Eigenvalues of Non-Commuting Polynomials involving Small and Large Rank Matrices and Their Fluctuations

par Benoit Collins

jeudi 26 sept. 2024, 14:00 → 15:00 Europe/Paris

435 (ENS de Lyon)

ATTENTION : Contrairement à ce qui avait été initialement annoncé, la séance aura lieu à l'ENS de Lyon, et non pas à Lyon 1.

 

Voiculescu’s asymptotic freeness theorem asserts that for independent families of random matrices with limiting distributions, a non-commuting polynomial in these matrices also exhibits a limiting distribution. We explore a variant of this scenario where at least one family of matrices has small rank, and the non-commuting polynomial consistently has a non-zero valuation in this small-rank family. Consequently, the random matrix model maintains small rank in the large $$ n $$ limit, described by a novel non-commutative probability theory known as cyclic monotone independence. After presenting the foundational results, we will introduce models for the limiting objects, analogous to free products, and address the eigenvalue fluctuation problem, which leads to numerous new limiting distributions. This presentation is based on collaborative work with Fujie, Hasebe, Leid, and Sakuma.