UNIVERSALITY IN SEVERAL-MATRIX MODELS VIA APPROXIMATE TRANSPORT MAPS

Abstract : We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices (i.e., they are given by the sine-kernel in the bulk and the Tracy-Widom distribution at the edge), and we show averaged energy universality (i.e., universality for averages of m-points correlation functions around some energy level E in the bulk). As a corollary, these results yield universality for self-adjoint polynomials in several independent GUE or GOE matrices which are close to the identity.