Real roots near the unit circle of random polynomials

Abstract

Let $f_n(z)={\sum }_{k=0}^n{\epsilon }_k{z}^k$ be a random polynomial where ${\epsilon }_0,\dots ,{\epsilon }_n$ are i.i.d. random variables with $\mathbb{E}{\epsilon }_1=0$ and $\mathbb{E}{\epsilon }_1^2=1$. Letting $r_1,r_2,\dots ,r_k$ denote the real roots of $f_n$, we show that the point process defined by $\{|r_1|-1,\dots ,|r_k|-1\}$ converges to a non-Poissonian limit on the scale of $n^{-1}$ as $n\to \infty$. Further, we show that for each $\delta >0$, $f_n$ has a real root within ${\Theta }_{\delta }(1/n)$ of the unit circle with probability at least $1-\delta$. This resolves a conjecture of Shepp and Vanderbei from 1995 by confirming its weakest form and refuting its strongest form.

19 pages