论文标题
定向晶格路径的轨迹
Trajectories of directed lattice paths
论文作者
论文摘要
通过确定戴克（Dyck）占用顶点的概率分布和吸附线性聚合物的投票路径模型的概率分布来建模沿着硬壁上的线性聚合物的单体分布。例如，Dyck路径与coordinates $（\ lfloorεn\ rfloor，\lfloorΔ\ sqrt {n} \ rfloor）$在平方点中的概率（\lfloorΔ随着路径$ n $接近无穷大的长度，沿戴克路径的顶点的概率密度：$ \ hbox {p} _r（ε，δ）= \ frac {4Δ^2} e^{ - δ^2/ε（1-ε）} \。$$硬壁的聚合物涂层的特性以及涂层中单体的密度或分布在诸如诸如聚合物或在诸如药物覆盖的plopt polapt oper oper oper of grol polymer growapt prolymer gropt的药物递送系统中稳定胶体分散的应用，这是相关的。
The distribution of monomers along a linear polymer grafted on a hard wall is modelled by determining the probability distribution of occupied vertices of Dyck and ballot path models of adsorbing linear polymers. For example, the probability that a Dyck path passes through the lattice site with coordinates $(\lfloor εn \rfloor,\lfloor δ\sqrt{n}\rfloor)$ in the square lattice, for $0 < ε< 1$ and $δ\geq 0$, is determined asymptotically as $n\to\infty$ and this uncovers the probability density of vertices along Dyck paths in the limit as the length of the path $n$ approaches infinity: $$\hbox{P}_r (ε,δ) = \frac{4δ^2}{\sqrt{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.
