Universality laws for random matrices via exchangeable counterparts

TL;DR

This paper provides a simpler proof of universality laws for random matrices, showing that their spectral statistics resemble those of Gaussian matrices, using exchangeable counterparts.

Contribution

It introduces an elementary proof technique for universality laws in random matrix theory based on exchangeable counterparts.

Findings

Spectral statistics of independent sums of random matrices match Gaussian matrices.

New proof method simplifies understanding of universality laws.

Supports broader application of universality principles in random matrix analysis.

Abstract

Recently, Brailovskaya & van Handel (GAFA, 2024) established a suite of nonasymptotic universality laws which demonstrate that the spectral statistics of an independent sum of random matrices mirror the spectral statistics of a Gaussian random matrix with the same first- and second-order moments. This paper develops a more elementary proof of their main results by means of a new implementation of the method of exchangeable counterparts.