Rutgers Discrete Mathematics Seminar

Title: How many real eigenvalues does a random matrix have?

Speaker: Van Vu, Yale

Date: Tuesday, April 2, 2013 2:00pm

Location: Hill Center, Room 124, Rutgers University, Busch Campus, Piscataway, NJ


Let M_n be a matrix whose entries are iid random variables with mean 0 and variance 1. As M_n is not symmetric, one expects that most of its eigenvalues are complex. In fact, the only case when it is clear that M_n should have a real eigenvalue is when n is odd. (Personally, I fail to see any other reason.) However, in 1995, Edelman et. al. proved that if the entries are gaussian, then in expectation there are about cn^{1/2} real eigenvalues. Later, Forrester et. al. showed that the variance (of the number of real eigenvalues) is also of order n^{1/2}. Both proofs rely heavily on properties of the gaussian distribution and cannot be extended to any other distribution. In this talk, we are going to show that one can obtain similar estimates for many other model of random matrices, by attacking a much more general problem about universality of local statistics of eigenvalues M_n. While universality is a basic problem in mathematical physics and probability, our leading idea is combinatorial and quite different from approaches used for symmetric matrices. If time allows, we will discuss several open questions. (Joint work with Terence Tao.)