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 $q,r\ge 2$. 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) ${\mathbb{Z}}_q\wr \mathbb{Z}$ with respect to a natural set of generators. We describe the full Martin compactification of these random walks on $DL$-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.

26 pages

Cited by in corpus (2)

• Brownian motion and Harmonic functions on Sol(p,q)
• Analysis of unbounded operators and random motion