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

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 an anchored bounded density property, ensuring not too many points can accumulate in large regions. The second is an anchored bounded 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.