| Issue |
ESAIM: PS
Volume 8, August 2004
|
|
|---|---|---|
| Page(s) | 169 - 199 | |
| DOI | https://doi.org/10.1051/ps:2004009 | |
| Published online | 15 September 2004 | |
Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster
1
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.
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.
Received:
4
September
2003
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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