Isotropic random walks in a tree

Abstract

Let T be an infinite homogeneous tree of order a+1. We study Markov chains {Xn} in T whose transition functions p(x, y)=A[d(x,y)] depend only on the shortest distance between x and y in the graph. The graph T can be represented as a symmetric space of a p-adic matrix group; we prove a series of results using essentially the spherical functions of this symmetric space. Theorem 1.d(Xn,x)∼β n a.s., where β>0 if A(0) ≠ 1, X0=x. Assuming {Xn} is strongly aperiodic, Theorem 2. p2(x, y)∼CRn/n3/2 for fixed x, y where R=∑φ(d) A(d)<1, and if E[d(X1, X0)2]