Seminar on Free Probability
and Random Matrices

Winter 2023

A newcomer to random matrix theory may find themselves wondering: where’s the Fourier transform? The method of characteristic functions, which brings such clarity to the classical limit theorems of probability, has not been successfully ported to the realm of high-dimensional random matrices. There are several reasons for this, but the core issue is that we lack tools for estimating high-dimensional Fourier matrix integrals. I will explain recent progress in developing such methods which indicates that a Fourier approach to large random matrices is just around the corner.

The development of free probability theory has drawn much inspiration from its deep and far reaching analogy with classical probability theory. The same holds for its operator-valued extension, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally led to operator-valued free analogues of key and fundamental limiting theorems such as the operator-valued free Central Limit Theorem due to Voiculescu and asymptotic distributions of matrices with operator-valued entries.

In this talk, I will show Berry-Esseen bounds for sums of operator-valued free, Boolean and monotone independent variables, in terms of the moments of the summands. The estimates are on the level of operator-valued Cauchy transforms and the Lévy distance. Our approach relies on a Lindeberg method that we develop for sums of free/Boolean/monotone independent random variables and push to the operator-valued infinitesimal setting. Among various applications, I will discuss the application of these results to the CLT in each of the above mentioned settings. Based on joint works with O. Arizmendi, T. Mai and P.L. Tseng

Most of the literature on noncommutative stochastic calculus treats stochastic processes with some kind of “independent increments” property, e.g., -Brownian motions.

In classical stochastic analysis, the “right” object to integrate against is a semimartingale: the sum of a local martingale and an FV (finite variation) process. We describe a noncommutative analogue of such processes and a corresponding theory of stochastic calculus.

The talk will focus on motivation and building a working understanding; in particular, no substantial technical background in classical stochastic calculus will be assumed. This is forthcoming joint work with D. Jekel and T. Kemp.

In this talk I will explain some recent advances in the theory of finite free probability, obtained together with Jorge Garza and Daniel Perales.

We will first describe a combinatorial formula for the finite free cumulants of $p(x)\, {\scriptstyle\boxtimes}_{\,d}\, q(x)$ and explain a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera.

Then I will give some applications of said formula, such as the study of the infinitesimal distribution of certain families of polynomials that include Hermite and Laguerre, a new brief and conceptual proof of a recent result [Steinerberger (2020), Hoskins and Kabluchko (2020)] which connects root distributions of polynomial derivatives with fractional powers of free additive convolution

If time allows, I will also explain recent results on additional limiting theorems for finite free multiplicative convolutions, recently discovered in a joint work with Katsunori Fujie and Yuki Ueda.

Permutations whose cycles can be drawn without crossing are central to free probability. Ones which are real symmetric and annular are central to infinitesimal distributions. I will give a formal power series relation connecting the moments and cumulants. This formula is related to Kerov's work on interlacing polynomials and work of Arizmendi, Garza-Vargas, and Perales on the asymptotic distribution of the zeros of Laguerre polynomials. This is ongoing joint work with Guillaume Cébron.

In free probability, a fundamental result of Voiculescu is that random unitary matrices are asymptotically free. A representative special case is the fact that sums $A + U B U^*$ and products $A U B U^*$ of large randomly rotated matrices approximate free additive and multiplicative convolutions. In 2015, Marcus, Spielman, and Srivastava realized that in the non-asymptotic setting, one can recover ``finite" analogues of these free convolutions by looking at the expected characteristic polynomials of $A + U B U^*$ or $A U B U^*$.

After reviewing these finite free convolutions, I will show how techniques from combinatorial representation theory can help to understand finite free convolutions, focusing on the problem (which I recently solved in arXiv:2209.00523) of describing the commutator of randomly rotated matrices in this context. Time permitting, I will discuss some questions and ongoing work related to real-rootedness and finite free cumulants which are raised by comparison with the commutator in free probability.

In a past talk I introduce the concept of higher order free cumulants. I mentioned that the first and second order are well known and showed that there is a similar behavior for the third order case, however, I didn't get into the proof of this result. For this talk I will talk in more detail around the proof of the third order case.

The existence of Voiculescu’s subordination functions in the context of non-tracial operator-valued C$^*$-probability spaces has been established using analytic function theory methods. We use a matrix construction to show that the subordination functions thus obtained also satisfy an appropriately modified form of subordination for conditional expectations. (joint work with Hari Bercovici, arXiv:2209.12710)

Let $X_N$ be a GUE ensemble and $Y_N$ be a finite rank constant matrix. Following the work by Sklyakhtenko (2018), it is known that $X_N$ and $Y_N$ are asymptotically infinitesimally free. We will present simple formulas for the limit infinitesimal law of $XY+YX$ and $i (XY-YX)$ where $X$ and $Y$ are infinitesimally free and follow the limit inf. law of $X_N$ and $Y_N$ respectively. This is joint work with Jamie Mingo.

Given X, a $N \times N$ Wigner matrix, it is well known that the limiting distribution of the expectation of its normalized trace converges to the Catalan numbers as $N$ goes to infinity. Similarly, it was found that the fluctuations moments have a large limiting distribution, moreover, these can be described by non-crossing pairings on the annulus. The latest results can be described in an easier way by using the free cumulants of first and second order. Inspired in these results we found both, the third order moments and free cumulants of the Wigner matrix. In this talk I will talk about the third order case, I will sketch the proof and show how they have a complete combinatorial description via the non-crossing partitioned permutations. The main result of this talk is based in the following preprint

Voiculescu's freeness emerges in computing the asymptotic of randomly rotated $N \times N$ random matrices with respect to the normalized trace.

In this talk, I will explain a recent extension of this result : the asymptotic conditional freeness (in the sense of Bozejko, Leinert and Speicher) of rotated matrices with respect to vector states.

It enlightens the (already known) location of the outlier eigenvalues in deformed models and give new matrix models for Boolean independence and monotone independence.

The result comes from the fact that asymptotic freeness holds in expectation up to $O(1/N^2)$. I will recall the proof of this fact and also speak about the next term of order $O(1/N^2)$ in the expansion.

Based on joint works with Dahlqvist, Gabriel (arXiv:2205.01926) and Gilliers (arXiv:2207.06249)