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Short-Range Spin Glasses and Random Overlap Structures

  • Louis-Pierre ArguinEmail author
  • Michael Damron
Article

Abstract

Properties of Random Overlap Structures (ROSt)’s constructed from the Edwards-Anderson (EA) Spin Glass model on ℤ d with periodic boundary conditions are studied. ROSt’s are ℕ×ℕ random matrices whose entries are the overlaps of spin configurations sampled from the Gibbs measure. Since the ROSt construction is the same for mean-field models (like the Sherrington-Kirkpatrick model) as for short-range ones (like the EA model), the setup is a good common ground to study the effect of dimensionality on the properties of the Gibbs measure. In this spirit, it is shown, using translation invariance, that the ROSt of the EA model possesses a local stability that is stronger than stochastic stability, a property known to hold at almost all temperatures in many spin glass models with Gaussian couplings. This fact is used to prove stochastic stability for the EA spin glass at all temperatures and for a wide range of coupling distributions. On the way, a theorem of Newman and Stein about the pure state decomposition of the EA model is recovered and extended.

Keywords

Spin glasses Edwards-Anderson model Metastate Stochastic stability 

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) CrossRefzbMATHMathSciNetADSGoogle Scholar
  2. 2.
    Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. Commun. Math. Phys. 130, 489–528 (1990) CrossRefzbMATHADSMathSciNetGoogle Scholar
  3. 3.
    Aldous, D.: Exchangeability and related topics. In: École d’été de Probabilités de Saint-Flour XIII. Lecture Notes in Mathematics, vol. 1117, pp. 1–198. Springer, Berlin (1985) CrossRefGoogle Scholar
  4. 4.
    Arguin, L.-P., Chatterjee, S.: Random overlap structures: properties and applications to spin glasses. arXiv:1011.1823 (2010)
  5. 5.
    Contucci, P.: Replica equivalence in the Edwards-Anderson model. J. Phys. A, Math. Gen. 36, 10961 (2003) CrossRefzbMATHADSMathSciNetGoogle Scholar
  6. 6.
    Contucci, P.: Stochastic stability: a review and some perspectives. J. Stat. Phys. 138, 543–558 (2010) CrossRefzbMATHADSMathSciNetGoogle Scholar
  7. 7.
    Contucci, P., Giardina, C.: Spin-glass stochastic stability: a rigorous proof. Ann. Inst. Henri Poincaré 5, 915–923 (2005) MathSciNetGoogle Scholar
  8. 8.
    Contucci, P., Giardina, C., Giberti, C.: Interaction-flip identities in spin glasses. J. Stat. Phys. 135, 1181–1203 (2009) CrossRefzbMATHADSMathSciNetGoogle Scholar
  9. 9.
    Dovbysh, L., Sudakov, V.: Gram-de Finetti matrices. J. Sov. Math. 24, 3047–3054 (1982) Google Scholar
  10. 10.
    Fisher, D., Huse, D.: Pure states in spin glasses. J. Phys. A, Math. Gen. 20, L997–1003 (1987) CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Külske, C.: Metastates in disordered mean-field models: random field and Hopfield models. J. Stat. Phys. 88, 1257–1293 (1997) CrossRefzbMATHADSGoogle Scholar
  12. 12.
    Newman, C.M.: Topics in Disordered Systems. Birkhäuser, Basel (1997), 88 pp. zbMATHCrossRefGoogle Scholar
  13. 13.
    Newman, C.M., Stein, D.L.: Non-mean-field behavior of realistic spin glasses. Phys. Rev. Lett. 76, 515–518 (1996) CrossRefADSGoogle Scholar
  14. 14.
    Newman, C.M., Stein, D.L.: The metastate approach to thermodynamic chaos. Phys. Rev. E 55, 5194–5211 (1997) CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Newman, C.M., Stein, D.L.: Thermodynamic chaos and the structure of short-range spin glasses. In: Bovier, A., Picco, P. (eds.) Mathematics of Spin Glasses and Neural Networks. Birkhaüser, Boston (1997) Google Scholar
  16. 16.
    Newman, C.M., Stein, D.L.: Metastates, translation ergodicity, and simplicity of thermodynamic states in disordered systems: an illustration. In: Sidoravicius, V. (ed.) New Trends in Mathematical Physics: Proceedings of the 2006 International Congress of Mathematical Physics, pp. 643–652. Springer, Heidelberg (2009) Google Scholar
  17. 17.
    Panchenko, D.: On the differentiability of the Parisi formula. Electron. Commun. Probab. 13, 241–247 (2008) zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Panchenko, D.: On the Dovbysh-Sudakov representation result. arXiv:0905.1524 (2009)
  19. 19.
    Panchenko, D.: The Ghirlanda-Guerra identities for mixed p-spin model. C. R. Math. 348, 189–192 (2010) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Panchenko, D.: Spin glass models from the point of view of spin distributions. arXiv:1005.2720 (2010)
  21. 21.
    Parisi, G.: Stochastic stability. AIP Conf. Proc. 553, 73–79 (2001) CrossRefADSGoogle Scholar
  22. 22.
    Simon, B.: The Statistical Mechanics of Lattice Gases, vol. 1. Princeton University Press, Princeton (1993), 520 pp. zbMATHGoogle Scholar
  23. 23.
    Talagrand, M.: Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models. Springer, Berlin (2003), 586 pp. zbMATHGoogle Scholar
  24. 24.
    Talagrand, M.: Construction of pure states in mean field models for spin glasses. Probab. Theory Relat. Fields 148, 601–643 (2009) CrossRefMathSciNetGoogle Scholar
  25. 25.
    van Hemmen, J.L., Palmer, R.G.: The thermodynamic limit and the replica method for short-range random systems. J. Phys. A, Math. Gen. 15, 3881 (1982) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Courant InstituteNYUNew YorkUSA
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA

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