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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 Publication JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS ISSN 1402-9251 Volume 26Issue: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. Keyword Hankel Determinant Smallest Eigenvalue Double Scaling DOI 10.1080/14029251.2019.1544786 URL View the original Indexed By SCI Language 英语 WOS Research Area Mathematics ; Physics WOS Subject Mathematics, Applied ; Physics, Mathematical WOS ID WOS:000451827400002 Publisher TAYLOR & FRANCIS LTD Fulltext Access Citation statistics Document Type Journal article Collection DEPARTMENT OF MATHEMATICS Affiliation 1.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 CitationGB/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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