Andrea Montanari et al J. Stat. Mech. (2008) P04004 doi:10.1088/1742-5468/2008/04/P04004
Andrea Montanari1,2, Federico Ricci-Tersenghi3 and Guilhem Semerjian4
Show affiliationsWe 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.
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.70.Fh Phase transitions: general studies
02.50.-r Probability theory, stochastic processes, and statistics
82C21 Dynamic continuum models (systems of particles, etc.)
82C26 Dynamic and nonequilibrium phase transitions (general)
Issue 04 (April 2008)
Received 29 February 2008, accepted for publication 12 March 2008
Published 8 April 2008
Andrea Montanari et al J. Stat. Mech. (2008) P04004
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