Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots

Abstract

Let $p_n$ be a random polynomial of degree $n$ whose roots are independent and identically distributed according to a rotationally invariant probability measure ${\mu }_0$ on the complex plane with finite logarithmic moment. If $k_n/n\to t\in (0,1)$, we prove that, as $n\to \infty$, the empirical zero measure of the $k_n$-th derivative of $p_n$ converges weakly in probability to a deterministic rotationally invariant probability measure. We describe the limiting measure explicitly in terms of the radial quantile function of ${\mu }_0$. This proves a conjecture of Hoskins and Kabluchko [Exp. Math. 32 (2023), no. 4].

arXiv comment29 pages, 2 figures; minor corrections and updated one reference