四元数观点下随机多项式的概率论
Probability Theory of Random Polygons from the Quaternionic Viewpoint
摘要：
我们在封闭空间和平面多边形上建立了一个新的概率测度。关键的构造是Knutson和Hausmann使用四元数上的Hopf映射给出的一个映射，从n-空间中的2-标架的复Stiefel流形到总长为2的3-空间中的闭n-标架空间。利用该映射将Stiefel流形上的Haar测度向前推进，定义了多边形空间上的概率测度.一个类似的建设产生一个概率测度的平面多边形，来自一个真正的Stiefel流形。根据我们的措施采样的多边形的边的分布服从β分布。这使得我们的多边形测量不同于通常研究的那些具有高斯或固定边的多边形测量。我们的措施的一个优点是，我们可以明确地计算的期望和力矩的弦长和回转半径。另一个原因是，根据我们的测量直接采样是快速的（在边缘数量上是线性的），并且易于编码。我们的一些方法将在研究多边形空间上的其他概率测度时具有独立的意义。我们定义一个边集集成（ESE）是一组多边形创建重新安排一组给定的n个边缘。一个关键定理给出了一个公式，用于在ESE上跳过k个顶点的弦的平方长度的平均值，关于k，n和系综的边数。这使得人们可以很容易地计算期望值的平方弦长和半径的回转任何概率测度多边形空间不变下重新安排的边缘。
英文摘要：
We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons which comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chordlengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chordlengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges.
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