Free infinite divisibility, fractional convolution powers, and Appell polynomials

Abstract

Initiated by a result of Gorin and Marcus (2020) and an observation of Steinerberger (2019), there has been a recent growing body of literature connecting repeated differentiation of real rooted 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 roots satisfy some bounds and the number of derivatives is such that the remaining degree is fixed, then it has been shown in various contexts that these high derivatives converge to the Hermite polynomials. In the context of convolution semigroups and finite free probability, where Hermite polynomials are the analogue of the Gaussian distribution, 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 consider the limits of Appell sequences, generalizing the well known fact that the roots of a Hermite polynomial, after being appropriately normalized, are asymptotically distributed according to the semicircle distribution. 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.