Singularity of harmonic measure on Poisson–Voronoi tessellations in hyperbolic spaces

Abstract

We prove that the harmonic measure of simple random walk on the Poisson–Voronoi tessellation in $\mathbb{H}^d$ is almost surely singular with respect to Lebesgue measure on the boundary. This answers a question of Benjamini, Paquette, and Pfeffer in the hyperbolic plane and establishes the analogous result in higher dimensions. Our proof is based on ergodic theory of rerooting relations and their boundary extensions.