Stationary random graphs and the hyperbolic Poisson Voronoi tessellation

We consider the hyperbolic Poisson Voronoi (HPV) tessellation, a triangulation of the hyperbolic plane whose vertices are given by a homogeneous Poisson point process. This triangulation fails to have a positive isoperimetric constant, however we show that it does have a positive "anchored" isoperimetric constant. HPV is an example of a stationary random graph, one which when viewed from the point of view of random walk, has the same law at all times. We review some of the theory of stationary random graphs and give some extensions that allow us to conclude random walk on HPV is ballistic and converges almost surely to a point on the boundary.
This is joint work with Itai Benjamini (Weizmann) and Josh Pfeffer (Harvard).