A Semicircle Law for Derivatives of Random Polynomials

Author:

Hoskins Jeremy G1,Steinerberger Stefan2

Affiliation:

1. Department of Statistics, University of Chicago, Chicago, IL 60637, USA

2. Department of Mathematics, University of Washington, Seattle, WA 98195, USA

Abstract

Abstract Let $x_1, \dots , x_n$ be $n$ independent and identically distributed real-valued 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, a suitable rescaling of $p_n^{(n-\ell )}$ starts behaving like the $\ell -$th probabilists’ Hermite polynomial subject to a random shift. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.

Funder

National Science Foundation

Alfred P. Sloan Foundation

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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