数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (2): 439-462.doi: 10.1016/S0252-9602(17)30013-9

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THE LARGEST EIGENVALUE DISTRIBUTION OF THE LAGUERRE UNITARY ENSEMALE

Shulin LYU, Yang CHEN   

  1. Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China
  • 收稿日期:2015-11-06 出版日期:2017-04-25 发布日期:2017-04-25
  • 通讯作者: Shulin LYU,E-mail:lvshulin1989@163.com E-mail:lvshulin1989@163.com
  • 作者简介:Shulin LYU,E-mail:lvshulin1989@163.com;Yang CHEN,E-mail:yangbrookchen@yahoo.co.uk

THE LARGEST EIGENVALUE DISTRIBUTION OF THE LAGUERRE UNITARY ENSEMALE

Shulin LYU, Yang CHEN   

  1. Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China
  • Received:2015-11-06 Online:2017-04-25 Published:2017-04-25
  • Contact: Shulin LYU,E-mail:lvshulin1989@163.com E-mail:lvshulin1989@163.com

摘要:

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in (0,t), that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely, αn(t) and βn(t), as well as three auxiliary quantities, denoted by rn(t), Rn(t), and σn(t). We establish the second order differential equations for both βn(t) and rn(t). By investigating the soft edge scaling limit when α=O(n) as n or→∞ or α is finite, we derive a P, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials Pn(z) corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n, asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy's equation for ρ(t)=Ξ'(t) with Ξ(t) satisfying the σ-form of PIV or PV.

关键词: Orthogonal polynomials, Painlevé, equations, differential equations

Abstract:

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in (0,t), that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely, αn(t) and βn(t), as well as three auxiliary quantities, denoted by rn(t), Rn(t), and σn(t). We establish the second order differential equations for both βn(t) and rn(t). By investigating the soft edge scaling limit when α=O(n) as n or→∞ or α is finite, we derive a P, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials Pn(z) corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n, asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy's equation for ρ(t)=Ξ'(t) with Ξ(t) satisfying the σ-form of PIV or PV.

Key words: Orthogonal polynomials, Painlevé, equations, differential equations