Universality of Poisson limits for moduli of roots of Kac polynomials

TL;DR

This paper provides a new proof confirming that the rescaled moduli of roots of Gaussian Kac polynomials follow a universal Poisson point process, extending previous results and conjectures in the field.

Contribution

The authors introduce a novel proof technique to establish the universality of Poisson limits for roots of Kac polynomials, confirming conjectures about their statistical behavior.

Findings

Rescaled roots form a Poisson point process

Poisson statistics are universal across different settings

New proof technique simplifies previous arguments

Abstract

We give a new proof of a recent resolution by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei that the moduli of roots of Gaussian Kac polynomials of degree $n$, centered at $1$ and rescaled by $n^2$, should form a Poisson point process. We use this new approach to verify a conjecture of Michelen and Sahasrabudhe that the Poisson statistics are in fact universal.