关于随机可满足性公式的解数集中问题
On the concentration of the number of solutions of random satisfiability formulas
摘要：
设$Z(F)$为随机$k$ -可满足性公式$F$的解的个数，其中变量为$n$，子句密度为$α$ .假设$F$不可满足的概率是$\e>0$的$O(1/\log(n)^{1+\e})$。我们表明，（可能不包括一个可数集的“例外”$α$的）解决方案的数量集中在对数尺度，即，存在非随机函数$ϕ(α)$，使得对于任何$δ>0$，$(1/n)\log Z(F)\in [ϕ-δ,ϕ+δ]$具有高概率。特别地，该假设对所有$α<1$都成立，这在随机$2$-SAT的整个可满足性范围内证明了上述浓度声明。我们还将这些结果扩展到广泛的一类约束满足问题。证明是基于从自旋玻璃理论的插值技术，并在应用Friedgut的定理的尖锐阈值的图形属性。
英文摘要：
Let $Z(F)$ be the number of solutions of a random $k$-satisfiability formula $F$ with $n$ variables and clause density $α$. Assume that the probability that $F$ is unsatisfiable is $O(1/\log(n)^{1+\e})$ for $\e>0$. We show that (possibly excluding a countable set of `exceptional' $α$'s) the number of solutions concentrate in the logarithmic scale, i.e., there exists a non-random function $ϕ(α)$ such that, for any $δ>0$, $(1/n)\log Z(F)\in [ϕ-δ,ϕ+δ]$ with high probability. In particular, the assumption holds for all $α<1$, which proves the above concentration claim in the whole satisfiability regime of random $2$-SAT. We also extend these results to a broad class of constraint satisfaction problems. The proof is based on an interpolation technique from spin-glass theory, and on an application of Friedgut's theorem on sharp thresholds for graph properties.
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