Clustering of solutions in the random satisfiability problem.

Using elementary rigorous methods we prove the existence of a clustered phase in the random K-SAT problem, for K > or = 8. In this phase the solutions are grouped into clusters which are far away from each other. The results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks. It can be generalized to other systems of both physical and computational interest.