The birth of the contradictory component in random 2-SAT

Abstract

We prove that, with high probability, the contradictory components of a random 2-SAT formula in the subcritical phase of the phase transition have only 3-regular kernels. This follows from the relation between these kernels and the complex component of a random graph in the subcritical phase. This partly settles the question about the structural similarity between the phase transitions in 2-SAT and random graphs. As a byproduct, we describe the technique that allows to obtain a full asymptotic expansion of the satisfiability in the subcritical phase. We also obtain the distribution of the number of contradictory variables and the structure of the spine in the subcritical phase.