Mathematics Department Colloquium: Atoms of free convolutions and their relevance to random matrices

Colloquium | October 17 | 4:10-5 p.m. | 60 Evans Hall

 Hari Bercovici, Indiana University Bloomington

 Department of Mathematics

It has been known for some time that the free convolution of two nontrivial probability measures on the real line has few point masses. In fact, every point mass of the convolution is uniquely written as the sum of two point masses of the original measures, and the two points in question are obtained as boundary values of the analytic subordination functions that arise in this context. This is easily interpreted in terms of sums of independent sufficiently symmetric large random matrices. Subordination is also useful in determining outlying eigenvalues of "spiked" matrix models. The consideration of more complicated functions of two (or more) independent random matrices requires the free convolution of operator-valued probability distributions and the study of the "point masses" of such distributions. We will discuss the background of these questions as well as some progress coming from joint work with S. Belinschi and W. H. Liu.