[La formule de Parisi pour le modèle SK, et sa démonstration récente par Guerra et Talagrand]
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.
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.
Keywords: verre de spin, modèle de Sherrington-Kirkpatrick, énergie libre, brisure de la symétrie des répliques
Mot clés : spin glass, Sherrington-Kirkpatrick model, free energy, replica symmetry breaking
@incollection{SB_2004-2005__47__349_0,
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},
series = {Ast\'erisque},
note = {talk:948},
pages = {349--377},
publisher = {Soci\'et\'e math\'ematique de France},
number = {307},
year = {2006},
mrnumber = {2296424},
zbl = {1125.82015},
language = {en},
url = {http://archive.numdam.org/item/SB_2004-2005__47__349_0/}
}
TY - CHAP AU - Bolthausen, Erwin TI - On the proof of the Parisi formula by Guerra and Talagrand BT - Séminaire Bourbaki : volume 2004/2005, exposés 938-951 AU - Collectif T3 - Astérisque N1 - talk:948 PY - 2006 SP - 349 EP - 377 IS - 307 PB - Société mathématique de France UR - http://archive.numdam.org/item/SB_2004-2005__47__349_0/ LA - en ID - SB_2004-2005__47__349_0 ER -
%0 Book Section %A Bolthausen, Erwin %T On the proof of the Parisi formula by Guerra and Talagrand %B Séminaire Bourbaki : volume 2004/2005, exposés 938-951 %A Collectif %S Astérisque %Z talk:948 %D 2006 %P 349-377 %N 307 %I Société mathématique de France %U http://archive.numdam.org/item/SB_2004-2005__47__349_0/ %G en %F SB_2004-2005__47__349_0
Bolthausen, Erwin. On the proof of the Parisi formula by Guerra and Talagrand, dans Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 948, pp. 349-377. http://archive.numdam.org/item/SB_2004-2005__47__349_0/
[1] , & - “Some rigorous results on the Sherrington-Kirkpatrick model”, Comm. Math. Phys. 112 (1987), p. 3-20. | DOI | MR | Zbl
[2] , & - “An extended variational principle for the SK spin-glass model”, Phys. Rev. B 68 (2003), p. 214-403. | DOI
[3] & - “Stability of the Sherrington-Kirkpatrick solution of spin glasses”, J. Phys. A 11 (1978), p. 983.
[4] & - “On a non-hierarchical version of the Generalized Random Energy Model”, Ann. Appl. Prob. 16 (2006), p. 1-14. | DOI | MR | Zbl
[5] & - “On Ruelle's probability cascades and an abstract cavity method”, Comm. Math. Phys. 197 (1998), p. 247-276. | DOI | MR | Zbl
[6] & - “Derrida's Generalized Random Energy Models I & II”, Annales de l'Institut Henri Poincaré 40 (2004), p. 439-495. | Numdam | MR | Zbl
[7] - “Random energy model: An exactly solvable model of disordered systems”, Phys. Rev. B 24 (1981), p. 2613-2626. | DOI | MR | Zbl
[8] -, “A generalization of the random energy model that includes correlations between the energies”, J. Physique. Lett 46 (1986), p. 401-407.
[9] - “Replica broken bounds in the mean field spin glass model”, Comm. Math. Phys. 233 (2003), p. 1-12. | DOI | MR | Zbl
[10] & - “Quadratic replica coupling in the Sherrington-Kirkpatrick mean field spin glass model”, J. Math. Phys. 43 (2002), p. 3704-3716. | DOI | MR | Zbl
[11] -, “The thermodynamic limit in mean field spin glass models”, Comm. Math. Phys. 230 (2002), p. 71-79. | DOI | MR | Zbl
[12] - “The Concentration of Measure Phenomenon”, vol. 89, AMS, 2002. | MR | Zbl
[13] , & - Spin Glass Theory and Beyond, World Scientific, 1987. | MR | Zbl
[14] - Statistical Physics of Spin Glasses and Information Processing, Oxford Science Publications, 1999. | Zbl
[15] - “A sequence of approximate solutions to the S-K model for spin glasses”, J. Phys. A 13 L-115 (1980).
[16] - “A mathematical reformulation of Derrida's REM and GREM”, Comm. Math. Phys. 108 (1987), p. 225-239. | DOI | MR | Zbl
[17] & - “Characterization of invariant measures at the leading edge for competing particle systems”, Ann. Prob. 33 (2005), p. 83-113. | DOI | MR | Zbl
[18] & - “Solvable model of a spin glass”, Phys. Rev. Lett. 35 (1972), p. 1792-1796. | DOI
[19] - Spin Glasses: A Challenge for Mathematicians, Springer, Heidelberg, 2003. | MR | Zbl
[20] -, “The Parisi formula”, Ann. Math. 163 (2006), p. 221-263. | DOI | MR | Zbl
