Local limit theorems on relatively hyperbolic groups with respect to virtually nilpotent subgroups

TL;DR

This paper classifies the asymptotic behavior of return probabilities for symmetric random walks on relatively hyperbolic groups with virtually nilpotent subgroups, providing a comprehensive understanding of local limit theorems in this setting.

Contribution

It offers a complete classification of local limit theorems for finitely supported symmetric measures on these groups, up to bounded error.

Findings

Classification of all possible local limit theorems

Asymptotic formulas for return probabilities

Extension of local limit theorems to relatively hyperbolic groups

Abstract

Given a probability measure on a finitely generated group, the local limit problem consists in finding asymptotics of $p_n(e,e)$, the probability that the random walk at time $n$ is at the origin. We give the classification of all possible local limit theorems, up to bounded error, for finitely supported, symmetric, admissible probability measures on a relatively hyperbolic group with respect to virtually nilpotent subgroups.