Journal of Theoretical Probability

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

Explicit Bounds for the Return Probability of Simple Random Walks

  • Karen Ball
  • Jacob Sterbenz


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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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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