{
  "id": "2011.12904",
  "version": "v1",
  "published": "2020-11-25T17:40:53.000Z",
  "updated": "2020-11-25T17:40:53.000Z",
  "title": "Connectedness of the Free Uniform Spanning Forest as a function of edge weights",
  "authors": [
    "Marcell Alexy",
    "Márton Borbényi",
    "András Imolay",
    "Ádám Timár"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected transitive graph $H$. It is shown in arXiv:2006.06387 that the Free Uniform Spanning Forest ($\\mathsf{FSF}$) of this graph may not be connected, but the dependence of this connectedness on $H$ remains somewhat mysterious. We study the case when a positive weight $w$ is put on the edges of the $H$-copies in $G$, and conjecture that the connectedness of the $\\mathsf{FSF}$ exhibits a phase transition. For large enough $w$ we show that the $\\mathsf{FSF}$ is connected, while for a large family of $H$ and $T$, the $\\mathsf{FSF}$ is disconnected when $w$ is small (relying on arXiv:2006.06387). Finally, we prove that when $H$ is the graph of one edge, then for any $w$, the $\\mathsf{FSF}$ is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-25T17:40:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free uniform spanning forest",
      "edge weights",
      "connectedness",
      "finite connected transitive graph",
      "cartesian product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}