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arXiv · 2004.13986

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

Matthieu Dussaule
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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.

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Matthieu Dussaule. 2020-04-29. Local limit theorems in relatively hyperbolic groups II : the non-spectrally degenerate case. https://arxiv.org/abs/2004.13986

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