Localization transition for a copolymer in an emulsion

In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The polymer 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, organized in large square blocks occurring with density p and 1-p, respectively, where p ¿ (0, 1). The polymer in the emulsion has an energy that is minus a times the number of AA-matches minus ß times the number of BB-matches, where a, ß ¿ R are interaction parameters. Symmetry considerations show that without loss of generality we may restrict our attention to the cone {(a, ß) ¿ R2 : a ?? |ß|}. We derive a variational expression for the quenched free energy per monomer in the limit as the length n of the polymer tends to infinity and the blocks in the emulsion have size Ln such that Ln ¿ 8 and Ln/n ¿ 0. To make the model mathematically tractable, we assume that the polymer can only enter and exit a pair of neighboring blocks at diagonally opposite corners. Although this is an unphysical restriction, it turns out that the model exhibits rich and physically relevant behavior. Let pc ˜ 0.64 be the critical probability for directed bond percolation on the square lattice. We show that for p ?? pc the free energy has a phase transition along one curve in the cone, which turns out to be independent of p. At this curve, there is a transition from a phase where the polymer is fully A-delocalized (i.e., it spends almost all of its time deep inside the A-blocks) to a phase where the polymer is partially AB-localized (i.e., it spends a positive fraction of its time near those interfaces where it diagonally crosses the A-block rather than the B-block). We show that for p <pc the free energy has a phase transition along two curves in the cone, both of which turn out to depend on p. At the first curve there is a transition from a phase where the polymer is fully A,B-delocalized (i.e., it spends almost all of its time deep inside the A-blocks and the B-blocks) to a partially BA-localized phase, while at the second curve there is a transition from a partially BA-localized phase to a phase where both partial BA-localization and partial AB-localization occur simultaneously. We derive a number of qualitative properties of the critical curves. The supercritical curve is nondecreasing and concave with a finite horizontal asymptote. Remarkably, the first subcritical curve does not share these properties and does not converge to the supercritical curve as p ¿ pc. Rather, the second subcritical curve converges to the supercritical curve as p ¿ 0.