Random three-dimensional tilings of Aztec octahedra and tetrahedra: an extension of domino tilings
Random three-dimensional tilings of Aztec octahedra and tetrahedra: an extension of domino tilings
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阿兹特克八面体和四面体的随机三维平铺：多米诺骨牌平铺的扩展
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发表时间：
2000
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通讯作者：
Gary D. Yngve
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作者：
Dana Randall;Gary D. Yngve
我们提出了一个扩展的多米诺平铺平面晶格区域的三维。镶嵌包括填充“阿兹特克”八面体和四面体区域与三角棱柱。所述平铺块集合双射地对应于平面区域的有序多米诺平铺块集合，其中所述多米诺平铺块被迫遵守基于高度函数表示的偏序。我们定义了一个自然的马尔可夫链上的一组平铺，并证明它是快速混合。这是第一个非平凡的证明快速混合的马尔可夫链的配置对应于一个4维高度函数。基于马尔可夫链的模拟表明，一类八面体和四面体区域将具有类似于北极圈定理的冻结区域，该定理指出，阿兹特克钻石随机镶嵌的非冻结区域收敛到一个圆。接下来，我们证明了对于第二类镶嵌，八面体和四面体镶嵌将具有相等的熵。这是令人惊讶的，因为它们分别对应于正方形和阿兹特克钻石区域的有序镶嵌，已知这两个区域在二维中具有不同的熵。
We present an extension of domino tilings of planar lattice regions to three dimensions. The tilings consist of filling "Aztec" octahedral and tetrahedral regions with triangular prisms. The set of tilings corresponds bijectively to a set of ordered domino tilings of planar regions, where the domino tilings are forced to respect a partial order based on a height function representation. We define a natural Markov Chain on the set of tilings and prove that it is rapidly mixing. This is the first nontrivial proof of rapid mixing for a Markov chain on configurations which correspond to a 4-dimensional height function. Simulations based on this Markov chain have shown that a class of octahedral and tetrahedral regions will have frozen regions akin to-the arctic circle theorem, which states that the nonfrozen regions of random tilings of the Aztec diamond converge to a circle. Next, we show that for a second class of tihngs, the octahedral and tetrahedral tilings will have equal entropy. This is surprising because they correspond to ordered tilings of square and Aztec diamond regions, respectively, which are known to have different entropy in two dimensions.