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We perform an extensive numerical study of the disordered Poland-Scheraga(PS) model for DNA denaturation in which self-avoidance is completely takeninto account. In complement to our previous work, we focus here on the finitesize scaling in terms of pseudo-critical temperatures. We find notably that themean value and the fluctuations of the pseudo-$T_c$ scale with the sameexponent, the correlation length exponent $\nu_r$ (for which we provide therefined evaluation $\nu_r=2.9 \pm 0.4$). This result (coherent with the typicalpicture that describes random ferromagnets, when disorder is relevant) is atvariance with numerical results reported in the literature for the PS modelwith self-avoidance, leading to an alternative scenario with a pseudo firstorder transition. We moreover introduce a crossover chain length $N^*$, whichwe evaluate, appropriate for characterizing the approach to the asymptoticregime in this model. Essentially, below $N^*$, the behaviour of the model inour study could also agree with such alternative scenario. Based on anapproximate prediction of the dependence of $N^*$ on the parameters of themodel, we show that following the choice of such parameters it could be notpossible to reach the asymptotic regime in practice. In such context it becomesthen possible to reconcile the apparently contradictory numerical studies.

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