Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System

We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy–Widom formulas for the Fredholm determinants, which are explicitly given in terms o...

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Bibliographic Details
Published in:Communications in mathematical physics 2019-01, Vol.365 (2), p.515-567
Main Authors: Xu, Shuai-Xia, Dai, Dan
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Determinants
Eigenvalues
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
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Summary:We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy–Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-018-3257-y