Random copolymers and the Morita approximation: polymer adsorption and polymer localization

We analyse directed walk models of random copolymer adsorption and localization. Ideally we would like to solve the quenched problem, but it appears to be intractable even for simple directed models. The annealed approximation is solvable, but is inadequate in the strong interaction regime for the adsorption problem and gives a qualitatively incorrect phase diagram for the localization problem. In this paper, we treat these directed models using an approximation suggested by Morita (1964 J. Math. Phys. 5 1401-5) in which the proportion of each comonomer is fixed. We find that the Morita approximation leads to behaviour that is closer to that of the quenched average model and this is particularly interesting in the localization problem where the phase diagram is (at least qualitatively) very similar to that of the quenched average problem. We also show that the phase boundaries in the Morita approximation are bounds on the locations of the phase boundaries of the quenched model.