Transience of percolation clusters on wedges

TL;DR

This paper investigates the transience of infinite percolation clusters on wedges in three-dimensional space, establishing conditions under which these clusters are transient and linking finite energy flows between the lattice and clusters.

Contribution

It proves that the transience of the wedge implies the transience of the percolation cluster and relates finite energy flows on the lattice to those on the cluster, answering open questions.

Findings

Infinite percolation clusters are transient on transient wedges.

Finite energy flows on Z^2 are equivalent to those on percolation clusters under certain conditions.

Addresses open problems posed by Haggstrom, Mossel, and Hoffman.

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. Haggstrom 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 solves a question of C. Hoffman.