Woess, Wolfgang Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions. (English) Zbl 1066.05075 Comb. Probab. Comput. 14, No. 3, 415-433 (2005). The lamplighter group over the integers \(\mathbb{Z}\) is the wreath product of \(\mathbb{Z}_q\) with \(\mathbb{Z}\). The Cayley graph of this group, with respect to a natural generating set, is the Diestel-Leader graph \(\text{DL}(q,q)\). The author studies harmonic functions for the “simple” Laplacian on \(\text{DL}(q,q)\) and, in general, for a class of random walks on \(\text{DL}(q,r)\), where \(q\), \(r\geq 2\). These graphs are horocyclic products of two trees, and the author describes all positive harmonic functions in terms of the boundaries of these two trees; in particular, he determines the set of minimal positive harmonic functions. Reviewer: Arthur T. White (Kalamazoo) Cited in 1 ReviewCited in 26 Documents MSC: 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) Keywords:Laplacian PDFBibTeX XMLCite \textit{W. Woess}, Comb. Probab. Comput. 14, No. 3, 415--433 (2005; Zbl 1066.05075) Full Text: DOI arXiv