Almost sure behavior of the critical points of random polynomials

TL;DR

This paper proves that the empirical distribution of critical points of random polynomials with i.i.d. roots converges almost surely to the underlying distribution, strengthening previous convergence in probability results.

Contribution

It establishes almost sure convergence of the critical points' empirical measure to the base measure, confirming a conjecture and advancing understanding of random polynomial critical points.

Findings

Almost sure convergence of critical points' empirical measure

Strengthens previous convergence in probability results

Answers a conjecture posed by Kabluchko

Abstract

Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $\mu$ and let $P_n(X):=\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\mathbb C[X]$. In [Kab15], the author established the conjecture stated by Pemantle and Rivin in [PR13] that the empirical measure $\nu_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $\mu$. In this note, we establish that the convergence in fact holds in the almost sure sense. Our result positively answers a question raised by Z. Kabluchko and formalized as a conjecture in the recent paper [MV22].