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

TL;DR

This paper demonstrates that certain random perturbations of lattice structures, such as FCC and HCP in 3D and near-lattice processes in 2D, exhibit long-range orientational order, extending previous results and establishing new measure existence results.

Contribution

It introduces a framework for analyzing long-range orientational order in random lattice perturbations and proves the existence of measures aligning with fixed lattice orientations in 2D.

Findings

Long-range orientational order exists in 3D lattice perturbations.

Infinite-volume measures align with fixed lattice orientations in 2D.

Results extend previous understanding of order in random lattice processes.

Abstract

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 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 arbitrary closely.