Distribution of pseudo-critical temperatures and lack of self-averaging in disordered Poland-Scheraga models with different loop exponents

According to recent progresses in the finite size scaling theory of disordered systems, thermodynamic observables are not self-averaging at critical points when the disorder is relevant in the Harris criterion sense. This lack of self-averageness at criticality is directly related to the distribution of pseudo-critical temperatures T c (i,L) over the ensemble of samples (i) of size L. In this paper, we apply this analysis to disordered Poland-Scheraga models with different loop exponents c, corresponding to marginal and relevant disorder. In all cases, we numerically obtain a Gaussian histogram of pseudo-critical temperatures T c (i,L) with mean T c av (L) and width ΔT c (L). For the marginal case c=1.5 corresponding to two-dimensional wetting, both the width ΔT c (L) and the shift [T c (∞)-T c av (L)] decay as L -1/2 , so the exponent is unchanged (ν random =2=ν pure ) but disorder is relevant and leads to non self-averaging at criticality. For relevant disorder c=1.75, the width ΔT c (L) and the shift [T c (∞)-T c av (L)] decay with the same new exponent L -1/νrandom (where ν random ∼2.7 > 2 > ν pure ) and there is again no self-averaging at criticality. Finally for the value c=2.15, of interest in the context of DNA denaturation, the transition is first-order in the pure case. In the presence of disorder, the width ΔT c (L) ∼L -1/2 dominates over the shift [T c (∞)-T c av (L)] ∼L -1 , i.e. there are two correlation length exponents ν=2 and $\tilde \nu=1$ that govern respectively the averaged/typical loop distribution. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005