The Aizenman-Sims-Starr scheme and Parisi formula for mixed p-spin spherical models

Wei-Kuo Chen (University of Chicago)

Abstract


The Parisi formula for the free energy in the spherical models with mixed even p-spin interactions was proven in Michel Talagrand. In this paper we study the general mixed p-spin spherical models including p-spin interactions for odd p. We establish the Aizenman Sims-Starr scheme and from this together with many well-known results and Dmitry Panchenko's recent proof on the Parisi ultrametricity conjecture, we prove the Parisi formula.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-14

Publication Date: October 29, 2013

DOI: 10.1214/EJP.v18-2580

References

  • Aizenman, M., Sims, R., Starr, S. (2003) An extended variational principle for the SK spin-glass model. Phys. Rev. B, 68, 214403.
  • Arguin, Louis-Pierre; Chatterjee, Sourav. Random overlap structures: properties and applications to spin glasses. Probab. Theory Related Fields 156 (2013), no. 1-2, 375--413. MR3055263
  • Baffioni, F., Rosati, F. (2000) Some exact results on the ultrametric overlap distribution in mean field spin glass models. Eur. Phys. J. B, 17, 439-447.
  • Crisanti, A., Sommers, H. J. (1982) The spherical p-spin interaction spin glass model: The statics. Z. Phys. B. Condensed Matter, 83 341-354.
  • Dovbysh, L. N.; Sudakov, V. N. Gram-de Finetti matrices. (Russian) Problems of the theory of probability distribution, VII. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 119 (1982), 77--86, 238, 244--245. MR0666087
  • Ghirlanda, Stefano; Guerra, Francesco. General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A 31 (1998), no. 46, 9149--9155. MR1662161
  • Guerra, Francesco. Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 (2003), no. 1, 1--12. MR1957729
  • Parisi, Giorgio. Order parameter for spin-glasses. Phys. Rev. Lett. 50 (1983), no. 24, 1946--1948. MR0702601
  • Parisi, G. (1980) A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A, 13, L-115.
  • Panchenko, Dmitry. Spin glass models from the point of view of spin distributions. Ann. Probab. 41 (2013), no. 3A, 1315--1361. MR3098679
  • Panchenko, Dmitry. The Parisi ultrametricity conjecture. Ann. of Math. (2) 177 (2013), no. 1, 383--393. MR2999044
  • Panchenko, D. (2011) The Parisi formula for mixed p-spin models. To appear in the Ann. of Probab.
  • Sherrington, D., Kirkpatrick, S. (1972) Solvable model of a spin glass. Phys. Rev. Lett., 35, 1792-1796.
  • Stroock, Daniel W. Probability theory, an analytic view. Cambridge University Press, Cambridge, 1993. xvi+512 pp. ISBN: 0-521-43123-9 MR1267569
  • Talagrand, Michel. On Guerra's broken replica-symmetry bound. C. R. Math. Acad. Sci. Paris 337 (2003), no. 7, 477--480. MR2023757
  • Talagrand, Michel. Free energy of the spherical mean field model. Probab. Theory Related Fields 134 (2006), no. 3, 339--382. MR2226885
  • Talagrand, Michel. The Parisi formula. Ann. of Math. (2) 163 (2006), no. 1, 221--263. MR2195134
  • Talagrand, M. (2011) Mean-Field Models for Spin Glasses. Ergebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge A Series of Modern Surveys in Mathematics, Vol. 54, 55. Springer-Verlag.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.