Limit theorems for linear eigenvalue statistics of overlapping matrices

TL;DR

This paper establishes limit theorems for linear eigenvalue statistics of overlapping Wigner and covariance matrices, revealing that Chebyshev polynomials diagonalize the covariance and depend only on the first two moments of entries.

Contribution

It introduces new limit theorems for eigenvalue statistics of overlapping matrices and shows the covariance structure simplifies with Chebyshev polynomials.

Findings

Covariance matrix is diagonalized by Chebyshev polynomials.

Covariance depends only on first two moments for high-degree polynomials.

Graph-theoretic approach links eigenvalue statistics to non-backtracking paths.

Abstract

The paper proves several limit theorems for linear eigenvalue statistics of overlapping Wigner and sample covariance matrices. It is shown that the covariance of the limiting multivariate Gaussian distribution is diagonalized by choosing the Chebyshev polynomials of the first kind as the basis for the test function space. The covariance of linear statistics for the Chebyshev polynomials of sufficiently high degree depends only on the first two moments of the matrix entries. Proofs are based on a graph-theoretic interpretation of the Chebyshev linear statistics as sums over non-backtracking cyclic paths