From gap probabilities in random matrix theory to eigenvalue expansions

Thomas Bothner*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

We present a method to derive asymptotics of eigenvalues for trace-class integral operators K LJ : ;d 2 ( )⥀l , acting on a single interval J Ì , which belongs to the ring of integrable operators (Its et al 1990 Int. J. Mod. Phys. B 4 1003–37). Our emphasis lies on the behavior of the spectrum i J { i 0 l ( )} = ¥ of K as ∣J∣  ¥ and i is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant det I K L J ( )∣ 2 - g ( ) as ∣J∣  ¥ and g  1 in a Stokes type scaling regime. Concrete asymptotic formulæ are obtained for the eigenvalues of Airy and Bessel kernels in random matrix theory.
Original languageEnglish
Article number075204
Number of pages78
JournalJournal of Physics A: Mathematical and Theoretical
Volume49
Issue number7
Early online date13 Jan 2016
DOIs
Publication statusE-pub ahead of print - 13 Jan 2016

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