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Spectral characterization of aging: the REM-like trap model. (English) Zbl 1086.60064

The first REM-like trap model studied in this paper is a continuous-time random walk in a random environment \(Y_N(t), t\geq 0\), on the complete graph on \(N\)-vertices with infinitesimal generator \(G_N:=(g_{ij})\) given by \[ g_{ii} = \frac{(N-1)}{N}e^{-E_i} \quad \text{and}\quad g_{ij} = -\frac{1}{N}e^{-E_i}, \;i\neq j. \] The energy landscape is given by a family \(\{E_i ; i\in \mathbb N\}\) of i.i.d. random variables exponentially distributed with parameter \(\alpha\), \(0<\alpha<1\). As for the other trap models introduced by Bouchaud and Dean [J. Phys. I France 5, 265–286 (1995)], the mean waiting times are not integrable (with respect to the environment). A complete description of the spectral decomposition of the generator is given. This allows a deeper analysis of the aging behavior of the walk than what was previously done by renewal methods. The representation of the time correlation function \[ \Pi_N(t,t_w) := P_N(Y_N(s) = Y_N(t_w) \;\forall s\in[ t_w ,t_w +t]) \] as a complex integral is used to identify the thermodynamic limit and to investigate the phenomenon of visiting states with large waiting times. In the second part, the energy landscape of the REM-trap model is constructed from a Poisson point process. The aymptotic behavior of the time correlation is described for three different time-rescalings.

MSC:

60K37 Processes in random environments
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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[1] Ahlfors, L. V. (1979). Complex Analysis . McGraw–Hill, New York. · Zbl 0395.30001
[2] Ben Arous, G., Bovier, A. and Gayrard, V. (2002). Ageing in the Random Energy Model. Phys. Rev. Lett. 88 087201.
[3] Ben Arous, G., Bovier, A. and Gayrard, V. (2003). Glauber dynamics of the Random Energy Model 1. Metastable motion on the extreme states. Comm. Math. Phys. 235 379–425. · Zbl 1037.82038 · doi:10.1007/s00220-003-0798-4
[4] Ben Arous, G., Bovier, A. and Gayrard, V. (2003). Glauber dynamics of the Random Energy Model 2. Ageing below the critical temperature. Comm. Math. Phys. 236 1–54. · Zbl 1037.82039 · doi:10.1007/s00220-003-0799-3
[5] Ben Arous, G. and Černý, J. (2002). Bouchaud’s model admits two different aging regimes in dimension one.
[6] Ben Arous, G., Černý, J. and Mountford, T. (2004). Aging in two dimensional Bouchaud’s model. WIAS Preprint 887. · Zbl 1089.82017
[7] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[8] Bouchaud, J.-P., Cugliandolo, L. F., 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.
[9] Bouchaud, J.-P. and Dean, D. S. (1995). Ageing on Parisi’s tree. J. Phys. I France 5 265–286.
[10] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002). Metastability and low-lying spectra in reversible Markov chains. Comm. Math. Phys. 228 219–255. · Zbl 1010.60088 · doi:10.1007/s002200200609
[11] Černý, J. (2003). On two properties of strongly disordered systems, aging and critical path analysis. Ph.D. thesis, EPFL.
[12] Davies, E. B. (1982). Metastable states of symmetric Markov semigroups, I. Proc. London Math. Soc. ( 3 ) 45 133–150. · Zbl 0498.47017 · doi:10.1112/plms/s3-45.1.133
[13] Davies, E. B. (1982). Metastable states of symmetric Markov semigroups, II. J. London Math. Soc. ( 2 ) 26 541–556. · Zbl 0527.47028 · doi:10.1112/jlms/s2-26.3.541
[14] Davies, E. B. (1983). Spectral properties of metastable Markov semigroups. J. Funct. Anal. 52 315–329. · Zbl 0525.47030 · doi:10.1016/0022-1236(83)90071-X
[15] Doetsch, G. (1971). Handbuch der Laplace-Transformation I . Birkhäuser, Basel. · Zbl 0242.44001
[16] Feller, W. (1966). An Introduction to Probability Theory and Its Applications II . Wiley, New York. · Zbl 0138.10207
[17] Fontes, L. R. G., Isopi, G. M., Kohayakawa, Y. and Picco, P. (1998). The spectral gap of the REM under Metropolis dynamics. Ann. Appl. Probab. 8 917–943. · Zbl 0935.60084 · doi:10.1214/aoap/1028903457
[18] Fontes, L. R. G., Isopi, G. 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. · Zbl 1015.60099 · doi:10.1214/aop/1023481003
[19] Fontes, L. R. G., Isopi, G. M. and Newman, C. M. (2001). Aging in 1D discrete spin models and equivalent systems. Phys. Rev. Lett. 87 110201–110205.
[20] Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems . Springer, Berlin. · Zbl 0522.60055
[21] Gaveau, B. and Schulman, L. S. (1998). Theory of nonequilibrium first-order phase transitions for stochastic dynamics. J. Math. Phys. 39 1517–1533. · Zbl 1056.82513 · doi:10.1063/1.532394
[22] Kato, T. (1980). Perturbation Theory for Linear Operators . Springer, Berlin. · Zbl 0342.47009
[23] Matthieu, P. (2000). Convergence to equilibrium for spin glasses. Comm. Math. Phys. 215 57–68. · Zbl 1018.82019 · doi:10.1007/s002200000292
[24] Mélin, R. and Butaud, P. (1997). Glauber dynamics and ageing. J. de Physique I 7 691–710. · doi:10.1051/jp1:1997185
[25] Reed, M. and Simon, B. (1980). Methods of Modern Mathematical Physics I . Academic Press, Orlando. · Zbl 0459.46001
[26] Vincent, E., Hammann, M., Ocio, M., Bouchaud, J.-P. and Cugliandolo, L. F. (1997). Slow dynamics and aging in spin-glasses. In Proceedings of the Sitges Conference (E. Rubi, ed.). Springer, New York.
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