Brazilian Journal of Probability and Statistics

Gibbs measures and phase transitions on sparse random graphs

Amir Dembo and Andrea Montanari

Full-text: Open access


Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We review this approach and provide some results towards a rigorous treatment of these problems.

Article information

Braz. J. Probab. Stat., Volume 24, Number 2 (2010), 137-211.

First available in Project Euclid: 20 April 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Random graphs Ising model Gibbs measures phase transitions spin models local weak convergence


Dembo, Amir; Montanari, Andrea. Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24 (2010), no. 2, 137--211. doi:10.1214/09-BJPS027.

Export citation


  • [1] Achlioptas, D. and Coja-Oghlan, A. (2008). Algorithmic barriers from phase transitions. In 49th Annual Symposium on Foundations of Computer Science. Philadelphia, PA.
  • [2] Achlioptas, D. and Friedgut, E. (1999). A sharp threshold for k-colorability. Random Structures & Algorithms 14 63–70.
  • [3] Achlioptas, D. and Friedgut, E. (1999). Sharp thresholds of graph properties and the k-SAT Problem, with an appendix by J. Bourgain. Journal of the American Mathematical Society 12 1017–1054.
  • [4] Achlioptas D. and Naor, A. (2004). The two possible values of the chromatic number of a random graph. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing 587–593. ACM, New York.
  • [5] Achlioptas, D., Naor, A. and Peres, Y. (2005). Rigorous location of phase transitions in hard optimization problems. Nature 435 759.
  • [6] Achlioptas, D. and Ricci-Tersenghi, F. (2006). On the solution-space geometry of random constraint satisfaction problems. In Proc. 38th ACM Symposium on Theory Of Computing, Seattle (USA) 130–139. ACM, New York.
  • [7] Aizenman, M. and Warzel, S. (2007). The canopy graph and level statistics for random operators on tree. Mathematical Physics, Analysis and Geometry 9 291–333.
  • [8] 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.) 1–27. Springer, Berlin.
  • [9] Alon, N. and Spencer, J. (2000). The Probabilistic Method. Wiley, New York.
  • [10] Berger, N., Kenyon, C., Mossel, E. and Peres, Y. (2005). Glauber dynamics on trees and hyperbolic graphs. Probability Theory and Related Fields 131 311.
  • [11] Bhamidi, S., Rajagopal, R. and Roch, S. (2009). Network delay inference from additive metrics. In Random Structures Algorithms 35. To appear.
  • [12] Bhatnagar, N., Vera, J., Vigoda, E. and Weitz, D. (2009). Reconstruction for colorings on trees. In SIAM Journal on Discrete Mathematics. To appear.
  • [13] Bleher, P. M., Ruiz, J. and Zagrebnov, V. A. (1995). On the purity of limiting Gibbs state for the Ising model on the Bethe lattice. Journal of Statistical Physics 79 473–482.
  • [14] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labeled regular graphs. European Journal of Combinatorics 1 311–316.
  • [15] Borgs, C., Chayes, J., Mossel, E. and Roch, S. (2006). The Kesten–Stigum reconstruction bound is tight for roughly symmetric binary channels. In Proc. of IEEE FOCS.
  • [16] Caracciolo, S., Parisi, G., Patarnello, S. and Sourlas, N. (1990). 3d Ising spin glass in a magnetic field and mean-field theory. Europhysics Letters 11 783.
  • [17] Cover, T. and Thomas, J. A. (1991). Elements of Information Theory. Wiley, New York.
  • [18] Daskalakis, C., Mossel, E. and Roch, S. (2006). Optimal phylogenetic reconstruction. In STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing 159–168. ACM, New York.
  • [19] Dembo, A. and Montanari, A. (2010). Ising models on locally tree-like graphs. The Annals of Applied Probability. To appear.
  • [20] Dembo, A. and Montanari, A. (2008). Bethe states for graphical models. Preprint.
  • [21] Dembo, A. and Montanari, A. (2008). Unpublished manuscript.
  • [22] Dyer, M., Sinclair, A., Vigoda, E. and Weitz, D. (2004). Mixing in time and space for lattice spin systems: A combinatorial view. Random Structures & Algorithms 24 461–479.
  • [23] Dyson, F. J. (1969). Existence of a phase-transition in a one-dimensional Ising ferromagnet. Communications in Mathematical Physics 12 91–107.
  • [24] Ellis, R. S. and Newman, C. M. (1978). The statistics of Curie–Weiss models. Journal of Statistical Physics 19 149–161.
  • [25] Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (2000). Broadcasting on trees and the Ising model. The Annals of Applied Probability 10 410–433.
  • [26] Franz, S. and Parisi, G. (1995). Recipes for metastable states in spin glasses. J. Physique I 5 1401.
  • [27] Georgii, H.-O. (1988). Gibbs Measures and Phase Transition. de Gruyter, Berlin.
  • [28] Gerschenfeld, A. and Montanari, A. (2007). Reconstruction for models on random graphs. In 48nd Annual Symposium on Foundations of Computer Science. Providence, RI.
  • [29] Ginibre, J. (1970). General formulation of Griffiths’ inequalities. Communications in Mathematical Physics 16 310–328.
  • [30] Griffiths, R. B., Hurst, C. A. and Sherman, S. (1970). Concavity of magnetization of an Ising ferromagnet in a positive external field. Journal of Mathematical Physics 11 790–795.
  • [31] Grimmett, G. (1999). Percolation. Springer, New York.
  • [32] Guerra, F. and Toninelli, F. L. (2004). The high temperature region of the Viana–Bray diluted spin glass model. Journal of Statistical Physics 115 531–555.
  • [33] Guionnet, A. and Zegarlinski, B. (2002). Lectures on logarithmic Sobolev inequalities. Séminaire de Probabilites de Strasbourg 36 1–134.
  • [34] Janson, S., Luczak, T. and Ruciński, A. (2000). Random Graphs. Wiley, New York.
  • [35] Krzakala, F. and Zdeborova, L. (2007). Phase transitions in the coloring of random graphs. Physical Review E 76 031131.
  • [36] Krzakala, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G. and Zdeborova, L. (2007). Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences of the United States of America 104 10318.
  • [37] Liggett, T. (1985). Interacting Particle Systems. Springer, New York.
  • [38] Martinelli, F., Sinclair, A. and Weitz, D. (2003). The Ising model on trees: Boundary conditions and mixing time. In Proc. of IEEE FOCS.
  • [39] Mézard, M. and Montanari, A. (2009). Information, Physics and Computation. Oxford Univ. Press, Oxford.
  • [40] Mézard, M. and Montanari, A. (2006). Reconstruction on trees and spin glass transition. Journal of Statistical Physics 124 1317–1350.
  • [41] Mézard, M., Mora, T. and Zecchina, R. (2005). Clustering of solutions in the random satisfiability problem. Physical Review Letters 94 197–205.
  • [42] Mézard, M. and Parisi, G. (1999). Thermodynamics of glasses: A first principles computation. Physical Review Letters 82 747–751.
  • [43] Mézard, M., Parisi, G. and Virasoro, M. A. (1987). Spin Glass Theory and Beyond. World Scientific, Teaneck, NJ.
  • [44] Mézard, M., Parisi, G. and Zecchina, R. (2002). Analytic and algorithmic solution of random satisfiability problems. Science 297 812–815.
  • [45] Montanari, A., Restrepo, R. and Tetali, P. (2009). Reconstruction and clustering in random constraint satisfaction problems. SIAM Journal on Discrete Mathematics. To appear. Available at arXiv:0904.2751.
  • [46] Montanari, A. and Semerjian, G. (2006). Rigorous inequalities between length and time scales in glassy systems. Journal of Statistical Physics 125 23–54.
  • [47] Mossel, E. and Peres, Y. (2003). Information flow on trees. The Annals of Applied Probability 13 817–844.
  • [48] Mossel, E., Weitz, D. and Wormald, N. (2008). On the hardness of sampling independent sets beyond the tree threshold. Probability Theory and Related Fields 142 401–439.
  • [49] Mulet, R., Pagnani, A., Weigt, M. and Zecchina, R. (2002). Coloring random graphs. Physical Review Letters 89 268701.
  • [50] Pittel, B., Spencer, J. and Wormald, N. (1996). Sudden emergence of a giant k-core in a random graph. Journal of Combinatorial Theory Series B 67 111–151.
  • [51] Richardson, T. and Urbanke, R. (2008). Modern Coding Theory. Cambridge Univ. Press, Cambridge.
  • [52] Simon, B. (1980). Correlation inequalities and the decay of correlations in ferromagnets. Communications in Mathematical Physics 77 111–126.
  • [53] Sly, A. (2008). Reconstruction of random colourings. Communications in Mathematical Physics 288 943–961.