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The continuum disordered pinning model
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Caravenna, Francesco, Rongfeng, Sun and Zygouras, Nikos (2014) The continuum disordered pinning model. Probability Theory and Related Fields, 164 (1). pp. 17-59. doi:10.1007/s00440-014-0606-4
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Official URL: http://dx.doi.org/10.1007/s00440-014-0606-4
Abstract
Any renewal processes on N with a polynomial tail, with exponent α∈(0,1), has a non-trivial scaling limit, known as the α-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 α∈(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 α-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 law of the CDPM is singular with respect to the law of the α-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 α∈(1/2,1)
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||
| Library of Congress Subject Headings (LCSH): | Probabilities | ||||||
| Journal or Publication Title: | Probability Theory and Related Fields | ||||||
| Publisher: | Springer Berlin Heidelberg | ||||||
| ISSN: | 0178-8051 | ||||||
| Official Date: | 17 December 2014 | ||||||
| Dates: |
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| Volume: | 164 | ||||||
| Number: | 1 | ||||||
| Page Range: | pp. 17-59 | ||||||
| DOI: | 10.1007/s00440-014-0606-4 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Open Access | ||||||
| Funder: | European Research Council (ERC), Singapore. Ministry of Education. , Engineering and Physical Sciences Research Council (EPSRC), Marie Curie International Reintegration Grants (IRG) | ||||||
| Grant number: | 267356 VARIS (ERC), AcRF Tier 1 Grant R-146-000-148-112, EP/L012154/1 (EPSRC), IRG-246809 |
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