The Annals of Applied Probability

Bouchaud’s model exhibits two different aging regimes in dimension one

Gérard Ben Arous and Jiří Černý

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Let Ei be a collection of i.i.d. exponential random variables. Bouchaud’s model on ℤ is a Markov chain X(t) whose transition rates are given by wij=νexp(−β((1−a)EiaEj)) if i, j are neighbors in ℤ. We study the behavior 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 behavior of these functions when β>1 and a∈[0,1].

Article information

Ann. Appl. Probab., Volume 15, Number 2 (2005), 1161-1192.

First available in Project Euclid: 3 May 2005

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60G18: Self-similar processes
Secondary: 60F17: Functional limit theorems; invariance principles

Aging singular diffusions random walk in random environment Lévy processes


Arous, Gérard Ben; Černý, Jiří. Bouchaud’s model exhibits two different aging regimes in dimension one. Ann. Appl. Probab. 15 (2005), no. 2, 1161--1192. doi:10.1214/105051605000000124.

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  • Ben Arous, G. (2002). Aging and spin-glass dynamics. In Proceedings of the ICM 3 3--14.
  • Ben Arous, G., Bovier, A. and Gayrard, V. (2002). Aging in the random energy model. Phys. Rev. Lett. 88 087201.
  • Ben Arous, G., Bovier, A. and Gayrard, V. (2003). Glauber dynamics of the random energy model, II. Aging below the critical temperature. Comm. Math. Phys. 235 379--425.
  • Ben Arous, G., Černý, J. and Mountford, T. (2005). Aging for Bouchaud's model in dimension two. Probab. Theory Related Fields. To appear.
  • Ben Arous, G., Dembo, A. and Guionnet, A. (2001). Aging of spherical spin glasses. Probab. Theory Related Fields 120 1--67.
  • Bouchaud, J. P., Cugliandolo, L., Kurchan, J. and Mézard, M. (1998). Out-of-equilibrium dynamics in spin-glasses and other glassy systems. In Spin-Glasses and Random Fields (A. P. Young, ed.). World Scientific, Singapore.
  • Dembo, A., Guionnet, A. and Zeitouni, O. (2001). Aging properties of Sinai's random walk in random environment. Preprint. Available at
  • Fontes, L. R. G., Isopi, M. and Newman, C. M. (1999). Chaotic time dependence in a disordered spin system. Probab. Theory Related Fields 115 417--443.
  • Fontes, L. R. G., Isopi, M. and Newman, C. M. (2002). Random walks with strongly inhomogeneous rates and singular diffusions: Convergence, localization and aging in one dimension. Ann. Probab. 30 579--604.
  • Mathieu, P. (2000). Convergence to equilibrium for spin glasses. Comm. Math. Phys. 215 57--68.
  • Rinn, B., Maass, P. and Bouchaud, J.-P. (2000). Multiple scaling regimes in simple aging models. Phys. Rev. Lett. 84 5403--5406.
  • Stone, C. (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 638--660.