Branching Process approach for 2-SAT thresholds

It is well known that, as n tends to infinity, the probability of satisfiability for a random 2-SAT formula on n variables, where each clause occurs independently with probability α/2n, exhibits a sharp threshold at α = 1. We provide a simple conceptual proof of this fact based on branching process arguments. We also study a generalized 2-SAT model in which each clause occurs independently but with probability αi/2n where i ∈ {0, 1, 2} is the number of positive literals in that clause. We use 2-type branching process arguments to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.