Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Aging in two-dimensional Bouchaud's model
Download PDF
Download PDF
  • Published: 10 February 2005

Aging in two-dimensional Bouchaud's model

  • Gérard Ben Arous1,2,
  • Jiří Černý3 &
  • Thomas Mountford4 

Probability Theory and Related Fields volume 134, pages 1–43 (2006)Cite this article

  • 164 Accesses

  • 37 Citations

  • Metrics details

Abstract

Let E x 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 w xy = ν exp (−βE x ) if x, y are neighbours in ℤ2. We study the behaviour of two correlation functions: ℙ[X(t w +t) = X(t w )] and ℙ[X(t') = X(t w ) ∀ t'∈ [t w , t w + t]]. We prove the (sub)aging behaviour of these functions when β > 1.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  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. 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. Ben Arous, G., Černý, J.: Bouchaud's model exhibits two aging regimes in dimension one. To appear in Annals of Applied Probability (2004)

  4. Ben Arous, G.: Aging and spin glass dynamics. Proceedings of Inter. Cong. Matematicians. Beijing 2002 III, 1–12 (2002)

    Google Scholar 

  5. Bertoin, J.: Lévy processes. Cambridge: Cambridge University Press, 1996

  6. Billingsley, P. Convergence of probability measures. second ed., John Wiley & Sons Inc., New York, 1999, A Wiley-Interscience Publication

  7. Bouchaud, J.-P., Mézard, M.: Universality classes for extreme-value statistics. J. Phys. A: Math. Gen. 30, 7997–8015 (1997)

    Article  MATH  Google Scholar 

  8. Bouchaud, J.-P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713 (1992)

    Article  Google Scholar 

  9. Černý, J.: On two properties of strongly disordered systems, aging and critical path analysis. Ph.D. thesis, EPF Lausanne, 2003

  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)

    MathSciNet  Google Scholar 

  11. Lawler, G.F.: Intersections of random walks. Birkhäuser Boston Inc., Boston, MA, 1991

  12. Monthus, C., Bouchaud, J.-P.: Models of traps and glass phenomenology. J. Phys. A 29, 3847–3869 (1996)

    MATH  Google Scholar 

  13. Rinn, B., Maass, P., Bouchaud, J.-P.: Multiple scaling regimes in simple aging models. Phys. Rev. Lett 84, 5403–5406 (2000)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. École Polytechnique, Fédérale de Lausanne, 1015 Lausanne, Switzerland

    Gérard Ben Arous

  2. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012-1185, USA

    Gérard Ben Arous

  3. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117, Berlin, Germany

    Jiří Černý

  4. Département de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Thomas Mountford

Authors
  1. Gérard Ben Arous
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří Černý
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Mountford
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gérard Ben Arous.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ben Arous, G., Černý, J. & Mountford, T. Aging in two-dimensional Bouchaud's model. Probab. Theory Relat. Fields 134, 1–43 (2006). https://doi.org/10.1007/s00440-004-0408-1

Download citation

  • Received: 20 November 2003

  • Revised: 26 October 2004

  • Published: 10 February 2005

  • Issue Date: January 2006

  • DOI: https://doi.org/10.1007/s00440-004-0408-1

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Key words or phrases

  • Aging
  • Trap model
  • Lévy process
  • Random walk
  • Time change

Mathematics Subject Classification (2000)

  • 82D30
  • 82C41
  • 60F17
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature