Matrix models at low temperature

TL;DR

This paper studies the spectral behavior of multi-matrix models at low temperatures, proving tightness of spectral distributions and characterizing their limits via Dyson-Schwinger equations, with focus on specific models like the strong single variable and commutator models.

Contribution

It establishes conditions for spectral tightness in multi-matrix models at low temperature and analyzes specific models, advancing understanding of their limiting spectral distributions.

Findings

Spectral distributions are tight at low temperature under confining potentials.

Limit points of spectral distributions satisfy Dyson-Schwinger equations.

Analysis of specific models like the strong single variable and commutator models.

Abstract

In this article we investigate the behavior of multi-matrix unitary invariant models under a potential $V_\beta=\beta U+W$ when the inverse temperature $\beta$ becomes very large. We first prove, under mild hypothesis on the functionals $U,W$ that as soon at these potentials are "confining" at infinity, the sequence of spectral distribution of the matrices are tight when the dimension goes to infinity. Their limit points are solutions of Dyson-Schwinger's equations. Next we investigate a few specific models, most importantly the "strong single variable model" where $U$ is a sum of potentials in a single matrix and the "strong commutator model" where $U = -[X,Y]^2$.