Exact statistical properties of the zeros of complex random polynomials

TL;DR

This paper investigates the exact statistical properties of zeros of complex Gaussian random polynomials, revealing their connections to quantum systems and comparing their correlations to those of complex Gaussian random matrices.

Contribution

It extends the interpretation of these polynomials as quantum wave functions and provides exact formulas for their zero correlations, including boundary behavior.

Findings

Zeros exhibit statistical properties similar to quantum chaotic systems.

Exact formulas for one and two-point distributions are derived.

Boundary correlation asymptotics are characterized.

Abstract

The zeros of complex Gaussian random polynomials, with coefficients such that the density in the underlying complex space is uniform, are known to have the same statistical properties as the zeros of the coherent state representation of one-dimensional chaotic quantum systems. We extend the interpretation of these polynomials by showing that they also arise as the wave function for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest Landau level. A study of the statistical properties of the zeros is undertaken using exact formulas for the one and two point distribution functions. Attention is focussed on the moments of the two-point correlation in the bulk, the variance of a linear statistic, and the asymptotic form of the two-point correlation at the boundary. A comparison is made with the same quantities for the eigenvalues of complex Gaussian random matrices.