A unified fluctuation formula for one-cut β-ensembles of random matrices

Abstract

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut β-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut β-ensemble. As particular cases of the main result we consider the classical β-Gaussian, β-Wishart and β-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian unitary ensemble and the enumeration of planar maps.