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

2005/04/08 by Bertrand Eynard, Eynard, Bertrand
Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Mathematics and Applications #Matrix Theory and Algorithms #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.math-ph/0504034

in french, latex, 80 pages

arxiv created 2005/04/08 · openalex publication_date 2005/04/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

Related

• Matrix biorthogonal polynomials in the unit circle: Riemann-Hilbert problem and matrix discrete Painleve II system
• Two Matrix Model, the Riemann Hypothesis and Master Matrix Obstruction
• Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis
• Aspects géométriques et intégrables des modèles de matrices aléatoires
• Duality of spectral curves arising in two-matrix models