论文标题
过度均匀性的真实测试
A genuine test for hyperuniformity
论文作者
论文摘要
我们设计了第一个严格的显着性测试，即使是一个样本，即使是敏感结果，我们也会设计出敏感结果。我们的起点是对$ \ mathbb {r}^d $上固定点过程的经验傅立叶变换的详细研究。对于大型系统尺寸，我们得出渐近协方差，并证明了多元中心极限定理（CLT）。然后将散射强度用作结构因子的标准估计器。上面的CLT适用于优选的大量点过程，每当情况下，散射强度也满足多元极限定理。因此，我们可以使用似然比原理来测试过度均匀性。值得注意的是，在超均匀性的无效假设下，所得测试统计量的渐近分布是普遍的。我们以非常高的精度从模拟中获得其明确形式。这种新型测试准确地保持了超明均模型的名义意义水平，并且即使在边界情况下，它也拒绝具有高功率的非横滑示例。此外，只有一个具有实际相关系统尺寸的样本，它确实这样做。
We devise the first rigorous significance test for hyperuniformity with sensitive results, even for a single sample. Our starting point is a detailed study of the empirical Fourier transform of a stationary point process on $\mathbb{R}^d$. 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. Moreover, it does so given only a single sample with a practically relevant system size.
