Journal of Nonlinear Mathematical Physics

Volume 26, Issue 1, December 2018, Pages 24 - 53

The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble

Authors
Shulin Lyu
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China, lvshulin1989@163.com
James Griffin
Department of Mathematics and Statistics, American University of Sharjah Sharjah, PO Box 26666, UAE, jgriffin@aus.edu
Yang Chen
Department of Mathematics, University of Macau Avenida da Universidade, Taipa, Macau, China, yangbrookchen@yahoo.co.uk
Received 24 January 2018, Accepted 24 May 2018, Available Online 6 January 2021.
DOI
https://doi.org/10.1080/14029251.2019.1544786How to use a DOI?
Keywords
Hankel Determinant, Smallest eigenvalue, Double scaling
Abstract

We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function w(x,t)=xαextx, x ∈ [0, ∞), α > −1, t ≥ 0, are greater than s. This probability is expressed as the quotient of Dn(s, t) and its value at s = 0, where Dn(s, t) denotes the determinant of the n dimensional Hankel matrices generated by the moments of w(x; t) on x ∈ [s, ∞). In this paper we focus specifically on the Hankel determinant Dn(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 → ∞, s → 0, and t → 0 such that S := 4ns and T := (2n + 1 + α)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 → ∞ with T fixed, S → 0 with T > 0 fixed, and T → ∞ 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.

Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 1
Pages
24 - 53
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2019.1544786How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Shulin Lyu
AU  - James Griffin
AU  - Yang Chen
PY  - 2021
DA  - 2021/01
TI  - The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble
JO  - Journal of Nonlinear Mathematical Physics
SP  - 24
EP  - 53
VL  - 26
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1544786
DO  - https://doi.org/10.1080/14029251.2019.1544786
ID  - Lyu2021
ER  -