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

Raphael Meiners (Universität Münster)
Anselm Reichenbachs (Ruhr-Universität Bochum)


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.

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Pages: 1-11

Publication Date: February 12, 2013

DOI: 10.1214/ECP.v18-2377


  • 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 MR2252929
  • Bovier, Anton; Kurkova, Irina; Löwe, Matthias. Fluctuations of the free energy in the REM and the $p$-spin SK models. Ann. Probab. 30 (2002), no. 2, 605--651. MR1905853
  • 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 MR1619036
  • Derrida, B. Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45 (1980), no. 2, 79--82. MR0575260
  • Derrida, Bernard. Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B (3) 24 (1981), no. 5, 2613--2626. MR0627810
  • Dorlas, T. C.; Wedagedera, J. R. Large deviations and the random energy model. Internat. J. Modern Phys. B 15 (2001), no. 1, 1--15. MR1811341
  • Eichelsbacher, Peter; Löwe, Matthias. Moderate deviations for i.i.d. random variables. ESAIM Probab. Stat. 7 (2003), 209--218 (electronic). MR1956079
  • 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 MR0793553
  • 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. MR2317646
  • Löwe, 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. MR0765963
  • 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 MR1993891

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