Transport Maps for β-Matrix Models and Universality

We construct approximate transport maps for non-critical beta-matrix models, that is, maps so that the push forward of a non-critical beta-matrix model with a given potential is a non-critical beta-matrix model with another potential, up to a small error in the total variation distance. One of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension. In addition, we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models.