Random cones in high dimensions II: Weyl cones

2021/06/14 by Thomas Godland, Godland, Thomas, Zakhar Kabluchko +3
Mathematics · #52A55 #60D05. Secondary: 52A23 #60F05 #60F10 #FOS: Mathematics #Geometry and complex manifolds #Metric Geometry (math.MG) #Point processes and geometric inequalities #Primary: 52A22 #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2106.07244

openalex publication_date 2021/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider two models of random cones together with their duals. Let Y₁,…,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 pos\Y₁-Y₂,…,Y_n-1-Y_n\, respectively pos\Y₁-Y₂,…,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.

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