Discrete random walks on the group Sol

TL;DR

This paper studies the harmonic measure on the boundary of the group Sol for discrete random walks, identifying conditions for absolute continuity or singularity, and constructing examples with various measure properties.

Contribution

It characterizes when harmonic measures are singular or absolutely continuous, introduces new examples of singular measures with small speed, and compares these to Bernoulli convolutions.

Findings

Countable non-abelian subgroups admit singular harmonic measures.

Some finitely supported random walks have regular harmonic measures.

Existence of random walks with small speed but singular harmonic measures.

Abstract

The harmonic measure $\nu$ on the boundary of the group $Sol$ associated to a discrete random walk of law $\mu$ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with respect to Lebesgue measure. By ratio entropy over speed, we show that any countable non-abelian subgroup admits a finite first moment non-degenerate $\mu$ with singular harmonic measure $\nu$. On the other hand, we prove that some random walks with finitely supported step distribution admit a regular harmonic measure. Finally, we construct some exceptional random walks with arbitrarily small speed but singular harmonic measures. The two later results are obtained by comparison with Bernoulli convolutions, using results of Erd\"os and Solomyak.