On the proof of the Parisi formula by Guerra and Talagrand
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 948, p. 349-377

The Parisi formula is an expression for the limiting free energy of the Sherrington-Kirkpatrick spin glass model, which had first been derived by Parisi using a non-rigorous replica method together with an hierarchical ansatz for the solution of the variational problem. It had become quickly clear that behind the solution, if correct, lies an interesting mathematical structure. The formula has recently been proved by Michel Talagrand based partly on earlier ideas and results by Francesco Guerra. The talk will try to explain why the problem is mathematically interesting, and sketch the ideas of Guerra and Talagrand. It should be emphasized that despite the fact that the formula is proved, many things remain still quite mysterious.

La formule de Parisi est une expression pour l'énergie libre limite pour le modèle des verres de spin de Sherrington-Kirkpatrick, qui a d'abord été obtenue par Parisi en utilisant une méthode des répliques non rigoureuse avec un ansatz hiérarchique pour la solution du problème variationnel. Il est devenu rapidement clair que derrière la solution, si elle était correcte, se trouvait une structure mathématique intéressante. Cette formule a récemment été démontrée par Michel Talagrand en utilisant des idées et des résultats antérieurs de Francesco Guerra. L'exposé tentera d'expliquer pourquoi le problème est mathématiquement intéressant, et esquissera les idées de Guerra et Talagrand. Il convient de souligner que, même si la formule est démontrée, beaucoup de choses restent mystérieuses.

Classification:  82B44,  60K37
Keywords: verre de spin, modèle de Sherrington-Kirkpatrick, énergie libre, brisure de la symétrie des répliques
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     author = {Bolthausen, Erwin},
     title = {On the proof of the Parisi formula by Guerra and Talagrand},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:948},
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     zbl = {1125.82015},
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Bolthausen, Erwin. On the proof of the Parisi formula by Guerra and Talagrand, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 948, pp. 349-377. http://www.numdam.org/item/SB_2004-2005__47__349_0/

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