Bourbaki - Mars 2014

Collection Bourbaki - Mars 2014

Organisateur(s)
Date(s) 29/03/2014 - 29/03/2014
URL associée https://www.bourbaki.fr/seminaires/2014/Prog_mars14.html
00:00:00 / 00:00:00
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[1082] Ultrametricity in mean-field spin glasses

De Erwin Bolthausen

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]

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Séminaire Bourbaki, 66ème année (2013-2014), n°1082, mars 2014 PDF

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