Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics
arXiv:math-ph/0403029 · doi:10.1016/j.anihpb.2004.11.002
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.
15 pages; 17 figures