Abstract.
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 Tc(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 Tc(i,L) with mean Tc av(L) and width ΔTc(L). For the marginal case c=1.5 corresponding to two-dimensional wetting, both the width ΔTc(L) and the shift [Tc(∞)-Tc av(L)] decay as L-1/2, so the exponent is unchanged (νrandom=2=νpure) but disorder is relevant and leads to non self-averaging at criticality. For relevant disorder c=1.75, the width ΔTc(L) and the shift [Tc(∞)-Tc av(L)] decay with the same new exponent L-1/νrandom (where νrandom ∼2.7 > 2 > νpure) 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 ΔTc(L) ∼L-1/2 dominates over the shift [Tc(∞)-Tc 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.
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Monthus, C., Garel, T. Distribution of pseudo-critical temperatures and lack of self-averaging in disordered Poland-Scheraga models with different loop exponents. Eur. Phys. J. B 48, 393–403 (2005). https://doi.org/10.1140/epjb/e2005-00417-7
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DOI: https://doi.org/10.1140/epjb/e2005-00417-7