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

To study the localization of random heteropolymers at an interface separati ng two selective solvents within the model of Garel, Huse, Leibler and Orla nd, [Europhys. Lett. 8, 9 (1989)], we propose a disorder-dependent real spa ce renormalization approach. This approach allows to recover that a chain w ith a symmetric distribution in hydrophobic/hydrophilic components is local ized at any temperature in the thermodynamic limit, whereas a dissymmetric distribution in hydrophobic/hydrophilic components leads to a delocalizatio n phase transition. It yields in addition explicit expressions for thermody namic quantities as well as a very detailed description of the statistical properties of the heteropolymer conformations in the high temperature limit . In particular, scaling distributions are given for the lengths of the blo bs in each solvent, for the polymer density, and for some correlation funct ions. In the case of a small dissymmetry in hydrophobic/hydrophilic compone nts, the renormalization approach yields explicit expressions for the deloc alization transition temperature and for the critical behaviors of various quantities: in particular, the free energy presents an essential singularit y at the transition (the transition is thus of infinite order), the typical length of blobs in the preferred solvent diverges with an essential singul arity, whereas the typical length of blobs in the other solvent diverges al gebraically. Finite-size properties are also characterized in details in bo th cases. In particular, we give the probability distribution of the deloca lization temperature for the ensemble of random chains of finite (large) le ngth L, and the distribution of the numbers of blobs for the chains that ar e still localized at a given temperature. Finally, we discuss the non-equil ibrium dynamics at temperature T starting from a zero-temperature initial c ondition.