Renewal theory for random walks on surface groups

TL;DR

This paper develops a renewal theory framework for analyzing random walks on surface groups, enabling the derivation of a central limit theorem and extending results to groups with infinitely many ends.

Contribution

It introduces a renewal structure for surface group random walks, generalizing CLT results to a broader class of groups with infinitely many ends.

Findings

Established a renewal structure for random walks on surface groups.

Proved a central limit theorem for the distance to the origin.

Showed analyticity of speed and asymptotic variance.

Abstract

We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enters a particular type of a cone and never leaves it again. As a consequence, the trajectory of the random walk can be expressed as an "aligned union" of i.i.d. trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.