Non-vanishing uniqueness threshold for hyperbolic Poisson-Voronoi percolation in dimension at least three

Abstract

We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space ${\mathbb{H}}^d$ for $d\ge 3$. By recent results of Grebík and Recke and d'Achille et al., this "uniqueness threshold" $p_u(\hat{I}»)$ tends to zero as the intensity $\hat{I}»$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products ${\mathbb{H}}^{d_1}\times \cdots \times {\mathbb{H}}^{d_k}$ with $k,d_1,\dots ,d_k\ge 2$. In contrast, for Poisson-Voronoi percolation on the hyperbolic plane ${\mathbb{H}}^2$, Benjamini and Schramm have shown that $p_u(\hat{I}»)$ tends to one as $\hat{I}»$ tends to zero, and $p_u(\hat{I}»)>1/2$ for all $\hat{I}»>0$. An unpublished argument of D'Achille and Curien shows that for Poisson-Voronoi percolation on ${\mathbb{H}}^d$ with $d\ge 3$, the uniqueness threshold satisfies $p_u(\hat{I}»)\le 1/2$ for all $\hat{I}»>0$. Here we will show that ${\inf }_{\hat{I}»>0}{p}_u(\hat{I}»)>0$ for Poisson-Voronoi percolation on ${\mathbb{H}}^d$ with $d\ge 3$. This answers a question of Grebík and Recke.