Delocalization transition of the selective interface model: distribution of pseudo-critical temperatures

According to recent progress in the finite size scaling theory of critical disordered systems, the nature of the phase transition is reflected in 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 the delocalization transition of an heteropolymeric chain at a selective fluid-fluid interface. The width $\mathrm{\Delta }{T}_c(L)$ and the shift $[T_c(\mathrm{\infty })-T_c^{av}(L)]$ are found to decay with the same exponent $L^{-1\mathrm{/}{\nu }_R}$, where $1\mathrm{/}{\nu }_R\sim 0.26$. The distribution of pseudo-critical temperatures $T_c(i,L)$ is clearly asymmetric, and is well fitted by a generalized Gumbel distribution of parameter $m\sim 3$. We also consider the free energy distribution, which can also be fitted by a generalized Gumbel distribution with a temperature dependent parameter, of order $m\sim 0.7$ in the critical region. Finally, the disorder averaged number of contacts with the interface scales at $T_c$ like $L^{\rho }$ with $\rho \sim 0.26\sim 1\mathrm{/}{\nu }_R$.