Ideal Poisson-Voronoi tessellations on hyperbolic spaces

We study the limit in low intensity of Poisson--Voronoi tessellations in hyperbolic spaces $\mathbb{H}_{d}$ for $d \geq 2$. 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.