CRITICAL TWO-POINT FUNCTIONS FOR LONG-RANGE STATISTICAL-MECHANICAL MODELS IN HIGH DIMENSIONS

Abstract

We consider long-range self-avoiding walk, percolation and the Ising model on ZdZd that are defined by power-law decaying pair potentials of the form D(x)≍|x|−d−αD(x)≍|x|−d−α with α>0α>0. The upper-critical dimension dcdc is 2(α∧2)2(α∧2) for self-avoiding walk and the Ising model, and 3(α∧2)3(α∧2) for percolation. Let α≠2α≠2 and assume certain heat-kernel bounds on the nn-step distribution of the underlying random walk. We prove that, for d>dcd>dc (and the spread-out parameter sufficiently large), the critical two-point function Gpc(x)Gpc(x) for each model is asymptotically C|x|α∧2−dC|x|α∧2−d, where the constant C∈(0,∞)C∈(0,∞) is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between α<2α<2 and α>2α>2. We also provide a class of random walks that satisfy those heat-kernel bounds.