论文标题
平面晶格的硬核模型：磁盘包装问题和高密度阶段
The hard-core model on planar lattices: the disk-packing problem and high-density phases
论文作者
论文摘要
我们研究了磁盘和相关吉布斯分布的致密堆积，这些分布代表了单位三角形，蜂窝和方形晶格的硬核模型中的高密度相。该模型的特征是欧几里得排除距离$ d> 0 $和fugacity $ u> 0 $的值。我们使用Pirogov-Sinai理论来研究$ u $大的$ d $的Gibbs分布：$ u> u_0（d）$。对于无限序列$ d $，我们描述了一个完整的高密度相图：它表现出许多共存的纯阶段，它们的数字增长为$ O（d^2）$。对于$ d $的其余值，除了滑动的值外，共存的纯阶段数量仍然是$ e（d）\ geq o（d^2）$的表格；但是，纯相的确切识别需要额外的分析。对许多典型示例进行了这样的分析，其中涉及计算机辅助的证据。因此，对于所有值$ d> 0 $，在不发生滑动的情况下，我们确定了相变的存在。 研究中的关键步骤是（i）确定周期基态和（ii）PEIERLS结合的验证。这是通过使用代数数理论的连接来完成的。特别是，已指定了所谓的滑动值的完整列表。作为副产品，我们在考虑的晶格上解决了磁盘包装问题。与$ \ Mathbb {r}^2 $的情况不同，最大密度包装的数量和结构取决于磁盘直径$ d $。
We study dense packings of disks and related Gibbs distributions representing high-density phases in the hard-core model on unit triangular, honeycomb and square lattices. The model is characterized by a Euclidean exclusion distance $D>0$ and a value of fugacity $u>0$. We use the Pirogov-Sinai theory to study the Gibbs distributions for a general $D$ when $u$ is large: $u>u_0(D)$. For infinite sequences of values $D$ we describe a complete high-density phase diagram: it exhibits a multitude of co-existing pure phases, and their number grows as $O(D^2)$. For the remaining values of $D$, except for those with sliding, the number of co-existing pure phases is still of the form $E(D)\geq O(D^2)$; however, the exact identification of the pure phases requires an additional analysis. Such an analysis is performed for a number of typical examples, which involves computer-assisted proofs. Consequently, for all values $D>0$ where sliding does not occur, we establish the existence of a phase transition. The crucial steps in the study are (i) the identification of periodic ground states and (ii) the verification of the Peierls bound. This is done by using connections with algebraic number theory. In particular, a complete list of so-called sliding values of $D$ has been specified. As a by-product, we solve the disk-packing problem on the lattices under consideration. The number and structure of maximally-dense packings depend on the disk-diameter $D$, unlike the case of $\mathbb{R}^2$.
