Mean-field critical behaviour for correlation length for percolation in high dimensions
We prove that the correlation length \(\xi\) of independent bond percolation models exhibits mean-field type critical behaviour (i.e. \(\xi (p)\sim (p_ c-p)^{-1/2}\) as \(p\nearrow p_ c)\) in two situations: i) for nearest-neighbour independent bond percolation models on a d- dimensional hypercubic lattice \({\mathbb{Z}}^ d\), 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 which was used previously to prove the triangle condition and hence mean-field behaviour of the critical exponents \(\gamma\),\(\beta\),\(\delta\),\(\Delta\) and \(\nu_ 2\) for the above two cases.
scientific article; zbMATH DE number 4123029
scientific article