论文标题
圆柱体的多米诺骨牌：翻转下连接的组件和扭曲的正态分布
Domino tilings of cylinders: connected components under flips and normal distribution of the twist
论文作者
论文摘要
我们认为多米诺骨牌的瓷砖为$ 3 $维二维的立方体区域。三维多米诺骨牌是2x2x1矩形核心。我们对$ r_n = d \ times [0，n] $的区域的区域特别感兴趣，其中$ d $是固定的四个磁盘。在Dimension 3中，扭曲与整数$ T $ T $ TW（T）$相关。我们证明，当$ n $进入无限时，转折将遵循正态分布。 翻转是局部移动：将两个相邻的平行多米诺骨牌移除并放回不同位置。扭曲是在翻转下不变的。如果每当两个瓷砖$ t_0 $和$ t_1 $ r_n $满足$ TW（T_0）= TW（T_1）$，$ T_0 $和$ T_1 $可以通过一系列flip，则可以通过一系列翻转，只要允许一些额外的垂直空间，就可以连接一个二次磁盘$ d $。许多大磁盘是常规的，包括矩形$ d = [0，l] \ times [0，m] $，均为$ lm $，$ l，m \ ge 3 $。对于常规磁盘，我们描述了该区域瓷砖集$ r_n = d \ times [0，n] $的较大连接组件。作为推论，令$ p_n $是两个随机瓷砖$ t_0 $和$ t_1 $ d \ d \ times [0，n] $的可能性，可以通过一系列flips来加入，条件是曲折相等的。然后，$ p_n $趋于1时，并且仅当$ d $是常规的。 在适当的等价关系下，一组瓷砖具有组结构，即多米诺骨牌。这些结果说明了一个事实，即多米诺骨牌组决定了圆柱体砖的空间的许多属性$ r_n = d \ times [0，n] $，尤其是对于大$ n $。
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 $N$.
