The problem of identifying the satisfiability threshold of random $3$-SAT formulas has received a lot of attention during the last decades and has inspired the study of other threshold phenomena in random combinatorial structures. The classical assumption in this line of research is that, for a given set of $n$ Boolean variables, each clause is drawn uniformly at random among all sets of three literals from these variables, independently from other clauses. Here, we keep the uniform distribution of each clause, but deviate significantly from the independence assumption and consider richer families of probability distributions. For integer parameters $n$, $m$, and $k$, we denote by $\DistFamily_k(n,m)$ the family of probability distributions that produce formulas with $m$ clauses, each selected uniformly at random from all sets of three literals from the $n$ variables, so that the clauses are $k$-wise independent. Our aim is to make general statements about the satisfiability or unsatisfiability of formulas produced by distributions in $\DistFamily_k(n,m)$ for different values of the parameters $n$, $m$, and $k$.

翻译：识别随机$3$-SAT公式的可满足性阈值问题在过去数十年间备受关注，并激发了随机组合结构中其他阈值现象的研究。这一研究领域中的经典假设是：对于给定的$n$个布尔变量集合，每个子句均从这些变量的所有三文字集合中均匀随机抽取，且各子句之间相互独立。本文在保持每个子句均匀分布的同时，显著偏离独立性假设，考虑更丰富的概率分布族。对于整数参数$n$、$m$和$k$，我们用$\DistFamily_k(n,m)$表示能生成包含$m$个子句公式的概率分布族，其中每个子句均从$n$个变量的所有三文字集合中均匀随机选取，且这些子句满足$k$阶独立性。我们的目标是对不同参数$n$、$m$和$k$取值下，由$\DistFamily_k(n,m)$中分布生成的公式的可满足性或不可满足性作出一般性论断。