Real roots near the unit circle of random polynomials

Abstract:

Let $f_n(z) = \sum _{k = 0}^n \varepsilon _k z^k$ be a random polynomial where $\varepsilon _0,\ldots ,\varepsilon _n$ are i.i.d. random variables with $\mathbb {E} \varepsilon _1 = 0$ and $\mathbb {E} \varepsilon _1^2 = 1$. Letting $r_1, r_2,\ldots , r_k$ denote the real roots of $f_n$, we show that the point process defined by $\{|r_1| - 1,\ldots , |r_k| - 1 \}$ 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.