Regularity and Convergence Properties of Finite Free Convolutions
Katsunori Fujie
TL;DR
This work develops finite free convolutions $\boxplus_d$ and $\boxtimes_d$ as polynomial-based analogues of classical free convolutions, links them to the geometry of polynomials, and leverages interlacing and real-rootedness to establish regularity properties. It proves triangle-type distance inequalities, explicit atom-transfer rules, and shows that empirical root distributions converge weakly to $\mu \boxplus \nu$ and $\mu \boxtimes \nu$ as $d \to \infty$, with a strengthened Kolmogorov-distance convergence that removes prior compact-support assumptions for the additive case and extends to the multiplicative case. The approach is notably elementary, relying on partial orderings of measures, cut-up/cut-down techniques, and polynomial bases, rather than analytic transforms. These results deepen the connection between finite free probability and its infinite counterpart, enabling robust convergence statements and providing tools for further exploration of the finite-to-infinite transition in noncommutative probability and random matrix theory.
Abstract
Finite free convolutions, $\boxplus_d$ and $\boxtimes_d$, are binary operations on polynomials of degree $d$ that are central to finite free probability, a developing field at the intersection of free probability and the geometry of polynomials. Motivated by established regularities in free probability, this paper investigates analogous regularities for finite free convolutions. Key findings include triangle inequalities for these convolutions and necessary and sufficient conditions regarding atoms of probability measures. Applications of these results include proving the weak convergence of $\boxplus_d$ and $\boxtimes_d$ to their infinite counterparts $\boxplus$ and $\boxtimes$ as $d \to \infty$, without compactness assumptions. Furthermore, this weak convergence is strengthened to convergence in Kolmogorov distance.