Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs. This general result implies the conjecture on the information-theoretic threshold in the disassortative stochastic block model [: Phys. Rev. E (2011)] and allows us to pinpoint the exact condensation phase transition in random constraint satisfaction problems such as random graph coloring, thereby proving a conjecture from [: PNAS (2007)]. As a further application we establish the formula for the mutual information in Low-Density Generator Matrix codes as conjectured in [Montanari: IEEE Transactions on Information Theory (2005)]. The proofs provide a conceptual underpinning of the replica symmetric variant of the cavity method, and we expect that the approach will find many future applications.
We determine the asymptotics of the independence number of the random @math -regular graph for all @math . It is highly concentrated, with constant-order fluctuations around @math for explicit constants @math and @math . Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs. Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems. In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions of statistical physics. Here we revisit one of these models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain "survey propagation model" associated to this problem. We believe that these methods may be applicable in a wide class of related problems. 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_ (k) ). We prove that for ( d 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 [Proc Natl Acad Sci 104(25):10318–10323, 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. We establish the satisfiability threshold for random k-SAT for all k ≥ k0. That is, there exists a limiting density αs(k) such that a random k-SAT formula of clause density α is with high probability satisfiable for α αs. The satisfiability threshold αs is given explicitly by the one-step replica symmetry breaking (1SRB) prediction from statistical physics. We believe that our methods may apply to a range of random constraint satisfaction problems in the 1RSB class.