Ising model and the positive orthogonal Grassmannian

Abstract

We completely describe by inequalities the set of boundary correlation matrices of planar Ising networks embedded in a disk. Specifically, we build on a recent result of M.~Lis to give a simple bijection between such correlation matrices and points in the totally nonnegative part of the orthogonal Grassmannian, which has been introduced in 2013 in the study of the scattering amplitudes of 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.

48 pages, 8 figures; v2: added a section on Dubédat's bosonization identities and some other references; v3: minor exposition improvements; v4: a new proof added to Section 6.1. Final version, to appear in Duke Math Journal

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Cited by in corpus (7)

• Regularity theorem for totally nonnegative flag varieties
• The orthogonal momentum amplituhedron and ABJM amplitudes
• Vector-relation configurations and plabic graphs
• Critical varieties in the Grassmannian
• Electrical networks and the Grove algebra
• Vertex electrical model: lagrangian and non-negative properties
• Superport networks