Distributional lattices on Riemannian symmetric spaces

A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion of lattice. Distributional lattices exist in any Riemannian symmetric space: the Voronoi tessellation of a stationary Poisson point process is an example. We show that for an appropriate notion of amenability, the amenability of a distributional lattice is equivalent to the amenability of the ambient space. Using this equivalence, we show that the simple random walk on any nonamenable distributional lattice has positive embedded speed. For nonpositively curved, simply connected spaces, we show that the simple random walk on a Poisson-Voronoi tessellation has positive graph speed by developing some additional structure for Poisson-Voronoi tessellations.