超均匀性的随机谱推断
Randomized Spectral Inference for Hyperuniformity
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中文总结 AI 辅助
本文提出一种基于随机频率采样的谱推断方法，用于从单次平稳点过程实现中检验超均匀性，通过双半径外推估计低频谱质量并给出置信界与单侧检验。
中文摘要 AI 辅助
我们研究从平稳点过程的一个大实现中推断超均匀性的问题。对于具有完全正熵的基本自由平移作用的点过程，在勒贝格几乎每个频率上，经验傅里叶统计量渐近服从复高斯分布，无需定量混合或累积量可加性假设。随机采样的频率因此为低频谱质量提供了一个易于处理的极限实验。若频率在半径为 $r$ 的欧几里得球 $B_r$ 上均匀分布，则极限平方傅里叶统计量的均值为 \\[ A_r=\frac{\sigma_\eta(B_r)}{\rho_\eta\lambda_d(B_r)}. \\] 超均匀性由 $A_r\to0$（当 $r\downarrow0$ 时）刻画，因此无需对原点处的结构因子作连续性假设。在 \\[ A_r=s+c r^\alpha+O(r^\beta),\qquad 0<\alpha<\beta, \\] 且余项有指定界以及归一化巴特利特密度局部平方可积的条件下，双半径外推法可消除 $r^\alpha$ 项，并以显式的 $O(r^\beta)$ 偏差界估计 $s$。这给出了置信界以及关于 $H_0:s=0$ 的单侧检验，该检验在任一固定零假设过程下的渐近拒绝概率不超过所选显著性水平。
英文摘要
We test hyperuniformity from one large realization of a stationary point process. Hyperuniformity is equivalent to the average $A_r$ of the structure factor over $B_r$ vanishing as $r\downarrow0$, so the target is a low-frequency average rather than the value of the structure factor at the origin, which need not exist. We estimate $A_r$ from squared Fourier coefficients at frequencies drawn uniformly from $B_r$; if these coefficients are asymptotically Gaussian at almost every fixed frequency, the squared coefficients converge to independent variables with mean $A_r$. Assuming $A_r=s+c r^α+\varepsilon_r$ with $|\varepsilon_r|\leq Lr^β$ and given exponents $0<α<β$, a two-radius extrapolation removes the $r^α$ term and estimates $s$ with deterministic error of order $r^β$. This gives confidence bounds and a one-sided test of $H_0:s=0$ with asymptotic level at most $γ$, consistent against every fixed $s>0$ as $R\to\infty$, then the number of sampled frequencies tends to infinity, and then $r\downarrow0$. Essentially free translation actions with completely positive entropy satisfy the Fourier assumption.
发表机构
- Chalmers University of Technology（查尔姆斯理工大学）
- University of Gothenburg（哥德堡大学）
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