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

Anton Bovier (1) , Véronique Gayrard (2)

1 Institut für Angewandte Mathematik
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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.

Domaines: Mathématiques [math]; Physique mathématique [math-ph]

Anton Bovier, Véronique Gayrard. Convergence of clock processes in random environments and ageing in the $p$-spin SK model. Annals of Probability, 2013, 41 (2), ⟨10.1214/11-AOP705⟩. ⟨hal-01256640⟩