Concentration of the number of solutions of random planted CSPs and Goldreich's one-way candidates

This paper shows that the logarithm of the number of solutions of a random planted $k$-SAT formula concentrates around a deterministic $n$-independent threshold. Specifically, if $F^*_{k}(\alpha,n)$ is a random $k$-SAT formula on $n$ variables, with clause density $\alpha$ and with a uniformly drawn planted solution, there exists a function $\phi_k(\cdot)$ such that, besides for some $\alpha$ in a set of Lesbegue measure zero, we have $ \frac{1}{n}\log Z(F^*_{k}(\alpha,n)) \to \phi_k(\alpha)$ in probability, where $Z(F)$ is the number of solutions of the formula $F$. This settles a problem left open in Abbe-Montanari RANDOM 2013, where the concentration is obtained only for the expected logarithm over the clause distribution. The result is also extended to a more general class of random planted CSPs; in particular, it is shown that the number of pre-images for the Goldreich one-way function model concentrates for some choices of the predicates.