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 breaks down strictly \emph{before} the condensation transition. Our
proof also yields slightly improved bounds on the threshold for random
hypergraph 2-colorability. We expect that our techniques can be extended to
other, related problems such as random k-SAT or random graph k-coloring
