Gaussian fluctuations for random matrices with correlated entries

TL;DR

This paper proves that for certain correlated random matrices with non-Gaussian entries, the centered traces of polynomials converge to a Gaussian process, extending classical results to correlated settings.

Contribution

It establishes Gaussian fluctuations for correlated random matrices with non-Gaussian entries, using combinatorial methods and Chebyshev polynomial basis.

Findings

Centered traces converge to a Gaussian process

Covariance matrix is diagonal in Chebyshev basis

Extension of Wigner's semicircle law to correlated entries

Abstract

For random matrix ensembles with non-gaussian matrix elements that may exhibit some correlations, it is shown that centered traces of polynomials in the matrix converge in distribution to a Gaussian process whose covariance matrix is diagonal in the basis of Chebyshev polynomials. The proof is combinatorial and adapts Wigner's argument showing the convergence of the density of states to the semicircle law.