Martin boundary of random walks with unbounded jumps in hyperbolic groups

TL;DR

This paper extends the understanding of Martin boundaries for random walks in hyperbolic groups, showing that with superexponential tails, the boundary matches the geometric boundary, unlike the pathological cases with exponential tails.

Contribution

It demonstrates that superexponential tail measures in hyperbolic groups have Martin boundaries coinciding with the geometric boundary, extending previous finite support results.

Findings

Martin boundary coincides with geometric boundary for superexponential tails

Existence of measures with exponential tails leading to pathological boundaries

Asymptotic behavior of transition probabilities for symmetric measures

Abstract

Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is known since the work of Ancona and Gou{\"e}zel-Lalley that the Martin boundary coincides with the geometric boundary. The goal of this paper is to weaken the finite support assumption. We first show that, in any non-amenable group, there exist probability measures with exponential tails giving rise to pathological Martin boundaries. Then, for probability measures with superexponential tails in hyperbolic groups, we show that the Martin boundary coincides with the geometric boundary by extending Ancona's inequalities. We also deduce asymptotics of transition probabilities for symmetric measures with superexponential tails.