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Using a Coulomb gas approach, we compute the generating function of thecovariances of power traces for one-cut $\beta$-ensembles of random matrices inthe limit of large matrix size. This formula depends only on the support of thespectral density, and is therefore universal for a large class of models. Thisallows us to derive a closed-form expression for the limiting covariances of anarbitrary one-cut $\beta$-ensemble. As particular cases of the main result weconsider the classical $\beta$-Gaussian, $\beta$-Wishart and $\beta$-Jacobiensembles, for which we derive previously available results as well as new oneswithin a unified simple framework. We also discuss the connections between theproblem of trace fluctuations for the Gaussian Unitary Ensemble and theenumeration of planar maps.

A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices