[Ultramétricité et identités de Ghirlanda–Guerra : une preuve élémentaire du cas discret]
Dans cette Note nous donnons une nouvelle preuve du fait quʼune matrice aléatoire infinie, qui satisfait lʼidentité Ghirlanda–Guerra et dont les coefficiants prennent leurs valeurs dans un ensemble fini, est ultramétrique avec probabilité un. La preuve utilise uniquement des conséquences algébriques élémentaires des identités Ghirlanda–Guerra et la représentation de Dovbysh–Sudakov.
In this Note we give another proof of the fact that a random overlap array, which satisfies the Ghirlanda–Guerra identities and whose elements take values in a finite set, is ultrametric with probability one. The new proof bypasses random change of density invariance principles for directing measures of such arrays and, in addition to the Dovbysh–Sudakov representation, is based only on elementary algebraic consequences of the Ghirlanda–Guerra identities.
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@article{CRMATH_2011__349_13-14_813_0,
author = {Panchenko, Dmitry},
title = {Ghirlanda{\textendash}Guerra identities and ultrametricity: {An} elementary proof in the discrete case},
journal = {Comptes Rendus. Math\'ematique},
pages = {813--816},
publisher = {Elsevier},
volume = {349},
number = {13-14},
year = {2011},
doi = {10.1016/j.crma.2011.06.021},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.06.021/}
}
TY - JOUR AU - Panchenko, Dmitry TI - Ghirlanda–Guerra identities and ultrametricity: An elementary proof in the discrete case JO - Comptes Rendus. Mathématique PY - 2011 SP - 813 EP - 816 VL - 349 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.06.021/ DO - 10.1016/j.crma.2011.06.021 LA - en ID - CRMATH_2011__349_13-14_813_0 ER -
%0 Journal Article %A Panchenko, Dmitry %T Ghirlanda–Guerra identities and ultrametricity: An elementary proof in the discrete case %J Comptes Rendus. Mathématique %D 2011 %P 813-816 %V 349 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.06.021/ %R 10.1016/j.crma.2011.06.021 %G en %F CRMATH_2011__349_13-14_813_0
Panchenko, Dmitry. Ghirlanda–Guerra identities and ultrametricity: An elementary proof in the discrete case. Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 813-816. doi : 10.1016/j.crma.2011.06.021. http://www.numdam.org/articles/10.1016/j.crma.2011.06.021/
[1] On the stability of the quenched state in mean-field spin-glass models, J. Stat. Phys., Volume 92 (1998) no. 5–6, pp. 765-783
[2] On the structure of quasi-stationary competing particles systems, Ann. Probab., Volume 37 (2009) no. 3, pp. 1080-1113
[3] Gram–de Finetti matrices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., Volume 119 (1982), pp. 77-86
[4] General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A, Volume 31 (1998) no. 46, pp. 9149-9155
[5] A connection between Ghirlanda–Guerra identities and ultrametricity, Ann. Probab., Volume 38 (2010) no. 1, pp. 327-347
[6] On the Dovbysh–Sudakov representation result, Electron. Commun. Probab., Volume 15 (2010), pp. 330-338
[7] The Ghirlanda–Guerra identities for mixed p-spin model, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 189-192
[8] Spin Glasses: A Challenge for Mathematicians, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics, vol. 43, Springer-Verlag, 2003
[9] Construction of pure states in mean-field models for spin glasses, Probab. Theory Related Fields, Volume 148 (2010) no. 3–4, pp. 601-643
[10] Mean-Field Models for Spin Glasses, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics, vols. 54, 55, Springer-Verlag, 2011
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