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

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{DL}(q,q)$\end{document} . More generally, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{DL}(q,r)$\end{document} (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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf {DL}$\end{document} -graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.