Linear Statistics of Matrix Ensembles in Classical Background

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 ${\mathbb{E}}_{\beta }(\exp (-\lambda {\sum }_{j}F(x_j))),$ 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 $F(\cdot ).$

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