Random Walks with Nonnearest Neighbor Transitions. I. Analytic 1-D Theory for Next-Nearest Neighbor and Exponentially Distributed Steps

Publisher Website
Google Scholar
Abstract
We present here exact analytic results for random walks on one‐dimensional lattices with nonnearest neighbor transitions. After deriving the generating function for such lattices with and without boundaries, we have calculated a number of moment properties (mean first passage times to absorption, mean recurrence times and their dispersion, mean excursion from the origin, etc.) for random walks with next‐nearest neighbor transitions and for random walks with exponentially distributed step length. In the latter case, variation of one of the parameters permits us to cover the whole range of step lengths from nearest neighbor transitions to steps of any finite length l. Since we have obtained explicit expressions for the generating function for these walks, any additional desired moment properties can readily be calculated. Among the interesting results of this study are: (1) The moment results for random walks with next‐nearest neighbor transitions differ from the analogous nearest neighbor results at most by a factor of O(1); (2) the one‐dimensional moment results for walks with arbitrary step length differ from the analogous one‐dimensional results for walks with nearest neighbor transitions by several orders of magnitude; (3) the mean time to absorption for a random walker with equal probabilities for steps of arbitrary length in one dimension agrees to within a factor of O(1) with the mean time for absorption for a random walker with nearest neighbor steps in three dimensions; (4) the mean time to absorption for a random walker with equal probabilities for steps of arbitrary lengths is independent of the dimensionality of the lattice.

This publication has 7 references indexed in Scilit: