The random $k$-SAT Gibbs uniqueness threshold revisited

TL;DR

This paper establishes that for random k-SAT formulas with clause-to-variable ratios below a certain threshold, the number of solutions aligns with physics predictions, and it improves bounds on this threshold, especially for small k.

Contribution

It proves the replica symmetric solution accurately predicts the number of solutions below the Gibbs uniqueness threshold and provides improved lower bounds for this threshold for all k ≥ 3.

Findings

Number of solutions matches the replica symmetric prediction below the threshold.

New lower bounds on the Gibbs uniqueness threshold are derived.

Bounds are particularly improved for small k.

Abstract

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 $k$.