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

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 $\mathrm{\Theta }(N^{-\frac{1}{2}}\mathrm{\mid }\mathcal{T}({\mathcal{R}}_N)\mathrm{\mid })$. 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 $\mathbb{Z}\oplus \mathbb{Z}\mathrm{/}(2)$; 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 $O(c^N\mathrm{\mid }\mathcal{T}({\mathcal{R}}_N)\mathrm{\mid })$ for some $c\in (0,1)$.