Decagonal random tilings and their quasi-unit cell cluster coverings

Electron microscopy images of decagonal quasicrystals obtained recently have been shown to be related to cluster coverings with a Hexagon–Bow–Tie decagon as single structural unit. Most decagonal phases show more complex structural orderings than models based on deterministic tilings like the Penrose tiling. We analyze different types of decagonal random tilings and their coverings by a Hexagon–Bow–Tie decagon.

Extreme boundary conditions and random tilings

Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.]

Domino Tilings of Cylinders: Connected Components under Flips and Normal Distribution of the Twist

We consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a $2\times 1\times 1$ rectangular cuboid. We are particularly interested in regions of the form $\mathcal{R}_N = \mathcal{D} \times [0,N]$ where $\mathcal{D} \subset \mathbb{R}^2$ is a fixed quadriculated disk. In dimension $3$, the twist associates to each tiling $\mathbf{t}$ an integer $\operatorname{Tw}(\mathbf{t})$. We prove that, when $N$ goes to infinity, the twist follows a normal distribution. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is invariant under flips. A quadriculated disk $\mathcal{D}$ is regular if, whenever two tilings $\mathbf{t}_0$ and $\mathbf{t}_1$ of $\mathcal{R}_N$ satisfy $\operatorname{Tw}(\mathbf{t}_0) = \operatorname{Tw}(\mathbf{t}_1)$, $\mathbf{t}_0$ and $\mathbf{t}_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. Many large disks are regular, including rectangles $\mathcal{D} = [0,L] \times [0,M]$ with $LM$ even and $\min\{L,M\} \ge 3$. For regular disks, we describe the larger connected components under flips of the set of tilings of the region $\mathcal{R}_N = \mathcal{D} \times [0,N]$. As a corollary, let $p_N$ be the probability that two random tilings $\mathbf{T}_0$ and $\mathbf{T}_1$ of $\mathcal{D} \times [0,N]$ can be joined by a sequence of flips conditional to their twists being equal. Then $p_N$ tends to $1$ if and only if $\mathcal{D}$ is regular. Under a suitable equivalence relation, the set of tilings has a group structure, the {\em domino group} $G_{\mathcal{D}}$. These results illustrate the fact that the domino group dictates many properties of the space of tilings of the cylinder $\mathcal{R}_N = \mathcal{D} \times [0,N]$, particularly for large $N$.

Highly symmetric aperiodic structures -INVITED

The symmetries of periodic structures are severely constrained by the crystallographic restriction. In particular, in two and three spatial dimensions, only rotational axes of order 1, 2, 3, 4 or 6 are possible. Aperiodic tilings can provide perfectly ordered structures with arbitrary symmetry properties. Random tilings can retain part of the aperiodic order as well the rotational symmetry. They offer a more flexible approach to obtain homogeneous structures with high rotational symmetry, and might be of particular interest for applications. Some key examples and their diffraction are discussed.

Tilings: mathematical models for quasicrystals

This chapter discusses tilings as mathematical models for quasicrystals. In a first approximation quasicrystals may be described as being space filling with copies of two or more types of tiles. This description gives a connection with the mathematical notion of tilings, which have been well studied. A brief introduction of tilings is presented in this chapter along with the method of substitution to create aperiodic tilings. The symmetry of the tilings is also treated in this chapter, as are model sets and random tilings. Quasiperiodic crystals often have approximants, that is, periodic structures that are close to the aperiodic ones. The relations between quasiperiodic crystals and approximants also is described in this chapter.