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Perturbed Hankel determinant, correlation functions and Painlevé equations
Min Chen1; Yang Chen1; Engui Fan2
2016-01-06
Source PublicationJournal of Mathematical Physics
ISSN0022-2488
Volume57Issue:2
Abstract

In this paper, we continue with the study of the Hankel determinant, generated by a Pollaczek-Jacobi type weight, w(x; t, α, β) := x(1 - x)e, x ∈ [0, 1], α > 0, β > 0, t ≥ 0. This reduces to the "pure" Jacobi weight at t = 0. It was shown in the work of Chen and Dai [J. Approximation Theory 162(2), 2149-2167 (2010)] that the logarithmic derivative of this Hankel determinant satisfies a Jimbo-Miwa-Okamoto σ-form of Painlevé V (P). We show that, under a double scaling, where n the dimension of the Hankel matrix tends to ∞ and t tends to 0, such that s := 2nt is finite, the double scaled Hankel determinant (effectively an operator determinant) has an integral representation in terms of a particular P. Expansions of the scaled Hankel determinant for small and large s are found. We also consider another double scaling with α = - 2n + λ, where n → ∞, and t tends to 0, such that s := nt is finite. In this situation, the scaled Hankel determinant has an integral representation in terms of a particular P, and its small and large s asymptotic expansions are also found. The reproducing kernel in terms of monic polynomials orthogonal with respect to the Pollaczek-Jacobi type weight under the origin (or hard edge) scaling may be expressed in terms of the solutions of a second order linear ordinary differential equation (ODE). With special choices of the parameters, the limiting (double scaled) kernel and the second order ODE degenerate to Bessel kernel and the Bessel differential equation, respectively.

DOI10.1063/1.4939276
URLView the original
Indexed BySCI
Language英语
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000371620000063
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Cited Times [WOS]:6   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorYang Chen
Affiliation1.Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, People’s Republic of China
2.School of Mathematical Science, Fudan University, Shanghai 200433, People’s Republic of China
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Min Chen,Yang Chen,Engui Fan. Perturbed Hankel determinant, correlation functions and Painlevé equations[J]. Journal of Mathematical Physics,2016,57(2).
APA Min Chen,Yang Chen,&Engui Fan.(2016).Perturbed Hankel determinant, correlation functions and Painlevé equations.Journal of Mathematical Physics,57(2).
MLA Min Chen,et al."Perturbed Hankel determinant, correlation functions and Painlevé equations".Journal of Mathematical Physics 57.2(2016).
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