CHAPTER 1

Introduction

1.1. Objectives and Problems

1.1.1. Ensembles. As was indicated in the Preface, we will deal mostly with

spectral aspects of random matrix theory. One of the main subjects of this part of

the theory is the large-n asymptotic form of various spectral characteristics of n×n

matrices, whose probability distribution is given in terms of the matrix elements.

In other words, the goal of the theory is to "transfer" the probabilistic information

from matrix elements to eigenvalues and eigenvectors. Formulated in so general

a form, the goal of random matrix theory is similar to that of random operator

theory (see e.g. [396]), in particular the spectral theory of Schrödinger operators

with random potential. However, in the latter the emphasis is put on the analysis of

spectral types (pure point, absolutely continuous, etc.), i.e., in fact, on the spatial

behavior of eigenfunctions (solutions of corresponding differential or finite-difference

equations), while in the former we are mainly interested in the asymptotic behavior

of eigenvalues as n → ∞, although statistical properties of eigenvectors are also of

considerable interest for a number of applications.

The goal of the theory, seen from the point of view of an analyst, is the study

of integrals of the form

En

Φn(Mn)Pn(dMn),

where

• En is a set of n × n matrices, for instance

– real symmetric Sn,

– hermitian Hn,

– unitary Un, etc.;

• Φn is a function from En to R or C, which is often orthogonal or unitary

invariant. For example, in the case of Sn,

Φn(OnMnOn

T

) = Φn(Mn), ∀On ∈ O(n);

• Pn is a probability measure on En.

One is often interested in the asymptotic behavior of integrals as the size n of

matrices tends to infinity.

From the probabilistic point of view one is interested in the asymptotic prop-

erties of random variables of the form Φn(Mn), defined on the probability space

(En, Pn) and invariant in the above sense.

We will often call the sequence {(En, Pn)}n the random matrix ensemble or

simply a random matrix.

1

http://dx.doi.org/10.1090/surv/171/01