{
  "id": "1812.03127",
  "version": "v1",
  "published": "2018-12-07T17:39:48.000Z",
  "updated": "2018-12-07T17:39:48.000Z",
  "title": "Induced graphs of uniform spanning forests",
  "authors": [
    "Russell Lyons",
    "Yuval Peres",
    "Xin Sun"
  ],
  "comment": "17 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Given a subgraph $H$ of a graph $G$, the induced graph of $H$ is the largest subgraph of $G$ whose vertex set is the same as that of $H$. Our paper concerns the induced graphs of the components of $\\operatorname{WSF}(G)$, the wired spanning forest on $G$, and, to a lesser extent, $\\operatorname{FSF}(G)$, the free uniform spanning forest. We show that the induced graph of each component of $\\operatorname{WSF}(\\mathbb Z^d$) is almost surely recurrent when $d\\ge 8$. Moreover, the effective resistance between two points on the ray of the tree to infinity within a component grows linearly when $d\\ge9$. For any vertex-transitive graph $G$, we establish the following resampling property: Given a vertex $o$ in $G$, let $\\mathcal T_o$ be the component of $\\operatorname{WSF}(G)$ containing $o$ and $\\overline{\\mathcal{T}_o}$ be its induced graph. Conditioned on $\\overline{\\mathcal{T}_o}$, the tree $\\mathcal T_o$ is distributed as $\\operatorname{WSF}(\\overline{\\mathcal{T}_o})$. For any graph $G$, we also show that if $\\mathcal T_o$ is the component of $\\operatorname{FSF}(G)$ containing $o$ and $\\overline{\\mathcal{T}_o}$ is its induced graph, then conditioned on $\\overline{\\mathcal{T}_o}$, the tree $\\mathcal T_o$ is distributed as $\\operatorname{FSF}(\\overline{\\mathcal{T}_o})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-07T17:39:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "induced graph",
      "free uniform spanning forest",
      "paper concerns",
      "vertex set",
      "lesser extent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}