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