Glauber dynamics of the random energy model. II: Aging below the critical temperature.
This paper is the second part of a mathematical study of aging phenomenon in glassy disordered systems. In the first part [Commun. Math. Phys. 235, No. 3, 379--425 (2003; Zbl 1037.82034)], the asymptotic behavior of a discrete time Glauber dynamics of the (REM) is carefully studied and established in a weak form sufficient to retrieve previous physicists' results and predictions. This paper proves that these asymptotics results allow also a rigorous description of an aging phenomenon. Aging, an important feature in the theory of the dynamics of disordered systems, is a particular way in which dynamic properties of the system change with time when relaxing towards equilibrium. The time scale at which the process evolves slows down in proportion to the elapsed time, roughly speaking the system ``ages''. In this model, this is expressed in terms of entrance laws and waiting times, and it is proved that for a suitably chosen time scale, diverging with the size of the system, the mean time of arrival in the set of states having waiting time of order \(T\) is itself of order \(T\). This is done by the use of a natural autocorrelation function and it is also rigorously expressed that the long-time asymptotics of this function coincide with those of another paradigm for spin-glasses, the ``REM-like trap model'' of Bouchaud et al. The techniques heavily relies on the results of the first paper.
scientific article; zbMATH DE number 1964928