Green's Functions for Random Walks on ZN

Abstract

Green's function G(x) of a zero mean random walk on the N-dimensional integer lattice N⩾2 is expanded in powers of 1/∣x∣ under suitable moment conditions. In particular, we find minimal moment conditions for G(x) to behave like a constant times the Newtonian potential (or logarithmic potential in two dimensions) for large values of ∣x∣. Asymptotic estimates of G(x) in dimensions N⩾4, which are valid even when these moment conditions are violated, are computed. Such estimates are applied to determine the Martin boundary of the random walk. If N= 3 or 4 and the random walk has zero mean and finite second moment, the Martin boundary consists of one point, whereas if N⩾ 5, this is not the case, because non-harmonic functions arise as Martin boundary points for a large class of such random walks. A criterion for when this happens is provided. 1991 Mathematics Subject Classification: 60J15, 60J45, 31C20.