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Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics

Ioana Dumitriu·Alan Edelman
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Abstract

In this paper we examine the zero and first order eigenvalue fluctuations for the $β$-Hermite and $β$-Laguerre ensembles, using the matrix models we described in \cite{dumitriu02}, in the limit as $β\to \infty$. We find that the fluctuations are described by Gaussians of variance $O(1/β)$, centered at the roots of a corresponding Hermite (Laguerre) polynomial. We also show that the approximation is very good, even for small values of $β$, by plotting exact level densities versus sum of Gaussians approximations.

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Ioana Dumitriu, Alan Edelman. 2004-03-16. Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics. https://doi.org/10.1016/j.anihpb.2004.11.002

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