Course 5 An introduction to mean field spin glas theory: Methods and results

Abstract

The chapter introduces the ferromagnetic model, and discusses the behavior and properties of the free energy in the thermodynamic limit, in this very elementary case, the comparison and interpolation methods that will be also exploited, in a different context, in the spin glass case. Cavity methods will be also introduced, by showing how effective they are in providing powerful bounds. The basic features of the mean field spin glass models are discussed. Some important formulae, concerning the derivation with respect to the strength of the interaction, and the Gaussian comparison and interpolation method are introduced for generic Gaussian interactions. Simple applications to the mean-field spin glass model are discussed, in particular to the existence of the infinite-volume limit of the quenched free energy. The chapter also recalls the main features of the Parisi representation, and states the main theorem concerning the free energy.