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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.

Random arrangements of points in the plane, interacting only through a simple hard-core exclusion, are considered. An intensity parameter controls the average density of arrangements, in analogy with the Poisson point process. It is proved that, at high intensity, an infinite connected cluster of excluded volume appears almost surely.

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Long-range orientational order of random near-lattice hard sphere and hard disk processes

Percolation of Hard Disks

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Long-range orientational order of random near-lattice hard sphere and hard disk processes

Percolation of Hard Disks