The random $\textit{k}$-SAT Gibbs uniqueness threshold revisited

Abstract We prove that for any $k\geq 3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$ -SAT formulas is given by the ‘replica symmetric solution’ predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. 76 (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $k\geq 3$ , we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)]. The improvement is significant particularly for small $k$ .