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Derrida’s Random Energy Models

From Spin Glasses to the Extremes of Correlated Random Fields
  • Nicola KistlerEmail author
Part of the Lecture Notes in Mathematics book series (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.

Keywords

Branching diffusions Cover times Extreme value theory Gaussian free fields Log-correlated random fields Multi-scale analysis Random energy models Random matrices Riemann ζ-function 

Notes

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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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für MathematikGoethe Universität FrankfurtFrankfurt am MainGermany

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