三角形和六边形格子上的高密度硬核模型
High-density hard-core model on triangular and hexagonal lattices
摘要：
我们进行了严格的研究的吉布斯统计的高密度硬核随机配置的单位三角形晶格$\mathbb{A}_2$和单位蜂窝图$\mathbb{H}_2$，任何值的（欧几里德）排斥直径$D>0$。只有可达到的$D$值是相关的，$D^2=a^2+b^2+ab$、$a, b \in\mathbb{Z}$（洛希数）。根据算术性质的$D^2$，我们确定，大逸度，纯相（极端吉布斯措施），并指定其对称性。答案取决于边长为$D$的等边三角形如何内接于$\mathbb{A}_2$或$\mathbb{H}_2$。在$\mathbb{A}_2$上，我们的方法适用于所有可达到的$D^2$ ;在$\mathbb{H}_2$上，我们必须排除$D^2 = 4, 7, 31, 133$，其中发生滑动现象，类似于单位正方形晶格$\mathbb{Z}^2$。对于所有的值$D^2$除了被排除的，我们证明了存在的一阶相变共存的纯相的数量增长至少为$O(D^2)$。证明是基于Pirogov-西奈理论，它需要非平凡的关键假设验证：有限的周期性基态和Peierls界的集合。为了建立Peierls界，我们开发了一个通用的方法的概念的基础上重新分布的区域Delaunay三角形。一些提出的证明是计算机辅助的。作为基态识别的副产品，我们解决了磁盘包装问题的$\mathbb{A}_2$和$\mathbb{H}_2$的任何值的磁盘直径$D$。
英文摘要：
We perform a rigorous study of the Gibbs statistics of high-density hard-core random configurations on a unit triangular lattice $\mathbb{A}_2$ and a unit honeycomb graph $\mathbb{H}_2$, for any value of the (Euclidean) repulsion diameter $D>0$. Only attainable values of $D$ are relevant, for which $D^2=a^2+b^2+ab$, $a, b \in\mathbb{Z}$ (Löschian numbers). Depending on arithmetic properties of $D^2$, we identify, for large fugacities, the pure phases (extreme Gibbs measures) and specify their symmetries. The answers depend on the way(s) an equilateral triangle of side-length $D$ can be inscribed in $\mathbb{A}_2$ or $\mathbb{H}_2$. On $\mathbb{A}_2$, our approach works for all attainable $D^2$; on $\mathbb{H}_2$ we have to exclude $D^2 = 4, 7, 31, 133$, where a sliding phenomenon occurs, similar to that on a unit square lattice $\mathbb{Z}^2$. For all values $D^2$ apart from the excluded ones we prove the existence of a first-order phase transition where the number of co-existing pure phases grows at least as $O(D^2)$. The proof is based on the Pirogov--Sinai theory which requires non-trivial verifications of key assumptions: finiteness of the set of periodic ground states and the Peierls bound. To establish the Peierls bound, we develop a general method based on the concept of a re-distributed area for Delaunay triangles. Some of the presented proofs are computer-assisted. As a by-product of the ground state identification, we solve the disk-packing problem on $\mathbb{A}_2$ and $\mathbb{H}_2$ for any value of the disk diameter $D$.
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