Almost sure behavior of the critical points of random polynomials

Abstract

Let be a sequence of independent and identically distributed complex random variables with common distribution and let be the associated random polynomial in . Kabluchko established the conjecture stated by Pemantle and Rivin that the empirical measure associated with the critical points of converges weakly in probability to the base measure . In this note, we establish that the convergence, in fact, holds in the almost sure sense. Our result positively answers a question raised by Kabluchko and formalized as a conjecture in the recent paper (Michelen and Vu [arXiv:2212.11867]).