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Microcanonical Analysis of the Random Energy Model in a Random Magnetic Field

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Abstract

We study the spin glass system consisting of a Random Energy Model coupled with a random magnetic field. This system was investigated by de Oliveira Filho et al. (Phys Rev E 74:031117, 2006) who computed the free energy. In this paper, we recover their result rigorously using elementary large deviations arguments and a conditional second moment method. Our analysis extends at the level of fluctuations of the ground states. In particular, we prove that the joint distribution of the minimal energies has the law of a Poisson process with exponential density after a recentering, which is random as opposed to the standard REM. One consequence is that the Gibbs measure of the model exhibits a one-step replica symmetry breaking as argued by de Oliveira Filho et al. using the replica method.

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Acknowledgments

We thank the referees and Roberto Persechino for their careful reading and comments that helped improve the first version of this article. L.-P. A. is supported by a NSERC Discovery Grant and a Grant FQRNT Nouveaux chercheurs.

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Authors and Affiliations

  1. Département de Mathématiques et Statistique, Université de Montréal, 2920 chemin de la Tour, Montreal, QC , H3T 1J4, Canada

    Louis-Pierre Arguin

  2. Department of Mathematics, College of Staten Island, City University of New York, 2800 Victory Boulevard, Staten Island, NY , 10314, USA

    Nicola Kistler

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  1. Louis-Pierre Arguin
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  2. Nicola Kistler
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Correspondence to Louis-Pierre Arguin.

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Arguin, LP., Kistler, N. Microcanonical Analysis of the Random Energy Model in a Random Magnetic Field. J Stat Phys 157, 1–16 (2014). https://doi.org/10.1007/s10955-014-1072-1

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