A review of exact results for fluctuation formulas in random matrix theory

Abstract

Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often large $N$ universal forms for this correlation after smoothing, which results in particularly simple limiting formulas for the fluctuation of the linear statistics. We review these limiting formulas, derived in the simplest cases as corollaries of explicit knowledge of the truncated two-point correlation. One of the large $N$ limits is to scale the eigenvalues so that limiting support is compact, and the linear statistics vary on the scale of the support. This is a global scaling. The other, where a thermodynamic limit is first taken so that the spacing between eigenvalues is of order unity, and then a scale imposed on the test functions so they are slowly varying, is the bulk scaling. The latter was already identified as a probe of random matrix characteristics for quantum spectra in the pioneering work of Dyson and Mehta.