Quasicrystalline Gibbs States in 4-Dimensional Lattice-Gas Models with Finite-Range Interactions

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 stacked 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 based on Toom’s model, ensure stability of the trajectories in the presence of noise.