Ising model and the positive orthogonal Grassmannian

TL;DR

This paper characterizes boundary correlation matrices of planar Ising networks using inequalities and establishes a bijection with the totally nonnegative orthogonal Grassmannian, revealing geometric and duality properties.

Contribution

It provides a complete description of boundary correlations in planar Ising models via a novel connection to the orthogonal Grassmannian, including an inverse problem solution.

Findings

Boundary correlation matrices are characterized by inequalities.

A bijection between correlation matrices and the orthogonal Grassmannian is established.

The inverse problem for edge parameters is solved.

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.