Percolation at the uniqueness threshold via subgroup relativization
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 quest...
Saved in:
Main Authors | , |
---|---|
Format | Journal Article |
Language | English |
Published |
18.09.2024
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | 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. |
---|---|
DOI: | 10.48550/arxiv.2409.12283 |