**Spring 2012**

**Meetings:** TR 9:30-10:45, Van Vleck B129

**Instructor:** Benedek Valkó

**Office:** 409 Van Vleck

**Phone:** 263-2782

**Email:** valko at math dot wisc dot edu

**Office hours:** Tu 11-12, 2:30-3:30 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.

**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,
beta-ensembles). 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.

A couple of useful references:

- M. Mehta: Random Matrices
- P.A. Deift: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach.
- P. Forrester: Log-gases and Random matrices
- G. Anderson, A. Guionnet and O. Zeitouni: An Introduction to
Random Matrices

available from the O. Zeitouni's webpage - Z. Bai, J. W. Silverstein: Spectral Analysis of Large Dimensional Random Matrices
- 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 - Lecture notes from other random matrix courses

- Terence Tao
- Fraydoun Rezakhanlou
- Manjunath Krishnapur
- Math
833 - 2009 Fall (notes taken by students)

- Wigner's semicircle law
- Exact computation of joint eigenvalue densities (Gaussian ensembles, Ginibre ensemble)
- Tridiagonal representation of the Gaussian beta-ensemble
- Bulk and edge 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

Covered topics:

1. week (1/24, 1/26): Empirical spectral measure, local and global limits, various random matrix ensembles, Wigner's semicircle law with the moment method (based mostly on the notes of Math 833 - 2009 Fall)

2. week (1/31, 2/2): Wigner's semicircle law cont., Hoffman-Wielandt lemma, cutoff argument for removing the high moment condition, Stieltjes transform

3. week (2/7, 2/9): Stieltjes transform method for proving the semicircle law (based mostly on Manjunath Krishnapur's notes)

4. week (2/14, 2/16): Exact eigenvalue distributions (Gaussian ensembles, Haar unitary), tridiagonal representation of the Gaussian ensembles

5. week (2/21): Tridiagonal matrix representation for the Gaussian beta ensemble (based mostly on the notes of Math 833 - 2009 Fall)

6. week (2/28, 3/1): Determinantal formulas for ß=2, kernel functions for GUE, CUE, complex Ginibre ensemble, moment formulas

7. week (3/6, 3/8): Quick overview of properties of determinantal point processes, examples, expectation and variance for linear statistics and counting function for CUE, Costin-Lebowitz thm, Circular law, sine process limit from CUE

8. week (3/13, 3/15): gap probabilities as Fredholm determinants, Hermite polynomials/functions, formal proof of the bulk scaling limit of GUE using the harmonic oscillator, bulk limit at 0 using the Laplace method

9. week (3/20, 3/22): guest lectures: Laguerre hard edge limit (Diane Holcomb), eigenvalue and eigenfunction estimates using the Green's function (Jun Yin)

10. week (3/27, 3/29): Soft edge limit of the GUE, the method of steepest descent. Brief intro to stochastic integrals and the Ito-formula.

11. week (4/10, 4/12): Dyson's Brownian Motion, the stationary version, finding the joint eigenvalue density with DBM, Johansson's formula for the joint eigenvalue density of A+\eps GUE

12. week (4/17, 4/19): Soft edge limit of the Gaussian beta ensemble, Edelmann-Sutton conjecture, the stochastic Airy operator

13. week (4/24, 4/26): Ricatti transformation, SDE description of the Airy-ß process

14. week (5/1, 5/3): Bulk limit of the Gaussian beta ensemble, the Sine-beta process, SDE limit of the phase function