Coja-Oghlan, Amin and Zdeborová, Lenka (2012) The condensation transition in random hypergraph 2-coloring. In: ACM-SIAM Symposium on Discrete Algorithms 2012, Kyoto, Japan, 17-19 Jan 2012. Published in: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms pp. 241-250.

Abstract

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, Zdeborová: 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. Copyright © SIAM.

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