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

We consider three-dimensional domino tilings of cylinders $R_N = D \times [0,N]$ where $D \subset \mathbb{R}^2$ is a fixed quadriculated disk and $N \in \mathbb{N}$. A flip is a local move in the space of tilings $\mathcal{T}(R_N)$: 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 $D$, called regular, two tilings of $R_N$ with the same twist can be joined by a sequence of flips once we add vertical space to the cylinder. We have that if $D$ is regular then the size of the largest connected component under flips of $\mathcal{T}(R_N)$ is $\Theta(N^{-\frac{1}{2}}|\mathcal{T}(R_N)|)$. The domino group $G_D$ captures information of the space of tilings. It is known that $D$ is regular if and only if $G_{D}$ 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 $D$ often implies irregularity. In many, but not all, of these cases, we also prove that $D$ is strongly irregular, i.e., that there exists a surjective homomorphism from $G_{D}^+$ (a subgroup of index two of $G_{D}$) to the free group of rank two. Moreover, we show that if $D$ is strongly irregular then the cardinality of the largest connected component under flips of $\mathcal{T}(R_N)$ is $O(c^N |\mathcal{T}(R_N)|)$ for some $c \in (0,1)$.Comment: 35 pages, 29 figure