Positive Harmonic Functions for Semi-isotropic Random Walks on Trees, Lamplighter Groups, and Dl-graphs

@inproceedings{Woess2005PositiveHF,
title={Positive Harmonic Functions for Semi-isotropic Random Walks on Trees, Lamplighter Groups, and Dl-graphs},
author={Wolfgang Woess},
year={2005}
}

We determine all positive harmonic functions for a large class of " semi-isotropic " random walks on the lamplighter group, i.e., the wreath product Z q ≀ Z, where q ≥ 2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel-Leader graph DL(q, q). More generally, 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 DL-graphs. This is based on a careful study of the… CONTINUE READING