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

Abstract

Completing a strategy of Gouëzel and Lalley [GL11], 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 CR −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.

Poisson kernel expansions for Schrödinger operators on trees

“…In that context, what we called "Poisson kernel" bears the name "Martin kernel", and allows to represent all non-negative eigenfunctions by an integral of the kernel over the boundary. For Gromov-hyperbolic graphs, the coincidence of the Martin boundary with the geometric boundary has been proven in [3,4] in the interior of the positive spectrum, in [16,15] at the top of the positive spectrum, which coincides with the bottom of the ℓ 2 -spectrum. This relies highly on the fact that we are in a region where the Green function is positive.…”

Statistically Convex-Cocompact Actions of Groups with Contracting Elements

“…In this case, we choose F = {a, b, a −1 } and 0 = 1. To the best of our knowledge, this result was first proved by Arzhantseva and Lysenok [3, Lemma 3] for hyperbolic groups, and reproved later in [38,Lemma 4.4] and [43,Lemma 2.4]. In joint work with Potyagailo [68], we have proved a version for relatively hyperbolic groups.…”

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

“…In fact, we expect Theorem 2 to hold in a much wider context, and that the emergence of the Brownian continuum random tree as a limit of large Brownian loops is a signature of non-compact, negatively curved spaces that are "close to homogeneous". The intuition behind this result comes from the recent advances [17,16] on local limit theorems for transition probabilities in hyperbolic groups. Namely, Gouëzel's results in [16] imply in particular that if G is a nonelementary Gromov-hyperbolic group, and if S is a finite symmetric subset of generators of G, then the number C n of closed paths of length n in the Cayley graph of G associated with S is asymptotically C n ∼ α β n n −3/2 (modulo the usual periodicity caveat) for some α = α(G, S) ∈ (0, ∞) and β = β(G, S) ∈ (1, ∞).…”

“…This is a first hint that a walk in G conditioned to come back at its starting point after n steps might approximate a tree in some sense. In fact, this general idea is present in the approach of [16].…”