ESAIM: Probability and Statistics

Table of Contents - September 2004 - Volume 8  

Research Article

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

Garet, Oliviera1 and Marchand, Réginea2

a1 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.

a2 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.

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.

(Received September 4 2003)

(Revised June 25 2004)

(Online publication September 15 2004)

Key Words:

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

Mathematics Subject Classification:

  • 60G15;
  • 60K35;
  • 82B43