Ma 191c-sec1:  Introduction to Random Matrices (Spring 2016-17)

ANNOUNCEMENTS

Registration for Spring term opens Thursday, February 23, 2016.

This page was last updated:  .


COURSE DESCRIPTION

This course will provide an introduction to random matrix theory. We will first discuss relevant concentration inequalities and the moment method. Subsequently we will investigate the matrix norm of a random matrix and provide two proofs of Wigner's semi-circle law. After an introduction to free probability, we will focus on Gaussian ensembles and the edge of the spectrum.


PREREQUISITES

Foundational knowledge of Probability theory and Analysis

 

SCHEDULE

TR, 10:30 - 11:55, 257 Sloan.


INSTRUCTORS

Lukas Schimmer
HARRY BATEMAN RESEARCH INSTRUCTOR
Sloan 380
626-395-2891


TA's

There will be no TAs for this course.


OFFICE HOURS

By appointment via email


POLICIES

Grades

pass/fail

Based on final presentation and class attendance.

Homework Policy

n/a


TOPICS COVERED


TEXTBOOKS

[1]  T. Tao, 'Topics in Random Matrix Theory', Graduate Studies in Mathematics, 132. American Mathematical Society, Providence, RI, 2012. x+282 pp. ISBN: 978-0-8218-7430-1 

[2] G. W. Anderson, A. Guionnet and O. Zeitouni, 'An introduction to random matrices',  Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5 


LECTURE NOTES

Date Description Reference
04/04/17  Course overview, Universality, Large deviations, Moment method [1] pp. 55-63
04/06/17 Moment generating function, Truncation method, Weak/Strong law of
large numbers, Talagrand's concentration inequality
[1] pp. 63-73
04/11/17 Proof of Talagrand's concentration inequality, Distance to subspaces,
Random Matrix ensembles, Operator norm
[1] pp. 73-75, 78
[1] pp. 105-107
04/13/17 Epsilon net argument, Concentration of measure, 4th moment method [1] pp. 107-110
[1] pp. 114-117
04/18/17 k-th moment method, Leading order term, Dyck words [1] pp. 117-122
04/20/17 Weak Bai--Yin theorem (lower/upper bound), Catalan numbers [1] pp. 122-130
04/25/17 Strong Bai--Yin theorem, Empirical spectral distribution, modes of convergence [1] pp. 130-135
04/27/17 Wigner's semicircular law, Reductions, Proof of convergence in expectation
using the moment method
Levy's theorem, Carleman continuity theorem
[1] pp. 135-140
[2] pp. 10-15
[1] p. 83, 90
05/02/17 Proof of convergence in probability and almost surely, Stieltjes transform [1] pp. 140-146
[2] pp. 10-15
05/04/17 Proof of semicircular law using Stietljes transform,
Free probability
[1] pp. 146-152
[1] pp. 152-155
05/09/17 Non-commutative probability spaces, Stieltjes transform [1] pp. 155-164
05/11/17 Spectral theorem for bounded self-adjoint elements,
Wigner's semicirle law rephrased, Free independence
[1] pp. 164-174
05/16/17 Asymptotic freeness of Wigner matrices, Free central limit theorem [1] pp. 174-181
05/18/17 Student talk: Andrei Cosmin Pohoata
N. Alon, M. Krivelevich, V. Vu, On the concentration of eigenavalues
of random symmetric matrices
, Israel J. Math. 131 (2002), 259-267

Gausian Unitary Ensembles, Unitary invariance, Ginibre formula




[1] pp. 182-184
05/23/17 Proof of Ginibre formula, Ginibre formula for Gaussian ensembles,
Mean field approximation
[1] pp. 184-193
05/25/17 The determinantal form, Hermite polynomials, Gaudin-Mehta formula [1] pp. 194-196
05/30/17 Bulk asymptotics of the joint eigenvalue distribution, Sine kernel, Semiclassical Analysis [1] pp. 196-205
[1] pp. 259-261
[2] pp. 90-94
06/01/17 Edge asymptotics of the joint eigenvalue distribution, Airy kernel, Tracy--Widom law [1] pp. 261-265
[2] pp. 132-139
06/06/17 Student talks: Ruoqi Shen & Matthew Jin
Log-Sobolev inequality
A symmetrisation argument
 
06/08/17  Student talks: Karlming Chen & Jeffrey Gu
Linbdeberg swapping trick

R. Latała, Some estimates of norms of random matrices, Proc. Amer. Math. Soc. 133 (2005),
no. 5, 1273-1282
 

HOMEWORK

n/a


EXAMS

The examination will take place in the form of a student talk on a related topic. A list of possible topics will be provided but suggestions are welcome.


PRESENTATION TOPICS

From the textbooks:

Research papers: