Higher-Order Analogues of the Tracy-Widom Distribution and the Painleve II Hierarchy

Claeys, Tom;Its, A.;Krasovsky, I.
(2010) Communications on Pure and Applied Mathematics — Vol. 63, n° 3, p. 362-412 (2010)

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  • Claeys, Tomorcid-logoUCLouvain
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  • Its, A.
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  • Krasovsky, I.
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Abstract
We study Fredholm determinants related to a family of kernels that describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher-order analogues of the Airy kernel and are built out of functions associated with the Painleve I hierarchy. The Fredholm determinants related to those kernels are higher-order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painleve II hierarchy. In addition, we compute large gap asymptotics for the Fredholm determinants. (c) 2009 Wiley Periodicals, Inc.
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Claeys, T., Its, A., & Krasovsky, I. (2010). Higher-Order Analogues of the Tracy-Widom Distribution and the Painleve II Hierarchy. Communications on Pure and Applied Mathematics, 63(3), 362-412. https://hdl.handle.net/2078.5/151381 (Original work published 2010)