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

TL;DR

This paper establishes a general criterion for the convergence of clock processes in complex random environments and applies it to demonstrate almost sure convergence to stable subordinators in the p-spin SK model, revealing aging phenomena.

Contribution

It introduces a new criterion for the convergence of dependent sums and applies it to prove almost sure convergence of clock processes in the p-spin SK model.

Findings

Clock process converges to a stable subordinator almost surely.

Time-time correlation converges to the arcsine law.

Improves previous results by establishing almost sure convergence.

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.