平面曲线的马尔可夫链采样器
A Markov Chain Sampler for Plane Curves
摘要：
一条平面曲线是一个纽结图，其中每个交叉点都被一个4价顶点所取代，因此与平面四边形的子集是对偶的。本文的目的是介绍一种通过对平面曲线进行采样来对图形进行采样的新工具。目前，最有效的抽样方法是拒绝抽样，但这种方法是低效的，即使是适度的大小。我们引入马尔可夫链，样本从空间的平面曲线使用本地移动的基础上Reidemeister移动。然后，通过将这些曲线上的顶点映射到交叉点，我们生成随机结图。结合该链与平面直方图方法，我们实现了一个有效的采样器的平面曲线和结图。通过分析数据，从这个链，我们能够估计一个给定的大小的结图的数量，也计算打结概率，因此调查他们的渐近行为。
英文摘要：
A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sampling, however that method is inefficient at even modest sizes. We introduce Markov chains that sample from the space of plane curves using local moves based on Reidemeister moves. By then mapping vertices on those curves to crossings we produce random knot diagrams. Combining this chain with flat histogram methods we achieve an efficient sampler of plane curves and knot diagrams. By analysing data from this chain we are able to estimate the number of knot diagrams of a given size and also compute knotting probabilities and so investigate their asymptotic behaviour.
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