Rigorous Results in the Sherrington-Kirkpatrick Model

Since the proposal of the solution of the SK model by Giorgio Parisi, the mathematical community has investigated the rigorous foundations of the results obtained by means of the replica method: this chapter focuses on some of the outcomes of such investigations. In the first part, Wei-Kuo Chen discusses the Parisi formula for the free energy of the SK model, the stability of the replica-symmetric solution, and the most recent results on disorder chaos. In the second part, Dmitry Panchenko reviews the central concept of ultrametricity, its relation with the Ghirlanda–Guerra identities in a large class of spin systems, and its implications on their landscape structure. Finally, in the last part, Francesco Guerra overviews the interpolating method for the SK model and the REM, and the emergent variational principle associated to these models.