bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

two critical periods in the evolution of random planar graphs let p n m be a graph chosen uniformly at random from the family of all labeled planar graphs with n vertices and m edges in the paper we study the component structure of p n m combining counting arguments with analytic techniques we show that there are two critical periods in the evolution of p n m the first one of width n 2 3 is analogous to the phase transition observed in the standard random graph models and takes place for m n 2 o n 2 3 when the largest complex component is formed then for m n o n 3 5 when the complex components covers nearly all vertices the second critical period of width n 3 5 occurs starting from that moment increasing of m mostly affects the density of the complex components not its size

the continuum limit of critical random graphs we consider the erd s renyi random graph g n p inside the critical window that is when p 1 n n 4 3 for some fixed lambda in mathbb r we prove that the sequence of connected components of g n p considered as metric spaces using the graph distance rescaled by n 1 3 converges towards a sequence of continuous compact metric spaces the result relies on a bijection between graphs and certain marked random walks and the theory of continuum random trees our result gives access to the answers to a great many questions about distances in critical random graphs in particular we deduce that the diameter of g n p rescaled by n 1 3 converges in distribution to an absolutely continuous random variable with finite mean

hopf bifurcation for equivariant conservative and time reversible systems we study the bifurcation of small periodic solutions at a non semi simple 1 1 resonance in equivariant conservative or equivariant time reversible systems by using an equivariant liapunov schmidt method and restricting to solutions with an appropriate isotropy we reduce the problem to a scalar bifurcation equation the analysis of this equation shows a bifurcation behaviour similar to that found for the hamiltonian hopf bifurcation

local kesten mckay law for random regular graphs we study the adjacency matrices of random d regular graphs with large but fixed degree d in the bulk of the spectrum 2 sqrt d 1 varepsilon 2 sqrt d 1 varepsilon down to the optimal spectral scale we prove that the green s functions can be approximated by those of certain infinite tree like few cycles graphs that depend only on the local structure of the original graphs this result implies that the kesten mckay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors our method is based on estimating the green s function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs

level spacings distribution for large random matrices gaussian fluctuations we study the level spacings distribution for eigenvalues of large n x n matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1 defining by 77n s the number of nearest neighbors spacings greater than s 0 smaller than s 0 we prove functional limit theorem for the process 77n s e n s n1 2 giving weak convergence of this distribution to some gaussian random process on 0 x the limiting gaussian random process is universal for all classical compact groups it is holder continuous with any exponent less than 1 2 similar results can be obtained for the n level spacings distribution

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

eigenvalues outside the bulk of inhomogeneous erdős renyi random graphs in this article an inhomogeneous erd s renyi random graph on 1 ldots n is considered where an edge is placed between vertices i and j with probability varepsilon n f i n j n for i le j the choice being made independently for each pair the integral operator i f associated with the bounded function f is assumed to be symmetric non negative definite and of finite rank k we study the edge of the spectrum of the adjacency matrix of such an inhomogeneous erd s renyi random graph under the assumption that n varepsilon n rightarrow infty sufficiently fast although the bulk of the spectrum of the adjacency matrix scaled by sqrt n varepsilon n is compactly supported the kth largest eigenvalue goes to infinity it turns out that the largest eigenvalue after appropriate scaling and centering converges to a gaussian law if the largest eigenvalue of i f has multiplicity 1 if i f has k distinct non zero eigenvalues then the joint distribution of the k largest eigenvalues converge jointly to a multivariate gaussian law the first order behaviour of the eigenvectors is derived as a byproduct of the above results the results complement the homogeneous case derived by 18

extremal eigenvalues and eigenvectors of deformed wigner matrices we consider random matrices of the form h w lambda v lambda in mathbb r where w is a real symmetric or complex hermitian wigner matrix of size n and v is a real bounded diagonal random matrix of size n with i i d entries that are independent of w we assume subexponential decay of the distribution of the matrix entries of w and we choose lambda sim 1 so that the eigenvalues of w and lambda v are typically of the same order further we assume that the density of the entries of v is supported on a single interval and is convex near the edges of its support in this paper we prove that there is lambda in mathbb r such that the largest eigenvalues of h are in the limit of large n determined by the order statistics of v for lambda lambda in particular the largest eigenvalue of h has a weibull distribution in the limit n rightarrow infty if lambda lambda moreover for n sufficiently large we show that the eigenvectors associated to the largest eigenvalues are partially localized for lambda lambda while they are completely delocalized for lambda lambda similar results hold for the lowest eigenvalues

a method of linearization for painleve equations painleve iv v abstract a method to linearize the initial value problem of the painleve equations iv v is given the procedure involves formulating a riemann hilbert boundary value problem on intersecting lines for the inverse monodromy problem this boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space schlesinger transformations allow one to completely cover the parameter space special solutions are constructed from special cases of the riemann problem as well

inferring hidden states in langevin dynamics on large networks average case performance we present average performance results for dynamical inference problems in large networks where a set of nodes is hidden while the time trajectories of the others are observed examples of this scenario can occur in signal transduction and gene regulation networks we focus on the linear stochastic dynamics of continuous variables interacting via random gaussian couplings of generic symmetry we analyze the inference error given by the variance of the posterior distribution over hidden paths in the thermodynamic limit and as a function of the system parameters and the ratio ensuremath alpha between the number of hidden and observed nodes by applying kalman filter recursions we find that the posterior dynamics is governed by an effective drift that incorporates the effect of the observations we present two approaches for characterizing the posterior variance that allow us to tackle respectively equilibrium and nonequilibrium dynamics the first appeals to random matrix theory and reveals average spectral properties of the inference error and typical posterior relaxation times the second is based on dynamical functionals and yields the inference error as the solution of an algebraic equation

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

diluted banded random matrices scaling behavior of eigenfunction and spectral properties we demonstrate that the normalized localization length of the eigenfunctions of diluted sparse banded random matrices follows the scaling law the scaling parameter of the model is defined as where is the average number of non zero elements per matrix row n is the matrix size and additionally we show that also scales the spectral properties of the model up to certain sparsity characterized by the spacing distribution of eigenvalues

the structure of a random graph at the point of the phase transition consider the random graph models g n edges m and g n prob edge p with m m n 1 n 1 3 n 2 and p p n 1 an 1 3 n for 1 1 define an i component of a random graph as a component which has exactly i more edges than vertices call an i component with i 1 a complex component for both models we show that when a is constant the expected number of complex components is bounded almost surely a s each of these components if any exist has size of order n2 3 and the maximum value of i is bounded in probability we prove that a s the largest suspended tree in each complex component has size of order n2 3 and deletion of all suspended trees results in a smoothed graph of size of order n 3 with the maximum vertex degree 3 the total number of branching vertices i e of degree 3 is bounded in probability thus each complex component is almost surely topologically equivalent to a 3 regular multigraph of a uniformly bounded size lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3 we find a relatively simple integral formula for the limit distribution of the numbers of complex components which implies in particular that all values of the complexity spectrum have positive limiting probabilities we also answer questions raised by erdos and renyi back in 1960 it is proven that there exists p a the limiting planarity probability with 0 p a 1 p oo 1 p oo 0 in particular g n m g n p resp is almost surely nonplanar iff m n 2 n 213 00 np i n 1 3 oo resp

percolation on finite graphs and isoperimetric inequalities consider a uniform expanders family gn with a uniform bound on the degrees it is shown that for any p and c 0 a random subgraph of gn obtained by retaining each edge randomly and independently with probability p will have at most one cluster of size at least c gn with probability going to one uniformly in p the method from ajtai komlos and szemeredi combinatorica 2 1982 1 7 is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth as well as a simple proof of a result of kesten about the critical probability for bond percolation in high dimensions several problems and conjectures regarding percolation on finite transitive graphs are presented

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

spectral similarity for barabasi albert and chung lu models abstract in the paper we have analyzed spectral similarity between barabasi albert and chung lu models we have shown the similarity of spectral distribution for sufficiently large barabasi albert parameter value contrary extreme eigenvalues and principal eigenvector are not similar for those model we provide applications of obtained results related to the spectral graph theory and efficiency of quantum spatial search

spectral statistics of bernoulli matrix ensembles a random walk approach i we investigate the eigenvalue statistics of random bernoulli matrices where the matrix elements are chosen independently from a binary set with equal probability this is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues an approach which is similar to dyson s brownian motion model but with important modifications in particular we show our process is described by a fokker planck equation up to an error margin which vanishes in the limit of large matrix dimension the stationary solution of which corresponds to the joint probability density function of certain well known fixed trace gaussian ensembles

the phase transition in a random hypergraph we show that in the evolution of the random d uniform hypergraph gd n m the phase transition occurs when m n d d 1 o n2 3 we also prove local limit theorems for the distribution of the size of the largest component of gd n m in the subcritical and in the early supercritical phase

signal plus noise matrix models eigenvector deviations and fluctuations summary estimating eigenvectors and low dimensional subspaces is of central importance for numerous problems in statistics computer science and applied mathematics in this paper we characterize the behaviour of perturbed eigenvectors for a range of signal plus noise matrix models encountered instatistical and random matrix theoretic settings we establish both first order approximation results i e sharp deviations and second order distributional limit theory i e fluctuations the concise methodology presented in this paper synthesizes tools rooted in two core concepts namely deterministic decompositions of matrix perturbations and probabilistic matrix concentration phenomena we illustrate our theoretical results with simulation examples involving stochastic block model random graphs

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

global dynamics of parametrically excited nonlinear reversible systems with nonsemisimple 1 1 resonance abstract in this paper we analytically investigate the global dynamics associated with the nonlinear reversible systems that exhibit hopf bifurcation in the presence of one to one nonsemisimple internal resonance the effect of periodic parametric excitations is examined on such systems near the principal subharmonic resonance in presence of dissipation the nonlinear and nonautonomous system is simplified considerably by reducing it to the corresponding four dimensional normal form the normal form associated with the reversible systems is obtained as a special case from the general normal form equations obtained in n sri namachchivaya m m doyle w f langford and n evans normal form for generalized hopf bifurcation with non semisimple 1 1 resonance z angew math phys zamp 45 1994 312 335 under small perturbations arising from parametric excitations and nonreversible dissipation two mechanisms are identified in such systems that may lead to chaotic dynamics explicit restrictions on the system parameters are obtained for both of these mechanisms which lead to this complex behavior finally the results are demonstrated through a two degree of freedom model of a thin rectangular beam vibrating under the action of a pulsating follower force

generating random networks that consist of a single connected component with a given degree distribution we present a method for the construction of ensembles of random networks that consist of a single connected component with a given degree distribution this approach extends the construction toolbox of random networks beyond the configuration model framework in which one controls the degree distribution but not the number of components and their sizes unlike configuration model networks which are completely uncorrelated the resulting single component networks exhibit degree degree correlations moreover they are found to be disassortative namely high degree nodes tend to connect to low degree nodes and vice versa we demonstrate the method for single component networks with ternary exponential and power law degree distributions

cluster tails for critical power law inhomogeneous random graphs recently the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank 1 type having finite variance but infinite third moment degrees was obtained in bhamidi et al ann probab 40 2299 2361 2012 it was proved that when the degrees obey a power law with exponent tau in 3 4 the sequence of clusters ordered in decreasing size and multiplied through by n tau 2 tau 1 converges as n rightarrow infty to a sequence of decreasing non degenerate random variables here we study the tails of the limit of the rescaled largest cluster i e the probability that the scaling limit of the largest cluster takes a large value u as a function of u this extends a related result of pittel j combin theory ser b 82 2 237 269 2001 for the erd s renyi random graph to the setting of rank 1 inhomogeneous random graphs with infinite third moment degrees we make use of delicate large deviations and weak convergence arguments

dyson brownian motion for general beta and potential at the edge in this paper we compare the solutions of the dyson brownian motion for general beta and potential v and the associated mckean vlasov equation near the edge under suitable conditions on the initial data and the potential v we obtain optimal rigidity estimates of particle locations near the edge after a short time t mathrm o 1 our argument uses the method of characteristics along with a careful estimate involving an equation of the edge with the rigidity estimates as an input we prove a central limit theorem for mesoscopic statistics near the edge which as far as we know has been done for the first time in this paper additionally combining our results with landon and yau edge statistics of dyson brownian motion arxiv 1712 03881 2017 we give a proof of the local ergodicity of the dyson brownian motion for general beta and potential at the edge i e we show the distribution of extreme particles converges to the tracy widom beta distribution in a short time

sparse general wigner type matrices local law and eigenvector delocalization we prove a local law and eigenvector delocalization for general wigner type matrices our methods allow us to get the best possible interval length and optimal eigenvector delocalization in the dense case and the first results of such kind for the sparse case down to p g n log nn with g n we specialize our results to the case of the stochastic block model and we also obtain a local law for the case when the number of classes is unbounded

random walks on the random graph we study random walks on the giant component of the erd s renyi random graph g n p g n p where p np n for 1 1 fixed the mixing time from a worst starting point was shown by fountoulakis and reed and independently by benjamini kozma and wormald to have order log2nlog2 n we prove that starting from a uniform vertex equivalently from a fixed vertex conditioned to belong to the giant both accelerates mixing to o logn o log n and concentrates it the cutoff phenomenon occurs the typical mixing is at d 1logn logn 1 2 o 1 d 1log n log n 1 2 o 1 where and dd are the speed of random walk and dimension of harmonic measure on a poisson poisson galton watson tree analogous results are given for graphs with prescribed degree sequences where cutoff is shown both for the simple and for the nonbacktracking random walk

continuum limit of critical inhomogeneous random graphs the last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models to gain insight into real world systems in this study we analyze the maximal components at criticality of one famous class of such models the rank one inhomogeneous random graph model norros and reittu adv appl probab 38 1 59 75 2006 bollobas et al random struct algorithms 31 1 3 122 2007 section 16 4 viewing these components as measured random metric spaces under finite moment assumptions for the weight distribution we show that the components in the critical scaling window with distances scaled by n 1 3 converge in the gromov haussdorf prokhorov metric to rescaled versions of the limit objects identified for the erd s renyi random graph components at criticality in addario berry et al probab theory related fields 152 3 4 367 406 2012 a key step is the construction of connected components of the random graph through an appropriate tilt of a fundamental class of random trees called mathbf p trees camarri and pitman electron j probab 5 2 1 18 2000 aldous et al probab theory related fields 129 2 182 218 2004 this is the first step in rigorously understanding the scaling limits of objects such as the minimal spanning tree and other strong disorder models from statistical physics braunstein et al phys rev lett 91 16 168701 2003 for such graph models by asymptotic equivalence janson random struct algorithms 36 1 26 45 2010 the same results are true for the chung lu model chung and lu proc natl acad sci 99 25 15879 15882 2002 chung and lu ann combin 6 2 125 145 2002 chung and lu complex graphs and networks 2006 and the britton deijfen martin lof model britton et al j stat phys 124 6 1377 1397 2006 a crucial ingredient of the proof of independent interest are tail bounds for the height of mathbf p trees the techniques developed in this paper form the main technical bedrock for the general program developed in bhamidi et al scaling limits of random graph models at criticality universality and the basin of attraction of the erd s renyi random graph arxiv preprint 2014 for proving universality of the continuum scaling limits in the critical regime for a wide array of other random graph models including the configuration model and inhomogeneous random graphs with general kernels bollobas et al random struct algorithms 31 1 3 122 2007

the random cluster model on the complete graph the random cluster model of fortuin and kasteleyn contains as special cases the percolation ising and potts models of statistical physics when the underlying graph is the complete graph onn vertices then the associated processes are called mean field in this study of the mean field random cluster model with parametersp n andq we show that its properties for any value ofq 0 may be derived from those of an erd s renyi random graph in this way we calculate the critical point c q of the model and show that the associated phase transition is continuous if and only ifq 2 exact formulae are given for c q the density of the largest component the density of edges of the model and the free energy this work generalizes earlier results valid for the potts model whereq is an integer satisfyingq 2 equivalent results are obtained for a fixed edge number random cluster model as a consequence of the results of this paper one obtains large deviation theorems for the number of components in the classical random graph models whereq 1

the equation δ u u α 1 u f for 0 ga soit 0 et f l loc 1 r n n 1 on considere le probleme suivant u u 1 u f dans r n u l loc 1 r n u 1 u l loc 1 r n on demontre que si f 0 et s il existe une solution positive alors on a une solution positive minimale u on etudie l existence de solutions positives pour 0 1

a phase transition in the random transposition random walk our work is motivated by bourque pevzner s simulation study of the effectiveness of the parsimony method in studying genome rearrangement and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on n elements starting from the identity let d t be the minimum number of transpositions needed to go back to the identity element from the location at time t d t undergoes a phase transition for 0 1 d cn 2 u c n where u is an explicit function satisfying u x x 2 moreover we describe the fluctuations of d cn 2 about its mean at each of the three stages subcritical critical and supercritical the techniques used involve viewing the cycles in the random permutation as a coagulation fragmentation process and relating the behavior to the erd s renyi random graph model

local semicircle law for random regular graphs we consider random d regular graphs on n vertices with degree d at least log n 4 we prove that the green s function of the adjacency matrix and the stieltjes transform of its empirical spectral measure are well approximated by wigner s semicircle law down to the optimal scale given by the typical eigenvalue spacing up to a logarithmic correction aside from well known consequences for the local eigenvalue distribution this result implies the complete isotropic delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity 2017 wiley periodicals inc

bulk universality for generalized wigner matrices with few moments in this paper we consider n times n real generalized wigner matrices whose entries are only assumed to have finite 2 varepsilon th moment for some fixed but arbitrarily small varepsilon 0 we show that the stieltjes transforms m n z of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale when eta mathfrak i z is almost of order n 1 as a consequence we establish bulk universality for local spectral statistics of these matrices at fixed energy levels both in terms of eigenvalue gap distributions and correlation functions meaning that these statistics converge to those of the gaussian orthogonal ensemble in the large n limit

revealing the microstructure of the giant component in random graph ensembles the microstructure of the giant component of the erd s renyi network and other configuration model networks is analyzed using generating function methods while configuration model networks are uncorrelated the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree degree correlations of all orders we present exact analytical results for the degree distributions as well as higher order degree degree correlations on the giant components of configuration model networks we show that the degree degree correlations are essential for the integrity of the giant component in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component to demonstrate the importance and broad applicability of these results we apply them to the study of the distribution of shortest path lengths on the giant component percolation on the giant component and spectra of sparse matrices defined on the giant component we show that by using the degree distribution on the giant component one obtains high quality results for these properties which can be further improved by taking the degree degree correlations into account this suggests that many existing methods currently used for the analysis of the whole network can be adapted in a straightforward fashion to yield results conditioned on the giant component

local law and tracy widom limit for sparse random matrices we consider spectral properties and the edge universality of sparse random matrices the class of random matrices that includes the adjacency matrices of the erd s renyi graph model g n p we prove a local law for the eigenvalue density up to the spectral edges under a suitable condition on the sparsity we also prove that the rescaled extremal eigenvalues exhibit goe tracy widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included for the adjacency matrix of the erd s renyi graph this establishes the tracy widom fluctuations of the second largest eigenvalue when p is much larger than n 2 3 with a deterministic shift of order np 1

reversible relative periodic orbits abstract we study the bundle structure near reversible relative periodic orbits in reversible equivariant systems in particular we show that the vector field on the bundle forms a skew product system by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves we also discuss possibilities for drifts along group orbits our results extend those recently obtained in the equivariant context by b sandstede et al 1999 j nonlinear sci 9 439 478 and c wulff et al 2001 ergodic theory dynam systems 21 605 635

fixed energy universality of dyson brownian motion abstract we consider dyson brownian motion for classical values of with deterministic initial data v we prove that the local eigenvalue statistics coincide with the goe gue in the fixed energy sense after time t 1 n if the density of states of v is bounded above and below down to scales t in a window of size l t our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known our methodology builds on the homogenization theory developed in 16 which reduces the microscopic problem to a mesoscopic problem as an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical dyson brownian motion

bulk universality of sparse random matrices we consider the adjacency matrix of the ensemble of erd s renyi random graphs which consists of graphs on n vertices in which each edge occurs independently with probability p we prove that in the regime pn 1 these matrices exhibit bulk universality in the sense that both the averaged n point correlation functions and distribution of a single eigenvalue gap coincide with those of the goe our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

on a random graph evolving by degrees abstract we consider a random multi graph growth process g m on a vertex set n which is a special case of a more general process proposed by laci lovasz in 2002 g 0 is empty and g m 1 is obtained from g m by inserting a new edge e at random specifically the conditional probability that e joins two currently disjoint vertices i and j is proportional to d i d j where d i d j are the degrees of i j in g m and 0 is a fixed parameter the limiting case is the erd s renyi graph process we show that whp g m contains a unique giant component iff c 2 m n c 1 and the size of this giant is asymptotic to n 1 c c where c c is the root of c c 2 c c 2 a phase transition window is proved to be contained essentially in c a n 1 3 c b n 1 4 and we conjecture that 1 4 may be replaced with 1 3 for the multigraph version mg m we show that mg m is connected whp iff m m n n 1 1 we conjecture that for 1 m n is the threshold for connectedness of g m itself

robustness of onionlike correlated networks against targeted attacks recently it was found by schneider et al proc natl acad sci usa 108 3838 2011 using simulations that scale free networks with onion structure are very robust against targeted high degree attacks the onion structure is a network where nodes with almost the same degree are connected motivated by this work we propose and analyze based on analytical considerations an onionlike candidate for a nearly optimal structure against simultaneous random and targeted high degree node attacks the nearly optimal structure can be viewed as a set of hierarchically interconnected random regular graphs the degrees and populations of whose nodes are specified by the degree distribution this network structure exhibits an extremely assortative degree degree correlation and has a close relationship to the onion structure after deriving a set of exact expressions that enable us to calculate the critical percolation threshold and the giant component of a correlated network for an arbitrary type of node removal we apply the theory to the cases of random scale free networks that are highly vulnerable against targeted high degree node removal our results show that this vulnerability can be significantly reduced by implementing this onionlike type of degree degree correlation without much undermining the almost complete robustness against random node removal we also investigate in detail the robustness enhancement due to assortative degree degree correlation by introducing a joint degree degree probability matrix that interpolates between an uncorrelated network structure and the onionlike structure proposed here by tuning a single control parameter the optimal values of the control parameter that maximize the robustness against simultaneous random and targeted attacks are also determined our analytical calculations are supported by numerical simulations

radial solutions of a semilinear elliptic problem we analyse the set of nonnegative global and radial solutions radial solutions for short of the equation u u p f in r n n 1 where 0 where graphics when f r c r2p 1 p h r with h r o r2p 1 p as r radial solutions continue to exist if h r is sufficiently small at infinity existence however breaks down if h r 0 graphics whenever they exist radial solutions are characterised in terms of their asymptotic behaviour as r

estimating mixed memberships with sharp eigenvector deviations we consider the problem of estimating community memberships of nodes in a network where every node is associated with a vector determining its degree of membership in each community existing prov

poisson structure of the reversible 1 1 resonance abstract vector fields that are reversible but non hamiltonian are considered in the neighborhood of a 1 1 resonance it is found that the normalized vector field has a hidden poisson structure on the orbit space that is reminiscent of the equations governing the rigid body in mechanics the theory is used to give a geometric characterization of the phase space in the neighborhood of a reversible 1 1 resonance certain similarities and also essential differences with the hamiltonian 1 1 resonance are found

edge universality of correlation matrices xij 0 and e e 2 1 m n m these entries have a sub exponential decay at the tails we will be working in the regime n m dn limn 1 dn 6 0 1 in this paper we prove the edge universality of correlation matrices x x where the rectangular matrix x called the standardized matrix is obtained by normalizing each column of the data matrix e x by its euclidean norm our main result states that asymptotically the k point k 1 correlation functions of the extreme eigenvalues at both edges of the spectrum of the correlation matrix x x converge to those of the gaussian correlation matrix i e tracy widom law and thus in particular the largest and the smallest eigenvalues of x x after appropriate centering and rescaling converge to the tracy widom distribution the asymptotic distribution of extreme eigenvalues of the gaussian correlation matrix has been worked out only recently as a corollary of the main result in this paper we also obtain that the extreme eigenvalues of gaussian correlation matrices are asymptotically distributed according to the tracywidom law the proof is based on the comparison of green functions but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix we achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest

percolation on dense graph sequences in this paper we determine the percolation threshold for an arbitrary sequence of dense graphs g n let n be the largest eigenvalue of the adjacency matrix of g n and let g n p n be the random subgraph of g n obtained by keeping each edge independently with probability p n we show that the appearance of a giant component in g n p n has a sharp threshold at p n 1 n in fact we prove much more if g n converges to an irreducible limit then the density of the largest component of g n c n tends to the survival probability of a multi type branching process defined in terms of this limit here the notions of convergence and limit are those of borgs chayes lovasz s6s and vesztergombi in addition to using basic properties of convergence we make heavy use of the methods of bollobas janson and riordan who used multi type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

delocalization at small energy for heavy tailed random matrices we prove that the eigenvectors associated to small enough eigenvalues of an heavy tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity the delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of l 4 and l 2 norms of vectors in contrast as a consequence of a previous result for random matrices with sufficiently heavy tails the eigenvectors associated to large enough eigenvalues are localized according to the same criterion the proof is based on a new analysis of the fixed point equation satisfied asymptotically by the law of a diagonal entry of the resolvent of this matrix

time reversibility and particle sedimentation this paper studies an ordinary differential equation ode model called the stokeslet model and describes sedimentation of small clusters of particles in a highly viscous fluid this model has a trivial solution in which the n particles arrange themselves at the vertices of a regular n sided polygon when n 3 hocking j fluid mech 20 1964 pp 129 139 and caflisch et al phys fluids 31 1988 pp 3175 3179 prove the existence of periodic motion in the frame moving with the center of gravity in the cluster in which the particles form an isosceles triangle the study of periodic and quasiperiodic solutions of the stokeslet model is continued with emphasis on the spatial and time reversal symmetry of the model time reversibility is due to infinite viscosity and spatial dihedral symmetry is due to the assumption of identical particles and the symmetry of the trivial solution for three particles the existence of a second family of periodic solutions and a family of quasiperiodic solu

bulk eigenvalue statistics for random regular graphs ak was partly supported by swiss national science foundation grant 144662 hty was partly supported by nsf grant dms 1307444 and a simons investigator fellowship

reversible equivariant hopf bifurcation in this paper we study codimension one hopf bifurcation from symmetric equilibrium points in reversible equivariant vector fields such bifurcations are characterized by a doubly degenerate pair of purely imaginary eigenvalues of the linearization of the vector field at the equilibrium point the eigenvalue movements near such a degeneracy typically follow one of three scenarios splitting from two pairs of imaginary eigenvalues to a quadruplet on the complex plane passing on the imaginary axis or crossing a quadruplet crossing the imaginary axis we give a complete description of the behaviour of reversible periodic orbits in the vicinity of such a bifurcation point for non reversible periodic solutions in the case of hopf bifurcation with crossing eigenvalues we obtain a generalization of the equivariant hopf theorem

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

linear processes in high dimensions phase space and critical properties in this work we investigate the generic properties of a stochastic linear model in the regime of high dimensionality we consider in particular the vector autoregressive var model and the multivariate hawkes process we analyze both deterministic and random versions of these models showing the existence of a stable phase and an unstable phase we find that along the transition region separating the two regimes the correlations of the process decay slowly and we characterize the conditions under which these slow correlations are expected to become power laws we check our findings with numerical simulations showing remarkable agreement with our predictions we finally argue that real systems with a strong degree of self interaction are naturally characterized by this type of slow relaxation of the correlations

isotropic self consistent equations for mean field random matrices we present a simple and versatile method for deriving an isotropic local laws for general random matrices constructed from independent random variables our method is applicable to mean field random matrices where all independent variables have comparable variances it is entirely insensitive to the expectation of the matrix in this paper we focus on the probabilistic part of the proof the derivation of the self consistent equations as a concrete application we settle in complete generality the local law for wigner matrices with arbitrary expectation

mesoscopic eigenvalue density correlations of wigner matrices we investigate to what extent the microscopic wigner gaudin mehta dyson wgmd or sine kernel statistics of random matrix theory remain valid on mesoscopic scales to that end we compute the connected two point spectral correlation function of a wigner matrix at two mesoscopically separated points in the mesoscopic regime density correlations are much weaker than in the microscopic regime our result is an explicit formula for the two point function this formula implies that the wgmd statistics are valid to leading order on all mesoscopic scales that in the real symmetric case there are subleading corrections matching precisely the wgmd statistics while in the complex hermitian case these subleading corrections are absent we also uncover non universal subleading correlations which dominate over the universal ones beyond a certain intermediate mesoscopic scale the proof is based on a hierarchy of schwinger dyson equations for a sufficiently large class of polynomials in the entries of the green function the hierarchy is indexed by a tree whose depth is controlled using stopping rules a key ingredient in the derivation of the stopping rules is a new estimate on the density of states which we prove to have bounded derivatives of all order on all mesoscopic scales

the spectral edge of some random band matrices we study the asymptotic distribution of the eigenvalues of random hermitian periodic band matrices focusing on the spectral edges the eigenvalues close to the edges converge in distribution to the airy point process if and only if the band is sufficiently wide w n 5 6 otherwise a different limiting distribution appears

the outliers of a deformed wigner matrix our results extend those of 21 by admitting overlapping outliers and by computing the joint distribution of all outliers in particular we give a complete description of the failure of universality first observed in 8 10 we also show that under suitable conditions outliers may be strongly correlated even if they are far from each other our proof relies on the isotropic local semicircle law established in 21 the main technical achievement of the current paper is the joint asymptotics of an arbitrary finite family of random variables of the form v h z 1w

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

mixing properties of the swendsen wang process on the complete graph and narrow grids we consider the mixing properties of the swendsen wang process for the two state potts model or ising model on the complete n vertex graph kn and for the q state model on an a n grid where a is bounded as n

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

local spectral statistics of the addition of random matrices we consider the local statistics of h v x v u y u where v and u are independent haar distributed unitary matrices and x and y are deterministic real diagonal matrices in the bulk we prove that the gap statistics and correlation functions coincide with the gue in the limit when the matrix size n rightarrow infty under mild assumptions on x and certain rigidity assumptions on y the latter being an assumption on the convergence of the eigenvalues of y to the quantiles of its limiting spectral measure which we assume to have a density our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to dyson brownian motion our method also applies to the case when v and u are drawn from the orthogonal group our proof relies on the local law for h proved in bao et al commun math phys 349 3 947 990 2017 j funct anal 271 3 672 719 2016 adv math 319 251 291 2017 as well as the dbm convergence results of landon and yau commun math phys 355 3 949 1000 2017 and landon et al adv math 346 1137 1332 2019

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

mixing time of near critical random graphs let mathcal c 1 be the largest component of the erd s renyi random graph mathcal g n p the mixing time of random walk on mathcal c 1 in the strictly supercritical regime p c n with fixed c 1 was shown to have order log2n by fountoulakis and reed and independently by benjamini kozma and wormald in the critical window p 1 e n where e3n is bounded nachmias and peres proved that the mixing time on mathcal c 1 is of order n however it was unclear how to interpolate between these results and estimate the mixing time as the giant component emerges from the critical window indeed even the asymptotics of the diameter of mathcal c 1 in this regime were only recently obtained by riordan and wormald as well as the present authors and kim in this paper we show that for p 1 e n with e3n and o n the mixing time on mathcal c 1 is with high probability of order n log2 in addition we show that this is the order of the largest mixing time over all components both in the slightly supercritical and in the slightly subcritical regime i e p 1 e n with as above

delocalization of eigenvectors of random matrices with independent entries author s rudelson mark vershynin roman abstract we prove that an n by n random matrix g with independent entries is completely delocalized suppose the entries of g have zero means variances uniformly bounded below and a uniform tail decay of exponential type then with high probability all unit eigenvectors of g have all coordinates of magnitude o n 1 2 modulo logarithmic corrections this comes a consequence of a new geometric approach to delocalization for random matrices

bounds for the stieltjes transform and the density of states of wigner matrices we consider ensembles of wigner matrices whose entries are up to the symmetry constraints independent and identically distributed random variables we show the convergence of the stieltjes transform towards the stieltjes transform of the semicircle law on optimal scales and with the optimal rate our bounds improve previous results in particular from erd s et al adv math 229 3 1435 1515 2012 electron j probab 18 59 1 58 2013 by removing the logarithmic corrections as applications we establish the convergence of the eigenvalue counting functions with the rate log n n and the rigidity of the eigenvalues of wigner matrices on the same scale these bounds improve the results of erd s et al adv math 229 3 1435 1515 2012 electron j probab 18 59 1 58 2013 gotze and tikhomirov 2013

anisotropic local laws for random matrices we develop a new method for deriving local laws for a large class of random matrices it is applicable to many matrix models built from sums and products of deterministic or independent random matrices in particular it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity for definiteness we present the method for sample covariance matrices of the form open image in new window where t is deterministic and x is random with independent entries we prove that with high probability the resolvent of q is close to a deterministic matrix with an optimal error bound and down to optimal spectral scales as an application we prove the edge universality of q by establishing the tracy widom airy statistics of the eigenvalues of q near the soft edges this result applies in the single cut and multi cut cases further applications include the distribution of the eigenvectors and an analysis of the outliers and bbp type phase transitions in finite rank deformations they will appear elsewhere we also apply our method to wigner matrices whose entries have arbitrary expectation i e we consider w a where w is a wigner matrix and a a hermitian deterministic matrix we prove the anisotropic local law for w a and use it to establish edge universality

localization and delocalization of eigenvectors for heavy tailed random matrices consider an n times n hermitian random matrix with above the diagonal independent entries with alpha stable symmetric distribution and 0 2 we give vanishing bounds on the l p norm of the eigenvectors normalized to have unit l 2 norm on the contrary when 0 alpha 2 3 we prove that these eigenvectors are localized

bulk universality and related properties of hermitian matrix models we give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models assuming that the potential that determines the model is globally c 2 and locally c 3 function see theorem 3 1 the proof as our previous proof in pastur and shcherbina in j stat phys 86 109 147 1997 is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials rather we obtain the sin kernel as a unique solution of a certain non linear integro differential equation that follows from the determinant formulas for the correlation functions of the model we also give a simplified and strengthened version of paper boutet de monvel et al in j stat phys 79 585 611 1995 on the existence and properties of the limiting normalized counting measure of eigenvalues we use these results in the proof of universality and we believe that they are of independent interest

rigidity and a mesoscopic central limit theorem for dyson brownian motion for general beta β and potentials we study dyson brownian motion with general potential v and for general beta ge 1 for short times t o 1 and under suitable conditions on v we obtain a local law and corresponding rigidity estimates on the particle locations that is with overwhelming probability the particles are close to their classical locations with an almost optimal error estimate under the condition that the density of states of the initial data is bounded below and above down to the scale eta ll t ll 1 we prove a mesoscopic central limit theorem for linear statistics at all scales eta with n 1 ll eta ll t

hamiltonian hopf bifurcation with symmetry in this paper we study the appearance of branches of relative periodic orbits in hamiltonian hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework the theoretical study is illustrated with several examples

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized