The Planar Ising Model and Total Positivity

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• 作者：Marcin Lis
• 关键词：Ising model ; Total positivity ; Random currents ; Alternating flows
• 刊名：Journal of Statistical Physics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：166
• 期：1
• 页码：72-89
• 全文大小：589KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics;
• 出版者：Springer US
• ISSN：1572-9613
• 卷排序：166

文摘

A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph G. Let \(a_1,\dots ,a_k,b_k,\dots ,b_1\) be vertices placed in a counterclockwise order on the outer face of G. We show that the \(k\times k\) matrix of the two-point spin correlation functions $$\begin{aligned} M_{i,j} = \langle \sigma _{a_i} \sigma _{b_j} \rangle \end{aligned}$$is totally nonnegative. Moreover, \(\det M > 0\) if and only if there exist k pairwise vertex-disjoint paths that connect \(a_i\) with \(b_i\). We also compute the scaling limit at criticality of the probability that there are k parallel and disjoint connections between \(a_i\) and \(b_i\) in the double random current model. Our results are based on a new distributional relation between double random currents and random alternating flows of Talaska [37].KeywordsIsing modelTotal positivityRandom currentsAlternating flowsMathematics Subject Classification82B2060C0505C501 IntroductionThe Ising model was introduced by Lenz with the intention to describe the behaviour of ferromagnets, and was first solved in dimension 1 by Ising [19]. Peierls later showed that the model does undergo a phase transition in dimensions 2 or more [32], and it has been since the subject of extensive study both in the physics and mathematics literature. Notable results in the planar case include the exact solution obtained by Onsager [31] and Yang [38], and the recent breakthrough of Smirnov et al. showing conformal invariance in the critical scaling limit [7, 8, 17, 18, 36].