Random cones in high dimensions II: Weyl cones

TL;DR

This paper studies the geometric properties of two models of high-dimensional random cones, proving limit theorems for their expected features and identifying a phase transition in their statistical dimension as dimensions grow.

Contribution

It introduces new models of random cones based on i.i.d. vectors and establishes their asymptotic geometric behavior, including phase transitions.

Findings

Limit theorems for expected number of k-faces

Limit theorems for conic quermassintegrals

Identification of a phase transition in statistical dimension

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.