In this thesis several classes of directed paths are considered as models of linear polymers in a dilute solution. We obtain the generating functions for each model by considering factorization arguments. Information about the polymer behaviour can be extracted from the singularity structure of the associated generating functions. By using modified versions of these models we study the adsorption and localization of polymer molecules, the behaviour of polymers subject to a tensile force, the effects of stiffness, as well as the behaviour of polymers in confined geometries. In each of these situations the resulting generating functions contain at least two physical singularities. We identify the phase transitions in these systems by a changeover in the dominant singularity of the generating function. In the study of localization and polymers subject to a force, we utilize both homopolymer and random copolymer models. For copolymers, the physically relevant properties are obtained by considering a quenched average of the free energy over all possible monomer sequences. This procedure is intractable even for the simplest models. By considering the Morita approximation for several walk models we obtain results which give a bound on the corresponding features of the quenched system. We use a mapping between a simple model of duplex DNA and an adsorbing Motzkin path in order to study the mechanical unzipping of duplex DNA. From this model, we obtain force-temperature diagrams which show re-entrant behaviour of the force. We also develop a simple low temperature theory to describe the behaviour of the force close to T=0 and find that the shape of the force-temperature curve is associated with entropy in the ground state of the system. We consider the effect of stiffness on polymer adsorption and find that the phase transition is second order for all finite stiffness parameters. For systems of polymers in confined geometries, we find that the behaviour of the polymer depends on the distance between the confining surfaces and the associated interactions with each surface. In this problem, there exist regimes where the polymer exerts a force on the surfaces which can be attractive, repulsive or zero.

In this thesis several classes of directed paths are considered as models of linear polymers in a dilute solution. We obtain the generating functions for each model by considering factorization arguments. Information about the polymer behaviour can be extracted from the singularity structure of the associated generating functions. By using modified versions of these models we study the adsorption and localization of polymer molecules, the behaviour of polymers subject to a tensile force, the effects of stiffness, as well as the behaviour of polymers in confined geometries. In each of these situations the resulting generating functions contain at least two physical singularities. We identify the phase transitions in these systems by a changeover in the dominant singularity of the generating function. In the study of localization and polymers subject to a force, we utilize both homopolymer and random copolymer models. For copolymers, the physically relevant properties are obtained by considering a quenched average of the free energy over all possible monomer sequences. This procedure is intractable even for the simplest models. By considering the Morita approximation for several walk models we obtain results which give a bound on the corresponding features of the quenched system. We use a mapping between a simple model of duplex DNA and an adsorbing Motzkin path in order to study the mechanical unzipping of duplex DNA. From this model, we obtain force-temperature diagrams which show re-entrant behaviour of the force. We also develop a simple low temperature theory to describe the behaviour of the force close to T=0 and find that the shape of the force-temperature curve is associated with entropy in the ground state of the system. We consider the effect of stiffness on polymer adsorption and find that the phase transition is second order for all finite stiffness parameters. For systems of polymers in confined geometries, we find that the behaviour of the polymer depends on the distance between the confining surfaces and the associated interactions with each surface. In this problem, there exist regimes where the polymer exerts a force on the surfaces which can be attractive, repulsive or zero.