Linear Statistics of Matrix Ensembles in Classical Background

TL;DR

This paper analyzes the statistical behavior of linear functions of eigenvalues in classical random matrix ensembles, deriving explicit formulas for their moment generating functions and asymptotic distributions.

Contribution

It provides explicit determinant formulas for the moment generating functions of linear statistics in Gaussian and Laguerre ensembles, including large N asymptotics.

Findings

Explicit formulas for moment generating functions in classical ensembles

Large N asymptotic behavior of linear statistics derived

Determinant representations involving scalar operators obtained

Abstract

Given a joint probability density function of $N$ real random variables, $\{x_j\}_{j=1}^{N},$ obtained from the eigenvector-eigenvalue decomposition of $N\times N$ random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely, $\sum_{j=1}^{N}F(x_j).$ For the jpdfs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper the moment generating function $\mathbb{E}_{\beta}({\rm exp}(-\lambda\sum_{j}F(x_j))),$ where $\mathbb{E}_{\beta}$ denotes expectation value over the Orthogonal ($\beta=1$) and Symplectic ($\beta=4)$ ensembles, in the form one plus a Schwartz function, none vanishing over $\mathbb{R}$ for the Gaussian ensembles and $\mathbb{R}^+$ for the Laguerre ensembles. These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large $N$ asymptotic of the linear statistics from suitably scaled $F(\cdot).$