Roots of random polynomials whose coefficients have logarithmic tails

TL;DR

This paper investigates how the roots of random polynomials behave when coefficients have heavy tails, revealing a transition from roots concentrating near the unit circle to a more dispersed pattern based on tail behavior.

Contribution

It extends previous results by characterizing the root distribution transition for coefficients with logarithmic tail decay using Poisson point process analysis.

Findings

Roots concentrate near the unit circle when coefficients have finite logarithmic moments.

Transition to dispersed roots occurs with heavy-tailed coefficients described by a specific tail decay.

The structure of roots is linked to the least concave majorant of a Poisson point process.

Abstract

It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^n\xi_kz^k$ with i.i.d. coefficients $\xi_0,\ldots,\xi_n$ concentrate a.s. near the unit circle as $n\to\infty$ if and only if ${\mathbb{E}\log_+}|\xi_0|<\infty$. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like $L({\log}|t|)({\log}|t|)^{-\alpha}$ as $t\to\infty$, where $\alpha\geq0$, and $L$ is a slowly varying function. Under this assumption, the structure of complex and real roots of $G_n$ is described in terms of the least concave majorant of the Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)}\,du\,dv$.