The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on $\Zd$ 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 points to a deterministic shape that does not depend on the random environment.As a special case of the previous result, we obtain an asymptotic shape theoremfor the chemical distance in supercritical Bernoulli percolation.We also prove a flat edge result. Some various examples are also given.
Publié le : 2004-07-05
Classification:
first-passage percolation,
chemical distance,
asymptotic shape,
random environment,
percolation,
60G15; 60K35; 82B43,
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
@article{hal-00000300,
author = {Garet, Olivier and Marchand, R\'egine},
title = {Asymptotic shape for the chemical distance and first-passage percolation in random environment},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00000300}
}
Garet, Olivier; Marchand, Régine. Asymptotic shape for the chemical distance and first-passage percolation in random environment. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00000300/