The Planar Ising Model and Total Positivity.
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| Title: | The Planar Ising Model and Total Positivity. |
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| Authors: | Lis, Marcin1 m.lis@statslab.cam.ac.uk |
| Source: | Journal of Statistical Physics. Jan2017, Vol. 166 Issue 1, p72-89. 18p. |
| Subjects: | Alternating current circuits, Planar graphs, Ising model, Nonnegative matrices, Boundary value problems |
| 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 $$k\times k$$ matrix of the two-point spin correlation functions 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]. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: The Planar Ising Model and Total Positivity. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Lis%2C+Marcin%22">Lis, Marcin</searchLink><relatesTo>1</relatesTo><i> m.lis@statslab.cam.ac.uk</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Statistical+Physics%22">Journal of Statistical Physics</searchLink>. Jan2017, Vol. 166 Issue 1, p72-89. 18p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Alternating+current+circuits%22">Alternating current circuits</searchLink><br /><searchLink fieldCode="DE" term="%22Planar+graphs%22">Planar graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Ising+model%22">Ising model</searchLink><br /><searchLink fieldCode="DE" term="%22Nonnegative+matrices%22">Nonnegative matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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 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]. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10955-016-1690-x Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 18 StartPage: 72 Subjects: – SubjectFull: Alternating current circuits Type: general – SubjectFull: Planar graphs Type: general – SubjectFull: Ising model Type: general – SubjectFull: Nonnegative matrices Type: general – SubjectFull: Boundary value problems Type: general Titles: – TitleFull: The Planar Ising Model and Total Positivity. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Lis, Marcin IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Text: Jan2017 Type: published Y: 2017 Identifiers: – Type: issn-print Value: 00224715 Numbering: – Type: volume Value: 166 – Type: issue Value: 1 Titles: – TitleFull: Journal of Statistical Physics Type: main |
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