Winter 2023
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
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.
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.
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 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)