The Parisi Formula

Abstract

The main goal of this chapter is to prove the celebrated Parisi formula for the free energy in the mixed p-spin models. The Ruelle probability cascades studied in the previous chapter will be used in a number of different ways, but their most important role will be as an approximation of the asymptotic Gibbs measures that generate the overlap matrix in the thermodynamic limit. As we explained in Sect.2.4, a link between the Gibbs measure and the Ruelle probability cascades can be established using the Ghirlanda–Guerra identities and in this chapter we will show how these identities arise in the setting of the mixed p-spin models. In addition, the proof of the Parisi formula will be based on several other essential ideas, such as the Talagrand positivity principle, the Guerra replica symmetry breaking interpolation, and the Aizenman–Sims–Starr scheme.