Chain of matrices, loop equations and topological recursion

TL;DR

This paper explores the mathematical framework of matrix integrals, loop equations, and topological recursion, clarifying their roles in random matrix theory and applications like polynomial studies and surface enumeration.

Contribution

It analyzes the relation between perturbative and non-perturbative matrix integrals and discusses recursive solutions to loop equations for Hermitean matrices.

Findings

Clarifies different definitions of matrix integrals in applications.

Derives recursive solutions for loop equations in Hermitean matrix chains.

Connects matrix models with topological recursion techniques.

Abstract

Random matrices are used in fields as different as the study of multi-orthogonal polynomials or the enumeration of discrete surfaces. Both of them are based on the study of a matrix integral. However, this term can be confusing since the definition of a matrix integral in these two applications is not the same. These two definitions, perturbative and non-perturbative, are discussed in this chapter as well as their relation. The so-called loop equations satisfied by integrals over random matrices coupled in chain is discussed as well as their recursive solution in the perturbative case when the matrices are Hermitean.