Disorder and denaturation transition in the generalized Poland-Scheraga model

TL;DR

This paper studies how disorder affects the DNA denaturation transition in a generalized Poland-Scheraga model, showing disorder's irrelevance for certain parameters and relevance for others, with implications for phase transition properties.

Contribution

It provides a rigorous analysis of the impact of disorder on the denaturation transition, confirming predictions about disorder relevance based on the loop exponent parameter.

Findings

Disorder is irrelevant when lpha<1, with quenched and annealed critical points coinciding.

Disorder is relevant when lpha>1, leading to different quenched and annealed critical points.

The nature of the phase transition depends on the loop exponent lpha, with different behaviors in different regimes.

Abstract

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 Giacomin, Khatib (Stoch. Proc. Appl., 2017), via a $2$-dimensional renewal approach, with a loop exponent $2+\alpha$ (${\alpha>0}$): it was found to undergo a localization/delocalization phase 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.