Domino tilings and flips in dimensions 4 and higher

In this paper we consider domino tilings of bounded regions in dimension $n \geq 4$. We define the twist of such a tiling, an elements of ${\mathbb{Z}}/(2)$, and prove it is invariant under flips, a simple local move in the space of tilings. We investigate which regions $D$ are regular, i.e. whenever two tilings $t_0$ and $t_1$ of $D \times [0,N]$ have the same twist then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. We prove that all boxes are regular except $D = [0,2]^3$. Furthermore, given a regular region $D$, we show that there exists a value $M$ (depending only on $D$) such that if $t_0$ and $t_1$ are tilings of equal twist of $D \times [0,N]$ then the corresponding tilings can be joined by a finite sequence of flips in $D \times [0,N+M]$. As a corollary we deduce that, for regular $D$ and large $N$, the set of tilings of $D \times [0,N]$ has two twin giant components under flips, one for each value of the twist.