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

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

2008-02-25

arxiv.org/abs/0802.3627v2

J. Stat. Mech. P04004 (2008)

Physics

Condensed Matter

Disordered Systems and Neural Networks

30 pages, 14 figures, typos corrected, discussion of appendix C expanded with a new figure

Scientific paper

10.1088/1742-5468/2008/04/P04004

We study the set of solutions of random k-satisfiability formulae 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 larger or equal than 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.

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