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

Matthieu Dussaule (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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 Gouëzel for hyperbolic groups.

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]; Théorie des groupes [math.GR]

Matthieu Dussaule. Local limit theorems in relatively hyperbolic groups II : the non-spectrally degenerate case. 2020. ⟨hal-02556780⟩