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Abstract: Recently the renormalization group predictions on the effect of disorder onpinning models have been put on mathematical grounds. The picture isparticularly complete if the disorder is -relevant- or -irrelevant- in theHarris criterion sense: the question addressed is whether quenched disorderleads to a critical behavior which is different from the one observed in thepure, i.e. annealed, system. The Harris criterion prediction is based on thesign of the specific heat exponent of the pure system, but it yields noprediction in the case of vanishing exponent. This case is called -marginal-,and the physical literature is divided on what one should observe for marginaldisorder, notably there is no agreement on whether a small amount of disorderleads or not to a difference between the critical point of the quenched systemand the one for the pure system. In a previous work arXiv:0811.0723 we haveproven that the two critical points differ at marginality of at leastexp-c-beta^4, where c>0 and beta^2 is the disorder variance, for beta in0,1 and Gaussian IID disorder. The purpose of this paper is to improve such aresult: we establish in particular that the exp-c-beta^4 lower bound on theshift can be replaced by exp-cb-beta^b, cb>0 for b>2 b=2 is the knownupper bound and it is the result claimed in Derrida, Hakim, Vannimenus, JSP1992, and we deal with very general distribution of the IID disordervariables. The proof relies on coarse graining estimates and on a fractionalmoment-change of measure argument based on multi-body potential modificationsof the law of the disorder.

Author: Giambattista Giacomin, Hubert Lacoin, Fabio Lucio Toninelli

Source: https://arxiv.org/

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