The Condensation Phase Transition in Random Graph Coloring

Victor Bapst, Amin Coja-Oghlan, Samuel Hetterich, Felicia Raßmann, Dan Vilenchik

Research output: Contribution to journalArticlepeer-review

35 Scopus citations


Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry of interactions is induced by a sparse random graph or hypergraph. One example of such a model is the graph coloring problem on the Erdős–Renyi random graph G(n, d/n), which can be viewed as the zero temperature case of the Potts antiferromagnet. The cavity method predicts that in addition to the k-colorability phase transition studied intensively in combinatorics, there exists a second phase transition called the condensation phase transition (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007). In fact, there is a conjecture as to the precise location of this phase transition in terms of a certain distributional fixed point problem. In this paper we prove this conjecture for k exceeding a certain constant k0.

Original languageEnglish
Pages (from-to)543-606
Number of pages64
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - 1 Jan 2016

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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