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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