Copolymer in an emulsion: supercritical and subcritical regime

In this talk we discuss a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density $\frac{1}{2}$. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p\in (0,1)$. The copolymer in the emulsion has an energy that is minus $\alpha$ times the number of $AA$-matches minus $\beta$ times the number of $BB$-matches. We will consider both the supercritical regime (oil droplets form an infinite cluster) and the subcritical regime (no infinite cluster).