# Dispersion models on a circle: universal properties and asymptotic results

Jean-François Marckert, Zoé Varin

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Jean-François Marckert, Zoé Varin. Dispersion models on a circle: universal properties and asymptotic results. 2025. hal-05138592

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# Dispersion models on a circle: universal properties and asymptotic results

Jean-François Marckert & Zoé Varin

Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400 Talence, France

## Abstract

Consider a sequence of masses $m_0, m_1, ...$ arriving uniformly at random at some points $u_0, u_1, ...$ on the unit circle $\mathbb{R}/\mathbb{Z}$ (or on $\mathbb{Z}/n\mathbb{Z}$, in the discrete version). Upon arrival, each mass undergoes a relaxation phase during which it is dispersed, possibly also at random. This process can model many physical phenomena, such as the diffusion of liquid in a porous medium. In the discrete case, it can model parking (related to additive coalescence and hashing with linear probing) in which the cars are permitted to follow random displacement policies.

The dispersion policies considered in the paper ensure that at time $k$, after the successive dispersions of $m_0, \cdots, m_{k-1}$, the total covered region has Lebesgue measure $m_0 + \cdots + m_{k-1}$. Furthermore, during the dispersion of a given mass, the covered domain increases continuously, except when it merges with another covered connected component (CC).

We show a very general exchangeability property for the sequence of covered CC. Additionally, we demonstrate a universal spacing property between these CC, and a notable general result: if the $(u_i)$ are independent and rotationally invariant, then the number of free (not covered) CC follows a binomial distribution whose parameters depend solely on the number and total mass of arrived particles. Furthermore, conditional on the number of CC, the sizes of the free CC follow a simple Dirichlet distribution in the continuous case, regardless of the dispersion policy considered and the values of the masses. We also characterize the distribution of the occupied space.

In the second part of the paper, we study the total cost associated with these models for various cost models, and establish connections with the additive coalescent. We also provide an asymptotic representation of the limiting covered space as the number of masses goes to infinity.

# 1 Introduction

The dynamics we are interested in are quite general, and represent physical systems in which some matter arrives successively at random on a given surface $S$. Upon arrival, the matter undergoes a relaxation phase, during which this matter is dispersed on the surface until it covers a new surface, whose area is proportional to its initial mass. These types of models are common in many fields of sciences, particularly physics, chemistry, biology, computer science, combinatorics, and probability theory. At this stage, we can use two main images to illustrate the phenomenon we want to study:

- We can imagine water droplets of various random sizes (as in the case of rain) falling randomly on a structure, and moistening it, according to a (possibly random) dispersion dynamic, depending on the external conditions (wind, temperature, etc.). In this case, the space is the structure, and the connected components (CC) of the moistened area constitute the “occupied domain”. The moistened area can be imagined as having an area proportional to the total mass of water deposited (or equal to it, up to a change of unity).
– We can imagine some cars (all of the unit size), arriving one after the other in a parking lot. When the $i$th car arrives, it chooses a place $c_i$ uniformly at random, and parks in the nearest closest available place. In this case, the problem is discrete: the space is the parking lot, and the occupied domain is the set of occupied places. This is the classical parking model (see Section 1.2 for more details).

In order to fully define our models, we need to define four main characteristics:

(a) the space/surface on which the matter arrives (it will be one-dimensional, discrete or continuous, and cyclic),

(b) the masses distribution; they can be random or not, of discrete nature or not,

(c) the arrival positions of the masses: they may be of discrete or continuous nature, but they will be random, and uniform on a well-chosen space,

(d) and the “dispersion” algorithm, which will be the dynamics, random or not, for a mass $m_i$ arriving at position $u_i$, according to which it will cover a part of the space with size $m_i$, and remain there eventually.

This paper is devoted to obtaining exact and asymptotic results regarding the distribution of the occupied and free components after dispersion of several masses. Among results, we present universality results concerning the distribution of the size of occupied (resp. free) components that are valid under very weak assumptions about the dispersion/covering policies. Roughly speaking, the distribution of the occupied connected component depends on the masses $m_0, \cdots, m_{k-1}$, while the distribution of the sizes of the free connected component depends only on $(k, \sum_{i=0}^{k-1} m_i)$. These distributions are independent of the chosen dispersion policy.

The next section provides an overview of the results obtained in the paper.

<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of a Continuous dispersion model. On the left, masses $m_0, m_1$ and $m_2$ arrived respectively at $u_0, u_1$ and $u_2$ and have been spread continuously. Mass $m_1$ has been spread on the right, while a proportion 2/3 of mass $m_0$ has been spread on the right (and the rest was spread on the left). A drop of mass $m_3$ arrives at $u_3$. On the right, mass $m_3$ has been spread. The component that contains $u_0$, $u_2$ and $u_3$ thus has size $m_0 + m_2 + m_3$. After the 4th mass has been spread, there are two intervals in $O^{(4)}$ and $F^{(4)}$.</figcaption></figure>
## 1.1 Content of the paper and organization

In Section 1.4 we define precisely the class of valid continuous dispersion models (CDM). The term “continuous” refers to the space which is $R/Z$, while the masses can be discrete or continuous, and random or not. A model will be said to be valid if it satisfies several conditions, including being invariant under rotation, and the fact that during the relaxation phase, the occupied CC which contains the point where the last mass arrival occurs, evolves independently of the rest of the configuration (except at the collision times with other occupied components, when it merges with them).

The main quantities of interest, which are the occupied space $O^{(k)}$ after dispersion of $k$ masses, and the free space $F^{(k)}$ are introduced. Each of them are union of intervals.

A list of examples of valid CDM is provided in Section 1.5. The “right diffusion at constant speed” is an important model that will be referred to throughout the paper. In words, when a mass $m$ arrives at $u$, it is pushed to the right, and is eroded while passing through a free space at constant speed (but not eroded in occupied space). Once the amount of free space $f$ on which it has been pushed reaches the Lebesgue measure $m$, no mass remains to be eroded, and $f$ becomes occupied. Other models such as the infinitesimal particle-like diffusion, the range of the Brownian path, and the short-sighted jam-spreader model, demonstrate the extent of the class of valid CDM.

In Section 1.6, we present the main universal results on CDM. These principles are easier to understand when the masses $m_0, \cdots, m_{k-1}$ are not random, but not necessarily equal. These principles will be used later for random masses.

Arguably, the main theorems of the paper show that after $k$-masses $m_0, \cdots, m_{k-1}$ have been dispersed:

* the joint law of $(O^{(k)}, F^{(k)})$ does not depend on the mass arrival order (it has same distribution for $m_{\sigma_0}, \cdots, m_{\sigma_{k-1}}$ for any permutation $\sigma$) (Lemma 1.5),

* the law of $(O^{(k)}, F^{(k)})$ is universal for all valid CDM chosen (Lemma 1.5). Hence, it suffices to analyze one of them to analyze all of them (and the right diffusion with constant speed is probably the simplest one),

* The law of the number $N_k$ of CC of $F^{(k)}$ (or of $O^{(k)}$) depends only on $k$ and on the sum of masses $\sum_{j=0}^{k-1} m_j$. Hence it does not depend on the chosen CDM, nor on the individual masses ($m_j$), nor on their order: $N_k - 1$ follows the binomial distribution with parameters $(k-1, \sum_{j=0}^{k-1} m_j)$ (Lemma 1.5). Given $N_k = n$, the joint distribution of the free interval length is also explicit: it is a Dirichlet distribution.

* The distribution of the lengths of the CC of $O^{(k)}$ still does not depend on the valid CDM considered, but depends on the masses, and the formula remains explicit (Lemma 1.8).

* The law of the process $(S(t), 0 \le t \le k)$ where $S(t)$ is the multiset of sizes of the elements of $O^{(t)}$ at time $t$, is the same for all valid CDM (Lemma 1.11).

In Section 2, we present analogous results for valid discrete dispersion model (DDM), which are diffusion models on the discrete circle $Z/nZ$. In these models, the masses are integers and arrive at integer positions (and are diffused in a discrete manner). To make the discrete case fit into the continuous settings and allow for comparisons, we prefer to work on an isomorphic version of $Z/nZ$ embedded in the initial circle $\mathcal{C} = \mathbb{R}/\mathbb{Z}$, so we work on the set $\mathcal{C}_n := \{k/n, 0 \le k \le n-1\}$ (seen as a subgroup of $\mathbb{R}/\mathbb{Z}$), and we consider that the masses arrive uniformly on $\mathcal{C}_n$. The results obtained are similar to the continuous
dispersion model cases (concerning the universality of the law of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ for all valid DDM considered, see Lemma 2.3, or the fact that we can permute the masses without changing the law of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$) and the distribution of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ can be described (see Section 1.6 for more details). The distribution of the number of CC is no longer binomial, but remains explicit and quite simple (see Section 2.2).

In Section 3, we present the first asymptotic results, when the number of masses goes to $+\infty$ (so that their sizes goes to zero, and also in the discrete settings as $n \to +\infty$). These problems present difficulties that have only been partially overcome in this paper. A few open questions remain for the interested reader.

In Section 3.2, we investigate the difference between DDM and CDM, by defining models involving masses that are multiples of $1/n$ which can be defined in both discrete and continuous settings, and for which comparison is possible. The parking case in which all masses are multiples of $1/n$ can be very precisely studied. In this case, Chassaing & Louchard [10, Theorem 1.5] have established a phase transition when $t_n(\lambda) := n - \lambda\sqrt{n}$ cars have been parked in the discrete parking (at this time, the largest CC contains a linear number of cars, and then, a macroscopic proportion in our settings). Our analysis shows that the total numbers of CC of $\mathbf{O}^{(t_n(\lambda))}$, $N_{t_n(\lambda)}^{(n)}/\sqrt{n}$ and $\mathbf{N}_{t_n(\lambda)}/\sqrt{n}$ in the discrete and continuous parking, respectively, have different limits in probability. Thus, they are very different, while large occupied CC are essentially the same in distribution (see Lemma 3.1). A generalization of this comparison is provided

<figure><img src="image_3.png" /><figcaption>Figure 2: In the standard parking problem, a car chooses a place, here 5 and parks on the first available parking slot, which is an interval of the type $[k, k+1]$, on the right of the chosen place. Here, [7,8].</figcaption></figure>

in Lemma 3.5, which treats the case of random discrete masses. In Section 3.3, more general random masses are treated (some results from the parking problem for caravans of Bertoin and Miermont [7] are recalled).

Finally, in Section 4, a rather general question is investigated for which only preliminary answers are obtained. One of the main questions of interest in some dispersion models may depend on its entire history. For example, the cost of constructing a table in the data structure called “hashing with linear probing” depends on its entire history (Knuth [22, Sec. 6.4]). This cost is defined as to the total displacement of the cars in the standard parking problem, which corresponds to the sum of $\lfloor U_i b_i \rfloor$, where the $(U_i)$ are independent and identically distributed (i.i.d.) uniform on $[0, 1]$ and $b_i$ is the size of the occupied component that received the $i$th car. In general, we may imagine that the cost corresponds to some energy dissipation of the model (or “work”, in physical terminology, to disperse matter). This cost appears to be a kind of integral of a functional of the sequence, indexed by $k$, of the size of the CC that received
the *k*th mass.

In the case of the parking construction, we provide limit results describing “all large block sizes” appearing during construction (Lemma 4.8). In Lemma 4.4, we prove that if the unitary cost of a single mass insertion into a block of size *k* satisfies certain assumptions, then, the limiting total cost can be expressed as a functional of the Brownian excursions. The general question is discussed in Section 4.3, but only preliminary results are stated.

## 1.2 Related works / Motivation

Here, we would like to discuss some related work and provide perspective on the phenomena at play in the paper, especially regarding the universality results obtained. Although diffusion processes form a large family of models encompassing many existing models, their global universality properties can be viewed as generalization, or infinitesimal generalization, of phenomena observed and utilized in various domains of combinatorics, and probability theory. A large part of combinatorics aims to understand macroscopic objects made of small components that respects certain rules. Often, the set of objects of size *n* can be equipped with a probability distribution induced by a given construction, or more simply, the uniform distribution. The same distribution can often be obtained by several different constructions. Most of the time, these identities in distribution between constructions cannot be proven directly. For example, Aldous & Broder construction of the uniform spanning tree is very different from Wilson's algorithm. They use different sources of randomness, and each of them can not be reduced to the other, even if they ultimately generate same distribution on the set of spanning trees.

In the present paper, the main reason for the universality is given in Section 1.8. Roughly speaking, it relies on the proof that infinitesimal deposition at a boundary of the relaxation interval containing the currently treated mass can be modified without distorting the configuration distribution after this infinitesimal time. Infinitesimal modifications to the dispersion policies alter the occupation seen as a time-indexed process, but do not change its distribution after *k* relaxations.

This type of “modification/displacement” of elementary constituents in a structure, while still controlling the resulting distribution, is central to many significant advances in research devoted to understanding random combinatorial models. In the rest of this section, we will give some key elements, growing in complexity, that exemplify this statement.

The most “elementary” result in this field consist in bijections made of step rearrangements.

For example, as illustrated in Figure 3, when $N = 2n$ is even, the number of meanders with $N$ steps (the set of paths starting at 0, with steps $\pm 1$ and staying non-negative) coincides with the number of size $N$ bridges (paths starting and ending at 0 with steps $\pm 1$). A bijection between these sets consists in turning over, in the bridge, the steps at which a new record minimum is reached. A more involved rearrangement allows to represents paths on a graph as pairs (SAW, EC) where SAW is a self-avoiding walk, and EC a heap of cycles supported by SAW (see Viennot [36], Zeilberger [37], Marchal [28], Marckert & Fredes [17]).

Rearranging the steps of structures formed by several paths is at the core of important results obtained during the last decades in probability theory: we can cite in that respect

- Propp and Wilson [34] sampling method of the uniform spanning tree,

- the result of Diaconis and Fulton [13] for a certain commutation property regarding the set of final position of random walks, when particles start successively from a sequence of vertices $s_1, \cdots, s_j$ (with possible multiplicity) and perform random walks stopped at a vertex visited for the first time (by one of the random walks): the final set of visited vertices has same distribution if the set $(s_1, \cdots, s_j)$ is permuted.
<figure><img src="image_4.png" /><figcaption>Figure 3: Illustration of a bijection between the set of paths with steps (+1, -1) ending at 0 with 2n steps (called bridges), and the set of meanders with same size (paths that stay non-negative all along). To convert a bridge into a meander, replace each downstep that leads to a new “left-to-right minimum” with an upstep. To convert a meander into a bridge, replace each upstep corresponding to the last passage of an edge of this level with a downstep. Do this for all $k+1$ smaller to half the final meander value.</figcaption></figure>

This result is of the prime importance in the study of internal DLA (see e.g. Lawler & al. [26]) and in the study of loop-erased random walks (see e.g. Lawler [25]).

A direct motivation of the present paper are the results obtained in Varin [35] devoted to studying of the golf process (defined by Fredes & Marckert [17] as a tool in a new proof of Aldous–Broder theorem). The golf process is defined on a graph in which some vertices initially contain a ball, some vertices contain a hole, and some vertices are neutral. Then, each ball moves according to a random walk and stops at the first hole it encounters. This hole then becomes a neutral vertex for subsequent balls, .

As for Diaconis and Fulton commutation property, the order in which the balls are activated does not affect the distribution of the remaining holes at the end of the process. Even better, on $Z/nZ$, if $b$ balls and $h$ holes are placed (according to a uniform distribution on the set of possible choices), then the final set of free holes $H$ is the same for a large class of ball displacement policies that largely encompass random walks model. As explained in [35] (and a similar remark has been done by Nadeau [31]), the distribution of $H^1$ is the same for all displacement policies under which a ball at a given position determines its trajectory distribution, using (at most) the data of its current position in the interval formed by the first hole to its left and right. This includes random walks that do not use this information, and many policies unrelated to random walks.

Hence, on $Z/nZ$, this universal distribution property is stronger than the commutation property of Diaconis and Fulton. This result suggested that a strong property of exchangeability should hold for general diffusion policies, including continuous ones, on spaces possessing some additional symmetries. This leads us to the present work.

In the standard parking model defined on $Z/nZ$, the cars move in the positive direction (to the right). The occupation distribution, when $k$ cars have been parked is known (see e.g. [24, 33, 10, 6]), and is invariant by the space symmetry $S: i \to n-i$. This means that if, after a car arrival, we perform the sym-
metry S with probability p, then the distribution of the parking is unchanged<sup>12</sup>. However, as a process indexed by k, this is not longer true, since in the standard parking model, large occupied blocks essentially grow from the right. For example, it implies that if the cars are directed either to the right, or to the left, with probabilities p and 1-p, then, finally, the occupation statistics is the same for each value of p at any fixed time k (this is false for the temporal process). As we will prove, the fact is that even if several cars are allowed to arrive simultaneously (say $n_i$ cars arrive at position $u_i$, uniform on $Z/nZ$, at time i), even if their policies may be complex and depend on the previous cars' choices, as long as during the evolution of the car positions, each displacement depends at most on their joint positions on the current CC of parked cars that contains them (or equivalently, to the distance to the interval formed by the first free place on the left and on the right), then, the distribution of the geometry of the final occupied places depends only on the vector ($n_1, \cdots, n_k$), and not on the particular dispersion policy considered (and it is the same for all permutation of the $n_i$). This remark is important because the asymptotic behavior of “these general parking” (that will be considered as discrete dispersion models here) coincides with parking in which the cars (or caravans) are right directed. Results for these cases are already known (Chassaing & Louchard [10], Bertoin & Miermont [7]).

The evolution of occupied blocks consists of the creation of CC (when a mass arrives in a free CC), the growth of blocks (during the mass dispersion), and the coalescence of existing blocks. For this reason, and because the (discrete) additive coalescence process is related to parking models ([10, 32, 9], coalescence considerations and representation will be important throughout the paper.

## 1.3 The nature of general dispersion models, and restriction to “valid ones”

It is difficult to define a general framework for dispersion models because they take very different forms in nature and in combinatorial or technical applications. The matter arriving on the medium can be discrete or continuous. The medium itself can be homogeneous or heterogeneous. The matter can be discrete or continuous and can flow continuously on the medium. Dispersion can be continuous over time. After deposition and stabilization, the layer of deposited matter can be non-constant and non-homogeneous, as with sand dunes.

To account for the wide range of mass arrival models, we could use a process ($A_t, t \ge 0$), where for each $t$, $A_t$ is a random Borelian measure on $\mathcal{C}$, and for a Borelian $b$, $A_t(b)$ would be the total mass that has arrived in $b$ before time $t$.

The state of the system at time $t$, could then be described by a second measure $L_t$ which would give the level of matter at time $t$. Again, for a Borelian $b$, $L_t(b)$ would be the quantity of matters, deposited on $b$, at time $t$. However, this would not be sufficient in many cases, since the simplest mechanical model would require additional data, such as local speed and forces.

<footnote n="1"><sup>1</sup>The reasoning can be extended: if after each unit displacement of a car, the symmetry is made with probability p, or even with a probability depending on this k previous moves and previous symmetries done, then finally, as long as the policy chosen ensures that the car a.s. parks, the distribution of the parking occupation after the parking of the kth car has same distribution as in the standard parking model. To prove that, by recurrence, it suffices to start with the parking already occupied by k-1 cars, placed using the standard model). When the kth car moves according to the any local policy which is independent from the parking outside the current occupied block it is moving in, it induces a distribution for the k-cars parking occupation which is a mixture on “standard right directed parking” and “standard left directed parking”, the two being equal at time k.</footnote>

<footnote n="2"><sup>2</sup>On a segment too, if the cars decide to “go to the right” with probability p and to the left with probability 1-p, independently, as explained by Durmić & al. [14], the probability for n-1 cars to park on the segment (that is, no car gets out of it), is independent of p. As we will discuss here, these phenomena are much more general.</footnote>
In some cases, the physical or technical phenomenon governing the relaxation phase can be well described by partial differential equations acting on a function $f$, where $f(t, x)$ represents the amount of matter at $x$ at time $t$. However, in other cases, a stochastic process is needed to account for the stochastic nature of the studied problem, and perhaps some control is necessary in other cases. In any case, the variety of imaginable models is vast. In many cases, an important enrichment of the probability space would be needed in order to accommodate the random physical phenomenon under consideration.

However, **the main conceptual result** of this paper is as follows. There is a huge class of dispersion models, that we will call **Valid continuous dispersion models**, that have the following characteristics:

(a) they are “discrete time processes” – the time arrivals of masses occur each slot of time,

(b) they produce a fixed “level of deposition”, in words, an abscissa $x$ in the medium is either free or occupied/covered (and once occupied it stays so forever),

(c) the masses are treated sequentially, in the sense that stabilization/relaxation of mass $m_k$ is done before the arrival of mass $m_{k+1}$,

(d) during the dispersion of the $k+1$th mass $m_k$ that arrives at $u_k$, the process ($\text{IR}_k(t)$, $k \le t \le k+1$) that describes the connected occupied component containing $u_k$ at time $t$ grows continuously (except when it merges with an already existing occupied CC of $O^{(k)}$ it hits) and the evolution of $\text{IR}_k$ on $[t, t+dt]$ is independent of the complement of $\text{IR}_k(t)$. In other words, their exists a “continuous time” Interval Relaxation process $\text{IR}_k$ for $t \in [k, k+1]$ which describes the evolution of this CC, such that $\text{Leb}(\text{IR}_k(t) \setminus O^{(k)})$ is continuous, and starts at zero at time $t=k$, grows at speed 1 until time $k+m_k$, and then stays unchanged until time $k+1$.

(e) the model is invariant under rotation (in particular, the arrival positions $u_k$ are uniform), and all these models, after stabilization of $(m_0, \cdots, m_{k-1})$, induce the same (tractable) distribution on the finally occupied domain $O^{(k)}$.

These properties, that are needed for the universal results described above, **are not properties of the complete dispersion models (which as already said, maybe, demand a much richer probability space), but only induced properties on occupied CC processes.**

Moreover in the analysis (a), (b), (d) and (e) are crucial, while (c) is not, but simplifies greatly the exposition, and allows us to define analyzable “cost functions” associated with the dispersion process.

In Section 1.4, we provide formal definitions. In Section 1.5, we present a list of examples (that can be understood without reading Section 1.4).

## 1.4 Valid continuous dispersion models (valid CDM) : formal addendum

Here, we provide more precise definitions of the conditions (a) – (e) that were discussed informally in the previous subsection.

**Notation.** For any $a, b \in \mathbb{R}/\mathbb{Z}$ denote by $\bar{a}$ and $\bar{b}$ the representatives of $a$ and $b$ in $[0, 1)$, respectively. Also, denote the distance from $a$ to $b$, in the canonical positive direction by $\vec{d}(a, b) = \inf\{x \ge 0 : \bar{a} + x - \bar{b} = 0 \text{ mod } 1\}$.

We denote by $[a, b]_-$ the circular interval: for $a, b \in \mathbb{R}/\mathbb{Z}$, $[a, b]_- = \{\bar{a} + \vec{x}, x \in [0, \vec{d}(a, b)]\}$.
Following the content of Section 1.3, we assume that “the physical dispersion phenomenon” takes place on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$,

■ Continuous space.

The medium on which the dispersion takes place is the (continuous) unit circle $\mathcal{C} := \mathbb{R}/\mathbb{Z}$.

■ Model of masses.

The sequence of masses is $(m_i, 0 \le i \le n)$ for a finite $n$ (in the paper). The masses are random or not, each mass is non-negative, and the total mass $\sum_{i=0}^n m_i \le 1$; observe that the mass $m = 0$ is permitted.

■ Uniform arrival positions, and independence between masses and positions.

The mass $m_i$ arrives at position $u_i$: the $(u_i, i \ge 0)$ are i.i.d., independent of the masses $(m_i, i \ge 0)$, and taken according to Uniform($\mathcal{C}$).

■ Valid Continuous Dispersion Models.

A valid dispersion model successively treats a sequence of mass arrival events $(m_i, u_i)$ where $i$ goes from 0 to $n-1$, and $u_i$ is the place at which the mass $m_i$ arrives. The set $\mathbf{O}^{(j)}$ denotes the occupied set after the first $j$ masses $m_0, \dots, m_{j-1}$ have been dispersed. The free space is denoted $\mathbf{F}^{(j)} = \mathcal{C} \setminus \mathbf{O}^{(j)}$.

At time 0, the occupied set $\mathbf{O}^{(0)}$ is the empty set. We will add hypotheses to the diffusion process so that – among other things – the set $\mathbf{O}^{(k)}$ is a finite set of closed intervals (its CC), and the sequence $(\mathbf{O}^{(k)})$ satisfies

$$ \mathbf{O}^{(k)} \supseteq \mathbf{O}^{(k-1)}, \quad \text{for all } k \ge 1. $$

The event $\mathbf{O}^{(k)} = \mathbf{O}^{(k-1)}$ is possible only when a zero mass $m_{k-1} = 0$ arrives in $\mathbf{O}^{(k-1)}$, in an already occupied CC. Moreover, the Lebesgue measure of occupied space at time $k$ corresponds to the total mass that has arrived before time $k$:

$$ \mathrm{Leb} \left( \mathbf{O}^{(k)} \right) = \sum_{i=0}^{k-1} m_i. \qquad (1.1) $$

As explained in the previous section, we will not define a “general dispersion model”, but rather, we will fix some of the properties that it must satisfy, regarding the expansion of the occupied CC that receives the new mass $(m_k, u_k)$, or, in other words, regarding the interval relaxation processes $(\mathrm{IR}_k)$ that it induces.

Assume that $\mathbf{O}^{(k)}$ has been constructed (using the $k$ elements $(u_0, m_0), \dots, (u_{k-1}, m_{k-1})$) and has the aforementioned properties. The hypotheses we need to specify concern the process $(\mathrm{IR}_k(t), k \le t \le k+1)$, which describes the evolution of the CC receiving the new mass $m_k$ at $u_k$, from the time $k$ it receives it until time $k+1$.

To be valid, the interval relaxation process $\mathrm{IR}_k := (\mathrm{IR}_k(t), k \le t \le k+1)$:

(i) has to be well defined a.s. for all set $\mathbf{O}^{(k)}$ of disjoint closed intervals in the support of $\mathbf{O}^{(k)}$, and for all mass arrival events $(u_k, m_k)$ in the support of the considered mass arrival model.

(ii) Given $(\mathbf{O}^{(k)}, u_k, m_k)$, the process $\mathrm{IR}_k$ can be deterministic, or random. For each $t \in [k, k+1]$, $\mathrm{IR}_k(t) = [a(t), b(t)]$, is an interval of $\mathcal{C}$, and we demand that $\mathrm{IR}_k$ takes its values a.s. in $D([k, k+1], \mathcal{C}^2)$ (the space of càdlàg processes indexed by $[k, k+1]$ with values in $\mathcal{C}^2$, equipped with the Skorokhod topology), where we identified the set of directed intervals on $\mathcal{C}$ with $\mathcal{C}^2$.

(iii) Define $\mathrm{IR}_k(k)$ as the CC of $\mathbf{O}^{(k)}$ containing $u_k$, and if there are no such component, set $\mathrm{IR}_k(k) = \{u_k\}$. We require that during the relaxation time $[k, k+m_k]$, $\mathrm{Leb}(\mathrm{IR}_k(t))$ grows outside $\mathbf{O}^{(k)}$, at unit speed, that is

$$ \mathrm{Leb} \left( \mathrm{IR}_k(k+t) \setminus \mathbf{O}^{(k)} \right) = t, \text{ for } t \in [0, m_k] \qquad (1.2) $$
and can jump (that is $\mathbb{R}_k(k+t) \neq \mathbb{R}_k(k+t-)$) only when the interval $\mathbb{R}_k(k+t)$ coalesces with one of the intervals present in $\mathbf{O}^{(k)}$ (contrary to appearances, this condition is not really a condition on the speed of dispersion! See Lemma 1.2).

(iv) For all $s \le k + m_k$, $(\mathbb{R}_k(t), t \in [k, s))$ is independent from $\mathbf{O}^{(k)} \setminus \mathbb{R}_k(s)$: in words, the evolution of the CC of $\mathbb{R}_k$ containing $u_k$ between time $t$ and $t+dt$ must be independent of the rest of the state of the medium, that is, from $\mathbf{O}^{(k)} \setminus \mathbb{R}_k(t)$. From time $k+m_k$ to time $k+1$, $\mathbb{R}_k(t)$ remains constant, and equal to $\mathbb{R}_k(k+m_k)$.

(v) The model is invariant under rotation, meaning that the (distribution of the) diffusion processes $\mathbb{R}_k(t)$ defined from time $k$ to time $k+m_k$ does not depend on its position on the circle. First, $u_k$ is uniform (and independent from the other random variables). Denotes by $r_\alpha$ the rotation by $\alpha \in \mathbb{R}/\mathbb{Z}$ in $\mathbb{R}/\mathbb{Z}$. For all $k$, the diffusion process $\mathbb{R}_k(t)$ defined on the time interval $[k, k+m_k]$ has a distribution which depends uniquely on $(\mathbf{O}^{(k)}, u_k, m_k)$. For all $\alpha \in \mathbb{R}/\mathbb{Z}$,

$$ \mathcal{L}((\mathbb{R}_k) | (\mathbf{O}^{(k)}, u_k, m_k)) = \mathcal{L}(r_{-\alpha}(\mathbb{R}_k) | (r_\alpha(\mathbf{O}^{(k)}), r_\alpha(u_k), m_k)). \quad (1.3) $$

We define $\mathbf{O}^{(k+1)}$ as the closed set $\mathbb{R}_k(k+1) \cup \mathbf{O}^{(k)}$ (by construction $\mathbb{R}_k(k+1)$ may contain zero, one or several CC of $\mathbf{O}^{(k)}$).

**Definition 1.1.** A continuous dispersion model (CDM) is said to be valid, if all the interval relaxation processes $(\mathbb{R}_k)$ are valid.

**Remark 1.2** (About the relaxation time of the $k$th mass). A good way to understand the extent of the class of valid continuous processes is to consider that the relaxation phase of the $k$th mass occurs within a time frame which is independent of the discrete times enumerating the mass arrival events $(m_k, u_k)$.

When one wants to encode an actual physical system (e.g. the diffusion of a droplet in a random medium), it is often natural to encode the relaxation time of the $k$th mass by a continuous time interval:

(a) for example, on $[k, k+1)$ (so that, everything is done for the arrival of the next $k+1$th mass,

(b) or on $[0, +\infty)$ to let as many time as needed for the dispersion time,

(c) or on $[k, k+m_k]$, and then assume that the mass deposition is done at unit speed.

Since we observe the states after the relaxation times (to define $\mathbf{O}^{(k)}, m_k, u_k...$), and since one can pass from any of the models (a), ..., (c) to the others by a change of time, these points of view are equivalent for our purposes. We then choose the model (c) to make easier the description of everything on the same clock.

When needed, we will refer to this property as “the constant-speed deposition property”.

**Remark 1.3** (About the invariance under rotation). Without the invariance under rotation hypothesis (v), the diffusion process could depend on the arrival points $u_i$. We could design some diffusion processes that would stop their diffusion at some special places (e.g. so that to produce occupied sets $\mathbf{O}^{(k)}$ such that each CC has a (random) extremity with a rational coordinate on $\mathcal{C}$). All universal results presented in Section 1.6 would not hold without condition (v).

Figure 1 illustrates the definition of the continuous dispersion model.

## 1.5 List of examples of valid continuous dispersion processes

In this section we provide a list of 8 valid continuous dispersion models.
### 1.5.1 Right Diffusion at Constant Speed (RDCS): an important cornerstone.

The Right Diffusion at Constant Speed is an important diffusion process that will serve as a cornerstone for the rest of the paper. It is defined for any masses ($m_0, \dots, m_k$) as long as their sum is in $[0, 1)$. We exclude the trivial case $\sum m_i = 1$, in which all the space is occupied, to avoid treating this special case which would add unnecessary complications.

Assume that there is a mass arrival $(m, u)$ at time $k$, on a configuration characterized by $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$. Initially, $\text{IR}_k(k) = [a, b]_{-}$ is the occupied CC of $\mathbf{O}^{(k)}$ receiving $u$ if it exists (it may be created if $u$ was not in $\mathbf{O}^{(k)}$, in which case $\text{IR}_k(k) = [a, b] := [u, u]$). Now, the mass $(m, u)$ is pushed to the right on $\mathbb{R}/\mathbb{Z}$ from $u$ and covers the space from there. At time $t \in [k, k+m_k]$, the CC is the smallest oriented interval $\text{IR}(t) = [a, b(t)]_{-}$ on $\mathbb{R}/\mathbb{Z}$ (that grows on its right extremity) and satisfies (1.2) (that is $\text{Leb}(\text{IR}_k(k+t)\mathbf{O}^{(k)}) = t$, for $t \in [0, m_k]$). See Figure 4 for an illustration.

<figure><img src="image_5.png" /><figcaption>Figure 4: The cycle $\mathbb{R}/\mathbb{Z}$ is represented as the segment $[0, 1)$. In the RDCS, the masses are pushed to the right. In the first column, two mass arrival events and their relaxations. A new mass is pushed to the right and covers the medium at the same speed as if it was eroded. The third picture shows that the erosion occurs only in free spaces, while the mass is pushed with no loss in occupied domains.</figcaption></figure>

The second column illustrates a phenomenon that will be proven later: the final occupied domain does not depend on the order of the sequence $(u_i, m_i)$. This is clear in the RDCS model, because, the quantity of “non-eroded mass” that passes above a given abscissa $x$ does not depend on “the identity of the mass” which is eroded at a given location.

**Remark 1.4.** The representation in Figure 4 appears in many papers, typically to encode paths with a deterministic slope -1. The randomness appears as positive jumps (notably in queuing theory with one server. Łukasiewicz paths are used to encode the remaining charge as a function of time. Customers arrive at some random times, and demand a random quantity of service. The server delivers service at a constant rate, so that the total amount of service remaining to be done at time $t$, seen as a process of $t$, can be represented by a path with a slope -1 and vertical jumps at the arrival times of customers. The value of each jump corresponds to the customer's demand. In Bertoin & Miermont [7] (see Section 3.3), the RDCS is called the “model of caravan”: “masses are drops of paint brushed to the right”. There is however a subtle difference: they encode the covered space $\mathbf{O}^{(k)}$ as a union of open sets, while we take them as closed. It allows us to consider covered set reduced to a point, which will be shown to significantly simplify our analysis.
Of course the Left Diffusion at Constant Speed (LDCS) can be defined analogously. It will be only used in the next three examples.

### 1.5.2 $(p, 1-p)$ proportion of the mass is diffused to the right/to the left

When a mass arrival event $(m, u)$ occurs, first, proceed to the RDCS of $(pm, u)$ (a proportion $p$ of the mass is diffused to the right), and then proceed to the LDCS of $((1-p)m, u)$. If we proceed to the RDCS and LDCS one after the other, the total time of diffusion is $m$. If we perform both tasks simultaneously, the final result is the same but occurs earlier, at time $\max\{p, 1-p\}m$. This is another instance in which we can observe that many details of the models are irrelevant when we are only interested in the final configuration.

### 1.5.3 With probability $p$ perform a right diffusion, with probability $1-p$ a left one

When a mass arrival event $(m, u)$ occurs, toss a Bernoulli($p$) coin. If the result is 1, then proceed to the RDCS of $(m, u)$; otherwise proceed to the LDCS of $(m, u)$.

### 1.5.4 Diffusion to the closest side

When a mass arrival event $(m, u)$ occurs: If $u$ does not belong to the current occupied domain $O^{(k)}$ then do a (1/2, 1/2) proportion to the right/left diffusion. Otherwise, $u \in [a, b]$ with $[a, b]$ a CC of $O^{(k)}$. Compare the distance $\vec{d}(a, u)$ with $\vec{d}(u, b)$. If the former is smaller then do an LDCS of $(m, u)$, else an RDCS.

### 1.5.5 Diffusion to the closest side with constant reevaluation

Do the same thing as in “diffusion to the closest side”, but after each infinitesimal time $dt$, recompute the distance to the sides of the relaxation interval. When the sides are at an equal distance, grow both sides at the same speed.

### 1.5.6 Infinitesimal particle-like diffusion

This model describes the asymptotic diffusion behavior of masses $m_i$ which are composed of “infinitesimal” particles of size $1/M$ (and the limit regime is taken for $M \to +\infty$), where these small particles perform independent random walks successively until the moment they exit the occupied interval. In doing so, they increase the size of the occupied domain by $1/M$ for subsequent infinitesimal particles composing $m_i$.

In order to describe these dynamics let us assume that a mass $m$ arrives at 0, in a block $[-a, b]_{-}$ (with $a, b > 0$). As usual, if the arrival location 0 is in the free space, then $[a(0), b(0)]_{-} = \{0\}$. Since the dynamics are invariant under rotation, it suffices to provide a full description of this case (as we will see, we specify the speeds at which $a$ and $b$ move, so that what we will say remains valid when CC merge: it suffices to modify $a$ and $b$ accordingly). For $B$, a Brownian motion starting at 0,

$$ \mathbb{P}(\inf\{t: B(t) = -a\} < \inf\{t: B(t) = b\}) = b/(a+b). $$
At the limit over $M$, the speed at which a boundary moves satisfies a simple differential equation which is the immediate limit of the preceding considerations; to understand how the interval changes over time, let us work on $\mathbb{R}$ for a moment (the transfer to $\mathbb{R}/\mathbb{Z}$ is just a formality).

If $(m, 0)$ is a mass arrival event, and $|\mathbb{R}_k(k+t)| = [-a(t), b(t)]$ is the occupied CC (with $-a[t] < 0 < b[t]$), then the relative side speeds should be proportional to $b(t)$ and $a(t)$, respectively<fnref n="3" /> and then, we get

$$b(t) = \frac{t + a(0) + b(0)}{2} + \frac{b(0)^2 - a(0)^2}{2(a(0) + b(0) + t)}, \quad a(t) = \frac{t + a(0) + b(0)}{2} + \frac{a(0)^2 - b(0)^2}{2(a(0) + b(0) + t)}. \quad (1.4)$$

Of course, upon collision with another occupied block, $|\mathbb{R}_k(t)|$ jumps, as usual.

We justify the approximation of the interval evolution by this fluid limit in Section 5.1.

### 1.5.7 Range of the Brownian path

Let us provide an example given by a random process (see Figure 5).

Let $R(u, t) := \{B_s^{(u)} \mod 1, 0 \le s \le t\}$ denote the range of a Brownian motion $B^{(u)}$ on $\mathbb{R}/\mathbb{Z}$, starting at $u$. Of course $t \mapsto R(u, t)$ is non-decreasing with respect to the inclusion partial order.

To define $\mathbf{O}^{(k+1)}$ in terms of $\mathbf{O}^{(k)}$, start a Brownian motion at $u_k$, and stop it at the (first random) time $\tau$ for which the newly visited space has the right size, that is, satisfies $\text{Leb}(R(u_k, \tau) \setminus \mathbf{O}^{(k)}) = m_k$. Then, set $\mathbf{O}^{(k+1)} = \mathbf{O}^{(k)} \cup R(u_k, \tau)$ as usual.

<figure><img src="image_6.png" /><figcaption>Figure 5: The range of a Brownian motion model: the first Brownian motion $B^{(u_0)}$ starts at $u_0$ and is killed at the first $t$ for which $\text{Leb}(\{B_s^{(u_0)}, s \le t\}) = m_0$. Its track is $\mathbf{O}^{(1)} = \{B_s^{(u_0)}, s \le t\}$. The Brownian motion $B^{(u_i)}$ starts at $u_i$ and is killed when $\text{Leb}(\{B_s^{(u_i)}, s \le t\} \setminus \mathbf{O}^{(i-1)}) = m_i$, that is when the set of points visited by $B^{(u_i)}$, but that have never been visited by the previous Brownian motions, has Lebesgue measure $m_i$.</figcaption></figure>

This is a case where the relaxation interval process (which grows at constant speed) can be described explicitly. For any $x \in [0, m_k]$, consider $\tau(x) = \inf\{s \ge 0 : \text{Leb}(R(u_k, s) \setminus \mathbf{O}^{(k)}) \ge x\}$. For all $0 \le x \le m_k$, define $|\mathbb{R}(k+x)|$ as the CC of $\mathbf{O}^{(k)} \cup R(u_k, \tau(x))$ containing $u_k$ (and let $|\mathbb{R}_k|$ be constant, as usual, between time $k+m_k$ and time $k+1$).

$$b'(t) = \frac{a(t)}{a(t) + b(t)}, \quad a'(t) = \frac{b(t)}{a(t) + b(t)},$$

<footnote n="3">³that is for a function $C(t)$ we should have $b'(t) = C(t)a(t)$ and $a'(t) = C(t)b(t)$, and since we opted for the unit-speed deposition, we choose $C(t)$ so that $a'(t) + b'(t) = 1$. We then have</footnote>
### 1.5.8 The short-sighted jam-spreader model

A short-sighted individual wants to spread some drops of jam that falls on a donut (identified with $\mathcal{C}$) according to a classical mass arrival events $((m_k, u_k), k \ge 0)$. During the spreading of drop $(m_k, u_k)$, the spreader cannot see where the jam is lacking on $\mathcal{C}$. He therefore acts randomly, progressively transforming the pile of non-deposited jam, that we could represent by a Borelian measure $(M_{k+t}, 0 \le t \le 1)$ on $\mathbb{R}_k(k+t)$ whose total mass at time $k+t$ is the mass $m_k - t$ of undeposited jam. We may assume that the spreader proceeds randomly, and that his actions continuously expand $\mathbb{R}_k(k+t)$, until all the jam that has arrived is finally spread, and a new portion of $\mathcal{C}$ with Lebesgue measure equal to $m_k$ is covered.

We assume that any individual's "strategy" is invariant under rotation, and that the $k$th policy he adopts for the drop $(u_k, m_k)$ is independent of the current sets $\mathbf{O}^{(k)}$ and $\mathbf{F}^{(k)}$, and depends only on the occupied block in which this drop falls (and possibly on the current measure $M_{k+t}$ (encoding the deposition process of $m_k$) he is spreading). The policies used for different $i$ may differ.

## 1.6 Main universality results for valid CDM: deterministic masses case

This section is devoted to the study of the statistical properties of a valid continuous dispersion model, when the sequence of masses $m[k] := (m_i, i = 0, \dots, k-1)$ that arrive in the system is formed by deterministic, non-negative masses, and satisfies only the total weight condition

$$W(m[k]) := \sum_{i=0}^{k-1} m_i < 1. \qquad (1.5)$$

We will turn to random masses in Section 3.1.

We insist on the fact that we allow the $m_i$ to be 0 too. Again, a zero mass arriving in an occupied CC has no effect. However, when it arrives in a free CC, it creates a new occupied CC reduced to the point $\{u\}$ at which the arrival took place.

The first aim of the paper is to describe the distribution of the occupied and free spaces $\mathbf{O}^{(k)}$ and $\mathbf{F}^{(k)}$ at time $k$ when $k$ masses $((m_i, u_i), 0 \le i \le k-1)$ have been dispersed according to a valid continuous dispersion model. In particular, we put in bold $u_i$ to distinguish its "random variable" type from that of the $(m_i, i \ge 0)$ simple non-negative real numbers, which are not random for the moment.

We let

$$\mathbf{N}_k = \#\mathbf{F}^{(k)} = \#\mathbf{O}^{(k)} + 1, \quad k=0, \qquad (1.6)$$

be the number of free CC when $k$ masses have been dispersed in the system. For $k \ge 1$, label the CC $\mathbf{O}_0^{(k)}, \dots, \mathbf{O}_{N_k-1}^{(k)}$ of $\mathbf{O}^{(k)}$, and $\mathbf{F}_0^{(k)}, \dots, \mathbf{F}_{N_k-1}^{(k)}$ those of $\mathbf{F}^{(k)}$, such that by turning around the circle $\mathbb{R}/\mathbb{Z}$, one finds successively (they are adjacent in this order),

$$\mathbf{L}^{(k)} := \mathbf{O}_0^{(k)}, \mathbf{F}_0^{(k)}, \dots, \mathbf{O}_{N_k-1}^{(k)}, \mathbf{F}_{N_k-1}^{(k)}, \qquad (1.7)$$

and moreover, the point

$$0 \in \mathbf{O}_0^{(k)} \cup \mathbf{F}_0^{(k)}. \qquad (1.8)$$

The point 0 of the circle is then used to determine which CC of $\mathbf{O}^{(k)}$ and $\mathbf{F}^{(k)}$ will be named $\mathbf{O}_0^{(k)}$ and $\mathbf{F}_0^{(k)}$, respectively: this introduces a bias which complicates the exposition slightly. The sequence of block lengths

$$|\mathbf{L}^{(k)}| := (|\mathbf{O}_0^{(k)}|, |\mathbf{F}_0^{(k)}|, \dots, |\mathbf{O}_{N_k-1}^{(k)}|, |\mathbf{F}_{N_k-1}^{(k)}|) \qquad (1.9)$$
together with the shift information

$$s_0 = -\min \mathbf{O}_0^{(k)}, \text{ and then } s_0 \in [0, |\mathbf{O}_0^{(k)} \cup \mathbf{F}_0^{(k)}|]$$

allows to reconstitute $\mathbf{L}^{(k)}$ (each of them characterizes the other one).

### 1.6.1 Universal properties

The next theorem is one of the paper's main results and gathers the properties that are universal for all valid CDM. In short, for a fixed $k$, the distribution of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ is an invariant of all valid CDM.

**Theorem 1.5.** Consider a valid CDM A, and some deterministic masses $m[k] = (m_0, \dots, m_{k-1})$, which are either positive or zero and satisfy the total weight condition $W(m[k]) < 1$. Let $(\mathbb{R}_j^A, 0 \le j \le k-1)$ denote the interval relaxation processes associated with A. We have

(i) For a fixed $k$, the law of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ does not depend on A (it is the same for all valid CDM that are designed to treat the masses $(m_0, \dots, m_{k-1})$.

(ii) The law of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ is invariant under the permutation of the masses: For any permutation $r \in \mathbf{S}_k$ (the symmetric group on $\{0, \dots, k-1\}$),

$$\mathcal{L}((\mathbf{O}^{(k)}, \mathbf{F}^{(k)}) | (m_0, \dots, m_{k-1})) = \mathcal{L}((\mathbf{O}^{(k)}, \mathbf{F}^{(k)}) | (m_{r(0)}, \dots, m_{r(k-1)}))$$

(where we have written, as a condition, the successive masses that are used).

(iii) For $k \ge 1$, the number of free CC, $\mathbf{N}_k$, has (almost) a binomial distribution:

$$\mathbf{N}_k \stackrel{(d)}{=} 1 + B(m[k]), \text{ where } B(m[k]) \sim \text{Binomial}(k-1, R_k) \qquad (1.10)$$

where $R_k = 1 - W(m[k])$ is the final free space available after the deposition of the first $k$ masses.

(iv) Take a uniform element $\theta$ in $\{0, \dots, b-1\}$ (independently of everything). Conditional on $\mathbf{N}_k = b$,

$$\left( \left| \mathbf{F}_{i+\theta \bmod b}^{(k)} \right|, 0 \le i \le b-1 \right) \stackrel{(d)}{=} R_k (\mathbf{D}_0, \dots, \mathbf{D}_{b-1}) \qquad (1.11)$$

where $(\mathbf{D}_0, \dots, \mathbf{D}_{b-1}) \sim \text{Dirichlet}(b; 1, \dots, 1)$. Hence, the sequence $(|\mathbf{F}_i^{(k)}|, 0 \le i \le b-1)$ taken according to a uniform independent rotation, is unbiased.

(v) The process $(\mathbf{N}_k, k \ge 0)$ is a Markov chain, and its transition matrix can be computed explicitly (see Section 1.9).

Removing the rotation in (iv) comes at a cost, due to the bias concerning the block containing 0 (see Lemma 1.10). The free set has an extra universal property that we prefer to state separately for a technical reason, appearing already in Theorem 1.5 (i).

Let $\mathbf{F}^{(k)}$ denote the free spaces random variable under the right diffusion with constant speed model (RDCS).
**Theorem 1.6.** Assuming the hypothesis of Lemma 1.5, for a fixed $k \ge 1$,

$$ \mathcal{L}(|\mathbf{F}^{(k)}| | (m_0, \dots, m_{k-1})) = \mathcal{L}\left(\left|\overrightarrow{\mathbf{F}}^{(k)}\right| | (W(m[k]), \underbrace{0, \dots, 0}_{k-1 \text{ zeroes}})\right); $$

that is, for all valid continuous dispersion models, the lengths of the free spaces are distributed as if the masses $(m_0, \dots, m_{k-1})$ were replaced by a “global mass” $W(m[k])$, and $k-1$ zero masses, and dispersed by the right diffusion with constant speed model.

Hence, even if a CDM is not made to deal with all masses, the distribution of the free spaces is the same as those for the RDCS, for which the masses $m_i$ can be permuted, or even rearranged the way we want as long as we keep their number and sum. This theorem is a Corollary of Lemma 1.5, in which by (iii) – (v), we see that the total free space distribution depends only on $(k, \sum_{i=0}^{k-1} m_i)$.

### 1.6.2 On the occupied blocks distribution

According to Theorem 1.5, to understand the distribution of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ at time $k$ for any given CDM, it suffices to describe the distribution of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ for the RDCS, also at time $k$. As we will see the distribution of $\mathbf{O}^{(k)}$ is more complex than that of $\mathbf{F}^{(k)}$.

A technical tool comes into play.

**Definition 1.7.** For some masses $m[k] := (m_0, \dots, m_{k-1})$ with sum in $[0, 1)$, the piling propensity$^4$ of $m[k]$ is defined by

$$ Q(m[k]) := W(m[k])^{k-1}. \qquad (1.12) $$

By Theorem 1.5,

$$ Q(m[k]) = \mathbb{P}(\mathbf{N}_k = 1), \qquad (1.13) $$

so that $Q(m[k])$ measures the propensity of $m[k]$ to form a single occupied CC, under the action of any valid CDM. The notation $Q(m[k])$ perhaps hides the fact that $Q$ depends on $k$ and of $W(m[k])$ only.

Given that $\mathbf{N}_k = 1$, $\mathbf{O}^{(k)}$ is reduced to a single (random) interval $[A, A + W[m[k]]]$, and by invariance under rotation $A$ is uniform on $\mathbb{R}/\mathbb{Z}$:

$$ \mathbb{P}(A \in da | \mathbf{N}_k = 1) = da $$

so that

$$ \mathbb{P}(A \in da, \mathbf{N}_k = 1) = W(m[k])^{k-1} da. \qquad (1.14) $$

This single-block consideration accounts for the fact that all $m_i$ fall within $[a, a + W(m[k])]$, with one of the $m_i$ arriving at $da$, and the others combining to cover exactly $[a, a + W(m[k])]$. This mundane observation allows to produce a multi-block formula at the double cost of fixing the identity of the masses participating to each block, and placing the blocks precisely on the circle so that they do not intersect (this suffices, since for any valid CDM the dispersion policy of a mass is independent of what is outside of the connected component containing that mass).

For a set of indices $J$, set

$$ W(m(J)) := \sum_{j \in J} m_j. $$

<footnote>$^4$This piling propensity is “close to” the probability that all the masses arrive within an interval of size $W(m[k])$ which is $W(m[k])^k$.</footnote>
Let $\mathbf{I}_j^{(k)}$ denote the indices of the masses among $m[k]$ that have been dispersed to form the occupied CC
$\mathbf{O}_j^{(k)} = [\mathbf{A}_j^{(k)}, \mathbf{B}_j^{(k)}]_-$ . We have, as a consequence of the single block formula:

**Theorem 1.8.** Assume the same hypothesis as in Lemma 1.5. For any $b \in \{1, \dots, k\}$, any partition $(J_0, \dots, J_{b-1})$ of the set $\{0, \dots, k-1\}$ with non-empty parts, any sequence $(a_0, \dots, a_{b-1})$ in $\mathscr{C}^b$ such that the $a_0, \dots, a_{b-1}$ are cyclically ordered around $\mathscr{C}$ (i.e. such that turning around the circle we get $a_0 \le 0 \le a_1 \le \dots \le a_{b-1} \le a_0 \le 0 \dots$) and such that

$$
\vec{d}(a_i, a_{i+1 \bmod b}) > W(m(J_i)) \quad \text{for } i \in \{0, \dots, b-1\},
$$

we have

$$
\mathbb{P} \left( \mathbf{A}_j^{(k)} \in \mathrm{d}a_j, \mathbf{I}_j^{(k)} = J_j, 0 \le j \le b-1 \right) = \prod_{j=0}^{b-1} Q(m(J_j)) \mathrm{d}a_j. \quad (1.15)
$$

• The condition $\vec{d}(a_i, a_{i+1 \bmod b}) > W(m(J_i))$ is needed because the interval $\mathbf{O}_i^{(k)}$ has length $W(m(J_i))$, starts at $a_i$ and must end before the start of the next occupied CC,

• In this formula, a bias appears under the condition $a_0 \le 0 \le a_1$.

• Notice that the event $\{\mathbf{A}_j^{(k)} \in \mathrm{d}a_j, \mathbf{I}_j^{(k)} = J_j, 0 \le j \le b-1\}$ characterizes entirely $\mathbf{O}^{(k)}$ as well as $\mathbf{F}^{(k)}$.
To extract the occupied block sizes, we need to “count” the number of ways to produce some given block sizes.

**Theorem 1.9.** *Take the same hypothesis as Lemma 1.5. Let $M_0, \dots, M_{b-1}$ be some non-negative masses. For all $a_0 \le 0 \le a_1 \le \dots \le a_{b-1} \le a_0 \le \dots$ and such that $\mathrm{Leb}([a_i, a_{i+1 \bmod b}]_-) > M_i$ for all $i \in \{0, \dots, b-1\}$,*

$$
\mathbb{P} \left( \mathbf{A}_j^{(k)} \in \mathrm{d}a_j, \left| \mathbf{o}_j^{(k)} \right| = M_j, 0 \le j \le b-1 \right) = \sum_{(J_0, \dots, J_{b-1}) \in \mathcal{P}(k,b)} \prod_{j=0}^{b-1} Q(m(J_j)) \mathbb{1}_{W(m(J_j))=M_j} \mathrm{d}a_j \quad (1.16)
$$

where $P(k, b)$ is the set of partitions of $\{0, \dots, k-1\}$ into $b$ parts.

In some cases, Formula (1.16) can be simplified. For instance, when the cardinalities of the parts $J_j$ of the partitions $(J_0, \dots, J_{b-1})$ compatible with $(M_0, \dots, M_{b-1})$, have a fixed number of elements. For example, if all the masses $m_j$ are equal to some constant $w$, then the possible sizes $M_j$ are multiples of $w$ (there exists $k_j \in \mathbb{N}$ such that $M_j = k_j w$), and then in this special case $Q(m(J_j)) = (M_j)^{k_j-1}$ and the number of partitions $(J_0, \dots, J_{b-1})$ in non-empty parts, such that, for all $i$, $W(m(J_i)) = M_i$ is $(M_i)^{k_j-1}$ (where the total mass $\sum_{i=0}^{b-1} M_i = kw$). In this special case, we then have

$$
\mathbb{P} \left( \mathbf{A}_j^{(k)} \in \mathrm{d}a_j, \left| \mathbf{o}_j^{(k)} \right| = M_j, 0 \le j \le b-1 \right) = \left( \begin{array}{c} k \\ k_0, \dots, k_{b-1} \end{array} \right) \prod_{j=0}^{b-1} (k_j w)^{k_j-1} \mathrm{d}a_j.
$$

Again, Lemma 1.9 characterizes the distribution of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$. Computing the “marginal” $|\mathbf{O}^{(k)}|$, which records only the block length sizes is also possible. To do so, we need to compute $\mathrm{T}(M[b])$ the volume of the “translation set” $(a_0, \dots, a_{b-1})$ described in Lemma 1.9:

$$
\mathbb{P}\left(\left|\mathbf{o}_j^{(k)}\right| = M_j, 0 \le j \le b-1\right) = \mathrm{T}(M[b]) \sum_{(J_0, \dots, J_{b-1}) \in \mathcal{P}(k,b)} \prod_{j=0}^{b-1} Q(m(J_j)) \mathbb{1}_{W(m(J_j))=M_j} \quad (1.17)
$$

where

$$
\mathrm{T}(M[b]) = M_0 \frac{(1 - W(m[k]))^{b-1}}{(b-1)!} + \frac{(1 - W(m[k]))^b}{b!} \quad (1.18)
$$
(see Section 5.2 for a proof which relies on the fact that this set has same measure as $S := \{(u, s_0, \dots, s_{b-2}), u \in [0, M_0 + s_0], 0 \le s_0 \le s_1 \le \dots s_{b-2} \le 1 - W(m[k])\}$ where the variable $s_j$ accumulates the free length spaces between the $\mathbf{O}_0^{(k)}$ and $\mathbf{O}_{j+1}^{(k)}$, and the variable $u$ is used to place zero in the interval $[0, M_0 + s_0]$).

The measure $T(M[b])$ depends only on $(M_0, W(m[k]))$. The presence of $M_0$ is due to the size bias.

**Remark 1.10** (bias of $\mathbf{O}_0^{(k)}$ and $\mathbf{F}_0^{(k)}$). Equations (1.17) and (1.18) show that $\mathbf{O}^{(k)}$ is biased, and the global bias of a configuration being proportional to $|\mathbf{O}_0^{(k)}| + |\mathbf{F}_0^{(k)}|$.

The block $\mathbf{F}_0^{(k)}$ is also biased, and while the rotation of the blocks in Theorem 1.5.(iv) allowed to avoid dealing with it, we can also characterize the distribution of $\mathbf{F}_0^{(k)}$:

We condition on $\mathbf{N}_k = b$. Recall that the entire process is invariant under rotation and that the occupied blocks are exchangeable:

- either 0 belongs to an occupied block (with probability $1 - R_k$), and $\mathbf{F}_0^{(k)}$ is the block that follows. Its size is $R_k \times \ell'$, where $\ell'$ is a $\beta(1, b-1)$ distributed random variable (thus $|\mathbf{F}_0^{(k)'}|$ has density $g_1(x) := (b-1)(1 - x/R_k)^{b-2}/R_k \mathbb{1}_{x \in [0, R_k]}$), this is the distribution of the first marginal in a Dirichlet($b$; $1, \dots, 1$) random variable),

- or 0 belongs to a free block (which is thus $\mathbf{F}_0^{(k)}$), in this case $|\mathbf{F}_0^{(k)}| = R_k \times \ell$ where $\ell$ a $\beta(2, b-1)$ random variable (and thus $|\mathbf{F}_0^{(k)}|$ has density $g_1(x) = b(b-1)(x/R_k)(1 - x/R_k)^{b-2}/R_k \mathbb{1}_{x \in [0, R_k]}$).

Finally, conditional on $\mathbf{N}_k = b$, the density of $\mathbf{F}_0^{(k)}$ is $(R_k g_1(x) + (1 - R_k) g_2(x))$.

Informally, a multiset $\{\{x_1, \dots, x_m\}\}$ is a set in which elements may have an arbitrary multiplicity$^5$.

The final universal result we would like to present concerns “the coalescence” process induced by the diffusion models we study.

**Theorem 1.11.** Let $(m_0, m_1, \dots, m_{k-1})$ be some masses with sum in $[0, 1)$. Consider, for $j \in \{0, \dots, k\}$, the multiset

$$S(j) := \{\{|\mathbf{O}_i^{(j)}|, 0 \le i \le \mathbf{N}_j - 1\}\}$$

which provides the sizes (with multiplicity) of the occupied CC at time $j$.

The distribution of the process $(S(j), 0 \le j \le k-1)$ is the same for all valid CDM.

**Remark 1.12.** Free spaces do not have the same property. There is a positive probability that using the RDCS from time $t$ to $t+1$ with a non zero-mass, a single free CC is reduced. However, when using the “$(p, 1-p)$ proportion to the right/left diffusion” of Section 1.5.2, this never happens.

**Remark 1.13.** Chassaing & Louchard [10], considered $S(t)$ the multiset of occupied block sizes at time $t$ in the parking process. The blocks correspond to a (maximal) set of consecutive occupied places along with the first free place to the right of them. They proved that the process $(S(t), t \ge 0)$ has the same law as in the additive coalescent process starting with $n$ particles of mass 1 observed at successive coalescence times. This process can be encoded by coalescent forests (Pitman [32]), in which tree sizes correspond to block sizes. The joint distribution of the tree sizes is known: suitably ordered they have the same distribution as i.i.d. Borel variables$^6$ conditioned to have a fixed sum (see e.g. Pitman [32, Prop.6], Bertoin [6, Cor. 5.8], [10, Proof of Prop. 5.1], Marckert & Wang [30]... ).

<footnote>$^5$Define an equivalence relation between sequences: two sequences are equivalent if they have the same length, and if they are equal up to a permutation of their terms. A multiset is an equivalence class for this relation.</footnote>

<footnote>$^6$The Borel law Borel($\lambda$), is the discrete distribution $p_\lambda$ with support $\{1, 2, 3, \dots\}$ defined by $p_\lambda(k) = (\lambda k)^{k-1} \exp(-k\lambda)/k!$.</footnote>
## 1.7 Reduction to the Right Diffusion with Constant Speed and excursion sizes

Lemma 1.5 tells us that for any valid CDM, given $m[k]$, the distribution of $(O^{(k)}, F^{(k)})$ can be studied using the RDCS. We recommend looking at Figure 6 before reading the following text.

<figure><img src="image_7.png" /><figcaption>Figure 6: On the first column: By Lemma 1.5, the masses can be treated in our preferred order. If we use the order 7,1,2,3,4,5,6 with the RDCS, we get the third picture. Since we are on the cycle $R/Z$, there are three occupied CC. The second picture represents the process $S$. To recover the excursions, there is a boundary effect. For example, the first excursion above the “minimum process” does not correspond to an occupied CC.</figcaption></figure>

In the second column, one can see that working with $\bar{S}$ which is obtained by pasting several trajectories of $S$ head to tail, then, starting from the abscissa $a$ of the minimum of $\argmin S$ on $[0, 1]$, one recovers the occupied block lengths as the excursion lengths above the minimum process of $\bar{S}$ on $[a, a+1]$.

The same property holds if one considers instead $a' = \argmin(t \to S_t + S_1 t)$ on $[0, 1]$.

Notice also that this encoding is not sufficient to encode zero masses, since they leave unchanged the collecting path, while possibly creating CC. The red portions of the paths correspond to the points where the left to right minimum decreases. Thus, excursions correspond to the portions of $\bar{S}$ between red points. The red portions of the paths do not correspond to red parts on $S$ that are represented on the left column because of a border effect at zero, to take into account to recover the excursion that straddles zero.

**Definition 1.14.** Define the “collecting path” process $S := (S_x, 0 \le x \le 1)$ by

$$S_x = -x + \sum_{j=0}^{k-1} m_j 1_{u_j \le x}, \quad \text{for } x \in [0, 1]. \qquad (1.19)$$

The extended collecting path $\bar{S} = (\bar{S}_x, x \in \mathbb{R})$ is a process indexed by $\mathbb{R}$, defined as on Figure 6 by concatenating copies of the collecting paths head to tail,

$$\bar{S}_x = S_{\{x\}} + [x] S_1, \quad \text{for } t \in \mathbb{R}$$

where $\{x\}$ is the fractional part of $x$.

The collecting path is named “profile” in Bertoin & Miermont [7] (but the term “profile” is used in many other contexts, with few commonalities to the present situation. The term “collecting path” seems more appropriate for its function of collecting and updating server tasks over time). For all levels $y \in \mathbb{R}$ set

$$\tau_y = \inf\{t: \bar{S}_t = y\}$$
as the first hitting time of y. To define the excursion of $\bar{S}$ at level y, set $\tau^y = \sup\{x: \min\{\bar{S}_x \in [\tau_y, x] = y\}\}$. Set $\ell_x := \tau^y - \tau_y$ the length of the excursion above level y. Let a be the first time that S reaches its minimum on $[0, 1]$:

$$a = \min \operatorname{argmin} S.$$

**Lemma 1.15.** The multiset of positive occupied CC sizes in the RDCS with mass arrival events $[(m_i, u_i), 0 \le i \le k-1]$, corresponds to the non-zero excursion lengths of the collecting path $\bar{S}$ for $x \in [a, a+1)$ above the current minimum process.

**Remark 1.16.** Zero masses leave unchanged the collecting paths and can create zero-size CC, which cannot be recovered in $\bar{S}$. Lemma 1.15 is proven in [7, Lemma 3] (and similar discrete results are present in [2], [10], ...). Studying the sizes of excursions in collecting paths is an important tool for analyzing cluster sizes in multiplicative and additive coalescence processes. The link between them is given by this Lemma.

**Remark 1.17 (CDM with random masses).** Hence, for some deterministic masses $m[k]$ with $\sum W(m[k]) < 1$, the 8 dispersion models introduced in Section 1.5 induce the same distribution for $(O^{(k)}, F^{(k)})$. This is also true, as a simple corollary, if they use the same random masses. The number of CC of $F^{(k)}$ is then always one plus a random variable with law Binomial($k-1, W(m[k])$). The law of their sizes, and other characteristics are the same in all cases. In particular, they can be studied using the RDCS model, and correspond then (up to the boundary effect discussed in Figure 6), to the excursion sizes of the collecting paths above the current minimum.

## 1.8 Proof of Lemma 1.5, Lemma 1.6 and Lemma 1.11

The proofs rely on some elementary principles. In this section, we introduce them step by step, describing what happens when we progressively add some mass arrival events on the circle.

Consider $k$ i.i.d. uniform points $u_0, \dots, u_{k-1}$ taken on the circle $\mathcal{C} = \mathbb{R}/\mathbb{Z}$.

### ■ (A) A static principle.

Take $I$ to be a measurable subset of $\mathcal{C}$ with positive Lebesgue measure, that can be random or not, but is chosen independently of the $u_i$.

If we condition the set $S := \{u_i, 0 \le i \le k-1\}$ to satisfy $|S \cap I| = m$ (i.e. to contain $m$ elements of $S$), then $S \setminus I$, the elements of $S$ that are not in $I$, have same law as a sample of $k-m$ uniform random variables on $\mathcal{C} \setminus I$. Of course, the cardinality of $|S \cap I|$ has law Binomial($k, \text{Leb}(I)$).

### ■ (B) A space eating principle.

We will “eat” the space around $u_0$ at constant speed. We introduce two continuous non-decreasing, non-negative functions $\alpha$ and $\beta$, such that $\alpha(0) = \beta(0) = 0$, and

$$\beta(t) + \alpha(t) = t, \text{ for } t \in [0, 1). \qquad (1.20)$$

Again, we allow $((\alpha(t), \beta(t)), t \ge 0)$ to be a random process, independent of the $u_i$. For this, define $s[t]$, “the swallowed space at time $t$ around $u_0$”, as the length $t$ interval on the circle,

$$s[t] := [u_0 - \alpha(t), u_0 + \beta[t]]_{-}.$$

Now, let us consider the non-eaten variables $u_j$ at time $t$:

$$X(t) := \{u_0, \dots, u_{k-1}\} \setminus s[t].$$
At any time $t \in [0, 1)$, the (remaining) non-eaten space $R(t) := \mathcal{C} \setminus s[t]$, satisfies

$$ \text{Leb}(R(t)) = 1 - t; $$

each of the $u_i$ is in $s[t]$ with probability $t$ (except $u_0$ which is surely in $s[t]$), so that

$$ |X(t)| \sim \text{Binomial}(k-1, 1-t) $$

and conditional on $|X(t)| = j$, the non-eaten variables $u_i$ are i.i.d. uniform in $R(t)$.

### (C) The Markovian dynamics of $|X(t)|$.

- the process $(|X(t)|, t \in [0, 1])$ is a Markov process independent of $(\alpha(t), \beta(t))$,

- the law of $(|X(t)|, t \in [0, 1])$ is the same regardless of the process $(\alpha(t), \beta(t))$.

Both statements are valid provided that $\alpha(t) + \beta(t) = t$ for all $t$. Indeed, from time $t$ to time $t+dt$, the swallowed space passes from a Lebesgue measure $t$ to $t+dt$, and this is true whatever are the relative speeds of $\alpha(t)$ and of $\beta(t)$ (since this speed must be independent of the uneaten points).

Each element of $X(t)$ survives up to time $t+dt$ with probability $\text{Leb}(R(t+dt))/\text{Leb}(R(t))$, and at time $t+dt$, each surviving random variable (included in $X(t+dt)$) will be uniform in $R(t+dt)$.

It is worth noticing that $|X(t)|$ is always a Markov process even if $(\alpha(t), \beta(t))$ is not!

More than that, we may allow $(\alpha(t+dt), \beta(t+dt))$ to depend on the number/identity of the swallowed $u_j$ before time $t$ (and also, of the absorption times): as long as at any time $t$, $(\alpha(t+dt), \beta(t+dt))$ is defined independently from the non-swallowed uniform points, the probability of absorption of a new point between time $t$ and time $t+dt$ does not depend on the choice of the distribution of $(\alpha, \beta)$, which implies that $|X(t)|$ is Markovian: the distribution of $(|X(t)|, t \ge 0)$ is universal for this class of processes $(\alpha, \beta)$.

### (D) Relation between continuous dispersion model and space-eating configurations

In the previous paragraphs we discussed the space-eating model, and we let an interval $s(t)$ grow around $u_0$, as in the continuous diffusion process. In a diffusion process where the points $u_1$ to $u_{k-1}$ have zero masses while $u_0$ is the only point having a positive mass, this mass would diffuse around, and cover a space $s(m_0)$ (with Lebesgue measure $m_0$), and cover/eat the other $u_j$ in $s(m_0)$. The conditions we took for $(\alpha(t), \beta(t))$, namely unit-speed deposition and independence with respect to $u_1, \dots, u_{k-1}$, are the conditions for a valid continuous dispersion model. Once again, the second item of (C) applies: as long as the eaten-space dynamics depend only on the eaten points and on the present eaten space $s(t)$, the distribution of the final configuration say, at time $m_0$, does not depend on the details of the definition of $(\alpha(t), \beta(t))$.

This eaten-space point of view is illustrated on Figure 7.

### (E) On the disappearance of the eaten space.

A topological solution for formalizing the disappearance of space on $\mathcal{C}$ consists in identifying all the points of $s(t)$. This amounts to designing an equivalence relation $\sim_{s(t)}$ on $\mathcal{C}$ where $x \sim_{s(t)} y$ if $x = y$ or if $x$ and $y$ are both in $s(t)$. The quotient space $\mathcal{C} / \sim_{s(t)}$ is isomorphic to $\mathbb{R}/(1-t)\mathbb{Z}$. In this latter space, the non-eaten random variables are, conditionally on their number $b-1$, uniform on $\mathbb{R}/(1-t)\mathbb{Z}$, while the eaten interval $s(t)$ corresponds to a uniform point $u_0(t)$, independent of the others, like a scar.

We can relabel by $v_0(t), \dots, v_{b-1}(t)$ these $b$ variables (with $v_0(t) = u_0(t)$, using the initial order of the labels between the surviving $u_j$ to determine the identities $1$ to $b-1$ of the vertices $v_1(t), \dots, v_{b-1}(t)$). If one is only interested in the joint position of the $v_j(t)$ on $\mathbb{R}/(1-t)\mathbb{Z}$, one can see that, given that they are $b$ of them in total, they are distributed as $b$ uniform points on $\mathbb{R}/(1-t)\mathbb{Z}$.
<figure><img src="image_8.png" /><figcaption>Figure 7: Illustration of the correspondence between the continuous dispersion model and the eaten-space dynamics. When mass $m_0$ has been spread in the CDM, a portion $m_0$ of $\mathcal{C}$ has been eaten in the eaten-space model. At time 1, there are two uniform points on $\mathcal{C} \setminus \sim_{s(m_0)} v_0$ (that contains all the points in $s(m_0)$, i.e. $u_0$ and $u_2$) and $v_1$ (that only contains $u_1$).</figcaption></figure>

Letting $\ell_0, \cdots, \ell_{b-1}$ be the sequence of lengths of the intervals of $(\mathbb{R}/(1-t)\mathbb{Z}) \setminus \{v_i(t), 0 \le i \le b-1\}$ formed by the $v_i(t)$, where these intervals are taken cyclically around $\mathbb{R}/(1-t)\mathbb{Z}$, starting from the intervals at the right 7 of $v_0(t)$, (conditionally on the fact that there are $b$ points), then

$$ (\ell_0, \cdots, \ell_{b-1}) \stackrel{(d)}{=} (1-t) [\mathbf{D}_0, \cdots, \mathbf{D}_{b-1}] \qquad (1.21) $$

where $[\mathbf{D}_0, \cdots, \mathbf{D}_{b-1}] \sim \text{Dirichlet}(b; 1, \cdots, 1)$.

On $\mathcal{C}$, if one interprets now $s(t)$, as well as all the other $\{u_j\}$, as occupied intervals that are not covered by $s(t)$, then, these $b$ elements form $\mathbf{O}^{(k)}$, and the intervals in between (that are isomorphic to those in $\mathbb{R}/(1-t)\mathbb{Z}$), the free space lengths $|\mathbf{F}^{(k)}|$, and then are distributed as $(1-t) [\mathbf{D}_0, \cdots, \mathbf{D}_{b-1}]$.

### ■ (F) Is it allowed in the CDM to place the points ($u_i, 0 \le i \le k-1$) beforehand?

As long as the dispersion of the masses ($m_i, u_i$) does not depend on the presence of the points with a higher index, the masses can be placed beforehand. Their presence makes it clear that, when a single mass is dispersed while the other ones are zeroes, the dispersion of this mass and the arrival of the others commute, if we are only interested in the distribution of the final configuration.

This allows us to disconnect somehow “the diffusion date” $i$, from the arrival place $u_i$. The presence of points all together in the system allows to compare the effect of diffusing a small quantity of matter $dm$ from one of these points or from another.

### ■ (G) Space eaten around two points / CDM diffusion.

Now assume that $u_0$ and $u_1$ have positive masses $m_0$ and $m_1$ while the other ones, $u_2$ to $u_{k-1}$, still have mass 0. Assume that $u_0$ is again equipped with two functions $(\alpha_0(t), \beta_0(t))$ satisfying $\alpha_0(t) + \beta_0(t) = t$ for all $t$, which defines an interval $s_0(t) = [u_0 - \alpha_0(t), u_0 + \alpha_0(t)]$. In the CDM perspective $u_0$ has invaded $s_0(m_0)$ at time $m_0$ on $\mathcal{C}$, while on the eaten-space perspective, $u_0(m_0)$ is now a point on $\mathbb{R}/((1-m_0)\mathbb{Z})$ and this point can now have been identified with other $u_i$, including the other massive point $u_1$.

From the CDM perspective, when $u_1$ starts, its environment has changed. There are two cases: either $u_1$ is in $s_0(m_0)$, or it is not.

<footnote n="7">7 we can start the labeling from any $v_i(t)$ as long as the labeling is chosen independently from the interval lengths</footnote>
In both cases, it will use some functions $\alpha_1(t), \beta_1(t)$, from its activation time, $t=1$, to eat the space or to diffuse. If $u_1$ is in $s_0(m_0)$ (which occurs with probability $m_0$ by the previous discussion), $u_1$ may use (random) processes $\alpha_1(t)$ and $\beta_1(t)$ depending on $(u_0, s_0(m))$, while if it is not in $s_0(m_0)$ (with probability $1-m_0$), $\alpha_1(t)$ and $\beta_1(t)$ must be independent of $s_0(m_0)$, at least before a possible collision.

Define $s_1(t) = [u_1 - \alpha_1(t), u_1 + \beta_1(t)]_-$ and we assume now that $\text{Leb}(s_1(t) \setminus s_0(t)) = (t-1)$ for $t \in [1, 1+m_1]$ so that $s_1$ grows at unit speed (out of $s_0(m_0)$).

Again, there are two points of view on the evolution of $s_1(t)$ as $t$ grows: one is as a CDM on $\mathcal{C}$ where $s_0(m_0)$ is already covered, and the other one is in $\mathbb{R}/((1-m_0)\mathbb{Z})$ where $u_0(m_0)$, the special point representing $s_0(m_0)$ behaves as the others.

Let us now discuss the free spaces evolution as $s_1(t)$ evolves.

■ Eaten-space perspective on $\mathbb{R}/((1-m_0)\mathbb{Z})$,

- if $u_1$ is in $s_0(m_0)$, then the evolution of $s_1(t)$ after time 1 is indistinguishable in distribution from the evolution of $s_0$ after time 1 (if we would let $s_0(m_0)$ resume its eating activity after time 1).

- if $u_1$ is not in $s_0(m_0)$, then, in terms of the free spaces, it is the same! That is, eating the space around $u_0(m_0)$ or around $u_1$ is the same in law, since the surviving points are distributed as i.i.d. uniform on $\mathbb{R}/((1-m_0)\mathbb{Z})$!

■ CDM perspective on $\mathcal{C}$: The idea is the same: when $u_1$ becomes active, either it is in $s_0(t)$ or it is not. The free spaces around $\mathcal{C}$ are distributed as Dirichlet($b$; $1, \dots, 1$) (under the condition that there are $b$ intervals if $u_0$ has covered all but $b-1$ other points). If $u_1$ is in $s_0(1)$, then in terms of free space, as we said before, only the size of $s_0(1+t)$ matters, not how it grows. Therefore, diffusing around $s_0$ or around $s_1$ is the same, in distribution for the final free space (at time $1+m_1$).

## Conclusion

Assume that we are back to the initial models of several masses $(u_0, m_0), \dots, (u_{k-1}, m_{k-1})$ and that a valid CDM is given. Assume for a moment that we are only interested in the free spaces $\mathbf{F}^{(k)}$.

Then, from the eaten-space point of view, during the relaxation phase of the $i$th point, the new mass extends around one of the surviving points, that are, conditional on their numbers, uniform points on $\mathbb{R}/((m_0 + \dots + m_{i-2})\mathbb{Z})$. The point around which the extension is done is not important, by invariance by relabeling at time $i$: everything would have been the same if all the space was eaten around the same point! Hence **Lemma 1.6** is a consequence of this fact: if one is only interested in free space, we can relocate the location where the relaxation is done, without consequence.

**Remark 1.18.** A Corollary of this analysis is that if we allow the relaxation phases to occur simultaneously, possibly with different speeds, as long as the procedure is invariant under rotation and independent between different occupied CC (before coalescence), then, the distribution of the free and occupied spaces would be the same (for any valid CDM).

The distribution of free space would resist to even more involved generalizations, as for example, transfer of non-deposited masses – between current occupied CC.

As **Theorem 1.5** also deals with occupied components, we can add some details to conclude.

Points (iii), (iv) and (v) can be stated as properties of the free CC only. They are consequences of the previous discussion (details for **Lemma 1.5**(v) are postponed in Section 1.9 below). For example for point (iii), the fact that $\mathcal{L}(N_k - 1) = \text{Binomial}(k-1, R_k)$ comes from the fact that we can rearrange the masses as we want when we are only interested in the free CC sizes and number, and in the case in which $m_0 = W$ while $m_1 = \dots = m_{k-1} = 0$, the result is obvious.
Clearly, the distribution of the CC of the occupied set $O^{(k)}$ depends on the masses. For example, two masses 0.8 and 0 differ from two masses 0.6 and 0.2.

To prove (i), we can add a recursive argument to the previous discussion. First, the position of the CC $s_0(m_0)$ has a distribution independent from $(\alpha_0(t), \beta_0(t))$, and of $(u_1, \dots, u_{k-1})$ (its position is uniform on $\mathcal{C}$). Now, assume that at time $i^{-}$ (just before the arrival of the $i+1$th mass $(u_i, m_i)$), the current state $(\mathbf{O}^{(i)}, \mathbf{F}^{(i)})$ has a distribution that does not depend on the choice of the valid CDM considered. There are two cases: either $u_i$ is in one the CC $O \in \mathbf{O}^{(i)}$, or it is in a free space $F \in \mathbf{F}^{(i)}$. Then the relaxation process $\mathrm{IR}(t) = (\alpha_i(t), \beta_i(t))$ (with dependence already discussed) will govern the growing of $s_i(t)$ around $u_i$.

The construction of the occupied and free spaces at time $i$ is independent from the position of the origin 0 on the circle, and this is due to the invariance of valid models under rotation (condition $(\nu)$). A dependence arises when one labels the occupied components (as explained in (1.8) and in Lemma 1.10), but the occupied and free sets at time $i$, $(\mathbf{O}^{(i)}, \mathbf{F}^{(i)})$, are invariant under rotation by any angle (chosen independently from the current state). The conjunction of events that lead to the position of occupied blocks at time $i$ has a local nature. Each occupied block has essentially grown its own territory. By symmetry, the probability of the conjunction of events that leads to the presence of a unique occupied CC in each of some fixed (deterministic) intervals with respective length $a_1, \dots, a_b$ separated by some free spaces with lengths $f_1, \dots, f_b$, is the same regardless of the order of these intervals or the order of the $f_i$. Moreover, as long as the $f_i$ are non-zero, this probability it does not depend neither on the precise value of the $(f_i)$, because the valid dispersion algorithms are independent of free spaces. Thus, the amount of space that $s_i(t)$ will cover before colliding with another CC does not depend on $(\alpha_i(t), \beta_i(t))$. In particular, the probability that a collision occurs before the end of the diffusion (at time $i+m_i$) is independent of $(\alpha_i(t), \beta_i(t))$. If a collision occurs, the time of the first collision has a distribution independent of $(\alpha_i(t), \beta_i(t))$. In this case, $s_i(t)$ will coalesce with a CC whose size has a law which does not depend on $(\alpha_i(t), \beta_i(t))$ (it has same distribution as if it was chosen uniformly among the other CC sizes!).

The arguments for the proof of (ii) have also been given: the sizes of the occupied CC depend on the masses that are diffused from each vertex, but not on the order of diffusion (see Lemma 1.18).

Finally, Lemma 1.11 is a consequence of the general exchangeability results obtained (universality of the probability of collision of the connected component that undergoes the relaxation phase during a time interval $dt$, and in case of collision, universality of the distribution of the size of the occupied CC hit).

## 1.9 Proof of Theorem 1.5($\nu$)

Markovian property of $(\mathbf{N}_j)$: We have

$$ \mathcal{L}\left(\mathbf{F}^{(k+1)} \mid (\mathbf{N}_j, 0 \le j \le k)\right) = (n_j, 0 \le j \le k)\right) = \mathcal{L}\left(\mathbf{F}^{(k+1)} \mid \mathbf{N}_k = n_k\right). \quad (1.22) $$

The reason is that conditional on $\mathbf{N}_k$, and also conditional on $(\mathbf{N}_j, j \le k)$, the uniformly rotated free spaces ($|\mathbf{F}_{i+\theta \bmod b}^{(k)}, 0 \le i \le b-1|$ (where $\theta$ is uniform on $\{0, \dots, b-1\}$ and independent of the $|\mathbf{F}_i^{(k)}|$) are distributed as $R_k \times (\mathbf{D}_1, \dots, \mathbf{D}_b)$ (by Theorem 1.5($i\nu$)), where the $(\mathbf{D}_1, \dots, \mathbf{D}_b)$ are Dirichlet($b; 1, \dots, 1$) distributed. This ensures that (1.22) is true. Let us now describe its transition matrix of $(\mathbf{N}_i, i \ge 0)$.

Now, assume that $\mathbf{N}_k = n$, and $m_0, \dots, m_k$ are given, with $W = W(m[k]) = m_0 + \dots + m_{k-1}$, $W' = W(m[k+1]) = m_0 + \dots + m_k$, and set $R = R_k = 1-W$, $R' = R_{k+1} = 1-W'$.

For a Dirichlet vector $\mathbf{D}^{(b)} = (\mathbf{D}_1^{(b)}, \dots, \mathbf{D}_b^{(b)}) \sim \text{Dirichlet}(b; 1, \dots, 1)$, denote by $\mathbf{S}_j^{(b)} = \mathbf{D}_1^{(b)} + \dots + \mathbf{D}_j^{(b)}$ (so
that $\mathbf{S}_0^{(b)} = 0$ and $\mathbf{S}_b^{(b)} = 1$). Observe that, for $x \in [0, 1]$,

$$ \mathbb{P}(\mathbf{S}_j^{(b)} \le x \le \mathbf{S}_{j+1}^{(b)}) = \binom{b-1}{j} x^j (1-x)^{b-1-j}, $$

because the Dirichlet random variable with these parameters, is the joint distribution of b intervals between the order statistics of $\hat{u}_1, \dots, \hat{u}_{b-1}$ uniform random variables on $[0, 1]$.

$$ \mathbb{P}(\mathbf{N}_{k+1} = n+1 | \mathbf{N}_k = n) = R_k \mathbb{P}(\mathbf{S}_1^{(n+1)} \le m_k / R_k). $$

Moreover, for $j \in \{0, \dots, n-1\}$,

$$ \begin{aligned} \mathbb{P}(\mathbf{N}_{k+1} = n - j | \mathbf{N}_k = n) &= R_k \mathbb{P}(\mathbf{S}_{j+1}^{(n+1)} \le m_k / R_k \le \mathbf{S}_{j+2}^{(n+1)}) + (1-R_k) \mathbb{P}(\mathbf{S}_j^{(n)} \le m_k / R_k \le \mathbf{S}_{j+1}^{(n)}) \\ &= R_k \binom{n}{j+1} \left(\frac{m_k}{R_k}\right)^{j+1} \left(1-\frac{m_k}{R_k}\right)^{n-1-j} + (1-R_k) \binom{n-1}{j} \left(\frac{m_k}{R_k}\right)^j \left(1-\frac{m_k}{R_k}\right)^{n-1-j}. \end{aligned} $$

The term starting with $R_k$ corresponds to the case $u_{k+1} \in \mathbf{F}^{(k)}$. To have $\mathbf{N}_{k+1} = n-j$, we need either $u_{k+1} \in \mathbf{F}^{(k)}$, and then that the component growing from $u_k$ absorbs $j+1$ blocks among the $n$ other blocks, or to have $u_{k+1} \in \mathbf{O}^{(k)}$, and then that the component growing from $u_k$ absorbs $j$ blocks among the $n-1$ other blocks.

**Remark 1.19.** We can also directly compute the distribution of $\mathbf{N}_{k_2}$ conditional on $\mathbf{N}_{k_1}$ (with $k_1 < k_2$). In fact, one can prove, using the same arguments as for Theorem 1.6, that assuming the hypothesis of Lemma 1.5,

$$ \mathcal{L}(|\mathbf{F}^{(k)}| | (m_0, \dots, m_{k_2-1})) = \mathcal{L} \left( \left| \overrightarrow{\mathbf{F}^{(k)}} \right| | \left( W(m[k_1]), \underbrace{0, \dots, 0}_{k_1-1 \text{ zeroes}}, W(m[k_2]) - W(m[k_1]), \underbrace{0, \dots, 0}_{k_2-k_1-1 \text{ zeroes}} \right) \right). $$

We let $M_1 = W(m[k_1])$, $M_2 = W(m[k_2]) - M_1$, $R_1 = 1 - M_1$ and $R_2 = 1 - M_1 - M_2$. Then, we consider the CDM with $m_0 = M_1$, $m_{k_1} = M_2$, and $m_j = 0$ for any $j \in [1, k_2 - 1] \setminus \{k_1\}$. Then,

$$ \begin{aligned} \mathbf{N}_{k_2} &\stackrel{(d)}{=} 1 + \binom{\mathbf{N}_{k_1} - 1, R_2/R_1}{1} + B(R_2) + \binom{\mathbf{k}_2 - \mathbf{k}_1 - 1, R_2}{1} \\ &\stackrel{(d)}{=} 1 + \binom{\mathbf{N}_{k_1} - 1, R_2/R_1}{1} + \binom{\mathbf{k}_2 - \mathbf{k}_1, R_2}{1}. \end{aligned} $$

In the first equality, $\binom{\mathbf{N}_{k_1} - 1, R_2/R_1}{1}$ is a binomial r.v. that counts the occupied blocks of size 0 at time $k$ that are not absorbed into the newly covered area at time $k_1 + 1$, after the addition of mass $M_2$. The variable $B(R_2)$ is a Bernoulli r.v. that equals 1 if the masses $M_1$ and $M_2$ form a single occupied CC. Finally, $\binom{\mathbf{k}_2 - \mathbf{k}_1 - 1, R_2}{1}$ counts the number of 0 masses that arrive between times $k_1 + 2$ and $k_2$ and do not fall into an already occupied CC.

## 2 Valid discrete dispersion models

We will not go into details about everything, but we will point out the differences with continuous dispersion models. Here, the masses arrive in the space $\mathscr{C}_n := \{0/n, \dots, (n-1)/n\} \subset \mathscr{C}$, which is considered as a subgroup of $\mathbb{R}/\mathbb{Z}$. The masses are now positive multiples of $1/n$, and zero masses are still allowed.
**To be valid, a discrete dispersion model** must satisfy the same conditions as continuous models, except that:

- The invariance under rotation (condition (1.3)) must hold only for the rotation by $1/n$, (in particular the arrival positions are uniform on $\mathcal{C}_n$ now),

- after each interval relaxation, the occupied set $\mathbf{O}^{(k)}$ is still formed by close intervals (some of them having possibly length zero), with extremities in $\mathcal{C}_n$.

The notion of occupation remains the same as in the continuous case, and the right diffusion with constant speed is still valid.

For example, if $n \ge 8$, if there are four mass arrival events $(m_0, u_0) = (3/n, 0)$, $(m_1, u_1) = (1/n, 5/n)$, $(m_2, u_2) = (0, 3/n)$, $(m_3, u_3) = (0, 7/n)$, then, using the RDCS: we have $\mathbf{O}^{(1)} = \{[0, 3/n]\}$, $\mathbf{O}^{(2)} = \{[0, 3/n], [5/n, 6/n]\}$, $\mathbf{O}^{(3)} = \mathbf{O}^{(2)}$, $\mathbf{O}^{(4)} = \{[0, 3/n], [5/n, 6/n], [7/n]\}$.

Again, the addition of a zero mass can leave the occupied set unchanged, or create the appearance of a point (as stated in Lemma 1.15 and Lemma 1.16). The distance between two occupied CC is at least $1/n$.

**Remark 2.1.** The number of points in $\mathcal{C}_n$ in a given CC with extremities in $\mathcal{C}_n$ is not proportional to the Lebesgue measure of this CC seen as an interval: if it has length $k/n$, it contains $k+1$ points of $\mathcal{C}_n$.

## 2.1 Main universality result for valid discrete dispersion models

To state the analogue of Lemma 1.5 for valid discrete dispersion models (DDM), we define a discrete analogue to the Dirichlet distribution. If the total free space is $R/n$ for some positive integer $R$, then there can be at most $R$ free CC, each of them with a positive length multiple of $1/n$.

Let $(k, R)$ be a pair of positive integers with $k \le R$. We call composition of $R$ in $k$ parts a sequence $(s_1, \dots, s_k)$ with positive integer coordinates, summing to $R$. Let $\text{Comp}(R, k)$ be the set of these compositions. We have

$$|\text{Comp}(R, k)| = \binom{R-1}{k-1} \qquad (2.1)$$

since we can associate bijectively with each increasing sequence $(x_1, \dots, x_{k-1})$ made of elements of $\{1, \dots, R-1\}$, the sequence $(x_1, x_2 - x_1, \dots, x_{k-1} - x_{k-2}, R - x_{k-1})$.

**Definition 2.2.** For all $(k, R)$ integers such that $1 \le k \le R$, we call Discrete Dirichlet distribution of $R$ in $k$ parts (and write $\text{DDirichlet}(R, k)$) the uniform distribution on $\text{Comp}(R, k)$.

The following analogue of Lemma 1.5 holds in the discrete setting:

**Theorem 2.3.** Let $n \ge 1$ be an integer. Consider a valid DDM A, and some deterministic masses $m[k] = (m_0, \dots, m_{k-1})$ all non-negative multiples of $1/n$, satisfying the total weight condition $W(m[k]) < 1$. Let $R_k = 1 - W(m[k])$.

(i) The law of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ is the same for all valid DDM A.

(ii) The law $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ is invariant under the permutation of the masses.

(iii) The number of free CC $\mathbf{N}_k^{(n)}$ has an explicit distribution $v(R_k, k)$ (see Section 2.2, where an analogue of Lemma 1.5 (v) can also be found).
(iv) Take a uniform element $\theta$ in $\{0, \dots, b-1\}$ (independently of everything else). Conditional on $\mathbf{N}_k^{(n)} = b$, the (unbiased) free-space sizes

$$ \left( \left| \mathbf{F}_{i+\theta \bmod b}^{(k)} \right|, 0 \le i \le b-1 \right) \stackrel{(d)}{=} C[b] := (C_0, \dots, C_{b-1}) \qquad (2.2) $$

where $nC[b]$ follows the DDirichlet($nR_k$, $b$) distribution.

The proof is similar to that of Lemma 1.5 (and can be proven using the same kind of principles). Once again, the following theorem is an important feature that allows one to perform precise computations:

**Theorem 2.4.** *Assuming the hypothesis of Lemma 2.3, the distribution of $\mathbf{F}^{(k)}$ only depends on $k$ and the total mass $W(m[k]):*

$$ \mathcal{L}\left(\mathbf{F}^{(k)} \mid (m_0, \dots, m_{k-1})\right) = \mathcal{L}\left(\overrightarrow{\mathbf{F}^{(k)}} \mid \left(W(m[k], \underbrace{0, \dots, 0}_{k-1 \text{ zeroes}})\right)\right). $$

**Discrete piling propensity**

**Lemma 2.5.** Let A be a valid DDM process on $\mathscr{C}_n$. If the masses $(m_0, \dots, m_{k-1})$ are non-negative multiple of $1/n$ and satisfy $W(m[k]) < 1$, then, the discrete piling propensity is

$$ Q^{(n)}(m[k]) = \mathbb{P}(\mathbf{N}_k^{(n)} = 1) = \left( W(m[k]) + \frac{1}{n} \right)^{k-1}. \qquad (2.3) $$

Moreover, for any $\ell \in \mathscr{C}_n$, by letting $A_j$ be the starting point of the block $j$, we have,

$$ \mathbb{P}\left(\mathbf{N}_k^{(n)} = 1, A_0 = \ell\right) = Q^{(n)}(m[k])/n. \qquad (2.4) $$

*Proof.* Using the rotation invariance, Formula (2.4) and (2.3) are equivalent. We will prove (2.4) which is a bit easier to prove, from a combinatorial perspective.

We remove the normalization by $n$, and consider the masses $nm[k]$ instead on $\mathbb{Z}/n\mathbb{Z}$ and also, first on $\mathbb{Z}/(M+1)\mathbb{Z}$ where $M = nW(m[k])$. The RDCS on $\mathbb{Z}/(M+1)\mathbb{Z}$ with these masses leaves a single vertex remains free at the end, and with probability $1/(M+1)$, the last vertex $M$ is free. This means that if mass $m_i$ is placed at $p_i$ for $i$ going from 0 to $k-1$ in $\mathbb{Z}/(M+1)\mathbb{Z}$, among the $(M+1)^k$ such map $p$, a fraction $1/(M+1)$ of them leaves the last place empty. Hence, the number of ways to arrange the $k$ masses on $\mathbb{Z}/n\mathbb{Z}$ such that, after relaxation (in the discrete RDCS model) the set (of classes) $\ell, \dots, \ell+M-1$ is occupied in $\mathbb{Z}/n\mathbb{Z}$ is also $(M+1)^{k-1}$. It suffices to multiply by $1/n^k$ which is the probability of each map $p$. $\square$

We could have also used Theorem 2.4 to compute $\mathbb{P}(\mathbf{N}_k^{(n)} = 1)$ in the case when there is one mass $W(m[k])$ (the $k-1$ others being zero): it covers a fraction $W(m[k]) + 1/n$ of $\mathscr{C}_n$, and $\mathbf{N}_k^{(n)} = 1$ if and only if every zero mass falls on one of these vertices. This happens with probability $(W(m[k]) + 1/n)^{k-1}$, since there are $k-1$ such masses.

There exists also an analogous of Lemma 1.8 and Lemma 1.9:

**Theorem 2.6.** Consider some masses $m_0, \dots, m_{k-1}$, non-negative multiple of $1/n$, such that $W(m[k]) < 1$. For any $b \in \{1, \dots, k\}$, any partition $(J_0, \dots, J_{b-1})$ of the set $\{0, \dots, k-1\}$ with non-empty parts, any sequence
$(a_0, a_1, \cdots, a_{b-1})$ in $\mathscr{C}_n^b$ such that the $a_0, a_1, \cdots, a_{b-1}$ are cyclically ordered around $\mathscr{C}_n$ (i.e. such that turning around the circle we get $a_0 \le 0 \le a_1 \le \cdots \le a_{b-1} \le a_0 \le \cdots$) and such that

$$ \mathrm{Leb}([a_i, a_{i+1} \bmod b]_-) > W(m(J_i)) \quad \text{for } i \in \{0, \cdots, b-1\}. $$

For a valid DDM, we have

$$ \mathbb{P}(\mathbf{A}_j^{(k)} = a_j, \mathbf{I}_j^{(k)} = J_j, 0 \le j \le b-1) = \prod_{j=0}^{b-1} \frac{Q^{(n)}(m(J_j))}{n}. \qquad (2.5) $$

Moreover, if we consider some non-negative masses $M_0, \cdots, M_{b-1}$,

$$ \mathbb{P}(\mathbf{A}_j^{(k)} = a_j, |\mathbf{O}_j^{(k)}| = M_j, 0 \le j \le b-1) = \sum_{(J_0, \cdots, J_{b-1}) \in \mathcal{P}(k, b)} \prod_{j=0}^{b-1} \frac{Q^{(n)}(m(J_j))}{n} \mathbb{1}_{W(m(J_j))=M_j} \quad (2.6) $$

where, as in Theorem 1.9, $P(k, b)$ is the set of partitions of $\{0, \cdots, k-1\}$ into b parts.

We have in the discrete case the analogue of Lemma 1.11:

**Theorem 2.7.** Let $(m_0, m_1, \cdots, m_{k-1})$ be masses multiple of $1/n$ with sum in $[0, 1)$. Then, the distribution of the (time indexed) process $(S(t), t \in \{0, \cdots, k\})$, recording the multiset of occupied CC cardinalities, defined by $S(t) := \{\{|\mathbf{O}_m^{(t)}|, 0 \le m \le \mathbf{N}_j - 1\}\}$ is the same for all valid DDM.

**Remark 2.8** (Reduction to the Right Diffusion with Constant Speed and excursion sizes). The same phenomenon occurs here. The law of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$, for a fixed $k$ is the same for all valid DDM, and then, can be computed for the simplest model.

## 2.2 Distribution of the number of CC $\mathbf{N}_k^{(n)}$.

The distribution of $\mathbf{N}_k^{(n)}$ is more complicated than in the continuous case, when it was just binomial. Consider $u_0, \cdots, u_{k-1}$ some random variables i.i.d. uniform on $\mathscr{C}_n$ and consider the set

$$ Z_k := \{u_0, \cdots, u_{k-1}\} $$

of arrival places. Note that when all the masses $m_0, \cdots, m_{k-1}$ are equal to zero, $\mathbf{O}^{(k)} = Z_k$. The support of the random variable $|Z_k|$ is $\{1, \cdots, \max\{k, n\}\}$ and the distribution of the set $Z_k$ is computable, since for $z$ a subset of $\mathscr{C}_n$ with cardinality $|z| > 0$

$$ \mathbb{P}(Z_k = z) = S(k, |z|)n^{-k} $$

where $S(a, b) := \sum_{j=0}^b (-1)^{b-j} \binom{b}{j} j^a$ is the number of surjections from $\{1, \cdots, a\}$ to $\{1, \cdots, b\}$. Hence, the law of $|Z_k|$ is also explicit

$$ \mathbb{P}(|Z_k| = m) = \binom{n}{m} S(k, m) n^{-k}, $$

and conditional on $\{|Z_k| = m\}$ (for $m \in \{1, \cdots, \max\{k, n\}\}$), $Z_k$ is a uniform subset of $\mathscr{C}_n$ with cardinality $m$.

We use Lemma 2.4, in order to describe $\mathbf{N}_k^{(n)}$ when the masses $m_0, \cdots, m_{k-1}$ have been treated with a valid DDM: it suffices to diffuse the mass $W(m[k])$ from $u_0$ and keep the other elements of $Z_k$ unchanged
(and observe how many of them are not covered). The diffusion from $u_0$ will reach $u_0$ and the $M := nW(m[k])$ next points of $\mathscr{C}_n$ on its right, so that

$$ \mathbf{N}_k^{(n)} = 1 + |Z_k \setminus \{u_0, u_0 + W(m[k])\}|. $$

and then, a random subset $V$ of $Z_k \setminus \{u_0\}$ with size $c$ remains free (not covered) with probability

$$ \mathbb{P}(\mathbf{N}_k^{(n)} = 1 + c) = \sum_{m=1+c}^{k} \mathbb{P}(|Z_k| = m) \frac{\binom{n-1-M}{c} \binom{M}{m-1-c}}{\binom{n-1}{m-1}} \quad (2.7) $$

where, here, the classical factor $\frac{\binom{n-1-M}{c} \binom{M}{m-1-c}}{\binom{n-1}{m-1}}$ is the probability that, considering a subset of $m-1$ elements in a set of cardinality $n-1$, exactly $c$ of them fall into a prescribed subset of size $n-1-M$.

We now consider the process $(\mathbf{N}_k^{(n)}, k \ge 0)$. It is a Markov chain for the same reason as in the continuous case. Let us describe its transition matrix. Given that $\mathbf{N}_k^{(n)} = b$, the occupied domain is constituted with $b$ components, with total mass $W(m[k])$ and then contains a proportion

$$ q_b := (nW(m[k]) + b)/n = W(m[k]) + b/n $$

of the elements of $\mathscr{C}_n$; let $p_b := 1 - q_b$ denote the proportion of $\mathscr{C}_n$ that is still free.

We now add a mass arrival event $(m_k, u_k)$, and will proceed to the RDCS of this mass from $u_k$. Initially, $u_k$ belongs to $\mathbf{O}^{(k)}$ with probability $q_b$, and to $\mathbf{F}^{(k)}$ with probability $p_b$. Hence, just after the insertion of the point $u_k$, we have $b+X$ free CC (possibly reduced to a single point), where $X$ is a Bernoulli($p_b$) random variable. The free spaces, up to a uniform rotation of indices (to disregard the influence of the labeling choice), after the insertion of the point $u_k$, form a uniform partition of $R := n(1 - W(m[k])) - b$, each free spaces having size at least $1/n$ (and multiple of $1/n$). Hence, conditional on $X = x$, these free spaces form a DDirichlet$(R, b+x)$-distributed vector.

We let $(\mathbf{D}_1^{\ell,R}, \dots, \mathbf{D}_{\ell}^{\ell,R})$ be a DDirichlet$(R, \ell)$ random variable. Denote again $\mathbf{S}_j^{\ell,R} = \mathbf{D}_1^{\ell,R} + \dots + \mathbf{D}_j^{\ell,R}$. We have

$$ \mathbb{P}(\mathbf{N}_{k+1} = b+1 | \mathbf{N}_k = b) = p_b \mathbb{P}(nm_k < \mathbf{S}_1^{b+1,R}) \quad (2.8) $$

and

$$ \mathbb{P}(\mathbf{N}_{k+1} = b-j | \mathbf{N}_k = b) = p_b \mathbb{P}(\mathbf{S}_{j+1}^{b+1,R} \le nm_k < \mathbf{S}_{j+2}^{b+1,R}) + q_b \mathbb{P}(\mathbf{S}_j^{b,R} \le nm_k < \mathbf{S}_{j+1}^{b,R}), \quad (2.9) $$

and the computation can be done since

$$ \mathbb{P}(S_j^{\ell,R} \le x < S_{j+1}^{\ell,R}) = \binom{x}{j} \binom{R-1-x}{\ell-1-j} / \binom{R-1}{\ell-1} $$

## 2.3 Examples of DDMs.

* **Parking model (or hashing with linear probing):** when masses all have weight $1/n$, if the dispersion policy is the RDCS then the model is exactly the classical parking model (introduced in Konheim and Weiss [24] and studied for example by Chassaing and Louchard [10]; see also [11]). We will come back to this model in Section 3.2; the cost analysis of the hashing with linear probing (in terms of total car displacement from their chosen place to their final park place), due the Flajolet & al. [16], and also to [10] are discussed in Section 4.1. However, our analysis shows that the
distribution of the occupied CC of the parking (at any time $k$) does not depend on the valid CDM used. (Janson [19] noticed the same property for three different policies in which a car may eject an already parked car, this ejected car searching a place on its right: for these three policies, if a car arrives in a block $[a, b]$, after insertion, the new block becomes $[a, b+1/n]$ (possibly merged with the next block).

* **Particle dispersion** The masses $m_i$ have weights proportional to $1/n$, and are composed by particles of size $1/n$. The particles perform independent random walks successively until they exit the occupied interval in which they are.

* **Discrete caravan type model:** The masses are proportional to $1/n$ and undergo the RDCS so that they progressively fill in the empty slots at the right of the arrival position.

# 3 Asymptotics

## 3.1 Discussion regarding random masses results

Theorems 1.5 and Lemma 1.6 hold for any valid CDM, for any fixed masses $(m_0, \dots, m_{k-1})$, as long as $\sum m_i < 1$. Here, we assume that the masses $\mathbf{m}[k] = (\mathbf{m}_0, \dots, \mathbf{m}_{k-1})$ are random and taken according to a distribution $\mu^{(k)}$ on $[0, 1)^k$, such that

$$ \mathbb{P}(W(\mathbf{m}[k]) < 1) = 1. \qquad (3.1) $$

Again, take some i.i.d. arrival positions $(\mathbf{u}_0, \dots, \mathbf{u}_{k-1})$, independent from the masses.

Since conditional on the event $\{\mathbf{m}_i = m_i, 0 \le i \le k-1\}$, the results of Theorems 1.5 hold, in particular, $\mathbf{N}_k \stackrel{(d)}{=} 1 + B$ where $\mathcal{L}(B | W(\mathbf{m}[k]) = W) = \text{Binomial}(k-1, 1-W)$. The occupied and free block results of Theorem 1.9 can be extended to random block size by integration (with respect to $\mu^{(k)}$). However, this general statement is insufficient for understanding or computing the typical behavior of the block sizes for a given distribution $\mu^{(k)}$ when $k$ becomes large. Important “partially tractable” examples correspond to the case for which the combinatorial terms can be understood. Other important examples correspond to the case where the right diffusion with constant speed representation allows to bypass exact representations.

However, we will need additional hypotheses on masses distribution to design limit theorems. We will come back on particular cases in Section 3.3.

## 3.2 Discrete masses: comparison between continuous and discrete spaces

The aim of this section is to consider the cases where discrete dispersion models and continuous dispersion models meet: when the masses are multiples of $1/n$. In this case, it is possible to define the right diffusion with constant speed on $\mathcal{C}$ and $\mathcal{C}_n$ respectively, and compare the statistics of the space occupation in both cases. As we will see, discrete dispersion model and continuous dispersion model defined using the same masses have some similarities, and some discrepancies.

One interesting case is the continuous model of parking, that we will call the lazy bulldozer parking model.

In the usual parking problem defined on the discrete cycle $\mathcal{C}_n$ (see [10] or Section 4.1), some cars (with size $1/n$) arrive successively, and the $i$th car chooses a uniform place $c_i$ in $\mathcal{C}_n$. It then parks at
the first available place among $c_i$, $c_i + 1/n$ mod 1, $c_i + 2/n$ mod 1, etc (the place where it parks is then not available for the subsequent cars). In terms of DDM, it occupies the interval $[\ell, \ell + 1/n]$, where $\ell$ is the first available place discovered.

The lazy bulldozer parking model is defined on the continuous parking $\mathcal{C}$. The bulldozers are each $1/n$ in length $1/n$ and choose uniform places ($u_i$) on $\mathcal{C}$ (where the $u_i$ are i.i.d.). The $i$th bulldozer arrives when bulldozers 0 to $i-1$ are already parked. It observes the parking and locates $y$, the first free point at the right of $u_i$ (and $y$ may belong to a free space smaller than $1/n$, as on Figure 8). Bulldozer then parks on $[y, y + 1/n]$, pushing to the right, if needed, the bulldozers already on $[y, y + 1/n]$, if any, as on Figure 8. More precisely, if the interval $[y, y + 1/n]$, intersects an occupied block $O$, then, this component is translated on its right (and then $O$ will possibly push the next occupied component, and this recursively, if needed), so that the length of $O$ is preserved, and after this action by the bulldozer, $O$ is at the right of $[y, y + 1/n]$, and adjacent to it.

In terms of occupied space, the lazy bulldozer parking model corresponds to the RDCS (see Section 1.5.1).

<figure><img src="image_9.png" /><figcaption>Figure 8: The bulldozer parks in the first free space after the chosen location and pushes what is on the way to make the room it needs. If the chosen place $u_\ell$ is already occupied, the bulldozer searches for the first free point $y$ at the right of $u_\ell$ and parks on $[y, y + 1/n]$, pushing what is needed to clear the place it wants.</figcaption></figure>

– Consider the number $N_k$ of blocks in the (continuous state) bulldozer model, and let $O^{(k)}, F^{(k)}$ denote the state at time $k$ (with $k$ cars of size $1/n$ treated).

– Consider the number $N_k^{(n)}$ of blocks in the (discrete state) parking model, and denote by $O^{(n,k)}, F^{(n,k)}$ the state at time $k$, with an additional exponent $n$ (with $k$ cars of size $1/n$ treated).

By (1.12) and (2.3), if all the $m_i$ are multiple of $1/n$, the discrete and continuous piling propensities are very similar:

$$Q(m[n]) = \mathbb{P}(N_k = 1) = W(m[k])^{k-1} \quad \text{and} \quad Q^{(n)}(m[n]) = \mathbb{P}(N_k^{(n)} = 1) = (1/n + W(m[k]))^{k-1}.$$

This similarity, as well as the resemblance between the block sizes in Theorems 1.9 and 2.6, may give the intuition that discrete and continuous diffusion processes should behave very similarly... However, as we will see, this is only partially true.

One first discrepancy is that the support of $N_k$ and $N_k^{(n)}$ are different for large $k$. Indeed, if $n-k$ cars (masses) have parked (in the parking problem), the remaining space has size $k/n$. Since every free
CC has size at least $1/n$, there can be at most $k$ occupied CC in $O^{(n,n-k)}$: hence the support of $N_{n-k}^{(n)}$ is $[1, \min\{k, n-k\}]$ (the number of CC is also smaller than the number of cars $n-k$).

In the bulldozer case, the free CC may be arbitrary small, so there is a positive probability that every bulldozer is isolated, so the support of $N_{n-k}$ is easily seen to coincide with $[1, n-k]$.

We will give more detailed comparison results in the remainder of this section. The two models exhibit significant similarities and discrepancies depending on the observed statistics.

The similarities concern large occupied CC, while the discrepancies concern small occupied CC and the number of occupied CC.

While comparing the two models, we focus on the phase transition that occurs at time

$$t_n(\lambda) := \lfloor n - \lambda \sqrt{n} \rfloor,$$

for some $\lambda > 0$, at which $t_n$ vehicles/masses have been treated (this phase transition has been identified in [10]). For any $k \ge 1$, and any integer $t \in \{0, \dots, n\}$, set

$$L^{(n),\downarrow}[t, k] := (L_i^{(n)}, 1 \le i \le k),$$

$$L^{\downarrow}[t, k] := (L_i, 1 \le i \le k)$$

respectively, the Lebesgue measures of the $k$ largest occupied CC of $O^{(n,t)}$ in the discrete parking and of $O^{(t)}$, in the continuous parking, both lists being sorted in decreasing order, and completed by zeroes, if there are less than $k$ CC.

**Proposition 3.1.** In the discrete parking,

$$\mathbf{N}_{t_n}^{(n)}/\sqrt{n} \xrightarrow{n} \lambda(1-e^{-1}) \qquad (3.2)$$

while, in the continuous parking

$$\mathbf{N}_{t_n}/\sqrt{n} \xrightarrow{n} \lambda. \qquad (3.3)$$

For any $k$, the following convergence in distribution holds in $\mathbb{R}^k$,

$$n^{-1} L^{(n),\downarrow}[t_n(\lambda), k] \xrightarrow{n} (\ell_i^{\downarrow}(\lambda, \mathbf{e}), 1 \le i \le k) \qquad (3.4)$$

$$n^{-1} L^{\downarrow}[t_n(\lambda), k] \xrightarrow{n} (\ell_i^{\downarrow}(\lambda, \mathbf{e}), 1 \le i \le k) \qquad (3.5)$$

where $(\ell_i^{\downarrow}(\lambda, \mathbf{e}), 1 \le i \le k)$ is the (random) vector formed by the $k$ largest excursion sizes of the process $t \mapsto (e_t - \lambda_t) - \min_{s \le t} (e_s - \lambda s)$, sorted in descending order (where $e$ is a normalized Brownian excursion).

**Remark 3.2.** In Bertoin [6, Lemma 5.11], the limiting fragment sizes sorted in decreasing order are described as the ranked atoms $(a_1, a_2, \dots)$ of a Poisson point measure on $(0, +\infty)$ with intensity $\frac{\lambda}{\sqrt{2\pi a^3}} da$, conditioned by $\sum_{i \ge 1} a_i = 1$. In Aldous & Pitman [3, Theo.4, Cor. 5], Chassaing & Louchard [10, Theo. 1.5], the description of the law of the blocks when sorted according to a size bias order are given.

**Remark 3.3.** Since there can only be a finite number of blocks of size at least $x n$ for any given $x$, the convergence stated in Lemma 3.1 is equivalent to the vague convergence on $(0, 1]$ of the measure $\sum_{O \in O^{(k)}} \delta_{\text{Leb}(O)}/n$ to the sum of the Dirac masses at the excursion lengths of $(e_t - \lambda_t) - \min_{s \le t} (e_s - \lambda s)$.
**Remark 3.4.** The discrepancy in the distribution of the occupied domains between discrete and continuous cases stems from the combined effects of the piling propensity and the “spacing propensity” which is a measure of the set of non-intersecting positioning of CC around the circle:

■ in the continuous case, the measure of the set of the possible distances between b occupied blocks when the remaining space is R, is

$$S(b, R) = R^{b-1}/(b-1),$$

independently from the masses, since this is the measure of the simplex $\{(x_1, \cdots, x_b) \in [0,1]^b, \sum x_i = R\}$,

■ in the discrete case, the number of ways of sharing k (unit) remaining spaces (each of them being of Lebesgue measure 1/n), into b non-empty (integer) blocks is (as explained in (2.1)) $S^n(b, R) = |\text{Comp}(nR, b)| = \binom{nR-1}{b-1}$. Of course, the two formulas of $S(b, R)$ and $S^n(b, R)$ have to be included in a larger picture – this is the goal of Theorems 1.9 and 2.6 – but the fact that these quantities do not have the same support, and are not (close to) “proportional” is the main message here.

*Proof of Proposition 3.1.* Proof of (3.3): We know that $N_{t_n(\lambda)} - 1$ follows the binomial distribution with parameter $(t_n(\lambda) - 1, 1 - t_n(\lambda)/n) = (t_n(\lambda) - 1, \lambda/\sqrt{n})$. Hence $N_{t_n(\lambda)}/\sqrt{n}$ has mean $\lambda + o(1)$ and variance $\lambda O(1/\sqrt{n})$, so that (3.3) follows by Bienaymé-Chebyshev inequality.

*Proof of (3.2)*: By Lemma 2.4, the number of occupied CC would be the same if we would place a “big” mass equals to $t_n(\lambda)/n = (1 - \lambda/\sqrt{n})$ (which by invariance under rotation can be placed wherever we want, say on $[0, t_n(\lambda)/n]$, and $(t_n(\lambda) - 1)$ masses 0. In this case $N_{t_n}^{(n)}$ would be 1 (big block) plus $B_n$, the number of sites in $S_n := [t_n(\lambda)/n + 1/n, (n-1)/n] \cap \mathscr{C}_n$, that receive at least one mass. A given site in $S_n$ receives a binomial $B(t_n(\lambda), 1/n)$ number of zero masses (this converges to a Poisson(1) r.v., in distribution). From this, we easily deduce that since $|S_n| \sim \lambda\sqrt{n}$, $E(B_n) \sim \lambda\sqrt{n}(1 - \exp(-1))$, and $B_n$ is approximately a Poisson distributed r.v.. To obtain (3.2), a concentration argument relying on a second moment method suffices.

We present just some elements on (3.4), (3.5), for which variants exist in the literature. First, to prove the asymptotic proximity between both largest occupied CC, the idea is to construct, on the probability space on which the continuous space parking is defined, with the data $((1/n, \mathbf{U}_i), 0 \le i \le t_n(\lambda))$ of mass arrival events, a discrete counterpart, by taking

$$\mathbf{U}_i^{(n)} := [n\mathbf{U}_i]/n, \qquad (3.6)$$

as location of the $i$th arrival. Hence $\mathbf{U}_i^{(n)}$ is a bit on the right of the continuous one $\mathbf{U}_i$, but, on interval of the form $(k/n, (k+1)/n]$ for some $k$, the numbers of arrivals in the discrete and continuous parking coincide perfectly. We construct the collecting paths (already seen in (1.19)) associated with both parking models. Set, for two discrete time $t$,

$$S^{(n)}(t, x) = -x + \sum_{i \le t} \mathbb{1}_{\mathbf{U}_i^{(n)} \le x}/n, \qquad (3.7)$$

$$S(t, x) = -x + \sum_{i \le t} \mathbb{1}_{\mathbf{U}_i \le x}/n. \qquad (3.8)$$

The term $-x$ represents the deposition on $[0, x]$ while $\sum_{i \le t} \mathbb{1}_{\mathbf{U}_i \le x}/n$, encodes the total mass that has arrived on the same interval. The erosion term contributes even when there is no mass to erode. This method has the advantage of making the occupied components appear under the form of excursions above the current minimum. These collecting paths<fnref n="8" /> encode the occupation of the parking (the idea of

<footnote n="8">⁸The continuous version $S(t, x)$ has been represented on Figure 6</footnote>
this coupling is borrowed to Chassaing & Marckert [11] and can be found also in Bertoin & Miermont [7, Section 5.]). Now, there are essentially two methods to conclude:

• either we prove that $\sqrt{n}S^{(n)}(t_n(\lambda), \cdot)$ converges in distribution in $C[0, 1]$, up to a rotation that we define below, to $e_\lambda$ defined as $e_\lambda: t \mapsto e(t) - \lambda t$, where $e$ is a standard Brownian excursion; and we recover the sizes of the occupied CC by taking into account Figure 6: in words, we need to define a rotated version of $S^{(n)}$ with respect to $a'_n = \min \operatorname{argmin}(t \mapsto S^{(n)}(t_n(\lambda), t) - S^{(n)}(t_n(\lambda), 1)t)$ (which amounts in Figure 6 to put the new origin at $(a'_n, S^{(n)}(t_n(\lambda), a'_n))$ and consider in this new basis, the obtained length 1 trajectory. Technically, this corresponds to set $\bar{S}^{(n)}(t_n(\lambda), \cdot) = \operatorname{Rot}(a'_n, S^{(n)}(t_n(\lambda), \cdot))$ where the rotation is defined more generally, for all function $f$ and $a \in [0, 1]$ by

$$ \operatorname{Rot}(a, f)(x) = \begin{cases} f(a+x) - (\min f) & \text{for } x \in [0, 1-a], \\ f(1) - (\min f) + f(x - (1-a)) & \text{for } x \in [1-a, 1]. \end{cases} $$

Then to complete the proof by the Skorokhod representation theorem, it is possible to find a probability space, containing copies of $S^{(n)}(t_n(\lambda), \cdot)$ and $e_\lambda$ (still denoted by the same names in the sequel), such that $\sqrt{n}\bar{S}^{(n)} \xrightarrow{n}^{(as.)} e_\lambda$, and since $e_\lambda$ reaches its minimum a.s. once (say in $a'$) and one has also on this space $a'_n \xrightarrow{n}^{(as.)} a'$, then $\sqrt{n}\bar{S}^{(n)}(t_n(\lambda), \cdot) \xrightarrow{n}^{(d)} \operatorname{Rot}(a', e_\lambda)$. Now, the convergence of the excursion sizes above the current minimum toward those of the limit is due to Aldous [2, Lemma 7].

• Or we may use the method of [10] which consists in working on the complete parking process conditioned on having the last vertex free when $n-1$ cars are parked. This has no incidence on the distribution of the occupied CC sizes at time $t_n(\lambda)$. In this case the collecting path $S^{(n)}(n-1, \cdot)$ converges in distribution to the Brownian excursion $e$ and,

$$ (\sqrt{n} S^{(n)}(t_n(\lambda), x), 0 \le x \le 1) \xrightarrow{n}^{(d)} (e(x) - \lambda x, 0 \le x \le 1) \quad (3.9) $$

where the convergence holds in $C[0, 1]$, and the convergence holds also jointly, for a finite number of $\lambda_j$ (the proof is [10], but other proofs can be given, e.g. using the argument in [29], and the convergence as a function of $(t, x)$ has also been given in [9], as recalled further in the paper, in (4.30)). From here, again by Aldous [2, Lemma 7], we get (3.4).

Now, it remains to explain why in continuous parking version, (3.5) holds, that is, why the result are the same in discrete and continuous parking, regarding large excursion sizes. We have, for a fixed $\lambda$,

$$ \sqrt{n} (S^{(n)}(t_n(\lambda), \cdot), S(t_n(\lambda), \cdot)) \xrightarrow{n}^{(d)} (b_\lambda, b_\lambda) \quad (3.10) $$

for the Skorokhod topology on $D([0, 1], \mathbb{R}^2)$. We have already explained the convergence of the first marginal, so it suffices to explain why the uniform distance between the two processes goes to zero in probability. This has already been proven in Chassaing & Marckert [11] and is simple: the two processes $(S(t, x), x \in [0, 1])$ and $(S^{(n)}(t, x), x \in [0, 1])$ coincide at the points $x = k/n$ (with $k$ integer). Let $R(k, t) = \{i: i \le t, U_i \in (k/n, (k+1)/n]\}$ be the binomial distributed number of cars arrived in $(k/n, (k+1)/n]$ at time $t$, and clearly

$$ \sup_{x \in [0,1]} |S^{(n)}(t,x) - S(t,x)| \le \max_{0 \le k \le n-1} R(k,t)/n \le \max_{0 \le k \le n-1} R(k,n)/n, $$

which is then the corresponding value in the completely full parking. Now, it is well known that there exists a constant $C > 0$ such that

$$ \mathbb{P}\left(\max_{0 \le k \le n-1} R(k, n) \ge C \log n\right) \le n \mathbb{P}(R(1, n) \ge C \log n) \le 1/n $$
for C large enough.

Again, Aldous [2, Lemma 7] allows us to deduce the convergence of the second marginal in (3.10).

□

### 3.2.1 Extension to more general discrete masses model

The general message, here, is that if we are given a model of masses ($m_i, 0 \le i \le t_n(\lambda)$), where the $m_i$ are multiple of $1/n$ (the $m_i$ being random or not), taken at some time $t_n(\lambda)$, random or not<fnref n="9" />. Again the collecting paths $S^{(n)}(t, \cdot)$ and $S(t, \cdot)$ associated with the discrete and continuous models as defined in (3.7) are some tool to consider first.

For all models for which the range of $S^{(n)}$ is typically of order larger than $1/n$ (globally), the large excursions of $S^{(n)}(t, \cdot)$ and $S(t, \cdot)$ above their current minima will be asymptotically identical. This can be proven using the arguments in the proof of Proposition 3.1 together with Aldous [2, Lemma 7] and Bertoin [5, Lemma4] (see also Bertoin & Miermont [7, Proof of Prop. 1]):

**Theorem 3.5.** Consider a sequence of DDM (or CDM) such that, for some sequence of times $(t_n, n \ge 0)$,

$$ (\alpha_n S^{(n)}(t_n, x), x \in [0, 1]) \xrightarrow[n]{(d)} (S^{\infty}(x), x \in [0, 1]) $$

where $\alpha_n/n \to 0$ and $\alpha_n \to +\infty$, with $S^{\infty}(1) < 0$, and where the convergence holds in $C[0, 1]$ (or in $D[0, 1]$). If $S^{\infty}$ reaches its minimum a.s. once on $[0, 1]$ at some $a \in [0, 1]$, and if moreover $S$ has a.s. the property that its excursions above the current minimum are strict (in the sense that there are no strict non-trivial sub-excursions of any excursion at the same level), then for $S = \text{Rot}(a, S^{\infty})$, the sequence formed by the $k$ largest clusters converges in distribution to the length of the excursions of $S$ above its current minimum

$$ n^{-1} L^{(n), \downarrow}[t_n, k] \xrightarrow[n]{(d)} (\ell_i^{\downarrow}(S), 1 \le i \le k) $$

$$ n^{-1} L^{\downarrow}[t_n, k] \xrightarrow[n]{(d)} (\ell_i^{\downarrow}(S), 1 \le i \le k) $$

* We refer to Bertoin [5] for the right way of defining excursions in the case where $S$ has some jumps,

* We could have stated convergence for the vague topology instead (on $(0, 1]$).

Both trajectories $S^{(n)}(t, \cdot)$ and $S(t, \cdot)$ coincides at the discrete points $(k/n, 0 \le k \le n)$. Moreover the evolution of the $-x$ term is $-1/n$ on such an interval. A small figure is sufficient to see that, if $\alpha_n S^{(n)}(t, \cdot)$ converges in distribution to a limit in $D[0, 1]$, then, if $\alpha_n = o(n)$, the largest excursion lengths of $S^{(n)}$ and of $S$ above their current minimum should be close asymptotically. The condition $\alpha_n = o(n)$ is necessary, because if $S^{(n)}$ has order $1/n$, then the block sizes are expected to be very small (of order $1/n$), and the asymptotic behavior of $\alpha_n S^{(n)}(t, \cdot)$ would not be an appropriate tool for studying them. As exemplified in the parking case, statistics concerning small occupied CC are expected to differ in DDM and CDM.

## 3.3 Random models (and caravans)

By Theorems 1.5, 1.8, 1.11, 2.3, 2.6 and 2.7, the study of occupied CC of all valid continuous or discrete diffusion processes reduces to that of right diffusion with constant speed, which are called “caravan” in Bertoin & Miermont [7] (up to the open/close representation of the occupied space, and the processing of zero masses, see Lemma 1.4).

<footnote n="9">⁹when the masses are taken as i.i.d. random variables, stopping the diffusion process just before the space overflow, that is, at time $\tau - 1$ with $\tau = \inf\{t: m_0 + \cdots + m_{t-1} > 1\}$, provides a random time as which it is natural to stop the diffusion process.</footnote>
### 3.3.1 Bertoin & Miermont results about caravans

To study the asymptotic behavior of valid CDM or DDM when the number of masses goes to $+\infty$, we need to make some hypotheses about the masses. The case in which all masses are the same size corresponds to parking$^{10}$ and was discussed in Section 3.2.

When the number $t_n$ of masses goes to $+\infty$ (with sum in $[0, 1)$) and the masses are random, the collecting path $W(t, \cdot)$ will have a limiting distribution under regularity assumptions on the masses (for example, the fact that they are i.i.d., have some moments, or a regular tail).

Bertoin & Miermont [7] consider masses ($m_i, i \ge 0$) that come from some normalized i.i.d. random variables ($l_i, i \ge 0$) with finite expectation $E(\ell) = \mu_1 < +\infty$ and either finite second moment ($\mu_2 = E(\ell^2)$, in this case, $\ell$ is said to be in $\mathcal{D}_2$), or have a regular tail:

$$ \mathbb{P}(\ell > x) \sim cx^{-\alpha} $$

for some $c \in (0, +\infty)$ and $\alpha \in (1, 2)$. In these cases, $\ell$ is in the domain of attraction of spectrally positive stable distribution with index $\alpha$, and said to be in $\mathcal{D}_\alpha$). Then, they consider for (small) $\varepsilon > 0$,

$$ \tau_{1/\varepsilon} = \inf\{i : \ell_0 + \cdots + \ell_i > 1/\varepsilon\} $$

and they work on a circle of size $1/\varepsilon$, with these masses, but in terms of the present paper, this amounts to taking $k = 1 + \tau_{1/\varepsilon}$ masses of size

$$ m_i = \varepsilon \ell_i, \text{ for } i < \tau_{1/\varepsilon}, $$

and with the last mass taken to complete “1”, that is $m_{\tau_{1/\varepsilon}} = 1 - W(m[\tau_{1/\varepsilon}])$. They then prove the convergence of the associated collecting path ((1/ε)$^{1/\alpha}$S_t, 0 ≤ t ≤ 1) toward an analogue of the Brownian bridge with a linear drift for stable processes, the standard stable loop. See the definition in Bertoin & Miermont [7], in Formula (2) and the surrounding text. From this, they deduce the finite-dimensional convergence in distribution, of the sizes of the largest occupied CC in their Theorem 1, in a critical time window defined in terms of $\alpha$.

Additional discussion concerning discrete masses arriving on the discrete circle $\mathbb{Z}/n\mathbb{Z}$ are provided in their Section 5, notably a coupling equivalent to (3.7) is introduced for describing the limiting discrete parking in terms of the continuous one.

### 3.3.2 Other conditioned tractable models

We can imagine two other classes of models of mass that lead to asymptotic tractable models: (1). i.i.d. random masses ($m_i$) conditioned by their sums. Their numbers $K_n$ can be random or not.

(2). Prescribed deterministic masses ($m_i$).

In case (2), in order to obtain convergence of the collecting paths, some regularities are required. For example, one can fix in the $n$th model, the proportion $p_j(n)$ of masses equal to $w_j(n)$, and then demand some regularities for $p_j(n)$ and $w_j(n)$ (for example, require that $p_j(n) \to p_j$ for each $j$, and $nw_j(n) \to w_j$, and $\sum p_j = 1$, $\sum p_j w_j \le 1$). In other words, we prescribe the proportion of masses of each type, and require some convergence (in general additional assumption are needed to get a limiting behavior, if the number of possible masses types “j” goes to $+\infty$), see e.g. [8].

<footnote>$^{10}$even if the size of the cars is $\varepsilon > 0$ and is not of the form of $1/n$, one may take a discretization of $[1/\varepsilon]$ parking places with an additional “incomplete place” which can clearly be ignored in the asymptotic range of Section 3.2</footnote>
To illustrate the case (1) without going into too much detail, we can for example, construct the masses using i.i.d. random variables ($X_i$) taking their values in $\mathbb{N}$ (this simplifies the condition), with common distribution $\mu$, mean $m = 1$ and variance $\sigma^2 \in [0, +\infty)$, and take as mass

$$m_i = X_i / n.$$

With this choice, the space is naturally totally occupied at time $K_n$ close to $n$ (with $\sqrt{n}$ fluctuations), and we may then require that, for example at time $K_n = n - a\sqrt{n}$ the free space is $b/\sqrt{n}$.

Instead, we propose requiring that it be completely full at time $n$, which allows us to let time go backwards without getting lost in the change of time or variable and without altering the nature of the phenomenon at play. We will then condition by the event

$$\mathcal{E}_n := \left\{ \sum_{i=0}^{n-1} m_i = 1 \right\} = \left\{ \sum_{i=0}^{n-1} X_i = n \right\}.$$

When one writes the final collecting path

$$S_n^{(n)}(x) = -x + \sum_{i=0}^{n-1} (X_i/n) \mathbb{1}_{u_i \le x},$$

one sees that there is a binomial number $B(n, x)$ masses that falls in $[0, x]$ (and if one is interested in the multidimensional distributions $S(x_i) - S(x_{i-1})$ where $x_0 = 0$ and $x_k = 1$, then one notices that

$$(|\{i : u_i \in (x_{j-1}, x_j]\}|, 1 \le j \le k) =: (\Delta_j(n), 1 \le j \le k) \quad (3.11)$$

$$\sim \text{Multinomial}(n, (x_j - x_{j-1}, 1 \le j \le k)). \quad (3.12)$$

This means that, in distribution, since the $X_i$ are exchangeable, if one defines

$$W_n(k) = \sum_{j=1}^{k} X_j / n, \quad k \in \{0, \dots, n\},$$

one has

$$\sqrt{n}(S_n(x_j), 1 \le j \le k) \stackrel{(d)}{=} \sqrt{n}(-x_j + W_n(\Delta_1(n) + \dots + \Delta_j(n)), 1 \le j \le k) \quad (3.13)$$

$$= \sqrt{n}\left(-x_j + \frac{\Delta_1(n) + \dots + \Delta_j(n)}{n}, 1 \le j \le k\right) \quad (3.14)$$

$$+ \sqrt{n}\left(W_n(\Delta_1(n) + \dots + \Delta_j(n)) - \frac{\Delta_1(n) + \dots + \Delta_j(n)}{n}, 1 \le j \le k\right). \quad (3.15)$$

We have

- If $\sigma^2 = 0$, then $X_i = 1$ a.s. (this is the parking case), $W_n(k) = k/n$, so that the term in (3.15) vanishes, and $\sqrt{n}S_n(\cdot)$ converges to the Brownian bridge in distribution.

- If $\sigma^2 > 0$, under the hypotheses we have on ($X_i$),

$$(w_n(t), t \in [0, 1]) := (\sqrt{n}(W_n(nt) - t), t \in [0, 1]) \xrightarrow{n} (\sigma b_t, t \in [0, 1])$$
in $C[0, 1]$, where $b$ is a Brownian bridge (we assume here that $W_n$ is interpolated linearly between integer positions).$^{11}$ Since $x_j^{(n)} := \frac{\Delta_1(n)+\cdots+\Delta_j(n)}{n} \xrightarrow{\text{(proba.)}} x_j$, the rhs of (3.15), which coincides with $(w_n(x_j^{(n)}), 1 \le j \le k)$ satisfies

$$ (w_n(x_j^{(n)}), 1 \le j \le k) \xrightarrow{n} \sigma(b_{x_1}, \cdots, b_{x_k}), $$

for $\max_{1 \le j \le k} |x_j^{(n)} - x_j| \xrightarrow{n} 0$ together with the tightness of $(\tilde{w}_n)$ implies that $(w_n(x_j^{(n)}) - w_n(x_j), 1 \le j \le k) \xrightarrow{n} \sigma(b_{x_1}, \cdots, b_{x_k})$.

On the other hand, by standard limit theorem on multinomial random variables

$$ \sqrt{n} \left( -x_j + \frac{\Delta_1(n) + \cdots + \Delta_j(n)}{n}, 1 \le j \le k \right) \xrightarrow{n} (\tilde{b}_{x_j}, 1 \le j \le k) $$

where $b$ and $\tilde{b}$ are two independent Brownian bridges. Hence, in $C[0, 1]$ $^{12}$,

$$ \sqrt{n} S_n(\cdot) \xrightarrow{n} \sqrt{1 + \sigma^2} b. $$

**Proposition 3.6.** For all compact interval $\Lambda \subset [0, +\infty)$

$$ \left( \sqrt{n} S_{n-\lambda\sqrt{n}}(x) - \sqrt{n} S_n(x), 0 \le x \le 1, \lambda \in \Lambda \right) \xrightarrow{n} (-\lambda x, 0 \le x \le 1, \lambda \in \Lambda) \quad (3.16) $$

where the convergence holds in $C([0, 1] \times \Lambda, \mathbb{R})$, and

$$ \left( \sqrt{n} S_{n-\lambda\sqrt{n}}(x), 0 \le x \le 1, \lambda \in \Lambda \right) \xrightarrow{n} \left( \sqrt{1+\sigma^2} b_x - \lambda x, 0 \le x \le 1, \lambda \in \Lambda \right) $$

for the same topology.

*Proof.* The proof of (3.16) here is simple because the convergence of the FDD is a consequence the concentration of a sum of random variables. The tightness comes from the fact that $(\lambda, x) \mapsto \sqrt{n}S_{n-\lambda\sqrt{n}}(x)$ is monotonous in both $\lambda$ and in $x$ and the limit is continuous. In the deterministic case, pointwise convergence toward a continuous function is uniform (on compact sets) as soon as the functions are monotonous. More details are given in [29], for “a similar correction phenomenon”, and additional elements on topology are given in the Appendix there. $\square$

In Aldous [4, p.168-169], general criteria for the tightness of “auto-normalized” processes with ex-changeable increments (such as those in the two previous examples) are given under the hypothesis that the sum of the increments are zero and sum of their square is 1. This hypothesis must be slightly adapted here, since we want the limiting trajectories to end at a negative height (while regarding this height as the rescaled limit of the number of occupied components).

In terms of excursion length above the current minimum, the limit is then the same as for the parking process.

<footnote>$^{11}$The proof of this fact is an exercise, several methods being available: the convergence of the finite dimensional distributions (FDD) can be proven by proving a local limit theorem for them, and this latter being a consequence of the central local limit theorem. The tightness can be proven to be a consequence of the tightness of the same family of processes without the eventual conditioning by $S_n = n-1$ (see Proof of Lemma 1 in [20]). An alternative proof (of the weak convergence) consists of using Kaigh's result [21] about the weak convergence in $C[0, 1]$, of random walks conditioned to hit -1 for the first time at the end, to Brownian excursion, and then use the fact that the random walk only conditioned by $S_n = -1$ are obtained by a random re-rooting of those conditioned to hit -1 at the end).</footnote>

<footnote>$^{12}$The Brownian bridge is a centered Gaussian process with covariance function $\text{cov}(b_s, b_t) = s(1-t)$ for $0 \le s \le t \le 1$. Hence, the sum $\bar{b} := c_1 b + c_2 \bar{b}$ for two independent Brownian bridges $b$ and $\bar{b}$, and two constants $c_1, c_2$ gives as a covariance function $\text{cov}(\bar{b}_s, \bar{b}_t) = (c_1^2 + c_2^2)(s(1-t))$</footnote>
# 4 Energy dissipation and other cost associated with dispersion model

This section addresses a natural question related to dispersion models: Assume that a mass $m_k$ arrives at some point $u_k$ within a partially occupied space $\mathcal{C}$. In a physical system, or in an abstract data structure (as are parking models, or hash tables), the mass – which may be composed of smaller particles – while it undergoes its dispersion process (until occupying a new space $\mathbf{O}^{(k)}\setminus\mathbf{O}^{(k-1)}$), have(s) moved or flowed, or maybe has lost some potential energy, or been pushed along a given distance, or been submitted to some other physical process during a particular time. In many situations, the main quantities of interest in a diffusion process are related to these additional considerations.

Assume that a “unitary cost” (a real number) can be associated with the dissipation of $(m_k, u_k)$. The “global cost” at time $t$ will then be defined as the sum of the unitary costs of the first $t$ masses. For example, in the RDCS, a mass $m$ can be seen as composed of tiny masses $dm$, and if a mass $m$ arrives at a point $u$ inside $\mathbf{O}^{(k)}$, if one assumes that transporting this mass $dm$ at distance $x$ has a cost $x dm$, then the resulting cost of the displacement is

$$ \int_{\mathbf{O}^{(k+1)}\setminus\mathbf{O}^{(k)}} \vec{d}(u, y)dy $$

(recall notation given in Section 1.4).

The problem we face is that the arrival of a single mass, on a CC $O$ of $\mathbf{O}^{(k)}$ can result in the coalescence of many occupied CC. In general it is quite intricate to study, and even to define such a cost model, since we allow CDM to depend on the current occupied interval during the dispersion.

The simplest models are the DDM in which the masses only have size $1/n$ on $\mathcal{C}_n$. These models are simpler because the dispersion of the $k$th mass stops when it gets out of the CC of $\mathbf{O}^{(k)}$ in which it arrived (on its left or on its right), so that the global cost is a function of the collections of arrival block sizes ($|B_j|, j \le t$). In this case, the cost is entirely determined by the “unitary cost model” which specifies the unitary cost distribution, for a single car arriving uniformly in an interval of length $\ell$, for all $\ell$. This type of models has been studied for “simple cost models”, for analysis of algorithm purpose mainly.

We first discuss at length this case, and will come back for partial results, in the general case, in Section 4.3.

## 4.1 Cost of parking construction for general unitary cost function

Before discussing general costs, let us review standard parking cost functions. To align with the standard representation, we will work with the set $\mathbb{Z}/n\mathbb{Z}$ and will come back to $\mathcal{C}_n$ later on. So, let us write $\mathbf{c}_0, \cdots, \mathbf{c}_{k-1}$ the arrival places of the cars, that are now i.i.d. uniform on $\mathbb{Z}/n\mathbb{Z}$ (with the letter “c” for the choice). In the standard parking problem, the car $i$ parks at the first available place among $\mathbf{c}_i, \mathbf{c}_i + 1 \mod n, \cdots$, say at place $\mathbf{c}_i + d_i \mod n$ (with $d_i \in \{0, \cdots, n-1\}$). Therefore, $d_i$ is the distance car $i$ has to do from its choice $c_i$ to its final parking spot.

Define

$$ \text{Cost}(k) = \sum_{i=0}^{k-1} d_i. \qquad (4.1) $$

**Remark 4.1.** In computer science, the hashing with linear probing is an algorithm devoted to store some data ($x_i, i \ge 0$), taken in a set $\mathcal{D}$ from more or less any type, in an array whose entries are labeled by $\mathbb{Z}/n\mathbb{Z}$. A hash function $h: \mathcal{D} \to \mathbb{Z}/n\mathbb{Z}$ is applied, and data $x_i$ is stored in the first available place among $h(x_i) + k \mod \mathbb{Z}/n\mathbb{Z}$, for $k$ ranging from 0 to $n-1$. Up to the vocabulary, this is the parking model under the
hypothesis that the hash function produces (almost) uniform random variables. Under this hypothesis, Cost(k) corresponds to the number of cell availability tests in the array needed to store the first k data $x_i$, and it is therefore a crucial parameter in the analyzing the cost of this algorithm.

Let us consider again the phase transition that occurs at time

$$t_n(\lambda) := \lfloor n - \lambda \sqrt{n} \rfloor,$$

**Proposition 4.2.** We have the following convergences in distribution, [Flajolet & Poblete & Viola [16], Janson [18], Chassaing & Marckert [11]],

$$\text{Cost}(n)/n^{3/2} \xrightarrow{n} \int_0^1 e(t)dt$$

and [Chassaing & Louchard<fnref n="13" /> [10]]

$$\text{Cost}(t_n(\lambda))/n^{3/2} \xrightarrow{n} \int_0^1 e_{\lambda}(s) - e_{\underline{\lambda}}(s)ds. \quad (4.2)$$

where e is again, the Brownian excursion, and $e_{\lambda}(s) = e(s) - \lambda s$, and $e_{\underline{\lambda}}(s) = \min_{t \le s} e_{\lambda}(t)$.

**The “cluster size insertion sequence”.**

Let us call an occupied cluster, a successive set of occupied places (as discussed above, an occupied CC $[a, b]$ on $\mathbb{R}/n\mathbb{Z}$, corresponds to the occupied cluster $\{a, a+1 \mod n, \dots, b-1 \mod n\}$ in the standard parking terminology.

When the $k$th car arrives in the parking there are two possibilities. Either:
- the choice $c_k$ is an element of an occupied cluster $C = \{a, a+1 \mod n, \dots, a+s-1 \mod n\}$ of size $s := s \ge 1$, and this occurs with probability $s/n$ for an occupied cluster of size $s$; in this case conditional on the fact that $c_k \in C$, then $c_k$ is uniform in $C$,

- or it arrives on a free site $f$ and parks there; its arrival cluster is then considered to have size $s_k = 0$. Each free place at time $k$ has probability $1/n$ to be the choice of car $c_k$.

Let UCost(s) denote the unitary cost corresponding to the cost arrival in a block of size $s$ (knowing that the arrival occurs in such a block, the position is uniform within it). In general, we have

$$\text{Cost}_n(t) = \sum_{k \le t} \text{UCost}_k^{(s_k)} \quad (4.3)$$

where again, $s_k$ is the size of the occupied cluster in which arrives the $k$th car, and the index $k$ at UCost_k(s_k) indicates that we take different and independent copies of UCost(s) if several blocks of the same size $s$ are considered. Two main factors influence the global cost Cost_n: the sequence of distributions of the unitary cost random variables (UCost(s), $s \ge 0$), and (s_k) the cluster size insertion sequence.

<footnote n="13">¹³In [10], (4.2) is not stated, but it is a consequence of their Lemma 3.7, in which they prove the convergence in distribution of the Łukasiewicz path encoding the parking at time $n - \lambda \sqrt{n}$, and normalized by $n^{1/2}$ and suitably change of origin, to $(e_{\lambda}(s) - e_{\underline{\lambda}}(s), s \in [0, 1])$. The cost being an integral of the Łukasiewicz path, the convergence stated in our Proposition 4.2 follows (this tool is used in a lot of places in the literature, including [11], [9], [30]).</footnote>
## Asymptotic behavior of the cluster size insertion sequence.

We already have some information on the cluster size sequence (see Lemma 3.1), and additional can be found in Pittel [33], Chassaing & Louchard [10], and in other cited references. At time $t_n(\lambda)$, only $\lambda\sqrt{n}$ cars are lacking, and the sequence $(s_k, k \le t_n(\lambda))$ is composed by $t_n(\lambda)$ elements, most of them being very small (by Pittel, for $c \in [0, 1)$ fixed, and some $C > 0$, at time $cn$, all the clusters have size $\le C \log n$ with probability converging to 1). However, many of them are large as well. First, we establish a limit theorem for the empirical measure associated with the “large blocks” in the cluster size insertion sequence. Let

<figure><img src="image_10.png" /><figcaption>Figure 9: Illustration of a Brownian excursion, the pair $(L(x, y), R(x, y))$ associated with a point $(x, y) \in \text{under}(e)$. The slanted dotted line has slope $S(x, y)$.</figcaption></figure>

$\mathcal{L}(e)$ be the set of abscissa of local minima of a Brownian excursion $e$ and let $\text{under}(e) = \{(x, y) : 0 < x < 1, 0 \le y \le e_x\}$ be the set of points under the Brownian excursion curve. We say that $e$ is above a linear map $\ell: [0, 1] \to \mathbb{R}$ on $[a, b]$, if, for any $x \in [a, b]$, $e_x \ge \ell(x)$ (see Figure 9). For any $(x, y) \in \text{under}(e)$ and $u \in \mathcal{L}(e)$ such that $u < x$, define the linear map $\ell_{u,x}: [0, 1] \to \mathbb{R}$ of the line passing by $(u, e_u)$ and $(x, y)$ (that is $\ell_{u,x}(u) = e_u$ and $\ell_{u,x}(x) = y$). Now, let $L(x, y)$ be the largest $u \in \mathcal{L}(e)$ such that $u < x$ and such that $e$ is above $\ell_{u,x}$ on $[0, x]$. Then, set $R(x, y)$ be the abscissa of the first intersection of $\ell_{u,x}$ with the graph of $e$, on $[x, 1]$. Let $S(x, y)$ be the slope of $\ell_x := \ell_{u,x}$.

It is easy to see that the map $(x, y) \to (L(x, y), R(x, y), S(x, y))$ is indeed measurable

**Lemma 4.3.** For any continuous function $f: [0, 1] \to \mathbb{R}$ such that $|f(x)| \le Cx$ for some constant $C$, set

$$ \langle f, M_{\lambda}(e) \rangle = \int_{0}^{1} \int_{0}^{e_x} \frac{2}{R(x, y) - L(x, y)} f(R(x, y) - L(x, y)) \mathbf{1}_{S(x,y) \ge \lambda} dy dx. \quad (4.4) $$

* Almost surely, $\langle f, M_{\lambda}(e) \rangle$ is well defined and finite,

* $M_{\lambda}(e)$ is a positive (random) measure on $[0, 1]$ (equipped with the Borelian $\sigma$-field).

* The measure $M_{\lambda}(e)$ has a.s. infinite total mass.

In the sequel we will often write $M_{\lambda}$ instead of $M_{\lambda}(e)$.

*Proof.* Notice that for $f := x \mapsto x$ (we will write sometimes $f = \text{Id}$)

$$ \langle f, M_{\lambda}(e) \rangle = 2 \int_{0}^{1} \int_{0}^{e_x} \mathbf{1}_{S(x,y) \ge \lambda} dy dx = 2 \int_{0}^{1} e_{\lambda}(x) - \underline{e}_{\lambda}(x) dx < +\infty \text{ a.s..} \quad (4.5) $$
This last property can be seen by a simple picture (the first integral is the Lebesgue measure of the points above some chords in e, and, in the graphical representation of e_λ - e_λ, the corresponding chords, correspond to the intervals that support the excursions above zero).

The fact that $M_λ$ has a.s. infinite total mass is proven in Lemma 4.7. □

**Theorem 4.4.** Suppose that for each $s$, $UCost^{(s)}$ is in $L^2$, and set

$$ \Psi(s) = \mathbb{E}(UCost^{(s)}), \quad V(s) = \mathrm{Var}(UCost^{(s)}). $$

Assume that there exists a sequence $(\alpha_n)$ with $\alpha_n \to +\infty$ such that the following conditions hold:

(i) for all $t > 0$, $\Psi(|nt|)/\alpha_n \to \Psi_\infty(t)$ uniformly on $[a, 1]$ for all $a > 0$, where $\Psi_\infty$ is continuous on $[0, 1]$ and satisfies $|\Psi_\infty(t)| \le Ct$ for some constant $C \ge 0$, in a neighborhood of zero.

(ii) For $\varepsilon > 0$

$$ \limsup_n \frac{1}{\alpha_n n^{1/2}} \left[ \frac{n}{2} \Psi(0) + 4n \sum_{k=1}^{n\varepsilon} \frac{\Psi(k)}{k^{3/2}} \right] = o(\varepsilon). \quad (4.6) $$

(iii) For $\varepsilon > 0$,

$$ \limsup_n \frac{1}{\alpha_n^2} \left[ \frac{V(0)}{2} + 4 \sum_{k=1}^{n\varepsilon} \frac{V(k)}{k^{3/2}} \right] = o(\varepsilon). \quad (4.7) $$

(iv) For all $\varepsilon > 0$

$$ \limsup_n \frac{1}{\sqrt{n}\alpha_n^2} V(|nx|) = 0. \quad (4.8) $$

Under these four conditions, for all $\lambda \ge 0$,

$$ \frac{\mathrm{Cost}_n(|n - \lambda\sqrt{n}|)}{\sqrt{n}\alpha_n} \xrightarrow{n} \langle \Psi_\infty, M_\lambda \rangle. \quad (4.9) $$

**Remark 4.5.** Some conditions, in the theorem, are probably stronger than needed. For example, the condition on $\Psi_\infty$ near zero is sufficient (but probably not necessary) to have $\langle 1_{[0,\varepsilon]}|\Psi_\infty|, M_\lambda \rangle \to 0$ in probability as $\varepsilon$ goes not 0, because

$$ \langle 1_{[0,\varepsilon]} | \Psi_\infty |, M_\lambda \rangle \le C \int_0^1 \int_0^{e_x} 1_{R(x,y)-L(x,y)\le\varepsilon} dy dx \xrightarrow{\varepsilon\to 0} 0. \quad (4.10) $$

### 4.1.1 Applications of Theorem 4.4

In each of the following case, the expectation of the limit can be computed thanks to the formulas of Section 5.3.

**1-Standard parking:** $UCost^{(k)}$ is $1+U_k$ where $U_k$ is uniform on $\{0, \dots, k-1\}$ (in general, the cost is defined as the set of “tries” a car needs to park, but this +1 can be dropped, if one prefers, it does not change the asymptotic results). We then have

$$ \Psi(k) = (k+1)/2, \quad V_k = (k^2-1)/12, \text{ for } k \ge 1 $$

and $\Psi(0) = 1, V(0) = 0$.

It follows the hypothesis of Theorem 4.4 holds, since:

- (i) with $\alpha_n = n$, $\Psi_\infty(t) = t/2$,
(ii) write $\frac{1}{\alpha_n n^{1/2}} \left( \frac{n}{2} \Psi(0) + 4n \sum_{k=1}^{n\epsilon} \frac{\Psi(k)}{k^{3/2}} \right) = \frac{1}{\alpha_n n^{1/2}} 4nO((n\epsilon)^{1/2}) = o(\epsilon)$.

(iii) write $\frac{1}{\alpha_n^2} \left( \frac{V(0)}{2} + 4 \sum_{k=1}^{n\epsilon} \frac{V(k)}{k^{3/2}} \right) = \frac{1}{n^2} O((n\epsilon)^{3/2}) = o(\epsilon)/\sqrt{n}$ – (iv), immediate.

Notice that (ii) is tight in this case. We then have

$$ \frac{\mathrm{Cost}_n(\lfloor n - \lambda \sqrt{n} \rfloor)}{n^{3/2}} \xrightarrow{n} \langle \Psi_\infty, M_\lambda \rangle. \qquad (4.11) $$

and we recover by (4.5), the Chassaing & Louchard results recalled in Lemma 4.2.

**2-Parking with random direction:** If upon arrival the drivers choose to go to the right with probability $p$, and to the left with probability $1-p$, the unitary cost has the same distribution as for the previous point: the same asymptotic result holds.

**3-Choosing the closest place strategy:** This strategy provides a cost $1+\min\{U_k, k-U_k\}$ for $U_k$ a uniform random variable on $\{0, \dots, k\}$. It follows that $\Psi(k)/k \to 1/3$ and $V(k)/k^2 \to 3/80$, since $(1+\min\{U_k, k-U_k\})/k$ is a sequence of uniformly bounded random variable converging in distribution to a $\beta(1,3)$ random variable, its mean and variance converges to those of the $\beta(1,3)$. Besides $V(k) = O(k^2)$ (the maximum variance for a random variable with support $\lfloor 1, k+1 \rfloor$). Hence, the standard parking (1) analysis above applies, with $\Psi_\infty(t) = t/3$ instead (so, asymptotically, a factor $(1/3)/(1/2) = 2/3$ of the usual parking cost).

**4-Making a p-random walk to find a place, with $p > 1/2$:** $UCost^{(k)}$ is the exit time of an interval of size $k$ by a p-random walk, starting from a uniform point. We skip the details since the asymptotic are the same for two costs that differs from a constant. In this case $UCost^{(k)}/k$ converges in distribution to $U/(2p-1)$ (the asymptotic speed of a p random walk is $1/(2p-1)$), and $U$ is uniform on $[0,1]$. The convergence of $\Psi(k)/k$ and $V(k)/k^2$ to $\mathbb{E}(U)/(2p-1) = 1/(2(2p-1))$ and to $V(U)/((2p-1)^2) = 1/(12(2p-1)^2)$ can be proven using the Hoeffding inequality which ensures exponentially rare deviation for $UCost^{(k)}/k$ from its expected values (at a smaller scale). Hence,

$$ \mathrm{Cost}(n - \lambda \sqrt{n})/n^{3/2} \xrightarrow{n} \int_0^1 \frac{e_\lambda(s) - e_\lambda(s)}{2p-1} ds \qquad (4.12) $$

**5-Making a p-random walk to find a place, with $p = 1/2$:** Here $UCost^{(k)}$ is the time $T^{(k)}$ needed for a centered random walk to exit an interval of time $k$, starting from a uniform position. It can be proven that

$$ \Psi(k)/k^2 \to 1/6, V(k)/k^4 \to 19/180. $$

There are two main methods to prove this result. The first one consists in noticing that $UCost^{(k)}/k^2 \xrightarrow{n} T_U$ where $T_U$ is the needed time needed for a Brownian motion $B^{(U)}$ starting from $U$ uniform on $[0,1]$ to exit the interval $[0,1]$, which is a consequence of Donsker. Since by Lépingle [27], for a Brownian motion $B^{(U)}$ starting at a fixed point $u \in [0,1]$, the exit time $T_u$ from $[0,1]$, satisfies

$$ \mathbb{E}[\exp(\alpha^2 T_u/2)] = \frac{\cos(\alpha(1-2u)/2)}{\cos(\alpha/2)} $$
for $0 \le \alpha \le \pi$. Follows that,

$$ \mathbb{E}[\exp(\alpha^2 T_U/2)] = \frac{2\sin(\alpha/2)}{\alpha \cos(\alpha/2)}, \qquad (4.13) $$

and then $T_U$ admits all its moments, and $\mathbb{E}(T_U) = 1/6$ as well as $\mathbb{E}(T_U^2) = 2/15$ can be obtained by expansion of $\mathbb{E}(\exp(xT_U))$ (set $\alpha^2/2 = x$ in (4.13)). Now, to prove that the moments of $T^{(k)}/k^2$ converges to those of $T_U$ some uniform integrability argument can be used, for example, Komlos, Major and Tushnády [23] strong convergence theorem (with speed), or again, the second part of Lépingle [27] paper, who computed the Laplace transform of the exit time of a random walk from an interval, which ensures the convergence of the Laplace transform of $T^{(k)}/k^2$ to that of $T_U$ (for positive arguments, in a neighborhood of zero), from which convergence of moments are immediate. A second proof, totally different, would rely on martingale argument: it is well known that the hitting time $T_{a,b}$ of $\{a, b\}$ for a random walk starting at 0, were $a < 0 < b$, satisfies $\mathbb{E}(T_{a,b}) = -ab$, and this can be proven by the martingale stopping time theorem, applied to the martingale $(W_k^2 - k, k \ge 0)$ where $W_k$ is the simple symmetric random walk. The other moments of $T_{a,b}$ can be computed using the family of martingales, indexed by $c$, $Z_n^{(c)} = e^{cW_n}(\cosh c)^{-n}$ (see also Lépingle [27], and Chung [12, Sec. II.3] for Brownian motion techniques that can be adapted to random walks). Now take $\alpha_n = n^2$. Since $\Psi(k)/k^2 \to a := 1/6$ and $V(k)/k^4 \to b := 19/180$

$$ \Psi(nt)/n^2 \to at^2, \limsup_n \sup_{x \ge \varepsilon} \frac{V(|nx|)}{\sqrt{n}\alpha_n^2} \to 0. $$

Since $\Psi(k)/k^2 \le C$ and $V(k)/k^4 \le C$ by convergence, for a well-chosen constant $C$, one can check that (ii), (iii) and (iv) hold. (The convergence in (i) follows the fact that $\mathbb{E}(T_{a,b}) = -ab$ from which the results follow. We then have

$$ \text{Cost}(n - \lambda \sqrt{n}, p)/n^{5/2} \xrightarrow{n}^{(d)} \langle g, M_{\lambda} \rangle = \frac{1}{3} \int_{0}^{1} \int_{0}^{e_x} (R(x, y) - L(x, y)) \mathbf{1}_{S(x,y) \ge \lambda} dy dx \quad (4.14) $$

where $g(x) = x^2/6$.

## 4.2 Proof of Lemma 4.4

This section consists of three subsections. First, we discuss about some properties of $M_\lambda$, and provide an alternative representation of this measure. Next, we introduce the measure $\Theta_n^{(\lambda)}$ that encodes the insertion length sequence $(s_k, 0 \le k \le t_n(\lambda))$. We then establish its convergence to $M_\lambda$ after normalization, and conclude with the proof of Lemma 4.4.

### 4.2.1 Some distribution identities concerning the measures $M_\lambda$

Take $\lambda > 0$, and a continuous function $f: [0, 1] \to \mathbb{R}$ such that $|f(x)| \le C \cdot x$, for some constant $C$ and all $x \in [0, 1]$. Consider the null set $Z_\lambda = \{x: e_\lambda(x) - \underline{e}_\lambda(x) = 0\}$. The complement of $Z_\lambda$ in $[0, 1]$ is an open set $L_\lambda$, whose CC are countably many open intervals $L_\lambda = \cup_{i \ge 1} L_\lambda(i)$. The null set $Z_\lambda$ has Lebesgue measure zero, a.s., so that a.s., for $\ell_\lambda(k) = \text{Leb}(L_\lambda(k))$, we have

$$ \sum_{k \ge 1} \ell_{\lambda}(k) = 1, $$

and then we let

$$ \rho_{\lambda} = \sum_{k \ge 1} \delta_{\ell_{\lambda}(k)}, \qquad (4.15) $$
and we define

$$ \langle \rho_{\lambda}, f \rangle = \sum_{k \ge 1} f(\ell_{\lambda}(k)). $$

It is well-defined, by Fubini, and finite. In particular, this is the case for $f(x) = x^{\beta}$ for $\beta \ge 1$.

A.s. the map $\lambda \mapsto \langle \rho_{\lambda}, f \rangle$ is Lebesgue measurable, and Lebesgue integrable on any compact of $\mathbb{R}^+$. To prove this, we take $f$ non-negative (which is sufficient), and a time interval $[t, t'] \subset [0, +\infty)$. For $\varepsilon > 0$ there exists $N$ such that the $N$ largest intervals $(\hat{\ell}_{t'}(1), \dots, \hat{\ell}_{t'}(N))$, sorted in decreasing order, satisfy $\sum_{k=1}^{N} \hat{\ell}_{t'}(k) \ge 1 - \varepsilon$. Since the intervals in $L_{t'}$ are obtained from those of $L_t$ by fragmentation, for any $t \le \lambda \le t'$, $\sum_{k=1}^{N} \hat{\ell}_{\lambda}(k) \ge 1 - \varepsilon$. Since $|f(x)| \le C \cdot x$, up to $\varepsilon > 0$, $\langle \rho_{\lambda}, f \rangle$ is determined by the $N$ largest intervals of $L_{\lambda}$ (because $\sum_{k>N} f(\hat{\ell}_{\lambda}(k)) \le C \sum_{k>N} \hat{\ell}_{\lambda}(k) \le C\varepsilon$).

Let $\bar{L}_{\lambda}(k)$ be the interval, at time $\lambda$, that contains the $k$th largest interval $L_{t'}(n_k)$ at time $t' \ge \lambda$ (whose length is $\hat{\ell}_{t'}(k)$). It can happen that $\bar{L}_{\lambda}(k) = \bar{L}_{\lambda}(j)$ for some $k \ne j$. If it is the case, then keep only the ones with smaller index and keep $(\bar{L}_{\lambda}(k), k \in I(\lambda))$ where $I(\lambda)$ is the corresponding set of indices.

For the corresponding interval lengths $(\bar{\ell}_{\lambda}(k), k \in I(\lambda))$, we also have $|\sum_{k \in I(\lambda)} f(\bar{\ell}_{\lambda}(k)) - \langle \rho_{\lambda}, f \rangle| < C\varepsilon$. The maps $\lambda \mapsto \bar{\ell}_{\lambda}(k)$ as $\lambda$ goes from $t'$ to $t$, are non decreasing (and may possibly vanish in case of coalescence with another interval of this same family with lower index). It follows that, for fixed $k$, $\lambda \mapsto f(\bar{L}_{\lambda}(k))$ is measurable (by taking the value zero in case of disappearance). The integrability is clear since $\langle \rho_{\lambda}, f \rangle \le C$. Thus

$$ \int_t^{t'} \langle \rho_{\lambda}, f \rangle d\lambda = \int_t^{t'} \sum_k f(\ell_{\lambda}(k)) d\lambda, \quad (4.16) $$

is well-defined and finite. This well-definedness can be extended to Lebesgue measurable functions $f$ satisfying $\sup\{|f(x)/x|, x \in [0, 1]\} < +\infty$, by density.

**Theorem 4.6.** For all bounded and Lebesgue measurable function $g: [0, 1] \to \mathbb{R}^+$,

$$ \langle g, M_{\lambda}(e) \rangle = \int_{\lambda}^{+\infty} \sum_{k \ge 0} g(\ell_{t}(k)) \ell_{t}(k) dt, \quad \text{a.s.} \quad (4.17) $$

In particular, for $f = \text{Id}/2$, we have:

$$ \langle f, M_{\lambda}(e) \rangle = \int_{0}^{1} e_{\lambda}(s) - e_{\lambda}(s) dx = \frac{1}{2} \int_{\lambda}^{+\infty} \sum_{k=1}^{+\infty} \ell_{t}(k)^{2} dt. \quad (4.18) $$

The first statement of the theorem can be rephrased as: for $G := x \mapsto xg(x)$, $\langle g, M_t(e) \rangle = \int_t^{+\infty} \langle \rho_{\lambda}, G \rangle d\lambda$.

The first equality is discussed above, see (4.5). See Section 5.3 for close formulas for $\mathbb{E}(\langle x \mapsto x^k, M_\lambda \rangle)$.

**Proof.** Decomposition of $\langle f, M_{\lambda}(e) \rangle$ as a sum over the abscissa of local minima of $e$ for general $f$. We may partition under(e) = $\cup_{m \in \mathcal{L}(e)} \mathcal{P}(m)$ where

$$ \mathcal{P}(m) = \{(x, y) \in \text{under}(e) : L(x, y) = m\}. $$

The sets $(\mathcal{P}(m), m \in \mathcal{L}(e))$ form a partition of under(e), and then any integral on the set under(e) can be written as a sum over $\mathcal{L}(e)$ (under usual conditions, as Fubini's theorem). We will show that:

$$ \langle g, M_{\lambda}(e) \rangle = \int_{\lambda}^{+\infty} \sum_{m \in \mathcal{L}(e)} \ell(m, t) g(\ell(m, t)) dt, \quad (4.19) $$
where $\ell(m, t)$ is the size of the segment with slope $t$, starting at $(m, e(m))$ and stopped when it passes above the graph of $e$, or, in other words, the length of the excursion of $x \to e_t(x) - \varepsilon_t(x)$ starting at $m$.

For any Borelian function $f: [0, 1] \to \mathbb{R}^+$,

$$ \begin{aligned} \langle f, M_{\lambda}(e) \rangle &= \int_{0}^{1} \int_{0}^{e_{x}} \frac{2f(R(x, y) - L(x, y))}{R(x, y) - L(x, y)} \mathbb{1}_{S(x,y) \ge \lambda} dydx \\ &= \sum_{m \in \mathcal{L}(e)} \int_{0}^{1} \int_{0}^{e_{x}} \frac{2f(R(x, y) - m)}{R(x, y) - m} \mathbb{1}_{S(x,y) \ge \lambda} \mathbb{1}_{(x,y) \in \mathcal{P}(m)} dydx. \end{aligned} $$

For any $(x, y) \in \mathcal{P}(m)$, by letting $t$ be the slope $t = S(x, y)$, one has $y = e(m) + t(x - m)$. We may then proceed to a change of variable $(x, y) \to (x, t)$. For this choice, $\mathbb{1}_{S(x,y) \ge \lambda}$ will be transferred to the integration domain, $\{t: t \ge \lambda\}$. We obtain

$$ \langle f, M_{\lambda}(e) \rangle = \sum_{m \in \mathcal{L}(e)} \int_{0}^{1} \int_{\lambda}^{+\infty} \frac{2f(R(x, e(m) + t(x-m)) - m)}{R(x, e(m) + t(x-m)) - m} (x-m) \mathbb{1}_{(x, e(m) + t(x-m)) \in \mathcal{P}(m)} dt dx \quad (4.20) $$

$$ = \sum_{m \in \mathcal{L}(e)} \int_{\lambda}^{+\infty} \frac{2f(\ell(m, t))}{\ell(m, t)} \int_{m}^{m+\ell(m, t)} (x-m)dxdt \quad (4.21) $$

$$ = \int_{\lambda}^{+\infty} \sum_{m \in \mathcal{L}(e)} \ell(m, t) f(\ell(m, t)) dt. \quad (4.22) $$

□

**Remark 4.7.** The first formula in Lemma 4.6 applied to the constant function $g$ equal to 1, gives the total mass of $M_{\lambda}(e)$: Hence this total mass is $\int_{\lambda}^{+\infty} \sum_{k \ge 0} \ell_t(k) dt$. As already said, a.s., for all $t$, $\sum_{k \ge 0} \ell_t(k) = 1$, so that the total mass is infinite.

### 4.2.2 A measure encoding the cluster size insertion sequence

We store in a measure $\text{Mes}_n^{(t)}$ the elements of the cluster size insertion sequence $(s_k, k \ge 0)$ viewed before time $t$ (with multiplicity) :

$$ \text{Mes}_n^{(t)} = \sum_{j \le t} \delta_{s_j}. \quad (4.23) $$

A consequence of (4.3), is that when some unitary cost laws (the laws of $\text{UCost}^s$) are fixed, then the total cost can be expressed as

$$ \text{Cost}_n(t) = \sum_k \sum_{j=1}^{\text{Mes}_n^{(t)}(\{k\})} \text{UCost}_j^{(k)}. \quad (4.24) $$

**Foreword.** In view of (4.23) and (4.24), we may expect that under some regularity assumptions on the unitary cost laws, the knowledge of the limiting behavior for $\text{Mes}_n^{(t)}$, after rescaling, when $n \to +\infty$, and for $\text{UCost}^{(k)}$ should lead to the convergence result for the global cost. While this reasoning is somewhat valid, to capture the asymptotic behavior of $\text{Mes}_n^{(t)}$, we need to apply a normalization that causes clusters with sub-linear sizes vanish at the limit. Those with linear sizes provide a limit random measure with infinite total mass. The difficulty that follows is that, for some cost functions, small clusters can play a role, while for others, only large ones matter. This is apparent in Lemma 4.4, where our hypotheses imply that only large clusters will characterize the limiting behavior of the cost, which is not true in general.
**Several normalizations are possible** for Mes<sub>n</sub><sup>(t)</sup>, depending on the values of t to be studied, and the unitary cost model. The normalization is mainly threefold, and it is natural to consider

$$ \frac{1}{\alpha_n} \mathrm{Mes}_n^{(t_n)} (\cdot / \beta_n). $$

Here, $\alpha_n$ addresses the total mass normalization, while $\beta_n$ is the block size normalization, and $t_n$ is the time at which we stop the construction.

**For example,** if $\mathrm{UCost}^{(k)} = 1$ for all $k \ge 0$, the total cost is $\mathrm{Cost}_n(t) = t$, it is not needed to search for a limit of Mes<sub>n</sub><sup>(t)</sup> to see that! Nevertheless when searching for a limit of Mes<sub>n</sub><sup>(t_n)</sup>, for the linear time $t = t_n := an$ (and $a \in (0, 1)$, the maximum $s_j$ for $j \le an$ has order $O(\log n)$ in probability by [33]). It means that, taking a limit over $n$,

$$ \frac{1}{n} \mathrm{Mes}_n^{(an)} (\cdot / n) \xrightarrow{n} a\delta_0, $$

so that the block size information essentially vanishes. This exemplifies a more general phenomenon. It might seem natural to take $\alpha_n = t_n$ to maintain a measure with total mass one. However, since most of the blocks are very small (at least for $t_n$ close to $n$) this leads to the (degenerated) Dirac mass at the limit. Thus, we must then change the normalization in order to analyze much of the cost model.

In order to describe the large blocks, $\alpha_n$ must be different from the total mass: the number of clusters that have a linear size has order $\sqrt{n}$ (for the critical time $t_n(\lambda) = \lfloor n - \lambda \sqrt{n} \rfloor$). We then consider the random Borelian measure $\Theta_n^{(\lambda)}$ characterized, for all interval $A \subset [0, 1]$, by

$$ \Theta_n^{(\lambda)}(A) := \frac{1}{\sqrt{n}} \mathrm{Mes}_n^{(t_n(\lambda))}(nA) \qquad (4.25) $$

$$ = \sum_{k:k \in nA} \frac{1}{\sqrt{n}} \mathrm{Mes}_n^{(t_n(\lambda))}(\{k\}) \qquad (4.26) $$

$$ = \int_0^n \frac{1}{\sqrt{n}} \mathrm{Mes}_n^{(t_n(\lambda))}([x]) \mathbf{1}_{\lfloor x \rfloor \in nA} dx \qquad (4.27) $$

$$ = n \int_0^1 \frac{1}{\sqrt{n}} \mathrm{Mes}_n^{(t_n(\lambda))}([ny]) \mathbf{1}_{\lfloor ny \rfloor / n \in A} dy. \qquad (4.28) $$

These four formulas help explain the applied normalization: integration of a continuous and bounded function against $\Theta_n^{(\lambda)}$ coincides then with

$$ \langle f, \Theta_n^{(\lambda)} \rangle = n \int_0^1 \frac{1}{\sqrt{n}} \mathrm{Mes}_n^{(t_n(\lambda))}([nx]) f(x) dx. \qquad (4.29) $$

The measure $\Theta_n^{(\lambda)}$ has total mass $\sqrt{n} - \lambda \to +\infty$ with, much of this mass close to the point zero. Let us consider the vague topology on the set of Borel measures on $\mathcal{M}(0, 1)$ (which, prosaically, allows us to ignore this degeneracy at zero, since the test functions for these topologies are those with compact support strictly included in $(0, 1)$).

**Theorem 4.8.** For any $\lambda > 0$, $\Theta_n^{(\lambda)} \xrightarrow{n} M_\lambda$ for the vague convergence topology on $\mathcal{M}(0, 1)$ (and the joint convergence $(\Theta_n^{(\lambda_i)}, 1 \le i \le \kappa) \xrightarrow{n} (M_{\lambda_i}, 1 \le i \le \kappa)$ holds too, for any times $0 < \lambda_1 < \dots < \lambda_\kappa$ and any $\kappa$).

*Proof.* We borrow to Chassaing & Louchard [10] some elements of their analysis.
Consider a random walk $S[n] := (S_0 = 0, S_1, S_2, \dots)$ with i.i.d. increments $(X_i - 1, i \ge 1)$, where the $X_i$ are Poisson(1) random variable. Denote by $\tau_{-1}(S) = \inf\{j : S_j = -1\}$.

The collecting path (recall Section 1.7) of the classical parking problem on a size $n$ parking, stopped when $n-1$ cars are parked, and conditional on the fact that the last place is empty, is distributed as $S[n]$ conditioned by $\tau_{-1}(S) = n$, where $\tau_{-1}(S) = \inf\{j : S_j = -1\}$. Moreover, the vector $(Y_1, \dots, Y_n)$ of cars that have chosen place 1 to $n$, satisfies $\mathcal{L}(Y_1, \dots, Y_n) = \mathcal{L}((X_1, \dots, X_n) | \tau_{-1}(S) = n)$. The corresponding state of this random walk at time $t_n(\lambda)$, that is when $\lambda\sqrt{n}$ cars are lacking, is denoted $S^{(t_n(\lambda))}$. Let

$$ (s_t^{n,(\lambda)}, t \in [0,1]) := (n^{-1/2} S_{nt}^{(t_n(\lambda))}, t \in [0,1]) $$

where $S^{(t_n(\lambda))}$ is seen as a continuous process, interpolated linearly between integer points. In Broutin & Marckert [9, Theo.8] is has been shown that

$$ (s_t^{n,(\lambda)}, t \in [0,1], \lambda \ge 0) \xrightarrow{n} (e_t - \lambda t) - \min_{s \le t} (e_s - \lambda s), t \in [0,1], \lambda \ge 0 \quad (4.30) $$

in $D(\mathbb{R}^+, (C[0,1], \mathbb{R}))$, while the convergence of FDD (that is, for a finite number of distinct times $\lambda_i$) was already present in [10] (we see $s^{n,(\lambda)}$ as a process indexed by $\lambda$, taking, for each $\lambda$ its values in $C([0,1], \mathbb{R}))$.

By the Skorokhod representation theorem, there exists a probability space on which there exists some copies $(\tilde{s}_t^{n,(\lambda)}, \lambda \ge 0, t \in [0,1])$ of $(s_t^{n,(\lambda)}, \lambda \ge 0, t \in [0,1])$ (for each $n$), and a copy $\tilde{e}$ of $e$, such that

$$ (\tilde{s}_t^{n,(\lambda)} - [\tilde{e}_t - \lambda t) - \min_{s \le t} (\tilde{e}_s - \lambda s], t \in [0,1], \lambda \ge 0) \xrightarrow{n} 0 $$

for the same topology.

Consider $\mathcal{E}$ the two following conjunction of events:

(i) $\mathcal{L}(e)$ is countable and each $x \in (0, 1) \cap \mathcal{L}(e)$ is a global minimum on a non-empty open interval,

(ii) if for $j$ from 1 to 3, the $p_j := (x_j, e_{x_j})$ are points on the curve of $e$, such that the $x_i$ are distinct elements of $\mathcal{L}(e)$, then the $p_j$ are not aligned.

The measure $\Theta_n^{(\lambda)}$ records only the insertion block sizes, but can be enriched to record also the position of these cluster (on $\tilde{s}_t^{n,(\lambda)}$): for $B$ a Borelian of $\mathbb{R}^2$, set

$$ \Xi_n^{(\lambda)}(B) = \frac{1}{\sqrt{n}} \sum_{k \le t_n(\lambda)} \delta_{[a_k/n, b_k/n] \in B} $$

where $[a_k, b_k]$ is the cluster, which size $s_k$, which contains the arrival position $u_k$ (and which was then an excursion position in the $k$th collecting path $S^{(k)}$). Now, recall the computation made in the proof of Lemma 4.6, and in particular the decomposition of the measure $M_\lambda$ as a sum over the abscissa of local minima of $e$, that is $M_\lambda = \sum_{m \in \mathcal{L}(e)} M_\lambda^{(m)}$ and

$$ \langle f, M_{\lambda}^{(m)} \rangle = \int_{0}^{1} \int_{0}^{e_x} \frac{2f(R(x, y) - m)}{R(x, y) - m} \mathbb{1}_{S(x, y) \ge \lambda} \mathbb{1}_{(x, y) \in \mathcal{P}(m)} dy dx, $$

then, this decomposition $M_\lambda$ can also be seen as a measure on $\mathbb{R}^2$ of the limiting block positions, and in $M_\lambda^{(m)}(A)$, the number $m$ is the left-hand side and $A$ measures the distance from the right-hand-side to $m$. Further, for $\lambda' > \lambda$,

$$ M_{\lambda}^{(m)}(A) := \langle \mathbb{1}_{A}, M_{\lambda}^{(m)} \rangle = \int_{0}^{1} \int_{0}^{e_x} \frac{2}{R(x, y) - m} \mathbb{1}_{S(x, y) \ge \lambda} \mathbb{1}_{(x, y) \in \mathcal{P}(m)} \mathbb{1}_{R(x, y) \in m+A} dy dx. $$
By the same computation as in (4.20), for $\lambda' \ge \lambda$,

$$ \langle f, M_{\lambda}^{(m)}(e) - M_{\lambda'}^{(m)}(e) \rangle = \int_{\lambda}^{\lambda'} \ell(m, t) f(\ell(m, t)) dt $$

so that for $f = 1_{[a,b]}$,

$$ \langle f, M_{\lambda}^{(m)}(e) - M_{\lambda'}^{(m)}(e) \rangle = \int_{\lambda}^{\lambda'} \ell(m, t) 1_{\ell(m,t) \in [a,b]} dt. $$

Let us see that this gives the limit number of insertions in clusters located “approximately” in $[a+m, b+m]$, normalized by $\sqrt{n}$, between the times $t_n(\lambda')$ and $t_n(\lambda)$. We already know that $\tilde{s}_t^{n,(\lambda)} \to e_t - \lambda t$ uniformly for $(\lambda, t) \in [\Lambda_1, \Lambda_2] \times [0, 1]$ (because the limit is a.s. continuous), and that there exists an interval $[m_a, m_b]$ containing $m$ in its interior on which $m$ is a global minimum (a.s.). Let $m_n$ be the value of $t$ at which $\tilde{s}_t^{n,(\lambda)}$ takes its minimum on $[m_a, m_b]$ for the last time. By defining the analogues $R_n(x, y)$, $S_n(x, y)$ and $\ell_n(m_n, t)$ of $R(x, y)$, $S(x, y)$ and $\ell(m, t)$ one sees that

$$ \langle f, M_{\lambda}^{m}(e) - M_{\lambda'}^{m}(e) \rangle = \lim_{n \to \infty} \int_{\lambda}^{\lambda'} \ell_{n}(m_{n}, t) 1_{\ell_{n}(m_{n}, t) \in [a, b]} dt, $$

for almost all $a, b, \lambda, \lambda'$, a.s.. It remains to be seen that this quantity gives also the limit number of insertions during the time interval $[t_n(\lambda'), t_n(\lambda)]$, in an occupied interval starting in $nm_n$ and of length belonging to $n[a, b]$, divided by $\sqrt{n}$ (call this value $\xi_n(\lambda, \lambda', m, a, b)$, and this is a sort of equivalent of $\Theta_n^{(\lambda)} - \Theta_n^{(\lambda')}$, enriched, to take into account the minimum $m$, and the start and length of insertion intervals). For some time $k \in [t_n(\lambda'), t_n(\lambda)]$, the probability that $u_k$ arrives in such an interval, conditional on $S^{(k)}$ is either 0 if there is no excursion starting in $m_n$ and such that $\ell_n(m_n, k) \in n[a, b]$, or $\ell_n(m_n, k)/n$ if such an excursion exists. Now, by a concentration argument for martingales with bounded increments (as Azuma's inequality), we have

$$ \xi_n(\lambda, \lambda', m, a, b) - \int_{\lambda}^{\lambda'} \ell_n(m_n, t) 1_{\ell_n(m_n, t) \in [a, b]} dt, $$

goes to zero in probability. This result implies the announced result. $\square$

**Lemma 4.9.** (i) For all $\lambda \ge 0$,

$$ X_n^{(\lambda)} := \max_{j \le t_n(\lambda)} \mathbf{s}_j / n \xrightarrow{n} X^{(\lambda)} $$

where $X^{(\lambda)}$ is a.s. in $(0, 1)$.

(ii) We have $X^{(\lambda)} \xrightarrow{(proba.)} 0$ when $\lambda \to +\infty$.

(iii) For all $\varepsilon > 0$, there exists $\Lambda$ such that for all $\lambda \ge \Lambda$, for all $n$ large enough

$$ \mathbb{P}(X_n^{(\lambda)} \ge \varepsilon) \le \varepsilon. $$

*Proof.* (i) This is a consequence of [10, Section 3.3] (which itself relies on Aldous [2, Lemma 7]), which allows us to deduce the convergence of the excursion size above the current minima of a process, which converges in distribution, to those of the limit under some natural assumptions.

(ii) We borrow this argument to [35] (many other proofs are possible but this one does not depend on the statistical properties of the process $e_t - \lambda_t$). Consider any continuous function $f: [0, 1] \to \mathbb{R}$ such that $f(0) = f(1) = 0$, and consider the maximal length $L_\lambda$ of an excursion of $\Psi_\lambda(f): x \mapsto f(x) - \lambda x$ above its current minimum. As a matter of fact the map $\Psi_\lambda(f)$ lies between the two lines with equations $y = -\lambda_x + \max f$ (above) and $y = -\lambda_x + \min f$ (below). A simple picture shows that the excursions above the current minima of a function in this band is $O(1/\lambda)$.
(iii) Observe that $\lambda \mapsto X_n^{(\lambda)}$ is nonincreasing. Take $\Lambda$ large enough such that $\mathbb{P}(X_n^{(\lambda)} \ge \varepsilon/2) \le \varepsilon$; by (ii), for $n$ large enough, $|\mathbb{P}(X_n^{(\lambda)} \ge \varepsilon/2) - \mathbb{P}(X_n^{(\lambda)} \ge \varepsilon/2)| \le \varepsilon/2$ which allows concluding (we also used the fact that $X_n^{(\lambda)}$ has an atomless distribution). $\square$

### 4.2.3 Proof of Theorem 4.4

Lemma 4.4 is a consequence of the three following Lemmas, and of Lemma 4.5 which explains that for all $\lambda > 0$ fixed, all $\delta, \delta' > 0$, for all small enough $\varepsilon > 0$,

$$ \mathbb{P}(|\langle 1_{[0,\varepsilon]} \Psi_{\infty}, M_{\lambda} \rangle| \ge \delta) \le \delta'. $$

Recall that $t_n(\lambda) := [n - \lambda\sqrt{n}]$, and from (4.24) (or (4.3)),

$$ C_n(\lambda) := \mathrm{Cost}_n(t_n(\lambda)) = \sum_k^{\mathrm{Mes}_n^{(t_n(\lambda))}(\{k\})} \sum_{j=1}^{\mathrm{UCost}_j^{(k)}} \mathrm{UCost}_j^{(k)}. $$

A strategy to analyze this sum consists in using the convergence of $\Theta_n^{(\lambda)}$, and then working conditionally on $\mathrm{Mes}_n^{(t_n(\lambda))}(\{k\})$, somehow. The convergence of $\Theta_n^{(\lambda)}$ captures only the blocks of linear size. By decomposing

$$ \mathrm{UCost}_j^{(k)} = \Psi(k) + (\mathrm{UCost}_j^{(k)} - \Psi(k)) $$

along its mean and its fluctuation around its mean, we naturally write $C_n(\lambda)$ as a sum of 4 terms:

For $\varepsilon \in (0, 1)$, $\lambda \ge 0$ fixed, set

$$ C_n(\lambda) := L_n(\lambda, \varepsilon) + S_n(\lambda, \varepsilon) + LF_n(\lambda, \varepsilon) + SF_n(\lambda, \varepsilon) $$

where

$$ \left\{ \begin{aligned} L_n(\lambda, \varepsilon) &= \sum_{k \ge n\varepsilon} \Psi(k) \mathrm{Mes}_n^{(t_n(\lambda))}(\{k\}), \\ S_n(\lambda, \varepsilon) &= \sum_{k < n\varepsilon} \Psi(k) \mathrm{Mes}_n^{(t_n(\lambda))}(\{k\}), \\ SF_n(\lambda, \varepsilon) &= \sum_{k < n\varepsilon} \sum_{j=1}^{\mathrm{Mes}_n^{(t_n(\lambda))}} (\mathrm{UCost}_j^{(k)} - \Psi(k)), \\ LF_n(\lambda, \varepsilon) &= \sum_{k \ge n\varepsilon} \sum_{j=1}^{\mathrm{Mes}_n^{(t_n(\lambda))}} (\mathrm{UCost}_j^{(k)} - \Psi(k)) \end{aligned} \right. \qquad (4.31) $$

(symbols $L$ and $S$ are chosen to represent “large” and “small” blocks contributions, and the $F$ for the “fluctuation terms”, conditional on $\mathrm{Mes}_n^{(t_n(\lambda))}$).

Using Theorem 4.8, by the Skorokhod representation theorem, there exists a probability space $(\overline{\Omega}, \mathcal{A}, \mathbb{P})$, on which are defined some copies $(\overline{\Theta}_n^{(\lambda)}, n \ge 0)$ of $(\Theta_n^{(\lambda)}, n \ge 0)$ that converges a.s. to a copy $\overline{M}_{\lambda}$ of $M_{\lambda}$ (for the vague topology).

**Lemma 4.10 (Analysis of $L_n(\lambda, \varepsilon)$).** Assuming the hypothesis of Lemma 4.4, for $\varepsilon > 0$ and $\lambda \ge 0$ fixed, on the probability space $(\overline{\Omega}, \mathcal{A}, \mathbb{P})$,

$$ \frac{L_n(\lambda, \varepsilon)}{\alpha_n n^{1/2}} \xrightarrow{n \to \infty} \langle 1_{[\varepsilon, 1]} \Psi_{\infty}, \overline{M}_{\lambda} \rangle. $$
*Proof.* Observe that since $\Psi_\infty$ is continuous on $(0, 1]$,

$$ \|\Psi_\infty\|_{\infty, \varepsilon} := \max_{x \in [\varepsilon, 1]} |\Psi_\infty(x)| < +\infty. $$

Therefore,

$$ \langle 1_{[\varepsilon, 1]} |\Psi_\infty|, M_\lambda \rangle = \int_0^1 \int_0^{e_x} \frac{1_{R(x,y)-L(x,y)\ge\varepsilon}}{R(x,y)-L(x,y)} |\Psi_\infty(R(x,y)-L(x,y))| dy dx \quad (4.32) $$

$$ \le \|\Psi_\infty\|_{\infty, \varepsilon} \int_0^1 \int_0^{e_x} (1/\varepsilon) dy dx \quad (4.33) $$

$$ \le \|\Psi_\infty\|_{\infty, \varepsilon} \int_0^1 e(x)/\varepsilon dx < +\infty \text{ a.s..} \quad (4.34) $$

With the same argument (taking somehow $\Psi_\infty = 1$), one sees that we have also

$$ \langle 1_{[\varepsilon, 1]}, M_\lambda \rangle < +\infty \text{ a.s..} \quad (4.35) $$

Now, using that $\text{Mes}_n^{(t_n(\lambda))}(nx)$ is zero when $nx$ is not an integer, we have (by (4.29)),

$$ \frac{L_n(\lambda, \varepsilon)}{n^{1/2} \alpha_n} = \sum_{k \ge n\varepsilon} \frac{\Psi(k)}{\alpha_n \sqrt{n}} \text{Mes}_n^{(t_n(\lambda))}(k) \quad (4.36) $$

$$ = \left\langle 1_{[\varepsilon, 1]} \Psi_{\infty}, \Theta_n^{(\lambda)} \right\rangle + \left\langle 1_{[\varepsilon, 1]} \left( \frac{\Psi(|nx|)}{\alpha_n} - \Psi_{\infty}(x) \right), \Theta_n^{(\lambda)} \right\rangle. \quad (4.37) $$

By the a.s. convergence of $\Theta_n^{(\lambda)}$ to $M_\lambda$ (for the vague topology on $(0, 1)$), the fact that $\Psi_\infty$ is continuous on $(0, 1]$ (and that $M_\lambda$ has a.s. no atom at $\varepsilon$, and by (4.34)),

$$ \left\langle 1_{[\varepsilon, 1]} \Psi_{\infty}, \Theta_n^{(\lambda)} \right\rangle \xrightarrow{n \to \infty} \left\langle 1_{[\varepsilon, 1]} \Psi_{\infty}, \overline{M}_{\lambda} \right\rangle. $$

Since $\frac{\Psi(|nx|)}{\alpha_n} \to \Psi_\infty(x)$ uniformly on $[\varepsilon, 1]$ and by (4.35),

$$ \left\langle 1_{[\varepsilon, 1]} \left( \frac{\Psi(|nx|)}{\alpha_n} - \Psi_{\infty}(x) \right), \Theta_n^{(\lambda)} \right\rangle \xrightarrow{n \to \infty} 0. $$

□

**Lemma 4.11** (Analysis of $S_n(\lambda, \varepsilon)$). Let $\lambda \ge 0$ and $\eta, \delta > 0$. If $\varepsilon$ is small enough, for all $n$ large enough,

$$ \mathbb{P} \left( \frac{S_n(\lambda, \varepsilon)}{\alpha_n n^{1/2}} \ge \eta \right) \le \delta. $$

*Proof.* It suffices to prove the Lemma for $\lambda = 0$ (since $\lambda \mapsto S_n(\lambda, \varepsilon)$ is non-increasing). Recall that $s_m$ is the size of cluster receiving the $m+1$th car: it is of size 0 if the chosen place $c_{m+1}$ is free. The length of the cluster is observed before the insertion of this $m+1$th car. We have

$$ \mathbb{P}(s_m = k) = \mathbb{1}_{k=0} \frac{n-m}{n} + \mathbb{1}_{0<k \le m} \binom{m}{k} n(k+1)^{k-1} (n-k-1)^{m-k} \frac{n-1-m}{n-k-1} \frac{1}{n^m} \frac{k}{n} \quad (4.38) $$

$$ = \mathbb{1}_{k=0} \frac{n-m}{n} + \mathbb{1}_{0<k \le m} \mathbb{P} \left( \text{Bin} \left( m, \frac{k+1}{n} \right) = k \right) \frac{n-1-m}{n-k-1} \frac{k}{k+1} \quad (4.39) $$

where $\text{Bin}(m, \frac{k+1}{n})$ is a binomial random variable with parameters $m$ and $(k+1)/n$.

The proof of the first formula is mainly combinatorial (while the passage to the other is algebraic), the
case $k = 0$ being clear, let us see the case $k > 0$. We will add the total weights of the parking histories leading to the event $\{\mathbf{s}_m = k\}$.

- The identities of the $k$ cars of the cluster of size $k$ is any subset of $\{1, \dots, m\}$ of size $k$: this the factor $\binom{m}{k}$.

- Now, let us place this cluster somewhere on $\mathbb{Z}/n\mathbb{Z}$ by fixing the empty place $e$ preceding it: $n$ choices.

- The $k$ cars we just talk about, must have fallen between $(e+1) \mod n, \dots, (e+k) \mod n$, and all of them must be finally parked on the same places: the number of such parking schemes is known to be $(k+1)^{k-1}$.

- The remaining $(m-k)$ cars must have arrived in $e+k+2 \mod n, \dots, e \mod n$ (in total $n-(k+1)$ places), and left the last place free. To count the number of compatible positions of arrivals of cars, we use the (already used) trick of working with the set $\mathbb{Z}/(n-k-1)\mathbb{Z}$ instead and demanding that position zero to be empty: $(n-k-1)^{n-m}$ is the number of ways to put $n-m$ cars on $n-k-1$ places, and $\frac{n-1-m}{n-k-1}$ is the proportion of those letting zero free (by invariance under rotation).

Finally, we introduce the weight: $\frac{1}{n^m}$ is the probability of each history of the first $m$ cars, and $k/n$ is the probability that the $m+1$th car arrives in the block of size $k$, we were talking about all along.

We have, by bounding $k/(k+1) \le 1$, and $1/(n-1-k) \le 1/(n(1-\epsilon))$,

$$ \mathbb{E}(S_n(1, \epsilon)) = \sum_{k<n\epsilon} \sum_{m=1}^{n} \Psi(k) \mathbb{P}(\mathbf{s}_m = k) \quad (4.40) $$

$$ \le \sum_{m=1}^{n} \frac{n-m}{n} \Psi(0) + \frac{1}{n(1-\epsilon)} \sum_{k=1}^{n\epsilon} \Psi(k) \sum_{m=1}^{n} \mathbb{P}\left(\text{Bin}\left(m, \frac{k+1}{n}\right) = k\right) (n-m) \quad (4.41) $$

Observe that the infinite sum $\sum_{m \ge 0} \mathbb{P}(\text{Bin}(m, \frac{k+1}{n}) = k) = 1 + n/(k+1)$ is equal to the mean time, a random walk $(S_j^{(k)}, j \ge 0)$ with Bernoulli increments with parameter $(k+1)/n$ passes at level $k$ (and the number of passages is a geometric random variables, plus one). Roughly speaking, the first time $\tau_k^{(k)}$ that such a random walk reaches $k$ is around time $m$ such that $m(k+1)/n$ is close to $k$, so, when $m$ is around $n$, the factor $n-m-1$ in the rhs of (4.41) can not be bounded by $n$, otherwise, the upper bound would be much too large, and useless. Rewrite

$$ \sum_{m=1}^{n} \mathbb{P}\left(\text{Bin}\left(m, \frac{k+1}{n}\right) = k\right) (n-m) = \mathbb{E}\left(\sum_{m=1}^{n} 1_{S_m^{(k)}=k} (n-m)\right) \quad (4.42) $$

$$ \le \mathbb{E}\left(\sum_{m=1}^{n} 1_{S_m^{(k)}=k} (n - \tau_k^{(k)})_+\right) \quad (4.43) $$

$$ \le \mathbb{E}\left((\tau_{k+1}^{(k)} - \tau_k^{(k)})(n - \tau_k^{(k)})_+\right) \quad (4.44) $$

$$ \le \frac{n}{k+1} \mathbb{E}((n - \tau_k^{(k)})_+) \quad (4.45) $$

$$ \le \frac{n}{k+1} \mathbb{E}(|n - \tau_k^{(k)}|), \quad (4.46) $$

where we define more generally $\tau_j^{(k)}$ as the first time at which the random walk (still with Bernoulli increments with parameter $(k+1)/n$) reaches $j$. The penultimate equality comes from the strong Markov property. Since $\tau_k^{(k)} \stackrel{(d)}{=} \sum_{j=1}^k G_j^{(k)}$ where the $G_j^{(k)}$ are geometric with parameter $(k+1)/n$; this is a negative binomial random variable with mean $\frac{kn}{k+1}$ and variance $k(1-(k+1)/n)/((k+1)/n)^2 = kn(n-k-1)/(k+1)^2$.
Write, using $\mathbb{E}[|X|] \le \sqrt{\mathbb{E}[X^2]}$,

$$
\begin{align*}
\mathbb{E}[|n - \tau_k^{(k)}|] & \le \frac{n}{k+1} + \mathbb{E}\left[|n \frac{k}{k+1} - \tau_k^{(k)}|\right] \\
& \le \frac{n}{k+1} + \mathbb{E}\left[\left(n \frac{k}{k+1} - \tau_k^{(k)}\right)^2\right]^{1/2} \\
& \le \frac{n}{k+1} + \left(\frac{kn(n-k-1)}{(k+1)^2}\right)^{1/2} \\
& \le \frac{n}{k+1} + \frac{n^{1/2}(n-k-1)^{1/2}}{\sqrt{k+1}} \\
& \le \frac{n}{k} + \frac{n}{\sqrt{k}} + \frac{2n}{\sqrt{k}}.
\end{align*}
$$

Now, we conclude by (4.46): for $\varepsilon \in (0, 1/2)$,

$$
\mathbb{E}(S_n(1, \varepsilon)) \le \frac{n}{2}\Psi(0) + \frac{1}{n(1-\varepsilon)}\sum_{k=1}^{n\varepsilon} \Psi(k) \frac{n}{k} \frac{2n}{\sqrt{k}} \quad (4.47)
$$

$$
\le \frac{n}{2} \Psi(0) + 4n \sum_{k=1}^{n\epsilon} \Psi(k) / k^{3/2} \quad (4.48)
$$

Condition (4.6) is designed so that, for $n$ large enough, this is $\lim_n \mathbb{E}(S_n/(\alpha_n n^{1/2}))$ is as small as wanted, up to take a small $\varepsilon > 0$; and then the Markov inequality allows us to conclude the proof. $\square$

**Fluctuation analysis.** Since the random variables $\text{UCost}_j^{(k)} - \Psi(k)$ are all centered, mutually independent, and independent of $\text{Mes}_n^{(t)}$ for all $t$, we have

$$
\mathbb{E}(SF_n(\lambda, \varepsilon)) = \mathbb{E}(LF_n(\lambda, \varepsilon)) = 0.
$$

Let us control the variance of $SF_n(\lambda, \varepsilon)$ and of $LF_n(\lambda, \varepsilon)$. We have $SF_n(\lambda, \varepsilon) = \sum_{j=1}^{t_n(\lambda)} X_j^{(s_j)} 1_{s_j \le n\varepsilon}$ where the $(X_\ell^{(b)}, b \ge 0, \ell \ge 0)$ are independent random variables, and for all $\ell$ and $b$,

$$
X_{\ell}^{(b)} \stackrel{(d)}{=} \mathrm{UCost}_{1}^{(b)} - \Psi(b);
$$

we have

$$
\mathrm{Var}(SF_n(\lambda, \varepsilon)) = \mathbb{E} \left[ \left( \sum_{k=1}^{t_n(\lambda)} X_k^{(s_k)} 1_{s_k \le n\varepsilon} \right)^2 \right]
$$

and by conditioning by $\text{Mes}_n^{(t_n(\lambda))}$, using that $\mathbb{E}[X_\ell^{(b)}] = 0, \text{Var}(X_\ell^{(b)}) = \mathbb{E}[(X_\ell^{(b)})^2] = V(b)$, for all $b$, and by independence of the $X_i^{(j)}$, we obtain

$$
\left\{
\begin{array}{l}
\mathrm{Var}(SF_n(\lambda, \varepsilon)) = \mathbb{E}\left[\sum_{k<n\varepsilon} \mathrm{Mes}_n^{(t_n(\lambda))} (\{k\}) V(k)\right], \\
\mathrm{Var}(LF_n(\lambda, \varepsilon)) = \mathbb{E}\left[\sum_{k \ge n\varepsilon} \mathrm{Mes}_n^{(t_n(\lambda))} (\{k\}) V(k)\right].
\end{array}
\right.
\qquad (4.49)
$$

**Lemma 4.12.** Under the hypothesis of Lemma 4.4, for all $\eta, \delta > 0$, if $\varepsilon > 0$ is small enough, for $n$ large enough,

$$
\mathbb{P}\left(\left|\frac{SF_n(\lambda, \varepsilon)}{\alpha_n n^{1/2}}\right| \ge \eta\right) \le \delta \qquad (4.50)
$$

and

$$
\frac{LF_n(\lambda, \varepsilon)}{\alpha_n n^{1/2}} \xrightarrow{n} 0. \qquad (4.51)
$$
**Proof.** **Control of SF<sub>n</sub>(λ, ε).** Since these variables are centered, it suffices to control properly the variance. The variance Var(SF<sub>n</sub>(λ, ε)) given in (4.49) has exactly the same form as E(SF<sub>n</sub>(λ, ε)) as studied in the proof of Lemma 4.11, with V(k) instead of Ψ(k). By the same analysis to that of E(S<sub>n</sub>(1, ε)) we get

$$ \mathrm{Var}(\mathrm{SF}_n(1, \varepsilon)) \leq \frac{n}{2} V(0) + 4n \sum_{k=1}^{n\varepsilon} \frac{V(k)}{k^{3/2}}. $$

Condition (4.7) is designed so that $\limsup_n \mathrm{Var}(\mathrm{SF}_n(\lambda, \varepsilon) / (\alpha_n n^{1/2})) = o(\varepsilon)$, and the Bienaymé-Chebyshev inequality allows to deduce (4.50).

**Control of LF<sub>n</sub>(λ, ε).** Consider the variable

$$ Y_{n,\varepsilon} := \mathbb{E} \left[ \mathrm{LF}_n(\lambda, \varepsilon)^2 \left| \mathrm{Mes}_n^{(t_n(\lambda))} \right. \right] = \mathbb{E} \left[ \left( \sum_{k=1}^{t_n(\lambda)} X_k^{(s_k)} \mathbf{1}_{s_k \ge n\varepsilon} \right)^2 \left| \mathrm{Mes}_n^{(t_n(\lambda))} \right. \right] \quad (4.52) $$

$$ = \sum_{k \ge n\varepsilon} \mathrm{Mes}_n^{(t_n(\lambda))} (\{k\}) V(k); \quad (4.53) $$

this variable is the variance of LF<sub>n</sub>(λ, ε) conditional on Mes<sub>n</sub><sup>(t<sub>n</sub>(λ))</sup>(\{k\}). We then have

$$ Y_{n,\varepsilon} \le Y'_{n,\varepsilon} := n \int_{x \ge \varepsilon} \mathrm{Mes}_n^{(t_n(\lambda))}(\{\lfloor nx \rfloor\}) V(\lfloor nx \rfloor) dx, \quad (4.54) $$

where the introduction of $Y'_{n,\varepsilon}$ is here only to treat the boundary effect near $\varepsilon$, so that,

$$ \frac{Y'_{n,\varepsilon}}{n\alpha_n^2} = \left\langle \frac{1}{\sqrt{n}\alpha_n^2} V(\lfloor n \rfloor), \Theta_n^{(\lambda)} \right\rangle. \quad (4.55) $$

By hypothesis $\lim_n \sup_{x \ge \varepsilon} \frac{1}{\sqrt{n}\alpha_n^2} V(\lfloor nx \rfloor) = 0$, and by Lemma 4.8, $\Theta_n^{(\lambda)} \xrightarrow{n} M_\lambda$ for the vague topology. This implies that $\frac{Y_{n,\varepsilon}}{n\alpha_n^2}$ converges to zero in probability, from what the conclusion follows. $\square$

## 4.3 Discussion: Cost for general diffusion processes

Lemma 4.4, which is valid uniquely in the parking case (where the diffusion stops when the car leaves the initial occupied CC containing the arrival points), shows that even in this simplest case, the analysis is not trivial. In general, defining of the “total cost” of a general diffusion process begins with the question of defining the unitary cost model, corresponding to the insertion and dispersion of a single mass. An important issue arises: the dispersion of a single mass may result in the coalescence of many occupied CC. Hence, there are mainly two kinds of “unitary cost functions”:

(a) those that depend uniquely of the size $s_k$ of the occupied CC that contains the arrival position $u_k$,

(b) those that depend of all the occupied CC involved in the dispersion of each mass (for example, by using all the information contained in interval relaxation process $IR_k$).

Cost models that falls in the category (a) can be studied as for the parking model (under the same conditions), if their collecting paths converge to the Brownian bridge $b_\lambda$ (as stated in Lemma 3.6); the lemmas and theorems we used for the parking model would need to be adapted to this new model<fnref n="14" />. The same method could be applied to the subset of models of type (b) for which, at the first order, the unitary cost depends on $s_k$ (for example, a cost that would depend on the size $s'_k$ of the occupied CC that

<footnote n="14">¹⁴For example, $\Theta_n^{(\lambda)}$ that encodes the insertion sequence ($s_k, 0 \le k \le t_n(\lambda)$) should converge toward $M_\lambda$ possibly up to a constant, since the mean of the masses play a role in the time normalization.</footnote>
contains $u_k$, but after dispersion, should behave as if the size was taken before insertion, for many cost models, since very few coalescence events concern several linear size occupied CC).

Many models of type (b) depend on small CC, and should not be accessible to this kind of analysis. For example, a unitary cost that depends on the product of each of the occupied CC that merged during the kth insertion would bring important additional problems.

# 5 Appendix

## 5.1 Justification of fluid approximation of Section 1.5.6

In order to justify the approximation of the interval evolution by this fluid limit, several methods are possible. We use the results on Pólya urns. Since the model is identical to a Pólya urn starting from an urn composition $(a,b)$ (number of balls red/blue, with replacement matrix $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$); this model is called in the literature the adverse-campaign model (Friedman) (see e.g. Section 2.2 in Flajolet & al. [15]) and it is accessible to exact computations, and in fact, it may be used as an exercise in martingale lectures. Letting $[a(0), b(0)]$ be given, at time $t$, $L(t) := b(t) + a(t) = t + (b(0) + a(0))$, and then knowing $b(t)$ is sufficient to characterize the configuration at time $t$. Since conditional on $b(t)$, $b(t+1) = b(t) + X_t$ where $X_t$ is a Bernoulli random variable with parameter $(L(t) - b(t))/L(t)$ we get that

$$ \mathbb{E}(b(t+1) | b(t)) = 1 + b(t)(1 - 1/L(t)). \quad (5.1) $$

From here, setting $m(t) := \mathbb{E}(b(t))$, by taking the expectation in (5.1), we have

$$ L(t)m(t+1) = L(t-1)m(t) + L(t) = \dots = L(0)m(1) + \sum_{j=1}^{t} L(j) = L(0)m(1) + tL(0) + (t+1)/2, $$

so that

$$ m(t+1) = \frac{L(0)m(1) + tL(0) + (t+1)/2}{t + L(0)}, $$

with $m(1) = b(0) + a(0)/L(0)$. Now, the concentration around the mean is a standard consequence of exponential bounds for martingales with bounded increments (for example, Azuma's inequality). Then, assume that the balls have mass $1/M$ with $M$ large, and that $a(0) = M\alpha_0, b(0) = M\beta_0$ corresponds to some large urns content, where $\alpha_0, \beta_0 > 0$. At time $tM$ for some $t > 0$, $m(tM+1)/M$ is the mean position of the right extremity of the segment in this rescaled process,

$$ \frac{m(tM+1)}{M} = \frac{1}{M} \frac{(\alpha_0 + \beta_0)M(\beta_0M + \alpha_0/(\alpha_0 + \beta_0)) + tM(\alpha_0 + \beta_0)M + (tM)(tM+1)/2}{tM + (\alpha_0 + \beta_0)M} $$

and we get

$$ \frac{m(tM+1)}{M} \underset{M \to +\infty}{\longrightarrow} \frac{(\alpha_0 + \beta_0)\beta_0 + t(\alpha_0 + \beta_0) + t^2/2}{t + \alpha_0 + \beta_0} $$

which can be seen to be identical to the formula giving $b(t)$ in (1.4) (in identifying $(\alpha_0, \beta_0)$ with $(a(0), b(0))$).

## 5.2 Justification of (1.18)

Consider the set $S := \{(u, s_0, \dots, s_{n-2}), u \in [0, M_0 + s_0], 0 \le s_0 \le s_1 \le \dots s_{n-2} \le 1 - W(m[k])\}$ (the variable $s_j$ cumulates the free length spaces between the $O_0^{(k)}$ and $O_{j+1}^{(k)}$); the map

$$ \Psi: \quad S \longrightarrow \mathscr{C}^n \\ (u, s_0, \dots, s_{n-2}) \longrightarrow (-u, -u + s_0 + M_0, \dots, -u + s_{n-2} + (M_0 + \dots + M_{n-2})) $$
is linear. Its Jacobian determinant is 1 (the linear map has determinant -1), and sends S onto the set $(a_0, \dots, a_{n-1})$ described in Lemma 1.9. We then have $\mathrm{TL}(M[n]) = \mathrm{Leb}(S)$, and then, as announced

$$ \mathrm{TL}(M[n]) = \int_{0<y_0<\dots<y_{n-2}<1-W(m[k])} (M_0+y_0)dy_0\cdots dy_{n-2} \quad (5.2) $$

$$ = M_0 \frac{(1-W(m[k]))^{n-1}}{(n-1)!} + \frac{(1-W(m[k]))^n}{(n)!}. \quad (5.3) $$

## 5.3 Computation of $\mathbb{E}(\langle x \to x^k, M_\lambda \rangle)$

We consider first the case when $f = \mathrm{Id}$, that is $\mathbb{E}(\langle x \to x^1, M_\lambda \rangle)$; we recall that Aldous CRT usual convention is to consider that the CRT is the continuous tree whose contour process is 2e.

According to Lemma 4.6, in this case,

$$ \langle \mathrm{Id}, M_\lambda(\mathrm{e}) \rangle = \int_\lambda^{+\infty} \sum_{m \in \mathcal{L}(\mathrm{e})} (\ell(m, t))^2 dt \quad (5.4) $$

and we know thanks to (4.5) that the right hand side is equal to $\int_0^1 2e_\lambda(x) - 2\underline{e}_\lambda(x) dx$.

We first discuss the case $\lambda = 0$. On the one hand, we know that

$$ \int_0^1 2e_0(x)dx = \int_0^1 2e(x)dx = \mathbb{E}_e[2e(U)] \stackrel{(d)}{=} \mathbb{E}_e[d_{2e}(U_1, U_2)] \quad (5.5) $$

where $U$ is uniform on $[0, 1]$ (independent of $e$), and $U_1$ and $U_2$ are two uniform points taken in the CRT $T_{2e}$ (in other words, using $2e$ as a contour process, the points $U_1$ and $U_2$ corresponds by the canonical projection $[0, 1]_{\sim 2e}$, to the image of independent uniform random variables $u_1, u_2$ on $[0, 1]$). The second equality stems from the fact that the RHS can be seen as the average height of $2e$ on $[0, 1]$. Moreover, in a continuous random tree characterized by $2e$, $2e(U)$ is also the distance between a uniform point of the tree and the root. Thus, up to a uniform re-rooting of the tree (operation that preserves the continuum random tree), it is also the distance between two uniform points.

On the other hand, at some time $\lambda$, and for $t \ge \lambda$, $\sum_{m \in \mathcal{L}(2e)} (\ell(m, t))^2 = \sum_{m \in \mathcal{L}(e)} (\ell(m, t))^2$ corresponds to the probability that two points chosen uniformly at random in the CRT $T_{2e}$ are in the same CC at time $t$ in the fragmentation (corresponding to the time reversal of the additive coalescent). As proven by Aldous & Pitman [3], in order to construct this fragmentation process, one may equip the CRT $T_{2e}$ with a Poisson point process with intensity 1 on the product $\mathrm{Skel}(T_{2e}) \times \mathbb{R}^+$, where $\mathrm{Skel}(T_{2e})$ is set of vertices of degree 2 of the tree (the skeleton can be seen as the spanning tree of a countable number of points taken uniformly at random in the CRT, and of its root). A point $\xi = (u, t)$ of this Poisson point process corresponds to a fragmentation event: the node $u$ is removed at time $t$.

Take two uniform points $U_1$ and $U_2$ in the CRT $T_{2e}$: the probability that they are still in the same component at time $t$ is, conditional on their distance $D = d_{2e}(U_1, U_2)$ in the CRT, $\mathbb{P}(\mathrm{Expo}(D) \ge t | D) = e^{-Dt}$ (for $t \ge 0$), where $\mathrm{Expo}(D)$ is an exponential random variable with intensity $D$. Hence, for any $t \ge 0$,

$$ \sum_{m \in \mathcal{L}(e)} \ell(m, t)^2 \stackrel{(d)}{=} \exp(-td_{2e}(U_1, U_2)) \quad (5.6) $$

since both sides give the probability that two uniform points are still in the same component of the fragmented CRT at time $t$, during the fragmentation process. As a process in $t$, the two sides of (5.6) are not
identical in distribution, since the right hand side is a deterministic function of its value at any time $t = 1$. But the relation (5.6) is sufficient to compute the mean:

$$ \mathbb{E}\left[\int_{\lambda}^{+\infty} \sum_{m \in \mathcal{L}(e)} \ell(m, t)^2 dt\right] = \mathbb{E}\left[\int_{\lambda}^{+\infty} \exp(-t d_{2e}(U_1, U_2)) dt\right] = \mathbb{E}\left[\frac{\exp(-\lambda d_{2e}(U_1, U_2))}{d_{2e}(U_1, U_2)}\right]. \quad (5.7) $$

This can be computed, because $d_{2e}(U_1, U_2)$ is Rayleigh distributed (see e.g. Aldous [1, Lemma 21], [3, Section 2.1]) which is the distribution with density $xe^{-x^2/2}1_{x \ge 0}$. The RHS in (5.7) is

$$ \int_{0}^{+\infty} \frac{\exp(-\lambda x/2)}{x/2} xe^{-x^2/2} dx = \sqrt{\pi/2} e^{\lambda^2/2} \left(1 - \erf\left(\lambda/\sqrt{2}\right)\right). $$

**Computation of** $\mathbb{E}(\langle x \to x^m, M_\lambda(e) \rangle)$ **for** $m \ge 2$ (where $m$ is an integer). We have, by Lemma 4.6,

$$ \mathbb{E}(\langle x \to x^m, M_\lambda(e) \rangle) = \mathbb{E}\left(\int_{\lambda}^{+\infty} \sum_{k \ge 0} \ell_k^{m+1}(t) dt\right) $$

and by the interpretation done before, $\sum_{k \ge 0} \ell_k^{m+1}(t) dt$ is the probability, given the process e, that $m + 1$ uniform random points $U_1, \dots, U_{m+1}$ are still in the same component of the Poisson fragmentation of the CRT at time $t$. When one takes $m + 1 \ge 2$ random points in the CRT, the total lengths $L_{m+1}$ of the tree spanned by these points satisfies

$$ \mathbb{P}(L_{m+1} \ge y) = \mathbb{P}(N(y^2/2) < m) = \sum_{j=0}^{m-1} \exp(-y^2/2)(y^2/2)^j / j!, \quad (5.8) $$

where $N(a)$ is a Poisson random variable with parameter $a$ (by [3, Theo. 8]). At time $t$, knowing $L_{m+1}$, the $U_1, \dots, U_{m+1}$ are still in the same component with probability $\exp(-tL_{m+1})$. Hence

$$ \mathbb{E}(\langle x \to x^m, M_\lambda(e) \rangle) = \mathbb{E}(\exp(-\lambda L_{m+1})). $$

which can be computed thanks to (5.8), and gives the explicit expression,

$$ \mathbb{E}(\langle x \to x^m, M_\lambda(e) \rangle) = \int_0^\infty \exp(-\lambda y) \frac{dy}{y} (1 - \mathbb{P}(L_{m+1} \ge y)) dy. $$

## Conclusion

The universal properties of the valid continuous dispersion models described here are one-dimensional results. In this case, the geometry of the CC are well encoded by their lengths.

As previously discussed, our results regarding the universality of the distribution of $(\mathbf{F}^{(k)}, \mathbf{O}^{(k)})$ for a fixed $k$, for all valid continuous dispersion model, can be extended to other diffusion models. For example, this can be applied to models that treat all masses simultaneously, as long as all along the diffusion, the connected components grow independently from the others.

When there is an infinite number of masses with total mass $< 1$, some of our results still apply, and a.s. the number of connected components of $\mathbf{O}^{(+\infty)}$ is infinite too. The analysis of the connected component sizes, or of the fractal dimensions of the final occupied set should be accessible for some special distribution of masses.

Finally, in 2D (on a torus $(\mathbb{R}/\mathbb{Z})^2$, for example), there are no analogue of Lemma 1.5(i) (with the same degree of generality) which states that in 1D, the distribution of $(\mathbf{O}^{(k)}, \mathbf{F}^{(k)})$ depends only on the masses,
and not on the details of the valid continuous dispersion model considered: the coalescence induced by a
continuous dispersion model producing Euclidean balls (say, making components grow on their bound-
ary, with the same speed everywhere) is nothing compared to continuous dispersion model which would
produce very thin hairs visiting a large part of $(\mathbb{R}/\mathbb{Z})^2$. However, we can still define, beyond the piling
propensity, the cluster distribution as the distribution of the dispersion of some masses conditioned to
form a single CC. The distribution of the configuration at time $k$ can still be written as a kind of product
of the clusters probabilities, weighted by the Lebesgue measure of the translation space (that measures
the "amount" of position at which we can place these clusters while avoiding intersection). This could
maybe be used to study the statistical properties of some continuous dispersion model.

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