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Universal covariance formula for linear statistics on random matrices.

Authors
  • Cunden, Fabio Deelan
  • Vivo, Pierpaolo
Type
Published Article
Journal
Physical Review Letters
Publisher
American Physical Society
Publication Date
Aug 15, 2014
Volume
113
Issue
7
Pages
70202–70202
Identifiers
PMID: 25170690
Source
Medline
License
Unknown

Abstract

We derive an analytical formula for the covariance cov(A,B) of two smooth linear statistics A=[under ∑]ia(λ_{i}) and B=[under ∑]ib(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.

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