The Mézard-Parisi formula

Abstract

This paper proves the Mézard-Parisi formula for the free energy of random $K$-sat and diluted $p$-spin models with even $p$, completing an approach developed in earlier work of the author and collaborators. The upper bound was established by Franz and Leone. Here the matching lower bound is proved for a general class of diluted models under symmetry and integrability assumptions. The main new contribution is a proof of an approximate form of the Mézard-Parisi reproducibility hypothesis without assuming that the overlap distribution has finite support.