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 $2(\alpha \wedge 2)$ for self-avoiding walk and the Ising model, and $3(\alpha \wedge 2)$ for percolation. Let $\alpha \ne 2$ 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 $C|x{|}^{\alpha \wedge 2-d}$, 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.

Published in at http://dx.doi.org/10.1214/13-AOP843 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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