Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models

TL;DR

This paper investigates the global spectral fluctuations of $eta$-Hermite and $eta$-Laguerre ensembles using matrix models, showing they follow a Gaussian process and extending results to broader matrix classes.

Contribution

It provides a detailed analysis of spectral fluctuations for these ensembles and extends the results to larger classes of random matrices.

Findings

Spectral fluctuations follow a Gaussian process on monomials.

Results apply to $eta$-Hermite and $eta$-Laguerre ensembles.

Extensions to broader classes of matrices achieved.

Abstract

We study the global spectrum fluctuations for $\beta$-Hermite and $\beta$-Laguerre ensembles via the tridiagonal matrix models introduced in \cite{dumitriu02}, and prove that the fluctuations describe a Gaussian process on monomials. We extend our results to slightly larger classes of random matrices.