turkmath.org

Free probability, introduced by Voiculescu in the early 1980's is a general method to study asymptotic statistics of random matrices. It also provides a parallel theory of probability with the notion of independence replaced by so called 'freeness'. Research over the the last three decades has shown the existence of 'free' analogues of results ranging from central limit theorems to the existence of Brownian motion to De Finetti type theorems in free probability. Freeness however does not exist in finite dimensions and as such free probability is inherently qualitative in the results it yields for random matrices. In 2015, Adam Marcus proposed a theory called 'finite free probability' that seems capable of providing non-asymptotic results. There are several questions that are wide open in this new setting and I will mention some current work of mine that seems to clarify at least one of them.