Strong spatial mixing (SSM) is an important quantitative notion of correlation decay for Gibbs distributions arising in statistical physics, probability theory, and theoretical computer science. A longstanding conjecture is that the uniform distribution on proper $q$-colorings on a $\Delta$-regular tree exhibits SSM whenever $q \ge \Delta+1$. Moreover, it is widely believed that as long as SSM holds on bounded-degree trees with $q$ colors, one would obtain an efficient sampler for $q$-colorings on all bounded-degree graphs via simple Markov chain algorithms. It is surprising that such a basic question is still open, even on trees, but then again it also highlights how much we still have to learn about random colorings. In this paper, we show the following: (1) For any $\Delta \ge 3$, SSM holds for random $q$-colorings on trees of maximum degree $\Delta$ whenever $q \ge \Delta + 3$. Thus we almost fully resolve the aforementioned conjecture. Our result substantially improves upon the previously best bound which requires $q \ge 1.59\Delta+\gamma^*$ for an absolute constant $\gamma^* > 0$. (2) For any $\Delta\ge 3$ and girth $g = \Omega_\Delta(1)$, we establish optimal mixing of the Glauber dynamics for $q$-colorings on graphs of maximum degree $\Delta$ and girth $g$ whenever $q \ge \Delta+3$. Our approach is based on a new general reduction from spectral independence on large-girth graphs to SSM on trees that is of independent interest. Using the same techniques, we also prove near-optimal bounds on weak spatial mixing (WSM), a closely-related notion to SSM, for the antiferromagnetic Potts model on trees.

翻译：强空间混合（SSM）是统计物理、概率论和理论计算机科学中衡量吉布斯分布相关衰减的重要定量概念。一个长期存在的猜想是：当$q \ge \Delta+1$时，$\Delta$-正则树上关于正则$q$-染色的均匀分布具有SSM性质。此外，普遍认为只要在有界度树上$q$种颜色满足SSM，就能通过简单的马尔可夫链算法为所有有界度图的$q$-染色问题构造高效采样器。令人惊讶的是，这个基础性问题至今仍未解决——即便在树上也是如此，但这恰恰凸显了我们对随机染色问题的理解仍存在重大空白。本文证明以下结论：（1）对任意$\Delta \ge 3$，当$q \ge \Delta + 3$时，最大度为$\Delta$的树上的随机$q$-染色具有SSM性质。这几乎完全解决了前述猜想。我们的结果显著改进了此前需要$q \ge 1.59\Delta+\gamma^*$（其中$\gamma^* > 0$为绝对常数）的最佳上界。（2）对任意$\Delta\ge 3$和围长$g = \Omega_\Delta(1)$，当$q \ge \Delta+3$时，我们建立了最大度为$\Delta$且围长为$g$的图上$q$-染色Glauber动力学的最优混合性。我们的方法基于一个具有独立意义的新泛化约简：将大围长图上的谱独立性约化为树上的SSM。利用相同技术，我们还证明了树上反铁磁Potts模型弱空间混合（WSM，与SSM密切相关的概念）的近似最优上界。