A Note on Percolation on Z: Isoperimetric Profile via Exponential Cluster Repulsion

We show that for all p > pc(Z) percolation parameters, the probability that the cluster of the origin is finite but has at least t vertices at distance one from the infinite cluster is exponentially small in t. Then we use this to give a very short proof of the important fact that the isoperimetric profile of the infinite cluster basically coincides with the profile of the original lattice. This implies for instance that simple random walk on the largest cluster of a finite box [−n,n]d with high probability has L∞-mixing time Θ(n), and that the heat kernel (return probability) on the infinite cluster a.s. decays like pn(o, o) = O(n−d/2). Versions of these results have been proven by Benjamini and Mossel (2003), Mathieu and Remy (2004), Barlow (2004) and Rau (2006). We also give a short proof of a theorem of Angel, Benjamini, Berger and Peres (2006): the infinite percolation cluster of a wedge in Z is a.s. transient whenever the wedge itself is transient.