## The Annals of Probability

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

#### Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 817-847.

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.aop/1362750943

Digital Object Identifier
doi:10.1214/11-AOP705

Mathematical Reviews number (MathSciNet)
MR3077527

Zentralblatt MATH identifier
1267.82114

#### Citation

Bovier, Anton; Gayrard, Véronique. Convergence of clock processes in random environments and ageing in the $p$-spin SK model. Ann. Probab. 41 (2013), no. 2, 817--847. doi:10.1214/11-AOP705. https://projecteuclid.org/euclid.aop/1362750943

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