Domino tilings of cylinders: connected components under flips and normal distribution of the twist

TL;DR

This paper studies 3D domino tilings of cylindrical regions, showing that the twist distribution becomes normal as the height increases, and explores the connectivity of tilings under flips, especially for regular disks.

Contribution

It introduces the concept of regular disks, analyzes the distribution of the twist, and describes the connected components of tilings under flips, linking these to the domino group.

Findings

Twist follows a normal distribution as N approaches infinity.

For regular disks, tilings with the same twist are connected via flips with high probability.

The domino group influences the structure and connectivity of tilings in cylindrical regions.

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 $N$.