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Explicit Bounds for the Return Probability of Simple Random Walks

Authors:	Karen?Ball author-information" > author-information__contact u-icon-before" > mailto:kball@ima.umn.edu" title=" kball@ima.umn.edu" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >Email author,Jacob?Sterbenz

Affiliation:	(1) Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, Minnesota;(2) Department of Mathematics, Princeton University, Princeton, New Jersey

Abstract:	We give an exact computation of the second order term in the asymptotic expansion of the return probability, P_2n^d(0,0), of a simple random walk on the d-dimensional cubic lattice. We also give an explicit bound on the remainder. In particular, we show that P_2n^d(0,0) < 2 (d/4n)^d/2 where n M=M(d) is explicitly given.

Keywords:	Simple random walk random walk return probability asymptotic probability
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