论文标题
随机锥的阈值现象
Threshold phenomena for random cones
论文作者
论文摘要
我们考虑在$ d $二维的欧几里得空间上具有均匀的概率分布，并具有将其分配给通过原点的任何超平面分配给零的属性。给定$ n $独立的随机向量，在这种分布的情况下，在它们不能积极地跨越整个空间的情况下，这些向量的正船体是一个随机的多面体锥（并且其与单位球的相交是随机的球形多型）。它首先是由Cover和Efron研究的。我们考虑这些随机锥的预期面部数，并描述尺寸$ d $和随机向量的数字$ n $ n倾向于无限时的阈值现象。以类似的方式，我们对待实体角度，更普遍地处理格拉曼角。当$ k $也倾向于无限时，我们进一步考虑了$ k $ faces的预期数量和索引$ d-k $的Grassmann角度。
We consider an even probability distribution on the $d$-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given $N$ independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension $d$ and the number $N$ of random vectors tend to infinity. In a similar way, we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of $k$-faces and of Grassmann angles of index $d-k$ when also $k$ tends to infinity.
