  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 and we show that the mutual
information holds the key to understanding certain important phase transitions
in random graph models. We work out several concrete applications of these
general results. For instance, we pinpoint the exact condensation phase
transition in the Potts antiferromagnet on the random graph, thereby improving
prior approximate results [Contucci et al.: Communications in Mathematical
Physics 2013]. Further, we prove the conjecture from [Krzakala et al.: PNAS
2007] about the condensation phase transition in the random graph coloring
problem for any number $q\geq3$ of colors. Moreover, we prove the conjecture on
the information-theoretic threshold in the disassortative stochastic block
model [Decelle et al.: Phys. Rev. E 2011]. Additionally, our general result
implies the conjectured formula for the mutual information in Low-Density
Generator Matrix codes [Montanari: IEEE Transactions on Information Theory
2005].
