Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. (Q1872334)

The authors consider various models on \(\mathbb{Z}^d\) (self-avoing walks, bond percolation, lattice trees, bond lattice animals) with long finite-range connections above their respective critical dimensions. Using the lace expansion, they prove that, provided the models are sufficiently spread-out, the 2-point function decays, at the critical point, as a multiple of \(| x| ^{2-d}\) as \(x\to\infty\).