Free Probability and Operators

Date: April 4, 2025

Time: 4:00PM - 5:00PM

Location: BLOC 306

Speaker: Daniel Perales, Texas A&M University

Title: Even Hypergeometric Polynomials and Finite Free Commutators

Abstract: The finite free convolutions are binary operations on polynomials that behave well with respect to the roots and can be understood as a discrete analogue of free probability. On the other hand, these operations can be realized as expected characteristic polynomials of adding (or multiplying) two randomly rotated matrices. We will focus on the class of even polynomials and their behavior with respect to these finite free convolutions. First, we will use rectangular finite free convolution to understand even real-rooted polynomials in terms of positive-rooted polynomials. Then, we will study some related operations, such as symmetrizations, and finite free commutators. We provide new examples using even hypergeometric polynomials, that include classical families of orthogonal polynomials (such as Laguerre, Hermite, and Jacobi). Finally, we relate the limiting root distributions of sequences of even polynomials with the corresponding symmetric measures that arise in free probability.