Ising models on locally tree-like graphs

Annals of Applied Probability

Volumn 20, Issue 2, 2010, Pages 565-592

  Dembo, Amir ^a   Montanari, Andrea ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

Belief propagation; Bethe measures; Cavity method; Ising model; Local weak convergence; Random sparse graphs

Indexed keywords

EID: 77949678834     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/09-AAP627     Document Type: Article

References (26)

1. ALDOUS, D. and LYONS, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454-1508 (electronic). MR2354165.
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. MR2023650.
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. MR2373815.
4. BOLLOBÁS, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311-316. MR595929.
5. DEMBO, A., GERSCHENFELD, A. and MONTANARI, A. (2010). Spin glasses on locally treelike 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. MR1619036.
7. DE SANCTIS, L. and GUERRA, F. (2008). Mean field dilute ferromagnet: High temperature and zero temperature behavior. J. Stat. Phys. 132 759-785. MR2430780.
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. MR0503332.
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. MR1768240.
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. MR0266507.
13. JANSON, S., ŁUCZAK, T. and RUCINSKI, A. (2000). Random Graphs. Wiley, New York. MR1782847.
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. MR1629170.
16. JONASSON, J. and STEIF, J. E. (1999). Amenability and phase transition in the Ising model. J. Theoret. Probab. 12 549-559. MR1684757.
17. KRZA̧KAŁA, 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). MR2317690.
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. MR776231.
20. LYONS, R. (2000). Phase transitions on nonamenable graphs. J. Math. Phys. 41 1099-1126. MR1757952.
21. MÉZARD, M. and MONTANARI, A. (2009). Information, Physics, and Computation. Oxford Univ. Press, Oxford. MR2518205.
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. MR589426.
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.