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

Summary Extending the method of 27], we prove that the corrlation length of independent bond percolation models exhibits mean-field type critical behaviour (i.e. (p(p _c –p)^–1/2 aspp _c ) in two situations: i) for nearest-neighbour independent bond percolation models on ad-dimensional hypercubic lattice ^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 , , , and ₂ for the above two cases.