Convergence of clock processes in random environments and ageing in the p-spin SK model

Details Convergence of clock processes in random environments and ageing in the p-spin SK model Convergence of clock processes in random environments and ageing in the p-spin SK model

2010-08-23

arxiv.org/abs/1008.3839v1

Mathematics

Probability

Scientific paper

We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [15,16], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [13]. We demonstrate the power of this criterion by applying it to the case of random hopping time dynamics of the p-spin SK model. We prove that on 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 results of Ben Arous et al. [1] that obtained similar convergence result in law with respect to the random environment.

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