The Annals of Applied Probability

A positive temperature phase transition in random hypergraph 2-coloring

Victor Bapst, Amin Coja-Oghlan, and Felicia Raßmann

Full-text: Open access


Diluted mean-field models are graphical models in which the geometry of interactions is determined by a sparse random graph or hypergraph. Based on a nonrigorous but analytic approach called the “cavity method”, physicists have predicted that in many diluted mean-field models a phase transition occurs as the inverse temperature grows from $0$ to $\infty$ [Proc. National Academy of Sciences 104 (2007) 10318–10323]. In this paper, we establish the existence and asymptotic location of this so-called condensation phase transition in the random hypergraph $2$-coloring problem.

Article information

Ann. Appl. Probab., Volume 26, Number 3 (2016), 1362-1406.

Received: October 2014
Revised: March 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40] 05C15: Coloring of graphs and hypergraphs

Discrete structures random hypergraphs phase transitions positive temperature


Bapst, Victor; Coja-Oghlan, Amin; Raßmann, Felicia. A positive temperature phase transition in random hypergraph 2-coloring. Ann. Appl. Probab. 26 (2016), no. 3, 1362--1406. doi:10.1214/15-AAP1119.

Export citation


  • [1] Achlioptas, D. and Coja-Oghlan, A. (2008). Algorithmic barriers from phase transitions. Proc. 49th FOCS 793–802.
  • [2] Achlioptas, D. and Moore, C. (2002). On the 2-colorability of random hypergraphs. In Randomization and Approximation Techniques in Computer Science. Lecture Notes in Computer Science 2483 78–90. Springer, Berlin.
  • [3] Achlioptas, D. and Theodoropoulos, P. (2014). Manuscript in preparation.
  • [4] Bapst, V., Coja-Oghlan, A., Hetterich, S., Raßmann, F. and Vilenchik, D. (2014). The condensation phase transition in random graph coloring. Preprint. Available at arXiv:1404.5513.
  • [5] Bayati, M., Gamarnik, D. and Tetali, P. (2013). Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. Ann. Probab. 41 4080–4115.
  • [6] Coja-Oghlan, A. and Zdeborová, L. (2012). The condensation transition in random hypergraph 2-coloring. In Proceedings of the Twenty-Third Annual ACM–SIAM Symposium on Discrete Algorithms 241–250. ACM, New York.
  • [7] Contucci, P., Dommers, S., Giardinà, C. and Starr, S. (2013). Antiferromagnetic Potts model on the Erdős–Rényi random graph. Comm. Math. Phys. 323 517–554.
  • [8] Dall’Asta, L., Ramezanpour, A. and Zecchina, R. (2008). Entropy landscape and non-Gibbs solutions in constraint satisfaction problems. Phys. Rev. E 77 031118.
  • [9] Janson, S., Łuczak, T. and Ruciński, A. (2000). Random Graphs. Wiley, New York.
  • [10] Kauzmann, W. (1948). The nature of the glassy state and the behavior of liquids at low temperatures. Chem. Rev. 43 219–256.
  • [11] Krzakala, F., Mézard, M., Sausset, F., Sun, Y. and Zdeborová, L. (2012). Probabilistic reconstruction in compressed sensing: Algorithms, phase diagrams, and threshold achieving matrices. J. Stat. Mech. P08009.
  • [12] 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. Proc. National Academy of Sciences 104 10318–10323.
  • [13] Krzakala, F. and Zdeborová, L. (2008). Potts glass on random graphs. Europhys. Lett. 81 57005.
  • [14] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics 195–248. Springer, Berlin.
  • [15] Mézard, M. and Montanari, A. (2009). Information, Physics and Computation. Oxford Univ. Press, London.