Local universality of zeroes of random polynomials

TL;DR

This paper proves that the local correlation structure of zeros of certain random polynomials is universal, depending only on the first two moments of coefficients, and applies this to classical questions about real zeros.

Contribution

It establishes a general universality principle for the zeros of random polynomials with independent coefficients, extending ideas from random matrix theory.

Findings

Correlation functions of zeros are approximately the same for polynomials with matching moments.

Provides bounds on the number of real zeros for various classes of random polynomials.

Introduces a replacement principle based on log-magnitude comparisons.

Abstract

In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\sum_{i=1}^n c_i \xi_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde \xi_i z^i$, where the $\xi_i$ and $\tilde \xi_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our results show, under some light conditions on the coefficients $c_i$ and the tails of $\xi_i, \tilde \xi_i$, that the correlation functions of the zeroes of $f$ and $\tilde f$ are approximately the same. As an application, we give some answers to the classical question `"How many zeroes of a random polynomials are real?" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions $f$ and $\tilde f$ if their log magnitudes $\log |f|, \log|\tilde f|$ are close in distribution, and if some non-concentration bounds are obeyed.