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

Abstract

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 ν^hom> 2 and relevant if ν^hom< 2, where ν^hom is the order of the homogeneous phase transition. The marginal case ν^hom = 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 ν^hom< 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.