The asymptotic $k$-SAT threshold

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems [Mezard, Parisi, Zecchina: Science 2002]. The cavity method predicts that the satisfiability threshold in the random $k$-SAT problem is $2^k\ln2-\frac12(1+\ln 2)+\epsilon_k$, with $\lim_{k\rightarrow\infty}\epsilon_k=0$ [Mertens, Mezard, Zecchina: Random Structures and Algorithms 2006]. This paper contains a proof of that conjecture.