| Summary: | 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.
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