Paper detail

Roots of random polynomials whose coefficients have logarithmic tails

It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^nξ_kz^k$ with i.i.d. coefficients $ξ_0,\ldots,ξ_n$ concentrate a.s. near the unit circle as $n\to\infty$ if and only if ${\mathbb{E}\log_+}|ξ_0|<\infty$. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like $L({\log}|t|)({\log}|t|)^{-α}$ as $t\to\infty$, where $α\geq0$, 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 $αv^{-(α+1)}\,du\,dv$.