Seminar on Free Probability
and Random Matrices

Winter 2015

In this talk, we will introduce the group of symmetries of a multiantenna channel. Based on this group and the Haar measure on it, we will derive a theorem that generalizes many capacity theorems in the literature. With this machinery, we will explore the question of why the isotropic input is “good” in many “natural” situations.

Free probability has been an important area of Operator Algebras since its inception by Voiculescu. Although originally motivated by analytic techniques, a combinatorial approach to free probability was developed by Speicher via the theory of non-crossing partitions. These two approaches to free probability each have their own advantages and many results may be demonstrated using either approach. Approximately two years ago, Voiculescu introduced the notion of bi-free pairs of algebras. Voiculescu was able to generalize many concepts in free probability to the bi-free setting using analytic techniques. The goal of this talk is to discuss the combinatorial aspects of bi-free independence. The main difference between the combinatorial structures of free and bi-free independence come from handling specific permutations. Consequently, bi-non-crossing partitions must be used in place of non-crossing partitions. Analyzing properties of bi-non-crossing partitions yields various notions of independences arising in bi-free probability, bi-free partial transforms, bi-multiplicative functions for operator-valued bi-free probability, and bi-matrix models.

In this talk, we will discuss the occurrence of $*$-freeness in tensor products collections $(a_{1;i}\otimes\cdots\otimes a_{1;K}:i\in I)$ of noncommutative random variables with applications in the study of asymptotic freeness for tensor products of large random matrices. We will also investigate connections of this work with the study of freeness in direct products of groups.

In this talk we address the problem of how many entries of a Haar distributed orthogonal matrix can be jointly approximated by i.i.d. normal random variables. In particular, we consider a gaussian random matrix $Y(n)$ of order n and apply to it the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar distributed orthogonal matrix $U(n)$. If $F_i^m$ denotes the vector formed by the first m-coordinates of the i-th row of $Y(n)-\sqrt n U(n)$ and $m/n$ tends to $\alpha$, our main result shows that the euclidean norm $F_i^m$ converges exponentially fast to $f(\alpha)\sqrt m$, up to negligible terms.

To show the extent of this result, we will see how to use it to study the convergence of the supremum norm $\epsilon_n(m)=\displaystyle\sup_{1\leq i \leq n, 1\leq j \leq m} \big|\,y_{i,j}- \sqrt{n}u_{i,j}\, \big|$ and we find a coupling that improves by a factor $\sqrt{2}$ the recently proved best known upper bound of $\epsilon_n(m)$. We will also discuss an application of our main result to Quantum Information Theory.

We derive the bi-free analogue of the Lévy-Hinčin formula for compactly supported planar probability measures which are infinitely divisible with respect to the additive bi-free convolution recently introduced by Voiculescu. We also provide examples of bi-free infinitely divisible distributions with their bi-free Lévy-Hinčin representations, and briefly discuss the bi-free Lévy processes. This is joint work with H.-W. Huang and J. Mingo.

In this third and final talk we will show that the channels introduced in the previous talk can be studied using operator- valued free probability theory. In particular, the capacity and capacity achieving input covariance matrix can be efficiently approximated using asymptotic results from random matrix theory and free probability. We will analyze the capacity of some concrete channels using this free probabilistic approach. Particular emphasis will be given to the free probabilistic modelling aspects. No previous knowledge of free probability is assumed. This is joint work with Victor Pérez-Abreu (CIMAT).

Resuming from the previous time, we will start with a theorem of Telatar in which the capacity of the so called canonical model is computed. After that, providing a sightly different proof, we will extend Telatar’s theorem for a wider class of channels. To finish the discussion on the classical work of Telatar, we will prove rigorously an observation of Telatar which stablished the link between asymptotic random matrix theory and wireless communications. The basic notions of random matrix theory will be explained as needed.

This is a report on a joint work with Todd Kemp and Antoine Dahlqvist. Biane prove that the unitary Brownian motion converges in distribution to the free unitary Brownian motion. We prove that this convergence is strong. The proof relies on combinatorics, stochastic calculus, and an ‘unfolding trick’. We will mention some applications of this result, too.

About 20 years ago, multiantenna systems appeared. Since then, random matrix theory has been one of the main tools for the design and analysis of these systems. With recent developments in free probability theory, it is now possible to study these systems using what we called operator-valued models. In particular, we are able to study the information theoretic capacity in a very efficient way.

In this first talk, we will provide a terse introduction to the communication and information theory required to settle down the main capacity theorems for multiantenna channels. We will emphasize the role of certain random matrix theory results in the aforementioned theorems. At the end, we will discuss some random matrix theory questions arising in this context.

We investigate the existence of a continuous eigenvalue for a Cantor minimal system, $(X, T)$, with regards to its dimension group, $K_0(X, T)$. In this context, the notion of irrational mixability for dimension groups is introduced and some (necessary and) sufficient conditions for this property will be given. The main property of these dimension groups is the absence of irrational values in the set of continuous spectrum of their realizations by Cantor minimal systems. Any realization of an irrationally mixable dimension group with cyclic rational subgroup is weakly mixing and cannot be (strong) orbit equivalent to a Cantor minimal system with non-trivial spectrum. The talk is based on a recent joint work with Theirry Giordano and David Handelman.

We will discuss infinitely divisible compactly supported probability measures relative to free convolution on the real line by using function theory. The content of the talk is based on the work of Bercovici and Voiculescu.

We will discuss infinitely divisible compactly supported probability measures relative to free convolution on the real line by using function theory. The content of the talk is based on the work of Bercovici and Voiculescu.

I will continue from last week.

Wishart matrices are a basic tool in multivariate analysis. Recently partially transposed Wishart matrices have become an object of interest in quantum information theory as G. Aubrun showed that a partially transposed Wishart matrix is asymptotically semi-circular; this means that all free cumulants above $\kappa_2$ vanish. M. Popa and I have shown that the full and partial transpose operation is asymptotically liberating. In this talk I will explain what second order cumulants are and how to compute them for a partially transposed Wishart matrix. It turns out that most, but not all, vanish.