Ideal Poisson--Voronoi tessellations for Finsler metrics on symmetric spaces

Abstract

Ideal Poisson--Voronoi tessellations are random tessellations that arise as limits of Voronoi diagrams for Poisson point processes as their intensity tends to zero. For the Riemannian metric on a symmetric space of higher rank, Frączyk, Mellick, and Wilkens (arXiv:2307.01194) proved that the corresponding ideal Poisson--Voronoi tessellation is almost surely completely connected, with an unbounded side between any two cells. This contrasts sharply with the geometry of ideal Poisson--Voronoi tessellations in hyperbolic spaces studied by D'Achille, Curien, Enriquez, Lyons, and Ünel (arXiv:2303.16831). In this paper we introduce and study ideal Poisson--Voronoi tessellations for Finsler metrics on higher rank symmetric spaces. We show that ideal Poisson--Voronoi tessellations with respect to polyhedral Finsler metrics can exhibit either behavior seen in these previous works: generically they resemble the hyperbolic case, where almost surely all cells have bounded sides. We further introduce the ideal Selberg--Poisson--Voronoi tessellation of the space of positive definite symmetric matrices, which has particularly nice properties.