The partition function of the Ising model of a graph $G=(V,E)$ is defined as $Z_{\text{Ising}}(G;b)=\sum_{\sigma:V\to \{0,1\}} b^{m(\sigma)}$, where $m(\sigma)$ denotes the number of edges $e=\{u,v\}$ such that $\sigma(u)=\sigma(v)$. We show that for any positive integer $\Delta$ and any graph $G$ of maximum degree at most $\Delta$, $Z_{\text{Ising}}(G;b)\neq 0$ for all $b\in \mathbb{C}$ satisfying $|\frac{b-1}{b+1}| \leq \frac{1-o_\Delta(1)}{\Delta-1}$ (where $o_\Delta(1) \to 0$ as $\Delta\to \infty$). This is optimal in the sense that $\tfrac{1-o_\Delta(1)}{\Delta-1}$ cannot be replaced by $\tfrac{c}{\Delta-1}$ for any constant $c > 1$ unless P=NP. To prove our result we use a standard reformulation of the partition function of the Ising model as the generating function of even sets. We establish a zero-free disk for this generating function inspired by techniques from statistical physics on partition functions of a polymer models. Our approach is quite general and we discuss extensions of it to a certain types of polymer models.

翻译：图$G=(V,E)$的Ising模型配分函数定义为$Z_{\text{Ising}}(G;b)=\sum_{\sigma:V\to \{0,1\}} b^{m(\sigma)}$，其中$m(\sigma)$表示满足$\sigma(u)=\sigma(v)$的边$e=\{u,v\}$的数量。我们证明：对于任意正整数$\Delta$和任意最大度不超过$\Delta$的图$G$，当复数$b$满足$|\frac{b-1}{b+1}| \leq \frac{1-o_\Delta(1)}{\Delta-1}$（其中$o_\Delta(1) \to 0$当$\Delta\to \infty$）时，$Z_{\text{Ising}}(G;b)\neq 0$。该结果在以下意义下是最优的：除非P=NP，否则$\tfrac{1-o_\Delta(1)}{\Delta-1}$不能被任意常数$c>1$对应的$\tfrac{c}{\Delta-1}$所替代。为证明该结果，我们采用Ising模型配分函数的标准重述形式，将其转化为偶集合的生成函数。受统计物理学中聚合物模型配分函数技术的启发，我们为该生成函数建立了一个无零圆盘。该方法具有广泛适用性，并讨论了其向特定类型聚合物模型的扩展。