Asymptotic Shape for the Chemical Distance and First-passage Percolation in Random Environment

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 the random environment. As a special case of the previous result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result. Some various examples are also given. First-passage percolation was introduced in 1965 by Hammersley and Welsh [HW65] as a model for the spread of a fluid in a porous medium. To each edge of the Z lattice is attached a nonnegative random variable which corresponds to the travel time needed by the fluid to cross the edge. When the passage times are independent identically distributed variables, Cox and Durrett showed that, under some moment conditions, the renormalized set of wet points at time t almost surely converges to a deterministic asymptotic shape. Deriennic [Kes86], and next Boivin [Boi90], progressively extended the result to the stationary ergodic case. In this paper, we want to study the analogous problem of spread of a fluid in a more complex medium. On the one hand, an edge can either be open or closed according to the local properties of the medium – e.g. according to the absence or the presence of non-porous particles. In other words, the Z lattice is replaced by a random environment given by the infinite cluster of a super-critical Bernoulli percolation model. On the other hand, as in the classical model, a random passage time is attached to each open edge. This random time corresponds to the local porosity of the medium – e.g. the density of the porous phase. Thus, our model can be seen as a combination between classical Bernoulli percolation and stationary first-passage percolation. Our goal is to prove in this context the convergence of the renormalized set of wet points to a deterministic shape that does not depend on the random environment. As a special case of the previous result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. In the first section, we give some notations adapted to this problem, precise the assumptions on the passage time and introduce the background of ergodic theory needed for the main proof. In section 2, we give some estimations on the chemical distance and the travel times. In section 3, we prove the existence of an asymptotic speed in a given direction and study its properties ( e.g. continuity, homogeneity and sub-additivity. . . ) Section 4 is devoted to the question of the positivity of these speeds, and in section 5 we prove the asymptotic shape result. In section 6, we study the existence of a flat edge in the asymptotic shape. In the last section, 1991 Mathematics Subject Classification. 60G15, 60K35, 82B43.