A renewal theory approach to periodic copolymers with adsorption
Résumé
We consider a general model of an heterogeneous polymer chain fluctuating in the proximity of an interface between two selective solvents. The heterogeneous character of the model comes from the fact that monomer units interact with the solvents and with the interface according to some charges that they carry. The charges repeat themselves along the chain in a periodic fashion. The main question on this model is whether the polymer remains tightly close to the interface, a phenomenon called localization, or there is a marked preference for one of the two solvents yielding thus a delocalization phenomenon. We propose an approach to this model, based on renewal theory, that yields sharp estimates on the partition function of the model in all the regimes (localized, delocalized and critical). This in turn allows to get a very precise description of the polymer measure, both in a local sense (thermodynamic limit) and in a global sense (scaling limits). A key point, but also a byproduct, of our analysis is the closeness of the polymer measure to suitable Markov Renewal Processes.