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

Siamak Taati, Jacek Miȩkisz

TL;DR

This work demonstrates that a finite-range four-dimensional lattice-gas model can host non-periodic, quasicrystalline Gibbs states at low temperatures. The authors integrate three components—Gács–Reif stacking to build noise-resilient CA, a NW-deterministic Ammann tile–based CA construction, and the PCA–Gibbs correspondence—to realize ground states corresponding to cloned Ammann tilings and guarantee their stability against fluctuations. They show that, for sufficiently low temperature (large $\beta$), there exist extremal Gibbs states $\nu_z^{(\beta)}$ that remain uniformly close to a standard non-periodic ground state $z$ and exhibit a range-$r$ sea-island pattern of disagreements, ensuring non-periodicity. This provides a concrete, constructive bridge between tiling theory, CA resilience, and Gibbsian statistical mechanics, offering a tractable model for thermally stabilized quasicrystals in discrete settings.

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.

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

TL;DR

This work demonstrates that a finite-range four-dimensional lattice-gas model can host non-periodic, quasicrystalline Gibbs states at low temperatures. The authors integrate three components—Gács–Reif stacking to build noise-resilient CA, a NW-deterministic Ammann tile–based CA construction, and the PCA–Gibbs correspondence—to realize ground states corresponding to cloned Ammann tilings and guarantee their stability against fluctuations. They show that, for sufficiently low temperature (large ), there exist extremal Gibbs states that remain uniformly close to a standard non-periodic ground state and exhibit a range- sea-island pattern of disagreements, ensuring non-periodicity. This provides a concrete, constructive bridge between tiling theory, CA resilience, and Gibbsian statistical mechanics, offering a tractable model for thermally stabilized quasicrystals in discrete settings.

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.
Paper Structure (10 sections, 4 theorems, 11 equations, 1 figure)

This paper contains 10 sections, 4 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Informal description
  3. Ingredients of the construction
  4. Gács--Reif stacking construction
  5. Cellular automaton based on deterministic Wang tiles
  6. PCA and space-time Gibbs states
  7. Construction: combining the above ingredients
  8. Quasicrystalline Gibbs states at low temperatures
  9. Discussion
  10. Acknowledgments.

Key Result

Theorem 3.1

Let $F\colon\Sigma^{\mathbb{Z}}\to\Sigma^{\mathbb{Z}}$ be an arbitrary CA and $\tilde{F}$ the stacked version of $F$ as described above. For every $\delta>0$, there exists $\varepsilon>0$ such that for every configuration $x\in\Sigma^{\mathbb{Z}}$, if $X$ is any $\varepsilon$-perturbed trajectory of

Figures (1)

  • Figure 1: A sample space-time trajectory of the CA associated with Ammann's aperiodic Wang tiles. The diagram is tilted to make it look like a tiling. Circles indicate blanks.

Theorems & Definitions (5)

  • Theorem 3.1: Gács--Reif reliable simulation GR1988
  • Theorem 3.2: PCA trajectories vs. Gibbs states DK1984GKL+1989
  • Theorem 5.1
  • proof
  • Theorem 5.2: Quasicrystalline Gibbs states