3D domino tilings: irregular disks and connected components under flips

TL;DR

This paper studies three-dimensional domino tilings of cylinders, introducing concepts of regularity and irregularity based on the structure of the tiling space and the domino group, revealing how certain geometric features influence connectivity.

Contribution

It characterizes regular and irregular disks for domino tilings, linking geometric properties to algebraic invariants and connectivity of tiling spaces, and establishes new results on the structure of these spaces.

Findings

Regular disks have domino groups isomorphic to Z ⊕ Z/2

Irregular disks often contain bottlenecks leading to irregularity

Strong irregularity implies exponential decay in connected component size

Abstract

We consider three-dimensional domino tilings of cylinders $\mathcal{R}_N = \mathcal{D} \times [0,N]$ where $\mathcal{D} \subset \mathbb{R}^2$ is a fixed quadriculated disk and $N \in \mathbb{N}$. A domino is a $2 \times 1 \times 1$ brick. A flip is a local move in the space of tilings $\mathcal{T}(\mathcal{R}_N)$: remove two adjacent dominoes and place them back after a rotation. The twist is a flip invariant which associates an integer number to each tiling. For some disks $\mathcal{D}$, called regular, two tilings of $\mathcal{R}_N$ with the same twist can be joined by a sequence of flips once we add vertical space to the cylinder. We have that if $\mathcal{D}$ is regular then the size of the largest connected component under flips of $\mathcal{T}(\mathcal{R}_N)$ is $\Theta(N^{-\frac{1}{2}}|\mathcal{T}(\mathcal{R}_N)|)$. The domino group $G_{\mathcal{D}}$ captures information of the space of tilings. A disk $\mathcal{D}$ is regular if and only if $G_{\mathcal{D}}$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}/(2)$; sufficiently large rectangles are regular. We prove that certain families of disks are irregular. We show that the existence of a bottleneck in a disk $\mathcal{D}$ often implies irregularity. In many, but not all, of these cases, we also prove that $\mathcal{D}$ is strongly irregular, i.e., that there exists a surjective homomorphism from $G_{\mathcal{D}}^+$ (a subgroup of index two of $G_{\mathcal{D}}$) to the free group of rank two. Moreover, we show that if $\mathcal{D}$ is strongly irregular then the cardinality of the largest connected component under flips of $\mathcal{T}(\mathcal{R}_N)$ is $O(c^N |\mathcal{T}(\mathcal{R}_N)|)$ for some $c \in (0,1)$.