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 τ n to return in a walk of finite duration n steps. For walks of Pólya type, τ n is found asymptotically proportional to `√ n, n log 2 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.
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