A Semicircle Law for Derivatives of Random Polynomials

TL;DR

This paper demonstrates that derivatives of large random polynomials with roots from i.i.d. variables exhibit universal behavior, converging to Hermite polynomials and roots following a Wigner semicircle distribution.

Contribution

It establishes a new universality law for derivatives of random polynomials, linking their behavior to Hermite polynomials and semicircle law.

Findings

Derivatives of random polynomials approximate Hermite polynomials.

Remaining roots follow the Wigner semicircle distribution.

Convergence to Gaussian distribution for certain variables.

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 $n \rightarrow \infty$, the $(n-\ell)-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval,$${n^{\ell/2}} \frac{\ell!}{n!} \cdot p_n^{(n-\ell)}\left( \frac{x}{\sqrt{n}}\right) \rightarrow He_{\ell}(x + \gamma_n),$$ where $He_{\ell}$ is the $\ell-$th probabilists' Hermite polynomial and $\gamma_n$ is a random variable converging to the standard $\mathcal{N}(0,1)$ Gaussian as $n \rightarrow \infty$. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.