Random cones in high dimensions II: Weyl cones

Abstract

We consider two models of random cones together with their duals. Let (Formula presented.) be independent and identically distributed random vectors in (Formula presented.) whose distribution satisfies some mild condition. The random cones (Formula presented.) and (Formula presented.) are defined as the positive hulls (Formula presented.), respectively, (Formula presented.), conditioned on the event that the respective positive hull is not equal to (Formula presented.). 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.