The support of the free additive convolution of multi-cut measures
Abstract
We consider the free additive convolution of two probability measures and , supported on respectively and disjoint bounded intervals on the real line, and derive a lower bound and an upper bound that is strictly smaller than , on the number of connected components in its support. We also obtain the corresponding results for the free additive convolution semi-group . Throughout the paper, we consider classes of probability measures with power law behaviors at the endpoints of their supports with exponents ranging from to . Our main theorem generalizes a result of Bao, Erd\H{o}s and Schnelli~[4] to the multi-cut setup.
Keywords
Cite
@article{arxiv.2201.05582,
title = {The support of the free additive convolution of multi-cut measures},
author = {Philippe Moreillon and Kevin Schnelli},
journal= {arXiv preprint arXiv:2201.05582},
year = {2022}
}
Comments
Corrected the proof and statement of Proposition 4.12 (Prop. 4.10 in v1). The upper bound in Theorem 1.5 accordingly changed to $2n_\alpha n_\beta$ from $n_\alpha n_\beta$, proof of Theorem 1.5 in Section 5 is now shorter. Extended Lemma 4.9 yielding a lower bound on the number of components in the support of the free additive convolution in Theorem 1.5. Corrected the proof of Proposition 4.10