In this paper we translate Talagrand’s solution of the $K$-sat model at high temperature into the language of asymptotic Gibbs measures. Using exact cavity equations in the infinite volume limit allows us to remove many technicalities of the inductions on the system size, which clarifies the main ideas of the proof. This approach also yields a larger region of parameters where the system is in a pure state and, in particular, for small connectivity parameter one can prove the replica symmetric formula for the free energy at any temperature.
"On the replica symmetric solution of the $K$-sat model." Electron. J. Probab. 19 1 - 17, 2014. https://doi.org/10.1214/EJP.v19-2963