Positive Harmonic Functions for Semi-isotropic Random Walks on Trees, Lamplighter Groups, and Dl-graphs
نویسنده
چکیده
We determine all positive harmonic functions for a large class of “semiisotropic” 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 DL(q, q). More generally, DL(q, r) (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 DL-graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.
منابع مشابه
Lamplighters, Diestel-Leader Graphs, Random Walks, and Harmonic Functions
The lamplighter group over Z is the wreath product Zq ≀ 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 ≥ 2. The DL-graphs are horocyclic products of two trees, and we give a full description of all posi...
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تاریخ انتشار 2005