Matrix random products with singular harmonic measure

TL;DR

This paper demonstrates that for any Zariski dense subgroup of SL(d,R), there exists a finitely supported symmetric random walk with a singular harmonic measure on the flag space, using new dimension estimates and explicit constructions.

Contribution

It introduces a novel upper bound for the Hausdorff dimension of harmonic measure projections and constructs specific random walks with bounded entropy and diverging Lyapunov exponents.

Findings

Harmonic measure on flag space can be singular for certain subgroups.

New estimates relate Hausdorff dimension to entropy and Lyapunov exponents.

Explicit random walk constructions achieve desired measure properties.

Abstract

Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in $R^d$ in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and Lyapunov exponents going to infinity.