Almost sure behavior of the critical points of random polynomials

Abstract

Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $μ$ 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 $ν_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $μ$. 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].

Theorems & Definitions (25)

• Theorem 1.1: Theorem 1.1 of MR3283656
• Theorem 1.2
• Remark 1.1
• Definition 2.1
• Remark 2.1
• Proposition 2.1
• Remark 2.2
• proof : Proof of Proposition \ref{['prop.kolmorogo']}
• Lemma 2.1
• proof : Proof of Lemma \ref{['lem.petrov']}