Dimension drop of harmonic measure for some finite range random walks on Fuchsian Schottky groups

TL;DR

This paper proves that for specific finite-range random walks on Fuchsian Schottky groups, the harmonic measure's dimension is strictly less than the Hausdorff dimension of the limit set, revealing a dimension drop phenomenon.

Contribution

It establishes a dimension drop result for harmonic measures associated with finite-range random walks on Fuchsian Schottky groups, a novel finding in this context.

Findings

Harmonic measure dimension is strictly less than the limit set's Hausdorff dimension.

Dimension drop occurs for certain finite-range random walks on Fuchsian Schottky groups.

Provides new insights into the geometric properties of harmonic measures in hyperbolic groups.

Abstract

We prove that the harmonic measures of certain finite range random walks on Fuchsian Schottky groups, have dimension strictly smaller than the Hausdorff dimension of the corresponding limit set.