Ising model and the positive orthogonal Grassmannian. (English) Zbl 1446.82008
Summary: 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.
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
Keywords:
planar Ising model; total positivity; totally nonnegative Grassmannian; orthogonal Grassmannian; Kramers-Wannier duality; ABJM scattering amplitudes; Griffiths’s inequalities; electrical networksReferences:
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