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Universal covariance formula for linear statistics on random matrices

We derive an analytical formula for the covariance $\mathrm{Cov}(A,B)$ of two smooth linear statistics $A=\sum_i a(λ_i)$ and $B=\sum_i b(λ_i)$ to leading order for $N\to\infty$, where $\{λ_i\}$ are the $N$ real eigenvalues of a general one-cut random-matrix model with Dyson index $β$. The formula, carrying the universal $1/β$ prefactor, depends on the random-matrix ensemble only through the edge points $[λ_-,λ_+]$ of the limiting spectral density. For $A=B$, we recover in some special cases the classical variance formulas by Beenakker and Dyson-Mehta, clarifying the respective ranges of applicability. Some choices of $a(x)$ and $b(x)$ lead to a striking \emph{decorrelation} of the corresponding linear statistics. We provide two applications - the joint statistics of conductance and shot noise in ideal chaotic cavities, and some new fluctuation relations for traces of powers of random matrices.