The Complexity of Approximating the Complex-Valued Ising Model on Bounded Degree Graphs

We study the complexity of approximating partition function $Z_{\mathrm{Ising}}(G; \beta)$ Ising model in terms relation between edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on maximum degree input graph $G$. Following recent trends both statistical physics algorithmic research, we allow to be any complex number. Many results focus parameters, because physical relevance key role case delineating tractability/intractability phase transition approximation problem. In this work establish new tractability intractability results. Our show that $Z_{\mathrm{Ising}}(-; has FPTAS when $\lvert \beta - 1 \rvert / \lvert + < \tan(\pi (4 \Delta 4))$. The core proof showing there are no inputs $G$ make $0$ range. result significantly extends known zero-free region (and hence results). it $\mathrm{\#P}$-hard multiplicatively approximate norm additively argument $\beta \in \mathbb{C}$ algebraic number such \not \mathbb{R} \cup \{i, -i\}$ 1\rvert > \sqrt{\Delta 1}$. These first $Z_{\mathrm{Ising}}(-, bounded graphs with $\beta$. Moreover, demonstrate situations zeros imply hardness model.