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

Garet, Olivier; Marchand, Régine

ESAIM: Probability and Statistics, Tome 8 (2004), pp. 169-199.

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on ${\mathbb{Z}}^d$ 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.

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