Energy of Flows on Percolation Clusters

It is well known for which gauge functions H there exists a flow on Z with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Z we let our (random) graph , p) be the graph obtained from Z by removing edges with probability 1-p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for 3, p > p c (Z ), simple random walk on , p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x energy e f(e) is finite. Levin and Peres (1998) sharpened this result, and showed that if d ), then , p) supports a nonzero flow f such that the x energy is finite for all q > d/(d 1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Z . In this paper we close the gap by showing that if d 3 and Z supports a flow of finite H energy then the infinite percolation cluster on Z also support flows of finite H energy. This disproves a conjecture of Levin and Peres