Ideal Poisson--Voronoi tessellations beyond hyperbolic spaces

Abstract

We construct and study the ideal Poisson--Voronoi tessellation of the product of two hyperbolic planes $\mathbb{H}_{2}\times \mathbb{H}_{2}$ endowed with the $L^{1}$ norm. We prove that its law is invariant under all isometries of this space and study some geometric features of its cells. Among other things, we prove that the set of points at equal separation to any two corona points is unbounded almost surely. This is analogous to a recent result of Fr\k{a}czyk-Mellick-Wilkens for higher rank symmetric spaces.