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 $λ>0$, the hard-core model on $G$ at activity $λ$ is the probability distribution $μ_{G,λ}$ on independent sets in $G$ given by $μ_{G,λ}(I)\propto λ^{|I|}$. As one of our main applications, we show that the hard-core model at activity $λ$ on the hypercube $Q_d$ exhibits a `structured phase' for $λ= Ω( \log^2 d/d^{1/2})$ in the following sense: in a typical sample from $μ_{Q_d,λ}$, 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 $λ=Ω(\log d/ d^{1/3})$. As another application, we establish a fully polynomial-time approximation scheme (FPTAS) for the hard-core model on a $d$-regular bipartite $α$-expander, with $α>0$ fixed, when $λ= Ω( \log^2 d/d^{1/2})$. This improves upon the bound $λ=Ω(\log d/ d^{1/4})$ 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：我们建立了由Galvin提出的图容器引理的一个改进版本，并讨论了其在二分扩展图硬核模型中的若干应用。给定图$G$与参数$λ>0$，$G$上活性为$λ$的硬核模型是定义在$G$的独立集上的概率分布$μ_{G,λ}$，满足$μ_{G,λ}(I)\propto λ^{|I|}$。作为主要应用之一，我们证明超立方体$Q_d$上活性为$λ$的硬核模型在$λ= Ω( \log^2 d/d^{1/2})$时呈现"结构化相"：在$μ_{Q_d,λ}$的典型样本中，大多数顶点位于$Q_d$二分划分的同一侧。这改进了Galvin先前对$λ=Ω(\log d/ d^{1/3})$情形得到的结果。作为另一应用，我们为$d$-正则二分$α$-扩展图（固定$α>0$）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$λ= Ω( \log^2 d/d^{1/2})$。这改进了由第一作者、Perkins和Potukuchi先前获得的$λ=Ω(\log d/ d^{1/4})$边界。我们还讨论了该方法对Galvin-Tetali、Balogh-Garcia-Li及Kronenberg-Spinka相关结果的类似改进。