In a seminal paper, Weitz showed that for two-state spin systems, such as the Ising and hardcore models from statistical physics, correlation decay on trees implies correlation decay on arbitrary graphs. The key gadget in Weitz's reduction has been instrumental in recent advances in approximate counting and sampling, from analysis of local Markov chains like Glauber dynamics to the design of deterministic algorithms for estimating the partition function. A longstanding open problem in the field has been to find such a reduction for more general multispin systems like the uniform distribution over proper colorings of a graph. In this paper, we show that for a rich class of multispin systems, including the ferromagnetic Potts model, there are fundamental obstacles to extending Weitz's reduction to the multispin setting. A central component of our investigation is establishing nonconvexity of the image of the belief propagation functional, the standard tool for analyzing spin systems on trees. On the other hand, we provide evidence of convexity for the antiferromagnetic Potts model.

翻译：在一篇开创性论文中，Weitz证明了对于二态自旋系统（如统计物理学中的Ising模型和硬核模型），树上的关联衰减意味着任意图上的关联衰减。Weitz约简中的核心构造方法在近似计数与采样的最新进展中发挥了关键作用，其应用范围从分析Glauber动力学等局部马尔可夫链，到设计用于估算配分函数的确定性算法。该领域长期存在的开放问题是如何为更一般的多自旋系统（如图的合法着色均匀分布）建立此类约简。本文证明，对于包含铁磁Potts模型在内的丰富多自旋系统类，将Weitz约简扩展到多自旋设定存在根本性障碍。我们研究的核心环节是确立置信传播泛函（分析树上自旋系统的标准工具）像集的非凸性。另一方面，我们为反铁磁Potts模型提供了凸性存在的证据。