Correlated Random Matrices: Band Rigidity and Edge Universality

TL;DR

This paper proves that a broad class of correlated Wigner matrices exhibit universal eigenvalue behavior at the edges of their spectra, demonstrating band rigidity and extending universality results.

Contribution

It establishes edge universality and band rigidity for correlated Wigner matrices with arbitrary expectations, including internal edges, which was not previously known.

Findings

Edge universality holds for correlated Wigner matrices.

Band rigidity excludes eigenvalue mismatches near edges.

Results apply to both real symmetric and complex Hermitian matrices.

Abstract

We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models.