  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 $c > 1$ 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.
