Information, Geometry, and Physics Seminar

Free probability is a theory of random variables which do not commute under multiplication, which can be realized as operators on a Hilbert space. Voiculescu showed that free probability describes the large N behavior of independent random N x N matrices in many situations. This talk discusses the analog of optimal transport theory for free probability, as well as the large N behavior of optimal transport for certain random matrix models. In the free setting, there are many obstructions to optimal transport that don't exist in the classical setting. However, there is still an analog of Monge-Kantorovich duality. Moreover, for certain classes of non-commutative random variables (those associated to free Gibbs laws), we have a very good understanding of the optimal transport maps and how they arise from the random matrix models in the large N limit.