Skip to main content
SpringerLink
Log in

The Parisi Formula

  • Chapter
  • First Online:
The Sherrington-Kirkpatrick Model

Part of the book series: Springer Monographs in Mathematics ((SMM))

  • 2860 Accesses

Abstract

The main goal of this chapter is to prove the celebrated Parisi formula for the free energy in the mixed p-spin models. The Ruelle probability cascades studied in the previous chapter will be used in a number of different ways, but their most important role will be as an approximation of the asymptotic Gibbs measures that generate the overlap matrix in the thermodynamic limit. As we explained in Sect.2.4, a link between the Gibbs measure and the Ruelle probability cascades can be established using the Ghirlanda–Guerra identities and in this chapter we will show how these identities arise in the setting of the mixed p-spin models. In addition, the proof of the Parisi formula will be based on several other essential ideas, such as the Talagrand positivity principle, the Guerra replica symmetry breaking interpolation, and the Aizenman–Sims–Starr scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aizenman, M., Contucci, P.: On the stability of the quenched state in mean-field spin-glass models. J. Statist. Phys. 92(5–6), 765–783 (1998)

    Google Scholar 

  2. Aizenman, M., Sims, R., Starr, S.L.: An extended variational principle for the SK spin-glass model. Phys. Rev. B 68, 214403 (2003)

    Google Scholar 

  3. Aldous, D.: Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11(4), 581–598 (1981)

    Google Scholar 

  4. Aldous, D.: Exchangeability and related topics. École d’été Probabilités de Saint-Flour, XIII-1983, Lecture Notes in Math., vol. 1117, pp. 1–198. Springer, Berlin (1985)

    Google Scholar 

  5. Arguin, L.-P., Aizenman, M.: On the structure of quasi-stationary competing particles systems. Ann. Probab. 37(3), 1080–1113 (2009)

    Google Scholar 

  6. Arguin, L.-P., Chatterjee, S.: Random overlap structures: properties and applications to spin glasses. Probab. Theory Related Fields (2012) doi: 10.1007/s00440-012-0431-6

    Google Scholar 

  7. Austin, T.: On exchangeable random variables and the statistics of large graphs and hypergraphs. Probab. Surv. 5, 80–145 (2008)

    Google Scholar 

  8. Austin, T.: Exchangeable random measures. Preprint (2013)

    Google Scholar 

  9. Baffioni, F., Rosati, F.: Some exact results on the ultrametric overlap distribution in mean field spin glass models. Eur. Phys. J. B 17, 439–447 (2000)

    Google Scholar 

  10. Bolthausen, E., Sznitman, A.-S.: On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197(2), 247–276 (1998)

    Google Scholar 

  11. Bovier, A., Kurkova, I.: Derrida’s generalized random energy models I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40(4), 439–480 (2004)

    Google Scholar 

  12. Carmona, P., Hu, Y.: Universality in Sherrington-Kirkpatrick’s spin glass model. Ann. Inst. H. Poincaré Probab. Statist. 42(2), 215–222 (2006)

    Google Scholar 

  13. Chen, W.-K., Panchenko, D.: An approach to chaos in some mixed p-spin models. Probab. Theory Related Fields (2012) doi: 10.1007/s00440-012-0460-1

    Google Scholar 

  14. Contucci, P., Giardina, C.: Spin-glass stochastic stability: a rigorous proof. Ann. Henri Poincaré 6(5), 915–923 (2005)

    Google Scholar 

  15. de Dominicis, C., Hilhorst, H.: Random (free) energies in spin glasses. J. Phys. Lett. 46, L909–L914 (1985)

    Google Scholar 

  16. Derrida, B.: Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45(2), 79–82 (1980)

    Google Scholar 

  17. Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B (3) 24(5), 2613–2626 (1981)

    Google Scholar 

  18. Derrida, B.: A generalization of the random energy model that includes correlations between the energies. J. Phys. Lett. 46, 401–407 (1985)

    Google Scholar 

  19. Derrida, B., Gardner, E.: Solution of the generalised random energy model. J. Phys. C 19, 2253–2274 (1986)

    Google Scholar 

  20. Derrida, B., Toulouse, G.: Sample to sample fluctuations in the random energy model. J. Phys. Lett. 46, L223–L228 (1985)

    Google Scholar 

  21. Diaconis, P., Janson, S.: Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28(1), 33–61 (2008)

    Google Scholar 

  22. Dovbysh, L.N., Sudakov, V.N.: Gram-de Finetti matrices. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 119, 77–86 (1982)

    Google Scholar 

  23. Dudley, R.M.: Real analysis and probability. In: Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  24. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1966)

    Google Scholar 

  25. Ghirlanda, S., Guerra, F.: General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A 31(46), 9149–9155 (1998)

    Google Scholar 

  26. Guerra, F., Toninelli, F.L.: The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230(1), 71–79 (2002)

    Google Scholar 

  27. Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233(1), 1–12 (2003)

    Google Scholar 

  28. Hestir, K.: A representation theorem applied to weakly exchangeable nonnegative definite arrays. J. Math. Anal. Appl. 142(2), 390–402 (1989)

    Google Scholar 

  29. Hoover, D.N.: Relations on probability spaces. Preprint (1979)

    Google Scholar 

  30. Hoover, D.N.: Row-column exchangeability and a generalized model for probability. In: Exchangeability in Probability and Statistics (Rome, 1981), pp. 281–291, North-Holland, Amsterdam (1982)

    Google Scholar 

  31. Kallenberg, O.: Foundations of Modern Probability, Probability and Its Applications. Springer, New York (1997)

    Google Scholar 

  32. Kingman, J.F.C.: Poisson Processes. Oxford University Press, New York (1993)

    Google Scholar 

  33. Ledoux, M.: The concentration of measure phenomenon. In: Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence (2001)

    Google Scholar 

  34. Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Combin. Theory Ser. B 96(6), 933–957 (2006)

    Google Scholar 

  35. Lovász, L., Szegedy, B.: Szemerédi’s lemma for the analyst. Geom. Funct. Anal. 17(1), 252–270 (2007)

    Google Scholar 

  36. Lovász, L., Szegedy, B.: Limits of compact decorated graphs. arXiv:1010.5155 (2010)

    Google Scholar 

  37. Mézard, M., Parisi, G., Sourlas, N., Toulouse, G., Virasoro, M.A.: On the nature of the spin-glass phase. Phys. Rev. Lett. 52, 1156 (1984)

    Google Scholar 

  38. Mézard, M., Parisi, G., Sourlas, N., Toulouse, G., Virasoro, M.A.: Replica symmetry breaking and the nature of the spin-glass phase. J. de Physique 45, 843 (1984)

    Google Scholar 

  39. Mézard, M., Parisi, G., Virasoro, M.: Random free energies in spin glasses. J. Phys. Lett. 46, L217–L222 (1985)

    Google Scholar 

  40. Mézard, M., Parisi, G., Virasoro, M.A.: Spin glass theory and beyond. In: World Scientific Lecture Notes in Physics, vol. 9. World Scientific, Teaneck (1987)

    Google Scholar 

  41. Panchenko, D.: A note on Talagrand’s positivity principle. Electron. Comm. Probab. 12, 401–410 (2007)

    Google Scholar 

  42. Panchenko, D.: On differentiability of the Parisi formula. Elect. Comm. in Probab. 13, 241–247 (2008)

    Google Scholar 

  43. Panchenko, D.: A connection between Ghirlanda-Guerra identities and ultrametricity. Ann. Probab. 38(1), 327–347 (2010)

    Google Scholar 

  44. Panchenko, D.: On the Dovbysh-Sudakov representation result. Electron. Comm. Probab. 15, 330–338 (2010)

    Google Scholar 

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

    Google Scholar 

  46. Panchenko, D.: Spin glass models from the point of view of spin distributions. To appear in Ann. of Probab. arXiv: 1005.2720 (2010)

    Google Scholar 

  47. Panchenko, D.: Ghirlanda-Guerra identities and ultrametricity: An elementary proof in the discrete case. C. R. Acad. Sci. Paris Ser. I 349, 813–816 (2011)

    Google Scholar 

  48. Panchenko, D.: The Parisi formula for mixed p-spin models. To appear in Ann. of Probab. arXiv:1112.4409 (2011)

    Google Scholar 

  49. Panchenko, D.: A unified stability property in spin glasses. Comm. Math. Phys. 313(3), 781–790 (2012)

    Google Scholar 

  50. Panchenko, D.: The Parisi ultrametricity conjecture. Ann. of Math. (2) 177(1), 383–393 (2013)

    Google Scholar 

  51. Parisi, G.: Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43, 1754–1756 (1979)

    Google Scholar 

  52. Parisi, G.: A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13, L-115 (1980)

    Google Scholar 

  53. Parisi, G.: Order parameter for spin glasses. Phys. Rev. Lett. 50, 1946 (1983)

    Google Scholar 

  54. Parisi, G., Talagrand, M.: On the distribution of the overlaps at given disorder. C. R. Math. Acad. Sci. Paris 339(4), 303–306 (2004)

    Google Scholar 

  55. Pastur, L.A., Shcherbina, M.V.: Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model. J. Statist. Phys. 62(1–2), 1–19 (1991)

    Google Scholar 

  56. Ruelle, D.: A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108(2), 225–239 (1987)

    Google Scholar 

  57. Schechter, M.: Principles of Functional Analysis. Academic, New York (1971)

    Google Scholar 

  58. Sherrington, D., Kirkpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett. 35, 1792–1796 (1975)

    Google Scholar 

  59. Talagrand, M.: Gaussian averages, Bernoulli averages, and Gibbs’ measures. Random Struct. Algorithms 21(3–4), 197–204 (2002)

    Google Scholar 

  60. Talagrand, M.: On Guerra’s broken replica-symmetry bound. C. R. Math. Acad. Sci. Paris 337(7), 477–480 (2003)

    Google Scholar 

  61. Talagrand, M.: On the meaning of Parisi’s functional order parameter. C. R. Math. Acad. Sci. Paris 337(9), 625–628 (2003)

    Google Scholar 

  62. Talagrand, M.: Spin Glasses: A Challenge for Mathematicians. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge A Series of Modern Surveys in Mathematics, vol. 43. Springer (2003)

    Google Scholar 

  63. Talagrand, M.: Parisi measures. J. Funct. Anal. 231(2), 269–286 (2006)

    Google Scholar 

  64. Talagrand, M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006)

    Google Scholar 

  65. Talagrand, M.: Construction of pure states in mean-field models for spin glasses. Probab. Theory Relat. Fields 148(3–4), 601–643 (2010)

    Google Scholar 

  66. Talagrand, M.: Mean-Field Models for Spin Glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge A Series of Modern Surveys in Mathematics, vol. 54, 55. Springer (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Dmitry Panchenko

Authors
  1. Dmitry Panchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Panchenko, D. (2013). The Parisi Formula. In: The Sherrington-Kirkpatrick Model. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6289-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation