On the number of distinct visited sites by a random walk on the infinite cluster of the percolation model

We consider random walk oil the infinite cluster of the percolation model on the edges of Z(d) (d >= 2) with law Q, in the surcritical case. We prove that the Laplace transformation of the number of visited sites up to time n, called N-n, has the same behaviour as the random walk oil Z(d). More precisely, we show for all 0 < alpha < 1, there exists some constants C-i, C-s > 0 such that for almost all realisations of the percolation such that the origin belongs to the infinite cluster and for large enough n, e(-Cind/(d+2)) <= E-0(omega)(alpha(Nn)) <= e(-Csnd/(d+2)). The main work is to get the upper bound. Our approach is based, first oil finding an isoperimetric inequality oil the infinite cluster and secondly to lift it on a wreath product, which enables us to get all tipper bound of the return probability of a particular random walk. The introduction of a wreath product is motivated by the fact that the return probability oil such graph is linked to the Laplace transform of distinct visited sites.