The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on 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.
Mots clés : percolation, first-passage percolation, chemical distance, infinite cluster, asymptotic shape, random environment
@article{PS_2004__8__169_0,
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://www.numdam.org/articles/10.1051/ps:2004009/}
}
TY - JOUR AU - Garet, Olivier AU - Marchand, Régine TI - Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster JO - ESAIM: Probability and Statistics PY - 2004 SP - 169 EP - 199 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2004009/ DO - 10.1051/ps:2004009 LA - en ID - PS_2004__8__169_0 ER -
%0 Journal Article %A Garet, Olivier %A Marchand, Régine %T Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster %J ESAIM: Probability and Statistics %D 2004 %P 169-199 %V 8 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2004009/ %R 10.1051/ps:2004009 %G en %F PS_2004__8__169_0
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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