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Derrida’s Random Energy Models

From Spin Glasses to the Extremes of Correlated Random Fields

  • Chapter
Correlated Random Systems: Five Different Methods

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2143))

Abstract

We discuss Derrida’s random energy models under the light of the recent advances in the study of the extremes of highly correlated random fields. In particular, we present a refinement of the second moment method which provides a unifying approach to models where multiple scales can be identified, such is the case for e.g. branching diffusions, the 2-dim Gaussian free field, certain issues of percolation in high dimensions, or cover times. The method identifies some universal mechanisms which seemingly play a fundamental role also in the behavior of the extremes of the characteristic polynomials of certain random matrix ensembles, or in the extremes of the Riemann ζ-function along the critical line.

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Notes

  1. 1.

    It goes without saying, if this is the case technicalities are naturally reduced by an order of magnitude.

  2. 2.

    One may follow analogous steps in order to identify the scales in a GREM(K), for generic K. In this case one simply decomposes telescopically into a sum of K terms ensuing from local projection on larger and larger neighborhoods.

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Acknowledgements

I am indebted to Erwin Bolthausen, who taught me all I know about the random energy models. It is a pleasure to thank Louis-Pierre Arguin, David Belius, Anton Bovier, Yan V. Fyodorov and Markus Petermann for the countless discussions on the topics of these notes. This work has been supported by the German Research Council in the SFB 611, the Hausdorff Center for Mathematics in Bonn, and Aix Marseille University/CIRM in Luminy through the Chair Jean Morlet. Hospitality of the University of Montreal where part of this work was done is also gratefully acknowledged.

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  1. Institut für Mathematik, Goethe Universität Frankfurt, Robert-Mayer Str. 10, 60054, Frankfurt am Main, Germany

    Nicola Kistler

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  1. Nicola Kistler
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  1. Institut de Mathématiques de Marseille, Aix-Marseille Université CNRS, Marseille Cedex 13, France

    Véronique Gayrard

  2. Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

    Nicola Kistler

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Kistler, N. (2015). Derrida’s Random Energy Models. In: Gayrard, V., Kistler, N. (eds) Correlated Random Systems: Five Different Methods. Lecture Notes in Mathematics, vol 2143. Springer, Cham. https://doi.org/10.1007/978-3-319-17674-1_3

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