Harmonic measures versus quasiconformal measures for hyperbolic groups

TL;DR

This paper derives a dimension formula for harmonic measures of random walks on hyperbolic groups, linking them to quasiconformal measures via the Green metric, and characterizes when this dimension is maximal.

Contribution

It introduces a new dimension formula for harmonic measures and characterizes maximal dimension cases using the Green metric in hyperbolic groups.

Findings

Dimension formula for harmonic measures established

Characterization of random walks with maximal harmonic measure dimension

Harmonic measures interpreted as quasiconformal measures on the boundary

Abstract

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as \qc measures on the boundary of the group.