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

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, $f(z)= \mathbb E [\frac 1 {z-\xi _k}]$ 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.