The random (2 + p)-SAT model has been proposed [18] to study the possible relation between the “order” of phase transitions and computational complexity. It was also claimed that there exists pc > 0, such that for p < pc the random (2 + p)-SAT instance behaves like 2-SAT. Later, Achlioptas et al. [3] obtained the first rigorous results that 0.4 ≤ pc ≤ 0.695, the methods they use are the first moment method and the simple Unit-Clause algorithm. In this paper, we try to optimize the local maximality condition of the truth assignments when implementing the first moment method. We prove that the phase transition point of clauses-to-variables ratio r (dependent on p) can be improved. Moreover, we show that the upper bound of pc can be reduced to 0.6846. This fact implies that, for a constant λ < 1, a random (2 + p)-SAT formula with λn 2-clauses and 2.17n 3-clauses is almost surely unsatisfiable.
SPONTANEOUS POLARIZATION IN THE CLUSTER FERROELECTRIC SYSTEM CLOSE TO THE PHASE TRANSITION POINT
