  We determine all positive harmonic functions for a large class of
"semi-isotropic" random walks on the lamplighter group, i.e., the wreath
product of the cyclic group of order q with the infinite cyclic group. 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) 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.
