We establish that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting “free energy density” which coincides with the (nonrigorous) Bethe prediction of statistical physics. Using this result, we characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs without the use of the second moment method employed in previous works on these questions. As a consequence, we show that for both the hard-core model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on $d$-regular graphs when the model has nonuniqueness on the $d$-regular tree. Together with results of Jerrum–Sinclair, Weitz, and Sinclair–Srivastava–Thurley, this gives an almost complete classification of the computational complexity of homogeneous two-spin systems on bounded-degree graphs.
"Counting in two-spin models on d-regular graphs." Ann. Probab. 42 (6) 2383 - 2416, November 2014. https://doi.org/10.1214/13-AOP888