The 2-matrix model, biorthogonal polynomials, Riemann-Hilbert problem, and algebraic geometry

Abstract

This preprint is the introduction of my habilitation thesis for Paris7 university. It is a sumary of a collection of works on the 2 matrix model. In an introduction, 3 different and unequivalent definitions of matrix models are given (convergent model, model with fixed filling fractions on contours, and formal model). Then, a sumary of properties of differential systems satisfied by biorthogonal polynomials, in particular spectral duality and Riemann-Hilbert problem. Then, a section on loop equations and algebraic geometry formulation of the large N expansion. Then, a conjecture for the asymptotics of biorthogonal polynomials.

in french, latex, 80 pages

References in corpus (5)

• All genus correlation functions for the hermitian 1-matrix model
• Rational Theories of 2D Gravity from the Two-Matrix Model
• Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions
• Bilinear semi-classical moment functionals and their integral representation
• Second and Third Order Observables of the Two-Matrix Model

Cited by in corpus (6)

• Algebraic methods in random matrices and enumerative geometry
• Universality in the two matrix model with a monomial quartic and a general even polynomial potential
• Mixed correlation function and spectral curve for the 2-matrix model
• Asymptotic analysis of the two matrix model with a quartic potential
• Some comments on the local curve (B-side)
• The Hermitian two matrix model with an even quartic potential