Abstract

I present a universal fluctuation formula for linear statistics on random matrices. Given two linear statistics A = \sum_j a(λ_j) and B = \sum_j b(λ_j) on the N eigenvalues λ of a one-cut β-ensemble of N x N random matrices, I present a formula that gives the covariance Cov(A, B) in the limit N → ∞. 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, one recovers in some special cases the classical variance formulas by Beenakker and Dyson-Mehta. I provide two applications - the joint statistics of conductance and shot noise in ideal chaotic cavities, and a unified fluctuation relation for traces of powers of random matrices (related to enumeration problems in combinatorics).
 
References:
 
[1] F. D. Cunden and PV, Phys. Rev. Lett. 113, 070202 (2014) 
[2] F. D. Cunden, F. Mezzadri and PV, [arXiv:1504.03526] (2015)