We establish the satisfiability threshold for random k-sat for all k ě k0. That is, there exists a limiting density αspkq such that a random k-sat formula of clause density α is with high probability satisfiable for α ă αs, and unsatisfiable for α ą αs. The satisfiability threshold αspkq is given explicitly by the one-step replica symmetry breaking (1rsb) prediction from statistical physics. We believe that our methods may apply to a range of random constraint satisfaction problems in the 1rsb class.

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We consider the random regular k-nae-sat problem with n variables each appearing in exactly d clauses. For all k exceeding an absolute constant k 0 , we establish explicitly the satisfiability threshold d ‹ " d ‹ pkq. We prove that for d ă d ‹ the problem is satisfiable with high probability while for d ą d ‹ the problem is unsatisfiable with high probability. If the threshold d ‹ lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krzakała et al. (2007). Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs.

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We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $\phi$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $\phi$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $\phi$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on $\phi$. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on $T$. In the special case that $T$ has a Galton-Watson law, this formula coincides with the nonrigorous "Bethe prediction" obtained by statistical physicists using the "replica" or "cavity" methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOP828 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org