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/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 window W(n, δ) = (α_(n, δ), a+(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 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 + ε)n, where ε may depend on n as long as |ε| is sufficiently small and |ε|n1/3 is 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|-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.
Random Structures and Algorithms
Bollobás, B., Borgs, C., Chayes, J., Kim, J., & Wilson, D. (2001). The scaling window of the 2-SAT transition. Random Structures and Algorithms, 18 (3), 201-256. https://doi.org/10.1002/rsa.1006