Energy and Cutsets in Infinite Percolation Clusters

Grimmett, Kesten and Zhang (1993) showed that for d ≥ 3, simple random walk on the infinite cluster C∞(Z, p) of supercritical percolation on Zd is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the 2–energy ∑ e f(e) 2 is finite. Here we sharpen this result, and show that if d ≥ 3 and p > pc(Z), then C∞(Z, p) supports a nonzero flow f such that the q–energy ∑ e |f(e)|q is finite for all q > d/(d − 1). As a corollary, we obtain that any sequence {Πn} of disjoint cutsets in C∞(Z, p) that separate a fixed vertex from infinity, must satisfy ∑ n |Πn| < ∞ for all β > 1/(d − 1). Our proofs are based on the method of “unpredictable paths”, developed by Benjamini, Pemantle and Peres (1998) and refined by Häggström and Mossel (1998).