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

I. V. Krasovsky
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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 $\sin z/(πz)$ 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.

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I. V. Krasovsky. 2004-04-22. Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. https://arxiv.org/abs/math/0401258

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