A Semicircle Law for Derivatives of Random Polynomials

Abstract

Let $x_1,\dots ,x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1,\dots ,x_n$. We prove that for $\ell \in \mathbb{N}$ fixed as $n\to \infty$, the $(n-\ell )-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval,$$n^{\ell /2}\frac{\ell !}{n!}\cdot p_n^{(n-\ell )}\left(\frac{x}{\sqrt{n}}\right)\to H{e}_{\ell }(x+{\gamma }_n),$$ where $H{e}_{\ell }$ is the $\ell -$th probabilists' Hermite polynomial and ${\gamma }_n$ is a random variable converging to the standard $\mathcal{N}(0,1)$ Gaussian as $n\to \infty$. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.