论文标题
具有一般边缘电势的随机图嵌入
Random graph embeddings with general edge potentials
论文作者
论文摘要
在本文中，我们研究了根据任何势能分布的聚合物网络的随机嵌入，这些能量可以根据单体对之间的距离表示。这包括自由接头的链，空间效应，Lennard-Jones电位，弯曲能和其他物理现实的模型。 $ \ Mathbb {r}^d $中$ n $单体的配置可以写成$ d $坐标向量的集合，每个集合中$ \ mathbb {r}^n $。我们的第一个主要结果是，即使它们是同一单体的不同坐标，来自不同坐标向量的条目也不相关。我们预测，该属性在逼真的模拟和实际聚合物配置中（在没有外部磁场的情况下）。 我们的第二个主要贡献是一个定理，该定理解释了复杂图的嵌入方式何时以及如何将概率分布推向更简单图的嵌入到分布中，以帮助计算。该结构基于同源理论中的链图的概念。我们使用它为幻影网络理论中的边缘协方差提供了一个新的公式，并计算了对自由关联网络的一些期望。
In this paper, we study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances between pairs of monomers. This includes freely jointed chains, steric effects, Lennard-Jones potentials, bending energies, and other physically realistic models. A configuration of $n$ monomers in $\mathbb{R}^d$ can be written as a collection of $d$ coordinate vectors, each in $\mathbb{R}^n$. Our first main result is that entries from different coordinate vectors are uncorrelated, even when they are different coordinates of the same monomer. We predict that this property holds in realistic simulations and in actual polymer configurations (in the absence of an external field). Our second main contribution is a theorem explaining when and how a probability distribution on embeddings of a complicated graph may be pushed forward to a distribution on embeddings of a simpler graph to aid in computations. This construction is based on the idea of chain maps in homology theory. We use it to give a new formula for edge covariances in phantom network theory and to compute some expectations for a freely-jointed network.
