Dimension jump at the uniqueness threshold for percolation in $\infty+d$ dimensions

TL;DR

This paper investigates percolation on products of trees and lattices, revealing a dimension jump at the uniqueness threshold and establishing critical threshold coincidences in nonamenable graphs.

Contribution

It proves a Hausdorff dimension jump at the percolation threshold and shows that certain critical thresholds coincide in nonamenable graph products, extending previous results.

Findings

Hausdorff dimension of cluster boundary points jumps from at most 1/2 to 1 at p_u

Various critical thresholds including p_{2→2} coincide with p_u in these models

Results apply to products of trees with arbitrary infinite amenable Cayley graphs and lamplighter groups

Abstract

Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which percolation has $0$, $\infty$, and $1$ infinite clusters a.s., and it was proven by Schonmann (1999) that there are infinitely many infinite clusters a.s. at the uniqueness threshold $p=p_u$. We strengthen this result by showing that the Hausdorff dimension of the set of accumulation points of each infinite cluster in the boundary of the tree has a jump discontinuity from at most $1/2$ to $1$ at the uniqueness threshold $p_u$. We also prove that various other critical thresholds including the $L^2$ boundedness threshold $p_{2\to 2}$ coincide with $p_u$ for such products, which are the first nonamenable examples proven to have this property. All our results apply more generally to products of trees with arbitrary infinite amenable Cayley graphs and to the lamplighter on the tree.