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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 PublicationRandom Matrices: Theory and Application
ISSN20103271 20103263
Volume6Issue: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.

KeywordHankel Determinant Ladder Operator Orthogonal Polynomials
DOIhttps://doi.org/10.1142/S2010326317400020
URLView the original
Indexed BySCI
Language英语
WOS Research AreaPhysics ; Mathematics
WOS SubjectPhysics, Mathematical ; Statistics & Probability
WOS IDWOS:000419266600003
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Cited Times [WOS]:6   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
AffiliationDepartment of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, P. R. China
First Author AffilicationUniversity 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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