Electronic Journal of Probability

A duality principle in spin glasses

Antonio Auffinger and Wei-Kuo Chen

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Abstract

We prove a duality principle that connects the thermodynamic limits of the free energies of the Hamiltonians and their squared interactions. Under the main assumption that the limiting free energy is concave in the squared temperature parameter, we show that this relation is valid in a large class of disordered systems. In particular, when applied to mean field spin glasses, this duality provides an interpretation of the Parisi formula as an inverted variational principle, establishing a prediction of Guerra [13].

Article information

Source
Electron. J. Probab. Volume 22 (2017), paper no. 61, 17 pp.

Dates
Received: 5 February 2017
Accepted: 21 May 2017
First available in Project Euclid: 22 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1500689052

Digital Object Identifier
doi:10.1214/17-EJP70

Subjects
Primary: 60F10: Large deviations 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
large deviation Legendre duality Parisi formula spin glass

Rights
Creative Commons Attribution 4.0 International License.

Citation

Auffinger, Antonio; Chen, Wei-Kuo. A duality principle in spin glasses. Electron. J. Probab. 22 (2017), paper no. 61, 17 pp. doi:10.1214/17-EJP70. https://projecteuclid.org/euclid.ejp/1500689052.


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References

  • [1] Arguin, L.-P., Kistler, N: Microcanonical analysis of the random energy model in a random magnetic field. J. Stat. Phys., 157, no. 1, 1–16 (2014)
  • [2] Auffinger, A., Chen, W.-K.: The Parisi formula has a unique minimizer. Comm. Math. Phys., 335, no. 3, 1429–1444 (2015)
  • [3] Auffinger, A., Chen, W.-K.: The Legendre structure of the Parisi formula. To appear in Comm. Math. Phys. (2015)
  • [4] Bolthausen, E., Kistler, N.: Universal structures in some mean field spin glasses and an application. J. Math. Phys., 49, 25–205 (2008)
  • [5] Bolthausen, E., Kistler, N.: A quenched large deviation principle and a Parisi formula for a Perceptron version of the GREM. Probability in Complex Physical Systems, Springer Proceedings in Mathematics, 11, 425–442 (2012)
  • [6] Chen, W.-K.: The Aizenman-Sims-Starr scheme and Parisi formula for mixed p-spin spherical models. Elec. Journal Probab., 18, no. 94, 1–14 (2013)
  • [7] Chen, W.-K.: Variational representation for the Parisi functional and the two-dimensional Guerra-Talagrand bound. arXiv:1501.06635 (2015)
  • [8] Chen, W.-K., Sen, A.: Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models. arXiv:1512.08492 (2015)
  • [9] Crisanti, A., Sommers, H.-J.: The spherical p-spin interaction spin glass model: the statics. Zeitschrift für Physik B Condensed Matter, 87, 341–354 (1992)
  • [10] Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. 2nd ed., Springer-Verlag, New York (2009)
  • [11] Ellis, R.: Entropy, Large Deviations, and Statistical Mechanics. 1st ed., Springer-Verlag, Berlin (2006)
  • [12] Guerra, F.: The phenomenon of spontaneous replica symmetry breaking in complex statistical mechanics systems. J. Phys.: Conf. Ser., 442, 012013 (2013)
  • [13] Guerra, F.: Legendre structures in statistical mechanics for ordered and disordered systems, Cambridge University Press, in Advances in disordered systems, random processes and some applications, Contucci, P. et al, eds, in print (2016)
  • [14] Mézard, M., Parisi, G., Virasoro, M.: Spin glass theory and beyond. World Scientific, 9, Singapore (2004)
  • [15] Panchenko, D.: Free energy in the generalized Sherrington-Kirkpatrick mean field model. Rev. Math. Phys., 17, no. 7, 793–857 (2005)
  • [16] Panchenko, D.: On differentiability of the Parisi formula. Electron. Commun. Probab., 13, 241–247 (2008)
  • [17] Panchenko, D.: The Sherrington-Kirkpatrick model. Springer Monographs in Mathematics. Springer, New York (2013)
  • [18] Panchenko, D.: The Parisi formula for mixed p-spin models. Ann. Probab., 42, no. 3, 946–958 (2014)
  • [19] Panchenko, D.: The free energy in a multi-species Sherrington-Kirkpatrick model. Ann. of Prob., 43, no. 6, 3494–3513 (2015)
  • [20] Panchenko, D.: Free energy in the Potts spin glass. arXiv:1512.00370 (2015)
  • [21] Panchenko, D.: Free energy in the mixed $p$-spin models with vector spins. arXiv:1512.04441 (2015)
  • [22] Parisi, G.: Infinite number of order parameters for spin-glasses. Phys. Rev. Lett., 43, 1754–1756 (1979)
  • [23] Rassoul-agha, F., Seppäläinen, T.: A course on large deviations with an introduction to Gibbs measures. Graduate Studies in Mathematics, American Mathematical Society (2015)
  • [24] Talagrand, M.: 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, 46, Springer-Verlag, Berlin (2003)
  • [25] Talagrand, M.: The Parisi formula. Ann. of Math. (2), 163, no. 1, 221–263 (2006)
  • [26] Talagrand, M.: The free energy of the spherical mean-field model. Probab. Theory Relat. Fields, 134, 339–382 (2006)
  • [27] Talagrand, M.: Mean field models for spin glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 55, Springer-Verlag, Berlin (2011)