Extension of level-spacing universality

In the theory of random matrices. several properties are known to be universal, i.e., independent of the specific probability distribution. For instance, Dyson's short-distance universality of the correlation functions implies the universality of P(s), the level-spacing distribution. We first briefly review how this property is understood for unitary invariant ensembles and consider next a Hamiltonian H = H-0 + V, in which H-0 is a given, nonrandom, N x N matrix, and V is an Hermitian random matrix with a Gaussian probability distribut ion. The standard techniques, based on orthogonal polynomials, which are the key for the understanding of the H-0 = 0 case, are no longer available. Then using a completely different approach, we derive closed expressions for the n-point correlation functions, which are exact for finite N. Remarkably enough the result may still be expressed as a determinant of an n x n matrix, whose elements are given by a kernel K(lambda, mu) as in the H-0 = 0 case. From this representation we can show that Dyson's short-distance universality still holds. We then conclude that P(s) is independent of H-0.