Universal covariance formula for linear statistics on random matrices

Abstract

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 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.

5 pages, 2 figures. This arXiv version: minor typos fixed in Table I and at p.3

References in corpus (7)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Systematic approach to statistics of conductance and shot-noise in chaotic cavities
• Nonlinear statistics of quantum transport in chaotic cavities
• Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry
• Full counting statistics of chaotic cavities with many open channels
• Statistics of thermal to shot noise crossover in chaotic cavities

Cited by in corpus (15)

• Wigner time delay and related concepts -- Application to transport in coherent conductors
• Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations
• Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices
• Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
• Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities
• Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel
• Joint statistics of quantum transport in chaotic cavities
• A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices
• An exact formula for the variance of linear statistics in the one-dimensional jellium mode
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices
• Conductance and thermopower fluctuations in interacting quantum dots
• Pair correlation function of the one-dimensional Riesz gas
• Full counting statistics of 1d short-range Riesz gases in confinement
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Large deviations of spread measures for Gaussian matrices