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The largest eigenvalue distribution of the laguerre unitary ensemble
Shulin LYU; Yang CHEN
Source PublicationActa Mathematica Scientia

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 α is finite, we derive a PII , 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 ̺(t) = Ξ′ (t) with Ξ(t) satisfying the σ-form of PIV or PV .

KeywordDifferential Equations Orthogonal Polynomials Painlev´e Equations
URLView the original
Indexed BySCI
WOS Research AreaMathematics
WOS SubjectMathematics
WOS IDWOS:000398760600012
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Cited Times [WOS]:6   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
Corresponding AuthorShulin LYU
AffiliationDepartment of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Shulin LYU,Yang CHEN. The largest eigenvalue distribution of the laguerre unitary ensemble[J]. Acta Mathematica Scientia,2017,37(2):439-462.
APA Shulin LYU,&Yang CHEN.(2017).The largest eigenvalue distribution of the laguerre unitary ensemble.Acta Mathematica Scientia,37(2),439-462.
MLA Shulin LYU,et al."The largest eigenvalue distribution of the laguerre unitary ensemble".Acta Mathematica Scientia 37.2(2017):439-462.
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