## The Annals of Probability

### Counting in two-spin models on d-regular graphs

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 30 September 2014

https://projecteuclid.org/euclid.aop/1412083628

Digital Object Identifier
doi:10.1214/13-AOP888

Mathematical Reviews number (MathSciNet)
MR3265170

Zentralblatt MATH identifier
1311.60117

#### Citation

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. https://projecteuclid.org/euclid.aop/1412083628

#### References

• [1] Achlioptas, D. and Naor, A. (2005). The two possible values of the chromatic number of a random graph. Ann. of Math. (2) 162 1335–1351.
• [2] Achlioptas, D., Naor, A. and Peres, Y. (2005). Rigorous location of phase transitions in hard optimization problems. Nature 435 759–764.
• [3] Achlioptas, D., Naor, A. and Peres, Y. (2007). On the maximum satisfiability of random formulas. J. ACM 54 Art. 10, 21 pp. (electronic).
• [4] Alimonti, P. and Kann, V. (1997). Hardness of approximating problems on cubic graphs. In Algorithms and Complexity (Rome, 1997). Lecture Notes in Computer Science 1203 288–298. Springer, Berlin.
• [5] Bethe, H. A. (1935). Statistical theory of superlattices. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 150 552–575.
• [6] Dembo, A. and Montanari, A. (2010). Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24 137–211.
• [7] Dembo, A. and Montanari, A. (2010). Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 565–592.
• [8] Dembo, A., Montanari, A., Sly, A. and Sun, N. (2014). The replica symmetric solution for Potts models on $d$-regular graphs. Comm. Math. Phys. 327 551–575.
• [9] Dembo, A., Montanari, A. and Sun, N. (2013). Factor models on locally tree-like graphs. Ann. Probab. 41 4162–4213.
• [10] Dommers, S., Giardinà, C. and van der Hofstad, R. (2010). Ising models on power-law random graphs. J. Stat. Phys. 141 638–660.
• [11] Dyer, M., Frieze, A. and Jerrum, M. (1999). On counting independent sets in sparse graphs. In 40th Annual Symposium on Foundations of Computer Science (New York, 1999) 210–217. IEEE Comput. Soc., Los Alamitos, CA.
• [12] Dyer, M., Frieze, A. and Jerrum, M. (2002). On counting independent sets in sparse graphs. SIAM J. Comput. 31 1527–1541.
• [13] Galanis, A., Ge, Q., Štefankovič, D., Vigoda, E. and Yang, L. (2011). Improved inapproximability results for counting independent sets in the hard-core model. In Approximation, Randomization, and Combinatorial Optimization (L. Goldberg, K. Jansen, R. Ravi and J. Rolim, eds.). Lecture Notes in Computer Science 6845 567–578. Springer, Heidelberg.
• [14] Galanis, A., Štefankovič, D. and Vigoda, E. (2012). Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. Available at arXiv:1203.2226v2.
• [15] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
• [16] Jerrum, M. and Sinclair, A. (1993). Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22 1087–1116.
• [17] Li, L., Lu, P. and Yin, Y. (2013). Correlation decay up to uniqueness in spin systems. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 47–66. Available at arXiv:1111.7064v2.
• [18] Luby, M. and Vigoda, E. (1999). Fast convergence of the Glauber dynamics for sampling independent sets. Random Structures Algorithms 15 229–241.
• [19] Montanari, A., Mossel, E. and Sly, A. (2012). The weak limit of Ising models on locally tree-like graphs. Probab. Theory Related Fields 152 31–51.
• [20] Mossel, E., Weitz, D. and Wormald, N. (2009). On the hardness of sampling independent sets beyond the tree threshold. Probab. Theory Related Fields 143 401–439.
• [21] Motwani, R. and Raghavan, P. (1995). Randomized Algorithms. Cambridge Univ. Press, Cambridge.
• [22] Sinclair, A., Srivastava, P. and Thurley, M. (2012). Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms 941–953. SIAM, Philadelphia, PA.
• [23] Sly, A. (2010). Computational transition at the uniqueness threshold. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science FOCS 2010 287–296. IEEE Comput. Soc., Los Alamitos, CA.
• [24] Weitz, D. (2006). Counting independent sets up to the tree threshold. In STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing 140–149. ACM, New York.
• [25] Zachary, S. (1983). Countable state space Markov random fields and Markov chains on trees. Ann. Probab. 11 894–903.