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Journal of Statistical Physics

, Volume 135, Issue 5–6, pp 1167–1180 | Cite as

Small Perturbations of a Spin Glass System

  • Louis-Pierre ArguinEmail author
  • Nicola Kistler
Article

Abstract

We show through a simple example that perturbations of the Hamiltonian of a spin glass which cannot be detected at the level of the free energy can completely alter the behavior of the overlap. In particular, perturbations of order O(log N), with N→∞ the size of the system, suffice to have ultrametricity emerge in the thermodynamical limit.

Keywords

Mean field spin glasses Extreme value theory Parisi theory Ultrametricity 

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References

  1. 1.
    Aizenman, M., Contucci, P.: On the stability of the quenched state in mean-field spin-glass models. J. Stat. Phys. 92, 765–783 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aizenman, M., Sims, R., Starr, S.: Extended variational principle for the Sherrington-Kirkpatrick spin-glass model. Phys. Rev. B 68, 214403 (2003) CrossRefADSGoogle Scholar
  3. 3.
    Amaro de Matos, J., Patrick, A., Zagrebnov, V.: Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model. J. Stat. Phys. 66, 139–164 (1992) zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Arguin, L.-P.: A remark on the infinite-volume Gibbs measure of spin glasses. J. Math. Phys. 49, 1–8 (2008) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Clarendon, Oxford (1992), 277 pp. zbMATHGoogle Scholar
  6. 6.
    Bolthausen, E., Kistler, N.: On a non hierarchical version of the generalized random energy model. Ann. Appl. Probab. 16, 1–14 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bolthausen, E., Kistler, N.: On a nonhierarchical version of the generalized random energy model, II. Ultrametricity. Stoch. Process. Appl. (2009). doi: 10.1016/j.spa.2008.12.002, arXiv:0802.3436
  8. 8.
    Bolthausen, E., Sznitman, A.S.: On Ruelle’s probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276 (1998) zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Bolthausen, E., Sznitman, A.S.: Ten Lectures on Random Media. DMV Seminar. Birkhäuser, Basel (2001), 132 pp. Google Scholar
  10. 10.
    Bovier, A., Kurkova, I.: Derrida’s generalized random energy models I & II. Ann. Inst. Henri Poincaré 40, 439–480 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bovier, A., Kurkova, I.: Gibbs measures of Derrida’s generalized random energy model and the genealogy of Neveu’s continuous state branching process. WIAS Preprint Google Scholar
  12. 12.
    Contucci, P., Giardina, C.: Spin-glass stochastic stability: a rigorous proof. Ann. Inst. Henri Poincaré 6, 915–923 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dovbysh, L., Sudakov, V.: Gram-de Finetti matrices. J. Sov. Math. 24, 3047–3054 (1982) Google Scholar
  14. 14.
    Ghirlanda, S.: F. Guerra, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A: Math. Gen. 31, 9149–9155 (1998) zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233, 1–12 (2003) zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Panchenko, D.: A connection between Ghirlanda-Guerra identities and ultrametricity. Preprint arXiv:0810.0743
  17. 17.
    Mézard, M., Parisi, G., Virasoro, M.: Spin glass theory and beyond. World Scientific Lecture Notes in Physics, vol. 9. World Scientific, Singapore (1987), 461 pp. zbMATHGoogle Scholar
  18. 18.
    Parisi, G., Talagrand, M.: On the distribution of the overlaps at given disorder. C.R.A.S. 339, 306–313 (2004) MathSciNetGoogle Scholar
  19. 19.
    Ruelle, D.: A mathematical reformulation of Derrida’s REM and GREM. Commun. Math. Phys. 108, 225–239 (1987) zbMATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Talagrand, M.: Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models. Springer, Berlin (2003), 586 pp. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Institute of Applied MathematicsUniversity of BonnBonnGermany

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