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March 2015 Critical two-point functions for long-range statistical-mechanical models in high dimensions
Lung-Chi Chen, Akira Sakai
Ann. Probab. 43(2): 639-681 (March 2015). DOI: 10.1214/13-AOP843


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.


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Lung-Chi Chen. Akira Sakai. "Critical two-point functions for long-range statistical-mechanical models in high dimensions." Ann. Probab. 43 (2) 639 - 681, March 2015.


Published: March 2015
First available in Project Euclid: 2 February 2015

zbMATH: 1342.60162
MathSciNet: MR3306002
Digital Object Identifier: 10.1214/13-AOP843

Primary: 60K35 , 82B20 , 82B27 , 82B41 , 82B43

Keywords: Critical behavior , Lace expansion , Long-range random walk , percolation , Self-avoiding walk , the Ising model , two-point function

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 2 • March 2015
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