Law of Large Numbers for Roots of Finite Free Multiplicative Convolution of Polynomials

TL;DR

This paper establishes a law of large numbers for the roots of polynomials under finite free multiplicative convolution, analyzing their asymptotic root distributions as polynomial degrees grow large.

Contribution

It introduces a law of large numbers for roots of polynomials with non-negative roots under finite free multiplicative convolution and studies their asymptotic distributions.

Findings

Law of large numbers for roots of polynomials under finite free multiplicative convolution

Asymptotic root distributions of limit polynomials analyzed as degree tends to infinity

Empirical root distributions converge to a specific limit distribution

Abstract

We provide the law of large numbers for roots of finite free multiplicative convolution of polynomials which have only non-negative real roots. Moreover, we study the empirical root distributions of limit polynomials obtained through the law of large numbers of finite free multiplicative convolution when their degree tends to infinity.