Extension of level-spacing universality

Abstract

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, $N\ifmmode\times\else\texttimes\fi{}N$ matrix, and $V$ is an Hermitian random matrix with a Gaussian probability distribution. 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 $n\ifmmode\times\else\texttimes\fi{}n$ matrix, whose elements are given by a kernel $K(\ensuremath{\lambda},\ensuremath{\mu})$ 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 ${H}_{0}$.