论文标题
多米诺骨牌砖和翻转4及更高的尺寸
Domino tilings and flips in dimensions 4 and higher
论文作者
论文摘要
在本文中，我们考虑了尺寸$ n \ geq 4 $的有限区域的多米诺骨牌瓷砖。我们定义了这样的瓷砖的扭曲，$ {\ mathbb {z}}/（2）$的元素，并证明它在翻转下是不变的，这是一个简单的本地移动。 我们调查了哪些区域$ d $是常规的，即每当两个瓷砖$ t_0 $和$ t_1 $的$ d \ times [0，n] $具有相同的扭曲，然后$ t_0 $和$ t_1 $可以通过一系列翻转加入，前提是允许一些额外的垂直空间。我们证明所有盒子都是常规的，除了$ d = [0,2]^3 $。 此外，鉴于常规的区域$ d $，我们表明存在一个值$ m $（仅取决于$ d $），因此，如果$ t_0 $和$ t_1 $是$ d \ times [0，n] $的瓷砖，那么相应的瓷砖可以通过$ d \ d \ times times times $ d \ times n n n+m]的有限范围来加入。作为推论，我们推断出，对于常规的$ d $和大型$ n $，$ d \ times [0，n] $的瓷砖集有两个双重巨型组件，一个twin the the the the Twist的值。
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.
