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:超强度是用大型窗口中Poisson次偏差的固定点过程的研究。 换句话说, 计数某个大区域下方超统一点过程的点点数, 就会对体积进行小变化的 Monte Carlo 估计。 超强点过程在统计物理学中受到了很多关注, 无论是对自然有组织的结构的调查还是对材料的合成。 不幸的是, 严格证明点过程是超强的通常困难的。 统计物理和化学中的一种常见做法是使用几个样本来估计光谱测量称为结构系数的值。 其零周围的衰减提供了超统一度诊断。 不同的应用字段使用不同的估计器, 以及每个字段的定位器进行重要的算法选择。 本文提供了对已知的结构或结构的自然估计器进行系统调查和推断。 我们还利用这些估计器的一致性来帮助首次对高度一致性统计测试。 我们用所有测量器和超统一度的诊断仪表和超强度诊断器来对一套示例进行诊断。 不同的应用字段使用不同的估计器, 提供系统化结构调查, 以及系统化工具, 提供我们系统化的系统化工具 。</s>