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

Abstract

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.