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

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Abstract

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∼(pc-p)-1/2 as p↗pc) in two situations: i) for nearest-neighbour independent bond percolation models on a d-dimensional hypercubic lattice ℤd, with d 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.

Original languageEnglish
Pages (from-to)337-385
Number of pages49
JournalProbability Theory and Related Fields
Volume86
Issue number3
DOIs
Publication statusPublished - Sep 1 1990
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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