Title: Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions
Speaker: Jorge Garza-Vargas (UC Berkeley)
Abstract: Since the seminal work of Voiculescu in the early 90’s, the connection between the asymptotic behavior of random matrices and free probability has been extensively studied. More recently, in relation to the solution of the Kadison-Singer problem, Marcus, Spielman, and Srivastava discovered a deep connection between certain classical polynomial convolutions and free probability, and this connection was further understood by Marcus, who introduced the notion of finite free probability.
In this talk I will present recent results on finite free probability with applications to the asymptotic analysis of real-rooted polynomials. Our approach is based on a careful combinatorial analysis of the finite free cumulants, from where we derive a topological expansion. In turn, we use this expansion to study the root distribution dynamics of polynomials after repeated differentiation, as well as the root distribution fluctuations for certain families of polynomials. This is joint work with Octavio Arizmendi and Daniel Perales: arXiv:2108.08489.