Olivier Garet

Learn More
We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Z d or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times that for distinct points x, y ∈ Z d , there is a strictly positive probability that {z ∈ Z d ; d(y, z) < d(x, z)} and {z(More)
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 behavior 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 0 ↔ x, D(0, x)(More)
We consider a random field (Xn) n∈Z d and investigate when the set A h = {k ∈ Z d ; X k ≥ h} has infinite clusters. The main problem is to decide whether the critical level h c = sup{h ∈ R; P (A h has an infinite cluster) > 0} is neither 0 nor +∞. Thus, we say that a percolation transition occurs. In a first time, we show that weakly dependent Gaussian(More)
The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on Z d to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet points to a deterministic shape that does not depend on(More)
Consider two epidemics whose expansions on 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(More)
We study the growth of a population of bacteria in a dynami-cal hostile environment corresponding to the immune system of the colonised organism. The immune cells evolve as subcritical open clusters of oriented per-colation and are perpetually reinforced by an immigration process, while the bacteria try to grow as a supercritical oriented percolation in the(More)
We study the possibility for branching random walks in random environment (BR-WRE) 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(More)