Classical theory

The main goal of random matrix theory is to study the distribution of eigenvalues  and (to some extend) the eigenvectors. In what follows, we consider random Hermtian matrices with independent entries.

The big picture is as follows:

When the entries have gaussian distributions (GOE and GUE),  most limiting distributions can be obtained  by direct (but by no mean simple) calculation thanks to the explicit joint distribution  of the eigenvalues.

In other cases, one expects that THE SAME limiting distribution holds. This is known as the universality phenomenon in random matrix theory. Very often, numerical experiments gave strong support to this phenomenon (see this survey).

In the last few years, remarkable progresses have been made towards several central problems concerning Universality. In Jan 2013, I organize a mini-course at the annual meeting of the AMS (San Diego) in order to introduce these new progresses to general audience.

Some representative publications (together with discussions)

(1)  Tao and Vu, Random matrices: Universality of the Local statistics, Acta Math.

For discussion, see this post at Tao’s blog.

(2) Tao and Vu, Random matrices: Universality of the Local statistics of non-Hermtian matrices, submitted.

(3) Tao and Vu (with appendix by Krisnapur), Random matrices: Universality of the ESDs and the Circular Law, Annals of Prob.

One of the key ideas in these papers is that the distributions of the eigenvalues depend only the moments of the entries. A famous example of such phenomenon is the Central Limit theorem.  In this case, we study the sum of atom variables. However, the eigenvalues are are very difficult (and not even explicit) functions of the atom variables (entries). But somewhat surprisingly, the same phenomenon continues to hold.The first few moments of the entries decide everything. If one care about global distribution (such as in  (3)), then 2 moments suffice. For local distributions (such as in (1) and (2)), 4 is the magic number.