Numerical study of DNA denaturation with self-avoidance: pseudo-critical temperatures and finite size behaviour

We perform an extensive numerical study of the disordered Poland-Scheraga (PS) model for DNA denaturation in which self-avoidance is completely taken into account. In complement to our previous work, we focus here on the finite size scaling in terms of pseudo-critical temperatures. We find notably that the mean value and the fluctuations of the pseudo-$T_c$ scale with the same exponent, the correlation length exponent $\nu_r$ (for which we provide the refined evaluation $\nu_r=2.9 \pm 0.4$). This result (coherent with the typical picture that describes random ferromagnets, when disorder is relevant) is at variance with numerical results reported in the literature for the PS model with self-avoidance, leading to an alternative scenario with a pseudo first order transition. We moreover introduce a crossover chain length $N^*$, which we evaluate, appropriate for characterizing the approach to the asymptotic regime in this model. Essentially, below $N^*$, the behaviour of the model in our study could also agree with such alternative scenario. Based on an approximate prediction of the dependence of $N^*$ on the parameters of the model, we show that following the choice of such parameters it could be not possible to reach the asymptotic regime in practice. In such context it becomes then possible to reconcile the apparently contradictory numerical studies.