Asymptotic shape for the chemical distance and first-passage percolation in random environment

Olivier Garet (1) , Régine Marchand (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 IECN - Institut Élie Cartan de Nancy

The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on $\Zd$ 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 points to a deterministic shape that does not depend on the random environment.As a special case of the previous result, we obtain an asymptotic shape theoremfor the chemical distance in supercritical Bernoulli percolation.We also prove a flat edge result. Some various examples are also given.

Mots clés: percolation; first-passage percolation; chemical distance; asymptotic shape; random environment

Domaines: Probabilités [math.PR]

Olivier Garet, Régine Marchand. Asymptotic shape for the chemical distance and first-passage percolation in random environment. ESAIM: Probability and Statistics, 2004, 8, p. 169-199. ⟨hal-00000300⟩