Energy of Flows on Percolation Clusters

It is well known for which gauge functions H there exists a flow in Z d 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 d we let our (random) graph cal C∞(Z d,p) be the graph obtained from Z d by removing edges with probability 1−p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d ≥ 3, p > pc(Z d), simple random walk on C cal ∞(Z d, 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 2 energy ∑e f(e)2 is finite. Levin and Peres (1998) sharpened this result, and showed that if d ≥ 3 and p > pc(Zd), then cal C∞(Zd, p) supports a nonzero flow f such that the x q 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 Zd. In this paper we close the gap by showing that if d ≥ 3 and Z d supports a flow of finite H energy then the infinite percolation cluster on Zd also support flows of finite H energy. This disproves a conjecture of Levin and Peres.