Probability Theory and Related Fields

, Volume 86, Issue 3, pp 337–385

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

  • Takashi Hara
Article

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∼(pcp)−1/2 asppc) 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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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Takashi Hara
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of PhysicsGakushuuin UniversityTokyoJapan

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