Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics

In this paper we examine the zero and first order eigenvalue fluctuations for the beta-Hermite and beta-Laguerre ensembles, using tridiagonal matrix models, in the limit as beta -> infinity. We prove that the fluctuations are described by multivariate Gaussians of covariance O(1/beta), centered at the roots of a corresponding Hermite (Lagueffe) polynomial. The covafiance matrix itself is expressed as combinations of Hermite or Laguerre polynomials respectively. We show that the approximations are of real value even for small beta; we can use them to approximate the true functions even for the traditional beta = 1, 2, 4 values. (c) 2005 Elsevier SAS. All rights reserved.