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The survival probability for critical spread-out oriented percolation above 4 + 1 dimensions. I. Induction
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  • Published: 03 November 2006

The survival probability for critical spread-out oriented percolation above 4 + 1 dimensions. I. Induction

  • Remco van der Hofstad1,
  • Frank den Hollander2,3 &
  • Gordon Slade4 

Probability Theory and Related Fields volume 138, pages 363–389 (2007)Cite this article

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Abstract

We consider critical spread-out oriented percolation above 4 + 1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n + 1) decays like 1/Bn 2 as \(n\to\infty\), where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the probability that the origin is connected to the hyperplane at time n) decays like 1/Bn as \(n\to\infty\). The latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster. The present paper is Part I in a series of two papers. In Part II, we derive a lace expansion for the survival probability, adapted so as to deal with point-to-plane connections. This lace expansion leads to a nonlinear recursion relation for the survival probability. In Part I, we use this recursion relation to deduce the asymptotics via induction.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600, MB, Eindhoven, The Netherlands

    Remco van der Hofstad

  2. EURANDOM, P.O. Box 513, 5600, MB, Eindhoven, The Netherlands

    Frank den Hollander

  3. Mathematical Institute, Leiden University, P.O. Box 9512, 2300, RA, Leiden, The Netherlands

    Frank den Hollander

  4. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Gordon Slade

Authors
  1. Remco van der Hofstad
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  2. Frank den Hollander
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  3. Gordon Slade
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Correspondence to Frank den Hollander.

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van der Hofstad, R., Hollander, F.d. & Slade, G. The survival probability for critical spread-out oriented percolation above 4 + 1 dimensions. I. Induction. Probab. Theory Relat. Fields 138, 363–389 (2007). https://doi.org/10.1007/s00440-006-0028-z

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  • Received: 03 October 2005

  • Revised: 10 July 2006

  • Published: 03 November 2006

  • Issue Date: July 2007

  • DOI: https://doi.org/10.1007/s00440-006-0028-z

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Keywords

  • Random Walk
  • Survival Probability
  • Recursion Relation
  • Extinction Probability
  • Contact Process
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