Math 833 - Random Matrices

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 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.

The course will not have an official textbook.
A couple of useful references:
Course Content: we (plan to) cover the following topics :
If time permits, we will also touch on the following topics: circular limit law, Dyson's Brownian motion, universality results from Erdős-Yau-... and Tao-Vu.

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

Histogram of the eigenvalues of a 1000X1000 symmetric matrix with i.i.d. standard Gaussian entries:

Eigenvalues of a 1000X1000 symmetric
      matrix with i.i.d. Gaussian entries

Eigenvalues of a 1000X1000 matrix with i.i.d. Gaussian entries

Eigenvalues of a 1000X1000 matrix with
      i.i.d. Gaussian entries