Central Limit Theorem for Linear Eigenvalue Statistics for Submatrices of Wigner Random Matrices

TL;DR

This paper establishes a Central Limit Theorem for linear eigenvalue statistics of submatrices of Wigner matrices, linking the covariance to correlated Gaussian Free Fields, under smooth test functions.

Contribution

It extends CLT results to submatrices of Wigner matrices and connects the covariance structure to Gaussian Free Fields.

Findings

Proves CLT for eigenvalue statistics of submatrices.

Connects covariance structure to Gaussian Free Fields.

Applicable under smooth test functions.

Abstract

We prove the Central Limit Theorem for finite-dimensional vectors of linear eigenvalue statistics of submatrices of Wigner random matrices under the assumption that test functions are sufficiently smooth. We connect the asymptotic covariance to a family of correlated Gaussian Free Fields.