On the proof of the Parisi formula by Guerra and Talagrand

Erwin Bolthausen

Séminaire Bourbaki (2004-2005)

  • Volume: 47, page 349-378
  • ISSN: 0303-1179

Abstract

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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.

How to cite

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Bolthausen, Erwin. "On the proof of the Parisi formula by Guerra and Talagrand." Séminaire Bourbaki 47 (2004-2005): 349-378. <http://eudml.org/doc/252149>.

@article{Bolthausen2004-2005,
abstract = {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.},
author = {Bolthausen, Erwin},
journal = {Séminaire Bourbaki},
keywords = {verre de spin; modèle de Sherrington-Kirkpatrick; énergie libre; brisure de la symétrie des répliques},
language = {eng},
pages = {349-378},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {On the proof of the Parisi formula by Guerra and Talagrand},
url = {http://eudml.org/doc/252149},
volume = {47},
year = {2004-2005},
}

TY - JOUR
AU - Bolthausen, Erwin
TI - On the proof of the Parisi formula by Guerra and Talagrand
JO - Séminaire Bourbaki
PY - 2004-2005
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 47
SP - 349
EP - 378
AB - 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 - eng
KW - verre de spin; modèle de Sherrington-Kirkpatrick; énergie libre; brisure de la symétrie des répliques
UR - http://eudml.org/doc/252149
ER -

References

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  17. [17] A. Ruzmaikina & M. Aizenman – “Characterization of invariant measures at the leading edge for competing particle systems”, Ann. Prob.33 (2005), p. 83–113. Zbl1096.60042MR2118860
  18. [18] D. Sherrington & S. Kirkpatrick – “Solvable model of a spin glass”, Phys. Rev. Lett.35 (1972), p. 1792–1796. 
  19. [19] M. Talagrand – Spin Glasses: A Challenge for Mathematicians, Springer, Heidelberg, 2003. Zbl1033.82002MR1993891
  20. [20] —, “The Parisi formula”, Ann. Math.163 (2006), p. 221–263. Zbl1137.82010MR2195134

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