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Infinite paths with bounded or recurrent partial sums
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  • Published: May 2001

Infinite paths with bounded or recurrent partial sums

  • Lorna Booth1 &
  • Ronald Meester1 

Probability Theory and Related Fields volume 120, pages 118–142 (2001)Cite this article

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  • 1 Citations

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Abstract.

We consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with probability p, or −1, with probability 1 −p. We ask whether an infinite path π exists, with the property that the partial sums of the ±1s along π are uniformly bounded, and whether there exists an infinite path π' with the property that the partial sums along π' are equal to zero infinitely often. The answers to these question depend on the type of path one allows, the value of p and the uniform bound specified. We show that phase transitions occur for these phenomena. Moreover, we make a surprising connection between the problem of finding a path to infinity (not necessarily self-avoiding, but visiting each vertex at most finitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points of a Poisson process in the plane. For the recurrence problem, we also show that the probability of finding such a path is monotone in p, for p≥½.

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Authors and Affiliations

  1. Department of Mathematics, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands. e-mail: booth@math.uu.nl; meester@math.uu.nl, , , , , , NL

    Lorna Booth & Ronald Meester

Authors
  1. Lorna Booth
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  2. Ronald Meester
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Received: 10 January 2000 / Revised version: 14 August 2000 / Published online: 9 March 2001

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Booth, L., Meester, R. Infinite paths with bounded or recurrent partial sums. Probab Theory Relat Fields 120, 118–142 (2001). https://doi.org/10.1007/PL00008775

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  • Issue Date: May 2001

  • DOI: https://doi.org/10.1007/PL00008775

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  • Mathematics Subject Classification (2000): 82B43, 82B26, 82B21, 82B20, 82B41
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