Roots of random polynomials whose coefficients have logarithmic tails

Details Roots of random polynomials whose coefficients have logarithmic tails Roots of random polynomials whose coefficients have logarithmic tails

2011-10-12

arxiv.org/abs/1110.2585v1

Mathematics

Probability

32 pages, 2 figures

Scientific paper

It has been shown by Ibragimov and Zaporozhets [to appear in Prokhorov Festschrift; arXiv:1102.3517] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^n \xi_k z^k$ with i.i.d.\ coefficients $\xi_0,...,\xi_n$ concentrate a.s.\ near the unit circle as $n\to\infty$ if and only if $\E \log_{+} |\xi_0|<\infty$. 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 $\alpha\geq 0$ 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 $\alpha v^{-(\alpha+1)} dudv$.

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