Combinatorial Theory

The partition function of the Ising model of a graph \(G=(V,E)\) is defined as \(Z_{\operatorname{Ising}}(G;b)=\sum_{\sigma:V\to \{0,1\}} b^{m(\sigma)}\), 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 \(\Delta\), \(Z_{\operatorname{Ising}}(G;b)\neq 0\) 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 polymer models. Our approach is quite general and we discuss extensions of it to certain types of polymer models.

Mathematics Subject Classifications: 05C31, 82B20, 68W25

Keywords: Ising model, partition function, even set, polymer model, Fisher zeros, approximate counting