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On the localization of random heteropolymers at the interface between two selective solvents

To study the localization of random heteropolymers at an interface separating two selective solvents within the model of Garel, Huse, Leibler and Orland, Europhys. Lett. {\bf 8} 9 (1989), we propose an approach based on a disorder-dependent real space renormalization procedure. This approach allows to recover that a chain with a symmetric distribution in hydrophobic/hydrophilic components is localized at any temperature in the thermodynamic limit, whereas a dissymmetric distribution in hydrophobic/hydrophilic components leads to a delocalization phase transition. It yields in addition explicit expressions for the thermodynamic quantities as well as a very detailed description of the statistical properties of the behaviors of the heteropolymers in the high temperature limit. For the case of a small dissymmetry in hydrophobic/hydrophilic components, the renormalization approach yields explicit predictions for the delocalization transition temperature and for the critical behaviors of various quantities : in particular, the free energy presents an essential singularity at the transition, the typical length of blobs in the preferred solvent diverges with an essential singularity, whereas the typical length of blobs in the other solvent diverges algebraically. Finite-size properties are also characterized in details for both cases. In particular, we give the probability distribution of the delocalization temperature for the ensemble of chains of finite (large) length $L$. Finally, we discuss the non-equilibrium dynamics at temperature $T$ starting from a zero-temperature initial condition.