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

2020/04/29 by Matthieu Dussaule, Dussaule, Matthieu · 1 citation
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Advanced Operator Algebra Research

paper · doi:10.48550/arxiv.2004.13986

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) ∼ 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 Gouëzel for hyperbolic groups.

Cited by

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

Related

• Local limit theorems in relatively hyperbolic groups I : rough estimates
• A local limit theorem for convergent random walks on relatively hyperbolic groups
• Local limit theorems on relatively hyperbolic groups with respect to virtually nilpotent subgroups
• Local limit theorem for symmetric random walks in Gromov-hyperbolic groups
• Exotic local limit theorems at the phase transition in free products