Trajectories of directed lattice paths

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 $n\to \mathrm{\infty }$ and this uncovers the probability density of vertices along Dyck paths in the limit as the length of the path $n$ approaches infinity: $${\text{P}}_r(\epsilon ,\delta )=\frac{4{\delta }^2}{\sqrt{\pi {\epsilon }^3(1-\epsilon {)}^3}}{e}^{-{\delta }^2\mathrm{/}\epsilon (1-\epsilon )}\mathrm{.}$$ 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.