Asymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals

The sine kernel K(x, y) := sin(x− y) π(x− y) arises in many areas of mathematics and mathematical physics. There is an extensive literature on the asymptotics of the eigenvalues of Ks, the operator with this kernel on an interval of length s, as s→ ∞, for example [4, 6, 8, 12], and asymptotic fomulas of various kinds were obtained. Some of these derivations were rigorous, others were more heuristic. The Fredholm determinant of the kernel is of particular interest. In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian matrices, the probability that an interval of length s contains no eigenvalues is equal to