Quasicrystalline Gibbs states in 4-dimensional lattice-gas models with finite-range interactions

TL;DR

This paper constructs a four-dimensional lattice-gas model exhibiting non-periodic, quasicrystalline Gibbs states at low temperatures, using cellular automata and error correction to ensure stability of these states.

Contribution

It introduces a novel 4D lattice-gas model with quasicrystalline Gibbs states, leveraging cellular automata and error correction mechanisms for stability.

Findings

Existence of non-periodic Gibbs states in 4D lattice-gas models

Construction of cellular automata based on Ammann's tiles

Stable quasicrystalline states at low temperatures

Abstract

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.