Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions

The lamplighter group over $\mathbb Z$ is the wreath product $\mathbb Z_q \wr \mathbb Z$. With respect to a natural generating set, its Cayley graph is the Diestel-Leader graph $DL(q,q)$. We study harmonic functions for the "simple" Laplacian on this graph, and more generally, for a class of random walks on $DL(q,r)$, where $q,r \ge 2$. The DL-graphs are horocyclic products of two trees, and we give a full description of all positive harmonic functions in terms of the boundaries of these two trees. In particular, we determine the minimal Martin boundary, that is, the set of minimal positive harmonic functions.