The continuum disordered pinning model

• 成果类型: Article
• 署名作者: Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos
• 署名单位: University of Milano-Bicocca; National University of Singapore; University of Warwick
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0606-4
• 发表日期: 2016
• 页码: 17-59
• 关键词: directed polymers; LIMIT-THEOREMS

摘要

Any renewal processes on N-0 with a polynomial tail, with exponent a is an element of (0, 1), has a non-trivial scaling limit, known as the alpha-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i. i. d. random environment, called disordered pinning models. We show that for alpha is an element of (1/2, 1) these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of R in a white noise random environment, with subtle features: Any fixed a.s. property of the alpha-stable regenerative set (e.g., its Hausdorff dimension) is also an a.s. property of the CDPM, for almost every realization of the environment. Nonetheless, the lawof theCDPMis singular with respect to the lawof the alpha-stable regenerative set, for almost every realization of the environment. The existence of a disordered continuum model, such as the CDPM, is a manifestation of disorder relevance for pinning models with alpha is an element of(1/2, 1).