We construct a four-dimensional lattice-gas model with finite-range interactions that has non-periodic, ``quasicrystalline'' Gibbs states at low temperatures. Such Gibbs states are probability measures which are small perturbations of non-periodic ground-state configurations corresponding to tilings of the plane with Ammann's aperiodic tiles. Our construction is based on the correspondence between probabilistic cellular automata and Gibbs measures on their space-time trajectories, and a classical result on noise-resilient computing with cellular automata. The cellular automaton is constructed on the basis of Ammann's tiles, which are deterministic in one direction, and has non-periodic space-time trajectories corresponding to each valid tiling. Repetitions along two extra dimensions, together with an error-correction mechanism, ensure stability of the trajectories subjected to noise.

翻译：我们构建了一个具有有限程相互作用的四维晶格气体模型，该模型在低温下存在非周期性的“准晶”吉布斯态。此类吉布斯态是概率测度，它们是对应于用Ammann非周期瓦片铺砌平面的非周期基态构型的小扰动。我们的构建基于概率性元胞自动机与其时空轨迹上的吉布斯测度之间的对应关系，以及一个关于使用元胞自动机进行抗噪声计算的经典结果。该元胞自动机基于Ammann瓦片构建，这些瓦片在一个方向上是确定性的，并且具有对应于每个有效铺砌的非周期性时空轨迹。沿两个额外维度的重复，结合一个纠错机制，确保了轨迹在噪声作用下的稳定性。