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Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite n Gaussian Unitary Ensembles
Man Cao1; Yang Chen1; James Griffin2
2014-08-08
Source PublicationJournal of Statistical Physics
ISSN0022-4715
Volume157Issue:2Pages:363-375
Abstract

In this paper we study the gap probability problem in the Gaussian unitary ensembles of n by n matrices : The probability that the interval J : = (–a, a) is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke, and Forrester and Witte on this subject, it has been shown that two Painlevé type differential equations arise in this context. The first is the Jimbo–Miwa–Okomoto σ-form and the second is a particular Painlevé IV. Using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted as σ(a),R(a) and r (a). We show that each one satisfies a second order Painlevé type differential equation as well as a discrete Painlevé type equation. In particular, in addition to providing an elementary derivation of the aforementioned σ-form and Painlevé IV we are able to show that the quantity r (a) satisfies a particular case of Chazy’s second degree second order differential equation. For the discrete equations we show that the quantity r (a) satisfies a particular form of the modified discrete Painlevé II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities R (a) and σ (a).

KeywordGap Probability Painlevé Equations Random Matrices
DOI10.1007/s10955-014-1076-x
URLView the original
Indexed BySCI
Language英语
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000341710800008
PublisherSPRINGER, 233 SPRING ST, NEW YORK, NY 10013 USA
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被引频次[WOS]:4   [WOS记录]     [WOS相关记录]
Document TypeJournal article
专题DEPARTMENT OF MATHEMATICS
Affiliation1.Department of Mathematics, Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira, Taipa Macau, China
2.Department of Mathematics and Statistics, American University of Sharjah, PO Box 26666, Sharjah, UAE
First Author AffilicationFaculty of Science and Technology
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Man Cao,Yang Chen,James Griffin. Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite n Gaussian Unitary Ensembles[J]. Journal of Statistical Physics,2014,157(2):363-375.
APA Man Cao,Yang Chen,&James Griffin.(2014).Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite n Gaussian Unitary Ensembles.Journal of Statistical Physics,157(2),363-375.
MLA Man Cao,et al."Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite n Gaussian Unitary Ensembles".Journal of Statistical Physics 157.2(2014):363-375.
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