THE LARGEST EIGENVALUE DISTRIBUTION OF THE LAGUERRE UNITARY

发布时间：2018-07-05 16:55

  本文选题：Orthogonal + polynomials　； 参考：《Acta Mathematica Scientia(English Series)》2017年02期

【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, 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 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.
[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, 伪 -n (t) and 尾 _ S _ n (t), as well as three auxiliary quantities, denoted by _ r _ n _ (t), _ n _ (t), and _ 蟽 _ n _ (t). We establish the second order differential equations for both 尾 s n (t) and rn (t). By investigating the soft edge scaling limit when 伪 = O (n) as n 鈭，

本文编号：2100991