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 \[ \min_{k} \zeta_k  1n^2 \rightarrow \mathrm{Exp}(1/6),\] 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.10869
 arXiv:
 arXiv:2010.10869
 Bibcode:
 2020arXiv201010869M
 Keywords:

 Mathematics  Probability;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Combinatorics;
 60G10;
 60G15;
 60F05;
 42A32;
 26C10