论文标题
在布尔晶格的中间两层中的独立集
Independent sets in the middle two layers of Boolean lattice
论文作者
论文摘要
对于奇数整数$ n = 2d-1 $，令$ \ mathcal {b}（n，d）$是两个最大层引起的hypercube $ q_n $的子图。在本文中，我们描述了$ \ Mathcal {b}（n，d）$中独立集的典型结构，并在其数量上提供精确的渐近性。这些证明使用Sapozhenko的图形容器方法以及最近开发的Jenssen和Perkins的方法，该方法将Sapozhenko的图形容器引理与来自统计物理学的聚合物模型的群集扩展相结合。
For an odd integer $n=2d-1$, let $\mathcal{B}(n, d)$ be the subgraph of the hypercube $Q_n$ induced by the two largest layers. In this paper, we describe the typical structure of independent sets in $\mathcal{B}(n, d)$ and give precise asymptotics on the number of them. The proofs use Sapozhenko's graph container method and a recently developed method of Jenssen and Perkins, which combines Sapozhenko's graph container lemma with the cluster expansion for polymer models from statistical physics.
