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Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble
Min,Chao1; Chen,Yang2
Source PublicationRandom Matrices: Theory and Application

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities Rn(t) and rn(t) and show that they satisfy the coupled Riccati equations, from which we find that Rn(t) satisfies the Painlevé V equation. Furthermore, we prove that σn(t), a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo-Miwa-Okamoto σ-form of the Painlevé V. In the end, by using Dyson's Coulomb fluid approach, we consider the large n asymptotic behavior of our problem at the soft edge, which gives rise to the Painlevé XXXIV equation.

KeywordAsymptotics Degenerate Laguerre Unitary Ensemble Hankel Determinant Ladder Operators Orthogonal Polynomials Painlevé Equations
URLView the original
Indexed BySCIE
WOS Research AreaPhysics ; Mathematics
WOS SubjectPhysics, Mathematical ; Statistics & Probability
WOS IDWOS:000527885800006
Scopus ID2-s2.0-85074520399
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Cited Times [WOS]:8   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
Corresponding AuthorChen,Yang
Affiliation1.School of Mathematical Sciences,Huaqiao University,Quanzhou,362021,China
2.Department of Mathematics,Faculty of Science and Technology,University of Macau,Macao
Corresponding Author AffilicationFaculty of Science and Technology
Recommended Citation
GB/T 7714
Min,Chao,Chen,Yang. Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble[J]. Random Matrices: Theory and Application,2020,9(2).
APA Min,Chao,&Chen,Yang.(2020).Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble.Random Matrices: Theory and Application,9(2).
MLA Min,Chao,et al."Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble".Random Matrices: Theory and Application 9.2(2020).
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