On the distribution patterns of zeros for random polynomials with regularly varying coefficients

2025/11/15 by Zakhar Kabluchko, Kabluchko, Zakhar, Boris A. Khoruzhenko +3 · 1 citation
Mathematics · #30B20 #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Primary: 26C10 #Probability (math.PR) #Secondary: 60F17

paper · pdf · doi:10.48550/arxiv.2511.12302

openalex publication_date 2025/11/15 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

Abstract

This paper investigates asymptotic distribution of complex zeros of random polynomials P_n(z):=∑_k=0ⁿb(k)ξ_k z^k, as n→∞, where b is a regularly varying function at infinity with index α∈ ℝ and (ξ_k)_{k≥ 0} is a sequence of independent copies of a complex-valued random variable ξ. The limiting distribution of zeros both inside and outside the unit disk is determined assuming 𝔼[log⁺|ξ|]<∞. Under the additional assumptions 𝔼[ξ]=0 and 𝔼[|ξ|²]<∞, local universality results for zeros near the boundary of the unit disk are established. Notably, it is shown that the point process of zeros undergoes a transition from liquid-like to crystalline phases as α crosses the critical value α_c = -1/2 from right to left. In the liquid phase (α> α_c), the limiting point process of zeros is universal. In the crystalline phase, it is universal if and only if α= α_c and ∑_k b²(k) = +∞ (the weak crystalline phase), and non-universal when ∑_k b²(k) < +∞ (the strong crystalline phase). The zeros of the so-called random self-inversive polynomials on the unit circle exhibit a similar phase transition.

Citations

• Real roots of random polynomials: asymptotics of the variance
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• SU(1,1) Random Polynomials

Cited by

• A simple proof of local universality for roots of Kac polynomials

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