We derive the strong spatial mixing property for the general 2-spin system from zero-free regions of its partition function. We view the partition function of the 2-spin system as a multivariate function over three complex parameters $(\beta, \gamma, \lambda)$, and we allow the zero-free regions of $\beta, \gamma$ or $\lambda$ to be of arbitrary shapes. As long as the zero-free region contains a positive point and it is a complex neighborhood of $\lambda=0$ when fixing $\beta, \gamma \in \mathbb{C}$, or a complex neighborhood of $\beta\gamma=1$ when fixing $\beta, \lambda\in \mathbb{C}$ or $\gamma, \lambda\in \mathbb{C}$ respectively, we are able to show that the corresponding 2-spin system exhibits strong spatial mixing on such a region. The underlying graphs of the 2-spin system are not necessarily of bounded degree, while are required to include graphs with pinned vertices. We prove this result by establishing a Christoffel-Darboux type identity for the 2-spin system on trees and using certain tools from complex analysis. To our best knowledge, our result is general enough to turn all currently known zero-free regions of the partition function of the 2-spin system where pinned vertices are allowed into the strong spatial mixing property. Moreover, we extend our result to obtain strong spatial mixing for the ferromagnetic Ising model (even with non-uniform external fields) from the celebrated Lee-Yang circle theorem.

翻译：我们通过配分函数的零自由区域推导出一般二自旋系统的强空间混合性质。将二自旋系统的配分函数视为关于三个复参数$(\beta, \gamma, \lambda)$的多元函数，并允许$\beta$、$\gamma$或$\lambda$的零自由区域具有任意形状。只要零自由区域包含一个正点，且在固定$\beta, \gamma \in \mathbb{C}$时为$\lambda=0$的复邻域，或在分别固定$\beta, \lambda\in \mathbb{C}$或$\gamma, \lambda\in \mathbb{C}$时为$\beta\gamma=1$的复邻域，我们即能证明对应的二自旋系统在该区域上表现出强空间混合性质。二自旋系统的底层图不要求有界度数，但需包含带钉扎顶点的图。我们通过在树上建立二自旋系统的Christoffel-Darboux型恒等式并运用复分析工具来证明该结果。据我们所知，该结果具有足够普遍性，可将当前已知的所有允许钉扎顶点的二自旋系统配分函数零自由区域转化为强空间混合性质。此外，我们还将结果扩展至铁磁伊辛模型（即便具有非均匀外场），从著名的李-杨圆定理推导出其强空间混合性质。