[HG_AP/HG_GGD]Singularity of harmonic measure on Poisson-Voronoi tessellations in hyperbolic spaces
ABSTRACT
Poisson-Vononoi tessellation provide natural examples of invariant random tilings of homogeneous spaces. Benjamini, Paquette, and Pfeffer proved that the simple random walk on the Poisson-Voronoi tessellation of the hyperbolic plane has positive speed and hence converges almost surely to the boundary circle. They conjectured that its hitting measure on the boundary is singular with respect to Lebesgue measure. I will explain how this follows from an ergodic-theoretic argument based on rerooting relations, and briefly discuss the higher-dimensional case as well.
