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

Sébastien Gouëzel (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Completing a strategy of Gouëzel 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.

Mots clés: Local limit theorem; random walk; hyperbolic group; spectral radius; Martin boundary; transfer operator

Domaines: Systèmes dynamiques [math.DS]

Sébastien Gouëzel. Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. Journal of the American Mathematical Society, 2014, 27 (3), pp.893-928. ⟨hal-00732340⟩