Solutions of loop equations are random matrices

TL;DR

This paper proves that solutions to loop equations in random matrix models correspond exactly to polynomial moments of certain matrix eigenvalue measures, extending to complex arcs and more general potentials.

Contribution

It establishes an isomorphism between solutions of loop equations and polynomial moments of matrix measures, generalizing to complex arcs, multi-matrix models, and non-polynomial potentials.

Findings

Solutions of loop equations are polynomial moments of matrix measures.

An isomorphism exists between homology of arcs and solutions of loop equations.

Generalizations include multi-matrix models and non-polynomial potentials.

Abstract

For a given polynomial $V(x)\in \mathbb C[x]$, a random matrix eigenvalues measure is a measure $\prod_{1\leq i<j\leq N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}dx_i$ on $\gamma^N$. Hermitian matrices have real eigenvalues $\gamma=\mathbb R$, which generalize to $\gamma$ a complex Jordan arc, or actually a linear combination of homotopy classes of Jordan arcs, chosen such that integrals are absolutely convergent. Polynomial moments of such measure satisfy a set of linear equations called "loop equations". We prove that every solution of loop equations are necessarily polynomial moments of some random matrix measure for some choice of arcs. There is an isomorphism between the homology space of integrable arcs and the set of solutions of loop equations. We also generalize this to a 2-matrix model and to the chain of matrices, and to cases where $V$ is not a polynomial but $V'(x)\in \mathbb C(x)$.