Vanishing uniqueness thresholds in Voronoi percolation on products

TL;DR

This paper investigates percolation thresholds in non-amenable product spaces, introducing a new method to show the vanishing of the uniqueness threshold at small intensities, with applications to various geometric and graph product structures.

Contribution

It develops a novel approach based on unbounded borders to prove the smallness of the uniqueness threshold in Poisson--Voronoi percolation on product spaces, extending previous results to new examples.

Findings

Smallness of the uniqueness threshold $p_u(\lambda)$ at small intensities $\lambda>0$

Application to products of regular trees and hyperbolic spaces

Construction of non-amenable Cayley graphs with unique infinite clusters

Abstract

We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold $p_u(\lambda)$ at small intensities $\lambda>0$ based on the unbounded borders phenomenon of their underlining ideal Poisson--Voronoi tessellation. We apply our method to several concrete examples in both the discrete and the continuum setting, including $k$-fold graph products of $d$-regular trees for $k\ge2,d\ge3$ and products of hyperbolic spaces $\mathbb H_{d_1}\times \ldots \times \mathbb H_{d_k}$ for $k\ge2, d_i\ge2$, complementing a recent result of the second and fourth author for symmetric spaces of connected higher rank semisimple real Lie groups with property (T). We also provide new examples of non-amenable Cayley graphs with the FIID sparse unique infinite cluster property, answering positively a recent question of Pete and Rokob.