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

We consider the free additive convolution ${\mu }_{\alpha }⊞{\mu }_{\beta }$ 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 $\{{\mu }^{⊞t}:t\ge 1\}$. 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.