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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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
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
Dates:	Date Event 3 November 2017 Published 4 January 2017 Accepted
Volume:	27
Number:	5
Page Range:	pp. 3050-3112
DOI:	10.1214/17-AAP1276
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
RIOXX Funder/Project Grant:	Project/Grant ID RIOXX Funder Name Funder ID UNSPECIFIED Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni [National Group for Mathematical Analysis, Probability and their Applications] UNSPECIFIED UNSPECIFIED Istituto Nazionale di Alta Matematica "Francesco Severi" http://dx.doi.org/10.13039/100009112 R-146-000-185-112 National University of Singapore http://dx.doi.org/10.13039/501100001352 EP/L012154/1 Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266
Open Access Version:	ArXiv

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