**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 16-17 (General beta bulk limit, recent developments)

Homework problems (if you are interested...)