We prove that for any $k\geq3$ 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. (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $k\geq3$, 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 $k$.

翻译：我们证明，对于任意k≥3，在对应Galton-Watson树的吉布斯唯一性阈值以下的子句/变量比范围内，随机k-SAT公式的可满足赋值数量由物理方法预测的'副本对称解'给出[Monasson, Zecchina: Phys. Rev. Lett. (1996)]。此外，虽然对于任意k≥3，吉布斯唯一性阈值的确切值仍未知，但我们推导了该阈值的新下界，改进了先前的研究成果[Montanari and Shah: SODA (2007)]。该改进对于较小的k值尤为显著。