Critical points of random polynomials with independent identically\n distributed roots

2012/06/28 by Zakhar Kabluchko, Kabluchko, Zakhar · 1 citation
Mathematics · #30C15 (Primary) 60G57 #60B10 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1206.6692

openalex publication_date 2012/06/28 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let X₁,X₂,... be independent identically distributed random variables\nwith values in C. Denote by \μ the probability distribution of X₁.\nConsider a random polynomial P_n(z)=(z-X₁)...(z-X_n). We prove a conjecture\nof Pemantle and Rivin [arXiv:1109.5975] that the empirical measure\n\μ_n:= frac 1n-1\∑_Pn'(z)=0 \δ_z counting the complex zeros of\nthe derivative P_n' converges in probability to \μ, as n\→\∞.\n

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