Asymptotic Shape for the Chemical Distance and First-passage Percolation on the Infinite Bernoulli Cluster

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 realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result in the case of dimension 2. Various examples are also given. Mathematics Subject Classification. 60G15, 60K35, 82B43. Received September 4, 2003. Revised June 25, 2004. First-passage percolation was introduced by Hammersley and Welsh [12] 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 [7] showed that, under some moment conditions, the renormalized set of wet vertices at time t almost surely converges to a deterministic asymptotic shape. Derriennic (cited by Kesten [13]), and next Boivin [3], progressively extended the result to the stationary ergodic case. Häggström and Meester [11] also proved that every symmetric compact set with nonempty interior can be obtained as the asymptotic shape of a stationary first-passage percolation model. In this paper, we want to study the analogous problem of spread of a fluid in a more complex medium. On one hand, an edge can either be open or closed according to the local properties of the medium – e.g. 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.