Finite N Fluctuation Formulas for Random Matrices

TL;DR

This paper derives exact formulas for the probability densities of linear statistics in Gaussian, Laguerre, and circular random matrix ensembles, demonstrating they follow a central limit theorem as matrix size grows.

Contribution

It provides explicit finite N formulas for linear statistic distributions in several random matrix ensembles and proves their convergence to normality as N increases.

Findings

Exact p.d.f.s for linear statistics in Gaussian and Laguerre ensembles.

Exact p.d.f.s for specific linear statistics in circular ensemble.

Validation of central limit theorem for these statistics as N approaches infinity.

Abstract

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic $\sum_{j=1}^N (x_j - <x>)$ is computed exactly and shown to satisfy a central limit theorem as $N \to \infty$. For the circular random matrix ensemble the p.d.f.'s for the linear statistics ${1 \over 2} \sum_{j=1}^N (\theta_j - \pi)$ and $- \sum_{j=1}^N \log 2|\sin \theta_j/2|$ are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem as $N \to \infty$.