Convergence of clock processes in random environments and ageing in the p-spin SK model
Abstract
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 [(2010), (2011), forthcoming], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [Ann. Probab. 6 (1978) 829-846]. 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, Bovier and Cerny [Comm. Math. Phys. 282 (2008) 663-695] that obtained similar convergence results in law, with respect to the random environment.
Keywords
Cite
@article{arxiv.1008.3839,
title = {Convergence of clock processes in random environments and ageing in the p-spin SK model},
author = {Anton Bovier and Véronique Gayrard},
journal= {arXiv preprint arXiv:1008.3839},
year = {2013}
}
Comments
Published in at http://dx.doi.org/10.1214/11-AOP705 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)