Asymptotics of the first-passage function on free and Fuchsian groups

TL;DR

This paper derives explicit asymptotic estimates for the first-passage function on free groups and demonstrates the singularity of the hitting measure for certain random walks on Fuchsian groups, advancing understanding of their boundary behaviors.

Contribution

It provides new explicit asymptotic estimates for the first-passage function and proves measure singularity on Fuchsian groups, linking random walk behavior to geometric group properties.

Findings

Explicit asymptotic estimates for first-passage functions on free groups

Proof of measure singularity for random walks on Fuchsian groups

Enhanced understanding of boundary behaviors in geometric group theory

Abstract

In this preprint we derive explicit estimates for the asymptotics of the first-passage function for a specific class of random walks on free groups and use them to prove the singularity of the hitting measure for a similarly defined class of random walks on Fuchsian groups.