Roots of random polynomials whose coefficients have logarithmic tails

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 $G_{n}(z)=\sum_{k=0}^{n}\xi_{k}z^{k}$ 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 $\alpha v^{-(\alpha+1)}\,du\,dv$.