Universal Singularity at the Closure of a Gap in a Random Matrix Theory

We consider a Hamiltonian H = H0 + V , in which H0 is a given nonrandom Hermitian matrix, and V is an N × 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 H0. We consider here the case in which the spectrum of H0 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 H0 is tuned so that the gap closes, a new class of universality appears for the energy correlations in the vicinity of this singular point. Typeset using REVTEX Unité propre du centre national de la Recherche Scientifique, Associée à l’Ecole Normale Supérieure et à l’Université de Paris-Sud