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 $\Delta T_c(L)$. For the marginal case $c=1.5$ corresponding to two-dimensional wetting, both the width $\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ decay as $L^{-1/2}$, so the exponent is unchanged ($\nu_{random}=2=\nu_{pure}$) but disorder is relevant and leads to non self-averaging at criticality. For relevant disorder $c=1.75$, the width $\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ decay with the same new exponent $L^{-1/\nu_{random}}$ (where $\nu_{random} \sim 2.7 > 2 > \nu_{pure}$) 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 $\Delta T_c(L) \sim L^{-1/2}$ dominates over the shift $[T_c(\infty)-T_c^{av}(L)] \sim L^{-1}$, i.e. there are two correlation length exponents $\nu=2$ and $\tilde \nu=1$ that govern respectively the averaged/typical loop distribution.