The recurrence coefficients of a semi-classical Laguerre polynomials and the large n asymptotics of the associated Hankel determinant | |
Han P.; Chen Y. | |
2017-10-01 | |
Source Publication | Random Matrices: Theory and Application |
ISSN | 20103271 20103263 |
Volume | 6Issue:4 |
Abstract | In this paper, we study the recurrence coefficients of a deformed or semi-classical Laguerre polynomials orthogonal with respect to the weight w(x,s) = w(x; := xe-N(x+s(x2-x)), 0 ≤ x < ∞. Here >-1, 0 ≤ s ≤ 1 and N > 0. We will describe this problem in terms of the ratio r:= n N, where ultimately r is bounded away from 0, and close to 1. From the ladder operator approach, and the associated compatibility conditions, the recurrence coefficients satisfy a second order ordinary differential equation (ODE) when viewed as functions of the parameter s in the weight. Then we work out the large degree asymptotics of their recurrence coefficients. We also discuss the associated Hankel determinant. We show that the logarithmic derivative of Dn(s) can be expressed in terms of the recurrence coefficients, and obtained the large degree asymptotics of ln[Dn(1)]. Based on this result, we compute the probability that an n × n (or rN × rN) random matrix from a generalized Gaussian Unitary Ensemble (gGUE) is positive definite. |
Keyword | Hankel Determinant Ladder Operator Orthogonal Polynomials |
DOI | https://doi.org/10.1142/S2010326317400020 |
URL | View the original |
Indexed By | SCI |
Language | 英语 |
WOS Research Area | Physics ; Mathematics |
WOS Subject | Physics, Mathematical ; Statistics & Probability |
WOS ID | WOS:000419266600003 |
Fulltext Access | |
Citation statistics | |
Document Type | Journal article |
Collection | DEPARTMENT OF MATHEMATICS |
Affiliation | Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, P. R. China |
First Author Affilication | University of Macau |
Recommended Citation GB/T 7714 | Han P.,Chen Y.. The recurrence coefficients of a semi-classical Laguerre polynomials and the large n asymptotics of the associated Hankel determinant[J]. Random Matrices: Theory and Application,2017,6(4). |
APA | Han P.,&Chen Y..(2017).The recurrence coefficients of a semi-classical Laguerre polynomials and the large n asymptotics of the associated Hankel determinant.Random Matrices: Theory and Application,6(4). |
MLA | Han P.,et al."The recurrence coefficients of a semi-classical Laguerre polynomials and the large n asymptotics of the associated Hankel determinant".Random Matrices: Theory and Application 6.4(2017). |
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