تقرير
Domino tilings and flips in dimensions 4 and higher
| العنوان: | Domino tilings and flips in dimensions 4 and higher |
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| المؤلفون: | Klivans, Caroline, Saldanha, Nicolau C. |
| سنة النشر: | 2020 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, 05B45, 52C20, 52C22, 05C70 |
| الوصف: | 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. Comment: 30 pages, 15 figures. Minor changes from previous version: added a new Remark and corresponding reference, added a figure to clarify a proof |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2007.08474 |
| رقم الأكسشن: | edsarx.2007.08474 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |