Free Convolution Powers via Roots of Polynomials

Abstract

Let $\mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power ${\mu }^{⊞k}$ for any real $k\ge 1$. The purpose of this short note is to give an elementary description of ${\mu }^{⊞k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.