The Full Martin Boundary of the Bi-Tree

Abstract

We determine the Martin boundary for aperiodic simple random walk on a bi-tree, that is, the Cartesian product of two homogeneous trees. This is obtained by first deriving a "renewal theorem," giving an asymptotic estimate of the Green kernel as the space variable tends to infinity. The basic tool is a result of Lalley that gives a uniform estimate of transition probabilities of nearest neighbour random walks on trees.