`CitationsPreviousUpNextArticleFrom References: 26 From Reviews: 2MR1824274 (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: &quot;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  &lt; - and is less than  for  &gt; + . 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. 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MR0595607 (82d:05070)Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.c Copyright American Mathematical Society 2002, 2008`

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