Correlations and Pairing Between Zeros and Critical Points of Gaussian Random Polynomials

TL;DR

This paper analyzes the asymptotic correlations between zeros and critical points of Gaussian random polynomials, revealing short-range decay and pairing behavior at microscopic scales.

Contribution

It provides an explicit asymptotic formula for the joint intensity of zeros and critical points, demonstrating their short-range correlations and pairing structure.

Findings

Correlations decay exponentially with distance as e^{-N|z-w|^2}

Zeros and critical points form rigid pairs at microscopic scales

Expected distance and angular dependence between paired zeros and critical points are tightly bounded

Abstract

We study the asymptotics of correlations and nearest neighbor spacings between zeros and holomorphic critical points of $p_N$, a degree N Hermitian Gaussian random polynomial in the sense of Shiffman and Zeldtich, as N goes to infinity. By holomorphic critical point we mean a solution to the equation $\frac{d}{dz}p_N(z)=0.$ Our principal result is an explicit asymptotic formula for the local scaling limit of $\E{Z_{p_N}\wedge C_{p_N}},$ the expected joint intensity of zeros and critical points, around any point on the Riemann sphere. Here $Z_{p_N}$ and $C_{p_N}$ are the currents of integration (i.e. counting measures) over the zeros and critical points of $p_N$, respectively. We prove that correlations between zeros and critical points are short range, decaying like $e^{-N\abs{z-w}^2}.$ With $\abs{z-w}$ on the order of $N^{-1/2},$ however, $\E{Z_{p_N}\wedge C_{p_N}}(z,w)$ is sharply peaked near $z=w,$ causing zeros and critical points to appear in rigid pairs. We compute tight bounds on the expected distance and angular dependence between a critical point and its paired zero.