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


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.


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 



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.


  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Volume 55 of National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington, DC (1964)Google Scholar
  2. 2.
    Ben Arous, G., Bogachev, L., Molchanov, S.: Limit theorems for sums of random exponentials. Probab. Theory Relat. Fields 132(4), 579–612 (2005)Google Scholar
  3. 3.
    Bena, I., Droz, M., Lipowski, A.: Statistical mechanics of equilibrium and nonequilibrium phase transitions: the Yang–Lee formalism. Int. J. Mod. Phys. B 19(29), 4269–4329 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Probab. 20(1), 137–151 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Biskup, M., Borgs, C., Chayes, J.T., Kleinwaks, L.J., Kotecký, R.: General theory of Lee–Yang zeros in models with first-order phase transitions. Phys. Rev. Lett. 84(21), 4794–4797 (2000)CrossRefGoogle Scholar
  6. 6.
    Bogachev, L.: Limit laws for norms of IID samples with Weibull tails. J. Theor. Probab. 19(4), 849–873 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bovier, A.: Statistical Mechanics of Disordered Systems. A Mathematical Perspective. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2006)Google Scholar
  8. 8.
    Bovier, A., Kurkova, I., Löwe, M.: Fluctuations of the free energy in the REM and the \(p\)-spin SK models. Ann. Probab. 30(2), 605–651 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cranston, M., Molchanov, S.: Limit laws for sums of products of exponentials of iid random variables. Israel J. Math. 148, 115–136 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Csörgő, S.: Limit behaviour of the empirical characteristic function. Ann. Probab. 9, 130–144 (1981)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Derrida, B.: Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45, 79–82 (1980)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B 24(5), 2613–2626 (1981)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Derrida, B.: The zeroes of the partition function of the random energy model. Phys A Stat. Mech. Appl. 177(1–3), 31–37 (1991)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Derrida, B., Evans, M., Speer, E.: Mean field theory of directed polymers with random complex weights. Comm. Math. Phys. 156(2), 221–244 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Dobrinevski, A., Le Doussal, P., Wiese, K.J.: Interference in disordered systems: a particle in a complex random landscape. Phys. Rev. E 83(6), 061116 (2011)CrossRefGoogle Scholar
  16. 16.
    Düring, G., Kurchan, J.: Statistical mechanics of Monte Carlo sampling and the sign problem. Europhys. Lett. 92(5), 50004 (2010)CrossRefGoogle Scholar
  17. 17.
    Eisele, T.: On a third-order phase transition. Comm. Math. Phys. 90(1), 125–159 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Feuerverger, A., Mureika, R.: The empirical characteristic function and its applications. Ann. Stat. 5, 88–97 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley Publishing Co., Reading, Massachusetts (1968). (Translated from the Russian, annotated, and revised by K. L. Chung)Google Scholar
  20. 20.
    Heathcote, C., Hüsler, J.: The first zero of an empirical characteristic function. Stoch. Process. Appl. 35(2), 347–360 (1990)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hough, B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes, volume 51 of University Lecture Series. American Mathematical Society, Providence (2009)Google Scholar
  22. 22.
    Hüsler, J.: First zeros of empirical characteristic functions and extreme values of Gaussian processes. In: Statistical Data Analysis and Inference. Invited Papers Presented at the International Conference, held in Neuchâtel, Switzerland, 1989, in honor of C. R. Rao, pp. 177–182. North-Holland (1989)Google Scholar
  23. 23.
    Ikeda, S.: Asymptotic equivalence of probability distributions with applications to some problems of asymptotic independence. Ann. Inst. Stat. Math. 15, 87–116 (1963)CrossRefzbMATHGoogle Scholar
  24. 24.
    Itô, K., Nisio, M.: On the convergence of sums of independent Banach space valued random variables. Osaka J. Math. 5, 35–48 (1968)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Janßen, A.: Limit laws for power sums and norms of i.i.d. samples. Probab. Theory Relat. Fields 146(3–4), 515–533 (2010)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kabluchko, Z.: Functional limit theorems for sums of independent geometric Brownian motions. Bernoulli 17(3), 942–968 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kabluchko, Z.: Critical points of random polynomials with independent identically distributed roots. Submitted (2012). Preprint available at
  28. 28.
    Kabluchko, Z., Zaporozhets, D.: Universality for zeros of random analytic functions. Submitted (2012). Preprint available at
  29. 29.
    Koukiou, F.: Analyticity of the partition function of the random energy model. J. Phys. A Math. Gen. 26(23), L1207–L1210 (1993)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Lee, T.D., Yang, C.N.: Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87(2), 410–419 (1952)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Meerschaert, M., Scheffler, H.-P.: Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice. Wiley, Chichester (2001)zbMATHGoogle Scholar
  32. 32.
    Moukarzel, C., Parga, N.: Numerical complex zeros of the random energy model. Phys. A Stat. Mech. Appl. 177(1–3), 24–30 (1991)CrossRefGoogle Scholar
  33. 33.
    Moukarzel, C., Parga, N.: Analytic determination of the complex field zeros of REM. J. Phys. I France 2(3), 251–261 (1992)CrossRefGoogle Scholar
  34. 34.
    Moukarzel, C., Parga, N.: The REM zeros in the complex temperature and magnetic field planes. Phys. A Stat. Mech. Appl. 185(1–4), 305–315 (1992)CrossRefGoogle Scholar
  35. 35.
    Nazarov, F., Sodin, M.: What is \(\ldots \) a Gaussian entire function? Notices Am. Math. Soc. 57(3), 375–377 (2010)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Obuchi, T., Takahashi, K.: Partition-function zeros of spherical spin glasses and their relevance to chaos. J. Phys. A 45(12), 125003 (2012)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Olivieri, E., Picco, P.: On the existence of thermodynamics for the random energy model. Comm. Math. Phys. 96(1), 125–144 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Petrov, V.V.: Limit Theorems of Probability Theory. Sequences of Independent random Variables, volume 4 of Oxford Studies in Probability. Oxford University Press, New York (1995)Google Scholar
  39. 39.
    Resnick, S.: Extreme Values, Regular Variation, and Point Processes, volume 4 of Applied Probability. Springer, New York (1987)CrossRefGoogle Scholar
  40. 40.
    Rvačeva, E.L.: On domains of attraction of multi-dimensional distributions. In: Select. Transl. Math. Statist. and Probability, vol. 2, pp. 183–205. American Mathematical Society (1962)Google Scholar
  41. 41.
    Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Stochastic Modeling. Chapman & Hall, New York (1994)Google Scholar
  42. 42.
    Takahashi, K.: Replica analysis of partition-function zeros in spin-glass models. J. Phys. A 44(23), 235001 (2011)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Talagrand, M.: Mean field models for spin glasses. Volume I, volume 54 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer, Berlin (2011)Google Scholar
  44. 44.
    Yang, C.N., Lee, T.D.: Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87(2), 404–409 (1952)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

Personalised recommendations