Critical points of random polynomials with independent identically distributed roots

TL;DR

This paper proves that the empirical measure of the zeros of the derivative of a random polynomial with i.i.d. roots converges to the distribution of the roots themselves, confirming a conjecture by Pemantle and Rivin.

Contribution

It establishes the convergence in probability of the empirical measure of critical points to the roots' distribution for i.i.d. roots, confirming a previously conjectured behavior.

Findings

Empirical measure of critical points converges to the roots' distribution

Confirmed Pemantle and Rivin's conjecture

Convergence holds in probability as degree increases

Abstract

Let $X_1,X_2,...$ be independent identically distributed random variables with values in $\C$. Denote by $\mu$ the probability distribution of $X_1$. Consider a random polynomial $P_n(z)=(z-X_1)...(z-X_n)$. We prove a conjecture of Pemantle and Rivin [arXiv:1109.5975] that the empirical measure $\mu_n:=\frac 1{n-1}\sum_{P_n'(z)=0} \delta_z$ counting the complex zeros of the derivative $P_n'$ converges in probability to $\mu$, as $n\to\infty$.