Abstract
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 cal ∞(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>p c(Z d), simple random walk on cal 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>p c(Z d), then cal C cal ∞(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 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 d. 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 Z d also support flows of finite H energy. This disproves a conjecture of Levin and Peres.
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Hoffman, C., Mossel, E. Energy of Flows on Percolation Clusters. Potential Analysis 14, 375–385 (2001). https://doi.org/10.1023/A:1011216004099
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DOI: https://doi.org/10.1023/A:1011216004099