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The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on Z 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 vertices to a deterministic shape that does not depend on the… (More)

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Z 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, there is a strictly positive probability that {z ∈ Z; d(y, z) < d(x, z)} and {z ∈ Z;… (More)

- Olivier Garet
- 2004

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

- Olivier Garet
- 2002

We consider a random field (Xn)n∈Zd and investigate when the set Ah = {k ∈ Zd; ‖Xk‖ ≥ h} has infinite clusters. The main problem is to decide whether the critical level hc = sup{h ∈ R; P (Ah 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 fields… (More)

The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on Zd 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)

- Olivier Garet
- 2006

Consider two epidemics whose expansions on Z 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)

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm μ on R 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 { t(x) μ(x) 6∈ [1 − ε, 1 + ε]}; these bounds are optimal for… (More)

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… (More)

- Olivier Garet
- 2008

We study a competition model on Z 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,− pc} such that coexistence can occur.