The Annals of Applied Probability

Ising models on locally tree-like graphs

Amir Dembo and Andrea Montanari

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We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the “cavity” prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.

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Ann. Appl. Probab., Volume 20, Number 2 (2010), 565-592.

First available in Project Euclid: 9 March 2010

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Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 82B23: Exactly solvable models; Bethe ansatz 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20] 05C05: Trees

Ising model random sparse graphs cavity method Bethe measures belief propagation local weak convergence


Dembo, Amir; Montanari, Andrea. Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 (2010), no. 2, 565--592. doi:10.1214/09-AAP627.

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  • [1] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508 (electronic).
  • [2] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures (H. Kesten, ed.). Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
  • [3] Bandyopadhyay, A. and Gamarnik, D. (2007). 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] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.
  • [5] Dembo, A., Gerschenfeld, A. and Montanari, A. (2010). Spin glasses on locally tree-like graphs. To appear.
  • [6] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
  • [7] De Sanctis, L. and Guerra, F. (2008). Mean field dilute ferromagnet: High temperature and zero temperature behavior. J. Stat. Phys. 132 759–785.
  • [8] Dorogovtsev, S. N., Goltsev, A. V. and Mendes, J. F. F. (2002). Ising model on networks with an arbitrary distribution of connections. Phys. Rev. E 66 016104.
  • [9] Ellis, R. S. and Newman, C. M. (1978). The statistics of Curie–Weiss models. J. Statist. Phys. 19 149–161.
  • [10] Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410–433.
  • [11] Gerschenfeld, A. and Montanari, A. (2007). Reconstruction for models on random graphs. In Proceedings of the 48th Annual Symposium on Foundations of Computer Science 194–204. IEEE, New York.
  • [12] Griffiths, R. B., Hurst, C. A. and Sherman, S. (1970). Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11 790–795.
  • [13] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
  • [14] Jerrum, M. and Sinclair, A. (1990). Polynomial-time approximation algorithms for the Ising model. In Automata, Languages and Programming. Lecture Notes in Computer Science 443 462–475. Springer, New York.
  • [15] Johnston, D. A. and Plecháč, P. (1998). Equivalence of ferromagnetic spin models on trees and random graphs. J. Phys. A 31 475–482.
  • [16] Jonasson, J. and Steif, J. E. (1999). Amenability and phase transition in the Ising model. J. Theoret. Probab. 12 549–559.
  • [17] Krz̧akała, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G. and Zdeborová, L. (2007). Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. USA 104 10318–10323 (electronic).
  • [18] Leone, M., Vázquez, A., Vespignani, A. and Zecchina, R. (2002). Ferromagnetic ordering in graphs with arbitrary degree distribution. Eur. Phys. J. B 28 191–197.
  • [19] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [20] Lyons, R. (2000). Phase transitions on nonamenable graphs. J. Math. Phys. 41 1099–1126.
  • [21] Mézard, M. and Montanari, A. (2009). Information, Physics, and Computation. Oxford Univ. Press, Oxford.
  • [22] Mooij, J. M. and Kappen, H. J. (2005). On the properties of the Bethe approximation and loopy belief propagation on binary networks. J. Stat. Mech. P11012.
  • [23] Salez, J. and Shah, D. (2009). Belief propagation: An asymptotically optimal algorithm for the random assignment problem. Available at arXiv:0902.0585.
  • [24] Sudderth, E., Wainwright, M. and Willsky (2008). Loop series and Bethe variational bounds in atractive graphical models. In Advances in Neural Information Processing Systems 1425–1432. MIT Press, Cambridge, MA.
  • [25] Simon, B. (1980). Correlation inequalities and the decay of correlations in ferromagnets. Comm. Math. Phys. 77 111–126.
  • [26] Tatikonda, S. and Jordan, M. I. (2002). Loopy belief propagation and Gibbs measures. In Proccedings of the 18th Conference on Uncertainty in Artificial Intelligence 493–450. Morgan Kaufmann, San Francisco, CA.