A simple proof of local universality for roots of Kac polynomials
Marcus Michelen, Oren Yakir
TL;DR
The paper proves local universality of the zeros of random Kac polynomials at microscopic scale $1/n$ around a fixed unit-circle point by identifying the scaling limit as a Gaussian analytic function $G(z)=\int_{0}^{1} e^{zt}\,dB_{\mathbb{C}}(t)$ with covariance $K(z,w)=\int_{0}^{1} e^{(z+\overline w)t}\,dt$, and showing the zeros of the local scaling converge to the zeros of $G$. It first demonstrates convergence in distribution of the local polynomials $F_n$ to $G$ in the space of entire functions, then upgrades to convergence of the $k$-point measures of zeros by establishing uniform integrability via Esseen's small-ball bound and Jensen's formula, together with Hurwitz continuity. The main result is that for each $k\ge 1$, the $k$-point measure $\alpha^{k}_{\mathcal{Z}_{F_n}}$ converges to $\alpha^{k}_{\mathcal{Z}_G}$ in the vague topology, confirming local universality under only finite second moments. This approach yields a simpler, self-contained proof compared to prior log-potential methods and suggests robustness of microscopic root statistics that may extend to other models of random analytic functions.
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.