On returns to the starting site in lattice random walks

Abstract

In sufficiently low-dimensional systems, the conditional mean time to return to the starting site (conditional upon return eventually occuring) is infinite. We examine the conditional mean time τn to return in a walk of finite duration n steps. For walks of Pólya type, τn is found asymptotically proportional to `√n, n log2 n, √n and log n in dimensions 1, 2, 3 and 4 respectively. Results are also given for walks with long-ranged transitions, and for a one-dimensional walk in a central potential. © 1986.