A local limit theorem for convergent random walks on relatively hyperbolic groups

TL;DR

This paper establishes a local limit theorem for convergent random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups, detailing the asymptotic behavior of return probabilities.

Contribution

It proves a precise local limit theorem for such random walks, classifying their asymptotic return probabilities based on the group's structure.

Findings

Return probability asymptotics: $p_n(e, e) \\sim CR^{-n}n^{-d/2}$

Classification of all possible behaviors of $p_n(e, e)$ on these groups

Extension of local limit theorems to relatively hyperbolic groups with specific subgroup conditions

Abstract

We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius.When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form $p_n(e, e)\sim CR^{-n}n^{-d/2}$, where $p_n(e, e)$ is the probability of returning to the origin at time $n$, $R$ is the inverse of the spectral radius of the random walk and $d$ is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate.This concludes the classification all possible behaviour for $p_n(e, e)$ on such groups.