随机测度的高阶超均匀性
Higher-order hyperuniformity of random measures
- Chalmers University of Technology（查尔姆斯理工大学）
- University of Gothenburg（哥德堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过局部k点模式方差定义高阶超均匀性条件，证明各阶条件不等价，给出切割投影过程的精确判据，并构造球窗反例，发现隐身点过程不满足二阶超均匀性。
AI中文摘要：
我们通过局部$k$点模式的大尺度方差来研究高阶超均匀性，并将所得条件记为$\HU_k$；其中$\HU_1$即通常的超均匀性。条件$\HU_k$不必彼此一致：对每个$k\geq1$，存在一个$\mathbb R$不变的弱混合简单点过程，它属于所有$j\leq k$的$\HU_j$，但不属于$\HU_{k+1}$。随机平移的格点属于每个$k$的$\HU_k$，而格点的足够小的非退化独立同分布扰动以及投影行列式点过程已经属于$\HU_1\setminus\HU_2$。对于正则欧几里得切割投影过程，我们给出了$\HU_k$的精确判据。对于球窗，在内部维度二和三中，$\HU_k$等价于$\HU_1$，而在每个内部维度$m\geq4$中，我们构造了属于$\HU_1\setminus\HU_2$的球窗例子。最后，非周期的Kurasov--Sarnak傅里叶准晶点过程是隐身的——其一阶谱在原点处有间隙——然而它不属于$\HU_2$。
英文摘要：
We study higher-order hyperuniformity through the large-scale variance of local $k$-point patterns, and write $\HU_k$ for the resulting condition; $\HU_1$ is ordinary hyperuniformity. The conditions $\HU_k$ need not coincide: for every $k\geq1$ there exists an $\mathbb R$-invariant weakly mixing simple point process that belongs to $\HU_j$ for all $j\leq k$ but not to $\HU_{k+1}$. Randomly translated lattices belong to $\HU_k$ for every $k$, whereas sufficiently small non-degenerate iid perturbations of lattices and projection determinantal point processes already belong to $\HU_1\setminus\HU_2$. For regular Euclidean cut-and-project processes we give an exact criterion for $\HU_k$. For ball windows, $\HU_k$ is equivalent to $\HU_1$ in internal dimensions two and three, whereas in every internal dimension $m\geq4$ we construct ball-window examples in $\HU_1\setminus\HU_2$. Finally, the nonperiodic Kurasov--Sarnak Fourier-quasicrystalline point process is stealthy---its first-order spectrum has a gap at the origin---yet does not belong to $\HU_2$.
