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Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions
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  • Published: 10 February 2005

Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions

  • Remco van der Hofstad1 &
  • Akira Sakai2 

Probability Theory and Related Fields volume 132, pages 438–470 (2005)Cite this article

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Abstract.

We consider self-avoiding walk and percolation in ℤd, oriented percolation in ℤd×ℤ+, and the contact process in ℤd, with p D(·) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y−x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point p c . We investigate the value of p c when d>6 for percolation and d>4 for the other models, and L≫1. We prove in a unified way that p c =1+C(D)+O(L−2d), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L−d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that p c =1+cL−d+O(L−d−1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models.

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Author information

Authors and Affiliations

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

    Remco van der Hofstad

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

    Akira Sakai

Authors
  1. Remco van der Hofstad
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  2. Akira Sakai
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Correspondence to Remco van der Hofstad.

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Hofstad, R., Sakai, A. Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions. Probab. Theory Relat. Fields 132, 438–470 (2005). https://doi.org/10.1007/s00440-004-0405-4

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  • Received: 19 February 2004

  • Revised: 20 October 2004

  • Published: 10 February 2005

  • Issue Date: July 2005

  • DOI: https://doi.org/10.1007/s00440-004-0405-4

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Keywords

  • Phase Transition
  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Mathematical Biology
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