Clustering of solutions in the random satisfiability problem

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Using elementary rigorous methods we prove the existence of a clustered phase in the random K-SAT problem, for K≥ 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.