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

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.