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, $f(z)= \mathbb E[1/(z-\xi_k)]$ 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.