Correlations between zeros of non-Gaussian random polynomials

TL;DR

This paper establishes the existence and universality of the scaling limit for zeros of non-Gaussian random polynomials away from the origin, extending known Gaussian results, and describes a crossover at the origin.

Contribution

It proves the existence of a universal scaling limit for non-Gaussian polynomial zeros away from the origin, and characterizes the non-universal behavior at the origin.

Findings

Scaling limit exists away from the origin

Scaling limit is universal away from the origin

Crossover from non-universal to universal behavior at the origin

Abstract

The existence of the scaling limit and its universality, for correlations between zeros of {\it Gaussian} random polynomials, or more generally, {\it Gaussian} random sections of powers of a line bundle over a compact manifold has been proved in a great generality in the works [BBL2], [Ha], [BD], [BSZ1]-[BSZ4], and others. In the present work we prove the existence of the scaling limit for a class of {\it non-Gaussian} random polynomials. Our main result is that away from the origin the scaling limit exists and is universal, so that it does not depend on the distribution of the coefficients. At the origin the scaling limit is not universal, and we find a crossover from the nonuniversal asymptotics of the density of the probability distribution of zeros at the origin to the universal one away from the origin.