Seminar on Free Probability
and Random Matrices

Fall 2015

This will be a continuation from November 26.

I will review the construction of second order cumulants and work out the semi-circular case. I will then show how to compute these for partially transposed Wishart matrices.

This will be a continuation from November 5.

This will be a continuation from the previous week.

Second order freeness is a feature exhibited in the large $N$ limit by random matrix ensembles such as the orthogonal Gaussian random matrices and the orthogonal Wishart random matrices. Its main purpose is to help determine the asymptotic behaviour of the covariance of traces of products of random matrices; this is comparable to the description of the asymptotic behaviour of the expectation of the trace of products of random matrices that we get from freeness. In this talk, we will give the definition of second order freeness and present some of the most relevant work related to it

It is well known that certain ensembles of unitary matrices deliver asymptotic freeness under suitable conditions. Related to the intuitive question about how random an ensemble must be in order to deliver freeness, Anderson and Farrell found a unitary ensemble which is “less random” than the Haar distributed ensemble originally proposed by Voiculescu. In this talk we propose to study the speed at which these ensembles deliver freeness as an alternative measure of “randomness”. This is joint work with Jamie Mingo.

This will be a continuation from last week.

In quantum information theory a frequently used test for entanglement is the positive partial transpose test. For a Wishart matrix Aubrun found the aspect ratio that guarantees a non-positive positive partial transpose and thus entanglement. Mihai Popa and I considered the interaction between a Wishart matrix and its partial transpose. We showed that the matrix, its full transpose and its partial transposes are asymptotically free. In this talk I will consider what happens when we consider orthogonal and unitary matrices.

I will discuss how matrix cumulants, defined by Capitaine and Casalis (2007) in terms of a convolution, connect to diagrammatic methods of calculating matrix integrals, and how these tools are particularly useful in the multi-matrix setting. I will focus on the real case, and consider also how the complex case is related. I will also discuss how this approach is useful in the quaternionic case, in particular in problems involving partial traces.

I will also discuss some of the tools useful for analyzing the non-orientable diagrams which appear in the real and quaternionic cases.