THE LARGEST EIGENVALUE DISTRIBUTION OF THE LAGUERRE UNITARY ENSEMALE

doi:10.1016/S0252-9602(17)30013-9

Abstract

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, α_n(t) and β_n(t), as well as three auxiliary quantities, denoted by r_n(t), R_n(t), and σ_n(t). We establish the second order differential equations for both β_n(t) and r_n(t). By investigating the soft edge scaling limit when α=O(n) as n or→∞ or α is finite, we derive a P_Ⅱ, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials P_n(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 P_IV or P_V.

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

Shulin LYU, Yang CHEN. THE LARGEST EIGENVALUE DISTRIBUTION OF THE LAGUERRE UNITARY ENSEMALE[J].Acta mathematica scientia,Series B, 2017, 37(2): 439-462.

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