{
  "id": "2106.07244",
  "version": "v1",
  "published": "2021-06-14T08:55:10.000Z",
  "updated": "2021-06-14T08:55:10.000Z",
  "title": "Random cones in high dimensions II: Weyl cones",
  "authors": [
    "Thomas Godland",
    "Zakhar Kabluchko",
    "Christoph Thäle"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.MG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-14T08:55:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A22",
      "60D05",
      "52A23",
      "52A55",
      "60F05",
      "60F10"
    ],
    "keywords": [
      "random cones",
      "high dimensions",
      "weyl cones",
      "limit theorems",
      "th conic quermassintegrals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}