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

Abstract

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 $N$.