Math 833 - Random Matrices

Fall 2009

Meetings: TR 9:30-10:45, Van Vleck B131
Instructor: Benedek Valkó
Office: 409 Van Vleck
Phone: 263-2782
Email: valko at math dot wisc dot edu
Office hours: Tuesday 13:30-14:30 or by appointment

I will use the class email list to send out corrections, announcements, please check your wisc.edu email from time to time.

Evaluation: Students taking the course for credit are expected to write up two or three lectures in consultation with the instructor. There will also be (non mandatory) homework exercises.

Course description:
The course is an introduction to random matrix theory. We will cover results on the asymptotic properties of various random matrix models (Wigner matrices, Gaussian ensembles, Dyson's beta-ensemble). We will investigate the limit of the empirical spectral measure both on a global and local scale.

Prerequisites: Basics in probability theory and linear algebra. Some knowledge of stochastic processes will also be helpful.

• M. Mehta: Random Matrices
• P.A. Deift: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach.
• P. Forrester: Log-gases and Random matrices (to appear)
  available from the author's webpage
• G. Anderson, A. Guionnet and O. Zeitouni: An Introduction to Random Matrices (to appear)
  available from the O. Zeitouni's webpage
• A. Guionnet: Large random matrices: lectures on macroscopic asymptotics. Lectures from the 36th Probability Summer School held in Saint-Flour, 2006.
  available from the author's webpage

• Wigner's semicircle law
• Marchenko-Pastur law for the spectral density of sample covariance matrices
• Joint eigenvalue density of certain classic random matrix ensembles (GOE, GUE)
• Tridiagonal representation of the Dyson's beta-ensemble
• Bulk scaling limit for the eigenvalue process of the Gaussian Unitary Ensemble
• Edge scaling limit of the beta-ensemble
• Bulk scaling limit of the beta-ensemble

• Lecture 1 (Examples for matrix models)
• Lectures 2-3 (Wigner's semicircle law)
• Lectures 2-5 (Wigner's semicircle law, general version, together with previous week's notes) 
• Lectures 6-7 (Marchenko-Pastur law) 
• Lectures 8-9 (Gaussian ensembles)
• Lectures 10-11 (Gaussian ensembles)
• Lectures 12-13 (Gaussian ensembles, local limits) 
• Lectures 14-15 (Bulk limit of GUE) 
• Lectures 16-17 (End of the Sine kernel proof, general beta edge scaling) 
• Lectures 18-19 (General beta edge scaling cont.)
• Lectures 20-22 (General beta edge, beta bulk ) 
• Lectures 23-24 (General beta bulk limit) 
• Lectures 25-26 (General beta bulk limit, recent developments)