On the localized phase of a copolymer in an emulsion: supercritical percolation regime

In this paper we study 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 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, where without loss of generality the interaction parameters can be taken from the cone ${(\alpha,\beta)\in\R^2\colon \alpha\geq |\beta|}$. To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In \cite{dHW06}, it was found that in the supercritical percolation regime $p \geq p_c$,with$p_c$ the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of $p$. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the $A$-blocks to a phase where it is partially localized near the $AB$-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase.