We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows $O(n\log{n})$ mixing time on any $n$-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity $\lambda$, we establish $O(n\log{n})$ mixing time for the Glauber dynamics on any $n$-vertex graph of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\Delta)$ where $\lambda_c(\Delta)$ is the critical point for the uniqueness/non-uniqueness phase transition on the $\Delta$-regular tree. More generally, for any antiferromagnetic 2-spin system we prove $O(n\log{n})$ mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain $O(n\log{n})$ mixing for $q$-colorings of triangle-free graphs of maximum degree $\Delta$ when the number of colors satisfies $q > \alpha \Delta$ where $\alpha \approx 1.763$, and $O(m\log{n})$ mixing for generating random matchings of any graph with bounded degree and $m$ edges.

翻译：我们证明了在多种设置下，单点位更新马尔可夫链（即Glauber动力学或吉布斯采样）的最优混合时间界。本研究改进了Anari等人（2020年）的谱独立性方法，表明当关联影响矩阵的最大特征值有界时，任意n顶点有界度图上的混合时间为$O(n\log{n})$。作为结果的应用：针对以逸度$\lambda$加权的独立集硬核模型，当$\lambda<\lambda_c(\Delta)$（其中$\lambda_c(\Delta)$为$\Delta$-正则树上唯一性/非唯一性相变临界点）时，我们证明了任意最大度$\Delta$为常数的n顶点图上的Glauber动力学混合时间为$O(n\log{n})$。更一般地，对于任意反铁磁双自旋系统，我们在对应树唯一性区域中证明了有界度图上Glauber动力学的$O(n\log{n})$混合时间。我们的结果具有更广泛的适用性：例如，当颜色数满足$q > \alpha \Delta$（其中$\alpha \approx 1.763$）时，我们得到最大度为$\Delta$的无三角形图上$q$着色的$O(n\log{n})$混合时间；对于任意有界度图及$m$条边的随机匹配生成，我们得到$O(m\log{n})$混合时间。