arXiv subjects

Cécile Mailler

Publications and source records attributed to Cécile Mailler.

At least 19 recordsLinked to original sources

arXiv2026

The forward-loop-erased ants walk

In this paper, we study a new variant of the ants walk: a model of reinforced random walks for the phenomenon of a colony of ants being able to find the shortest path between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes in a finite graph. The ants perform successive random walks on this graph, started at the nest and stopped when they first hit the food source; the transition probabilities of each random walk depends on the trajectories of the previous ants, through some edge-weights that represent the amount pheromones deposited on each edge. In our new variant of the model, the ant (1) forgets loops made on the way to the food as soon as they are made, and (2) comes back to the nest following this loop-erased version of its forward trajectory, and deposits pheromone on the edges it crosses by doing so (i.e. adds one to these edges' weights). We say that the ants "find the shortest path" if, almost surely as the number of ants tends to infinity, the proportion of ants that travel along a graph-geodesic path from the nest to the food tends to one and the proportion of ants that travel along the remaining edges tends to zero. We are able to prove that this new variant of the model finds the shortest path when the graph distance between the nest and the source is at most 2. The strength of our result is that there is no other assumption on the shape of the graph. We prove this result using stochastic approximation theory, the electrical network method for random walks on weighted graphs, and the theory of weighted uniform spanning trees (WUSTs). In particular, the link with WUSTs is unique to the model studied in this paper due to the way the ant erases loops on the outward journey.

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arXiv2026

A random recursive tree model with doubling events

We introduce a new model of random tree that grows like a random recursive tree, except at some exceptional "doubling events" when the tree is replaced by two copies of itself attached to a new root. We prove asymptotic results for the size of this tree at large times, its degree distribution, and its height profile. We also prove a lower bound for its height. Because of the doubling events that affect the tree globally, the proofs are all much more intricate than in the case of the random recursive tree in which the growing operation is always local.

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arXiv2026

The two-nest ants process on triangle-series-parallel graphs

The ants process is a stochastic process introduced by Kious, Mailler and Schapira as a model for the phenomenon of ants finding shortest paths between their nest and a source of food (seen as two marked nodes in a finite graph), with no other means of communications besides the pheromones they lay behind them as they explore their environment. The ants process relies on a reinforcement learning mechanism. In this paper, we modify the ants process by having more than one ants nest (and still one source of food). For technical reasons, we restrict ourselves to the case when there are two nests, and when the graph is a triangle between the two nests and the source of food, whose edges have been replaced by series-parallel graphs. In this setting, using stochastic approximation techniques, comparison with Pólya urns, and combinatorial arguments, we are able to prove that the ants process converges and to describe its limit.

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arXiv2025

A localisation phase transition for the catalytic branching random walk

We show the existence of a phase transition between a localisation and a non-localisation regime for a branching random walk with a catalyst at the origin. More precisely, we consider a continuous-time branching random walk that jumps at rate one, with simple random walk jumps on $\mathbb Z^d$, and that branches (with binary branching) at rate $λ>0$ everywhere, except at the origin, where it branches at rate $λ_0>λ$. We show that, if $λ_0$ is large enough, then the occupation measure of the branching random walk localises (i.e. when normalised by the total number of particles, it converges almost surely without spatial renormalisation), whereas, if $λ_0$ is close enough to $λ$, then the occupation measure delocalises, in the sense that the proportion of particles in any finite given set converges almost surely to zero. The case $λ= 0$ (when branching only occurs at the origin) has been extensively studied in the literature and a transition between localisation and non-localisation was also exhibited in this case. Interestingly, the transition that we observe, conjecture, and partially prove in this paper occurs at the same threshold as in the case $λ=0$. One of the strengths of our result is that, in the localisation regime, we are able to prove convergence of the occupation measure, whilst existing results in the case $λ= 0$ give convergence of moments instead.

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arXiv2025

Multi-drawing Pólya urns via labelled random DAGs

A Pólya urn of replacement matrix $R=(R_{i,j})_{1\leq i,j\leq d}$ is a Markov process that encodes the following experiment: an urn contains balls of $d$ different colours and at every time-step, a ball is drawn uniformly at random in the urn, and if its colour is $i$, then it is replaced in the urn with an additional $R_{i,j}$ balls of colour $j$, for all $1\leq i, j\leq d$. We study a natural extension of this model in which, instead of drawing one ball at each time-step, we draw a set of $m\geq 2$ balls: in this case, the replacement matrix becomes a replacement tensor. Because of the multi-draws, this process can no longer be seen as a branching process, which makes its analysis much more intricate than in the classical Pólya urn case. Partial results proved by stochastic approximation techniques exist in the literature. In this article, we introduce a new approach based on seeing the process as a stochastic process indexed by a random directed-acyclic graph (DAG) and use this approach, together with the theory of stochastic tensors, to prove a convergence theorem for these multi-drawing Pólya urns, with assumptions that are straightforward to check in practice.

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arXiv2024

Central limit theorems for the monkey walk with steep memory kernel

The monkey walk is a stochastic process defined as the trajectory of a walker that moves on $\mathbb R^d$ according to a Markovian generator, except at some random "relocation" times at which it jumps back to its position at a time sampled randomly in its past, according to some "memory kernel". The relocations make the process non-Markovian and introduce a reinforcement effect (the walker is more likely to relocate in a Borel set in which it has spent a lot of time in the past). In this paper, we focus on "steep" memory kernels: in these cases, the time sampled in the past at each relocation time is likely to be quite recent. One can see this as a way to model the case when the walker quickly "forgets" its past. We prove limit theorems for the position of the walker at large times, which confirm and generalise the estimates available in the physics literature.

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arXiv2024

A two-table theorem for a disordered Chinese restaurant process

We investigate a disordered variant of Pitman's Chinese restaurant process where tables carry i.i.d. weights. Incoming customers choose to sit at an occupied table with a probability proportional to the product of its occupancy and its weight, or they sit at an unoccupied table with a probability proportional to a parameter $θ>0$. This is a system out of equilibrium where the proportion of customers at any given table converges to zero almost surely. We show that for weight distributions in any of the three extreme value classes, Weibull, Gumbel or Fréchet, the proportion of customers sitting at the largest table converges to one in probability, but not almost surely, and the proportion of customers sitting at either of the largest two tables converges to one almost surely.

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arXiv2023

The trace-reinforced ants process does not find shortest paths

In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes $N$ and $F$ in a finite graph $\mathcal G$. The ants perform a sequence of random walks on this graph, starting from the nest and stopped when first hitting the source of food. At each step of its random walk, the $n$-th ant chooses to cross a neighbouring edge with probability proportional to the number of preceding ants that crossed that edge at least once. We say that {\it the ants find the shortest path} if, almost surely as the number of ants grow to infinity, almost all the ants go from the nest to the source of food through one of the shortest paths, without loosing time on other edges of the graph. Our contribution is three-fold: (1) We prove that, if $\mathcal G$ is a tree rooted at $N$ whose leaves have been merged into node $F$, and with one edge between $N$ and $F$, then the ants indeed find the shortest path. (2) In contrast, we provide three examples of graphs on which the ants do not find the shortest path, suggesting that in this model and in most graphs, ants do not find the shortest path. (3) In all these cases, we show that the sequence of normalised edge-weights converge to a {\it deterministic} limit, despite a linear-reinforcement mechanism, and we conjecture that this is a general fact which is valid on all finite graphs. To prove these results, we use stochastic approximation methods, and in particular the ODE method. One difficulty comes from the fact that this method relies on understanding the behaviour at large times of the solution of a non-linear, multi-dimensional ODE.

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arXiv2023

Large deviations principle for a stochastic process with random reinforced relocations

Stochastic processes with random reinforced relocations have been introduced in the physics literature to model animal foraging behaviour. Such a process evolves as a Markov process, except at random relocation times, when it chooses a time at random in its whole past according to some ``memory kernel'', and jumps to its value at that random time. We prove a quenched large deviations principle for the value of the process at large times. The difficulty in proving this result comes from the fact that the process is not Markov because of the relocations. Furthermore, the random inter-relocation times act as a random environment.

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arXiv2023

Scaling limit of critical random trees in random environment

We consider Bienaymé-Galton-Watson trees in random environment, where each generation $k$ is attributed a random offspring distribution $μ_k$, and $(μ_k)_{k\geq 0}$ is a sequence of independent and identically distributed random probability measures. We work in the ``strictly critical'' regime where, for all $k$, the average of $μ_k$ is assumed to be equal to $1$ almost surely, and the variance of $μ_k$ has finite expectation. We prove that, for almost all realizations of the environment (more precisely, under some deterministic conditions that the random environment satisfies almost surely), the scaling limit of the tree in that environment, conditioned to be large, is the Brownian continuum random tree. The habitual techniques used for standard Bienaymé-Galton-Watson trees, or trees with exchangeable vertices, do not apply to this case. Our proof therefore provides alternative tools.

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arXiv2022

Dynamical Models for Random Simplicial Complexes

We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula encompasses results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scale-free nature of Complex Quantum Network Manifolds in dimensions $d > 2$, and special types of Network Geometry with Flavour models studied in the physics literature by Bianconi, Rahmede [$\mathit{Sci. Rep.} \; \mathbf{5},\text{ 13979 (2015) and }\mathit{Phys. Rev. E} \; \mathbf{93},\text{ 032315 (2016)}$].

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arXiv2021

Fluctuations of balanced urns with infinitely many colours

In this paper, we prove convergence and fluctuation results for measure-valued Pólya processes (MVPPs, also known as Pólya urns with infinitely-many colours). Our convergence results hold almost surely and in $L^2$, under assumptions that are different from that of other convergence results in the literature. Our fluctuation results are the first second-order results in the literature on MVPPs; they generalise classical fluctuation results from the literature on finitely-many-colour Pólya urns. As in the finitely-many-colour case, the order and shape of the fluctuations depend on whether the "spectral gap is small or large". To prove these results, we show that MVPPs are stochastic approximations taking values in the set of measures on a measurable space $E$ (the colour space). We then use martingale methods and standard operator theory to prove convergence and fluctuation results for these stochastic approximations.

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arXiv2021

Multiple drawing multi-colour urns by stochastic approximation

A classical Pólya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$). We are interested in multi-drawing Pólya urns, where the replacement rule depends on the random drawing of a set of $m$ balls from the urn (with or without replacement). This generalisation has already been studied in the literature, in particular by Kuba & Mahmoud (ArXiv:1503.09069 and 1509.09053), where second order asymptotic results are proved for $2$-colour urns under the balanced and the affinity assumptions. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to remove the affinity hypothesis of Kuba & Mahmoud and generalise the result to more-than-two-colour urns. We also give some partial results in the two-colour non-balanced case.

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arXiv2021

Parametrised branching processes: a functional version of Kesten \& Stigum theorem

Let $(Z_n,n\geq 0)$ be a supercritical Galton-Watson process whose offspring distribution $μ$ has mean $λ>1$ and is such that $\int x(\log(x))_+ dμ(x)<+\infty$. According to the famous Kesten \& Stigum theorem, $(Z_n/λ^n)$ converges almost surely, as $n\to+\infty$. The limiting random variable has mean~1, and its distribution is characterised as the solution of a fixed point equation. \par In this paper, we consider a family of Galton-Watson processes $(Z_n(λ), n\geq 0)$ defined for~$λ$ ranging in an interval $I\subset (1, \infty)$, and where we interpret $λ$ as the time (when $n$ is the generation). The number of children of an individual at time~$λ$ is given by $X(λ)$, where $(X(λ))_{λ\in I}$ is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that $\mathbb E(X(λ))=λ>1$ for all $λ\in I$. This allows us to define $Z_n(λ)$ the number of elements in the $n$th generation at time $λ$. Set $W_n(λ)= Z_n(λ)/λ^n$ for all $n\geq 0$ and $λ\in I$. We prove that, under some moment conditions on the process~$X$, the sequence of processes $(W_n(λ), λ\in I)_{n\geq 0}$ converges in probability as~$n$ tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.

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arXiv2021

Voronoi cells in random split trees

We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are essentially all of order $n\exp(-\mathrm{const}\sqrt{\log n})$. This discrepancy persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different "influence" parameters (called "speeds" in the paper) to each of the $k$ vertices. Our findings are in contrast to corresponding results on random uniform trees and on the continuum random tree, where it is known that the vector of the relative sizes of the $k$ Voronoi cells is asymptotically uniformly distributed on the $(k-1)$-dimensional simplex.

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arXiv2020

Finding geodesics on graphs using reinforcement learning

It is well-known in biology that ants are able to find shortest paths between their nest and the food by successive random explorations, without any mean of communication other than the pheromones they leave behind them. This striking phenomenon has been observed experimentally and modelled by different mean-field reinforcement-learning models in the biology literature. In this paper, we introduce the first probabilistic reinforcement-learning model for this phenomenon. In this model, the ants explore a finite graph in which two nodes are distinguished as the nest and the source of food. The ants perform successive random walks on this graph, starting from the nest and stopped when first reaching the food, and the transition probabilities of each random walk depend on the realizations of all previous walks through some dynamic weighting of the graph. We discuss different variants of this model based on different reinforcement rules and show that slight changes in this reinforcement rule can lead to drastically different outcomes. We prove that, in two variants of this model and when the underlying graph is, respectively, any series-parallel graph and a 5-edge non-series-parallel losange graph, the ants indeed eventually find the shortest path(s) between their nest and the food. Both proofs rely on the electrical network method for random walks on weighted graphs and on Rubin's embedding in continuous time. The proof in the series-parallel cases uses the recursive nature of this family of graphs, while the proof in the seemingly-simpler losange case turns out to be quite intricate: it relies on a fine analysis of some stochastic approximation, and on various couplings with standard and generalised Pólya urns.

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arXiv2020

Competing growth processes with random growth rates and random birth times

Finding the most powerful node in a dynamic random network, the largest set in a partition-valued stochastic process, or the largest family in an evolving population at a given time, can be a very difficult problem. This is particularly the case when the underlying stochastic process has complex dependencies and the individual strength of an object has an impact that only plays out over time. We propose a novel technique to deal with such problems and show how it can be applied to a broad range of examples where it produces new insight and surprising results. The method relies on two steps: In the first step, which is highly problem dependent, the problem is embedded into continuous time so that the evolution of the sizes of objects after their individual birth times become approximately independent while we only need minimal control over the birth times themselves. Once such an embedding is achieved, the second step is to apply a Poisson limit theorem that allows a comparison of object sizes in a critical window and therefore allows a description of features of extremal objects. In this paper we prove such a versatile limit theorem, based on extreme value theory, and show how the technique can be used to study extremal behaviour in different types of preferential attachment networks with fitness, branching processes with selection and mutation, and random permutations with random cycle weights.

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arXiv2020

Stochastic approximation on non-compact measure spaces and application to measure-valued Pólya processes

Our main result is to prove almost-sure convergence of a stochastic-approximation algorithm defined on the space of measures on a non-compact space. Our motivation is to apply this result to measure-valued Pólya processes (MVPPs, also known as infinitely-many Pólya urns). Our main idea is to use Foster-Lyapunov type criteria in a novel way to generalize stochastic-approximation methods to measure-valued Markov processes with a non-compact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject. From the MVPPs point of view, our result implies almost-sure convergence of a large class of MVPPs, this convergence was only obtained until now for specific examples, with only convergence in probability established for general classes. Furthermore, our approach allows us to extend the definition of MVPPs by adding "weights" to the different colors of the infinitely-many-color urn. We also exhibit a link between non-"balanced" MVPPs and quasi-stationary distributions of Markovian processes, which allows us to treat, for the first time in the literature, the non-balanced case. Finally, we show how our result can be applied to designing stochastic-approximation algorithms for the approximation of quasi-stationary distributions of discrete- and continuous-time Markov processes on non-compact spaces.

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