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，$\Delta$-正则树上正常$q$-染色均匀分布具有SSM性质。此外，广泛认为只要在最大度有界树上对$q$色染色满足SSM，就能通过简单马尔可夫链算法为所有最大度有界图的高效$q$-染色采样器。令人惊讶的是，即使对于树这种基本情形，该问题仍未解决，这也凸显出我们对随机染色仍知之甚少。本文证明了以下结论：(1) 对于任意$\Delta \ge 3$，当$q \ge \Delta + 3$时，最大度为$\Delta$的树上随机$q$-染色满足SSM。这几乎完全解决了前述猜想。我们的结果显著改进了此前最优界（需要$q \ge 1.59\Delta+\gamma^*$，其中$\gamma^* > 0$为绝对常数）。(2) 对于任意$\Delta\ge 3$和围长$g = \Omega_\Delta(1)$，当$q \ge \Delta+3$时，我们在最大度为$\Delta$且围长为$g$的图上建立了Glauber动力学的$q$-染色最优混合。我们的方法基于从大围长图上的谱独立性到树上SSM的通用新归约，该归约本身具有独立意义。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）的近最优界。