The random walk on the random connection model

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like $|x-y|^{-\alpha}$ with $\alpha>d$, where $d$ denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits the same properties as classical discrete long-range percolation models studied in [3] with regard to recurrence and transience of the random walk. The recurrence results are valid for every intensity of the Poisson point process while the transience results hold for large enough intensity. Moreover, we address a question which is related to a conjecture in [16] for this graph.Comment: New version of the manuscript with some extension

Marches au hasard sur des graphes géométriques aléatoires engendrés par des processus ponctuels

Random walks on random graphs embedded in Rd appear naturally in problems arisingfrom statistical mechanics such that the description of flows, molecules or heat diffusionsin random and irregular environments. The general idea is to extend known results forrandom walks on Zd or on random perturbations of the grid to results for random walkson graphs generated by point processes in Rd.In this thesis, we consider nearest neighbor random walks on graphs depending on thegeometry of a random infinite locally finite set of points. More precisely, given a realisationof a simple stationary point process in Rd, a connected infinite and locally finite graph G isconstructed. This graph is then possibly equipped with a conductance function C, that is apositive function defined on its edge set. Examples of graphs studied in this manuscript arethe Delaunay triangulation, the Gabriel graph, the creek-crossing graphs and the skeletonof the Voronoi tiling generated by the point process. We study properties of the simplerandom walk or of a random walk associated with the conductance C on such graphs.The main results concern the characterisation of the recurrence or transience of therandom walks and the description of their diffusive scaling limits. Under suitable assumptionson the underlying point process and the conductance function, we show that therandom walks on the Delaunay triangulation, the Gabriel graph and the skeleton of theVoronoi tiling generated by almost every realisation of the point process are recurrent ifd = 2 and transient if d 3. We state an annealed invariance principle for simple randomwalks starting from the origin on the Delaunay triangulation, the Gabriel graph and thecreek-crossing graphs generated by Palm measures of suitable point processes. Finally,we show a quenched invariance principle for simple random walks on random Delaunaytriangulations.This thesis uses tools from both stochastic geometry (point processes, Palm measures,random graphs ...) and the theory of random walks (links with electrical networks theory,the environment seen from the particle,...).Les marches aléatoires sur des graphes aléatoires plongés dans R^d apparaissent naturellementdans de nombreux problèmes issus de la mécanique statistique tels que la descriptionde flux, de diffusions de molécules ou de chaleur dans des milieux aléatoires et irréguliers.L’idée générale est d’étendre des résultats connus sur la grille Z^d ou des perturbationsaléatoires de celle-ci à des graphes engendrés par des processus ponctuels dans R^d.Dans cette thèse, on considère des marches au plus proche voisin sur des graphesdépendant de la géométrie d’un ensemble aléatoire et infini de points. Plus précisément,étant donnée une réalisation d’un processus ponctuel simple et stationnaire dans R^d, ungraphe G, connexe, infini et localement fini, est construit. Ce graphe est ensuite muni´eventuellement d’une fonction de conductance C, c’est-`a-dire une fonction strictement positived´efinie sur son ensemble d’arˆetes. Les exemples de graphes g´eom´etriques ´etudi´es dansce manuscrit sont la triangulation de Delaunay, le graphe de Gabriel, les creek-crossinggraphs et le squelette de la mosaïque de Voronoï engendrés par le processus ponctuel. Onétudie les propriétés la marche simple et la marche associée à la conductance C sur de telsgraphes.Les principaux résultats portent sur la caractérisation de la récurrence ou de la transiencepresque sûre des marches aléatoires et sur la description de leurs limites diffusives.On montre que, sous des hypothèses convenables sur le processus ponctuel sous-jacentet la fonction de conductance, les marches aléatoires sur la triangulation de Delaunay, legraphe de Gabriel et le squelette de la mosaïque de Voronoï engendrés par presque touteréalisation de ce processus ponctuel sont récurrentes si d = 2 et transitoires si d 3. On´etablit aussi un principe d’invariance annealed (ou en moyenne) pour les marches simplespartant de l’origine sur la triangulation de Delaunay et le graphe de Gabriel engendr´espar les mesures de Palm de certains processus ponctuels ainsi qu’un principe d’invariancequenched (ou presque sûr) pour les marches simples sur des triangulations de Delaunayengendrées par des processus ponctuels.Cette thèse exploite à la fois des outils de géométrie aléatoire (processus ponctuels,mesures de Palm, mosaïques et graphes aléatoires...) et de la théorie des marches aléatoires(liens avec les réseaux électriques, l’environnement vu par la particule)

Annealed Invariance Principle for Random Walks on Random Graphs Generated by Point Processes in R-d

International audienceWe consider simple random walks on random graphs embedded in R-d and generated by point processes such as Delaunay triangulations, Gabriel graphs and the creek-crossing graphs. Under suitable assumptions on the point process, we show an annealed invariance principle for these random walks. These results hold for a large variety of point processes including Poisson point processes, Matern cluster and Matern hardcore processes which have respectively clustering and repulsiveness properties. The proof relies on the use the process of the environment seen from the particle. It allows to reconstruct the original process as an additive functional of a Markovian process under the annealed measure

Recurrence or transience of random walks on random graphs generated by point processes in Rd\mathbb{R}^d

To appear in Stochastic Processes and their ApplicationsWe consider random walks associated with conductances on Delaunay triangulations, Gabriel graphs and skeletons of Voronoi tilings which are generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on point processes and conductances, we show that, for almost any realization of the point process, these random walks are recurrent if $d=2$ and transient if $d\geq 3$. These results hold for a large variety of point processes including Poisson point processes, Matérn cluster and Matérn hardcore processes which have clustering or repulsive properties. In order to prove them, we state general criteria for recurrence or almost sure transience which apply to random graphs embedded in $\mathbb{R}^d$

The longest edge of the one-dimensional soft random geometric graph with boundaries

International audienceThe object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this article, we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant a, which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result, we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power a and we recover a phase transition which coincides with exactly the same phases in Benjamini and Berger[ 2]

The bi-dimensional directed IDLA forest

Publié dans Annals of Applied Probability, Vol. 33, No. 3, p. 2247-2290, 2023We investigate three types of Internal Diffusion Limited Aggregation (IDLA) models. These models are based on simple random walks on Z 2 with infinitely many sources that are the points of the vertical axis I (∞) = {0} × Z. Various properties are provided, such as stationarity, mixing, stabilization and shape theorems. Our results allow us to define a new directed (w.r.t. the horizontal direction) random forest spanning Z 2 , based on an IDLA protocol, which is invariant in distribution w.r.t. vertical translations

Compound Poisson approximation for simple transient random walks in random sceneries

Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(\xi(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(\xi(S_n))_{n\geq 0}$. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit