Random polynomials: the closest roots to the unit circle

2020/10/21 by Marcus Michelen, Julian Sahasrabudhe, Michelen, Marcus +1
Mathematics · #26C10 #42A32 #60F05 #60G10 #60G15 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2010.10869

openalex publication_date 2020/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f = ∑_k=0ⁿ ε_k z^k be a random polynomial, where ε₀,… ,ε_n are iid standard Gaussian random variables, and let ζ₁,…,ζ_n denote the roots of f. We show that the point process determined by the magnitude of the roots \ 1-|ζ₁|,…, 1-|ζ_n| \ tends to a Poisson point process at the scale n^-2 as n→ ∞. One consequence of this result is that it determines the magnitude of the closest root to the unit circle. In particular, we show that min_k ||ζ_k| - 1|n² → Exp(1/6), in distribution, where Exp(λ) denotes an exponential random variable of mean λ^-1. This resolves a conjecture of Shepp and Vanderbei from 1995 that was later studied by Konyagin and Schlag.

Related

• Real roots near the unit circle of random polynomials
• The Statistical Distribution of the Zeros of Random Paraorthogonal Polynomials on the Unit Circle
• Critical points of random polynomials with independent identically\n distributed roots
• Asymptotic Results for Random Polynomials on the Unit Circle
• Limit law for root separation in random polynomials