Course 18177, Spring 2013
The focus of this course will be random matrices.
Random matrices first appeared in the work of Wishart to model data arrays.
It was introduced into the theoretical physics community by Wigner in the 1950s to describe the statistics of energy levels of heavy nuclei. In an independent development in the early 1970s, Montgomery conjectured that the spacings between the zeros of the Riemann zeta function on the critical line behaves like the eigenvalues of a random matrix. At about the same time, 't Hooft and Br\'ezin, Itzykson, Parisi and Zuber, showed that random matrix integrals are generating functions for the enumeration of graphs embedded into surfaces which are sorted by their genus (the so-called topological expansion). Since that time an extraordinary variety of
mathematical, physical and engineering systems have been related with Random Matrix Theory; it has emerged as an interdisciplinary scientific activity par excellence. It is nowadays recognized as one of the hottest topic in probability theory, with many mathematical breakthroughs and rigorous advances during the last twenty years.
In this course, we will study some of the most classical results concerning the spectrum of random matrices such as Wigner's theorem, concentration of measures, or stochastic processes and Dyson Brownian motion, as well as most recent ones including
heavy tailed matrices, and universality. We will also study topological expansion and
their applications in combinatorics. Considering several random matrices together, we will approach free probability theory and the combinatorics of colored graphs.
This course is a graduate course which will require no particular prerequisite, but some basics in linear algebra and probability theory. It will be independent but complimentary to A. Edelman course 18.338. We encourage students to follow both courses. Grading will be based on based on homeworks, projects or equivalents.
Office hours: Tuesday 2PM-4PM
Lecture notes:
Lectures notes on the global behaviour of the spectrum of Wigner matrices
Lectures notes on free probability and the enumeration of maps: last lectures
More details can be found in the book
"An introduction to Random Matrices" by Greg Anderson, Ofer Zeitouni and myself. It can be found on
Ofer Zeitouni's webpage
Homework 1, due March 7
Homework 2, due April 4
Homework 3, due May 6. This homework is optional.
Bibliography:
G. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices,
Cambridge Studies in Advanced Mathematics, vol 118, Cambridge University Press
ZD. Bai and J. Silverstein, Jack W., Spectral analysis of large dimensional random matrices, Springer Series in Statistics, Springer, NY 2010
A. Guionnet, Large random matrices: lectures on macroscopic asymptotics, Lecture Notes in Mathematics, Vol. 1957, Lectures from the 36th Probability Summer School held in Saint-Flour, 2006
T. Tao, Topics in random matrix theory, Graduate Studies in Mathematics, 132,
American Mathematical Society, Providence, RI, 2012.