Extending the method of , 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 , 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.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty