Random cones in high dimensions II: Weyl cones

Abstract

We consider two models of random cones together with their duals. Let Y 1 , ⋯ , Y n $Y_1,\dots ,Y_n$ be independent and identically distributed random vectors in R d $\mathbb {R}^d$ whose distribution satisfies some mild condition. The random cones G n , d A $G_{n,d}^A$ and G n , d B $G_{n,d}^B$ are defined as the positive hulls pos { Y 1 − Y 2 , ⋯ , Y n − 1 − Y n } $\mbox{pos}\lbrace Y_1-Y_2,\dots ,Y_{n-1}-Y_n\rbrace$ , respectively, pos { Y 1 − Y 2 , ⋯ , Y n − 1 − Y n , Y n } $\mbox{pos}\lbrace Y_1-Y_2,\dots ,Y_{n-1}-Y_n,Y_n\rbrace$ , conditioned on the event that the respective positive hull is not equal to R d $\mathbb {R}^d$ . 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 kth 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.