{
  "id": "2402.07509",
  "version": "v2",
  "published": "2024-02-12T09:30:44.000Z",
  "updated": "2025-02-10T10:41:07.000Z",
  "title": "First-order behavior of the time constant in non-isotropic continuous first-passage percolation",
  "authors": [
    "Anne-Laure Basdevant",
    "Jean-Baptiste Gouéré",
    "Marie Théret"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider $\\Xi$ a homogeneous Poisson point process on $\\mathbb{R}^d$ ($d\\geq 2$) with unit intensity with respect to the Lebesgue measure. For $\\varepsilon\\geq 0$, we define the Boolean model $\\Sigma_{p, \\varepsilon}$ as the union of the balls of volume $\\varepsilon$ for the $p$-norm ($p\\in [1,\\infty]$) and centered at the points of $\\Xi$. We define a random pseudo-metric on $\\mathbb{R}^d$ by associating with any path a travel time equal to its $p$-length outside $\\Sigma_{p,\\varepsilon}$. This defines a continuous model of first-passage percolation, that has been studied in \\cite{GT17,GT22} for $p=2$, the Euclidean norm. For $p=1$, this model is expected to share common properties with the classical first-passage percolation on the graph $\\mathbb{Z}^d$ with a distribution of passage times of the form $\\varepsilon \\delta_0 + (1-\\varepsilon) \\delta_1$. The exact calculation of the time constant of this model $\\tilde \\mu_{p,\\varepsilon} (x)$ is out of reach. We investigate here the behavior of $\\varepsilon \\mapsto \\tilde \\mu_{p,\\varepsilon} (x)$ near $0$, and enlighten how the speed at which $\\| x \\|_p - \\tilde \\mu_{p,\\varepsilon} (x) $ goes to $0$ depends on $x$ and $p$. For instance, for $p\\in (1,\\infty)$, we prove that $\\| x \\|_p - \\tilde \\mu_{p,\\epsilon} (x)$ is of order $\\varepsilon ^{\\kappa_p(x)}$ with $$\\kappa_p(x): = \\frac{1}{d- \\frac{d_1(x)-1}{2} - \\frac{d-d_1 (x)}{p}}\\,,$$where $d_1(x)$ is the number of non null coordinates of $x$. The exact order of $\\| x \\|_p - \\tilde \\mu_{p,\\epsilon} (x)$ is also given for $p=1$ and $p=\\infty$. Related results are also discussed, about properties of the geodesics, and analog properties on closely related models.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-02-10T10:41:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-isotropic continuous first-passage percolation",
      "time constant",
      "first-order behavior",
      "non null coordinates",
      "share common properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}