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We investigate the generalized Poland-Scheraga model, which is used in the bio-physical literature to model the DNA denaturation transition, in the case where the two strands are allowed to be non-complementary (and to have different lengths). The homogeneous model was recently studied from a mathematical point of view in [35, 7], via a 2-dimensional renewal approach, with a loop exponent 2+α (α > 0): it was found to undergo a localization/delocalization phase transition of order ν = min(1, α) −1 , together with – in general – other phase transitions. In this paper, we turn to the disordered model, and we address the question of the influence of disorder on the denaturation phase transition, that is whether adding an arbitrarily small amount of disorder (i.e. inhomogeneities) affects the critical properties of this transition. Our results are consistent with Harris' predictions for d-dimensional disordered systems (here d = 2). First, we prove that when α d/2), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order ν = α −1. On the other hand, when α > 1, disorder is relevant: we prove that the quenched and annealed critical points differ. Moreover, we discuss a number of open problems, in particular the smoothing phenomenon that is expected to enter the game when disorder is relevant.

The Poland-Scheraga model for DNA denaturation, besides playing a central role in applications, has been widely studied in the physical and mathematical literature over the past decades. More recently a natural generalization has been introduced in the biophysics literature to overcome the limits of the original model, namely to allow an excess of bases -- i.e. a different length of the two single stranded DNA chains -- and to allow slippages in the chain pairing. The increased complexity of the model is reflected in the appearance of configurational transitions when the DNA is in double stranded form. In a previous work of two of the authors the generalized Poland-Scheraga model has been analyzed thanks to a representation in terms of a bivariate renewal process. In this work we exploit this representation farther and fully characterize the path properties of the system, making therefore explicit the geometric structures -- and the configurational transitions -- that are observed when the polymer is in the double stranded form. What we prove is that, when the excess of bases is not absorbed in a homogeneous fashion along the double stranded chain, then it either condensates in a single macroscopic loop or it accumulates into an unbound single strand free end., Comment: 29 pages, 5 figures