Anchored expansion of Delaunay complexes in real hyperbolic space and stationary point processes

TL;DR

This paper establishes conditions under which Delaunay complexes in hyperbolic space exhibit positive anchored expansion, with applications to stationary Poisson and determinantal point processes, including the introduction of Berezin point processes.

Contribution

It provides new sufficient conditions for anchored expansion in hyperbolic Delaunay complexes and introduces Berezin point processes, expanding understanding of geometric and probabilistic properties in hyperbolic space.

Findings

Stationary Poisson--Delaunay graphs have positive anchored expansion.

Delaunay graphs from stationary determinantal point processes also exhibit positive anchored expansion.

Introduction and partial characterization of Berezin point processes in hyperbolic space.

Abstract

We give sufficient conditions for a discrete set of points in any dimensional real hyperbolic space to have positive anchored expansion. The first condition is a bounded mean density property, ensuring not too many points can accumulate in large regions. The second is a bounded mean vacancy condition, effectively ensuring there is not too much space left vacant by the points over large regions. These properties give as an easy corollary that stationary Poisson--Delaunay graphs have positive anchored expansion, as well as Delaunay graphs built from stationary determinantal point processes. We introduce a family of stationary determinantal point processes on any dimension of real hyperbolic space, the Berezin point processes, and we partially characterize them. We pose many questions related to this process and stationary determinantal point processes.