ASYMPTOTICS FOR THE FREDHOLM DETERMINANT OF THE SINE KERNEL ON A UNION OF INTERVALS

In the bulk scaling limit of the Gaussian Unitary Ensemble of hermitia n matrices the probability that an interval of length s contains no ei genvalues is the Fredholm determinant of the sine kernel sin(x-y)/pi(x -y) over this interval. A formal asymptotic expansion for the determin ant as s tends to infinity was obtained by Dyson. In this paper we rep lace a single interval of length s by sJ, where J is a union of m inte rvals and present a proof of the asymptotics up to second order. The l ogarithmic derivative with respect to a of the determinant equals a co nstant (expressible in terms of hyperelliptic integrals) times s, plus a bounded oscillatory function of s (zero if m = 1, periodic if m = 2 , and in general expressible in terms of the solution of a Jacobi inve rsion problem), plus o(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 orthogon al polynomial theory.