Appears in collection : Bourbaki - Mars 2014

Ultrametricity lies at the core of the Parisi theory of spin glasses, particularly for the Sherrington-Kirkpatrick model. In a vague sense, it claims that the Gibbs measure is hierarchically organized. This picture was crucial for the original derivation by Parisi of the free energy using the non-rigorous replica method, and also in the later developed cavity method by Mézard and Parisi. However, the first rigorous proof by Talagrand of the Parisi formula completely avoided a discussion of ultrametricity, and in fact, it was not possible to prove ultrametricity by Talagrand’s method. In a recent development, this point was clarified to a large extent, at least for the SK-model and related ones. It is based on a proof that a slightly perturbed SK-model satisfies the so-called Ghirlanda-Guerra identities, and then in the proof by Panchenko that these identities imply ultrametricity. This then leads also to a new proof of the Parisi-formula for the free energy, which is conceptually very close to the original physicists picture of mean-field type spin glasses.

[After Dmitry Panchenko]