The Annals of Probability

The Parisi formula for mixed $p$-spin models

Dmitry Panchenko

Full-text: Open access

Abstract

The Parisi formula for the free energy in the Sherrington–Kirkpatrick and mixed $p$-spin models for even $p\geq2$ was proved in the seminal work of Michel Talagrand [Ann. of Math. (2) 163 (2006) 221–263]. In this paper we prove the Parisi formula for general mixed $p$-spin models which also include $p$-spin interactions for odd $p$. Most of the ideas used in the paper are well known and can now be combined following a recent proof of the Parisi ultrametricity conjecture in [Ann. of Math. (2) 177 (2013) 383–393].

Article information

Source
Ann. Probab. Volume 42, Number 3 (2014), 946-958.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1395838120

Digital Object Identifier
doi:10.1214/12-AOP800

Mathematical Reviews number (MathSciNet)
MR3189062

Zentralblatt MATH identifier
1292.82020

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Sherrington–Kirkpatrick model free energy ultrametricity

Citation

Panchenko, Dmitry. The Parisi formula for mixed $p$-spin models. Ann. Probab. 42 (2014), no. 3, 946--958. doi:10.1214/12-AOP800. https://projecteuclid.org/euclid.aop/1395838120.


Export citation

References

  • [1] Aizenman, M., Sims, R. and Starr, S. (2003). An extended variational principle for the SK spin-glass model. Phys. Rev. B 68 214403.
  • [2] Arguin, L.-P. and Aizenman, M. (2009). On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 1080–1113.
  • [3] Arguin, L. P. and Chatterjee, S. (2014). Random overlap structures: Properties and applications to spin glasses. Probab. Theory Related Fields 156 375–413.
  • [4] Baffioni, F. and Rosati, F. (2000). Some exact results on the ultrametric overlap distribution in mean field spin glass models. Eur. Phys. J. B 17 439–447.
  • [5] Bolthausen, E. and Sznitman, A. S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276.
  • [6] Dovbysh, L. N. and Sudakov, V. N. (1982). Gram-de Finetti matrices. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 119 77–86, 238, 244–245.
  • [7] Ghirlanda, S. and Guerra, F. (1998). General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A 31 9149–9155.
  • [8] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
  • [9] Mézard, M., Parisi, G., Sourlas, N., Toulouse, G. and Virasoro, M. (1984). Replica symmetry breaking and the nature of the spin glass phase. J. Physique 45 843–854.
  • [10] Mézard, M., Parisi, G., Sourlas, N., Toulouse, G. and Virasoro, M. A. (1984). On the nature of the spin-glass phase. Phys. Rev. Lett. 52 1156.
  • [11] Panchenko, D. (2007). A note on Talagrand’s positivity principle. Electron. Commun. Probab. 12 401–410 (electronic).
  • [12] Panchenko, D. (2010). A connection between the Ghirlanda–Guerra identities and ultrametricity. Ann. Probab. 38 327–347.
  • [13] Panchenko, D. (2010). On the Dovbysh–Sudakov representation result. Electron. Commun. Probab. 15 330–338.
  • [14] Panchenko, D. (2010). The Ghirlanda–Guerra identities for mixed $p$-spin model. C. R. Math. Acad. Sci. Paris 348 189–192.
  • [15] Panchenko, D. (2011). Ghirlanda–Guerra identities and ultrametricity: An elementary proof in the discrete case. C. R. Math. Acad. Sci. Paris 349 813–816.
  • [16] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer, New York.
  • [17] Panchenko, D. (2013). Spin glass models from the point of view of spin distributions. Ann. Probab. 41 1315–1361.
  • [18] Panchenko, D. (2013). The Parisi ultrametricity conjecture. Ann. of Math. (2) 177 383–393.
  • [19] Parisi, G. (1979). Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43 1754–1756.
  • [20] Parisi, G. (1980). A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13 L–115.
  • [21] Ruelle, D. (1987). A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 225–239.
  • [22] Sherrington, D. and Kirkpatrick, S. (1972). Solvable model of a spin glass. Phys. Rev. Lett. 35 1792–1796.
  • [23] Talagrand, M. (2003). On Guerra’s broken replica-symmetry bound. C. R. Math. Acad. Sci. Paris 337 477–480.
  • [24] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
  • [25] Talagrand, M. (2010). Construction of pure states in mean field models for spin glasses. Probab. Theory Related Fields 148 601–643.
  • [26] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume I. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 54. Springer, Berlin.