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

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∼(p c −p)−1/2 asp↗p c ) in two situations: i) for nearest-neighbour independent bond percolation models on ad-dimensional hypercubic lattice ℤ d , 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.