Random matrices

TL;DR

The text offers a self-contained entry into random matrix theory, articulating three central approaches—Coulomb gas with its spectral-curve geometry, loop equations and topological recursion, and orthogonal polynomials with ties to integrable systems. It emphasizes the spectral curve as a unifying geometric object encoding universal large-N behavior and the full perturbative and some nonperturbative structure of matrix models. It also discusses formal versus convergent matrix integrals, ribbon graphs and the combinatorics of maps, and peripheral subjects such as angular integrals and matrix models for knot theory. The work aims to equip theorists and mathematicians with concrete tools, from contour-deformation and Riemann–Hilbert methods to algebraic-geometry constructions, to analyze both macroscopic and microscopic spectral properties and their universal manifestations. Finally, it highlights connections to integrable systems, topological string theory, and random surfaces, underlining the broad relevance of spectral-curve methods in mathematical physics.

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.