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Escape probabilities for slowly recurrent sets
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  • Published: March 1992

Escape probabilities for slowly recurrent sets

  • Gregory F. Lawler1 

Probability Theory and Related Fields volume 94, pages 91–117 (1992)Cite this article

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Summary

A setA⊂Z d (d>-3) is defined to be slowly recurrent for simple random walk if it is recurrent but the probability of enteringA∩{z:n<|z|<-2n} tends to zero asn→∞. A method is given to estimate escape probabilities for such sets, i.e., the probability of leaving the ball of radiusn without entering the set. The methods are applied to two examples. First, half-lines and finite unions of half-lines inZ 3 are considered. The second example is a random walk path in four dimensions. In the latter case it is proved that the probability that two random walk paths reach the ball of radiusn without intersecting is asymptotic toc(lnn)−1/2, improving a result of the author.

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References

  1. Duplantier, B.: Intersections of random walks: a direct renormalization approach. Commun. Math. Phys.117, 279–330 (1987)

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  2. Kesten, H.: Some caricatures of multiple contact diffusion-limited aggregation and the η-model. In: Stochastic analysis. Barlow, M., Bingham, N. (eds.), pp. 179–227. Cambridge: Cambridge University Press 1991

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  3. Lawler, G.: Intersections of random walks: Boston: Birkhäuser 1991

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  4. Lawler, G.: L-shapes for the logarithmic η-model for DLA in three dimensions. Seminar on stochastic processes. Boston: Birkhäuser 1991

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Authors and Affiliations

  1. Department of Mathematics, Duke University, 27706, Durham, NC, USA

    Gregory F. Lawler

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  1. Gregory F. Lawler
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Research partially supported by the National Science Foundation

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Cite this article

Lawler, G.F. Escape probabilities for slowly recurrent sets. Probab. Th. Rel. Fields 94, 91–117 (1992). https://doi.org/10.1007/BF01222512

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  • Received: 23 April 1991

  • Revised: 23 March 1992

  • Issue Date: March 1992

  • DOI: https://doi.org/10.1007/BF01222512

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Mathematics Subject Classification

  • 60 J 15
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