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: TBA
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.
The course will not have an official textbook.
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
Course Content: we (plan to) cover the following topics :
- 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
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
2. week (1/31, 2/2): Wigner's
semicircle law cont., Hoffman-Wielandt lemma, cutoff argument for
removing the high moment condition, Stieltjes transform
Histogram of the eigenvalues of a 1000X1000 symmetric matrix with
i.i.d. standard Gaussian entries:
Eigenvalues of a 1000X1000 matrix with i.i.d. Gaussian entries
