Energy and Cutsets in In nite Percolation Clusters

Grimmett, Kesten and Zhang (1993) showed that for d 3, simple random walk on the innnite cluster C 1 (Z d ; p) of supercritical percolation on Z d is a.s. transient. Their result is equivalent to the existence of a nonzero ow f on the innnite cluster such that the 2{energy P e f(e) 2 is nite. Here we sharpen this result, and show that if d 3 and p > p c (Z d), then C 1 (Z d ; p) supports a nonzero ow f such that the q{energy P e jf(e)j q is nite for all q > d=(d ? 1). As a corollary, we obtain that any sequence f n g of disjoint cutsets in C 1 (Z d ; p) that separate a xed vertex from innnity, must satisfy P n j n j ? < 1 for all > 1=(d ? 1). Our proofs are based on the method of \unpredictable paths", developed by Benjamini, Pemantle and Peres (1998) and reened by HH aggstrr om and Mossel (1998).