We establish a refined version of a graph container lemma due to Galvin and discuss several applications related to the hard-core model on bipartite expander graphs. Given a graph $G$ and $\lambda>0$, the hard-core model on $G$ at activity $\lambda$ is the probability distribution $\mu_{G,\lambda}$ on independent sets in $G$ given by $\mu_{G,\lambda}(I)\propto \lambda^{|I|}$. As one of our main applications, we show that the hard-core model at activity $\lambda$ on the hypercube $Q_d$ exhibits a `structured phase' for $\lambda= \Omega( \log^2 d/d^{1/2})$ in the following sense: in a typical sample from $\mu_{Q_d,\lambda}$, most vertices are contained in one side of the bipartition of $Q_d$. This improves upon a result of Galvin which establishes the same for $\lambda=\Omega(\log d/ d^{1/3})$. As another application, we establish a fully polynomial-time approximation scheme (FPTAS) for the hard-core model on a $d$-regular bipartite $\alpha$-expander, with $\alpha>0$ fixed, when $\lambda= \Omega( \log^2 d/d^{1/2})$. This improves upon the bound $\lambda=\Omega(\log d/ d^{1/4})$ due to the first author, Perkins and Potukuchi. We discuss similar improvements to results of Galvin-Tetali, Balogh-Garcia-Li and Kronenberg-Spinka.

翻译：我们建立了Galvin提出的图容器引理的一个精细化版本，并讨论了其在二分扩展图上硬核模型相关的若干应用。给定图$G$与参数$\lambda>0$，$G$上活性为$\lambda$的硬核模型是定义在$G$的独立集上的概率分布$\mu_{G,\lambda}$，满足$\mu_{G,\lambda}(I)\propto \lambda^{|I|}$。作为主要应用之一，我们证明超立方体$Q_d$上活性为$\lambda$的硬核模型在$\lambda= \Omega( \log^2 d/d^{1/2})$时呈现"结构化相"：在$\mu_{Q_d,\lambda}$的典型样本中，绝大多数顶点位于$Q_d$二分划分的同一侧。该结果改进了Galvin先前对$\lambda=\Omega(\log d/ d^{1/3})$情形的结论。作为另一应用，我们为$d$正则二分$\alpha$-扩展图（固定$\alpha>0$）上的硬核模型建立了完全多项式时间近似方案（FPTAS），其适用范围为$\lambda= \Omega( \log^2 d/d^{1/2})$。这改进了第一作者、Perkins与Potukuchi先前获得的$\lambda=\Omega(\log d/ d^{1/4})$界限。我们进一步讨论了该方法对Galvin-Tetali、Balogh-Garcia-Li及Kronenberg-Spinka相关结果的类似改进。