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One dimensional pinning models have been widely studied in the physical andmathematical literature, also in presence of disorder. Roughly speaking, theyundergo a transition between a delocalized phase and a localized one. Inmathematical terms these models are obtained by modifying the distribution of adiscrete renewal process via a Boltzmann factor with an energy that containsonly one body potentials. For some more complex models, notably pinning modelsbased on higher dimensional renewals, it has been shown that other phases maybe present. We study a generalization of the one dimensional pinning model inwhich the energy may depend in a nonlinear way on the contact fraction: thisclass of models contains the circular DNA case considered in the bio-physicsliterature. We give a full solution of this generalized pinning model inabsence of disorder and show that another transition appears. In fact thesystems may display up to three different regimes: delocalization, partiallocalization and full localization. What happens in the partially localizedregime can be explained in terms of the "big-jump" phenomenon for sums of heavytail random variables under conditioning. We then show that disorder completelysmears this second transition and we are back to the delocalization versuslocalization scenario. In fact we show that the disorder, even if arbitrarilyweak, is incompatible with the presence of a big-jump.

Localization, big-jump regime and the effect disorder for a class of generalized pinning models