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

Olivier Garet; Régine Marchand

ESAIM: Probability and Statistics (2004)

- Volume: 8, page 169-199
- ISSN: 1292-8100

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topGaret, Olivier, and Marchand, Régine. "Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster." ESAIM: Probability and Statistics 8 (2004): 169-199. <http://eudml.org/doc/244937>.

@article{Garet2004,

abstract = {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.},

author = {Garet, Olivier, Marchand, Régine},

journal = {ESAIM: Probability and Statistics},

keywords = {percolation; first-passage percolation; chemical distance; infinite cluster; asymptotic shape; random environment; Percolation},

language = {eng},

pages = {169-199},

publisher = {EDP-Sciences},

title = {Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster},

url = {http://eudml.org/doc/244937},

volume = {8},

year = {2004},

}

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

PB - EDP-Sciences

VL - 8

SP - 169

EP - 199

AB - 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.

LA - eng

KW - percolation; first-passage percolation; chemical distance; infinite cluster; asymptotic shape; random environment; Percolation

UR - http://eudml.org/doc/244937

ER -

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