Distances between zeroes and critical points for random polynomials with i.i.d. zeroes

TL;DR

This paper investigates the proximity of critical points to i.i.d. zeroes of random polynomials, showing they are typically very close with a Gaussian-distributed difference, and extends to conjectures on dependent zeroes.

Contribution

It provides a precise asymptotic description of the distribution of the difference between zeroes and critical points for large degree random polynomials with i.i.d. zeroes.

Findings

Critical points are very close to zeroes with high probability.

The difference between a zero and its critical point is approximately Gaussian.

The mean and variance of this difference are explicitly characterized.

Abstract

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi_0,\xi_1,\ldots,\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to\infty$, with high probability there is a critical point of $Q_n$ which is very close to $\xi_0$. We localize the position of this critical point by proving that the difference between $\xi_0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\xi_0))$ and variance of order $\log n \cdot n^{-3}$. Here, $f(z)= \mathbb E[1/(z-\xi_k)]$ is the Cauchy-Stieltjes transform of the $\xi_k$'s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.