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 $\mathbb{Z}^{d}$ that are defined by power-law decaying pair potentials of the form $D(x)\asymp|x|^{-d-\alpha}$ with $\alpha>0$. The upper-critical dimension $d_{\mathrm{c}}$ is $2(\alpha\wedge2)$ for self-avoiding walk and the Ising model, and $3(\alpha\wedge2)$ for percolation. Let $\alpha\ne2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{\mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{\mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{\alpha\wedge2-d}$, where the constant $C\in(0,\infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $\alpha<2$ and $\alpha>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.