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Painlevé VI, Painlevé III, and the Hankel determinant associated with a degenerate Jacobi unitary ensemble
Min,Chao1; Chen,Yang2
2020-10-01
Source PublicationMathematical Methods in the Applied Sciences
ISSN0170-4214
Volume43Issue:15Pages:9169-9184
Abstract

This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a nonlinear second-order differential equation, which turns out to be the Jimbo–Miwa–Okamoto σ-form of the Painlevé VI equation by a translation transformation. We also show that, after a suitable double scaling, the differential equation is reduced to the Jimbo–Miwa–Okamoto σ-form of the Painlevé III. In the end, we obtain the asymptotic behavior of the Hankel determinant as t→1 and t→0 in two important cases, respectively.

KeywordDegenerate Jacobi Unitary Ensemble Double Scaling Analysis Hankel Determinant Ladder Operators Painlevé Equations Random Matrix Theory
DOI10.1002/mma.6609
URLView the original
Indexed BySCIE
Language英語English
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000544585200001
Scopus ID2-s2.0-85087295116
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Cited Times [WOS]:2   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorMin,Chao
Affiliation1.School of Mathematical Sciences,Huaqiao University,Quanzhou,China
2.Department of Mathematics,Faculty of Science and Technology,University of Macau,Macao
Recommended Citation
GB/T 7714
Min,Chao,Chen,Yang. Painlevé VI, Painlevé III, and the Hankel determinant associated with a degenerate Jacobi unitary ensemble[J]. Mathematical Methods in the Applied Sciences,2020,43(15):9169-9184.
APA Min,Chao,&Chen,Yang.(2020).Painlevé VI, Painlevé III, and the Hankel determinant associated with a degenerate Jacobi unitary ensemble.Mathematical Methods in the Applied Sciences,43(15),9169-9184.
MLA Min,Chao,et al."Painlevé VI, Painlevé III, and the Hankel determinant associated with a degenerate Jacobi unitary ensemble".Mathematical Methods in the Applied Sciences 43.15(2020):9169-9184.
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