Finite N Fluctuation Formulas for Random Matrices

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.&#39;s for the linear statistics ${1 \over 2} \sum_{j=1}^N (θ_j - π)$ and $- \sum_{j=1}^N \log 2|\sin θ_j/2|$ 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 $N \to \infty$.