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Ideal Poisson-Voronoi tessellations on hyperbolic spaces

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arxiv 2303.16831 v3 pith:QCMJ3VIH submitted 2023-03-29 math.PR

Matteo D'Achille , Nicolas Curien , Nathanaël Enriquez , Russell Lyons , Meltem Ünel This is my paper · ORCID
classification math.PR
keywords hyperbolicidealintensitymathbbmathcalspacestessellationtessellations
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Convergence towards Ideal Poisson--Voronoi tessellations with a focus on Diestel--Leader graphs

    math.PR 2026-06 unverdicted novelty 8.0 of 10

    Necessary and sufficient conditions for convergence of low-intensity Poisson–Voronoi diagrams to a unique ideal tessellation, applied to symmetric spaces and Diestel–Leader graphs.

  2. Ideal Poisson--Voronoi tessellations beyond hyperbolic spaces

    math.PR 2024-12 conditional novelty 6.0 of 10

    The low-intensity Poisson-Voronoi tessellation of H2 × H2 with the L1 metric converges to an isometry-invariant ideal tessellation whose cell ends are unions of boundary circles, and equal-separation loci between coro...

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