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Régine Marchand

Publications and source records attributed to Régine Marchand.

18 recordsLinked to original sources
arXiv2026

The Contact Process Can Survive on a Slightly Subcritical Dynamical Percolation Cluster

The contact process on dynamic edges (CPDE) is a contact process evolving on a dynamic environment given by a dynamical percolation on the edges of Z d\,: each edge updates its state to open or closed with respective rates vp and v(1 -p). By coupling a well-chosen subset of once infected sites in the CPDE with a cluster of some supercritical percolation on the edges of Z d , we prove that, for every dimension d $\ge$ 2, we can find some slightly subcritical p < pc(d) such that for every update speed v > 0, the contact process with large enough infection rate can survive. This extends the result for dimension 1 proved by Linker and Remenik in [LR20].

math.PR↗
arXiv2025

Richardson's model and the contact process with stirring: long time behavior

We study two famous interacting particle systems, the so-called Richardson's model and the contact process, when we add a stirring dynamics to them. We prove that they both satisfy an asymptotic shape theorem, as their analogues without stirring, but only for high enough infection rates, using couplings and restart techniques. We also show that for Richardson's model with stirring, for high enough infection rates, each site is forever infected after a certain time almost surely. Finally, we study weak and strong survival for both models on a homogeneous infinite tree, and show that there are two phase transitions for certain values of the parameters and the dimension, which is a result similar to what is proved for the contact process.

math.PR↗
arXiv2022

Corner percolation with preferential directions

Corner percolation is a dependent bond percolation model on Z^2 introduced by Bálint Tóth, in which each vertex has exactly two incident edges, perpendicular to each other. Gábor Pete has proven in 2008 that under the maximal entropy probability measure, all connected components are finite cycles almost surely. We consider here a regime where West and North directions are preferred with probability p and q respectively, with (p,q) different from (1/2,1/2). We prove that there exists almost surely an infinite number of infinite connected components, which are in fact infinite paths. Furthermore, they all have the same asymptotic slope (2q-1)/(1-2p).

math.PR↗
arXiv2021

Percolation and first-passage percolation on oriented graphs

We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices $\Z^d$ that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for the existence of an infinite cluster may be direction-dependent. Then, we prove that the phase transition in a given direction is sharp, and study the links between percolation and first-passage percolation on these oriented graphs.

math.PR↗
arXiv2016

Continuity of the time and isoperimetric constants in supercritical percolation

We consider two different objects on super-critical Bernoulli percolation on $\mathbb{Z}^d$ : the time constant for i.i.d. first-passage percolation (for $d\geq 2$) and the isoperimetric constant (for $d=2$). We prove that both objects are continuous with respect to the law of the environment. More precisely we prove that the isoperimetric constant of supercritical percolation in $\mathbb{Z}^2$ is continuous in the percolation parameter. As a corollary we prove that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdroff metric. Concerning first-passage percolation, equivalently we consider the model of i.i.d. first-passage percolation on $\mathbb{Z}^d$ with possibly infinite passage times: we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbf{P}[t(e)<+\infty] >p_c(d)$. We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property previously proved by Cox and Kesten for first passage percolation with finite passage times.

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

Growth of a population of bacteria in a dynamical hostile environment

We study the growth of a population of bacteria in a dynamical hostile environment corresponding to the immune system of the colonised organism. The immune cells evolve as subcritical open clusters of oriented percolation and are perpetually reinforced by an immigration process, while the bacteria try to grow as a supercritical oriented percolation in the remaining empty space. For appropriate values of the parameters, we prove that the population of bacteria grows linearly. In this perspective, we build general tools to study dependent percolation models issued from renormalization processes.

q-bio.PE↗
arXiv2012

The critical branching random walk in a random environment dies out

We study the possibility for branching random walks in random environment (BRWRE) to survive. The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. As noted by Comets and Yoshida, the BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity $Ψ$ called the free energy is well studied. Comets and Yoshida proved that there is no survival when $Ψ<0$ and that survival is possible when $Ψ>0$. We proved here that, except for degenerate cases, the BRWRE always die when $Ψ=0$. This solves a conjecture of Comets and Yoshida.

math.PR↗
arXiv2012

Asymptotic shape for the contact process in random environment

The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if H_t denotes the set of already occupied sites at time t, we show that for almost every environment, when the contact process survives, the set H_t/t almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.

math.PR↗
arXiv2012

Large deviations for the contact process in random environment

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $μ$ on $\Rd$ such that the hitting time $t(x)$ is asymptotically equivalent to $μ(x)$ when the contact process survives. We provide here exponential upper bounds for the probability of the event $\{\frac{t(x)}{μ(x)}\not\in [1-ε,1+ε]\}$; these bounds are optimal for independent random environment. As a special case, this gives the large deviation inequality for the contact process in a deterministic environment, which, as far as we know, has not been established yet.

math.PR↗
arXiv2009

La forme asymptotique du processus de contact en environnement aléatoire

The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if H_t denotes the set of already occupied sites at time t, we show that for almost every environment, when the contact process survives, the set H_t/t almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.

math.PR↗
arXiv2009

Déviations modérées de la distance chimique

In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.

math.PR↗
arXiv2009

Moderate deviations for the chemical distance in Bernoulli percolation

In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.

math.PR↗
arXiv2007

Large deviations for the chemical distance in supercritical Bernoulli percolation

The chemical distance D(x,y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $μ$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{μ(x)}\notin (1-ε, 1+ε) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.

math.PR↗
arXiv2006

First-passage competition with different speeds: positive density for both species is impossible

Consider two epidemics whose expansions on $\mathbb{Z}^d$ are governed by two families of passage times that are distinct and stochastically comparable. We prove that when the weak infection survives, the space occupied by the strong one is almost impossible to detect: for instance, it could not be observed by a medium resolution satellite. We also recover the same fluctuations with respect to the asymptotic shape as in the case where the weak infection evolves alone. In dimension two, we prove that one species finally occupies a set with full density, while the other one only occupies a set of null density. We also prove that the Häggström-Pemantle non-coexistence result "except perhaps for a denumerable set" can be extended to families of stochastically comparable passage times indexed by a continuous parameter.

math.PR↗
arXiv2005

Competition between growths governed by Bernoulli Percolation

We study a competition model on $\mathbb{Z}^d$ where the two infections are driven by supercritical Bernoulli percolations with distinct parameters $p$ and $q$. We prove that, for any $q$, there exist at most countably many values of $p<\min(q, \overrightarrow{p\_c})$ such that coexistence can occur.

math.PR↗