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Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics. (English) Zbl 1079.15014
This paper examines the zero and first order eigenvalue fluctuations for the $$\beta$$-Hermite and $$\beta$$-Laguerre ensembles, using tridiagonal matrix models, in the limit as $$\beta\to\infty$$. The main results say that the fluctuations are described by multivariate Gaussians of covariance $$O(1/\beta)$$, centered at the roots of a corresponding Hermite (Laguerre) polynomial. The covariance matrix itself is expressed as combinations of Hermite or Laguerre polynomials respectively. It is also proven that the approximations are of real value even for small $$\beta$$; one can use them to approximate the true functions even for the traditional $$\beta=1,2,4$$ values.

##### MSC:
 15A22 Matrix pencils
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