Lectures on random matrix models. The Riemann-Hilbert approach

Abstract

This is a review of the Riemann-Hilbert approach to the large $N$ asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the Riemann-Hilbert approach to the large $N$ asymptotics of orthogonal polynomials and its applications to the problem of universality in random matrix models, the double scaling limits, the large $N$ asymptotics of the partition function, and random matrix models with external source.

84 pages, 23 figures, to appear in the CRM volume on "Random Matrices", Springer, 2008

References in corpus (21)

• Level-Spacing Distributions and the Airy Kernel
• Universal singularity at the closure of a gap in a random matrix theory
• Spectral form factor in a random matrix theory
• Level Spacing of Random Matrices in an External Source
• The Pearcey Process
• Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model
• Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem
• Random hermitian matrices in an external field
• Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices
• Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions
• Universal correlations for deterministic plus random Hamiltonians
• Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase
• Square ice, alternating sign matrices and classical orthogonal polynomials
• Partition functions for Matrix Models and Isomonodromic Tau functions
• Hermitian vs. Anti-Hermitian 1-Matrix Models and Their Hierarchies
• Symmetry Breaking in the Double-Well Hermitian Matrix Models
• Loop Equations and the Topological Phase of Multi-Cut Matrix Models
• Matrix Model Combinatorics: Applications to Folding and Coloring
• A concise expression for the ODE's of orthogonal polynomials
• On the proof of universality for orthogonal and symplectic ensembles in random matrix theory
• Multi-critical unitary random matrix ensembles and the general Painleve II equation

Cited by in corpus (10)

• Universality of the Pearcey process
• On the origins of Riemann-Hilbert problems in mathematics
• Asymptotics for a special solution to the second member of the Painleve I hierarchy
• Gaussian matrix model in an external field and non-intersecting Brownian motions
• On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials
• On the singular sector of the Hermitian random matrix model in the large N limit
• Ladder relations for a class of matrix valued orthogonal polynomials
• An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model
• How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave
• Aspects géométriques et intégrables des modèles de matrices aléatoires