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Article Dans Une Revue Bulletin of the London Mathematical Society Année : 2023

Almost sure behavior of the critical points of random polynomials

Résumé

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].

Dates et versions

hal-03952902 , version 1 (23-01-2023)

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Jürgen Angst, Dominique Malicet, Guillaume Poly. Almost sure behavior of the critical points of random polynomials. Bulletin of the London Mathematical Society, 2023, ⟨10.1112/blms.12963⟩. ⟨hal-03952902⟩
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