Directed Paths in a Wedge

TL;DR

This paper analyzes directed lattice paths modeling linear polymers confined in a wedge, determining their generating function and asymptotic behavior, especially for the case p=2, revealing detailed growth rates and constants.

Contribution

It provides an exact generating function for directed paths in a wedge and detailed asymptotics for the case p=2, advancing understanding of polymer models in confined geometries.

Findings

Exact generating function for p=2 wedge paths obtained

Asymptotic growth rate of paths determined

Constant factor in path count approximated to high precision

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 $$ c_n = [0.67874...]\times 2^{n-1}(1+(-1)^n) + O((4/3^{3/4})^{n+o(n)}) + o((4/3^{3/4})^n) $$ where we can determine the constant $0.67874...$ to arbitrary accuracy with little effort.