Belief propagation on the random k-SAT model

Corroborating a prediction from statistical physics, we prove that the belief propagation message passing algorithm approximates partition function of random k-SAT model well for all clause/variable densities and inverse temperatures which modest absence long-range correlations condition is satisfied. This known as “replica symmetry” in physics language. From this result deduce replica symmetry breaking phase transition occurs at low temperature below but close to satisfiability threshold.