Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In this talk, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the “tensor distribution” limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of independence, which we term “tensor freeness”. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables. This is joint work with Ion Nechita.

In physics, the phase transition between localized and delocalized phases in disordered systems, often called the Anderson transition, has attracted significant interest. Several intriguing models display this behavior, including random Schrödinger operators, random band matrices, and sparse random matrices. Heavy-tailed random matrices similarly capture this phase transition, making them a crucial class of models in understanding localization phenomena. In this talk, we will discuss the phase transition of the right singular vector associated with the smallest singular value of a rectangular random matrix. This work is in collaboration with Zhigang Bao (University of Hong Kong) and Xiaocong Xu (University of Southern California).

We study the time evolution of an initial product state in a system of almost-bosonic-extended-anyons in the large-particle limit. We show that the dynamics of this system can be well approximated, in finite time, by a product state evolving under the effective Chern–Simons–Schrödinger equation. Furthermore, we provide a convergence rate for the approximation in terms of the radius of the extended anyons. These results establish a rigorous connection between the microscopic dynamics of almost-bosonic-anyon gases and the emergent macroscopic behavior described by the Chern–Simons–Schrödinger equation.This talk is based on a work with Théotime Girardot