We investigate the statistical properties of translation invariant random fields (including point processes) on Euclidean spaces (or lattices) under constraints on their spectrum or structure function. An important class of models that motivate our study are hyperuniform and stealthy hyperuniform systems, which are characterised by the vanishing of the structure function at the origin (resp., vanishing in a neighbourhood of the origin). We show that many key features of two classical statistical mechanical measures of randomness—namely, fluctuations and entropy, are governed only by some particular local aspects of their structure function. We obtain exponents for the fluctuations of the local mass in domains of growing size, and show that spatial geometric considerations play an important role—both the shape of the domain and the mode of spectral decay. In doing so, we unveil intriguing oscillatory behaviour of spatial correlations of local masses in adjacent box domains. We describe very general conditions under which we show that the field of local masses exhibit Gaussian asymptotics, with an explicitly described limit. We further demonstrate that stealthy hyperuniform systems with joint densities exhibit degeneracy in their asymptotic entropy per site. In fact, our analysis shows that entropic degeneracy sets in under much milder conditions than stealthiness, as soon as the structure function fails to be logarithmically integrable.

我们研究了欧几里得空间（或晶格）上的平移不变随机场（包括点过程）在其谱或结构函数的约束下的统计特性。激发我们研究的一类重要模型是超均匀和隐形超均匀系统，其特征是结构函数在原点消失（分别在原点附近消失）。我们表明，随机性的两个经典统计力学度量的许多关键特征——即波动和熵，仅受其结构函数的某些特定局部方面的控制。我们获得了尺寸不断增长的域中局部质量波动的指数，并表明空间几何考虑起着重要作用——域的形状和光谱衰减模式。在这样做时，我们揭示了相邻盒域中局部质量空间相关性的有趣振荡行为。我们描述了非常一般的条件，在这些条件下，我们表明局部质量场表现出高斯渐近，并具有明确描述的限制。我们进一步证明，具有联合密度的隐形超均匀系统在每个站点的渐近熵中表现出简并性。事实上，我们的分析表明，只要结构函数不能对数可积，熵简并就会在比隐身更温和的条件下开始。我们描述了非常一般的条件，在这些条件下，我们表明局部质量场表现出高斯渐近，并具有明确描述的限制。我们进一步证明，具有联合密度的隐形超均匀系统在每个站点的渐近熵中表现出简并性。事实上，我们的分析表明，只要结构函数不能对数可积，熵简并就会在比隐身更温和的条件下开始。我们描述了非常一般的条件，在这些条件下，我们表明局部质量场表现出高斯渐近，并具有明确描述的限制。我们进一步证明，具有联合密度的隐形超均匀系统在每个站点的渐近熵中表现出简并性。事实上，我们的分析表明，只要结构函数不能对数可积，熵简并就会在比隐身更温和的条件下开始。