Disordered stealthy many-particle systems in $\mathbb{R}^d$ are exotic states of matter that suppress single scattering events for a finite range of wavenumbers around the origin in reciprocal space. We derive analytical formulas for the nearest-neighbor functions of disordered stealthy systems. First, we analyze asymptotic small-$r$ approximations and bounding expressions of the nearest-neighbor functions based on the pseudo-hard-sphere ansatz. We then determine how many of the standard $n$-point correlation functions are needed to determine the nearest neighbor functions, and find that a finite number suffice. Via theoretical and computational methods, we compare the large-$r$ behavior of these functions for disordered stealthy systems to those belonging to crystalline lattices. Such ordered and disordered stealthy systems have bounded hole sizes. However, we find that the approach to the critical-hole size can be quantitatively different. We argue that the probability of finding a hole close to the critical-hole size should decrease as a power law with an exponent only dependent on the space dimension $d$ for ordered systems, but that this probability decays asymptotically faster for disordered systems. This implies that holes close to the critical-hole size are rarer in disordered systems. The rarity of observing large holes in disordered systems creates substantial numerical difficulties in sampling the nearest neighbor distributions near the critical-hole size. This motivates both the need for new computational methods for efficient sampling and the development of novel theoretical methods. We also devise a simple analytical formula that accurately describes these systems in the underconstrained regime for all $r$. These results provide a foundation for the analytical description of the nearest-neighbor functions of stealthy systems in the disordered, underconstrained regime.

中文翻译：

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无序隐形超均匀多粒子系统的最近邻函数

$\mathbb{R}^d$ 中的无序隐形多粒子系统是奇异的物质状态，可以抑制倒易空间中原点周围有限波数范围内的单个散射事件。我们推导出无序隐身系统的最近邻函数的解析公式。首先，我们分析了基于伪硬球 ansatz 的最近邻函数的渐近小 $r$ 近似和边界表达式。然后我们确定需要多少个标准的 $n$ 点相关函数来确定最近邻函数，并发现一个有限数就足够了。通过理论和计算方法，我们将无序隐形系统的这些函数的大 $r$ 行为与属于晶格的函数进行了比较。这种有序和无序的隐形系统具有有限的孔尺寸。然而，我们发现临界孔尺寸的方法在数量上是不同的。我们认为，对于有序系统，找到接近临界孔尺寸的孔的概率应该作为幂律下降，而指数仅取决于空间维度 $d$，但对于无序系统，该概率逐渐衰减得更快。这意味着接近临界孔尺寸的孔在无序系统中很少见。在无序系统中观察大孔的罕见性在对临界孔尺寸附近的最近邻分布进行采样时造成了大量的数值困难。这激发了对有效采样的新计算方法的需求和新理论方法的发展。我们还设计了一个简单的分析公式，可以准确地描述所有 $r$ 在欠约束状态下的这些系统。这些结果为无序、欠约束状态下隐身系统的最近邻函数的分析描述提供了基础。