How to test for hyperuniformity

Kurzfassung

Despite the quickly growing interest in hyperuniformity, a reliable detection of hyperuniformity has remained difficult for a long time because it is a long-range property. This talk presents the first rigorous significance test for hyperuniformity with sensitive results even for a single sample and at practically relevant system sizes. Our starting point is a detailed study of the empirical Fourier transform of a stationary point process. For large system sizes, we derive the asymptotic covariances and prove a multivariate central limit theorem (CLT). 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.