Seminar on Free Probability
and Random Matrices

Fall 2017

In this talk, we will focus on a single type B variable, and introduce the corresponding infinitesimal law. In addition, we will also define the free additive convolution for infinitesimal laws, and to see the relation between the type B laws and the infinitesimal laws.

In this talk, we will start from an infinitesimal non-commutative probability space, and define the freeness, and additive convolution for infinitesimal laws. Then, we will also establish the relation among type A free convolution, type B free convolution, and infinitesimal free convolution.

We will continue our discussion of type B R-transform. Then we are going to introduce the free independence of type B, and establish the relation between type B freeness and vanishing mixed cumulants in type B.

In this talk, we are going to introduce the framework of type B non-commutative probability space. Then, we will give the definition of the cumulants of type B and show the relation between the cumulants of type A and the cumulants of type B.  Finally, we will discuss the corresponding moments series and R-transform of type B.

The discussion of non-commutative independence revolves around what are known as universal products on non-commutative probability spaces. These were defined by Roland Speicher in a paper showing that there are only three such products, yielding free independence, tensor (classical) independence and Boolean independence. We will expand Speicher's definition of independence to more general types of probability spaces focusing much of our discussion around the notion of traffic free independence.

In this talk, we will show that conjugation by certain random unitary liberating matrices delivers the bounded cumulants property.

The discussion of non-commutative independence revolves around what are known as universal products on non-commutative probability spaces. These were defined by Roland Speicher in a paper showing that there are only three such products, yielding free independence, tensor (classical) independence and Boolean independence. We will expand Speicher's definition of independence to more general types of probability spaces focusing much of our discussion around the notion of traffic free independence.

In this talk, we will show that conjugation by certain random unitary liberating matrices delivers the bounded cumulants property.

I will continue from last week.

I will continue from last week.

Like tensor independence, free independence gives us rules for doing calculations. With random matrix models, we usually need tensor independence of the entries and some kind of group invariance of the joint distribution of the entries to get the (asymptotic) freeness necessary to apply the tools of free probability.

A few years ago Mihai Popa and I found that the transpose also produces asymptotic freeness, i.e. a matrix could be asymptotically free from its own transpose. Since that we have expanded this work to the case of the partial transposes that arise in quantum information theory.

In this talk I will explain what happens when one transposes certain R-cyclic operators.