Energy of Ows on Z 2 Percolation Clusters

Grimmett, Kesten, and Zhang (1993) showed that for d 3 and p > p c (Z d) simple random walk on the innnite percolation cluster on Z d is a.s. transient. This is equivalent to the existence of a ow of nite energy on the innnite cluster. Benjamini, Pemantle, and Peres (1998) gave an alternate proof of this result. Levin and Peres adapted the approach of Benjamini, Pemantle, and Peres to show that if d 3 and p > p c (Z d), then the innnite cluster supports a nonzero ow f with nite q energy for all q > d=(d ? 1) and d 3. In this paper we use the method of Grimmet, Kesten, and Zhang to extend this result of Levin and Peres's to d = 2. We also extend another result of Levin and Peres to show that the innnite cluster on Z 2 supports a nonzero ow f with nite H energy for all > 2. As an application we exhibit a graph that has transient percolation clusters, but does not admit exponential intersection tails. This answers a question asked by Benjamini, Pemantle, and Peres.