Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics

TL;DR

This paper studies the eigenvalue fluctuations of Hermite and Laguerre ensembles as the parameter beta grows large, showing they are Gaussian with variance decreasing as 1/beta, and validating the approximation for small beta.

Contribution

It provides a detailed analysis of eigenvalue fluctuations in beta-ensembles, connecting them to Hermite and Laguerre polynomial roots, and demonstrates the accuracy of Gaussian approximations.

Findings

Eigenvalue fluctuations are Gaussian with variance O(1/beta).

Centered at roots of Hermite and Laguerre polynomials.

Gaussian approximation remains accurate even for small beta.

Abstract

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