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：在任何 $q \ge \Delta+1$ 的条件下，$\Delta$-正则树上正常 $q$-着色均匀分布具有SSM性质。更广泛地认为，只要在$q$种颜色下有界度树满足SSM，就能通过简单马尔可夫链算法为所有有界度图的有效$q$-着色采样器提供支撑。令人惊讶的是，即使对于树这一基础结构，此类基本问题仍悬而未决，这凸显了我们对随机着色问题的认知鸿沟。本文证明如下结果：(1) 对所有 $\Delta \ge 3$，当 $q \ge \Delta + 3$ 时，最大度为 $\Delta$ 的树上随机 $q$-着色具有SSM性质。该结果几乎完全解决了上述猜想，显著改进了此前要求 $q \ge 1.59\Delta+\gamma^*$（其中 $\gamma^*>0$ 为绝对常数）的最优边界。(2) 对所有 $\Delta\ge 3$ 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 $q$-着色Glauber动力学的最优混合性。我们的方法基于一项具有独立价值的新通用归约：将大围长图的谱独立性转化为树上的SSM。运用相同技术，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。