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Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis
Min Chen; Yang Chen
2015-06
Source PublicationJournal of Mathematical Physics
ISSN0022-2488
Volume56Issue:6Pages:063506
Abstract

We continue with the study of the Hankel determinant, defined by D-n(t, alpha) = det. (integral(infinity)(0) x(j+k)w(x; t, alpha)dx)(j, k= 0)(n-1), generated by a singularly perturbed Laguerre weight, w(x; t, alpha) = x(alpha)e(-x)e(-t/x), x is an element of R+, alpha > 0, t > 0, and obtained through a deformation of the Laguerre weight function, w(x; 0, alpha) = x(alpha)e(-x), x is an element of R+, alpha > 0, via themultiplicative factor e(-t/x). An earlier investigation was made on the finite n aspect of such determinants, which appeared in Chen and Its [J. Approx. Theory 162, 270-297 (2010)]. It was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular Painleve III (P-III, for short) transcendent and its t derivatives. In this paper, we show that under a double scaling, where n, the size of the Hankel matrix tends to infinity, and t tends to 0(+), the scaled-and therefore, in some sense, infinite dimensional-Hankel determinant has an integral representation in terms of a C potential. The second order non-linear ordinary differential equation satisfied by C, after a change of variables, is another P-III transcendent, albeit with fewer number of parameters. Expansions of thedouble scaled determinant for small and large parameters are obtained. (C) 2015 AIP Publishing LLC.

DOIhttps://doi.org/10.1063/1.4922620
Language英语
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000357615900035
PublisherAMER INST PHYSICS, 1305 WALT WHITMAN RD, STE 300, MELVILLE, NY 11747-4501 USA
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Cited Times [WOS]:10   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorYang Chen
AffiliationDepartment of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China.
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Min Chen,Yang Chen. Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis[J]. Journal of Mathematical Physics,2015,56(6):063506.
APA Min Chen,&Yang Chen.(2015).Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis.Journal of Mathematical Physics,56(6),063506.
MLA Min Chen,et al."Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis".Journal of Mathematical Physics 56.6(2015):063506.
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