Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics

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• 作者：Ioana Dumitriu ; Alan Edelman
• 关键词：Fractional Brownian motion ; Stochastic integration ; Malliavin calculus ; Symmetric integral
• 刊名：Annales de l'Institut Henri Poincare (B) Probability and Statistics
• 出版年：2005
• 出版时间：November-December 2005
• 年：2005
• 卷：41
• 期：6
• 页码：1083-1099
• 全文大小：418 K

文摘

In this paper we examine the zero and first order eigenvalue fluctuations for the β-Hermite and β-Laguerre ensembles, using tridiagonal matrix models, in the limit as β→∞. We prove that the fluctuations are described by multivariate Gaussians of covariance O(1/β), centered at the roots of a corresponding Hermite (Laguerre) polynomial. The covariance matrix itself is expressed as combinations of Hermite or Laguerre polynomials respectively.

We show that the approximations are of real value even for small β; we can use them to approximate the true functions even for the traditional β=1,2,4 values.