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

Olivier Garet; Régine Marchand

ESAIM: Probability and Statistics (2010)

- Volume: 8, page 169-199
- ISSN: 1292-8100

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topGaret, Olivier, and Marchand, Régine. "Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster." ESAIM: Probability and Statistics 8 (2010): 169-199. <http://eudml.org/doc/104317>.

@article{Garet2010,

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.; random environment},

language = {eng},

month = {3},

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/104317},

volume = {8},

year = {2010},

}

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

DA - 2010/3//

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.; random environment

UR - http://eudml.org/doc/104317

ER -

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