Probability Theory and Related Fields

, Volume 134, Issue 1, pp 1–43 | Cite as

Aging in two-dimensional Bouchaud's model

  • Gérard Ben Arous
  • Jiří Černý
  • Thomas Mountford
Article

Abstract

Let Ex be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on ℤ2 is a Markov chain X(t) whose transition rates are given by wxy = ν exp (−βEx) if x, y are neighbours in ℤ2. We study the behaviour of two correlation functions: ℙ[X(tw+t) = X(tw)] and ℙ[X(t') = X(tw) ∀ t'∈ [tw, tw + t]]. We prove the (sub)aging behaviour of these functions when β > 1.

Key words or phrases

Aging Trap model Lévy process Random walk Time change 

Mathematics Subject Classification (2000)

82D30 82C41 60F17 

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References

  1. 1.
    Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. I. Metastable motion on the extreme states. Comm. Math. Phys. 235 (3), 379–425 (2003)Google Scholar
  2. 2.
    Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. II. Aging below the critical temperature. Comm. Math. Phys. 236 (1), 1–54 (2003)Google Scholar
  3. 3.
    Ben Arous, G., Černý, J.: Bouchaud's model exhibits two aging regimes in dimension one. To appear in Annals of Applied Probability (2004)Google Scholar
  4. 4.
    Ben Arous, G.: Aging and spin glass dynamics. Proceedings of Inter. Cong. Matematicians. Beijing 2002 III, 1–12 (2002)Google Scholar
  5. 5.
    Bertoin, J.: Lévy processes. Cambridge: Cambridge University Press, 1996Google Scholar
  6. 6.
    Billingsley, P. Convergence of probability measures. second ed., John Wiley & Sons Inc., New York, 1999, A Wiley-Interscience PublicationGoogle Scholar
  7. 7.
    Bouchaud, J.-P., Mézard, M.: Universality classes for extreme-value statistics. J. Phys. A: Math. Gen. 30, 7997–8015 (1997)CrossRefMATHGoogle Scholar
  8. 8.
    Bouchaud, J.-P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713 (1992)CrossRefGoogle Scholar
  9. 9.
    Černý, J.: On two properties of strongly disordered systems, aging and critical path analysis. Ph.D. thesis, EPF Lausanne, 2003Google Scholar
  10. 10.
    Fontes, L.R.G., Isopi, M., Newman, C.M.: Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Probab. 30 (2), 579–604 (2002)MathSciNetGoogle Scholar
  11. 11.
    Lawler, G.F.: Intersections of random walks. Birkhäuser Boston Inc., Boston, MA, 1991Google Scholar
  12. 12.
    Monthus, C., Bouchaud, J.-P.: Models of traps and glass phenomenology. J. Phys. A 29, 3847–3869 (1996)MATHGoogle Scholar
  13. 13.
    Rinn, B., Maass, P., Bouchaud, J.-P.: Multiple scaling regimes in simple aging models. Phys. Rev. Lett 84, 5403–5406 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Gérard Ben Arous
    • 1
    • 2
  • Jiří Černý
    • 3
  • Thomas Mountford
    • 4
  1. 1.École PolytechniqueFédérale de LausanneSwitzerland
  2. 2.Courant Institute of Mathematical SciencesNew YorkUSA
  3. 3.Weierstrass Institute for Applied Analysis and Stochastics (WIAS)BerlinGermany
  4. 4.Département de MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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