A simple proof of local universality for roots of Kac polynomials
Abstract
Let f n f_n be a random polynomial of degree n n with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of f n f_n cluster uniformly around the unit circle as n 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 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 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.
Fields of Study & Subjects
Geometry and complex manifolds
Random Matrices and Applications
Mathematical functions and polynomials
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 11
- Pages
- 4947–4957
- Publication date
- 20 May 2026
- Language
- EN