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We derive an analytical formula for the covariance $\mathrm{Cov}(A,B)$ of twosmooth linear statistics $A=\sum_i a(\lambda_i)$ and $B=\sum_i b(\lambda_i)$ toleading order for $N\to\infty$, where $\{\lambda_i\}$ are the $N$ realeigenvalues of a general one-cut random-matrix model with Dyson index $\beta$.The formula, carrying the universal $1/\beta$ prefactor, depends on therandom-matrix ensemble only through the edge points $[\lambda_-,\lambda_+]$ ofthe limiting spectral density. For $A=B$, we recover in some special cases theclassical variance formulas by Beenakker and Dyson-Mehta, clarifying therespective ranges of applicability. Some choices of $a(x)$ and $b(x)$ lead to astriking \emph{decorrelation} of the corresponding linear statistics. Weprovide two applications - the joint statistics of conductance and shot noisein ideal chaotic cavities, and some new fluctuation relations for traces ofpowers of random matrices.

Universal covariance formula for linear statistics on random matrices