Open Access
November 2014 Counting in two-spin models on d-regular graphs
Allan Sly, Nike Sun
Ann. Probab. 42(6): 2383-2416 (November 2014). DOI: 10.1214/13-AOP888


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.


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Allan Sly. Nike Sun. "Counting in two-spin models on d-regular graphs." Ann. Probab. 42 (6) 2383 - 2416, November 2014.


Published: November 2014
First available in Project Euclid: 30 September 2014

zbMATH: 1311.60117
MathSciNet: MR3265170
Digital Object Identifier: 10.1214/13-AOP888

Primary: 60K35 , 68Q87 , 82B20

Keywords: anti-ferromagnetic Ising model , Bethe free energy , Gibbs uniqueness threshold , hard-core model , independent sets , locally tree-like graphs

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.42 • No. 6 • November 2014
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