On the asymptotic behaviour of correlators of multi-cut matrix models

We consider invariant ensembles of n×n Hermitian random matrices (known also as the matrix models) in the case where the support of the limiting density of states (DOS) consists of several intervals. By using recent results on the asymptotics of orthogonal polynomials, we find first that the amplitudes of the leading terms of the correlator of the normalized traces of resolvent of random matrices and of their densities of states are quasi-periodic functions of n, whose frequencies are the integrals of the limiting DOS over the intervals of the support, and whose form is uniquely determined by the edges of the support. This suggest a certain parametrization of the universality classes of the correlator. Second we show that the leading terms of these correlators can be expressed correspondingly via the matrix elements of the resolvent and of the spectral kernel of a certain quasi-periodic Jacobi matrix whose coefficients are determined by the same frequencies.