Positive Harmonic Functions for Semi-Isotropic Random Walks on Trees, Lamplighter Groups, and DL-Graphs

Abstract

We determine all positive harmonic functions for a large class of “semi-isotropic” random walks on the lamplighter group, i.e., the wreath product ℤq≀ℤ, where q≥2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel–Leader graph \(\mathsf{DL}(q,q)\) . More generally, \(\mathsf{DL}(q,r)\) (q,r≥2) is the horocyclic product of two homogeneous trees with respective degrees q+1 and r+1, and our result applies to all \(\mathsf {DL}\) -graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.