Green's Functions for Random Walks on Z N

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.