Probability Theory and Related Fields

, Volume 157, Issue 1–2, pp 251–283 | Cite as

Convergence to extremal processes in random environments and extremal ageing in SK models

Article

Abstract

This paper extends recent results on ageing in mean field spin glasses on short time scales, obtained by Ben Arous and Gün (Commun Pure Appl Math 65:77–127, 2012) in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in (Gayrard in Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM, 2010; Electron J Probab 17(58): 1–33, 2012) and naturally complements (Bovier and Gayrard in Ann Probab, 2012).

Keywords

Ageing Spin glasses Random environments Clock process  Lévy processes Extremal processes 

Mathematics Subject Classification (2000)

82C44 60K35 60G70 

References

  1. 1.
    Ben Arous, G., Bovier, A., Černý, J.: Universality of the REM for dynamics of mean-field spin glasses. Comm. Math. Phys. 282(3), 663–695 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arous Ben, G., Gün, O.: Universality and extremal aging for dynamics of spin glasses on subexponential time scales. Commun. Pure Appl. Math. 65, 77–127 (2012)CrossRefMATHGoogle Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)MATHGoogle Scholar
  4. 4.
    Bouchaud, J.-P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713 (1992)CrossRefGoogle Scholar
  5. 5.
    Bouchaud, J.-P., Dean, D.S.: Aging on Parisi’s tree. J. Phys. I (France) 5, 265 (1995)CrossRefGoogle Scholar
  6. 6.
    Bovier, A., Gayrard, V.: Convergence of clock processes in random environments and aging in the p-spin SK model. Ann. Probab. (2012) (to appear)Google Scholar
  7. 7.
    Durrett, R., Resnick, S.I.: Functional limit theorems for dependent variables. Ann. Probab. 6(5), 829–846 (1978)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gayrard, V.: Aging in reversible dynamics of disordered systems II. Emergence of the arcsine law in the random hopping time dynamics of the REM. LAPT,Université d’Aix-Marseille, Marseille (2010)Google Scholar
  9. 9.
    Gayrard, V.: Convergence of clock process in random environments and aging in Bouchaud’s asymmetric trap model on the complete graph. Electronic J. Probab. 17(58), 1–33 (2012)MathSciNetGoogle Scholar
  10. 10.
    Kasahara, Y.: Extremal process as a substitution for “one-sided stable process with index \(0\)”. In: Stochastic Processes and their Applications (Nagoya, 1985). Lecture Notes in Mathematics,vol. 1203, pp. 90–100. Springer, Berlin (1986)Google Scholar
  11. 11.
    Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. Springer, New York (1983)CrossRefGoogle Scholar
  12. 12.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. In: Propp, J. G., Wilson, D. B. (eds.) American Mathematical Society, Providence, RI (2009)Google Scholar
  13. 13.
    Mori, T., Oodaira, H.: A functional law of the iterated logarithm for sample sequences. Yokohama Math. J. 24(1–2), 35–49 (1976)MathSciNetMATHGoogle Scholar
  14. 14.
    Neveu, J.: Processus ponctuels. In: École d’Été de Probabilités de Saint-Flour, VI–1976. Lecture Notes in Mathematics, vol. 598, pp. 249–445. Springer, Berlin (1977)Google Scholar
  15. 15.
    Resnick, S.: Extreme values, regular variation, and point processes. In: Applied Probability. A Series of the Applied Probability Trust, vol. 4. Springer, New York (1987)Google Scholar
  16. 16.
    Talagrand, M.: Mean field models for spin glasses, vol. I. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 54 [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (2011)Google Scholar
  17. 17.
    Talagrand, M.: Mean field models for spin glasses, vol. II. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 55 [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anton Bovier
    • 1
  • Véronique Gayrard
    • 2
  • Adéla Švejda
    • 1
  1. 1.Institut für Angewandte MathematikRheinische Friedrich-Wilhelms-UniversitätBonnGermany
  2. 2.CMI, LAPTUniversité de ProvenceMarseille Cedex 13France

Personalised recommendations