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

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

2004-01-20

arxiv.org/abs/math/0401258v2

Int.Math.Res.Not. 2004 (2004), no.25, 1249-1272

Mathematics

Functional Analysis

20 pages, 1 figure, small changes

Scientific paper

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 $\sin z/(\pi z)$ 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.

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