We consider Motzkin path models for polymers confined to a slit.The path interacts with each of the two confining lines and we define parameters a and b to characterize the strengths of the interactions with the two lines. We consider the cases where (i) each vertex in a confining line contributes to the energy and (ii) where each edge in a confining line contributes to the energy.For the vertex and edge versions of Motzkin paths we can find the generating functions and give rigorous explicit expressions for the free energy at some special points in the (a, b)-plane, and asymptotically (ie for large slit widths) elsewhere.We find regions where the force between the lines is long-range and repulsive, short-range and repulsive and short-range and attractive. Our results indicate that the general form of the phase diagram is model independent, and similar to previous results for a Dyck path model, although the details do depend on the underlying configurational model.We also contrast the method used here to find the generating function with the transfer matrix and heap methods.

We consider several types of inhomogeneous polymer adsorption. In each case, the inhomogeneity is regular and resides in the surface, in the polymer or in both. We consider two different polymer models: a directed walk model that can be solved exactly and a self-avoiding walk model which we investigate using Monte Carlo methods. In each case, we compute the phase diagram. We compare and contrast the phase diagrams and give qualitative arguments about their forms.

In this paper we examine the phases of a directed path model of a copolymer attached to a surface under the influence of a pulling force. The simplest model of an adsorbing directed polymer, attached at the one end to a surface and pulled from the surface by the other end, is reviewed-its phase diagram includes free, adsorbed and ballistic phases. In contrast to this model, we consider an adsorbing directed block copolymer attached at both endpoints to the adsorbing surface, and pulled away from the surface at its middle vertex. The phase diagram of this model includes ballistic, adsorbed and free phases, as well as a quasi-ballistic phase which has characteristics of both an adsorbed and a ballistic phase. We also generalize our methods to a model of directed block star copolymers and show that such directed copolymers have numerous adsorbed and quasi-ballistic phases. The phase transitions in this system are continuous in some cases and first order in others.

We consider several directed path models of semi-flexible polymers. In each model we associate an energy parameter for every pair of adjacent collinear steps, allowing for a model of a polymer with tunable stiffness. We introduce weightings for vertices or edges in a distinguished plane to model the interaction of a semi-flexible polymer with an impenetrable surface. We also investigate the desorption of such a polymer under the influence of an elongational force and study the order of the associated phase transitions. Using a simple lowtemperature theory, we approximate and study the ground state behaviour of the models.

Let L be the d-dimensional hypercubic lattice and let L 0 be an s-dimensional sublattice, with 2 ≤ s < d. We consider a model of inhomogeneous bond percolation on L at densities p and σ, in which edges in L\L 0 are open with probability p, and edges in L 0 open with probability σ. We generalizee several classical results of (homogeneous) bond percolation to this inhomogeneous model. The phase diagram of the model is presented, and it is shown that there is a subcritical regime for σ < σ * (p) and p < p c (d) (where p c (d) is the critical probability for homogeneous percolation in L), a bulk supercritical regime for p > p c (d), and a surface supercritical regime for p < p c (d) and σ > σ * (p). We show that σ * (p) is a strictly decreasing function for p ∈ [0, p c (d)], with a jump discontinuity at p c (d). We extend the Aizenman-Barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase. We prove that the cluster size distribution decays exponentially in the subcritical phase, and sub-exponentially in the supercritical phases. For a model of lattice animals with a defect plane, the free energy is related to functions of the inhomogeneous percolation model, and we show how the percolation transition implies a non-analyticity in the free energy of the animal model. Finally, we present simulation estimates of the critical curve σ * (p).