Volume 8, August 2004
|Page(s)||169 - 199|
|Published online||15 September 2004|
Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster
Laboratoire de Mathématiques, Applications et Physique Mathématique d'Orléans UMR 6628, Université d'Orléans, BP 6759, 45067 Orléans Cedex 2, France;
2 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.
Revised: 25 June 2004
The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on 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.
Mathematics Subject Classification: 60G15 / 60K35 / 82B43
Key words: Percolation / first-passage percolation / chemical distance / infinite cluster / asymptotic shape / random environment.
© EDP Sciences, SMAI, 2004
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