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 ( 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.
For many random constraint satisfaction problems such as random satisfiability or random graph or hypergraph coloring, the best current estimates of the threshold for the existence of solutions are based on the first and the second moment method. However, in most cases these techniques do not yield matching upper and lower bounds. Sophisticated but non-rigorous arguments from statistical mechanics have ascribed this discrepancy to the existence of a phase transition called condensation that occurs shortly before the actual threshold for the existence of solutions and that affects the combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time that a condensation transition exists in a natural random CSP, namely in random hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment method applied to the number of 2-colorings breaks down strictly before the condensation transition. Our proof also yields slightly improved bounds on the threshold for random hypergraph 2-colorability.