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Probability Theory and Related Fields

, Volume 158, Issue 1–2, pp 159–196 | Cite as

Complex random energy model: zeros and fluctuations

  • Zakhar Kabluchko
  • Anton Klimovsky
Article

Abstract

The partition function of the random energy model at inverse temperature \(\beta \) is a sum of random exponentials \( \mathcal{Z }_N(\beta )=\sum _{k=1}^N \exp (\beta \sqrt{n} X_k)\), where \(X_1,X_2,\ldots \) are independent real standard normal random variables (=random energies), and \(n=\log N\). We study the large N limit of the partition function viewed as an analytic function of the complex variable \(\beta \). We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex \(\beta \), both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.

Keywords

Random energy model Sums of random exponentials  Zeros of random analytic functions Central limit theorem Extreme value theory  Stable distributions Logarithmic potentials 

Mathematics Subject Classification (2000)

Primary 60G50 Secondary 82B44 60E07 30B20 60F05 60F17 60G15 

Notes

Acknowledgments

We are grateful to Mikhail Sodin and Avner Kiro for pointing out a mistake in the first version of the paper. We also thank the anonymous referee for useful comments, as well as Dmitry Zaporozhets for useful discussions. AK thanks the Institute of Stochastics of Ulm University for kind hospitality.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of StochasticsUlm UniversityUlmGermany
  2. 2.Mathematical InstituteLeiden UniversityLeidenThe Netherlands

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