Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals

Abstract

In the bulk scaling limit of the Gaussian Unitary Ensemble of hermitian matrices the probability that an interval of lengths contains no eigenvalues is the Fredholm determinant of the sine kernel$\frac{{\sin (x - y)}}{{\pi (x - y)}}$ over this interval. A formal asymptotic expansion for the determinant ass tends to infinity was obtained by Dyson. In this paper we replace a single interval of lengths bysJ, whereJ is a union ofm intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect tos of the determinant equals a constant (expressible in terms of hyperelliptic integrals) timess, plus a bounded oscillatory function ofs (zero ifm=1, periodic ifm=2, and in general expressible in terms of the solution of a Jacobi inversion problem), pluso(1). Also determined are the asymptotics of the trace of the resolvent operator, which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none. The proofs use ideas from orthogonal polynomial theory.