Abstract
We study random walks on supercritical percolation clusters on wedges in $Z^3$, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on $Z^2$ is equivalent to the (a.s.) existence of a finite energy flow on the supercritical percolation cluster. This answers a question of C. Hoffman.
Highlights
For simple random walk in the Zd lattice, Polya [21] showed in 1920 that the transition from recurrence to transience occurs when d increases from 2 to 3
For an increasing positive function h, the wedge Wh is the subgraph of Z3 induced by the vertices
We identify μ with the flow that it induces, i.e. the expectation of a unit flow through a path chosen by μ
Summary
For simple random walk in the Zd lattice, Polya [21] showed in 1920 that the transition from recurrence to transience occurs when d increases from 2 to 3. In [14] Hoffman and Mossel (refining an earlier result of Levin and Peres [15]) proved that the same is true for the infinite cluster of supercritical percolation in Zd, provided that d ≥ 3. In [12] Hoffman proved that on the infinite cluster in Z2 there are flows with finite (2, 2 + ε)energy. Hoffman asks whether there are flows with finite (2, 1 + ε)-energy on the infinite percolation cluster. The infinite cluster of supercritical bond percolation in Z2 a.s. supports a flow of finite (2, 1 + ε)-energy. This proof relies on connectivity properties of Zd instead of the Antal-Pisztora Theorem and can be extended to other graphs.
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