Trajectories of directed lattice paths

Abstract

The distribution of monomers along a linear polymer grafted on a hard wall is modelled by determining the probability distribution of occupied vertices of Dyck and ballot path models of adsorbing linear polymers. For example, the probability that a Dyck path passes through the lattice site with coordinates $(\lfloor \epsilon n\rfloor ,\lfloor \delta \sqrt{n}\rfloor )$ in the square lattice, for $0<\epsilon <1$ and $\delta \ge 0$, is determined asymptotically as $n\to \infty$ and this uncovers the probability density of vertices along Dyck paths in the limit as the length of the path $n$ approaches infinity: $$P_r(\epsilon ,\delta )=\frac{4{\delta }^2}{\sqrt{\pi {\epsilon }^3(1-\epsilon {)}^3}}{e}^{-{\delta }^2/\epsilon (1-\epsilon )}.$$ The properties of a polymer coating of a hard wall and the density or distribution of monomers in the coating is relevant in applications such as the stabilisation of a colloid dispersion by a polymer or in a drug delivery system such as a drug-eluding stent covered by a grafted polymer.

References in corpus (3)

• A directed walk model of a long chain polymer in a slit with attractive walls
• Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions
• The pressure exerted by adsorbing directed lattice paths and staircase polygons