## Electronic Communications in Probability

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

#### 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

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

Rights

#### 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

#### 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; Là¸£à¸–we, 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; Là¸£à¸–we, 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.
• Là¸£à¸–we, 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