  One dimensional pinning models have been widely studied in the physical and
mathematical literature, also in presence of disorder. Roughly speaking, they
undergo a transition between a delocalized phase and a localized one. In
mathematical terms these models are obtained by modifying the distribution of a
discrete renewal process via a Boltzmann factor with an energy that contains
only one body potentials. For some more complex models, notably pinning models
based on higher dimensional renewals, it has been shown that other phases may
be present. We study a generalization of the one dimensional pinning model in
which the energy may depend in a nonlinear way on the contact fraction: this
class of models contains the circular DNA case considered in the bio-physics
literature. We give a full solution of this generalized pinning model in
absence of disorder and show that another transition appears. In fact the
systems may display up to three different regimes: delocalization, partial
localization and full localization. What happens in the partially localized
regime can be explained in terms of the "big-jump" phenomenon for sums of heavy
tail random variables under conditioning. We then show that disorder completely
smears this second transition and we are back to the delocalization versus
localization scenario. In fact we show that the disorder, even if arbitrarily
weak, is incompatible with the presence of a big-jump.
