Energy of flows on percolation clusters

It is well known for which gauge functions H there exists a flow in Z(d) wi th finite H energy. In this paper we discuss the robustness under random th inning of edges of the existence of such flows. Instead of Z(d) we let our (random) graph C-infinity(Z(d),p) be the graph obtained from Z(d) by removi ng edges with probability 1-p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d greater than or equal to3,p >p(c)(Z(d)) , simple random walk on C-infinity(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 Sigma (e)f(e)(2) is finite. Levin and Peres (1998 ) sharpened this result, and showed that if d greater than or equal to3 and p >p(c)(Z(d)), then C-infinity(Z(d),p) supports a nonzero flow f such that the x(q) energy is finite for all q >d/(d-1). However, for general gauge f unctions, 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 flo ws for Z(d). In this paper we close the gap by showing that if d greater th an or equal to3 and Z(d) supports a flow of finite H energy then the infini te percolation cluster on Z(d) also support flows of finite H energy. This disproves a conjecture of Levin and Peres.