  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.
