{
  "id": "1902.01110",
  "version": "v1",
  "published": "2019-02-04T10:23:42.000Z",
  "updated": "2019-02-04T10:23:42.000Z",
  "title": "Regarding the Probabilistic Representation of the Free Effective Resistance of Infinite Graphs",
  "authors": [
    "Tobias Weihrauch",
    "Stefan Bachmann"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "On a finite weighted graph $G$, the effective resistance between two vertices $x$ and $y$ admits the probabilistic representation \\[ R(x,y) = \\frac{1}{c_x \\cdot \\mathbb{P}_x[\\tau_y < \\tau_x^+]} \\] using the the random walk $\\mathbb{P}_x$ on $G$ starting in $x$. We show that the same representation does not hold in general for the free effective resistance of an infinite graph. More precisely, we show that a transient graph admits such a representation if and only if it is a subgraph of an infinite line.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-04T10:23:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C63",
      "05C81",
      "60J10",
      "05C12"
    ],
    "keywords": [
      "free effective resistance",
      "infinite graph",
      "probabilistic representation",
      "transient graph admits",
      "finite weighted graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}