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 corona points are unbounded almost surely.
Ideal Poisson-Voronoi tessellations on hyperbolic spaces
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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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Ideal Poisson--Voronoi tessellations beyond hyperbolic spaces
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 corona points are unbounded almost surely.