Local limit theorems in relatively hyperbolic groups II : the non-spectrally degenerate case

TL;DR

This paper establishes precise asymptotic probabilities for return times of non-spectrally degenerate random walks in relatively hyperbolic groups, extending classical results to a broader class of groups using thermodynamic formalism.

Contribution

It provides the first detailed asymptotic formula for return probabilities in non-spectrally degenerate relatively hyperbolic groups, generalizing prior results for free products and hyperbolic groups.

Findings

Proves $p_n(e, e) \,\sim\, CR^{-n} n^{-3/2}$ for non-spectrally degenerate cases

Extends classical return probability results to a wider class of groups

Uses thermodynamic formalism combined with Green function estimates

Abstract

This is the second of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this second paper, we restrict our attention to non-spectrally degenerate random walks and we prove precise asymptotics of the probability $p_n(e, e)$ of going back to the origin at time $n$. We combine techniques adapted from thermodynamic formalism with the rough estimates of the Green function given by the first paper to show that $p_n(e, e) \sim CR^{-n} n^{-3/2}$ , where $R$ is the spectral radius of the random walk. This generalizes results of W. Woess for free products and results of Gou{\"e}zel for hyperbolic groups.