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Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities
Zhu, Mengkun1,2; Wang, Dan2; Chen, Yang2
2021-09-01
Source PublicationAnalysis and Mathematical Physics
ISSN1664-2368
Volume11Issue:3
Abstract

In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles w(z,t)=Aθ(z-t)e-z2+tz,here θ(x) is the Heaviside function, where A> 0 , t> 0 , and z∈ [0 , ∞). We derive the ladder operators and its interrelated compatibility conditions. By using the ladder operators, we show two auxiliary quantities R(t) and r(t) satisfy the coupled Riccati equations, from which we also prove that R(t) satisfies a particular Painlevé IV equation. Even more, σ(t) , allied to R(t) , satisfies both the discrete and continuous Jimbo–Miwa–Okamoto σ-form of the Painlevé IV equation. Finally, we consider the situation when n gets large, the second order linear differential equation satisfied by the polynomials P(x) orthogonal with respect to the semi-classical weight turn to be a particular bi-confluent Heun equation.

KeywordAsymptotics Ladder Operators Laguerre Unitary Ensembles Orthogonal Polynomials Painlevé Equations
DOI10.1007/s13324-021-00560-x
URLView the original
Indexed BySCIE
Language英語English
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000667679300001
Scopus ID2-s2.0-85108970374
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Cited Times [WOS]:1   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorWang, Dan; Chen, Yang
Affiliation1.School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China
2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Avenida da Universidade, Macao
First Author AffilicationFaculty of Science and Technology
Corresponding Author AffilicationFaculty of Science and Technology
Recommended Citation
GB/T 7714
Zhu, Mengkun,Wang, Dan,Chen, Yang. Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities[J]. Analysis and Mathematical Physics,2021,11(3).
APA Zhu, Mengkun,Wang, Dan,&Chen, Yang.(2021).Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities.Analysis and Mathematical Physics,11(3).
MLA Zhu, Mengkun,et al."Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities".Analysis and Mathematical Physics 11.3(2021).
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