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

2004/01/31 by I. V. Krasovsky · 1 citation
Mathematics · Physics and Astronomy · #math.FA #math-ph #math.MP

paper · pdf

published as Int.Math.Res.Not. 2004 (2004), no.25, 1249-1272 · 20 pages, 1 figure, small changes

arxiv created 2004/04/22 · arxiv updated 2009/12/01

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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