Abstract. We construct approximate transport maps for non-critical β-matrix models, that is, maps so that the push forward of a non-critical β-matrix model with a given potential is a non-critical β-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 which are uniform in the dimension. In addition, we find a very useful asymptotic expansion for such maps which allow us to deduce that local statistics have the same asymptotic behavior for both models.

We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or β-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.

We prove that the linear statistics of eigenvalues of β-log gasses satisfying the onecut and off-critical assumption with a potential V ∈ C 6 (R) satisfy a central limit theorem at all mesoscopic scales α ∈ (0; 1). We prove this for compactly supported test functions f ∈ C 5 (R) using loop equations at all orders along with rigidity estimates.

We use the transport methods developped in [3] to obtain universality results for local statistics of eigenvalues in the bulk and at the edge for β-matrix models in the multi-cut regime. We construct an approximate transport map inbetween two probability measures from the fixed filling fraction model discussed in [6] and deduce from it universality in the initial model.

No abstract

No abstract