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 $A=\sum _{i}a({\lambda }_i)$ and $B=\sum _{i}b({\lambda }_i)$ to leading order for $N\to \infty$, where $\{{\lambda }_i\}$ are the $N$ real eigenvalues of a general one-cut random-matrix model with Dyson index $\beta$. The formula, carrying the universal $1/\beta$ prefactor, depends on the random-matrix ensemble only through the edge points $[{\lambda }_{-},{\lambda }_{+}]$ 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.

Figure 1

Covariance of the dimensionless conductance $G$ and shot noise $P$ [Eq. (7)] as a function of $\alpha$. The dashed blue ($\beta =1$) and the solid black ($\beta =2$) lines are the analytical results (7), in very good agreement with the numerical diagonalization of ${10}^4$ Jacobi matrices of size $N=30$ (points).

Figure 2

The covariance $\mathrm{cov}(\mathrm{Tr}{\mathcal{W}}^n,\mathrm{Tr}{\mathcal{W}}^m)$ as a function of $m/n$, where $\mathcal{W}$ is a complex Wishart matrix $(\beta =2)$. For the simulation, we sampled ${10}^4$ $\mathcal{W}$ matrices of size $N=800$. Main: In the simulation (circles), $n$ is fixed to 50 and $m$ varies. The solid curve is Eq. (8). Inset: The numerical simulations (squares) show the convergence of the rescaled variance to the limit value ${\lambda }_{+}^2=16$.