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.

中文翻译：

---

布尔格子中间两层中的独立集

对于一个奇数整数 $ñ=2d-\mathrm{1个}$，让 $\mathcal{乙}（ñ，d）$ 成为超立方体的子图 ${问}_{ñ}$由两个最大的层诱导。在本文中，我们描述了独立集的典型结构$\mathcal{乙}（ñ，d）$并给出精确的渐近线。证明使用Sapozhenko的图容器方法以及Jenssen和Perkins的最新开发的方法，该方法将Sapozhenko的图容器引理与用于统计物理的聚合物模型的聚类扩展相结合。