Proof Methods in Random Matrix Theory

TL;DR

This survey introduces the method of moments and the Stieltjes transform method in random matrix theory, demonstrating their application to fundamental laws like the semicircle and Marchenko-Pastur laws.

Contribution

It thoroughly develops and pedagogically presents two proof methods in random matrix theory, making these techniques accessible for learners with measure-theoretic probability background.

Findings

Derivation of the semicircle law using the methods

Derivation of the Marchenko-Pastur law using the methods

Clear pedagogical presentation of proof techniques

Abstract

In this survey article, we give an introduction to two methods of proof in random matrix theory: The method of moments and the Stieltjes transform method. We thoroughly develop these methods and apply them to show both the semicircle law and the Marchenko-Pastur law for random matrices with independent entries. The material is presented in a pedagogical manner and is suitable for anyone who has followed a course in measure-theoretic probability theory.