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 2x2x1 rectangular cuboid. We are particularly interested in regions of the form $R_N = D \times [0,N]$ where $D$ is a fixed quadriculated disk. In dimension 3, the twist associates to each tiling $t$ an integer $Tw(t)$. 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 $D$ is regular if, whenever two tilings $t_0$ and $t_1$ of $R_N$ satisfy $Tw(t_0) = Tw(t_1)$, $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. Many large disks are regular, including rectangles $D = [0,L] \times [0,M]$ with $LM$ even and $L,M \ge 3$. For regular disks, we describe the larger connected components under flips of the set of tilings of the region $R_N = D \times [0,N]$. As a corollary, let $p_N$ be the probability that two random tilings $T_0$ and $T_1$ of $D \times [0,N]$ can be joined by a sequence of flips conditional to their twists being equal. Then $p_N$ tends to 1 if and only if $D$ is regular. Under a suitable equivalence relation, the set of tilings has a group structure, the domino group. These results illustrate the fact that the domino group dictates many properties of the space of tilings of the cylinder $R_N = D \times [0,N]$, particularly for large $N$.