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 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(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.
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