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On returns to the starting site in lattice random walks

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  • Hughes, Barry D.

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 τn to return in a walk of finite duration n steps. For walks of Pólya type, τn is found asymptotically proportional to `√n, nlog2 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.

Suggested Citation

  • Hughes, Barry D., 1986. "On returns to the starting site in lattice random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 134(2), pages 443-457.
    • Handle: RePEc:eee:phsmap:v:134:y:1986:i:2:p:443-457
    DOI: 10.1016/0378-4371(86)90058-0
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    References listed on IDEAS

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    1. West, Bruce J. & Shlesinger, Michael, 1984. "Random walk model of impact phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 127(3), pages 490-508.
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