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.


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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