Abstract
We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_{\rm sat}(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha\lt \alpha_{\rm sat}$, and unsatisfiable for $\alpha>\alpha_{\rm sat}$. We show that the threshold $\alpha_{\rm sat}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the $1$-RSB universality class.
Citation
Jian Ding. Allan Sly. Nike Sun. "Proof of the satisfiability conjecture for large $k$." Ann. of Math. (2) 196 (1) 1 - 388, July 2022. https://doi.org/10.4007/annals.2022.196.1.1
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