Transport maps for β-matrix models in the multi-cut regime

We use the transport methods develop in [F. Bekerman, A. Figalli and A. Guionnet, Transport maps for beta-matrix models and universality, Commun. Math. Phys. 338(2) (2013) 589-619] to obtain universality results for local statistics of eigenvalues in the bulk and at the edge for beta-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 [G. Borot, A. Guionnet and K. K. Kozlowski, Large-N asymptotic expansion for mean field models with coulomb gas interaction, Int. Math. Res. Not. (2015)] and deduce from it universality in the initial model.