CLT for spectra of submatrices of Wigner random matrices

TL;DR

This paper establishes a central limit theorem for the spectra of submatrices of Wigner matrices, revealing their Gaussian fluctuations and connections to Gaussian Free Fields under certain moment conditions.

Contribution

It introduces a CLT for spectra of submatrices of Wigner matrices and links the limiting process to Gaussian Free Fields when off-diagonal moments are GOE/GUE-like.

Findings

Spectra of submatrices follow a Gaussian distribution in the limit.

Limiting process is a collection of correlated Gaussian Free Fields.

Results apply to real symmetric and Hermitian Wigner matrices.

Abstract

We prove a CLT for spectra of submatrices of real symmetric and Hermitian Wigner matrices. We show that if in the standard normalization the fourth moment of the off-digonal entries is GOE/GUE-like then the limiting Gaussian process can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian Free Fields.