Application de la r\'ecurrence topologique aux int\'egrales de matrices al\'eatoires et aux syst\`emes int\'egrables

TL;DR

This paper explores the applications of topological recursion to random matrix integrals and integrable systems, demonstrating its effectiveness in asymptotic expansions and connections to Painlevé equations.

Contribution

It provides a comprehensive review of topological recursion applications in matrix integrals and integrable systems, including new insights into asymptotic analysis and universality phenomena.

Findings

Topological recursion recovers all sub-leading orders in matrix integral asymptotics.

Method extends to integrals with hard edges, unitary matrices, and β-ensembles.

Explicit connection established between topological recursion and Painlevé equations.

Abstract

The goal of this "Habilitation \`a diriger des recherches" is to present two different applications, namely computations of certain partition functions in probability and applications to integrable systems, of the topological recursion developed by B. Eynard and N. Orantin in 2007. Since its creation, the range of applications of the topological recursion has been growing and many results in different fields have been obtained. The first aspect that I will develop deals with the historical domain of the topological recursion: random matrix integrals. I will review the formalism of the topological recursion as well as how it can be used to obtain asymptotic $\frac{1}{N}$ series expansion of various matrix integrals. In particular, a key feature of the topological recursion is that it can recover from the leading order of the asymptotic all sub-leading orders with elementary computations. This method is particularly well known and fruitful in the case of hermitian matrix integrals, but I will also show that the general method can be used to cover integrals with hard edges, integrals over unitary matrices and much more. In the end, I will also briefly mention the generalization to $\beta$-ensembles. In a second chapter, I will review the connection between the topological recursion and the study of integrable systems having a Lax pair representation. Most of the results presented there will be illustrated by the case of the famous six Painlev\'e equations. Though the formalism used in this chapter may look completely disconnected from the previous one, it is well known that the local statistics of eigenvalues in random matrix theory exhibit a universality phenomenon and that the encountered universal systems are precisely driven by some solutions of the Painlev\'{e} equations. As I will show, the connection can be made very explicit with the topological recursion formalism.