# Catching the k-NAESAT threshold

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## Abstract

The best current estimates of the thresholds for the existence of solutions in random constraint satisfaction problems ('CSPs') mostly derive from the first and the second moment method. Yet apart from a very few exceptional cases these methods do not quite yield matching upper and lower bounds. According to deep but non-rigorous arguments from statistical mechanics, this discrepancy is due to a change in the geometry of the set of solutions called condensation that occurs shortly before the actual threshold for the existence of solutions (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). To cope with condensation, physicists have developed a sophisticated but non-rigorous formalism called Survey Propagation (Mezard, Parisi, Zecchina: Science 2002). This formalism yields precise conjectures on the threshold values of many random CSPs. Here we develop a new Survey Propagation inspired second moment method for the random k-NAESAT problem, which is one of the standard benchmark problems in the theory of random CSPs. This new technique allows us to overcome the barrier posed by condensation rigorously. We prove that the threshold for the existence of solutions in random $k$-NAESAT is $2^{k-1}\ln2-(\frac{\ln2}2+\frac14)+\eps_k$, where $|\eps_k| \le 2^{-(1-o_k(1))k}$, thereby verifying the statistical mechanics conjecture for this problem.

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