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The scaling window of the 2-SAT transition. (English) Zbl 0979.68053
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 $$x\vee y$$, 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 $$m/m\to\alpha$$, 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 $$m= (1+\varepsilon)n$$, where $$\varepsilon$$ may depend on $$n$$ as long as $$|\varepsilon|$$ is sufficiently small and $$\varepsilon n|n^{1/3}$$ is sufficiently large, we show that the probability of satisfiability decays like $$\exp(-\Theta(n\varepsilon^3))$$ above the window, and goes to one like $$1-\Theta(n^{- 1}|\varepsilon|^{- 3})$$ below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in $$n$$ both inside an 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.

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 05C80 Random graphs (graph-theoretic aspects)
##### Keywords:
random 2-satisfiability
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