Universal singularity at the closure of a gap in a random matrix theory

We consider a Hamiltonian ${H=H}_0+V$, in which $H_0$ is a given nonrandom Hermitian matrix, and $V$ is an $N\times N$ Hermitian random matrix with a Gaussian probability distribution. We had shown before that Dyson’s universality of the short-range correlations between energy levels holds at generic points of the spectrum independently of $H_0$. We consider here the case in which the spectrum of $H_0$ is such that there is a gap in the average density of eigenvalues of $H$ which is thus split into two pieces. When the spectrum of $H_0$ is tuned so that the gap closes, a new class of universality appears for the energy correlations in the vicinity of this singular point.