Proof of the satisfiability conjecture for large k

TL;DR

This paper proves the existence and explicit value of the satisfiability threshold for large k in random k-SAT, confirming predictions from statistical physics and introducing a new analytic method for moment calculations.

Contribution

It establishes the satisfiability threshold for all large k and develops a novel analytic approach applicable to various random CSPs within the 1-RSB class.

Findings

Threshold $oxed{ ext{exists and is explicitly characterized}}$ for large k

Method based on analyzing tree recursions for high-dimensional optimization

Potential applicability to other random constraint satisfaction problems

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