Electronic Communications in Probability

A note on Talagrand's positivity principle

Dmitriy Panchenko

Full-text: Open access


Talagrand's positivity principle states that one can slightly perturb a Hamiltonian in the Sherrington-Kirkpatrick model in such a way that the overlap of two configurations under the perturbed Gibbs' measure will become typically nonnegative. In this note we observe that abstracting from the setting of the SK model only improves the result and does not require any modifications in Talagrand's argument. In this version, for example, positivity principle immediately applies to the setting of replica symmetry breaking interpolation. Also, abstracting from the SK model improves the conditions in the Ghirlanda-Guerra identities and as a consequence results in a perturbation of smaller order necessary to ensure positivity of the overlap.

Article information

Electron. Commun. Probab., Volume 12 (2007), paper no. 38, 401-410.

Accepted: 21 October 2007
First available in Project Euclid: 6 June 2016

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]
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Talagrand's positivity principle Ghirlanda-Guerra identities

This work is licensed under aCreative Commons Attribution 3.0 License.


Panchenko, Dmitriy. A note on Talagrand's positivity principle. Electron. Commun. Probab. 12 (2007), paper no. 38, 401--410. doi:10.1214/ECP.v12-1326. https://projecteuclid.org/euclid.ecp/1465224981

Export citation


  • Aizenman, M., Sims, R., Starr, S. (2003) An extended variational principle for the SK spin-glass model. Phys. Rev. B, 68, 214403.
  • Contucci, Pierluigi; Giardinà, Cristian. The Ghirlanda-Guerra identities. J. Stat. Phys. 126 (2007), no. 4-5, 917–931.
  • 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.
  • Guerra, Francesco. Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 (2003), no. 1, 1–12.
  • Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces.Isoperimetry and processes.Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9
  • Panchenko, Dmitry. Free energy in the generalized Sherrington-Kirkpatrick mean field model. Rev. Math. Phys. 17 (2005), no. 7, 793–857.
  • Parisi, Giorgio; Talagrand, Michel. On the distribution of the overlaps at given disorder. C. R. Math. Acad. Sci. Paris 339 (2004), no. 4, 303–306.
  • Sherrington, D., Kirkpatrick, S. (1972) Solvable model of a spin glass. Phys. Rev. Lett. 35, 1792-1796.
  • Talagrand, Michel. Spin glasses: a challenge for mathematicians.Cavity and mean field models. 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], 46. Springer-Verlag, Berlin, 2003. x+586 pp. ISBN: 3-540-00356-8
  • Talagrand, Michel. On Guerra's broken replica-symmetry bound. C. R. Math. Acad. Sci. Paris 337 (2003), no. 7, 477–480.
  • Talagrand, M. (2006) Parisi formula. Ann. of Math. (2) 163, no. 1, 221-263.
  • Talagrand, M. (2007) Large deviations, Guerra's and A.S.S. Schemes, and the Parisi hypothesis. Lecture Notes in Mathematics, Vol. 1900, Eds: E. Bolthausen, A. Bovier.