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=∑ᵢa(λᵢ) and B=∑ᵢb(λᵢ) to leading order for N→∞, where λᵢ 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.