Disorder and denaturation transition in the generalized Poland–Scheraga model

Berger, Quentin; Giacomin, Giambattista; Khatib, Maha

Annales Henri Lebesgue, Tome 3 (2020), pp. 299-339.

Nous étudions ici le modèle de Poland–Scheraga généralisé, qui est utilisé dans la litérature bio-physique pour modéliser la transition de dénaturation de l’ADN, dans le cas où les deux brins ne sont pas forcément complémentaires (et peuvent avoir des longueurs différentes). La version homogène du modèle a été étudiée récemment d’un point de vue mathématique dans [BGK18, GK17], en utilisant un processus de renouvellement bidimensionnel, possédant un exposant de boucle $2+\alpha$ ($\alpha >0$) : une transition de phase de type localisation/délocalisation y a été mise en évidence (correspondant à la transition de dénaturation), d’ordre $\nu =min{(1,\alpha )}^{-1}$, ainsi que d’autres transitions de phase (en général). Dans cet article, nous nous tournons vers la version désordonnée du modèle, et nous traitons la question de l’influence du désordre sur la transition de dénaturation, c’est-à-dire de savoir si l’ajout de désordre d’intensité arbitrairement petite (i.e. d’inhomogénéités) possède un effet sur les propriétés critiques de cette transition de phase. Nos résultats sont en accord avec les prédictions de Harris pour les systèmes désordonnés $d$–dimensionnels (ici, $d=2$). Tout d’abord, nous montrons que quand $\alpha <1$ (i.e. $\nu >d/2$), alors le désordre est non-pertinent : les points critiques gelés et recuits sont égaux, et la transition de phase du modèle désordonné est aussi d’ordre $\nu ={\alpha }^{-1}$. D’autre part, quand $\alpha >1$, le désordre est pertinent : nous montrons que les points critiques gelés et recuits diffèrent. De plus, nous commentons un certains nombre de problèmes ouverts, notamment concernant le phénomène de lissage de la transition de phase que l’on s’attend à observer lorsque le désordre est pertinent.

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 [BGK18, GK17], via a $2$–dimensional renewal approach, with a loop exponent $2+\alpha$ ($\alpha >0$): it was found to undergo a localization/delocalization phase transition (which corresponds to the denaturation transition) of order $\nu =min{(1,\alpha )}^{-1}$, 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 $\alpha <1$ (i.e. $\nu >d/2$), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order $\nu ={\alpha }^{-1}$. On the other hand, when $\alpha >1$, 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.

Classification : 60K35, 82D60, 92C05, 60K05, 60F10
Mots clés : DNA denaturation, disordered polymer pinning model, critical behavior, disorder relevance, two-dimensional renewal processes

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