Random polynomials: the closest roots to the unit circle

Abstract

Let $f = \sum_{k=0}^n \varepsilon_k z^k$ be a random polynomial, where $\varepsilon_0,\ldots ,\varepsilon_n$ are iid standard Gaussian random variables, and let $\zeta_1,\ldots,\zeta_n$ denote the roots of $f$. We show that the point process determined by the magnitude of the roots $\{ 1-|\zeta_1|,\ldots, 1-|\zeta_n| \}$ tends to a Poisson point process at the scale $n^{-2}$ as $n\rightarrow \infty$. One consequence of this result is that it determines the magnitude of the closest root to the unit circle. In particular, we show that \[ \min_{k} ||\zeta_k| - 1|n^2 \rightarrow \mathrm{Exp}(1/6),\] 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.