The support of the free additive convolution of multi-cut measures

TL;DR

This paper investigates the structure of the support of free additive convolutions of measures supported on multiple disjoint intervals, providing bounds on the number of connected components and generalizing previous results.

Contribution

It extends existing results to multi-cut measures, establishing bounds on support components for free additive convolutions and their semi-groups.

Findings

Derived bounds on the number of support components

Generalized previous single-cut results to multi-cut measures

Analyzed measures with power law behaviors at endpoints

Abstract

We consider the free additive convolution $\mu_\alpha\boxplus\mu_\beta$ of two probability measures $\mu_\alpha$ and $\mu_\beta$, supported on respectively $n_\alpha$ and $n_\beta$ disjoint bounded intervals on the real line, and derive a lower bound and an upper bound that is strictly smaller than $2n_\alpha n_\beta$, on the number of connected components in its support. We also obtain the corresponding results for the free additive convolution semi-group $\{\mu^{\boxplus t}\,:\, t\ge 1\}$. Throughout the paper, we consider classes of probability measures with power law behaviors at the endpoints of their supports with exponents ranging from $-1$ to $1$. Our main theorem generalizes a result of Bao, Erd\H{o}s and Schnelli~[4] to the multi-cut setup.