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<Subscript>c</Subscript>(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<Subscript>c</Subscript>(i,L) with mean T<Subscript>c</Subscript> <Superscript>av</Superscript>(L) and width ΔT<Subscript>c</Subscript>(L). For the marginal case c=1.5 corresponding to two-dimensional wetting, both the width ΔT<Subscript>c</Subscript>(L) and the shift [T<Subscript>c</Subscript>(∞)-T<Subscript>c</Subscript> <Superscript>av</Superscript>(L)] decay as L<Superscript>-1/2</Superscript>, so the exponent is unchanged (ν<Subscript>random</Subscript>=2=ν<Subscript>pure</Subscript>) but disorder is relevant and leads to non self-averaging at criticality. For relevant disorder c=1.75, the width ΔT<Subscript>c</Subscript>(L) and the shift [T<Subscript>c</Subscript>(∞)-T<Subscript>c</Subscript> <Superscript>av</Superscript>(L)] decay with the same new exponent L<Superscript>-1/νrandom</Superscript> (where ν<Subscript>random</Subscript> ∼2.7 > 2 > ν<Subscript>pure</Subscript>) 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<Subscript>c</Subscript>(L) ∼L<Superscript>-1/2</Superscript> dominates over the shift [T<Subscript>c</Subscript>(∞)-T<Subscript>c</Subscript> <Superscript>av</Superscript>(L)] ∼L<Superscript>-1</Superscript>, i.e. there are two correlation length exponents ν=2 and <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\tilde \nu=1$</EquationSource> </InlineEquation> that govern respectively the averaged/typical loop distribution. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005