# Proof of the satisfiability conjecture for large $k$

### 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 $\alpha_{\mathrm{SAT}}(k)$ such that a random k-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha\lt\alpha_{\mathrm{SAT}}$, and unsatisfiable for
$\alpha>\alpha_{\mathrm{SAT}}$. We show that the threshold $\alpha_{\mathrm{SAT}}(k)$ 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.

## Authors

Jian Ding

University of Pennsylvania, Philadelphia, PA

Allan Sly

Princeton University, Princeton, NJ

Nike Sun

Massachusetts Institute of Technology, Cambridge, MA