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

TL;DR

This paper analyzes the critical two-point functions of long-range statistical-mechanical models in high dimensions, establishing their asymptotic behavior and crossover phenomena based on the decay parameter and model specifics.

Contribution

It proves the asymptotic form of the critical two-point function for long-range models in high dimensions, including crossover behavior between different decay regimes.

Findings

Critical two-point functions decay as |x|^{α∧2 - d} in high dimensions.

Established crossover between α<2 and α>2 regimes.

Provided conditions and examples of random walks satisfying heat-kernel bounds.

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.