A near-optimal zero-free disk for the Ising model

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 $\mathrm{\Delta }$ and any graph $G$ of maximum degree at most $\mathrm{\Delta }$, $Z_{\text{Ising}}(G;b)\mathrm{\ne }0$ for all $b\in \mathbb{C}$ satisfying $\mathrm{\mid }\frac{b-1}{b+1}\mathrm{\mid }\le \frac{1-o_{\mathrm{\Delta }}(1)}{\mathrm{\Delta }-1}$ (where $o_{\mathrm{\Delta }}(1)\to 0$ as $\mathrm{\Delta }\to \mathrm{\infty }$). This is optimal in the sense that $\frac{1-o_{\mathrm{\Delta }}(1)}{\mathrm{\Delta }-1}$ cannot be replaced by $\frac{c}{\mathrm{\Delta }-1}$ for any constant $c>1$ subject to a complexity theoretic assumption. 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.