We show that a point process of hard spheres exhibits long-range orientational order. This process is designed to be a random perturbation of a three-dimensional lattice that satisfies a specific rigidity property; examples include the FCC and HCP lattices. We also define two-dimensional near-lattice processes by local geometry-dependent hard disk conditions. Earlier results about the existence of long-range orientational order carry over, and we obtain the existence of infinite-volume measures on two-dimensional point configurations that turn out to follow the orientation of a fixed triangular lattice arbitrarily closely.

中文翻译：

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随机近晶格硬球和硬盘过程的长程定向顺序

我们表明硬球的点过程表现出长程定向顺序。该过程被设计为满足特定刚性属性的三维晶格的随机扰动；例子包括 FCC 和 HCP 晶格。我们还通过局部几何相关的硬盘条件定义了二维近晶格过程。早期关于长程定向有序存在的结果得到了延续，我们在二维点配置上获得了无限体积测量的存在，结果证明它们任意紧密地遵循固定三角形晶格的方向。