Random $k$-SAT: The Limiting Probability for Satisfiability for Moderately Growing $k$

Abstract

We consider a random instance $I_m=I_{m,n,k}$ of $k$-SAT with $n$ variables and $m$ clauses, where $k=k(n)$ satisfies $k-\log_2 n\to\infty$. Let $m=2^k(n\ln 2+c)$ for an absolute constant $c$. We prove that $$\lim_{n\to\infty}\Pr(I_m\hbox{ is satisfiable})=e^{-e^{-c}}.$$