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 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 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 matrix with i.i.d. Gaussian entries
