AI 中文总结
研究WSK动力学混合时间，通过相对熵收缩开发新工具，引入ACTE和ACSE标准及新归纳方法，在弦图和外平面图上为其建立最优混合时间，改进已知界限。
AI 中文摘要
我们研究了用于均匀采样恰当\(q\)着色的Wang-Swendsen-Kotecký（WSK）动力学的混合时间。WSK动力学在统计物理中广泛用于从反铁磁Potts模型采样，可视为翻转动力学的全局对应物。尽管其重要性，但分析此类动力学的工具有限。我们开发新工具，通过相对熵收缩来分析WSK动力学的混合时间。引入多自旋分布的新标准：熵的近似颜色方向张量化（ACTE）和熵的近似颜色方向次可加性（ACSE）。还为特定类型图建立这些标准开发新的归纳方法。具体应用中，在弦图和外平面图上为WSK动力学建立了最优\(O_q(\log n)\)混合时间。
英文摘要
We study the mixing time of Wang-Swendsen-Kotecký (WSK) dynamics for uniformly sampling proper $q$-colorings. The WSK dynamics is widely used in statistical physics for sampling from the antiferromagnetic Potts model and can be considered a global counterpart of the flip dynamics, which currently yields the state-of-the-art bounds for sampling colorings in general graphs (Carlson and Vigoda, SODA 2025). However, despite its importance, the tools for analyzing such dynamics remain limited. We develop new tools that enable us to analyze the mixing time of the WSK dynamics through the lens of relative entropy contraction. We introduce new criteria for multi-spin distributions: approximate colorwise tensorization of entropy (ACTE) and approximate colorwise subadditivity of entropy (ACSE). These criteria provide a colorwise counterpart to standard vertex-wise entropy factorization principles, and expose a form of color symmetry beyond coordinate-wise analyses. We also develop new inductive approaches for establishing such criteria on specific types of graphs, which can be viewed as local-to-global arguments for proving high-dimensional functional inequalities in a graph-theoretic sense. As concrete applications, we establish an optimal $O_q(\log n)$ mixing time for the WSK dynamics on chordal and outerplanar graphs, down to the optimal number of colors. Because trees and line graphs of trees are chordal, the result covers both vertex and edge colorings of trees. Our results work in a regime that bypasses the irreducibility threshold for Glauber dynamics while also improving the best known mixing time bounds (Carlson, Chen, Feng and Vigoda, SODA 2025).
