Asymptotic eigenvalue distribution of polynomials in independent random matrices and free probability theory
Free probability theory was invented around 1985 by D. Voiculescu,
originally as a tool for the theory of operator algebras, but later on,
it became prominent also for its deep connections to random matrix
theory. This is due to the remarkable fact that many types of
independent random matrices become asymptotically free if their
dimension tends to infinity.
In my talk, I will explain how free probability provides in such cases
an effective algorithm for the calculation of the asymptotic eigenvalue
distribution of arbitrary polynomials in the considered random matrices.
Moreover, I will discuss what free probability can say about the
regularity of those limiting distributions.