{
  "id": "1611.02005",
  "version": "v1",
  "published": "2016-11-07T11:55:41.000Z",
  "updated": "2016-11-07T11:55:41.000Z",
  "title": "First passage percolation in Euclidean space and on random tessellations",
  "authors": [
    "Sebastian Ziesche"
  ],
  "comment": "27 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "There are various models of first passage percolation (FPP) in $\\mathbb R^d$. We want to start a very general study of this topic. To this end we generalize the first passage percolation model on the lattice $\\mathbb Z^d$ to $\\mathbb R^d$ and adapt the results of \\cite{boivin1990first} to prove a shape theorem for ergodic random pseudometrics on $\\mathbb R^d$. A natural application of this result will be the study of FPP on random tessellations where a fluid starts in the zero cell and takes a random time to pass through the boundary of a cell into a neighbouring cell. We find that a tame random tessellation, as introduced in the companion paper \\cite{ziesche2016bernoulli}, has a positive time constant. This is used to derive a spatial ergodic theorem for the graph induced by the tessellation. Finally we take a look at the Poisson hyperplane tessellation, give an explicit formula to calculate it's FPP limit shape and bound the speed of convergence in the corresponding shape theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-07T11:55:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60D05"
    ],
    "keywords": [
      "euclidean space",
      "shape theorem",
      "first passage percolation model",
      "tame random tessellation",
      "fpp limit shape"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}