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 2-cut case. From it we rederive by a direct and easy meanfield-like method the 2-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.

In this article, we study the q-state Potts random matrix models extended to branched polymers, by the equations of motion method. We obtain a set of loop equations valid for any arbitrary value of q. We show that, for q = 2 − 2 cos l r π (l, r mutually prime integers with l < r ), the resolvent satisfies an algebraic equation of degree 2r − 1 if l + r is odd and r − 1 if l + r is even. This generalizes the presently-known cases of q = 1, 2, 3. We then derive for any 0 ≤ q ≤ 4 the Potts-q critical exponents and string susceptibility.

In this letter, we show how one can solve easily the Potts-3 + branching interactions and Potts-∞ matrix models, by the means of the equations of motion (loop equations). We give an algebraic equation for the resolvents of these models, and their scaling behaviour. This shows that the equations of motion can be a useful tool for solving such models.