  Recently the renormalization group predictions on the effect of disorder on
pinning models have been put on mathematical grounds. The picture is
particularly complete if the disorder is 'relevant' or 'irrelevant' in the
Harris criterion sense: the question addressed is whether quenched disorder
leads to a critical behavior which is different from the one observed in the
pure, i.e. annealed, system. The Harris criterion prediction is based on the
sign of the specific heat exponent of the pure system, but it yields no
prediction in the case of vanishing exponent. This case is called 'marginal',
and the physical literature is divided on what one should observe for marginal
disorder, notably there is no agreement on whether a small amount of disorder
leads or not to a difference between the critical point of the quenched system
and the one for the pure system. In a previous work (arXiv:0811.0723) we have
proven that the two critical points differ at marginality of at least
exp(-c/beta^4), where c>0 and beta^2 is the disorder variance, for beta in
(0,1) and Gaussian IID disorder. The purpose of this paper is to improve such a
result: we establish in particular that the exp(-c/beta^4) lower bound on the
shift can be replaced by exp(-c(b)/beta^b), c(b)>0 for b>2 (b=2 is the known
upper bound and it is the result claimed in [Derrida, Hakim, Vannimenus, JSP
1992]), and we deal with very general distribution of the IID disorder
variables. The proof relies on coarse graining estimates and on a fractional
moment-change of measure argument based on multi-body potential modifications
of the law of the disorder.
