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

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.