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.