A genuine test for hyperuniformity

TL;DR

This paper presents a rigorous, sensitive statistical test for hyperuniformity in point processes, capable of reliably analyzing a single large sample using Fourier transform analysis and a universal asymptotic distribution.

Contribution

It introduces a novel significance test for hyperuniformity based on empirical Fourier transforms and a multivariate CLT, applicable to a wide class of point processes.

Findings

The test accurately maintains the nominal significance level.

It effectively rejects non-hyperuniform models with high power.

The method works reliably with only a single large sample.

Abstract

We introduce a rigorous and sensitive significance test for hyperuniformity that yields reliable results even from a single sample. Our approach is based on a detailed analysis of the empirical Fourier transform of a stationary point process in $\mathbb{R}^d$. For large system sizes, we derive the asymptotic covariances and establish a multivariate central limit theorem (CLT) for these empirical Fourier transforms. Their absolute square value, the scattering intensity, is then used as the standard estimator of the structure factor. The above CLT holds for a preferably large class of point processes, and whenever this is the case, the scattering intensity satisfies a multivariate limit theorem as well. Hence, we can use the likelihood ratio principle to test for hyperuniformity. Remarkably, the asymptotic distribution of the resulting test statistic is universal under the null hypothesis of hyperuniformity. We obtain its explicit form from simulations with very high accuracy. The novel test precisely keeps a nominal significance level for hyperuniform models, and it rejects non-hyperuniform examples with high power even in borderline cases. Moreover, it does so given only a single sample with a practically relevant system size.