Trajectories of directed lattice paths

TL;DR

This paper analyzes the asymptotic distribution of vertices along directed lattice paths, modeling monomer positions in grafted polymers, with implications for colloid stabilization and drug delivery systems.

Contribution

It provides a new asymptotic probability density function for vertices along Dyck paths, linking lattice path models to polymer monomer distribution.

Findings

Derived the asymptotic probability density for vertices in Dyck paths

Connected lattice path models to polymer coating density

Applicable to colloid stabilization and drug delivery systems

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 $\delta\geq 0$, 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: $$\hbox{P}_r (\epsilon,\delta) = \frac{4\delta^2}{\sqrt{\pi\,\epsilon^3(1-\epsilon)^3}} \, e^{-\delta^2/\epsilon(1-\epsilon)}\ .$$ 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.