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    Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs

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    We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph DL(q,r)DL(q,r), where q,r≥2q,r \ge 2. The latter is the horocyclic product of two homogeneous trees with respective degrees q+1q+1 and r+1r+1. When q=rq=r, it is the Cayley graph of the wreath product (lamplighter group) Zq≀Z{\mathbb Z}_q \wr {\mathbb Z} with respect to a natural set of generators. We describe the full Martin compactification of these random walks on DLDL-graphs and, in particular, lamplighter groups. This completes and provides a better approach to previous results of Woess, who has determined all minimal positive harmonic functions.Comment: 26 page
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