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Jean-Baptiste Gouéré

Publications and source records attributed to Jean-Baptiste Gouéré.

At least 19 recordsLinked to original sources
arXiv2026

The $K$-th nearest neighbor random walk on a Poisson point process gets trapped

The $K$-th nearest neighbor random walk $(X_n)_{n \geq 0}$ on a homogeneous Poisson point process $χ$ on $\R^d$ ($d\geq 1$), starts at the origin and at each step picks its next Poisson point among its closest neighbors according to i.i.d. labels having the same distribution as $K$. Our main result (Theorem 1) states that the number of Poisson points visited by $(X_n)_{n \geq 0}$ admits an exponential decay whenever the random variable $K$ has a bounded support (BS). In particular, the $K$-th nearest neighbor random walk visits finitely many Poisson points if and only if $K$ satisfies Assumption (BS). To prove it, we introduce the key notion of pioneer point which allows us to deal with the region of $\R^d$ already explored by $(X_n)_{n \geq 0}$. Still under Assumption (BS), we also prove an exponential decay for the Euclidean length of the trajectory performed by $(X_n)_{n \geq 0}$ (Theorem 2). Finally, and quite surprisingly, we exhibit an example of label distribution with bounded support for which the $K$-th nearest neighbor random walk discovers new Poisson points after a number of steps whose tail distribution is at least polynomial (Theorem 3).

math.PR↗
arXiv2025

First-order behavior of the time constant in non-isotropic continuous first-passage percolation

Consider $Ξ$ a homogeneous Poisson point process on $\mathbb{R}^d$ ($d\geq 2$) with unit intensity with respect to the Lebesgue measure. For $\varepsilon\geq 0$, we define the Boolean model $Σ_{p, \varepsilon}$ as the union of the balls of volume $\varepsilon$ for the $p$-norm ($p\in [1,\infty]$) and centered at the points of $Ξ$. We define a random pseudo-metric on $\mathbb{R}^d$ by associating with any path a travel time equal to its $p$-length outside $Σ_{p,\varepsilon}$. This defines a continuous model of first-passage percolation, that has been studied in \cite{GT17,GT22} for $p=2$, the Euclidean norm. For $p=1$, this model is expected to share common properties with the classical first-passage percolation on the graph $\mathbb{Z}^d$ with a distribution of passage times of the form $\varepsilon δ_0 + (1-\varepsilon) δ_1$. The exact calculation of the time constant of this model $\tilde μ_{p,\varepsilon} (x)$ is out of reach. We investigate here the behavior of $\varepsilon \mapsto \tilde μ_{p,\varepsilon} (x)$ near $0$, and enlighten how the speed at which $\| x \|_p - \tilde μ_{p,\varepsilon} (x) $ goes to $0$ depends on $x$ and $p$. For instance, for $p\in (1,\infty)$, we prove that $\| x \|_p - \tilde μ_{p,ε} (x)$ is of order $\varepsilon ^{κ_p(x)}$ with $$κ_p(x): = \frac{1}{d- \frac{d_1(x)-1}{2} - \frac{d-d_1 (x)}{p}}\,,$$where $d_1(x)$ is the number of non null coordinates of $x$. The exact order of $\| x \|_p - \tilde μ_{p,ε} (x)$ is also given for $p=1$ and $p=\infty$. Related results are also discussed, about properties of the geodesics, and analog properties on closely related models.

math.PR↗
arXiv2024

Absence of percolation for infinite Poissonian systems of stopped paths

The state space of our model is the Euclidean space in dimension d = 2. Simultaneously, from all points of a homogeneous Poisson point process, we let grow independent and identically distributed random continuum paths. Each path stops growing at time t \> 0 if it hits the trace of the other curves realized up until time t. Such dynamic is well-defined as long as the distribution of paths has a finite second moment at each time t \> 0. Letting the time runs until infinity so that each path reaches its stopping curve, we study the connected property of the graph formed by all stopped curves. Our main result states the absence of percolation in this graph, meaning that each cluster consists of a finite number of curves. The assumptions on the distribution of paths are very mild, with the main one being the so-called 'loop assumption' which ensures that finite clusters (necessarily containing a loop) occur with positive probability. The main issue in this model comes from the long-range dependence arising from long sequences of causalities in the hitting/stopping procedure. Most methods based on block approaches fail to effectively address the question of percolation in this setting.

math.PR↗
arXiv2024

Disjoint finite geodesics in first-passage percolation

We investigate first-passage percolation on the lattice $\Z^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $τ$. We only assume $¶[τ=0]0$ is explicit) for the probability of having two disjoint geodesics between two pairs of neighbouring vertices at distance $n$. Additionally, under more specific assumptions on the distribution of $τ$, we obtain similar lower bounds for the probability of having two disjoint geodesics (except for their starting and ending points) between the same two vertices.

math.PR↗
arXiv2021

First-order behavior of the time constant in Bernoulli first-passage percolation

We consider the standard model of first-passage percolation on $\mathbb{Z}^d$ ($d\geq 2$), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter $1-ε$. These passage times induce a random pseudo-metric $T_ε$ on $\mathbb{R}^d$. By subadditive arguments, it is well known that for any $z\in\mathbb{R}^d\setminus \{0\}$, the sequence $T_ε(0,\lfloor nz \rfloor) / n$ converges a.s. towards a constant $μ_ε(z)$ called the time constant. We investigate the behavior of $ε\mapsto μ_ε(z)$ near $0$, and prove that $μ_ε(z) = \| z\|_1 - C (z) ε^{1/d_1(z)} + o ( ε^{1/d_1(z)}) $, where $d_1(z)$ is the number of non null coordinates of $z$, and $C(z)$ is a constant whose dependence on $z$ is partially explicit.

math.PR↗
arXiv2020

Percolation in the Boolean model with convex grains in high dimension

We investigate percolation in the Boolean model with convex grains in high dimension. For each dimension d, one fixes a compact, convex and symmetric set K $\subset$ R d with non empty interior. In a first setting, the Boolean model is a reunion of translates of K. In a second setting, the Boolean model is a reunion of translates of K or $ρ$K for a further parameter $ρ$ $\in$ (1, 2). We give the asymptotic behavior of the percolation probability and of the percolation threshold in the two settings.

math.PR↗
arXiv2020

Continuity of the time constant in a continuous model of first passage percolation

For a given dimension d $\ge$ 2 and a finite measure $ν$ on (0, +$\infty$), we consider $ξ$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $ν$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $Σ$ = $\cup$ (c,r)$\in$$ξ$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $Σ$ and at infinite speed inside $Σ$. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like $μ$ x when x goes to infinity, where $μ$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of $μ$ as a function of the measure $ν$ associated with the underlying Boolean model.

math.PR↗
arXiv2018

Equivalence of some subcritical properties in continuum percolation

We consider the Boolean model on $\R^d$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let $\ell$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $E(|C|)$ is finite if and only if there exists $A,B >0$ such that $¶(\ell \ge n) \le Ae^{-Bn}$ for all $n \ge 1$.

math.PR↗
arXiv2017

Positivity of the time constant in a continuous model of first passage percolation

We consider a non trivial Boolean model $Σ$ on ${\mathbb R}^d$ for $d\geq 2$. For every $x,y \in {\mathbb R}^d$ we define $T(x,y)$ as the minimum time needed to travel from $x$ to $y$ by a traveler that walks at speed $1$ outside $Σ$ and at infinite speed inside $Σ$. By a standard application of Kingman sub-additive theorem, one easily shows that $T(0,x)$ behaves like $μ\|x\|$ when $\|x\|$ goes to infinity, where $μ$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the positivity of $μ$. More precisely, under an almost optimal moment assumption on the radii of the balls of the Boolean model, we prove that $μ\textgreater{}0$ if and only if the intensity $λ$ of the Boolean model satisfies $λ\textless{} \widehatλ\_c$, where $ \widehatλ\_c$ is one of the classical critical parameters defined in continuum percolation.

math.PR↗
arXiv2015

The Number of Open Paths in Oriented Percolation

We study the number $N\_n$ of open paths of length $n$ in supercritical oriented percolation on $\Zd \times \N$, with $d \ge 1$. We prove that on the percolation event $\{\inf N\_n\textgreater{}0\}$, $N\_n^{1/n}$ almost surely converges to a positive deterministic constant. We also study the existence of directional limits. The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation.

math.PR↗
arXiv2014

Non-optimality of constant radii in high dimensional continuum percolation

Consider a Boolean model $Σ$ in $\R^d$. The centers are given by a homogeneous Poisson point process with intensity $λ$ and the radii of distinct balls are i.i.d.\ with common distribution $ν$. The critical covered volume is the proportion of space covered by $Σ$ when the intensity $λ$ is critical for percolation. Previous numerical simulations and heuristic arguments suggest that the critical covered volume may be minimal when $ν$ is a Dirac measure. In this paper, we prove that it is not the case in sufficiently high dimension.

math.PR↗
arXiv2013

Branching brownian motion seen from its left-most particule

This paper is a review of the recent progresses on the branching brownian motion seen from its left-most particule. We describe in particular the recent works by Arguin-Bovier-Kistler and Aïdékon-Berestycki-Brunet-Shi. This is the text in french of a Bourbaki seminar given in March 2013.

math.PR↗
arXiv2013

The shape of large balls in highly supercritical percolation

We exploit a connection between distances in the infinite percolation cluster, when the parameter is close to one, and the discrete-time TASEP on $\mathbb{Z}$. This shows that when the parameter goes to one, large balls in the cluster are asymptotically shaped near the axes like arcs of parabola.

math.PR↗
arXiv2013

Continuum percolation in high dimensions

Consider a Boolean model $Σ$ in $\R^d$. The centers are given by a homogeneous Poisson point process with intensity $λ$ and the radii of distinct balls are i.i.d.\ with common distribution $ν$. The critical covered volume is the proportion of space covered by $Σ$ when the intensity $λ$ is critical for percolation. Previous numerical simulations and heuristic arguments suggest that the critical covered volume may be minimal when $ν$ is a Dirac measure. In this paper, we prove that it is not the case at least in high dimension. To establish this result we study the asymptotic behaviour, as $d$ tends to infinity, of the critical covered volume. It appears that, in contrast to what happens in the constant radii case studied by Penrose, geometrical dependencies do not always vanish in high dimension.

math.PR↗
arXiv2012

Monotonicity in first-passage percolation

We consider standard first-passage percolation on $\Z^d$. Let $e_1$ be the first coordinate vector. Let $a(n)$ be the expected passage time from the origin to $ne_1$. In this short paper, we note that $a(n)$ is increasing under some strong condition on the support of the law of the passage times on the edges.

math.PR↗
arXiv2011

Percolation in a multiscale Boolean model

We consider percolation in a multiscale Boolean model. This model is defined as the union of scaled independent copies of a given Boolean model. The scale factor of the $n^{\textrm{th}}$ copy is $ρ^{-n}$. We prove, under optimal integrability assumptions, that no percolation occurs in the multiscale Boolean model for large enough $ρ$ if the rate of the Boolean model is below some critical value.

math.PR↗
arXiv2010

Survival probability of the branching random walk killed below a linear boundary

We give an alternative proof of a result by N. Gantert, Y. Hu and Z. Shi on the asymptotic behavior of the survival probability of the branching random walk killed below a linear boundary, in the special case of deterministic binary branching and bounded random walk steps. Connections with the Brunet-Derrida theory of stochastic fronts are discussed.

math.PR↗
arXiv2010

Brunet-Derrida behavior of branching-selection particle systems on the line

We consider a class of branching-selection particle systems on $\R$ similar to the one considered by E. Brunet and B. Derrida in their 1997 paper "Shift in the velocity of a front due to a cutoff". Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size $N$ of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate $(\log N)^{-2}$. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of $N$ independent branching random walks killed below a linear space-time barrier.

math.PR↗