Seminar on Free Probability
and Random Matrices

Fall 2019

This term we will be having a learning seminar Math 922 on Free Probability and Random Matrices from the book by Mingo & Speicher.

This is a continuation of the last talk in which I explained what Helgason proved in 1957 for random Fourier series on compact groups. His original proof was involved with detailed analysis on unitary groups and spheres, but I am going to talk about an operator algebraic proof for this. Indeed, non-commutative Grothendieck's inequality, Khintchine inequalities and some standard functional analytic arguments are enough to get the conclusion.

The random Fourier series is obtained by changing signs of Fourier coefficients of a given $L^1$ function. One interesting theorem that was proved by Littlewood in 1926 is that, if all random Fourier series of f are $L^1$-functions, then the given $L^1$ function $f$ is automatically square integrable. The main aim of this talk is to introduce some basics of Fourier analysis on compact groups and to explain a natural analogue of the above theorem within the framework of compact (quantum) groups.

The Kesten-McKay law describes the spectrum of the adjacency for random regular graphs. I will compute the free cumulants of this distribution.

Nielsen characterized the convertibility of two finite-dimensional bipartite pure states via local operations and classical communication (LOCC) using majorization. This important result, which has seen many applications in quantum information, describes the LOCC-transfer of entanglement between bipartite pure states. In this talk, we present a version of Nielsen's theorem in the commuting operator framework using a generalized class of LOCC operations and the theory of majorization in von Neumann algebras. As a corollary, we obtain an operational interpretation of maximal entanglement relative to von Neumann factors of type II$_1$. This is joint work with David Kribs, Rupert Levene and Ivan Todorov.