Clusters of solutions and replica symmetry breaking in random k-satisfiability
Abstract
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.
Keywords
Cite
@article{arxiv.0802.3627,
title = {Clusters of solutions and replica symmetry breaking in random k-satisfiability},
author = {Andrea Montanari and Federico Ricci-Tersenghi and Guilhem Semerjian},
journal= {arXiv preprint arXiv:0802.3627},
year = {2009}
}
Comments
30 pages, 14 figures, typos corrected, discussion of appendix C expanded with a new figure