We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both $2$-spin and multi-spin systems. As applications for this approach: $\bullet$ We prove the optimal $O(n)$ relaxation time for the Glauber dynamics of random $q$-list-coloring on an $n$-vertices triangle-tree graph with maximum degree $\Delta$ such that $q/\Delta > \alpha^\star$, where $\alpha^\star \approx 1.763$ is the unique positive solution of the equation $\alpha = \exp(1/\alpha)$. This improves the $n^{1+o(1)}$ relaxation time for Glauber dynamics obtained by the previous work of Jain, Pham, and Vuong (2022). Besides, our framework can also give a near-linear time sampling algorithm under the same condition. $\bullet$ We prove the optimal $O(n)$ relaxation time and near-optimal $\widetilde{O}(n)$ mixing time for the Glauber dynamics on hardcore models with parameter $\lambda$ in $\textit{balanced}$ bipartite graphs such that $\lambda < \lambda_c(\Delta_L)$ for the max degree $\Delta_L$ in left part and the max degree $\Delta_R$ of right part satisfies $\Delta_R = O(\Delta_L)$. This improves the previous result by Chen, Liu, and Yin (2023). At the heart of our proof is the notion of $\textit{coupling independence}$ which allows us to consider multiple vertices as a huge single vertex with exponentially large domain and do a "coarse-grained" local-to-global argument on spin systems. The technique works for general (multi) spin systems and helps us obtain some new comparison results for Glauber dynamics.

翻译：我们开发了一个新框架来证明无界度数自旋系统上Glauber动力学混合时间或弛豫时间。该框架适用于一般自旋系统，包括二自旋与多自旋系统。作为此方法的应用：$\bullet$ 我们证明了在满足 $q/\Delta > \alpha^\star$ 条件下（其中 $\alpha^\star \approx 1.763$ 是方程 $\alpha = \exp(1/\alpha)$ 的唯一正解），$n$ 个顶点的三角树图上随机 $q$-列表着色Glauber动力学具有最优的 $O(n)$ 弛豫时间，该图的最大度数为 $\Delta$。这改进了Jain、Pham和Vuong（2022）先前工作中获得的 $n^{1+o(1)}$ 弛豫时间。此外，在相同条件下，我们的框架还能给出一个近线性时间的采样算法。$\bullet$ 我们证明了在参数 $\lambda$ 满足 $\lambda < \lambda_c(\Delta_L)$ 且右部最大度数 $\Delta_R$ 满足 $\Delta_R = O(\Delta_L)$ 的$\textit{平衡}$二分图上，硬核模型的Glauber动力学具有最优的 $O(n)$ 弛豫时间与近最优的 $\widetilde{O}(n)$ 混合时间，其中 $\Delta_L$ 为左部最大度数。这改进了Chen、Liu和Yin（2023）的先前结果。我们证明的核心是$\textit{耦合独立性}$的概念，它允许我们将多个顶点视为一个具有指数大定义域的巨型单顶点，并在自旋系统上进行“粗粒度”的局部到全局论证。该技术适用于一般（多）自旋系统，并帮助我们获得了Glauber动力学的一些新比较结果。