Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle

Abstract

We obtain uniform asymptotics for polynomials orthogonal on a fixed and varying arc of the unit circle with a positive analytic weight function. We also complete the proof of the large s asymptotic expansion for the Fredholm determinant with the kernel sinz/(π z) on the interval [0,s], verifying a conjecture of Dyson for the constant term in the expansion. In the Gaussian unitary ensemble of random matrices, this determinant describes the probability for an interval of length s in the bulk scaling limit to be free from the eigenvalues.