{
  "id": "2206.04941",
  "version": "v1",
  "published": "2022-06-10T08:31:31.000Z",
  "updated": "2022-06-10T08:31:31.000Z",
  "title": "The extremal position of a branching random walk in the general linear group",
  "authors": [
    "Ion Grama",
    "Sebastian Mentemeier",
    "Hui Xiao"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a branching random walk $(G_u)_{u\\in \\mathbb T}$ on the general linear group $\\textrm{GL}(V)$ of a finite dimensional space $V$, where $\\mathbb T$ is the associated genealogical tree with nodes $u$. For any starting point $v \\in V \\setminus\\{0\\}$, let $M^v_n=\\max_{|u| = n} \\log \\| G_u v \\|$ denote the maximal position of the walk $\\log \\| G_u v \\|$ in the generation $n$. We first show that under suitable conditions, $\\lim_{n \\to \\infty} \\frac{M_n^v }{n} = \\gamma_{+}$ almost surely, where $\\gamma_{+}\\in \\mathbb R$ is a constant. Then, in the case when $\\gamma_+=0$, under appropriate boundary conditions, we refine the last statement by determining the rate of convergence at which $M_n^v$ converges to $-\\infty$. We prove in particular that $\\lim_{n \\to \\infty} \\frac{M_n^v}{\\log n} = -\\frac{3}{2\\alpha}$ in probability, where $\\alpha >0$ is a constant determined by the boundary conditions. Similar properties are established for the minimal position. As a consequence we derive the asymptotic speed of the maximal and minimal positions for the coefficients, the operator norm and the spectral radius of $G_u$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-10T08:31:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60B20",
      "60J05"
    ],
    "keywords": [
      "general linear group",
      "branching random walk",
      "extremal position",
      "minimal position",
      "appropriate boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}