The hardcore model is one of the most classic and widely studied examples of undirected graphical models. Given a graph $G$, the hardcore model describes a Gibbs distribution of $λ$-weighted independent sets of $G$. In the last two decades, a beautiful computational phase transition has been established at a precise threshold $λ_c(Δ)$ where $Δ$ denotes the maximum degree, where the task of sampling independent sets transitions from polynomial-time solvable to computationally intractable. We study the critical hardcore model where $λ= λ_c(Δ)$ and show that the Glauber dynamics, a simple yet popular Markov chain algorithm, mixes in $\tilde{O}(n^{4+O(1/Δ)})$ time on any $n$-vertex graph of maximum degree $Δ\geq3$, significantly improving the previous upper bound $\tilde{O}(n^{12.88+O(1/Δ)})$ by the recent work arXiv:2411.03413. Our improvement comes from an optimal bound on the $\ell_\infty$-spectral independence for the hardcore model at all subcritical fugacity $λ< λ_c(Δ)$.

翻译：硬核模型是无向图模型中最经典且被广泛研究的示例之一。给定图 $G$，硬核模型描述了 $G$ 上 $λ$ 加权的独立集的吉布斯分布。过去二十年间，研究在精确阈值 $λ_c(Δ)$ 处（其中 $Δ$ 表示最大度）建立了一个优美的计算相变：在该阈值处，独立集采样任务从多项式时间可解转变为计算不可行。我们研究临界硬核模型，即 $λ= λ_c(Δ)$ 的情形，并证明 Glauber 动力学（一种简单而流行的马尔可夫链算法）在任何最大度 $Δ\geq3$ 的 $n$ 顶点图上的混合时间为 $\tilde{O}(n^{4+O(1/Δ)})$，这显著改进了近期工作 arXiv:2411.03413 所给出的先前上界 $\tilde{O}(n^{12.88+O(1/Δ)})$。我们的改进源于对硬核模型在所有亚临界逸度 $λ< λ_c(Δ)$ 下 $\ell_\infty$-谱独立性的最优界。