Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λc($\mathbb{T}_{\Delta}$) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented anFPTASfor the partition function when λ < λc($\mathbb{T}_{\Delta}$) for graphs with constant maximum degree Δ. In contrast, Sly showed that for all Δ ⩾ 3, there exists εΔ> 0 such that (unless RP = NP) there is noFPRASfor approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λc($\mathbb{T}_{\Delta}$) < λ < λc($\mathbb{T}_{\Delta}$) + εΔ.We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava and Thurley (2014) extended Weitz's approach to the antiferromagnetic Ising model, yielding anFPTASfor the partition function for all graphs of constant maximum degree Δ when the parameters of the model lie in the uniqueness region of the infinite Δ-regular tree. We prove the complementary result for the antiferromagnetic Ising model without external field, namely, that unless RP = NP, for all Δ ⩾ 3, there is noFPRASfor approximating the partition function on graphs of maximum degree Δ when the inverse temperature lies in the non-uniqueness region of the infinite tree$\mathbb{T}_{\Delta}$. Our proof works by relating certain second moment calculations for random Δ-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.