Ideal Poisson-Voronoi tessellations on hyperbolic spaces

Abstract

We study the limit in low intensity of Poisson--Voronoi tessellations in hyperbolic spaces ${\mathbb{H}}_d$ for $d\ge 2$. In contrast to the Euclidean setting, a limiting nontrivial ideal tessellation ${\mathcal{V}}_d$ appears as the intensity tends to $0$. The tessellation ${\mathcal{V}}_d$ is a natural, isometry-invariant decomposition of ${\mathbb{H}}_d$ into countably many unbounded polytopes, each with a unique end. We study its basic properties, in particular, the geometric features of its cells.

Second revised version, 51 pages, 14 figures. Accepted for publication on Annals of Probability