Domino tilings and flips in dimensions 4 and higher

TL;DR

This paper explores domino tilings in four or more dimensions, introducing a twist invariant, analyzing regular regions, and demonstrating the connectivity of tilings via flips, revealing a structure of giant components distinguished by the twist.

Contribution

It introduces the twist invariant for domino tilings in higher dimensions and characterizes regular regions, showing how tilings connect through flips and the existence of giant components.

Findings

All boxes are regular except [0,2]^3.

Tilings of regular regions can be connected with a bounded number of flips.

Two giant components exist for large N, distinguished by twist.

Abstract

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.