Random cones in high dimensions II: Weyl cones

Abstract

We consider two models of random cones together with their duals. Let $Y_1,\dots,Y_n$ be independent and identically distributed random vectors in $\mathbb R^d$ whose distribution satisfies some mild condition. The random cones $G_{n,d}^A$ and $G_{n,d}^B$ are defined as the positive hulls $\text{pos}\{Y_1-Y_2,\dots,Y_{n-1}-Y_n\}$, respectively $\text{pos}\{Y_1-Y_2,\dots,Y_{n-1}-Y_n,Y_n\}$, conditioned on the event that the respective positive hull is not equal to $\mathbb R^d$. We prove limit theorems for various expected geometric functionals of these random cones, as $n$ and $d$ tend to infinity in a coordinated way. This includes limit theorems for the expected number of $k$-faces and the $k$-th conic quermassintegrals, as $n$, $d$ and sometimes also $k$ tend to infinity simultaneously. Moreover, we uncover a phase transition in high dimensions for the expected statistical dimension for both models of random cones.