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


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.


Mean field spin glasses Extreme value theory Parisi theory Ultrametricity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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/, 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

Personalised recommendations