Preface

In the winter of 2010, I taught a topics graduate course on random matrix

theory, the lecture notes of which then formed the basis for this text. This

course was inspired by recent developments in the subject, particularly with

regard to the rigorous demonstration of universal laws for eigenvalue spacing

distributions of Wigner matrices (see the recent survey [Gu2009b]). This

course does not directly discuss these laws, but instead focuses on more

foundational topics in random matrix theory upon which the most recent

work has been based. For instance, the first part of the course is devoted

to basic probabilistic tools such as concentration of measure and the cen-

tral limit theorem, which are then used to establish basic results in random

matrix theory, such as the Wigner semicircle law on the bulk distribution of

eigenvalues of a Wigner random matrix, or the circular law on the distribu-

tion of eigenvalues of an iid matrix. Other fundamental methods, such as

free probability, the theory of determinantal processes, and the method of

resolvents, are also covered in the course.

This text begins in Chapter 1 with a review of the aspects of prob-

ability theory and linear algebra needed for the topics of discussion, but

assumes some existing familiarity with both topics, as well as a first-year

graduate-level understanding of measure theory (as covered for instance in

my books [Ta2011, Ta2010]). If this text is used to give a graduate course,

then Chapter 1 can largely be assigned as reading material (or reviewed as

necessary), with the lectures then beginning with Section 2.1.

The core of the book is Chapter 2. While the focus of this chapter is

ostensibly on random matrices, the first two sections of this chapter focus

more on random scalar variables, in particular, discussing extensively the

concentration of measure phenomenon and the central limit theorem in this

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