Seminar on Free Probability
and Random Matrices

Fall 2016

After giving an elementary approach to the construction of unitary representations of the Thompson group F, I show how the underlying ideas transfer to the setting of free probability.

Roughly speaking, this transfer yields a Markov type perturbation of shifts on infinite free products as they arise from quantum spreadability. My talk is in parts based on joint work with Gwion Evans, Rajarama Bhat, Rolf Gohm, and Stephen Wills.

Bi-free probability is a generalization of free probability to study pairs of left and right faces in a non-commutative probability space. In this talk, I will demonstrate a characterization of bi-free independence inspired by the “vanishing of alternating centred moments” condition from free probability. I will also show how these ideas can be used to introduce a bi-free unitary Brownian motion and a liberation process which asymptotically creates bi-free independence.

About a decade ago, Mingo, Nica, and Speicher studied the fluctuations of the moments of Gaussian matrices from a combinatorial perspective. Based on their work, in this talk we will study the fluctuations of the moments of block Gaussian matrices. In particular, we will find a semi-explicit formula for a matricial version of the second-order Cauchy transform. Using the linearization technique, this formula will provide the second-order Cauchy transform of polynomials in Gaussian matrices. This is joint work with Serban Belinschi and James Mingo.

Given two analytic functions $f$ and $g$, we say that $f$ is subordinate to $g$, if we can find $h$ such that $f = g \circ h$ for some analytic function $h$ (defined on a suitable domain). We investigate this when $f$ and $g$ are the Cauchy transforms of probability measures on $\mathbb{R}$; and give a combinatorial interpretation of this using the non-commutative derivative.

Freeness was defined in terms the centredness of certain moments. There is also a formulation in terms the vanishing of mixed cumulants which is frequently much easier to work with. In many extensions of freeness this is the only formulation. I will show the equivalence of freeness and the vanishing of mixed free cumulants.

We show that in the free case the relation between the ordinary generating functions of moments and free cumulants is given by the Cauchy transform.

We show that the exponential generating functions of the classical cumulants and moments are related by the logarithm.

We use our understanding of the combinatorics of a single Gaussian random matrix to study the interaction of several Gaussian random matrices.

We use our understanding of the combinatorics of a single Gaussian random matrix to study the interaction of several Gaussian random matrices.

Our understanding of the combinatorics of Gaussian random variables will be applied to Gaussian random matrices.

Our understanding of the combinatorics of Gaussian random variables will be applied to Gaussian random matrices.

Our understanding of the combinatorics of Gaussian random variables will be applied to Gaussian random matrices.

A new phenomenon in random matrix theory was found by Voiculescu twenty five years ago, now called the theory of free probability. I will give the first few lectures of a learning seminar starting with the combinatorics of Gaussian random variables. The seminar will follow the first chapter of a new book by Roland Speicher and me on random matrix theory and free probability.

Free probability is a non-commutative probability theory where the classical notion of independence is replaced by free independence and operator-valued free probability is a generalization of free probability where the field of complex numbers is replaced by a unital algebra. On the other hand, c-free probability is an extension of free probability where the notion of c-free independence is given with respect to two states instead of one, and bi-free probability is an extension of free probability where systems of left and right random variables are considered simultaneously. In this talk, we will review the operator-valued generalization of bi-free probability and discuss a possible extension where the corresponding notion of independence is given with respect to two conditional expectations, hence generalizing everything mentioned above. This is joint work with P. Skoufranis.

A Wigner random matrix is a self-adjoint random matrix with i.i.d. entries. It is known that independent Wigner matrices are asymptotically free and that Wigner matrices and constant matrices are asymptotically free.

In this talk I shall show that this is only partially true at the level of fluctuations. Thus an extension to second order freeness is required to capture this phenomenon. While the precise statement remains open it is clear the definition should involve the action of a graphical operad on an algebra.

I will also demonstrate the second order relation between Wigner and constant matrices. This is joint work with Roland Speicher.