The theory of random matrices is an amazingly rich topic in mathematics.
Beside being interesting in its own right, random matrices play a fundamental role
in various areas such as statistics, mathematical physics, combinatorics, theoretical
computer science, number theory and numerical analysis, to mention a few. A
famous example is the work of the physicist Eugene Wigner, who used the spectrum
of random matrices to model energy levels of atoms, and consequently discovered
the fundamental semi-circle law which describes the limiting distribution of the
eigenvalues of a random hermitian matrix.
Special random matrices models where the entries are iid complex or real gauss-
ian random variables (GUE, GOE or Wishart) have been studied in detail. How-
ever, much less was known about general models, as the above mentioned study
relies very heavily on properties of the gaussian distribution. In the last ten years
or so, we have witnessed considerable progresses on several fundamental problems
concerning general models, such as the Circular law conjecture or Universality con-
jectures. More importantly, these new results are proved using novel and robust
approaches which seem to be applicable to many other problems. Surprising con-
nections to the emerging field of free probability have also been made and fortified.
Equally surprising is the discovery that many practical tricks for numerical prob-
lems (to make the computation of eigenvalues faster or more reliable, say) can also
be used as powerful theoretical tools to study spectral limits.
Another area where we see rapid progressions is the theory of computing and
applications (which includes numerical analysis, theoretical computer science, ma-
chine learning and data analysis). Here properties of random matrices have been
used for the purpose of designing and analyzing practical algorithms. As already
realized by von Neumann and Goldstine at the dawn of the computer era, bounds
on the condition number of large random matrices would play a central role in a
vast number of numerical problems. Their questions were posed 70 years ago, but
effective ways to estimate this number have only been found in recent years. As
a model for random noise/error, random matrices enter all problems concerning
large data, perhaps one of the most talked about subjects in applied science in
recent years. Today, random matrices are studied not only for their own mathe-
matical beauty, but also for a very real purpose of making digital images sharper
or computer networks more reliable. These new goals have motivated new lines of
research, such as non-asymptotic or large deviation theory for random matrices.
This volume contains surveys by leading researchers in the field, written in
introductory style to quickly provide a broad picture about this fascinating and