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We show that for all p>p_c(\Z^d) percolation parameters, the probability thatthe cluster of the origin is finite but has at least t vertices at distance onefrom the infinite cluster is exponentially small in t. We use this to give ashort proof of the strongest version of the important fact that theisoperimetric profile of the infinite cluster basically coincides with theprofile of the original lattice. This implies, e.g., that simple random walk onthe largest cluster of a finite box [-n,n]^d with high probability hasL^\infty-mixing time \Theta(n^2), and that the heat kernel (return probability)on the infinite cluster a.s. decays like p_n(o,o)=O(n^{-d/2}). Versions ofthese results have been proven by Benjamini and Mossel (2003), Mathieu and Remy(2004), Barlow (2004) and Rau (2006). For general infinite graphs, we provethat anchored isoperimetric properties survive supercritical percolation,provided that the probability of the cluster of the origin being finite withlarge boundary decays rapidly; this is the case for a large class of graphswhen $p$ is close to 1. As an application (with the help of some entropyinequalities), we give a short conceptual proof of a theorem of Angel,Benjamini, Berger and Peres (2006): the infinite percolation cluster of a wedgein \Z^3 is a.s. transient whenever the wedge itself is transient.

A note on percolation on $Z^d$: isoperimetric profile via exponential cluster repulsion