Vanishing uniqueness thresholds in Voronoi percolation on products

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 $p_u(\hat{I}»)$ at small intensities $\hat{I}»>0$ 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 $k\ge 2,d\ge 3$ and products of hyperbolic spaces ${\mathbb{H}}_{d_1}\times \dots \times {\mathbb{H}}_{d_k}$ for $k\ge 2,d_i\ge 2$, 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.

32 pages + 5 pages appendix, 1 figure. Comments welcome!