  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. Moreover, we prove universality for the local statistics of eigenvalues
of self-adjoint polynomials in several independent GUE or GOE matrices which
are close to the identity.

Probability