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Title:	CRITICAL TWO-POINT FUNCTIONS FOR LONG-RANGE STATISTICAL-MECHANICAL MODELS IN HIGH DIMENSIONS
Authors:	陳隆奇 Chen, Lung-Chi Sakai, Akira
Contributors:	應數系
Date:	2015-02
Issue Date:	2016-01-13 16:23:13 (UTC+8)
Abstract:	We consider long-range self-avoiding walk, percolation and the Ising model on ZdZd that are defined by power-law decaying pair potentials of the form D(x)≍\|x\|−d−αD(x)≍\|x\|−d−α with α>0α>0. The upper-critical dimension dcdc is 2(α∧2)2(α∧2) for self-avoiding walk and the Ising model, and 3(α∧2)3(α∧2) for percolation. Let α≠2α≠2 and assume certain heat-kernel bounds on the nn-step distribution of the underlying random walk. We prove that, for d>dcd>dc (and the spread-out parameter sufficiently large), the critical two-point function Gpc(x)Gpc(x) for each model is asymptotically C\|x\|α∧2−dC\|x\|α∧2−d, where the constant C∈(0,∞)C∈(0,∞) is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between α<2α<2 and α>2α>2. We also provide a class of random walks that satisfy those heat-kernel bounds.
Relation:	The Annals of Probability, 43(2), 639-681.
Data Type:	article
DOI link:	http://dx.doi.org/10.1214/13-AOP843
DOI:	10.1214/13-AOP843
Appears in Collections:	[Department of Mathematical Sciences] Periodical Articles

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