## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 43, Number 2 (2015), 639-681.

Dates
First available in Project Euclid: 2 February 2015

https://projecteuclid.org/euclid.aop/1422885572

Digital Object Identifier
doi:10.1214/13-AOP843

Mathematical Reviews number (MathSciNet)
MR3306002

Zentralblatt MATH identifier
1342.60162

#### Citation

Chen, Lung-Chi; Sakai, Akira. Critical two-point functions for long-range statistical-mechanical models in high dimensions. Ann. Probab. 43 (2015), no. 2, 639--681. doi:10.1214/13-AOP843. https://projecteuclid.org/euclid.aop/1422885572

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