多项式镶嵌模型

摘要：

给定图$G$和子图$T$（称为瓦片）的集合，我们考虑用$T$中的瓦片的副本覆盖$G$，使得每个顶点$v\in G$被预定的重数覆盖。多项式镶嵌模型是此类配置的自然概率测度（它是$G$相应“放大”的标准镶嵌的统一度量）。在大重数的极限下，我们计算了多项式分块数的渐近增长率。我们表明，个别瓦片密度趋于高斯场相对于一个相关的离散拉普拉斯算子。当我们改变多重数时，我们也发现了一个精确的离散库仑气体极限。对于${\mathbb Z}^d$的平铺与翻译的一个单一的瓷砖和一个小密度的缺陷，我们研究了结晶现象时，缺陷密度趋于零，并给出了自然发生的准晶在这个框架内的例子。

英文摘要：

Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural probability measure on such configurations (it is the uniform measure on standard tilings of the corresponding "blow-up" of $G$). In the limit of large multiplicities we compute asymptotic growth rate of the number of multinomial tilings. We show that the individual tile densities tend to a Gaussian field with respect to an associated discrete Laplacian. We also find an exact discrete Coulomb gas limit when we vary the multiplicities. For tilings of ${\mathbb Z}^d$ with translates of a single tile and a small density of defects, we study a crystallization phenomena when the defect density tends to zero, and give examples of naturally occurring quasicrystals in this framework.

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