We present a simple combinatorial framework for establishing approximate tensorization of variance and entropy in the setting of spin systems (a.k.a. undirected graphical models) based on balanced separators of the underlying graph. Such approximate tensorization results immediately imply as corollaries many important structural properties of the associated Gibbs distribution, in particular rapid mixing of the Glauber dynamics for sampling. We prove approximate tensorization by recursively establishing block factorization of variance and entropy with a small balanced separator of the graph. Our approach goes beyond the classical canonical path method for variance and the recent spectral independence approach, and allows us to obtain new rapid mixing results. As applications of our approach, we show that: 1. On graphs of treewidth $t$, the mixing time of the Glauber dynamics is $n^{O(t)}$, which recovers the recent results of Eppstein and Frishberg with improved exponents and simpler proofs; 2. On bounded-degree planar graphs, strong spatial mixing implies $\tilde{O}(n)$ mixing time of the Glauber dynamics, which gives a faster algorithm than the previous deterministic counting algorithm by Yin and Zhang.

翻译：我们提出一种简单的组合框架，用于在自旋系统（即无向图模型）中基于底层图的平衡分割建立方差与熵的近似张量化。此类近似张量化的结果直接推论出相关吉布斯分布的许多重要结构性质，尤其是用于采样的格劳伯动力学的快速混合性。我们通过递归建立图的小平衡分割上的方差与熵的块分解来证明近似张量化。我们的方法超越了经典的方差规范路径方法和最近的谱独立性方法，使我们能够获得新的快速混合结果。作为我们方法的应用，我们表明：1. 在树宽为 $t$ 的图上，格劳伯动力学混合时间为 $n^{O(t)}$，这以改进的指数和更简单的证明重现了Eppstein和Frishberg的最新结果；2. 在有界度平面图上，强空间混合性蕴含格劳伯动力学 $\tilde{O}(n)$ 的混合时间，这比Yin和Zhang之前的确定性计数算法提供了更快的算法。