Journal of Theoretical Probability

, Volume 18, Issue 2, pp 317–326 | Cite as

Explicit Bounds for the Return Probability of Simple Random Walks



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/4πn) d/2 where nM=M(d) is explicitly given.


Simple random walk random walk return probability asymptotic probability 


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  1. 1.
    Felker, J.L., Lyons, R. 2003High-precision entropy values for spanning trees in latticesJ. Phys. A.3683618365Google Scholar
  2. 2.
    Olver, F. W. J. (1974). Asymptotics and Special Functions, Computer Science and Applied Mathematics, Academic Press.Google Scholar
  3. 3.
    Ritzmann, C. (2002). Good Local Bounds for Simple Random Walks. available at Scholar
  4. 4.
    Spitzer, F. 1976Principles of Random Walks. 2nd ed., Graduate Texts in Mathematics, Vol.~34.Springer,New York, Heidelberg.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Institute for Mathematics and its ApplicationsUniversity of Minnesota
  2. 2.Department of MathematicsPrinceton University

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