Directed Paths in a Wedge

Abstract

Directed paths have been used extensively in the scientific literature as a model of a linear polymer. Such paths models in particular the conformational entropy of a linear polymer and the effects it has on the free energy. These directed models are simplified versions of the self-avoiding walk, but they do nevertheless give insight into the phase behaviour of a polymer, and also serve as a tool to study the effects of conformational degrees of freedom in the behaviour of a linear polymer. In this paper we examine a directed path model of a linear polymer in a confining geometry (a wedge). The main focus of our attention is $c_n$, the number of directed lattice paths of length $n$ steps which takes steps in the North-East and South-East directions and which is confined to the wedge $Y=\pm X/p$, where $p$ is an integer. In this paper we examine the case $p=2$ in detail, and we determine the generating function using the iterated kernel method. We also examine the asymtotics of $c_n$. In particular, we show that $$c_n=[\mathrm{0.67874...}]\times 2^{n-1}(1+(-1{)}^n)+O((4/3^{3/4}{)}^{n+o(n)})+o((4/3^{3/4}{)}^n)$$ where we can determine the constant $\mathrm{0.67874...}$ to arbitrary accuracy with little effort.

References in corpus (1)

• Two Non-holonomic Lattice Walks in the Quarter Plane