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 $\Delta$, $Z_{\text{Ising}}(G;b)\ne 0$ for all $b\in \mathbb{C}$ satisfying $|\frac{b-1}{b+1}|\le \frac{1-o_{\Delta }(1)}{\Delta -1}$ (where $o_{\Delta }(1)\to 0$ as $\Delta \to \infty$). This is optimal in the sense that $\frac{1-o_{\Delta }(1)}{\Delta -1}$ cannot be replaced by $\frac{c}{\Delta -1}$ 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.

12 pages; we have added a few propositions in Section 2 and reorganized the section to clarify the proof of Lemma 3.1. Some other small modifications have also been made as per suggestion of two referees