2- SAT转换的缩放窗口
The Scaling Window of the 2-SAT Transition
摘要：
我们考虑随机2-可满足性问题，其中每个实例都是一个公式，该公式是m个子句的形式（x或y）的合取，从n个布尔变量及其否定的所有2-子句中随机均匀选择。当m和n在m/n --> alpha中趋于无穷大时，已知问题在alpha_c = 1处有一个相变，低于该相变，公式可满足的概率趋于1，高于该相变，公式可满足的概率趋于0。我们确定了关于这个过渡的有限大小的缩放，即最大窗口W（n，delta）=（alpha_-（n，delta），alpha_+（n，delta））的缩放，使得对于alpha alpha_+，可满足性的概率大于1-delta< alpha_- and is less than delta for alpha >。我们证明了W（n，delta）=（1-Theta（n^{-1/3}），1 +Theta（n^{-1/3}）），其中Theta中隐含的常数依赖于delta.我们还确定在窗口的边界处可满足性的概率接近1和0的速率。也就是说，对于m=（1+ ε）n，其中ε可以依赖于n，只要|epsilon|足够小，并且|epsilon| *n^（1/3）足够大，我们证明了可满足性的概率在窗口之上像exp（-Theta（n* ε ^3））那样衰减，在窗口之下像1-Theta（1/（n* |epsilon| ^3））那样衰减。我们证明了这些结果，通过定义一个序参数的过渡，并建立其缩放行为在n的内部和外部的窗口。利用这个序参量，我们证明了2-SAT相变是连续的，序参量临界指数为1。我们还确定了另外两个临界指数的值，表明2-SAT的指数是相同的随机图。
英文摘要：
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.
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