Singularities of the Green function of a random walk on a discrete group

Abstract

LetX be a countable discrete group and let μ be an irreducible probability onX. The radius of convergence ϱ of the Green function $$G\left( {x;z} \right) = \sum {_{n = 0}^\infty \mu ^{ * n} } \left( x \right)z^n $$ is finite, and independent ofx. Let $$d = \gcd \left\{ {n \geqslant 1:\mu ^{ * n} \left( e \right) > 0} \right\}$$ be the period of μ. We show that for eachx∈X the singularities of the analytic functionz→G(x; z) on the circle {z∈ℂ:|z|=ϱ} are precisely the points ϱe 2πik/d k=0, ...,d−1. In particular, ϱ is the only singularity on the circle in the aperiodic cased=1 (which occurs, for example, when μ(e)>0). This affirms a conjecture ofLalley [5]. When μ is symmetric, i.e., μ(x −1)=μ(x) for allx∈X, d is either 1 or 2. As another particular case of our result, we see that-ϱ is then a singularity ofz→G (x; z) if and only ifd=2, in which caseX is “bicolored”. This answers a question ofde la Harpe, Robertson andValette [2].