From gap probabilities in random matrix theory to eigenvalue expansions

Abstract

We present a method to derive asymptotics of eigenvalues for trace-class integral operators $K:L^2(J;d\lambda )\circlearrowleft$, acting on a single interval $J\subset \mathbb{R}$, which belong to the ring of integrable operators \cite{IIKS}. Our emphasis lies on the behavior of the spectrum $\{{\lambda }_i(J){\}}_{i=0}^{\infty }$ of $K$ as $|J|\to \infty$ and $i$ is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant $\det (I-\gamma K){|}_{L^2(J)}$ as $|J|\to \infty$ and $\gamma \uparrow 1$ in a Stokes type scaling regime. Concrete asymptotic formulæ\, are obtained for the eigenvalues of Airy and Bessel kernels in random matrix theory.

50 pages, 10 figures. To appear in J. Phys. A: Mathematical and Theoretical. Version 2 corrects typos and updates literature

References in corpus (4)

• The Pearcey Process
• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Universality of a double scaling limit near singular edge points in random matrix models
• On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I

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• Transition asymptotics for the Painlevé II transcendent
• The sine process under the influence of a varying potential