Application de la récurrence topologique aux intégrales de matrices aléatoires et aux systèmes intégrables

The goal of this “Habilitation à 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. In relation with these applications, the formalism of the topological recursion has been precised and stabilized. Thus it appears that it is the right time to present an up to date review of the possibilities offered by the topological recursion in both probability and integrable systems. Since the scope of the applications of the topological recursion is quite large, I have decided to focus only on two important subjects in relation to my research activities even though it does not cover the full extent of my research. The first aspect that I will develop in this document 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 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 (known in the field as a “spectral curve”) 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 β-ensembles and its challenges. 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 Painlevé 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 Painlevé equations. As I will show, the connection can be made very explicit with the topological recursion formalism. The use of the topological recursion in integrable systems is developing rapidly at the moment and many results are expected in the next few years. In particular, the construction of a “quantum” version of the spectral curve in relation with integrable systems is presently an open and hot topic. At the border between probability and integrable systems, the topological recursion with the loop equation method is a very effective tool in theoretical physics, topological string theory and conformal field theory in dimension 2. With a wide generalization potential and a quite customizable formalism, it certainly provides an excellent tool in all those fields.