K-processes, scaling limit and aging for the trap model in the complete graph. (English) Zbl 1154.60073

Fontes, L. R. G.; Mathieu, P.

Ann. Probab. 36, No. 4, 1322-1358 (2008).

The paper departs from trap models in the complete graph (symmetric continuous time random walk, typically in a regular graph, finite or infinite), employed in the study of scaling limits of various mean-field theory spin-glass systems which may exhibit an aging property (one or multiple aging regimes). Phenomenological models, like that of the Glauber dynamics for the random energy model, usually give rise to aging as a property of a given two-time correlation function, while the authors attempt to prove that it may arise as a scaling limit of the full dynamics. Here, in the scaling limit, the aging is a dynamical phenomenon at vanishing times.
The major technical tool are the so-called K-processes whose first examples were given by A. N. Kolmogorov [On the differentiability of the transition probabilities in stationary Markov processes with a denumerable number of states. Moskov. Gos. Univ. Učenye Zapiski Matematika 148, 53–59 (1951)] and their classifications among Markov systems with a denumerable state space was set by P. Lévy [Ann. Sci. Éc. Norm. Supér., III. Sér. 68, 327–381 (1951; Zbl 0044.33803)]. An approach based on Dirichlet forms is supplemented by an explicit probabilistic construction. Arguments in favor of an equivalence of both routes are given. Links with the Fukushima [cf. Zh.-Q. Chen, M. Fukushima, J. Funct. Anal. 252, No. 2, 710–733 (2007; Zbl 1129.60072)] extension of certain Markov processes beyond a killing time are mentioned.

Reviewer: Piotr Garbaczewski (Opole)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60K37	Processes in random environments
82C44	Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics

Keywords:

spin glasses; aging; trap models; scaling; random energy model; Glauber dynamics; Markov processes; denumerable state space; complete graph; Dirichlet form; Kolmogorov (K) process; Levy characterisation

Citations:

Zbl 0044.33803; Zbl 1129.60072

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