The scaling window of the 2‐SAT transition

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Abstract
We consider the random 2‐satisfiability (2‐SAT) problem, in which each instance is a formula that is the conjunction ofmclauses of the formx∨y, chosen uniformly at random from among all 2‐clauses onnBoolean variables and their negations. Asmandntend to infinity in the ratiom/n→α, 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 windowW(n, δ)=(α−(n,δ), α+(n,δ)) such that the probability of satisfiability is greater than 1−δ for α<α−and is less than δ for α>α+. We show thatW(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, form=(1+ε)n, where ε may depend onnas long as |ε| is sufficiently small and |ε|n1/3is sufficiently large, we show that the probability of satisfiability decays like exp(−Θ(nε3)) above the window, and goes to one like 1−Θ(n−1|ε|−3below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior innboth 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

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