We revisit Sapozhenko's classic proof on the asymptotics of the number of independent sets in the discrete hypercube ${\{0,1\}}^d$ and Galvin's follow‐up work on weighted independent sets. We combine Sapozhenko's graph container methods with the cluster expansion and abstract polymer models, two tools from statistical physics, to obtain considerably sharper asymptotics and detailed probabilistic information about the typical structure of (weighted) independent sets in the hypercube. These results refine those of Korshunov and Sapozhenko and Galvin, and answer several questions of Galvin.

中文翻译：

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重访超立方体中的独立集

我们回顾一下Sapozhenko关于离散超立方体中独立集合数量渐近性的经典证明 ${\{0，\mathrm{1个}\}}^d$以及Galvin关于加权独立集的后续工作。我们将Sapozhenko的图容器方法与统计物理中的两个工具—聚类展开和抽象聚合物模型相结合，以获取更清晰的渐近性和有关超立方体中（加权）独立集的典型结构的详细概率信息。这些结果完善了Korshunov，Sapozhenko和Galvin的结果，并回答了Galvin的几个问题。