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 path and Dyck meander models of adsorbing linear polymers. For example, the probability that a Dyck path passes through the lattice site with coordinates <?CDATA $(\lfloor \epsilon n\rfloor ,\lfloor \delta \sqrt{n}\rfloor )$?> ( ⌊ ϵ n ⌋ , ⌊ δ n ⌋ ) in the square lattice, for 0 < ϵ < 1 and δ ≥ 0, is determined asymptotically as n → ∞ and this uncovers the probability density of vertices along Dyck paths in the limit as the length of the path n approaches infinity: <?CDATA \begin{eqnarray*}{{\mathbb{P}}}^{(D)}(\epsilon ,\delta )=\displaystyle \frac{4{\delta }^{2}}{\sqrt{\pi \,{\epsilon }^{3}{\left(1-\epsilon \right)}^{3}}}\,{e}^{-{\delta }^{2}/\epsilon (1-\epsilon )}.\end{eqnarray*}?> P ( D ) ( ϵ , δ ) = 4 δ 2 π ϵ 3 1 − ϵ 3 e − δ 2 / ϵ ( 1 − ϵ ) . 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.

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