Matrix models at low temperature

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 $U=-[X,Y{]}^2$.