Free Central Limit Theorem, Semicircular Element, and Dyck Paths
Date:
I cooperated with undergraduate student Yujun Che, and received generous guidance from graduate student Sam McKeown. We primarily use the book Lectures on the Combinatorics of Free Probability.
Description: Free probability is a field which attempts to apply ideas from probability to more abstract settings, especially where the variables don’t commute. In these settings we might not even be able to talk about ‘probabilities’, but thinking about expectations and distributions (suitably translated) can still tell us a lot. For example, it can tell us what the eigenvalues of a random matrix look like. Some funny things happen in the translation, like the role of the bell curve being taken by the semicircle (see Wigner’s Semicircle Law for a related result). The material involves an interplay between things like operator algebras, combinatorics and probability, and would be interesting mostly to students who enjoy probability and didn’t mind taking abstract algebra.