Issue |
ESAIM: Proc.
Volume 51, October 2015
Modélisation Aléatoire et Statistique - Journées MAS 2014
|
|
---|---|---|
Page(s) | 74 - 88 | |
DOI | https://doi.org/10.1051/proc/201551005 | |
Published online | 12 October 2015 |
Influence of Disorder For the Polymer Pinning Model
LPMA, Université Pierre et Marie
Curie, Case courrier 188 - 4 Place
Jussieu, 75252
Paris Cedex 05 -
France
When studying physical systems, the influence of disorder on the phase transition is a central question: one wants to determine whether an arbitrary small amount of randomness modifies the critical properties of the system, with respect to the non-disordered case. We present here an overview of the mathematical results obtained to answer that question in the context of the polymer pinning model. In the IID case, the picture of disorder relevance/irrelevance is by now established, and follows the so-called Harris criterion: disorder is irrelevant if νhom> 2 and relevant if νhom< 2, where νhom is the order of the homogeneous phase transition. The marginal case νhom = 2 has been subject to controversy in the physics literature in the context of pinning models, but has recently been fully settled. In the correlated case, Weinrib and Halperin predicted that, if the two-point correlations decay as a power law with exponent ξ> 0, then the Harris criterion would be modified if ξ< 1: disorder should be relevant whenever νhom< 2 max (1,1 /ξ). It turns out that this prediction is not accurate: the key quantity is not the decay exponent ξ, but the occurrence of rare regions with atypical disorder. An infinite disorder regime may appear, in which the relevance/irrelevance picture is crucially modified. We also mention another recent approach to the question of the influence of disorder for the pinning model: the persistence of disorder when taking the scaling limit of the system.
© EDP Sciences, SMAI 2015
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.