Copolymers near a Random Selective Interface

Abstract

In this chapter we consider a different version of the directed copolymer model analyzed in Chapter 9, namely, one where the linear interface is replaced by a random interface. In particular, rather than putting the oil and the water in two halfplanes, we place them in large square blocks in a random percolation-type fashion. This is a crude model of a copolymer in an emulsion, consisting of oil droplets floating in water (see Fig. 10.1). We will see that this model exhibits a remarkably rich critical behavior, with two phases in a supercritical percolation regime and four phases in a subcritical percolation regime. Large deviations and partial annealing estimates will again play a central role, together with coarse-graining, entropy estimates and variational calculus.