Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics

Abstract

In this paper we examine the zero and first order eigenvalue fluctuations for the ${\hat{I}}^2$-Hermite and ${\hat{I}}^2$-Laguerre ensembles, using the matrix models we described in \cite{dumitriu02}, in the limit as ${\hat{I}}^2\to \infty$. We find that the fluctuations are described by Gaussians of variance $O(1/{\hat{I}}^2)$, centered at the roots of a corresponding Hermite (Laguerre) polynomial. We also show that the approximation is very good, even for small values of ${\hat{I}}^2$, by plotting exact level densities versus sum of Gaussians approximations.

15 pages; 17 figures