The Free Energy and Gibbs Measure

Abstract

In Sect.1.1, we will introduce the Sherrington–Kirkpatrick model and a family of closely related mixed p-spin models and give some motivation for the problem of computing the free energy in these models. A solution of this problem in Chap.?? will be based on a description of the structure of the Gibbs measure in the thermodynamic limit and in this chapter we will outline several connections between the free energy and Gibbs measure. At the same time, we will introduce various ideas and techniques, such as the Gaussian integration by parts, Gaussian interpolation, and Gaussian concentration, that will play essential roles in the key results of this chapter and throughout the book. In the last section, we will prove the Dovbysh–Sudakov representation for Gram-de Finetti arrays, which will allow us to define a certain analogue of the Gibbs measure in the thermodynamic limit. As a first step, we will prove the Aldous–Hoover representation for exchangeable and weakly exchangeable arrays. In Sect.1.4, we will give a classic probabilistic proof of this result for weakly exchangeable arrays and, for a change, in the Appendix we will prove the representation for exchangeable arrays using a different approach, based on more recent ideas of Lovasz and Szegedy in the framework of limits of dense graph sequences. We will describe another application of the Aldous–Hoover representations for exchangeable arrays in Chap.??.