Clusters of solutions and replica symmetry breaking in random k-satisfiability

We study the set of solutions of random k-satisfiability formulas through the cavity method. It is known that, for an interval of the clause- to-variables ratio, this decomposes into an exponential number of pure states (clusters). We refine substantially this picture by: (i) determining the precise location of the clustering transition; (ii) uncovering a second ‘condensation’ phase transition in the structure of the solution set for k ? 4. These results both follow from computing the large deviation rate of the internal entropy of pure states. From a technical point of view our main contributions are a simplified version of the cavity formalism for special values of the Parisi replica symmetry breaking parameter m (in particular for m = 1 via a correspondence with the tree reconstruction problem) and new large-k expansions.