6 research outputs found
Belief Propagation on the random -SAT model
Corroborating a prediction from statistical physics, we prove that the Belief
Propagation message passing algorithm approximates the partition function of
the random -SAT model well for all clause/variable densities and all inverse
temperatures for which a modest absence of long-range correlations condition is
satisfied. This condition is known as "replica symmetry" in physics language.
From this result we deduce that a replica symmetry breaking phase transition
occurs in the random -SAT model at low temperature for clause/variable
densities below but close to the satisfiability threshold
Charting the replica symmetric phase
Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous ‘cavity method’, physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007). In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the k-XORSAT model and the diluted k-spin model for even k. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention (Decelle et al. in Phys Rev E 84:066106, 2011)