# The matrix Dyson equation and its applications for random matrices

## TL;DR

This paper introduces recent techniques centered around the Matrix Dyson Equation (MDE) to analyze local spectral universality in a broad class of random matrices, including those with correlated or non-identically distributed entries.

## Contribution

It extends existing methods by focusing on the stability analysis of the MDE for generalized random matrices, broadening the scope of spectral universality proofs.

## Key findings

- Stability properties of the MDE are crucial for understanding spectral behavior.
- The techniques handle matrices with correlated and non-identically distributed entries.
- The approach generalizes previous results on Wigner matrices.

## Abstract

These lecture notes are a concise introduction of recent techniques to prove local spectral universality for a large class of random matrices. The general strategy is presented following the recent book with H.T. Yau. We extend the scope of this book by focusing on new techniques developed to deal with generalizations of Wigner matrices that allow for non-identically distributed entries and even for correlated entries. This requires to analyze a system of nonlinear equations, or more generally a nonlinear matrix equation called the Matrix Dyson Equation (MDE). We demonstrate that stability properties of the MDE play a central role in random matrix theory. The analysis of MDE is based upon joint works with J. Alt, O. Ajanki, D. Schr\"oder and T. Kr\"uger that are supported by the ERC Advanced Grant, RANMAT 338804 of the European Research Council. The lecture notes were written for the 27th Annual PCMI Summer Session on Random Matrices held in 2017. The current edited version will appear in the IAS/Park City Mathematics Series, Vol. 26.