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The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble
Lyu, Shulin1; Griffin, James2; Chen, Yang3
2019-01-02
Source PublicationJOURNAL OF NONLINEAR MATHEMATICAL PHYSICS
ISSN1402-9251
Volume26Issue:1Pages:24-53
Abstract

We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function , are greater than s. This probability is expressed as the quotient of D-n(s,t) and its value at s = 0, where D-n(s,t) denotes the determinant of the n dimensional Hankel matrices generated by the moments of w(x;t) on x is an element of [s, infinity). In this paper we focus specifically on the Hankel determinant D-n(s,t) and its properties. Based on the ladder operators adapted to the monic polynomials orthogonal with respect to w(x;t), and from the associated supplementary conditions and a sum-rule, we show that the log-derivative of the Hankel determinant, viewed as a function of s and t, satisfies a second order sixth degree partial differential equation, where n appears as a parameter. In order to go to the thermodynamic limit, of infinitely large matrices, we envisage a scenario where n -> infinity, s -> 0, and t -> 0 such that S := 4ns and T := (2n + 1 + alpha)t are finite. After such a double scaling, the large finite n equation reduces to a second order second degree equation, in the variables S and T, from which we derive the asymptotic expansion of the scaled Hankel determinant in three cases of S and T : S -> infinity with T fixed, S -> 0 with T > 0 fixed, and T -> infinity with S > 0 fixed. The constant term in the asymptotic expansion is shown to satisfy a difference equation and one of its solutions is the Tracy-Widom constant.

KeywordHankel Determinant Smallest Eigenvalue Double Scaling
DOI10.1080/14029251.2019.1544786
URLView the original
Indexed BySCI
Language英语
WOS Research AreaMathematics ; Physics
WOS SubjectMathematics, Applied ; Physics, Mathematical
WOS IDWOS:000451827400002
PublisherTAYLOR & FRANCIS LTD
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Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Affiliation1.Sun Yat Sen Univ, Sch Math Zhuhai, Zhuhai 519082, Guangdong, Peoples R China;
2.Amer Univ Sharjah, Dept Math & Stat, POB 26666, Sharjah, U Arab Emirates;
3.Univ Macau, Dept Math, Ave Univ, Taipa, Macao, Peoples R China
Recommended Citation
GB/T 7714
Lyu, Shulin,Griffin, James,Chen, Yang. The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble[J]. JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS,2019,26(1):24-53.
APA Lyu, Shulin,Griffin, James,&Chen, Yang.(2019).The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble.JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS,26(1),24-53.
MLA Lyu, Shulin,et al."The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble".JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS 26.1(2019):24-53.
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