随机锥的阈值现象

摘要：

考虑$d$维欧几里得空间上的一个偶概率分布，它具有将测度零赋给通过原点的任何超平面的性质。给定具有这种分布的$N$独立随机向量，在它们不正跨越整个空间的条件下，这些向量的正壳是一个随机多面体圆锥（并且它与单位球面的交点是一个随机球面多面体）。它首先由Cover和Efron研究。我们考虑这些随机锥的期望面数，并描述了一个阈值现象时，尺寸$d$和随机向量的数量$N$趋于无穷大。以类似的方式，我们处理立体角，以及更一般的格拉斯曼角。我们进一步考虑了当$k$趋于无穷大时，$k$ -面和指数$d-k$的格拉斯曼角的期望数.

英文摘要：

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.

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