Random polynomials: the closest roots to the unit circle

Abstract

Let $f={\sum }_{k=0}^n{\epsilon }_k{z}^k$ be a random polynomial, where ${\epsilon }_0,\dots ,{\epsilon }_n$ are iid standard Gaussian random variables, and let ${\zeta }_1,\dots ,{\zeta }_n$ denote the roots of $f$. We show that the point process determined by the magnitude of the roots $\{1-|{\zeta }_1|,\dots ,1-|{\zeta }_n|\}$ tends to a Poisson point process at the scale $n^{-2}$ as $n\to \infty$. One consequence of this result is that it determines the magnitude of the closest root to the unit circle. In particular, we show that \[ \min_{k} ||ζ_k| - 1|n^2 \rightarrow \mathrm{Exp}(1/6),\] in distribution, where $\mathrm{Exp}(\lambda )$ denotes an exponential random variable of mean ${\lambda }^{-1}$. This resolves a conjecture of Shepp and Vanderbei from 1995 that was later studied by Konyagin and Schlag.