Aspects g\'eom\'etriques et int\'egrables des mod\`eles de matrices al\'eatoires

TL;DR

This thesis explores the geometric and integrable properties of random matrix models, linking algebraic geometry, differential equations, and string theory applications, including universal limits and quantum algebraic structures.

Contribution

It introduces new connections between matrix models, integrable systems, and algebraic geometry, extending existing theories to non-hermitian models and quantum curves.

Findings

Universal Painlevé II equations near eigenvalue merging

Link between matrix models and Jimbo-Miwa-Ueno theory

Explicit computation of Gromov-Witten invariants via matrix models

Abstract

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of integrable systems. The variety of these applications shows why matrix models are important from a mathematical point of view. First, the thesis will focus on the study of the merging of two intervals of the eigenvalues density near a singular point. Specifically, we will show why this special limit gives universal equations from the Painlev\'e II hierarchy of integrable systems theory. Then, following the approach of (bi) orthogonal polynomials introduced by Mehta to compute partition functions, we will find Riemann-Hilbert and isomonodromic problems connected to matrix models, making the link with the theory of Jimbo, Miwa and Ueno. In particular, we will describe how the hermitian two-matrix models provide a degenerate case of Jimbo-Miwa-Ueno's theory that we will generalize in this context. Furthermore, the loop equations method, with its central notions of spectral curve and topological expansion, will lead to the symplectic invariants of algebraic geometry recently proposed by Eynard and Orantin. This last point will be generalized to the case of non-hermitian matrix models (arbitrary $\beta$) paving the way to "quantum algebraic geometry" and to the generalization of symplectic invariants to "quantum curves". Finally, this set up will be applied to combinatorics in the context of topological string theory, with the explicit computation of an hermitian random matrix model enumerating the Gromov-Witten invariants of a toric Calabi-Yau threefold.