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MR1824274 (2002a:68052) 68Q25 Bollob´ s, B´ la (1-MEMP); Borgs, Christian (1-MSFT); Chayes, Jennifer T. (1-MSFT); a e Kim, Jeong Han (1-MSFT); Wilson, David B. [Wilson, David Bruce] (1-MSFT) The scaling window of the 2-SAT transition. (English summary) Random Structures Algorithms 18 (2001), no. 3, 201­256. Summary: "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 , the problem is known to have a phase transition at 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, ) = (- (n, ), + (n, )) such that the probability of satisfiability is greater than 1 - for < - and is less than for > + . We show that W (n, ) = (1 - (n-1/3 ), 1 + (n-1/3 )), where the constants implicit in depend on . 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 + )n, where may depend on n as long as || is sufficiently small and ||n1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(-(n3 )) above the window, and goes to one like 1 - (n-1 ||-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." References 1. D. Achlioptas, Setting 2 variables at a time yields a new lower bound for random 3-SAT (extended abstract), Proc. 32nd ACM Symposium on Theory of Computing, (2000), pp. 28­ 37. MR2114514 2. D. J. Aldous. A random walk construction of uniform spanning trees and uniform labelled trees, SIAM J Discrete Math, 3 (1990), 450­465. MR1069105 (91h:60013) 3. D. Achlioptas and M. Molloy, personal communication (1998). 4. B. Aspvall, M. F. Plass, and R. E. Tarjan, A linear-time algorithm for testing the truth of certain quantified Boolean formulas, Inform Process Lett 8 (1979), 121­123. MR0526451 (80b:68050) 5. N. Alon and J. Spencer. The Probabilistic Method, John Wiley & Sons, New York 1992. MR1140703 (93h:60002) 6. D. Achlioptas and G. B. Sorkin, Optimal myopic algorithms for random 3-SAT, Proc. 41st Symp the Foundations of Computer Science, 2000, pp. 590­600. MR1931856 7. B. Bollob´ s, C. Borgs, J. T. Chayes, and J. H. Kim, Lecture at the Workshop on the Interface a between Statistical Physics and Computer Science, Torino, Italy, (1998) unpublished.

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