The Edge Universality of Correlated Matrices

TL;DR

This paper proves the universality of extreme eigenvalues for correlated Gaussian matrices with power law decay, using local laws, eigenvalue concentration, and Dyson-Brownian motion techniques.

Contribution

It establishes edge universality for correlated matrices with power law decay, extending previous results to a broader class of random matrices.

Findings

Proved local laws for correlated matrices with power law decay.

Established eigenvalue concentration around the spectral edge.

Demonstrated universality of extreme eigenvalues using Dyson-Brownian motion.

Abstract

We consider a Gaussian random matrix with correlated entries that have a power law decay of order $d>2$ and prove universality for the extreme eigenvalues. A local law is proved using the self-consistent equation combined with a decomposition of the matrix. This local law along with concentration of eigenvalues around the edge allows us to get an bound for extreme eigenvalues. Using a recent result of the Dyson-Brownian motion, we prove universality of extreme eigenvalues.