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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)
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)
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)
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)
In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance D(x, y) 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(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)