Geodesic
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

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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.

DOI : 10.1051/ps:2004009
Classification : 60G15, 60K35, 82B43
Keywords: percolation, first-passage percolation, chemical distance, infinite cluster, asymptotic shape, random environment 
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     author = {Garet, Olivier and Marchand, R\'egine},
     title = {Asymptotic shape for the chemical distance and first-passage percolation on the infinite {Bernoulli} cluster},
     journal = {ESAIM: Probability and Statistics},
     pages = {169--199},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2004},
     doi = {10.1051/ps:2004009},
     mrnumber = {2085613},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2004009/}
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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

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