CONVERGENCE OF CLOCK PROCESSES IN RANDOM ENVIRONMENTS AND AGEING IN THE p-SPIN SK MODEL

• 成果类型: Article
• 署名作者: Bovier, Anton; Gayrard, Veronique
• 署名单位: University of Bonn; Aix-Marseille Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP705
• 发表日期: 2013
• 页码: 817-847
• 关键词: random energy-model; glauber dynamics; REGIMES; limit

摘要

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.