Free infinite divisibility, fractional convolution powers, and Appell polynomials
Andrew Campbell
TL;DR
The work builds a cohesive analytic bridge between repeated differentiation of real-rooted polynomials and free probability, centering on Appell sequences and the Laguerre–Pólya class. By leveraging finite free and rectangular free transforms, it identifies precise domains of attraction and limits where root distributions converge to free infinitely divisible laws, and it demonstrates a deep correspondence between Appell structures and free-ID distributions. The results extend to rectangular (finite) free probability, establishing fractional convolution powers and Appell-like polynomials that converge to boxplus_λ-ID laws, with a robust R-transform framework that unifies derivatives, diffusions, and root geometry. The paper also outlines a broader program toward an analytic theory of finite free probability, connections to beta-corners processes, and potential applications to random matrices and the Riemann Hypothesis via Jensen polynomials. Overall, it advances a tractable, transform-centered approach to understanding how polynomial root configurations encode free-probabilistic limits and new forms of infinite divisibility.
Abstract
Initiated by a result of Gorin and Marcus [Int. Math. Res. Not., (3):883--913, 2020] and an observation of Steinerberger [Proc. Amer. Math. Soc., 147(11):4733--4744, 2019], there has been a recent growing body of literature connecting repeated differentiation of polynomials to free additive convolution semigroups in free probability. Roughly, this connection states that in the large degree limit the empirical measure of the roots after many derivatives is, up to a rescaling, the original empirical measure of the roots raised to a free additive convolution power. If the original real roots satisfy some bounds and the number of derivatives is such that the remaining degree is fixed, then these high derivatives converge to the Hermite polynomials. In the context of convolution semigroups and finite free probability these results have a natural interpretation as a central limit theorem for repeated differentiation. We consider the case when these root bounds are removed and identify the potential limits of repeated differentiation as the real rooted Appell sequences. We prove that a sequence of polynomials is in the domain of attraction of an Appell sequence exactly when the empirical measures of the roots are, up to a rescaling, in the domain of attraction of a free infinitely divisible distribution naturally associated to the Appell sequence. We additionally extend these notions of infinite divisibility and fractional convolution semigroups to rectangular finite free probability. Our approach is based on the finite free R-transform of the polynomials, providing a step towards an analytic theory of finite free probability. These transforms provide a clear connection between Appell polynomials and free infinitely divisible distributions, where the finite free R-transform of a real rooted Appell polynomial is a truncated version of the R-transform of an infinitely divisible distribution.