Seminar on Free Probability
and Random Matrices

Fall 2014

We will discuss some theorems which characterize freely infinitely divisible laws in terms of Voiculescu transforms. Also to be discussed are freely infinitely divisible laws are characterized via the free Lévy-Hinčin formula.

We will discuss some theorems which characterize freely infinitely divisible laws in terms of Voiculescu transforms. Also to be discussed are freely infinitely divisible laws are characterized via the free Lévy-Hinčin formula.

This will be a continuation from November 3.

M. Popa and I have shown that a random matrix can be free from its transpose if it is unitarily invariant. We have recently taken this one step further and have shown that for a Wishart matrix one gets an asymptotically free family consisting of the original matrix, two partial transposes (left and right), and the full transpose.

The theory of positive definite matrices, positive definite functions, and positive linear maps is rich in content. The notion of infinitely divisible matrices, a subclass of positive definite matrices, also has been studied for a long time by R. Horn and R. Bhatia. These subjects are closely related to moment sequences, infinitely divisible laws with respect to classical convolution, and non-commutative analysis. We will present some of these results and discuss the applications in free probability theory.

I will continue from last week.

Let $M$ be a finite non-empty set. Suppose that for each $n \in \mathbb{N}$ we are given a family $\{B_{n,m}\}_{m \in M}$ of $n$-by-$n$ complex matrices such that $\displaystyle{ \sup_{n \in \mathbb{N} } \sup_{m\in M}} \Vert B_{n,m} \Vert < \infty $, and the limit $\displaystyle{ \lim_{n\rightarrow \infty } } \frac{1}{n} \mathbf{tr} [ (B_{n,m})^{k} ]$ exists for all $k\in \mathbb{N}$ and all $m \in M$. Can we modify the sequence $\{ \{B_{n,m}\}_{m \in M}\}_{n\in \mathbb{N}}$ of families of complex matrices such that the resulting modified sequence is asymptotically free? For instance, a well-known result in free probability states that if $\{U_{n,m}\}_{m\in M}$ is a family of independent $n$-by-$n$ Haar-distributed random unitary matrices for each $n \in \mathbb{N}$, then the sequence $\{ \{U_{n,m}^{*}B_{n,m}U_{n,m}\}_{m\in M} \}_{n \in \mathbb{N}}$ is asymptotically free. In this talk, we first present the notion of asymptotically liberating sequences of families of random unitary matrices. Then, we show that such sequences can be used to the same end as the sequence of families of independent Haar-distributed random unitary matrices above, i.e., we show that if $\{\{V_{n,m}\}_{m\in M}\}_{n \in \mathbb{N}}$ is asymptotically liberating, then $\{ \{V_{n,m}^{*}B_{n,m}U_{n,m}\}_{m\in M} \}_{n \in \mathbb{N}}$ is asymptotically free. After, we state and prove a theorem on sufficient conditions for a sequences of families of random unitary matrices in order to be asymptotically liberating. Finally, if time permits we provide an example of an asymptotically liberating sequence derived from uniformly distributed random signed permutation matrices and Hadamard matrices.

If one takes an analytic self-map of the complex upper half-plane and consider it iterations, then the resulting dynamical system is a normal family and hence nothing happens dynamically. Yet, if we look at the action of the iterations on the real line, then interesting dynamics occurs. In this talk, we shall discuss the connections between such a dynamics and non-commutative probability theory. The key notion here is that of monotone convolution introduced by Muraki around 2000.

In this talk we will study certain elements in $L^{\infty-}(\Omega,{\mathcal F},{\mathbb P}) \otimes \textrm{M}_d({\mathcal C})$ where $({\mathcal C},\varphi)$ is a non-commutative probability space and $(\Omega,{\mathcal F},{\mathbb P})$ is a classical probability space. In particular, a numerical algorithm to compute the mean $\textrm{ M}_d({\mathbb C})$-valued Cauchy transform of elements of the form $A\circ {\bf X}$ with $A$ a $d\times d$ selfadjoint random matrix and ${\bf X}$ a $d\times d$ selfadjoint operator-valued matrix with free circular entries will be analyzed. Here $\circ$ denotes the Hadamard or pointwise multiplication. Also a central limit theorem for a more general class of elements in $L^{\infty-}(\Omega,{\mathcal F},{\mathbb P}) \otimes \textrm{M}_d({\mathcal C})$ will be derived.