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

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 part I to show that$p_n(e,e)\sim CR^{-n}n^{-3/2}$, where$R$is the inverse of the spectral radius of the random walk. This both generalizes results of Woess for free products and results of Gouëzel for hyperbolic groups.