Higher order Morita approximations for random copolymer localization

The Morita approximation is a constrained annealing procedure which yields upper bounds on the quenched average free energy for models of quenched randomness. In this article we consider a bilateral Dyck path model, first introduced by S.G. Whittington and collaborators, of the localization of a random copolymer at the interface between two immiscible solvents. The distribution of comonomers along the polymer chain is initially determined by a random process and once chosen it remains fixed. Morita approximations in which we control correlations to various orders between neighbouring monomers along the polymer chain are applied to this model. Although at low orders the Morita approximation does not yield the correct path properties in the localized region of the phase diagram, we show that this problem can be overcome by including sufficiently high-order correlations in the Morita approximation. In addition by comparison with an appropriate lower bound, we show that well-within the localized phase the Morita approximation provides a relatively tight upper bound on the limiting quenched average free energy for bilateral Dyck path localization.