We consider spin systems on general $n$-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen--Wang dynamics for the $q$-state ferromagnetic Potts model on graphs of maximum degree $\Delta$, where $\Delta$ is allowed to grow with $n$, converges in $O((\Delta \log n)^c)$ steps where $c > 0$ is a constant independent of $\Delta$ and $n$. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is $O(\Delta^c \log n)$ for a constant $c>0$ independent of $\Delta$ and $n$. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. Our result implies optimal $O(\log n)$ mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called $k$-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph.

翻译：我们考虑一般 $n$ 顶点无界度图上的自旋系统，并探究谱独立性对全局马尔可夫链收敛至平衡态速率的影响。谱独立性是一种量化自旋系统模型中相关性衰减的新方法，它极大地推进了自旋系统马尔可夫链的研究。我们证明：当谱独立性成立时，最大度为 $\Delta$（$\Delta$ 可随 $n$ 增长）的图上 $q$ 态铁磁 Potts 模型的流行 Swendsen--Wang 动力学在 $O((\Delta \log n)^c)$ 步内收敛，其中 $c>0$ 为独立于 $\Delta$ 和 $n$ 的常数。我们还证明了在谱独立性假设下，一般自旋系统的块动力学具有类似的混合时间界。最后，对于单调自旋系统（如二分图上的 Ising 模型和硬核模型），我们证明谱独立性意味着系统扫描动力学的混合时间为 $O(\Delta^c \log n)$，其中常数 $c>0$ 独立于 $\Delta$ 和 $n$。系统扫描动力学应用广泛但极难分析。我们的结果表明：在树唯一性阈值以内的一般图上，铁磁 Ising 模型的任意系统扫描动力学均达到最优 $O(\log n)$ 混合时间界。我们的主要技术贡献是对熵泛函的改进分解：这是所有证明的共同出发点。具体而言，我们建立了所谓的熵的 $k$ 部分解，其常数依赖于图最大度的多项式函数。