The scaling window of the 2‐SAT transition

We consider the random 2-satisfiability (2-SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨y, chosen uniformly at random from among all 2-clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/nα, the problem is known to have a phase transition at αc=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite-size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α−(n,δ), α+(n,δ)) such that the probability of satisfiability is greater than 1−δ for α α+. We show that W(n,δ)=(1−Θ(n−1/3), 1+Θ(n−1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+e)n, where e may depend on n as long as |e| is sufficiently small and |e|n1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(−Θ(ne3)) above the window, and goes to one like 1−Θ(n−1|e|−3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2-SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2-SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001