Percolation at the uniqueness threshold via subgroup relativization

TL;DR

This paper investigates percolation on nonamenable groups at the critical threshold, establishing new non-uniqueness results and properties of infinite clusters using subgroup relativization techniques.

Contribution

It introduces a novel method of subgroup relativization to prove non-uniqueness at the threshold and explores inheritance properties of non-uniqueness in nonamenable groups.

Findings

Non-uniqueness at $p_u$ for groups with amenable, $wq$-normal subgroups of exponential growth

Inheritance of non-uniqueness properties from subgroups to entire groups

Resolution of a question on intersections of random walks with percolation clusters

Abstract

We study percolation on nonamenable groups at the uniqueness threshold $p_u$, the critical value that separates the phase in which there are infinitely many infinite clusters from the phase in which there is a unique infinite cluster. The number of infinite clusters at $p_u$ itself is a subtle question, depending on the choice of group, with only a relatively small number of examples understood. In this paper, we do the following: 1. Prove non-uniqueness at $p_u$ in a new class of examples, namely those groups that contain an amenable, $wq$-normal subgroup of exponential growth. Concrete new examples to which this result applies include lamplighters over nonamenable base groups. 2. Prove a co-heredity property of a certain strong form of non-uniqueness at $p_u$, stating that this property is inherited from a $wq$-normal subgroup to the entire group. Remarkably, this co-heredity property is the same as that proven for the vanishing of the first $\ell^2$ Betti number by Peterson and Thom (Invent. Math. 2011), supporting the conjecture that the two properties are equivalent. Our proof is based on the method of subgroup relativization, and relies in particular on relativized versions of uniqueness monotonicity, the equivalence of non-uniqueness and connectivity decay, the sharpness of the phase transition, and the Burton-Keane theorem. As a further application of the relative Burton-Keane theorem, we resolve a question of Lyons and Schramm (Ann. Probab. 1999) concerning intersections of random walks with percolation clusters.