A simple proof of local universality for roots of Kac polynomials

TL;DR

This paper provides a straightforward proof of local universality for the roots of Kac polynomials, showing their correlation functions near a fixed point on the unit circle converge to those of a Gaussian analytic function.

Contribution

It introduces a simple, self-contained approach that directly compares scaled Kac polynomials to Gaussian functions, bypassing complex potential analysis.

Findings

Roots cluster uniformly around the unit circle as degree increases

Correlation functions at microscopic scale converge to those of a Gaussian analytic function

Proof uses basic complex analysis and anti-concentration bounds

Abstract

Let $f_n$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_n$ cluster uniformly around the unit circle as $n$ grows large. We give a simple and self-contained proof of local universality for the correlation functions of the roots at the microscopic scale $1/n$ around a fixed point on the circle. While previous proofs of local universality were focused on studying the logarithmic potential of $f_n$, we instead directly compare the scaled random polynomial to a limiting Gaussian analytic function, and establish convergence of correlations via a soft argument, using only basic complex analysis and an anti-concentration bound of Esseen.