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

Published online by Cambridge University Press:  15 September 2004

Olivier Garet  and
Régine Marchand
Olivier Garet
Affiliation:
Laboratoire de Mathématiques, Applications et Physique Mathématique d'Orléans UMR 6628, Université d'Orléans, BP 6759, 45067 Orléans Cedex 2, France; Olivier.Garet@labomath.univ-orleans.fr.
Régine Marchand
Affiliation:
Institut Elie Cartan Nancy (mathématiques), Université Henri Poincaré Nancy 1, Campus Scientifique, BP 239, 54506 Vandœuvre-lès-Nancy Cedex, France; Regine.Marchand@iecn.u-nancy.fr.
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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.

Keywords

Percolationfirst-passage percolationchemical distanceinfinite clusterasymptotic shaperandom environment.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 8 , August 2004 , pp. 169 - 199
Copyright
© EDP Sciences, SMAI, 2004

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