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

DOI : 10.1051/ps:2004009
Classification : 60G15, 60K35, 82B43
Mots clés : percolation, first-passage percolation, chemical distance, infinite cluster, asymptotic shape, random environment
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Garet, Olivier; Marchand, Régine. Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM: Probability and Statistics, Tome 8 (2004), pp. 169-199. doi : 10.1051/ps:2004009. http://www.numdam.org/articles/10.1051/ps:2004009/

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