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 . 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 ¿ (0, 1). The copolymer 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 without loss of generality the interaction parameters can be taken from the cone . 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 [7], a variational expression was derived for the quenched free energy per monomer in the limit as the length n of the copolymer tends to infinity and the blocks in the emulsion have size L n such that L n ¿ 8 and L n /n ¿ 0. Under this restriction, the free energy is self-averaging with respect to both types of randomness. It was found that in the supercritical percolation regime p = 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. In the subcritical percolation regime p <p c , the phase diagram is much more complex. This regime will be treated in a forthcoming paper