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Mean-field critical behaviour for correlation length for percolation in high dimensions
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  • Published: September 1990

Mean-field critical behaviour for correlation length for percolation in high dimensions

  • Takashi Hara1 nAff2 

Probability Theory and Related Fields volume 86, pages 337–385 (1990)Cite this article

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Summary

Extending the method of [27], we prove that the corrlation length ξ of independent bond percolation models exhibits mean-field type critical behaviour (i.e. ξ(p∼(p c −p)−1/2 asp↗p c ) in two situations: i) for nearest-neighbour independent bond percolation models on ad-dimensional hypercubic lattice ℤd, withd sufficiently large, and ii) for a class of “spread-out” independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents γ, β, δ, Δ and ν2 for the above two cases.

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Author notes
  1. Takashi Hara

    Present address: Department of Physics, Gakushuuin University, Toshima-ku, 171, Tokyo, Japan

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, 10012, New York, NY, USA

    Takashi Hara

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  1. Takashi Hara
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Hara, T. Mean-field critical behaviour for correlation length for percolation in high dimensions. Probab. Th. Rel. Fields 86, 337–385 (1990). https://doi.org/10.1007/BF01208256

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  • Received: 02 August 1989

  • Revised: 21 February 1990

  • Issue Date: September 1990

  • DOI: https://doi.org/10.1007/BF01208256

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Keywords

  • Stochastic Process
  • Probability Theory
  • High Dimension
  • Mathematical Biology
  • Correlation Length
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