Open Access
March 2013 Convergence of clock processes in random environments and ageing in the $p$-spin SK model
Anton Bovier, Véronique Gayrard
Ann. Probab. 41(2): 817-847 (March 2013). DOI: 10.1214/11-AOP705


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 Černý [Comm. Math. Phys. 282 (2008) 663–695] that obtained similar convergence results in law, with respect to the random environment.


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Anton Bovier. Véronique Gayrard. "Convergence of clock processes in random environments and ageing in the $p$-spin SK model." Ann. Probab. 41 (2) 817 - 847, March 2013.


Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1267.82114
MathSciNet: MR3077527
Digital Object Identifier: 10.1214/11-AOP705

Primary: 60G70 , 60K35 , 82C44

Keywords: Aging , clock process , Lévy processes , random dynamics , random environments , Spin glasses

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 2 • March 2013
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