Random matrices

TL;DR

This paper offers a comprehensive introduction to random matrices, focusing on three main approaches—Coulomb gas, loop equations, and orthogonal polynomials—and their geometric interpretations, with applications in counting surfaces and integrals.

Contribution

It presents a unified, self-contained overview of key methods in random matrix theory, emphasizing their geometric and algebraic structures, and connecting them to combinatorial and integral computations.

Findings

Detailed explanation of Coulomb gas and algebraic geometry in random matrices

Solution of loop equations via topological recursion

Relation between orthogonal polynomials and integrable systems

Abstract

We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.