setting. These facts will be used repeatedly when we then turn our attention
to random matrices, and also many of the proof techniques used in the scalar
setting (such as the moment method) can be adapted to the matrix context.
Several of the key results in this chapter are developed through the exercises,
and the book is designed for a student who is willing to work through these
exercises as an integral part of understanding the topics covered here.
The material in Chapter 3 is related to the main topics of this text, but
is optional reading (although the material on Dyson Brownian motion from
Section 3.1 is referenced several times in the main text).
This text is not intended as a comprehensive introduction to random
matrix theory, which is by now a vast subject. For instance, only a small
amount of attention is given to the important topic of invariant matrix
ensembles, and we do not discuss connections between random matrix theory
and number theory, or to physics. For these topics we refer the reader
to other texts such as [AnGuZi2010], [DeGi2007], [De1999], [Fo2010],
[Me2004]. We hope, however, that this text can serve as a foundation for
the reader to then tackle these more advanced texts.
I am greatly indebted to my students of the course on which this text
was based, as well as many further commenters on my blog, including Ah-
met Arivan, Joshua Batson, Florent Benaych-Georges, Sivaraman Balakr-
ishnan, Alex Bloemendal, Kaihua Cai, Andres Caicedo, Emmanuel Cand´es,
J´ erˆ ome Chauvet, Brian Davies, Ben Golub, Stephen Heilman, John Jiang, Li
Jing, Rowan Killip, Sungjin Kim, Allen Knutson, Greg Kuperberg, Choong-
bum Lee, George Lowther, Rafe Mazzeo, Mark Meckes, William Meyerson,
Samuel Monnier, Andreas Naive, Srivatsan Narayanan, Giovanni Peccati,
Leonid Petrov, Anand Rajagopalan, Brian Simanek, James Smith, Mads
Sørensen, David Speyer, Ambuj Tewari, Luca Trevisan, Qiaochu Yuan, and
several anonymous contributors, for comments and corrections. These com-
ments, as well as the original lecture notes for this course, can be viewed
The author is supported by a grant from the MacArthur Foundation, by
NSF grant DMS-0649473, and by the NSF Waterman award.
Last, but not least, I am indebted to my co-authors Emmanuel Cand´es
and Van Vu, for introducing me to the fascinating world of random matrix
theory, and to the anonymous referees of this text for valuable feedback and