Mathematics – Combinatorics
Scientific paper
1999-09-05
Random Structures and Algorithms 18(3):201--256, 2001
Mathematics
Combinatorics
57 pages. This version updates some references
Scientific paper
We consider the random 2-satisfiability problem, in which each instance is a formula that is the conjunction of m clauses of the form (x or 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 --> 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 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,delta) = (alpha_-(n,delta),alpha_+(n,delta)) such that the probability of satisfiability 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 delta. 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 sufficiently small and |epsilon|*n^(1/3) is sufficiently large, we show that the probability of satisfiability decays like exp(-Theta(n*epsilon^3)) above the window, and goes to one like 1-Theta(1/(n*|epsilon|^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.
Bollobas Bela
Borgs Christian
Chayes Jennifer T.
Kim Jeong Han
Wilson David B.
No associations
LandOfFree
The Scaling Window of the 2-SAT Transition does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Scaling Window of the 2-SAT Transition, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Scaling Window of the 2-SAT Transition will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-534872