Free Probability and Brown measure

Abstract: In 1980, Voiculescu introduced free probability theory, which lets us study non-commutative random variables, such as random matrices. In particular, in the free probability CLT the Gaussian limit is replaced by a semi-circular limit, which implies the semi-circular law. So far, we have been computing interesting quantities ( such as the empirical eigenvalue distribution) associated to an NxN matrix and taken N to infinity. It's reasonable to ask if there is a limiting object that captures the limit of the entire matrix model. We will talk about what that limiting object is in the context of free probability and how is relates to the Brown measure.