The Annals of Probability

The free energy in a multi-species Sherrington–Kirkpatrick model

Dmitry Panchenko

Full-text: Open access


The authors of [Ann. Henri Poincaré 16 (2015) 691–708] introduced a multi-species version of the Sherrington–Kirkpatrick model and suggested the analogue of the Parisi formula for the free energy. Using a variant of Guerra’s replica symmetry breaking interpolation, they showed that, under certain assumption on the interactions, the formula gives an upper bound on the limit of the free energy. In this paper we prove that the bound is sharp. This is achieved by developing a new multi-species form of the Ghirlanda–Guerra identities and showing that they force the overlaps within species to be completely determined by the overlaps of the whole system.

Article information

Ann. Probab., Volume 43, Number 6 (2015), 3494-3513.

Received: February 2014
Revised: August 2014
First available in Project Euclid: 11 December 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G09: Exchangeability 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Spin glasses Sherrington–Kirkpatrick model


Panchenko, Dmitry. The free energy in a multi-species Sherrington–Kirkpatrick model. Ann. Probab. 43 (2015), no. 6, 3494--3513. doi:10.1214/14-AOP967.

Export citation


  • [1] Aizenman, M., Sims, R. and Starr, S. L. (2003). An extended variational principle for the SK spin-glass model. Phys. Rev. B 68 214403.
  • [2] Auffinger, A. and Chen, W.-K. (2015). The Parisi formula has a unique minimizer. Comm. Math. Phys. 335 1429–1444.
  • [3] Barra, A., Contucci, P., Mingione, E. and Tantari, D. (2015). Multi-species mean-field spin-glasses. Rigorous results. Ann. Henri Poincaré 16 691–708.
  • [4] Chen, W.-K. (2013). The Aizenman–Sims–Starr scheme and Parisi formula for mixed $p$-spin spherical models. Electron. J. Probab. 18 14.
  • [5] 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.
  • [6] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
  • [7] Guerra, F. and Toninelli, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 71–79.
  • [8] Panchenko, D. (2013). The Parisi ultrametricity conjecture. Ann. of Math. (2) 177 383–393.
  • [9] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer, New York.
  • [10] Panchenko, D. (2014). The Parisi formula for mixed $p$-spin models. Ann. Probab. 42 946–958.
  • [11] Parisi, G. (1979). Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43 1754–1756.
  • [12] Parisi, G. (1980). A sequence of approximate solutions to the S–K model for spin glasses. J. Phys. A 13 L–115.
  • [13] Parisi, G. and Talagrand, M. (2004). On the distribution of the overlaps at given disorder. C. R. Math. Acad. Sci. Paris 339 303–306.
  • [14] Ruelle, D. (1987). A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 225–239.
  • [15] Sherrington, D. and Kirkpatrick, S. (1975). Solvable model of a spin glass. Phys. Rev. Lett. 35 1792–1796.
  • [16] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
  • [17] Talagrand, M. (2011). Mean-Field Models for Spin Glasses. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge a Series of Modern Surveys in Mathematics 54. Springer, Berlin.
  • [18] Toninelli, F. L. (2002). About the Almeida–Thouless transition line in the Sherrington–Kirkpatrick mean-field spin glass model. Europhysics Letters 60 764.