Work

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

Year: 2026

Type: article

Abstract: We prove that for any k greater than or equals 3 k ≥ 3 $k\geq 3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random k k $k$ -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 greater than or equals 3 k ≥ 3 $k\geq 3$ , 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 k $k$ .

Source: Combinatorics Probability Computing

DOI: 10.1017/s0963548326100406