An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots

TL;DR

This paper investigates the asymptotic behavior of critical points of random polynomials with i.i.d. roots, revealing their close pairing with roots and describing the distribution of their fluctuations, thus refining the classical Gauss-Lucas theorem.

Contribution

It provides the first asymptotic analysis of critical point fluctuations for random polynomials with radially symmetric roots, extending the classical Gauss-Lucas theorem.

Findings

Largest critical points closely paired with largest roots

Fluctuations follow Gaussian or heavy-tailed stable distributions

Refinement of Gauss-Lucas theorem for random polynomials

Abstract

If $p:\mathbb{C} \to \mathbb{C}$ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when $p$ is a random polynomial of degree $n$ with roots chosen independently from a radially symmetric, compactly supported probably measure $\mu$ in the complex plane. We show that the largest (in magnitude) critical points are closely paired with the largest roots of $p$. This allows us to compute the asymptotic fluctuations of the largest critical points as the degree $n$ tends to infinity. We show that the limiting distribution of the fluctuations is described by either a Gaussian distribution or a heavy-tailed stable distribution, depending on the behavior of $\mu$ near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials.