Seminar on Free Probability
and Random Matrices

Winter 2014

In these lecture series, we will discuss a recent paper by Greg W. Anderson and Brendan Farrell which exhibits some systems of random unitary matrices that, when used for conjugation, lead to freeness.

According to Paul Turán special functions are defined as useful mathematical functions. Clearly, the mathematical functions $e^{az}$ and $\Gamma(z)$ are fundamental special functions. In mathematics a function $f(q,z)$ is an $q$-analogue of $g(z)$ means that $g(z)$ could be obtained as $q\to1$ limit of $f(q,z)$ under suitable scalings. A special function may have many interesting $q$-analogues, for example, $q$-special functions $e_{q}(z),$ $E_{q}(z)$ $A_{q}(z)$ and $\mathcal{E}_{q}(\cos\alpha,\cos\beta;t)$ are all $q$-analogues of $e^{az}$. $q$-analogues of classical special functions have many important applications in mathematics and physics. The Euler's $q$-exponential function \[ E_{q}(-z) = (z;q)_{\infty} = \prod_{k=0}^{\infty} \left(1-zq^{k} \right), \quad q \in (0,1), z \in \mathbb{C} \] is an $q$-analogue of $e^{-z}$ and Jackson's $q$-gamma function \[ \Gamma_{q}(z) = \frac{(q;q)_{\infty}} {(q^{z};q)_{\infty}} (1-q)^{1-z} \quad q \in (0,1), z \in \mathbb{C} \] is an $q$-analogue of $\Gamma(z)$. These two $q$-functions are the cornerstones of the theory of basic hypergeometric series (a.k.a $q$-series). In this talk we present a derivation for the complete asymptotic expansions of $(z;q)_{\infty}$ and $\Gamma_{q}(z)$ via an Mellin transform. These asymptotic formulas are valid throughout the entire complex plane, uniformly on compact subsets. The formula for $(z;q)_{\infty}$ suggests that the modular properties of Dedekind eta function is a consequence of vanishing of odd Bernoulli numbers.

In these lecture series, we will discuss a recent paper by Greg W. Anderson and Brendan Farrell which exhibits some systems of random unitary matrices that, when used for conjugation, lead to freeness. Specifically, in this lecture we will introduce the functions of the χχ-class and present some of their properties related to sequences of complex matrices.

We will continue discussing the regularity properties of the free additive convolution. Specifically, a probability measure $\mu$ on $\mathbb{R}$ is said to have the property (H) if the density of $\mu\boxplus\nu$ is positive and analytic everywhere on $\mathbb{R}$ for any probability measure $\nu$ on $\mathbb{R}$. We will provide necessary and sufficient conditions on $\mu$ so that $\mu$ has the property (H) when it is $\boxplus$-infinitely divisible. We will also give some examples which have the property (H).

We review the notion of numerical range, and show that it behaves in an almost deterministic way for very general examples of random matrix models. By passing, we obtain norm estimates for DT-random matrix models introduced by Dykema and Haagerup. Joint work with Sasha Litvak, Karol Zyckowski, Piotr Gawron.

Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}$. We will discuss some regularization properties of free additive convolution $\mu\boxplus\nu$, where $\mu$ is $\boxplus$-infinitely divisible. More precisely, we will provide necessary and sufficient conditions on $\mu$ so that for any $\nu$ the density of $\mu\boxplus\nu$ is positive and analytic everywhere on $\mathbb{R}$. We will also give necessary and sufficient conditions so that the density is analytic at points at which the density vanishes. This is a joint work with Jiun-Chau Wang.

We will first finish the proof from last week, then we will briefly discuss a new concept in free probability, called bi-freeness, which was introduced by D. Voiculescu last year. On a free product of Hilbert spaces with specified unit vector, there are two actions of the operators of the initial spaces, corresponding to a left and to a right tensorial factorization, respectively. The notion of bi-free independence (or bi-freeness) arises when algebras of left-operators and algebras of right-operators on the free product space are considered at the same time.

We will continue from last week and give an alternative description for the ``special'' set of partitions that was introduced last time. The definition arises from the concept of a double-ended queue used in theoretical computer science which, in a certain way, describes the joint action of the left-right canonical operators on full Fock space. If time permits, we will prove the main theorem stated last week.

Let $\mathcal{T}$ be the full Fock space over $\mathbb{C}^d$ and consider a $(2d)$-tuple $A_1, \dots, A_d, B_1, \dots, B_d$ of canonical operators on $\mathcal{T}$, where $A_1, \dots, A_d$ act on the left and $B_1, \dots, B_d$ act on the right. The joint moments of the $(2d)$-tuple can be computed using the family of $(l, r)$-cumulant functionals, which enlarges the family of free cumulant functionals. Moreover, let $f$ and $g$ be the joint $R$-transforms of $(A_1, \dots, A_d)$ and $(B_1, \dots, B_d)$ with respect to the vacuum-state defined on $\mathcal{B}(\mathcal{T})$, then it turns out that every joint moment of the combined $(2d)$-tuple can be written in a canonical way as a sum of products of coefficients of $f$ and $g$ combined. This talk is based on a recent paper by M. Mastnak and A. Nica.

This series of lectures, which are divided in three parts, are based on the paper of Verbovetskyi and Vinnikov "Foundations of Noncommutative Function Theory".

This series of lectures, which are divided in three parts, are based on the paper of Verbovetskyi and Vinnikov "Foundations of Noncommutative Function Theory". Part I: Taylor-Taylor (TT) Formula Part II: Some Analytical Aspects Part III: Application Part I. In this part we will introduce the noncommutative functions and their difference-differential operator. Motivated by the latter, we will define the so called higher order noncommutative functions. Finally, we will derive the TT formula, the analogue of the classical Taylor formula in the case of noncommutative functions.