Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

Abstract

Abstract. The aim of this paper is to extend 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 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. 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 d 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.

50 Years of First-Passage Percolation

“…When the edge weights are either 1 or ∞, the passage time is also known as the chemical distance. In this setting, the benchmark is the work of Garét-Marchand [80], where the analogues of Theorems 2.5, 2.16 were proven under a moment condtion Eτ [80]. Their results are also valid for stationary ergodic passage times, in the spirit of the work of Boivin [27].…”

Continuity for the Rate Function of the Simple Random Walk on Supercritical Percolation Clusters

“…Garet-Marchand have investigated the asymptotic behavior of the chemical distance. The following proposition is one of their results [10], which states that the chemical distance is asymptotically equivalent to a deterministic norm on R d . The next proposition is our objective of this subsection.…”

Overall Properties of a Discrete Membrane with Randomly Distributed Defects

“…. σ (s( j + 1)) with 0 j r , and this piece is contained in B(w( j)) by (19). Since σ contains at most ηN black edges, by assumption, (22) and it contains only white edges.…”