Universality and Random Matrices
Few weeks ago, I gave a survey talk at IAS on random matrices. The topic is the Universality Phenomenon (click on the link to see an informal discussion by my coauthor T. Tao) which has been one of the main foci of my research in the last 10 years.
We focus on random matrices with iid entries. The idea is that limiting distributions concerning eigenvalues should not depend very much on the distribution of the entries (in other words, these distributions are universal). This contains, as a subtopic, the universality problem coming from the math physics literature, in which universality usually refers to the universality of the correlation functions.
A basic example of universality results is the Central Limit Theorem, which asserts that the limiting distribution of the sum , normalized by , is gaussian, where are idd random variables with mean 0 and variance 1. It does not matter if we take to be gaussian itself, or to be, say, the random variable. The hardness of universality problems in random matrix theory lies in the fact that most spectral parameters are defined implicitly in term of the entries, in contrast to the central limit theorem, where the function is very explicit (simply the normalized sum of the atom variables). For example, it is easy to define the middle eigenvalue of a large hermitian matrix, but there is no reasonable way to write down this quantity in an explicit form in term of the entries. Without such a form, there is simply nothing to compute with. However, we believe (and can prove under certain conditions) that the limiting distribution of this middle eigenvalue (after a proper normalization) is gaussian, regardless the nature of the distribution of the entries.
The talk (see this link) surveys recent developments in understanding universality, with many open questions, some of which are extremely simple to state. Here is an example. Let be a random (non-symmetric) matrix of size whose entries are random with probability . Prove that with probability tending to one, as tends to infinity, the matrix has at least 2 real eigenvalues.