Seminar on Free Probability
and Random Matrices

Winter 2017

The free probability perspective on random matrices is that the large size limit of random matrices is given by some (usually interesting) operators on Hilbert spaces and corresponding operator algebras. The prototypical example for this is that independent GUE random matrices converge to free semicircular operators, which generate the free group von Neumann algebra. The usual convergence in distribution has been strengthened in recent years to a strong convergence, also taking operator norms into account. All this is on the level of polynomials. In my talk I will recall this and then go over from polynomials to rational functions (in non-commuting variables). Unbounded operators will also play a role.

I will continue from Friday's talk.

I will continue from Monday's talk.

In this talk we will discuss some recent developments in second-order free probability theory. In particular, we will present some results concerning the second-order Cauchy transform and the covariance of the linear statistics of random matrices.

We discuss some analytic properties of the additive bi-free convolution, both scalar-valued and operator-valued. We show that using the one-variable subordination functions associated with the additive free convolution, simple formulas for additive bi-free convolutions can be derived. As an application, we prove a result about atoms of the additive bi-free convolution.

We discuss some analytic properties of the additive bi-free convolution, both scalar-valued and operator-valued. We show that using the one-variable subordination functions associated with the additive free convolution, simple formulas for additive bi-free convolutions can be derived. As an application, we prove a result about atoms of the additive bi-free convolution.

This will be a continuation from last week.

Last time we showed that every complex polynomial in non-commutative variables can be linearized into a linear polynomial with matricial coefficients. In this talk we will show that this is also true for a non-commutative rational function.

In this talk we will show that every complex polynomial in non-commutative variables can be linearized into a polynomial with matricial coefficients. This linearization technique, also knows as 'descriptor realizations' in the control theory community, has important consequences in the realm of free probability theory.

I will continue from last week.

In this talk, we present some estimations for the asymptotic behaviour of the covariance of (unnormalized) traces of random matrices, when conjugated by asymptotically liberating random unitary matrices.

Since Voiculescu introduced free independence 35 years ago, many variants have appeared: Boolean, monotone, type B, second order, higher order, real, quaternionic, infinitesimal, and bi-free independence (plus combinations of the above) to name a few. Most of the constructions are given combinatorially, but some have an interpretation in terms of analytic functions. I will discuss the 2003 paper of Biane, Goodman, and Nica, which introduced freeness of type B.

This will be the first of two lectures. This lecture will describe free cumulants of type B and type B freeness. The second lecture will explain how the hyperoctahedral group comes into play and hence why this is called type B freeness.

Since Voiculescu introduced free independence 35 years ago, many variants have appeared: Boolean, monotone, type B, second order, higher order, real, quaternionic, infinitesimal, and bi-free independence (plus combinations of the above) to name a few. Most of the constructions are given combinatorially, but some have an interpretation in terms of analytic functions. I will discuss the 2003 paper of Biane, Goodman, and Nica, which introduced freeness of type B.

This will be the first of two lectures. This lecture will describe free cumulants of type B and type B freeness. The second lecture will explain how the hyperoctahedral group comes into play and hence why this is called type B freeness.