Charged Polymers

Abstract

In this chapter we consider a model introduced in Kantor and Kardar [203], where each monomer carries a random charge, and each self-intersection of the polymer is rewarded when the two charges of the associated monomers have opposite sign and is penalized when they have the same sign. This model is a variation on the weakly self-avoiding walk described in Chapters 3 and 4, with a random self-interaction driven by the charges. We will focus on the annealed path measure, of the type defined in (1.5). We will show that the annealed charged polymer is in a collapsed phase, irrespective of its overall charge distribution, and is subdiffusive with a scaling limit that can be computed explicitly, namely, Brownian motion conditioned to stay inside a finite ball. The free energy will be different for neutral and for non-neutral charged polymers, even though the scaling limit is the same. Once more local times will prove to be useful. In particular, the large deviation behavior of the local times of SRW will play a crucial role in the identification of the scaling limit.