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

Compositio mathematica

article

Matthieu Dussaule¹ 

[1] Faculté des Sciences et Techniques, Université de Tours

关键词: random walks;    thermodynamic formalism;    transfer operator;    relative Ancona inequalities;    20F65;    05C81;    60F15;    37C30;   

DOI  :  10.1112/S0010437X22007448

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

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.

【 授权许可】

CC BY