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Universality in marginally relevant disordered systems
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Caravenna, Francesco , Sun, Rongfeng and Zygouras, Nikos (2017) Universality in marginally relevant disordered systems. Annals of Applied Probability, 27 (5). pp. 3050-3112. doi:10.1214/17-AAP1276
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Official URL: http://doi.org/10.1214/17-AAP1276
Abstract
We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2+1), the long-range directed polymer model with Cauchy tails in dimension (1+1) and the disordered pinning model with tail exponent 1/2. We show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit: a log-normal random field with a multi-scale correlation structure, which undergoes a phase transition as the disorder strength varies. As a by-product, we show that the solution of the two-dimensional Stochastic Heat Equation, suitably regularized, converges to the same limit. The proof, which uses the celebrated Fourth Moment Theorem, reveals an interesting chaos structure shared by all models in the above class.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||
| Divisions: | Faculty of Science > Statistics | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Order-disorder models, Polymers -- mathematical models, Stochastic differential equations | |||||||||||||||
| Journal or Publication Title: | Annals of Applied Probability | |||||||||||||||
| Publisher: | Institute of Mathematical Statistics | |||||||||||||||
| ISSN: | 1050-5164 | |||||||||||||||
| Official Date: | 3 November 2017 | |||||||||||||||
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| Volume: | 27 | |||||||||||||||
| Number: | 5 | |||||||||||||||
| Page Range: | pp. 3050-3112 | |||||||||||||||
| Identifier: | 10.1214/17-AAP1276 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
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