The scaling window of the 2-SAT transition

We consider the random 2-satisfiability (2-SAT) problem, in which each inst ance is a formula that is the conjunction of m clauses of the form x boolea n OR y, chosen uniformly at random from among all 2-clauses on n Boolean va riables and their negations. As m and n tend to infinity in the ratio m/n - -> alpha, the problem is known to have a phase transition at alpha (c) = 1, below which the probability that the formula is satisfiable tends to one a nd above which it tends to zero. We determine the finite-size scaling about this transition, namely the scaling of the maximal window W(n, delta) = (a lpha (-)(n, delta), alpha (+)(n, delta)) such that the probability of satis fiability is greater than 1 - delta for alpha < alpha (-) and is less than delta for alpha > alpha (+). We show that W(n, delta) = (1 - Theta (n(-1/3) ), 1 + Theta (n(-1/3))), where the constants implicit in Theta depend on de lta. 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 + epsilon )n, where epsilon may depend on n as long as \ epsilon \ is suf ficiently small and \ epsilon \n(1/3) is sufficiently large, we show that t he probability of satisfiability decays like exp(-Theta (n epsilon (3))) ab ove the window, and goes to one like 1 - Theta (n(-1)\ epsilon \ (-3)) belo w the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outs ide the window. Using this order parameter, we prove that the 2-SAT phase t ransition 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. (C) 2001 Jo hn Wiley & Sons, Inc.