## The Annals of Probability

### Factor models on locally tree-like graphs

#### Abstract

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $\phi$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $\phi$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $\phi$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on $\phi$.

In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on $T$. In the special case that $T$ has a Galton–Watson law, this formula coincides with the nonrigorous “Bethe prediction” obtained by statistical physicists using the “replica” or “cavity” methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.

#### Article information

Source
Ann. Probab., Volume 41, Number 6 (2013), 4162-4213.

Dates
First available in Project Euclid: 20 November 2013

https://projecteuclid.org/euclid.aop/1384957785

Digital Object Identifier
doi:10.1214/12-AOP828

Mathematical Reviews number (MathSciNet)
MR3161472

Zentralblatt MATH identifier
1280.05119

#### Citation

Dembo, Amir; Montanari, Andrea; Sun, Nike. Factor models on locally tree-like graphs. Ann. Probab. 41 (2013), no. 6, 4162--4213. doi:10.1214/12-AOP828. https://projecteuclid.org/euclid.aop/1384957785

#### References

• [1] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.
• [2] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
• [3] Bandyopadhyay, A. and Gamarnik, D. (2006). Counting without sampling. New algorithms for enumeration problems using statistical physics. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms 890–899. ACM, New York.
• [4] Bandyopadhyay, A. and Gamarnik, D. (2008). Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models. Random Structures Algorithms 33 452–479.
• [5] Bayati, M., Gamarnik, D. and Tetali, P. (2010). Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. In STOC’10—Proceedings of the 2010 ACM International Symposium on Theory of Computing 105–114. ACM, New York.
• [6] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic).
• [7] Biskup, M., Borgs, C., Chayes, J. T. and Kotecký, R. (2000). Gibbs states of graphical representations of the Potts model with external fields. J. Math. Phys. 41 1170–1210.
• [8] Cook, S. A. (1971). The complexity of theorem-proving procedures. In STOC ’71 Proceedings of the Third Annual ACM Symposium on Theory of Computing 151–158. ACM, New York.
• [9] Dembo, A. and Montanari, A. (2010). Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24 137–211.
• [10] Dembo, A. and Montanari, A. (2010). Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 565–592.
• [11] Dembo, A., Montanari, A., Sly, A. and Sun, N. (2012). The replica symmetric solution for Potts models on $d$-regular graphs. Preprint. Available at http://arxiv.org/abs/1207.5500.
• [12] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
• [13] Dommers, S., Giardinà, C. and van der Hofstad, R. (2010). Ising models on power-law random graphs. J. Stat. Phys. 141 638–660.
• [14] Dyson, F. J. (1969). Existence of a phase-transition in a one-dimensional Ising ferromagnet. Comm. Math. Phys. 12 91–107.
• [15] Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410–433.
• [16] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random-cluster model. I. Introduction and relation to other models. Physica 57 536–564.
• [17] Franz, S. and Leone, M. (2003). Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111 535–564.
• [18] Franz, S., Leone, M. and Toninelli, F. L. (2003). Replica bounds for diluted non-Poissonian spin systems. J. Phys. A 36 10967–10985.
• [19] Galanis, A., Štefankovič, D. and Vigoda, E. (2012). Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. Preprint. Available at http://arxiv.org/abs/1203.2226v2.
• [20] 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.
• [21] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
• [22] Greenhill, C. (2000). The complexity of counting colourings and independent sets in sparse graphs and hypergraphs. Comput. Complexity 9 52–72.
• [23] Grimmett, G. (2009). Correlation inequalities of GKS type for the Potts model. Preprint. Available at arXiv:0901.1625.
• [24] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften 333. Springer, Berlin.
• [25] Guerra, F. and Toninelli, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 71–79.
• [26] Hammersley, J. M. (1962). Generalization of the fundamental theorem on sub-additive functions. Proc. Cambridge Philos. Soc. 58 235–238.
• [27] Higuchi, Y. (1977/78). Remarks on the limiting Gibbs states on a $(d+1)$-tree. Publ. Res. Inst. Math. Sci. 13 335–348.
• [28] Karp, R. M. (1972). Reducibility among combinatorial problems. In Complexity of Computer Computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, NY, 1972) 85–103. Plenum, New York.
• [29] Kelly, F. P. (1985). Stochastic models of computer communication systems (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 47 379–395, 415–428.
• [30] Lawler, E. L., Lenstra, J. K. and Rinnooy Kan, A. H. G. (1980). Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9 558–565.
• [31] Liggett, T. M. (2005). Interacting Particle Systems. Springer, Berlin.
• [32] Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931–958.
• [33] Mézard, M. and Montanari, A. (2009). Information, Physics, and Computation. Oxford Univ. Press, Oxford.
• [34] 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.
• [35] 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.
• [36] Panchenko, D. and Talagrand, M. (2004). Bounds for diluted mean-fields spin glass models. Probab. Theory Related Fields 130 319–336.
• [37] Riva, V. and Cardy, J. (2006). Holomorphic parafermions in the Potts model and stochastic Loewner evolution. J. Stat. Mech. Theory Exp. 12 P12001, 19 pp. (electronic).
• [38] Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. W. A. Benjamin, New York-Amsterdam.
• [39] Simon, B. (1979). A remark on Dobrushin’s uniqueness theorem. Comm. Math. Phys. 68 183–185.
• [40] Sly, A. (2010). Computational transition at the uniqueness threshold. 2010 IEEE 51st Annual Symposium on Foundations of Computer Science 287–296. Las Vegas, USA.
• [41] Sly, A. and Sun, N. (2012). The computational hardness of counting in two-spin models on $d$-regular graphs. Preprint. Available at http://arxiv.org/abs/1203.2602.
• [42] Sokal, A. D. (2005). The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In Surveys in Combinatorics 2005. London Mathematical Society Lecture Note Series 327 173–226. Cambridge Univ. Press, Cambridge.
• [43] Spitzer, F. (1975). Markov random fields on an infinite tree. Ann. Probab. 3 387–398.
• [44] 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.
• [45] Wu, F. Y. (1982). The Potts model. Rev. Modern Phys. 54 235–268.
• [46] Yedidia, J. S., Freeman, W. T. and Weiss, Y. (2005). Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. Inform. Theory 51 2282–2312.
• [47] Zachary, S. (1983). Countable state space Markov random fields and Markov chains on trees. Ann. Probab. 11 894–903.
• [48] Zinn-Justin, J. (1993). Quantum Field Theory and Critical Phenomena, 2nd ed. International Series of Monographs on Physics 85. The Clarendon Press Oxford Univ. Press, New York.
• [49] Zuckerman, D. (2006). Linear degree extractors and the inapproximability of max clique and chromatic number. In STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing 681–690. ACM, New York.