Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

TL;DR

This paper proves a local limit theorem for symmetric random walks on Gromov-hyperbolic groups, describing the asymptotic behavior of return probabilities and extending boundary theory results.

Contribution

It establishes a local limit theorem for symmetric finitely supported measures on Gromov-hyperbolic groups and extends Ancona's boundary results to the spectral radius.

Findings

Return probability behaves like C R^{-n} n^{-3/2}

Martin boundary coincides with the geometric boundary

Symmetry assumption can be removed for surface groups

Abstract

Completing a strategy of Gou\"ezel and Lalley, we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non-elementary Gromov-hyperbolic group: denoting by $R$ the inverse of the spectral radius of the random walk, the probability to return to the identity at time $n$ behaves like $C R^{-n}n^{-3/2}$. An important step in the proof is to extend Ancona's results on the Martin boundary up to the spectral radius: we show that the Martin boundary for $R$-harmonic functions coincides with the geometric boundary of the group. In an appendix, we explain how the symmetry assumption of the measure can be dispensed with for surface groups.