Electronic Communications in Probability

On the accuracy of the normal approximation for the free energy in the Random Energy Model

Raphael Meiners and Anselm Reichenbachs

Full-text: Open access

Abstract

In the present paper we consider the fluctuations of the free energy in the random energy model (REM) on a moderate deviation scale. We find that for high temperatures the normal approximation holds only in a narrow range of scalings away from the CLT. For scalings of higher order, probabilities of moderate deviations decay faster than exponentially.

Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 12, 11 pp.

Dates
Accepted: 12 February 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315551

Digital Object Identifier
doi:10.1214/ECP.v18-2377

Mathematical Reviews number (MathSciNet)
MR3033595

Zentralblatt MATH identifier
1308.60030

Subjects
Primary: 60F10: Large deviations
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Random Energy Model free energy moderate deviations large deviations

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

Citation

Meiners, Raphael; Reichenbachs, Anselm. On the accuracy of the normal approximation for the free energy in the Random Energy Model. Electron. Commun. Probab. 18 (2013), paper no. 12, 11 pp. doi:10.1214/ECP.v18-2377. https://projecteuclid.org/euclid.ecp/1465315551


Export citation

References

  • Bovier, Anton. Statistical mechanics of disordered systems. A mathematical perspective. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2006. xiv+312 pp. ISBN: 978-0-521-84991-3; 0-521-84991-8
  • Bovier, Anton; Kurkova, Irina; Lรถwe, Matthias. Fluctuations of the free energy in the REM and the $p$-spin SK models. Ann. Probab. 30 (2002), no. 2, 605–651.
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
  • Derrida, B. Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45 (1980), no. 2, 79–82.
  • Derrida, Bernard. Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B (3) 24 (1981), no. 5, 2613–2626.
  • Dorlas, T. C.; Wedagedera, J. R. Large deviations and the random energy model. Internat. J. Modern Phys. B 15 (2001), no. 1, 1–15.
  • Eichelsbacher, Peter; Lรถwe, Matthias. Moderate deviations for i.i.d. random variables. ESAIM Probab. Stat. 7 (2003), 209–218 (electronic).
  • Ellis, Richard S. Entropy, large deviations, and statistical mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 271. Springer-Verlag, New York, 1985. xiv+364 pp. ISBN: 0-387-96052-X
  • Fedrigo, M.; Flandoli, F.; Morandin, F. A large deviation principle for the free energy of random Gibbs measures with application to the REM. Ann. Mat. Pura Appl. (4) 186 (2007), no. 3, 381–417.
  • Lรถwe, Matthias; Meiners, Raphael. Moderate deviations for Random Field Curie-Weiss Models, Journal of Statistical Physics 149 (2012), 701–721.
  • Olivieri, Enzo; Picco, Pierre. On the existence of thermodynamics for the random energy model. Comm. Math. Phys. 96 (1984), no. 1, 125–144.
  • Reichenbachs, Anselm. Moderate deviations for a Curie-Weiss model with dynamical external field, ESAIM: Probability and Statistics eFirst (2012).
  • 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