Breakdown of universality in multi-cut matrix models

We solve the puzzle of the disagreement between orthogonal polynomials methods and mean-field calculations for random N × N matrices with a disconnected eigenvalue support. We show that the difference does not stem from a ℤ 2 symmetry breaking, but from the discreteness of the number of eigenvalues. This leads to additional terms (quasiperiodic in N ) which must be added to the naive mean-field expressions. Our result invalidates the existence of a smooth topological large- N expansion and some postulated universality properties of correlators. We derive the large- N expansion of the free energy for the general two-cut case. From it we rederive by a direct and easy mean-field-like method the two-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.