Ideal Poisson-Voronoi tessellations on hyperbolic spaces

TL;DR

This paper investigates the limiting behavior of Poisson-Voronoi tessellations in hyperbolic spaces as intensity approaches zero, revealing a unique ideal tessellation with specific geometric properties.

Contribution

It introduces and analyzes a new ideal tessellation in hyperbolic spaces that emerges in the low-intensity limit, contrasting with Euclidean cases.

Findings

Existence of a nontrivial ideal tessellation in hyperbolic spaces at low intensity

The tessellation is isometry-invariant and decomposes hyperbolic space into unbounded polytopes

Each cell has a unique end and specific geometric features

Abstract

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.