "Introduction", "From the beginning of the past decade, the study and application of Deep Neural Network (DNN) models have sky-rocketed in every research field.", "It is currently being used for computer vision [9], [8], speech recognition[12], [22], medical image analysis[28], [39], natural language processing[50], [32], and many more.", "DNN’s are capable of solving large scale complex problems with relative ease which, tends to be difficult for regular statistical machine learning models.", "That is why in many real-world applications, DNN’s are used profoundly and explicitly.", "In a study by C. Szegedyszegedy2013intriguing, showed that DNN models are not that robust in the image domain.", "They are in fact, quite easy to fool and can be tampered in such a way to obey the will of an adversary.", "This study caused an uproar in the researcher community and researchers started to explore this issue in other research areas as well.", "Different researchers worked tirelessly and showed that DNN models were vulnerable in object recognition systems [18], audio recognition[10], malware detection [19], and sentiment analysis systems [14] as well.", "An example of the adversarial attacks is shown in figure 1.", "The number of studies of adversarial attacks and defenses in the image domain outnumbers the number of studies performed in textual data [44].", "In Natural Language Processing (NLP), for various applications like sentiment analysis, machine translations, question-answering, and in many others, different attacks and defense have been employed.", "In the field of NLP, Papernot papernot2016crafting paved the way by showing that adversarial attacks can be implemented for textual data as well.", "After that, various research has been performed to explore adversarial attacks and defense in the textual domain.", "Adversarial attacks are a security concern for all real-world applications that are currently running on DNNs.", "It is the same scenario for NLP as well.", "There are many real-world programs launched, which is based on DNNs like sentiment analysis [33], text question-answering [20], machine translation [47], and many others.", "Users in the physical world use these applications in their lives to get suggestions about a product, movies, or restaurant or to translate texts.", "An Adversary could easily use attack techniques for ill-will and provide wrong recommendations and falsify texts.", "Since these attacks are not observed by the models due to the lack of robustness these programs would lose values.", "Thus, the study of adversarial attacks and defense with respect to text is of utmost importance.", "Figure: Adversarial attacks on image and audio data In this review, our major contributions can be summarized as follows.", "We provide a systematic analysis and study of different adversarial attacks and defense techniques which are shown in different research projects related to classification, machine translations, question-answering, and many others.", "We present here adversarial attack and defense techniques by considering different attack levels.", "After going through all the research works we have tried to answer which attack and defense technique has the advantage over other techniques.", "Finally, we present some exciting findings after going through all the research works and point out challenges that need to be overcome in the future.", "Our manuscript is organized as follows.", "Related research works are mentioned in section 2.", "In section 3 we start by providing preliminary information about adversarial machine learning concerning both image and textual data.", "Here we also provide a classification of different attack and defense approaches as well.", "Following this, in section 4 we discuss various attack techniques based on the taxonomy presented in section 3.", "To defend models from these attack techniques we discuss different defense techniques in section 5.", "We provide an in-depth discussion of our findings in these topics and some challenges in section 6.", "We conclude our manuscript by providing a conclusion in section 7." ], [ "Related Works", "For textual data this topic has not been explored that much but for the image domain, it has been explored much more.", "Since this topic hasn’t been explored that much small number of publications about it have been found and a smaller number of review works have been done.", "We were able to go through three review papers related to this topic [5], [48], [51].", "In the manuscript of Belinkovbelinkov2019analysis, they mainly studied different analysis in NLP, visualization and what type of information neural networks capture.", "They introduced adversarial examples to explain that traditional DNN models are weak.", "Xu xu2019adversarial explained adversarial examples in all domains i.e.", "image, audio, texts etc.", "It was not specialized for text but mostly related to image domain.", "Zhang zhang2020adversarial in their manuscript described different publications related to adversarial examples.", "Unfortunately it was focused largely on attack strategies and shed little light on defense technique.", "They mainly discussed data augmentation, adversarial training and one distillation technique proposed by [35].", "Table 1 provides a comparative analysis between these survey papers and ours.", "Table: Comparison with recent surveys.", "(No.", "of stars represent how much that topic is discussed)" ], [ "Adversarial Machine Learning", "Modern machine learning and deep learning has achieved a whole new height because of its high computational power and fail-proof architecture.", "However, recent advances in adversarial training have broken this illusion.", "A compelling model can misbehave by a simple attack by adversarial examples.", "An adversarial example is a specimen of input data that has been slightly transformed in such a way that can fool a machine learning classifier resulting in misclassification.", "The main idea behind this attack is to inject some noise to the input to be classified that is unnoticeable to the human eye so that the resulting prediction is changed from actual class to another class.", "Thus we can understand the threat of this kind of attack to classification models." ], [ "Definition", "For a given input data and its labels $(x,y)$ and a classifier $F$ which is capable of mapping inputs $x$ to its designated labels $y$ in the general case we can define them as $F(x) = y$ .", "However, for adversarial attack techniques apart from input data a small perturbation $\\delta $ is also added to the classifier $F$ .", "Note that, this perturbation is imperceptible to human eyes and it is limited by a threshold $||\\delta ||<\\epsilon $ .", "In this case, the classifier is unable to map it to the original labels.", "Hence, $F(x+\\delta ) \\ne y$ .", "The concept of imperceptibility is discussed in length in section $5.2$ .", "A robust DNN model should be able to look beyond this added perturbation and be able to classify input data properly.", "i.e.", "$F(x+\\delta ) = y$ ." ], [ "Existence of Adversarial Noises", "Since adversarial examples have been uncovered, a growing and difficult question have been looming over the research community.", "Why adversarial examples exist in real-life examples.", "Several hypotheses have been presented in an attempt to answer this question.", "However, none of them have achieved a unanimous agreement of the overall researcher community.", "The very first explanation comes from C. Szegedy’s paper szegedy2013intriguing.", "In it, he says that adversarial examples exist because there are too much non-linearity and the network is not regularized properly.", "Opposing this hypothesis I. Goodfellow says that it exists because of too much linearity in machine learning and deep learning models goodfellow2014explaining.", "Various activation functions that we use today like ReLU and Sigmoid are straight lines in the middle parts.", "He argues that since we want to protect our gradients from vanishing or exploding we tend to keep our activation functions straight.", "Hence, if a small noise is added to the input because of the linearity it perpetuates in the same direction and accumulates at the end of the classifier and produces miss-classification.", "Another hypothesis that is present today is called tilted boundary [42].", "Since the classifier is never able to fit the data exactly there is some scope for the adversarial examples to exist near the boundaries of the classifier.", "A recent paper argues that adversarial examples are not bugs but they are features and that is how deep neural networks visualize everything [23].", "They classified the features into two categories called robust and non-robust features and showed that by adding small noises non-robust features can make the classifier to provide a wrong prediction." ], [ "Classification of Adversarial Examples", "Here, we provide a basic taxonomy of adversarial attacks and adversarial defense techniques based on different metrics.", "For adversarial attack techniques, we can classify different attacks based on how much the adversary has knowledge about the model.", "We can divide it into two types.", "White-Box Attacks: In order to execute these types of attacks the adversary needs to have full access to the classifier model.", "Using the model parameters, architectures, inputs, and outputs the adversary launch the attack.", "These types of attacks are the most effective and harmful since it has access to the whole model.", "Black-Box Attacks: These attacks represent real-life scenarios where the adversary has no knowledge about the model architecture.", "They only know about the input and output of the model.", "In order to obtain further information, they use queries.", "We can classify attacks based on the goal of the adversary as well.", "We can classify it into two types.", "Non-Targeted Attack: Adversary in this scenario does not care about the labels that the model produces.", "They are only interested in reducing the accuracy of the model.", "i.e.", "$F(x+\\delta ) \\ne y$ .", "Targeted Attack: In targeted attacks, the adversary forces the model to produce a specific output label for given images.", "i. e. $F(x+\\delta ) = y*$ .", "These are the general classifications of different attacks.", "However, for NLP tasks, we can classify attacks differently.", "Since the data in text-domain is different from the data in the image or audio domain attack strategy and attack types are somewhat different.", "Based on which components are modified in the text we can classify attack techniques into four different types.", "They are called character-level attacks, word-level attacks, sentence-level attacks, and multi-level attacks.", "In these adversarial attacks text data are generally inserted, removed, swapped/replaced, or flipped.", "Though not all of these options are explored in different levels of attacks.", "Character-Level Attack: Individual characters in this attack are either modified with new characters, special characters, and numbers.", "These are either added to the texted, swapped with a neighbor, removed from the word, or flipped.", "Word-Level Attack: In this attacks words from the texts are changed with their synonyms, antonyms, or changed to appear as a typing mistake or removed completely.", "Sentence-Level Attack: Generally new sentences are inserted as adversarial examples in these types of attacks.", "No other approach has been explored yet.", "Multi-Level Attack: Attacks which can be used in a combination of character, word, and sentence level are called multi-level attack.", "Figure: Adversarial attacks classificationAdversarial defense techniques have been studied in mainly three directions [1].", "They are Modified Training/Input: In these cases, the model is trained differently to learn more robust features and become aware of adversarial attacks.", "During testing, inputs are also modified to make sure no adversarial perturbation is added to it.", "Modifying Networks: By adding more layers or sub-networks and changing loss or activation functions defense is sought in this scenario.", "Network Add-on: Using external networks as additional sections for classifying unseen data." ], [ "Adversarial Attacks", "Most of the literature is about attack techniques that are where we start our discussion.", "In this section, we will be analyzing different attack techniques published in recent years in detail.", "In order to provide a clear understanding, we are diving our explanations based on the taxonomy for the NLP that we mentioned in section 2." ], [ "Character-Level Attack", "As mentioned before character level attacks includes attack schemes which try to insert, modify, swap, or remove a character, number, or special character.", "Ebrahimi ebrahimi2018adversarial in his paper worked with generating adversarial examples for character-level neural machine translation.", "They provided white and black box attack techniques and showed that white-box attacks were more damaging than black-box attacks.", "They proposed a controlled adversary that tried to mute a particular word for translation and targeted adversary which aimed to push a word into it.", "They used gradient-based optimization and in order to edit the text, they performed four operations insert, swap two characters, replace one character with another and delete a character.", "For black-box attack, they just randomly picked a character and made necessary changes.", "Belinkov belinkov2017synthetic worked with character-based neural machine translation as well.", "In their paper, they didn’t use or assume any gradients.", "They relied on natural and synthetic noises for generating adversarial noises.", "For natural noises, they collected different errors and mistakes from various datasets and replaced correct words with wrong ones.", "In order to generate synthetic noises they relied on swapping characters, randomized characters of a word except the first and last one, randomized all the characters, and replaced one character with a character from its neighbor in the keyboard.", "Another black-box attack was proposed by Gao gao2018black.", "They worked with Enron spam emails and IMBD dataset for classification tasks.", "Since in the black-box settings, an adversary does not have access to gradients they proposed a two-step process to determine which words are the most significant ones.", "Temporal score and temporal tail scores are to be calculated to determine the most significant word.", "This approach was coined as DEEPWORDBUG by the authors.", "To calculate the temporal score, they checked how much effect each word had on the classification result.", "The Temporal tail score is the complement of temporal scores.", "For temporal tail score, they compared results for two trailing parts of sentences, one had a particular word and another did not have it.", "TEXTBUGGER is both a white-box and black-box attack framework that was proposed by Li li2018textbugger.", "For generating bugs or adversarial examples they focused on five kinds of edits: insertion, deletion, swapping, substitution with a visually similar word, substitution with semantically similar meaning.", "For white-box attacks, they proposed a two-step approach.", "The first step is to determine which words are most significant with the help of determining the Jacobian matrix.", "Then generate all five bugs and choose the one which is the most optimal for reducing accuracy.", "In order to generate black-box attacks in this framework, they propose a three-step approach.", "Since there is no access to the gradient thus they propose to determine first which sentence is the most important one.", "Then determine which word is the most significant and finally generate five bugs for it and choose which one is the most optimal.", "Gil gil2019white was able to transform a white-box attack technique to a black-box attack technique.", "They generated adversarial examples from a white-box attack technique and then trained a neural network model to imitate the overall procedure.", "They transferred the adversarial examples generation by the HotFlip approach to a neural network.", "They coined these distilled models as DISTFLIP.", "Their approach had the advantages of not being dependent on the optimization process which made their adversarial example generation faster.", "Perspective is an API that is built by Google and Jigsaw to detect toxicity in comments.", "Hosseini hosseini2017deceiving showed that it can be deceived by modifying inputs.", "They didn’t have any calculated approach, mainly modified toxic words by adding a dot (.)", "or space between two characters or swapping two characters, and thus they showed the API got lower toxicity score than before." ], [ "Word Level Attack", "Papernot papernot2016crafting was the first one to generate adversarial examples for texts.", "They used a computational graph unfolding technique to calculate the forward derivative and with its help the Jacobian.", "It helps to generate adversarial examples using the FGSM technique.", "The words they choose to replace with are chosen randomly so the sentence doesn’t keep original meaning or grammatical correctness.", "To change a particular text classification label with the minimum number of alterations Samanta samanta2017towards proposed a model.", "In their model, they either inserted a new word or deleted one or replaced one.", "They at first determined which words are highly contributing to the classifier.", "They determined a word is highly contributing if removing it changes the class probability to a large extend.", "To replace the words they created a candidate pool based on synonyms, typos that produce meaningful words and genre-specific words.", "They changed a particular word based on the following conditions.", "Word is an adverb and highly contributing remove it Choose a word from the candidate pool the word is an adjective and candidate word is adverb Insert replace particular word with candidate word Liang liang2017deep proposed a white-box and black-box attack strategy based on insertion, deletion, and modification.", "To generate adversarial examples they used natural language watermarking technique [3].", "To perform a white-box attack they provided a concept of Hot Training Phrase (HTP) and Hot Sample Phrase (HSP).", "These are obtained with the help of back-propagation to get all the cost gradients of each character.", "HTP helps to determine what needs to be inserted while HSP helps to determine where to insert, delete, and modify.", "For black-box attacks, they borrowed the idea of fuzzing technique [40] for implementing a test to get HTPs and HSPs.", "To preserve syntactical and semantic meaning Kuleshov kuleshovadversarial used thought vectors.", "They took the inspiration from [6], [30] which mapped sentences to vectors.", "Those who had similar meanings were placed together.", "To ensure semantic meaning they introduce syntactic constraint.", "Their approach was iterative and in each iteration, they replaced only one word with their nearest neighbor which changed the objective function the most.", "Genetic algorithm based black-box attack techniques are proposed by Alzanot alzantot2018generating.", "They tried to generate adversarial examples that were semantically and syntactically similar.", "For a particular sentence, they randomly select a word and replace it with a suitable replacement word that fits the context of the sentence.", "For this, they calculate the first few nearest neighbor words according to the GloVe embedding space.", "Next using Google’s 1 billion words language model they try to remove any words which do not match the context.", "After that, they select a particular word which maximizes the predication.", "This word is then inserted into the sentence.", "An improvement of the genetic algorithm based attack was proposed by Wang wang2019natural.", "They modified it by allowing a single word of a particular sentence to be changed multiple times.", "To ensure that the word is indeed a synonym of the original word it needs to be fixed.", "A sememe based word substitution method using particle swarm optimization technique was proposed by [49].", "A sememe is the minimum semantic unit in human language.", "They argued that word embedding and language model based substitution methods can find many replacements but they are not always semantically correct or related to the context.", "They compared their work with [2] attack technique and showed their approach was better." ], [ "Sentence Level Attack", "In the domain of question answering Robin Jia jia2017adversarial introduced two attack techniques called ADDSENT and ADDANY.", "They also introduced two variants of this ADDONESENT and ADDCOMMON randomly.", "Here, ADDONESENT is a model-independent attack i.e.", "black-box attack.", "Using these attacks they generated an adversarial example that does not contradict the original answer and inserts it at the end of the paragraph.", "To show the effectiveness of ADDSENT and ADDANY they used it on 16 different classifiers and showed that all of them got reduced F1 score.", "Figure 5. provides a visual representation of how ADDSENT and ADDANY attack is used for generating texts.", "Figure: ADDANY and ADDSENT Attack Generation There is another group of researchers named Yicheng Wang wang2018robust who have worked on the modification of the ADDSENT model.", "They proposed two modifications of the ADDSENT model and named their model as ADDSENTDIVERSE.", "ADDSENT model creates fake answers that are semantically irrelevant but follows similar syntax as a question.", "In ADDSENTDIVERSE, they targeted to generate adversarial examples with a higher variance where distractors will have randomized placements so that the set of fake answers will be expanded.", "Moreover, to address the antonymstyle semantic perturbations that are used in ADDSENT, they added semantic relationship features enabling the model to identify the semantic relationship among contexts of questions with the help of WordNet.", "The paper shows that the ADDSENTDIVERSE model beats ADDSENT trained model by an average improvement of 24.22% in F1 score across three different classifiers indicating an increase in robustness.", "Zhao zhao2017generating Proposed a new framework utilizing Generative Adversarial Networks (GAN) on Stanford Natural Language Interface (SNLI) dataset to generate grammatically legible and natural adversarial examples that are valid and semantically close to the input and can detect local behavior of input by searching in semantic space of continuous data representation.", "They have implemented these adversaries in different applications such as image classification, machine translation, and textual entailment to evaluate the performance of their proposed approach on black-box classifiers such as ARAE ( Adversarially Regularized Autoencoder), LSTM and TreeLSTM.", "By their work, they have proved that their model is successful to generate adversaries that can pass common-sense reasoning by logical inference and detect the vulnerability of the Google Translate model during machine translation.", "Cheng cheng2019robust worked with neural machine translation and proposed a gradient-based white-box attack technique called AdvGen.", "Guided by the training loss they used a greedy choice based approach to find the best solution.", "They also used the language model into it as well because it is computationally easy for solving an intractable solution and it also retains somewhat semantic meaning.", "Their research paper is based on using adversarial examples for both attack generation and using these adversarial examples to improve the robustness of the model.", "Michaelmichel2019evaluation worked with neural machine translation as well and in their manuscript, they proposed a natural criterion for untargeted attacks.", "It is “adversarial examples should be meaning preserving on the source side but meaning destroying on the target side”.", "From it, we can see that they are focusing on the point about preserving the meaning of the sentences while pushing adversarial examples into it.", "They propose a white-box attack using the gradients of the model which replaces one word from the sentences to maximize the loss.", "To preserve the meaning of the sentences they used KNN to determine the top 10 words which are similar to a given word.", "This approach has the advantage of preserving the semantic meaning of the sentence.", "They allowed swapping characters to create substitute words but if the word is out of the vocabulary then they repeated the last character to generate the substitute word." ], [ "Multi-level Attack", "HotFlip is a very popular, fast, and simple attack technique that was proposed by Ebrahimi ebrahimi2017hotflip.", "This is a white-box gradient-based attack.", "In the core of the attack lies a simple flip operation which is based on the directional derivatives of the model with respect to one-hot encoding input.", "Only one forward and backward pass is required to predict the best flip operation.", "This attack can also include insertion and deletion as well if they are represented as character sequences.", "After estimating which changes ensure the highest classification errors a beam search algorithm finds a set of manipulations that works together to ensure the classifier is confused.", "Their original manuscript was on character level adversarial attack but they also showed that their approach can be extended to word-level attack as well.", "Since flipping a word to another has the possibility of losing its original value they flipped a word only if it satisfied certain conditions.", "They flipped a word if the cosine similarity of the word embeddings were higher than a given threshold and if they were members of the same parts of speech.", "They didn’t allow stop words to be removed.", "On the topic of question answering system Blohm blohm2018comparing implemented word and sentence level white-box and black-box attacks.", "They started by achieving the state of the art score on the MovieQA dataset and then investigate different attacks effect.", "Word-level Black-box Attack: For this type of attack the authors substituted the words manually by choosing lexical substitutions that preserved their meanings.", "To ensure the words were inside of the vocabulary they only switched words which were included in the pretrained GloVe embeddings.", "Word-level White-box Attack: With the help of the attention model they used for classification they determined which sentence and which word was the most important one.", "This had a huge impact on the prediction results.", "Sentence-level Black-box Attack: Adopting the strategy of ADDANY attack proposed by [24] they initialized a sentence with ten common English words.", "Then each word is changed to another word which reduces the prediction confidence the most.", "Sentence-level White-box Attack: Similar to the word-level white-box attack they target the sentence which has the highest attention i.e.", "the plot sentence.", "They removed the plot sentence to see if the classifier was indeed focusing on it and its prediction capability.", "Wallace wallace2019universal proposed a technique in which they added tokens at the beginning or end of a sentence.", "They attempt to find universal adversarial triggers that are optimized based on the white-box approach but which can also be transferred to other models.", "At the very beginning, they start by choosing trigger lengths as this is an important criterion.", "Longer triggers are more effective but more noticeable than shorter ones.", "To replace the current tokens they took inspiration from the HotFlip approach proposed by [14].", "They showed that for text classification tasks the triggers caused targeted errors for sentiment analysis and reading comprehension task triggers can cause paragraphs to generate arbitrary target prediction." ], [ "Adversarial Defense", "As mentioned earlier most of the researchers focused on attacking DNN models in the field of NLP few focused on defending it.", "Here we divide our studied manuscripts into two sections one being the most common approach called adversarial training found in [18].", "In the second section, we include all the research papers that try to tackle attacks by working on a specific defense technique." ], [ "Adversarial Training", "Adversarial training is the process of training a model on both the original dataset and also adversarial example with correct labels.", "The general idea behind this approach is that, since the classifier model is now introduced to both original and adversarial data the model will now look beyond the perturbations and recognize the data properly.", "Belinkov belinkov2017synthetic in their experiments showed that training the model with different types of mixed noises improves the model's robustness to different kinds of noises.", "In the experiments of Li li2018textbugger they also showed for TEXTBUGGER attack adversarial training can improve model performance and robustness against adversarial examples.", "In the experiments of Zang zang2019textual they showed that their sememe based substitution and PSO based optimization improved classifiers' robustness to attacks.", "By using CharSwap during adversarial training on their attack Micheal showed that adversarial training can also improve the robustness of the model.", "Ebrahimi ebrahimi2017hotflip in their manuscript of HotFlip also performed adversarial training.", "During their testing phase, they implemented beam search which wasn’t used for training hence the adversary in the training wasn’t strong as the testing ones.", "This reflects in their adversarial training experimental results as well.", "Though after training with adversarial examples the model attains certain robustness its accuracy isn’t as high as the original testing." ], [ "Topic Specific Defense Techniques", "One of the major problems with adversarial training is that during training different types of attacks need to be known.", "Since adversaries don’t publicize their attack strategies adversarial training is limited by the users’ knowledge.", "If a user tries to perform adversarial training against all attacks known to him then the model would not be able to perform classification properly as it would have very low information on the original data.", "In the research work of Alzanot alzantot2018generating they found that their attack approach which was based on genetic algorithm was indifferent to adversarial training.", "A good reason for this would be that since their attack diversified the input so much adversarial training did not affect them.", "To protect models from synonym based attack techniques Wang wang2019natural proposed the synonym encoding method (SEM) which puts an encoder network before the classifier model and checks for perturbations.", "In this approach, they cluster and encode all the synonyms to a unique code so that they force all the neighboring words to have similar codes in the embedding space.", "They compared their approach with adversarial training on four different attack techniques and showed that the SEM-based technique was better in the synonym substitution attack method.", "Adversarial spelling mistakes were the prime concern of Pruthi pruthi2019combating in their research work.", "Through their approach, they were able to handle adversarial examples which included insertion, deletion, swapping of characters, and keyboard mistakes.", "They used a semi character-based RNN model with three different back-off strategies for a word recognition model.", "They proposed three back-off strategies pass-through, back-off to a neutral word, back-off to the background model.", "They tested their approach against adversarial training and data augmentation based defense and found out that ScRNN with pass-through back-off strategy provided the highest robustness.", "A defense framework was proposed by Zhou zhou2019learning to determine whether a particular token is a perturbation or not.", "The discriminator provides some candidate perturbations and based on the candidate perturbations they used an embedding estimator to restore the original word and based on the context using the help of KNN search.", "The authors named this framework as DISP.", "This discriminator is trained on the original corpus during training time for figuring out which one is the perturbation.", "Token of the embedding corpus is fed to the embedding estimator to train it and recover the original word.", "During the testing phase, the discriminator provides candidate tokens which are perturbations, and for each of the candidate perturbation, the estimator provides an approximate embedding vector and attempts to restore the word.", "After this, the overall restored text can be passed to the model for prediction.", "To evaluate their frameworks' ability to identify perturbation tokens they compared their results against spell checking technique on three character level attacks and two-word level attacks.", "Results show that their approach was more successful in achieving better results.", "To test the robustness of their approach they compared against adversarial training, spell checking, and data augmentation.", "Their approach was able to perform in this experiment as well." ], [ "Discussion", "We will be providing a discussion on some of the interesting findings that we found while studying different manuscripts.", "We are also going to shed some light on the challenges in this area." ], [ "Interesting Findings", "Based on the papers that we had studied we summarize and list out here some interesting findings.", "Character-level Perturbations: It can be astounding to see that changing a single character can affect the model's prediction.", "Hosseini showed that adding dots(.)", "or space in words can be enough to confuse perspective api[21].", "Not only this in HotFlip we have seen that the authors swapped a character based on the gradients to fool the model.", "So, while designing defense strategies a mere character level manipulation needs to be considered as well.", "Research Direction: From the papers that we studied we found out that most of the papers were based on different attack strategies.", "Very few papers were focused on defending the model.", "The same can be said for multi-stage attacks as well.", "Only a few researchers produced manuscripts for multi-stage attacks.", "Adversarial Example Generation: Through the studies of different manuscripts we found that to generate adversarial examples most of the researchers followed a two-step approach.", "The first being finding out which word was the most significant in providing prediction and the second step was to replace it with suitable candidates that benefit the adversary." ], [ "Challenges", "In our study, we found several challenges in this field.", "They are mentioned below.", "This phenomenon was first found in existence in the image domain and it has gained a lot of attention.", "Many research works have been published on it.", "It can be an easy assumption that we can use it in the text domain as well.", "However, there is a significant difference between them.", "In the image domain, the data is continuous but text data is discrete.", "Hence, the attacks proposed in the image domain cannot be utilized in text-domain.", "Another limitation of textual data is the concept of imperceptibility.", "In the image domain, the perturbation can often be made virtually imperceptible to human perception, causing humans and state-of-the-art models to disagree.", "However, in the text domain, small perturbations are usually clearly perceptible, and the replacement of a single word may drastically alter the semantics of the sentence and be noticeable to human beings.", "So, the structure of imperceptibility is an open issue.", "Till now no defense strategy can handle all different types of attacks that were mentioned here.", "Each defense strategy worked on a single type of attack approach.", "For example, for spelling mistakes, we can use the defense technique proposed by [36].", "For synonym based attacks we can use the SEM model.", "A unified model that can tackle all these issues has not been proposed yet.", "The concept of universal perturbation has still not been introduced for textual data.", "In the image domain, researchers established a method that was able to generate a single perturbation that can fool the model.", "Whenever a new attack technique is proposed researchers use different classifiers and datasets as there is no benchmark.", "From table 6 we can see for a particular application different datasets are used no ideal dataset is being used for attack generation.", "Since there is no benchmark it is not easy to compare different attack and defense strategies with each other.", "Lacking such a benchmark is a big gap in this field of research.", "There is no standard toolbox that can be used to easily reproduce different researchers' work.", "There are many toolbox which can be used in the image domain like cleverhans [34], art [31], foolbox [37] etc but there is no standard toolbox for text-domain." ], [ "Conclusion", "In this review, we discussed the adversarial attack and defense techniques for textual data.", "Since the inception of adversarial examples, it has been a very important research topic for many aspects of deep learning applications.", "DNNs perform very well on a standard dataset but perform poorly in the presence of adversarial examples.", "We tried to present an accumulated view of why they exist, different attack and defense strategies based on their taxonomy.", "Also, we pointed out several challenges that can be tended to for getting future direction about research works in the future." ] ]

[ [ "The landscape of disk outflows from black hole - neutron star mergers" ], [ "Abstract We investigate mass ejection from accretion disks formed in mergers of black holes (BHs) and neutron stars (NSs).", "The third observing run of the LIGO/Virgo interferometers provided BH-NS candidate events that yielded no electromagnetic (EM) counterparts.", "The broad range of disk configurations expected from BH-NS mergers motivates a thorough exploration of parameter space to improve EM signal predictions.", "Here we conduct 27 high-resolution, axisymmetric, long-term hydrodynamic simulations of the viscous evolution of BH accretion disks that include neutrino emission/absorption effects and post-processing with a nuclear reaction network.", "In the absence of magnetic fields, these simulations provide a lower-limit to the fraction of the initial disk mass ejected.", "We find a nearly linear inverse dependence of this fraction on disk compactness (BH mass over initial disk radius).", "The dependence is related to the fraction of the disk mass accreted before the outflow is launched, which depends on the disk position relative to the innermost stable circular orbit.", "We also characterize a trend of decreasing ejected fraction and decreasing lanthanide/actinide content with increasing disk mass at fixed BH mass.", "This trend results from a longer time to reach weak freezout and an increasingly dominant role of neutrino absorption at higher disk masses.", "We estimate the radioactive luminosity from the disk outflow alone available to power kilonovae over the range of configurations studied, finding a spread of two orders of magnitude.", "For most of the BH-NS parameter space, the disk outflow contribution is well below the kilonova mass upper limits for GW190814." ], [ "Introduction", "The Advanced LIGO interferometer has completed three observing runs – with Advanced Virgo joining part of the way – resulting in the official detection of 11 binary black hole (BH) mergers and two neutron star (NS) mergers [2], [3], , with many more in candidate status at the time of this writing.", "The increased sensitivity of the third observing run also yielded an event that can be either a BH-BH or a BH-NS merger (GW190814) (e.g., [4], [19], [5], [6], , ).", "Only one of these events (GW170817), however, has had electromagnetic (EM) counterparts detected [1].", "While multiple reasons can account for the lack of an EM detection (such as a large distance, large localization area, galactic extinction, or Sun constraints; e.g.", "[33]), the possibility remains that these sources were intrinsically fainter than the kilonova from GW170817.", "BH-NS mergers can lead to a wide range of ejected masses depending on whether the NS is tidally disrupted by the BH.", "The outcome depends on the masses of the ingoing BH and NS, as well as on the spin of the BH and the compactness of the NS (e.g., [39]).", "The dynamical ejecta emerges as a very neutron-rich, equatorial tidal tail that quickly leaves the system.", "The nucleosynthesis properties and contribution to the kilonova transient are mostly set at the time of ejection (e.g., ), and its properties can be parameterized by direct comparison with dynamical merger simulations (e.g., , ).", "The accretion disk, on the other hand, ejects mass on a longer timescale, as angular momentum is transported initially by gravitational torques and later by magnetohydrodynamic (MHD) turbulence (see, e.g., [28], for reviews).", "This longer evolutionary timescale allows weak interactions to modify the composition, resulting in a different $r$ -process yield (and possibly a different kilonova color) than the dynamical ejecta.", "The complexity of this evolution makes it very expensive to realistically model the disk, however.", "Only a handful of three-dimensional general-relativistic (GR) MHD simulations of disks around BHs have been carried out including at least some important microphysics or neutrino effects , , , [32], , [18].", "Furthermore, all of these simulations either focus on a narrow subset of parameter space and/or do not evolve the system for long enough to achieve completion of mass ejection.", "More extensive studies of BH accretion disks have been carried out using axisymmetric hydrodynamic simulations with a wide variety of approximations to the physics [27], , [29], [43].", "None of these studies covers a significant fraction of all the possible BH accretion disk configurations, however.", "Despite missing the magnetic field, hydrodynamic simulations can provide a good description of the late-time thermal component of the outflow that arises when weak interactions freeze out and heating of the disk by viscous stresses (in lieu of MHD turbulence) is unbalanced.", "Close comparison between GRMHD and hydrodynamic simulations show that the latter provide a lower limit to the fraction of the disk ejected, with magnetic enhancements dependent on the strength of the initial poloidal field in the disk [32], [18].", "Here we carry out an extensive set of long-term hydrodynamic simulations of accretion disks around BH remnants, with the aim of sampling the entirety of the parameter space resulting from BH-NS mergers, and thereby improving parameter estimation models that take disk outflow properties as input (e.g., [8], , [19]).", "Along the way, a broad probe of parameter space allows to identify trends in the disk ejection physics, helping to focus on areas where improvements in the physics (e.g.", "neutrino transport) have the most impact.", "Finally, by providing a lower limit to the disk mass ejected, we are also estimating the lower limit to the raw radioactive heating available to power kilonova transients.", "The structure of the paper is as follows.", "Section presents our methods, including our choice of initial conditions from the plausible parameter space of BH-NS mergers.", "Section presents our results, divided into mass ejection, outflow composition, and implications for EM counterparts.", "We close with a summary and discussion in Section .", "In order to sample representative initial disk masses for our simulations, we map the parameter space of BH-NS merger remnants using analytical formulae that are calibrated to numerical relativity simulations.", "For a given ingoing BH mass $M_{\\rm bh(in)}$ and NS mass $M_{\\rm ns}$ , we uniformly sample the range $9-13$  km for NS radii and $0-0.7$ for the ingoing BH spin, and compute distributions of (1) the remnant baryon mass left outside the BH using the formula of [39], (2) the disk mass using the formula of for the dynamical ejecta, and (3) the post-merger BH mass $M_{\\rm bh(out)}$ and its spin using the formulae of and the output from steps (1)-(2).", "This approach is intended to be agnostic about the properties of the EOS of dense matter and initial BH spin distribution, within plausible limits.", "The resulting cumulative distributions of initial disk masses are shown in Figure REF for two NS mases $\\lbrace 1.35,1.45\\rbrace \\,M_\\odot $ and ingoing BH masses such that the median value of $M_{\\rm bh(out)}$ is $\\lbrace 3,5,8,10,15\\rbrace \\,M_\\odot $ .", "The lowest post-merger BH mass is chosen to explore the hypothetical case of a very low-mass ingoing BH, or the prompt collapse of a NS-NS system.", "The highest BH mass is chosen such that at least $10\\%$ of mergers result in disruption.", "Median disk masses range from $0.1 M_\\odot $ for the $3 M_\\odot $ post-merger BH, to $0.02 M_\\odot $ for the $15M_\\odot $ BH.", "In most cases, sensitivity to the specific choice of NS mass does alter the shape of the histogram but not the extreme values.", "The post-merger BH spin distributions have medians in the range $0.85-0.9$ , except for the lowest mass BHs considered, which have spins $>0.95$ .", "For each post-merger BH mass, Figure REF shows the disk masses sampled in our study.", "The lowest disk mass in all cases is taken to be $0.01M_\\odot $ , which is optically thin to neutrinos, while the largest disk mass is such that it is at the uppermost end of plausible values.", "In all of our models, we take the spin of the post-merger BH to be $0.8$ , which while somewhat lower than the median values of the distributions obtained, lies within the range of plausible values (except for $M_{\\rm bh(out)}=3M_\\odot $ ) and does not demand a prohibitively small time step.", "Furthermore, the effect of BH spin on the disk outflow properties is known, with more mass with higher average electron fraction being ejected for higher BH spins [29], [43].", "Our simulations start from idealized equilibrium tori (§REF ) which approach a Keplerian angular velocity distribution after a few orbits.", "This equilibrium configuration requires more parameters in addition to the disk mass: a radius of maximum density, an entropy (internal energy content), and an electron fraction (composition).", "While these parameter choices would not be necessary if we mapped the disk directly from a merger simulation, our broad coverage of possible BH-NS combinations would be limited by accessible merger simulation data.", "The remaining initial disk parameters are therefore chosen by inspecting the output of numerical relativity simulations of BH-NS mergers and making educated guesses about these values in regimes not covered by simulations.", "We adopt initial disk radii that roughly follow the location of density maxima outside of the BH in published BH-NS simulations [25], , [35], [36], [37], , , [38], [15].", "This radius of maximum density is not well defined, however, since it depends on (1) the time at which it is measured, (2) the metric, and (3) on whether the local density, surface density, or enclosed mass are reported.", "Since there is no consistency across the literature for this quantity, we adopt a fiducial set of initial disk radii $R_{\\rm d}=\\lbrace 50,50,60,90,120\\rbrace $  km for post-merger BH masses $\\lbrace 3,5,8,10,15\\rbrace \\,M_\\odot $ , respectively.", "The entropy of all disks is taken to be $8\\,k_{\\rm B}$ per baryon, which results in ratios of isothermal sound speed to orbital speed $\\sim 10-30\\%$ at the point of maximum density.", "While this choice has some effect on the amount of mass ejected, we use a constant value for uniformity.", "Finally, the default electron fraction of the disks is set to $Y_{e,\\rm ini} = 0.2$ , although we vary this parameter in our simulations given that it is dependent on the quality of the neutrino transport implementation in the merger simulation, and has a non-negligible effect on the disk outflow composition." ], [ "Hydrodynamic Simulations", "We perform time-dependent hydrodynamic simulations with FLASH version 3.2 [42], [23], with the modifications described in [27], , [29], and .", "The code solves the Euler equations in axisymmetric spherical polar coordinates $(r,\\theta )$ , subject to source terms that include the pseudo-Newtonian gravitational potential of a spinning BH [7] without disk self-gravity, shear viscosity with an $\\alpha $ parameterization , and a leakage scheme for neutrino emission, with absorption included as a disk-like light bulb [27], .", "We only include electron type neutrinos/antineutrinos interacting with nucleons via charged-current weak interactions.", "The code employs the equation of state (EOS) of with the abundances of neutrons, protons, and alpha particles in nuclear statistical equilibrium (NSE) above a temperature $T=5\\times 10^9$  K and accounting for the nuclear binding energy of these particles.", "The electron-positron quantities are extended above the high-density limit of the table using analytic expressions [14], [13].", "The initial condition is an equilibrium torus with constant angular momentum, entropy, and electron fraction, with mass fractions assumed to be in NSE (e.g., [27]).", "Parameters are chosen according to §REF .", "The floor of density is set to 10 g cm$^{-3}$ at $r=4R_{\\rm d}$ , and has an initial radial dependence $r^{-2}$ .", "For $r\\le 4R_{\\rm d}$ , the radial exponent of the floor is smoothly brought to zero on a timescale of 40 orbital times at $r=R_{\\rm d}$ , reaching a flat floor in this region ([32], see also ).", "The initial ambient density is set at $1.1$ times the floor.", "The computational domain extends from an inner radius $r_{\\rm in}$ midway between the radius of the innermost stable circular orbit (ISCO) $r_{\\rm isco}$ and the BH horizon, to an outer radius $r_{\\rm out} = 10^4 r_{\\rm in}$ , with the polar angle spanning the range $[0,\\pi ]$ .", "The grid is discretized logarithmically in radius, using 128 cells per decade, and a polar grid equispaced in $\\cos \\theta $ using 112 cells.", "On the equatorial plane, this results in a spacing $\\Delta r /r \\simeq 1.8\\% \\simeq 1^\\circ \\simeq \\Delta \\theta $ .", "This resolution is double that of the models in [31], equivalent to that of the high-resolution models of [27] and [29], and the same as in [26] and the hydrodynamic models of [32].", "The boundary conditions are set to outflow in radius and reflecting in $\\theta $ ." ], [ "Nuclear Reaction Network Post-Processing", "Passive tracer particles are initially placed in the disk following the density distribution.", "For each hydrodynamic simulation we employ $10^4$ particles, each representing an equal amount of mass.", "Particles are advected with the flow and record various kinematic and thermodynamic quantities as a function of time.", "Particles that are ejected with positive Bernoulli parameter beyond a radius of $10^9$  cm by the end of the simulation are considered to be part of the disk outflow.", "Outflow trajectories are post-processed with the nuclear reaction network SkyNet , using the same settings as in .", "The network employs 7843 nuclides and more than $1.4\\times 10^5$ reactions, including strong forward reaction rates from the REACLIB database [20] with inverse rates computed from detailed balance; spontaneous and neutron-induced fission rates from [41], , , and ; weak rates from [44], , , and the REACLIB database; and nuclear masses from the REACLIB database, which includes experimental values were available, or otherwise theoretical masses from the finite-range droplet macroscopic model (FRDM) of .", "The rates of electron neutrino/antineutrino absorption/emission recorded by the trajectory are included in the evolution of the proton and neutron fraction.", "Likewise, the temperature and entropy are evolved self-consistently by accounting for nuclear heating from the network, as well as viscous heating and neutrino heating/cooling in the hydrodynamic simulation as recorded by the trajectory.", "Processing begins when the trajectory reaches 10 GK for the last time, or when the temperature is maximal if lower than 10 GK at all times.", "For the portion of the evolution in which the temperature is higher than 7 GK, abundances are evolved in NSE, subject to neutrino interactions, while full network integration is carried out at lower temperatures.", "Trajectories are extended beyond the end of the simulation ($12-25$  s) by assuming that the density decays with time as $t^{-3}$ , to allow $r$ -process nuclei with long half-lives to decay.", "Since the $r$ -process is complete by the time this transition is made, most of the nuclear heating has already been deposited and the exact time dependence of the density decay is not important.", "While trajectories are evolved until 30 yr, information is extracted at $t=1$  day and $t=1$ week to estimate the properties of the kilonova at peak." ], [ "Models Evolved", "Table REF shows all of the hydrodynamic models we evolve.", "As a baseline set, we take disks with initial conditions as described in §REF : black hole and disk masses as in Figure REF , initial entropy $8k_{\\rm B}$ per baryon, and initial electron fraction $Y_{\\rm e, ini}= 0.2$ .", "Model names follow the convention bXXdYY, where XX and YY refer to the BH mass and disk mass, respectively.", "Tori are constructed as an equilibrium solution to the momentum equation with constant Bernoulli parameter, constant angular momentum, and the pseudo-Newtonian potential of the BH [27].", "The torus shape is controlled by a dimensionless distortion parameter $d$ which is solved for by fixing the entropy, $Y_e$ , and torus mass.", "The distortion parameter is related to the torus initial Bernoulli parameter $b_{\\rm ini}$ , black hole mass $M_{\\rm bh}\\equiv M_{\\rm bh(out)}$ , and radius of initial density peak $R_{\\rm d}$ by $b_{\\rm ini} = -\\frac{1}{2d}\\frac{GM_{\\rm bh}}{R_d}.$ The baseline set is evolved with a viscosity parameter $\\alpha =0.03$ To assess the effects of initial composition, we evolve a few models with lower initial electron fraction than the baseline set $Y_{\\rm e, ini}= \\lbrace 0.10,0.15\\rbrace $ .", "Likewise, we evolve two models with higher viscosity parameter, $\\alpha =0.1$ .", "All hydrodynamic models are evolved for $5,000$ orbits at the initial density peak radius, which corresponds to $\\simeq 12-25$  s of physical time (Table REF ).", "This time is chosen such that the mass ejection from the disk is mostly complete.", "Tracer particles from each simulation are then post-processed with the nuclear reaction network as described in § REF Table: Hydrodynamic models evolved and input parameters.", "Columns from left to right show model name,black hole mass, disk mass, radius of initial disk density peak, initial electron fraction, torusdistortion parameter, viscosity parameter, and maximum evolution time." ], [ "Results", "The overall evolution of neutrino cooled accretion disks follows well-known stages (e.g., , , [22], , [17], ).", "Depending on the initial disk mass, neutrinos can be trapped or escape freely.", "In the former case, an initial optically thick phase ensues until the density has decreased sufficiently for transparency.", "Thereafter, neutrino cooling is important compared to viscous heating, the inner disk is not too thick vertically, and accretion proceeds efficiently.", "After about a viscous time $R_{\\rm d}^2/(\\alpha c_i^2/\\Omega _{\\rm K})\\sim $ few 100 ms (with $c_i$ the isothermal sound speed and $\\Omega _{\\rm K}$ the orbital frequency; ), the density becomes low enough that weak interactions freeze out, shutting down cooling (e.g., ).", "At this point, the disk is radiatively inefficient, with viscous heating and nuclear recombination of $\\alpha $ particles being unbalanced, thus an outflow is launched until no more mass is available to be ejected.", "In the absence of magnetic fields, this thermal outflow is the only relevant mass ejection channel when a BH sits at the center, as neutrino-driven winds are weak given that self-irradiation is not efficient (e.g., ).", "A comparison with long-term GRMHD simulations shows that the outflow from hydrodynamic simulations is of similar quantity and has similar velocities as the analog process occurring due to dissipation of MHD turbulence [32].", "The magnetic field provides for additional, faster components that can eject a comparable amount of mass than the thermal outflow.", "In the following, we discuss mass ejection properties across the range of models we evolve, the composition of these ouflows, and the implications for EM counterparts of BH-NS mergers.", "Table: Summary of results.", "Columns from left to right show model name,disk compactness (eq.", "),ejected mass, fraction of initial disk mass ejected M ej /M d M_{\\rm ej}/M_{\\rm d},average ouflow velocity, average outflow electron fraction,ejecta mass with X La + Ac <10 -4 X_{\\rm La+Ac}< 10^{-4} (M blue M_{\\rm blue}),ejecta mass with X La + Ac >{10 -4 ,10 -3 ,10 -2 }X_{\\rm La+Ac}> \\lbrace 10^{-4},10^{-3},10^{-2}\\rbrace ({M -4 ,M -3 ,M -2 }\\lbrace M_{-4},M_{-3},M_{-2}\\rbrace , respectively),and radioactive heating power (in units of 10 40 10^{40} erg s -1 ^{-1}) at 1 day and 7 days, ignoringthermalization efficiency.Figure: Left: Fraction of the initial disk mass ejected with positiveBernoulli parameter (eq.", ")as a function of BH mass.", "Different symbols and colors correspond to differentdisk masses, as labeled.", "The gray number above each symbol column correspondsto the disk compactness parameter C d C_{\\rm d}(eq. ).", "Right: Fraction of the initialdisk mass ejected as a function of disk compactness parameter, using the samecolor and symbol coding as in the left panel.", "The gray numbers above eachsymbol column denote the corresponding BH mass.", "The red dotted line is a linearfit to the ejected fraction for disks with M d =0.03M ⊙ M_{\\rm d}=0.03M_\\odot .", "GRMHD effectscan enhance the ejected fraction and average velocity by up toa factor of ∼2\\sim 2 relative to hydrodynamic models ." ], [ "Mass ejection", "The disk mass ejection rate in all directions $\\dot{M}_{\\rm out}$ is measured at a radius $r_{\\rm out}=10^9$  cm, far enough outside the disk that mostly complete ejection is achieved before the disk viscously spreads to that radius.", "Material is considered to be unbound when its Bernoulli parameter $b = \\frac{1}{2}\\left[v_r^2 + v_\\theta ^2 + \\left(\\frac{j}{r\\sin \\theta }\\right)^2 \\right] + e_{\\rm int} + \\frac{p}{\\rho } + \\Phi $ is positive.", "In equation (REF ), $v_r$ and $v_\\theta $ are the radial and meridional velocities, $e_{\\rm int}$ is the specific internal energy, $p$ is the pressure, $\\rho $ is the density, $j$ is the specific angular momentum, and $\\Phi $ is the gravitational potential.", "Table REF shows the ejected outflow mass $M_{\\rm ej}$ – the time integral of $\\dot{M}_{\\rm out}$ over the simulation – for all models.", "The fraction of the initial disk mass ejected is shown in Figure REF for the baseline model sequence.", "This fraction ranges from $4\\%$ for the heaviest disk with $M_{\\rm bh}=8M_\\odot $ to $21\\%$ for the lightest disk around the $M_{\\rm bh}=3M_\\odot $ BH.", "The most important trend in mass ejection apparent from Table REF is a monotonically decreasing ejected fraction with strength of gravity at the disk, $\\propto M_{\\rm bh}/R_{\\rm d}$ .", "For convenience, we define a disk compactness parameter as $C_{\\rm d} = \\left(\\frac{M_{\\rm bh}}{5M_\\odot }\\right)\\left(\\frac{50\\,\\textrm {km}}{R_{\\rm d}}\\right).$ Figure REF also shows that the dependence of the ejected fraction with compactness is approximately linear, although at low BH masses the disk mass becomes an additional factor.", "This dependence on the strength of gravity has previously been documented in [27], and is also apparent from the results of , [31], and [43], although this is the first time that it is sampled over an extended region of parameter space.", "The second mass ejection trend in all models of the baseline sequence is a monotonic decrease in the ejected fraction with increasing disk mass at constant compactness.", "Figure REF shows that the strength of this dependence on disk mass is itself a function of disk compactness, with low-compactness disks being the most sensitive to the initial disk mass, while in high compactness systems this property has a smaller impact on the ejected fraction.", "The physical origin of these trends in mass ejection can be traced back to the nature of the ejection mechanism.", "Most of the outflow is launched once weak interactions freeze out in the disk, removing the source of cooling.", "The fraction of the disk mass available to be ejected depends on how much has already been lost to accretion onto the BH by the time freezout occurs.", "This interplay is illustrated in Figure REF , which shows the evolution of the mass accretion rate ($\\dot{M}_{\\rm isco}$ ) and cumulative mass accreted at the ISCO ($M_{\\rm acc}$ ), mass ejection rate at large radii ($\\dot{M}_{\\rm out}$ ), and the electron neutrino luminosity.", "In the model with the highest compactness (b08d03), accretion starts much earlier when measured in orbital times than in the lower-compactness models.", "By the time weak interactions freeze out (steep plummet in neutrino luminosity at about 100 orbits) a significant fraction of the disk ($85\\%$ ) has already been accreted to the BH.", "This earlier onset of accretion, despite having the same viscosity parameter, is due to the disk being closer to the ISCO.", "In terms of dimensionless numbers: $r_{\\rm isco}/R_{\\rm d}=\\lbrace 0.26,0.57\\rbrace $ for $C_{\\rm d}=\\lbrace 0.60,1.33\\rbrace $ , respectively.", "Figure: Top: Mass accretion rate at the ISCO (M ˙ isco \\dot{M}_{\\rm isco}, dotted lines), cumulative accretedmass at the ISCO (M acc M_{\\rm acc}, dashes lines), and mass outflow rate in unbound material at r=10 9 r=10^9 cm(M ˙ out \\dot{M}_{\\rm out}, solid lines) as a function of time for models b08d03 (high-compactness),b03d01 (low compactness, low disk mass), and b03d30 (low compactness, high disk mass), as labeled.To facilitate comparison, the data from models b08d03 and b03d30 has been normalized to a diskmass of 0.01M ⊙ 0.01M_\\odot (as in model b03d01).", "Bottom: Electron neutrino luminosityfor models b08d03, b03d01, and b03d30, as labeled.", "The fraction of the disk ejected isrelated to the fraction of the disk accreted at the time when weak interactionsfreeze out.For disks of the same compactness, the evolution of the accretion rate is very similar.", "Figure REF shows that mass ejection begins later in the more massive disk, which also takes longer time to reach freezout of weak interactions.", "More massive disks are more optically thick to neutrinos owing to their higher initial density, and for the same strength of viscosity, it takes more orbits for the density to decrease to a level where neutrino processes are no longer effective in cooling the disk.", "At the time when the neutrino luminosity reaches $10^{50}$  erg s$^{-1}$ , models b03d01 and b03d30 have accreted $60\\%$ and $78\\%$ of their initial disk masses, respectively.", "Figure: Distribution of net specific heat gained by tracer particles due to source terms(eq. )", "in the hydrodynamic evolution of models b08d03, b03d01, and b03d30, as labeled.The heat gain is normalized to the initial Bernoulli parameter in the disk(eq.", ").Table: Average time-integrated heat gain or loss per unit mass (eqns.", "-),and change in Y e Y_e (eqns.", "-), due to various processes actingon tracer particles during the hydrodynamic evolution of selected models.", "The specific heat gain due toprocess ii is normalized as Δ ¯q i,19 =Δ ¯q i /(10 19 erg g -1 )\\bar{\\Delta } q_{i,19} = \\bar{\\Delta } q_i/(10^{19}\\,\\textrm {erg\\,g}^{-1}),and Δ ¯\\bar{\\Delta } denotes average over all outflow particles.", "Each quantity is separatelyrounded for clarity, net sums match when including all significant digits.The density dependence of the neutrino optical depth is not the only factor influencing the freezout time.", "Table REF and equation (REF ) show that to keep the entropy constant, initial equilibrium disks with higher masses also have a higher internal energy content and therefore higher temperatures.", "This effect is stronger for lower compactness configurations.", "While disk material is more weakly bound, it can also radiate neutrinos at relevant levels for a longer time, and so this acts in the direction of delaying the onset of mass ejection.", "The post-merger entropy of the disk is thus an important parameter to keep track of in dynamical merger simulations given its effect on mass ejection efficiency.", "Using the outflow trajectories from hydrodynamic simulations we can further analyze the energetics of mass ejection.", "In particular, we can quantify the strength and importance of different processes that change the heat content of the fluid: viscous heating, neutrino heating/cooling, and nuclear recombination of alpha particles.", "For each outflow trajectory, we compute the time integral of the local energy source term, yielding the heat gained or lost by the fluid element per unit mass: $\\Delta q_{\\rm visc} & = & \\int _0^{t_{\\rm max,p}} \\dot{q}_{\\rm visc} dt\\\\\\Delta q_\\nu & = & \\int _0^{t_{\\rm max,p}} \\dot{q}_{\\rm \\nu } dt\\\\\\Delta q_\\alpha & = & \\frac{B_\\alpha }{m_\\alpha }\\left[X_{\\rm alpha}(t_{\\rm max,p}) - X_\\alpha (0) \\right]\\\\\\Delta q_{\\rm net} & = & \\Delta q_{\\rm visc} + \\Delta q_\\nu + \\Delta q_\\alpha ,$ where $\\lbrace \\dot{q}_{\\rm visc},\\dot{q}_{\\rm \\nu }\\rbrace $ stand for viscous heating and neutrino heating/cooling, respectively, ${B_\\alpha }/{m_\\alpha }\\simeq 6.8\\times 10^{18}$  erg g$^{-1}$ is the specific nuclear binding energy of alpha particles, $X_\\alpha $ is the mass fraction of alpha particles, $t_{\\rm max,p}$ is the maximum time for the particle evolution, either when it leaves the outer boundary of the computational domain or when the simulation ends, whichever is shorter, and $\\Delta q_i$ is the heat gained or lost from process $i$ .", "The resulting quantities are shown in Table REF for selected models.", "Figure: Distribution of specific heat gained or lost by tracer particles due to source terms(eqns.", "-), as labeled, in the hydrodynamic evolutionof model b08d03.", "The inset shows a snapshot of theinitial positions of outflow particles in the disk, with gray/black particles correspondingto the shaded/dotted subsets of the neutrino and viscous heat gain histograms,respectively (see also fora larger plot of the initial particle distribution).Figure REF shows the distribution of net heat gained $\\Delta q_{\\rm net}$ (eq. )", "by outflow particles for the same set of simulations shown in Figure REF .", "Specific energies are shown in units of the initial Bernoulli parameter of the disk (equation REF ).", "In all three models, the distribution shows a peak at around $60\\%$ of $|b_{\\rm ini}|$ , with a tail toward high gain that is more extended for less compact models and higher disk masses.", "For a large fraction of the outflow, unbinding is not completely achieved by absorption of heat alone, which means that a significant part of the energy gain comes from adiabatic work.", "Disks with higher compactness are also less effective at absorbing heat, despite the fact that the absolute value of the net heat gain is higher (Table REF ).", "It is worth emphasizing that while disks with a higher compactness are more gravitationally bound in an absolute sense ($b_{\\rm ini} \\propto M_{\\rm bh}/R_{\\rm d}$ in eq.", "REF ), they also undergo more net heating than lower compactness models (Table REF ).", "Thus the absolute value of the gravitational potential (in a Newtonian sense), while correlating negatively with the fractional amount of heat absorbed, does not by itself explain the efficiency of mass ejection without accounting for how close the disk is to the ISCO radius.", "For individual models, viscous heating and nuclear recombination contribute with net heating, while neutrinos primarily cool the disk (Table REF ).", "This is illustrated in Figure REF , which shows the distribution of the individual heating/cooling terms for the outflow from model b08d03.", "While there is a non-negligible fraction of particles for which neutrinos provide net heating, the magnitude of this heating ($\\sim 10^{17}$  erg g$^{-1}$ ) is dynamically negligible when compared to the dominant source terms ($\\sim 10^{19}$  erg g$^{-1}$ ).", "The integrated heat gain due to $\\alpha $ particle recombination in the hydrodynamic simulation is sub-dominant compared to that from viscous heating.", "Despite its low global value, however, the heating due to $\\alpha $ recombination is deposited over a short amount of time as the outflow is launched, and it can become comparable or even exceed the rate of viscous heating in this phase.", "The bulk of neutrino heating and cooling takes place before weak freezout.", "Figure REF shows that the nuclear recombination gain is narrowly distributed around the mean value, $3-4\\times 10^{18}$  erg g$^{-1}$ (Table REF ).", "This value can be understood from the fact that upon expansion and cooling, all fluid elements achieve the maximal alpha particle mass fraction set by charge conservation, $X_{\\alpha ,\\rm max} = 2Y_e$ .", "The average electron fraction of the outflow $\\langle Y_e\\rangle \\simeq 0.3$ (Table REF ), then sets the average amount of energy gained.", "Figure REF also shows that the tracer particles in model b08d03 follow bimodal distributions of viscous heating and neutrino cooling.", "This bimodality can be traced back to the initial positions of the particles in the disk.", "These particles originate from two regions: (1) the equatorial plane of the disk, and (2) regions above the equatorial plane around the initial density peak (see also ).", "The first group of particles experiences stronger viscous heating and neutrino cooling, while the second group experiences heating or cooling with less intensity.", "The latter group includes all the particles that experience net neutrino heating.", "The initial position of the particles is related to the way the disk overturns in the poloidal direction due to viscous heating, and may differ from that obtained when MHD turbulence transports angular momentum.", "The fraction of the disk mass ejected and average outflow velocity are nearly insensitive to the initial $Y_e$ of the disk except for very massive disks in low-compactness systems, where differences of a few percent of the disk mass can arise (Table REF ).", "Changes in the viscosity parameter, on the other hand, result in important changes to both ejected fraction and outflow velocity.", "The average outflow radial velocity is in the range $0.03-0.04$  c for all models that use $\\alpha =0.03$ , while this average increases to $0.05-0.06$  c for the models with $\\alpha =0.1$ .", "The fraction of the disk mass ejected increases by $\\sim 30\\%$ for the model with low compactness (b03d01-v10) and by a factor of 2 for the high compactness model (b08d03-v10).", "Table REF shows that the increase in the viscosity parameter results in more viscous heating in the high-compactness model b08d03-v10 and less neutrino cooling in the least compact model b03d01-v10, in both cases increasing the net heat gain of the outflow.", "Figure: Top: Distribution of lanthanide and actinide mass fractions at 1 day, fornuclear-network-processed particles from models b08d01-b08d20, as labeled.Since each particle represents an equal mass element in the disk, a sum over the bins yields thefraction of the mass with a given Lanthanide and Actinide fraction (M blue M_{\\rm blue} and{M -4 ,M 3 ,M -2 }\\lbrace M_{-4},M_{\\-3},M_{-2}\\rbrace , c.f.", "eqns.", "()-() andTable ).", "The histograms continue to mass fractionslower than 10 -12 10^{-12} with similar slope,and were truncated for clarity.", "Bottom: Isotopic abundances at 1 day for nuclear-network-processedparticles from models b08d01-b08d20, as labeled.", "Abundances are normalized such that their massfractions add up to unity.", "The solar system rr-process abundances from are normalizedto model b08d03 at A=130A=130.", "Note that the dynamical ejecta is rich in elements with A>130A>130, andin combination with the disk outflow can supply the entire range of rr-process elements (e.g., )." ], [ "Outflow Composition", "Table REF shows that the average electron fraction of the disk outflow is in the range $0.25-0.35$ for all simulated models.", "These values are just above the critical transition at which the nucleosynthesis changes from rich to poor in elements with mass numbers $A>130$ (e.g., , ).", "For the purposes of predicting kilonova properties, the mass fraction in lanthanides ($57\\le Z \\le 72$ , with $Z$ the atomic number) and actinides ($89\\le Z\\le 104$ ) is the most important, since these species have an outsize influence on the ejecta opacity – and therefore on the kilonova color, luminosity, and duration – given their atomic complexity , , [9], [34], .", "We therefore refine our diagnostic of the outflow composition by analyzing the output of post-processed tracer particles with SkyNet (§REF ).", "For each simulation, we report in Table REF the fraction of the outflow particles with lanthanide and actinide mass fraction $X_{\\rm La}+X_{\\rm Ac}< 10^{-4}$ and define it as the `blue mass' $M_{\\rm blue} = \\int _{0}^{10^{-4}}\\frac{dM_{\\rm ej}}{d(X_{\\rm La}+X_{\\rm Ac})}\\,d(X_{\\rm La}+X_{\\rm Ac}).$ The value of $10^{-4}$ is small enough that the outflow opacity is indistinguishable from that dominated by iron-group elements .", "Likewise, we define three `red' masses $M_{-k} = \\int _{10^{-k}}^1 \\frac{dM_{\\rm ej}}{d(X_{\\rm La}+X_{\\rm Ac})}\\,d(X_{\\rm La}+X_{\\rm Ac}).$ using $-k=\\lbrace -4,-3,-2\\rbrace $ .", "These four numbers provide a description of the incidence of heavy $r$ -process elements in the ejecta that, while coarser than detailed abundances, is more informative than just the electron fraction.", "Figure REF shows the distribution of mass fractions at 1 day in the outflow particles from models b08d01-b08d20 (highest compactness), along with the regions encompassed by $M_{\\rm blue}$ and $M_{-k}$ , and the isotopic abundances for comparison.", "At 1 day, the mass fractions of actinides are in general 10 times smaller than that of lanthanides in our models, and thus we lump them together when discussing composition effects (for more detailed studies on actinide production see [24] or ).", "The most robust composition trends from Table REF are that (1) the majority of the disk outflow mass is lanthanide poor ($M_{\\rm blue}$ ), and that (2) the fraction of lanthanide poor material is a monotonically increasing function of the disk mass, for constant BH mass.", "The dependence on disk mass is clearly illustrated in Figure REF : the fraction of particles with high lanthanide abundance is a steep function of the disk mass, as is the abundance of of elements with $A > 130$ (this dependence has also been reported by and [43]).", "While there is some dependence on the compactness $C_{\\rm d}$ at constant disk mass, this dependence is not fully monotonic, and is weaker than the dependence on disk mass at fixed compactness.", "Therefore, this variation with compactness is more likely to be dependent on the details of how the disk evolution is modeled.", "The trend of more lanthanide poor ejecta with increasing disk mass is apparent from the distribution of electron fraction (Figure REF ).", "Despite having a very similar average value, the $Y_e$ distribution of model b08d20 extends to significantly higher values than in model b08d01.", "Naively, one would expect that higher electron fraction is associated with a higher equilibrium $Y_e$ arising from a higher abundance of positrons given higher entropies (e.g., [11]).", "However, the entropy distributions of models b08d01 and b08d20 show that the average entropy decreases with higher disk masses.", "The lower entropy can be understood from the fact that the temperature is primarily set by the strength of the gravitational potential once accretion is established, and is similar in both models.", "On the other hand, the densities are higher at any given time for a higher disk mass, with a correspondingly lower entropy than at lower disk masses.", "Figure: Mass histograms of electron fraction (left) and entropy (right) for modelsb08d01 (low disk mass), b08d20 (high disk mass), and b08d03-v10 (high viscosity), obtainedby the end of the hydrodynamic simulation at r=10 9 r=10^{9} cm.", "To facilitate comparison,histogram masses have been normalized to that of model b08d01.", "The hatched region is thesubset of the histogram of model b08d03-v10 for times t<1t<1 s .Figure: Average electron fraction change in particles due to neutrino emission or absorption(eqns.", "- and , Table ),during the hydrodynamic evolution of models b08d01-b08d20, as labeled.", "The blackhorizontal lines denote the net change in Y e Y_e when including all processes.", "The relative importanceof neutrino absorption increases as the disk mass increases, all else being equal, accountingfor the higher Y e Y_e of the outflow with lower overall entropies (Figure ).We can further analyze the evolution of the electron fraction distribution of the outflow by computing the time-integrated contribution of the different neutrino processes that change $Y_e$ .", "The net rate of change of $Y_e$ due to neutrino emission and absorption arises from the following reactions: $\\Gamma ^{\\rm em, \\nu _e}: && e^- + p \\rightarrow n + \\nu _e\\\\\\Gamma ^{\\rm em, \\bar{\\nu }_e}: && e^+ + n \\rightarrow p + \\bar{\\nu }_e\\\\\\Gamma ^{\\rm abs, \\nu _e}: && \\nu _e + n \\rightarrow e^- + p\\\\\\Gamma ^{\\rm abs, \\bar{\\nu }_e}: && \\bar{\\nu }_e + p \\rightarrow e^+ + n\\\\{\\smallskip }\\Gamma ^{\\rm em} & = & \\Gamma ^{\\rm em, \\bar{\\nu }_e} - \\Gamma ^{\\rm em, \\nu _e}\\\\\\Gamma ^{\\rm abs} & = & \\Gamma ^{\\rm abs, \\nu _e} - \\Gamma ^{\\rm abs, \\bar{\\nu }_e}\\\\\\Gamma ^{\\rm net} & = & \\Gamma ^{\\rm em} + \\Gamma ^{\\rm abs}$ with the net rate setting the overall evolution of the electron fraction in the hydrodynamic simulation: $\\partial Y_e/\\partial t + \\mathbf {v}\\cdot \\nabla Y_e = \\Gamma ^{\\rm net}$ .", "For each tracer particle, we compute a separate time integral for each of the rates above, obtaining a contribution to the change in electron fraction: $\\Delta Y_e^i = \\int _0^{t_{\\rm max,p}} \\Gamma ^i dt,$ where again $t_{\\rm max,p}$ is the maximum time of the particle in the simulation.", "Table REF shows that the rates of neutrino emission dominate over neutrino absorption for all models, in line with the overall dominance of neutrino cooling over neutrino heating (c.f.", "Figure REF ).", "However, the change in $Y_e$ is set by differences in the rates of neutrino/antineutrino emission and absorption.", "If the two emission rates are closer in magnitude than the two absorption rates, the latter can dominate the change in $Y_e$ despite being smaller in magnitude than the former.", "Figure REF shows the average change in $Y_e$ for tracer particles as a function of disk mass in the baseline sequence with $M_{\\rm bh}=8M_\\odot $ , along with the breakdown of this change between emission and absorption of electron neutrinos/antineutrinos.", "At low disk masses, emission processes dominate the change in $Y_e$ , with decreasing relative importance with increasing disk mass.", "Emission processes act toward driving $Y_e$ to its local equilibrium value set by the entropy, and this equilibrium is lower at higher disk masses given the lower entropy (Figure REF ).", "At the highest disk mass (model b08d20), this change in $Y_e$ due to emission is even negative.", "Absorption, on the other hand, is set by the ambient flux of incident neutrinos and the mass fractions of neutrons and protons.", "At higher disk masses, neutrino/antineutrino luminosities are higher and stay high for a longer time (Figure REF ) thus increasing the ambient neutrino flux and the thus the magnitude of absorption terms.", "The asymmetry in the neutron-proton mass fraction then results in different absorption efficiencies and a net change in $Y_e$ which always acts in the direction of increasing it (because the reaction in eqn.", "occurs more frequently).", "Figure REF shows that at high disk masses, absorption dominates the evolution of $Y_e$ in the high-compactness models shown.", "Table REF shows that this trend is also present in models with the lowest compactness (b03d01-b03d30), with an even stronger effect of absorption on $Y_e$ than in the high compactness sequence.", "Models with lower initial $Y_e = \\lbrace 0.10,015\\rbrace $ eject a higher proportion of lanthanides and actinides, as expected, although the change is at a $\\lesssim 10\\%$ level by mass relative to our baseline parameters (Figure REF ).", "While the $Y_e$ distributions extend to slightly lower electron fractions than the default models, the bulk of the outflow has $Y_e > 0.2$ in all cases.", "Table REF shows that these changes are driven primarily by neutrino emission processes, which adjust to provide the required change in $Y_e$ toward equilibrium.", "Figure: Distribution of lanthanide and actinide mass fractions at 1 day, fornuclear-network-processed particles from models b08d03 (baseline), b08d03-y10 and b08d03-y15(varying initial Y e Y_e), and b08d03-v10 (high viscosity), as labeled.", "The bin size is the same isin the top panel of Figure .", "The lowest bin contains all the particleswith X La +X ac <10 -14.5 X_{\\rm La}+X_{\\rm ac} < 10^{-14.5}.Models with higher viscosity have a similar average $Y_e$ than their low-viscosity counterparts, but a significantly higher fraction of material rich in lanthanide and actinides.", "The electron fraction distribution of model b08d03-v10 has a tail to low $Y_e$ that extends well below the lower limit of the distribution of model b08d01.", "Figure REF shows that the material with the lowest $Y_e$ is ejected at the earliest times in the high-viscosity model, in line with the general expectation that the faster the evolution of a disk, the stronger the sensitivity of the outflow composition to initial conditions.", "This is consistent with the results of GRMHD simulations, which show even stronger memory of initial conditions given their faster evolution; [32].", "Figure: Total radioactive heating luminosity at 1 day as a function of disk compactness(eq.", "), for various disk masses, as labeled.The BH mass is shown in gray under each symbol column.", "The crosses denote the heatingrates interpolated to the median disk masses from Figure .The total radioactive heating rate is an upper limit to the bolometric luminosity of thekilonova, being subject to thermalization efficiency and radiative transfer effects." ], [ "Implications for EM counterpart searches", "The key question we are interested in is how does the disk outflow contribute to a kilonova transient.", "The answer depends on the amount of mass ejected and its velocity, its composition, as well as how efficiently does the radioactive heating thermalize (e.g., ).", "The mass ejected, and the composition to a lesser extent, determines how much radioactive power is available for a kilonova.", "Figure REF shows the total radioactive heating luminosity at 1 day as a function of compactness parameter $C_{\\rm d}$ and initial disk mass.", "For fixed compactness, the total radioactive heating is proportional to the ejected mass, since the average radioactive heating per unit mass is close to $2\\times 10^{10}$  erg g$^{-1}$  s$^{-1}$ for most models, given their similar composition.", "The dependence of ejected fraction on compactness results in an additional variation of a factor $\\sim 5$ between the least and most compact models, for fixed disk mass.", "The raw radioactive heating at 1 day ranges from $2\\times 10^{40}$  erg s$^{-1}$ for the lightest and most compact disk (b08d01) to $1.5\\times 10^{42}$  erg s$^{-1}$ for the heaviest disk in the least compact configuration (b03d30).", "Thermalization efficiency can result in a decrease in these values by a factor $\\sim 2$ [10], , , , while radiative transfer effects (dependent on the opacity and thus on composition) set whether this power can readily escape the ejecta or is trapped until later times.", "Table REF shows that the radioactive power at 1 week is about 10 times smaller than that at 1 day for most models.", "If we take the median disk masses from Figure REF as representative values for each BH mass and interpolate the ejected masses from Table REF , we obtain disk outflow masses $\\lbrace 2.00, 0.70, 0.22, 0.25, 0.14\\rbrace \\times 10^{-2}M_\\odot $ for BH masses $\\lbrace 3, 5, 8, 10, 15\\rbrace M_\\odot $ , respectively.", "These values are subject to enhancement by a factor $\\lesssim 2$ if GRMHD effects were to be included [32], [18].", "In the case of GW190425, for which our lowest BH mass model would be applicable, the median disk outflow mass would be similar to the total ejecta from GW170817 within a factor of two, and hence it would have been detected with good sky coverage .", "A BH-NS merger with a low-mass BH and high-mass NS is most efficient at tidally disrupting the NS and most inefficient at producing dynamical ejecta, with most mass ejection coming from the disk [40].", "In contrast, two massive NS that collapse promptly to a BH are the least efficient configuration for ejecting mass and forming a disk (e.g., , but see , [12] for the case of an asymmetric mass ratio NS-NS merger generating a more massive BH accretion disk than a symmetric binary).", "Regarding the BH-NS merger candidate GW190814, which had a much smaller localization area and deeper EM coverage relative to other events, constraints on the total mass ejection are in the range $0.02-0.1M_\\odot $ depending on viewing angle, composition, and distance.", "(e.g., [6], , , ).", "Our results indicate that, with the exception of a very low-mass BH and/or very high disk masses, most BH-NS merger systems would not have generated sufficient disk outflow for a detectable kilonova.", "An additional factor influencing the kilonova appearance is the relative masses of the disk and the dynamical ejecta.", "In a BH-NS merger, the dynamical ejecta is produced along the equatorial plane, and it blocks only part of the viewing angles.", "It is expected to be very rich in lanthanides and thus have a high opacity that blocks the light from the disk toward most equatorial directions.", "The results of [30] and [31] – who studied the combined long-term evolution of these two components mapped from dynamical merger simulations – show that the bulk of fallback accretion mixes in with the disk before weak freezout occurs and the disk outflow is launched.", "The net outflow has a very similar composition as the dynamical ejecta and disk when evolved separately, but with an added component that has intermediate electron fraction values.", "The expected kilonova color has an important dependence on viewing angle, as the bluer disk emission would only be detectable from directions not obscured by the dynamical ejecta (see, e.g., [16], [21], , for more recent work on viewing angle dependencies of different ejecta configurations).", "In hydrodynamic simulations of BH accretion disks, the spatial dependence of the composition is quite generic, with the highest $Y_e$ material being ejected first along intermediate latitudes, and then wrapping around the outermost edge of the disk outflow [29].", "This configuration would guarantee the existence of a blue spike at early times (albeit a faint one) if the merger remnant is viewable from polar latitudes.", "All of our models are such that at least $50\\%$ of the disk outflow is lanthanide-poor, and in some cases this fraction can reach even $100\\%$ (Table REF ).", "GRMHD models show, however, that magnetic fields begin to eject matter much earlier than weak freezout, thus adding material that has not been sufficiently processed by weak interactions from the initial post-merger composition (e.g., , [32], , [18]).", "This will likely increase the lanthanide-rich fraction at the leading edge of the disk outflow and thus modify the color of the disk-contributed kilonova." ], [ "Summary and Discussion", "We have performed axisymmetric hydrodynamic simulations of the viscous evolution of accretion disks formed in BH-NS mergers.", "Our models include the effects of neutrino emission and absorption on the outflow composition, and the results are post-processed with a nuclear reaction network for a more precise diagnostic of the nucleosynthesis yield.", "These hydrodynamic models provide a lower-limit to the amount of mass in the disk outflow relative to three-dimensional GRMHD models, and hence provide a lower-limit to the contribution of disk outflows to the kilonova transient.", "Our simulations cover, for the first time, a large fraction of the plausible parameter space of BH and disk masses expected from these mergers (Figure REF ).", "Our main results are the following: 1.", "– The fraction of the initial disk mass ejected as an unbound outflow has an approximately inverse linear scaling with the initial compactness of the disk, and can vary by a factor of $\\sim 4$ (Figure REF and Table REF ).", "While this dependence on compactness was implicit in previous work, this is the first time that it is systematically probed over a wide parameter space.", "The origin of this dependence can be traced back to the earlier onset of accretion in more compact disks, as they are located closer to the ISCO.", "Compared to lower compactness disks, a higher fraction of the initial mass is accreted by the time weak interactions freeze out and the outflow is launched (Figure REF ).", "2.", "– At constant compactness, the fraction of the disk mass ejected decreases for higher disk masses.", "This effect is related to the longer time to reach weak freezout in more massive disks, which delays the onset of mass ejection to a time when more mass has been accreted to the BH (Figure REF ).", "The dependence on initial disk mass is weaker for higher compactness systems (Figure REF and Table REF ).", "The initial density and entropy of the disk are the key variables regulating this mass dependency of the ejection efficiency (§REF ).", "3.", "– The disk outflow is more lanthanide/actinide-poor for higher disk masses (Figure REF ), at constant compactness (this trend has also been found in previous studies).", "This effect can be traced back to neutrino absorption becoming more important relative to emission in increasing $Y_e$ for more massive disks (Figure REF and Table REF ).", "Stronger absorption counteracts the action of neutrino emission in lowering $Y_e$ given the lower entropy of the outflow from more massive disks (Figure REF ).", "While our disk outflows are $50-100\\%$ lanthanide-poor by mass, magnetically driven contributions – not included here – can add a significant amount of lanthanide-rich matter, hence the net composition of disk outflows requires simulations with more complete physics for reliable predictions.", "4.", "– The ejected fraction and final composition are sensitive to the viscosity parameter of the disk, as known from previous work, with more mass ejected, at higher velocities, and with a higher lanthanide-rich fraction for higher viscosity parameter (Table REF ).", "The most neutron-rich material is produced at the earliest times in the simulation (Figure REF ) and is thus related to the shorter evolution time of these disks.", "5.", "– In most cases, the initial $Y_e$ of the disk has a negligible effect on the amount of mass ejected, with the exception of massive disks in low-compactness systems, where the effect modifies the ejection efficiency by a few percent of the disk mass (similar to numerical resolution).", "Hence, the ejected fraction is mostly robust to variations in the initial composition (Table REF ).", "The final composition does depend on the initial conditions, with corrections of the order of $10\\%$ to the fraction of the outflow mass that is lanthanide-rich (Figure REF ) 6.", "– The range of ejecta masses from the disk outflow can result in a range of two orders of magnitude in raw radioactive luminosity over the BH-NS parameter space probed (Figure REF ).", "Except for very low-mass BHs and/or very high disk masses, most BH-NS mergers should generate disk outflows that are below constraints for the total ejecta mass from the BH-NS merger candidate GW190814 (§REF ).", "Our results are consistent with previous hydrodynamic models of BH accretion disks.", "Model t-a80-hr from [29] is equivalent to our model b03d01 but evolved until $3,000$ orbits.", "By that time, the total and unbound (positive energy) mass ejected in their model are $19\\%$ and $12\\%$ , respectively, while here we obtain $21\\%$ and $14\\%$ , respectively.", "The $\\sim 10\\%$ difference can be attributed to the lower ambient and floor of density used here, and on improvements in the neutrino implementation as reported in .", "Similarly, model Fdisk of [31] falls in between models b08d03 and b08d10 in terms of compactness, but has a slightly higher BH spin (0.86).", "The higher ejected fraction in their model ($8\\%$ ) can be partially accounted for by the higher value of the BH spin, and also by a more extended initial density distribution with radius in the torus mapped from the merger simulation (covering a wider range in compactness) We have made specific choices for the disk entropy (§REF ), which can have implications for the sensitivity of the ejected fraction to initial disk mass.", "Similarly, the choice of viscosity parameter is on the low end of values that bracket the amount of angular momentum transport seen in GRMHD simulations of equivalent accretion disks [32].", "More realistic values for these two parameters must come from direct mapping of the outcome of GRMHD simulations of BH-NS mergers that include neutrino transport, where the magnetic field geometry and strength replaces the viscosity parameter.", "A direct mapping would also avoid the need to make well-motivated but ultimately arbitrary choices for an initial torus radius, which is required by an equilibrium initial condition and directly enters the compactness parameter (equation REF ).", "Mapping from merger simulations would also inform the initial $Y_e$ distribution which, while not crucial for determining the amount of mass ejected, has implications for the outflow composition at the level of detail needed for accurate predictions of kilonovae and nucleosynthesis yields.", "Compact object merger simulations that account for both MHD and neutrino effects are few and only implement the latter via leakage schemes (e.g., , ).", "Simulations with more advanced (e.g., M1) neutrino transport do not yet include MHD effects.", "Our results suggest that GRMHD simulations carried out with no neutrino absorption (e.g., , [32], [18]) can give reasonable $Y_e$ distributions only for very low mass disks $< 0.01M_\\odot $ , for which the absorption contribution to $Y_e$ is subdominant.", "For the configuration studied by , [32], and [18] ($M_{\\rm bh}=3M_\\odot $ and $M_{\\rm d}=0.03M_\\odot $ ), we find that absorption is already more important than emission in setting the $Y_e$ distribution (Table REF ), in line with the results of who showed a significant increase in the $Y_e$ of the outflow when including neutrino absorption.", "Further GRMHD studies of BH accretion disks over a wide region of parameter space are needed to determine whether the ejected fraction has the same dependence on compactness as in pure hydrodynamic models, and whether the fraction of the outflow that is lanthanide-rich is significantly larger than what we find here.", "Both of these questions are crucial to improve predictions of EM counterparts to BH-NS and NS-NS sources, and require (1) models evolved for long timescales, (2) realistic initial field strengths, geometries, entropies, and electron fractions, and (3) inclusion of neutrino absorption.", "Such simulations remain challenging given current algorithms and computational resources." ], [ "Acknowledgements", "We thank the anonymous referee for helpful comments that improved the presentation of the paper.", "RF acknowledges support from the National Sciences and Engineering Research Council (NSERC) of Canada through Discovery Grant RGPIN-2017-04286, and from the Faculty of Science at the University of Alberta.", "FF gratefully acknowledges support from the U.S. National Science Foundation (NSF) through grant PHY-1806278, from the National Aeronautics and Space Administration (NASA) through grant 80NSSC18K0565, and from the U.S. Department of Energy CAREER grant DE-SC0020435.", "This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory.", "Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No.", "89233218CNA000001).", "This work was assigned report number LA-UR-20-23877.", "The software used in this work was in part developed by the U.S. Department of Energy NNSA-ASC OASCR Flash Center at the University of Chicago.", "This research was enabled in part by support provided by WestGrid (www.westgrid.ca), the Shared Hierarchical Academic Research Computing Network (SHARCNET, www.sharcnet.ca), Calcul Québec (calculquebec.ca), and Compute Canada (www.computecanada.ca).", "Computations were performed on Graham, Cedar, and Béluga.", "This research also used storage resources of the U.S. National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the U.S. Department of Energy under Contract No.", "DE-AC02-05CH11231 (repository m2058).", "Graphics were developed with matplotlib .", "The data underlying this article will be shared on reasonable request to the corresponding author." ] ]

[ [ "Non-linear soliton confinement in weakly coupled antiferromagnetic spin\n chains" ], [ "Abstract We analyze the low-energy dynamics of quasi one dimensional, large-$S$ quantum antiferromagnets with easy-axis anisotropy, using a semi-classical non-linear sigma model.", "The saddle point approximation leads to a sine Gordon equation which supports soliton solutions.", "These correspond to the movement of spatially extended domain walls.", "Long-range magnetic order is a consequence of a weak inter-chain coupling.", "Below the ordering temperature, the coupling to nearby chains leads to an energy cost associated with the separation of two domain walls.", "From the kink-antikink two-soliton solution, we compute the effective confinement potential.", "At distances large compared to the size of the solitons the potential is linear, as expected for point-like domain walls.", "At small distances the gradual annihilation of the solitons weakens the effective attraction and renders the potential quadratic.", "From numerically solving the effective one dimensional Schr\\\"oedinger equation with this non-linear confinement potential we compute the soliton bound state spectrum.", "We apply the theory to CaFe$_{2}$O$_{4}$, an anisotropic $S=5/2$ magnet based upon antiferromagnetic zig-zag chains.", "Using inelastic neutron scattering, we are able to resolve seven discrete energy levels for spectra recorded slightly below the N\\'eel temperature $T_\\textrm{N}\\approx 200$~K.", "These modes are well described by our non-linear confinement model in the regime of large spatially extended solitons." ], [ "Introduction", "Confinement and deconfinement of particles, topological defects or fractionalized excitations are recurring motifs in many areas of physics.", "A famous example is the quark-gluon plasma, which is predicted to form at extremely high temperatures.", "In this new state of matter the quarks and gluons, which under normal conditions are strongly confined in atomic nuclei, behave as asymptotically free particles[1].", "Another example of a confinement-deconfinement transition is the Berenskii-Kosterlitz-Thouless transition[2], [3] in two-dimensional XY magnets that is driven by an unbinding of thermally excited vortex-antivortex pairs.", "Spin-charge separation in one dimension[4], [5], [6] can be viewed as a fractionalization of the electrons into holons and spinons, carrying the charge and spin degrees of freedom, respectively.", "If local repulsions lead to charge localization, the insulating system is well described by the antiferromagnetic $S=1/2$ Heisenberg model.", "In the presence of Ising exchange anisotropy, spinons can be viewed as domain walls in the antiferromagnetic order and are created in pairs by a single spin flip (see Fig.", "REF (a)).", "They are therefore fractionalized excitations that carry half of the spin-1 quantum of a magnon excitation.", "[7] If spinons are free to propagate, these pairs are expected to form a triplet excitation continuum.", "Such continua are predicted theoretically,[8], [9], [10] building on the analytical Bethe Ansatz solution,[11] and observed experimentally in a number of quasi one-dimensional $S=1/2$ antiferromagnets.", "[12], [13], [14], [15] Staggered $g$ -tensors and Dzyaloshinskii-Moriya interactions can lead to an unusual field dependence, such as an induced gap,[16], [17] $\\Delta \\sim H^{2/3}$ , and field dependent soft modes at incommensurate wave vectors,[18], [19] as predicted by spinon and Bethe Ansatz descriptions.", "[20], [21], [22] Through a procedure of bosonization, the dynamics of such systems can be shown to be governed by the quantum sine-Gordon model which admits soliton and breather solutions, corresponding to propagating and oscillating domain walls, respectively.", "[17], [23] This suggests that spinons can be viewed as quantum solitons [24] and therefore exhibit chirality, which was indeed confirmed by polarized neutron scattering.", "[25] Soliton and breather modes were identified in neutron-scattering [26], [27] and electron-spin-resonance [28], [29] experiments.", "The effect of a weak interchain interaction is twofold.", "Firstly, it sets the temperature scale $T_\\textrm {N}$ at which two- or three-dimensional long-range order develops.", "Secondly, it generates an effective attraction between spinons below $T_\\textrm {N}$ since the separation of domain walls will frustrate interchain interactions with an associated energy cost that grows linearly with their distance.", "Such a linear confinement potential gives rise to spinon bound states, leading to a quantization of the excitation continuum into discrete energy levels, as observed in BaCo$_2$ V$_2$ O$_8$ ,[30] SrCo$_2$ V$_2$ O$_8$ ,[31], [13] and Yb$_2$ Pt$_2$ Pb.", "[14] These systems all consist of weakly coupled Ising-Heisenberg antiferromagnetic (XXZ) chains of $S=1/2$ moments and the measured spinon bound-state energies are almost perfectly described by the eigenvalues of a one-dimensional Schrödinger equation with an attractive linear potential.", "Linear confinement due to weak interchain coupling is not specific to spinons in $S=1/2$ quantum antiferromagnets but occurs generically for any type of kink-like domain-wall excitations.", "In CoNb$_2$ O$_6$ , a quasi one-dimensional Ising ferromagnet, the two-kink continuum breaks up into discrete bound-state excitations below the magnetic ordering temperature, with the same characteristic level spacing as in the spinon case.", "[32] In this paper we analyze the domain-wall confinement in large-$S$ spin-chain antiferromagnets with easy axis, single-ion anisotropy.", "Our work is motivated by the observation of discrete energy levels in the anisotropic antiferromagnet CaFe$_{2}$ O$_{4}$ ,[33] a spin-5/2 system consisting of weakly-coupled zig-zag chains.", "As expected for confinement due to frustrated interchain coupling, the bound states form below the the Néel temperature $T_\\textrm {N}\\approx 200$  K. However, the energy levels do not follow the negative zeroes of the Airy function, as predicted for a linear confinement potential.", "In the large-$S$ limit, the low-energy effective field theory of the quantum antiferromagnet is the non-linear $\\sigma $ model.", "Starting from this semi-classical description, Haldane demonstrated that the spin dynamics of the one-dimensional quantum antiferromagnet with easy-axis anisotropy is governed by a sine-Gordon equation which supports soliton solutions.", "[34] Hence the domain walls in the antiferromagnetic chain are chiral solitons.", "In these spin textures the staggered magnetization rotates between the two favored orientations in a clockwise or anti-clockwise direction over a typical distance $\\xi $ (see Fig.", "REF (b)).", "Since the overall chirality in the system is conserved, the domain walls are created in pairs of soliton (kink, $\\textrm {K}$ ) and anti-soliton (anti-kink, $\\overline{\\textrm {K}}$ ).", "Here we compute the confinement potential $V(y)$ from the $\\textrm {K}\\overline{\\textrm {K}}$ two-soliton solution of the sine-Gordon equation and show that the extended nature of semi-classical solitons gives rise to a crossover as a function of the domain-wall separation $|y|$ .", "At large separations, $|y|\\gg \\xi $ , the solitons can be considered as point-like objects, giving rise to a linear confinement potential, $V(y) \\sim |y|$ .", "For $|y|<\\xi $ the soliton and anti-soliton overlap, leading to a gradual annihilation of the defects and preventing the staggered magnetization between domain walls from fully rotating to the other easy direction.", "This reduces the interchain-frustration energy, corresponding to a weakening of the effective confinement potential.", "We find that at small distances, $|y|\\ll \\xi $ , the confinement potential is rendered quadratic, $V(y)\\sim y^2$ .", "The bound-state spectrum is obtained from the numerical solutions of a one-dimensional Schrödinger equation with the computed potential $V(y)$ .", "Because of the crossover in $V(y)$ , the energies of tightly-bound states are almost equidistant, as expected for a harmonic oscillator, while for the weakly-bound states at higher energies they approach Airy function behavior as predicted for linear confinement.", "In order to test our theory, we compare computed spectra to those obtained in inelastic neutron scattering experiments on high-quality single crystals of CaFe$_{2}$ O$_{4}$ .", "Slightly below the Néel ordering temperature, we are able to resolve seven bound states which are well described by our theory of non-linear confinement of spatially extended solitons.", "The outline of this paper is as follows.", "In Sec.", "we introduce a generic spin Hamiltonian and resulting low energy, non-linear $\\sigma $ model description of a system of weakly coupled antiferromagnetic chains with single-ion Ising anisotropy.", "We show that the saddle-point approximation results in a sine-Gordon equation and briefly review the one and two-soliton solutions.", "In Sec.", "we compute the energy of a single spin chain with a pair of domain walls from the kink-antikink solution, treating the interchain coupling at mean-field level.", "The bound state energies are obtained from numerical solutions of the effective Schrödinger equation with the effective non-linear confinement potential.", "Experimental details and results of our inelastic neutron-scattering experiments on CaFe$_{2}$ O$_{4}$ are presented in Sec. .", "We demonstrate that the measured bound-state energies are well described by our theoretical model.", "Finally, in Sec.", "we summarize and discuss our results.", "Figure: Staggered magnetizations of antiferromagnetic spin chains with Ising anisotropy in the presence of two domain walls (red).", "(a) For the S=1/2S=1/2 chain, a spin-flipexcitation fractionalizes into a pair of spinons.", "The energy cost due to the coupling to nearby chains scales with the number of spins between the domain walls,giving rise to a linear confinement potential, V(y)∼|y|V(y)\\sim |y|.", "(b) For large SS spin chains the domain walls are semi-classical chiral solitons of size ξ\\xi .", "Shown are different timeinstances of the collision of a soliton (K\\textrm {K}) and anti-soliton (K ¯\\overline{\\textrm {K}}) obtained from the KK ¯\\textrm {K}\\overline{\\textrm {K}} two-soliton solution of the sine-Gordonequation.", "The spatial extent of the domain walls causes them to annihilate gradually, rendering the effective confinement potential quadratic at small distances, V(y)∼y 2 V(y)\\sim y^2." ], [ "Theoretical Model", "Our starting point is a generic spin model of weakly coupled chains with antiferromagnetic Heisenberg couplings $J$ between nearest neighbor along the chains and $J_\\perp \\ll J$ between the chains.", "Each spin is subject to a single-ion, easy axis anisotropy $\\alpha >0$ .", "The Hamiltonian of the system is given by $\\hat{\\mathcal {H}} & = & J \\sum _{i,m} \\hat{\\mathbf {S}}_{i,m} \\hat{\\mathbf {S}}_{i+1,m}-\\alpha \\sum _{i,m}\\left(\\hat{S}^{z}_{i,m}\\right)^{2}\\nonumber \\\\& & + J_\\perp \\sum _{i,\\langle m,n\\rangle } \\hat{\\mathbf {S}}_{i,m} \\hat{\\mathbf {S}}_{i,n},$ where $i$ labels the positions in the chains, $m,n$ the different chains, and $\\langle m,n\\rangle $ denotes nearest neighbor bonds between adjacent chains.", "In this minimal model, we neglect longer-range exchanges and assume the interchain couplings to be the same in all directions.", "For simplicity, we have neglected exchange anisotropy between different spin components and Dzyaloshinskii-Moriya interactions.", "Such terms are not relevant in the case of Calcium Ferrite ($S=5/2$ , $L=0$ ) because of the lack of any orbital degrees of freedom.", "A discussion of single-ion anisotropy in systems with quenched orbital moment can be found in Ref.", "[Yosida10]." ], [ "Non-linear $\\sigma $ model", "Let us first focus on an isolated antiferromagnetic chain and drop the chain index for brevity.", "The effective long-wavelength, non-linear $\\sigma $ model is obtained using a path integral in imaginary time $\\tau \\in [0,\\beta ]$ , $\\beta = 1/(k_\\textrm {B} T)$ , and resolving the identities between adjacent time slices in terms of over-complete spin-coherent states, $| \\mathbf {N}_i(\\tau ) \\rangle $ .", "These states are parametrized by unit vectors $\\mathbf {N}_i(\\tau )$ and have the property $\\langle \\mathbf {N}_i(\\tau ) | \\hat{\\mathbf {S}}_i | \\mathbf {N}_i(\\tau ) \\rangle = S \\mathbf {N}_i(\\tau )$ .", "In order to perform a spatial continuum limit, we introduce the staggered Néel order-parameter field $\\mathbf {n}_i(\\tau )$ through the relation $\\mathbf {N}_i(\\tau ) = (-1)^i \\mathbf {n}_i(\\tau ) + a \\mathbf {L}_i(\\tau )$ , where $a$ denotes the lattice constant and $\\mathbf {L}_i(\\tau )$ describes the spin fluctuations perpendicular to $\\mathbf {n}_i(\\tau )$ .", "The latter fluctuations are massive and can therefore be integrated out.", "After taking the continuum limit, this procedure leads to the non-linear $\\sigma $ model,[34], [36], [37] $S=\\frac{\\rho _{S}}{2} \\int _0^\\beta \\mathrm {d}\\tau \\int _{-\\infty }^\\infty \\mathrm {d}x\\Bigg \\lbrace \\left(\\partial _{x}\\mathbf {n}\\right)^{2}+\\frac{1}{c^{2}}\\left(\\partial _\\tau \\mathbf {n}\\right)^{2}-\\kappa n_{z}^{2} \\Bigg \\rbrace ,$ with spin-stiffness $\\rho _S$ , spin-wave velocity $c$ and easy-axis anisotropy $\\kappa $ .", "These parameters are related to the microscopic parameters in the spin Hamiltonian (REF ), $\\rho _{S} = JS^{2}a, \\quad c = \\sqrt{2}JSa, \\quad \\textrm {and} \\;\\; \\kappa = \\frac{2\\alpha }{a^{2}J}.$ In the absence of anisotropy, $\\kappa =0$ , the relativistic field theory gives rise to a linear dispersion $\\omega = c k$ , corresponding to spin-wave excitations of the antiferromagnet.", "This is also reflected by the saddle-point approximation $\\delta S/\\delta \\mathbf {n}(x,t) = 0$ in real time $t=-i\\tau $ , which gives rise to the classical wave equation $\\partial ^2_x \\mathbf {n} -\\frac{1}{c^2} \\partial ^2_t \\mathbf {n} =0$ ." ], [ "Sine-Gordon equation and soliton solutions", "In the presence of anisotropy, it is useful to express the unit vector field $\\mathbf {n}(x,\\tau )$ in terms of spherical coordinates, $\\mathbf {n} =(\\sin \\theta \\cos \\phi ,\\sin \\theta \\sin \\phi ,\\cos \\theta )$ since the anisotropy only depends on the polar-angle field $\\theta (x,\\tau )$ , $S & = & \\frac{\\rho _{S}}{2} \\int _0^\\beta \\mathrm {d}\\tau \\int _{-\\infty }^\\infty \\mathrm {d}x\\Bigg \\lbrace \\left(\\partial _{x}\\theta \\right)^{2}+\\frac{1}{c^{2}}\\left(\\partial _\\tau \\theta \\right)^{2}\\nonumber \\\\& & +\\mathrm {sin}^{2}\\theta \\Big [\\left(\\partial _{x}\\phi \\right)^{2}+\\frac{1}{c^{2}}\\left(\\partial _\\tau \\phi \\right)^{2}\\Big ]-\\kappa \\, \\mathrm {cos}^{2}\\theta \\Bigg \\rbrace .$ The equations of motion are obtained from the saddle-point equations $\\delta S/\\delta \\phi (x,t) = 0$ and $\\delta S/\\delta \\theta (x,t) = 0$ .", "For the azimuthal angle we obtain a classical wave equation, $\\partial ^2_x \\phi -\\frac{1}{c^2} \\partial ^2_t \\phi =0$ .", "Since we are interested in soliton excitations and not in spin waves we will assume that $\\phi (x,t)=\\textrm {const}$ .", "In a system with $z$ -axis Ising anisotropy the free energy is independent of the choice of this constant.", "This removes all dependence of the action (REF ) on $\\phi $ and the dynamics for the polar angle is governed by the sine-Gordon equation, $\\partial ^2_x\\theta -\\frac{1}{c^2}\\partial ^2_t \\theta = \\frac{1}{2}\\kappa \\, \\textup {sin}(2\\theta ),$ which is known to admit soliton solutions.", "[38], [39] In terms of dimensionless length and time, $\\tilde{x}:= \\sqrt{\\kappa } x, \\quad \\textrm {and}\\; \\tilde{t}:=\\sqrt{\\kappa } c t,$ the 1-soliton solutions are given by $\\theta _{1,\\textrm {K}/\\overline{\\textrm {K}}}(\\tilde{x},\\tilde{t}) = 2 \\arctan \\left[ e^{\\pm \\gamma (\\tilde{x}-\\tilde{v}\\tilde{t})+\\delta _0}\\right],$ where $\\gamma =1/\\sqrt{1-\\tilde{v}^2}$ denotes the Lorentz factor and $\\tilde{v}=v/c$ the velocity of the relativistic soliton excitation in units of the spin-wave velocity $c$ , which plays the role of the speed of light.", "The different signs in the exponent correspond to kink ($\\textrm {K}$ ) and antikink ($\\overline{\\textrm {K}}$ ), respectively.", "$\\delta _0$ is a constant that is determined by the initial conditions.", "New soliton solutions can be generated from known solutions via transformations from one pseudo-spherical surface to another.", "[40] By application of such a transformation, known as a Bäcklund transformation, one can generate multiple-soliton solutions from the single soliton.", "[41] Important for our analysis is the “kink-antikink\" ($\\textrm {K}\\overline{\\textrm {K}}$ ), 2-soliton solution[42] $\\theta _{2,\\textrm {K}\\overline{\\textrm {K}}}(\\tilde{x},\\tilde{t}) = 2 \\arctan \\left[ \\frac{\\sinh \\left( \\frac{\\tilde{v}\\tilde{t}}{\\sqrt{1-\\tilde{v}^2}} \\right)}{\\tilde{v}\\cosh \\left( \\frac{\\tilde{x}}{\\sqrt{1-\\tilde{v}^2}} \\right)} \\right],$ where he have chosen the initial conditions such that $\\theta _{2,\\textrm {K}\\overline{\\textrm {K}}}(\\tilde{x},\\tilde{t}=0)=0$ , corresponding to a perfectly ordered chain $\\mathbf {n}(x)\\equiv \\hat{\\mathbf {e}}_z$ with no defects.", "The $\\textup {K}\\overline{\\textup {K}}$ solution (REF ) therefore describes the creation of a soliton and anti-soliton at $x=0$ at $t=0$ that propagate outwards in opposite directions for $t>0$ .", "This situation is therefore similar to the creation of two spinons by a single spin flip in the $S=1/2$ antiferromagnetic chain.", "The staggered magnetizations for outwards propagating solitons are shown in Fig.", "REF and compared with point-like domain walls.", "Another class of 2-soliton solutions that satisfy the sine-Gordon equation (REF ) are the breathers.", "[39], [42] These can be obtained directly from the $\\overline{\\textrm {K}}\\textrm {K}$ solution by analytic continuation to imaginary values of the velocity $\\tilde{v}$ .", "By doing so, one arrives at the breather solution $\\theta _{2,\\textrm {B}}(\\tilde{x},\\tilde{t}) = 2 \\arctan \\left[ \\frac{\\sqrt{1-\\tilde{\\omega }^{2}}}{\\tilde{\\omega }} \\frac{\\sin \\left(\\tilde{\\omega }\\tilde{t} \\right)}{\\cosh \\left( \\tilde{x}\\sqrt{1-\\tilde{\\omega }^2} \\right)} \\right].$ Such semi-classical breathers correspond to two domain walls which oscillate anharmonically within a maximum distance.", "Crucially, both the breather and $\\textrm {K}\\overline{\\textrm {K}}$ solutions have spatially extended domain walls and so the annihilation of a soliton and an anti-soliton happens gradually (see Fig.", "REF b)." ], [ "Soliton Confinement", "The theory of linear confinement of spinons in weakly coupled $S=1/2$ antiferromagnetic chains with XXZ-Ising exchange anisotropy[30], [31], [13], [14] or of domain walls in quasi one-dimensional Ising ferromagnets[32] is based on the assumption that domain walls are point-like.", "In this case, the interchain-frustration energy cost associated with the separation of two domain walls is simply proportional to the number of spins $N_y=|y|/a$ between two domain walls with distance $|y|$ .", "This gives rise to a linear confinement potential $V(y)\\simeq J_\\perp S^2 n_\\perp |y|/a$ , where $n_\\perp $ denotes the number of neighboring chains and $J_\\perp $ is the nearest-neighbor interchain coupling.", "The bound-state spectrum obtained from a one-dimensional Schrödinger equation with an attractive linear potential indeed gives a convincing description of the experimental data.", "[30], [31], [13], [14], [32] Here we generalize this approach to describe the non-linear confinement of spatially extended soliton domain walls.", "Our semi-classical path integral approach allows us to treat the finite-width of domain walls and to drop the assumption of Ising alignment.", "As we will see, in the limit of strong Ising anisotropy, the theory of linear confinement is recovered." ], [ "Effective confinement potential", "For a given spin profile along the chain, described by a field $\\theta (\\tilde{x})$ and constant $\\phi (\\tilde{x})=\\phi _0$ , the energy of the chain is given by $E_\\parallel = \\frac{\\rho _S \\sqrt{\\kappa }}{2}\\int ^{\\infty }_{-\\infty }\\mathrm {d}\\tilde{x}\\Big \\lbrace \\left(\\partial _{\\tilde{x}}\\theta \\right)^{2}-\\left(\\cos ^2\\theta -1\\right)\\Big \\rbrace ,$ where we subtracted the energy of a fully polarized chain ($\\theta (\\tilde{x})\\equiv 0$ ), which diverges in the thermodynamic limit.", "We treat the the interchain coupling at mean-field level, introducing the staggered magnetization $M= \\left| \\langle \\hat{S}_{i,m}^z \\rangle \\right|$ .", "The resulting energy contribution per chain is given by $E_\\perp = \\frac{\\rho _{S}}{2\\sqrt{\\kappa }}g_{\\perp }\\int ^{\\infty }_{-\\infty }\\mathrm {d}\\tilde{x}\\Big \\lbrace 1-\\cos \\theta \\Big \\rbrace ,$ where we have again subtracted the contribution for a fully polarized chain and defined the coupling $g_{\\perp }= \\frac{2n_{\\perp } M J_{\\perp }}{a^{2}S J},$ with $n_\\perp $ the number of neighboring chains and $J_\\perp $ the interchain coupling.", "Because of the dependence on the magnetic order parameter $M$ , the coupling $g_\\perp $ vanishes above $T_\\textrm {N}$ .", "The effective confinement potential $V(y)$ between a soliton and an anti-soliton can be obtained by evaluating the total energy $E_\\parallel +E_\\perp $ for the $\\textrm {K}\\overline{\\textrm {K}}$ solution (REF ) at given times $t_0$ corresponding to a distance $y = 2 v t_0$ between the domain walls.", "Note that $\\theta _{2,\\textrm {K}\\overline{\\textrm {K}}}$ is obtained for $g_\\perp = 0$ , neglecting the feedback of the interchain coupling on the soliton dynamics of the spin chain.", "This approximation is justified in the limit $J_\\perp \\ll J$ or slightly below the ordering temperature where $M\\ll 1$ .", "For larger $g_\\perp $ one would have to self-consistently determine the soliton solutions in the presence of the mean field from ordered neighboring chains.", "In this case the equation of motion is a double sine-Gordon equation which is not, in general, integrable but nonetheless can be solved numerically.", "[43], [44] Using the solution $\\theta _{2,\\textrm {K}\\overline{\\textrm {K}}}$ of the isolated chain, the confinement potential $V(y) = V_\\parallel (y)+V_\\perp (y)$ can be computed analytically.", "As a function of the dimensionless separation $\\tilde{y} = \\sqrt{\\kappa } y$ we obtain $\\frac{V_\\parallel (\\tilde{y})}{E_0} & = & \\sqrt{1-\\tilde{v}^2} \\frac{A(\\tilde{y})^{2}}{1+A(\\tilde{y})^{2}}\\left( 1+\\frac{\\mathrm {arcsinh}A(\\tilde{y})}{A(\\tilde{y})\\sqrt{1+A(\\tilde{y})^{2}}} \\right)\\nonumber \\\\& & + \\frac{1}{\\sqrt{1-\\tilde{v}^2}}\\left(1-\\frac{\\mathrm {arcsinh}A(\\tilde{y})}{A(\\tilde{y})\\sqrt{1+A(\\tilde{y})^{2}}} \\right), \\\\\\frac{V_\\perp (\\tilde{y})}{E_0} & = & \\frac{g_{\\perp }}{\\kappa }\\sqrt{1-\\tilde{v}^2}\\frac{A(\\tilde{y})\\mathrm {arcsinh} A(\\tilde{y})}{\\sqrt{1+A(\\tilde{y})^{2}}},$ where we have normalized by the rest energy $E_0=mc^{2}=2 \\rho _{S}\\sqrt{\\kappa }$ of a single soliton and defined the function $A(\\tilde{y})=\\tilde{v}^{-1} \\mathrm {sinh}(\\tilde{y} /2\\sqrt{1-\\tilde{v}^{2}})$ .", "Figure: Optimum dimensionless soliton velocity, v ˜=v/c\\tilde{v}=v/c, as a function of dimensionless domain wall separation y ˜\\tilde{y}, obtained by minimizing V ∥ (y ˜)V_{\\parallel }(\\tilde{y})with respect to v ˜\\tilde{v}.The effective potential still depends on the dimensionless velocity $\\tilde{v}$ .", "This parameter can be expressed as a function of the domain-wall separation $\\tilde{y}$ if we minimize the energy of the isolated chain, $E_\\parallel =V_\\parallel (\\tilde{y})$ , with respect to $\\tilde{v}$ .", "The resulting function $\\tilde{v}(\\tilde{y})$ is determined numerically and plotted in Fig.", "REF .", "While for $\\tilde{y}\\rightarrow 0$ the velocity approaches a constant $\\tilde{v}_0\\approx 0.725$ , for large domain-wall separations the velocity decays exponentially, $\\tilde{v}\\simeq 2.85 \\exp (-|\\tilde{y}|/2)$ .", "Let us first investigate the asymptotic behavior of the contributions $V_\\parallel $ (REF ) and $V_\\perp $ () to the potential.", "At large distances ($|\\tilde{y}|\\rightarrow \\infty $ ), the intra-chain contribution $V_\\parallel (\\tilde{y})$ approaches the energy $2 E_0$ of two free solitons at rest, while the inter-chain contribution grows linearly, $\\frac{V_\\perp (\\tilde{y})}{E_0}\\approx \\frac{g_\\perp }{\\kappa } |\\tilde{y}|.$ This is the same behavior as for point-like domain walls.", "This is expected since at large distances the spatial extent $\\xi $ of the solitons becomes irrelevant.", "Expressed in terms of the microscopic parameters, using Eqs.", "(REF ), (REF ) and the definition of $E_0$ (REF ), we can express the asymptotic result in terms of the microscopic parameters to recover $V\\sim n_\\perp J_\\perp |y|/a$ .", "At small separations ($\\tilde{y}\\ll 1$ ), both contributions are quadratic, $\\frac{V_\\parallel (\\tilde{y})}{E_0} & \\approx & \\frac{4-3\\tilde{v}_0^2}{6\\tilde{v}_0^2\\sqrt{1-\\tilde{v}_0^2}^3} \\tilde{y}^2\\approx 2.35 \\, \\tilde{y}^2,\\\\\\frac{V_\\perp (\\tilde{y})}{E_0} & \\approx & \\frac{1}{4\\tilde{v}_0^2\\sqrt{1-\\tilde{v}_0^2}} \\frac{g_\\perp }{\\kappa }\\tilde{y}^2 \\approx 0.69 \\frac{g_\\perp }{\\kappa }\\tilde{y}^2,$ which is the result of the gradual annihilation of the extended soliton and anti-soliton.", "The intra-chain contribution $V_\\parallel (\\tilde{y})$ and the full confinement potential $V(\\tilde{y})=V_\\parallel (\\tilde{y})+V_\\perp (\\tilde{y})$ are shown in Fig.", "REF as a function of the dimensionless domain-wall separation $\\tilde{y} = \\sqrt{\\kappa } y$ .", "They display the asymptotic behavior discussed above.", "The crossover from linear to quadratic behavior of $V(\\tilde{y})$ occurs at $\\tilde{y} =1$ .", "Since the crossover is expected to occur when the solitons start to overlap (see Fig.", "REF b), we can identify the size of the solitons as $\\xi \\simeq \\frac{1}{\\sqrt{\\kappa }} = a\\sqrt{\\frac{J}{2\\alpha }}.$ This equation shows that the size of the solitons is controlled by the relative strength of the Ising anisotropy, $\\alpha /J$ .", "In the case of strong Ising anisotropy, the size of the solitons is of the order of the lattice spacing $a$ .", "On the other hand, in systems with very weak anisotropy, the spatial extent of soliton domain walls can be of the order of hundreds of lattice spacings.", "Figure: (a) In-chain KK ¯\\textup {K}\\overline{\\textup {K}} potential V ∥ (y ˜)V_\\parallel (\\tilde{y}) as a function of dimensionless separationy ˜=κy\\tilde{y}=\\sqrt{\\kappa }y.", "At large separations, V ∥ V_\\parallel approaches the energy 2E 0 2 E_0 of two free solitons.", "Due to the gradual destructive interference of the solitons,V ∥ V_\\parallel is rendered quadratic at small distances.", "The crossover occurs at y ˜=1\\tilde{y}=1, corresponding to a soliton size ξ=1/κ\\xi = 1/\\sqrt{\\kappa }.", "(b) The same crossover is found in the effective confinement potential V(y ˜)=V ∥ (y ˜)+V ⊥ (y ˜)V(\\tilde{y})=V_\\parallel (\\tilde{y})+V_\\perp (\\tilde{y}).", "At large separations thepotential is linear, V(y ˜)/E 0 ≈(g ⊥ /κ)|y ˜|V(\\tilde{y})/E_0\\approx (g_\\perp /\\kappa ) |\\tilde{y}|, while at small separations the potential is quadratic due to the gradual annihilation of the extended solitons." ], [ "Bound-State Spectrum", "The gradual destructive interference of extended solitons at separations $y<\\xi $ weakens the confinement potential and renders it quadratic.", "In the following we will consider the solitons as point-like particles interacting with the effective non-linear potential $V(y)$ and determine the discrete bound-state spectrum from the solution of the one-dimensional Schrödinger equation $-\\frac{\\hbar ^{2}}{2 \\mu } \\frac{\\mathrm {d}^{2}\\psi }{\\mathrm {d}y^{2}}+V(y)\\psi =\\epsilon \\psi $ for the effective one-body problem for the relative coordinate $y$ of the soliton pair.", "Here $\\mu = m/2$ denotes the reduced mass in terms of the single-soliton mass $m$ .", "As a point of reference, let us first consider the limit of very strong Ising anisotropy.", "In this case the potential is linear down to lattice scale, $V(y)=\\lambda |y|$ , and the theory of linear confinement[30], [31], [13], [14], [32] applies.", "The resulting bound-state energies are given by[31] $\\epsilon _j^> = 2E_0 + \\xi _j \\lambda ^{2/3} \\left(\\frac{\\hbar ^2}{\\mu }\\right)^{1/3},$ where $\\xi _j$ are the negative zeroes of the Airy function, $\\textrm {Ai}(-\\xi _j)=0$ , $\\xi _1\\approx 2.338$ , $\\xi _2\\approx 4.088$ , $\\xi _3\\approx 5.520$ , $\\ldots $ .", "In the limit of very weak anisotropy on the other hand, the confinement potential is quadratic over a significant range, $V(y) \\simeq \\frac{1}{2} \\mu \\omega ^2 y^2$ , giving rise to equidistant energy levels $\\epsilon _j^< = \\hbar \\omega \\left(j+\\frac{1}{2} \\right).$ Due to the crossover of $V(y)$ from quadratic behavior at short distances to linear behavior at large distances, we expect to a related crossover in the energy level spacing of the bound states.", "The strongly bound states at low energies will be almost equidistant, as described by $\\epsilon _j^<$ (REF ), while the weakly bound states at higher energies will approach the sequence $\\epsilon _j^>$ (REF ).", "This crossover is controlled by the strength of the Ising anisotropy $\\alpha /J$ .", "Figure: (a) Soliton-antisoliton bound-state energiesϵ j \\epsilon _j in units of the single soliton energy E 0 E_0 for different ratios g ⊥ /κg_\\perp /\\kappa and S=5/2S=5/2.The corresponding confinement potentials are shown in the inset.", "(b) Same spectra but normalized by the energy ϵ 1 \\epsilon _1 of the first bound state.", "For larger values ofg ⊥ /κg_\\perp /\\kappa the level spacing becomes more harmonic oscillator like (dashed line).In order to obtain the bound-state spectrum for the full confinement potential we transform the Schrödinger equation (REF ) to dimensionless units, $-\\frac{1}{2 S^2} \\frac{\\mathrm {d}^{2}\\psi }{\\mathrm {d}\\tilde{y}^{2}}+\\tilde{V}(\\tilde{y})\\psi =\\tilde{\\epsilon }\\psi ,$ $\\tilde{y}=\\sqrt{\\kappa } y$ , $\\tilde{\\epsilon }=\\epsilon /E_0$ and $\\tilde{V}(\\tilde{y})=V_\\parallel (\\tilde{y})/E_0+V_\\perp (\\tilde{y})/E_0$ (REF ,), and then numerically solve the equation, using the finite-element method implemented in Mathematica.", "[45] In Fig.", "REF (a) the resulting bound-state energies $\\epsilon _j/E_0$ for $S=5/2$ (value for CaFe$_{2}$ O$_{4}$ ) and different values of $g_\\perp /\\kappa $ are shown.", "In the regime of large $g_\\perp /\\kappa $ , the dominant contribution to the confinement potential comes from the frustrated inter-chain coupling.", "The tightly bound states have almost equidistant energy levels with spacing $\\Delta \\epsilon /E_0 \\approx (1.17/S) \\sqrt{g_\\perp /\\kappa }$ , as expected for the asymptotic quadratic form of the potential at small distances, $\\tilde{V}(\\tilde{y})\\approx \\tilde{V}_\\perp (\\tilde{y})\\approx 0.69 (g_\\perp /\\kappa )\\tilde{y}^2$ .", "At higher energies, the level spacing is reduced because of the crossover of the potential to a linear form at large distances.", "Normalizing the energies by the energy $\\epsilon _1$ of the first bound state (see Fig.", "REF (b)), it is apparent that the spectrum becomes more like that of a harmonic oscillator if the value of $g_\\perp /\\kappa $ is increased." ], [ "Application to Calcium Ferrite", "In this section we will apply our theory of non-linear soliton confinement to the $S=5/2$ antiferromagnet CaFe$_{2}$ O$_{4}$ .", "Recent neutron scattering experiments[33] found signatures of solitary magnons in this material with a sequence of nine quantized excitations below the magnetic ordering transition at $T_\\textrm {N}\\approx 200$  K. CaFe$_{2}$ O$_{4}$ has a complex magnetic phase diagram due a competition between two different spin arrangements, termed the $A$ and $B$ phases.", "[46] The magnetic structure of the $B$ phase, which dominates at high temperatures, consists of antiferromagnetic zig-zag chains along the $b$ axis (see Fig.", "REF ).", "The moments are oriented along $b$ due to a small easy-axis anisotropy.", "The $A$ phase might coexist with the $B$ phase over the full temperature range but becomes clearly visible only below 170 K, which has been identified as its onset temperature in early studies.", "[46] The two phases are distinguished by their $c$ -axis stacking of ferromagnetic $b$ -axis stripes: the $B$ phase consists of stripes with antiferromagnetic alignment within the zig-zag chain, $(\\uparrow \\downarrow )(\\uparrow \\downarrow )$ , while in the $A$ phase the zig-zag chains are ferromagnetic with stacking $(\\uparrow \\uparrow )(\\downarrow \\downarrow )$ along $c$ .", "[46] It has been suggested[33] that the gradual increase of the $A$ phase component is linked to anti-phase domain boundaries along $c$ , combined with a continuous change of the Fe-O-Fe bond angle which controls the strength and sign of the super-exchange[47], [48] between the two legs forming the zig-zag chain.", "This scenario is supported by the presence of diffuse scattering rods along the $L$ direction and spin-wave excitations that show magnetic order in the $ab$ -plane with short-ranged correlations along $c$ .", "[33] Figure: Magnetic structure in the high-temperature B phase of CaFe 2 _{2}O 4 _{4}, showing antiferromagnetic zig-zag chains along the bb axis.The system exhibits a weak easy-axis anisotropy along bb.", "Calcium Ferrite is based upon an orthorhombic unit cell (space group 62 PnmaPnma) with dimensionsa=9.230a=9.230 Å, b=3.017b=3.017 Å, and c=10.689c=10.689 Å., From now on we focus on the $B$ phase that completely dominates at high temperatures where the discrete excitations are observed.", "As pointed out in Ref.", "[Stock+16], the level spacing of the excitations cannot be explained based on the linear confinement picture.", "This led the authors to speculate that the discrete nature of the excitations is not due to interaction-driven bound-state formation but instead a result of spatial confinement along the $c$ axis.", "Here we show that an effective non-linear interaction potential arising from the extended nature of solitons in CaFe$_{2}$ O$_{4}$ would lead to a bound-state spectrum that is consistent with the data.", "Let us first inspect the discrete energy-level spectrum presented in Ref.", "[Stock+16] more closely.", "The excitations can only be observed above the spin wave anisotropy gap, which shows a strong temperature dependence.", "The gap opens below $T_\\textrm {N}\\approx 200$  K and saturates to a value of $\\Delta \\approx 3$  meV below 100 K. For this reason, the lowest energy excitation can only be resolved slightly below $T_\\textrm {N}$ where strong fluctuations almost completely fill in the gap.", "The data at 200 K show six discrete energy levels below 2 meV.", "At 150 K the spin-wave gap almost completely masks this energy range.", "Instead three energy levels become visible above around 1.8 meV.", "In Ref.", "[Stock+16] it was assumed that the discrete excitations energies have a negligible temperature dependence and that the three levels observed at 150 K are the continuation of the energy sequence at 200 K. If the discrete excitations were due to soliton bound-state formation one would expect the excitation spectrum to depend upon temperature.", "Based on our theory, we expect that the main temperature dependence enters through the effective mean-field coupling $g_\\perp $ to neighboring chains.", "Since $g_\\perp $ is proportional to the magnetization of the system, it increases as temperature is lowered.", "This would explain why the bound states at 150 K have a larger level spacing than those at 200 K. Moreover, magnetoelastic effects and small changes to the Fe-O-Fe bond angle close to the threshold at which the superexchange would change sign could give rise to a non-negligible temperature dependence of magnetic exchange couplings.", "[51] Finally, the gradual onset of the $A$ phase could give rise to additional effects which might obscure the soliton signal in the neutron scattering experiment" ], [ "Experimental Results", "Here we present previously unpublished data that were collected alongside those published in Ref. [Stock+16].", "Instead of combining measurements at different temperatures we focus on $T=200$  K, allowing us to trace the excitations down to very low energies.", "Our experiments were performed on single crystals of CaFe$_{2}$ O$_{4}$ grown using a mirror furnace.", "High momentum and energy resolution data was obtained using the OSIRIS backscattering spectrometer located at the ISIS Neutron and Muon Source.", "[52] A white beam of neutrons is incident on the sample and the final energy of the scattered neutrons is fixed at $E_f=1.84$  meV using cooled graphite analyzers.", "A cooled Beryllium filter was used on the scattered side to reduce background.", "The default configuration is set for a symmetric dynamic range of $\\pm 0.5$  meV, however by shifting the incoming energy band width using a chopper the dynamic range was extended into the inelastic region.", "For this experimental setup, the elastic energy resolution (full-width) was $2\\delta E=0.025$  meV.", "Due to kinematic constraints, we focussed our measurements around $\\mathbf {Q} =(2, 0, 0)$  (r.l.u) so that the quantized excitations could be tracked up to energy transfers of $\\sim 3$  meV.", "Figure: (a) High resolution low energy data recorded on OSIRIS at T=200T=200 K, showing seven clearly discernible excitations ϵ j \\epsilon _j at 𝐐=(2,0,0)\\mathbf {Q}=(2,0,0) (r.l.u.).", "The modesshow a weak quadratic dispersion along LL, highlighted by dashed yellow lines.", "(b) At T=125T=125 K the spin-wave gap masks excitations below 2 meV.", "Above this energy, threeadditional excitations ϵ ˜ j \\tilde{\\epsilon }_j are visible.", "(c) Scattering intensityat 𝐐=(2,0,0)\\mathbf {Q}=(2,0,0) as a function of energy.", "Peaks at ϵ 2 ,...,ϵ 7 \\epsilon _2,\\ldots ,\\epsilon _7 are clearly resolved.", "The energy ϵ 1 \\epsilon _1 is below the elastic line.As shown in Fig.", "REF (a), at 200 K we find seven discrete excitations in low energy scattering data below 2 meV, located at $\\mathbf {Q}=(2,0,0)$ and with a weak quadratic dispersion along $L$ .", "The intensities are integrated over a small window of $2\\pm 0.05$ r.l.u.", "in the $H$ direction.", "The excitations have an almost linear level spacing $\\Delta \\epsilon \\approx 0.3$  meV, in very good agreement with previous results.", "[33] In comparison, at 125 K the spin-wave gap masks the excitations below 2 meV but three discrete excitations at $\\tilde{\\epsilon }_1$ , $\\tilde{\\epsilon }_2$ and $\\tilde{\\epsilon }_3$ are visible above this energy (Fig.", "REF (b)).", "The modes are at slightly higher energies than those identified at 150 K in Ref.", "[Stock+16], suggesting that there might exists a non negligible temperature dependence.", "In the following we will discard the excitations above 2 meV since they cannot be resolved at 200 K. In Fig.", "REF (c), the scattering intensity at 200 K as a function of energy at $\\mathbf {Q}=(2,0,0)$ is shown.", "Peaks at the energy levels $\\epsilon _2,\\ldots \\epsilon _7$ are very clearly visible.", "The first excitation $\\epsilon _1$ is beneath the incoherent background in the OSIRIS data and cannot be resolved in the energy cut.", "However, the energy $\\epsilon _1$ can be estimated thanks to the weak quadratic dispersion along $L$ (see dashed yellow lines in Fig.", "REF (a))." ], [ "Fitting to Non-Linear Confinement Model", "We now investigate whether the seven discrete excitations measured at 200 K can be explained in terms of soliton bound-state formation.", "The quantized excitations $\\epsilon _j$ extracted from the neutron scattering experiment are shown as open circles in Fig.", "REF .", "For the levels $j=2,\\ldots ,7$ we estimate the experimental error $\\delta \\epsilon _j$ from the full peak width at half maximum.", "For the lowest energy state, which is masked by the incoherent background of the elastic line, we assume a larger uncertainty of $\\delta \\epsilon _1\\approx 0.15$  meV.", "As point of reference, we first assume a linear confinement potential.", "In this case the soliton bound-state energies would be given by $\\epsilon _j = A + B\\xi _j$ , where $\\xi _j$ are the negative zeroes of the Airy function and the energies $A$ and $B$ are related to the soliton rest mass and the slope of the linear potential, as defined in Eq.", "(REF ).", "Here we use $A$ and $B$ as free fitting parameters, not imposing any additional constraints.", "The resulting best case scenario for the linear-confinement model (dashed magenta line in Fig.", "REF ) strongly deviates from the data, showing that the discrete excitations in CaFe$_{2}$ O$_{4}$ cannot be understood in terms of a linear confinement of solitons.", "Figure: Comparison between the measured excitation energies (open circles) and the soliton bound state energies calculated from the non-linear confinement model.", "The figure showsbest fits to the data for different values of g ⊥ /κg_\\perp /\\kappa .", "The quality of the fits improves with increasing values of g ⊥ /κg_\\perp /\\kappa and decreasing soliton energy E 0 E_0.", "Good agreementis achieved for g ⊥ /κ≥50g_\\perp /\\kappa \\ge 50.", "For comparison, the best fit of the linear-confinement model is shown in magenta.The bound-state spectra obtained from the effective non-linear confinement potential depend on two parameters, the soliton rest energy $E_0=2\\rho _S\\sqrt{\\kappa }$ and the dimensionless ratio $g_\\perp /\\kappa $ .", "For a given value of $g_\\perp /\\kappa $ we obtain the best fit to the data $\\lbrace \\epsilon _j \\pm \\delta \\epsilon _j\\rbrace $ by minimizing $\\chi ^2 = \\sum _j \\left(\\frac{\\epsilon _j^\\textrm {th}(E_0)-\\epsilon _j}{\\delta \\epsilon _j}\\right)^2$ with respect to $E_0$ , where $\\lbrace \\epsilon _j^\\textrm {th}\\rbrace $ refers to the spectrum obtained from our theoretical model.", "As shown in Fig.", "REF , the fits improve with increasing values of $g_\\perp /\\kappa $ , corresponding to decreasing optimum values of $E_0$ .", "A good description of our data is obtained for $g_\\perp /\\kappa =50$ and $E_0=0.061$  meV.", "Although for larger values of $g_\\perp /\\kappa $ the fits continue to improve slightly, the soliton size $\\xi =1/\\sqrt{\\kappa } = 2\\rho _S/E_0$ would eventually become too large for our theoretical description to be valid.", "For $g_\\perp /\\kappa =50$ the levels are almost equidistant, showing that the first 7 levels fall in the harmonic potential regime.", "To check consistency, we calculate the average mean-square displacement of the soliton bound states, $d_j = \\sqrt{ \\langle \\hat{y}^2 \\rangle _j}$ , using the approximate quadratic potential (REF ) at small distances, $y<\\xi $ .", "For the highest level resolved experimentally we obtain $d_7/\\xi \\approx 0.55 <1$ , indicating a significant overlap of the bound solitons.", "The parameters $\\rho _S$ , $g_\\perp $ and $\\kappa $ describe the long-wavelength, low energy behavior of the system.", "This effective continuum description is completely generic and applies to any system of weakly coupled antiferromagnetic spin chains in the the large-$S$ limit.", "For illustrative purposes, we have considered a minimal spin model (REF ) and established how the effective parameters in the continuum field theory depend on the exchange couplings and single-ion anisotropy of the lattice Hamiltonian (see Eqs.", "(REF ),(REF )).", "However, this model is too simplistic for CaFe$_{2}$ O$_{4}$ , e.g.", "it neglects the ferromagnetic exchange along the legs of the zig-zag chains, which is likely to be rather strong.", "Unfortunately, spin-wave excitations, which could be used to determine a more realistic spin model, have not been measured in the $B$ phase, but only at 4 K where the competing $A$ phase dominates.", "[33] On the other hand, close to the Néel transition collective fluctuations are very strong, leading to universal behavior detached from microscopic details.", "The spin stiffness is expected to vanish continuously at $T_\\textrm {N}$ , satisfying Josephson scaling $\\rho _S\\sim (T_\\textrm {N}-T)^{(d-2)\\nu }$ [Goldenfeld,Chubukov+94], where $\\nu $ is the correlation-length exponent and $d$ the spatial dimension.", "The bound states are observed slightly below $T_\\textrm {N}$ where the stiffness is strongly reduced.", "If we assume $\\rho _S/a\\approx 3$  meV, which is of the order of the gap and about a tenth of the spin-wave bandwidth at low temperature, we would obtain a soliton size of about 100 lattice constants, $\\xi /a = 2(\\rho _S/a)/E_0\\approx 100$ .", "As suggested in Ref.", "[Stock+16], quantized excitations in CaFe$_{2}$ O$_{4}$ could also arise from anti-phase boundaries along the $c$ axis that separate the two competing magnetic phases and lead to spatial confinement.", "This mechanism is unlikely to be relevant close to $T_\\textrm {N}$ where the phase boundaries are dynamic and the $A$ phase is almost completely absent.", "At low temperatures, however, the anti-phase domain boundaries become static and carry an uncompensated moment that can be tuned by a magnetic field.", "[55] The presence of uncompensated spins at phase or domain boundaries is also confirmed by thin-film experiments.", "[56] Isolated clusters of such orphan spins would provide a natural explanation of the discrete magnetic excitations observed at very low temperatures below the spin-wave gap.", "[55]" ], [ "Discussion", "To summarize, we have developed a theory for the confinement of solitons in weakly coupled, large-spin antiferromagnetic chains with easy-axis anisotropy.", "Below the Néel transition the frustrated interchain coupling generates an attractive potential that leads to the formation of soliton-antisoliton bound states.", "This mechanism is analogous to the confinement of spinons in $S=1/2$ antiferromagnetic XXZ chains[30], [31], [13], [14] or of domain-wall kinks in ferromagnetic Ising chains.", "[32] But while for these systems the domain-wall defects can be considered as point like, leading to a linear confinement potential, semi-classical solitons have a significant spatial extent.", "This renders the effective confinement potential quadratic on length scales smaller than the size of the solitons, giving rise to a crossover in the energy level spacing of the bound states.", "The $S=5/2$ antiferromagnet CaFe$_{2}$ O$_{4}$ is a good candidate system to test our theory since this material shows a sequence of discrete low-energy excitations[33] below $T_\\textrm {N}$ and exhibits a magnetic structure that consists of antiferromagnetic zig-zag chains, subject to a weak Ising anisotropy.", "[46] Our inelastic neutron scattering experiments, performed slightly below $T_\\textrm {N}$ , confirmed the existence of seven discrete excitations below 2 meV with an almost linear level spacing.", "Our analysis shows that the quantized excitations can be explained well by the non-linear confinement of large, spatially extended solitons.", "We argue that strong collective fluctuations close to $T_\\textrm {N}$ play a crucial role, collapsing the anisotropy gap and strongly reducing the spin stiffness.", "There are many possible ways in which our theory can be extended to describe a rich variety of physical systems.", "To model materials with strong interchain coupling one can include the feedback of the effective field from neighboring chains on the soliton dynamics.", "Such a staggered field changes the equation of motion to a double sine-Gordon equation which is no longer integrable but nonetheless can be solved numerically.", "[43], [44] Staggered fields could also be generated by applying external fields in systems with staggered $g$ tensors.", "[16], [17] Since solitons and antisolitons have opposite chirality it would be interesting to study the effects of a weak Dzyaloshinskii-Moriya interaction which would introduce chirality in the antiferromagnetic background.", "Finally, one might include finite-lifetime effects due to collisions of bound soliton pairs and the interactions with spin-wave excitations.", "Thanks to recent advances in crystal growth and neutron scattering technology it is now possible to resolve soliton bound states at very low energies.", "The relevant theoretical parameters in the effective long-wavelength description, such as the spin stiffness, spin-wave velocity and staggered magnetization, vanish at the continuous Néel transition, showing characteristic power-law behavior.", "The measurement of soliton bound states close to the transition could therefore provide a novel route to study universal critical behavior in inelastic neutron scattering experiments.", "Acknowledgements The authors thank E. Christou, A.", "Green, A. James and M. Songvilay for useful discussions.", "F.K.", "acknowledges financial support from EPSRC under Grant No.", "EP/P013449/1.", "S.W.C was supported by the DOE under Grant No.", "DOE: DE-FG02-07ER46382.", "C.S.", "and H.L.", "wish to thank the EPSRC and the STFC for funding.", "H.L.", "was co-funded by the ISIS facility development studentship programme.", "Experiments at the ISIS Neutron and Muon Source were supported by a beamtime allocation RB1510445 (DOI: 10.5286/ISIS.E.RB1510445) from the Science and Technology Facilities Council." ] ]