Skip to main content
Log in

Large Deviations, Guerra’s and A.S.S. Schemes, and the Parisi Hypothesis

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We investigate the problem of computing

$$\lim_{N \to \infty}\frac{1}{aN}\log EZ_N^a$$

for any value of a, where Z N is the partition function of the celebrated Sherrington-Kirkpatrick (SK) model, or of some of its natural generalizations. This is a natural “large deviation” problem. Its study helps to get a fresh look at some of the recent ideas introduced in the area, and raises a number of natural questions. We provide a complete solution for a ≥ 0.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Aizenman, R. Sim and S. L. Starr, An extended variational principle for the SK model. Phys. Rev. B 68:214403 (2003).

    Google Scholar 

  2. A. Crisanti and H. J. Sommers, The spherical p-spin interaction spin glass model: The statics Z. Phys. B. condensed. Matter. 83:341–354 (1992).

    Google Scholar 

  3. V. Dotsenko, S. Franz and M. Mézard, preprint, 2004.

  4. F. Guerra, Replica broken bounds in the mean field spin glass model. Commun. Math. Phys. 233:1–12 (2003).

    Google Scholar 

  5. F. Guerra, About the cavity fields in mean field spin glass models/Cond-math 0307673 (2003).

  6. F. Guerra and F. Toninelli, Quadratic replica coupling in the Sherrington-Kirkpatrick mean field spin glass model. J. Math. Phys. 43(7):3704–3716 (2002).

    Google Scholar 

  7. S. Ghirlanda and F. Guerra, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A31:9149–9155 (1998).

    Google Scholar 

  8. M. Mézard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, 1987).

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

  10. D. Panchenko, A question about the Parisi functional, Elect. Com. Probab. 10 (to appear).

  11. J. Pitman and M. Yor, The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25:855–900 (1997).

    Google Scholar 

  12. M. Talagrand, Spin Glasses, a challenge for mathematicians (Springer Verlag, New York, 2003).

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

    Google Scholar 

  14. M. Talagrand, Free energy of the spherical model, Probab. Theor. Relat. Fields 134:471–486 (2006)

    Google Scholar 

  15. M. Talagrand, Some obnoxions problems on spin glasses, to appear in the proceedings of the Monte Veritas conference.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michel Talagrand.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Talagrand, M. Large Deviations, Guerra’s and A.S.S. Schemes, and the Parisi Hypothesis. J Stat Phys 126, 837–894 (2007). https://doi.org/10.1007/s10955-006-9108-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-006-9108-9

Keywords

Navigation