Hyperuniformity is the study of stationary point processes with a sub-Poisson variance in a large window. In other words, counting the points of a hyperuniform point process that fall in a given large region yields a small-variance Monte Carlo estimation of the volume. Hyperuniform point processes have received a lot of attention in statistical physics, both for the investigation of natural organized structures and the synthesis of materials. Unfortunately, rigorously proving that a point process is hyperuniform is usually difficult. A common practice in statistical physics and chemistry is to use a few samples to estimate a spectral measure called the structure factor. Its decay around zero provides a diagnostic of hyperuniformity. Different applied fields use however different estimators, and important algorithmic choices proceed from each field's lore. This paper provides a systematic survey and derivation of known or otherwise natural estimators of the structure factor. We also leverage the consistency of these estimators to contribute the first asymptotically valid statistical test of hyperuniformity. We benchmark all estimators and hyperuniformity diagnostics on a set of examples. In an effort to make investigations of the structure factor and hyperuniformity systematic and reproducible, we further provide the Python toolbox structure_factor, containing all the estimators and tools that we discuss.

翻译：超均匀性研究的是在大窗口内具有亚泊松方差的平稳点过程。换言之，对超均匀点过程落入给定大区域内的点数进行计数，可得到体积的小方差蒙特卡洛估计。超均匀点过程在统计物理学中备受关注，既用于研究自然界中的有序结构，也用于材料合成。然而，严格证明一个点过程具有超均匀性通常十分困难。统计物理学和化学中的常见做法是利用少量样本估计一种称为结构因子的谱测度，其在零点附近的衰减可作为超均匀性的诊断指标。然而，不同应用领域使用不同的估计量，关键算法选择也源于各领域的经验传统。本文对已知或自然的结构因子估计量进行了系统综述与推导。我们还利用这些估计量的一致性，首次提出了具有渐近有效性的超均匀性统计检验方法。我们在一组示例上对所有估计量和超均匀性诊断方法进行了基准测试。为使结构因子与超均匀性的研究系统化且可复现，我们进一步提供了Python工具箱structure_factor，其中包含本文讨论的所有估计量和工具。