Comptes Rendus
no. 13-14
Probability Theory/Mathematical Physics
Ghirlanda–Guerra identities and ultrametricity: An elementary proof in the discrete case
[Ultramétricité et identités de Ghirlanda–Guerra : une preuve élémentaire du cas discret]

Présenté par : Michel Talagrand

Dmitry Panchenko1
1 Department of Mathematics, Texas A&M University, 77843 College Station, TX, USA
Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 813-816.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.06.021
Dmitry Panchenko 1

1 Department of Mathematics, Texas A&M University, 77843 College Station, TX, USA 
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Dmitry Panchenko. Ghirlanda–Guerra identities and ultrametricity: An elementary proof in the discrete case. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 813-816. doi : 10.1016/j.crma.2011.06.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.06.021/

[1] M. Aizenman; P. Contucci 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] L.-P. Arguin; M. Aizenman On the structure of quasi-stationary competing particles systems, Ann. Probab., Volume 37 (2009) no. 3, pp. 1080-1113

[3] L.N. Dovbysh; V.N. Sudakov Gram–de Finetti matrices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., Volume 119 (1982), pp. 77-86

[4] S. Ghirlanda; F. Guerra 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] D. Panchenko A connection between Ghirlanda–Guerra identities and ultrametricity, Ann. Probab., Volume 38 (2010) no. 1, pp. 327-347

[6] D. Panchenko On the Dovbysh–Sudakov representation result, Electron. Commun. Probab., Volume 15 (2010), pp. 330-338

[7] D. Panchenko The Ghirlanda–Guerra identities for mixed p-spin model, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 189-192

[8] M. Talagrand 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] M. Talagrand 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] M. Talagrand 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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