Universal Covariance Formula for Linear Statistics on Random Matrices

We derive an analytical formula for the covariance cov ( A , B ) of two smooth linear statistics A = ∑ i a ( λ i ) and B = ∑ i b ( λ i ) to leading order for N → ∞ , 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 by Dyson and Mehta, clarifying the respective ranges of applicability. Some choices of a ( x ) and b ( x ) lead to a striking 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.