SINGULARITIES OF THE GREEN-FUNCTION OF A RANDOM-WALK ON A DISCRETE GROUP

Let X be a countable discrete group and let mu be an irreducible probability on X. The radius of convergence rho of the Green function G(x; z) = SIGMA(n=0)infinity-mu*n(x)z(n) is finite, and independent of x. Let d = gcd {n greater-than-or-equal-to 1: mu*n(e) > 0) be the period of mu. We show that for each x is-an-element-of X the singularities of the analytic function z bar arrow pointing right G (x; z) on the circle {z is-an-element-of C: \z\ = rho) are precisely the points rho-e2-pi-ik/d, k = 0, ..., d - 1. In particular, rho is the only singularity on the circle in the aperiodic case d = 1 (which occurs, for example, when mu(e) > 0). This affirms a conjecture of LALLEY [5]. When mu is symmetric, i.e., mu (x-1) = mu(x) for all x is-an-element-of X, d is either 1 or 2. As another particular case of our result, we see that -rho is then a singularity of z bar arrow pointing right G(x; z) if and only if d = 2, in which case X is "bicolored". This answers a question of DE LA HARPE, ROBERTSON and VALETTE [2].