  In this paper we study multi-matrix models whose potentials are small
perturbations of the quadratic potential associated with independent GUE random
matrices. More precisely, we compute the free energy and the expectation of the
trace of polynomials evaluated in those matrices. We prove an asymptotic
expansion in the inverse of the matrix dimension to any order. Out of this
result we deduce new formulas for map enumeration and the microstates free
entropy. The approach that we take is based on the interpolation method between
random matrices and free operators developed in [8,29].
