Work

Regularity and convergence properties of finite free convolutions

Year: 2025

Type: article

Abstract: Finite free convolutions, [Formula: see text] and [Formula: see text], are binary operations on polynomials of degree [Formula: see text] 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 [Formula: see text] and [Formula: see text] to their infinite counterparts ⊞ and ⊠ as [Formula: see text], without compactness assumptions. Furthermore, this weak convergence is strengthened to convergence in Kolmogorov distance.

Source: International Journal of Mathematics

DOI: 10.1142/s0129167x26500102

Author: Katsunori Fujie

Institution: Kyoto University

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Japan Society for the Promotion of Science

Awards: G7433082671, 自由確率論における極限定理とランダム行列への応用