Trajectories of directed lattice paths

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