Ising model and the positive orthogonal grassmannian

Pavel Galashin, Pavlo Pylyavskyy

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We use inequalities to completely describe the set of boundary correlation matrices of planar Ising networks embedded in a disk. Specifically, we build on a recent result of Lis to give a simple bijection between such correlation matrices and points in the totally nonnegative part of the orthogonal Grassmannian, which was introduced in 2013 in the study of the scattering amplitudes of Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. We also show that the edge parameters of the Ising model for reduced networks can be uniquely recovered from boundary correlations, solving the inverse problem. Under our correspondence, the Kramers-Wannier high/low temperature duality transforms into the cyclic symmetry of the Grassmannian, and using this cyclic symmetry, we prove that the spaces under consideration are homeomorphic to closed balls.

Original languageEnglish (US)
Pages (from-to)1877-1942
Number of pages66
JournalDuke Mathematical Journal
Issue number10
StatePublished - Jul 15 2020

Bibliographical note

Funding Information:
Pylyavskyy’s work was partially supported by National Science Foundation grants DMS-1148634 and DMS-1351590.

Publisher Copyright:
© 2020 Duke University Press. All rights reserved.


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