Abstract
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 ktimes k matrix of the two-point spin correlation functions Mi,j=⟨σaiσbj⟩\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} M_{i,j} = \\langle \\sigma _{a_i} \\sigma _{b_j} \\rangle \\end{aligned}$$\\end{document}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].
Highlights
The 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
A seemingly unrelated notion is that of totally positive matrices characterized by having all their minors positive. They appear in various areas of mathematics and physics including oscillations in mechanical systems [12,13], stochastic processes and statistical mechanics [9,11,22], quantum groups [28,29,30], and algebraic geometry [33,34]
Summary
The 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]. The most notable example is the result of Aizenman, Duminil-Copin and Sidoravicius [4] who studied percolation properties of double random currents to show continuity of magnetization for a wide range of Ising models on Zd (including the d = 3 case). Their argument was later generalized by Björnberg to the setting of quantum Ising models [5].
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