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Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions. (English) Zbl 1066.05075

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.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Keywords:

Laplacian
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