楔形中的有向路径
Directed Paths in a Wedge
摘要：
有向路径在科学文献中被广泛用作线性聚合物的模型。这种路径模型特别是线性聚合物的构象熵及其对自由能的影响。这些定向模型是自避免行走的简化版本，但它们仍然可以洞察聚合物的相行为，并且也可以作为研究线性聚合物行为中构象自由度的影响的工具。在本文中，我们研究了有向路径模型的线性聚合物在一个封闭的几何形状（楔）。我们关注的主要焦点是$c_n$，长度为$n$步的有向晶格路径的数量，这些路径在东北和东南方向上采取步骤，并且被限制在楔形$Y=\pm X/p$中，其中$p$是整数。在本文中，我们研究的情况下，$p=2$详细，我们确定的生成函数使用迭代核方法。我们还研究了$c_n$的对称性。特别是，我们表明，$$ c_n = [0.67874...]\times 2^{n-1}(1+(-1)^n) + O((4/3^{3/4})^{n+o(n)}) + o((4/3^{3/4})^n) $$在这里，我们可以确定常数$0.67874...$，以任意精度，很少的努力。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
