Stability of the Matrix Dyson Equation and Random Matrices with Correlations

TL;DR

This paper establishes local laws and universality for eigenvalue statistics of correlated random matrices, using a novel stability analysis of the matrix Dyson equation to handle general correlation structures.

Contribution

It introduces a detailed stability analysis of the matrix Dyson equation for correlated matrices, extending universality results to broader correlation types.

Findings

Proves local laws for resolvents of correlated matrices.

Establishes universality of eigenvalue statistics in the bulk.

Handles correlations with fast decay and general form.

Abstract

We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.