Free Convolution Powers Via Roots of Polynomials

ABSTRACT

Let μ 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 a formal proof of an elementary description of ${\mu }^{⊞k}$ in terms 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.

KEYWORDS:

• Free probability
• roots of polynomials
• free additive convolution

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 46L54
• 26C10

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

S.S. is supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation.