Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs
Probability
2012-12-05 v1 Group Theory
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 of the cyclic group of order q with the infinite cyclic group. 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) is the horocyclic product of two homogeneous trees with respective degrees and , and our result applies to all DL-graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.
Cite
@article{arxiv.math/0501440,
title = {Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs},
author = {Sara Brofferio and Wolfgang Woess},
journal= {arXiv preprint arXiv:math/0501440},
year = {2012}
}