Critical two-point functions for long-range statistical-mechanical models in high dimensions

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