Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation

Abstract

In this article, we consider random walk on the infinite cluster of bond percolation on ${\mathbb{Z}}^d(d\ge 2)$. We show that the Laplace transformation of the number of visited points $N\_n$, has a behaviour as the random walk was on ${\mathbb{Z}}^d$. More precisely, for all $0<\alpha <1$, we proved that there exist constants $C\_i$ and $C\_s$ such that for all infinite cluster that contains the origin, we have: $$ e^{-C\_i n^{\frac{d}{d+2}}} \leq \E\_0^ω (α^{N\_n}) \leq e^{-C\_sn^{\frac{d}{d+2}}}.$$ Our approach is based on finding an isoperimetric inequalities on the infinite cluster, lifted on a wreath product which give good behaviour. The problem of the isoperimetry on wreath product was already raised by A.Ershler.

38 pages, 3 figures