Abstract
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 ) , where q , r ⩾ 2 . The latter is the horocyclic product of two homogeneous trees with respective degrees q + 1 and r + 1 . When q = r , it is the Cayley graph of the wreath product (lamplighter group) Z q ≀ Z with respect to a natural set of generators. We describe the full Martin compactification of these random walks on DL -graphs and, in particular, lamplighter groups. This completes previous results of Woess, who has determined all minimal positive harmonic functions.
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