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

Abstract

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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.

How to cite

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Garet, 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 -

References

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