[Construction rigoureuse des états purs pour certains modèles de verre de spin]
Pour une large classe d'Hamiltoniens Gaussiens, comprenant le modèle de Sherrington–Kirkpatrick, le modèle à p-spin pour p pair, et bien d'autres, nous prouvons à toute temperature (sous des conditons très peu restrictives) l'existence des « états purs » prédits par les physiciens. Génériquement, à désordre donné, l'espace des configurations se décompose de façon essentiellement unique en une suite de sous-ensembles sur lesquels le recouvrement de deux configurations prend essentiellement sa valeur maximale.
For a large class of Gaussian Hamiltonians, that includes the Sherrington–Kirkpatrick model, the p-spin interaction model for even p, and many others, we prove at each temperature (under a very mild and essentially necessary condition) that generically the “pure states” predicted by physics do exist. The configuration space decomposes in an essentially unique manner in a sequence of subsets on which the overlap takes essentially its maximum value.
Accepté le :
Publié le :
@article{CRMATH_2008__346_17-18_1003_0,
author = {Talagrand, Michel},
title = {Rigorous construction of the pure states for certain spin glass models},
journal = {Comptes Rendus. Math\'ematique},
pages = {1003--1005},
publisher = {Elsevier},
volume = {346},
number = {17-18},
year = {2008},
doi = {10.1016/j.crma.2008.07.008},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.07.008/}
}
TY - JOUR AU - Talagrand, Michel TI - Rigorous construction of the pure states for certain spin glass models JO - Comptes Rendus. Mathématique PY - 2008 SP - 1003 EP - 1005 VL - 346 IS - 17-18 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.07.008/ DO - 10.1016/j.crma.2008.07.008 LA - en ID - CRMATH_2008__346_17-18_1003_0 ER -
%0 Journal Article %A Talagrand, Michel %T Rigorous construction of the pure states for certain spin glass models %J Comptes Rendus. Mathématique %D 2008 %P 1003-1005 %V 346 %N 17-18 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.07.008/ %R 10.1016/j.crma.2008.07.008 %G en %F CRMATH_2008__346_17-18_1003_0
Talagrand, Michel. Rigorous construction of the pure states for certain spin glass models. Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 1003-1005. doi : 10.1016/j.crma.2008.07.008. http://www.numdam.org/articles/10.1016/j.crma.2008.07.008/
[1] Spin Glass Theory and Beyond, World Scientific, Singapore, 1987
[2] Spin Glasses, A Challenge to Mathematicians, Springer-Verlag, 2003
[3] Parisi measures, J. Funct. Anal., Volume 231 (2006) no. 2, pp. 269-286
[4] Free energy of the spherical mean field model, Probab. Theory Related Fields, Volume 134 (2006) no. 3, pp. 339-382
[5] M. Talagrand, Mean Field Models for Spin Glasses, book in preparation
Cité par Sources :
