Higher order Morita approximations for random copolymer adsorption

Random linear copolymers are linear polymers with two or more types of monomer where the monomer sequence is determined by some random process. Once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness. Even for simple configurational models the quenched model is too difficult to solve analytically. The Morita approximation is a partial annealing procedure which yields upper bounds on the quenched average free energy. In this paper we consider higher order Morita approximations in which we control correlations to various orders between neighbouring monomers along the polymer chain. We consider different approaches for incorporating correlations and apply these to Motzkin and Dyck path models of the adsorption of a random copolymer at a surface. We also present lower bounds which, along with the Morita bounds, determine the limiting quenched average free energy for adsorption very precisely at low temperatures.