The Annals of Probability

Counting in two-spin models on d-regular graphs

Allan Sly and Nike Sun

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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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Ann. Probab., Volume 42, Number 6 (2014), 2383-2416.

First available in Project Euclid: 30 September 2014

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40]

Hard-core model independent sets anti-ferromagnetic Ising model locally tree-like graphs Bethe free energy Gibbs uniqueness threshold


Sly, Allan; Sun, Nike. Counting in two-spin models on d -regular graphs. Ann. Probab. 42 (2014), no. 6, 2383--2416. doi:10.1214/13-AOP888.

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