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 $P\left(s\right)\mathrm{}$, 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, $N\times N$ 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 $H_0=0\mathrm{}$ 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 $n\times n$ matrix, whose elements are given by a kernel $K(\lambda ,\mu )$ as in the $H_0=0\mathrm{}$ case. From this representation we can show that Dyson’s short-distance universality still holds. We then conclude that $P\left(s\right)\mathrm{}$ is independent of $H_0$.