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Convergence of clock processes in random environments and ageing in the \(p\)-spin SK model. (English) Zbl 1267.82114

Summary: We derive a general criterion for the convergence of clock processes in random dynamics in random environments which is applicable in cases in which correlations are not negligible, extending recent results [the second author, Electron. J. Probab. 17, Paper No. 58 (2012; Zbl 1252.82091); “Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM”, Preprint, arXiv:1008.3849], based on general criterion for convergence of sums of dependent random variables due to R. Durrett and S. I. Resnick [Ann. Probab. 6, 829–846 (1978; Zbl 0398.60024)]. We demonstrate the power of this criterion by applying it to the random hopping time dynamics of the \(p\)-spin SK model. We prove that, for a wide range of time scales, the clock process converges to a stable subordinator almost surely with respect to the environment. We also show that a time-time correlation function converges to the arcsine law for this subordinator, almost surely. This improves recent similar convergence results in law with respect to the random environment (see [G. Ben Arousand and the authors, Commun. Math. Phys. 236, No. 1, 1–54 (2003; Zbl 1037.82039)]).

MSC:

82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G70 Extreme value theory; extremal stochastic processes
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References:

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